An Introduction to Global Spectral Modeling, Second Edition (Atmospheric and Oceanographic Sciences Library, 35)

An Introduction to Global Spectral Modeling Second Revised and Enlarged Edition

ATMOSPHERIC AND OCEANOGRAPHIC SCIENCES LIBRARY VOLUME 35

Editors Lawrence A. Mysak, Department of Atmospheric and Oceanographic Sciences, McGill University, Montreal, Canada Kevin Hamilton, International Pacific Research Center, University of Hawaii, Honolulu, HI, U.S.A. Editorial Advisory Board L. Bengtsson A. Berger P.J. Crutzen J.R. Garratt G. Geernaert M. Hantel A. Hollingsworth H. Kelder T.N. Krishnamurti P. Lemke P. Malanotte-Rizzoli S.G.H. Philander D. Randall J.-L. Redelsperger S.H. Schneider F. Schott G.E. Swaters J.C. Wyngaard

Max-Planck-Institut fur ¨ Meteorologie, Hamburg, Germany Universit´e Catholique, Louvain, Belgium Max-Planck-Institut fur ¨ Chemie, Mainz, Germany CSIRO, Aspendale, Victoria, Australia DMU-FOLU, Roskilde, Denmark Universitat ¨ Wien, Austria European Centre for Medium Range Weather Forecasts, Reading, UK KNMI (Royal Netherlands Meteorological Institute), De Bilt, The Netherlands The Florida State University, Tallahassee, FL, U.S.A. Alfred-Wegener-Institute for Polar and Marine Research, Bremerhaven, Germany MIT, Cambridge, MA, U.S.A. Princeton University, NJ, U.S.A. Colorado State University, Fort Collins, CO, U.S.A. METEO-FRANCE, Centre National de Recherches M´et´eorologiques, Toulouse, France Stanford University, CA, U.S.A. Universitat ¨ Kiel, Kiel, Germany University of Alberta, Edmonton, Canada Pennsylvania State University, University Park, PA, U.S.A.

The titles published in this series are listed at the end of this volume.

An Introduction to Global Spectral Modeling Second Revised and Enlarged Edition

T.N. Krishnamurti Department of Meteorology Florida State University Tallahassee, FL, USA

H.S. Bedi Delhi, India

V.M. Hardiker Houston, TX, USA

L. Ramaswamy Department of Meteorology Florida State University Tallahassee, FL, USA

T.N. Krishnamurti Department of Meteorology Florida State University Tallahassee, FL 32306 USA

H.S. Bedi c/o Vinny Bedi 84 Mausam Bhawan Delhi 11051 India

V.M. Hardiker Bell South Houston, TX 75063 USA

L. Ramaswamy Department of Meteorology Florida State University Tallahassee, FL 32306 USA

Library of Congress Control Number: 2005935659 ISBN-10: 0-387-30254-9 ISBN-13: 978-0387-30254-6 Printed on acid-free paper.
C 2006 Springer Science+Business Media, Inc. First edition
C 1998 Oxford University Press All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springer.com

(TB/SBA)

Preface This is an introductory textbook on global spectral modeling designed for senior-level undergraduates and possibly for first-year graduate students. This text starts with an introduction to elementary finite-difference methods and moves on towards the gradual description of sophisticated dynamical and physical models in spherical coordinates. Computational aspects of the spectral transform method, the planetary boundary layer physics, the physics of precipitation processes in large-scale models, the radiative transfer including effects of diagnostic clouds and diurnal cycle, the surface energy balance over land and ocean, and the treatment of mountains are some issues that are addressed. The topic of model initialization includes the treatment of normal modes and physical processes. A concluding chapter covers the spectral energetics as a diagnostic tool for model evaluation. This revised second edition of the text also includes three additional chapters. Chapter 11 deals with the formulation of a regional spectral model for mesoscale modeling which uses a double Fourier expansion of data and model equations for its transform. Chapter 12 deals with ensemble modeling. This is a new and important area for numerical weather and climate prediction. Finally, yet another new area that has to do with adaptive observational strategies is included as Chapter 13. It foretells where data deficiencies may reside in model from an exploratory ensemble run of experiments and the spread of such forecasts. These classroom lectures emerged from discussions with a large number of former colleagues that include: Masao Kanamitsu, Richard Pasch, Hua-Lu Pan, Steve Cocke, Chia Bo Chang, John Molinari, Naomi Surgi, Lahouari Bounoua, Fred Carr, Simon Low-Nam, Takeo Kitade, Masato Sugi, Mukut Mathur, and Jishan Xue. Many others who were part of Krish’s laboratory at the Florida State University also contributed in many ways towards the material presented here. In addition, we owe a great deal to the United States funding agencies who supported our research in this area: NSF (Pamela Stephens), ONR (Scott Sandgathe), and NOAA (Kenneth Mooney).

v

Contents Preface

v

1

Introduction

1

2

An Introduction to Finite Differencing 2.1 Introduction 2.2 Application of Taylor’s Series to Finite Differencing 2.3 Forward and Backward Differencing 2.4 Centered Finite Differencing 2.5 Fourth-Order Accurate Formulas 2.6 Second-Order Accurate Laplacian 2.7 Fourth-Order Accurate Laplacian 2.8 Elliptic Partial Differential Equation in Meteorology 2.9 Direct Method 2.10 Relaxation Method 2.11 Sequential Relaxation Versus Simultaneous Relaxation 2.12 Advective Nonlinear Dynamics 2.13 The 5-Point Jacobian 2.14 Arakawa Jacobian 2.15 Exercises

4 4 5 6 7 8 11 15 19 19 23 27 29 33 33 39

3

Time-Differencing Schemes 3.1 Introduction 3.2 Amplification Factor 3.3 Stability 3.4 Shallow-Water Model

40 40 40 43 54

4

What Is a Spectral Model? 4.1 Introduction 4.2 The Galerkin Method 4.3 A Meteorological Application 4.4 Exercises

60 60 60 63 64

5

Lower-Order Spectral Model 5.1 Introduction 5.2 Maximum Simplification 5.3 Conservation of Mean-Square Vorticity and Mean Kinetic Energy 5.4 Energy Transformations 5.5 Mapping the Solution 5.6 An Example of Chaos

65 65 66 68 71 72 73

vii

Contents

viii 5.7 Exercises

74

6

Mathematical Aspects of Spectral Models 6.1 Introduction 6.2 Legendre Equation and Associated Legendre Equation 6.3 Laplace’s Equation 6.4 Orthogonality Properties 6.5 Recurrence Relations 6.6 Gaussian Quadrature 6.7 Spectral Representation of Physical Fields 6.8 Barotropic Spectral Model on a Sphere 6.9 Shallow-Water Spectral Model 6.10 Semi-implicit Shallow-Water Spectral Model 6.11 Inclusion of Bottom Topography 6.12 Exercises

76 76 79 82 83 87 89 95 100 104 108 110 110

7

Multilevel Global Spectral Model 7.1 Introduction 7.2 Truncation in a Spectral Model 7.3 Aliasing 7.4 Transform Method 7.5 The x-y-ı Coordinate System 7.6 A Closed System of Equations in ı Coordinates on a Sphere 7.7 Spectral Form of the Primitive Equations 7.8 Examples

112 112 112 115 115 119 126 136 142

8

Physical Processes 8.1 Introduction 8.2 The Planetary Boundary Layer 8.3 Cumulus Parameterization 8.4 Large-Scale Condensation 8.5 Parameterization of Radiative Processes

146 146 146 155 167 173

9

Initialization Procedures 9.1 Introduction 9.2 Normal Mode Initialization 9.3 Physical Initialization 9.4 Initialization of the Earth’s Radiation Budget

192 192 192 202 210

10 Spectral Energetics 10.1 Introduction 10.2 Energy Equations on a Sphere 10.3 Energy Equations in Wavenumber Domain 10.4 Energy Equations in Two-Dimensional Wavenumber Domain

213 213 213 228 241

Contents

ix

11 Limited Area Spectral Model 11.1 Introduction 11.2 Map Projection 11.3 Model Equations 11.4 Orography and Lateral Boundary Relaxation 11.5 Spectral Representation and Lateral Boundary Conditions 11.6 Spectral Truncation 11.7 Model Physics and Vertical Structure 11.8 Regional Model Forecast Procedure

