Physics of the Sun and Its Atmosphere: Proceedings of the National Workshop (India) on ''Recent Advances in Solar Physics'' Meerut College, Meerut, India 7 - 10 November 2006

Physics of the Sun and its Atmosphere Proceedings of the National Workshop (India) on “Recent Advances in Solar Physics”

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Editors

B. N. Dwivedi Banaras Hindu University, India

U. Narain Meerut College, India

Phy SICS of

the Sun and its

A tmospher

Proceedings of the National Workshop (India) on "Recent Advances in Solar Physics" Meerut College, Meerut, India 7-10 November 2006

World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

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PHYSICS OF THE SUN AND ITS ATMOSPHERE Proceedings of the National Workshop (India) on “Recent Advances in Solar Physics” Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-283-271-9 ISBN-10 981-283-271-8

Printed in Singapore.

ChianYang - Phys of the Sun.pmd

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9/18/2008, 1:31 PM

PREFACE

This book emerged from a successful workshop on “Recent Advances in Solar Physics”, held at Meerut College, Meerut in November 2006. We thought a book covering the modern view of the Sun from its interior to its exterior from a pedagogical viewpoint will be a valuable input to the beginners pursuing solar physics. With this objective, we decided to publish this volume by World Scientific, Singapore. In this process, we invited most of the leading experts who lectured at the workshop apart from inviting some internationally reputed scientists to make this volume more valuable. It is satisfying to see this book in print at a time when one of us (Dr. Udit Narain) superannuates after pursuing an active solar physics research for over three decades while teaching physics at Meerut College. All this could be possible with the kind and generous support of our esteemed colleagues from all over and above all, the authors of this book for which we cannot thank them enough.

B.N. Dwivedi & Udit Narain

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ACKNOWLEDGEMENTS

It has been the voice of my soul to organize a National Solar Workshop at Physics Dept, Meerut College before my superannuation on 31 July 2007. Prof. B.N. Dwivedi (IT-BHU) was the first person to support me and the national workshop on “RECENT ADVANCES IN SOLAR PHYSICS” was held at Meerut College premises entirely through his efforts. I carried on his advice and programmes mechanically since the preparation for the workshop started. Prof. N.K. Dadhich, Director, IUCAA wanted to hold the solar workshop at BHU under the leadership of Prof. Dwivedi who convinced him to let it take place at Meerut College in view of my superannuation. Dr H.P. Mittal, Head of Physics Dept and Dr S.K. Agarwal, Principal, Meerut College kindly allowed and supported it without any financial support. I gratefully acknowledge the SOC members: S.M. Chitre, B.N. Dwivedi (chair), R. Jain, P.K. Manoharan, J. Singh, W. Uddin and P. Venkatakrishnan. Prof. Dwivedi started looking after the academic part, namely Scientific Organising Committee, speakers, schedule of lectures, publication of proceedings etc. I started looking after local organization part, namely LOC, financial aspects, accommodation, transport etc. The application for finacial assistance was submitted to IUCAA, DST, UGC, CSIR, ISRO and INSA. The request for financial assistance was also made to IIA, PRL, and ARIES. IUCAA provided 25000/=; IIA 30,000/=; CSIR 20,000/=; INSA 10,000/= and UGC 50,000/=. PRL allowed four speakers, ARIES and RAC/TIFR one each with their travel expenses. I am very grateful to them as it was not possible to organize the workshop without their help. Individuals, namely Late Prof. Rajkumar, Dr Mukul Kumar, Dr Rakesh Kumar Sharma, Dr Sushil Kumar, Mr Nishant Mittal and Mr Joginder Sharma provided crucial financial assistance for which I am very grateful to them. I am very much grateful to Prof. S.S. Hasan, Director, IIA, Bangalore for inaugurating and delivering the keynote address at the workshop which was highly appreciated.

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Acknowledgements

Prof. S.P. Khare (Former Head, Pro-VC, CCS Univ., Meerut) has been very kind in providing guidance and help from beginning to the end and I wish to express my gratitude to him. Help and support from Dr H.P. Mital and other faculty members, non-teaching staff, research scholars and PG students of Physics Dept, are greatly appreciated. Special thanks are due to Dr Ajay Chauhan for his help in all computer related problems and presentations. The help, cooperation and encouragement by Dr S.K. Agarwal is highly appreciated. I also thank Prof. V.K. Rastogi (CCS Univ, Meerut) for help, support and his personal involvement. Last but not the least, I wish to express my gratitude to the retired and active faculty members and individuals of Meerut who enlivened the proceedings of the workshop by their participation.

Meerut

Udit Narain

CONTENTS

Preface

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Acknowledgements

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Chapter 1: Recent Advances in Solar Physics B.N. Dwivedi

1

Chapter 2: Overview of the Sun S.S. Hasan

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Chapter 3: Seismic View of the Sun S.M. Chitre and B.N. Dwivedi

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Chapter 4: Solar Magnetism P. Venkatakrishnan and S. Gosain

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Chapter 5: Waves and Oscillations in the Solar Atmosphere R. Erdélyi

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Chapter 6: VUV Spectroscopy of Solar Plasma A. Mohan

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Chapter 7: Active Region Diagnostics H.E. Mason and D. Tripathi

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Chapter 8: Hall Effect and Ambipolar Diffusion in the Lower Solar Atmosphere V. Krishan

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Contents

Chapter 9: On Solar Coronal Heating Mechanisms K. Pandey and U. Narain

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Chapter 10: Coronal Mass Ejections (CMEs) and Associated Phenomena N. Srivastava

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Chapter 11: The Radio Sun P.K. Manoharan

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Chapter 12: The Solar Wind P.K. Manoharan

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Chapter 13: The Sun-Earth System: Our Home in Space J.L. Lean

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CHAPTER 1 RECENT ADVANCES IN SOLAR PHYSICS

B.N. DWIVEDI Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India

1.1. Introduction For millennia, the Sun (and the universe) has been viewed in the visual light. As the bestower of light and life, the ancients made God out of the Sun. With the Babylonians, or with the multiple origins with the Chinese, Egyptians and Indians, quoting the Rig Veda: “All that exists was born from Sūrya, the God of gods.”, we have come a long way to understanding the Sun. In the early seventeenth century, however, Galileo showed that the Sun was not an immaculate object. Thus began our scientific interests in our nearest stellar neighbour, the Sun (cf., Figure 1.1.), with its sunspots and the related solar activity. The observations of the Sun and their interpretations are of universal importance for at least two reasons: First, the Sun is the source of energy for the entire planetary system and all aspects of our life have direct impact on what happens on the Sun; and second, the Sun’s proximity makes it unique among the billions of stars in the sky of which we can resolve its surface features and study physical processes at work. Observations of the solar atmosphere led to the development of the theory of radiative transfer in stellar atmospheres and the discovery of the element helium. Moreover, the Sun is the principal magnetohydrodynamic (MHD) laboratory for large magnetic Reynolds numbers, exhibiting the totally unexpected phenomena of magnetic fibrils, sunspots, prominences, flares, coronal loops, coronal mass ejections (CMEs), the solar wind, the X-ray corona, and irradiance variations etc. It is the physics of these exotic phenomena, collectively making up variations of solar activity, with which we are confronted today. The activity affects the terrestrial environment, from occasionally knocking out power grids to space weather and most probably general climate. 1

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Beginning with the first solar ultraviolet light from space in 1946, X-rays in 1948, hard X-rays and γ-rays in 1958; many experiments have been conducted or being conducted using balloons, rockets and satellites (e.g., OSOs, Skylab, SMM, Yohkoh, SOHO, TRACE, RHESSI, Hinode and STEREO etc.). Artificial satellites have provided the unique opportunity to have uninterrupted observations of the Sun from the vantage points, such as the Sun-Earth Lagrangian point L1 (e.g., SOHO), or from outside the ecliptic plane (e.g., Ulysses), or in stereoscopic modes using different orbits (e.g., STEREO). All these have provided a rich source of data, unlocking the secrets of the Sun and addressing some of its outstanding riddles (e.g., coronal heating, solar wind acceleration etc.)

Figure 1.1. Brief view of the Sun: The Sun’s energy derives from nuclear reactions that occur in its core which is at temperatures of 15 million degrees Kelvin. This energy moves outward, first in the form of electromagnetic radiation (e.g., X-rays and γ-rays) in the radiative zone. Energy then moves upward in photon-heated solar gas through convection in the convective zone (outer 200 000 km). Because of tremendous pressure, this energy is continually absorbed and re-absorbed and may take millions of years to reach the surface of the Sun. Convection motions in the Sun’s interior generate magnetic fields, emerging at the Sun’s surface as sunspots and loops of hot gas called prominences. Most solar energy finally escapes from a thin layer of the Sun’s atmosphere called the photosphere, which is the part of the Sun observable to the naked eye. Image credit: NASA.

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Ground-based observations suffer from the effects of the Earth’s atmosphere such as atmospheric extinction resulting in the limited radiative spectrum of the Sun, and turbulence resulting in image distortions. None the less, making use of adaptive optics system, solar images with resolution of about 0.13" (90 km on the Sun), or even smaller structures down to 60 km, have been obtained by the Swedish 1-meter Solar Telescope (SST) on La Palma. Neutrino detectors have provided a unique tool for probing the Sun’s interior by comparing the emitted flux with the predictions of the standard solar models. Helioseismology from space and from ground (e.g., GONG) have revolutionised our understanding of the workings of the Sun. Against this brief background, it is intended to bring some of these developments in a limited but pedagogical and updated way to help beginners pursue solar physics research. 1.2. Main Contents This book contains 13 chapters beginning with a glance of main contents of each chapter as follows: Chapter 2: Overview of the Sun (Hasan): This chapter begins with how solar physics is going through an exciting period, particularly due to new insights obtained from space and ground observations. These have contributed significantly to improving our understanding of fundamental processes occurring in the solar atmosphere, from the interior to the heliosphere. By combining the information obtained through observations with theoretical developments, a holistic view is slowly beginning to emerge of the physical mechanisms taking place on the Sun. Chapter 3: Seismic view of the Sun (Chitre and Dwivedi): This chapter presents the solar seismology which probes the internal structure and dynamics of the Sun using hundreds of thousands of accurately measured frequencies of solar oscillations. With the accumulation of the helioseismic data obtained with Global Oscillation Network Group (GONG) and Michelson Doppler Imager (MDI) instruments over the past solar cycle, it is possible to study temporal variations that occur within the solar interior with the progress of the cycle. Chapter 4: Solar Magnetism (Venkatakrishnan and Gosain): This chapter is basically divided into two parts. In the first part, the important properties of the solar magnetic field are summarized. The discussion begins with a simple

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introduction to solar magnetohydrodynamics, describing the current status of the solar dynamo theory. Some very curious and interesting results on magnetic helicity and force-free fields are then presented in very basic terms. Finally, the application of this theoretical frame-work to the problems of coronal heating, solar flares and coronal mass ejections are developed in a simple unified scheme, based on a hierarchy of physical conditions. The second part of this chapter consists of a tutorial on magnetographs. It begins with a description of polarization of light from very fundamental notions of coherence of light. This is followed by simple but comprehensive explanations of the Zeeman and Hanle effects along with the necessary basic ideas of quantum physics of scattering of light. Then the working of a few important magnetographs is outlined, with special emphasis on a solar vector magnetograph developed for USO, to provide a “hands on” perspective. The chapter concludes with a few brief remarks on the possible future directions for research in the domain of solar magnetism. Chapter 5: Waves and Oscillations in the Solar Atmosphere (Erdélyi): This chapter introduces waves and oscillations in the solar atmosphere in a lucid manner. Recent satellite and ground-based imaging and spectral instruments have observed a wide range of wave and oscillatory phenomena in the visible, EUV, X-ray and radio wavelengths in the solar atmosphere. Because in most cases these waves and oscillations are tied to the complex magnetic structure of the solar atmosphere, these oscillatory and wave phenomena are interpreted in terms of magnetohydrodynamic (MHD) waves. Waves and oscillations are crucial in the understanding of the diagnostics and dynamics of the magnetised solar atmosphere, as these periodic motions contain information about the medium they occur in. Using undergraduate tools of applied mathematics, the basic properties of MHD waves and oscillations are described. The theoretical description is then strongly linked to the latest observational findings with applications to the wealth of MHD wave phenomena present in the solar atmosphere. Observed MHD waves propagating from the lower solar atmosphere into the higher, often very dynamic regions of the magnetized corona, have the potential to provide an excellent insight into the physical processes at work at the coupling point between these apparently different regions of the Sun. High-resolution wave observations combined with advanced forward MHD modelling can give an unprecedented insight into the connectivity of the magnetized solar atmosphere, which further provides us with a realistic chance to reconstruct the

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structure of the magnetic field in the solar atmosphere. This type of solar exploration is termed as atmospheric magneto-seismology. Some new trends in the observational study of waves and oscillations, discussing their origin, and their propagation through the atmosphere are summarised. Finally, the role of the leakage of photospheric motions, whether coherent (e.g., p-modes), random (e.g., granular buffetting), or casual (e.g., footpoint nano-scale energy release) on the dynamics present in the solar atmosphere is addressed. Chapter 6: VUV Spectroscopy of Solar Plasma (Mohan): Electron densities, temperatures, elemental abundances and emission measures of the space plasma are the basic parameters to give the informations regarding the generation and transport of mass, momentum and energy. The fundamental property of hot solar plasmas is their inhomogeneity. Using the spectroscopic diagnostic techniques for the temperature and density structures of hot optically thin plasmas, the solar atmosphere and its composition have been thoroughly examined. As an illustration, the potential for plasma diagnostics of forbidden transitions from ground levels in the nitrogen-like ions has been presented. Some of the lines considered in the present chapter have been measured by SUMER for the first time. Also using the SUMER spectra, electron density, temperature and abundance anomalies in the off-limb solar corona are discussed. In particular, the behaviour of the solar FIP (first ionization potential) effect with height above an active region observed at the solar limb is presented. Chapter 7: Active Region Diagnostics (Mason and Tripathi): Recent observations from SOHO, Yohkoh and TRACE clearly demonstrate the complex and dynamic nature of the solar atmosphere. In order to explore the nature of solar active regions, it is important to determine the local plasma parameters (electron density, temperature, emission measure distribution, element abundances, flows, non-thermal line broadening etc.). This can only be reliably achieved using simultaneous imaging and spectroscopic observations. The Hinode and STEREO spacecrafts, launched in autumn 2006, are providing some spectacular new observations and insights. This chapter focuses on what has been learnt about active regions in particular from recent observations using spectroscopic diagnostics in the UV and X-ray wavelength ranges. Chapter 8: Hall Effect and Ambipolar Diffusion in the Lower Solar Atmosphere (Krishan): This chapter highlights the realistic importance of incorporating multi-fluid system in the Sun’s atmosphere to understand the

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physical processes in operation. The lower solar atmosphere is a partially ionized plasma consisting of electrons, protons and predominantly hydrogen atoms. The discrete structures such as the sunspots, the prominences and the spicules also consist of the three main species of particles. This essentially forms a three fluid system and therefore, it is mandatory to go beyond the single fluid magnetohydrodynamic studies. One must include the Hall effect which arises from the treatment of the electrons and the protons as two separate fluids and the ambipolar diffusion arising due to the inclusion of neutrals as the third fluid. The Hall effect and the ambipolar diffusion have been shown to be operational in a region beginning from below the photosphere up to the chromosphere. In this three fluid system, the magnetic induction is subjected to the ambipolar diffusion and the Hall effect, in addition to the usual resistive dissipation caused by the electron-proton and electron-neutral collisions. These effects produce novel modifications in the equilibrium configurations of the flows and the fields, the wave phenomena and the magnetic field transport processes. A first principle derivation of these effects in a three fluid system along with an account of their role in the characterization of the lower solar atmosphere is given in this chapter. Chapter 9: On Solar Coronal Heating Mechanisms (Pandey and Narain): The million degree temperature of the solar corona has been an outstanding astrophysical problem since 1943. A number of mechanisms, such as accretion of intergalactic matter, acoustic waves, magnetoacoustic waves, Alfvén waves, currents/magnetic fields, spicules, magnetic flux emergence, velocity filtration etc have been offered as possible explanation. Alfvén waves may heat coronal holes as well as coronal loops in the solar corona. These structures can also be heated by currents/magnetic fields (as nano- and micro-flares) generated by slow photospheric foot point motions. The expected behaviour of X-ray and EUV intensities from Alfvén waves and nano- and micro-flares are quite similar. Only suitable experimental techniques can discriminate between the two main mechanisms. Velocity filtration does not require any source of energy but it requires the existence of highly energetic particles (mechanism still not known) at the base of corona. Other mechanisms do contribute to the energy budget of the solar corona but they cannot resolve the coronal heating problem individually. Chapter 10: Coronal Mass Ejections (CMEs) and Associated Phenomena (Srivastava): Coronal mass ejections are spectacular expulsions of mass from the Sun that display a three-part structure comprising a leading edge, a dark

Recent Advances in Solar Physics

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cavity and a bright knot. Since the discovery of coronal mass ejections (CMEs) in 1970s, our knowledge about CMEs has improved considerably. This is largely due to multi-wavelength observations with several instruments dedicated to the observations of the Sun both from ground and space. Our understanding of CMEs took a giant leap with the unprecedented observations obtained by several instruments aboard SOHO designed to probe different layers of the Sun. This chapter expounds physical and kinematic properties of CMEs and associated phenomena, such as flares, eruptive prominences, sigmoids and EIT waves. It also examines the links between these phenomena and the physical processes that lead to eruption of CMEs. Finally, the observations of CMEs from the recently launched STEREO and Hinode are highlighted and the problems that these missions might be able to address and resolve. Chapter 11: The Radio Sun (Manoharan): Solar radio observations from ground-based and space-based instruments have contributed a unique perspective on the physical phenomena occurring on the Sun. In particular, radio observations have played a key role in probing the different altitudes of the corona and provided the possibility to trace the three-dimensional structure of the coronal magnetic field. Moreover, the comparison of radio observations with other multi-wavelength data (e.g., X-ray, EUV, and optical) has clearly shown specific advantages and allowed for a deeper understanding of solar flares and coronal mass ejections and the physics behind the fundamental processes of the solar radio emission mechanism. This chapter gives the overview of radio observations of the quiet and active Sun and physics of the explosive energy release. Chapter 12: The Solar Wind (Manoharan): This chapter reviews the evolution of the solar wind, with the particular emphasis on the properties of the solar wind within about 1 AU of the Sun. To start with, a brief discussion of coronal heating is given followed by the energy balance in the solar atmosphere and the formation of the solar wind. The solar wind measurement using the interplanetary scintillation technique is explained in detail. The results on the large-scale properties and long-term changes of quasi-stationary structures of the solar wind are presented. The solar wind disturbances resulting from the solar phenomena and their heliospheric evolution in space and time are reviewed based on radio scintillation technique. The solar cycle changes of the solar wind in the three-dimensional heliosphere are reviewed in the aspect of space weather effects of the solar wind. The final part also includes the turbulence characteristics of the quasi-stationary and transient solar wind.

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Chapter 13: The Sun-Earth System: Our Home in Space (Lean): The concluding chapter of this book addresses the Sun-Earth system and our home in space. Energy flowing from the Sun heats the Earth, structures its atmosphere and organizes the surrounding space environment. Changes in this energy occur continually, with myriad terrestrial impacts, some of which have societal consequences involving climate change, the ozone layer and space-based enterprise. 1.3. Concluding Remarks A pedagogical updated modern view of the Sun from its interior to its exterior as well as the Sun-Earth system in this book by eminent solar physicists present a rich menu to motivate graduate students who wish to pursue solar physics research career.

CHAPTER 2 OVERVIEW OF THE SUN S.S. HASAN Indian Institute of Astrophysics Bangalore-560034, India

2.1. Introduction The Sun plays a central role in two important respects: firstly, it provides a cosmic laboratory for investigating processes that cannot be simulated in the terrestrial environment and secondly, because of its relative closeness it serves as a proxy for understanding conditions in other stars. Formed about 4.6 billion years from a cloud of gas dust and frozen ice, the Sun at the current epoch in its life is a normal main-sequence star of spectral classification G2 with an average surface temperature of around 5700 K. It displays an astounding range of phenomena on myriad spatial and temporal scales that have traditionally defied comprehension. The Sun’s magnetic field, that varies on a 22-year cycle, triggers activity and powerful eruptions that affect regions extending from the Earth’s atmosphere to the distant edges of the solar system. Despite the inherent complexities of these processes, some progress has been achieved in understanding them through recent spectacular advances in observational techniques coupled with theoretical modelling. The aim of the present review is to provide a broad overview of some modern developments in solar physics that have had a significant impact on the subject. We begin by discussing in Sect. 2.2 the internal structure of the Sun and the processes taking place in the interior. Section 2.3 deals with magnetic fields and their influence on the dynamics, heating and activity in the solar atmosphere. In Sect. 2.4 we discuss processes in the outer atmosphere that includes eruptive phenomena such as CMEs and flares. The final Sect. 2.5 looks at future perspectives and directions.

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2.2. Internal Processes The source of the Sun’s energy derives from thermonuclear reactions, which occur in the core, a region at a temperature of about 107 K that extends from the centre to about 0.28 R
, where R
is the solar radius. These reactions convert hydrogen into helium, principally through the proton-proton chain. This energy is transported through radiation up to about 0.7 R
beyond which this process is no longer effective. From this radius up to the surface, the dominant mechanism for energy transport is through convection, similar to the fluid motions that arise when water is heated from below. The solar interior is opaque to optical radiation in view of its high opacity: in fact, visible light from the Sun emanates from a thin layer of around 100 kilometres thickness in the surface layers known as the photosphere. However, neutrinos generated in the core by nuclear reactions can travel unimpeded into space since they interact very weakly with matter. The mismatch between the neutrino flux determined from theory and its experimental verification constituted what has been commonly referred to as the solar neutrino problem. Recently, a resolution of this conflict has been found by invoking a mechanism that permits neutrinos emitted from the Sun to change “flavours” during their passage to the Earth. A novel experiment at the Sudbury Neutrino Observatory in Ontario Canada1 that can measure all flavours of neutrinos was able to achieve an impressive agreement with the theoretical prediction (cf., Chitre & Dwivedi, Chapter 3, this volume for more details). 2.2.1. Helioseismology and rotation The thermodynamic state of the solar interior is now known to a high level of precision mainly due to helioseismology, which in simple terms uses the properties of acoustic oscillations with periods of around 5 min. to probe the deeper layers that are inaccessible to direct observations. The accuracy with which the sound speed can be inferred using this technique is higher than a fraction of a percent. Due to the thermal stratification, the sound speed increases with depth from the surface. As a consequence, an acoustic wave, propagating downward will be refracted and eventually reach a level where it will turn around and travel upwards. On reaching the surface, the wave encounters a sharp decrease in density and is reflected back into the interior. In this way acoustic modes are trapped in a cavity with an extension that depends on the horizontal wavelength of the mode. Longer wavelength modes penetrate deeper into the Sun and can be used as a diagnostic for probing conditions close to the core, whereas modes of short wavelengths provide information about physical

Overview of the Sun

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conditions just below the surface layers. This subject is dealt in greater detail in Chapter 3 of this volume. In addition to revealing the sound speed, helioseismology can also be used to deduce the rotation profile of the plasma in the solar interior because rotation produces a splitting of the acoustic modes. A careful analysis of this splitting makes it possible to determine the rotation rate at each depth and latitude. From observations it is known that surface layers of the Sun rotate differentially with a period that increases from 25 days at the equator to about 35 days at the poles. From helioseismology investigations it is established that apart from a narrow region below the photosphere, the differential rotation preserves approximately the same form as the surface in the interior layers. This behaviour continues to the base of the convection zone, below which the rotation character changes abruptly from differential to rigid body in a thin shear layer, known as the tachocline. Below the tachocline, material in the radiative interior rotates homogenously with an intermediate period of around 27 days. 2.2.2. Magnetic field generation As mentioned earlier, the solar magnetic field has a profound influence on practically all observed phenomena. What is the mechanism for generating and maintaining this field? The solar plasma in the interior is in the form of a fully ionized conducting gas that is “frozen” into the magnetic field. It is now generally accepted that the magnetic field is generated through a dynamo action at the base of the convection zone in the tachocline, which has a high velocity shear. The mechanism involves the interplay of differential rotation and convective motions. In the first step a weak poloidal (i.e. oriented principally in the north-south direction) field is twisted by the differential rotation to generate an azimuthal (toroidal) component (ω -effect), which is gradually amplified due to the rotation. The next step in the process is the regeneration of the poloidal component from the toroidal field. Parker2 proposed a workable model for this process that incorporates the effects of helical motions arising from Coriolis forces. This is the so-called α-effect, which can be estimated more formally using mean-field magnetohydrodynamics3,4. By suitably fine tuning adjustable parameters, such models could reproduce many aspects of the “butterfly diagram” that depicts the sunspot distribution with time over a solar cycle. However, in order to accomplish this, the angular velocity in the interior needs to increase with depth by up to 40%, which is inconsistent with the results of helioseismology. Other difficulties with this model include the problem of storing strong toroidal magnetic fields for a significant fraction of a sunspot

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cycle to overcome their premature escape to the surface due to magnetic buoyancy. Furthermore, detailed numerical simulations yielded magnetic patterns on the surface that were inconsistent with observations. A phenomenological model of the solar dynamo developed by Babcock5 and Leighton6,7 that was remarkably successful in explaining several observational features such as Hale’s polarity reversal, migration of sunspots and the solar cycle has recently been revived8-13. The key elements in the model involve advection by meridional circulation of poloidal magnetic fields from the surface downwards in to the convection zone where they are both sheared by the differential rotation as well as transported towards the equator. The newly created toroidal flux rises due to magnetic buoyancy and produces sunspots at the surface. The decay of these spots generates poloidal flux and the cycle repeats again. A general difficulty with this class of models is that the surface fields at the poles tend to be too strong13. Incidentally, the majority of dynamo models to date are kinematic, i.e. they involve solely the amplification of field through fluid motions but ignore the back reaction of the field on the fluid motions. Some efforts to produce full dynamic models of the solar dynamo have recently started with varying degrees of success14 but they still have a long way to go before they can match observations. Despite the impressive progress that has been made in dynamo models, particularly in recent years through detailed numerical simulations, there are still many open questions such as: (a) Is the dynamo action global or local (i.e. does it occur in a narrow region at the interface between the convection and radiative zones)? (b) What limits the amplitude of the solar magnetic field; (c) Can models reliably predict the strengths of future cycles? Some aspects of these problems are under active examination and hopefully will be resolved in the near future. 2.3. Solar Magnetism Above the photosphere, the temperature decreases gradually reaching a minimum value of around 4200 K at a height of 500 km above the surface. This is the base of a region known as the chromosphere. The temperature now increases with height, slowly at first to a value of about 104 K, and thereafter extremely rapidly from a height of 2000 km to a value of 105 K in a thin layer (known as the transition region with an extension of some 100 km). Above this region is the corona, in which the temperature increases gradually to few million degrees. The focus of this section is on processes in the photosphere and chromosphere, while the following section will discuss phenomena in the corona.

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White light observations of the solar surface have revealed the presence of cellular motions that at the largest scales of convection defines a pattern known as supergranulation with a typical horizontal size of about 30,000 km. The boundaries of the supergranules15,16 are the sites of intense magnetic fields and define what is commonly referred to as the magnetic network. The regions between the supergranules with strong magnetic fields are also associated with jets of gas known as spicules that protrude high in to the chromosphere. Spicules play an important role in the dynamics and energy balance of the chromosphere, though not contributing to mass loss from the Sun. The mechanism responsible for producing spicules is unfortunately not fully understood, although there is strong evidence that it involves the magnetic field in a fundamental way. High resolution observations also reveal a cellular pattern on a smaller scale of about a 1000 km known as granulation. Ground-based telescopes equipped with adaptive optics and space telescopes with an angular resolution better than 1 arc sec like the Solar Optical Telescope on Hinode (a Japanese spacecraft launched in 2006) can easily resolve this pattern. Recent numerical simulations17 have been fairly successful in reproducing the broad observational properties associated with granulation. 2.3.1. Quiet Sun magnetism In the “quiet Sun” i.e. the regions outside sunspots and other centres of activity, the magnetic field broadly falls into two broad categories: network and internetwork fields. Various studies from the 1970s onwards confirmed that more than 90% of the quiet Sun flux is in the form of discrete bundles or flux tubes with field strengths in the kilogauss range and with diameters of the order of 100 km or less18,19. Observational techniques for magnetic field measurements are discussed by Venkatakrishnan (Chapter 4, this volume). Magnetic network elements can be identified with bright points in G-band (430.05 nm) images20. High-resolution observations indicate that these network bright points, (NBPs) located in “lanes” at the boundaries of granules, are in a highly dynamical state due to continuous buffeting by random convective motions in the subphotosphere. With the availability of new telescopes at excellent sites and sophisticated image reconstruction techniques it is now possible to examine NBPs with a resolution better than 70 km and investigate their structure in unprecedented detail21. This picture of the network is corroborated by recent observations22 images in G band of the molecular bandhead of CH (430.5 nm) and the line of Ca II H (396.8 nm). The G band and Ca II H lines are formed in the photosphere and chromosphere respectively. These observations reveal a

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S.S. Hasan

picture of a network region consisting of bright multiple flux elements that are hotter than their surroundings. The physical processes that heat the network are still not fully identified. Is the network heated by wave dissipation and if so, what is the nature of these waves? These and other aspects related to the nature of waves in magnetic fields are discussed by Fay-Siebenburgen (Chapter 5, this volume). Let us now turn to the region in the interior of the supergranule cells that is often referred to as the internetwork (hereafter IN). Till recently it was believed that the IN magnetic field was weak with a mean field strength of a few Gauss. However, recent observations using high polarimetric sensitivity have provided new information on the nature of IN magnetism and revealed a field distribution with strengths ranging from a few Gauss to kilogauss23. Stronger fields have been identified with bright points in the cell interior23. In addition, observations have shown that a large fraction of magnetic fields in the cell interiors reside in small-scale weaker fields with a typical range of 20-150 G (ref. 24), a finding that is supported by the recent high resolution observations from the Hinode spacecraft25. Despite the impressive advances in observational techniques, there are many aspects about the internetwork magnetic field that need to be elaborated. For instance, the origin of this field is still unknown. Is the field a remnant produced from the decay of active regions or is it created by a turbulent dynamo26 ? 2.3.2. Sunspots 2.3.2.1. Structure Sunspots, in contrast to the thin flux tubes associated with NBPs, are much larger structures (with typical thickness in the range of 10000 to 20000 km) that are strongly magnetized (around 3000 G in the central regions). They are visible on the solar surface as dark features with a dark core called the umbra with a much lower temperature (around 4000 K) than the ambient atmosphere and surrounded by a lighter region called the penumbra. The darkness of sunspots has traditionally been attributed to suppression of convective energy transport (relative to the surrounding photosphere) by the strong magnetic field. The orientation of the magnetic field is mainly vertical in the centre of the umbra and becomes increasingly inclined with radial distance to about 70° (with respect to the vertical) at the edge of the penumbra, where the field strength drops to about 1000 G. The penumbra displays radial filaments along which fluid motions with speeds of several kilometres per second occur. This is the well-known Evershed

Overview of the Sun

15

effect, discovered in Kodaikanal, India in 1909, the origin of which is still being debated. Sunspot umbrae also reveal fine structure in the form of bright points or umbral dots with a typical diameter of about 150 km and brightness comparable to the photosphere. It was earlier believed that the magnetic field in umbral dots is reduced compared to the background umbra but recent observations do not indicate a decrease in field strength27. The physical mechanism responsible for their formation is most likely related to convection in a vertical magnetic field. Recent high-resolution observations have shown that the penumbral magnetic field exhibits an “interlocking comb structure”, consisting of two distinct groups of field lines associated with: (a) inclined bright filaments, and (b) almost horizontal dark filaments28,29. From a theoretical viewpoint this dual topology is also not well understood. It has been suggested that buoyancy and downward pumping of magnetic may contribute to creating and maintaining such structures30. 2.3.2.2. Solar cycle The number of sunspots and sunspot groups (sunspot number) present on the solar surface changes with time and exhibit a cyclic behaviour with an approximately 11 year period. The amplitude of the cycle (often called the strength of the cycle) varies from one cycle to another. Sunspots occur typically in the latitude range ± 35° and drift in latitude towards the equator as the cycle progresses (Spörer’s law). In recent years evidence has accumulated that the solar cycle has a long-term modulation consisting of epochs of hyperactivity (most recent being the Medieval maximum in the 12th century) as well as spells without sunspots (Maunder minimum during 1645-1715). These periods of abnormal activity are without explanation. Incidentally, the total solar irradiance (the energy from the Sun observed at Earth per unit area per unit time and unit wavelength interval) also exhibits a 11 year cycle which is in phase with the sunspot cycle and has implications for the terrestrial climate (see Lean, Chapter 13, this volume). 2.3.3. Active regions Active regions are areas on the Sun with enhanced magnetic fields and where magnetic flux has erupted through the photosphere into the chromosphere and corona. They are visible in white light, enhanced line emission (such as in Calcium), X-rays and radio wavelengths. Sunspots are ubiquitous features of active regions and are associated with eruptive phenomena such as solar flares,

16

S.S. Hasan

filaments, radio bursts, enhanced coronal heating, and coronal mass ejections. In addition ephemeral active regions that are bipolar in nature represent another group of regions on the solar surface. Considerable efforts have gone in to understand the birth, evolution, and decay of active regions, as well as the underlying mechanisms responsible for the above processes. Recently a new method for imaging active regions on the far side of the Sun has been developed31 using local helioseismology techniques. This will enable detection in advance of activity before these regions are accessible to direct observations. Detailed information on their properties has revealed by space and high resolution ground observations, which are discussed by Tripathi and Mason (Chapter 7, this volume). Possible scenarios for enhanced heating in active regions, particularly in the corona, are dealt by Mohan (Chapter 6, this volume). 2.4. Processes in the Corona 2.4.1. Eruptive phenomena Magnetic fields in active regions are responsible for a large number of dynamic processes that produce copious radiation over the full electromagnetic spectrum as well as acceleration of the solar plasma. Many of these occur on a short time scale of minutes to hours and generate effects that produce disturbances in the terrestrial environment and beyond. 2.4.1.1. Flares The most dramatic eruptive phenomenon in the atmosphere of the Sun is the solar flare producing radiation often over a wide range of frequencies (radio, optical, X-rays and even gamma rays) as well as energetic mass motions. The energy released in a large event is typically in the range of 1028 – 1033 erg. Smaller events called microflares (1026 erg) and nanoflares (1022 erg) have also been recently included in the category of flares. There have been suggestions that the latter are largely responsible for heating the corona32. Flares have traditionally been observed in the H-alpha (656.3 nm) line as bright kernels located on opposite sides of dark filaments that outline a region where the magnetic polarity changes sign (the so called neutral line). This suggests that magnetic field lines of opposite polarity must be involved in the physical mechanism that produces flares. Many flares begin with an eruption of the filament (filament activation). However, there are also events, particularly associated with small flares, which often involve an interaction of two loops and

Overview of the Sun

17

the release of energy through magnetic reconnection (i.e. the “annihilation” of magnetic field lines of opposite polarity). There is yet another class of flares in which pre-existing structures in the form of loops erupt – such events often produce the most powerful flares. It is generally accepted that the source of energy for flares comes from the magnetic field. However, what is still not firmly established are the precise mechanisms responsible for these phenomena. Considerable efforts have gone in to develop elaborate models to identify the processes that result in the sudden release of enormous energy. Most theoretical models are based on magnetic reconnection but the precise details are still under intense investigation. 2.4.1.2. Coronal Mass Ejections (CMEs) Coronal mass ejections (CMEs) are amongst the most energetic events in active regions involving the ejection of solar plasma into interplanetary space with speeds ranging from a few hundred to thousands of kilometers per second. They are often associated with flares, but there are events closely related to filament eruption without flares. CMEs have an important bearing on space weather and the Sun-Earth connection. A wealth of space and radio observations in recent years has provided considerable information on this phenomenon. Theoretical models for CMEs are under active study: in addition to specifying the mechanism that triggers CMES, models need to identify observable signatures, the necessary conditions for their onset and their relation to flares. It is generally agreed that they originate in regions of closed magnetic field lines. More details on CMEs and the associated radio bursts can be found in Srivastava (Chapter 10, this volume) and Manoharan (Chapter 11, this volume) respectively. 2.4.2. Other phenomena The outer solar atmosphere consisting of the corona is at a temperature between 1-2 × 106 K, making it visible in EUV and soft X-rays. Despite persistent efforts, there is no broad agreement on how the corona is heated (see Pandey & Narain, Chapter 9, this volume, for further details). Embedded in this hot tenuous atmosphere are the relatively cool dense prominences. New generation of telescopes from the ground as well as from space have revealed fine structure in prominences and provided valuable diagnostic information. A critical input for theoretical models is a precise knowledge of the vector magnetic field as well as of the thermodynamic quantities in prominences. Above active regions, loops are often observed, which are believed to be magnetically confined closed structures

18

S.S. Hasan

with an enhanced temperature (around 2.5 × 106) with a typical size between 104 km and 105 km. Smaller loops show up as bright points in X-rays. In addition the corona exhibits open magnetic structures known as coronal holes with a reduced density compared to the surrounding atmosphere. Dark regions in coronal holes are associated with the high-speed (600 – 800 km s-1) solar wind, which originates from magnetic funnels between supergranules (see Manoharan, Chapter 12, this volume). 2.5. Future Perspectives Solar physics has made impressive progress in recent years, particularly due to a good synergy between observations and theory. In the 1990s space missions such as YOKOH, SOHO and TRACE were launched which provided observations of solar features in unprecedented detail and contributed significantly to our knowledge of processes on the Sun. New missions launched recently such as STEREO and HINODE are supporting these efforts by providing multiwavelength information with high spatial and temporal resolution. The Solar Dynamics Observatory (SDO), the first major initiative of NASA as part of the Living With a Star (LWS) programme that is scheduled for launch during 2008, will focus on the causes of solar variability and its influence on the Earth’s environment. Long terms plans in the next decade include Solar Orbiter to study the Sun at a distance of 45 solar radii in an orbit co-rotating with the Sun. It will enable a study for the first time of the Sun’s polar regions. Another mission that is under study is Solar Probe that will come as close as 3 solar radii to the Sun’s surface in order to understand coronal heating and solar wind acceleration. A follow-up programme to SDO under LWS called the Solar Sentinels is also being considered. It consists of several spacecrafts at various distances from the Sun to study the origin of the solar winds and carry in situ measurements of energetic particles and interplanetary disturbances. In addition to space missions, several ground-based facilities have come up and many more are underway. The new developments in adaptive optics have provided a major fillip to the quality of ground-based observations. In fact, current ground-based telescopes at good sites and equipped with adaptive optics can provide angular resolution in optical wavelengths that is higher than is presently available from space. The Swedish Solar Telescope at La Palma and the German Vacuum Telescope in Tenerife are good examples of such facilities where a spatial resolution better than 0.2” (the limit on the Solar Optical Telescope on Hinode) can be achieved using adaptive optics. Two new facilities viz. the German 1.5-m GREGOR telescope at Tenerife and the 1.6-m NST (New

Overview of the Sun

19

Solar Telescope) of the Big Bear Observatory will shortly be commissioned. The Indian Institute of Astrophysics, Bangalore is planning a 2-m class National Large Solar Telescope (NLST) at a high altitude site, possibly in the Leh-Ladakh region. The main aims of this telescope is to resolve the fundamental scale in the solar atmosphere such as the pressure scale height, carry out spectroscopic observations with high spectral and spatial resolution of features close to the diffraction limit of the telescope and spectro-polarimetry in the visible and infrared to an accuracy of at least 0.1% to accurately derive vector magnetic fields. Proposals for larger facilities such as the Advanced Technology Solar Telescope (ATST) in the U.S.A. and the European Solar Telescope (EST) involving 4-m class instruments are under active consideration. A Frequency-Agile Solar Radio Telescope (FASR), a multifrequency (0.03-0 GHz) imaging radio interferometer consisting of 100 antennas, is also likely to come up in coming decade. Solar physics is going through an exciting phase with a large number of new programmes currently under way and several more on the anvil. These will provide multiwavelength information on a range of spatial and temporal scales that was unattainable earlier. Developments in observations and theory are beginning to yield a holistic picture of solar phenomena. References 1. Ahmad, Q. R., Allen, R. C., Andersen, T. C. et al., Phys. Rev. Lett. 87, 71301 (2001). 2. Parker, E. N., Astrophys. J. 122, 293 (1955). 3. Steenback, M., Krause, F. and Rädler, K. H., Z. Naturforsch. 21a, 369 (1966). 4. Stix, M., The Sun, Springer Verlag, Berlin (1989). 5. Babcock, H. W., Astrophys. J. 133, 572 (1961). 6. Leighton, R. B., Astrophys. J. 140, 1547 (1964). 7. Leighton, R. B., Astrophys. J. 156, 1 (1969). 8. Wang, Y.-M., Sheeley Jr, N. R. and Nash, A. G., Astrophys. J. 383, 431 (1991). 9. Durney, B. R., Solar Phys. 160, 213 (1995). 10. Dikpati, M. and Charbonneau, P., Astrophys. J. 518, 508 (1999). 11. Nandy, D. and Choudhuri, A. R., Astrophys. J. 551, 576 (2001). 12. Nandy, D. and Choudhuri, A. R., Science 296, 1671 (2002). 13. Charbonneau, P., Living Rev. Solar Phys., http://www.livingreviews.org/lrsp2005-2 (2005).

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14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

24. 25. 26. 27. 28. 29. 30. 31. 32.

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Brun, A. S., Miesch, M. S. and Toomre, J., Astrophys. J. 614, 1073 (2004). Simon, G. W. and Leighton, R. B., Astrophys. J. 140, 1120 (1964). Wang, H., Solar Phys. 117, 343 (1988). Stein, R. F. and Nordlund, Å., Solar Phys. 192, 91 (2000). Zwaan (1987). Solanki, S. K., Space Sci. Rev. 63, 1 (1993). Berger, T. E., Rouppe van der Voort, L. H. M., Löfdahl, M. G. et al., Astron. Astrophys. 428, 613 (2004). Rouppe van der Voort, L. H. M., Hansteen, V. H., Carlsson, M. et al. (2005). Rutten, R., Bettonivil, J. F. C. M., Hammerschlag, R. H. et al., in “Multiwavelength investigations of solar activity”, Eds. A. V. Stepanov, E. E. Benevolenskaya and A. G. Kosovichev, IAU Symp. (Cambridge) 223, 597 (2005). Sánchez Almeida, J., in “The Solar-B mission and the forefront of solar physics”, Eds. T. Sakurai and T. Sekii, ASP Conf. Ser. (San Francisco) 325, 115 (2004). Trujillo Bueno, J., Shchukina, N. and Asensio Ramos, A., Nature 430, 326 (2004). Orozco Suárez, D., Bellot Rubio, L. R., del Toro Iniesta, J. C. et al., Astrophys. J. 670L, 610 (2007). Petrovay, K. and Szakaly, G., Astron. Astrophys. 274, 543 (1993). Lites, B. W., Bida, T. A., Johannesson, A. et al., Astrophys. J. 373, 683 (1991). Degenhardt, D. and Wiehr, E., Astron. Astrophys. 252, 821 (1991). Thomas, J. H. and Weiss, N. O., Annual Rev. Astron. Astrophys. 42, 517 (2004). Weiss, N. O., Space Sci. Rev. 124, 13 (2006). Lindsey, C. and Braun, D. C., Solar Phys. 192, 261 (2000). Parker, E. N., Astrophys. J. 330, 474 (1988).

CHAPTER 3 SEISMIC VIEW OF THE SUN

S.M. CHITRE UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai-400 098, India B.N. DWIVEDI Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India

3.1. Introduction The Sun has played a major role in the development of mathematics and physics over the past centuries and has been widely described as the “Rosetta Stone” of Astronomy. This is undoubtebly an apt description of a celestial object whose internal and external layers provide an ideal laboratory for testing atomic and nuclear physics, high-temperature plasma physics and magnetohydrodynamics, neutrino physics and general theory of relativity. The proximity of our star to Earth has enabled us to make a close scrutiny of its surface regions and the overlying atmosphere. It has provided a wealth of information of high spatial resolution about its surface features which is evidently not possible for other stars. Indeed, from very ancient times the Chinese and Greek astronomers had not failed to notice the dark spots on the otherwise immaculate surface of the Sun. Solar astronomers have, in fact, maintained systematic records of the appearance of these vivid regions on the visible disk of the Sun, hoping to understand the processes that drive the solar cycle and possibly shed some light on its role in influencing the terrestrial climate. The interior of the Sun is clearly not directly accessible to observations as the internal layers are shielded by the solar material beneath the photosphere. But nevertheless, it is possible to infer the physical conditions prevailing inside the Sun with the help of equations governing its structure together with boundary conditions provided by observations.

21

22

S.M. Chitre and B.N. Dwivedi

For a spherical star the governing equations for mechanical equilibrium are the following:

d P (r ) GM (r ) = − ρ (r ) , dr r2 d M (r ) = 4π r 2 ρ (r ) . dr Here P(r) is the pressure, ρ (r) the density and M(r) mass interior to the radius r for a spherically symmetric Sun. For maintaining thermal equilibrium, the energy radiated by the Sun, as measured by its luminosity, L(r), must be balanced by the nuclear energy generated throughout the solar interior,

d L( r ) = 4π r 2 ρ (r ) ε , dr

ε is the energy generation rate per unit mass and L(r ) = 4π r 2 ( Frad + Fconv ), Frad and Fconv being respectively the radiative

where

and convective flux of energy. The energy generation takes place in the central regions of the Sun by the thermonuclear reactions converting hydrogen into helium mainly by the protonproton chain which contributes over 98% to the energy production with the rest less than 2% from the CNO cycle reaction. The energy generated by these reaction networks is transported from the centre to the solar surface by radiative processes within the inner two-thirds of the Sun and in the outer third by convective processes. The radiative flux is given by Frad = − K rad

dT , dr

( K rad =

4acT 3 is the radiative conductivity) , 3κρ

where a is the Stefan-Boltzmann constant, c the speed of light and κ the opacity of solar material, and the convective flux modeled in the framework of a local mixing-length formalism can be expressed as

Fconv = − K turb ρT

dS (r ) , dr

where K turb is the turbulent diffusivity and S the entropy.

Seismic View of the Sun

23

The outstanding question is how to check the correctness of the theoretically constructed models of the Sun. It turns out that the solar interior is, indeed, transparent to neutrinos resulting from thermonuclear processes in the solar core and also to seismic waves generated through bulk of the body of the Sun. These serve as complementary probes which provide valuable information about the structure and dynamics of the solar interior. The standard solar model (SSM) used for this purpose1 is based on the simplest set of assumptions: The Sun is assumed to be a spherically symmetric body with negligible effects of rotation, magnetic field, tidal forces and mass loss on its global properties. It is supposed to be in quasi-mechanical and thermal equilibrium with the energy generation taking place in the central regions by nuclear reactions which convert hydrogen into helium mainly by the proton-proton chain. The energy is transported outwards principally by radiative processes, but in the top third by radius the energy flux is carried largely by convection modeled in the framework of a mixing-length prescription, and there is no other process for transporting energy such as any wave motion. There is supposed to be no mixing of nuclear reaction production beyond the convection zone, except for the slow gravitational settling of helium and heavy elements by a slow diffusion beneath this zone into the radiative interior. It is presumed that the standard nuclear and neutrino physics is applicable for constructing theoretical solar models satisfying the observed constraints, namely mass, radius, luminosity and the ratio of chemical abundances Z/X, where X and Z refer respectively to the fractional abundance by mass of hydrogen and elements heavier than helium.

3.2. Solar Neutrinos The early investigations in Solar Physics were largely concerned with an extensive collection of spectroscopic data for inferring the temperature, density and chemical composition in the surface layers of the Sun. Since the mid-1960s there have been experiments set up to measure the flux of neutrinos released by the nuclear reaction network operating in the solar core2. The neutrino count rate is highly sensitive to the temperature and composition profile in the central regions of the Sun. It was hoped that the steep temperature dependence of some of the nuclear reaction rates involved would enable a determination of the Sun’s central temperature to an accuracy of better than a few percent. The motivation for setting up the neutrino experiment was “to see into the interior of a star and thus verify directly the hypothesis of nuclear energy generation in stars” and it was thought that “The use of a radically different observational probe may reveal wholly unexpected phenomena; perhaps, there is some great surprise in store for

24

S.M. Chitre and B.N. Dwivedi

us when the first experiment in neutrino astronomy is completed”3. Indeed, there have been valiant efforts since the mid-1960s to set up experiments that are designed to undertake the exceedingly difficult measurement of neutrinos from the Sun.

PROTON-PROTON CHAIN

p + p → d + e + + νe −

p + e + p → d + νe 3

p+d → pp-I:

pp-II:

pp-III:

He + 3 He →

3

He + p →

3

He + 4 He → −

Be + e

7

Li + p →

4

4

7 7

→ 8

He + 2 p

He + e + + νe

(≤ 18.8 MeV)

Be + γ

Li + νe

(0.38, 0.86 MeV)

Be + γ

8

Be → 2 He

3

He + 4 He →

7

Be + p →

8

8

B →

8

(1.44 MeV)

He + γ

3

7

(≤ 0.42 MeV)

4

8

7

Be + γ

B + γ

Be + e + + νe

(≤ 14.6 MeV)

4

Be → 2 He

CNO CYCLE 12

C+ p →

13

N →

13

13

C + e + + νe

13

C+ p →

14

14

N+ p →

15

15

15

15

O →

N+ p →

N+ γ

N+ γ

O+ γ

N + e + + νe 12

(≤ 1.2 MeV)

4

C + He

(≤ 1.7 MeV)

Seismic View of the Sun

25

or 15 16

O+ p →

16

O+ γ

O+ p →

17

F+ γ

17

O + e + + νe

17

F →

17

O +p →

14

(≤ 1.7 MeV)

N + 4 He

Davis’s Chlorine experiment was located some 1480 m underground in the Homestake gold mine in South Dakota. It has a tank containing 615 tons of liquid perchloroethylene (C2Cl4) which is sensitive to intermediate and high energy neutrinos. In this experiment the Chlorine nuclei serve as solar neutrino absorbers according to the reaction 37

Cl + ν
→ 37Ar + e– (threshold = 0.814 MeV).

The count rate is dominated by the high-energy 8B neutrinos contributing 5.9 SNU, with 7Be neutrinos making a contribution of 1.1 SNU (1 SNU = 10–36 captures per target atom per second). The theoretically predicted capture rate for SSM for the Chlorine experiment is 7.6 ± 1.2 SNU4. Davis, however, reports measurement of the solar neutrino count rate of 2.56 ± 0.023 SNU which clearly shows a puzzling deficit by nearly a factor of 3 over the SSM prediction. Throughout the experimental runs in the Homestake mine, Davis has been consistently reporting a count rate which is significantly lower than that predicted by the standard solar model. This is the celebrated solar neutrino problem which has been haunting the community of solar and neutrino physicists for nearly four decades or so. There have been a number of ingenious suggestions5 which have been proposed to lower the central temperature of the Sun. These have included proposals invoking partial mixing in the solar core which can bring additional fuel of hydrogen and helium to the centre, thus maintaining the nuclear energy production at a slightly lower temperature; the presence of a small admixture of Weakly Interacting Massive Particles (WIMPs) in the central regions which would effectively diminish the temperature gradient as a result, contributing an increase in the thermal conductivity; the rapidly rotating solar core; the centrally concentrated magnetic field; lower heavy element abundance in the core. All these proposals lead to a slight reduction in the central temperature resulting in a lowering of the flux of high energy neutrinos. A quarter of a century after the Chlorine experiment, the Japanese experiment consisting of a 680 ton of ordinary water tank was located about a

26

S.M. Chitre and B.N. Dwivedi

kilometer underground in the Kamiokande mine. This experiment was designed to detect charged particles by measuring Cérenkov light through the elastic scattering reaction

νx + e– → ν′x + e– (threshold = 5 MeV). The Kamiokande and the later upgraded Super-Kamiokande experiment are sensitive to the count of the high-energy 8B neutrinos released in the reaction network. The measured flux from the Super-Kamiokande experiment again shows deficiency by a factor of 2 over the total flux predicted by SSM. It is clear that the Chlorine and Super-Kamiokande experimental measurements are inconsistent with the proposition of resolving the solar neutrino problem by lowering the central temperature. Such a reduction of the central temperature will cause even a larger suppression of the high-energy 8B neutrino flux to which the Super-Kamiokande is exclusively sensitive, while the Chlorine experiment which detects the intermediate as well as the high energy neutrinos shows even a larger deficit in the neutrino counting rate! This is a paradoxical situation which leads to the conclusion that a cooler solar core is not a viable solution of the solar neutrino puzzle. Besides these two experiments, there are three other radiochemical experiments, GALLEX, SAGE and GNO that use gallium detector for capturing the lower energy neutrinos via the reaction: 71

Ga + νe → 71Ge + e–

(threshold = 0.233 MeV).

The counting rate for the gallium experiments is on an average 74.7 ± 5.0 SNU, while the SSM prediction of the neutrino counting rate is 128 ± 8 SNU, again showing a deficit in the measured neutrino flux. A possible resolution of this conundrum is to endow neutrinos with a tiny mass enabling them to transform their flavour during propagation. Thus, the electron neutrinos released in the reaction network could get converted into neutrinos of a different flavour while transiting through the interior of the Sun and of the Earth and along their flight path through the interplanetary space. The first compelling evidence for such oscillations of neutrino flavours came from the Super-Kamiokande’s analysis of data on high energy cosmic ray produced neutrinos in the Earth’s atmosphere. The asymmetry in the measured up and down fluxes of neutrinos produced by cosmic ray interactions with the terrestrial atmosphere would be the result of passage of the upward moving neutrinos through the solid mantle of Earth, while the downward moving neutrinos coming

27

Seismic View of the Sun

from overhead are generated afresh in the Earth’s atmosphere and are less likely to undergo any flavour oscillations. The recent measurements by the Sudbury Neutrino Observatory (SNO) appears to provide convincing evidence for solar neutrino oscillations. The SNO experiment located at a depth of over 6000 meters of water equivalent in Sudbury (Canada) uses 1000 ton of heavy water containing the isotopes of deuterium. In both heavy water tank in Sudbury and the ordinary water detector at Super-Kamiokande, neutrinos can elastically scatter electrons to produce Cérenkov radiation, but such an electron scattering may be caused by any of the three neutrino flavours (x = e -, µ - and τ - neutrinos):

νe + d → p + p + e − (charged current) ν x + e − → ν′x + e −

(elastic scattering)

ν x + d → ν′x + p + n

(neutral current)

SNO’s heavy water detector is capable of isolating electron neutrinos via the charged current (CC) reaction. The neutral current (NC) reaction is equally sensitive to all the neutrino flavours, while the elastic scattering (ES) has significantly lower sensitivity to µ – and τ – neutrinos. SNO has reported the elastic scattering count rate which equals Super-Kamiokande’s event rate to within experimental errors. It is noteworthy that SNO’s count of the charged current reaction which is sensitive exclusively to the electron – neutrinos is lower than the count rate of Super-Kamiokande and SNO. Interestingly, the total 8 B neutrino flux as measured by the NC reaction is (5.09 ± 0.62) × 106 cm-2 s-1, in agreement with the prediction of SSM. Table 1. Solar neutrino experimental results. Expt.

Chlorine

Gallium

Superkamiokande

Sudbury

Threshold (MeV)

0.834

0.233

5

5

R

0.33 ± 0.03

0.55 ± 0.03

0.465 ± 0.015

0.36 ± 0.015

(0.36 ± 0.015)*

(1.0 ± 0.1)**

R = * **

Measured neutrino flux

.

Predicted model neutrino flux = Neutral current corrected neutrino flux. = Total neutrino flux measured by neutral current reaction.

28

S.M. Chitre and B.N. Dwivedi

These experimental measurements are a reassurance to Solar Physicists that the resolution of solar neutrino problem should be sought in the realm of Particle Physics and that non-standard neutrino physics is responsible for the deficit in the measured neutrino fluxes, thus validating the bold proposal made by Gribov and Pontecorvo6 that the discrepancy between theoretically predicted neutrino count rate and Davis’s experiment could be due to our inadequate understanding of neutrino physics!

3.3. Seismic Sun The surface of the Sun undergoes a series of mechanical vibrations which manifest as Doppler shifts in a spectral line. These solar oscillations were discovered by Leighton et al.7 when they measured the velocity at some point on the solar disk to find an oscillatory pattern centred around a period of 5 minutes. The nature of these oscillations was clarified8,9 as acoustic modes of pulsation of the whole solar body that are trapped below the surface. Subsequently Deubner10 established that the power in oscillations is concentrated along a series of ridges in the frequency – wavenumber diagram in exact accordance with the prediction of theoretical models for acoustic modes. The frequencies of oscillations representing a superposition of millions of independent modes with amplitude of the order of a few cm/s have been determined to an accuracy of better than 1 part in 105 largely because of continuous observations extending over very long periods of time achieved with the help of ground-based networks (GONG, BiSON, TON) and satellite – borne instruments (MDI) on board SOHO. The accurately measured oscillation frequencies have provided very stringent constraints on the admissible solar models. The oscillatory modes are characterized by the spherical harmonic degree, ℓ, the azimuthal order m and the radial order n. Were the Sun to be spherically symmetric, the frequencies would be independent of m, but on account of asphericities due to effects of rotation, magnetic field, thermal perturbation, velocities, indeed, the frequencies depend on m. Since the departures from spherical symmetry are ≤ 10-5, the oscillation frequencies may be conveniently expressed as

ν nℓm = ν nℓ + ∑ a knℓ p kℓ (m) . k

Here ν nℓ is the mean frequency determined from the spherically symmetric structure of the solar interior; a knℓ the splitting coefficients and pkℓ (m) are orthogonal polynomials of degree k in m.

Seismic View of the Sun

29

There are two main classes of waves generated inside the Sun: highfrequency acoustic p-modes which are driven by pressure forces and lowfrequency gravity g-modes for which buoyancy is the main controlling force, and in between are the f-modes which are surface gravity modes which are essentially independent of stratification. The helioseismic data of oscillation frequencies may be analyzed in two ways: i) Forward method, ii) Inverse method. In the forward method, an equilibrium standard solar model is perturbed in a linearized theory to determine the eigen-frequencies of solar oscillations and these are compared with the precisely measured p-mode frequencies11. The fit is naturally seldom perfect, but a comparison of the frequencies indicates that the thickness of the convection zone is close to 200,000 km and the helium abundance by mass Y, in the solar envelope is about 0.25. The direct method has had only a limited success, although it led to an improvement of the input microphysics such as opacities and emphasized the role of diffusion of helium and heavy elements beneath the convection zone into the radiative interior12. The limitations of the forward technique prompted the use of inversion techniques13 which have proved quite effective in inferring the acoustic structure of the Sun using only equation of mechanical equilibrium. While adopting the inversion method, it is convenient to write the adiabatic equations of solar oscillations in the variational form14 which may be linearized to express the differences between the frequencies obtained from the reference model and those measured for the Sun by relating to the differences in the sound speed cs and density ρ, say, as R δν nℓ R nℓ δ c2 F (ν nℓ ) δρ = ∫ κ c 2 , ρ (r ) 2s (r ) dr + ∫ κ ρnℓ,c 2 (r ) (r ) dr + , s ν nℓ 0 s E nℓ ρ cs 0

where the kernels κ cn2ℓ, ρ (r ) and κ ρnℓ,c 2 (r ) are determined by the eigenfunctions s

s

given by the reference model and δν nℓ , δc s2 and δρ represent the difference between the Sun and a solar model. The mode inertia is Enℓ and F (ν nℓ ) is the surface term. A triumph of the inversion technique has been a very reliable inference about the internal acoustic structure of the Sun15,16. The profile of the sound  ∂ ℓn P   being the adiabatic index, can be speed, c s = Γ1 P / ρ , Γ1 ≡   ∂ ℓn ρ  s determined through bulk of the solar interior to an accuracy better than 0.1% and

30

S.M. Chitre and B.N. Dwivedi

the profiles of density and adiabatic index to a somewhat lower accuracy. There is a remarkable agreement between the profiles of the sound speed (and also density) deduced from seismic inversions and SSM, except for a pronounced discrepancy near the base of the convection zone and a noticeable difference in the energy generating core (Fig. 3.1).

Figure 3.1. Relative difference in sound speed profile between the Sun (as inferred by seismic inversions) and a standard solar model (SSM).

The hump near the base of the convection zone may be attributed to a sharp change in the gradient of helium abundance profile almost certainly resulting from diffusion, and a moderate amount of rotationally-induced mixing just beneath the convection zone can smooth out this feature17. The dip in the relative sound speed difference around 0.2 R
may be due to ill-determined chemical composition profiles in the SSM, possibly resulting from inadequate understanding of the diffusion process or from the use of inaccurate nuclear reaction rates. The adiabatic index Γ1, which is normally equal to 5/3, is decreased below this value in the hydrogen and helium ionization zones and the extent of reduction is determined by the equation of state and by chemical abundances. The dimensionless ratio of acoustic and gravitational acceleration, represented by,

W (r ) =

1 dcs2 g dr

Seismic View of the Sun

31

has a value of ≈ –2/3 through bulk of the convection zone (cf. Fig. 3.2) and shows features resulting from the dip in the adiabatic index Γ1 in the helium ionization zone. Thus, the peak in the plot of W(r) around r = 0.98 R
may be calibrated to determine the helium abundance in the convective envelope18,19 to obtain Y = 0.249 ± 0.003. The large peak in W(r) near the surface (r ≈ 0.99R
) arises from hydrogen and singly-ionized helium ionization zones, and near r = 0.7R
. The striking discontinuity in the gradient of W(r) marks the base of the convection zone which can be used to determine the depth of the convection zone. The base of the convection zone from the seismic data comes out to be (0.7315 ± 0.0005)R
20. It is also possible to estimate seismically the heavy element abundance in a manner similar to that adopted for determination of helium abundance, since different elements leave separate imprints on W(r) below the He II ionization zone. The signatures are necessarily small because of the low abundance of heavy elements, but it becomes possible to determine Z with this technique to obtain Z = 0.0172 ± 0.00221. The extent of overshoot of convective eddies beneath the base of the convection zone can also be surmised from the oscillatory signal in frequency differences. It is found to be consistent with no overshoot, with an upper limit of 0.05 Hp (Hp = local pressure scale height)22,23.

Figure 3.2. The function W(r) for a solar model is shown by the continuous line, while the dashed line represents the same for the Sun using the inverted sound speed profile.

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S.M. Chitre and B.N. Dwivedi

The recent downward revision of solar abundances of oxygen and other heavy elements by Asplund et al.24,25 from Z = 0.017 to Z = 0.012 has led to a serious discordance between SSM constructed with new abundances of heavy elements and the seismically inferred solar models. There have been a number of attempts to resolve this vexatious problem (e.g. increasing the diffusion rate of helium and heavy elements beneath the convection zone; increasing the opacity near base of the convection zone by ~ 15-20%, increasing the neon abundance), but none of them have had a universal acceptance. The global helioseismic structure of the Sun discussed hitherto is based on the equations of mechanical equilibrium. However, in order to determine the temperature and chemical composition profiles in the solar interior, it is necessary to supplement the seismically inferred structure through primary inversions by the equations of thermal equilibrium, together with the auxiliary input physics such as the opacity, equation of state and nuclear energy generation rates26,27,28. It turns out that the inverted sound speed, density, temperature and composition profiles, and consequently the resulting neutrino fluxes come close to those predicted by SSM. In general, the total solar luminosity computed from the seismically inferred profiles would not necessarily match the observed total luminosity (L
≈ 3.8 × 1033 erg s-1). The discrepancy between the computed and observed solar luminosity can, in fact, be effectively used to provide a diagnostic test of input nuclear physics; in particular, it can be demonstrated that the proton-proton cross-section, S11 needs to be increased slightly to S11 = (4.06 ± 0.07) × 10-25 MeV barn29. The calculations also enable us to set limits on the heavy element abundance, Z in the solar core30. The seismic models lead to a determination of the central temperature of Sun ~ (15.6 ± 0.4) × 106 K, allowing for an uncertainty of up to 10% in the opacities31. It is also possible to infer the helium abundance profile Y, assuming the heavy element abundance, Z. The resulting helium abundance profile comes to a fairly close agreement with that obtained with SSM which includes diffusion, except in the regions just below the convection zone where the abundance profile is essentially flat27. This is again indicative of some sort of a mixing, possibly resulting from a rotationally- induced instability. It is noteworthy that the temperature at base of the convection zone is ≤ 2.2 × 106 K which is not adequate to burn lithium. However, if there is some amount of mixing that extends beyond the base to a radial distance of 0.68R
, temperatures will exceed 2.5 × 106 K at which lithium can be destroyed by nuclear burning to explain the low lithium abundance at the solar surface.

Seismic View of the Sun

33

Figure 3.3. The inferred rotation as a function of depth inside the Sun at different solar latitudes.

The surface rotation of the Sun has been measured through observations of sunspots and other features. It is well known that the Sun has differential latitudinal rotation with the equatorial layers rotating faster than the polar regions. The rotation rate in the Sun’s interior could be determined from the frequency splittings of various modes, since each mode of solar oscillation is trapped in a different region and it is possible to infer both the radial and latitudinal variation of the rotation rate inside the Sun using the accurately measured frequency splittings of acoustic modes. The Coriolis force gives the first-order contribution from rotation with the resulting splittings having only odd powers of the azimuthal order m. These odd order splitting coefficients, a1, a3, a5, … enable a determination of the rotation rate as a function of radius and latitude. The centrifugal force which is of second order in rotation contributes only to even order splitting coefficients, a2, a4, a6, …. The rotation rate obtained by averaging the GONG and MDI data sets, as a function of fractional solar radius and latitude is displayed in Fig. 3.3. It is evident that the observed surface differential rotation persists through the solar convection zone, with the radiative interior rotating almost uniformly32. The transition region near the base of the convection zone, called the tachocline, is centred at a radial distance of r = (0.6916 ± 0.0019)R
has a half-thickness of

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S.M. Chitre and B.N. Dwivedi

(0.0065 ± 0.0013)R
33. There is evidence for a shear layer just beneath the solar surface extending to r ≈ 0.95R
with the rotation rate increasing with depth. The helioseismically inferred rotation rate in the solar interior is consistent with the measured solar oblateness of approximately 10-5 (ref. 34). The resulting quadrupole moment turns out to be (2.18 ± 0.06)10-7 (ref. 35) which leads to a precession of perihelion of the orbit of planet Mercury of ~ 0.03 arcsec/century. This is clearly consistent with the framework of general theory of relativity.

3.4. Inconstant Sun The accumulation of helioseismic data by GONG and MDI projects over this past solar cycle 23 has enabled a study of the temporal variations in the solar structure and dynamics with the activity cycle. Earlier works11,36 of Libbrecht & Woodard36, and Elsworth et al.11 (1990) had, in fact, established temporal variations of the solar oscillation frequencies with the shift in the frequencies of acoustic modes up to 0.4 µHz during the course of the cycle, with maximum frequency occurring at the peak of solar activity. It was also demonstrated by Bhatnagar et al.37 that the frequency variation is well correlated with various solar activity indices. Furthermore, the frequency shift scaled by mode inertia depends on frequency alone, thus indicating that the frequency variations probably occur near the surface layers of the Sun and that the temporal variations in the structure are confined to the outermost layers below the photosphere and there is hardly any variation discerned in the solar interior. Likewise, temporal variations of the even splitting coefficients appear to be well correlated with the corresponding component of the surface magnetic flux38 and the surface term is found to incorporate most of the observed time variations. The time variations in the frequencies of f-modes which are surface gravity modes, however, seem to have two components: an oscillatory component with a period of almost exactly 1 year and a secular component well correlated to solar activity. The former is very likely reflecting the orbit period of Earth in data analysis, while the latter is probably on account of the magnetic field located in the superficial sub-surface layers varying with the solar activity cycle. One of the most striking observations about solar rotation rate was made by Howard and LaBonte39 to demonstrate that there is a temporal variation of the surface rotation rate with the solar activity cycle. These observations established the existence of faster- and slower- than average bands migrating towards the equator. This so called “torsional oscillation” was found to be highly correlated to the migrating field pattern in the ‘butterfly diagram’40. The helioseismic data accumulated over the solar cycle has now established that this zonal band pattern

Seismic View of the Sun

35

persists through the solar convection zone41,42. It is found that at mid-latitudes the bands of fast and slow rotation migrate equatorwards like the sunspot butterfly plot, while at high latitudes, they migrate towards the poles resembling the pattern of magnetic features observed at the surface43,44 (Fig. 3.4). Interestingly, helioseismic observations over the solar cycle indicate that the polar regions speed up while the equatorial latitudes slow down with the buildup of the activity cycle45. Through bulk of convection zone it appears that the temporally varying kinetic energy of rotation shows an increasing trend in high latitudes and a decreasing trend in the low latitudes with progress of the solar cycle. The residual kinetic energy of rotation and magnetic energy are, in fact, out of phase in the equatorial band and the magnetic field tends to show an increase from minimum to the maximum phase of the solar cycle. It is, therefore, tempting to speculate that angular momentum is being transported from the equatorial to polar regions during the course of the solar cycle, perhaps by meridional circulations serving as a conveyor belt !

Figure 3.4. Migrating zonal bands.

Solar irradiance is known to vary on at least two time-scales – one related to the period of solar rotation and the other on the longer time-scale of activity cycle with the amplitudes of both these variations ≥ 0.1% (ref. 46). The short time-scale changes (~ 1 month) are probably a consequence of surface magnetic features (e.g. sunspots, faculae, magnetic network), while the longer time-scale

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S.M. Chitre and B.N. Dwivedi

(~ 11 years) variations which are in phase with the solar activity cycle may be primarily due to changes in the solar luminosity. It is likely that the solar irradiance variation arises partly from the competing effects of darkening due to sunspots and brightening due to faculae and also the network and also partly from cyclic changes in the luminosity caused by fluctuations in the thermal energy content in the outer layers of the Sun. During the course of the activity cycle, there is a slight brightening and darkening of the Sun and its total energy which is modulated periodically and gets channelled into different reservoirs such as potential, rotational or magnetic. One of the outstanding problems in Solar Physics is to identify a plausible underlying mechanism that is responsible for simultaneous temporal variation with solar activity cycle of oscillation frequencies, solar rotation, magnetic field and total solar irradiance.

Acknowledgements We thank H.M. Antia for valuable comments and for supplying the Figures.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

Bahcall, J.N. and Pinsonneault, H.M., Rev. Mod. Phys. 67, 781 (1995). Davis, R., Phys. Rev. Lett. 12, 303 (1964). Bahcall, J.N., Astrophys. J. 149, L7 (1967). Bahcall, J.N., Basu, S. and Pinsonneault, H.M., Phys. Lett. B433, 1 (1998). Chitre, S.M., Bull. Astron. Soc. India 23, 379 (1995). Gribov, V. and Pontecorvo, B., Phys. Lett. B28, 493 (1969). Leighton, R.B., Noyes, R.W. and Simon, G.W., Astrophys. J. 135, 474 (1962). Ulrich, R.K., Astrophys. J. 162, 993 (1970). Leibacher, J.W. and Stein, R.F., Astrophys. J. 7, L191 (1971). Deubner, F.-L., Astron. Astrophys. 44, 371 (1975). Elsworth, Y., Howe, R., Isaak, G.R., McLeod, C.P. and New R., Nature 347, 536 (1990). Christensen-Dalsgaard, J., Proffitt, C.R. and Thompson, M.J., Astrophys. J. 403, L75 (1993). Gough, D.O. and Thompson, M.J., in Solar Interior and Atmosphere (eds. Cox, A.N., Livingston, W.C., Mathews, S.M.), Univ. Arizona Press, p. 519 (1991). Chandrasekhar, S., Astrophys. J. 139, 664 (1964). Gough, D.O. et al., Science 272, 1296 (1996).

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16. Kosovichev, A.G. et al., Solar Phys. 170, 43 (1997). 17. Brun, A.S., Turck-Chièze, S. and Zahn, J.-P., Astrophys. J. 525, 1032 (1999). 18. Gough, D.O., Mem. Soc. Astron. Ital. 55, 13 (1984). 19. Däppen, W., Gough, D.O., Kosovichev, A.G. and Thompson, M.J., Lecture Notes in Phys., Vol. 388, 111 (1991). 20. Basu, S., Monthly Notices Roy. Astron. Soc. 298, 719 (1998). 21. Antia, H.M. and Basu, S., Astrophys. J. 644, 1292 (2006). 22. Basu, S., Antia, H.M. and Narasimha, D., Monthly Notices Roy. Astron. Soc. 585, 553 (2003). 23. Monteiro, M. J. P. F. G., Christensen-Dalsgaard, J. and Thompson, M. J., Astron. Astrophys. 283, 247 (1994). 24. Asplund, M., Greversse, N., Sauval, A.J., Allende Prieto, C. and Kiselman, Astron. Astrophys. 417, 751 (2004). 25. Asplund, M., Grevesse, N. and Sauval, A.J., in Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, ASP Conf. Ser. 336, p. 25 (2005). 26. Gough, D.O. and Kosovichev, A.G., in Inside the Sun, IAU Coll. 121, 327 (1990). 27. Antia, H.M. and Chitre, S.M., Astron. Astrophys. 339, 239 (1998). 28. Takata, M. and Shibahashi, H., Astrophys. J. 504, 1035 (1998). 29. Brun, A.S., Antia, H.M., Chitre, S.M. and Zahn, J.-P., Astron. Astrophys. 391, 725 (2002). 30. Antia, H.M. and Chitre, S.M., Astron. Astrophys. 393, L95 (2002). 31. Antia, H.M. and Chitre, S.M., Astrophys. J. 442, 434 (1995). 32. Thompson, M.J., Toomre, J.; Anderson, E., Antia, H.M., Berthomieu, G., Burtonclay, D., Chitre, S.M., Christensen-Dalsgaard, J., Corbard, T., Derosa, M. and 16 coauthors, Science 272, 1300 (1996). 33. Basu, S. and Antia, H.M., Astrophys. J. 585, 553 (2003). 34. Kuhn, J.R., Bush, R.I., Scherrer, P. and Scheick, X., Nature 392, 155 (1998). 35. Pijpers, F.P., Monthly Notices Roy. Astron. Soc. 297, L76 (1998). 36. Libbrecht, K.G. and Woodard, M.F., Nature 345, 779 (1990). 37. Bhatnagar, A., Jain, K. and Tripathy, S.C., Astrophys. J. 521, 885 (1999). 38. Antia, H.M., Basu, S., Hill, F., Howe, R.W. and Schou, J., Mon. Not. R. Astron. Soc. 327, 1029 (2001). 39. Howard, R. and LaBonte, B.J., Astrophys. J. 239, L33 (1980). 40. Snodgrass, H.B., Astrophys. J. 383, L85 (1991).

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41. Howe, R., Christensen-Dalsgaard, J., Hill, F., Komm, R.W., Larsen, R.M., Schou, J., Thompson, M.J. and Toonre, J., Astrophys. J. 533, L163 (2000). 42. Vorontsov, S.V., Christensen-Dalsgaard, J. Schou, V.N., Strakhov, V.N. and Thompson, M.J., Science 296, 101 (2002). 43. Leroy, J.-L. and Noens, J.-C., Astron. Astrophys. 120, L1 (1983). 44. Makarov, V.I. and Sivaraman, K.R., Solar Phys. 123, 367 (1989). 45. Antia, H.M., Chitre, S.M. and Gough, D.O., Astron. Astrophys. 477, 657 (2008). 46. Fröhlich, C. and Lean, J., Astron. Astrophys. Rev. 12, 273 (2004).

CHAPTER 4 SOLAR MAGNETISM P. VENKATAKRISHNAN and SANJAY GOSAIN Udaipur Solar Observatory, Physical Research Laboratory, P. Box 198, Dewali, Udaipur 313001, Rajasthan, India

INTRODUCTION This chapter is basically divided into 2 parts. In the first part, the important properties of the solar magnetic field are summarized. The discussion begins with a simple introduction to solar magneto hydrodynamics. This introduction will be sufficient to understand the current status of the solar dynamo theory that follows. Some very curious and interesting results on force free fields are then presented in very basic terms. Finally, the application of this theoretical framework to the problems of coronal heating, solar flares and coronal mass ejections are developed in a simple unified scheme, based on a hierarchy of physical conditions. The second part consists of a tutorial on magnetographs. It begins with a description of polarization of light from very fundamental notions of coherence of light. This is followed by simple but comprehensive explanations of the Zeeman and Hanle effects along with the necessary basic ideas of quantum physics of scattering of light. Then the working of a few important magnetographs is outlined, with special emphasis on a solar vector magnetograph developed for USO, to provide a “hands on” perspective. The article concludes with a few brief remarks on the possible future directions for research in the domain of solar magnetism. PART 1. FUNDAMENTALS There is a theorem in astrophysics called the Vogt-Russell theorem, which states very confidently that the structure of a star can be completely determined once we know its mass and chemical composition. Alas, this confidence is not fully justified. The reason is that the atmospheres of two stars with the same mass, 39

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P. Venkatakrishnan and S. Gosain

luminosity, and chemical composition are sometimes seen to be very much different from each other. What causes such differences? The answer is not completely understood by astrophysicists. What we are somewhat sure about is the fact that the study of the Sun’s atmosphere and its relationship with the features seen on the Sun’s surface will eventually provide the key to solve this puzzle. 4.1. Sunspots and the Eleven Year Sunspot Cycle The first inkling of the surprises in store for astronomers came with the invention of the telescope. Galileo turned a telescope at the Sun, and found that the Sun’s face was not pure white, but had several dark spots on it. These blemishes on the face of a heavenly body caused a lot of confusion to the religious leaders of that time who always imagined that heavenly bodies were free of defects. The dark sunspots remained a curiosity for many years. Careful recording of the sunspots’ positions, day after day, for many years by many scientists revealed a curious waxing and waning of the number of sunspots with a rhythm of about eleven years. The names of Schwabe, Carrington, Wolf, Maunder and Sporer, are linked with the exciting story of the discovery of the sunspot cycle1, shown in figure 1. What made these spots increase and decrease in number? Why are the spots dark? These were some of the provocative questions that arose at that time. Even though some progress has been made towards answering these questions, we are far away from a complete understanding. The first step towards a physical understanding of sunspots became possible soon after Zeeman discovered, in 1896, that spectral lines formed in magnetic fields split into many components and was awarded the Nobel Prize for this discovery. George Ellery Hale, of Mount Wilson Observatories in the United States of America, had noticed that the picture of sunspots taken in a spectral line of hydrogen showed whirlpool-like structures. He was reminded of the distribution of iron filings around a magnet. The Zeeman effect now gave Hale a way to detect magnetic fields in sunspots by looking for splitting of spectral lines in the light coming from sunspots. Indeed, he did see such a splitting in 1908, which confirmed that the sunspots had strong magnetic fields present in them (Hale, 1908). John Evershed, from Kodaikanal observatory wanted to see whether the field was produced by a whirlpool motion of gases within the sunspot. He went about measuring the motions of the gases using a spectrograph, and was surprised to see no whirlpool motion, but a radial outward motion from sunspots (Evershed, 1909). This discovery, in 1909, is yet to be satisfactorily explained. Evershed’s intuitive ideas about the possible ways of producing

Solar Magnetism

41

magnetic fields in electrically conducting fluids were much ahead of his times. The science of conducting fluids became fully developed only in middle of the 20th century, culminating in the discovery of hydro-magnetic waves by Hannes Alfven (Alfven 1942), for which he received the Nobel Prize in 1970. This new branch of physics began to be vigorously applied to the problem of magnetic field production in the Sun. In the regions of strong magnetic fields, the movement of plasma across the fields sets up an inductive electric field, much in the same way as the flow of induced current along a coil of wire that is rapidly moved in a magnetic field. The induced electric current in the solar plasma produces a magnetic field opposing the original motion of the fluid, which set up the induced current in the first place.

Figure 1. Plot of monthly averaged sunspot number showing the 11 year cycle.

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4.2. Magnetic Cycle Judging by the very “broken up” nature of the magnetic field on the solar surface (figure 2), we might even wonder whether one can talk about the magnetic field of the whole sun. However, there are several indications that there is something global about the magnetic field. First, we must realize that magnetic dynamos depend on the pattern of fluid motion. Evershed failed to find local dynamos for the individual sunspots, thereby indicating that the origin of the spot magnetic fields is related to some other pattern of fluid motions, not related to the individual spots. One such global motion is the rotation of the Sun, which Carrington first noticed from the systematic movements of the sunspots across the face of the Sun. Hale and Nicholson (1938) noticed that all the pairs of sunspots having opposite magnetic polarity behaved in a systematic way during every eleven year sunspot cycle (figure 3). First, these bipolar spots invariably had their axis almost parallel to the solar equator (as defined by the solar rotation). Each bipolar pair of spots had their magnetic polarity pointed the same way during every eleven years. After completion of eleven years, the pairs appearing in the new cycle had their polarities swapped with respect to that of the previous cycle. Further, the pattern in the southern hemisphere was opposite to that in the northern hemisphere of the Sun. This type of systematic behavior over a long time clearly indicates a global origin for the spot magnetic fields as can be visualized in a “magnetic butterfly diagram” in figure 4.

Figure 2. A full-disk magnetogram showing line-of-sight magnetic flux distribution, observed by SoHO/MDI satellite.

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43

Figure 3. A Cartoon of the Hale’s polarity rule. At T=0, (sunspot minimum) the spots of a new cycle appear at high latitudes. The polar field has maximum strength during sunspot minimum. The polarity of the polar field is the same as that of the leading spot of the corresponding hemisphere. The size of the spot cartoon is a measure of the number of spots. As we proceed into the cycle (T=2.75 years), the spots increase in number and appear at lower latitudes. At the same time, the strength of the polar field decreases (depicted by decreasing area). At solar maximum (T=5.5 years), the polar field goes through a minima reversing its sign, while sunspots appear at mid latitudes. At the next sunspot minimum (T=11 years), the old cycle spots are close to the equator while new cycle spots appear at high latitudes with their signs reversed. This continues through the new cycle maximum (T=16.5 years), until T=22 years, when magnetic cycle is completed.

Figure 4. Magnetic butterfly diagram indicates the global solar surface magnetic field over one complete magnetic cycle. The magnetic field is shown as gray scale vs. latitude and time as constructed from a sequence of Carrington maps obtained at NSO KP. White areas represent positive polarity; darker areas represent negative polarity.

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A simple model to explain this behavior of magnetic field was proposed by Babcock (1961). He assumed that the Sun has a seed field that stretches from one pole to the other, like what one would encounter in a bar magnet, for example. If the Sun rotated about its axis with the same period at all the latitudes, then this seed field would merely move round and round like the wires of a wicker basket would move around if we rotated the basket around (see figure 5). However, the Sun is known to rotate faster at the equator than at the poles. If we continued our experiments with the wicker basket and made the middle portion of the basket rotate faster than its end portions, then we will end up twisting up the basket into a sorry shape. The uneven rotation of the Sun will do very much the same thing to the seed field, producing large twists in the field. The twist in the lines of force can go unchecked, but nature has a safety valve. The twisted field has a tendency to push material out of the knotted places, making these portions lighter than their surroundings. The knots in the field then rise up to the surface of the Sun and produce the sunspots.

Figure 5. Generation of toroidal field from poloidal field by the action of differential rotation: (a) Purely poloidal field (b) & (c) progressive winding of the field lines (d) Purely toroidal fields.

A part of the knotted field gets straightened out again because of untwisting cyclonic motion of the gases in the convection zone, and we get back some of the seed field.

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If the twisting motions were to be stronger than the untwisting motions, then eventually all the seed field would get twisted up leaving no more seed field and the process would have to stop. If the untwisting motions were more effective, then we will have only the seed field with no sunspots. Thus, there is a balance between the two kinds of motions giving a continuing process. The sunspot fields eventually are fragmented into weaker fields that move to the poles and are destroyed. The game begins all over again, except that the sunspot fields change their polarity. This explanation by Babcock is only a description of the observed changes in the sunspot field using the movements of the solar plasma as a basis. Surprisingly no one has been able to give a proper explanation despite a lot of study about the generation of magnetic fields. In what follows, will present the current theoretical understanding of the solar dynamo after reviewing the basic magnetohydrodynamic equations. 4.3. Basic MHD Magnetohydrodynamics is the combination of electrodynamics with fluid dynamics. The Maxwell’s equations (Eqs. 1a-1d) provide the inter-relationship between the electric and magnetic fields in any medium, including a vacuum. The first equation expresses the electric field E for a given distribution of the electric charge density ρ , where ε ο is the permittivity of free space. The second equation denotes the non-existence of magnetic monopoles. The third equation is Ampere’s law for the creation of magnetic field B due to the distribution of current density j, with Maxwell’s contribution of the displacement current appearing as the second term of the R.H.S. Here µ is the permeability in a vacuum. The last equation is the Faraday’s law of electromagnetic induction. For magnetohydrodynamics, a simplified form of these equations is used as described in the following section. These simplified equations inter-relate the fluid velocity with the magnetic field through the induction equation. Finally, inclusion of the Lorentz force in the equation of motion of the fluid provides the generator for fluid motion caused by the magnetic field. This, basically, is the essence of magnetohydrodynamics. ∇ ⋅ D = ρ∗

(4.1a)

∇⋅B = 0 ∇ × H = j + ∂D / ∂t

(4.1b) (4.1c)

∇ × E = −∂ B / ∂ t

(4.1d)

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Here, B = µH and D = εE , µ is the permeability and ε is the permittivity of the medium. 4.4. MHD Approximation If L is a typical length scale and T is a typical time scale of the process, then MHD deals with the situation where L/T << c. The induction equation can be used to estimate E as, E ~ LB/T. Then curl B is ~ B/L, while the displacement current is ~ E/(c2T) ~ (B/L) (L/T)2/c2, hence ∇ × B >> displacement current. Thus, the displacement current can be neglected in the MHD approximation and Ampere’s law simplifies to curl B = µj. The second assumption for the MHD approximation is charge neutrality. From Gauss’s law, we have E=Le(n+ - n-)/ε, or n+ - n- = εE/(Le) = εB/(Te). For charge neutrality, n >> n+ - n-, or n >> εB/(Te). Finally, we need to modify the Ohm’s law for a conducting fluid in motion. A fluid moving in a magnetic field is subject to an electric field v × B in addition to the electric field E at rest. Ohm’s law then applies to the total field as j = σ(E + v×B). Using the above three ingredients of the MHD approximation (viz., Ampere’s law, j =∇×B/µ, charge neutrality, and the modified Ohm’s law E = j/σ–(v×B), we can arrive at the induction equation in magneto-hydrodynamics as ∂B/∂t = ∇× (v × B) –∇× ((∇×B) /σ). This is the fundamental equation that is used in solar dynamo theory for generation of magnetic field. 4.5. Solar Dynamo The aim of any solar dynamo theory is to explain the sustained cycle of the global magnetic field with the period of 22 years. A further goal would be to understand the slow variation in the strength of a cycle over a time period, which is much larger than 22 years. It is well known that any solenoidal vector field (which has zero divergence) like the magnetic field can be expressed as the sum of a poloidal and a toroidal field. In the case of the sun, we can imagine the process to begin with a completely poloidal field (as manifested during a sunspot minimum) which gets transformed into a chiefly toroidal field (as manifested during a sunspot maximum) through the action of differential rotation. This toroidal field needs to be converted back into a poloidal field, but with reversed polarity, at the end of eleven years. A similar half cycle is repeated to complete the 22 year magnetic cycle. Parker proposed the first theory in terms of a mean field kinematic dynamo (Parker 1955). Essentially, Parker separated the fluid variables into a slowly varying component and a fluctuating component. After

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taking an ensemble average over the corresponding 2 components of the induction equation, Parker obtained an induction equation in which the rate of change of the mean magnetic field is proportional to the mean field with a coefficient containing the ensemble average of the fluctuation of kinetic helicity. Similar to the situation in terrestrial cyclones, we expect this ensemble average to have opposite signs in the two hemispheres. Parker finally obtained an oscillatory solution for mean field in terms of the correlations of the flow field. The chief parameters which determine the functioning of the dynamo are 1) Ω parameter which is proportional to the gradient of rotational profile and 2) the Parker’s α parameter which is proportional to the chirality of the vortex motions of the rising eddies in the convection zone. The Ω parameter is mainly responsible for converting the poloidal field into the sunspot-forming toroidal field, while the α parameter is chiefly responsible for converting the toroidal field back into a poloidal field. By assuming suitable values for these parameters, Parker was able to obtain a period for the oscillation that was in the region of tens of years. The product of α and Ω had to be negative for the existence of an oscillatory dynamo. Later, Radler, Krause and Stix, made extensive numerical calculations and were able to simulate the various properties of the real solar dynamo with suitable models of the flow field (Stix, 1976). Further, other variants of this α-Ω dynamo, e.g. the α2 and α2-Ω dynamos were also proposed . A “wake up call” occurred for solar dynamo theorists when Peter Gilman completed a dynamical model of the solar dynamo, which calculated the flow field generating the magnetic field from the numerical simulation of magnetoconvection in a rotating sun (Gilman 1986). Gilman’s flow field followed essentially the Taylor-Proudman theorem, which states that the rotation of a slowly rotating fluid sphere will be constant on cylinders co-axial with the axis of rotation. However, the resulting “butterfly diagram” had spots moving towards the poles, opposite to the equator-ward movement seen in the actual sun! The solution to this paradox became clear when helioseismology enabled solar physicists to map the depth and latitude dependence of the solar internal rotation. The solar rotation was found to decrease with depth until the base of the convection zone, at which there was a sharp gradient of the rotation followed by a regime of rigid rotation. This gradient was named the tachocline. The problem of why the sun does not follow the Taylor-Proudman theorem is a topic of contemporary study. The tachocline is an ideal site for generating a strong toroidal field from the poloidal field. However, there are two basic problems associated with the

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tachocline. The first is that the gradient of the rotation profile is strongly positive at the tachocline. This, combined with the sign of Parker’s alpha parameter violates Parker’s criterion for an oscillatory dynamo. Secondly, the vigour of overshooting convection at the base of the convection zone is not strong enough to distort the very strong toroidal fields, that are necessary to exhibit Joy’s law according to the mechanism proposed by D’Silva and Choudhuri (1993). Thus, Parker’s suggestion for generating the poloidal fields from toroidal fields by the action of cyclonic eddies in the convection zone, would not work in the tachocline. A combination of Joy’s law and diffusion of magnetic flux on the solar surface does seem to generate the poloidal fields as demonstrated by the so-called flux transport mechanisms (Sheeley et al 1989). But these models do not seem to predict the strength of the poloidal field at sunspot minimum from a given amount of toroidal field seen at sunspot maximum. Further, the amount of “open magnetic flux” predicted from these models is not compatible with the latitudinal and temporal variation of the observed interplanetary magnetic field. Thus, there is plenty of scope for serious theoretical studies in this branch of solar magnetism. 4.6. Force-free Fields The main forces acting on a fluid parcel in the solar atmosphere are the Lorentz force, the gradient of plasma pressure and the weight of the material. The Lorentz force has two parts, 1) the tension force given by (B. ∇ )B/4 π and the pressure force given by ∇ (B2/8 π ). The plasma pressure and density decrease exponentially with height, while the magnetic field decreases less sharply as a polynomial function (Spruit,1983). Thus, the magnetic forces completely dominate the dynamics in the solar chromosphere and corona. At these heights, the magnetic tension and magnetic pressure force can no longer be individually balanced by the plasma pressure gradient and gravity force. Hence, for equilibrium, the magnetic tension and pressure forces have to balance each other, making the net Lorentz force zero. This configuration of magnetic field with zero Lorentz force is called a force free field. In a force free field, the current is aligned with the field and is proportional to it. Thus, ∇ × B = α B. This implies that (B. ∇ ) α = 0, or α is constant along a field line. This condition leads to the following problem of compatibility at the two foot-points of a magnetic field. We know that the force-free parameter alpha is determined by the 3 components of the photospheric vector magnetic field at each foot-point. However, the vector field at each foot point is independently specified by the sub-photospheric flow beneath each point. Since there is no

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physical condition to prevent arbitrary flow parameters at the foot points, the value of alpha can be quite different at the foot points. This violates the constancy of alpha for a force free field. Parker sought a way out of this dilemma, by invoking the spontaneous creation of discontinuities in the corona (Parker, 1994). These discontinuities could well produce electric fields and currents on small scales. These small scale currents are easily dissipated, but the discontinuity in alpha will be maintained by the photospheric driver. In fact, this is indeed Parker’s suggestion for coronal heating by the dissipation of small scale coronal currents which are maintained by the photospheric motions of the foot-points. This mode of coronal heating can be considered as a stationary stochastic process, as long as the rate of photospheric driving is equal to the rate of dissipation of the coronal currents. 4.7. Solar Eruptions The twisting action of the sub-photospheric flow can produce magnetic energy build-up on large scales. When the rate of energy supply is faster than the rate of energy dissipated by field discontinuities in the corona, the free energy or nonpotential energy of the magnetic field gets stored in the azimuthal component of the magnetic field. This component of the magnetic field can be detected only by a vector magnetograph. Several studies have shown that flares are chiefly produced in active regions possessing a large amount of free energy. It is also fairly certain that the interaction of existing field with newly emerging field acts as a trigger to initiate the onset of non-equilibrium of magnetic field that results in a solar flare. In the canonical picture of a solar flare evolved from concepts of Carmichael (1964), Sturrock (1966), Hirayama (1974), Kopp and Pneuman (1976), called the CSHKP model, the action begins in the corona (figure 6). Reconnection of coronal magnetic fields at the cusped summit of a loop system accelerates electrons (and rarely, protons) which stream down to the base of the chromosphere and generate Bremhstrahlung radiation in the hard X-ray part of the electromagnetic spectrum. H-alpha is also emitted due to recombination of the ionized hydrogen atoms. The process takes place so quickly, that the plasma has no time to relax to thermodynamic equilibrium. Thus, the hard X-ray as well as the H-alpha emission induced by electron precipitation are both non-thermal processes. The H-alpha emission is initiated in small kernels, which quickly spread out in the form of flare ribbons on either side of the magnetic polarity inversion line. By the time plasma relaxes to thermodynamic equilibrium, the intense heating of the plasma leads to explosive evaporation. The evaporation is explosive because of a thermal instability of hydrogen plasma which sets in at

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~ 10000 degrees K and catastrophically sets the temperature at 1 million degree K. Hot plasma expands quickly and fills the overlying coronal loops. The thermal radiation in the hot plasma is in the form of soft X-rays. Thus, soft X-ray onset will coincide with H-alpha flare and the hard X-ray spike, but soft X-ray peak emission occurs a little later, when the entire loop is filled and starts cooling. The source of energy for a flare is the non-potential magnetic energy of the active region, while the trigger is most likely to be a local re-arrangement of field lines caused by magnetic reconnection when a newly emerging flux system impacts on an existing system. Thus study of magnetic field of active regions is crucial for understanding flares.

Figure 6. Cartoon of the standard flare model, also known as CSHKP model.

4.8. Coronal Mass Ejections According to the most recent understanding of the CME phenomenon, the physical processes responsible for a CME start right from the solar interior, where the solar magnetic field is generated. This magnetic field emerges into the photosphere and later into the corona in the form of active region fields.

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Although the magnetic flux changes sign during each sunspot cycle, the magnetic helicity does not and keeps accumulating in the corona. Some part of the nonpotential or free energy is released through magnetic reconnection in the form of flares. The rest remains associated with the helicity in the form of a magnetic flux rope, which is confined by the remainder of the original magnetic field that escaped annihilation by the magnetic reconnection. The flux rope is also anchored by the weight of the associated prominence material. When the accumulated helicity attains a critical value, the confinement fails, either by reconnection above the rope (so-called break-out model of Antiochos, De Vore and Klimchuk 1999) or by reconnection below the rope (tether cutting model of Moore and La Bonte 1980) The flux rope then escapes from the sun in the form of a CME, with the canonical three part structure of the bright front (disturbed coronal streamer), followed by the cavity (the flux rope) and the core (part of the associated prominence; Zhang & Low 2005). PART 2. MEASUREMENT TECHNIQUES 4.9. Polarization of Light Visible light is part of the electromagnetic spectrum with frequencies in the range 1015 to 3 × 1014 Hz. Thus, the electric field of the electromagnetic wave oscillates over one cycle in 10-15 s. The source of the wave is any atom of the solar plasma, which is subjected to collisional and radiative excitation and deexcitation at typically 108 times per second. Moreover, several atoms undergo these processes at different times with no correlation between each other. The result is a set of broken wave trains with random phases. A light detector will detect a superposition of these randomly generated wave trains. Hence there will be no preferential plane of vibration for the electric field and the light is said to be unpolarized. However, any process which breaks this symmetry can produce a difference between the intensities of light measured in any two orthogonal planes around the direction of propagation, as well as a net phase difference between the oscillations of the electric field in the two directions. This results in the polarization of light. Mathematically, Stokes (1852) formulated the four parameters which fully characterize the polarization state of the light. These are given by I = < Ex.Ex*> + < Ey.Ey*> Q = < Ex.Ex*> - < Ey.Ey*> U = 2 Re <Ex.Ey*> V = 2 Im <Ex.Ey*>,

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where Ex and Ey are the complex amplitudes of the electric field and < > denotes ensemble average. Experimentally, I parameter is obtained by adding the intensities transmitted through a Polaroid whose axis is placed along x- and ydirections respectively, Q from the difference of the two intensities, U from difference in intensities along 45 and 135 degrees respectively to the x-axis, while V is obtained by doing a Q measurement after inserting a quarter wave retarder before the Polaroid. 4.10. Zeeman Effect As mentioned earlier, Zeeman detected a splitting of spectral lines when the source of light is placed in an external magnetic field. The so-called normal Zeeman effect shows right and left circularly polarized line components on the 2 sides of the central line and no line at the original position, when the direction of magnetic field is parallel to the light propagation direction. When the field is transverse to the propagation direction, there is a central or pi component which is linearly polarized in a direction parallel to the field direction, while the sigma components on either side of the pi component are linearly polarized, with the direction of polarization perpendicular to the magnetic field direction. The Zeeman effect can be explained in the following way. If at least one level of the atomic transition (producing the spectral line) is degenerate, then the degeneracy is lifted in the presence of the magnetic field. This is because the interaction energy of the magnetic field with the spin induced magnetic moment of the electron depends on the azimuthal quantum number. Thus, we now have the possibility of transitions between the various m-states of each level. However, the selection rule for change in m quantum number selects only certain types of transitions. When ∆ m = ± 1, circularly polarized light is emitted, while ∆ m = 0 transitions yield linearly polarized light The separation between the split components of the spectral lines depends on the strength of the magnetic field and splitting cannot be observed if this separation is smaller than the thermal broadening of the line. For such weak fields, one could use the variation of the polarization along the line profile to infer the strength and direction of the magnetic field. Typically, with a thermal broadening of 25 mÅ in visible Fe I lines of solar photosphere, the splitting is detectable only for fields stronger than about 1000 G. But splitting can be observed for weaker fields (300 G) in the infra-red Fe I lines at 15600 Å.

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Figure 7. Schematic of the classical picture of the Zeeman effect.

4.11. Hanle Effect In the Zeeman effect, the individual transitions from different m-levels of the excited state to the lower energy state produce separate spectral lines with slightly different wavelengths. In the case of a scattering, there is a transition from a lower energy level to a higher energy level followed by transition back to the lower energy level. Since the transitions occur in the same atom, there is stable phase relation between the incident and scattered photon. Thus, the process of scattering produces polarized radiation. This is called resonance scattering polarization. However, when the atom is immersed in an isotropic radiation field, the net polarization is zero. On the other hand, if the incident radiation field is anistropic, as for example near the solar limb, then we do detect the polarization in the scattered spectral line. When the scattering involves a degenerate upper level of the atomic transition, then a photon can excite all the degenerate states together in a coherent superposition of states. When deexcitation takes place, the resulting photon carries information about all the degenerate upper levels through a process of quantum interference. The emitted (or scattered) photon will continue to bear coherency relationships with the absorbed (or incident) photon. Now, if we introduce a magnetic field, then the degeneracy is lifted and we expect the quantum interference to be completely destroyed. This is not the case, as long as the interaction energy of the electron with the external magnetic field is smaller than the uncertainty in energy produced by the finite life-time of the upper energy level. Thus, the presence of a magnetic field that is weak enough to satisfy the above condition will modify the

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quantum interference and produce detectable effects like a) reduction in degree of polarization and b) rotation of the plane of polarization. These effects, which were first detected by Hanle in 1924 in laboratory sources, are together called the Hanle effect (Hanle, 1924). Since the Hanle effect is independent of the velocity of the frame of reference, we can detect this effect even in highly broadened lines, even for coronal emission lines. For the typical conditions in the solar photosphere, this effect can be used to detect fields as small a few tens of gauss in the solar photosphere. 4.12. Solar Magnetographs A solar magnetograph is an instrument that measures the polarization of sunlight emanating from a specific portion of the solar disk and within a narrow band of the solar spectrum. Older magnetographs exploited the Zeeman effect, where the polarization is a definite function of the position on the line profile. By using the values of the Stokes parameters at different positions on the line profile, we can estimate the chief parameters of the magnetized atmosphere at that particular region of the sun, e.g. the strength, inclination and azimuth of the magnetic field, as well as the temperature, fill factor or stray light component and few other parameters. Obviously, the more information we have, the better will our estimates approximate the “real” situation. In practice, there are different kinds of difficulties for each type of measurement and the observer often has to compromise on a few parameters and concentrate on those parameters, which make up his or her primary science goal. For example, people interested in the pre-flare evolution of magnetic fields would prefer a 2-D record of the magnetic field with high cadence. In this case, it is difficult to measure the Stokes parameters at several spectral positions. Hence, a filtergraph is used which records the 2-D image of the sun through a narrow spectral band. Different devices can be used to isolate the spectral band and we will give 2 examples below, one of which uses a Michelson interferometer, while the other uses a Fabry-Perot tunable etalon. Generically, the magnetograph will consist of a light collector, and an imaging system. There would be a polarization modulator followed by an analyzer, which forms the polarimeter. There would also be a spectral isolator, as mentioned earlier. 4.13. GONG Magnetograph The Global Oscillations Network Group consists of 6 telescopes distributed at convenient longitudes around the globe, so that the oscillations of the solar

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surface can be recorded continuously without data breaks. This is essential to avoid windowing effects while determining the Fourier frequencies and enhances the accuracy of the frequency measurements. Along with velocity oscillation measurements, the GONG instrument measures the line-of-sight magnetic field distribution over the entire solar disk with a cadence of 1 minute. It thus provides an automatic record of magnetic field evolution prior and after every solar flare, and has provided an extremely useful data-base to solar physicists interested in the dynamics and energetics of solar active regions that lead to solar flares. The light collector of the GONG magnetograph (Harvey et al 1998) consists of 2 flat mirrors which can be rotated along axes which are perpendicular to each other, forming an alt-alt system. The 75 mm beam is sent horizontally towards the North direction and into a laboratory building which houses the rest of the instrumentation. The alt-alt mount is robotic and moves to the sun’s direction automatically every day according to an almanac stored within the computer system. The telescope at each GONG site can be controlled remotely from the central GONG-OPS station located at NOAO, Tucson, Arizona, USA. The heart of the GONG magnetograph is a Michelson Interferometer (MI). The idea of using an MI to measure the spectral line profile is attributed to Michelson himself. The first complete implementation of this idea for solar work was done by Title and Ramsey (1980). The earliest users of MIs for measuring solar Doppler shifts were Brown (1980) and Kozhevatov (1983). A Lyot filter with a fixed bandpass of 0.1 nm (FWHM transmission) acts as the pre-filter to isolate the Ni I line at 676.8 nm. A polarizing, temperature-compensated, wide-field MI acts as the tunable filter. It has a “channel” transmission function in polarized light with a period equal to the typical width of the spectral line. The principle of the measurement process is given schematically in figure 8. The MI produces a modulation of the light intensity that is a function of the path delay. In GONG, the MI is a fixed interferometer with both arms transmitting orthogonal linear polarization states. A quarter wave retarder, combines both the orthogonal polarizations and encodes wavelength into intensity by having a linear polarization vector whose azimuth varies linearly with wavelength. A rotating analyzer then transmits different wavelengths as a function of the angle of the transmission axis. In this way, the channel spectrum of the MI is swept across bandwidth of the pre-filter. The presence of the absorption line produces the modulation. This modulated signal is sampled 3 times per rotation period and integrates the light from 3 different parts of the line profile as shown in the bottom right corner of the figure. The mean signal value is proportional to the intensity of the light transmitted by the pre-filter (including the spectral line).

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The amplitude of the modulation is a measure of the strength of the spectral line, while the phase of the modulation is proportional to the center of intensity or effective wavelength position of the line profile. The difference of this effective wavelength measured in right and left circularly polarized light is proportional to the line-of-sight magnetic field.

Figure 8. Schematic of the GONG magnetograph.

4.14. USO Solar Vector Magnetograph The Fabry-Perot (FP) based tunable filters are very useful in solar astronomy as imaging spectrometers. These filters basically isolate a single interference order (transmission maxima) of the FP etalon by blocking other orders with a suitable interference filter. The bandwidth (FWHM) of the interference filter is chosen to be half the Free Spectral Range (FSR), i.e., wavelength separation between two adjacent interference orders of the FP. The advantage of these filters is that they have higher throughput for a given spectral resolution R, compared to other type of spectrometers like prism or grating spectrograph (Chabbal and Jacquinot, 1955). Modern FPs use highly reflective multi-layer dielectric coatings deposited over substrates with surface flatness of about λ/200. These substrates are kept parallel to each other within λ/1000 using servo control system (Hicks et al 1976). The tuning of the FP pass-band is achieved by applying 12 bit digital voltage to piezo-electric crystals, which hold the reflecting glass plates, thereby

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adjusting the cavity length. Such filters deliver good spectral resolution, rapid scanning and good repeatability in their performance. The magnetographs based on such filters have the advantage of high SNR (due to higher throughput), simultaneous spectrometry of large field-of-view and large spectral range (by using different interference filters).

Figure 9. Schematic of the Solar Vectoer Magnetograph at Udaipur Solar Observatory.

Last decade has seen development of many solar magnetographs or imaging polarimeters, as some would prefer to call them, designed around a Fabry-Perot filter. Some examples are (i) Imaging Vector Magnetograph (IVM) of Haleakala, Hawaii (Mickey et al 1996), (ii) Tenerif Infra-red Polarimeter (TIP) at the Canary Islands, Spain (Martinez Pillet et al 1999), and (iii) Digital Vector Magnetograph (DVM) at the Big Bear Solar Observatory (BBSO) California, USA (Denker et al 2002). Here we give a brief description of newly built Solar Vector Magnetograph (SVM) at Udaipur Solar Observatory (Gosain et al 2006). The schematic of the instrument is given in figure 9. These instrument records polarized spectra of the selected active region in Fe I 630.25 nm line. The schematic layout of the instrument is shown in figure above. The essential components of a spectropolarimeter are a spectrometer and a polarization modulator and analyzer. The

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SVM uses as a spectrometer, an air-gap, piezo-tuned and servo-controlled FabryPerot etalon, similar to one described above. For polarization analysis of the light it uses a modulator unit with two rotatable quarter wave-plates. The instrument is fed by a Schmidt-Cassegrain telescope of eight-inch aperture. To avoid heating of the reflective coatings of the telescope mirrors due to excessive solar flux, the telescope is quipped with an eight-inch aperture interference filter in front. This filter limits the bandpass of the incoming solar radiation to a narrow range of about 15 nm, centered at Fe I 630.25 nm line. This also prevents the heating of the subsequent optics in the optical train. The instrument has a symmetric-design and avoids any oblique reflection for minimal instrumental polarization. The entire optical train is mounted on a computerized GermanEquatorial mount. The field-of-view of 4 arc-min diameter can be selected at the prime focus of the telescope. The subsequent optics consists of a polarimeter package, a FP spectrometer, the analyzer assembly and the CCD camera system. A polarimeter calibration unit can be inserted just after the prime focus for polarimeter calibration. There are two ways one can modulate the polarized light, namely spatial and temporal modulation. As the name suggests the former is sequential and latter is parallel. The temporal modulation is useful if the modulation frequency is faster than the frequency at which the atmospheric seeing changes. Although, modulators with very fast, kHz range modulation frequency, (much above frequency of seeing variation ~100 Hz) are available, the detectors at corresponding frequencies are not available. Thus one is forced to use a spatial or combination of both modulation schemes. In the latter case all the available photons are utilized for building up the SNR of the measurement. The SVM uses a spatial modulation scheme using two crossed calcites as the analyzer. The two beams polarized in orthogonal directions are recorded simultaneously on the same CCD chip. The retrieval of magnetic field vector using these spectro-polarimetric observations is done by fitting the observations with theoretical Stokes profiles generated using analytical solutions of radiative transfer equation in a model atmosphere. These analytical solutions were first derived by Unno (1956) and Rachkovsky (1962a, 1962b). It can be seen that in order to retrieve magnetic field vector one need to go through several steps. In order to expedite these numerous steps, that is, observing, reducing and calibrating the raw data, and deriving the magnetic field vector by fitting observed Stokes profiles a considerable amount of automation is required. The SVM is supported by many automated softwares with graphical user interface (GUI) to expedite these processes. The SVM is geared to measure magnetic fields during entire solar cycle 24 and shall contribute to the study of evolution

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of solar magnetic fields and also complement observations from space based instruments like HINODE and SDO in different spectral lines. 4.15. Future Directions It is very difficult to predict future directions in the study of solar magnetism. Solar physicists will be basically pre-occupied by two kinds of studies. The first kind would be on the long term and solar cycle modulation of active region parameters as well as the polar field evolution. For the former, we require more accurate polarimetry for precise determination of helicity related parameters, and further development of the dynamo theories. For the latter, we need to employ the Hanle effect over different spatial scales near the polar regions. The second type of study will be high resolution studies of the magnetic field on the smallest possible scales, for identification of the exact mechanism of magnetic heating of the atmosphere as well as the detailed behaviour of magnetic fields during the onset of solar eruptions. Both type of studies require great progress in observational as well as theoretical methods which will continue to occupy the attention of future generations of scientists. References Alfven, H., 1942, Nature, 150, 405. Antiochos, S. K., DeVore, C. R. and Klimchuk, J. A., 1999, ApJ, 510, 485. Babcock, H. W., 1961, ApJ, 133, 572. Brown, T. M., 1984, Bull. Astron. Soc. Amer., 16, 978. Carmichael, H., 1964, Proceedings of the AAS-NASA Symposium, Edited by Wilmot N. Hess., p.451. Chabbal, R. and Jacquinot, P., 1955, Nuovo Cim. 2, 661. Denker, C., Ma, J., Wang, J., Didkovsky, L., Varsik, J., Wang, H., Goode, P. R., 2002, Proc. SPIE, 4853, 223. D’Silva, S. and Choudhari, A. R., 1993, A&A, 272, 621. Evershed, J., 1909, MNRAS, 69, 454. Gilman, P. A., 1986, in P. A. Sturrock (ed) Physics of the Sun, vol I, Reidel, 95. Gosain, S., Venkatakrishnan, P. and Venugopalan, K., 2006, JAA, 27, 285. Hale, G. E., 1908, ApJ, 28, 315. Hale, G. E., Nicholson, S. B., 1938, Publ. Carnegie Inst. 498, Washington. Hanle, W., 1924, Z. Physik., 30, 93. Harvey, J. and GONG Team, 1998, Bull. Astron. Soc. India, 26, 135. Hicks, T. R., Reay, N. K. and Stephens, C. L., 1976, A&A, 51, 367.

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Hirayama, T., 1974, Solar Phys., 34, 323. Kopp, R. A. and Pneuman, G. W., 1976, Solar Phys., 50, 85. Kozhevatov, I. E., 1983, ISSLEDOVANIE. GEO. AERO. FIZ. SOLNSTA NO. 64, 42. Mart´ınez Pillet, Collados, S´anchez Almeida et al., 1999, in ASP Conference Series 183, Rimelli et al. (eds), p264. Mickey, D. L., Canfield, R. C., LaBonte, B. J., Leka, K. D., Waterson, M. F. et al., 1996, Solar Phys., 168, 229. Moore, R. L. and LaBonte, B., 1980, in IAU Symp. 91, Solar and Interplanetary Dynamics, ed. M. Dryer & E. Tandberg-Hanssen (Dordrecht:Reidel), 207. Parker, E. N., 1955, ApJ, 121, 491. Parker, E. N., 1994, Spontaneous Discontinuities in Magnetic Fields, Oxford Univ. Press. Rachkovsky, D. N., 1962a, Izv. Krym. Astrofiz. Obs., 27, 148. Rachkovsky, D. N., 1962b, Izv. Krym. Astrofiz. Obs., 28, 259. Schwabe, S. H., Astron. Nachr. 21, 2 (1844). Sheeley, N. R., Wang, Y. M. and Harvey, J. W., 1989, Solar Physics, 119, 323. Spruit, H. C., 1983, IAU Symp., 102, 41. Stix, M., 1976, IAU Symp., 71, 367. Stokes, G. G., 1852, Trans. Cambridge Phil. Soc., 9, 399. Sturrock, P. A., 1966, Nature, 211, 695. Title, A. and Ramsey, H., 1980, Applied Optics, 19, 2046. Unno, W., 1956, Publ. Astron. Soc. Japan, 8, 108. Zhang, M. and Low, B. C., 2005, Annual Re. of Astron. & Astroph., vol. 43, Issue 1, pp.103-137.

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CHAPTER 5 WAVES AND OSCILLATIONS IN THE SOLAR ATMOSPHERE ´ ROBERT ERDELYI Solar Physics & Space Plasma Research Centre (SP2 RC), Department of Applied Mathematics, University of Sheffield, S3 7RH, Sheffield, U.K. E-mail: [email protected]

1. Introduction The actual operating heating process that generates and sustains the hot solar corona has so far defied a quantitative understanding despite efforts spanning over half a century. Particular attention is paid here towards the exploration of the coronal heating problem from the perspectives of MHD waves and oscillations. Do MHD waves play any role in the heating of the solar atmosphere? In order to attempt answering this question, first we need do embark on the key properties of the heating of the solar atmosphere. Space observations, from Skylab in the 70th through SMM, Yohkoh and in very present times SoHO, TRACE, RHESSI and Hinode have investigated the solar atmosphere with unprecedented spatial and temporal resolution covering wavelengths from (E)UV, through soft and hard X-ray to even gamma rays. These high-resolution imaging and spectroscopic observations contributed to many discoveries in the solar atmosphere. The solar atmospheric zoo, to the best of our knowledge today, consists of features from small-scale X-ray bright points to very large coronal loops (Figure 1a). For an excellent textbook on the corona see, e.g. Ref. 1. Soon after the discovery of the approximately few MK hot plasma of the solar corona theoreticians came up with various physical models trying to explain the apparently controversial behaviour of the temperature in the atmosphere. The key point is the observed distribution of temperature: the 61

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Fig. 1. Left: The very inhomogeneous and dynamic solar atmosphere. The full disk image is taken by SOHO/EIT in He II at 304 ˚ A. Right: Solar atmospheric temperature and density distributions as a function of height. The formation of some popular lines for observations is indicated by dots on curve T. Note the logarithmic scales.

solar energy is produced by thermonuclear fusion in the very hot (approximately 14 MK) internal core of the Sun. This vast amount of energy then propagates outwards, initially in the form of radiation (radiation zone) up to about 0.72R⊙ and later by convection (convective zone) right to the solar surface (photosphere) continuously cooling the solar plasma. Surprisingly, after reaching its minimum at the top of the photosphere, the temperature starts to rise slowly throughout the entire chromosphere (up to around 20,000 K), followed by a very steep and sharp increase in the narrow transition region (few 100,000 K) up to around 2 MK in the corona (Figure 1b). Although going continuously away from the energy producing solar core, instead of a temperature decrease, the tendency of temperature increase was found (Figure 1b). Maintaining this high temperature requires some sort of input of energy because without it the corona would cool down by thermodynamic relaxation on a minute-scale. Surprisingly, this non-thermal energy excess to sustain the solar corona is just a reasonably small fraction of the total solar output (see Table 1). It is relatively straightforward to Table 1.

Average coronal energy losses (in erg cm−2 sec−1 ).

Loss mechanism Conductive flux Radiative flux Solar wind flux Total flux

Quiet Sun 105

2× 105 < 5 × 104 3 × 105

Active region 105

107

5 × 106 < ×105 107

Coronal hole 6 × 104 104 7 × 105 8 × 105

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estimate the entire energy budget needed for the solar corona: approximately just a tiny 10−4 fraction of the Sun’s total energy output is needed giving, at least in theory, a fairly easy task for theoreticians to put forward various mechanisms that could divert 0.01% of the total solar output into heating the corona. The question is today not where does the coronal non-thermal energy come from, but rather how is this energy actually transferred into the corona and how does it dissipate efficiently there. 1.1. Importance of Atmospheric Magnetism With increasing spatial and time resolution large-scale structures like sunspots, complex active regions, prominences, coronal loops, coronal holes are observed in great details. The improved resolution allowed to reveal fine structures like the magnetic pores, dark mottles, spicules, supergranular cells, filaments, X-ray and EUV bright points, etc. Since the discoveries of the solar cycle, the Hale’s polarity law, the butterfly diagram for sunspots and the cyclic variations in sunspot numbers the role of solar magnetic fields became a central theme. Skylab observations made it clear for the first time that the X-ray emitting hot and bright coronal regions and the underlying surface magnetic field concentrations are strongly correlated suggesting that coronal heating and solar magnetism are intimately linked (Figure 2). Today

Fig. 2. Approximately concurrently taken magnetogram, UV, EUV and X-ray full disk images. Observe that the locci of magnetic field concentrations at photospheric level coincide with the locii of high emissions in (E)UV and X-ray. Image credit: SOHO and Yohkoh.

it is evident that the solar atmosphere is highly structured and is most likely that various heating mechanisms operate in different atmospheric magnetic structures.2,3 In closed structures, e.g. in active regions, temperatures may reach up to 8 − 20 × 106 K, while in open magnetic regions like coronal holes maximum temperatures may only be around 1 − 1.5 × 106 K. Observations also show that density and magnetic field are highly inhomogeneous. Fine structures (e.g. filaments in loops) may have 3-5 times higher densities than

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in their environment. The fluctuating brightness and the associated velocities, as opposed to the quasi-static nature of the corona, are far-reaching observational constraints what are not yet modelled adequately. There is also little known about how the heating depends on magnetic field strength, structure size (length, radius, expansion) and age. 1.2. Atmospheric Heating Mechanisms In order to explain the solar (and stellar) atmospheric heating mechanism(s) models have to provide a mechanism or mechanisms that result(s) in a steady supply of energy not necessarily on a steady way. Random energy releases that produce a statistically averaged steady state are allowed for to balance the atmospheric (chromospheric and coronal) energy losses and these models became more viable.4 -7 Testing a specific heating mechanism observationally may be rather difficult because several mechanisms may operate at the same time. Ultimate dissipation occurs on very small spatial scales, sometimes of the order of a few hundred metres that even with current high spatial resolution satellite techniques cannot (and will not for a while!) be resolved. A distinguished signature of a specific heating mechanism could be obliterated during the thermalisation of the input energy.4 We should, instead, predict the macroscopic consequences of a specific favoured heating mechanism2 and confirm these signatures by observations.8 For example one could predict the generated flows9 or specific spectral line profiles or line broadenings.10,11 The heating process is usually split into three phases: (i) the generation of a carrier of energy; (ii) the transport of energy from the locii of generation into the solar atmospheric structures; and finally (iii) the actual dissipation of this energy in the various magnetic or non-magnetic structures of the atmosphere. Without contradicting observations it is usually not very hard to come up with a theory that generates and drives an energy carrier. The most obvious candidate is the magneto-convection right underneath the surface of the Sun. Neither seems the literature to be short of transport mechanisms. There is, however, real hardship and difficulty in how the transported energy is dissipated efficiently on a time-scale such that the corona is not relaxed thermally. A brief and schematic summary of the most commonly accepted heating mechanisms is given in Table 2, see also Refs. 12-14. The operating heating mechanisms in the solar atmosphere can be classified whether they involve magnetism or not. For magnetic-free regions (e.g. in the chromosphere of quiet Sun) one can suggest a heating mechanism

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Summary of various popular heating mechanisms, see also Refs. 13-14.