252 252 253 254 258 258 259 260 261

12 Ensemble Forecasting 12.1 Introduction 12.2 Monte Carlo Method 12.3 National Center for Environmental Prediction Method 12.4 Florida State University Method 12.5 European Center for Medium Range Forecasts Method 12.6 Superensemble Methodology and Results

263 263 263 265 268 273 277

13 Adaptive Observational Strategies 13.1 Introduction 13.2 Techniques for Targeted Observations

295 295 296

Appendix A

303

Appendix B

304

References

305

Index

313

Chapter 1

Introduction The spectral modeling approach to numerical weather prediction is being practiced in many parts of the world. Historically, the spectral approach came into atmospheric sciences from studies of geomagnetism, where it was introduced by Elsasser in the late nineteenth century. The first attempt of spectral representation of data sets and its use via simple vorticity conserving models came in the 1940s. The works of Neamtan (1946) from the University of Manitoba and Haurwitz (1940) and Craig (1945) from New York University were pioneering during this era. These were still close to linear problems, lacking any formalism for addressing the nonlinear advective dynamics. It was in the late 1950s when we saw the emergence of formal proposals for the solution of the nonlinear barotropic vorticity equation. Pioneering work from the University Chicago (Platzman 1960; Baer 1964) explored what is sometimes called the interaction coefficients approach for the nonlinear problem. Around the same time at MIT we saw the elucidation of what are now called the low-order systems. This pioneering work of Lorenz (1960b) and Saltzman (1959) brought to us the first exposure to simple nonlinear systems and the concept of chaos. These simple three-component systems demonstrated some of the essentials of nonlinear dynamics and the growth of errors arising from initial state uncertainties. The interaction coefficients approach to the solution of the weather forecast problem led to unmanageably large memory requirements that were not easily amenable to the then available, or even to the present, memory of computers. It was during the mid-1950s when the Cooley-Tukey algorithm (1965) emerged and provided a break though via the fast Fourier transform. This was exploited and demonstrated to provide accurate representations of the quadratic terms for fast computation of the nonlinear advective dynamics by contributions from Eliasen et al. (1970) and Machenhauer (1974) from the University of Copenhagen and from Orszag (1970) at MIT. Thereafter, we saw a rapid development of global spectral models in many parts of the world, especially Australia, Canada, England, Japan, and the United States. Noteworthy contributions on the multilevel framework came from Machenhauer and Daley (1972) in Copenhagen, Bourke (1974) from the Bureau of Meteorology Research Center in Melbourne, and Robert (1966), Daley et al. (1976), and Merilees (1968) from the research Provision Numerique in Montreal. Numerous others have contributed to these developments. Currently there are as many as 30 global modeling groups that are active in different parts of the world.

1

2

An Introduction to Global Spectral Modeling

There are several components to the global spectral modeling of the weather prediction problem. In this introductory text, we only provide an exposure to this approach. Knowledge of simple finite differences in the space and time domains are still useful ingredients in the overall construction of spectral models. Chapters 2 and 3 provide an introduction to finite-differencing and timedifferencing procedures. The definition of a spectral model is provided in Chapter 4, where we introduce the concept of the Galerkin techniques. Chapter 5 addresses the Lorenz-type low-order systems with an introduction to chaotic systems. The use of spherical harmonics as basis functions for the casting of meteorological equations in the spectral space is provided in Chapter 6. This chapter also provides the recurrence relations for the accurate computation of associated Legendre functions and their derivatives using the Gaussian quadrature. Spectral relationships for the kinematics of the atmospheric variables are also provided in this chapter. Using the above principles, the construction of simple single-level barotropic and shallow-water spectral models is also presented here. This entails the use of the semi-implicit algorithm and solutions of Helmholtz-type equations. The use of Fourier-Legendre transforms and inverse transforms is an integral part of these models, and this is brought out throughout the chapter. The multilevel spectral weather prediction model is elaborated on in Chapter 7, where the use of a vertical coordinate system following the earth’s surface forms the basis for the definition of variables in the vertical. The vorticity and divergence equations replace the conventional momentum equations. The closed system includes conservation laws for momentum, mass, moisture, and heat. The Fourier-Legendre transform of this basic closed system of equations leads to a coupled system of nonlinear of ordinary differential equations. A reasonably sized weather prediction model carries 100 waves in the zonal and meridional directions over some 20 vertical layers of the atmosphere. Computationally, this amounts to close to 1.5 million coupled ordinary differential equations. These equations can be solved by various methods, such as those described in Chapter 7. These model equations include a number of physical processes, such as the effects of cumulus convection, nonconvective precipitation, surface fluxes of heat, moisture, and momentum, the planetary boundary layer, land-surface processes, radiative transfer, cloud radiative interactions, diurnal change, surface energy balance, effects of orography, and the effects of oceans, snow cover, and ice cover. This is a rather comprehensive list. A brief treatment of the physical processes is provided in Chapter 8. Data initialization issues are addressed in Chapter 9. Here, the emphasis is on two currently popular themes: one called normal mode initialization and the other called physical initialization. The former deals with the suppression of gravity-inertia oscillations arising from initial data imbalances in the mass and motion fields. The latter discusses the issue of improving the rain rates of the model’s initial state. Spectral energetics is a topic that deals with model output diagnostics. Kinetic and potential energy are exchanged among zonal flows and different wave components and are also mutually exchanged among the different waves. Saltzman (1957) laid the foundation for these types of inquiries in the late 1950s and carried out such studies to

Introduction

3

completion in the late 1970s. The theoretical basis for these formulations is presented in Chapter 10. In Chapter 11, the workings of a limited area regional spectral model are provided. Chapter 12 deals with a new area in weather and climate forecasting. This includes a description of the multimodel superensemble that appears to carry higher forecast skills compared to member models of a suite. Chapter 13 addresses a new and upcoming area for forecast modeling called ‘adaptive observational strategies’. This design addresses finding regions where observations are needed for improving the skill of a specific forecast.

Chapter 2

An Introduction to Finite Differencing 2.1

Introduction

This chapter on finite differencing appears oddly placed in the early part of a text on spectral modeling. Finite differences are still traditionally used for vertical differencing and for time differencing. Therefore, we feel that an introduction to finite-differencing methods is quite useful. Furthermore, the student reading this chapter has the opportunity to compare these methods with the spectral method which will be developed in later chapters. One may use Taylor’s expansion of a given function about a single point to approximate the derivative(s) at that point (Fig. 2.1). Derivatives in the equation involving a function are replaced by finite difference approximations. The values of the function are known at discrete points in both space and time. The resulting equation is then solved algebraically with appropriate restrictions. Suppose u is a function of x possessing derivatives of all orders in the interval  x  n'x, x  n'x . Then we can obtain the value of u at points x r n'x , where n is any integer, in terms of the value of the function and its derivatives at point x , that is, u  x and its higher derivatives. For example: u ( x  'x) u ( x) 



d nu dx n

du dx

'x  x

 'x x

n!

d 2u dx 2

n



 'x x

d n 1 u dx n 1

2

 ˜˜˜

2!

n 1

 'x , T  n  1 !

where 'x is finite increment for the value of x and d nu / dx x

2.1)

x

is the value of the nth

derivative at point x . It should be noted that 'x may be negative. However, for convenience we let 'x ! 0 . Furthermore, we assume that x  T  x  'x . Similarly, we may write

u ( x  'x) u ( x) 

du dx

'x  x

4

d 2u dx 2

 'x x

2!

2

 ˜˜˜

(2.2)

An Introduction to Finite Differencing

5

x x  2'x

x x x x x  ' x  x' x  2 'x 

x  

x (2.1) (2.2)   

  

(2.4)

Figure 2.1. equations.

(2.3)

Representation of grid spacing and corresponding Taylor series expansion

  1

n

d nu dx n

 'x x

n

n!

  1

n 1

d n 1u dx n 1

n 1

 'x , T  n  1 !

where x  'x  T  x . Likewise, du dx

u ( x  2'x) u ( x)  d nu  n dx

x

2 'x  x

d 2u dx 2

 2'x T

(2'x) n d n 1u  n 1 n! dx

2

 ˜˜˜

2!

n 1

 2'x T  n  1 !