Energy carrier Dissipation mechanism Hydrodynamic heating mechanisms Acoustic waves (P < Pacoustic cutoff ) Shock dissipation Pulsation waves (P > Pacoustic cutoff ) Shock dissipation Magnetic heating mechanisms 1. Alternating current (AC) or wave mechanisms Slow waves Shock damping, resonant abs. Longitudinal MHD tube waves Fast MHD waves

Landau damping

Alfv´ en waves (transverse, torsional)

Current sheets

Mode coupling, res. heating, phase mixing, compressional viscous heating, turb. heating, Landau damping, res. absorption 2. Direct current (AC) mechanisms Reconnection (e.g. turbulent or wave heating)

that yields within the framework of hydrodynamics. Such heating theories can be classified as hydrodynamic heating. Examples of hydrodynamic heating are, among others, e.g. acoustic waves and pulsations. However, if the plasma is embedded in magnetic fields as it is in most parts of the solar atmosphere, the framework of MHD may be the appropriate approach. These coronal heating theories are called MHD heating mechanisms; for reviews see, e.g. Refs. 14-21 and 22. The ultimate dissipation in MHD models invoke Joule heating or, in a somewhat less extent, viscosity. Examples of energy carrier of magnetic heating are the slow and fast MHD waves, Alfv´en waves, magnetoacoustic-gravity waves, current sheets, etc. There is an interesting concept put forward by, e.g. Ref. 23, where the direct energy coupling and transfer from the solar photosphere into the corona is demonstrated by simulations and TRACE observations, see also Ref. 24. For a recent review on MHD waves and oscillations see, e.g. Refs. 25-27. Finally, a popular alternative MHD heating mechanism is the selective decay of a turbulent cascade of magnetic field.28 -30 Based on the times-scales involved an alternative classification of the heating mechanism can be constructed. If the characteristic time-scale of the perturbations is less than the characteristic times of the back-reaction, in a non-magnetised plasma acoustic waves are good approximations describing the energy propagation; if, however, the plasma is magnetised and perturbation time-scales are small we talk about alternating current (AC- )

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heating mechanisms, e.g. MHD waves.21,25,31 On the other hand, if perturbations have low frequencies hydrodynamic pulses may be appropriate in a non-magnetised plasma, while if the external driving forces (e.g. photospheric motions) operate on longer times-cales compared to dissipation and transit times very narrow current sheets are built up resulting in direct current (DC-) heating mechanisms in magnetised plasmas.20 After it was discovered that the coronal plasma is heavily embedded into magnetic fields the relevance of the hydrodynamic heating mechanisms for the corona part of the atmosphere was re-evaluated. It is believed today that hydrodynamic heating mechanisms could still contribute to atmospheric heating of the Sun but only at lower layers, i.e. possibly in the chromosphere and up to the magnetic canopy.32 -34 At least as a first approximation the plasma is considered frozen-in in the various magnetic structures in the hot solar atmosphere. The magnetic field plays a central and key role in the dynamics and energetics of the solar corona (see Figure 2). High-resolution satellite observations show the magnetic building blocks that seem to be in the form of magnetic flux tubes (Figure 3) in the solar atmosphere. These flux tubes expand rapidly

Fig. 3. TRACE images of the highly structured solar corona where the plasma is frozen in semi-circular shaped magnetic flux tubes. Left: The magnetic field in the solar atmosphere shapes the structures that we see, as the emitting gas can generally only move along the field. Courtesy Charles Kankelborg. Right: The image shows the evolution of loop system: an increasing number of loops appears in the 1 MK range. Courtesy TRACE (http://vestige.lmsal.com/TRACE/Public/Gallery/Images/TRACEpod.html).

in height because of the strong drop in density. Magnetic fields fill almost entirely the solar atmosphere at about 1,500 km above the photosphere. This environment is well described within the approximation of MHD.

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2. Equations of Ideal and Dissipative MHD The MHD equations are discussed in many excellent books, see, e.g. Refs. 35-40. Here we only give a short account of the MHD equations and their main properties necessary for grasping the key points of MHD waves and the heating processes. We consider a collision-dominated plasma with isotropic kinetic pressure. The state of the plasma is described by the density ρ, the pressure p, the temperature T , the velocity v, the magnetic induction B, the electrical field E, the electrical current density j, and the electrical charge density ρe . We use the SI system of units. In this system ρ is measured in kg m−3 , p in kg m−1 s−2 , T in K, v in m s−1 , B in tesla = kg s−2 A−1 = 104 G (A is amper), E in Vm−1 , j in Am−2 , and ρe in A s m−3 . The quantities B, E, j, and ρe are related by Maxwell equations ∇ × B = µ0 j,

(1)

∂B = −∇ × E, ∂t

(2)

∇ · E = ǫ−1 0 ρe ,

(3)

∇ · B = 0,

(4)

where we neglected the displacement current in Eq. (1). Here µ0 (= 4π × 10−7 kg m s−2 A−2 ) and ǫ0 (= 8.854×10−7 m−1 s AV−1 ) are the magnetic and electrical permeability of vacuum. Eq. (1) is called the Ampere’s equation. The quantities B, E, j, and v are, in addition, related by the Ohm’s law, which we take in its classical from j = σ(E + v × B),

(5) −1

where σ is the electrical conductivity measured in m−1 AV . The derivation of Eq. (5) and the discussion of the framework of its applicability can be found elsewhere, see, e.g. Ref. 38. Since Eq. (5) expresses E in terms of B, j, and v, Eq. (3) can be considered as the equation determining ρe . This equation is almost never used in magnetohydrodynamics. We substitute Eq. (5) into Eq. (2) and use Eq. (1) to arrive at ∂B = ∇ × (v × B) + η∇2 B, ∂t

(6)

where η = (σµ0 )−1 is the magnetic diffusivity measured in m2 s−1 . Equation (6) is called the induction equation.

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Plasma motions are described by the continuity, momentum, and energy equations. The continuity equation takes the form ∂ρ + ∇ · (ρv) = 0. ∂t

(7)

This equation expresses mass conservation. The momentum equation is   ∂v ρ + (v · ∇)v = −∇p + j × B + ρg + Fvis , (8) ∂t where g is the gravity acceleration and Fvis is the viscous force. This equation represents the second Newton law since its left-hand side is the acceleration of a unit volume of the plasma multiplied by its mass, while its right-hand side represents the force acting on this volume due to the pressure gradient, the Lorentz force, the gravity force, and viscosity. With the aid of Eq. (1) this equation can be re-written in the alternative form   1 ∂v ρ + (v · ∇)v = −∇p + (∇ × B) × B + ρg + Fvis . (9) ∂t µ0 The energy equation can be written as   ∂e ρ + v · ∇e + p∇ · v = −Q, ∂t

(10)

where e = cv T is the internal energy, cv the specific heat at constant volume, and Q the energy loss function describing the net effect of all the sinks and sources of energy. The first term on the right-hand side of Eq. (10) is the time derivative of the energy of a moving unit volume of plasma and the second term is the rate of work done by pressure on this volume. The temperature, pressure and density are related by the Clapeiron law p=

e R ρT, µ ˜

(11)

e = kB /mp is the gas constant, µ where R ˜ the mean atomic weight (the average mass per particle in units of the proton mass mp ), and kB the Boltzmann constant. For fully ionized plasmas consisting of protons and electrons only µ ˜ = 0.5. In what follows we also use the specific heat at constant pressure cp and the ratio of specific heats γ = cp /cv . The quantities e µ R, ˜, cv , and cp are related by cp = cv +

e R . µ ˜

(12)

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With the use of Eqs. (7), (8), and (12) we can rewrite Eq. (10) in the form     ∂ p p + v · ∇ = −(γ − 1)ρ−γ Q. (13) ∂t ργ ργ Since the entropy per unit mass of plasma is cv log(p/ργ ) + const, this equation is called the entropy equation. The term η∇2 B on the right-hand side of Eq. (6), the term Fvis on the right-hand side of Eq. (8), and the term Q on the right-hand side of Eq. (10) are dissipative terms. Their presence results in irreversible transfer of mechanical and magnetic energy into plasma heating. If we drop all these terms we arrive at the ideal MHD equations. The ideal momentum, induction, and entropy equations take the form   1 ∂v + (v · ∇)v = ∇p + (∇ × B) × B + ρg, (14) ρ ∂t µ0 ∂B = ∇ × (v × B), ∂t ∂ ∂t



p ργ



+v·∇



p ργ



(15)

= 0.

(16)

Equation (16) expresses the constancy of entropy in an elemental moving volume of plasma and it is called isentropic or adiabatic equation. 2.1. Linear MHD Equations Let us write all dependent variables in the MHD equations in the form f = f0 + f ′ , where f represents any dependent variable, the subscript ‘0’ indicates an equilibrium quantity, and the prime indicates the perturbation of a quantity. Then we substitute variables in this form into the MHD equations and obtain equations that contain terms of three types: terms that contain only equilibrium quantities; terms that are linear with respect to perturbations; and, terms that are nonlinear with respect to perturbations. Since the equilibrium quantities satisfy the MHD equations the terms of the first type cancel out. Let us now assume that |f ′ | ≪ f0 for any dependent variable. Then the terms of the third type can be neglected in comparison with the terms of the second type. As a result we arrive at the linear MHD equations ∂ρ′ + ∇ · (ρ0 v) = 0, ∂t

(17)

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∂v 1 lin = −∇p′ + [(∇ × B′ ) × B0 + (∇ × B0 ) × B′ ] + ρ′ g + Fvis , (18) ∂t µ0 ∂B′ = ∇ × (v × B0 ) + η∇2 B′ , ∂t

(19)

∇ · B′ = 0,

(20)

∂ ′ (p − c2S ρ′ ) + ργ0 v · ∇ ∂t



p0 ργ0



= −(γ − 1)Qlin ,

p′ T′ ρ′ = + . p0 T0 ρ0

(21)

(22)

Here c2S = γp0 /ρ0 is the square of the sound speed and the superscript ‘lin’ indicates a linearized quantity. When deriving these equations we have assumed that an equilibrium state is static, i.e. v0 = 0. Because of this assumption we have dropped the prime when writing the velocity perturbation. 2.2. MHD Waves in Ideal Uniform Plasmas Let us look for solutions to the ideal MHD equation in the form of normal modes and take the perturbations to be proportional to exp[i(k · r − ωt)] with r = (x, y, z) and k = (kx , ky , kz ) in the Cartesian coordinates x, y, z. If we assume that ω 6= 0, we arrive at an equation for the displacement ξ as 2 2 )k(k · ξ) − vA kk(b0 · ξ) cos ϕ ω 2 ξ = (c2S + vA 2 2 2 kb0 (k · ξ) cos ϕ. + vA k ξ cos2 ϕ − vA

(23)

Here vA is the Alfv´en speed and ϕ the angle between the equilibrium magnetic field and the wave vector k. They are determined by 2 vA =

B02 k · b0 , cos ϕ = . µ0 ρ0 k

(24)

Here b0 is the unit vector in the direction of the equilibrium magnetic field. It is straightforward to rewrite Eq. (23) as one scalar equation for the quantity ξ⊥ = ξ · (b0 × k)/k, which is the component of the vector ξ perpendicular to both k and b0 , and two coupled scalar equations for (k · ξ) and (b0 · ξ). These equations are 2 2 (ω 2 − vA k cos2 ϕ)ξ⊥ = 0,

(25)

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2 2 3 [ω 2 − (c2S + vA )k 2 ](k · ξ) + vA k (b0 · ξ) cos ϕ = 0,

(26)

c2S k(k · ξ) cos ϕ − ω 2 (b0 · ξ) = 0.

(27)

First we study the case where ϕ 6= 0. Equation (25) has non-trivial solution when 2 2 ω 2 = vA k cos2 ϕ.

(28)

The waves with the frequency determined by Eq. (28) are called Alfv´en waves. We see that only the component of the plasma displacement perpendicular to both k and b0 is non-zero in an Alfv´en wave. The set of two linear Eqs. (26) and (27) has non-trivial solution when its determinant is zero. This condition results in the dispersion equation 2 4 2 k cos2 ϕ = 0. )ω 2 k 2 + c2S vA ω 4 − (c2S + vA

The solution to this equation is n o 1 2 2 2 2 ω 2 = a2± k 2 ≡ 2 k 2 c2S + vA ± [(c2S + vA ) − 4c2S vA cos2 ϕ]1/2 .

(29)

(30)

The signs ‘+’ and ‘−’ in this expression correspond to fast and slow magnetoacoustic waves. When waves propagate along the equilibrium magnetic field (ϕ = 0) it is straightforward to check that ω+ = vA k when cS < vA and ω− = vA k when cS > vA . Hence ω = vA k is the double root of the dispersion equation. It still corresponds to Alfv´en waves. The plasma displacement in Alfv´en waves can now be in any direction perpendicular to B0 . The other root of the dispersion equation is ω = cS k. It corresponds to a sound wave propagating along the equilibrium magnetic field. MHD waves play a very important role in coronal physics. Flux tubes are shaken and twisted by photospheric motions (i.e. by both granular motion and global acoustic oscillations, the latter being called p-modes). Magnetic flux tubes are excellent waveguides (see e.g. the coupling from photosphere to the transition region as observed by Ref. 24). If the characteristic time of these photospheric footpoint motions is much less than the local Alfv´enic transient time the photospheric perturbations propagate in the form of various MHD tube waves (e.g. slow and fast MHD waves; Alfv´en waves). The dissipation of MHD waves is manifold: these waves couple with each other, interact non-linearly, resonantly interact with the closed waveguide (i.e. coronal loops) or develop non-linearly (e.g. solitons or shock waves can form), etc. For an extensive review on the observations of MHD waves, see, e.g. Ref. 8, while on theory see, e.g. Ref. 26.

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In an inhomogeneous and magnetised plasma there are two particular dissipation mechanisms of MHD waves that received extensive attention in the past decades: resonant absorption and phase mixing. Although there are major theoretical advances on these two particular dissipation mechanisms unfortunately we still have only indirect evidences that they may actually operate under solar circumstances. Thanks to the fantastic imaging capabilities of TRACE, plenty of observations of MHD wave damping in coronal loops are available8 and some of these cases may be an excellent candidate of resonant absorption. Further, it is less likely that phase mixing operates in closed magnetic structures, like solar coronal loops.

2.3. MHD Waves in Ideal Inhomogeneous Plasmas The analysis so far has been assumed a uniform medium. The mathematical advantage of this assumption is that the PDEs readily reduced to algebraic equations for the dispersion relation ω = ω(kx , ky , kz ). However this is only valid provided the wavelength is much smaller than the length-scale of the inhomogeneity. When the wavelength λ ≥ l0 , the length-scale of inhomogeneity, then the inhomogeneous nature of the medium determines the behaviour of the disturbances. In the solar atmosphere the principal causes of inhomogeneity are gravity and the structured magnetic field (magnetic pores, sunspots, chromospheric canopy, arcades or prominences, coronal loops, plumes, etc). Gravity creates a vertical stratification in plasma density and pressure, and the magnetic field can cause the plasma pressure to increase in a direction normal to the field. These stratifications introduce a few significant effects that affect MHD wave propagation: (i) amplification of the wave amplitude (e.g. chromospheric shocks); evanescence - regions in which waves, otherwise oscillating spatially, may decay exponentially (solar global f /p/g-mode oscillations); waveguide modes - a discontinuity (magnetic, density, ...) in the medium may give rise to waves guided by the structure (e.g. surface or body modes in a flux tube).

2.3.1. MHD waves at magnetic interface Let us first ignore gravity and focus interest on the effect of magnetic structuring on MHD wave propagation. Assume that in the basic state the plasma is permeated by a magnetic field B0 (x)ˆz, working in Cartesian coordinates x, y, z (see Fig. 4b). Then the pressure and density are structured by the x-dependence of the magnetic field and the basic state is found

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to be p0 = p0 (x),

ρ0 = ρ0 (x),

d dx

  B02 p0 + = 0, 2µ

(31)

for the pressure, density and total (magnetic + gas) pressure. Linearised perturbations from this state are taken and the equations of continuity, momentum, the induction and energy equations in ideal MHD give: ∂ρ + divρ0 v = 0, ∂t   ∂v 1 1 1 ρ0 = −∇ p + B0 · b + (B0 · ∇) b + (b · ∇) B0 , ∂t µ µ µ

(32)

(33)

∂b = curl (v × B0 ) , ∂t

(34)

  ∂p 2 ∂ρ + v · ∇p0 = c0 + v·∇ρ0 . ∂t ∂t

(35)

After Fourier analysis, e.g., (vx = vˆx (x)ei(ωt+ly+kz) , p = pˆ(x)ei(ωt+ly+kz) ) a single ODE for vˆx (x) is obtained. Consider the magnetic interface defined by  Be , x > 0, B0 (x) = (36) B0 , x < 0, with B0 and Be both constants. Pressure continuity at x = 0 gives pe +

Be2 B2 = p0 + 0 . 2µ 2µ

(37)

It is then found that the equation governing vx (x) is simply d2 vˆx − m20 vˆx (x) = 0 dx2

for x < 0,

(38)

d2 vˆx − m2e vˆx (x) = 0 dx2

for x > 0,

(39)

and

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where m20 (x) =

2 (k 2 c20 (x) − ω 2 )(k 2 vA (x) − ω 2 ) 2 (x))(k 2 c2 (x) − ω 2 ) , (c20 (x) + vA T 2 c20 (x)vA (x) 2 2 (x) , c0 (x) + vA way to m20 except that

c2T (x) =

(40) (41)

and m2e is defined in a similar the Alfv´en and sound speeds appropriate to x > 0 are taken. It is the presence of the discontinuity in B0 (x) that is responsible for the existence of surface waves which may arise if m20 and m2e are both real and positive. Solving (38) and (39) for vˆx (x) gives  αe e−me x , x > 0, vˆx (x) = (42) α0 em0 x , x < 0.

In writing this solution we are excluding laterally propagating waves, so only the modes with non-vanishing amplitudes (end energy) near the discontinuity arise. Then, using the continuity of vˆx (x) and of the total pressure perturbation pˆT (x) across the interface at x = 0 the general dispersion relation for the magnetic interface is found: 2 2 ρ0 (k 2 vA − ω 2 )me + ρe (k 2 vAe − ω 2 )m0 = 0,

(43)

valid for m20 and m2e both positive and for l = 0. From the inspection of the dispersion relation (43) one is able to conclude that the existence of a magnetic interface supports the propagation of surface waves as shown by, e.g. Ref. 41, and what we follow here closely. In particular, slow MHD surface waves exist if one side of the interface is field-free. If the gas in the magnetised region is cooler than the field-free medium the discontinuity may also support fast MHD surface waves. In general slow and fast MHD surface waves may arise if there is a sharp transition in the background slow and/or Alfv´en speeds. 2.3.2. Waves in magnetic slab First consider a magnetic slab with zero field surrounding it so that  B0 , |x| < x0 , B0 (x) = 0, |x| > x0 ,

(44)

with pressure p0 and density ρ0 inside the slab, pe and ρe outside. The two regions are related by  2 1 2  c0 + 2 γvA B2 p e = p 0 + 0 , ρe = ) ρ0 (45) 2µ c2e

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where co and ce are the sound speeds inside and outside the slab and vA is the Alfv´en speed in the slab. Again, attention is confined to two-dimensional disturbances so that the velocity perturbation component vy and wavenumber l are assumed to be zero. Analogue to the case of the magnetic interface it is found that d2 vˆx − m20 vˆx = 0 f or |x| < x0 , (46) dx2 and d2 vˆx − m2e vˆx = 0 f or |x| > x0 , (47) dx2 for m0 as above and me by analogous comparison. Again, the boundary conditions, vˆx (x) and pˆT (x) being continuous across the boundary x = ±x0 are used to obtain the general dispersion relation, as shown by, e.g. Refs. 42-43:     tanh ρe 2 − ω 2 )me = ω 2 m0 m0 x0 , (48) (k 2 vA ρ0 coth

valid for ω 2 < k 2 c2e . Dispersion relation (48) describes the existence of slow and fast magneto-acoustic waves that could be either body or surface waves, depending on their structure within the slab (i.e. evanescent for surface and oscillatory for body waves, respectively). Perturbations that are symmetric about the vertical axis of the slab are called sausage oscillations, while perturbations that are anti-symmetric are called kink oscillations. When the slab is considered thin in comparison to the wavelength (long wavelength approximation that is of interest for photospheric and coronal conditions) the kink mode vibrates as a single thin string and the sausage mode vibrates as both surface and body waves. 2.3.3. MHD waves in magnetic cylinder The magnetic building blocks in the solar atmosphere are the magnetic flux tubes. In a pioneering work,44 using cylindrical coordinates, it was derived the dispersion relations of MHD waves propagating in cylindrical magnetic flux tubes. The main obstacle to be overcome when introducing the concept of flux tubes is the conversion from Cartesian to cylindrical coordinates. This change results involving Bessel functions in the dispersion relation which are not yet possible to be solved analytically without simplification, e.g. through incompressibility or long and short wavelength approximations. Let us summarise here the key steps.44 Consider a uniform magnetic cylinder of magnetic field B0ˆ z confined to a region of radius a,

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Fig. 4. Left: Magnetic flux tube showing a snapshot of Alfv´ en wave perturbation propagating in the longitudinal z-direction along field lines at the tube boundary. At a given height the Alf´ enic perturbations are torsional oscillations, i.e. oscillations are in the ϕ-direction, perpendicular to the background field. Right: Snapshot showing Alfv´ en waves propagating along a magnetic discontinuity. Again, the key feature to note is that Alfv´ enic perturbations are within the magnetic surface (yz-plane) at the discontinuity, perpendicular to the background field (y-direction), while the waves themselves propagate along the field lines (z-direction).

surrounded by a uniform magnetic field Beˆ z (see Figure 4a). To simplify the MHD equations we assume again zero gravity, there are no dissipative effects and all the disturbances are linear and isentropic. Pressure (plasma and magnetic) balance at the boundary implies that p0 +

B02 B2 = pe + e . 2µ0 2µ0

(49)

Linear perturbations about this equilibrium give the following pair of equations valid inside the tube, ∂2 ∂t2

! 2 ∂2 2 2 2 2 2 ∂ − (c + v )∇ ∆ + c v ∇2 ∆ = 0, 0 A 0 A ∂t2 ∂z 2

(50)

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! 2 ∂2 ∂ 2 − vA Γ = 0, ∂t2 ∂z 2

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(51)

where ∇2 is the Laplacian operator in cylindrical coordinates (r, θ, z) and ∆ ≡ divv,

Γ=ˆ z · curlv

(52)

for velocity v = (vr , vθ , vz ). A similar pair of equations to (50) and (51) are valid outside the tube. Fourier analysing we let ∆ = R(r)exp[i(ωt + nθ + kz).

(53)

Then equations (50) and (51) give Bessel’s equation satisfied by R(r) as follows d2 R 1 dR + − dr2 r dr

! 2 n m20 + 2 R = 0, r

(54)

where m20 =

2 (k 2 c20 − ω 2 )(k 2 vA − ω2) 2 )(k 2 c2 − ω 2 ) . (c2e + vAe Te

(55)

We have used the notation cT for the the characteristic tube speed (sub2 −1/2 Alfv´enic), where cT = c0 vA /(c20 + vA ) . To obtain a solution to (54) bounded at the axis (r = 0) we must take R(r) = A0



In (m0 r), m20 > 0 Jn (n0 r), n20 = −m20 > 0



(r < a),

(56)

where A0 is an arbitrary constant and In , Jn are Bessel functions45 of order n. For a mode locked to the waveguide it is required that no energy propagates to or from the cylinder in the external region, i.e. the waves are evanescent outside the flux tube. Therefore we take R(r) = A1 Kn (me r),

r > a,

(57)

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where A1 is a constant and m2e =

2 (k 2 c2e − ω 2 )(k 2 vAe − ω2) 2 )(k 2 c2 − ω 2 ) , (c2e + vAe Te

(58)

which is taken to be positive (no leaky waves). Since we must have continuity of velocity component vr and total pressure at the cylinder boundary r = a, this yields the dispersion relations 2 ρ0 (k 2 vA − ω 2 )me

Kn′ (me a) I ′ (m0 a) 2 = ρe (k 2 vAe − ω 2 )m0 n , Kn (me a) In (m0 a)

(59)

for surface waves (m20 > 0) and 2 ρ0 (k 2 vA − ω 2 )me

Kn′ (me a) J ′ (m0 a) 2 = ρe (k 2 vAe − ω 2 )m0 n , Kn (me a) Jn (m0 a)

(60)

for body waves (m20 = −n20 < 0). The well-observed axisymmetric sausage mode is given by n = 0, while the kink mode (non-axisymmetric) is given by n = 1. Modes with n > 1 are called flute modes. Although the dispersion relations (59) and (60) are complicated, finding the phase speed for e.g. kink waves with coronal parameters simplifies matters considerably. In the corona one can assume B0 ≈ Be , vAe , vA > ce , c0 and ρ0 > ρe . This means that only fast and slow body waves may occur and there are no longer surface waves present (see Fig. 5). Also, for coronal loops the thin flux tube/long wavelength approximation (| k | a ≪ 1) is a good approximation since the observed loop length is always much larger than loop width. For instance, for fundamental standing mode kink oscillations the wavelength is twice the loop length. In this limit, also called the slender tube limit, the fast sausage mode does not even exist, while the fast kink wave propagates with the kink phase speed ck , given by ck =

2 2 ρ0 vA + ρe vAe ρ0 + ρe

!1/2

.

(61)

Fast kink modes, when vAe > vA , are sustained in dense loops with periods on an Alfv´enic timescale and it is found that these body waves have a low wavenumber cut off implying that only wavelengths shorter than the diameter of a loop can propagate freely. The sausage mode, however, has a much shorter period, approximately one tenth of that of the kink mode. Sausage and kink fast body modes exist only in high density loops. However, the slow modes appear in both high and low density cylinders.

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Fig. 5. The solution of the dispersion relations (59)-(60) in terms of phase speed (ω/k) of modes under coronal conditions vAe > vA > c0 > ce (all speeds are in km/s). The slow band is zoomed, and only the two first harmonics of a mode are shown (lower panel).

Under photospheric conditions, characterised by ce > vA > c0 , representative for sunspots or pores both the slow and fast bands have surface and body modes, respectively. The slow waves are in a narrow band since c0 ≈ cT . The slow body waves are almost non-dispersive, whereas the almost identical slow surface sausage and kink modes are weakly dispersive (bottom zoomed out panel in Fig. 6). 2.3.4. MHD waves in magnetically twisted cylinder Granular shear motions, differential rotation or meridional circulation in the photosphere can introduce a twist to the flux tubes from pores to sunspots. Erupting prominences or CMEs, with their footpoints anchored in the dense sub-photosphere, often appear to have twisted field lines. It is natural and practical to extend the investigations of MHD wave modes in twisted magnetic flux tubes. Twisted tubes have been studied before but mainly in terms of stability. Here we briefly summarise the current status on results

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Fig. 6. The solution of the dispersion relations (59)-(60) in terms of phase speed (ω/k) of modes under photospheric conditions ce > vA > c0 (all speeds are in km/s). The slow band is zoomed (lower panel).

on MHD waves in magnetically twisted solar atmospheric tubes, see, e.g. Refs. 46-48. Let a uniformly twisted flux tube embedded in a straight magnetic field (Fig. 7) given by: B=



(0, Ar, B0 ), (0 , 0 , Be ),

r < a, r > a.

(62)

In cylindrical equilibrium the magnetic field and plasma pressure satisfy the pressure-balance equation in the radial direction: d dr

2 2 B0ϕ + B0z p0 + 2µ

!

+

2 B0ϕ = 0. µr

(63)

Here, the second term in the brackets represents magnetic pressure and the third term derives from magnetic tension due to the azimuthal component of the equilibrium magnetic field, B0ϕ . For the sake of simplicity the plasma is taken incompressible, with the field and plasma pressure being structured in the radial direction only. Again, using continuity of total pressure pT and perturbation velocity vr across r = a and seeking for a bounded solution,

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Fig. 7. Straight, vertical, uniformly twisted magnetic flux tube in an ambient magnetic field. At the boundary of the flux tube there is a jump in the magnetic twist.

at r = 0 and r → ∞, leads to the dispersion relation ′

2 (ω 2 −ωA0 )

m0 aIm (m0 a) A −2mωA0 √ Im (m0 a) µρ0

2 )2 −4ω 2 (ω 2 −ωA0 A0

A2 µρ0

′

=

|kz |aKm (|kz |a) Km (|kz |a)

′

ρe 2 A2 |kz |aKm (|kz |a) 2 (ω −ωAe )+ ρ0 µρ0 Km (|kz |a)

.

(64) In this equation the dash denotes derivative with respect to the argument of the Bessel function and 1 ωA0 = √ (mA + kz B0 ), µρ0  2 2 m0 = kz 1 −

kz Be ωAe = √ , µρe

2 4A2 ωA0 2 )2 µρ0 (ω 2 − ωA0



(65) .

Eq. (64) is the dispersion relation for waves in an incompressible flux tube with uniform magnetic twist, embedded in a straight magnetic environment. For an incompressible tube without magnetic twist there are no body waves. However, magnetic twist introduces an infinite band of body waves. A dual nature of the mode is also discovered where a body wave exist for long wavelengths but surface wave characteristics are displayed when the eigenfunction is plotted for shorter wavelengths.

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2.3.5. MHD oscillations in annular magnetic cylinders Sub-resolution flux tube structure is still more a matter of some speculations. It is anticipated, that Hinode/SOT may advance the research on the internal structure of solar magnetic flux tubes. As a specific example, let as recall an interesting earlier observation:49 combined data of May 13 1998 from both the EIT instrument on SOHO and from TRACE show the simultaneous observation of two slow magnetosonic waves propagating along a perceived coronal loop with speeds of 95 and 110 km s−1 . This observation was interpreted by the authors as temperature differences within the observed loop hinting at a substructure of perhaps either concentric shells of different temperatures or of thin strands within the same loop at different temperatures. There is no conclusive proof disputing these possible flux tube structures nor preference given towards one in particular. Here let us shall assume that a flux tube consists of a central core surrounded by a shell or annulus layer, all embedded in uniform magnetic field (Fig. 8). For details see, e.g. Refs. 50-53. We restrict our investigation

Bi

B0 Be

a R

Fig. 8. The equilibrium configuration of a magnetic cylinder consisting of a core, annulus and external regions, all with straight magnetic field.

to an incompressible plasma for which the phase speeds of the slow and Alfv´en waves, in the limit of incompressibility, become indistinguishable, the modes are discernable through direction of perturbation only (Alfv´en wave perturbations are perpendicular to both magnetic field and propagation vector while slow wave perturbations are in the same plane). The fast waves are removed from the system. We consider a longitudinal magnetic

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field in each region of a magnetic annulus so that  r < a,  Bi = (0, 0, Bi ), B = B0 = (0, 0, B0 ), a ≤ r ≤ R,  Be = (0, 0, Be ), r > R,

83

(66)

where Bi , B0 , Be are constant. We take the densities in the core, annulus and external regions as ρi , ρ0 , ρe , respectively, and similarly denote the pressure in each region as pi , p0 , pe . The pressure balance at the boundaries r = a and r = R gives the relations Bi2 B2 = p0 + 0 , 2µ 2µ

pi +

p0 +

B02 B2 = pe + e , 2µ 2µ

(67)

and we denote vAi = Bi /(µρi )1/2 , vA0 = B0 /(µρ0 )1/2 and vAe = Be /(µρe )1/2 as the Alfv´en speeds in the internal, annulus and external regions, respectively. Taking linear perturbations of the ideal MHD equations about this equilibrium and Fourier-decomposing the total Lagrangian pressure pT (= p + B.b/µ for perturbed field b and plasma pressure p) and normal component of Lagrangian displacement ξr like (pT , ξr ) ∼ (ˆ pT (r), ξˆr (r)) ei(mθ+kz z−ωt) ,

(68)

and omitting the hat of the Fourier decomposed perturbations for the sake of simplicity of notation, we find that pT (r) satisfies the Bessel equation   1 dpT m2 d2 pT 2 + − kz + 2 pT = 0, (69) dr2 r dr r where m is the azimuthal wavenumber (see Eq. 24). To obtain a dispersion relation for the configuration of a magnetic annulus we require the solutions for pT in Eq. (69) to satisfy the continuity of total Lagrangian pressure pT and the continuity of the normal displacement perturbation, ξr , across the boundaries r = a and r = R. After some algebra and using these boundary conditions we arrive at: ′ ′ Qi0 Km (kz a) − (Im (kz a)Km (kz a)/Im (kz a)) ′ (k a)(Qi − 1) Im z 0

=

′ Km (kz R)(Qe0 − 1) . ′ (k R) − (K ′ (k R)I (k R)/K (k R)) Qe0 Im z m z m z m z

(70)

in which we have defined Qi0 =

2 ρi (ω 2 − ωAi ) , 2 2 ρ0 (ω − ωA0 )

Qe0 =

2 ρe (ω 2 − ωAe ) . 2 2 ρ0 (ω − ωA0 )

(71)

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Eq. (70) is the dispersion relation for wave propagation in an incompressible straight magnetic flux tube structure consisting of a core tube and an annulus of radii a and R, respectively, embedded in an ambient magnetic environment with straight field lines. For the limiting case of no core a → 0 (or equally a → R) we recover a dispersion relation derived earlier44 for a single straight monolithic tube with external Alfv´en speed vAe and internal Alfv´en speed vA0 (vAi ). There are two surface mode solutions to the disper-

Fig. 9. Dispersion curves for m = 0 sausage (dotted) and m = 1 kink modes (solid) giving the phase speed ω/k as function of the dimensionless wavenumber ka for a magnetic annulus with a/R = 0.8 modelling left photospheric conditions, and right the a flux tube with a dense core.

sion relation (70) arising for each the sausage and kink modes, respectively, for the annulus-core model. These modes propagate along the two natural surfaces of the system, i.e. at r = a and r = R. In the incompressible approximation, for a straight magnetic field, body modes are removed from the system. The modifications to the phase speeds, introduced by the annulus, as compared to a monolithic loop are significant and dependent not only on the Alfv´en speed in each region but also on the ratio a/R of the core and annulus radii. 2.4. Magneto-Seismology: Inhomogeneous Magnetic Field Post-flare transversal coronal loop oscillations have been observed many times using the high-resolution EUV imager onboard TRACE, see, e.g. Refs. 54-58. These oscillations were identified as the fundamental mode of the standing fast kink wave from MHD wave theory.44 The basic theory models a coronal loop as a straight magnetic cylinder with different external and internal plasma densities, both of which are taken to be constants (see the previous sections).

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Using emission measures, there is observational evidence that there is density stratification in coronal loops. In younger active regions there have been measurements of super-hydrostatic density scale heights that are up to four times higher than expected.59,60 On the other hand, loops have been observed in older active regions that are close to hydrostatic equilibrium61 with density scale heights that can be explained by gravitational stratification. The implications of this for coronal loop oscillations have been considered, e.g. Refs. 62 and 63 through numerical modelling of damping of loop oscillations in the framework of dissipative and radiative MHD. To complicate matters further, significant dynamical behaviour has also been observed in loops, e.g., flows64 -66 and cooling events.67,68 This is an important point because background flows can cause complex interactions between MHD waves. Theoretically, the effect of steady state flows on MHD waves in a uniform magnetic slab-geometry was investigated by, e.g. Refs. 69-71. They found the dispersion relation for such steady states and also have shown the presence of negative energy waves. Refs. 72-74 generalised the slab studies to flux tubes but their derivation is valid only for limited parameters. A detailed and comprehensive derivation of steady flow effects on uniform MHD waveguides in cylindrical geometry (with stratification due to gravity ignored) can be found, e.g. Ref. 75. In light of the exciting observations from TRACE, much work has been done developing more realisitic theory of fast kink waves in coronal loops. E.g., models have been developed with inhomogeneous plasma density equilibria. Firstly, spatial variation of density in the radial direction has been included in the analysis leading to a change in period and damping of the MHD waves.76 -80 Secondly, spatial variation of density in the longitudinal direction has been included in the analysis leading to changes in the ratios of the periods of the overtone modes to that of the fundamental mode and to deviations of the eigenfunctions from a single sine term in the longitudinal direction.81 -89 To develop a more complete theory of fast kink waves coronal loops, here we quantify the effects of both inhomogeneous plasma density and magnetic equilibria. At present, the structure of the magnetic field along coronal loops is probably even less well understood from observation than plasma density stratification.90 The indirect observational evidence so far has been rather puzzling. A study of TRACE loops91 has shown that the cross-sectional width remains relatively constant with increasing height above the photosphere. The flux tube interpretation suggests that magnetic field is therefore almost constant along loops but this contradicts potential and force-free

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field extrapolations using data from the Michelson Doppler Imager (MDI) onboard SOHO, where the field lines always diverge with height. It was also suggested92 that by twisting a loop this could reduce the amount of width expansion with height. It is therefore crucial that theoretical models are developed which can predict how different magnetic field structures in loops affect the properties of loop oscillations. It was summarised above the work that has already been done, particularly regarding the effect of magnetic twist on loop oscillations, e.g. Refs. 46-48 and 93. Furthermore, the effect of a twisted shell, i.e., tube within a tube, was studied by Refs. 51-53. It is hoped that models of this type that have more complex magnetically structured tubes can be tested against observations and help further advance the field of magneto-seismology.

2.4.1. Magnetic field and plasma density equilibrium Using cylindrical coordinates (r, θ, z), a magnetic flux tube of length 2L is modelled with arbitrary external and internal plasma densities ρe (z) and ρi (z). To model a magnetic field equilibrium that decreases in strength with height above the photosphere, we construct an expanding flux tube with rotational symmetry (see Fig. 10). To do this one must have ~ = Br (r, z)~er + Bz (r, z)~ez B

(72)

so that the solenoidal and force-free (potential) conditions are satisfied.

2.4.2. Governing equation and analysis By Fourier analysing the linear MHD equation and using the thin flux tube approximation in cold plasma and the fact the Br ≪ Bz , it can be shown that the governing equation of radial motion at the tube boundary (where all quantities can be expressed as functions of z only) for the observed fast kink mode (m = 1) is ′′

(Bz vr ) +

+

 Br ′ + 4ro′ (Bz vr ) Bz # 2  ′  ′ 2 ω 1 Br ro ro′′ + + + Bz vr = 0. ck 2ro Bz ro ro

1 2ro "



(73)

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Fig. 10.

87

The equilibrium plasma density and expanding magnetic field.

Assuming constant densities, ρe and ρi , Eq. (73) is equivalent to h z  z i z h z i a1 cosh2 + a2 cosh + a3 vr′′ + sinh a4 cosh + a5 vr′ L L L L z i h z + a6 cosh2 + a7 cosh + a8 + a9 ω 2 vr = 0, (74) L L where an are constants. Unfortunately, we know of no analytical solution to equations of general type Eq. (74). However, Eq. (74) is trivial to solve numerically by e.g., the shooting method. Solving Eq. (74) for the fundamental mode and first harmonic, the observable signatures of magnetic stratification are plotted in Fig. 11 for a small expansion Γ ∈ [1, 2]. In contrast to the case of density stratification with constant magnetic field, see, e.g. Ref. 89, the anti-node shift of the first harmonic is towards the loop apex (see Fig. 11a). It was argued94 that for Γ approximately less than 1.5 there is almost a linear relationship with node and anti-node shifts. In further contrast to the case of density stratification with constant magnetic field, see, e.g. Ref. 95, the frequency ratio of the first harmonic to the fundamental mode, ω2 /ω1 is greater than 2 (see Fig. 11b). There have been various studies to calculate the value of Γ for coronal loops in both soft X-ray and EUV. Using Yohkoh data, it was found96 that the mean value of Γ for a sample of 43 soft X-ray loops was 1.30.

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A

anti−node shift

Γ=1

(a)

Amplitude

Γ = 1.5

0

−A anti−node shift

−L

0 z

L

4 (c)

2

ω /ω

1

3

2

1

0 1

1.2

1.4

Γ

1.6

1.8

2

Fig. 11. (a) Comparison of 1st harmonic amplitude profiles with constant magnetic field (Γ = 1) and magnetic stratification (Γ = 1.5). (b) Frequency ratio of 1st harmonic and fundamental mode ω2 /ω1 against Γ.

In another study using EUV TRACE data91 it was also found that the mean Γ value for post-flare loops was 1.13. However there may have been large uncertainties in these results. Errors could also have been introduced by e.g., incorrect background subtraction and line of sight effects. Even allowing for a relatively small expansion factor of Γ = 1.13, this should give measurable observable effects. E.g., a loop half length L = 100 Mm and fundamental mode period 5 minutes, Γ = 1.13 will give an anti-node shift of 3.5 Mm and a change in the period of the first harmonic of -6.23 seconds. Certainly, spatial changes to the amplitude profile of a few Mm is within the current resolution of TRACE. Measuring changes in frequency down to the order of seconds may be possible with the fastest time cadences of the planned EUV imagers onboard SDO and SO (signal to noise ratio permitting). 2.5. Mechanism of Resonant Absorption Let us consider an ideal inhomogeneous vertical magnetic flux tube embedded in a magnetic free plasma such that the Alfv´en speed has a maximum

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at the axis of the tube and the Alfv´en speed is monotonically decreasing to zero as a function of the radial coordinate (Figure 12a). Suppose that there

Fig. 12. Left: Schematic sketch of resonant absorption. Right: Phase-mixing of surface waves caused by gradients in the background magnetic field (or Alfv´ en speed) where the footpoints of the field lines are shaken in the y-direction.

is a sound wave continuously impinging at the boundary of this flux tube. If the phase speed of this impinging (or driving) wave matches the local Alfv´en speed at a given location of the radius, say at rA , the driving wave is in resonance with the local Alfv´en waves at the magnetic surface at rA . In ideal MHD this would result in infinite amplitudes of the perturbations resulting in large gradients. However, once the gradients of perturbations become large, one cannot assume any longer the plasma is ideal, i.e. dissipative effects (e.g. resistivity, viscosity) have to be considered at least within the vicinity of such resonant location leading to energy dissipation. Such dissipation, i.e. energy absorption of the driving wave, will result in heating of the plasma converting the energy of the driving wave into localised thermal heating.97 -99 Resonant absorption, originally considered by plasma physicists as means of excess heating source for thermonuclear fusion, seems to work very well when modelling e.g. the interaction of solar global oscillations with sunspots;101,102 when applied to explain the damping of coronal loop oscillations,31,103 resonant flow instabilities, e.g. Ref. 70, etc. 2.6. Process of Phase Mixing There was proposed104 another interesting mechanism that is in a way fairly similar to resonant absorption. There is a magnetised plasma that is inhomogeneous in the x-direction of the xz-plane where the magnetic field lines are parallel to the z-axis (Figure 12b). We perturb each field line in a

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coherent (e.g. sinusoidal) way in the y-direction. Along each of the field lines an Alfv´en wave will develop and will propagating in the z-direction with a speed characteristic to that field line. Since the plasma is inhomogeneous the Alfv´en speed at two adjacent field lines is different and neighbouring oscillating field lines will be soon out of phase after some time resulting in large gradients of perturbations. At a given point when the gradients reach a critical value it is not correct anymore to assume that the plasma is ideal and dissipative effects have to be included in the analysis (just like in the case of resonant absorption) resulting in local heating. This dissipation of the initial perturbations is called phase mixing. Phase mixing is an excellent candidate for MHD wave energy dissipation in open magnetic regions like coronal funnels, plumes, solar wind. 3. MHD Waves in the Lower Solar Atmosphere It is a natural question, whether from the intrinsic periodic motions of the magnetic structures one would be able to derive crucial diagnostic information, e.g. density stratification both along and across a coronal loop; derive constrain on fine structure (single monolithic vs multi-thread magnetic structuring); obtain information about the geometry (curvature, inclination, expansion rate, cross-sectional formation) and topology (magnetic connectivity), etc. An application of the above approach to the solar corona was first put forward105,106 introducing the term of coronal seismology, what was reviewed for standing waves in the previous sections. The method of coronal seismology, however can be further extended and generalised to the entire solar atmosphere from the partially magnetised photosphere to magnetically dominated corona. In order to reduce the number of various and sometimes confusing labelling, but at the same time to express the strong relation and overlap of the methodology of helioseismology and coronal seismology, the unifying term of solar magneto-seismology is introduced. The terminology serves well the rapid emergence of seismic and diagnostic studies of the magnetised solar atmosphere, and, at the same time expresses the major future direction of developments in local helio(and astero)seismology. An interesting and promising application of magneto-seismology is to specifically investigate oscillations in the lower part of the solar atmosphere, where periodic motions maybe mostly but not exclusively in the form of propagating waves. The various and by large concentrated magnetic structures at photospheric to low-TR and coronal heights serve as excellent waveguides to the propagation of perturbations excited at foot-

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point regions. A typical and characteristic lower atmospheric waveguide, in a first approximation, could possibly be considered as an isolated magnetic flux tube with practically no magnetic field in their environment, resulting in a hat-like functional dependence of the Alfv´en speed variation across the MHD waveguide. Another feature that distinguishes these lower atmospheric waveguides from their coronal counterparts is that the density, pressure and magnetic strength scale heights inside the waveguide (and in the surrounding environment except for magnetic fields) may be comparable and of the order to the horizontal (i.e. radial) dimension of the magnetic structures, and/or, also comparable to the characteristic wavelength of the periodic motions they support. Even for the simplest mathematical approximation this latter feature leads to the introduction of a linear term to the governing equation of wave perturbations resulting in a Klein-Gordon-type of equation.26,107 Before we dwell on the recent developments made on observations of wave leakage and progressive waves in the (lower) solar atmosphere, we refer the reader to related reviews.8,11,33,108 -115 3.1. Magneto-Seismology in the Lower Boundary Layer To carry out magneto-seismology in the lower partially magnetised solar atmosphere, one of the first tasks is to understand what is the role of the presence of the boundary layer between the solar interior (β ≫ 1) and the magnetically dominated corona (β ≪ 1). The transition between the solar interior and the corona occurs in a rather narrow layer. This boundary layer, that includes the photosphere, chromosphere and TR is around 2-3 Mm thick and contains coherent and random magnetic and velocity fields, giving a very difficult task to describe even in simple approximate terms the wave perturbations. Random flows (e.g. turbulent granular motion), coherent flows (meridional flows or the near-surface component of the differential rotation), random magnetic fields (e.g. the continuously emerging small-scale magnetic flux, often referred to as magnetic carpet) and coherent fields (large loops and their superposition of the magnetic canopy region) each have their own effect on wave perturbations. Some of these effects may be more relevant and deterministic than the others. From practical helioseismology perspectives the main task here is to estimate the magnitude of these corrections one by one. It is strongly suspected that, both coherent and random magnetic and velocity fields may contribute to line widths or frequency shifts of the global acoustic oscillations on a rather equal basis. In helioseismology the corrections from this boundary layer

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are summarised in the surface term34,111,116 and in many helioseismologic diagnostic modelling the surface term is taken in some ad-hoc functional form. Now we dwell on the question: how photospheric motions (e.g. coherent internal acoustic oscillations or casually generated pulses) penetrate into the solar atmospheric boundary layer, and, what observable dynamic consequences this leakage and penetration may have on the magnetic structures of the solar atmosphere. 3.2. Wave Leakage from Photosphere Spicules and moss oscillations, detected by TRACE and by SUMER on board SOHO may bring us closer to the origin of the running (propagating) waves of coronal loops. The correlations on arcsecond scales between chromospheric and transition region emission in active regions were studied.118 The discovery of active region moss,117 i.e, dynamic and bright upper transition region emission at transition region heights above active region (AR) plage, provides a powerful diagnostic tool to probe the structure, dynamics, energetics and coupling of the magnetized solar chromosphere and transition region. It was also studied118 the possibility of direct interaction of the chromosphere with the upper TR, by searching for correlations (or lack thereof) between emission at varying temperatures using concurrently taken EUV lines emitted from the low chromosphere (Ca II K-line), the middle and upper chromosphere (Hα), the low transition region (C IV4 1550 ˚ A at 0.1 MK), and from the upper transition region (Fe IX/X 171 ˚ A at 1 MK and Fe XII 195 ˚ A at 1.5 MK). The high cadence (24 to 42 seconds) data sets obtained with the Swedish Vacuum Solar Telescope (SVST, La Palma) and TRACE allowed to them find a relation between upper transition region oscillations and low-laying photospheric oscillations. Intensity oscillations were analysed24 in the upper TR above AR plage. They suggested the possible role of a direct photospheric driver in TR dynamics, e.g. in the appearance of moss (and spicule) oscillations. Wavelet analysis of the observations (by TRACE) verifies strong (∼ 5 - 15%) intensity oscillations in the upper TR footpoints of hot coronal loops. A range of periods from 200 to 600 seconds, typically persisting for about 4 to 7 cycles were found. A comparison to photospheric vertical velocities (using the Michelson Doppler Imager onboard SOHO) revealed that some upper TR oscillations show a significant correlation with solar global acoustic p-modes in the photosphere. In addition, the majority of the upper TR oscillations are directly associated with upper chromospheric oscillations observed in

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Hα, i.e., periodic flows in spicular structures. The presence of such strong oscillations at low heights (of order 3,000 km) provides an ideal opportunity to study the direct propagation of oscillations from photosphere and chromosphere into the TR32 and low magnetic corona see, for example Ref. 23. These type of measurements can also help us to (i) understand atmospheric magnetic connectivity that is so crucial for diagnostic reconstruction in the chromosphere and TR, and, shed light on the dynamics of the lower solar atmosphere, e.g. the source of chromospheric mass flows such as spicules;32 (ii) explore the dynamic and magnetised lower solar atmosphere using the method of magneto-seismology.33,34 3.2.1. Global resonant acoustic waves in a stratified atmosphere Acoustic waves have often been invoked as possible candidates for the heating of solar and stellar chromospheres and coronae (see, e.g. Ref. 27 for the latest review on observations; and Refs. 14, 100 on theory). Until recently it was thought that high frequency waves could be responsible for the heating of the non-magnetic chromosphere. On the other hand, low frequency waves were believed to play little role as far as the dynamics and energetics of the atmosphere are concerned due to reflection from regions with steep temperature gradients. Recent works have changed these views. It was established that the power of the observed high frequency propagating (> 5 mHz) acoustic waves is not enough to balance the radiative losses in the chromosphere.119 On the other hand, new observations have shown that the energy flux carried by the low frequency (< 5 mHz) acoustic waves into the chromosphere is about a factor of 4 greater than that carried by high frequency waves.120 It was argued that these low frequency waves could propagate and carry their energy into the higher layers of the atmosphere through portals formed by the inclined magnetic field lines.23,32 These and other results have prompted renewed strong interest in the theory of low frequency acoustic wave propagation in stratified media. The vertical propagation of acoustic waves in a stratified atmosphere (either plasma or gaseous) can be demonstrated in a two-layer model (Fig. 13). The waves are described by the Klein-Gordon (KG) equation. The key point here to note is the resonance occurring at low frequencies which extends into the entire unbounded atmosphere as was first shown by Refs. 121 and 122. This previously unknown resonance may be responsible for the transfer of wave energy which could have dynamic consequences and heat the higher atmospheric layers. The KG equation is widely used in a range of fields such as atmospheric physics, cosmology, quantum field

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theory, solid state physics, solar/stellar physics. Therefore, the results121,122 also have wider applicability in distinct areas of physics and astrophysics. A

Fig. 13. Two-layer model depicting a stratified solar atmosphere. The lower part of the atmosphere (index 1) is separated from the upper part (index 2) by a density and temperature discontinuity at z = L. Waves are launched at z = 0 and propagate in the vertical z-direction.

necessary condition for the existence of a resonance is L/2Λ2 > 1 in a stratified atmosphere. The waves are resonantly amplified when

ω=

c2 L

r

L − 1. Λ2

(75)

The physical mechanism responsible for wave amplification is the following: the decreasing temperature results in an increasing acoustic cut-off frequency Ω = Ω(z) which forms a potential barrier similar to the one in quantum mechanics.123 Low frequency waves with Ω0 < ω < Ω2 driven at z = 0 are reflected back from the barrier and trapped in the lower layer 1. When the driver frequency matches the natural frequency of the cavity where the waves are trapped a standing wave is set up and amplified resonantly. In the case of a thin layer, only the fundamental mode is present with a frequency given by Eq. (75). The frequency of the fundamental mode decreases and higher harmonics appear as the thickness L increases. The resonance affects the evanescent tail of the waves in the upper atmosphere leading to a global resonance.

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3.3. Propagating Waves into Corona In the pre-SOHO/TRACE era probably, the first observations of MHD waves in the corona were reported124 with GSFC extreme-ultraviolet spectroheliograph on OSO-7 (the spatial resolution was few arcsec, the cadence time was 5.14 s). In Mg VII, Mg IX and He II emission intensity periodicity at about 262s was detected. The importance of this early work is that within the range of low-frequencies an analogy to photospheric and chromospheric oscillations was found, and, it was further speculated that the photospheric and chromospheric evanescent waves become vertically propagating, gravity-modified acoustic waves at that height in the chromosphere where temperature rise admits propagation again. Using Harvard College Observatory EUV spectroheliometer on Skylab,125 oscillations were detected in the C II, O IV, and Mg X emission intensity with periods of 117s and 141s. They suggested that the intensity fluctuation of the EUV lines is caused by small amplitude waves, propagating in the plasma confined in the magnetic loop, and that size of the loop might be important in determining its preferential heating in the active region. A final example from that era, though in a much shorter wavelength, is the observations,126 who detected with the Hard X-ray Imaging Spectrometer on-board SMM soft X-ray (3.5-5.5 keV) pulsations of period 24 min lasting for six hours. The periodicity was thought to be produced by a standing wave or a traveling wave packet which exists within the observed loop. It was concluded the candidates for the wave are fast or Alfv´en MHD modes of Alfv´enic surface waves. The situation by the launches of SOHO and TRACE have considerably changed our views since abundant evidences merged for MHD wave phenomena, in particular for propagating waves. In what follows we give an account of the observed propagating waves, and, we overview the attempts made to link these progressive waves to solar global (photospheric) motions, where the latter is accounted for being the driver behind these periodic coronal motions.

3.3.1. Observations of progressive waves Progressive waves may propagate in open (e.g. plumes) and closed (e.g. loops) coronal magnetic structures. The first undoubtable detection of progressive slow MHD waves was made by UVCS/SOHO (Ultraviolet Coronagraph Spectrometer). Observations of slow waves in an open magnetic structure, i.e. high above the limb

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of coronal holes127,128 were carried out. Analysing EIT/SOHO (Extremeultraviolet Imaging Telescope) data of polar plumes similiar compressive distrubances were detected129 with linear amplitudes of the order of 1020% and periods of 10-15 minutes. The observed compressive longitudinal distrubances were identified130,131 as progressive slow MHD waves. The detected damping of slow progressive waves was attributed to compressive viscosity. In closed structures, using EIT/SOHO132 it was reported first on slow modes. Following the success of SOHO, observers using TRACE also searched successfully for quasi-periodic disturbances of coronal loops.133 -135 A detailed overview of the observed properties of these propagating intensity perturbations is given by, e.g. Refs. 136 and 137 and we here only summarise the main fetures.110 An overview of the periods and propagation speeds found by various authors is given in Table 3. In all the reported Table 3. Overview of the periodicities and propagation speeds of propagating slow MHD waves detected in coronal loops. Adopted from Ref. 110. Nightingale et al. (1999) Schrijver et al. (1999) Berghmans & Clette (1999) De Moortel et al. (2000) Robbrecht et al. (2001) Berghmans et al. (2001) De Moortel et al. (2002a) De Moortel et al. (2002b) King et al. (2003)

Period (s) 300 ∼600 180–420 (282 ± 93) 172 ± 32 (sunsp.) 321 ± 74 (plage) 120–180 & 300-480

Speed (km/s) 130–190 70–100 75–200 70–165 65–150 ∼300 122 ± 43 25–40

Wavelength 171 & 195 195 195 171 171 & 195 SXT 171 171 171 171 & 195

cases the phase speed is of the order of the coronal sound speed. Since the progressive waves are observed as intensity oscillations, hence they are likely to be candidates of compressive disturbances. No significant acceleration or deceleration was observed. The combination of all these facts leads to the most plausible conclusion that the observed progressive waves are indeed slow MHD waves. 3.3.2. Source of progressive waves In order to answer the question of what is the source of progressive coronal waves, and, inspired by the observational findings of similarities between photospheric and TR oscillations, there was developed32 the general framework of how photospheric oscillations can leak into the atmosphere along inclined magnetic flux tubes. In a non-magnetic atmosphere p-modes are

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evanescent and cannot propagate upwards through the temperature minimum barrier since their period P (∼ 200 − 450 s) is above the local acoustic cut-off period Pc ≈ 200 s. However, in a magnetically structured atmosphere, where the field lines with some natural inclination θ -where θ is measured between the magnetic guide channelling the oscillations and the √ vertical- the acoustic cut-off period takes the form Pc ∼ T / cos θ with the temperature T . This inclination will allow to tunnel some non-propagating evanescent wave energy through the temperature minimum into the hot chromosphere of a the waveguide, where propagation is once again possible because of higher temperatures (Pc > 300 s). The authors have shown that inclination of magnetic flux tubes (well applicable e.g. to plage regions) can dramatically increase tunnelling, and can even lead to direct propagation of p-modes along inclined field lines as plotted in Fig. 14. a

b

Fig. 14. Leakage of evanescent photospheric p-mode power into chromosphere. Distribution of wavelet power (in arbitrary units, independent for each height)(for cases a and b, resp. θ = 0◦ and 50◦ ) as a function of wave period for different heights above the photosphere. Vertical flux tubes (a) allow minimal leakage of p-modes with periods of 300 s (> Pc ∼ 220 s), so that only oscillations with lower periods (< 250 s) can propagate and grow with height to dominate chromospheric dynamics. Inclined flux tubes (b) show an increased acoustic cut-off period Pc , allowing enhanced leakage and propagation of normally evanescent p-modes. Adapted from Ref. 32.

A perfectly natural generalisation of the above idea was put forward,23 who proposed that a natural consequence of the leakage of photospheric oscillations is that the spicule driving quasi-periodic shocks propagate into the low corona, where they may lead to density and thus intensity oscillations with properties similar to those observed by TRACE in 1 MK coronal

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loops. In other words, the origin of the propagating slow MHD waves detected in coronal loops (see a recent review on their properties by e.g. Ref. 110) was linked to wave energy leakage of solar global standing oscillations. It was highlighted23 that oscillations along coronal loops associated with AR plage have many properties that are similar to those of the moss oscillations: (i) the range of periods is from 200 to 600 seconds, with an average of 350 ± 60 s and 321 ± 74 s, for moss and coronal oscillations, respectively; (ii) the spatial extent for coherent moss oscillations is about 1-2′′ , whereas for coronal waves, the spatial coherence is limited to ∼′′ in the direction perpendicular to that of wave propagation. They also point out that although the oscillations in moss and corona have similar origins, they are results of different physical mechanisms: moss oscillations occur because of periodic obscuration by spicules, and coronal oscillations arise from density changes associated with the propagating magneto-acoustic shocks that drive the periodic spicules. 4. Where Magneto-Seismology and the Coronal Heating Enigma Meet The coronal heating enigma has challenged solar physicists for over half a century. None of the theories proposed so far has yet been the answer to teh coronal heating problem. The real task now is to establish which of these theories (if any) represents the the actual heating mechanism or perhaps a combination of various models. It is now becoming increasingly evident that predicting observational signatures and footprints of various heating models through forward modelling, and their direct comparisons with currently available high resolution observations is the way to proceed. Forward modelling and inversion of observed data represent two complementary aspects of solving the heating enigma. A combination of these two approaches will, however, provide an integrated and most efficient currently available route of tackling coronal. Section 4 briefly outlines this concept. In particular, we focus on the problem of inversion and plasma heating diagnostics using MHD waves. 4.1. Inversion and Diagnostics with MHD Waves Forward modelling is used to derive observational signatures and footprints predicted by a theoretical model. The reverse procedure of deriving from observations the actual footprints and physical background quantities characterising the magnetised plasma is a difficult task in general.

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A well-known example of the controversy in solar data inversion is the analysis of a single Yohkoh data set from which three different temperature and heating profiles along a magnetic loop were derived by three different authors: uniform,138 footpoint60 and apex139 heating. The discrepancy between the results was mainly attributed to the different ways in which the background was subtracted. Other authors have criticized the assumption of isothermal approximation which is implicit in the conventional filter ratio analysis.140,141 An alternative approach to the inversion problem is the use of MHD waves. MHD waves are an excellent diagnostic tool and their full potential has yet to be explored. We have already demonstrated that MHD waves could be used for a number of purposes: measuring the magnetic field strength, the fine structure, the transport coefficients in the solar atmosphere, understanding the processes responsible for the damping of these waves and so on.47,51,52,58,76,77,81,83,87,89,95,135,142 -146 For more details on the observed MHD waves, their damping, etc.27,113 Standing acoustic waves were detected147,148 in hot active region loops could be quite useful in quantifying the heating function. The damping of these oscillations was studied by Refs. 6, 62, 63, 149, 150. Observations show that these waves are usually preceded by footpoint brightenings. It haa been determined151 the mathematical form of the heating pulse required to rapidly set up a standing wave: the duration of the heating pulse must match approximately the period of the standing oscillations. The results were applied152 to an active region. An 86 Mm long loop underwent heating to T ≈ 7 MK followed by cooling to T ≈ 2 MK in less than two hours. The heating was followed by rapidly damped standing longitudinal oscillations with a period of about 8 mins. The maximum initial Doppler shift, observed by SUMER, was about 30 km s−1 . Forward modelling was carried out, using the parameters of the observed Yohkoh SXT loop, in dissipative hydrodynamics including thermal and radiative losses. The numerical results were then converted into observable quantities by applying spectral line synthesis. Figure 15 compares the simulated line profiles with the actual SUMER measurements from 3 to 7 MK. Note, the oscillations only appear in the Doppler shift as opposed to intensity. The initial negative blue shift of about 32 km/s is followed by damped oscillations. It was concluded152 that the observed oscillations are the fundamental mode standing acoustic wave; the standing waves are excited by a microflare occurring at one of the loop footpoints; they also determined the evolution of the heating rate along the loop.

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(a)

red

blue

black

red

blue

(b) blue black red

Time on 2000 Sep 17 (UT)

Fig. 15. Left - SUMER observations: the top panel represents the time series of the Fe xix (black) line intensity along the slit. The overlaid contours represent the Ca xv (λ 1098, red) and Ca xiii (blue) intensity time series. The contour levels are 70, 80, and 90% of the peak intensity; the bottom panel is the average time profile of the line-integrated intensity along the cut in the top panel. Right - synthesized observations: the top panel shows the forward modelled time profiles of the line intensities along the slit cut. The black, red and blue lines correspond to Fe xix , Ca xv (λ 1098), and Ca xiii intensities; the bottom panel shows the corresponding Doppler shift from Ref. 152.

The next task now is to show an example of data inversion. A reliable inversion may come from the analysis of Doppler shift time series. The idea, proposed by Ref. 153, is borrowed from helioseismology where the use of such time series has become a routine method for one of the most precise diagnostic measurements in astrophysics. The new method does not require the presence of coherent standing waves. The only underlying assumption is that loops (or their individual and unresolved components, called strands) are heated randomly both in time and in space (an argument supported strongly by solar observations). A linear ideal 1D loop, heated by random pulses, will respond to energy perturbations (depositions) by an infinite number of peaks in the velocity power spectrum corresponding to the frequencies of standing waves. The inclusion of dissipative processes (e.g. radiative losses, thermal conduction) and nonlinearity introduces noise in the power spectrum. The most prominent peak corresponding to the fundamental mode is always present regardless of the random heating function and the heliographic position of the loop on the solar disk. This peak could therefore be used to determine the average temperature of the plasma inside

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Line Shift (km s-1)

the loop! The peak corresponding to the second harmonic only appears in the case of uniformly random heating. Peaks corresponding to higher harmonics do not show up due to their small amplitudes, losses and nonlinearity. The results of the wavelet analysis for such a loop are displayed in Figure 16. Forward modelling shows that loops heated near their footpoints a) Doppler Shift Time Series for Fe X WAVELET ANALYSIS

5 0 -5 -10 0

500

1000

1500 Time (min)

2000

2500

3000

red

b) Wavelet Power Spectrum

c) Global

Frequency (mHz)

10

10

1

1

90% 99% 0

500

blue

1000

1500 Time (min)

2000

2500

3000

0

40 80 120 160 200 Power (km2s-2)

Fig. 16. Wavelet analysis of temporally randomly but spatially evenly heated 30 Mm long loop. The top panel (a) shows the Doppler shift time series in the Fe x line. The bottom left panel (b) displays the wavelet power spectrum. The red color represents high power and the blue color corresponds to low power. The bottom right panel (c) is the global wavelet spectrum. In the wavelet spectrum diagram (b), regions with 90% significance level are outlined in black. In the global wavelet diagram, the dotted lines indicate 99% significance level.

only display the fundamental mode. This is mainly due to the large values of thermal conduction around the maximum of the second harmonic which results in strong damping. The phenomenon is explained in more details by Ref. 153. On the other hand the power spectra (i.e. inversion) are also sensitive to the temporal distribution of heating and, as a result, they could be used to estimate the average amount of energy involved in a single heating event. Interestingly, the power peaks do not show up when the same analysis is applied to the intensity time series. This effect as well as the phase shifts in the synthesized EIS observations are analytically explained by Ref. 154. Finally, it is now concluded that the power spectrum analysis

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(i.e. data inversion) could therefore be used to establish the distribution of the random heating, i.e. determine the nature of the heating process. 5. Summary Let us now summarise the key points made in this chapter. MHD waves and oscillations serve as excellent diagnostic tools in gaining diagnostic information about the structure of the solar atmosphere. MHD waves are important in the energy balance of the atmosphere from photosphere to corona. Forward modeling of wave propagation and inversion of wave observations are two complementary approaches which allow the heating problem to be tackled in an integrated and comprehensive manner. The new generation satellites (TRACE, Hinode, STEREO, SDO, SOLO) have (or will) provided new constraints on the forward modelling of the signatures and observable consequences of the feasible theoretical heating scenarios. Coordinated efforts of the diagnostic capabilities of these missions are most likely to improve our understanding of the solar atmospheric magnetised plasma from the photosphere to the corona, and beyond, in the interplanetary space. Analogue to helioseismology, standing MHD waves in magnetic structures are important for quantifying the unknown heating function and the internal structure of the MHD waveguide. The analysis of Doppler shift and Doppler width time series is a new and very efficient tool for determining the spatial and temporal distribution of the heating function, magnetic field geometry, dissipative coefficients, etc. The method does not require the presence of individual coherent waves. It is strongly anticipated that new inversion methods, based on MHD waves, must be developed in order to make fundamental progress. MHD inversion methods will have the potential to compete and serve as real alternative to more conventional techniques, and, hopefully will close a chapter adequately in modern astrophysics. Acknowledgments Solar physics research at the Dept. of Applied Mathematics, Univ. of Sheffield is supported by the Science and Technology Facilities Council (STFC) of the UK. The author would like to thank M.S. Ruderman and Y. Taroyan for giving advise and help when preparing this chapter. He is

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grateful to V. Fedun for his immense readiness in preparing some of the figures, and also is grateful to B. Carter, M. Douglas and G. Verth for their help putting together this Chapter. RE also acknowledges M. K´eray for patient encouragement and is grateful to NSF Hungary (OTKA K67746). References 1. Golub, L. & Pasachoff, J.M. The Solar Corona, CUP (1997). 2. Cargill, P. in (eds.) J.L. Birch & J.H. Waite, Jr., Solar System Plasma Physics: Resolution of Processes in Space and Time, (1993). 3. Zirker, J.B., Solar Phys., 148, 43 (1993). 4. Mendoza-Brice˜ no, C.A., Erd´elyi, R. & Sigalotti, L.D.G. Astrophys. J., 579, 49 (2002). 5. Mendoza-Brice˜ no, C.A., Sigalotti, L.D.G., Erd´elyi, R. AdSpR, 32, 995 (2003). 6. Mendoza-Briceno, C.A., Sigalotti, L.D.G. & Erd´elyi, R. Astrophys. J., 624, 1080 (2005). 7. Mendoza-Briceno, C.A. & Erd´elyi, R. Astrophys. J., 648, 722 (2006). 8. Aschwanden, M. Review of Coronal Oscillations, in (eds.) R. Erd´elyi et al., Turbulence, Waves and Instabilities in the Solar Plasma, NATO Science Ser., 124, 215 (2003). 9. Ballai, I., Erd´elyi, R. & Ruderman, M.S. Phys. Plasmas, 5, 2264 (1998). 10. Erd´elyi, R., Doyle, J.G., Perez, E.P. & Wilhelm, K. Astron. Atrophys., 337, 213 (1998). 11. Taroyan, Y. in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-seismology, IAU Symposium, 247, 186 (2008). 12. Narain, U. & Ulmschneider, P. Space Sci. Rev., 75, 453 (1996). 13. Ulmschneider, P. in (eds.) J.-C. Vial et al., Lect. Notes in Phys., 507, 77 (1998). 14. Erd´elyi, R. Astron. & Geophys., 45, p.4.34 (2004). 15. Erd´elyi R., & Ballai, I. Astron. Nacht., 328, 726 (2007). 16. Browning, P.K. Plasma Phys. and Controlled Fusion, 33, 539 (1991). 17. Gomez, D.O. Fund. Cosmic Phys., 14, 131 (1990). 18. G´ omez D., & Dmitruk P., in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magnetoseismology, IAU Symposium, 247, 271 (2008). 19. Hollweg, J.V. in (eds.) P. Ulmschneider et al., Mechanisms of Chromospheric and Coronal Heating, Springer-Verlag, Berlin p.423 (1991). 20. Priest, E.R. & Forbes, T. Magnetic Reconnection, CUP (2000). 21. Roberts, B. & Nakariakov, V.M., in (eds.) R. Erd´elyi et al., Turbulence, Waves and Instabilities in the Solar Plasma, NATO Science Ser., 124, 167 (2003). 22. Walsh, R.W. & Ireland, J. Astron. Astrophys. Rev., 12, 1 (2003). 23. De Pontieu, B., Erd´elyi, R. & De Moortel, I. Astrophys. J., 624, 61 (2005). 24. De Pontieu, B., Erd´elyi, R., de Wijn, A.G.: 2003, Astrophys. J., 595, 63.

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25. Roberts, B. Solar Phys., 193, 139 (2000). 26. Roberts, B. in (sci. eds.) R. Erd´elyi, J.L. Ballester & B. Fleck SOHO 13 Waves, Oscillations and Small-Scale Transient Events in the Solar Atmosphere: A Joint View from SOHO and TRACE, ESA-SP, 547, 1 (2004). 27. Banerjee, D., Erd´elyi, R., Oliver, R. & O’Shea, E. Solar Phys., 246, 3 (2007). 28. Gomez, D.O., Dmitruk, P.A. & Milano, L.J. Solar Phys., 195, 299 (2000). 29. Hollweg, J.V. Adv. Space Res., 30, 469 (2002). 30. van Ballegooijen, A.A. Astrophys. J., 311, 1001 (1986). 31. Erd´elyi, R. in (eds.) J.L. Ballester & B. Roberts, MHD Waves in Astrophysocal Plasmas, UIB press, p.69 (2001). 32. De Pontieu, B., Erd´elyi, R. & James, S.P. Nature, 430, 536 (2004). 33. De Pontieu, B. & Erd´elyi, R. Phil. Trans. Roy. Soc. A., 364, 383 (2006). 34. Erd´elyi, R. Phil. Trans. Roy. Soc. A., 364, 351 (2006a). 35. Landau, L.D., Lifshitz, E.F. & Pitaevskii, L.P. Electrodynamics of Continuous Media Pergamon Press, Oxford (1984). 36. Cowling, T.G. Magnetohydrodynamics Interscience (1960). 37. Alfv´en, H. & Falthammar, G.G. Cosmical Electrodynamics, Oxford U.P. (1962). 38. Priest, E.R. Solar magnetohydrodynamics, D. Reidel, Dordrecht (1982). 39. Goedbloed, J.P. Lecture notes on ideal magnetohydrodynamics, Rijnhuizen Report 83-145 (1983). 40. Choudhuri, A.R. The physics of fluids and plasmas, Cambridge Univ. Press (1998). 41. Roberts, B. Solar Phys., 69, 27 (1981a). 42. Roberts, B. Solar Phys., 69, 39 (1981b). 43. Edwin, P.M. & Roberts, B. Solar Physics, 76, 239 (1982). 44. Edwin, P.M. & Roberts, B. Solar Physics, 88, 179 (1983). 45. Abramowitz, M. & Stegun, A. Handbook of Mathematical Functions, (New York : John Wiley and Sons), 1967. 46. Bennett, K., Roberts, B. and Narian, U. Solar Phys., 185, 41 (1999). 47. Erd´elyi, R. & Fedun, V. Solar Phys., 238, 41 (2006). 48. Erd´elyi, R. & Fedun, V. Solar Phys., 246, 101 (2007). 49. Robbrecht, E., Verwichte, E., Berghmans, D., Hochedez, J.F., Poedts, S., et al. Astron. Astrophys., 370, 591 (2001). 50. Mikhalyaev, B.B. and Solov’ev, A.A. Solar Phys. 227, 249 (2005). 51. Erd´elyi, R. and Carter, B.K., Astron. Astrophys., 455, 361 (2006). 52. Carter, B.K. and Erd´elyi, R., Astron. Astrophys., 475, 323 (2007). 53. Carter, B.K. and Erd´elyi, R., Astron. Astrophys., 481, 239 (2008). 54. Aschwanden, M.J., Fletcher, L., Schrijver, C.J. & Alexander, D., 1999a, ApJ 520, 880. 55. Aschwanden, M.J., De Pontieu, B., Schrijver, C.J. & Title, A.M. 2002, Solar Phys. 206, 99. 56. Nakariakov, V.M., Ofman, L., DeLuca, E.E., Roberts, B. & Davila, J.M. Science, 285, 862 (1999). 57. Verwichte, E., Nakariakov, V.M., Ofman, L. & Deluca, E.E. Solar Phys., 223, 77 (2004).

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CHAPTER 6 VUV SPECTROSCOPY OF SOLAR PLASMA A. MOHAN Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India

6.1. Introduction VUV spectroscopic diagnostics make use of information contained in the measured intensities and profiles of the spectral lines. Access to images and spectra of hot plasma in UV, EUV, and X- ray regions have provided basic tools to address the fundamental problems in the outer solar atmosphere. Observations of coronal lines have been recognized as important means of investigating the physical properties of coronal plasma and addressing the questions, which are in the forefront of space physics, such as coronal heating and solar wind acceleration. Diagnostics of solar plasma in the temperature range from 105 K to above 106 K carries the radiation signatures of chromosphere - corona transition region and the corona. Many solar ions belonging to various iso-electronic sequences have been systematically presented. Pre-SOHO era, the EUV spectra in the spectral range 170-450 Å observed by Solar Extreme Ultraviolet Research Telescope and Spectrograph (SERTS) have been used, and later, the high resolution ultraviolet observations obtained with Coronal Diagnostic Spectrometer (CDS, 150-600 Å) and Solar Ultraviolet Measurements of Emitted Radiation (SUMER, 465-1610 Å) onboard SOHO are used. A full description of the SUMER spectrograph and its performance are available1-3. 6.2. Atomic Processes For the atomic processes involved in hot (Te > 2×104 K) and low density (Ne < 1013 cm-3) plasma, we assume that the spectral lines are optically thin, which is valid for the outer atmosphere of the Sun and other stars. 109

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6.2.1. Emission lines Taking account of atomic excitation mechanisms, the line emissivity (per unit volume per unit time) for an optically thin spectral line is given by the expression:

ξ (λij ) = N j A ji

hc

λij

, ( j >i ), (erg cm-3 sec-1)

(6.1)

where Aji is spontaneous radiative transition probability, h is Planck’s constant, c is velocity of light, λij is wavelength for the transition i – j, and Nj is number density of level j. Thus the atomic physics problem reduces to the calculation of the population density of the upper excited level j. The case of low density optically thin plasmas are treated here. The number density Nj can be further parameterized as:

(

Nj X

+p

)

( ) ( ( )

)

N j X + p N X + p N ( X ) N (H ) = N e cm-3 N X + p N ( X ) N (H ) N e

where X+P is the pth ionization stage of the element; Nj(X+P)/N(X+P) is the population of level j relative to the total N(X+P) number density of the ion X+P and is a function of the electron temperature and density; N(X+P)/N(X) is the ionization ratio of the ion X+P which is predominantly a function of temperature; N(X)/N(H) is the element abundance relative to hydrogen which may vary in different astrophysical plasmas and also in different solar features; N(H)/Ne is the hydrogen abundance relative to electron density which is assumed to be 0.8 for a fully ionized plasma. The flux at the Earth of a spectral line is given by

I (λij ) =

1 ξ (λij )dV erg cm-2 sec-1 sr-1 4πR 2 ∫

where V is the volume of emission and R is Earth-to-object distance. The collisional excitation processes are generally faster than ionization and recombination time scales in low density optically thin plasmas. Therefore, the collisional excitation is dominant over ionization and recombination in producing excited states. Thus, the population density Nj of the upper excited level j must be calculated by solving the statistical equilibrium equations for a number of low-lying levels and taking account of all the important collisional and radiative excitation and de-excitation mechanisms.

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6.2.2. Coronal model approximation In coronal model approximation the assumption is made that the population of the upper level of transition j occurs mainly via collision excitation from the ground level g and the radiative decay overwhelms any other depopulation process. The statistical equilibrium equations can be solved as a two-level system for each transition

(

)

N g X + m N eC gje = N j Ajg If Ajg >> Ne Cgje, then the population of the upper level j is negligible in comparison with the ground level g, that means, Ng(X+m)/N(X+m) ≈ 1. Ni (X+P) Ne Cije = Nj Aji, where, Cgje and Cije are the electron collision excitation rate coefficient (cm3 sec-1) remembering that Ni (X+P)/N(X+P) = 1, and with substitutions in Eq. (6.1), we get

ξ (λij ) =

(

)

N X + p N ( X ) N (H ) e hc 2 Cij N N ( X ) N (H ) N e λij e

(6.2)

Thus the line intensity, which is the integration of the line emissivity over the emitting volume, is given by

I (λij ) = ∫ G (Te )N e2 dV

(6.3)

where G(Te) can be calculated from the ionization ratio, element abundance and atomic parameters. The electron density can be crudely deduced assuming that the spectral line is emitted over a homogeneous volume estimated from images in that line. 6.3. Plasma Diagnostics Without the knowledge of electron densities, temperatures and elemental abundances of space plasma, almost nothing can be said regarding the generation and transport of mass, momentum and energy. A fundamental property of hot solar plasmas is their inhomogeneity. The emergent radiances of spectral lines from optically-thin plasmas are determined by integrals along the line of sight through the plasma. Spectroscopic diagnostics of the density and temperature structures using emission-line intensities is described here. 6.3.1. Electron density diagnostics The first question which one might justifiably ask is why so much effort has been put into the development of electron density diagnostics? Obviously the electron

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pressure (NeTe) is an important parameter in any theoretical model for the plasma, but why not simply deduce the electron density from the total line emission and an estimate of the emitting volume? This can and has been done in numerous analyses. However, this assumes knowledge of the volume for the emission. If the spatial resolution for the observations was good enough and the emitting material was homogeneously distributed throughout the volume, then the electron density estimate would be reasonable. Solar plasma, however, is characterized by unresolved filamentary structures even with the best spatial resolution observations currently available. The determination of electron density from spectral line ratios from the same ion, makes no assumption about the size of the emitting volume, ionic fractions or the element abundance value, providing a powerful diagnostic for the plasma conditions. Line emissivity ratio of two spectral lines is expressed as,

ξ (λij ) Aji N j (X + p ) λkl = ξ (λkl ) Alk N l (X + p ) λij

(6.4)

The density-sensitive line ratios give the information about the density of the emitting region. 6.3.2. Electron temperature diagnostics The simplest but crudest method of deducing the plasma temperature is to assume ionization equilibrium. Since many ions are formed over the same range of temperature, line ratios can be plotted as a function of temperature of the emitting isothermal plasma. A more accurate determination of electron temperature can be obtained from the intensity ratio of two allowed lines excited from the ground level i but with significantly different excitation energy.

I (λij ) I (λik )

=

 ∆Eik − ∆Eij exp ∆Eik γ ik kTe  ∆Eij γ ij

  

(6.5)

The ratio is sensitive to the change in electron temperature if [(∆Eik - ∆Eij) / kTe] » 1, assuming that the lines are emitted by the same isothermal volume with the same electron density. 6.4. Nitrogen-Like Ions In this section the diagnostic applications of N–like ions are discussed. As shown in Fig. 6.1 the logarithmic temperatures of formation (TM) for the nitrogen-like

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coronal ions Al VII, Si VIII, P IX, S X, Ar XII, K XIII, and Ca XIV are 5.8, 5.9. 6.0, 6.1, 6.4, 6.4, and 6.5. respectively4. The lines from these ions can be used for plasma diagnostics in both quiet and active solar coronae. Here the densitydependence of these ions is presented, whose forbidden transitions within the levels in their ground configurations fall within the SUMER bandpass, and compare their emissivities and line ratios with a set of observations taken in the quiet-Sun and active regions. Some of the lines used in the present study are measured for the first time (Al VII 1056.77 Å and both the P IX line pairs). The higher resolution and enhanced sensitivity of the SUMER spectrograph allow for the identification and measurement of weaker lines than in any previous experiment, at positions higher off the solar limb. The Si VIII n = 3 → n = 3 lines observed by SUMER in the 900-1250 Å wavelength range, have been identified by SUMER for the first time in the solar spectrum. New calculations for n = 3 → n = 3 transitions of Si VIII have been performed by Bhatia & Landi5. These new atomic data allow to investigate the diagnostic potential of line ratios within the n = 3 configuration and in conjunction with ground forbidden n = 2 lines also observed by SUMER.

Fig. 6.1 Ion fractions of N-like ions as a function of log T

The availability of such a complete data set for the forbidden N-like lines allows, for the first time, a systematic assessment of the density diagnostic potential of these transitions along the N-like sequence. Version 4.0 of the CHIANTI database6,7 is used to calculate theoretical line ratios. In Version 4, CHIANTI data for the N-like sequence have been entirely renewed, by substituting more recent relativistic distorted wave (DW) calculations from Zhang & Sampson8 for the older DW calculations from Bhatia & Mason9. The

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former provide atomic data and DW transition probabilities for the first 15 finestructure levels, while the latter provide data only for the first 13 fine-structure levels. Moreover. R-matrix data have been used for the forbidden transitions within the ground configuration of Si VI and S X10,11. Si VIII and Ar XII levels include additional data for another 57 energy levels taken from other DW calculations5,13. CHIANTI, Version 4, also includes proton excitation rates for Si VIII transitions within the ground configuration, taken from Bhatia & Landi5. The effect of resonances in R-matrix transition probabilities, of radiative cascades from higher excitation levels, and of proton rates and photoexcitation from photospheric radiation may influence level populations of the ground configuration levels and alter intensity ratios from the forbidden transitions that originate from them and are considered here. They have been investigated in detail in order to assess their importance. This work also allows us to make a complete assessment of the quality of CHIANTI atomic data that are necessary to evaluate theoretical line emissivities for the considered lines. We study the effects of photoexcitation, proton collisional excitation, n = 3 levels, and resonances in collision rates on level populations. 6.4.1. Effects of different processes on level populations The populations of the Si VIII ground levels have been calculated with and without proton excitation rates, in order to check the effects of “proton rates” on the results12. The ratios between level populations of the ground 2D and 2P levels calculated in both ways have been plotted as a function of the electron density, for a few sample values of the electron temperature (log Te = 5.6, 5.9, and 6.1, Te in K). This shows that for the excited levels in the ground configuration, the proton rates are of moderate importance. At the temperature where Si VIII is most abundant (log Te = 5.9) the change in relative population is within 5%. However, the proton rate effects are larger at log Te = 6.2, so that for higher temperatures they need to be taken into account. Proton rates are more important at active region densities around l09 cm-3 and become negligible at very high densities. So, it is concluded that where quiet-Sun or moderate active region observations are used, the lack of proton excitation rates in CHIANTI and in the literature is not a source of significant uncertainty. Radiative cascades from higher energy levels can be an important process in the population of the lower lying levels. In order to assess the importance of the “n = 3” levels on level populations for N-like ions, we have compared level populations for the ground configuration calculated in two ways: by including

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the n = 2 levels only (corresponding to 15 energy levels), and by including the 2s22p33l (l = s,p,d) levels also (corresponding to 72 energy levels). It is important to note that the reduced atomic model (15 levels) is the one used for decades in plasma diagnostic studies involving N-like ions. The comparisons are made for Si VIII and Ar XII at three temperatures, log Te = 5.6, 5.9 and 6.2 for Si VIII and log Te = 6.1, 6.4 and 6.7 for Ar XII. It is found that the inclusion of n = 3 levels has greatly affected the level populations of the ground configuration and hence the N-like line emissivities. The effects are larger in Ar XII, where cascades from n = 3 levels cause alteration of the level population up to 35% at low densities. The effect is more marked at higher temperatures, while in cooler plasmas it is more limited. The inclusion of n = 3 levels in N-like atomic models provides two benefits: it takes into account the additional population process of cascades from higher levels, which can be non-negligible, and allows predictions of line emissivities for a number of n = 3 → n = 3 transitions that can be observed in the UV spectral range. Level populations are also calculated in the presence of a blackbody radiation field with temperature at 6000 K to see the effects of “photospheric radiations” on the level populations. This radiation field simulates the physical conditions in atmospheres of the Sun and late-type stars. This background radiation can potentially be an important process for radiative excitation within the ground levels. The geometrical dilution factor for the radiation field was taken to be 0.4, typical of the inner solar corona, where the N-like coronal lines are expected to be stronger. The differences in level populations and line emissivities are always smaller than 3%, so that this excitation mechanism has no real importance in the N-like ions and can be neglected. The effects of “resonances in collisional excitation” within the ground levels can be investigated by comparing populations calculated from collision rates derived using the R-matrix approximation14,15 and those obtained using the DW approximation16,17. R-matrix collision rates are available only for Si VIII and S X. A comparison is carried on their level populations. Results obtained using the Bell et al.10 for Si VIII and Bell & Ramshottom11 for S X effective collision strengths for transitions within the ground levels have been compared with populations obtained with the collisional data calculated using the DW approximation by Zhang & Sampson8. The ratios are plotted between level populations calculated using DW data and using R-matrix data, and no proton rates have been used in the calculation12. It is observed that the resonances play an important role in the level population process for the two ions considered.

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Effects are relevant for densities smaller than 1012 cm-3 and are larger for Si VIII. The study gives stress for the need of R-matrix data for the N-like sequence and demonstrates that their absence from the literature and from CHIANTI is a source of uncertainties in the calculation of N-like line emissivities. 6.4.2. Result and discussion Fig. 6.2 demonstrates that the (4S3/2 - 2P3/2 ) / (4S3/2 - 2P1/2) line intensity ratio can be effectively used as a tool for density diagnostics in the solar corona in both quiet and active regions, as ratios of ions from Mg VI to Si VIII are density sensitive in the density range typical of the solar corona. With the only exception of Mg VI, these line ratios are suggested for density diagnostics for the first time. Heavier elements (P IX, S X, and Ar XII), on the contrary, present nearly constant ratios up to 1010 cm-3 and are density sensitive only at higher densities. The (4S3/2 – 2D3/2) / (4S3/2 – 2D5/2) ratio, shown in Fig. 6.3 constitutes an even more useful tool for density diagnostics, as its density sensitivity is much more marked. Ratios from Si VIII can be used to infer electron density in low-density plasmas, such as those observed in off-disk coronal regions far from the solar surface. P IX and S X can also be used in coronal holes, quiet-Sun regions, and moderate active regions. Ar XII and K XIII can be excellent density indicators in active regions, while Ca XIV can diagnose even denser plasma. Line intensities and intensity ratios, along with measured electron densities from the SUMER

Fig. 6.2 Line intensity ratios involving the ground forbidden transitions (4S3/2 – 2P3/2) / (4S3/2 – 2P1/2) as a function of the electron density along the N-like sequence

Fig. 6.3 Line intensity ratios involving the ground forbidden transitions (4S3/2 – 2D3/2) / (4S3/2 – 2D5/2) as a function of the electron density along the N-like sequence

117

VUV Spectroscopy of Solar Plasma Table 6.1 Observed Line Intensities and Intensity Ratios Ion

log Te Wavelength Transition Å

Al VII

5.8

Si VIII

5.9

P IX

6.0

SX

6.1

Ar XII

6.4

Si VIII

5.9

P IX

6.0

SX

6.1

Ar XII

6.4

K XIII

6.4

Ca XIV

6.5

1053.84 1056.77 944.38 949.22 853 54 861.10 776.37 787.43 469.14 670.35 1440.49 1445.75 1307.57 1317.65 1196.2 1212.93 1018.89 1054.57 945.83 994.52 880.35 943.70

4

S3/2 – 2P3/2 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4

Intensity photons cm-2 s-1 arcsec-2

Ratio

0.089 ± 0.013 0.031 ± 0.005 4.9 ± 0.7 1.7 ± 0.3 0.069 ± 0.010 0.021 ± 0.003 2.2 ± 0.3 1.1 ± 0.2 2.7 ± 0.4 0.44 ± 0.07 0.91 ± 0.14 11.1 ± 1.7 0.027 ± 0.004 0.30 ± 0.05 4.1 ± 0.6 9.5 ± 1.4 3.6 ± 0.5 10.8 ± 1.6 0.98 ± 0.15 1.8 ± 0.3 19.2 ± 2.9 26.8 ± 4.0

2.9 ± 0.6 1.0 2.9 ± 0.7 2.9 ± 0.7 3.3 ± 0.7 1.0 2.0 ± 0.5 1.0 6.1 ± 1.3 1.0 1.0 12.2 ± 2.6 1.0 11.1 ± 2.5 1.0 2.3 ± 0.5 1.0 3.0 ± 0.6 1.0 1.84 ± 0.4 1.0 1.4 ± 0.3

log Ne

Solar Region

≤ 9.9

Quiet Sun Quiet Sun ≤ 9.7 Quiet Sun Quiet Sun 10.7 ± 0.5 Quiet Sun Quiet Sun ≤ 8.0 Quiet Sun Quiet Sun Obs. hig Active region Active region 8.2 ± 0.1 Quiet Sun Quiet Sun 8.95 ± 0.15 Quiet Sun Quiet Sun 8.10 ± 0.15 Quiet Sun Quiet Sun 9.6 ± 0.2 Active region Active region 9.6 ± 0.2 Active region Active region <9.8 Active region Active region

Data are from off-disk observations of a quiet-Sun region (Al VIII to S X) and of an active region (Ar XII to Ca XIV). “Obs. high” means that the observed intensity ratio is higher than predicted for any value of the electron density.

observations, are reported in Table 6.1. Quiet-Sun densities from different ratios are expected to be very similar since the plasma was shown to be isothermal and unstructured. Active region results, on the contrary, are likely to show larger variations as the emitting active region included many individual loop-like structures along the line of sight. 6.5. FIP Effect Measurements in the Off-Limb Corona This section deals with the investigations of electron density, temperature and the abundance anomalies in the off-limb solar corona using spectra obtained from the SUMER. Systematic differences between elemental abundances in the photosphere and in the corona have long been acknowledged in the Sun. These abundance anomalies are correlated with the first ionization potentials (FIPs) of the elements. This correlation is called FIP effect. The so-called low-FIP

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elements (FIP ≤ 10 eV) show enhancements by a factor around 4 in the corona, while the abundances for the high-FIP elements (FIP ≥ 10 eV) remain constant between the photosphere and the corona. So far no theoretical model has been able to explain this effect satisfactorily. The knowledge of the composition of the corona is an important key to our understanding of the physical processes in the solar atmosphere. The coronal elemental abundances can help to investigate the mechanisms that transport, accelerate and heat the solar coronal plasma and the wind. The composition of the corona must also be known in order to interpret observations of coronal emission spectra. Elements’ relative abundances must be known also to determine temperatures from the intensity ratios of the coronal lines of different elements, while the absolute abundances must be known to determine emission measure and radiation losses. Extensive research regarding the abundances of elements in the solar atmosphere has been reviewed by Feldman18 and an update on the most recent results can be found in the recent review by Feldman & Laming18. Dwivedi et al.20,21 carried out an observing sequence based on a theoretical study by Dwivedi & Mohan22, with intercombination and forbidden Ne VI and Mg VI lines, which are formed at essentially the same temperature5. The FIPs of Ne and Mg are 21.6 and 7.6 eV, respectively; they form a low- FIP/high-FIP pair. This investigation is extended taking account of other low-FIP/high-FIP pairs such as K/Ar, Si/Ar and S/Ar present in the spectra. The use of the recently identified K XIII line at 994.58 Å allows23 to investigate the amount of FIP bias also for the very low-FIP element K (FIP = 4.3 eV), so that it is possible to investigate whether the FIP bias of the elements with FIP ≤ 10 eV is constant or depends on the FIP value of each element, as suggested by Feldman & Laming19. The FIP bias of the S, Si and K elements is also investigated as a function of the height above the photosphere. 6.5.1. Instrument and observations The observations were made with the SUMER spectrograph on 1996 June 20 above the active region NOAA 7974 at the solar east limb, starting at 20:11 UT. Fig. 6.4 shows the position and extension of the SUMER raster superimposed on the He II 304 Å image of the eastern limb of the Sun taken at 19:41 UT with the EIT ultraviolet imager24. In the observations analyzed in the present work a slit size of 4‫ ״‬x 300‫ ״‬was used. The raster started 40‫ ״‬off-limb, above the position of the bright He II protrusion seen in the EIT image, and the slit was stepped eastward for 133‫״‬. At each position, two 40 Å wide spectra were obtained, one

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centered on the Ne VI 999 Å line, and the other on the Mg VI 1191 Å line. The integration time per spectrum was increased exponentially with position from 250 s at 40‫ ״‬to 867 s at 173‫ ״‬The total raster time was about 10 hr. The lines used in the present study are reported in Table 6.2.

raster direction Fig. 6.4 SUMER raster superimposed on a section of the 304 Å EIT image

Table 6.2 Observed Intensities for Si VIII lines from SUMER Wavelength Å

Intensity Photons cm-2 s-1 arcsec-2

Transition

944.38 949.22 1440.49 1445.75 982.18 983.56 988.21 994.59 1182.48 1184.00 1189.51 1199.50 1231.37

4.9 ± 0.7 1.7 ± 0.3 0.91 ± 0.14 11.1 ± 1.7 0.033 ± 0.005 0.069 ± 0.010 0.083 ± 0.012 0.20 ± 0.03 0.017 ± 0.003 0.25 ± 0.04 0.66 ± 0.10 0.14 ± 0.02 0.055 ± 0.008

2s22p3 4S3/2 – 2s22p3 2P3/2 2s22p3 4S3/2 – 2s22p3 2P1/2 2s22p3 4S3/2 – 2s22p3 2D5/2 2s22p3 4S3/2 – 2s22p3 2D3/2 2s22p2 3p 4D1/2 – 2s22p23d 4F3/2 2s22p2 3p 4D3/2 – 2s22p23d 4F5/2 2s22p2 3p 4D5/2 – 2s22p23d 4F7//2 2s22p2 3p 4D7/2 – 2s22p23d 4F9//2 2s22p2 3s 4P1/2 – 2s22p23p 4D3//2 2s22p2 3s 4P3/2 – 2s22p23p 4D5//2 2s22p2 3s 4P5/2 – 2s22p23p 4D7/2 2s22p2 3s 4P1/2 – 2s22p23p 4D1/2 2s22p2 3s 4P5/2 – 2s22p23p 4D5/2

Note: Data are from off-disk observations of quiet-Sun region at a height of 35,000 - 49,000 km above the limb.

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The observations include both plasma from active region structures and outside these structures. Fig. 6.5 shows the intensity maps obtained from S X and Mg VI lines. Mg VI shows a highly structured behaviour, indicating that this ion is present only inside in a relatively cool structured area belonging to the active region at the limb. Outside this area the residual Mg VI emission is mainly due to scattered light. S X presents a fairly uniform map with no signature of the active region. In this study the low temperature plasma inside the structures is referred as “prominence plasma” and the hotter coronal plasma as “active region corona.

Fig. 6.5 1196 Å (top) and 1191 Å (bottom) intensity maps of the emitting region

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6.5.2. Density and temperature measurements 6.5.2.1. Prominence plasma Making use of density-sensitive 1190/1191 Mg VI line ratio, the electron density (log value) was found to vary from 9.0 to 9.7 (Ne in cm-3) inside the bright structures (Fig. 6.6). Uncertainties at each pixel position are 0.15 dex. The random fluctuation of the measured density between 10‫ ״‬and 40‫ ״‬is probably due to strong contribution of the scattered light to line intensities and to the weakness of true line emission.

1005 Å intensity map of structures (in black portion line intensity is due mainly to instrument-scattered light).

Temperature map from the Mg VII/Mg VI (1189/1191) line ratio.

Density map from the Mg VI (1190/1191) line ratio.

Ne VI/Mg VI line ratio, yielding an estimate of the Ne/Mg abundance.

Fig. 6.6 Diagnostic results in the active region structures

The temperature, determined via the Mg VII/Mg VI 1189/1191 line ratio, is also displayed in Fig. 6.6. It is found that, where the Mg VI and Ne VI emission is strongest, the ratio is almost constant with a value of about 5-6 x 105 K. Outside the brightest positions, the ratio increases and shows a greater noise, due to the weakness of the diagnostic lines and to the increasing importance of the scattered light component to line intensity. The theoretical Mg VII/Mg VI ratio has been calculated in each pixel adopting the measured density value, to take

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into account the density sensitivity of the Mg VI 1191.64 Å line. The prominence plasma is thus relatively cooler and denser compared to the surrounding unstructured plasma. The pixel-to-pixel variation of the electron density suggests the presence of many unresolved plasma structures, with very similar electron temperatures and different densities. 6.5.2.2. Active region corona Temperature and density for the off-limb plasma are displayed in Fig. 6.7. Density-sensitive 1196/1213 S X line ratio is used to determine Ne outside the structure.

Fig. 6.7(a) Density vs height from S X (1196/1213) line ratio

Fig. 6.7(b) Temperature vs height from Ar XII / S X (1018/1196+1213) line ratio

When the plasma density is fairly low, the photoexcitation of ground levels by photospheric radiation is taken into account in calculating level populations. There is a critical density below which ground levels’ populations are affected by such a process. In the dataset Ne is sufficiently low to let photoexcitation play an important role for S X density diagnostics. Using CHIANTI and taking account of this process, the inferred log Ne values as a function of height are shown in Fig. 6.7(a). Uncertainties increase with height, and the last 5 density values are only an estimate of an upper limit, due also to the “flattening” of the theoretical ratio. The electron temperature Te outside the structure from Ar XII 1018/S X (1196 + 1213) line ratio as a function of height is measured and shown in Fig. 6.7(b). It is to be noted here that such a measurement could be biased to any problem in the relative Ar/S abundance. Argon is a high-FIP element while sulfur is just at the border (FIP is 10.4 eV), so this can provide additional uncertainty to the results.

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From this ratio it is found that the electron temperature remains more or less constant with height with log Te values between 6.24 and 6.26, assuming photospheric abundances. These values are slightly higher than those provided by Feldman et al.25 and Allen et al.26 in the quiet off-limb solar corona. 6.5.3. Relative element abundance 6.5.3.1. Prominence plasma Making use of the intensity maps from NeVI 1005.70 Å and Mg VI 1191.64 Å, the Ne/Mg ratio across the whole field of view is determined and compared with the Ne VI 1005.70 Å intensity map, which shows the complex plasma distribution of the active region, as shown in Fig. 6.6, and, for solar X = 45‫״‬, in Fig. 6.7, where also the uncertainties are reported. The variation of the Mg/Ne relative abundance seems to be strongly correlated with the plasma structures in the emitting source. From Figs. 6.5 and 6.6, some structures with strong Ne VI and Mg VI line emission indicate a normal FIP-bias in Mg/Ne varying from 1.6 to 3.2. However, in regions with strong Mg VI and weak Ne VI line emission the FIP-bias ranges from 3.1 to 8.8. These values show that the complex plasma structuring of the active region can have a strong effect on element abundances. Although the FIP bias is present in all the active region, its variability seem to suggest that the plasma in each individual structure might have its own peculiar composition. Such a possibility has been already suggested by a number of authors, who report similar abundance variations in active regions27-30. It is important to note that the peak ion fraction for Ne VI and Mg VI are quite different, which causes the relative intensity ratio between lines of these two ions to be temperature dependent, so that it is necessary to take into account the plasma electron temperature when these two ions are used for measurements of Ne/Mg relative abundance. 6.5.3.2. Active region corona In the unstructured, hot coronal plasma the low-FIP/high-FIP pairs S/Ar, Si/Ar and K/Ar and the height dependence of their intensity ratios is discussed. The FIPs for K, Si, S and Ar are 4.3, 8.2, 10.4 and 15.8 eV respectively. S X/Ar XII, Si XI/Ar XII and K XIII/Ar XII ratios are investigated, determining the FIP variation with height in the corona outside the structures. The results are displayed in Fig. 6.8. The results for the K/Ar, Si/Ar and S/Ar abundances outside the structures show different behaviours. The coronal relative abundance of S/Ar shown in Fig. 6.8 (top) stays almost at its photospheric value.

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Si/Ar shown in Fig. 6.8 (middle) varies between 18 and 26 compared to its photospheric value of 9. Thus a FIP of factor of 2 to 3 is found for Si/Ar in the corona. It is interesting to note that, apart from the last three positions, the Si/Ar relative abundance seems to decrease with height. The difference is slightly greater than the uncertainties, and show that the Si abundance tends to decrease relative to the Ar value.

Fig. 6.8 S, Si, K element abundances relative to Ar photosperic abundances Fig. 6.8 S, Si, K element abundances relative to Ar photosperic abundances

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The K/Ar relative abundance, Fig. 6.8 (bottom) varies from 0.7 to 1.6 in the off- limb plasma, showing a clear, strong FIP bias. The values of the FIP bias varies from 16 to 36, stronger than any FIP bias result quoted in the literature. The evident height dependence of the K FIP bias is a new observational fact whose magnitude (a factor around 2) is larger than the combined experimental and theoretical uncertainties. 6.5.4. Conclusion In the present study, the SUMER observations of a solar active region at the limb are used to investigate the FIP bias in both cool plasma confined in the core of an active region, and in the more diffuse, unstructured hot corona surrounding them. The lines from the high-FIP Ne VI, Ar XII ions, the intermediate-FIP S X ion, and from the low-FIP Mg VI, Mg VII, Si XI and K XIII ions are used and it is found that the Mg/Ne relative abundance is highly variable in the complex, cool core of the active region, and seems to be strongly correlated to line intensity patterns; this might suggest that each (spatially unresolved) plasma structure might have its own peculiar composition. Mg abundance enhancements relative to Ne reach up to a factor of 8.8. The analysis of the diffuse corona seems to suggest that in off-limb active region plasma the FIP bias inside the low-FIP elements’ class is dependent on the FIP value, being higher for the verylow FIP element K, although the presence of a Si VIII blending line makes this conclusion uncertain. References 1. 2. 3. 4. 5. 6. 7. 8.

Wilhelm, K., Curdt, W., Marsch, E. et al., Solar Phys. 162, 189 (1995). Wilhelm, K., Lemaire, P., Curdt, W. et al., Solar Phys. 170, 75 (1997). Lemaire, P. et al., Solar. Phys. 170, 105 (1997). Mazzotta, P. et al., A&AS. 133, 403 (1998). Bhatia, A. K. & Landi, E., ApJS. 147, 409 (2003). Dere, K. P., Landi, E. et al., A&AS. 125, 149 (1997). Young, P. R., Del Zanna, G. et al., ApJS. 144, 135 (2003). Zhang, H. L. & Sampson, D. H., At. Data Nucl.Data Tables. 72, 153 (1999). 9. Bhatia, A. K. & Mason, H. E., MNRAS. 190, 975 (1980). 10. Bell, K. L., Matthews, A. & Ramsbottom, C. A., MNRAS. 322, 779 (2001). 11. Bell, K. L. & Ramsbottom, C. A., At. Data Nucl. Data Tables. 76, 176 (2000).

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12. Mohan, A., Landi, E. & Dwivedi, B. N., ApJ. 582, 1162 (2003). 13. Bhatia, A. K., Seely, J. F. & Feldman, U., At. Data Nucl.Data Tables. 43, 99 (1989). 14. Burke, P. G., Hibbert, A. & Robb, W. D., J. Phys. B. 4, 153 (1971). 15. Berrngton, K. A. et al., Comput. Phys. Commun. 14, 367 (1978). 16. Eissner, W. & Seaton, M. J., J. Phys. B. 5, 2187 (1972). 17. Eissner, W., Comput. Phys. Commun. 114, 295 (1998). 18. Feldman, U., Phys. Scripta, 46, 202 (1992). 19. Feldman, U. & Laming, J. M., Phys. Scripta. 61, 222 (2000). 20. Dwivedi, B. N., Curdt, W. & Wilhelm, K., paper presented at the JDI9/IAU, Kyoto, August 26-27 (1997). 21. Dwivedi, B. N., Curdt, W. & Wilhelm, K., ApJ. 517, 516 (1999). 22. Dwivedi, B. N. & Mohan, A., Solar Phys. 157, 135 (1995). 23. Mohan, A., Dwivedi, B. N. & Landi, E., JAA. 21, 407 (2000). 24. Delaboudiniere, J. P. et al., Solar Phys. 162, 291 (1995). 25. Feldman, U., Phys.Scripta. 46, 202 (1992). 26. Allen, R. et al., A&A. 358, 332 (2000). 27. Young, P. R. & Mason, H. E., Solar Phys. 175, 523 (1997). 28. Widing, K. G. & Feldman, U., ApJ. 416, 392 (1993). 29. Widing, K.G. & Feldman, U., ApJ. 442, 446 (1995). 30. Landi, E. & Landini, M., A&A. 340, 265 (1998).

CHAPTER 7 ACTIVE REGION DIAGNOSTICS H.E. MASON and D. TRIPATHI Department of Applied Mathematics and Theoretical Physics, Center for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK

7.1. Introduction Active regions present us with complex and dynamic structures, which look relatively simple, but are in fact very difficult to model and explain. An active region is a region on the Sun where the magnetic field breaks through the surface (photosphere and chromosphere) to form giant arches of hot plasma. The magnetic field is stronger than in surrounding regions and the plasma is hotter, so emits strongly in UV and X-ray radiation. Figures 7.1 and 7.2 show UV images of the Sun taken with the Transition Region and Coronal Explorer (TRACE) spacecraft and the Extreme Ultraviolet Imaging Telescope (EIT) aboard the Solar and Heliospheric Observatory (SOHO) spacecraft. The hot plasma (around a million degree) traces out the magnetic field structures.

Figure 7.1: TRACE observations (UV image) of an active region on the limb, showing beautiful array of hot coronal loops. 127

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Movies from TRACE show how dynamic these active region loops can be. Indeed they are the source region for many energetic phenomena, such as solar flares, and Coronal Mass Ejections (CME’s, Tripathi et al.1). These phenomena can have far reaching effects on the Earth’s environment, and are termed ‘space weather’. The mechanism for such events is thought to be a sudden transfer of magnetic energy into other forms, such as thermal and kinetic energy. A better understanding of active regions would not only shed light on the coronal heating problem, but also on space weather phenomena. In order to model the physical processes prevalent in the solar atmosphere, it is necessary to have quantitative measurements of the physical properties of the plasma. Imaging instruments provide morphological information about active regions, often with high spatial and temporal resolution. They do not usually provide accurate plasma measurements. For these, we really need spectroscopic observations. It is the combination of simultaneous imaging and spectroscopic observations, which provides the most powerful insights into the nature of active regions. In this chapter we shall discuss recent spectroscopic and imaging observations of active regions in the UV and X-ray wavelength ranges. We will indicate (with examples) how plasma parameters (such as electron densities, temperature, elemental abundances and flows) have been derived.

Figure 7.2: Image obtained with the SOHO-EIT 171 Å filter, which is sensitive to emission lines from Fe IX and Fe X (at temperatures of around 1 MK).

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Several reviews have covered earlier work2-4. The aim of this chapter is to update, rather than repeat, what was previously reviewed, in particular with reference to observatories over the past decade: the Solar and Heliospheric Observatory (SOHO; launched in 1995), TRACE (launched in 1998), YOHKOH (launched in 1991), Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI; launched in 2002) and much more recently: Hinode and the Solar TErrestrial RElation Observatory (STEREO; both launched in autumn 2006). This chapter presents details of plasma diagnostic techniques and their practical application. 7.2. SKYLAB A good starting point for the study of solar active regions is the 1981 Monograph from the Skylab Workshop III on Solar Active Regions, edited by F. Q. Orrall1,4. This discusses the early work with Skylab. In particular, the Naval Research Laboratory’s S082A instrument (often affectionately called the ‘overlap-ogram’) on the Skylab’s Apollo Telescope Mount had an exceptionally good spatial resolution (better than 2"). This spectroscopic instrument covered a wavelength range 170-630 Å.

Figure 7.3: A drawing of active region structures by Cheng et al.5. This shows cool loops (Ne VII, 6 x 105 K) as solid lines and hotter structure (Fe XVI, 2 x 106 K) as shaded areas.

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The Skylab NRL S082A observations (Fig. 7.3) indicated that the hot loops (seen in Fe XVI and X-rays) were mostly low lying, compact and closely-packed. They connected directly across the neutral line. The cooler loops (seen in Ne VII) were large and slender. There were also many bright patches of emission or ‘kernals’ (shown by arrows in Fig. 7.3) in strong magnetic regions on either side of the neutral line (within bright chromospheric plage areas). These could be identified as the foot points of the Fe XVI and X-ray loops. Also in the Skylab Monograph, Dere and Mason2 wrote a review of Spectroscopic Diagnostics of the Active Region: Transition Zone and Corona and provided a summary of diagnostic line ratios, including electron density diagnostics. They also gave a comprehensive review of the atomic processes and data. They summarized electron density values derived for active regions, which ranged from 0.7 – 6.0 x 109cm-3 for a temperature range of 1.2-2 MK.

Figure 7.4: X-ray image of the Sun from Yohkoh/SXT, showing very hot plasma, over a million degrees.

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7.3. Yohkoh Yohkoh (Japanese for “sunbeam”) was a very successful joint Japanese, UK, USA satellite. The instruments on Yohkoh were able to observe the Sun in X-ray and Gamma Ray emission. The spacecraft carried four instruments: the Soft X-Ray Telescope (SXT), Bragg Crystal Spectrometer (BCS), and Wide Band Spectrometer (WBS), and the Hard X-Ray Telescope (HXT). The SXT instrument was widely used to study the plasma properties of active regions and solar flares. The SXT imaged the Sun in X-rays in the 0.25 – 4.0 keV range. Metallic filters were used to acquire images in restricted portion of this energy range, sensitive to different ranges of temperature. Yohkoh/SXT could resolve features down to 2.5 arcsec in size. Information about the temperature and density of the plasma emitting the observed X-rays was obtained by comparing images recorded using different filters. Fig. 7.4 shows a full disk image of the Sun recorded using the SXT. For details of the SXT instrument see Tsuneta et al.6. Yohkoh operated successfully for a decade. 7.4. SOHO The Solar and Heliospheric Observatory (SOHO, see Fig. 7.5) is a mission of international collaboration between ESA and NASA. SOHO, which was launched in 1995, comprises of 12 different instruments and is orbiting around the Sun located at the first Lagrange point. There are four instruments on board which are designed to observe the solar atmosphere in the ultraviolet wavelength range.

Figure 7.5: SOHO spacecraft.

Figure 7.6: EIT image at 195 Å.

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The EUV Imaging Telescope (EIT, Delaboudinière et al.7) provides images of the Sun in four different wavelength bands, which are sensitive to different temperatures. Fig. 7.6 shows an EIT image in the 195 Å filter which is sensitive to plasma emission at around 1.2 MK (from Fe XII). The active regions are bright and show loop like structures. The dark regions are coronal holes where the magnetic field is open to the interplanetary medium. EIT has other filters sensitive to lower coronal temperatures (171 Å, Fe IX/Fe X, 1MK) and higher temperatures (284 Å, Fe XV, 2MK). Fig. 7.7 shows the contrast in the SOHO/EIT images close to solar minimum (left hand side) and solar maximum (right hand side).

Figure 7.7: SOHO/EIT images in 195 Å (1.2 MK) close to solar minimum (left) and solar maximum (right).

The Coronal Diagnostic Spectrometer (CDS, Harrison et al.8) was designed to probe the solar corona by detecting the extreme-ultraviolet radiation from the Sun. The spectra recorded by the CDS provide an opportunity to measure plasma properties such as temperature, density, elemental composition in coronal and transition region plasma. The CDS is a complex instrument with a Grazing Incidence Spectrometer (GIS) and a Normal Incidence Spectrometer (NIS).

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The NIS has a 4′ long stigmatic slit, which produces spectra in the wavelength ranges 308-381 Å and 513-633 Å (as in Fig. 7.8, upper panel) simultaneously. Each position along the slit is spatially resolved. The slit can be stepped over a feature on the Sun (such as an active region) to build up a raster (as in Fig 7.8, lower panel). Spectra are obtained at each spatial location, so that the plasma properties can be determined. The range of temperatures for spectral lines shown in Fig. 7.8 goes from 20,000 degrees (584 Å, He I) up to 2MK (360 Å, Fe XVI). The bottom panel of Fig. 7.8 reveals the multi-thermal structure of active regions. The active region appears completely different at different temperature.

Figure 7.8: CDS spectrum (top) and raster (bottom) of an active region.

Temperature distributions (Emission Measure (EM), and Differential Emission Measure (DEM)) can be derived from the observation of several spectral lines (or filters). These can be compared with simulated emission measures derived from theoretical models. Figure 7.9 shows a sample DEM for an active region. Spectroscopic observations provide the opportunity of measuring plasma flows (as seen in Fig. 7.10) and also any non-thermal broadening in the spectral line profiles.

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Figure 7.9: Differential Emission Measure (DEM) of an Active Region derived from CDS data (From Landi and Landini9 ).

Figure 7.10: CDS raster and flows see in OV(0.1MK).

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The Solar Ultraviolet Measurements of Emitted Radiation (SUMER) instrument on SOHO has provided some fascinating diagnostics and results. SUMER is a powerful UV instrument capable of making reliable measurements of bulk motions in the chromosphere, TR and low corona with an accuracy better than 2 km s-1, with a spatial resolution of 1 arcsec across the slit (Wilhelm et al.10) and 2 arcsec along the slit (Lemaire et al.11). The instrument can also provide monochromatic images of selected areas on the Sun. In a very recent study, SUMER has provided with observations of red shifts along active region loops, as seen in Figure 7.11 (Dammasch et al.12).

Figure 7.11: SUMER observation of redshifts close to the footpoints of active region loops. The left panel shows an intensity image in Ne VIII. The middle and right panels show Doppler maps in Ne VIII and OIV respectively (From Dammasch et al. 2007).

7.5. TRACE The TRACE is a NASA SMall EXplorer (SMEX) mission to image the solar corona and transition region at high angular and temporal resolution. TRACE enables us to study the connections between fine-scale magnetic fields and the associated plasma structures on the Sun. TRACE has a very high spatial resolution of one arcsec (Fig. 7.12). TRACE also provides observations of the corona using filters at different narrow wavelength bands. Filter ratios can be

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used to determine the temperatures and estimate densities in the solar coronal features. For details about TRACE see Handy et al.13.

Figure 7.12: An active region seen by TRACE.

7.6. The CHIANTI Atomic Database CHIANTI is an assessed atomic database that allows the calculation of synthetic spectra and plasma diagnostics for optically thin plasmas, such as the solar and stellar coronae. It is a project of international collaboration between USA, UK and Italy. CHIANTI (http://wwwsolar.nrl.navy.mil/chianti.html) was the first freely available atomic database and is extensively used by solar physicists. The first release in 1996 (Dere et al.14) contained energy levels, radiative data, electron and proton collision rates. CHIANTI is regularly updated, and has an extensive suite of useful diagnostic routines. CHIANTI, v5, was released in 2003 (Young et al.15). This included extensive improvements to the atomic data, in particular for the coronal iron ions and for X-ray spectroscopy. Figure 7.13 shows an active region spectrum as observed by the GIS/CDS (top panel) and a synthetic spectrum produced by using CHIANTI (bottom panel). Note the remarkable similarities between two. CHIANTI, v6 will be released soon and will include ionization and recombination rates. A lot of work has been carried by G. Del Zanna and colleagues to assess and benchmark the data available for the coronal iron ions. In particular, Fe XII is an important coronal ion which has been observed by many solar instruments.

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Figure 7.13: SOHO/CDS GIS spectrum (top panel) compared to CHIANTI synthetic spectrum (bottom panel) for an active region (Mason et al.16).

7.7. Heating and Cooling of Active Region Loops Coronal loops are the building blocks of solar corona. The heating in a coronal loop i.e. spatial distribution of heat deposition along the loop is the most important quantity to identify and study in order to understand the heating mechanism of active regions and the solar corona in general. Almost three decades ago, Rosner et al.17 developed a model of solar corona, based on coronal loops, and compared this model with observations recorded by the X-Ray Telescope aboard Skylab. Starting with the assumption that quiescent loop structures seen in X-rays are in hydrostatic equilibrium they demonstrated that the quasi-static loops have a temperature maximum located at their apex. Moreover they were able to derive a unique relationship, with absolutely no free parameters, between the loop temperature, pressure and loop size which fitted the X-ray observation of coronal structures. This derived relation is known as

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RTV scaling law for coronal loops. Later, Serio et al.18 developed some generalized scaling laws in which they included loops with lengths greater than the pressure scale height. They found that loops with lengths greater than two to three times the pressure scale height must have a temperature minimum at the top. However, every static model predicts a hydrostatic stratification, which is in conflict (by up to four times) with the extended scale heights observed in the bright EUV loops. The TRACE loops, which are large and cool (1MK) were found to be ‘over dense’ and their lifetimes are too long. They cannot be sustained under hydrostatic models. The work of Priest et al.19,20 showed that the foot-point to apex temperature profile, calculated under the assumption of static equilibrium within the loop, is highly sensitive to the spatial distribution of heat deposition along it. They compared predicted intensities from various theoretical models with Yohkoh observations and found that a uniform heating model was most appropriate. Their analysis was later shown to have some severe limitations. However, it did start a long and fruitful debate on the heating mechanism for coronal loops and set us off in right direction – that of attempting to match theory and observations. Aschwanden et al.21 studied many TRACE loops and concluded that heating was predominantly at the foot-points. They found the loops to be almost isothermal. However, DEM analysis using CDS spectroscopic data showed that the loops were multi-thermal (Schmeltz22). There was a discrepancy between temperatures derived from filter ratio methods with TRACE and those derived from CDS observations. It should be noted that TRACE is an imaging instrument whereas CDS is a spectrometer. The discrepancy between the temperatures of coronal loops derived from CDS and TRACE was identified as a major issue. Understanding the temperature and density along coronal loops as well as across the loops is of fundamental importance in order to pin point the mechanisms responsible for heating. Using different instruments for different coronal structure studies have led to some misunderstandings and misinterpretations. Del Zanna and Mason23 tackled the problem rather differently i.e. using imaging and spectroscopic instruments together for the same coronal structure and complementing one by the other. This is the best way of using different spectroscopic and imaging instruments. Del Zanna and Mason23 used TRACE and CDS to study plasma parameters along a quiescent active region. In particular they studied one of the 1MK loops, which are best seen in the TRACE, 173 Å pass band. These loops are high lying structures, which can remain visible and stable over a long time scale. Figure 7.14 shows the TRACE emission for the 173 Å filter. The loop, which was analyzed, is at the very top of

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the image (in fact only half the loop is observed). This was chosen because it was relatively isolated which made the background subtraction easier. They found that these loops are isothermal across the loops. They found the loop temperature was about 0.7 - 1.1 MK which is much smaller than the temperature for the surrounding diffuse active region emission (T ~ 2 MK). They showed how important it is to correctly allow for background and foreground emission. The earlier discrepancies were mainly due to inaccurate or neglected background subtraction in the data analysis. Temperatures and densities were also derived by Del Zanna and Mason23, Del Zanna24 along the loop. The electron density was about 109 cm-3, and the temperature rose from 0.7 MK at the foot-point to 0.9 MK higher up the loop. Although TRACE with 1" spatial resolution, seems to resolve loops, it is possible that these loop structures could be made up of many bundles of ‘strands’, each of which is impulsively heated – possibly due to small scale magnetic reconnections, such as nano-flares (Parker25).

Figure 7.14: Active region on limb seen by TRACE.