,

(2.3)

 ˜˜˜

(2.4)

where x  T  x  2'x , and

u ( x  2'x) u ( x)    1

du dx n

2'x  x

d nu dx n

x

d 2u dx 2

 2'x x

2

2!

n 1 (2'x) n u n 1 d   1 n! dx n 1

n 1

 2'x T  n  1 !

,

where x  2'x  T  x . Notice that the right hand sides of u  x  'x and u  x  2'x

contain alternating signs. The value of 'x is taken to be small ('x  1) , such that ('x) n would be even smaller than 'x for n t 2 . We can then approximate the series on the right-hand side by truncating the higher-order terms. As a result, we incorporate some error which is known as truncation error. In general, the more terms we keep on the right-hand side, then the better and more accurate the value of u (x r n'x) would be.

2.2

Application of Taylor’s Series to Finite Differencing

If a function u is defined by an array of points in a single dimension, then by Taylor series expansion (2.1) we get du dx

2 2§d u 'x u ( x  'x)  u ( x)   'x ¨ 2 ¨ dx x ©

x

1 d 3u  2! dx 3

x

· 'x  h.o.t. ¸ , ¸ 3! ¹

where h.o.t. stands for the higher-order terms. Dividing throughout by 'x we obtain

An Introduction to Global Spectral Modeling

6

du dx

x

§ d 2u u ( x  'x)  u ( x)  'x ¨ 2 ¨ dx 'x ©

x

1 d 3u  2! dx 3

x

· 'x  h.o.t. ¸ . ¸ 3! ¹

(2.5)

If we introduce the notation O('x) , which means of the order of 'x , then (2.5) can be written as du dx

u ( x  'x)  u ( x)  O  'x , 'x

x

or

du dx

# x

u ( x  'x)  u ( x) , 'x

(2.6)

where § d 2u O  'x 'x ¨ 2 ¨ dx ©

x

1 d 3u  2! dx3

· 'x  h.o.t. ¸ . ¸ 3! ¹

x

For finding the value of du / dx x , we are only using the values of the function u at points

x and x  'x . All other values are neglected in this calculation. Also, the order of the truncation error involved is O('x) , which is not very desirable.

2.3

Forward and Backward Differencing

Given the values of a function u at discrete points in one dimension, two methods are presented to obtain the approximation for the derivative at a given point. Using (2.1), we can write

du dx

§ d 2u 'x u ( x  'x)  u ( x)  ¨ 2 ¨ dx x ©

 'x x

2

d 3u dx 3



2!

·  h.o.t. ¸ . ¸ 3! ¹

(2.7)

·  h.o.t. ¸ . ¸ 3! ¹

(2.8)

 'x x

3

Furthermore, using (2.2), we can write

du dx

§ d 2u 'x u ( x)  u ( x  'x)  ¨ 2 ¨ dx x ©

 'x x

2

2!



d 3u dx 3

 'x x

3

Dividing the above two equation by 'x , we obtain

du dx

du dx

x

x

u ( x  'x)  u ( x) ¨§ d 2u  ¨ dx 2 'x © u ( x)  u ( x  'x) ¨§ d 2u  ¨ dx 2 'x ©

x

x

x

'x d 3 u  2! dx3

x

Neglecting the terms in the parentheses, we have

·  h.o.t. ¸ , ¸ 3! ¹ 2  'x  h.o.t.¸· . ¸ 3! ¹

 'x

'x d 3 u  2! dx3

2

An Introduction to Finite Differencing

du dx du dx

7

#

u ( x  'x)  u ( x) , 'x

(2.9)

#

u ( x)  u ( x  'x) . 'x

(2.10)

x

x

Equation (2.9) is known as forward finite differencing, and (2.10) is known as backward finite differencing. These are also called first-order accurate one-sided differences. In both of the above, the truncation error is of the order of 'x , that is, O('x) , which corresponds to the largest term we neglected in approximating du / dx x .

Weather forecasters are continually making truncation errors which in turn have an impact on the weather forecast. For example, in the atmosphere advection can occur on small scales. However, the grid spacing in our finite difference representation may be on the order of 100 km, which is too coarse to resolve this advection. This introduces errors into our finite differencing equations that will compound as our forecast proceeds.

2.4

Centered Finite Differencing

In the previous section we obtained the approximation to du / dx x with a truncation error

of O('x) . In this section we will obtain a better approximation for du / dx x based on the centered finite-difference approximation, which has a truncation error of O('x) 2 . By adding (2.7) and (2.8) and dividing by 'x , we obtain 2

du dx du dx

# x

x

§ d 3u u ( x  'x)  u ( x  'x)  2('x) 2 ¨ 3 'x © dx

x

· 1  h.o.t. ¸ , 3! ¹

u ( x  'x)  u ( x  'x) #  O('x) 2 . 2'x

(2.11)

Here the order of the truncation error is ('x) 2 , that is, O('x) 2 , which is the largest term omitted in approximating du / dx x . Higher-order derivatives, namely d nu / dx n x for n t 2 , can also be found using finite-difference methods. Adding (2.1) and (2.2) gives

u ( x  'x)  u ( x  'x)

2u ( x)  2

d 2u dx 2

 'x x

2

2!

where terms with derivatives of the order d nu / dx n

2 x

d 4u dx 4

 'x x

4

 h.o.t. ,

4!

for n=1, 3, 5, … have cancelled

out. This can be written as 2

d 2u dx 2

 'x x

2

§ d 4u u ( x  'x)  u ( x  'x)  2u ( x)  ('x) 4 ¨ 2 4 © dx

2! 2

Now dividing by ('x) throughout, we obtain

x

· 1  h.o.t. ¸ . 4! ¹

An Introduction to Global Spectral Modeling

8

Table 2.1. Some finite difference formulas, along with their accuracy and order of truncation error. Formula

du dx

x

du dx

x

du dx

Accuracy

O('x)

First-order forward

#

u ( x)  u ( x  'x) 'x

O('x)

First-order backward

#

u ( x  'x)  u ( x  'x) 2'x

O('x) 2

Second-order centered

O('x) 2

Second-order centered

x

d 2u dx 2

O(error)

u ( x  'x)  u ( x) # 'x

# x

u ( x  'x)  u ( x  'x)  2u ( x) ('x) 2

d 2u dx 2

§ d 4u u ( x  'x)  u ( x  'x)  2u ( x)  ('x) 2 ¨ 2 4 2 ('x) © dx

x

x

· 1  h.o.t. ¸ . 4! ¹

This can be expressed as

d 2u dx 2

# x

u ( x  'x)  u ( x  'x)  2u ( x)  O('x) 2 , ('x) 2

(2.12)

which is the second-order accurate, second derivative formula. Here the truncation error in the leading term is of the order of ('x) 2 , that is, O('x) 2 . In summary, we may write the first-order forward and backward finite difference formulas and the second-order centered difference formulas as in Table 2.1. We next derive the fourth-order accurate formulas for du / dx x and d 2u / dx 2 x , respectively.

2.5

Fourth-Order Accurate Formulas

2.5.1

First Derivative

The fourth-order finite-differencing schemes can be obtained by an appropriate linear combination of (2.1) to (2.4) such that the terms of order ('x) 2 , ('x)3 , and ('x) 4 are eliminated in the sum and the leading error term is O('x) 4 . That is,

Au ( x)  B >u ( x  'x)  u ( x  'x) @  C >u ( x  2'x)  u ( x  2'x) @ du dx

'x  O ('x)5 .

(2.13)

x

To determine the coefficients A, B, and C, consider the following Taylor series representations:

9

An Introduction to Finite Differencing

u ( x  'x)  u ( x  'x)

2

du dx

'x  x

d 3u dx3

 'x

3



3

x

d 5u dx5

 'x x

5

(2.14)

60

and u ( x  2'x)  u ( x  2'x)

2

du dx

2 'x  x

 2'x

d 3u dx3

3



3

x

 2'x

d 5u dx5

5

.