The main problem in understanding the active region heating and solar coronal heating in general is to pin point the physical mechanisms of energy release. However, this is still an open question. From theoretical point of view Parker25 has shown that frequent small scale magnetic reconnections, producing nano-flares – could be responsible for heating the solar corona. Basically these

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small scale magnetic reconnections help dissipate the magnetic energy stored in the magnetic field lines due to photospheric motions into thermal energies. We do have observational evidences for magnetic reconnections in the corona and dissipation of magnetic energy into thermal energy in form of micro-flares, flares and CMEs. However, detection of nano-flares has not been possible so far. On the other hand the new observations recorded from TRACE reveal that magnetic field configurations in the corona are not as entangled as they should be if we take into account the foot point motions of these coronal loops. That suggests that there must be some form of magnetic reconnection at some altitude which prevents indefinite tangling of coronal field lines (Klimchuk26). If we assume that a coronal loop is not a single entity and made up of many fine strands (Cargill and Klimchuk27), magnetic reconnection could be occurring at any altitude in the corona. In order to verify the mechanism mentioned above, it is necessary to investigate the observational evidence. It is understood that magnetic reconnection leads to energetic particle acceleration, although the mechanism is not completely understood. These energetic particles then hit the thick chromosphere leading to the evaporation of chromospheric material into the corona. Later this plasma flows back down along the loops as it cools. However there have been no firm observations of up-flows in the coronal loops to date.

Figure 7.15: Joint observations of plasma condensations from TRACE (C IV emission) and CDS (He I, OV, and Fe XVI). Courtesy D. Müller.

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Müller et al.28 carried out some very interesting work on radiative instabilities in coronal loops, which lead to coronal condensations. They also include radiation in a self-consistent way using the code TTRANZ (developed by V. Hansteen). The runaway cooling in the loop leads to plasma condensation and the formation of micro-prominences. The rapid cooling leads to shocks, which can trigger a second condensation, which reaches much higher velocities (on the order of 100 km/s). Strong brightenings (a factor of 100 increase in intensity) occur when the blob hits the transition region. The depleted loop then re-heats. These calculations are consistent with the observations of cool loops, which emit at transition region temperatures. Figure 7.15 shows an example of joint observations of TRACE (filter which includes C IV emission, SUMER and CDS, He I, O V and Fe XVI lines. 7.8. The Evolution of Active Region Density and Temperature Structures As well as studying the physical properties of spatially resolved features such as coronal loops, it is also important to study the overall characterstics of active regions. This will help us to understand the evolution of active regions and the associated emission. There have been various attempts to measure active region plasma properties using instruments such as the Coronal Diagnostic Spectrometer (CDS) on board SOHO and also using early X-ray observations and eclipse observations29-30. Mason et al.31 studied an active region on limb using the observations recorded through CDS. Based on the Si X (346, 356) diagnostic line ratios they found that the core of the active region was hot and dense (greater than 2.3 x 109 cm-3). Figure 7.16 shows images of an active region taken in different spectral lines: Mg IX (Log T = 6.0, left panel, top image), Fe XVI (Log T = 6.4, left panel, bottom image) and Si X (Log T = 6.1, right panel, top image). In the images obtained at lower temperatures, such as those in Mg IX and Si X, lots more structures are seen in the active region than in the image taken in Fe XVI where only the very bright core is evident. This basically suggests that the core of the active region is very hot in comparison to the surroundings. The bottom images in the right panel of Figure 7.16 display the density map of the active region. The density map as derived using the Si X (356/347) line ratios. The density map of the active region suggests that the density is highest in the core of the active region which is seen in the Fe XVI raster. Another study of active region characterstics and plasma properties with CDS was carried out by Millgan et al.32 who found similar results.

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Figure 7.16: CDS observations of a limb active region. Left Panel: Intensity images taken in Mg X (top) and Fe XVI (bottom). Right Panel: Intensity images in Si X (top) and derived density map (bottom) from Si X (356/347). From Mason et al.31.

In a further study, Tripathi et al.33 tracked the evolution of density and temperature of an active region for five continuous days from CDS observations. The evolution of the active region was further combined with that of photospheric magnetic field recorded by the Michelson Doppler Imager (MDI; Scherrer et al.34) also aboard SOHO. Fig. 7.17 shows the correlation between the density (left panels), temperature (right panels) and magnetic field (overplotted contours, blue: negative polarities; yellow: positive polarities) covering four days of its evolution as it crosses the central meridian. The temperature of the active region was derived using the line ratios from Si XII (520Å) and Fe XVI (360Å). Usually deriving temperature of an active region using line ratios does not provide the complete temperature structure of the active region. The line ratios favours the temperature at which the emission function, G(T), of two lines overlaps. The best method for obtaining the full temperature structure would be to compute the differential emission measure. Nonetheless, the line ratios technique provides some useful information if there is plasma at the temperature between the peak formation temperature of these two lines. Producing temperature maps of the active region helps us locate the hotter and cooler locations in the active region. As is evident from the Figure 7.17, the hottest region of the active region is in general densest and vice versa. The overplotted magnetic field contours are from the line of sight magnetic fields which are obtained from MDI magnetograms. It is also evident from the figure that hotter

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and denser portions of the active region correspond to regions of strong magnetic field. A closer examination of the evolution of magnetic field and the hot, dense knots shows that these these regions are closely associated with emerging and submerging magnetic flux.

Figure 7.17: Density and temperature evolution of an on-disk active region. Left panel: Density maps derived from Si X (356/347) line ratios. Right panel: Corresponding temperature maps derived from Si XII (520) and Fe XVI (360) line ratios. Over plotted are the magnetic field contours (Yellow: Positive, Blue: Negative) as observed from MDI magnetogram.

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7.9. Active Region ‘Moss’ A specific type of emission known as “active region moss” was clearly visible in the 173Å pass-band of the TRACE (Berger et al.35; de Pontieu et al.36). Although this type of emission was mentioned in early observations taken by Skylab (Cheng et al.5), the physical understanding for this phenomena was limited. TRACE provided the first detailed observations of moss. Berger et al.35 studied the morphological aspects of moss regions and concluded that moss is the emission predominantly from the upper transition region. Based on a comparative study between observation taken by TRACE and Yohkoh/SXT it was conluded that the moss emission seen in the upper transition region TRACE images correspond to the foot points of the hot loops seen in the Yohkoh/SXT images (Berger et al.35, Zhao et al.37, see Figure 7.18).

Figure 7.18: Left panel: Image taken by Yohkoh/SXT of the same portion of the active region as in the left panel using the filter Al-Mg which shows the structure at temperature of about 2-3 MK. Right panel: An image obtained with TRACE using 173Å filter showing structures with temperature around 1MK. Loops can clearly be seen in addition to the mottled emission called “moss”.

Later, using a joint observations between TRACE and CDS/SOHO, Fletcher and de Pontieu38 showed that the moss regions have a temperature range 0.6-1.5 x 106 K and the density was 2-5 x 109 cm-3 at a temperature of about 1.3 x 106 K.

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7.10. Hinode The Hinode spacecraft (formerly called Solar-B, Fig. 7.19) was launched in September 2006. It carries three instruments: the Solar Optical Telescope (SOT), the X-ray Telescope (XRT) and the EUV Imaging Spectrometer (EIS). Hinode is a collaborative project between Japan (ISAS), the USA (NASA) and the UK (STFC). The aim of Hinode is to study the connections between fine magnetic field elements in the photosphere and the structure and dynamics of the entire solar atmosphere. In this review we mainly focus on the observations taken from the Hinode/EIS. The EIS telescope has an off-axis paraboloid design with multilayer, toroidal gratings which disperse the spectrum onto two detectors, 160-211 Å (CCD-B) and 246-292 Å (CCD-A). EIS covers spectral lines for a very wide range of temperatures: log T = 4.7, 5.4, 6.0 - 7.3 K. Simultaneous observations can be made in 25 spectral windows, covering many spectral lines. EIS is designed to have an angular resolution of 2”. Rasters can be made with a 1" or 2" slit. In addition a 40" slot provides an opportunity to carry out high cadence observations. There is also a 250” slot. The raster size is 360" x 512" and the center of the field of view can be moved East/West by ± 15". Detailed information regarding EIS instrumentation can be found in Culhane et al.39

Figure 7.19: Hinode spacecraft: An artistic view.

The spectral resolution of EIS allows Doppler velocities of 3 km/s and nonthermal velocities of 20 km/s to be determined from the line shifts and line profiles respectively. The temporal resolution (in spectroscopic mode) is less than 1s for dynamic events and around 10 s for active regions.

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Figure 7.20 shows spectrum of active region for both detectors. The overplotted dashed lines are the effective areas of the detectors. For the detector CCD-B the effective area curve peaks at around 195 Å whereas for the detector CCD-A the peak is at 274 Å. The complete active region spectrum comprises many spectral lines which are very useful and essential for plasma diagnostic. Figure 7.21 displays images obtained by EIS using 1" slit simultaneously at in many different lines at a wide range of temperature.

Figure 7.20: EIS spectrum of an active region (Young et al.40).

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Figure 7.21: Monochromatic images obtained from EIS spectral scan at wide range of temperature.

One of the many excellent features EIS has is sensitivity over broad range of temperatures (log T = 5.8 – 6.7) including many density sensitive lines. This provides an excellent opportunity to study electron density distribution and filling factors in coronal structures. Figure 7.22 displays intensity and density maps of an active region observed on 1st May 2007. The densities are derived using the line raios of Mg VII 278 and 280 (top row) at log T = 5.8, Fe XII 186 and 195 (2nd row) at log T = 6.0, Fe XIII 202 and 203 (3rd row) at log T = 6.2 and Fe XIV 264 and 274 (bottom row) at log T = 6.3. For the first time, we are able to reliably estimate electron densities at many different temperature simultaneously (Tripathi et al.41-42). Many more new and exciting results are anticipated from Hinode observations over the next few years.

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Figure 7.22: Intensity and density maps of an active region observed on May 01, 2007 using EIS/Hinode, 2 arcsec slit. Left and middle columns: intensity maps in Mg VII (top row), Fe XII (2nd row), Fe XIII (3rd row) and Fe XIV (bottom row). Right column displays the respective density maps derived using the line ratio technique. (From Tripathi et al.41)

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7.11. Summary In order to achieve our goal of understanding the coronal heating mechanisms, we must take an integrated approach to theory and observations. This means taking the theoretical models and simulating the observations. It is essential to combine imaging and spectral observations, in order to use the strengths of both types of instruments. The forward modeling approach must allow for dynamic and non-equilibrium processes. This is indeed a challenge for the next decade of theory and observations! Acknowledgments We acknowledge support from STFC. References 1. Tripathi, D., Bothmer, V. and Cremades, H., A&A, 422, 337 (2004). 2. Dere, K. P. and Mason, H. E., In: Solar active regions: A monograph from Skylab Solar Workshop III., Ed. F. Q. Orral, 129 (1981). 3. Gabriel, A. H. and Mason, H. E., In: Applied atomic collision physics, Volume 1, 345 (1982). 4. Mason, H. E. and Monsignori Fossi, B. C., A&A Reviews, 6, 123 (1994). 5. Cheng, C.-C., Tandberg-Hanssen, E. and Smith, J. B., Jr., Solar Phys., 67, 259 (1980). 6. Tsuneta, S., Acton, L., Bruner, M. et al., Solar Phys., 136, 37 (1991). 7. Delaboudinière, J.-P., Artzner, G. E., Brunad, J. et al., Solar Phys., 162, 291 (1995). 8. Harrison, R. A., Sawyer, E. C., Carter, M. K. et al., Solar Phys., 192, 233 (1995). 9. Landi, E. and Landini, M., A&A, 340, 265 (1998). 10. Wilhelm, K., Curdt, W., Marsch, E. et al., Solar Phys., 162, 189 (1995). 11. Lemaire, P., Wilhelm, K., Curdt, W. et al., Solar Phys., 170, 105 (1997). 12. Dammasch, I. E., Curdt, W., Dwivedi, B. N. and Parenti, S., Annales Geophysicae, In Press. 13. Handy, B. N., Acton, L. W., Kankelborg, C. C. et al., Solar Phys., 187, 229 (1999). 14. Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C. and Young, P. R., A&AS, 125, 149 (1997). 15. Young, P. R., Del Zanna, G., Landi, E. et al., ApJS, 144, 135 (2003). 16. Mason, H. E., Young, P. R., Pike, C. D. et al., Solar Phys., 170, 143 (1997).

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17. Rosner, R., Tucker, W. H. and Vaiana, G. S., ApJ, 220, 643 (1987). 18. Serio, S., Peres, G., Vaiana, G. S., Golub, L. and Rosner, R., ApJ, 243, 288 (1981). 19. Priest, E. R., Foley, C. R., Heyvaerts, J. et al., Nature, 393, 545 (1998). 20. Priest, E. R., Foley, C. R., Heyvaerts, J. et al., ApJ, 539, 1002 (2000). 21. Aschwanden, Markus J., Schrijver, Carolus J. and Alexander, David, ApJ, 550, 1036 (2001). 22. Schmelz, J. T., ApJL, 161 (2002). 23. Del Zanna, G. and Mason, H. E., A&A, 406, 1089 (2003). 24. Del Zanna, G., A&A, 406L, 5(2003). 25. Paeker, E. N., ApJ, 330, 474 (1988). 26. Klimchuk, J. A., Solar Phys., 234, 41 (2006). 27. Cargill, P. J. and Klimchuk, J. A., ApJ, 478, 799 (1997). 28. Müller, D. A. N., Peter, H. and Hansteen, V. H., A&A, 424, 289 (2004). 29. Gabriel, A. H. and Jordan, C., MNRAS, 173, 397 (1975). 30. Webb, D. F., In: Solar active regions: A monograph from Skylab Solar Workshop III. Ed. F. Q. Orral, (1981). 31. Mason, H. E., Landi, E., Pike, C. D. and Young, P. R., Solar Phys., 189, 129 (1999). 32. Milligan, R. O., Gallagher, P. T., Mathioudakis, M., Keenan, F. P. and Bloomfield, D. S., MNRAS, 363, 259 (2005). 33. Tripathi, D., Mason, H. E. and Young, P. R., In SoHO-17: 10 Years of SoHO and beyond, 617 (2006). 34. Scherrer, P. H., Bogart, R. S., Bush, R. I. et al., Solar Phys., 162, 129 (1995). 35. Berger, T. E., de Pontieu, B., Fletcher, L. et al., Solar Phys., 190, 409 (1999). 36. de Pontieu, B., Berger, T. E., Schrijver, C. J. and Title, A. M., Solar Phys., 190, 419 (1999). 37. Zhao, X. P., Hoeksema, J. T., Kosovichev, A. G., Bush, R. and Scherrer, P. H., Solar Phys., 193, 219 (2000). 38. Fletcher, L. and de Pontieu, B., ApJL, 520, 135 (1999). 39. Culhane, J. L., Harra, L. K., James, A. M. et al., Solar Phys., 243, 19 (2007). 40. Young, P. R., Del Zanna, G., Mason, H. E. et al., PASJ, 59, 857 (2007). 41. Tripathi, D., Mason, H. E., Young, P. R., Chifor, C. and Del Zanna, G., In Announcing first results from Hinode, Ed. S. Mathews, In press. 42. Tripathi, D., Mason, H. E., Young, P. R. and Del Zanna, G., A&A letters, In Press.

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CHAPTER 8 HALL EFFECT AND AMBIPOLAR DIFFUSION IN LOWER SOLAR ATMOSPHERE

V. KRISHAN Indian Institute of Astrophysics, Bangalore-560034, India and Raman Research Institute, Bangalore-560080, India

8.1. Introduction The sun is endowed with a variety of temperature variations resulting from a combination of thermal and radiative equilibria and departures therefrom. The thermal and the nonthermal nature of processes then translates into a plasma with varying degrees of ionization. Thus the ionization fraction α = ρρni could vary over several orders of magnitude (Figure 8.1) where ρi and ρn are respectively the ion and the neutral hydrogen mass densities. Discrete structures such as sunspots, prominences and spicules contain plasmas with varying degrees of ionization. The support of the neutral component against gravity is a major concern in the stability of these structures. Although the ideal magnetohydrodynamics (MHD) is often used as a starting point of an investigation, a partially ionized system dominated by the charged particle-neutral collisions and the neutral particle dynamics necessitates a 3-fluid treatment. For a low degree of ionization one can define a weakly ionized plasma by the condition1 that the electronB T 0.5 is much larger than neutral collision frequency νen ∼ 10−15 nn ( 8K πmen ) 2 the electron-ion collision frequency νei ∼ 6 × 10−24 ni ΛZ 2 (KB T )−1.5 . This translates into the ionization1 fraction np /nn < 5 × 10−11 T 2 where n
s are the particle densities and T is the temperature in Kelvin. A major part of the solar photosphere qualifies as a weakly ionized plasma. The strong charge particle-neutral coupling endows the neutral fluid with some of the properties of a conducting fluid. The neutral fluid is thus subjected to the 151

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Lorentz force along with the usual pressure gradient force. This attribute has been invoked to find the support for the neutral component of the partially ionized cold and dense solar prominences.3 The dynamical equations of a 3-component weakly ionized plasma including the Hall effect and the ambipolar diffusion are established in section 8.2. 108

nH/np

106 104 102 100 −500

0

Fig. 8.1.

500

1000 h (km)

1500

2000

2500

Fractional ionization in the solar atmosphere

The Hall- MHD equilibrium of the system is discussed in section 8.3 wherein a class of equilibria with unit magnetic Prandtl number and equal values of the fractional ion mass density α = ρi /ρn and the Hall parameter H = λi /L are shown to emerge. Here ρ’s are the uniform mass densities, λi is the ion inertial scale and L is the scale of the equilibrium structure. A scaling law between the ionization fraction and the scale of the structure is found. This scaling law accounts well for the sizes of the filamentary structures on the solar atmosphere. A few simple examples show that the linear and the nonlinear force free magnetic configurations along with nonlinear Beltrami flow field seem to be the general features. In section 8.4, the exact nonlinear three dimensional solution for the Hall waves including dissipation, is obtained in a uniformly rotating weakly ionized plasma such as exists in the photospheric flux tubes. The ω −k relation of the Hall-Alfv´en waves demonstrates the dispersive nature of the waves, introduced by the Hall-effect, at large axial and radial wavenumbers. It is found that purely axially propagating lineraly polarized nonlinear modes cannot exist in the cylindrical geometry. The partially ionized plasma supports lower frequency modes, lowered by the factor α = ratio of the ion density and the neutral particle density, as compared to the fully ionized plasma. The relation between the velocity and the magnetic field fluctuations departs significantly from the equipartition found in the Alfv´en waves.

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The evolution of the magnetic induction in a weakly ionized plasma governed by the Hall effect and the ambipolar diffusion along with the ohmic resistivity furnished predominantly by the electron-neutral collisions is investigated in section 8.5. The ambipolar diffusion preferentially dissipating the perpendicular (to the magnetic field) component of the current density J produces a state with J  B, an important issue in the conversion of a non-force free field into a force free field on the solar atmosphere.4 The Hall effect and the ambipolar diffusion being nonlinear functions of the magnetic field strength enable the formation of high current density structures often invoked for plasma heating and acceleration purposes through magnetic reconnection type processes.5 In section 8.6 we consider the Taylor-like relaxation of the weakly ionized plasma in the incompressible limit. The invariance of the total energy, magnetic helicity and the generalized helicity for this system are investigated. The magnetic helicity survives the onslaught of the nonideal effects due to the ambipolar diffusion and the Hall effect, the generalized helicity does in a rather restricted way and the energy is a complete non-survivor. With the energy as the most dissipative and the variant quantity, its minimization should guarantee the emergence of the relaxed state. 8.2. Weakly Ionized Magnetoplasma We begin with the three component partially ionized plasma consisting of electrons (e), ions (i) of uniform mass density ρi and neutral particles (n) of uniform mass density ρn . The equation of motion of the electrons can be written as:

 ∂V e Ve×B + (V e · ∇)V e = −∇pe − ene E + me n e ∂t c −me ne νen (V e − V n )

(8.1)

On neglecting the electron inertial force, the electric field E is found to be: E=−

Ve×B ∇pe me − νen (V e − V n ) − c ene e

(8.2)

This gives us Ohm’s law. For α = (ρi /ρn )  1 the ion dynamics can be ignored. The ion force balance then becomes:
 Vi×B (8.3) 0 = −∇pi + eni E + − νin ρi (V i − V n ) c

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where νin is the ion-neutral collision frequency and the ion-electron collisions have been neglected for the low density ionized component. Substituting for E from Eq. (8.2) we find the relative velocity between the ions and the neutrals: ∇(pi + pe ) J ×B − , (8.4) Vn−Vi = aνin ρi acνin ρi J = −ene (V e − V i )

(8.5)

e νen where a = 1 + m mi νin . The equation of motion of the neutral fluid is:
 ∂V n + (V n · ∇)V n = −∇pn − νni ρn (V n − V i ) − νne ρn (V n ρn ∂t

−V e ) + µ∇2 V n

(8.6)

where µ is the kinematic viscosity. Substituting from Eq. (8.4) and using νin ρi = νni ρn we find:
 ∂V n J ×B + (V n · ∇)V n = −∇p + + µ∇2 V n (8.7) ρn ∂t ac where p = (pn + ab (pi +pe )) and b = 1+ ννne . Observe that the neutral fluid is ni subjected to the Lorentz force as a result of the strong ion-neutral coupling due to their collisions. The Faraday law of induction, on substituting for the electric field from Eq. (8.2) and the relative velocity of the ion and the neutral fluid from Eq. (8.4), becomes:
   J J ×B ∂B =∇× Vn− + (8.8) × B + η∇2 B ∂t ene acνin ρi 2

e νen c where η = m4πe is the electrical resistivity predominantly due to electron 2n e - neutral collisions. One can easily identify the Hall term, (J /ene ), and the ambipolar diffusion term, (J × B). The Hall term is much larger than the ambipolar term for large neutral particle densities or for νin  ωci where ωci is the ion cyclotron frequency. In this system the magnetic field is not frozen to any of the fluids. Equations (8.7) and (8.8) along with the mass conservation

∇·Vn = 0

(8.9)

form the basis of our investigation. We write the equations in a dimensionless form . The magnetic and the velocity fields are respectively normalized √ by a uniform field B0 and the Alfv´en speed VA = B0 / 4πρi . The time and the space variables are normalized, respectively, with the Alfv´en travel

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time τA = L/VA , and a scale length L. The resistivity η is normalized by (LVA ) and the current density J by (cB0 /4πL). In these units the following dimensionless equations: ∂B = ∇ × [V n × B − H J × B + A (J × B) × B] + η∇2 B ∂t

(8.10)

∂(∇ × V n ) = ∇ × [V n × (∇ × V n ) − αB × (∇ × B) ∂t −µ∇ × (∇ × V n )]

(8.11)

describe the weakly ionized plasma in the incompressible limit. Here A = (aνin τA )−1 , H = λi /L = c/ωpi L where ωpi = (4πe2 ni /mi )1/2 is the ion plasma frequency and λi is the ion inertial length. Eq. (8.11) has been obtained by taking the curl of the equation of motion of the neutral fluid. We are now equipped with the tools necessary to understand the various properties of a weakly ionized plasma. The ion dynamics becomes important as the fractional ion mass density α increases. The condition for neglecting the ion dynamics in a partially ionized plasma is that the electron-neutral collision frequency remains larger than the electron-ion collision frequency. The electron-neutral collision frequency2 νen and the electron-ion collision frequency1 νei are given as : νen = 8.28 × 10−10 nn T 1/2 s−1

(8.12)

νei = 16ni T −3/2 s−1

(8.13)

The condition νen > νei gives ni < 5 × 10−11 T 2 = 5 × 10−5 T32 nn

(8.14)

Thus α must be smaller than 6.4 × 10−4 at T3 = 1 for mi = mn . We find that the criterion for the weakly ionized plasma is satisfied in the solar atmosphere6 from a height Z of -10 Km. to nearly 800 Km. 8.3. The Hall Equilibrium The study of plasmas with flows7 is extremely important for the space8 and astrophysical9 plasmas.10 The flows are often associated with convective processes and or large scale atmospheric circulation. Plasmas in a gravitational field acquire the Keplerian rotation profile along with the additional possibility of the outflows in the form of jets. The interrelation of the flow and the magnetic field is of the paramount importance in the stability and

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the transport processes of these plasmas. The flow field is relatively more accessible in space plasmas through the Doppler effect but an equivalent measure and form of the magnetic field is rather difficult notwithstanding the fact that the magnetic field is absolutely necessary to account for the whole gamut of the observed phenomena such as the magnetic activity and the polarized radiation. There are some indicators for the permissible magnitude of the magnetic field such as the Zeeman effect and the equipartition of energy, the choice of its configuration often necessitates theoretical input. In the weakly ionized plasmas found in the lower solar atmosphere the non-ideal effects such as the Hall effect and the ambipolar diffusion become important contributors.4 In this section we investigate the coupled equilibria of the incompressible flow and the magnetic field of partially ionized plasmas including only the Hall effect11 as it operates in a density regime that is different from that in which the ambipolar diffusion dominates. The condition for the Hall effect to dominate the ambipolar diffusion ( ωci < νin ) in the weakly ionized part of the solar atmosphere6 requires that the magnetic field B at Z = −10Km. must be less than 105 G and at Z = 800Km., it must be less than 114G. The Hall effect introduces the characteristic ion-inertial length scale λi in an otherwise scale free ideal MHD system. Here, we derive, the equilibrium equations describing the velocity and the magnetic field profiles along with the Bernoulli relation for the pressure profile. A few instructive examples of the Hall - equilibrium are discussed. We show that the equality of the two small parameters: 1) H = λi /L and 2) the fractional ion mass density, α = ρi /ρn , produces a special case of the equilibrium with unit magnetic Prandtl number.11 Adding equations (8.10) and (8.11) and using the equilibrium condition ∂ ( ∂t = 0 begets): ∂(B + ∇ × V n ) = ∇ × [V n × (B + ∇ × V n ) − (α − H )B × (∇ × B) ∂t − µ∇ × (∇ × V n ) − η∇ × B] which for the equilibrium condition,

∂ ∂t

(8.15)

= 0, is satisfied if

[V n × (B + ∇ × V n ) − (α − H )B × (∇ × B) − µ∇ × (∇ × V n ) − η∇ × B] = ∇ψ2

(8.16)

The equilibrium solution of Eq. (8.10) is : (V n − H ∇ × B) × B − η∇ × B = ∇ψ1

(8.17)

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Substitution of the equilibrium conditions (Eqs. 8.16 and 8.17) in the ∂ = 0) force balance of the neutral fluid (Eq. 8.7) furnishes stationary ( ∂t the generalized Bernoulli relation (in the dimensionless form): ∇(

Vn2 + 0.5αβ + ϕg ) = ∇ψ2 − ∇ψ1 . 2 = V n × (∇ × V n ) − αB × (∇ × B) − µ∇ × (∇ × V n )

(8.18)

where β = 8πp/B02 and ψj are the potentials corresponding to the energy density of the system. We observe that a variety of equilibrium structures can exist. A flowless neutral (Vn = 0) component can exist for the hydrostatic condition ∇(pe + pi + pn + ϕg ) = 0 with a uniform or a force free magnetic configuration for µ = 0, η = 0. 1. For uniform pressure and the gravitational potential ϕg = −GM/r, a nonlinear Beltrami flow ∇ × V n = λ(r)V n , λ(r) = ±

1 2r

(8.19) (8.20)

with Vnr = 0, Vnθ = (GM/r)1/2 , Vnz = ±(GM/r)1/2 exists for a uniform or a force free magnetic field for µ = 0, η = 0. Here M is the mass generating the gravitational potential. This equilibrium consists of a helical flow. 2. Let us consider a case of viscous equilibrium again with V n = µ = 0. The Bernoulli relation Eq. (8.18) becomes:  ∇

Vn2 + 0.5αβ + ϕg 2



= µ∇2 V n

√ αB and

(8.21)

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with a possible solution: µ Vnr = − , r

(8.22)

Vnθ = constant, Vnz = constant d (0.5αβ + ϕg ) = 0 (8.23) dr The radial inflow in the presence of viscosity emerges as an essential component of the flow. 3. Another interesting case obtains for V n = H ∇ × B, ∇ψ1 = 0 = ∇ψ2 and µ = 0 = η. From Eqs. (8.16) and (8.17) we get 2H ∇2 B − αB = a1 ∇ × B

(8.24)

where a1 is a constant. The solution is a superposition of fields at two −1 different spatial scales (λ−1 + , λ− ) with B = c+ B + + c− B − ,

(8.25)

∇ × B + = λ+ B + ,

(8.26)

∇ × B − = λ− B −

(8.27)

a1 1 √ ± 2 (a21 − 4α2H ) 22H 2H

(8.28)

where c’s are constants and λ± = −

This is also known as the double Beltrami solution representing a nonforce free state in which the Hall effect removes the degeneracy of the force free or the so called Taylor state. 8.4. Formation of Small Scale Magnetic Structures Let us consider a special case for which α = H and µ = η = 0, equations (8.10) and (8.11) have an identical form of the type: ∂Ωj = ∇ × [Uj × Ωj ] ∂t

(8.29)

where j = 1, 2 and Ω1 = B,

U 1 = V n − H ∇ × B,

Ω2 = B + ∇ × V n ,

U2 = V n

(8.30) (8.31)

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Note that the two “vorticities”, Ω1 and Ω2 , differ by the vorticity (∇× V n ) of the neutral fluid and the two “velocities” U 1 and U 2 differ by the Hall velocity V H = −J/ene . Equation (8.31) exhibits that the vorticity ∇ × V n and the magnetic field B share the same status and the “field” Ωj is frozen to the “flow” U j in the absence of dissipation . The steady state of the system contained in Eq. (8.29) can be described as: U j × Ωj = ∇ψj

(8.32)

Since the inhomogeneous energy densities (∇ψ1 , ∇ψ2 ) would lead to nonequilibrium situations we assume ∇ψ1 = 0, ∇ψ2 = 0. The equilibrium, therefore, is given by V n − H ∇ × B = β1 B

(8.33)

B + ∇ × V n = β2 V n

(8.34)

and

The condition α = H determines the scale of the equilibrium structure for a given neutral fluid density and the ionization fraction in the form of a scaling law given as:  −1 ni −1/2 cm (8.35) L = 2 × 102 ni10 nn where the masses are taken to be mi = mp = mn with mp as the mass of a proton and ni = ni10 × 1010 cm−3 . The characteristic scale L of the equilibrium structure at different heights in the solar atmosphere are dispalyed in Fig. (8.2). Thus the Hall- equilibrium supports structures of different sizes at different heights with the smallest size of ∼ 0.1 Km at a height of 20 Km. and the largest size of ∼ 3921 Km at a height of 450 Km. It must be remembered, however, that these numbers pertain to the region of weakly ionized plasma wherein the ion dynamics could be neglected. By eliminating the velocity from Eqs. (8.33 and 8.34) the magnetic field determined from H ∇ × ∇ × B + (β1 − H β2 )∇ × B + β1 (1 − β2 )B = 0

(8.36)

is a superposition of force free fields at two different spatial scales λ−1 + and such that λ−1 − B = c+ B + + c− B − ,

(8.37)

∇ × B + = λ+ B + ,

(8.38)

∇ × B − = λ− B −

(8.39)

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where c’s are constants and λ± = −

(β1 − H β2 ) 1 √ ± ((β1 − H β2 )2 − 4β1 H (1 − β2 )) 2H 2H

(8.40)

The corresponding velocity field is found to be: V n = (H λ+ + β1 )c+ B + + (H λ− + β1 )c− B −

(8.41)

An example of the fields can be given in the cylindrical geometry retaining only the radial variation and with the boundary conditions : Bz (r = 0) = 1 and the axial current density Jz (r = 0) ≡ I as Bz =

1 [(λ− − I)J0 (λ+ r) + (I − λ+ )J0 (λ− r)] , (λ− − λ+ )

(8.42)

Bθ =

1 [(λ− − I)J1 (λ+ r) + (I − λ+ )J1 (λ− r)] (λ− − λ+ )

(8.43)

where J0 , J1 are the ordinary Bessel functions. Thus the magnetic field is not force free as the superposition of the two force free magnetic fields produces a non- force free magnetic field. However the two spatial scales −1 λ−1 + and λ− can be far apart depending upon the values of the constants β1 and β2 which are a measure of the global quantities of the system such as the total energy and the total magnetic helicity. One observes that for β1 ∼ 0, λ+ ∼ 0 and λ− ∼ β2 . The field (Eq. 8.36) reduces to a force free field for λ− ∼ β2 ∼ I. In this case the flow velocity Vnz becomes a fraction H λ− of the Alfven speed VAz . For typical values: ni ∼ 1011 cm−3 , B0 ∼ 2KG, VAz ∼ 109 cm.s−1 , Vnz ∼ 0.1Km.s−1 , L ∼ 100Km. the scale λ−1 − ∼ 10 Km., a scale not very much smaller than the resolving potential of the present day solar observational facilities.12 Another important outcome of the Hall equilibrium is the existence of the cross field flows. From Eq. (8.41), it is seen that V n × B = H (λ+ − λ− )c+ c− (B + × B − )

(8.44)

demonstarating the Hall effect as the direct cause of the cross field flows essential for accounting umbral and penumbral motions in sunspots. In the −1 case of the large difference between the scales λ−1 + and λ− , the field at the shorter scale would suffer much stronger resistive damping than the one on the larger scale and the total field could be approximated to a force free field. This process of conversion into a force free field is different from the one based on the ambipolar diffusion4 wherein a preferential damping of the perpendicular to the magnetic field) current density results into a state with predominantly field aligned current density. The velocity field

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for β1 ∼ 0, λ+ ∼ 0 reduces to V n = IH B − . Thus the flow becomes proportional to the Hall parameter H in addition to the axial current density I. In order to examine the second condition of the equality of the kinematic viscosity µ and the electrical resistivity η or the unit magnetic Prandtl number, we take that the electrical resistivity is due to the electron-neutral collisions and is give by : η = 234(nn /ne )T 1/2 cm2 s−1

(8.45)

The kinematic viscosity is give by : T 5/2 cm2 s−1 (8.46) ρn Thus for the parameters in the solar atmosphere the viscosity is found to be far too small to be equal to the resistivity. This could be taken as a pointer to the existence of the anomalous viscosity. The Hall effect in weakly ionized rotating plasmas plays a decisive role in determining the equilibrium flows, magnetic fields and pressure profiles. A variety of velocity and magnetic field profiles including the so called double Beltrami emerge in the equilibrium. A few examples, although by no means exhaustive, have been given. The inclusion of the gravitational potential necessarily leads to an axial flow along with the Keplerian rotation. This indicates the likely generation of jet structures. Inward radial flow emerges as an essential feature of the viscous equilibrium. It is possible, sometimes, to arrive at the stability characteristics from the very nature of the equilibrium solution. For example a linear force free magnetic field along with the linear Beltrami flow represents the minimum energy state and is therefore stable whereas the nonlinear force free magnetic field with linear or nonlinear Beltrami flow does not represent the minimum energy state and thus liable to instability. The full implications of the Bernoulli relation (8.18) along with the equilibrium conditions (8.16) and (8.17) remain to be explored. µ = 2.2 × 10−16

8.5. Nonlinear Hall Waves The Alfv´en waves13 have been invoked in various astrophysical situations for their diverse roles.14 They can, accelerate particles such as the cosmic rays, heat plasmas15 such as the solar corona provide models for magnetohydrodynamic (MHD) turbulence16 such as in the solar wind.10 The Alfv´en waves of large amplitude are required for all these purposes. The nonlinear couplings of the waves transfer energy to shorter spatial scales so that

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10 4

LM (km)

10 3 10 2 10 1 10 0 −1

10

0

200

Fig. 8.2.

400 h (km)

600

800

Characteristic size of the equilibrium structure at different heights

they can resonantly interact with the plasma particles resulting in their heating.17 The heating processes become particularly favourable at short wavelengths. One way of obtaining short wavelength modes is to go beyond the ideal MHD and include nonideal effects such as the Hall effect. It is well known that the Alfv´en waves are the exact solutions18 of the ideal MHD system. We have recently reported the existence of normal modes of arbitrary amplitude even in the presence of the Hall effect in fully ionized plasmas.19 In this section we explore such a possibility for partially ionized uniformly rotating plasmas with electron-neutral collisions, such as might exist in flux tubes with uniform axial magnetic field and uniform densities of the three species of particles. It is demonstrated that the linear dispersion relation is also valid in the nonlinear regime. The dispersion relation of the arbitrarily large amplitude waves is found in the Hall limit for for different axial and radial wavenumbers. Significant departures from the well known equipartition Alfv´enic relation between the velocity and the magnetic field fluctuations of the Hall Waves are also exhibited. We work with the following dimensionless equations of the incompressible Hall-MHD ∂B = ∇ × [(Vn − H ∇ × B) × B − η∇ × B] ∂t

(8.47)

∂(∇ × V n ) = ∇ × [V n × (∇ × V n ) − αB × (∇ × B)] ∂t

(8.48)

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The equilibrium of the system is described by the uniform field B 0 = ez , eθ and constant densities. We dimensionless rotational velocity V 0 = VrΩ Ai split the fields into their ambient and the fluctuating parts: B = ez + b;

Vn =V0+v

(8.49)

and substitute in Eqs. (8.47) and (8.48) to get : ∂b = ∇ × [(v − H ∇ × b) × ez + V 0 × b + (v − H ∇ × b) × b ∂t − η∇ × b] ,

(8.50)

∂ (∇ × v) = ∇ × [v × (∇ × v) + v × (∇ × V 0 ) ∂t + V 0 × (∇ × v) + α(∇ × b) × ez + α(∇ × b) × b] .

(8.51)

The linearized equations in cylindrical geometry turn out to be:

where

∂ b = ( ez · ∇)[v − H ∇ × b] − ΩZ + η∇2 b, ∂t

(8.52)

∂ ∇ × v = ( ez · ∇)[2Ωv + α∇ × b] − ΩY ∂t

(8.53)


 ∂ ∂ ∂ br + eθ bθ + ez bz Z = er ∂θ ∂θ ∂θ

(8.54)

and Y has the same form except that the components of b are replaced by those of (∇ × v). In order to solve these equations we assume: b = b(r) exp(ikz + imθ − iωt),

(8.55)

v = v(r) exp(ikz + imθ − iωt).

(8.56)

and substitute in equations (8.52) and (8.53) to arrive at: η ωm v − H ∇ × b = − b + i ∇2 b, k k 2Ωv + α∇ × b = −

ωm ∇×v k

(8.57) (8.58)

where ωm = (ω − mΩ). Combining equations (8.57) and (8.58), we find: ∇×∇×b−

2Ω η 2 (ωm 2 − β − iωm η∇2 ) ∇×b− (1 − i ∇ )b = 0 H ωm k H ωm

(8.59)

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where β = (2ΩH + α)k 2 . The solution of equation (8.59) is: ∇ × b = λb

(8.60)

with eigenfunctions as the standard Chandrasekhar-Kendall functions given by:     imλbz ∂ mkbz ∂ + ik bz ; bθ = µ−2 −λ bz − br = µ−2 ; bz = AJm (µr) r ∂r ∂r r (8.61) where µ2 = λ2 − k 2

(8.62)

and µ is to be interpreted as a radial wavenumber. The linear dispersion relation is then found to be:
 2Ωk 2 − λH k + iηλ2 − β + 2iΩηλk = 0. ωm + ωm (8.63) λ 2 . It is instructive to realize that the radial variwhere, now η = c2 νen /ωpe ation can be neglected only if one retains the azimuthal variation and the azimuthal mode number m in this case must be equal to ±1. This can be ∂ = 0. We see that easily seen by writing the components of Eq. (8.60) for ∂r the only consistent solution is bz = 0, λ = ±k, m = ±1; the corresponding dispersion relation being:
 iηk 2 H k 2 2 + ωm + 2Ωωm 1 − − [(1 + 2ΩH )k 2 − 2iΩηk 2 ] = 0 (8.64) 2Ω 2Ω

eθ ] exp(−iωt + iθ + ikz) repreand the eigenfunctions are: b = const[ er + i senting circularly polarized waves. It can be easily checked that the solution, Eq. (8.60), along with the linearized Eqs. (8.57) and (8.58) for the fluctuations v and b are also the solution of the complete nonlinear Eqs. (8.50) and (8.51). The nonlinear terms in Eqs. (8.50) and (8.51) are (v − H ∇ × b) × b and (v × (∇ × v) + (∇ × b) × b). One can easily check that the nonlinear terms vanish for the relationships of b and v given in Eqs. (8.57), (8.58) and (8.60). Thus we conclude that the dispersion relation, Eq. (8.63) and the eigenfunctions Eq. (8.61) represent the exact solutions of the incompressible Hall-MHD of the weakly ionized uniformly rotating plasma for fluctuations of arbitrary amplitudes . The exact nonlinear solutions offer a better representation of the turbulent10 fluctuations.9 The flux tubes are characterized by the exponentially decreasing magnetic field and particle densities along the tube and no variations across

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their crosssections. Here, in order to bring out the role of the Hall effect, the dissipation and the rotation on the properties of the Hall waves, we assume an idealized model of a flux tube with uniform magnetic field and particle densities, neglecting the boundary effects, and study the waves in an essentially infinite medium. This could be either considered as a purely theoretical idealization or relevant for flux tubes of sizes smaller than the local scale height. We recall that the criterion for the weakly ionized plasma is satisfied in the solar atmosphere6 from a height Z of -10 Km. to nearly 800 Km. The condition for the Hall effect to dominate the ambipolar diffusion (ωci < νin ) in this region requires that the magnetic field B at Z = −10Km.must be less than 105 G and at Z = 800Km., it must be less than 114G. We notice that in the absence of the rotation (Ω = 0) and the Hall effect (H = 0) the dispersion relation of the waves in a weakly (α  1) √ ionized plasma becomes ω = ± αkVAi with the corresponding normalized √ cross helicity hc ≡ (v.b) | b |−2 = ∓ α in contrast to the case of the fully ionized plasma wherein the Alfv´en wave is described by ω = ±kVAi and (v.b) | b |−2 = ∓1. The Hall effect dominates dissipation for ωce > νen . Although this condition is satisfied in the region under consideration, we will retain the dissipation term since it is necessary for any possible heating. This gives us an idea of the structures which might support the Hall waves. With all these idealizations it is not fruitful to use actual solar atmosphere parameters to display the dispersion and dissipation characteristics of the Hall waves. It would suffice, at present, to elucidate the properties of the waves in a qualitative manner. It can be noticed from the dispersion relation Eq. (8.63) that a sufficiently large rotation can quench the Hall effect, though this may not happen in small flux tubes with moderate rotation. The quenching is brought about by the rigid rotation which forces the various species to rotate together in contrast to the Hall effect which thrives on the relative motion between electrons and ions. One must also appreciate that the damping is caused by the electron-neutral collisions and not by the ion-neutral collisions. We now neglect rotation and determine the real and the imaginary parts of the frequency of the Hall modes. In the weak damping limit we find for the real (ωR ) and the imaginary (ωI ) parts of the frequency:  2 2 2 1/2  k λ H kλ ± H + αk 2 ωR± = (8.65) 2 4 and   ηλ2 H kλ ωI± = −i (8.66) 1± 2 2 2 2 (H k λ + 4αk 2 )1/2

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In the Hall limit i.e. the short wavelength limit we find: ωR+ H kλ, ωI+ −iηλ2 k

2

(8.67)

2 VAi

or recovering the dimensions, ωR+ λk ωci , very different from the standard Alfven wave. The second root, in the Hall limit, becomes ωR−

− λk αωci and ωI− = −i ηα 2 , independent of the wavelength. Thus we observe H that the degenerate oppositely propagating alfven waves become completely distinct in their dispersion and dissipation characteristics on the inclusion of the Hall effect. The Hall effect removes the degeneracy with respect to the radial wavenumber µ and the dispersion increases with an increase of γ = µ2 k −2 . It is also clear that the equipartition of the pure Alfv´en wave is long lost with the magnetic fluctuations being much larger than the velocity fluctuations. This has important consequences for observing strategies which are generally based on velocity measurements through the Doppler shifts. Since the system has been shown to support waves of arbitrarily large amplitudes, the magnetic field fluctuations need not be small. The Zeeman profiles should carry the signatures of magnetic field fluctuations associated with the Hall-Alfv´en waves. The Hall-effect introduces a new spatial scale, the ion inertial length λi , in an otherwise scale free ideal MHD system. Here, we believe, for the first time, an exact three dimensional nonlinear solution of the Hall-Alfv´en wave in a weakly ionized, cylindrical and rotating plasma with collisions, a system of much interest for photospheric flux tubes, has been found. Although the ion inertial length is too small to be directly observable, the modified spectral characteristics such as dispersion and dissipation and the v, b relation can be discerned from the high resolution observations of the magnetic and the velocity fluctuations and their interrelationship. The effect is accentuated towards high frequencies. These short wavelength modes are extremely important for plasma heating in the solar atmosphere. The dissipation of the waves in a weakly ionized plasma is caused predominantly by the electron- neutral collisions. A realistic study, including plasma stratification and the associated phenomena of cutoffs and other propagation characteristics in a bounded system will reveal the other consequences of the inclusion of the Hall effect. 8.6. Magnetostatic Structures The formation of the flowless magnetic structures can be studied from the static solution of the induction equation in the presence of the Hall effect and the ambipolar diffusion.20,21 Splitting into the poloidal and the toroidal

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components the magnetic field can be written as B = B P +B T such that in the local Cartesian geometry, assumed for a spherical object of large radius, B = (Bx (x, z), 0, Bz (x, z)) and B T = (0, By (x, z), 0). Here (x, y) represent the latitudinal and the azimuthal coordinates with z being the radial coor∂ = 0) profile of the toroidal component is to be dinate. The equilibrium ( ∂t determined from   By2 ∂By ∂ 2 By ∂ p ∂ 2 B +η =0 (8.68) + − 2 ∂x y ∂x2 ∂x 4(1 + q)πne mi νin ∂x where p=

c ∂ne νen me c2 νen me , η = ,q = 2 2 4πne ∂z 4πe ne νin mi

(8.69)

Here we have retained only the vertical gradient of the electron density and ∂ By = assumed ne = ni . The solution with boundary conditions By = B0 , ∂x 0 at x = 0 is found to be: ω2 ω2 1 by − 1 ωci x (1 + ci2 ) log + ci2 by = +A 2 qνin by + 1 qνin 2qνin Lz

(8.70)

−1 ∂ne where by = By /B0 , L−1 z = ne ∂z is the density scale height and A is an integration constant. It is important to note that it is the electron-neutral collisions that dissipate the magnetic field. One can easily check that the Hall effect dominates the ambipolar effect for ωci  νin . It is instructive to examine the magnetic field profile retaining only one effect at a time before discussing the nature of the solution under the combined effects of the Hall, the ambipolar and the dissipation terms. In the presence of the Hall and the dissipation terms, the magnetic profile is given by:

by =

c ix exp( qνωin Lz )

c ix 2 − exp( qνωin Lz )

(8.71)

One notices that the characteristic spatial scale of the magnetic field is found to be LB = qνin Lz /ωci and could be much smaller than the vertical characteristic scale. In the presence of the ambipolar diffusion and the dissipation, one recovers the solution first discussed by Parker:22 ηby +

2 ωci 3 2 by = A3 x 3qνin

(8.72)

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where the second constant of integration is made to vanish by shifting the origin. This solution varies linearly with x for small x and as x1/3 at large x with a steep rise in the middle. 8.7. Taylor-like Relaxed State The emergence of large scale structures are studied by a variational process in which a constrained minimisation of a global property of the system determines a class of flows and fields. Thus the minimization of the magnetic energy in an ideal MHD system under the constancy of the magnetic helicity produced the Taylor state23 which got validated in the spheromaks and the reverse field pinches.24 A variant of the Taylor hypothesis has become a promising mechanism of the solar coronal heating.25 By including the flows Montgomery26 et al. demonstrated the invariance of the cross helicity in addition to the total energy and the magnetic helicity and derived a relaxed state characterized by the superposition of Beltrami flows and force free magnetic fields. Steinhauer and Ishida27 established the invariance of the self helicities of the compressible two-fluid system and investigated the emergence of the relaxed state with finite pressure and shear flows. Yoshida and Mahajan28 raised concerns about the validity of the variational process in which the energy is minimised keeping the two helicities constant when all the three quantities are the exact invariants in the absence of dissipation suggesting thereby the need for an additional coercive target functional for minimization. Here, we29 consider, in the incompressible limit, the invariance of the total energy, magnetic helicity and the generalized helicity for this system. The magnetic helicity survives the onslaught of the nonideal effects due to the ambipolar diffusion and the Hall effect, the generalized helicity does in a rather restricted way and the energy is a complete non-survivor. With the energy as the most dissipative and variant quantity, its minimization should guarantee the emergence of the relaxed state. It is a straight forward procedure to show that the time rate of change of the total energy E, for a partially ionized plasma, for rigid boundary conditons, is found to be:   dE = − A J 2⊥ B 2 d3 x − η J 2 d3 x (8.73) dt  1 V2 (8.74) E= ( n + B 2 )d3 x 2 α

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Thus the energy is not an invariant in the presence of the ambipolar diffusion even for zero resistivity. The rate of change of the magnetic helicity Hm is given by:  dHm = − ηJ .Bd3 x (8.75) dt  1 A.Bd3 x (8.76) Hm = 2 where A is the vector potential. So the magnetic helicity remains an invariant even in the presence of the ambipolar diffusion and the Hall effect for η = 0. The time rate of change of the generalized helicity Hg is:  dHg = A (J × B).(B × Ωn )d3 x + (8.77) dt   (α − H )(J × B).Ωn d3 x − η J .Ωn d3 x,  1 (A + V n ).(B + Ωn )d3 x (8.78) 2 where Ωn = ∇ × V n . The generalized helicity is an invariant only when α = H , a condition relating the ionization fraction and the characteristic spatial scale L, and BΩn , a condition known as an aligned rotator in astrophysics. Thus the only rugged invariant is the magnetic helicity. For A  η the energy dissipates much faster than the magnetic helicity and one does not need to invoke additional arguments of selective decay in order to set up the variational principle in which the energy is minimised while keeping the magnetic helicity a constant. The variational principle reads: Hg =

δE − λm δHm = 0

(8.79)

Vn = 0, α

(8.80)

∇ × B − λm B = 0

(8.81)

to get

The system of equations (8.80-8.81) gives the force free magnetic field along with zero flow for the neutral fluid, in sharp contrast with the MHD case. The ionic fluid has a velocity V i = −(νin ρi )−1 ∇(pi + pe ). Indeed the circumstances A  η exist for the weakly ionized solar plasmas wherein η

A ∼ 0.04. For the sake of completeness it must be pointed out that the aligned rotator along with the condition α = H essentially relaxes to an Alfvenic

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state with force free magnetic field as J × B = Ω × B = 0 and the Beltrami flow V n . The condition α = H is especially important as it allows the formation of organized structures11 larger than the ion inertial scale, larger by the ionization fraction (ρi /ρn )−1 . In conclusion, in a partially ionized plasma, the ambipolar diffusion provides a differential source of dissipation, dissipating the energy while preserving the magnetic helicity. Thus, there is no need to look for a more coercive target function or resort to cascading arguments in order to set up a variational process for the formation of organized structures. 8.8. Conclusion A few novel features that result from the three- fluid description of the partially ionized solar photospheric plasmas including nonideal effects such as the Hall effect and the ambipolar diffusion have been highlighted. The subject is rich with possibilities and a lot remains to be done. Acknowledgments The author express her deep gratitude to Dr. Baba Varghese for his help in the preparation of the manuscript. References 1. H. Alfve´ n and C.-G.Fa¨lthammer, Cosmic Electrodynamics, p180,Oxford, at the Clarendon Press (1962). 2. M.L.Khodachenko, T.D.Arber, H.O.Rucker and A. Hanslmeier, Astron. Astrophys., 422, 1073, (2004). 3. H.R.Gilbert, V.H.Hansteen and T.E.Holzer, Ap.J., 577, 464, (2002). 4. J.E.Leake and T.D.Arber, Astron. Astrophys., 450, 805, (2006). 5. A. Brandenburg and E.G.Zweibel, Ap.J., 427, L91, (1994). 6. A.N.Cox, Allens Astrophysical Quantities, 368, (2000). 7. K.Staciewicz, P.K.Shukla, G.Gustaffson, S.Buchert, B.Lavraud, B.Thide and Z.Klos, Phys. Rev. Lett., 90, 085002 (2003). 8. K.Staciewicz, Phys. Rev. Lett.,93, 125004 (2004). 9. V.Krishan, and S.M.Mahajan, Solar Physics, 220, 29 (2004). 10. V.Krishan, and S.M.Mahajan, J.G.R., 109, A11105 (2004). 11. V.Krishan and Z.Yoshida, Phys. Plasmas, 13, 092303, (2006). 12. G.B. Scharmer et al., Nature, 420, 151, (2002). 13. A.H. Nye, A.H. and J.V. Hollweg, : Solar Phys., 68, 279, (1980). 14. D.J. Mullan and A.J.Owens, : 1984, Ap.J., 280, 346, (1984). 15. Bogdan, T.J. : 2000, Solar Phys., 192, 373. 16. Goldstein, M.L., Roberts D.A. and Matthaeus W.H., : 1995, Annual Rev. Astron. Astrophys., 33, 283.

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17. Y. Uchida and O. Kaburaki : Solar Phys., 35, 451, (1974). 18. E.N. Parker, : Cosmical Magnetic Fields, Clarendon Press, Oxford, p104 (1979). 19. S.M. Mahajan and V.Krishan, MNRAS 359, L27, (2005). 20. S.M. Chitre and V. Krishan, MNRAS, 323, L23, (2001). 21. V. Krishan and S.M. Chitre, preprint (2007). 22. E.N. Parker, ApJS, 8, 177, (1963). 23. J.B.Taylor, Phys. Rev. Lett., 33, 1139, (1974). 24. J.B.Taylor, Rev. Mod. Phys., 58, 741, (1986). 25. J. Hayvaerts and E.R. Priest, Astron. Astrophys., 137, 63, (1984). 26. D. Montgomery, L. Turner and G. Vahala, Phys. Fluids, 757, (1978). 27. L.C. Steinhauer and A.Ishida, Phys. Rev. Lett., 79, 3423, (1997). 28. Z. Yoshida and S.M. Mahajan, Phys. Rev. Lett., 88, 095001, ( 2002). 29. V.Krishan and Z.Yoshida,”Taylor-like relaxation in weakly ionized plasmas” preprint, (2007).

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CHAPTER 9 ON SOLAR CORONAL HEATING MECHANISMS K. PANDEY and U. NARAIN Astrophysics Research Group, Meerut College, Meerut-250001, India

9.1. Introduction It seems worthwhile to explain some basic concepts and terms, in brief, before describing the coronal heating problem and various efforts made in this direction so far. 9.1.1. Some Definitions and Concepts The electron kinetic temperature Te, is defined on the basis of kinetic theory by the following equation: kbTe = ½ mev2

(9.1)

where kb is the Boltzmann constant, me is the mass of the electron and v is the velocity of the electron. Thus by measuring electron velocities in the solar atmosphere the corresponding temperatures can be found. Another important quantity, namely, “plasma beta”, denoted by β, is defined by β = pg/B2/8π

(9.2)

where pg is the gas (plasma) pressure and B is the magnetic induction. It has different values in the different regions of the solar atmosphere, e.g. in photosphere β >1, in chromosphere β ≈ 1 and in solar corona β <1. In solar coronal environment the basic dissipative agents are the following: i. ii.

Resistivity (η) which is the inverse of the electrical conductivity of the medium. Viscosity (νvis) (quite similar to friction) which comes into play when relative motion between layers of a fluid exists. 173

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We can define time scale for different dissipation processes as follows: Suppose L is the characteristic length of a feature in the corona then the ohmic time scale is defined by tohm ~ 4 πL2/c2 η

(9.3)

where c is the speed of light in vacuum. Similarly the time scale (tvis) for viscous dissipation is defined as tvis ~ L2/ν1

(9.4)

where ν1 is the coefficient of viscosity. When tohm is shorter than tvis then ohmic dissipation is dominant and vice versa. It is quite clear from Eq. (9.3) that ohmic dissipation is large if L is small and η is large, that is in shorter features the ohmic dissipation could be large provided the resistivity η is large. Similar is the case with viscous dissipation.

Fig. 9.1 An average model of Sun and its atmosphere1

A model based structure of the Sun and its atmosphere1 is shown in Fig. 9.1. The interior of the Sun is divided into three regions, namely the core (0-0.25 RΘ ), intermediate (or radiative) zone and convection zone (0.86-1 RΘ ), where RΘ is the radius of the Sun. The core contains almost half of the mass of

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the Sun in about one fiftieth of its volume but it generates 99% of the energy through the main thermonuclear reaction 41H
4He + 2e+ +2ν +26.7 MeV

(9.5)

where 1H is hydrogen, 4He is helium, e+ is the positron and ν is the neutrino. The energy is transferred slowly outwards across the intermediate zone by radiative diffusion; since the photons are absorbed and emitted many times (a photon takes about 10 million years from core to reach the surface). In convection zone the temperature gradient is too high for the material to be in static equilibrium and convection starts. It transports energy because each blob of plasma carries heat as it rises and then gives up some part of it before falling. It is believed that in this region the Sun’s magnetic field is generated. The solar atmosphere consists of three regions with different physical properties. The lowest is an extremely thin layer of gaseous plasma (known as photosphere) which is relatively dense and opaque and emits most of the (visible) solar radiation. Above photosphere lies the rarer and more transparent region called chromosphere. Whereas the width of the photosphere is 500 km only that of the chromosphere is about 2500 km. During total solar eclipse the chromosphere appears as a bright red crescent because of dominant emission of H-alpha (λ 6563) line of hydrogen.

Fig. 9.2 Sketch of the variation of electron temperature Te and density ne with height for an average model solar atmosphere2

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Fig. 9.2 gives temperature distribution in solar atmosphere2 under static conditions (but it is not static). It is clear from this figure that the temperature decreases to a minimum value in the low chromosphere and then rises again. It becomes million degrees in the corona. In between the chromosphere and corona there is transition region whose thickness is about 300 km. In this region the temperature rises from 0.1MK to more than 1MK so that the temperature gradient is relatively very steep. Table 9.1: Energy losses in the solar corona (in erg/cm2/s)3 Quiet Sun

Coronal Hole

Active Region

Conductive flux

2x105

6x104

105-107

Radiative flux

105

104

5x106

Solar Wind flux

≤ 5x104

7x105

<105

Total flux

3x105

8x105

107

Energy losses in the corona: in quiet, coronal hole (open-magnetic field) and active regions are given in Table 9.1. These energy losses are due to conduction (from higher temperature to lower temperature regions), radiation and solar wind (energetic particles coming out from the Sun). Clearly the losses from active regions are much higher than those from coronal hole or quiet regions. The thermal conductivity of coronal plasma is given by4 Kth = 1.8x10-6 T5/2 ergs cm-1K-1

(9.6) 9

with T ≈ 1MK in corona, Eq. (9.6) gives Kth (corona) = 1.8x10 cgs units. At room temperature silver has thermal conductivity of the order of unity. Thus the thermal conductivity of solar corona is extremely high. 9.1.2. The Coronal Heating Problem It is clear from the matter presented above that solar corona lies mainly in between two cooler regions, namely chromosphere and interplanetary space (having temperature smaller than 0.1 MK). It loses energy via conduction, radiation and solar wind. To prevent cooling of coronal plasma to chromospheric temperature some source of heating must be present.

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The source of heating cannot be thermal because within few seconds due to extremely high thermal conductivity of the solar corona the temperatures will soon equalize. Further the source cannot be radiative because solar corona consists of highly ionized ions of hydrogen, helium, iron, calcium, nickel, cobalt etc and they can not absorb enough photospheric radiation to maintain such a high temperature. 9.2. Source(s) of Heating The source of heating could be external or internal. It could be mechanical or electromagnetic or a combination of both. The external source is from outside the Sun whereas the internal source is from inside the Sun. The ultimate source of energy from inside the Sun is the convection zone. Another classification of heating mechanisms exists. It is based on Alfvén transit time. The Alfvén waves propagate with Alfvén velocity (vA) in a medium. The time taken by Alfvén waves to cover the distance (l) between the two end points of a coronal structure (such as a magnetic flux loop) is called Alfvén transit time (tA). Thus tA = l/ vA

(9.7)

suppose t is the characteristic time of an event such as photospheric motions twisting the foot-points of a magnetic flux loop. If these motions are fast so that t > tA then MHD waves are generated. Heating by these waves comes under A.C. heating category. Slow foot-point motions (t < tA) generate currents/magnetic fields. Heating by these currents/magnetic fields belongs to D.C. heating category. Magnetic reconnection is the process involved in D.C. type of heating. In this process the magnetic field lines diffuse through coronal plasma and reconnect. First of all, the magnetic energy is converted to kinetic energy of outflowing plasma and then to thermal energy via viscosity of the medium. Various sources of heating are briefly described in the following5-7. 9.2.1. Heating via Accretion (Accretion Hypothesis) It was proposed by Hoyle and his associates and has been summarized by Hoyle8. According to this hypothesis, the Sun captures interstellar particles in its passage through galaxy. Assuming proton-proton interaction, falling matter loses all free fall energy (that is converted to thermal energy) at a height of 1,30,000 km above photosphere. So the layer of maximum temperature lies at this height.

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Heating below the layer of maximum temperature takes place through a special convection process in which the heat comes downward rather than upward. The upward and downward streams differ in temperature by a factor of 2 or 3. By a suitable choice of the rate of incident particle flux, it has been possible to deduce temperature and density distributions which agreed reasonably well with the observations available at that time. A major advantage is that it suggests a very simple process by which a temperature of the order of 1MK may be obtained in the corona while the lower lying gas remains cooler. Further the chemical composition of the corona seems closer to that of extra solar material than of the photosphere. The disadvantage is that the inverted convective process and the particle trapping process, mentioned above do not seem highly probable. The arguments put forward are not strong enough. The height of the layer of the maximum, temperature presented by this theory is considerably lower than that indicated by the emission line data. Further the theory does not explain the very high temperature low lying yellow line regions. It may be remarked that accretion hypothesis was proposed before the discovery of the solar wind and no provision was made for the inward and outward flow of material in interplanetary space. Now it is believed that accretion of matter is not important for the Sun. 9.2.2. Heating by Acoustic Waves This process was first described by Biermann9 to explain the enhanced temperature in the chromosphere. Schatzman10 explained the high temperature of the corona by the same process. In the convection zone (Fig. 9.1) cells of hot gas move upward and generate acoustic waves in the photosphere lying above it. The velocity fluctuation in such a wave is amplified as the wave passes through the chromosphere into corona. In the photosphere the velocity amplitudes are small compared to the thermal velocity, hence dissipation is small. But in the chromosphere and corona the velocity amplitudes become comparable to thermal velocities and the waves become shocks and dissipate rapidly. The mechanical power being transported through each unit area of the photosphere is given by Fs = ρu2cs

(9.8)

where ρ is the density of the photospheric material, u is the material velocity and cs is the adiabatic velocity of sound in the photosphere and is given by cs = (γp/ρ)0.5

(9.9)

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where γ = Cp/Cv is the ratio of specific heats and p is the pressure in the photosphere11. Typical values for the Sun are ρ = 3x10-7 g cm-3, u = 1km/s, γ = 5/3 and p = 1.2x104 Pa. These, together with equations (9.8) and (9.9) give: cs = 8 km/s and Fs = 2x109 erg cm-2s-1. Thus there is adequate amount of acoustic wave energy flux to account for the heating of the corona. It emerges that the acoustic waves become shocks, because of decreasing density in the lower chromosphere and lose most of their energy very rapidly via viscosity and the thermal conductivity of the intervening medium. Thus they are not able to reach corona. Hence corona can not be heated by acoustic waves. 9.2.3. Heating by Magnetoacoustic (slow and fast mode) Waves There is magnetic field distribution inside and outside the Sun. In presence of magnetic fields a number of wave-modes are excited in the solar atmosphere, e.g. in a homogeneous magnetic field three types of waves are generated. They are Alfvén-mode, fast-mode and slow-mode waves1,12. If we neglect the effect of gravity the fast-mode and slow-mode, together, are known as magneto-acoustic waves. Waves are always present on the Sun, because it is a dynamic body containing features that are always in motion over a wide range of scales. The fast-mode has higher frequency whereas the slow-mode has lower frequency. These waves become MHD shocks in the upper chromosphere and lose their energy via resistivity and viscosity of the medium. Thus, they can not be held responsible for the heating of the solar corona12. 9.2.4. Heating by Alfvén Waves These waves satisfy the following dispersion relation13: ω = k11vA

(9.10)

where vA, the Alfvén wave velocity, is given by vA = B0/(4πρ)0.5

(9.11)

in which B0 is the intervening uniform magnetic field strength, ρ is the matter density and k11 is the propagation vector along the magnetic field. Waves obeying Eq. (9.10) are also known as shear Alfvén waves. In strong field regions (B > 20 G) they suffer little dissipation unless strong gradients in magnetic field or density exist in such regions. Normally they can reach corona without much dissipation in photosphere and chromosphere.

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The energy flux in these waves is given by FA = ρ u2vA -7

(9.12) 3

-1

As already said, in photosphere ρ = 3x10 g/cm , u = 1 km s , B0 = 50 G, so that Eq. (9.11) gives vA = 4.5x103cm/s and FA = 1.3x107 erg cm-2 s-1. Thus Alfvén wave flux is of the same order as the losses in the solar corona (Table 9.1). In regions with nonuniformity in density or magnetic field, Alfvén waves dissipate via following mechanisms: (a) Resonant Absorption Because of non-uniformity in density or magnetic field the Alfvén waves have different velocity (wavelength). Each structure has its own natural frequency of oscillation. When the Alfvén wave frequency and that of the structure matches the wave energy is converted to thermal energy through resistivity and viscosity of the medium. This mechanism is very effective in heating coronal loops14,15. (b) Phase-mixing In this case in the beginning the Alfvén waves travel in a medium in the same phase. As soon as they approach a region of non-uniformity they start getting out of phase. Ultimately the amplitude of these waves goes on decreasing and the wave energy is converted to thermal energy. Thus the region of non-uniformity gets heated. The surrounding region is heated via thermal conduction, radiation or convection. This process is known as phase-mixing16-18. (c) Non-linear mode-coupling Kleva and Drake19 proposed a direct and fundamental nonlinear mechanism for the dissipation of Alfvén waves. Here the nonlinear coupling of parent Alfvén waves generates daughter Alfvén waves which have a shorter wavelength than the parent Alfvén waves. These nonlinearly generated short wavelength modes dissipate energy through Coulomb collisions much more rapidly than do the longer wavelength parent waves because the resistive damping rate of a mode with wave number 2π/λ is proportional to λ-2. In this mechanism the shear Alfvén wave (acting as a perturbation) couples nonlinearly with a spectrum of Alfvén waves. When the electron-ion collision frequency is small and the amplitude of the spectrum of the wave is large it is found that the nonlinear dissipation is enormously enhanced. This is because the spectrum of Alfvén waves convects plasma across the magnetic field by means

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of the ExB drift and as a consequence of large amplitude the particle motion becomes stochastic (random) in time, say in the (x,y) plane and also the field line motion becomes stochastic, say in z-direction. Under certain conditions the nonlinear dissipation gets enhanced by a factor of about 107. (d) Intermittent magnetic levitation The open magnetic field lines define a coronal hole funnel down into the underlying photosphere. The Alfvén waves are generated by giggling of these field lines by microflares and spicules in the transition region and chromosphere in the network and by the turbulent photospheric granulation under the network. Each network microflare is a separate random event, largely independent of other microflares. Spicules and granules also have random character. Because of this, the Alfvén waves on any particular open field line should be generated largely independently of the Alfvén waves on other open field lines that are rooted a few thousand kilometers away in the photosphere. In view of above one may consider the waves in a flux-tube that extends up through a coronal hole. The coronal plasma is lifted up during intervals of enhanced Alfvén wave generation so that it gains potential energy. This energy is mechanically transferred to the plasma from the reflecting Alfvén waves by the upward push from the reflection. That is the plasma is magnetically levitated by the reflecting Alfvén waves. When a decrease in the Alfvén wave flux occurs, the coronal plasma falls down. The extra potential energy gained during the previous interval of enhanced wave push now goes into thermal energy through compression of plasma. In this way the energy is converted from the kinetic and magnetic energy of the reflecting Alfvén waves to gravitational potential energy by the magnetic levitation. Now this energy is converted to thermal energy when this levitated plasma settles back and compresses during lapses in magnetic levitation. This process has been termed intermittent magnetic levitation20. The above mechanisms seem to be quite efficient under above-mentioned conditions. It remains still unclear whether Alfvén waves are generated in adequate amount or not21,22. The next section deals with heating by currents (magnetic fields). 9.2.5. Heating by Currents/Magnetic Fields There are two types, namely, heating by field aligned currents which may or may not involve reconnection of magnetic field and heating by currents flowing perpendicular to magnetic field (particularly in neutral sheets, which generally

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involves reconnection). Both types of currents ultimately are due to the distortion of the coronal magnetic field by photospheric motions. Field-aligned currents appear where the plasma beta is smaller than the unity (β<<1) whereas transverse currents (such as current sheets) appear when β>>1. In both the cases, the stored magnetic energy is converted into heat and kinetic energy of outflowing material. We discuss them below: (a) Field-aligned DC currents These are generated by photospheric motions involving intergrannular vorticity. Such a current may dissipate directly by Joule heating through classical or anomalous resistivity or by a chain of events involving reconnection. If the direct Joule heating j2η, is to match the coronal radiative losses, the current density, j, must be sufficiently large (▼х B = 4πj/c). This implies that the current must flow in very narrow filaments (about 100m diameters) which occupy a small fraction of the total coronal volume. To heat a typical coronal loop about 104 such filaments are required. According to Spicer23 such field-aligned filaments could heat the corona without any reconnection. But such a system is inherently unstable so that ultimately reconnection of field lines takes place and the system collapses. We expect impulsive release of heat and non-thermal electrons producing X-ray bursts, when above happens24. (b) Neutral current sheets Parker25 demonstrated that static equilibrium of magnetized plasma of zero resistivity requires a uniform pattern of twisting along the magnetic field lines. In presence of resistivity, a large scale magnetic field will develop tangential discontinuities (current sheets) between regions of dissimilar twisting. He suggested that Joule heating via magnetic reconnection would occur at these current sheets. It is termed as topological dissipation. For the coronal events Parker used the term nanoflares in which about 1024 ergs (about 109 times less than the largest flare) energy is released. Many such nanoflares are responsible for the heating of solar corona. Van Ballegooijen26, in contrast to Parker25 proposed that a slowly braided field develops steep gradients in a cascade of energy from large to small linear scales. Several authors have simulated the initial stages of such a cascade, but a final equilibrium between dissipation at small scales and input at large scale, has not been demonstrated24.

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Parker27 proposed that corona is heated via flaring on nano-, micro- and normal- flare scales. He predicted length-scales of the order of 1 km or less. Following type of geometrical structures are quite well known where current dissipation takes place (cf. Fig. 9.3).

Fig. 9.3 Current sheet, current sheath, and filament

Moore et al.28 present a scenario in which coronal holes or coronal loops are heated by microflares on the basis of the observations of Porter et al.29. These microflares occur sporadically throughout quiet regions and are seated in small magnetic bipoles embedded in the magnetic network along the edges of the supergranules as shown in the Fig. 9.4. Following initial energy release in low loops spanning small bipoles in the magnetic network, energy is transferred to corona via secondary neutral sheets where the expanding low-lying loops push against the legs of adjacent coronal loops or holes. The birth rate of microflares over the entire Sun is estimated to be of the order of 103 s-1 and the energy release in every microflare is such that it corresponds to an energy flux of 106 erg cm-2 s-1 over the whole surface of the Sun. Thus quiet and non-hole coronal regions may be heated by network microflares if much of the energy they release goes to coronal heating. The network microflares generate Alfvén waves which may heat coronal holes by intermittent magnetic levitation and phase mixing. This is in addition to direct heating by microflares.

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Fig. 9.4 Depiction of the heating of large coronal structures magnetically rooted near microflares in the chromospheric magnetic network28

9.2.6. Heating via Miscellaneous Mechanisms Some of the proposed mechanisms, in addition to earlier ones, are as follows: (a) Heating by spicules, (b) Heating by magnetic flux emergence, (c) Heating by velocity filtration, etc. These are described, in brief, below: (a) Heating by Spicules This mechanism was proposed by Athay and Holzer30. In spicules solar material goes out with high speed from the solar surface. The kinetic energy of spicular material is converted to thermal energy by the viscosity of the medium through which this solar material moves. After losing much of its energy the material from spicules falls down to the surface because of the solar gravity. Such events occur frequently over the whole solar surface. On an average, about 104 erg cm-2 s-1 energy is available for heating which is insufficient to heat different coronal regions (cf., Table 9.1). Thus it does contribute to the energy budget of corona but it is not a primary mechanism of heating.

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(b) Heating by magnetic flux emergence Book31 considers a bubble-like object, composed of magnetic flux and plasma, emerging from the surface of the Sun. If the initial magnetic pressure B2/8π exceeds the average ambient pressure then the structure expands as it rises, this continues until pressure balance is reached and reconnection annihilates the magnetic flux. The work ∆w done on the surrounding plasma is about ∆VB2/8π, where ∆V is the initial volume of the bubble. A mass ∆m= ρ ∆V is injected into the corona at the same time. Here ρ is the mass density when the bubble emerges. If structures of this type are continuously generated and the average volume per unit time is dV/dt then the power balance implies (dV/dt) B2/8π = (Prad + Pwind + Pcond) As

(9.13a)

where Prad, Pwind, Pcond denote losses due to radiation, solar wind and thermal conduction along the field lines to the cooler chromosphere and As = 4πRs2 is the surface area of the Sun. Using Prad ≈ 105 As erg s-1, Pwind ≈ 2x105 As erg s-1 and Pcond ≈ 5x104 As. (dV/dt)B2/8π = 3.5x105 As erg s-1

(9.13b)

In the above dV/dt ~ 0.1 fAs υ, where υ is the velocity of emerging structure and f is some fraction (≤1) of the Sun’s bipolar magnetic regions. He finds that υ≤ 108 cms-1 for B≤ 50G. Book31 concludes that, bubbles or other structures containing magnetic flux with field strengths ~20 G and having transverse dimensions like those of the initial stages of bipolar magnetic regions, can supply heat to the solar atmosphere through expansion in amounts sufficient to explain the observed temperatures. Gokhale32 has already used similar idea to explain the heating of X-ray bright points. (c) Heating by velocity filtration This mechanism was proposed by Scudder33,34. In this case no outside heating is required. It is assumed that the base of the corona is overpopulated with high energy electrons which form the tail of a non-Maxwellian population distribution. These high speed tail electrons overcome gravity (because they are highly energetic) and reach corona to populate it. Thus the temperature of the corona becomes automatically as high as million degree Kelvin. No mechanical, Joule (ohmic) or wave heating is required. The scattering cross section for

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high-speed electrons is quite small (Fig. 9.5) so they are not scattered appreciably during their motion from the base of corona upwards. The low velocity electrons get scattered back and thus velocity filtration takes place.

Fig. 9.5 The general nature of scattering cross section

The existence of a non-Maxwellian tail, however, seems quite arbitrary assumption which requires an unknown acceleration mechanism in the chromosphere that energises electrons/ions to thermal velocities of 200-400 km s-1. According to Aschwanden35, there are following two predictions namely, the linear temperature increase with height, and the density scale heights (dh) of different ion species are reciprocal to the ion masses. Thus using the relation dh = 2kb Ti/mi gsun

(9.13c)

for iron ion with atomic mass mi = 56mp (where mp is the mass of proton) the gravitational scale height dh is 56 times shorter than that of a hydrogen atom. gsun is the acceleration due to gravity for the Sun. Observations of 171 Å emission line show that the scale height for iron ions and a hydrogen helium atmosphere are the same which contradicts the second prediction. The first prediction is also not consistent with observations which show an isothermal temperature profile in the coronal part of EUV loops. In fact the time scales of intermittent heating, flows, and radiative cooling are much

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shorter than the time scale for gravitational settling of the heavy ions in the corona hence the theory of velocity filtration can not be tested in the highly dynamic corona36. 9.3. Numerical Simulation It is quite valuable to use this technique to investigate problems connected with heating of solar corona. An important parameter α, is introduced below: 9.3.1. Power Law Distribution Parker37 proposed that the energy dissipation of the stressed magnetic structure takes place in a large number of small events, called nanoflares, which are spiky in space as well as in time. Heating by spiky events can be related to global heating as follows: If Et be the total heating rate contributed by all the events in the energy range Emin, Emax then Et = ƒ Emin EmaxP (E) dE

(9.14)

where P(E) is the number of events per unit energy range and time and is given by P(E) = AE-α

(9.15)

where A and α are some constants. On combining equations (9.14) and (9.15) we get38: Et = A (Emax 2-α - Emin 2-α) / (2-α)

(9.16)

Two cases arise: 1. When α<2, Eq. (9.16) gives Et1 = A Emax 2-α / (2-α)

(9.17)

That is, heating is dominated by high energy events, large and intermediate energy flares. 2. When α>2, Eq. (9.16) gives Et2 = A Emin 2-α / (α-2)

(9.18)

That is, heating is dominated by the low energy events, such as micro-, nanoflares or still lower energy events.

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9.3.2. Some Valuable Contributions Georgoulis and Vlahos39 and Georgoulis et al.40 developed a cellular automation self-organized critical (SOC) model for transients. They obtain two distinct power laws (cf. Eq. 9.15). The low energy (weaker) events have shorter and steeper power law with exponent α ≈ 3.26 whereas the large and intermediate events have α ≈ 1.73. Also the weaker events are responsible for almost 90% of the total magnetic energy released and thus support coronal heating by nanoflares. Karpen et al.41 and Antiochos et al.42 performed numerical simulation of the interaction between two bipoles through magnetic reconnection in the lower solar atmosphere in which the magnetic field is sheared asymmetrically and the bipoles have unequal field strengths. They find that the random nature of reconnection process creates a distribution of current sheets throughout the region occupied by reconnected field lines. Reconnection between the larger and smaller flux systems leads to long-lived current sheets that decay slowly and yield the observed X-ray structures. Another interesting and important result comes from Moriyasu et al.43 who examine the behaviour of hot corona in an initially cool loop as a result of nonlinear Alfvén waves. Here the dissipation takes place via mode-coupling with slow- and fast-mode waves which are balanced by conduction and radiative cooling. They find that the resulting corona is very dynamic and full of shocks so that the X-ray and EUV intensities show many nanoflare type events, quite similar to what is actually observed. The intensities obey power law distribution with index α~1.7 for X-ray and ~1.39 for EUV. Thus, the actual observed timevariation of the X-ray and EUV fluxes in the corona and in the chromosphere may not be evidence of small scale magnetic reconnections but could actually be due to Alfvén waves. Cargill and Klimchuk44 have investigated the radiative signatures of the nanoflare heated corona and speculate that if an observed coronal loop contains many small strands then continual heating and cooling of strands would lead to a corona having a multi-thermal structure. The heating and cooling makes the loops have lower and higher densities at high and low temperatures, respectively. Such behaviour has been reported in the analyses of coronal data by Winebarger et al.45. 9.4. Summary and Conclusions The million degree Kelvin hot solar corona lies in between two cooler regions (chromosphere and interplanetary space) although it loses energy via conduction,

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radiation and solar wind flows. Because of its extremely high thermal conductivity and very low density it can not maintain its high temperature unless some source of energy replenishes the losses. External and internal sources have been made available. Accretion of interstellar matter during the passage of the Sun in the galaxy is an external source and is not important for the Sun. The solar convection zone is responsible for all internal sources which can be divided into A.C. and D.C. mechanisms of heating. These mechanisms involve photospheric foot point motions which produce Acoustic waves, magnetoacoustic waves, Alfvén waves and currents. Acoustic waves seem to heat lower chromosphere, magnetoacoustic waves the upper chromosphere and Alfvén waves the coronal holes and coronal loops. Currents/magnetic fields dissipate via magnetic reconnection and heat the corona at all scales, namely, nano-, micro- and normal- flares. The primary agents for heating solar corona are Alfvén waves and small-scale magnetic reconnection processes in the transition region and lower corona. Some numerical simulations show that actual observed time variation of X-ray and EUV fluxes in the solar atmosphere may not be evidence of small scale magnetic reconnections but could actually be MHD shocks due to Alfvén waves. Generation mechanism and generated fluxes of Alfvén waves need precise study. Velocity filtration mechanism does not require any local source but process of producing high energy electrons/ions in the chromosphere/transition region is not convincing enough. The predictions of the theory also do not agree with observations. Perhaps it is not possible to verify them because the corona is highly dynamic (changes on shorter scales) in comparison with gravitational effects. Most of the mechanisms of heating do contribute to the energy budget of the solar corona. Only observations are capable of providing crucial test of the primary source of coronal heating. The Hinode (Solar-B) mission, launched in 2006, should be helpful when the results become available. We are fast approaching towards a solution of this important astrophysical puzzle. Acknowledgements The authors are thankful to college authorities for their cooperation and to Mr. Nishant Mittal. Thanks are due to the learned referee (Prof. B.N. Dwivedi) for improving the manuscript through his valuable comments.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

E.R. Priest, Solar Magnetohydrodynamics, D. Reidel (1982). R.W. Walsh and J. Ireland, Astron. Astrophys. Rev., 12, 1 (2003). G.L. Withbroe and R.W. Noyes, A.R.A&A, 15, 363 (1977). D.E. Billings, A guide to the solar corona, Academic press, New York (1966). U., Narain and P. Ulmschneider, Space Sci. Rev., 54, 377 (1990). U., Narain and P. Ulmschneider, Space Sci. Rev., 75, 453 (1996). B.N. Dwivedi, J. Astrophys. Astr, 27, 125 (2006). F. Hoyle, Some Recent Researches in Solar Physics, Cambridge Univ., Press (1949). L. Biermann, Z. Ap., 25, 161 (1948). E. Schatzman, Ann. d’ Ap., 12, 203 (1949). M. Stix, The Sun, Springer-Verlag, 2nd ed. (1991). D.E. Osterbrock, Ap.J, 134, 347 (1961). H. Alfvén, Mon. Not. Roy. Astr. Soc., 107, 211 (1947). J.A. Ionson, Ap.J, 226, 650 (1978). A.N. Wright and G.J. Rickard, Ap.J, 444, 458 (1995). J. Heyvaerts and E.R. Priest, Astron. Astrophys., 117, 220 (1983). A.W. Hood, J. Ireland and E.R. Priest, A&A 318, 957 (1997a). A.W. Hood, J. Ireland and E.R. Priest, A&A, 324, 11 (1997b). R.G. Kleva and J.F. Drake, Ap.J, 395, 697 (1992). R.L. Moore, R. Hammer, Z.E. Musielak, S.T. Suess and C.H. An, Ap.J, 397, L55 (1992). E.N. Parker, Ap.J, 376, 355 (1991). W. Collins, Ap.J, 384, 319 (1992). D.S. Spicer, in mechanisms of chromospheric and Coronal Heating, Springer, p. 547 (1991). Z.B. Zirker, Solar Phys., 148, 43 (1993). E.N. Parker, Ap.J, 174, 499 (1972). A.A. Van Ballegooijen, Ap.J, 311, 1001 (1986). E.N. Parker, Ap.J, 407, 342 (1993). R.L. Moore, Z.E. Musielak, S.T. Suess and C.H. An, Ap.J, 378, 347 (1991). J.G. Porter, R.L. Moore, E.J. Reichmann, O. Engvold and K.L. Harvey, Ap.J, 323, 380 (1987). R.G. Athay and T.E. Holzer, Ap.J, 255, 273 (1982). D.L. Book, Comments Plasma Phys. Cont. Fusion, 6, 193 (1981). M.H. Gokhale, Sol. Phys., 41, 381 (1975).

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Scudder, Ap.J, 398, 299 & 319 (1992). Scudder, Ap.J, 427, 446 (1994). M.J. Aschwanden, Physics of the solar corona, Springer, Chap. 9 (2004). S.W. Anderson, J.C. Raymond and A. Van Ballegooijen, Ap.J, 457, 939 (1996). E.N. Parker, Ap.J, 330, 474 (1988). U., Narain and K. Pandey, J. Astrophys. Astron., 27, 93 (2006). M.K. Georgoulis and L. Vlahos, Ap.J, 367, 326 (2001). M.K. Georgoulis, N. Vilmer and N.B. Crosby, A&A, 469, L135 (1996). J.T. Karpen, S.K. Antiochos and C.R. De Vore, Ap.J, 460, L73 (1996). S.K. Antiochos, J.T. Karpen and C.R. De Vore, Ap.J, 575, 570 (2002). S. Moriyasu, T. Kudoh, T. Yokoyama and K. Shibata, Ap.J, 601, L107 (2004). P.J. Cargill and J.A. Klimchuk, Ap.J, 605, 911 (2004). A.R. Winebarger, H.P. Warren and J.T. Mariska, Ap.J, 587, 439 (2003).

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CHAPTER 10 CORONAL MASS EJECTIONS AND ASSOCIATED PHENOMENA NANDITA SRIVASTAVA Udaipur Solar Observatory, Physical Research Laboratory, Udaipur -313001, India

10.1. Introduction Coronal mass ejections or CMEs were first classified as a distinct class of solar phenomena soon after the launch of Skylab mission in space in 1973 which carried a coronagraph onboard. They were identified as large-scale expulsion of plasma clouds from the Sun which tend to display a beautiful 3-part structure (Figure 10.1). With the success of the Skylab mission, a series of space-based coronagraphs were launched such as SOLWIND, SMM and LASCO. It may be important to mention here that an early definition of a CME based on observations was provided by Hundhausen et al.1 They defined a CME to be “an observable change in coronal structure that occurs on a time scale of a few minutes and several hours and 2) involves the appearance (and outward motion) of a new, discrete, bright, white light feature in the coronagraph field of view. This definition is unique as it still holds good, even after the new observations from recent coronagraphs like LASCO aboard SOHO spacecraft have been made available. Moreover, this definition does not specify the origin of the feature, nor its nature, be it the ejecta or the effects driven by them. Therefore, this definition is still the most appropriate one2-3. 10.2. Observations of coronal mass ejections The observations of CMEs are possible only with coronagraphs which artificially occult the solar disk as in the case of a solar eclipse and enable imaging of the solar corona. After the invention of coronagraphs by Lyot4, the design of these telescopes has undergone significant changes so as to image various parts of the solar corona. Basically, there are two types of coronagraphs, namely internally 193

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occulted and externally occulted coronagraphs. The former rejects the photospheric light internally through a hole in the telescope while the latter uses an external occulting disk which is placed in front of the objective lens of the telescope. The optics used with these telescopes may be reflective or refractive depending upon the desired resolution and field of view of the telescope5. 10.2.1 Ground-based coronagraphs: The ground-based instruments for imaging the corona are complementary to the space-based ones because they can achieve a better temporal resolution and are not limited by the telemetry rate. However, they are limited by intensity and temporal variability of the sky. The main operating ground-based coronagraphs today, include Mark IV coronagraph at Mauna Loa Solar Observatory (MLSO) which images the Sun in continuum or white light and has a field of view from 1.08-2.85 R
with a temporal resolution of 3 minutes. The other coronagraphs observe the solar corona in emission lines, for example in FeXIV 5303 Å and FeX 6374 Å. Examples of such instruments are Mirror Coronagraph for Argentina (MICA) and Norikura coronagraph in Japan. There are other coronagraphs operating in Hα at MLSO and Pic du Midi and in He 10830 at MLSO. 10.2.2 Space-based coronagraphs: Following the design of Newkirk and Bohlin6 which uses an external occulter, a number of orbital coronagraphs were flown in space which included Skylab, OSO-7, P78-1, the Solar Maximum Mission (SMM), and SPARTAN 201. SOHO spacecraft carried a set of three coronagraphs with nested field of view which imaged the Sun from 1.1 up to 32 R
. Out of these three coronagraphs, the first one was an internally occulted coronagraph designed to image the innermost corona from 1.1-3 R
in emission lines. The other two coronagraphs were externally occulted and designed to image the outer corona from 2 to 32 R
5. Apart from this set of LASCO coronagraphs, another operating in UV was also launched aboard SOHO which obtains slit spectroscopic observations between 1.2 to 10 R
7. Very recently, two sets of identical coronagraph packages namely, the Sun Earth Connection Coronal Heliospheric Investigation (SECCHI) were flown on two spacecrafts called Solar TErrestrial Relations Observatory (STEREO) A and B in October 20068. The SECCHI package consists of coronagraphs having unprecedented field of view ranging from 1.4-4 R
(COR1), 2-15 R
(COR2), 12-84 R
(HI1) and 66-318 R
(HI2) in white light. These coronagraphs have been designed to provide three dimensional view of a CME in the inner and outer corona.

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10.3. Physical properties of coronal mass ejections The success of space-borne coronagraphs lies in the fact that a large number of CMEs have been observed and recorded by now which have provided an opportunity to study not only the morphological properties but also the physical and kinematical properties of CMEs on a statistical basis. Some of these properties are described in detail in this section.

Leading front

Core

Figure 10.1: Image of the corona taken by LASCO-C2 coronagraph aboard SoHO display the three part structure of a CME on 20 December 2001, observed in white light as marked in left panel (Adapted from Gopalswamy et al.9). The right panel shows an image of the corona taken in emission line (FeXIV) by LASCO-C1 coronagraph, which also displays the three part structure in the lower corona (Adapted from Srivastava et al.10).

10.3.1 Morphology: A classical picture of a CME is that of three-part structure as observed in continuum or white light by any coronagraph (Figure 10.1). It displays a bright leading edge which contains the frontal and the material swept by the CME. The leading edge is followed by a darker cavity which is due to its low density but has high magnetic field. The cavity is then followed by a bright knot or core which is mostly due to material density of the associated prominence with a CME. Occasionally, the three part structure may also be observed in other wavelengths such as in FeXIV line as shown in Figure 10.1 (right panel).

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Figure 10.2: This figure shows the time-lapse images of a limb CME recorded by LASCO-C3 coronagraph on October 20, 1997. The observations reveal that the cone angle defined by the angle between the opposite flanks of a limb CME is maintained at approximately 650 through 12 hours of the passage of the CME in the field of view (Adapted from Schwenn et al.11).

Recent temporal observations obtained from LASCO coronagraphs also reveal that the cone angle and the general shape of a CME are maintained during the passage through the LASCO field of view. That is, CMEs in general maintain “self-similarity”12. This has implications for the propagation models of CMEs and extremely useful for the prediction of the space weather. 10.3.2 Types of CMEs: Observations reveal that CMEs appear in a variety of shapes as revealed by the data of the space-based coronagraphs. One can broadly define three types of CMEs: (i) Halo CMEs which refer to those CMEs that occur close to the disk center and often appear to surround the occulting disk of the coronagraph like a halo13. Halos can be front-sided or back-sided, which can be distinguished by obtaining simultaneous disk observations. (ii) Partial halos are CMEs with apparent widths between 120◦ and 360◦. (iii) Limb CMEs which occur above the limb and have an angular width less than 1200. However, this difference in appearance is mainly because of the projection effect. A typical CME seen above the limb with an angular width of 600 would appear as a halo or a partial halo CME depending upon whether it is oriented along the Sun-Earth direction or 400 away from the Sun-Earth direction, respectively.

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Figure 10.3: The above figure shows an image of full halo observed on 28 October 2003 (left panel), partial halo observed on 12 May 1998 (middle panel) and a limb CME observed on 2 June 1998 (right panel) observed by LASCO-C2 coronagraph.

10.3.3 Widths of CMEs: The white-light coronagraph images have been used to measure the angular widths or the angular span of the CMEs14. The average width of LASCO/SoHO CMEs is found to be 400 taking into account a data-set of limb CMEs which have an angular span less than 1200. A study of variation of the measured values of the angular-widths of the CMEs by Yashiro et al.15 reveals that the average width is relatively smaller (470) during solar activity minimum compared to that measured during solar maximum (610). For measuring the widths, they considered only the limb or non-halo CMEs which give better estimates of the true widths of the CMEs16-17.

Figure 10.4: The distribution of apparent width of CMEs recorded by LASCO during 1996-2003 (Adapted from Yashiro et al.15).

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These values obtained are in general agreement with those obtained by previous space-based coronagraphs. However, on inclusion of the CMEs with angular span larger than 1200 in the data-set the average width is found to be slightly larger, i.e. 670 approximately14. 10.3.4 CME occurrence rates: In the pre-SOHO era, it was generally known that the rate of occurrence of CME during solar minimum is smaller than during the solar maximum. In particular, Webb and Howard18 found that this rate corresponds to 0.3-0.7 CMEs/day during solar minimum and increased with solar activity cycle reaching a value of 1.75-3.11 CMEs during the solar maximum. Owing to the increased sensitivity and higher dynamic range, of the LASCO coronagraphs aboard SoHO the rate of occurrence of CMEs was found to be on the higher side14,19. They found that the rate of occurrence was lower than 1 soon after the launch of SoHO i.e. around solar minimum which increased to 6 in 2002 close to the solar maximum (Figure 10.5).

Figure 10.5: The rate of occurrence of CMEs as recorded by LASCO coronagraphs for the period 1996-2004. The graph shows that the rate is significantly lower in the solar minimum and rises at the time of the solar maximum (Adapted from Gopalswamy et al.14).

10.3.5 CME mass estimates: The mass of a CME is derived from the whitelight images obtained from a coronagraph and is based on a well established fact that the white-light emission of the corona is mainly due to the Thomson scattering of the photospheric light by the electrons in the corona20. This therefore requires estimation of the coronal density from a series of time-lapse images of a CME by the coronagraph seen as enhanced intensity in the images. The technique used to determine the mass is same as that of Poland et al.21 and

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Howard et al.22. This technique involves the measurement of the electron density, which is computed from the excessive brightness after removing the preCME brightness. The method is based on the assumption that a single electron at a certain point in the atmosphere will scatter a known amount of solar disk intensity. Thus, by measuring the intensity and assuming that all of the mass is in a single volume element, the number of electrons can be computed. Then, the mass of the CME is calculated assuming charge neutrality. Using a large sample of Solwind CMEs, which were tracked up to a distance of 10 R
, Howard et al.23 reported that the total mass of a CME ranges from 2 × 1014–4 × 1016 gm with an average value of 4.1 × 1015 gm. Using LASCO observations, the mass of CMEs were estimated and found to lie in the range between 1014 to 1016 gm24. The mass in the flux-rope CMEs remains fairly constant for some events while others exhibit a significant mass increase at lower heights and tend to a constant value in the outer corona, above about 10-15 R
. This observation suggests that there is a pile-up of pre-existing material observed only in some flux-rope CMEs. It is also interesting to examine how the mass increase close to the Sun relates to interplanetary snowplowing observations25. As the leading edge propagates outward, the total mass of the CME increases because more of CME material begins to appear in the field of view of coronagraph from behind the occulter10.

Figure 10.6: The variation of total mass (on log scale) of the CME of June 21-22, 1998 with distance as obtained from LASCO-C3 coronagraph, white light images.

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10.4. Kinematics 10.4.1 Speeds and acceleration: With increased sensitivity, field of view and time cadence, it was possible to study the kinematics of the CMEs recorded by LASCO in a better way. On the basis of height-time profiles measured for many CMEs, some authors have argued or reported in favour of at least two (or more) kinds of coronal mass ejections26-30. One class belongs to slowly evolving gradual CMEs, with balloon-like shapes, accelerating slowly over large distances to speeds in the range 300 to 600 km s-1 and another class belongs to the impulsive CMEs, often associated with flares, accelerated already lower in the corona up to speeds of 2000 km s-1. The gradual CMEs are more commonly found to occur during solar minimum and mostly associated with eruptive prominences11,27. It is now believed that these two different kinds of CMEs represent the extremes of otherwise continuous spectrum of speeds attained by CMEs11.

Figure 10.7: CME initiation phase: A height–time plot up to 4 R
showing the evolution of different features of a CME observed on June 21, 1998. The plot shows evolution of the initial phase of the CME in lower corona. The arrows (from left to right) mark the timings of actual ascension of the CME (a), the initiation of the brightening in X-rays (b), and the occurrence of a C1.7 flare in the nearby active region (c) (Adapted from Srivastava et al.10).

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10.4.2 Initiation of CMEs: Zhang et al.31-32 described the initiation of CMEs in a three-phase scenario: the initiation phase, the impulsive acceleration phase and the propagation phase. The initiation phase (some tens of minutes) always occurs before the onset of an associated flare, and the impulsive phase coincides well with the flare’s rise phase. The acceleration ceases with the peak of soft X-ray flares. Study of kinematics of CMEs by several authors like St. Cyr et al.33, Sheeley et al.26, Srivastava et al.10, Zhang et al.31-32, Alexander et al.34 and Gallagher et al.35 suggests that CME acceleration can vary by 3 orders of magnitude ranging from several m s-2 to several thousand m s-2 while the acceleration duration can also vary by the same magnitude i.e. from a few minutes to more than one thousand minutes. The final velocity of a CME is a combined effect of the acceleration duration and the acceleration magnitude. In case of gradual CMEs, the exact time of initiation of a CME is difficult to define as the slow rise phase precedes the eruption of the CME several hours ahead10. 10.5. CME models A variety of observations of CMEs from different ground-based instruments and space-based missions prompted theoreticians to invoke several mechanisms/ models that would possibly explain the initiation and eruption of a CME. An early theory suggesting that thermal pressure associated with solar flare could be responsible for a CME initiation (Dryer et al.36) was ruled out with the observations that showed that flares occurred after the CME initiation37. Presently, it is commonly believed that it is the magnetic energy which drives a CME. A variety of models of CME initiation have been proposed which can be classified based on their physical attributes38. These have been briefly described below. 10.5.1 Break-out model: The break-out model was initially proposed by Antiochos39-40. In this model, the initiation of a CME occurs in multipolar topological configuration wherein reconnection between a sheared arcade and neighboring flux system triggers the eruption. The term ‘break-out’ refers to this process of reconnection which removes the unsheared field above the low-lying sheared core flux allowing it to burst open (Figure 10.8). The magnetic break-out model is quite successful in explaining observed properties of CME. Firstly, the observation of very low-lying magnetic field lines down to the neutral line which can be opened towards infinity during an eruption. Secondly, the model supports the idea that the eruption is solely driven by magnetic energy stored in a closed sheared arcade.

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Figure 10.8: Schematic of break-out model40. The first sketch shows magnetic field configuration at an early stage and the second at a later stage of eruption. A force-free current is created by shearing the arcade (thick lines) at the equator but a current layer horizontal to the solar surface is also created as the sheared region bulges outward. The process of reconnection of magnetic field lines in this layer allows opening of the sheared field line outward to infinity (Adapted from Forbes41).

10.5.2 Tether cutting model: The tether-cutting model proposed by Sturrock et al.42, is based on reconnection which occurs in initially sheared bipolar arcades, leading to formation of a magnetic island or plasmoid, which is then ejected.

Figure 10.9: Standard model for the magnetic field explosion in single-bipole eruptive solar events (Adapted from Moore et al.43). The dashed curve is the photospheric neutral line, bright patches are ribbons of flare in the chromosphere at the feet of reconnected field lines. The diagonally lined feature above the neutral line in the top left panel is the filament of chromospheric temperature plasma.

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Subsequently, closed post-flare loops are formed underneath the erupting flux rope, which is an important observational signature to test the model. These post-flare loops form a new arcade that grows with time and sustained reconnection. The reconnection process is considered essential for the onset of magnetic explosion in this model. 10.5.3 Flux rope model: In this model, it is presumed that the initiation of a CME consists of two phases i.e. photospheric shearing and flux emergence. This leads to formation of a twisted flux tube. The pre-eruption configuration consists of an infinitely long flux rope and overlying arcade which starts to rise in the initial phase, set of magnetic field lines then form an island through which runs the twisted flux rope closing down below with field lines reconnecting region and finally a set of arcades close to the boundary that reforms with sustained reconnection. 10.5.4 Flux-injection model: The magnetic configuration of a CME is that of a flux rope with footpoints anchored below the photosphere. The eruption of such a configuration can be brought by “flux injection” process or a rapid increase in poloidal flux. This mechanism is quite successful in reproducing not only the observed features close to the Sun but in the interplanetary medium of a CME44.

Figure 10.10: Flux rope geometry showing minor radius ‘a’ within which the toroidal current flows, poloidal field Bp which extends beyond the minor radius, and ambient coronal field Bc. Prominence material is also indicated as vertical line (adapted from Chen45).

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10.5.5 Mass loading: In this model, slow build-up of magnetic stress begins prior to eruption. One way of attaining this is to load the field with mass. Various configurations have been suggested by several authors46-49. In this model, pre-eruption magnetic field is metastable so that sufficient energy is built up in the system as a result of mass loading. The mass loading can occur by a high density coronal material or extremely dense prominence material. A destabilization of such a “mass loaded” system would occur with a large and sudden perturbation. Eruptive prominences appear to play a key role in a CME as suggested by Low49-50. However, it is to be noted that observations suggest that not all CMEs are accompanied by eruptive prominences, hence this model is sufficient in explaining only a sub-set of observed CMEs. 10.6. Associated phenomena CMEs are often associated with a variety of phenomena in different layers of the solar atmosphere. While the chromosphere displays Hα flares, Moreton waves associated with the CMEs, the corona has many phenomena to display for example, dimming, post-eruptive arcade formation, prominence eruptions, EIT waves, metric radio bursts. These phenomena are crucial to understanding the initial conditions that lead to triggering of CMEs, their propagation in the interplanetary medium and their arrival at the earth if directed earthwards. In order to study these associated phenomena, it is therefore important that one records the chromospheric and coronal activity in different wavelengths corresponding to different layers on the Sun. Such a multiwavelength study is a key to CME research. 10.6.1 Sigmoids: Sigmoids are S-shaped structures normally seen in coronal images as in X-ray images by YOHKOH and EIT on SoHO51. These are normally located above an active region and are seen 2-3 days prior to flare. They normally display an S (and inverse S) shaped pattern because the magnetic field lines that comprise them are twisted, but not so densely coiled. Although the field lines are three-dimensional, but they look like an S, in projection. Sigmoids are aligned approximately along a neutral line and some areas of it are brighter than others. The sigmoids are typically larger than a sunspot and typically 50000 to 150000 km long, but they vary widely in size. The sunspots and sigmoids have a link between them. At times, whorls are observed in sunspots, which are related to sigmoids. They also show same sense of the slinky-type structure. Although sigmoids were first discovered in X-ray images

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Figure 10.11: Evolution of an active region sigmoid which erupted on 12 February 2007 as observed by XRT telescope on Hinode.

taken by the SXT telescope on YOHKOH, they were also observed in EUV wavelengths, in images taken by the EIT telescope on SoHO. However they appear weaker, than SXR sigmoids. This is possibly due to the fact that they correspond to plasma of higher temperature ranging between one to several million Kelvin, while the plasma seen in EUV corresponds to a temperature of 1.5 million Kelvin. A study by Leamon et al.52 showed that active region exhibiting S or reverse S shaped sigmoids have greater tendency to erupt and are also related to CMEs that can produce at least moderate to strong geomagnetic storms. Thus, these features assume importance from space weather forecasting perspective53. During the initiation of a CME a sigmoid undergoes transformation to arcade structure as observed in the X-ray observations and is strongly suggestive that an untwisting or topological restructuring of the magnetic field occurs51. 10.6.2 EIT waves: Observations obtained from the EIT telescope aboard SoHO revealed a new phenomena namely, the EIT waves on the solar disk. The EIT waves become conspicuous in the running difference images taken by EIT telescope in 195 Å54,55. An example of an EIT wave observed recently is shown in Figure 10.12. These images show an expanding wavefront moving outward from the center of an active region seen in the form of an almost circular rim of enhanced coronal brightness at a speed ranging between 200-300 km s-1. These waves were interpreted as fast mode MHD waves analogous to Moreton waves observed in chromospheric lines namely Hα56. A statistical study by Klassen et al.57 indicates that the typical velocities of EIT waves range from 170 to

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350 kms-1. EIT waves were also observed by the Transition Region And Coronal Explorer (TRACE) in the same passband58.

Figure 10.12: An EIT wave recorded by EIT telescope aboard SoHO in 195 Å as seen in running difference images of January 7, 2008.

10.6.3 Post Eruptive Arcades: One of the most important phenomena associated with CMEs and observable in the lower corona is bright loops or arcades on the solar disk after the eruption of a CME59-62. They were first observed in X-rays as by Skylab59 and Yohkoh60. However, the first observation of post-eruptive arcades or PEAs in extreme ultraviolet wave-lengths especially in 195Å came in after the launch of SoHO by the EIT telescope. The PEAs appear as transient brightening of large-scale loop systems and last over a period of several hours62. They serve as reliable disk tracers of disk CMEs and generally represent the restructuring of the magnetic field in the aftermath of a CME. A detailed study of 236 PEAs during 1997-2002 was made by Tripathi et al. (2004) who found that average lifetime of PEAs is about 7 hours. The heliographic positions of the PEAs matched with the location of the active region belts in both the hemispheres with none above 600 N or S. The orientation of the PEAs follow Joy’s law, being commonly NE to SW oriented in the Sun’s northern hemisphere and SE to NW in the southern hemisphere. For some of the PEA events the corresponding CMEs were found to cause major geomagnetic storms63-64. In these studies, the in-situ magnetic field properties of the CMEs matched the ones predicted from the photospheric field patterns of the CMEs’ source regions according to the scenario presented by Bothmer and

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Schwenn65,66 and Bothmer and Rust67. Thus, PEAs signatures and their imprints on the interplanetary magnetic field may serve as a useful tool for prediction of the orientation of the magnetic clouds or ICMEs in general, making it convenient to predict the space weather.

Figure 10.13: A post-eruptive arcade (PEA) observed by TRACE satellite on 4 November 2003 at 22:35 UT in EUV (195 Å) line62.

10.6.4 Eruptive prominences: Eruptive prominences very frequently accompany CMEs. Early observations of CMEs from Skylab showed that 70% of CMEs are associated with eruptive prominences68. On the other hand, a relatively recent study by Gilbert et al. (2000) found that 72% of Hα eruptive prominences (EPs) are associated with CMEs. A separate study based on microwave observations (at 17 GHz) was carried out by Gopalswamy et al.69 who found that 73% of EPs have associated CMEs, 16% are not associated with any CMEs and remaining 11% were associated with streamer changes. Thus, there appears to be a conspicuous association between the EPs and the CMEs. In fact, Gilbert et al.70 showed that all Hα prominences that attain a height of 1.2 R
were associated with CMEs which is consistent with the earlier results obtained by Munro et al.68. The source locations of CMEs and EPs extend to all latitudes towards the solar maximum in a similar way. While the central position angle of CMEs tend to cluster around the equator, that of eruptive prominences are confined to latitudes of active region belt.

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Recently, few studies of eruptive prominences observed in He 304 Å by the EIT telescope aboard SoHO showed that the eruption of a prominence occurs in two distinct phases, one is that of a slow rise prior to several hours of eruption. The other is that of eruptive phase which experiences larger acceleration ranging between 10-80 ms-2 (71). These values coincide with those obtained for the core of the CMEs observed in white light, implying a one- to-one correspondence of the features of EPs with those observed in the white-light coronagraph field of view. The CMEs associated with such slow rising eruptive prominences fall under the category of gradual CMEs as reported by Srivastava et al.10.

Figure 10.14: Time-lapse images of eruptive prominence observed in He 304 Å by EIT telescope aboard SoHO on July 1, 2000 (Adapted from Joshi and Srivastava71).

Generally, these CMEs do not show any observational evidence for magnetic reconnection or impulsive energy release such as that associated with flares. The study of activation of prominences/filaments prior to eruption is important as it may provide a clue to the properties of the associated CME. 10.6.5 Solar Flares: Solar flares are sudden and huge explosions in the sun’s atmosphere releasing energies of the order of 1032 ergs in a time-scale of few minutes. The flares take place in the coronal and chromospheric layers. Generally, they occur in active regions close to sunspots and are driven by magnetic energy. Statistical studies have shown that 40% of the CMEs are associated with flares observed in Hα as shown by Munro et al.68 and Webb and Hundhausen72 based on Skylab and SMM observations of CMEs, respectively. Although prior to mid 1980s, it was commonly believed that the flares are the primary drivers of the CMEs and that the latter are a coronal response to the flare activity73, our present perception of CME is that of a phenomena which

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results due to loss of equilibrium of large scale magnetic field74. Another view is that a CME occurs as a result of relaxation of highly stressed magnetic fields which have been driven to complex state by photospheric motions. In case of an active region loop a continuous shearing activity succeeds in generating sites of magnetic reconnection resulting in particle acceleration and heating to 10 million degrees thus leading to a solar flare75. In larger structure, the response may be an outward expansion of coronal loop leading to non-equilibrium and instability, thus resulting in a CME. Observations also provide strong evidence that is an asymmetry between the CME and flares sites and CME-related flares can occur any time within many tens of minutes of the CME onset, either before, coincident with or after the CME76-77.

Figure 10.15: A 1F/X1.7 flare in NOAA AR 9393 recorded in Hα at Udaipur Solar Observatory on March 29, 2001 (Adapted from Srivastava and Venkatakrishnan78). This flare was associated with a large CME observed by LASCO coronagraphs aboard SoHO.

When both flare and a CME occur concurrently the energetic process responsible for their acceleration and energy release is the same at the onset32. These results thus confirm the conclusions of Harrison79 that flares and CMEs both are a consequence of the same magnetic disease and are closely related. Their characteristics are the results of the local initial conditions and therefore one observes a spectrum of flare and ICME properties that seem to be unrelated. 10.7. Summary Observations reveal that CMEs originate in the lower corona. Our understanding of the pre-CME conditions of the corona and the initiation mechanism of CMEs depends mainly on high spatial and temporal observations of the CMEs and the

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associated phenomena in multi-wavelengths covering the chromospheric, lower coronal to outer coronal layers of the solar atmosphere. The previous missions like SoHO have contributed significantly to our present understanding of CMEs, in general. It is expected that recently launched missions like Hinode and STEREO will help us take a giant leap in understanding the source regions of CME and initiation mechanism especially with the 3-D observations of the Sun, made possible for the first time. References 1. Hundhausen, A. J., C. B. Sawyer, L. House, R. M. E. Illing and W. J. Wagner, J. Geophys. Res., 89, 2639 (1984). 2. Schwenn, R., Astrophys. and Space Sci., 243, 187 (1996). 3. Srivastava, N. and R. Schwenn, in The outer heliosphere: beyond the planets. Eds. K. Scherrer, H. Fichtner, E. Marsch, Katlenburg-Lindau, Germany: Copernicus-Gesellschaft. ISBN 3-9804862-3-0, 12 (2000). 4. Lyot, B., CR Acad. Sci. Paris, 191, 834 (1930). 5. Brueckner, G. E. et al., Solar Phys., 162, 357 (1995). 6. Newkirk, G. Jr. and D. Bohlin, Appl. Optics, 2, 131 (1963). 7. Kohl, J. et al., Solar Phys., 162, 313 (1995). 8. Howard, R., D. Moses, D. Socker, J. Cook, J. Davila, J. Lemen, R. Harrison, C. Eyles, N. Waltham and J.-M. Defise, Proc. 35th COSPAR Scientific Assembly held 18 - 25 July 2004, in Paris, France., p. 3893 (2004). 9. Gopalswamy, N., Z. Mikić, D. Maia, D. Alexander, H. Cremades, P. Kaufmann, D. Tripathi and Y.-M. Wang, Space Sci. Rev., 123, 1-3, 303 (2006). 10. Srivastava, N., R. Schwenn, B. Inhester, S. F. Martin and Y. Hanaoka, The Astrophys. J., 534, Issue 1, pp. 468 (2000). 11. Schwenn, R., A. dal Lago, E. Huttunen and W. D. Gonzalez, Ann. Geophys. 23, 1033 (2005). 12. Low, B. C., J. Geophys Res., 106, 25,141 (2001). 13. Howard, R. A., D. J. Michels, N. R., Jr. Sheeley and M. J. Koomen, Astrophys. J., 263, L101 (1982). 14. Gopalswamy, N., S. Yashiro, S. Krucker, G. Stenborg and R. A. Howard, J. Geophys. Res., 109, 12105 (2004). 15. Yashiro, S., N. Gopalswamy, G. Michalek, G., O. C. St. Cyr, S. P. Plunkett, N. B. Rich and R. A. Howard, J. Geophys. Res., 109, CiteID A07105 (2004).

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16. Burkepile, J. T., A. J. Hundhausen, A. L. Stanger, O. C. St. Cyr and J. A. Seiden, J. Geophys. Res., 109, Issue A3, CiteID 10.1029/2003JA010149 (2004). 17. Cremades, H. and V. Bothmer, Astron. Astrophys., 422, 307 (2004). 18. Webb, D. F. and R. A. Howard, J. Geophys. Res., 99, 4201 (1994). 19. St. Cyr, O. C. R. A. Howard, N. R. Sheeley, S. P. Plunk.ett, D. J. Michels, S. E. Paswaters, M. J. Koomen, G. M. Simnett, B. J. Thompson, J. B. Gurman and 4 coauthors, J. Geophys. Res., 105, 18,169 (2000). 20. Minnaert, M., Zeitschrift für Astrophysik, 1, 209 (1930). 21. Poland, A. I., R. A. Howard, M. J. Koomen, D. J. Michels and N. R. Jr. Sheeley, Solar Phys., 69, 169 (1981). 22. Howard, R. A. et al., in Coronal Mass Ejections, eds. N. Crooker, J. Joselyn and J. Feynman (Washington DC: Amer. Geophys. Union), p. 17 (1997). 23. Howard, R. A., N. R., Jr. Sheeley, D. J. Michels and M. J. Koomen, J. Geophys. Res., 90, 817 (1985). 24. Vourlidas, A., D. Buzasi, R. A. Howard and E. Esfandiari, in Solar variability from core to outer frontiers, Proc. of The 10th European Solar Physics Meeting, 9 Ed. A. Wilson. ESA SP-506, 91 (2002). 25. Webb, D. F., R. A. Howard and B. V. Jackson, AIP Conf. Proc., No. 382, p. 540 (1996). 26. Sheeley, N. R., Jr., J. H. Walters, Y.-M. Wang and R. A. Howard, J. Geophys. Res., 104, 24, 739 (1999). 27. Srivastava, N., R. Schwenn and G. Stenborg, in 8th SOHO Workshop: Plasma Dynamics and Diagnostics in the Solar Transition Region and Corona, ESA Special Publ. 446. Eds J.-C. Vial and B. Kaldeich-Schümann., p. 621 (1999a). 28. Srivastava, N., R. Schwenn, B. Inhester, G. Stenborg and B. Podlipnik, Space. Sci. Rev., 87, 303 (1999b). 29. Švestka, Z., Space Sci. Rev., 95, Issue 1/2, 135 (2001). 30. Moon, Y.-J., G. S. Choe, H. M. Wang, Y. D. Park, N. Gopalswamy, G. Yang and S. Yashiro, Astrophys. J., 581, 694 (2002). 31. Zhang, J., K. P. Dere, R. A. Howard, M. R. Kundu and S. M. White, Astrophys. J., 559, 452 (2001). 32. Zhang, J., K. P. Dere, R. A. Howard and A. Vourlidas, Astrophys. J., 604, 420 (2004). 33. St. Cyr, O. C., J. T. Burkepile, A. J. Hundhausen and A. R. Lecinski, J. Geophys. Res., 104, 12493 (1999).

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34. Alexander, D., T. R. Metcalf and N. V. Nitta, Geophys. Res. Lett., 29, Issue 10, DOI 10.1029/2001GL013670 (2002). 35. Gallagher, P. T., G. R. Lawrence and B. R. Dennis, Astrophys. J., 588, L53 (2003). 36. Dryer, M., S. T. Wu, R. S. Steinolfson and R. M. Wilson, Astrophys. J., 227, 1059 (1979). 37. Hundhausen, A. J., in The Many Faces of the Sun, Eds. K. T. Strong, J. L. R. Saba, B. M. Haisch and J. T. Schmelz, Springer, New York, p. 143 (1999). 38. Klimchuk, J. A., in Space Weather (AGU Monograph Series), edited by P. Song, G. Siscoe and H. Singer, AGU, Washington, p. 143 (2001). 39. Antiochos, S. K., Astrophys. J., 502, L181 (1998). 40. Antiochos, S., C. DeVore and J. Klimchuk, Astrophys. J., 510, 485 (1999). 41. Forbes, T., J. Geophys. Res., 105, 23153 (2000). 42. Sturrock, P. A., P. Kaufman, R. L. Moore and D. F. Smith, Solar Phys., 94, 341 (1984). 43. Moore, R. L., A. C. Sterling, H. S. Hudson and J. R. Lemen, Astrophys. J., 552, 833 (2001). 44. Krall, J., J. Chen and R. Santoro, Astrophys. J., 539, 964 (2000). 45. Chen, J., J. Geophys. Res., 101, 27499 (1996). 46. Low, B. C. and D. F. Smith, Astrophys. J., 410, 412 (1993). 47. Chou, Y.-P. and P. Charbonneau, Solar Phys., 166, 333 (1996). 48. Wolfson, R. and S. Saran, Astrophys. J., 499, 496 (1998). 49. Low, B. C., Solar Wind Nine, Proceedings of the Ninth International Solar Wind Conference, Nantucket, MA, October 1998. Eds S. R. Habbal, R. Esser, J. V. Hollweg and P. A. Isenberg, AIP Conf. Proc. 471, 109 (1999). 50. Low, B. C., Solar Phys., 167, 217 (1996). 51. Sterling, A. C. and H. S. Hudson, Astrophys. J., 491, L55 (1997). 52. Leamon, R. J., R. C. Canfield and A. Pevtsov, J. Geophys. Res., 107, DOI 10.1029/2001JA000313 (2002). 53. Canfield, R. C., H. S. Hudson and D. E. McKenzie, Geophys. Res. Lett., 26, 627 (1999). 54. Moses, D., F. Clette, J.-P. Delaboudinere and 32 co-authors, Solar Phys., 175, 371 (1997). 55. Thompson, B. J., S. P. Plunkett, J. B. Gurman, J. S. Newmark, O. C. St. Cyr and D. J. Michels, Geophys. Res. Lett., 25, 2465 (1998). 56. Moreton, G. E. and H. E. Ramsey, Pub. Astron. Soc. Pac., 72, 357 (1960).

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57. Klassen, A., H. Aurass, G. Mann and B. J. Thompson, Astron. and Astrophys. Supp., 141, 357 (2000). 58. Wills-Davey, M. J. and B. J. Thompson, Solar Phys., 190, 467 (1999). 59. Kahler, S., Astrophys. J., 214, 891 (1977). 60. Hudson, H. S. and D. F. Webb, in Coronal Mass Ejections, eds. N. Crooker, J. Joselyn and J. Feynman (Washington DC: Amer. Geophys. Union), p. 27 (1997). 61. Zarro, D. M., A. C. Sterling, B. T. Thompson, H. S. Hudson and N. Nitta, Astrophys. J., 520, L139 (1999). 62. Tripathi, D., V. Bothmer and H. Cremades, Astron. and Astrophys., 422, 337 (2004). 63. Bothmer, V., in ESA Publ. ESA SP-535, Ed. A. Wilson, ISBN 92-9092845-X, 419 (2003). 64. Yurchyshyn, V. B., H. Wang, P. R. Goode and Y. Deng, Astrophys. J., 563, 381 (2001). 65. Bothmer, V. and R. Schwenn, Space Sci. Rev., 70, 1-2, 215 (1994). 66. Bothmer, V. and R. Schwenn, Annales Geophys., 16, 1 (1998). 67. Bothmer, V. and D. M. Rust, in Coronal Mass Ejections, Vol. 99. AGU Press, Washington, DC. Eds. N. Crooker, J. Joselyn and J. Feynman, p. 139 (1997). 68. Munro, R. H., J. T. Gosling, E. Hildner, R. M. MacQueen, A. I. Poland and C. L. Ross, Solar Phys., 61, 201 (1979). 69. Gopalswamy, N., M. Shimojo, W. Lu, S. Yashiro, K. Shibasaki and R. A. Howard, Astrophys. J., 586, 562 (2003). 70. Gilbert, H. R., T. E. Holzer, J. T. Burkepile and A. J. Hundhausen, Astrophys. J., 537, 503 (2000). 71. Joshi, V. and N. Srivastava, Bull. Astron. Soc. India, 35, 4, 447 (2007). 72. Webb, D. F. and A. J. Hundhausen, Solar Phys., 108, 383 (1987). 73. Kahler, S., Ann. Rev. Astron. Astrophys., 30, 113 (1992). 74. Low, B. C., Ann. Rev. Astron. Astrophys., 28, 10 (1990). 75. Moore, R. L., Solar Phys., 113, 121 (1987). 76. Harrison, R. A., Adv. Space Res., 11 (1), 25 (1991). 77. Harrison, R. A., Astron. Astrophys., 304, 585 (1995). 78. Srivastava, N. and P. Venkatakrishnan, Geophys. Res. Lett., 29, 10, 1287 (2002). 79. Harrison, R. A., Solar Phys., 166, 441 (1996).