60

x

(2.15)

Using equation (2.13) and substituting in the right hand sides of (2.14) and (2.15) we obtain ª du d 3u Au ( x)  B « 2 'x  3 dx « dx x ¬ ª du d 3u C «2 2'x  3 dx « dx x ¬

ª du Au ( x)  B « 2 ¬ dx

ª d 3u C «8 3 «¬ dx

º 5  O  'x »  3 » ¼ 3  2'x  O 'x 5 º» # du  3 » dx ¼

 'x x

x

º ª du 'x »  C « 4 x ¼ ¬ dx

º ª d 3u 'x »  B « 3 x ¬ dx ¼

º du 5 »  O  'x # 3 » dx ¼

 'x x

Au ( x)  (2 B  4C )

3

du dx

 'x

5

'x  O  ' x , x 3

3

x



3

'x  ( B  8C ) x

'x , x

d 3u dx3

 'x x

3

3 5

 O  'x #

du dx

'x . x

Equating coefficients of u ( x) , du / dx x 'x , and [d 3u / d x3 ('x)3 ] / 3 , we obtain x

A = 0,

2B + 4C = 1,

B + 8C = 0

A = 0,

B = 2/3,

C = -1/12.

or Using (2.13) and the values of constants A, B, and C, we obtain du dx

2 u ( x  'x)  u ( x  'x) 1 u ( x  2'x)  u ( x  2'x) 4   O  'x , 3 12 'x 'x

x

or by neglecting the terms O('x) 4 , we obtain du dx

# x

4 u ( x  'x)  u ( x  'x) 1 u ( x  2'x)  u ( x  2'x)  . 3 2'x 3 4'x

(2.16)

In deriving (2.16), we have neglected terms O('x)5 and higher in the Taylor series. Therefore this expression for du / dx is accurate to the fourth order.

10

An Introduction to Global Spectral Modeling

2.5.2

Second Derivative

We want to write a general form invoking coefficients A, B, and C such that

Au ( x)  B >u ( x  'x)  u ( x  'x) @  C >u ( x  2'x)  u ( x  2'x) @ #

d 2u dx 2

('x) 2 .

(2.17)

x

We next determine coefficients A, B, and C such that terms of order ('x)3 , ('x) 4 , and

('x)5 are zero. Consider the two equations which come about from the addition of (2.1) and (2.2) and the addition of (2.3) and (2.4), respectively, u ( x  'x)  u ( x  'x)

u ( x  2'x)  u ( x  2'x)

2u ( x) 

d 2u dx 2

2u ( x)  4

2

 'x  x

d 2u dx 2

 'x x

2



d 4u dx 4

 'x x

4 d 4u 3 dx 4

4 6

 O  'x ,

12

 'x

4

(2.18) 6

 O  'x ,

(2.19)

x

so that (2.17) can be written as § · d 2u d 4u ('x) 4 Au ( x)  B ¨ 2u ( x)  2 ('x) 2  4  O('x)6 ¸ ¨ ¸ dx x dx x 12 © ¹ 2 4 § · d 2u d u 4d u 4 C ¨ 2u ( x)  4 2 ('x) 2   'x  O('x)6 ¸¸ # 2 4 ¨ dx x 3 dx x © ¹ dx

 'x

2

.

x

This equation can be rearranged as

( A  2 B  2C )u ( x)  ( B  4C )

d 2u dx 2

4 § B 4C · d u ¨  ¸ 4 © 12 3 ¹ dx

 'x x

 'x x

Equating coefficients of u ( x) , d 2u / dx 2

2

x

4

6

 O  'x #

d 2u dx 2

 'x

x

.

x

('x 2 ) , and d 4u / dx 4

gives A  2 B  2C 0 , B  4C 1 , and B /12  4C / 3 0 . B 4 / 3 , and C 1/12 , so that d 2u dx 2

2

x

('x 4 ) on both sides

This gives A 5 / 2 ,

1 § 5 4 ¨  u ( x )  > u ( x  'x )  u ( x  ' x ) @ ('x) 2 © 2 3 

1 4 · [u ( x  2'x)  u ( x  2'x)] ¸  O  'x . 12 ¹

(2.20)

11

An Introduction to Finite Differencing

Figure 2.2. A 5-point diamond stencil.

This is a fourth-order accurate representation of the second derivative. It can be shown that fourth-order accurate finite difference representation more closely mirrors the exact derivative than second-order accurate finite difference representation.

2.6

Second-Order Accurate Laplacian

2.6.1

5-Point Diamond Stencil

We start with Laplace’s equation, ’ 2\

0,

(2.21)

where ’ 2 is the three-dimensional or two-dimensional Laplacian operator in the Cartesian coordinate system, that is,

’2

w2 w2 w2  2 2, 2 wx wy wz ’2

Let \

w2 w2  2. 2 wx wy

\  x, y be any scalar function of x and y. The finite-difference analog of the

second-order accurate Laplacian will be discussed with reference to the 5-point stencil shown in Fig. 2.2. Let us suppose that we want to calculate ’ 2\ , where \ is the streamfunction. Using Taylor’s expansion (see Appendix A) for a function of two variables about point ( I , J ) , we can write the Taylor expansion of the right point with respect to the central point, that is,

\  I  1, J \  I , J 

w\ wx

'x  I ,J

w 2\ wx 2

 'x I ,J

2!

2

!

12

An Introduction to Global Spectral Modeling



w n\ wx n

 'x I ,J

n



n!

w n 1\ wx n 1

n 1

 'x

(n  1)!

T ,J

,

(2.22)

where (T , J ) is a point on the line joining points ( I  1, J ) and ( I , J ) and 'x is the grid spacing in the x-direction. The Taylor expansion of the left point with respect to the central point is given by

w\ wx

\  I  1, J \  I , J    1

n

w n\ wx n

w 2\ wx 2

'x  I ,J

 'x

  1

2

!

2!

I ,J

n

n!

I ,J

 'x n 1

w n 1\ wx n 1

(2.23)

 'x T ,J

n 1

(n  1)!

,

where (T , J ) is a point on the line joining points ( I  1, J ) and ( I , J ) . Similarly,

\  I , J  1

w\ \ I, J  wy



I ,J

w 2\ 'y  2 wy

 'y

w n\ wy n

n

n!

I ,J



w n 1\ wy n 1

 'y

2

!

2!

I ,J

 'y I ,T

n 1

(n  1)!

,

(2.24)

where ( I ,T ) is a point on the line joining points ( I , J  1) and ( I , J ) and ' y is the grid spacing in the y-direction. Also,

\  I , J  1

w\ \ I, J  wy   1

n

I ,J

w n\ wy n

w 2\ 'y  2 wy

 'y I ,J

n!

 'y 2!

I ,J

n

  1

n 1

2

!

w n 1\ wy n 1

(2.25)

 'y I ,T

n 1

(n  1)!

,

where ( I ,T ) is a point on the line joining points ( I , J  1) and ( I , J ) . In general, we can have 'x z 'y . For simplicity, assume that we have an even mesh, that is, 'x 'y ' . Adding (2.22), (2.23), (2.24), and (2.25), we obtain

\  I  1, J  \  I  1, J  \  I , J  1  \  I , J  1  4\  I , J ’ 2\

I ,J

§ w 4\ ' 4 w 4\ ' 4 · '2  2 ¨ 4  4 ¸ © wx 4! wy 4! ¹

 h.o.t.

(2.26)

I ,J

After dividing by ' 2 and neglecting terms of order ' 2 and higher, we obtain the secondorder accurate Laplacian

’ 2\

I ,J

# ª¬\  I  1, J  \  I  1, J  \  I , J  1  \  I , J  1  4\  I , J º¼ / ' 2 . (2.27)

Weight assigned to the five points ( I  1, J ) , ( I  1, J ) , ( I , J  1) , ( I , J  1) , and ( I , J ) in this 5-point stencil are 1, 1, 1, 1 and -4, respectively.

13

An Introduction to Finite Differencing

Figure 2.3. A 5-point square stencil.