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CHAPTER 11 THE RADIO SUN

P.K. MANOHARAN Radio Astronomy Centre National Centre for Radio Astrophysics Tata Institute of Fundamental Research P.O. Box 8, Udhagamandalam (Ooty) 643 001, India E-mail: [email protected]

11.1. Introduction The Sun is the most powerful radio waves emitting object in the sky. The first documented recognition of the reception of radio waves from the Sun was made in 1942 by Hey.15 Since then solar radio observations, from ground-based and space-based instruments, have played a major role in understanding the physics of the Sun and fundamental physical processes of the solar radio emitting phenomena. The solar radio emissions can be classified into three categories: (i) constant background continuum emission from the quiet Sun caused by the thermal electrons in the chromosphere and corona, (ii) slowly varying emission associated with sunspots and plages, and (iii) intense sudden burst-like emission caused by explosive and eruptive flare events on the Sun. Each category can cover a wide range of wavelengths and intensities. In particular, the radio emissions during a flare event are extremely intense to be of thermal origin and last for a short time interval. The intense flares in general are associated with the fast ejection of mass and magnetic field, i.e., coronal mass ejections (CMEs). Several mechanisms produce radio waves. They are: (i) free-free emission (bremsstrahlung), (ii) gyro-synchrotron emission, (iii) plasma emission, and (iv) gyroresonance, which could also be due to the heating. Other emission mechanisms may also be observed on the Sun, such as cyclotron maser,37 radiation caused by electrons moving in a strong electric field,53 and emission 215

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Fig. 11.1. Full disk images of the Sun observed on November 7, 2004 at about 12 UT in Halpha, extreme ultra violet (EUV), and soft X-ray (top row). The Halpha image is from Meudon spectroheliograph and EUV image and magnetogram (bottom left) are obtained from the EIT and MDI on-board SOHO spacecraft. The soft X-ray image is from the SXI Telescope on-board GOES satellite. In the X-ray and EUV images, coronal holes appear dark due to the less emission. The coronal holes are low density regions having open magnetic field configuration. They are sources of high-speed solar wind (refer to Manoharan, this volume). The X-ray emission is intense above the active region (also refer to Figure 11.3). The partial frame images shown at the bottom are MDI magnetogram and EIT image observed at 195 ˚ A to illustrate the magnetic field configuration and associated large-scale field lines surrounding the active regions.

by the transition of electron in a turbulent plasma.11 Simultaneously, one or more mechanisms can occur on the Sun. The radio observations are however better apprehended when combined with other wavelength measurements. Based on many observational studies and theoretical investigations of radio emission from the Sun, several comprehensive review articles and books are available. The readers may refer to them for more details.5,12,17,23,24,35,55 11.2. Solar Multi-Wavelength Observations Since the frequency of electromagnetic emission is directly linked to the change in the kinetic energy of charged particles, observations of the Sun in different parts of the electromagnetic spectrum can reveal various temperature levels as well as physical processes happening on the solar atmosphere. The photosphere is seen in visible light (T ≈ 6×103 K). The chromospheric

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and coronal features are observed in ultra violet emission (∼104 K). The hot regions of the corona are observable in extreme ultra violet wavelengths (∼105 K). The hottest part of the corona is observed in X-rays (∼106 K). The quiet corona produces a steady radio emission at meter wavelengths and the observed radio flux density from the tenuous corona confirms its temperature of about million K. However, the radio emission during an eruptive solar flare is often energetic and lasts only for a short time. Such radio bursts are extremely intense, more than million times the quiet Sun or background emission. These high energies can not be coupled with the temperature of the flare site or corona (i.e., they are known as non-thermal energies). Therefore, the non-thermal emission associated with high-energy particles observed at various heights of the corona provides an important tool to diagnose the coronal magnetic field involved in the energy release. In addition, the acceleration of particles in front of CME-driven and flaredriven shocks also produce the most dramatic solar energetic particle events observed within the heliosphere. The hard-X-ray is produced due to the interaction of high-energy particles with the surrounding plasma. Thus, the kinetic processes, which are governed by the interaction between magnetic field lines and plasma, play a major role in the generation of radio and X-ray waves. Figure 11.1 shows the images of the Sun seen in optical, ultra violet and in X-ray wavelengths. The Halpha image shows the photospheric features, such as prominences (observed as dark lines, which represent cool plasma embedded in the magnetic field lines), active regions (bright, intense emission regions). The EIT image observed in extreme ultra violet (EUV) represents the atmosphere above the photosphere, i.e., the chromospheric and transition region. The bright and spectacular structures seen in the X-ray image are produced by the release of energy in the hot corona. Large and short loops seen in X-ray are coronal magnetic field lines, which confine hot plasma within them. A system of loops can undergo continuous changes in all spatial and temporal scales. However, when there is a large-scale change or eruption of magnetic field structure, as seen in the case of solar flare, a huge amount of energy is released. Therefore, the most crucial part of the flare process is the localization of the energy release and particle acceleration site. The X-ray observations made with the Yohkoh satellite, both in soft and harder part of the X-ray spectrum, have revealed that the energy release is caused by the magnetic reconnection occurring in an X-shaped or Y-shaped current sheet above the flare loop (also refer to Figure 11.7).3,18,42,52,57

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Fig. 11.2. Daily averages of solar flux density for the period between 1988 and 2004 at frequencies 245, 410, 660, 2695, 4950, 8800, and 15400 MHz. These plots include two maximum phases of solar cycles 22 and 23, and a minimum phase during the year 1996. Data for these plots have been obtained from the Solar Geophysical Data Centre (http://spidr.ngdc.noaa.gov/spidr/). The solar cycle variations are evident in each frequency plot.

11.3. Radio Emission from the Sun The quiet Sun emits radio waves of slowly varying intensity, which essentially depends on the overall changes in the magnetic features of the Sun. Since the solar atmosphere is optically thick due to the free-free thermal emission from the hot corona at heights >1 R⊙ above the photosphere, the quiet Sun flux density can be estimated assuming a brightness temperature of ∼106 K at solar offsets of 1–2 R⊙ , depending on the observing frequency (i.e., at frequencies less than 500 MHz, flux density is proportional to the square of the observing frequency and at higher frequencies, it increases linearly with frequency). The radio flux density is denoted by Jansky (Jy), and 1 Jy = 10−26 Wm−2 Hz−1 . The solar flux density is normally expressed in solar flux unit (sfu) = 104 Jy.

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11.3.1. Solar Cycle Effects The quiet Sun radio flux considerably varies with the solar cycle. For example, the radio flux density from the entire solar disk at a frequency of 2800 MHz (λ = 10.7 cm) is recorded routinely and used as a fairly reliable indicator of the long-term behavior of the Sun. At this frequency, radio emission originates from atmospheric layers high in the solar chromosphere and low in the corona and changes gradually from day to day in response to the number of sunspot groups on the disk. The low flux density dominates during the minimum phase of the solar cycle and rises as the cycle proceeds. Figure 11.2 shows the variations of daily solar flux densities in the frequency range of 245 MHz (bottom) to 15.4 GHz (top). The modulation of the 11-year sunspot cycle is clearly evident in the radio emission. The marked increase in the daily occurrence of large radio events around the maximum phase of each solar cycle can also be seen. The double-peak structures observed in the number of radio burst events in both 22nd and 23rd cycles are clearly evident, respectively, during 1989-1992 and 2000-2003 and a similar signature is also observed in the occurrence rate of CMEs.13 Another note worthy point in these plots is that during the minimum phase of the solar cycle, the peak-to-peak variation of flux densities at each frequency is much less than the fluctuations observed near the maximum of the cycle. Moreover, these fluctuations tend to increase with the frequency (i.e., as the photosphere is approached; see next section).

11.3.2. Solar Radio Bursts and their Frequencies During a solar flare, the radio emission occurs in bursts, which are brief, energetic and explosive in nature. The intensity of such outbursts can increase up to a million times of the normal intensity of the Sun in just a few seconds and a solar flare can outshine the entire solar disk at radio wavelengths. For example, Figure 11.3 shows the full disk measurements of the Sun in X-ray and radio wavelengths during November 3–11, 2004. These plots explain the intense flare/CME events (compare with Figure 11.2 profiles) occurred in the active regions 695 and 696, shown in Figure 11.1. Radio bursts do not occur simultaneously at different radio frequencies, but instead drift to later arrival times at lower frequencies. A disturbance, traveling out through the progressively more rarefied layers of the solar atmosphere, makes the local electrons in the corona to vibrate at their natural frequency of oscillation, called the ‘plasma frequency’, fp . Since the bursts

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Fig. 11.3. Soft X-ray intensity of the whole Sun observed by the GOES spacecraft (1–8 and 0.5–4 ˚ A) during November 3–11, 2004 (top panel). The flare classification is marked on the right-hand side of the plot. The bottom panel shows the radio flux densities observed at 245 and 410 MHz. The recorded solar activities have been produced by the active regions (ARs) #695 and #696 (see Figure 11.1). The above data have been obtained from http://spidr.ngdc.noaa.gov/spidr/.

drift to later arrival times at longer wavelengths (smaller frequencies), a frequency-time (or wavelength-time) spectrum is useful to study various layers of the solar atmosphere from upper chromosphere to the corona and all the way to the interplanetary medium. The drift of radio emission from low corona to the interplanetary medium is governed by the refractive index, n, of the solar atmosphere as given by,

n=

s

1−



fp f

2

,

(11.1)

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Fig. 11.4. Solar radio spectrum observed in the frequency range of 25–2200 MHz (from Hiraiso Solar Radio Observatory, NICT, Japan, http://hirweb.nict.go.jp).

where f and fp are, respectively, frequency of radio wave and local plasma (or critical) frequency. The value of fp depends on the electron density of the medium, s p Ne e2 fp = = 9 Ne Hz , (11.2) 4π 2 ǫ0 me

where the electron density (Ne ), the electron charge (e) and mass (me ), the permittivity of free space (ǫ0 ) are in mks units. When the radio waves pass from a tenuous to a dense region, for a given refractive index, n, value less than unity (fp
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Figure 11.4 shows the typical solar radio dynamical spectrum, observed in the frequency range 25–2500 MHz. In this ‘frequency-time’ plot, the enhanced level of radio intensity is normally plotted by the grey-scale (or color code). The high-intensity features seen in the plot are radio bursts and they are classified based on: (1) duration, (2) drifting characteristics, and (3) frequency coverage. The below sections include brief description of bursts, which are commonly observed. 11.3.3.1. Type III Radio Bursts The fast drifting radio bursts are called ‘type III’ bursts. They indicate the rapid particle acceleration during a flare process and are generated by electron beams propagating along the open magnetic field lines and exciting Langmuir waves at each height of the corona and in the interplanetary medium (i.e., according to equation 11.2). The radio emission occurs at the fundamental or at the second harmonic of the plasma frequency, fp . Type III bursts typically last for a few minutes during when the hard X-ray emission is produced. A highly intense emission of type III burst suggests the stream of high-speed non-thermal electrons in the corona (i.e., speeds in the range of ∼0.1–0.3 c). In Figure 11.4, type III bursts can be seen as vertical lines in the time interval 20:21 - 20:23 UT. Type III bursts can have velocity gradient25 and explanation of such microstructures requires a complete understanding of the ambient solar wind. Further, long-duration type III bursts associated with fast CMEs show intense particle events.34 11.3.3.2. Type II Radio Bursts Type II bursts are generated by the propagating MHD shock waves. Naturally they have a much lower frequency drift rate, which corresponds to speed in the range of V ≈ 300–2000 kms−1 . The electron beams produced in front of the shock excite Langmuir waves (as in the case of type III), which are converted into escaping radio waves.36 The plasma emission generally appears in the fundamental and second harmonic bands. In Figure 11.4, intense type II bursts, both fundamental and second harmonic emissions, are seen in the time interval 20:25 to 20:31 UT at frequencies below 200 MHz. Type II bursts are grouped according to the frequency range (or wavelength range) covered by them: (i) metric (m), (ii) decametric to hectrometric (DH), and (iii) kilometric (km) bursts. Recent studies based on radio data from the WAVES/Wind experiment6 suggest that CME events associated with metric type II bursts are more energetic (i.e., width and

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speed of the CME are larger than the general population CMEs) and however less energetic than the type II bursts extending to metric to kilometric regions.14 Thus, the energy possessed by the CME determines the drift rate, duration, and frequency range of the type II burst. 11.3.3.3. Stationary and Moving Type IV Radio Bursts This class of radio burst is also known as the flare continuum. As shown in the HiRAS spectrum (Figure 11.4, in the time interval 20:25 – 20:37 UT and frequency range 500-2000 MHz), type IV bursts are long lasting, stationary, broadband emission, and start normally after the onset of type II bursts. The emission is caused by the non-thermal electrons trapped in the post-flare loops. A moving type IV burst is the signature of ascending prominence or plasmoid or closed magnetic cloud ejected during a CME19,30 and it can provide the height-time evolution of the coronal transient. 11.3.3.4. Type I Bursts and other Types of Bursts A type I noise storm may be observed at the active region site for hours to days and it is confined to low-frequency range (i.e., ∼25–500 MHz). The comparison of starting frequency of type III and type I storm, suggests a likely emission at the plasma frequency. However, the directivity properties of type I and type III bursts are quite different. When a type I storm is observed in association with type IV bursts, both of them arise from the trapped electrons within the loops (or moving plasmoid). Other types of radio bursts are U-type and J-type bursts. These names have been given based on their shapes. The U-type bursts are fast drifting bursts; but at a certain frequency, they turnover and display reverse drift rate, i.e., U-shape is seen in the dynamical spectrum. The different starting and turnover heights indicate that this type of burst is caused by electron traveling along the curved field lines.58 The J-shaped bursts are considered to be the modified form of U bursts. It is likely that the field lines of the descending arm have different physical properties than that of the ascending arm, resulting in a weak emission. 11.3.3.5. Coronal Magnetic Field The spectral line measurements from Zeeman and Hanle effects are mostly employed to derive the quantitative information on the photospheric magnetic fields of the cool plasma and prominences. However, the coronal lines

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Fig. 11.5. Direct detection of radio emission from a coronal mass ejection at 164 MHz. Images are 10 sec NRH data and are all linearly scaled between ∼30,000 and 1.5 million K. After the scaling, in order to enhance the loop brightness, images are multiplied by a radial filter at distances >1 R⊙ from the center of the Sun, i.e., a point at 3 R⊙ looks three times brighter than if it were below 1 R⊙ . This procedure increases the noise, but it helps to see the loop. Before this step the images are quite ‘clean’, with a dynamic range ∼200.27

are rather broad to allow Zeeman splitting. In order to obtain the topology of the coronal magnetic field, force-free magnetic extrapolation techniques are applied to the observed photospheric fields (since these techniques strongly depend on the model parameters, the likely error is high for the highly non-potential field conditions). However, a direct estimation of the coronal magnetic field is possible with the radio observations. Since the typical strength of the magnetic field and density of coronal plasma, respectively, are governed by the gyrofrequency and plasma frequency in the corona, following radio measurements are useful to get the estimates of coronal magnetic field: (i) polarization measurements of thermal free-free emission at centimeter wavelengths, and (ii) cyclotron emission measurements at centimeter wavelengths above the sunspot in the coronal regions. Thus, the radio measurements are sensitive to the radial density profile of the corona.9 The interplanetary type III measurements at a range of frequencies by the Ulysses spacecraft could track coronal disturbances as they propagate along the interplanetary magnetic field in the inner heliosphere.45 11.3.4. Solar Radio Imaging Instruments The simultaneous radio imaging and spectroscopy measurements are essential to exploit the radio emitting properties of the Sun. For any detail study, high-angular resolution images and spectrum must be obtained over a wide range of frequencies with best temporal and spectral resolutions.

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In this section, some of the radio telescopes, which are capable of making multi-frequency images of the Sun, are described in brief. The Nobeyama Radioheliograph (NoRH) is an array of radio telescopes dedicated for solar observations. It consists of 84 parabolic antennas of each 80 cm diameter, placed along the east-west arm of 490 m long and in northsouth arm of 220 m wide. Observing frequencies of NoRH are 17 GHz (right and left circular polarizations) and 34 GHz (intensity only), respectively, having resolutions of 10 and 5 arc sec. On routine basis, the full disk image of the Sun is made every second and on the flare mode, 0.1 sec resolution is achievable. The Nancay Radio Heliograph (NRH)20 is a dedicated solar radio heliograph operating in the frequency range of 150–450 MHz. It consists of 44 antennas of size ranging from 2–10 m, spread over two arms (EW and NS) of respective lengths 3200 m and 2440 m. It can observe Stokes I and V images of the Sun for about 7 to 8 hours per day at 10 frequencies in the above frequency range. The resolution of two-dimensional image and the field of view depend on the frequency and position of the Sun in the sky. An example of direct detection of radio emission from a CME event on April 20, 1998, is shown in Figure 11.5. These images, observed with the NRH at 164 MHz, show the expansion of the CME-associated radio lobe. The flare occurred behind the solar limb so that the bright emission from low height is occulted. The extended lobe-type emission outlines the CME observed simultaneously with the LASCO/SOHO coronagraph. The observed radio emission from the CME is a broadband continuum with a rapidly falling steep spectrum, suggesting synchrotron radiation from electrons accelerated at the leading edge of the CME.27 The Gauribidanur Radioheliograph (GRH) operates in the frequency range of 30-150 MHz.43,44 The basic receiving element of this instrument is an array of 192 log-periodic dipoles arranged in a ‘T’ configuration. The current spatial resolution of GRH is ≥5 arc min and data can be recorded at a maximum sampling rate of 256 ms for about 8 hours per day around the local transit of the Sun. Images of the Sun in the above low-frequency range are useful to study the CME events, particularly emission caused by the thermal bremsstrahlung at the frontal loop of the CME at an altitude range of 1.2–2 R⊙ from the center of the Sun. The Giant Metrewave Radio Telescope (GMRT) is one of the sensitive aperture synthesis telescopes, operating in the frequency range 150-1450 MHz.8,50 It consists of 30 fully steerable parabolic dishes (each 45 m diameter), spread over distances of up to 25 km. Fourteen dishes are located

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Fig. 11.6. Images demonstrate the capability of combining radio visibilities from two instruments, i.e., GMRT and NRH. Snapshot images are for 17 sec integration around 09:04 UT. The left image is made with the GMRT data only, the middle one with the NRH only and the right image with GMRT and NRH visibilities combined. For the GMRT+NRH map, the resolution is 49 arc sec and the rms dynamic range is 283.38

randomly in a compact central array in a region of about 1 sq km and the remaining dishes are spread out along the 3 arms of an approximately Y-shaped configuration over a much larger region, with the longest interferometric baseline of about 25 km. The array operates in six frequency bands centered around 50, 153, 233, 325, 610 and 1420 MHz, with dual polarization modes. The highest angular resolution achievable is in the range from ∼60 arc sec at the lowest frequencies to ∼2 arc sec at 1.4 GHz. Observing proposals are routinely invited and depending on the requirement, the GMRT is allotted for observing the Sun at discrete frequencies in the above frequency range. Figure 11.6 illustrates the result of an interesting study of combining visibilities from the GMRT and the NRH to produce composite snapshot images of the Sun at meter wavelength.38 This study yields high dynamic range snapshot images when the solar corona has structures with scales ranging from the image resolution of 49 arc sec to the size of the whole Sun. It emphasizes that snapshot images of a complex object such as the Sun, obtained by combining data from both instruments, are far better than images from either instrument alone, because their uv-coverages are very complementary. 11.4. Flare Energy and Particle Acceleration The important physical processes of a flare include energy build-up and release, particles acceleration, injection, and transportation. The basic understanding of these processes requires details of timings and spectral signatures of thermal and non-thermal particles in hard and soft X-rays, gamma rays, and radio wavelengths. Observations by Yohkoh and TRACE satellites have shown that the magnetic reconnection plays a key role in the

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Fig. 11.7. The synthesized geometry of the relative heights of particle acceleration, coronal and chromospheric hard X-ray sources, based on observations from Yohkoh satellite. The magnetic topology indicates upward and downward electron beams, in radio and hard X-ray. The panel on the right illustrates a dynamic radio spectrum with radio bursts plotted in the ‘frequency-time’. The acceleration site is located in a low-density region (in the cusp) with a density of ne ≈ 109 cm−3 . At this height, electrons are accelerated in the upward direction producing type III radio bursts and in downward direction causing reverse-slope bursts. Downward precipitating electrons produce pulses of chromospheric ‘thick-target’ bremsstrahlung emission, possible intercepting chromospheric upward flows. The loops that are filled with heated chromospheric plasma seen bright in soft X-ray and have density of about 2 orders more than the acceleration site above mentioned. There is a filling delay of soft X-ray loops, during which magnetic reconnection point rises higher, widening the hard X-ray emitting footpoints (figure reproduced with permission3 ).

rapid energy release. The reconnection is an efficient means of converting magnetic energy to thermal and bulk kinetic energies and to accelerate particles. For example, during a flare, unstable stressed magnetic field changes into a more stable configuration by magnetic reconnection and/or by changing the topology of field lines. This change provides free magnetic energy, which can heat and accelerate particles to high energies in several ways, such as reconnection associated electric field and shock wave and particle interaction. The main objective thus is to answer the following questions: (i) when and where the particles are accelerated in the flaring volume, and (ii) how the wave resonance scatters the particles. The important issue is to determine the pre-eruption magnetic field configuration and plasma condition. Recent in-situ measurements have indicated that a large fraction of CMEs have magnetic flux rope structures.29 But, it is important to figure out that

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how they are generated. In order to have a complete picture of the physical origin of solar flare/CME events, it is essential to have quantitative information on the coronal magnetic field strength and topology. Further, the geometry of the post-reconnection field lines is important to understand the transportation of particles to larger coronal heights and into the interplanetary medium. 11.4.1. Flare Models From the historical perspectives, it is known that flares occur near the magnetic neutral points and current sheets are formed by an instability near the neutral point, where field lines break or rejoin, i.e., the reconnection takes place. The reconnection sites of many gradual soft X-ray flares thus can be characterized by a ‘cusp-shaped’ or ‘helmet streamer’ geometry.7,16,22,40,46,47,51 Figure 11.7 illustrates the relative heights of particle acceleration sources and X-ray emitting regions. The hard and soft X-ray images from the Yohkoh satellite have provided strong observational evidences33,54,56 for the cusp-shaped model, which is presently known as the ‘Carmichael-Sturrock-Hirayama-Kopp-Pneuman (CSHKP model)’ reconnection model. These observations have also showed the important feature of ‘loop-top’ hard X-ray source, which is bright and short lived. The time variation of this source matches with those of foot-point sources. These findings clearly show that the energy release takes place in the neutral sheet above the cusp structure at coronal heights in the range 2–4 ×104 km.21 The reconnection process of the arcade of bipolar loops, as in the case of CSHKP model, can be applied to compact as well as to long-duration flare events. In such models, reconnection occurs at a considerable height in an inverted-Y-shaped neutral current sheet and energy propagates downwards in the form of energetic particles. In addition, the system of flare loops formed by reconnection is observed at the flare site. However, in the literature, several magnetic morphologies associated with flare events have been discussed based on different observing techniques. These events represent a wide variety of categories, but not limited to the following cases: (i) reconnection taking place in the low corona within closed loop system; (ii) with or without mass ejections; (iii) activities connected to neighboring magnetic regions or complex magnetic configuration; (iv) filament lying over the neutral line and held by closed field lines; and (v) the reconnection between the filament and overlying fields by gradual eroding to weaken the tethering field lines, etc.4,18 The comparison of models and observations and the resulted mismatches have pushed the researchers to sophisticate existing models and to refine observations.

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11.4.2. Flares and CMEs: Magnetic Reconnection In the case of a flare-associated CME event, the accumulated magnetic energy is released through the partial opening of the field lines, leading to the CME. The opening of field lines cannot be reached from a force-free closed state to the open field configuration, which requires additional energy, since the force-free open field represents a state of the highest energy.48 Further, CME studies have clearly indicated the association of both sheared arcades and twisted magnetic flux ropes during the eruption. Therefore, the increase in coronal magnetic energy can only be achieved by building stress in the magnetic field. It is however not yet clear that the field lines emerge through the photosphere possessing the pre-build-in stress or the stress is caused by the photospheric motion of field-line footpoints. It is likely that both mechanisms contribute to the build-up of magnetic energy. The magnetic stress exceeding a critical threshold as shown by theoretical MHD models can lead to the destabilization of large-scale coronal arcades.39 The partial opening of field lines (and re-closing of field) initiates the CME eruption. In a multi-polar topology, the reconnection between the sheared arcade and surrounding magnetic flux systems can trigger the eruption.2 In this breakout scenario, which requires complex multi-polar magnetic configuration, the low lying sheared core flux near the neutral point breaks open the overlying weak unsheared field lines. It is also emphasized that the opening of field lines and reconnection with overlying fields in a complex three-dimensional configuration can also play a key role in adding the stress (twist or shear).1,10

11.4.3. Radio Observations of CMEs CMEs are often associated with eruptive prominences or disappearing filaments on the solar disk. Their speeds can range from few tens to in excess of 3000 kms−1 . CMEs may be associated with several other manifestations, such as intense X-ray emission often followed by type II and IV metric radio bursts, Moreton waves, intense microwave emission, solar energetic particle events in the interplanetary space, coronal dimming, and post-flare loops and prolonged type I emission at the CME site.18 For example, microwave imaging combined with the meter-wavelength radio observations is useful to investigate the conditions before and during the eruption.49 As discussed in the previous section, the release of energy due to reconnection and/or breaking of the coronal magnetic field lines leads to the formation of the CME and particle acceleration. For example, the radio

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Fig. 11.8. North-west quarter of MDI, EIT, and radio images of the Sun during the pre-eruption of a partial-halo CME on 2001 April 2. The MDI image on the right shows the view of the magnetic configuration of the active region when it was close to the central meridian on March 29, 2001. During the eruption, the radiation at the metric wavelengths was dominated by the emission from the radio source A, which was located radially above the magnetic null point.30

imaging study of the Bastille Day event (July 14, 2000) showed systematic appearance of radio emitting region in space/time indicating the sequence of destabilization or interaction of loop system at heights above the flare site, leading to the opening of coronal field lines.31 The similar behavior has also been observed for a limb CME event on November 6, 1997.26 The radio observations are useful to link particle acceleration and the associated magnetic configuration.55 Furthermore, an investigation by Manoharan et al.32 using the X-ray images from Yohkoh satellite and radio observations from the NRH, shows that magnetic reconnection may not be confined to the area surrounding to the active region or above the chromospheric bright location, but it may involve large-scale loop systems at different heights in the corona and opening of field lines at high latitudes allows particles to escape.32,41 The radio observations, when compared with the Halpha images and extrapolated magnetic field geometry, suggest the release of energy at the magnetic null point.30 The ejection of plasmoids (speeds in the range 100– 200 kms−1 ) seen in the Halpha images, followed by the CME onset in radio further evidently shows the eruption starting at the chromospheric or low coronal level.30 Figure 11.8 shows the complexity of the magnetic configuration and the evolution of radio emitting regions at heights above

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the surface of the Sun. In this case, the evolution of the CME in the inner heliosphere has also been studied using white-light images from the LASCO and radio interplanetary scintillation images made at Ooty Radio Telescope and it suggests that the magnetic energy is utilized in the propagation of CME up to a distance of ∼100 R⊙ .28 11.5. Summary It is emphasized that the coordinated investigation using the radio measurements in association with other energetic solar phenomena such as solar X-rays, gamma rays, and energetic particle acceleration are gaining importance in the present day solar physics research. The essential requirements are to understand the spatial and temporal evolution of the source of acceleration of particles, to determine the energy associated with electrons and ions in an explosive solar event, and to trace their transportation into the interplanetary medium. Acknowledgments This work utilizes images obtained from EIT and MDI systems on-board SOHO spacecraft. SOHO is a project of international cooperation between ESA and NASA. Author acknowledges Hiraiso Solar Radio Observatory for the radio spectrum and Meudon group for the Halpha image. Author thanks M. Aschwanden, D. Maia, and P. Subramanian for providing figures. The author is pleased to thank the observing/engineering team and research students of Radio Astronomy Centre for the help in making observation and data reduction. Author thanks G. Agalya for the proof-reading of the manuscript. This work is partially supported by the CAWSES–India Program, which is sponsored by the Indian Space Research Organisation (ISRO). References 1. 2. 3. 4. 5.

T. Amari, J.F. Luciani, Z. Mikic and J.A. Linker, ApJ, 518, L-60 (1999). S.K. Antiochos, C.R. DeVore and J.A. Klimchuk, ApJ, 510, 485 (1999). M.J. Aschwanden and A.O. Benz, ApJ, 480, 825 (1997). M.J. Aschwanden, ApJ, 580, L79 (2002). T.S. Bastian, A.O. Benz and D.E. Gary, Annu. Rev. Astron. Astrophys., 36, 131 (1998). 6. J.L. Bougeret et al., Space. Sci. Rev., 71, 231 (1995). 7. H. Carmicheal, AAS-NASA Symp. on the Physics of Solar Flares, Greenbelt, MD, W.N. Hess (Ed), 451-456, NASA SP-50. (1964).

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8. J.N. Chengalur, Y. Gupta and K.S. Dwarakanath (eds.), Low Frequency Radio Astronomy (2004). 9. P. Demoulin and K.-L. Klein, Lecture Notes in Physics, 553, 99 (2000). 10. C.R. DeVore and S. Antiochos, ApJ, 539, 954 (2000). 11. G.D. Fleishman and S.W. Kahler, ApJ, 394, 688 (1992). 12. M.V. Goldman and D.F. Smith, Physics of the Sun, ed: P.A. Sturrock, D. Reidel Publishing Company, Dordrecht, 2, 325 (1986). 13. N. Gopalswamy, S. Nunes, S. Yashiro and R.A. Howard, Adv. Space. Res., 34, 391 (2004). 14. N. Gopalswamy, E. Aguilar-Rodrigues, S. Yashiro, S. Nunes, M.L. Kaiser and R.A. Howard, J. Geophys. Res., 110, 12507 (2005). 15. J.S. Hey, Nature, 157, 47 (1946). 16. T. Hirayama, Sol. Phys., 34, 323 (1974). 17. H. Hudson and J. Ryan, Annu. Rev. Astron. Astrophys., 33, 239 (1995). 18. H. Hudson, Solar Physics with Radio Observations, Proc. of the Nobeyama Symp., Kiyosato, Japan, eds: T. Bastian et al., 159 (1999). 19. B. Joshi, P.K. Manoharan, A.M. Veronig, P. Pant and K. Pandey, Sol. Phys., 242, 143 (2007). 20. A. Kerdraon and J.M. Delouis, In: Coronal Physics from Radio and Space Observations, Lect. Notes Phys., G. Trottet (Ed), Berlin: Springer-Verlag, 192 (1997). 21. K.L. Klein, In: Advances in Solar Physics, G. Belvedere, M. Rodono, G.M. Simnett (Eds), Lect. Notes in Physics, 432, 261 (1994). 22. R.A. Kopp and G.W. Pneuman, Sol. Phys., 50, 85 (1976). 23. A. Kruger, Introduction to Solar Radio Astronomy and Radio Physics, D. Reidel Publishing Company, Dordrecht (1979). 24. M.R. Kundu, Solar Radio Astronomy, Wiley - Interscience, New York (1965). 25. R.P. Lin, W.K. Levedahl, W. Lotko, D.A. Gurnett and F.L. Scarf, ApJ, 308, 954 (1986). 26. D. Maia, A. Vourlidas, M. Pick, R. Howard, R. Schwenn and A. Magalhaes, J. Geophys. Res., 104, 1254 (1999). 27. D. Maia, M. Pick, A. Vourlidas and R. Howard, ApJ, 528, L49 (2000). 28. P.K. Manoharan, Sol. Phys., 235, 345 (2006). 29. P.K. Manoharan, N. Gopalswamy, S. Yashiro, A. Lara, G. Michalek and R.A. Howard, J. Geophys. Res., 109, 6109 (2004). 30. P.K. Manoharan and M.R. Kundu, ApJ, 592, 597 (2003). 31. P.K. Manoharan, M. Tokumaru, M. Pick, P. Subramanian, F.M. Ipavich, K. Schenk, M.L. Kaiser, R.P. Lepping and A. Vourlidas, ApJ, 559, 1180 (2001). 32. P.K. Manoharan, L. van Driel-Gesztelyi, M. Pick and P. Demoulin, ApJ, 468, L73 (1996). 33. S. Masuda, T. Kosugi, H. Hara and S. Tsuneta, Nature, 371, 495 (1994). 34. R.J. MacDowall, A. Lara, P.K. Manoharan, N.V. Nitta, A.M. Rosas and J.L. Bougeret, GRL, 30, 8018 (2003). 35. D.J. McLean and N.R. Labrum, Solar Radiophysics, Cambridge University Press, Cambridge (1985).

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36. D.B. Melrose, Plasma Astrophysics, Gordon and Breach, New York (1980). 37. D.B. Melrose and G.A. Dulk, ApJ, 259, 844 (1982). 38. C. Mercier, P. Subramanian, A. Kerdraon, M. Pick, S. Ananthakrishnan and P. Janardhan, Astronomy and Astrophysics, 447, 1189 (2006). 39. Z. Mikic and J.A. Linker, In: Coronal Mass Ejection, Geophysical Monograph, N. Crooker, N.A. Joselyn, J. Feynman (Eds), 99, 57 (1997). 40. E.N. Parker, Phys. Rev., 999, 177 (1963). 41. M. Pick et al., GRL, 22, 3377 (1995). 42. E.R. Priest and T. Forbes, Magnetic Reconnection - MHD Theory and Applications, Cambridge University Press, Cambridge (2000). 43. R. Ramesh, K.R. Subramanian and Ch.V. Sastry, Astron. Astrophys. Suppl., 139, 179 (1999). 44. R. Ramesh, M.S. Sundararajan and Ch.V. Sastry, Experimental Astronomy., 21, 31 (2006). 45. M.J. Reiner, J. Fainberg and R.G. Stone, Science, 270, 461 (1995). 46. P.A. Sturrock, Nature, 211, 695 (1966). 47. P.A. Sturrock and E.T. Woodbury, In: Plasma Astrophysics, edited by P.A. Sturrock, Academic Press (1969). 48. P.A. Sturrock, ApJ, 380, 655 (1991). 49. P. Subramanian, S. Ananthakrishnan, P. Janardhan, M.R. Kundu, S.M. White and V.I. Garaimov, Sol. Phys., 218, 247 (2003). 50. G. Swarup, S. Ananthakrishnan, V.K. Kapahi, A.P. Rao, C.R. Subrahmanya and V.K. Kulkarni, Current Science, 60, 95 (1991). 51. P.A. Sweet, IAU Symp., 6, 123 (1958). 52. T. Tajima and K. Shibata, Plasma Astrophysics, Perseus Publishing, Cambridge, Massachusetts (2002). 53. T. Tajima, A.O. Benz, M. Thacker and J.N. Leboeuf, ApJ, 353, 667 (1990). 54. T. Takakura, M. Inda, K. Makishima, T. Kosugi, T. Sakao et al., Publ. Astron. Soc. Jpn., 45, 737 (1993). 55. G. Trottet et al., Astronomy and Astrophysics, 334, 1099 (1998). 56. S. Tsuneta, H. Hara, T. Shimizu, L.W. Acton, K.T. Strong et al., Publ. Astron. Soc. Jpn., 44, L63 (1992). 57. S. Tsuneta, ApJ, 483, 507 (1997). 58. J.P. Wild, Proc. Astron. Soc. Australia, 1, 365 (1970).

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P.K. MANOHARAN Radio Astronomy Centre National Centre for Radio Astrophysics Tata Institute of Fundamental Research P.O. Box 8, Udhagamandalam (Ooty) 643 001, India E-mail: [email protected]

12.1. Introduction The solar atmosphere consists of three main regions, the photosphere, the chromosphere and the corona. The photosphere is a thin layer of low-density gas and it allows visible photons to escape into space. The current of gas rising from the beneath the surface of the Sun causes the formation of granulations at the photosphere. In addition, the magnetic fields threading outward from the solar interior give rise to active regions, sunspots, coronal holes, and various structures at the solar atmosphere. These largescale magnetic structures can store huge amounts of energy, which may be released gradually or explosively. The chromosphere lies immediately above the photosphere. It glows faintly relative to the photosphere. The reddish in color chromosphere can seen only during total solar eclipses (the red color indicates the strong emission of Balmer Hα spectrum). The thickness of the chromosphere is ∼3000–5000 km. In this region, the temperature is about 5000 K at the photosphere side, but it rises to more than 5×105 K at the top, where the chromosphere merges with the corona. The rising (and sinking) of hot gas from the solar interior along the edges of huge convection cells causes jets of gas, spicules, from the chromosphere to the corona. The corona, the outermost layer of the solar atmosphere above the chromosphere, extends many solar radii from the Sun. As in the case of 235

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Fig. 12.1. The image of the solar corona observed in soft X-ray by the Yohkoh spacecraft on May 8, 1992. The coronal hole appears dark due to the less emission. The coronal holes are low density regions having open magnetic field configuration. They are sources of high-speed solar wind (refer to Section 12.6). The bright features are active regions with closed magnetic configuration (image courtesy http://solar.physics.montana.edu/).

chromosphere, the corona is visible from the ground only during a total solar eclipse (or an artificial eclipse created by a coronagraph). It is composed of a very low-density gas, but extremely hot, 3×106 K, which is inferred from the emission lines of highly ionized atoms (e.g., iron atoms in the +16 charge state). The high temperature of the corona is thought to be associated with the continuous transport of energy by the magnetic field from the lower regions of the Sun to the corona. However, the details of coronal heating are yet to be fully resolved. Figure 12.1 shows the corona observed in soft X-ray by the Yohkoh spacecraft on May 8, 1992. Because of the high temperature, the corona shines bright in X-ray. The coronal hole appears dark due to the less emission and in the above figure, it extends from the north pole to the nearequatorial region of the Sun. The coronal holes are low density regions having open magnetic field configuration and are sources of high-speed solar wind (refer to Section 12.6). The intense brightness features seen in the image are active regions of closed magnetic loop structures, where the emission is strong. The flow of plasma along these field lines provides the complex appearance to the corona and coronal structures, like prominences, helmet streamers, and loops stretch several thousands of kilometers into the extended corona. The strong interactions of multi-polar magnetic field lines cause violent explosive flare and coronal mass ejection (CME) events, which hurtle plasma, magnetic field, energetic particles, and radiation outward into interplanetary space. The corona and its magnetic field lines extend out into the entire heliosphere as part of the solar wind.

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Fig. 12.2. Hourly averages of solar wind speed (V ), density (N ), temperature (T ), and magnetic field intensity (|B|), measured in situ near the Earth’s orbit for three Carrington Rotations, CR#1936 (May 11 to June 7, 1998), CR#1964 (June 13 to July 10, 2000), and CR#2043 (May 8 to June 4, 2006). These rotations cover, respectively, minimum, ascending, and maximum phases of solar cycle #23 (also refer to Figure 12.7). The vertical lines indicate the time of interplanetary shocks observed at 1 AU. These strong shocks are associated with intense flare/CME events. The plotted data sets have been obtained from http://cdaweb.gsfc.nasa.gov/.

12.2. Solar Wind The solar wind originates at the base of the hot corona and is accelerated to supersonic speeds near the Sun. It blows continuously and inflates the

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heliosphere. It is composed of approximately equal numbers of ions and electrons. The ion component consists predominantly of protons (95%), with a small amount of doubly ionized helium and trace amounts of heavier ions. The solar wind varies in density, speed, temperature, and strength and orientation of the magnetic field embedded within the flow. The properties of the solar wind also change in phase with the 11-year solar cycle. Figure 12.2 displays the solar wind speed (V ), density (N ), temperature (T ), and magnetic field (B), observed at 1 AU for three different Carrington Rotations, CR#1936 (May 11 to June 7, 1998), CR#1964 (June 13 to July 10, 2000), and CR#2043 (May 8 to June 4, 2006), respectively, at the ascending phase, at the maximum, and at the minimum of the solar cycle #23. (1 Astronomical Unit (AU)≈1.5×108 km ≈ 215 solar radii, R
). Each plot covers a 27-day period (i.e., one solar rotation as viewed from the Earth). The low-speed wind tends to be cool and dense while the high-speed wind is hotter and more tenuous. These plots include interplanetary signatures of CMEs and their associated shocks as observed at the near-Earth environment (vertical lines drawn in CRs 1936 and 1964 indicate the arrival of interplanetary shocks). The observed differences in the flow properties are the consequences of different types of origin, acceleration mechanism, and evolution of the solar wind in the inner heliosphere. 12.2.1. Coronal Heating The formation of the solar wind is closely linked to the hot corona. Recent in situ measurements at the Earth’s orbit and remote observations at the UV wavelengths by SOHO spacecraft have provided clues for the heating of ion by the ion-cyclotron waves.25 The generation of wave energy may be continuous or episodic. Therefore, the estimation of intensity spectrum of magnetic turbulence in the corona and its evolution with time and solar offset is required to ascertain the nature of wave sources and to clarify whether the ions are heated and accelerated by cyclotron waves or by an entirely different mechanism, such as plasma jets caused by nanoflares. 12.2.2. Thermally-Driven Solar Wind In the open field corona, the energy is carried by the solar wind in the form of kinetic energy, which is the work done against the gravity of the Sun. However, the generation of the solar wind with a given density, temperature, speed, and embedded magnetic field in the quasi-steady corona as well as during the episodic explosive flare and CME events is still an open

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question. Several theories have been developed to explain the mechanism of the acceleration of the solar wind and readers are directed to review articles and references in them.13,20,24 Parker50 prediction is that the open corona cannot exist in hydrostatic equilibrium with respect to the local interstellar medium and can only be balanced by the continuous outward expansion of the corona into space. Since the rarefied corona is a good thermal conductor,51 its temperature can be maintained nearly at the same level even at sufficiently large distance from the Sun (i.e., the temperature gradient is very small). However, the decreasing density with increasing heliocentric distance produced by the gravity of the Sun causes a steep gradient in the gas pressure. As a result, the hydrostatic equilibrium can not be maintained in the corona and the pressure gradient aids to push the coronal material outward and solar wind flows radially outward in all directions and fills the whole interplanetary space. In a steady state, the equation of motion and the equation of continuity for the spherically symmetric flow of solar wind are given by, N mH V

d N mH GM
dV + (2N kT ) + =0, dr dr r2 N V r2 = N0 V0 r02 ,

(12.1) (12.2)

where N is the number of proton-electron pairs per unit volume, mH is the mass of the proton (electron mass is neglected), V is velocity, r is distance from the center of the Sun, k the Boltzmann constant, T is temperature, M
mass of the Sun, and G the gravitational constant. The factor 2 in the pressure gradient is to account for the pressure exerted by electrons and protons. In addition, one has an energy equation and the energy per unit mass of the coronal plasma is the sum of the thermal, kinetic and gravitational energies, E=

1 GM
3kT . + V2 − mH 2 r

(12.3)

Parker51 also investigated a generalized function for the variation of temperature with distance from the Sun, T (r) = T0 r−b .

(12.4)

At the base of the corona, the sum of energies, E, is negative and the system remains stable. With increasing distance, the gravitational potential

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decreases as 1/r and the thermal energy, governed by the temperature gradient, T (r), declines rather gradually with distance for power-law indices, b < 1. For a realistic value of b ≈ 0.3, the energy, E, becomes positive at distances beyond r ≈ 10R
, and solar wind flows with the supersonic speed. One can show that the gravitational field of the Sun acts as a nozzle (like in a rocket engine) for the flow. It may be noted that equation (12.3) has a critical point for a solar wind speed close to the sound speed, Vs , at GR2

r = V 2
and the acceptable solutions of the above equations provide an s approximate view of the acceleration of the solar wind.

12.2.3. MHD Waves Driven Solar Wind The thermal conduction alone cannot however adequately account for the flow speeds observed in the high-speed solar wind and other non-thermal processes must play a role in the coronal expansion and to accelerate the solar wind. That is, the additional supply of energy to generate the highspeed wind (as well as to explain the speed at 1 AU or beyond) could come from the heating, or by the work done on the plasma, or both. The spectral broadening measurements and radio sounding experiments at the base of the corona indicate the substantial increase of turbulence (or energy), which is attributed to the presence of Alfv´en waves.46 Therefore, the ‘waveheating’ component in a model should address the dissipation/damping of Alfv´en waves. Models considering the energy generation solely by the conduction of heat by electrons (i.e., one-fluid model) resulted in very cold proton near 1 AU. Thus, the electrons do not carry sufficient energy to fully support the coronal expansion and acceleration is likely to be associated with the ion thermal pressure gradient and with outward pressure gradient of Alfv´en waves.32,49,57,66,31 The next natural question is how and at what height the energy is added to the plasma. The addition of energy at height above the critical radius GR2

(r = V 2
), where the speed of the plasma equals the sound speed (Vs ), s provides the favorable location to produce faster flows.30 The experimental and modeling methods are to answer the following fundamental questions about the coronal heating and solar wind acceleration: (1) solar wind measurements of ion and electron distribution functions, (2) waves present in the near-Sun region, (3) model to produce/explain reasonable speed of the wind at 1 AU for a given set of reasonable parameters at the base of the corona, and (4) model-independent measurements at the base of the corona and in the inner heliosphere at all latitudes.

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Fig. 12.3. The ‘frozen-in’ solar magnetic field dragged by the solar wind plasma forms a spiral pattern in the interplanetary medium in the ecliptic plane. The above plot has been obtained for a constant solar wind speed of ∼425 kms−1 . The circle indicates the location of the orbit of the Earth.

12.3. Interplanetary Magnetic Field The solar wind plasma carries embedded (i.e., frozen-in) solar magnetic field lines and forms the interplanetary magnetic field (IMF). The frozenin IMF takes the shape of the Archimedian spiral pattern, because of the radial out flow of the solar wind, the rotation of the Sun, and the azimuthal symmetry of the solar wind with respect to the rotation axis of the Sun. At the orbit of the Earth, a solar wind speed of ∼400 kms−1 gives a spiral angle of 45◦ with respect to the radial flow direction. The radial component of the field varies as Br ∼ r−2 and at large distance, it merges with the azimuthal component, which falls off linearly with distance, Bθ ∼ 1/r.21,52 The IMF effectively shapes the three-dimensional view of the interplanetary medium. During solar minimum, the solar magnetic field becomes very regular, and covers large unipolar field regions of opposite polarities at poles. The field at one pole points outward and inward in the other. These unipolar regions are separated by a wavy band of closed field lines near the equatorial plane across which the average field change the polarity. The active regions and their associated closed-field configuration located along the equatorial belt of the Sun define the structure of this wavy interplanetary field

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surface.19 However, as the solar cycle progresses to the maximum phase, the solar magnetic field goes through a rigorous change effected by the field reversal and closed magnetic configurations are formed at higher latitudes. The solar field becomes complex and the frequency of explosive flares and CMEs increases. 12.3.1. Magnetic Storms Since the radius of curvature of IMF spiral is different for the fast and slow solar wind flows, the fast wind catches up and interacts with the slow wind.43 Such interaction results in solar wind compression region, where pressure, density, temperature, and magnetic field are enhanced. It can also produce moderate to strong interplanetary shock. The magnetized solar wind plasma also interacts with planets and other solar system bodies and influences the evolution of planets and their atmospheres. The merging or interaction of IMF with the magnetosphere of a planet is complex and it disturbs and energizes the magnetospheric plasma, leading to the release of magnetic energy, which causes magnetospheric storms and substorms. For example, the interaction between the solar wind and Earth’s magnetosphere injects energetic charged particles into the Earth’s atmosphere. The excitation of atoms and molecules by the collision of charge particles results in emission at various wavelengths, which include the spectacular aurora observed in the visible range. However, adverse space weather effects, such as the disruption of communication, navigation, and power grid, occur at times of severe geo-magnetic storms, which are triggered by extreme solar wind conditions associated with explosive solar events (e.g., Earth-directed CMEs). 12.4. Solar Wind Measurements 12.4.1. Spacecraft Data The coronal observations, to date, have been made: (1) remotely, through imaging, radio sounding, spectroscopy, and (2) directly by in situ sampling of the downstream plasma. Since 1960’s a large number of spacecraft have successfully conducted routine in situ sampling of the solar wind (e.g., speed, density, temperature, composition, and magnetic field measurements) at near-Earth space and beyond. However, in situ measurements are limited to the one-dimensional scan during the fortuitous encounter with spacecraft at the near-Earth space. The spacecraft observations have been made only down to a heliocentric distance of 0.3 AU (≈ 64R
), which is the

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perihelion distance of the Helios satellite.55 Moreover, spacecraft sampling is confined to the ecliptic plane and Ulysses is the first and only space mission to probe the high latitude heliosphere (http://swoops.lanl.gov/).53 ,67 12.4.2. Remote-Sensing Techniques The solar wind measurements in three-dimensional space at consecutive heliocentric distances are however essential to improve our knowledge on properties of the solar wind and the dynamics of turbulence generated by the interaction of the solar wind (e.g., interactions of high- and low-speed flows and ambient wind with propagating transients). Moreover, when a CME releases energy through a violent disruption of large-scale unstable magnetic configuration, it adds a considerable amount of mass and field to the large portion of the coronal volume. Although such transient disturbances pushed to the interplanetary medium are about a solar radius in size at the near-Sun region, they evolve to a significant structure of an AU at a distance of ∼1 AU from the Sun. They also involve: (1) traveling interplanetary shock waves, (2) enhanced energy flux in the solar wind, and (3) large-scale expanding intense magnetic flux ropes embedded in the solar wind. It is therefore important to consider the unique techniques and observations other than in situ that would be beneficial and provide great deals of information on the three-dimensional solar wind. It should be noted that various remote sensing measurements of the solar corona, when combined together, have provided a potentially powerful tool to understand the constantly varying corona and solar wind. In particular, the radio sounding techniques, e.g., phase scintillation, intensity scintillation, Doppler scintillation, and Faraday rotation, which measure fluctuations of radio signals after the passage through the solar corona, have provided the clearest picture to date on the characteristics of solar wind plasma at closer solar offsets and in the entire inner heliosphere.3,28,44,64,68 The technique to measure the intensity scintillation of natural radio source, as known as interplanetary scintillation (IPS) technique, is one among them.17 Several results obtained from the IPS measurements are discussed in this chapter and a brief description of the IPS technique is given in the next section. 12.5. Interplanetary Scintillation The interplanetary scintillation (IPS) technique exploits the scattering of radiation from distant point-like radio sources (e.g., quasars or radio

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galaxies of angular diameter ≤ 0.4 arcsec) by the electron-density irregularities in the solar wind. The plane wavefront, passing through the solar wind, gets phase modulated by the refractive index variations caused by the density fluctuations in the solar wind.58 The resultant diffraction pattern caused by the scattered radio waves, which drifts past the observer with the velocity of the solar wind, produces temporal intensity fluctuations on the ground. The spectrum of intensity scintillation, P (f ), contains details of solar wind speed, shape of density turbulence spectrum, and structure of the radio source (refer to equation 12.6). 12.5.1. Radial Evolution of Density Turbulence The intensity scintillation, produced by the density turbulence in the solar wind, is characterized by the scintillation index, m=

rms of intensity fluctuations , mean source intensity

(12.5)

which increases with decreasing distance (R) between the Sun and the lineof-sight to the radio source, until it reaches a maximum at R ≈ 40 R
(refer to Figure 12.4). In the weak-scintillation region (i.e., at R ≥ 40 R
), the intensity scintillation is linearly related to the electron-density fluctuations.36 The decline in scintillation at distances away from the transition point is caused by the fall of density (as well as density turbulence) approximately as a function of inverse square of heliocentric distance. However, in the strong-scintillation region (R < 40 R
), the relationship between the scintillation and the density fluctuations is not straight forward and as the radio source approaches the Sun, the scintillation index decreases steeply, which is due to the smearing caused by the angular size of the radio source as well as by the complex nature of the turbulence spectrum.41 Figure 12.4 demonstrates the radial dependence of scintillation index observed at 327 MHz for two radio quasars 1148-001 and 0138+136. At a given heliocentric distance, the largest scintillation is observed for an ideal point source, compared with other structured radio sources (i.e., the source size attenuates the scintillation). Thus, the shape of the scintillation index curve and its peak value are directly linked to the density turbulence and the size of the compact component of the radio source.36 In the case of a point source (e.g., 1148-001), the scintillation index peaks at m ∼1 and other broader sources attain peaks at m<1. The above plots have been made using measurements taken over 1986–2004 and include any systematic variations in the density turbulence caused by the solar cycles changes.

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Fig. 12.4. Scintillation index as a function of heliocentric distance observed using the Ooty Radio Telescope at 327 MHz for two radio quasars 1148-001 and 0138+136. The quasar 1148-001 has an equivalent angular diameter of ∼15 milli arcsec and it represents a nearly point source for which the peak of scintillation rises up to about unity. The other radio source 0138+136 has an angular diameter of ∼60 milli arcsec. The plot of 1148-001 has been made using measurements taken during the period 1986–2004 and it therefore includes the solar cycle changes of the density turbulence. The average expected scintillation index, for each source, is shown by the best-fit curve (drawn by solid line).

The average expected scintillation index for each source is shown by the best-fit curve. At a given heliocentric distance, a large deviation above (or below) the mean curve seen in the scintillation indicates the excess level of turbulence due to transients (or depletion of turbulence) in the solar wind. The normalized scintillation index, g, can eliminate the effects of source size and the radial dependence imposed on the scintillation (see subsection 12.5.3). 12.5.2. Temporal Spectrum of Scintillation In the weak-scintillation regime, the Born approximation is applicable. Thus the extended medium between the radio source and the observer can be considered to consist of thin layers perpendicular to the line of sight, the observed intensity fluctuations being the sum of contributions from all layers. The contribution from each layer is weighted by the local level of turbu2 (R), which decreases rapidly with distance from the Sun,36,39 lence, CN e 2 CNe (R) ∼ R−4 . Such a steep gradient means that most of the scattering power occurs at the point of closest approach of the line of sight to the Sun, and IPS measurements are heavily weighted to the solar wind in the region of closest solar approach. Since the drift rate of solar wind density inhomogeneities across the line of sight to the source causes the scintillation, the speed of the solar wind, Vp (z), and the wavenumber, κx , determine the temporal frequency,

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f = Vp (z)κx /2π, where Vp (z) is the projected velocity of the solar wind along the x-axis of a scattering layer at a distance z from the ground. In other words, the width of the IPS spectrum is linked to the bulk velocity of density structures in the solar wind. The observed temporal spectrum of scintillation is due to the contribution from each layer of the solar wind along the line of sight to the radio source and the integration is to be done over the x–y plane as well as along the propagation path, z. For a given observing wavelength, λ, the standard model temporal frequency spectrum can be given as,
∞
∞ dz 2 2 dκy CN ΦNe (κx , κy , z) P (f ) = (2πre λ) e |Vp (z)| −∞ 0 ×Fdiff (κx , κy , z)Fsource (κx , κy , z) ,

(12.6)

where re is the classical electron radius and λ the wavelength of observation. Fdiff (κx , κy , z) = 4 sin2 (κ2 zλ/4π) is the Fresnel propagation filter, which attenuates wavenumbers smaller than κf ≈ (2π/λz)1/2 and it does not alter the shape of temporal spectrum at large wavenumbers.39 The term Fsource is the squared modulus of the radio source visibility function, given by exp(−κ2 z 2 θ02 ) for a symmetrical Gaussian brightness distribution of half maximum diameter of the radio source, Θs = 2.35θ0 and it cuts off the spectrum at wavenumbers above κs = 1/(zθ0 ). The spatial spectrum of density fluctuations is of power-law nature and contains inner-scale term, ΦNe (κ) = κ−α exp(−κ2 /κ2i ), where Si ≈ 3/κi is the inner-scale or cut-off scale of the turbulence (refer to subsection 12.5.4 and Figure 12.6).39,42 12.5.3. Normalized Scintillation and Solar Wind Speed The normalized scintillation index, g, of a given source is obtained by normalizing the measured level of scintillation using the long-term average of ‘index–distance’ curve of the same source (in Figure 12.4, the long-term average curves are shown by best-fit solid lines). Such a normalization removes the systematic variation of index with solar distance and source size effect and enables easy comparison of turbulence level measured by a large number of IPS sources. Thus, in the case of a propagating transient (i.e., a CME or an interaction region), IPS measurements on a grid of several sources can easily detect the volume of excess of turbulence (i.e., enhancement of the normalized index, g), produced by the transient in the solar wind.22,40,42,59 For each IPS source, the temporal power spectrum, P (f ), can also be suitably calibrated (i.e., equation 12.6) to estimate the speed of the solar wind and shape of the density spectrum.35,39,60

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12.5.3.1. Scintillation Measurements on Grid of Radio Sources Additionally, the importance of the IPS technique increases when the dayto-day monitoring of the heliosphere is made on a grid of large number of radio sources, whose lines of sight cut across different parts of the heliosphere. At Ooty, the daily monitoring of the scintillation on a large number of radio sources (i.e., about 800 – 1000 radio sources) is made.37 Although IPS measurements are integrated along the path of the radio signal, the image processing of the normalized scintillation indices (g-values) and the estimated speeds for the grid of sources provides the three-dimensional view of the ambient solar wind as well as the turbulent regions associated with the propagating disturbances in the IPS field of view. A tomographic analysis of the above measurements will reveal the sharp boundary between the transient and background solar wind.27,28,29 12.5.3.2. Single and Multi-antenna IPS Measurements A regular monitoring of IPS on a given radio source using a single-antenna system of good sensitivity such as the Ooty Radio Telescope, at 327 MHz (λ = 0.92 m), can provide the speed and density turbulence of the solar wind at heliocentric distances R ≈ 20–225 R
in the three-dimensional heliosphere.35,41,60 It is to be noted that the transition of the weak to strong scattering in the solar wind (i.e., peak of the scintillation index curve refer to Figure 12.4) moves inward or outward from the Sun, respectively, for short and long observing wavelengths. Particularly high-frequency measurements, such as IPS at 933 MHz by EISCAT antenna, provide the characteristics of the solar wind in the near-Sun regions.12 In the case of a multi-antenna system, operated at the Solar-Terrestrial Environment Laboratory (STEL), Nagoya University, Japan, the speed of the solar wind is directly estimated by cross correlating IPS signals from pair of antennas.6,26,54 At the STEL, a small antenna is being replaced with a large one of better sensitivity (M. Kojima, private communication). The longitude difference between STEL, Japan (138◦ E) and Ooty, India (76◦ E) provides an opportunity to monitor an interplanetary disturbance at an earlier epoch at the STEL (before the rise time of the monitoring source at Ooty) and at also a later time over the Ooty sky. 12.5.4. Density Turbulence Spectrum 2 (R)∼([δNe (R)]2 ) varies apThe density turbulence in the solar wind (CN e −4.4±0.2 and this radial profile in the ecliptic plane shows proximately as R

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Fig. 12.5. Three-dimensional variations of density turbulence in the solar wind. These plots give contours of constant density turbulence (δNe (R)) during (a) solar minimum and (b) solar maximum of activities. At a given distance, during solar activity minimum, the δNe value decreases from equatorial region to the pole by factor of ∼2.5, whereas the δNe distribution is nearly spherically symmetric near the maximum of activity.36

no significant change with the solar activity cycle (refer to Figure 12.4).36 However, during the minimum phase of the solar cycle for a given heliocentric distance, the increase in heliographic latitude causes a decrease in density turbulence (refer to Figure 12.7). The turbulence measurements close to the Sun, < 10R
, by Coles et al.7 have shown similar changes with heliographic latitude. Figure 12.5 shows the contours of constant δNe (R) around the Sun during solar minimum and maximum periods.36 The peak of the ‘scintillation–distance’ curve for a given radio source corresponds to a fixed level of δNe (R) and the above plots have been made using the measurements (similar to that showed in Figure 12.4) on several radio sources near the minimum and maximum of the solar activities. They illustrate that when the solar activity is minimum, a given value of the δNe (R) appears closer to the Sun at poles than at the equatorial region and the contour of the constant δNe (R) appears like an ellipse around the Sun. During the maximum of solar activity, however, at a given distance from the Sun, on an average a uniform distribution of δNe (R) is seen at all heliographic latitudes and its constant value contour appears like a circle. During the solar cycle minimum, at ∼ 40R
from the Sun, the value of δNe (R) is about 2.5 times larger in the equatorial region than that in the polar region. The coronagraph observations have also suggested a similar trend that at a given

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Fig. 12.6. Shape of the density turbulence spectrum derived from the IPS and spacecraft measurements for the low-speed and high-speed flows (left panel from Manoharan et al.39 ) and for plasma dominated by the CME-generated disturbance (right panel from Manoharan et al.42 ). The spectral breaks are indicated, respectively, by tick marks (left) and arrow symbols (right).

heliocentric distance, the average density in the solar corona is smaller in the polar region than in the equatorial region.7 The spectral characteristics of density turbulence in the solar wind differ significantly between low- and high-speed flows. In general, the shape of the spectrum approaches that of fully developed turbulence, that is, a Kolmogorov spectrum, ΦNe ∼ κ−11/3 , where κ is the spatial wavenumber (see subsection 12.5.2). Spacecraft and IPS data, in the spatial scale range of 1/κ ≈ 10–106 km, have revealed that the prevailing spectral shape of turbulence differs in three distinct scale-size ranges. In the case of highspeed plasma, as shown in Figure 12.6(a), nearly a Kolmogorov spectrum of α ≈ 11/3 (where α is the power law index) is observed at spatial scale > 105 km, followed by a flattening, α ≈ 3, in the range of 103 –105 km, and a steeper spectrum, α ≥ 3.8, at scales less than 1000 km. The steepening at small-spatial scale of the spectrum occurs on scales of the order of ion (or proton) inertial length, rp = Va /ωp , where Va is the Alfv´en speed and ωp is the proton cyclotron frequency. The spectral flattening at mid scales suggests an enhancement associated with one of the possible plasma instabilities, (e.g., ion cyclotron or magnetoacoustic) at high wavenumbers.8,39,69 The steepening, at scales smaller than 1000 km, may be attributed to an increase in the Alfv´en speed and the low-level of the density in the solar wind from the coronal hole resulting in large inertial- or inner-scale −1/2 . In the case of the size. That is, the size of the inner scale goes as Np

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low-speed solar wind, the spectral characteristics differ at the mid-range and small scales, respectively, having slopes of α ≈ 3.5 and 2.8. Figure 12.6(b) shows the plots of density turbulence spectra for CME disturbed solar wind and ambient flow. The spectrum dominated by the disturbance plasma significantly differs from that of the ambient plasma. In the spatial wavenumber range 0.015–0.1 km−1 (i.e., 65–10 km), the spectral density of the disturbance plasma follows a power law of κ−2.8 and a break is observed at a spatial wavenumber of ∼0.1 km−1 (i.e., cutoff scale size ∼10 km), which indicates an inner scale smaller than ∼10 km.42 In the case of ambient solar wind, however, the spectrum is steeper, κ−3.3 , in the smaller wavenumber portion of the spectrum (κ<0.03 km−1 ) and has a larger inner scale cutoff at ∼100 km.