2.6.2

5-Point Square Stencil

There is yet another 5-point stencil which also gives second-order accurate results. This uses the points shown in Fig. 2.3. To obtain the finite-difference expression for the ’ 2 operator, we use the following equations:

§ w\ w\ ·  ¸ © wx wy ¹ I , J ' 2 § w 2\ w 2\ · w 2\  ¨ 2  2 ¸  '2 wy ¹ I , J wxwy 2! © wx § w\ w\ ·  \ I, J  '¨ ¸ © wx wy ¹ I , J

\  I  1, J  1 \  I , J  ' ¨

\  I  1, J  1

2 ' 2 § w 2\ w 2\ · 2 w \ ¨ 2  2 ¸ ' wy ¹ I , J wxwy 2! © wx § w\ w\ ·  \ I, J  '¨ ¸ © wx wy ¹ I , J

(2.28)  !, I ,J

(2.29)



\  I  1, J  1

2 ' 2 § w 2\ w 2\ · 2 w \ ¨ 2  2 ¸ ' wy ¹ I , J wxwy 2! © wx § w\ w\ · \ I, J  '¨  ¸ © wx wy ¹ I , J

 !, I ,J

(2.30)



\  I  1, J  1



' 2 § w 2\ w 2\ ·  ¨ ¸ 2! © wx 2 wy 2 ¹

 '2 I ,J

w 2\ wxwy

! , I ,J

(2.31)

! . I ,J

After adding the above four equations, we obtain

\  I  1, J  1  \  I  1, J  1  \  I  1, J  1  \  I  1, J  1  4\  I , J § w 2\ w 2\ · 2' 2 ¨ 2  2 ¸ wy ¹ © wx

 I ,J

' 4 § w 4\ w 4\ w 4\ · ¨ 4  4 6 2 2 ¸ 6 © wx wy wx wy ¹

 h.o.t. I ,J

By taking differences along the diagonal, (2.32) can also be written as

(2.32)

An Introduction to Global Spectral Modeling

14

Figure 2.4. Diamond and square finite-difference representations of the Laplacian and their ability to represent an area of low-pressure.

\  I  1, J  1  \  I  1, J  1  \  I  1, J  1  \  I  1, J  1  4\  I , J ’ 2\

2’ 2\

I ,J

2 '

I ,J

2

2

4

 2 ' ’ \ ! 2 1

1

2

4

4!

1 ' 2  ’ 4\' 4 !. 3

(2.33)

Dividing both sides by 2' 2 , we can write

’ 2\

1 ª\  I  1, J  1  \  I  1, J  1  \  I  1, J  1 2' 2 ¬ 2 \  I  1, J  1  4\  I , J º¼  O  ' .

(2.34)

This gives the finite-difference expression for the second-order Laplacian represented by the 5-point square stencil. Which 5-point stencil is more accurate, and why? Using a 5-point stencil with diamond configuration of points is slightly more accurate than a 5-point square stencil. This is because the distance from the center point ( I , J ) to the other points is less in the diamond stencil (') than the corresponding distance in the square stencil (21/2'). It is important to note that the analytical Laplacian is invariant with respect to the rotation of the coordinate system, i.e. ’ 2 has one and only one value. However, finite difference representations (diamond or square) of the Laplacian are not invariant to coordinate transformations. That is, we will get different answers for different finite difference schemes. For example, suppose we have a 5-point diamond stencil with an area of low-pressure located to the northwest of the grid points (Fig. 2.4). In this particular case, the diamond stencil will not allow us to catch this low-pressure area in our finite difference representation. However, by using a 5-point square stencil we were able to catch this area. 2.6.3 9-Point Stencil

There is yet another second-order accurate Laplacian which is based on a 9-point stencil. These nine points are shown in Fig. 2.5, with the corresponding weights written in

An Introduction to Finite Differencing

15

Figure 2.5. A 9-point stencil used for the second-order Laplacian with the weights assigned to each node given in parentheses.

parentheses. It should be noted that this configuration is made up of the two 5-point stencils discussed earlier. Consider now the Taylor expansion of the eight outer points with respect to the central point ( I , J ) . Combining (2.26) and (2.33) linearly in such a way that the terms O(') 2 in the two equations are neglected at the same accuracy, we obtain

\  I  1, J  1  \  I  1, J  1  \  I  1, J  1  \  I  1, J  1 4 ¬ª\  I , J  1  \  I  1, J  \  I , J  1  \  I  1, J ¼º  20\  I , J

6' 2’ 2\

I ,J

2  ' 4’ 4\ 3

I ,J

 h.o.t.

(2.35)

Dividing both sides by 6' 2 , we obtain 1 ^\  I  1, J  1  \  I  1, J  1  \  I  1, J  1  \  I  1, J  1 6' 2 2 4 ª¬\  I , J  1  \  I  1, J  \  I , J  1  \  I  1, J º¼  20\  I , J  O  ' . (2.36)

’ 2\

I ,J

This formulation uses more information around the central point. Therefore the results can be locally more accurate than the 5-point stencil given by (2.27) or (2.34).

2.7

Fourth-Order Accurate Laplacian

Let us attempt to formulate a fourth-order accurate Laplacian using the two different 5point stencils discussed earlier. We make use of the gridpoint representation in Fig. 2.6. Using (2.26), we can write

An Introduction to Global Spectral Modeling

16

Figure 2.6. The nine grid points for a fourth-order accurate Laplacian.

\  I  1, J  \  I  1, J  \  I , J  1  \  I , J  1  4\  I , J ’ 2\

I ,J

'2  2

' 4 § w 4\ w 4\ ·  ¨ ¸ 4! © wx 4 wy 4 ¹

2 I ,J

' 6 § w 6\ w 6\ ·  ¨ ¸ 6! © wx 6 wy 6 ¹

 h.o.t.

(2.37)

I ,J

Similarly, using the five grid points ( I  1, J  1) , ( I  1, J  1) , ( I  1, J  1) , ( I  1, J  1) , and ( I , J ) , we can write

\  I  1, J  1  \  I  1, J  1  \  I  1, J  1  \  I  1, J  1  4\  I , J 2

4

 2 ' §¨ w \  w \ ·¸ 2 1

2

4

 2 ' 4! © wx  2 ' §¨ w \  w \ ·¸  h.o.t. 2 1

’ 2\

2

I ,J

1

4

4

wy 4 ¹

I ,J

6

2

6

6!

© wx

6

6

wy 6 ¹

(2.38)

I ,J

Note that here the distance between grid point ( I , J ) and its surrounding points is 21 2 ' . To eliminate the terms O(' 4 ) from the above two equations, we multiply (2.37) by 4, from the resultant equation subtract (2.38), and divide by 2' 2 to obtain ’ 2\

I ,J

1 4 ª\  I , J  1 \  I  1, J  \  I , J  1 2' 2 ¬ \  I  1, J ¼º  ¬ª\  I  1, J  1  \  I  1, J  1

^

\  I  1, J  1  \  I  1, J  1 º¼  12\  I , J `



' 4 § w 6\ w 6\ ·  ¨ ¸ 180 © wx 6 wy 6 ¹

 h.o.t. I ,J

An Introduction to Finite Differencing

17

Figure 2.7. Another 9-point stencil for a fourth-order accurate Laplacian.

Hence the fourth-order accurate Laplacian obtained by using two 5-point stencils (in all we use nine grid points) is expressed as

’ 2\

I ,J

1 4 ª\  I , J  1 \  I  1, J  \  I , J  1 2' 2 ¬ \  I  1, J ¼º  ¬ª\  I  1, J  1  \  I  1, J  1

^

(2.39)

\  I  1, J  1  \  I  1, J  1 º¼  12\  I , J `  O  ' . 4

Continuing on the same idea, we will next formulate a fourth-order Laplacian using the nine-grid-point representation in Fig. 2.7. First, using the inner diamond grid points ( I  1, J ) , ( I  1, J ) , ( I , J  1) , ( I , J  1) , and ( I , J ) , we can write

\  I  1, J  \  I , J  1  \  I  1, J  \  I , J  1  4\  I , J ' 4 § w 4\ w 4\ ·  ¨ ¸ 4! © wx 4 wy 4 ¹ ' 6 § w 6\ w 6\ · 2 ¨ 6  6 ¸  h.o.t. wy ¹ I , J 6! © wx ’ 2\

I ,J

'2  2

I ,J

(2.40)

Next using the outer diamond grid points ( I  2, J ) , ( I  2, J ) , ( I , J  2) , ( I , J  2) , and ( I , J ) , which have grid spacing 2' we can write

\  I , J  2  \  I  2, J  \  I , J  2  \  I  2, J  4\  I , J

An Introduction to Global Spectral Modeling

18

(2') 4 § w 4\ w 4\ ·  ¨ ¸ 4! © wx 4 wy 4 ¹ (2')6 § w 6\ w 6\ · 2  ¨ ¸  h.o.t. 6! © wx 6 wy 6 ¹ I , J ’ 2\

I ,J

 2'

2

2

We seek a representation for ’ 2\

I ,J

I ,J

(2.41)

without the term of order ' 2 . This is

achieved by multiplying (2.40) by 16, then subtracting (2.41) from the resulting equation, and dividing by 12' 2 to obtain ’ 2\

I ,J

1 16 ª¬\  I  1, J \  I , J  1  \  I  1, J 12' 2 \  I , J  1 º¼  ª¬\  I , J  2 \  I  2, J  \  I , J  2

^

\  I  2, J º¼  60\  I , J 

' 4 § w 6\ w 6\ ·  ¨ ¸ 90 © wx 6 wy 6 ¹

 h.o.t. , I ,J

or ’ 2\

I ,J

1 16 ª¬\  I  1, J \  I , J  1  \  I  1, J 12' 2 \  I , J  1 º¼  ª¬\  I , J  2 \  I  2, J  \  I , J  2

^



\  I  2, J º¼  60\  I , J  O ' 4 .

(2.42)

Equation (2.42) represents yet another fourth-order accurate Laplacian. In general, this idea of adding and subtracting finite difference analogs can be used to develop finitedifferencing schemes of various higher orders. In review, the various finite difference representations of the Laplacian and their corresponding equations are given below in order from most accurate to least accurate representation: Square stencil 9-point Fourth-order accurate

Interspersed diamonds 9-point Fourth-order accurate

Square stencil 9-point Second-order accurate

x x x

x x x

x x x

x x x x x x x x x x x x

x x x

x x x

Equation (2.39)

Equation (2.42)

Equation (2.36)

An Introduction to Finite Differencing

Diamond stencil 5-point Second-order accurate

x

x x x x

Equation (2.27)

x

Equation (2.34)

x

Square stencil 5-point Second-order accurate

2.8

19

x x

x

Elliptic Partial Differential Equation in Meteorology

The most commonly occurring elliptic partial differential equations in atmospheric modeling are of the type

’ 2I ’ 2I  H I

G G

Poisson’s equation, Helmholtz equation.

(2.43) (2.44)

These equations occur in the relationship between vorticity and streamfunction, that is, ’ 2\ ] , where \ is the streamfunction and ] is the relative vorticity. A similar type of relationship occurs between divergence and velocity potential, or when going from I to \ and from \ to I in a balance equation, or when obtaining time tendencies in the quasigeostrophic prediction models or in the solution of the omega (Z) equation (Holton 1992). There are two ways in which we can solve these equations. One is based on the direct solution of simultaneous equations, which works efficiently for small domains but is computationally heavy for large domains. The other is called the relaxation method, which is more efficient for large domains.

2.9

Direct Method

For solving the three-dimensional elliptic boundary value problem of the type (2.43) and (2.44), one uses a similarity transform method involving matrices. A general outline of the method is to first write the given equation in finite difference form for each of the N vertical levels. This gives rise to a set of N linear algebraic equations. This set of N linear algebraic equations is then written in matrix form and solved. Consider the even-mesh domain shown in Fig. 2.8 on a horizontal plane with 'x 'y ' . Here i and j are nodes in x and y, respectively. i 1(1) K and j 1(1) L , that is, i and j vary from 1 to K and L, respectively, with increments of 1. Assume that I and G are functions of x, y, and z. Then (2.44) can be written as

’ 22I 

w 2I  HI wz 2

G.

The finite-difference representation for w 2I / wz 2 can be written as

(2.45)

An Introduction to Global Spectral Modeling

20

Figure 2.8. A horizontal mesh with 'x

w 2I wz 2

Ik 1  2Ik  Ik 1 ('z ) 2

'y

'.

,

(2.46)

where 'z is the distance between levels k and k  1 . The finite-difference form of (2.45) can be written as

’ 22Ik 1  2’ 22Ik  ’ 22Ik 1 Ik 1  2Ik  Ik 1   H k Ik '2 ( 'z ) 2

Gk

(2.47)

for k  2(1)( N  1) where ' 'x 'y . Thus the three-dimensional elliptic partial differential equation is transformed into a two-dimensional equation. For the top layer (k 1) we have ’ 22I2  2’ 22I1 I2  2I1   H1I1 ('z ) 2 '2 For the bottom layer (k

G1 .

(2.48)

N ) we have

2’ 22I N  ’ 22I N 1 I N 1  2I N   H NIN '2 ('z ) 2

GN ,

(2.49)

where we assume I0 I N 1 0 . This is equivalent to saying that the geopotential anomaly vanishes at the top and bottom. Rewriting (2.47), (2.48), and (2.49) in matrix form, we obtain

’ 22  A)  B)

G* ,

(2.50) *

where A and B are N u N coefficient matrices while ) and G are column vectors, that is,

An Introduction to Finite Differencing

21

ª I1 º «I » « 2 » « I3 » « » « # » «¬ I N »¼

)

and

ª G1 º «G » « 2» « G3 » . « » « # » «¬GN »¼

G*

Here Ik and Gk are the values of I and G at a particular level k and a fixed grid point in the horizontal. After premultiplying (2.50) by the matrix B 1 , we obtain

B 1’ 22  A)  B 1 B)

B 1G * .

(2.51)

2 2

Also, using the distributive property of the ’ operator, we obtain





B 1’ 22  A) ’ 22 B 1 A)  A)’ 22 B 1

and ’ 22 B 1

(2.52)

0 , since B 1 is a matrix of numbers which are constant. Noting also that

B 1 B 1 , we then have





’ 22 B 1 A)  )

B 1G * .

(2.53)

Let B 1 A C , a new matrix of order N u N . Then (2.53) can be written as ’ 22  C )  )

B 1G * .

(2.54)

One can now diagonalize matrix C using a similarity transformation. This calls for two matrices U and U 1 such that UCU 1

D,

(2.55)

where D is a diagonal matrix. After rearranging, we obtain C U 1 DU ,

(2.56)

where D is a diagonal matrix shown in Fig. 2.9. Each d k , k 1(1) N is an eigenvalue of matrix C, that is, CX k

dk X k

for some column vector X k . Substituting (2.56) into (2.54), we obtain

(2.57)

An Introduction to Global Spectral Modeling

22

Figure 2.9. The elements of the diagonal matrix D.





’ 22 U 1 DU )  )

B 1G * ,

or premultiplying by U, we can write

’ 22  DU )  U ) UB 1G * . Let us define U ) V (a column vector) and UB 1 (2.58) can be rewritten as

V  ’ 22  DV

(2.58) F (an N u N matrix). Then

FG* .

(2.59)

Equation (2.59) is the equivalent matrix representation for a set of N linearly independent Helmholtz equations. Solutions of (2.59) will provide us with the values of V, that is,

V

ª V1 º «V » « 2» « V3 » . « » « #» «¬VN »¼

Since we have U ) V , we can write

) U 1V . Thus if the value of V is known, then by premultiplying V by U determine ). Now ) is known, that is,

)

ª )1 º «) » « 2» « )3 » . « » « # » «¬) N »¼

(2.60) 1

we can easily

An Introduction to Finite Differencing

23

Next we obtain matrix U. Let d k X k , k 1(1) N be the eigenpairs of matrix C; in other words, d k is the eigenvalue and X k is the corresponding eigenvector. It should be noted that there is the possibility that we have more than one eigenvector corresponding to a single eigenvalue, but to simplify matters we assume that there are N distinct eigenpairs d k X k , k 1(1) N . Since d k X k is an eigenpair of C, we have CX k d k X k such that CX

XD ,

(2.61)

where X is the matrix consisting of the eigenvectors of C. Postmultiplying (2.61) by X 1 gives

C

XDX 1 .

(2.62)

X.

(2.63)

From (2.56) and (2.62), we obtain U 1

Hence U is the inverse matrix of X.

2.10 Relaxation Method This method is commonly used for solving Poisson- and Helmholtz-type equations. Given the 5-point stencil, the Laplacian of any function I is calculated as

’ 2I

Ii 1, j  Ii 1, j  Ii , j 1  Ii , j 1  4Ii , j '2

,

(2.64)

where Ii , j is the value of I at the node point (i, j ) . Also assume that 'x

'y

' . The

Helmholtz equation ’ 2I  PI

F,

(2.65)

can then be written as 2 Ii 1, j  Ii 1, j  Ii , j 1  Ii , j 1  ª 4  P  'x º Ii , j

¬

¼

2

Fi , j  'x .