Fig. 12.7. Two-dimensional synoptic images of density turbulence (left) and speed (right) of the solar wind plasma for the Carrington Rotations (CRs) 1936, 1964, and 2043. For these rotations in situ measurements of speed, density, and proton temperature at 1 AU are displayed in Figure 12.2. These CRs represent, respectively, ascending, maximum, and minimum phases of the solar cycle #23.

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12.6. Solar Cycle Dependence of Solar Wind The energy release and particle acceleration, arising due to change in magnetic field lines, on the solar atmosphere can manifest in a range of temperatures, from slight increase due to the heating (i.e., thermal emission produced by the particles possessing temperature of about a few times that of the photosphere) to thousands of fold increase in seconds or less. Each of these solar events modify the flow properties of the solar wind.

12.6.1. Steady-state Solar Wind The steady-state solar wind, normally, is organized by the flow speed. The fundamental differences between the fast wind (V ≥ 600 kms−1 ) and slow wind (V ≤ 400 kms−1 ) are due to the magnetic field configuration at the base of the corona.43 Figure 12.7 shows examples of results obtained from the Ooty IPS measurements, namely, two-dimensional synoptic images of the solar wind density turbulence (left panels) and speed (right panels) traced backward/forward from the measurement point onto the heliosphere of radius ∼ 100 R
. These images correspond to the same Carrington Rotations that are plotted in Figure 12.2, i.e., CRs 1936, 1964, and 2043 and depict conditions of the solar wind during three phases, respectively, at the ascending, at the maximum, and at the minimum phases of the solar cycle #23. These synoptic maps illustrate that the two-dimensional density and speed structures go through large-scale changes and indicate the evolution of the corona over the course of the solar cycle. A comparison between images of density turbulence and speed reveals association as well as difference in their large-spatial structures. As illustrated in the next section, the study at a finer scale is required to understand the source region and the acceleration mechanism of the solar wind. In the ascending phase of the solar cycle, while the reversal of field polarity takes place, the coronal holes migrate from the poles of the Sun to mid and low latitudes and the high-speed streams from a mid-latitude coronal hole can interact with the low-speed wind from the other closed field regions of the Sun (refer to Figure 12.7, CR#1936 maps). As these high-speed regions co-rotate with the Sun, the heliosphere is dominated by the co-rotating interaction regions (CIRs), which cause shocks of weak to moderate strength (refer to Section 12.7). At the maximum phase of the cycle, the regular structures vanish and rather variable speed and density structures are observed at all latitudes. The high speed regions shrink toward the polar caps.

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During solar minimum, the overall shape of the heliosphere is organized with respect to the equatorial belt and speeds of ∼800 kms−1 are observed at the polar regions (open magnetic field) of the Sun (refer to Figure 12.7, CR#2043 maps). In contrast, the middle- and low-latitude regions are dominated by a high-density ‘streamer-belt’, where the closed field largely confines the coronal plasma and the low-speed wind originates near the streamer belt. Moreover, the streamer belt, which marks the equatorial region of the north and south poles (i.e., the dipole equator), evolves into an interplanetary (or heliospheric) current sheet (HCS) that separates the flows originating in the two hemispheres. 12.6.2. Solar Wind Acceleration above Coronal Funnel In a recent study to understand the link between magnetic fields and highspeed solar wind from a coronal hole, Chinese-German team compared the small-scale features observed in the Doppler-shifted spectral emission (77 nm) images (SUMMER/SOHO data) from the coronal hole region with the corresponding magnetic field lines extrapolated from the simultaneous photospheric magnetogram (MDI/SOHO image).32,62 They found that the geometry of the coronal magnetic field played a key role in determining the origin and flow properties of the solar wind. The fast solar wind originates (at speeds ∼10-20 kms−1 ) and is accelerated above magnetic funnels formed by the intense open unipolar field lines at a height of ∼10–20×103 km. Although large closed loops are absent in the coronal hole, the short loops observed in the coronal hole confine the plasma and no significant flow occurs above them. The shape of the funnel is also determined by the crowding and pushing of the neighboring short loops. Figure 12.8 illustrates the location and geometry of the funnel-shaped three-dimensional magnetic field above the coronal hole, where fast solar wind is formed and accelerated (http://www.mpg.de/). 12.6.3. Low-speed Solar Wind The origin of slow speed flows (associated with both active regions and isolated mid-latitude small coronal holes) has been studied by several authors.1,9,28,47 In general at the maximum phase of the solar cycle, excluding transients flows, the low-speed wind is observed at all latitudes. At this phase, the closed-field structures and their associated loop-top plasma concentration dominate the entire Sun and the coronal parts above these active regions show high plasma density as well as high level of density

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Fig. 12.8. This picture is constructed from measurements on September 21, 1996, obtained from the Solar Ultraviolet Measurements of Emitted Radiation spectrometer (SUMER/SOHO), which provides Doppler spectroscopy of the coronal plasma. The field lines shown in the rectangular images are the extrapolated magnetic field lines obtained from the photospheric magnetogram observed with the Michelson Doppler Imager (MDI). The funnel-shaped open field lines and the dark solid closed lines are evident in these images. The image of the Sun shown in the left corner is from the Extreme ultraviolet Imaging Telescope (EIT). Image courtesy by Max Planck Institute for Solar System Research (MPS), http://www.mpg.de/.

turbulence, which correlates with the intense coronal emissions in Fe XIV spectral line and soft X-ray from the top of the closed-loop system.18 In Figure 12.7, the high-turbulence regions (high g-values), associated with the maximum phase of the cycle (refer to Figure 12.7, CR#1964), tends to map back along the edge of the low-speed region and not exactly on top of the low-speed patch. It suggests that the low-speed wind is likely to originate at the edge of the active regions site (i.e., near the leg of the closed-loop system). 12.6.4. Source of Slow Solar Wind The above inference is in agreement with the findings by Kojima and his co-workers28 that the source of low-speed wind lies away from the magnetic neutral line. Figure 12.9 gives an example of compact low-speed region observed near the equatorial belt of the Sun during the minimum phase of the solar cycle (i.e., CR 1913, August–September 1996) at 327 MHz using the multi-antenna IPS system at the STEL, Nagoya University (M. Kojima, private communication). The measured speed is projected on to a source surface at 2.5 R
. The plot also includes the corresponding photospheric

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Fig. 12.9. The solar wind map observed at STEL, Nagoya University, projected on to a source surface of 2.5 R
(top map). The magnetic field lines, obtained by extrapolating the photospheric field measurements from the Kitt Peak Observatory, show the location of the slow speed wind region.14 The slow speed wind coincides with the open field lines originating at the edge of the closed loops28 (M. Kojima, private communication).

magnetic field image obtained from the Kitt Peak Observatory and extrapolated potential field lines to the source surface at 2.5 R
. The portion of the image under consideration corresponds to speeds ≤400 kms−1 . It is evident that the low-speed part is not located above the closed-loop system (i.e., helmet structure), but it is well connected to the open-field region of the corona at the edge of the loop system. 12.6.5. Coronal Expansion Factor and Solar Wind Speed In Figure 12.7, at the minimum of the solar activity (CR#2043), the coronal hole regions at the south and north poles are dominated by the open field configuration, which allows the solar wind plasma to expand rapidly into interplanetary space. In the polar coronal hole regions, flow speeds of ∼800 kms−1 are observed. However, the size of the coronal hole and the intensity of the magnetic field associated with it essentially determine the flow properties.11,48,65 It has been shown by Wang and Sheeley65 that the inverse of magnetic flux expansion factor, 1/f (f is defined as the ratio between the magnetic field intensities at the photosphere and at the source surface, ∼2.5 R
), correlates with the flow speed above the coronal hole.

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cc=0.88

Velocity (km/s)

CH ce CH ce 1

CH 1 CH 1 CH 2

CH 2

B/f (gauss) Fig. 12.10. The observed solar wind speed plotted as a function of B/f factor. This plot includes coronal holes of variable size observed at different latitudes on the Sun. The high correlation coefficient 0.88 indicates that the magnetic energy available over a given area determines the speed of the flow. The solid lines are speed derived from B/f and B/f derived from the speed. The different symbols are: triangle - equatorial coronal hole associated with active region, square - isolated mid-latitude coronal hole, diamond - extension of polar coronal hole with smaller expansion rate, and circle - polar coronal hole.29

This model gives a physical picture that the reconnection energy is deposited in an expanding flux tube and it is utilized to accelerate the solar wind. Since variable size coronal holes appeared at the different phases of the solar cycle causing a range of speeds in the inner heliosphere and the intensity of the magnetic field (B) at the base of the corona is also linked to the energy supplied to the solar wind via magnetic reconnection,11 Kojima et al.29 proposed an empirical model to explain the speed in terms of B/f ratio. Figure 12.10 illustrates the correlation between the flow speed and B/f, where B is the intensity of magnetic field. This plot includes small and large size coronal holes, located at different latitudes on the Sun. The high correlation (88%) suggests that the flow speed is organized not only by the magnetic energy but also by the area over which this energy is available to accelerate the solar wind. 12.7. Solar Cycle #23: Solar Wind Large-scale Structures Figure 12.11 gives the latitudinal features of the solar wind density turbulence and speed observed at Ooty over the solar cycle #23. In a sense, these plots are ‘equivalent’ to the ‘magnetic butterfly’ diagram obtained from the

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Fig. 12.11. The top panel is the image of photospheric magnetic field observed during years 1995 to 2006. The middle and bottom panels are, respectively, density turbulence and speed maps of the solar wind observed over the solar cycle #23. It is to be noted that in the top plot, the magnetic field intensity is shown as a function of sine latitude.

magnetic field measurements of the photosphere (see top panel in Figure 12.11). It is evident in the ‘heliolatitude–year’ density distribution that during the minimum phase of the solar cycle, low-density turbulence dominates polar regions of the heliosphere (high-speed solar wind, i.e., ∼800 kms−1 ; refer to CR#2043 v-map in Figure 12.7). Apart from the systematic change of high speed flows at the polar regions to the low and variable speeds at low and mid latitudes, the equatorial-belt region (i.e., ±35◦ ) shows significant structures and their rapid evolution in the course of the solar cycle.

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Fig. 12.12. Sample co-rotating interaction regions (CIRs) and their associated geomagnetic storms observed at 1 AU for CRs 2013 and 2014 (February–April 2004). The solar wind and Dst data sets are, respectively, 1-min and 1-hour measurments, available at http://cdaweb.gsfc.nasa.gov/.

The drifting of density structures from high to low latitudes, as shown by the IPS images, is effected by the slow and gradual movement of the concentrated magnetic field regions on the corona. The latitudinal results are consistent with the evolution of warping of current sheet between the ascending and declining phases of the solar cycle.19 The structural evolution of the heliosphere during these phases of the solar cycle (i.e., around the years 2001–2002) is strongly linked to the polarity reversal, which gradually leads to the magnetically complex corona. The complexity peaks near the maximum of the cycle and the amplitude of the current sheet warping is as high as 60–70◦ latitude for the solar cycle #23 (http://wso.stanford.edu)

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Fig. 12.13. The portion of ‘latitude-year’ plot shows the density turbulence dominated by the interaction regions (i.e., caused by the coronal holes) and it radial evolution with heliocentric distance. The plot includes images at distances 75, 100, and 150 R
. It is evident that the interaction becomes intense with distance from the Sun.

and the frequency of occurrence of violent phenomena, such as flares and CMEs, is high.70 In the speed plot, the period between years 2002 and 2003 low-speed solar wind is observed centered around the equatorial region. During this period, occurrence of extremely low-density (N <2 cm−3 ) and speed (V <400 kms−1 ) solar wind at 1 AU has been recorded frequently. Another less turbulence latitudinal structure is observed around mid 1999 (covering ±60◦ heliolatitude range; Figure 12.11, middle panel). This period seems to be associated with phenomena of disappearance of the solar wind.23 Most of the large-scale features observed in the solar wind thus seem to result from the migration of magnetically concentrated regions of coronal holes from high to low latitudes and vice versa. The spreading of CIRs to the high latitudes in the heliosphere indicates the tilt of the magnetic dipole. Figure 12.12 shows sample plots of solar wind and Dst parameters associated with CIRs observed during Carrington rotations 2013 and 2014.

12.7.1. Evolution of Solar Wind Interaction Regions In Figure 12.13, density turbulence images observed at heliocentric distances, respectively, 75, 100, and 150 R
, are plotted to indicate the dominant interaction observed during year 2003.4,61 It is clear from these images that the interaction region as it moves away from the Sun develops into an intense shock. However, as pictured in Figure 12.11, the interaction regions occurred during the year 2002 seems to produce less significant shocks (or compression regions) within about 1 AU. But, as recorded by the Ulysses

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Fig. 12.14. Ooty scintillation images of halo and partial halo CMEs observed on November 5–6, 2003. The enhancements associated with these CMEs are at different heliocentric distances (between ∼50 and 225 R
). The concentric circles are at radii 50, 100, 150, and 200 R
.

at heliocentric distance >2.5 AU, the shocks have indeed been developed at the larger distances, suggesting the less efficient drivers (or heavily opposing ambient solar wind conditions to form the shock) associated with these interaction regions during the year 2002. 12.8. Coronal Mass Ejection Associated Transients A typical CME can involve energies (kinetic and magnetic energies) in the range ∼1030 – 1032 erg, initial speeds in the range 10–3000 kms−1 and carries plasma and magnetic field from the Sun to the interplanetary medium.42 The speed profile of each CME goes through changes in the inner heliosphere depending on the work done by the CME in the ambient solar wind or energy supplied to the CME by the background solar wind. It is important to note that the occurrence rate of CMEs has a strong solar cycle dependence.56,70 A CME, as it moves outward from the Sun, exhibits mixtures of closed, open, and disconnected magnetic topologies and their intrinsic variety and evolving three-dimensional structures are important to understand its effects on the Earth’s magnetosphere. A fast moving CME with speed in excess with respect to various MHD modes can produce shock transient in the near-Sun region (i.e., coronal shock) and/or in the interplanetary medium (i.e., interplanetary shock). Each CME carries information about its coronal origin into the heliosphere and can mix/modify the solar wind surrounding it. Further, CMEs (even from a same active region) can produce radically different effects at different heliocentric distances, latitudes, and longitude, which are determined by the internal energy possessed by the CMEs and the properties of the solar wind encountered by them on the

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Fig. 12.15. A sample ‘height-time’ plot of a CME and its associated disturbance obtained from white-light images, Wind/WAVES data, IPS images, and near-Earth in-situ measurements.

way to larger radial distances. Further, it is established that CMEs are the primary drivers of heliospheric disturbances with the greatest impact on the near-Earth environment.45 Therefore, multi-point measurements of CMEs as well as the surrounding solar wind are required as a function of heliocentric distance, latitude and longitude to understand their three-dimensional evolution.

12.8.1. Propagation of Coronal Mass Ejections The size evolution of CMEs obtained from white-light and IPS images in the Sun–1-AU distance range suggests a pressure balance maintained between the CME driver gas and background solar wind.44 Figure 12.14 shows examples of scintillation images obtained from the Ooty IPS measurements for halo and partial-halo CMEs during November 5–6, 2003. A sample height-time plot is shown in Figure 12.15.44 This plot is an excellent example for the coordinated study of CME and it has been made using observations obtained from white-light images (LASCO), interplanetary radio bursts (Wind), IPS measurements (STEL), and IPS images (from Ooty IPS measurements on a grid of radio sources).

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Fig. 12.16. Speed-distance, V(R), profiles for CMEs having average initial speeds between ∼300 kms−1 (bottom) and ∼2500 kms−1 (top).37 The vertical line indicates the Earth’s orbit.

12.8.2. Speed Evolution of Coronal Mass Ejections The speed evolution of CMEs demonstrates that the deceleration of the CMEs does not follow a simple radial law over the entire distance range in the inner heliosphere, but, indicates a two-level deceleration: (1) a low decline in speed at distances within or about 100 R
, and (2) a rapid decrease at larger distances from the Sun (Figure 12.16). These plots clearly show that the input energy associated with the CME eruption as well as the dynamics of the ambient solar wind shape the speed profile. The average speed of the ambient solar wind naturally divides transients into fast and slow categories, where fast ones get decelerated and slow CMEs are accelerated or carried away by the ambient solar wind flow.37 Manoharan37,44,45 has suggested that the magnetic energy of the flux rope associated with the CME would be utilized in assisting the propagation

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of the CME into the interplanetary medium. At large distances from the Sun (e.g., > 100 R
), when the stored magnetic energy is significantly reduced, the transient is expected to go through a rapid decline in speed, which is in fact shown by the rapid drop of the observed speed-distance curves. This study also suggests that most of the CMEs tend to attain the speed of the ambient flow at 1 AU or further out in the heliosphere. Numerical simulations, such as time-dependent, multi-dimensional MHD models, have also provided details on the evolution of solar-generated disturbances and their heliospheric consequences.10,15 For example, some of the simulated MHD models15 have indicated the compression and deflection of magnetic cloud against the radial direction of the solar wind.5,63 12.9. Summary This chapter gives an overview of evolution of solar wind in the inner heliosphere. The results discussed indicate that the solar magnetic field plays a major role in shaping the heliosphere. The interplanetary scintillation observations are vital to understand the energy transfer between the solar disturbances and the surrounding solar wind. The techniques and measurements discussed in this review emphasis that the coordinated studies are important in the solar physics. It is to be underscored that the improved solar observations, combined with the interplanetary scintillation images, white-light images, and the current STEREO mission would provide more insight and complete view of the coronal dynamics and their interplanetary consequences. Acknowledgments This work utilizes data obtained by the LASCO system on board SOHO spacecraft. SOHO is a project of international cooperation between ESA and NASA. Author acknowledges the Yohkoh team for the image of the Sun and Max Planck Institute for Solar System Research for the coronal funnel plot. Author thanks M. Kojima for valuable discussions and for providing plots (Figures 12.9 and 12.10) to include in this paper. Author acknowledges the support from the observing/engineering team and research students of Radio Astronomy Centre for the help in making observation and data reduction. Author also thanks G. Agalya and Preethi Manoharan for the help in preparing the list of references. This work is partially supported by the CAWSES–India Program, which is sponsored by the Indian Space Research Organisation (ISRO).

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References 1. C. N. Arge, K. L. Harvey, H. S. Hudson, and S. W. Kahler, in Solar Wind Ten AIP Conf. Proc. 679, 202 (2003). 2. U. Anzer, in Proc. IAU Symp. M. Dryer and E. Tandberg-Hanssen (Eds.), 91, 263 (1980). 3. M. K. Bird and P. Edenhofer, Physics of the Inner Heliospher - I R. Schwenn and E. Marsch, (Eds.), Springer–Verlag, Berlin (1990). 4. L. F. Burlaga & A. F.-Vi¨ nas, J. Geophys. Res. 109, A12107 (2004). 5. P. J. Cargill, J. Chen., D. S. Spicer and S. T. Zalesak, J. Geophys. Res. 101, 4855 (1996). 6. W. A. Coles and J. J. Kaufman, Radio Science 13, 591 (1978). 7. W. A. Coles, R. R. Grall, M. T. Klinglesmith, and G. Bourgois, J. Geophys. Res. 100, 17069 (1995). 8. W.A. Coles, W. Liu, J. K. Harmon, & C. L. Martin, J. Geophys. Res. 96, 1745 (1991). 9. N. U. Crooker, J. A. Joselyn and J. Feynman, (Eds.), Coronal Mass Ejections, American Geophysical Union monograph 99, 4673 (1997). 10. M. Dryer, Space Sci. Rev. 67, 363 (1994). 11. L. A. Fisk, N. A. Schwadron, and T. H. Zurbuchen, J. Geophys. Res. 104, 19765 (1999). 12. R. R. Grall, W. A. Coles, M. T. Klinglesmith, A. R. Breen, P. J. S. Williams and R. Esser, Nature 379, 429 (1996). 13. S.R. Habbal, R. Esser, J.W. Holweg, and P.A. Isenberg. Solar Wind Nine, AIP Conf. Proc. 471 (1999). 14. K. Hakamada, Sol. Phys. 181, 73 (1998). 15. S. M. Han, S. T. Wu and M. Dryer, Computer Fluids 16, 81 (1988). 16. K. L. Harvey & F. Recely, Sol. Phys. 211, 31 (2002). 17. A. Hewish, P. F. Scott and D. Wills, Nature 203, 1214 (1964). 18. P. L. Hick, B. V. Jackson, S. Rappaport, G. Woan, G. Slater, K. Strong, and Y. Uchida, Geophys. Res. Lett. 22, 643 (1995). 19. J. T. Hoeksema, and P. H. Scherrer, Rep. UAG-94, NOAA, Boulder, Colo. (1986). 20. J. V. Hollweg, Sol. Phys. 56, 305 (1978). 21. P. A. Isenberg, Geomagnetism 4, 1 (1990). 22. P. Janardhan, V. Balasubramanian, S. Ananthakrishnan, M. Dryer, A. Bhatnagar and P. S. McIntosh, Sol. Phys. 166, 379 (1996). 23. P. Janardhan, K. Fujiki, M. Kojima, M. Tokumaru, K. Hakamada, J. Geophys. Res. 110, A08101 (2005). 24. J. L. Kohl and S. R. Cranmer (Eds.), Coronal Holes and Solar Wind Acceleration, Kluwer Academic Publishers (1999). 25. J. L. Kohl, G. Noci, S. R. Cranmer, & J. C. Raymond, Astron. Astrophys. Rev. 13, 31 (2006). 26. M. Kojima and T. Kakinuma, Space Sci. Rev. 53, 173 (1990). 27. M. Kojima, M. Tokumaru, H. Watanabe, A. Yokobe, K. Asai, B. V. Jackson and P. L. Hick, J. Geophys. Res. 103, 1981 (1998).

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28. M. Kojima, K. Fujiki, T. Ohmi, M. Tokumaru, A. Yokobe and K. Hakamada, J. Geophys. Res. 104, 16993 (1999). 29. M. Kojima, M. Tokumaru, K. Fujiki, H. Itoh, T. Murakami, K. Hakamada, New Solar Physics with Solar-B Mission ASP Conference Series 369, 549 (2007). 30. E. Leer, T. E. Holzer, and T. Fla, Space Sci. Rev. 33, 161 (1982). 31. Øystein Lie-Svendsen & Ruth Esser, ApJ 618, 1057 (2005). 32. E. Marsch, Living Review in Solar Physics 3, 1 (2006). 33. N. Meyer-Vernet, A. Mangeney, M. Maksimovic, F. Pantellini and K. Issautier, Proceedings of the Tenth International Solar Wind Conference, AIP Conf. Proc. 679, 232 (2003). 34. W. H. Matthaeus, P. Dmitruk, S. Oughton and D. Mullan, Proceedings of the Tenth International Solar Wind Conference, AIP Conf. Proc. 679, 427 (2003). 35. P. K. Manoharan and S. Ananthakrishnan, MNRAS 244, 691 (1990). 36. P. K. Manoharan, Sol. Phys. 148, 153 (1993). 37. P. K. Manoharan, Sol. Phys. 235, 345 (2006). 38. P. K. Manoharan, in this volumne. 39. P. K. Manoharan, M. Kojima and H. Misawa, J. Geophys. Res. 99, 23411 (1994). 40. P. K. Manoharan, Bull. Astr. Soc. India 23, 399 (1995). 41. P. K. Manoharan, S. Ananthakrishnan, M. Dryer, T. R. Detman, H. Leinbach, M. Kojima, T. Watanabe and J. Kahn, Sol. Phys. 156, 377 (1995). 42. P. K. Manoharan, M. Kojima, N. Gopalswamy, T. Kondo and Z. Smith, ApJ 350, 1061 (2000). 43. P. K. Manoharan, in Lectures on Solar Physics, H. M. Antia et al. (Eds.), Springer Verlag, Heidelberg, 619, 299 (2003). 44. P. K. Manoharan, M. Tokumaru, M. Pick, P. Subramanian, F. M. Ipavich, K. Schenk, M. L. Kaiser, R. P. Lepping and A. Vourlidas, ApJ 559, 1180 (2001). 45. P. K. Manoharan, N. Gopalswamy, S. Yashiro, A. Lara, G. Michalek and R. A. Howard, J. Geophys. Res. 109, 6109 (2004). 46. J. T. Mariska, Annual Review Astr. Astrophy. 24, 23 (1986). 47. M. Neugebauer, R. J. Forsyth, A. B. Galvin, K. L. Harvey, J. T. Hoeksema, A. J. Lazarus, R. P. Lepping, J. A. Linker, Z. Mikic, J. T. Steiberg, R. Von Steiger, Y.-M. Wang, and R. F. Wimmer-Schweingruber, J. Geophys. Res. 90, 14587 (1998). 48. J. T. Notle, A. S. Krieger, A. F. Tomothy, R. E. Gold, E. C. Roelof, G. Vaiana, A. J. Lazarus, J. D. Sullivan, and P.S. McIntosh, Sol. Phys. 46, 303 (1976). 49. L. Ofman, Space Sci. Rev. 120, 67 (2005). 50. E. N. Parker, ApJ 128, 677 (1958). 51. E. N. Parker, ApJ 139, 72 (1964). 52. M. P¨ atzold, M. K. Bird, H. Volland, G. S. Levy, B. L. Seidel, and C. T. Stelzried, Sol. Phys. 109, 91 (1987).

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53. J. L. Phillips, A. Balogh, S. J. Bame, B. E. Goldstein, J. T. Gosling, J. T. Hoeksema, D. J. McComas, M. Neugebauer, N. R. Sheeley and Y.-M. Wang, J. Geophys. Res. 21, 1105 (1994). 54. B. J. Rickett and W. A. Coles, J. Geophys. Res. 96, 1717 (1991). 55. R. Schwenn, Space Sci. Rev. 34, 85 (1983). 56. W. Song, X. Feng, & Y. Hu, ApJ 667, L101 (2007). 57. Suzuki and K. Takeru, MNRAS 349, 1227 (2004). 58. V. I. Tatarski, Wave Propagation in a Turbulent Medium, McGraw-Hill, New York (1961). 59. M. Tokumaru, M. Kojima, K. Fijiki, M. Yamashita and A. Yokobe, J. Geophys. Res. 101, 1220 (2003). 60. M. Tokumaru, H. Mori, T. Tanaka, T. Kondo, H. Takaba and Y. Koyama, J. Geomag. and Geoelectr. 43, 619 (1991). 61. B. T. Tsurutani et al., J. Geophys. Res. 111, A07S01 (2006). 62. C.-Y. Tu C. Zhou, E. Marsch, L-D. Xia, L. Zhao, J-Z. Wang, & K. Wilhelm, Science 308, 519 (2005). 63. M. Vandas, S. Fischer, M. Dryer, Z. Smith and T. Detman, J. Geophys. Res. 101, 15645 (1996). 64. H. Volland, M. K. Bird, G. S. Levy, C. T. Stelzried, and B. L. Seidel, J. Geophys. Res. 42, 659 (1977). 65. Y.-M. Wang, and N. R. Sheely, Jr., ApJ 372, L45 (1991). 66. L. L. Williams, ApJ 424, L143 (1994). 67. J. Woch, W. I. Axford, U. Mall, B. Wilken, S. Livi, J. Geiss, G. Gloeckler, R. J. Forsyth, Geophy. Res. Lett. 24, 2885 (1997). 68. R. Woo, J. Geophys. Res. 93, 3919 (1988). 69. Y. Yamauchi, M. Tokumaru, M. Kojima, P. K. Manoharan, and R. Esser, J. Geophys. Res. 103, 6571 (1998). 70. S. Yashiro, N. Gopalswamy, G. Michalek, O. C. St Cyr, S. P. Plunkett, N. B. Rich and R. A. Howard, J. Geophys. Res. 109, 7105 (2004).

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CHAPTER 13 THE SUN-EARTH SYSTEM: OUR HOME IN SPACE

JUDITH L. LEAN Space Science Division Naval Research Laboratory, Washington DC 20375

13.1. Introduction Although separated by 150 million km, the Earth is inextricably linked to the Sun by an energy flow that both sustains and imperils life. A 15.8 million Kelvin nuclear furnace in the innermost one-third of the Sun is the source of this energy, which photons, particles and fields disseminate throughout the solar system. Figure 13.1 depicts major components of the Sun-Earth system, and Figure 13.2 summarizes the energy flow. Radiative then convective processes transfer energy from the Sun’s core to its “surface”, a gaseous shell that radiates electromagnetic energy primarily at visible wavelengths. Solar photons reach Earth in eight minutes where almost 99% are absorbed and reflected near the surface. This energy flow from the surface of the Sun to the surface of the Earth enables life by heating our planet, fueling photosynthesis and powering the interactions among oceans, land and atmosphere that generate weather and climate1. Encapsulating Earth’s thin life-sustaining surface region is an atmosphere of mainly nitrogen and oxygen gases, itself enveloped by belts of magnetic field anchored in Earth’s poles. Ultraviolet photons emitted from the Sun’s atmosphere establish the primary thermal structure of the Earth’s atmosphere, where they are absorbed and scattered2,3. A supersonic wind stretches the Sun’s outer atmosphere throughout the solar system, confining and deforming the outer regions of the Earth’s magnetic field to form an elongated region called the magnetosphere (Figure 13.1), which in turn traps impinging particles4,5. Though of much smaller magnitude, the aggregate energy from short wavelength photons, particles and fields maintains an infrastructure within the Sun-Earth system that protects the ecosphere from deleterious radiations, and both enables and impacts advanced global technologies on which society increasingly relies. 267

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Figure 13.1. Shown are the main components of the Sun-Earth system. Photons, particles and plasma flow from the Sun, their variations driven by dynamo action in the convection zone that produces sunspots and bright active regions. Photons from the Sun’s surface and atmosphere reach the Earth’s surface and atmosphere while the magnetosphere intercepts the particles and plasma. Eruptive events such as coronal mass ejections (shown emerging from the Sun’s atmosphere into the solar wind) perturb the magnetosphere, allowing energetic particles to penetrate Earth’s atmosphere in the polar regions (where the field lines are anchored).

The physical Sun-Earth system is sufficiently stable to have hosted some form of life for a few billion years1. By astrophysical standards, the Sun is hardly dynamic at all6,7. So assured was its quiescence that total solar radiative output was initially termed the solar “constant”, even though a prominent 11 year cycle in the dark sunspots appearing on the solar disk (shown in Figure 13.1) forewarned of its inconstancy. Extensive solar observations acquired in the past few decades, primarily from space, reveal a breadth and complexity of the Sun’s variability that Galileo could scarcely have imagined when observing sunspots in 1613. Nor could those who, for more than a century, uncovered correlations of sunspots with such things as grain prices and famine in India, have envisioned the ensuing controversies their early evidence presaged. Dismissed as recently as 1978 as “experiments in autosuggestion,” Sun-climate connections are newly

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apparent in a plethora of high fidelity paleoclimate indicators, and in the contentious - but unlikely - claim that the Sun, rather than human-produced greenhouse gases, is the cause of global warming in the past century. On shorter time scales, the raw power of the Sun to disrupt human endeavor was witnessed as early as 1859, when a solar eruption induced ground currents that burned U.S. telegraph wires, causing large fires; in 1989, a major solar storm again induced ground currents that triggered a blackout of northeast America and disrupted spacecraft orbits and operations. Today, sophisticated space-based navigation, communication and defense systems harness the Sun-Earth system for human advantage, a hundred years after Marconi, in 1901, first used the ionosphere to bounce radio waves around the globe. The radio waves that transit the space environment near Earth, and the hardware assets that reside there, are susceptible still to solar-driven space “weather” cycles and storms.

Figure 13.2. The flow of energy from the Sun to the Earth is compared for photons in four different wavelength bands, energetic particles and the plasma wind. The upper numbers are approximate energies and variations during an 11-year solar activity cycle, in Wm-2. Visible radiation connects the surfaces of the Sun and Earth while ultraviolet radiation connects their atmospheres. Particles and plasma connect the outer solar atmosphere primarily with Earth’s magnetosphere and high latitude upper atmosphere. Shown on the right are approximate temperatures of the various Sun and Earth regimes.

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13.2. Electromagnetic Radiation – Heating the Earth’s Surface and Structuring the Atmosphere Paramount for sustaining our habitat is the photon energy transfer that establishes Earth’s surface temperature as a balance between incoming radiation from the Sun’s 5770 Kelvin surface (with peak flux near 500 nm) and outgoing radiation from the Earth’s much cooler surface (with peak flux near 10 micron)1. The Sun’s total radiative output of approximately 1365 Watt per m2 delivers a globally averaged 341 Watt per m2 to Earth which reflects some 31%. Absorption of the remaining 69% heats the surface and lowest atmosphere. Warmed to 255 Kelvin, Earth’s surface then emits infrared photons that atmospheric gases (such as carbon dioxide) absorb and reradiate. This “greenhouse” effect traps additional energy that raises the surface temperature to a more hospitable global average 288 Kelvin. Atoms in the Sun’s atmosphere imprint upon the emergent black body spectrum signatures of hydrogen, helium and a host of minor heavier elements in various states of ionization6,7. Densely packed absorption lines, identified by Fraunhofer in 1817, deplete the Sun’s surface radiation whereas hot ionized gases emit thousands of lines and a few weak continua, raising the extreme ultraviolet flux of photons by many orders of magnitude above that of a 5770 Kelvin black body. At a few thousand km above the Sun’s surface, the million Kelvin corona sustains a shroud of highly ionized gases whose emissions crowd the extreme ultraviolet and X-ray region of the solar spectrum. Solar ultraviolet and X-ray photons link the atmosphere of the Sun with the atmosphere of the Earth. Although only ∼1.5% of total radiative output, the Sun’s photon energy at wavelengths shorter than 300 nm is the primary source of heating for the entire terrestrial atmosphere at altitudes from 10 to more than 500 km. This energy transfer approximately maps increasingly higher layers of the Sun’s atmosphere to increasingly higher altitudes of the Earth’s atmosphere: Photons with wavelengths between 300 and 170 nm are emitted from the Sun’s photosphere and absorbed in the Earth’s stratosphere and mesosphere, where they produce the ozone layer; photons at wavelengths shorter than 125 nm are emitted from the Sun’s chromosphere and corona and absorbed in the Earth’s thermosphere, creating within it layers of ionized gases that compose the ionosphere. Atoms and molecules in the Earth’s atmosphere control the deposition of solar ultraviolet photon energy at Earth. Molecular oxygen plays a singularly important role in this process because it absorbs photons with energies greater than 5 eV (242 nm) over a wide range of terrestrial altitudes2,3. The ensuing photodissociation produces atoms that combine with oxygen molecules to create

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ozone between 15 and 50 km. Ozone’s absorption of solar ultraviolet radiation at wavelengths between 200 and 300 nm heats the atmosphere, reversing its cooling trend away from the surface (Figure 13.2), and driving radiative and dynamical processes that couple the middle and lower atmospheres. At higher altitudes, above about 100 km, the absorption of solar photons by both molecular and atomic oxygen, as well as by molecular nitrogen, heats the entire thermosphere. Additionally, photons with energies greater than about 12 eV (wavelengths less than ∼100 nm) ionize these gases. Conducting ionospheric layers of electrons and ions are embedded in the neutral upper atmosphere from 50 to 1,000 km; outward flowing ions populate the plasmasphere, an intermediary between Earth’s ionosphere and magnetosphere that extends outward to a few earth radii (Figure 13.1)4,5. 13.3. Particles and Magnetic Fields - Controlling the Space Environment Near the Earth Charged particles with energies in the range of a few to billions of electron volts and magnetic fields with strengths from a few to tens of nano-Tesla populate space between the outermost atmospheres of the Sun and the Earth. Interactions of the magnetic fields, which originate within both the Sun and the Earth, structure geospace for hundreds of Earth radii, guiding and containing the particles which also have both solar and terrestrial origins4,5. A wind of mainly protons flows from the Sun’s continually expanding outer atmosphere. At a typical speed of 450 km per sec, the particles reach Earth in 4 or so days. They exit large dark “holes” in the Sun’s corona where magnetic fields extend into space instead of looping back to the solar surface6,7. This “open” magnetic flux is frozen into the low density solar wind plasma, which carries it past Earth. In deflecting the solar wind, Earth’s dipole magnetic field distorts to form the magnetosphere around the cooler plasmasphere. Sunward of Earth, the magnetosphere is a compressed bulge terminated around 10 Earth radii by the bow shock (Figure 13.1); downwind it is a stretched teardrop extending for many hundreds of earth radii. Although the solar wind typically transports less than one millionth of the Sun’s electromagnetic energy (Figure 13.2), it transfers to the magnetosphere and ionosphere an amount of electrical and kinetic energy comparable to man-made production4,5. Within the magnetosphere two concentric donut-shaped belts of magnetic field surround the Earth. Named after Van Allen, who discovered them in 1958, the belts are populated with charged particles of much higher energies than those in the solar wind. Many of the particles are protons ejected by collisions of

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galactic cosmic rays with atmospheric gases; others are produced from ionization of the Earth’s neutral atmosphere by solar EUV photons; still others come from the outer solar atmosphere. Some of these particles deposit their energy in collisions with gases in the Earth’s atmosphere, mostly in the vicinity of 100 to 400 km at high latitudes where magnetic field lines connect the magnetosphere to the denser lower thermosphere. The collisions excite oxygen and nitrogen molecules which then decay, emitting auroral light2,4. Very energetic particles of both solar and galactic origin – can penetrate to the Earth’s lower atmosphere where they alter the chemistry of stratospheric and tropospheric gases, depleting ozone and producing isotopes of many species. On reaching Earth, galactic cosmic rays initiate a complex chain of interactions that produce the 14C and 10 Be isotopes, whose archives in, respectively, tree-rings (as 14CO2 in the global carbon cycle) and ice (from aerosol precipitation), contain information about the local space environment8. 13.4. Sun-Earth System Variability in the Space Era Fluctuations in the Sun’s energy ceaselessly agitate the Sun-Earth system with a fundamental cadence near 11 years6,7. This quasi-periodic forcing arises from the significant photon and plasma changes that accompany cycles in solar “activity”, which in turn relates intimately to variations in magnetic flux in the Sun’s atmosphere. The driver of solar activity – and hence of Sun-Earth system variability - is a dynamo near the bottom of the outer one third of the Sun’s interior (Figures 13.1 and 13.2). Strong magnetic fields generated by differential rotation emerge at the Sun’s surface in the form of active regions; these fields are transported pole ward by a combination of turbulent convection and meridional flow, thereby reversing and regenerating the Sun’s polar fields. Magnetic flux is not distributed smoothly over the Sun’s surface nor throughout its atmosphere. Rather, it is concentrated in discrete structures that modify the local temperature and density, thereby altering the Sun’s radiative output6,7. Space-based radiometers record a net increase of 0.1% in total solar brightness during activity maxima, the result of a 1 Watt per m2 reduction by dark sunspots and a 2 Watt per m2 enhancement due to bright photospheric features (called faculae)9. Figure 13.3 (upper left) illustrates these competing effects in recent times. Ultraviolet radiation also increases when solar activity is high because of enhanced emission in active regions overlying faculae. The impact of active regions on the Sun’s photon output increases with height of the photon production in the solar atmosphere: upper photospheric radiation at 200 nm increases by 8% during the 11-year activity cycle whereas coronal X-rays increase by more than an order of magnitude6,9.

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Figure 13.3. Earth’s monthly mean global surface temperature is shown in the bottom panel. A statistical multiple regression model reproduces significant temperature variance by combining variations in solar irradiance (top panel), a secular trend (possibly anthropogenic, second panel), volcanic aerosols (third panel) and an index of the El Nino Southern Oscillation (fourth panel). ENSO and volcanoes cause changes of 0.2 to 0.3 Kelvin on timescales of months whereas solar irradiance causes a 0.1 Kelvin decadal cycle. The irradiance cycle arises from the competing effects of sunspots and faculae, features evident in the respective solar images (upper left). Their daily effects on irradiance are shown on the upper left.

Terrestrial responses to solar activity increase with altitude, eventually dominating other natural and anthropogenic influences. The temperature near the Earth’s surface increases by less than one tenth Kelvin in response to one Wm-2 increase in total solar brightness10-13. Figure 13.3 illustrates an empirical identification of this modest solar component in the past twenty six years, in comparison with a secular trend (attributed to changing concentrations of greenhouse gases and industrial aerosols), volcanic aerosols and the El Nino Southern Oscillation internal oscillation. An order of magnitude larger cycle of solar ultraviolet radiation initiates still modest but definitive changes that reach one Kelvin at 50 km. Solar-induced stratospheric ozone changes are a few percent, comparable to a secular decrease (primarily from increasing chlorofluorocarbon concentrations) and the influence of the internal quasibiennial oscillation (QBO) in equatorial stratospheric winds. Figure 13.4

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illustrates total ozone variability (the concentration integrated over all heights, globally) produced by the solar UV irradiance cycle, volcanic aerosols, the QBO and chloroflurocarbobon gas concentrations14-16. Near 500 km, temperature increases a dramatic 500 Kelvin in response to factors of two and more fluctuations in extreme UV and X-ray photons17-19. As Figure 13.5 shows, the accompanying increases in neutral and ion densities are an order of magnitude or more, greatly exceeding the decrease of a few percent per decade associated with cooling by increasing greenhouse gas concentrations19. Identifying and quantifying the diverse causes of contemporary change throughout the Sun-Earth system are ongoing challenges.

Figure 13.4. Earth’s monthly mean ~global (50S to 50N) deseasonalized total ozone concentration is shown in the bottom panel. A statistical multiple regression model reproduces significant ozone variance by combining variations in solar ultraviolet irradiance (top panel), a secular trend (possibly anthropogenic from chlorofluorocarbons, second panel), volcanic aerosols (third panel) and an index of the quasi biennial oscillation (QBO) in equatorial winds (fourth panel). The QBO and volcanoes causes changes of 2 to 4 Dobson Units (DU) on timescales of months to years, solar irradiance causes a cycle of 6 DU compared with the long term changes in anthropogenic gases cause a decrease of 7 DU. The UV irradiance cycle arises from the occurrence of bright faculae in the solar atmosphere. These features are evident in the solar image (upper left) at solar maximum, but absent during solar minimum.

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Figure 13.5. Fluctuations at 500 km in Earth’s upper atmosphere are shown over Boulder CO, at noon, estimated by a semi-empirical model17,19. Without solar activity, the noon-time temperature is approximately 750 ± 50 Kelvin. Solar EUV photon energy, whose variations are shown in the upper left, increases the temperature by more than 500 Kelvin during high solar activity (second panel). Particles and plasma that disrupt Earth’s magnetic environment produce smaller fluctuations, up to 100 Kelvin in daily means (third panel). The net changes in temperature at 500 km (fourth panel) are accompanied by variations in the total mass density of the upper atmosphere at 500 km (fifth panel) and in peak electron density (sixth panel). The changes in neutral and electron densities respectively impact the drag on spacecraft in low Earth orbit (including the Space Station at 400 km) and the critical frequency for radio wave propagation.

Sporadic solar energy outbursts punctuate quasi-cyclic changes and can modify the Sun-Earth system rapidly and substantially, especially in the regime where the extended atmosphere of the Sun connects with the space environment near Earth20,21. Such events occur when coronal magnetic fields abruptly reconnect, typically in sites overlying large dynamic sunspots, possibly instigated by the motions of the foot points that anchor the fields to the Sun’s surface. The flux of high energy photons from the Sun’s atmosphere then rapidly increases by large factors, causing total electron densities in the Earth’s atmosphere to also rapidly increase by many tens of percent. Ejected coronal mass propels billions of tons of charged particles through the solar wind at three to four times its normal speed. This fast-moving plasma drives shocks that produce near-relativistic energetic particles which reach the Earth within an hour, followed a day or so later by the shock and its driver plasma. Figure 13.6 illustrates this sequence for the “Halloween” (October 2003) event.

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Figure 13.6. Shown is the sequence of events that occurred during a major disturbance in the SunEarth system on 28 Oct 2003, denoted the “Halloween” storm25. The event originated in the vicinity of a large sunspot near the center of the solar disk, seen in the white light solar image on the upper left, made by the Big Bear Solar Observatory. It commenced near 13:19 hours with a bright flare recorded in images made by the Extreme-ultraviolet Imaging Telescope (EIT) onboard the Solar Heliospheric Observatory (SOHO) (top left images) and a coronal mass ejection, recorded by the Large Angle and Spectrometric Coronagraph (LASCO), also on SOHO, shown in the bottom left image. The high energy photons emitted in the flare produced an X class flare which saturated the X-ray detectors on NOAA’s Geostationary Operational Environmental Satellites (GOES) when they arrived at Earth about 8 minutes later (upper right panel). A surge of energetic particles reached Earth many hours later, where they saturated the LASCO detector, corrupting the image in the bottom left, and producing a major increase in the flux of energetic protons recorded by GOES (bottom right panel). The X–ray fluxes and energetic proton fluxes are from the NOAA National Weather Service http://www.sec.noaa.gov/.

Substantial field reconnection occurs subsequently, this time near Earth. Initially the reconfiguration of solar and terrestrial magnetic fields erodes the upwind magnetosphere, then the geomagnetic fields reconnect to restore the magnetosphere’s pre-storm configuration. The rapid motions of the fields accelerate particles toward Earth where their precipitation produces aurora and heating in polar regions, and depletes ozone. This localized heating slowly diffuses to lower latitudes over a period of days, altering both neutral and ion

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densities on global scales. In recent years, an unprecedented suite of spacecraft have traced the energy flow from the Sun to the Earth in some of the most severe storms of the space age. Ongoing analyses of the “Bastille Day” (July 2000) and “Halloween” (October 2003) events are revealing new insights on processes that facilitate abrupt Sun-Earth system change22-25. Agreement between observations and model simulations of Sun-Earth system variability differs markedly among different regimes. A major enigma is the immutable climate predicted by general circulation climate models in response to decadal solar variability when compared with solar signals in surface temperatures, cloud cover, drought, rainfall, tropical cyclones and forest fires26-29. For example, when forced with the observed 11-year cycle in total radiative output, modeled surface temperature changes are a factor of five smaller than those deduced from empirical deconstruction of the surface temperature record (Figure 13.3). Either the empirical evidence is deceptive or the models inadequate, for example in their parameterizations of feedbacks such as cloud processes and atmosphere-ocean couplings, or their neglect of indirect responses by the stratosphere and amplification of internal climate variability modes. The reverse situation challenges understanding of the coupled thermosphere and ionosphere, where general circulation models predict dramatic responses to changing solar energy inputs (Figure 13.5), but a lack of global datasets precludes comprehensive validation17-19. Attempts are underway to model large fractions of the Sun-Earth system30-32. General circulation climate models are being extended to cover the Earth’s entire atmosphere from the surface to a few hundred km. Initial results demonstrate the importance of vertical coupling in propagating and amplifying forcing from below (by ENSO and gravity waves) as well as from above (by solar activity). This “lifting the lid” of climate models is expected to improve understanding of solar-induced stratospheric influences on the troposphere. In a separate development, models are integrating the plasma environments of the solar wind, magnetosphere and ionosphere. Ultimate integration of the atmosphere and plasma models promises a new quantitative depiction of the whole Sun-Earth system. 13.5. Understanding Sun-Earth System Variability in the Past Since its formation about 4.5 billion years ago, the Sun-Earth system has evolved through epochs both warmer and colder than present. Earth’s changing aspect to the Sun generates “Milankovitch” cycles of ice ages, evident in the last few

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million years by global temperatures 5 to 7 Kelvin cooler than the current interglacial epoch of the past 10,000 years1. A variety of high resolution paleoclimate records in ice-cores, tree-rings, lake and ocean sediment cores and corals suggest that changes in the energy output of the Sun itself may have contributed to Sun-Earth system variability33-36. Many geographically diverse records of past climate are temporally coherent, with periods near 2400, 208 and 90 years that are also present in the 14C and 10Be cosmogenic archives. Reconstructions of actual climate forcings and surface temperatures in the past millennium suggest that solar and volcanic activity produced changes of a few tenths Kevin, together accounting for significant pre-industrial global surface temperature variance37. The most recent and best documented example of pre-industrial climate response to solar and volcanic radiative perturbations is an epoch from 1450 to 1850 when temperatures were 0.3 to 1 Kelvin colder than present. The beginning of this “Little Ice Age” coincided with anomalously low solar activity (in the Spörer and Maunder minima); the latter part coincided with both low solar activity (the Dalton minimum) and the Tamboora and Coseguina volcanic eruptions. But many uncertainties preclude a definitive understanding of the Sun’s role in the Little Ice Age26-28. The actual reduction of total radiative output is uncertain. Attempts have been made to relate past irradiance variations to proxy records of solar activity such as cosmogenic isotopes and inferred from Sun-like stars. These initial approaches suggested reductions of a few tenths percent38,39, as shown in Figure 13.7. More recently, simulations of the evolution of magnetic flux on the solar surface using a model of flux transport by differential rotation, diffusion and meridional circulation (also shown in Figure 13.7) suggest that the Maunder Minimum irradiance reduction may have been as small as 0.05%16,40. While cosmogenic isotopes were indeed elevated, the very low level of solar activity is insufficient to account for the large cosmic ray modulation suggested by 10Be. The overall cooler European temperatures suggest a solar-induced stratospheric influence on the natural Arctic oscillation with its primary expression in the North Atlantic32. Or, might the Little Ice Age be simply the most recent cool episode of millennial climate oscillation cycles? Another puzzle of past climate is the apparent sensitivity of drought and rainfall to solar variability, especially in tropical locations vulnerable to altered circulation patterns. For example, high solar activity corresponds to drier conditions in the Yucatan and equatorial East Africa33,36 and wetter conditions in

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Oman34. Model simulations and analysis of contemporary climate do suggest that higher solar activity strengthens regional precipitation regimes, including the monsoons41 (affecting famine in India?). Causes other than solar variability, such as inherent natural cycles, are postulated, since solar signals are absent in some climate records and intermittent in others. Nor are climate models generally able to reproduce the plethora of Sun-climate correlations. In fact, simulations with climate models yield decadal and centennial variability even in the absence of external forcing. Arguably, this very sensitivity of the climate system to unforced oscillation and stochastic “noise” predisposes it to non-linear responses to small forcings such as by the Sun.

Figure 13.7. A dynamo near the base of the Sun’s convection zone (bottom left) drives the Sun’s activity cycle6,7, altering the amount of magnetic flux that erupts onto the surface (recorded by the magnetogram, upper left). The three cartoons depict the transport of surface magnetic flux by the Sun’s differential rotation, poleward meridional flow and diffusion (courtesy of Y.-M. Wang and solarscience.msfc.nasa.gov). Shown in the figure (bottom right) are the variations in total solar irradiance accompanying the long-term evolution of the closed flux that generates bright faculae16,40 simulated with a flux transport model. In comparison, the upper envelope of the shaded region is the variations arising from the 11-year activity cycle, while the lower envelope is the total irradiance reconstructed with the long-term changes inferred from brightness changes in Sun-like stars39.

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A crucial challenge for understanding the Sun-Earth system in the past is the correct interpretation of the 14C and 10Be cosmogenic isotopes which correlate with many paleoclimate records8,33-36. The isotopes actually reflect solar-induced changes that impede the passage of cosmic rays through the plasma environment near the Earth, not the closed magnetic fields in sunspots and faculae that modulate irradiance38-40. Some suggest that cosmic rays themselves cause climate change (for example, by acting as cloud nucleation sites), even though the energy change associated with cosmic ray variability is one millionth that of solar electromagnetic energy change. Since climate itself can affect the deposition of the cosmogenic isotopes in ice and trees, the cosmogenic indicators may also indicate ocean (rather than solar) variability. Ultimately, a complete specification of the physical processes within the entire Sun-Earth system will be needed to resolve these issues. 13.6. Societal Consequences of Sun-Earth System Variability Perturbations to the Sun-Earth system are of practical concern when habitats are threatened by changes in climate and the ozone layer42,43 or when “weather” in the upper atmosphere and space environment interferes with society’s increasingly complex technological infrastructure44. Archeological and paleoclimate records suggest that since the earliest civilizations formed around 12,000 years ago climate change, possibly linked with the Sun, may have contributed to societal deterioration. For example, after a few hundred years of intermittent drought accompanied by famine and political unrest, the Mayan civilization of the Yucatan peninsular declined around 800 A.D. The droughts occurred with an approximate 208 year cycle, in phase with decreased levels of cosmogenic isotopes, indicating higher solar activity. Indigenous cultures in equatorial East Africa also prospered during wetter times that coincided with the Sporer and Maunder minima33-36. Contemporary habitat pressure is primarily from human – not solar activity. Doubled concentrations of “greenhouse” gases are projected to warm Earth’s surface by 4.2 Kelvin. Solar-driven surface temperature changes, which are unlikely to exceed 0.5 Kelvin and may be as small as 0.1 Kelvin (e.g., Figure 13.3), are substantially less. Nevertheless, they must be reliably specified so that policy decisions on global change have a firm scientific basis. Furthermore, climate encompasses more than surface temperatures, and future “surprises” are possible, perhaps involving the hydrological cycle and the Sun. Chlorofluorocarbons, used as refrigerants and industrial solvents, though inert in the troposphere, are dissociated in the stratosphere by solar ultraviolet

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photons to produce radicals that deplete ozone. A thinning ozone layer from increasing levels of chlorofluorocarbons since the 1970s (Figure 13.4) exposes biological organisms to increasing levels of UV-B radiation, which may increase rates of skin cancer and cataracts and deplete immune systems. Ozone levels are expected to recover in about 50 years as a result of reductions in chlorofluorocarbons specified by the Montreal Protocol. The solar-driven ozone cycle, which is comparable to the anthropogenic decline, complicates the early detection of this recovery14-16. Humans and hardware deployed beyond Earth’s surface are susceptible to more immediate impacts of Sun-Earth system variability17-19,44. Following solar eruptions (Figure 13.6), energetic particles can damage technological systems deployed in space and threaten the health of astronauts and aircraft passengers over polar regions when the particles penetrate the Earth’s atmosphere. Induced ground currents can debilitate the electrical grid, as in March 1989. Solar cycles and eruptions also threaten human safety and national security when ionospheric disturbances interrupt wireless navigation and communication. Management of communication frequencies relies on knowledge of the electron density in the ionosphere, which, as Figure 13.5 illustrates, depends critically on solar activity. Solar-driven atmospheric density fluctuations, also illustrated in Figure 13.5, can alter the orbits of the tens of thousands of objects in low earth orbit (LEO). Thus, scientific curiosity and societal utility both call for a robust understanding of the Sun-Earth system. How much of Earth’s recent surface warming is induced by solar, rather than man-made forcings? When will the recovery of the ozone layer be truly discernible against the backdrop of solarinduced changes? Will an eruptive solar event be benign or deadly, for space instruments and astronauts alike? Might resultant effects on navigation, communication and Earth-orbiting objects compromise security when their impacts are mistaken for adversaries? Can we mitigate and predict these effects? In seeking answers to these and other questions, once-disparate fields are coalescing slowly and a new paradigm is emerging - of the Sun and the Earth as one unified system, our home in space that extends well beyond the surface where we live. Acknowledgements Funding by NASA and ONR is gratefully acknowledged, as are interactions and collaborations with many Sun and Earth scientists, especially those in NRL’s Space Science Division, and D. Rind at NASA GISS. Much of the content of this paper was published in Physics Today, “Living with a Variable Sun”, June,

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2005. Parts of Figures 13.4 and 13.7 were published in Solar Physics, “SORCE contributions to new understanding of global change and solar variability”, 230, pages 27-53, 2005. References 1. W. F. Ruddiman, Earth’s Climate Past and Future, W. H. Freeman and Company, New York, 2001. 2. P. M. Banks and G. Kockarts, Aeronomy, Academic Press, New York, 1973. 3. G. Brasseur and S. Solomon, Aeronomy of the Middle Atmosphere, D. Reidel Publishing Co., Dordrecht, 1984. 4. J. K. Hargreaves, The Solar-Terrestrial Environment, Cambridge University Press, 1992. 5. Lyon, J. G., Science 288, 1987 (2000). 6. P. V. Foukal, Solar Astrophysics, John Wiley & Sons, Inc., New York, 1990. 7. D. G. Wentzel, The Restless Sun, Smithsonian Library of the Solar System, Smithsonian Institute Press, Washington, D.C., 1989. 8. E. Bard, G. M. Raisbeck, F. Yiou and J. Jouzel, Earth and Plan. Sci. Lett. 150, 453 (1997). 9. C. Fröhlich and J. Lean, Astron. Astrophys. Rev. 12, 273 (2004). 10. D. Douglass and B. D. Clader, Geophys. Res. Lett. 29 (2002); doi:10.1029/2002GL015345. 11. H. Gleisner and P. Thejll, Geophys. Res. Lett. 30, 1711 (2003). 12. W. B. White, M. D. Dettinger and D. R. Cayan, J. Geophys. Res. 108, 3248 (2003). 13. J. L. Lean, Geophys. Res. Lett. 33, L15701 (2006). 14. W. Steinbrecht, H. Claude and P. Winkler, J. Geophys. Res. 109, D02308 (2004); doi:10.1029/2003JD004284. 15. J. P. McCormack, L. L. Hood, R. Nagatani, A. J. Miller, W. G. Planet and R. D. McPeters, Geophys. Res. Lett. 24, 22, 2729 (1997). 16. J. Lean, G. Rottman, J. Harder and G Kopp, Solar Phys. 230, 27 (2005). 17. J. M. Picone, A. E. Hedin, D. P. Drob and A. C. Aikin, J. Geophys. Res. (2002). 18. D. J. Gorney, Rev. Geophys. 28, 315 (1990). 19. J. T. Emmert, J. M. Picone, J. L. Lean and S. H. Knowles, J. Geophys. Res. 109, A02301 (2004); 2003JA010176. 20. J. L. Burch, The Fury of Space Storms, Scientific American, April, 2001.

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