(2.66)

In the relaxation method, the values of F are specified at each of the interior points of the domain. It is desired to find the values of Ii , j at every point which satisfy (2.66) and the given boundary conditions. There are three types of boundary conditions that are commonly used. One is where the value of I is prescribed at the boundaries (called the Dirichlet boundary condition). Another is where the value of the normal derivative wI wn is specified at the boundaries (called the Neumann boundary condition). The last one is the mixed boundary condition, that is, wI / wn  EI is specified at the boundaries. These are three possible boundary conditions. Usually any other prescription defines an over specified system. Next we describe the procedure to obtain the solution using relaxation techniques. To begin, we assume a first guess field for I, say I0 , over the domain and ask, does I0

An Introduction to Global Spectral Modeling

24

satisfy the Helmholtz equation and the boundary conditions? If yes, then we have a solution. Otherwise, find the difference ’ 2I0  PI0  F . This is the residual between the actual solution and the first guess, that is, ’ 2I0  PI0

F R,

(2.67)

where R is residual. Now we aim to change I0 iteratively such that R is minimized to an acceptable degree of tolerance for every point in the domain. There are two types of relaxation schemes, the simultaneous relaxation scheme and the sequential relaxation scheme. The simultaneous relaxation scheme uses the original values of Ii , j from the previous iteration to calculate values at the next iteration. The sequential relaxation scheme uses new values of Ii , j for calculating values at the next iteration. As will be shown below, the sequential relaxation scheme turns out to be faster than the simultaneous relaxation. The analysis presented here follows a procedure described by Thompson (1961). We now address the simultaneous relaxation scheme, assuming that we are at level m of iteration at grid point (i, j ) . Then

Iim1, j  Iim1, j  Iim, j 1  Iim, j 1   4  H Iim, j where H

P d 2 and d 2

mth iteration. Note that R

(2.68)

('y ) 2 . Rim, j is the residual at grid point (i, j ) at the

('x) 2 m i, j

d 2 Fi , j  Rim, j ,

2

Rd and has the dimensions of I. We change the value of

Ii , j such that the residual vanishes locally in the above equation. If Iim, j is changed to Iim, j1 for iteration m  1 , then we obtain

Iim1, j  Iim1, j  Iim, j 1  Iim, j 1   4  H Iim, j1

d 2 Fi , j .

(2.69)

If we subtract (2.69) from (2.68), we obtain

GIim, j

Iim, j1  Iim, j

Rim, j 4 H

,

(2.70)

where GIim, j denotes the change in Ii , j necessary to make the residual vanish from (2.68). We next address the convergence of the above interation procedure. Let us assume that Ii , j is the solution. Then

Ii 1, j  Ii 1, j  Ii , j 1  Ii , j 1   4  H Ii , j

d 2 Fi , j .

(2.71)

It should be noted that there is no residual term Ri , j in the above equation since Ii , j is the exact solution. Subtracting (2.71) from (2.68), the error equation at the mth iteration is given by im1, j  im1, j  im, j 1  im, j 1 (4  H ) im, j Rim, j , where im, j is the error in Ii , j at the mth iteration. From the governing Helmholtz equation we have

(2.72)

25

An Introduction to Finite Differencing

ˆ 2 m  P d 2 m R m , ’ i, j i, j i, j

(2.73)

where ˆ 2         4  . ’ i, j i 1, j i 1, j i , j 1 i , j 1 i, j Furthermore, when Iim, j is changed to Iim, j1 so as to make the residual term vanish from the above equation, we obtain Rim, j

im, j 1  im, j

.

4 H

(2.74)

Using (2.73) and (2.74), we obtain

1 ˆ 2 m  P d 2 m ’ i, j i, j 4 H



im, j 1 im, j 



(2.75)

and the ratio

im, j 1

ˆ 2 m  4 m ’ i, j i, j

im, j

(4  H )im, j

.

(2.76)

If (im, j 1 im, j )  1 , then the error decreases as we increase the number of iterations, and the

scheme converges. Now we ask ourselves, what is the condition that will guarantee convergence? To examine this we express the error function at grid point (k , l ) at iteration m by a Fourier expansion, that is, mk ,l

¦¦ A

m i ( pk 'x ) i ( ql 'y ) p ,q

p

e

e

,

q

where p and q denote the east-west and north-south wavenumber, respectively, and m is the number of iterations. We use here the grid-point index (k , l ) instead of (i, j ) to avoid confusion with i (1)1 2 . Substituting the above into (2.75), we obtain

¦¦ A

m 1 i ( pk 'x ) i ( ql 'y ) p ,q

p

e

e

q

¦¦ A

m i ( pk 'x ) i ( ql 'y ) p ,q

p



1 4 H

e

e

q

§ 2 m ˆ  P d 2 ¨¨ ’ i, j ©

¦¦ A

m i ( pk 'x ) i ( ql 'y ) p ,q

p

e

q

e

· ¸¸ . ¹

(2.77)

ˆ 2 m equal to Furthermore, the expression for ’ k ,l ˆ 2 m ’ k ,l

¦¦ A  e m p ,q

p

i[ p ( k 1) 'x ] i ( ql 'y )

e

 ei[ p ( k 1) 'x ] ei ( ql'y )  ei ( pk 'x ) ei[ q ( l 1) 'y ]

q



 ei ( pk 'x ) ei[ q ( l 1) 'y ]  4 ei ( pk 'x ) ei ( ql 'y ) .

For the sake of simplicity, we have again assumed that ' x ' y .

(2.78)

An Introduction to Global Spectral Modeling

26

ˆ 2 m into (2.77) and equating the Substituting the above expression for ’ k ,l coefficients of ei ( pk 'x ) ei ( ql 'y ) , we can write Apm,q1

1 § · u eip'x  e  ip'x  eiq'y  4  P d 2 ¸ . Apm,q ¨1  H 4  © ¹





(2.79)

Using the Euler relation ei x  e  i x , 2

cos x

(2.80)

we obtain Apm,q1

1 § · Apm,q ¨1  2 cos p ' x  2 cos q ' y  4 P d 2 ¸ . H  4 © ¹





Furthermore, using cos 2 x 1  2sin 2 x and H Apm,q1

ª 1 Apm,q «1  ¬ 4 H

P d 2 , we obtain

q' y § ·º 2 p'x  2  4sin 2  4  H ¸» , ¨ 2  4sin 2 2 © ¹¼

or

Apm,q1

ª 1 Apm,q «1   H 4 ¬

q' y § ·º 2 p'x  4sin 2  H ¸» , ¨ 4sin 2 2 © ¹¼

(2.81)

so that Apm,q1 Apm,q

1

1 4 H

q' y § · 2 p'x  4sin 2 H¸. ¨ 4sin 2 2 © ¹

(2.82)

Let v be defined by

Q2

sin 2

p' x p' y  sin 2 d 2. 2 2

(2.83)

Hence we obtain

Apm,q1 Apm,q

1

1 4Q 2  H . 4 H





(2.84)

The ratio on the left-hand side of the above equation can satisfy one of the following possible scenarios. If Apm,q1 Apm,q

 1,

then we have strong convergence. If

Apm,q1 Apm,q

1,

then this is the neutral or nonamplifying case. If

An Introduction to Finite Differencing

27

Apm,q1 Apm,q

! 1,

then the scheme will not converge. We can increase the convergence role of a relaxation scheme by the use of an appropriate relaxation factor. The change in Ii , j from iteration m to m  1 is

Iim, j1  Iim, j

Rim, j

GIim, j

4 H

.

(2.85)

ˆ m , where In the case of over relaxation or underrelaxation, we change Iim, j by GI i, j ˆ m DGI m D GI i, j i, j

Rim, j 4 H

.

(2.86)

Here D is a relaxation factor. Next we show that this relaxation factor will result in faster convergence. If D 1 , we are back to GIim, j Rim, j /(4  H ) . For D z 1 and for larger scales, p'x and

q'y are small. The term Q 2 is very small if we choose D ! 1 . Then (2.84) becomes Apm.q1

1

m p ,q D

A

D 4 H

(4Q 2  H ) .

(2.87)

We can select the values of the relaxation factor D such that Apm.q1 m p ,q D

A

where Apm,q1 / Apm,q

D



Apm.q1 Apm,q

,

(2.88)

denotes that we are using the relaxation factor D which is not equal to

1. For large scales (i.e., for small p and q) in which 4Q 2  H  1 , 4 H an overrelaxation factor (D ! 1) would give rise to a faster rate of convergence. However, if scales are small (i.e., p and q are larger), then overrelaxation does not work. In that case, an underrelaxation coefficient (D  1) is better suited. An overrelaxation coefficient with values around 1.2 to 1.7, depending up the domain, is generally helpful for fast convergence. The scale of the wave depends on the domain. The larger the domain, then generally the larger the wavenumbers p and q will be.

2.11 Sequential Relaxation Versus Simultaneous Relaxation Here we answer the question, is sequential relaxation more efficient than simultaneous relaxation? We consider a Helmholtz equation in one dimension only, i.e.,

An Introduction to Global Spectral Modeling

28

w 2I  PI wx 2

F.

(2.89)

To make the problem simple, we consider the domain consisting of points i  2 , i  1 , i  1 , and i  2 , where i  2 and i  2 are boundary grid points with known prescribed values of Ii  2 and Ii  2 . We can express w 2I / wx 2 i in its finite-difference form as

I  I  2Ii w 2I # i 1 i 1 2 , ('x) wx 2 i

(2.90)

where 'x is the distance between two successive grid points and Ii is the value of I at the ith grid point. Substituting this into (2.89), we obtain 2

Ii 1  Ii 1  2Ii  P  'x Ii If H

P ('x) 2 and ('x) 2

2

Fi  'x .

(2.91)

d 2 , then we obtain

Ii 1   2  H Ii  Ii 1

Fi d 2 .

(2.92)

First consider simultaneous relaxation. At the mth level of iteration we have

Iim1   2  H Iim  Iim1

Fi d 2  Rim .

(2.93)

At the (m + 1)th iteration, Iim is changed to Iim 1 so that Rim vanishes. This gives

Iim1   2  H Iim 1  Iim1

Fi d 2 .

(2.94)

The corresponding error equations are im1   2  H im 1  im1 0 at grid point i,

(2.95)

0  2  H 

(2.96)

m 1 i 1

 2  H  m i

m i



m 1 i 1

0 at grid point i - 1,

0 at grid point i + 1.

(2.97)

It should be noted that there are no terms containing im 2 and im 2 , since Ii  2 and Ii  2 are known boundary values and only Ii 1 , Ii , and Ii 1 are to be determined. One additional iteration at point i results in im11   2  H im  2  im11 0 . After eliminating im11 and im11 with the help of the error equations (2.96) and (2.97) at grid points (i  1) and (i  1) , we obtain im m   2  H im  2  i 2 H 2 H 2 m 2 m 2 i   2  H i 0,

0,

(2.98) (2.99)

An Introduction to Finite Differencing

im  2

29

2 im . (2  H ) 2

(2.100)

Next consider the sequential relaxation procedure. Here we use the new values of the function at the previously corrected points during an iteration. The corresponding error equation is im11   2  H im 1  im1 0

at grid point i.

(2.101)

We have used the changed value of I at the (i -1)th grid point. Furthermore,

0   2  H im11  im 0

at grid point i - 1.

(2.102)

Similarly, we obtain im 1   2  H im11 0 0

at grid point i + 1.

(2.103)

The error at the (i -1)th grid point is related to the error at the ith grid point by im . 2 H

im11

(2.104)

Substituting the above value of im11 into (2.101) gives im   2  H im 1  im1 0 , 2 H

or 2

im   2  H im 1   2  H im1 0 .

(2.105)

From the error equation at the (i  1)th grid point at the mth iteration, we can get im1 im /(2  H ) . Substituting into (2.105), we obtain 2

im   2  H im 1  im 0 ,

or

im 1

2 im . (2  H ) 2

(2.106)

Thus, the convergence given by (2.106) is twice as fast as that given by (2.100). In conclusion, sequential relaxation converges faster than simultaneous relaxation.

2.12 Advective Nonlinear Dynamics Advective nonlinear dynamics is one of the most difficult processes to resolve. It creates a lot of noise in a forecast. Arakawa was the first to show how to correctly use finite differencing for nonlinear dynamics. He explained that to understand the numerics of nonlinear advective dynamics you must first be able to compute the finite difference of the Jacobian, which is simple nonlinear advective dynamics. The barotropic vorticity

An Introduction to Global Spectral Modeling

30

equation allows us to understand the properties of the Jacobian from which we can then ensure that the numerics meet these requirements. 2.12.1 Barotropic Vorticity Equation Consider the horizontal momentum equations and the nondivergent continuity equation in the (x, y, p) coordinate system which make up the primitive form of the barotropic vorticity equation: wu wu wu wz fv  g , u v wt wx wy wx wv wv wv wz u v  fu  g , wt wx wy wx wu wv  0. wx wy

(2.107) (2.108)

(2.109)

Here we have three equations with three unknowns: u, v, and z. For nondivergent flow we let u w\ / wy , v w\ / wx , and ’ 2\ wv / wx  wu / wy . By differentiating (2.107) with respect to y and (2.108) with respect to x and subtracting the two resulting equations, we get the barotropic vorticity equation: w 2 ’\ wt





 J \ , ’ 2\  E

w\ , wx

(2.110)

which consists of the time tendency of vorticity, the horizontal advection of vorticity represented by the Jacobian function, and the beta term, respectively, where E wf / wy . In this equation, the Jacobian is a nonlinear term and all other terms are linear. Also differentiating (2.107) with respect to x and (2.108) with respect to y and adding, we obtain the nonlinear balanced equation: ’ 2I

§ w\ w\ · ’ ˜ f ’\  2 J ¨ , ¸, © wx wy ¹

(2.111)

where we have neglected the time-derivative terms. The Jacobian function J is defined as

J  A, B

wA wB wA wB  . wx wy wy wx

Here we have two equations and two unknowns, \ and I. Arakawa showed that the barotropic vorticity model (2-D) has several integral constraints. These constraints that are maintained are kinetic energy, total vorticity, and total square vorticity. That is, the kinetic energy of nondivergent flow and the absolute vorticity have the following invariant properties over a closed domain: domain invariant :

u 2  v2 , ] a , ] an . 2

31

An Introduction to Finite Differencing

In this case, ] a i.e.,

]  f , n is a real number, and ( ) denotes the mean over the domain,

³³ Q dx dy . ³³ dx dy

Q

Conservation of these properties precludes nonlinear computational instability. Conservation of enstrophy ( ] a2 ) and kinetic energy is called quadratic invariance. If one satisfies quadratic invariance, then ] an for n ! 2 also seems to remain well bounded. The parcel-invariant physical quantities are ] a and ] an . If we set n = 2 in the domaininvariant quantity ] an , then we obtain ] a2 as invariant. These are important invariants (both parcel and domain). Next we prove the domain invariance for ] a2 . We start with barotropic vorticity equation (2.110), which can be written as

w ’ 2\  f wt









 J \ , ’ 2\  f .

(2.112)

By multiplying (2.112) by ’ 2\  f , we obtain 2

§ ’ 2\  f  J ¨\ , ¨ 2 ©





J ( A, B)

wA wB wA wB  wx wy wy wx

2 w ’\  f wt 2





2

· ¸ ¸ ¹

since

wA w B 2 wA w B 2  wx wy 2 wy wx 2

B J ( A, B)



J A, B

2

2

.

Integrating over the closed domain D leads to



2 w ’\  f 2 D wt

³³



2

dx dy



§ ’ 2\  f J ¨\ , ¨ 2 D ©



³³



2

· ¸ dx dy ¸ ¹

0,

since the integral of a Jacobian over a closed domain vanishes. Hence w (’ 2\  f ) 2 wt 2

0.

(2.113)

An Introduction to Global Spectral Modeling

32

Figure 2.10. A 5-point stencil for the Jacobian using the vector identity.

Therefore

(’ 2\  f ) 2 2 is invariant with time. Thus the square of absolute vorticity is conserved over a closed domain. In the same manner, it can be shown that all powers of absolute vorticity (i.e., ] an ) are time-invariant. Next we show the conservation of k

(u 2  v 2 ) / 2 . Multiply (2.112) by \ to

obtain

\

§\ 2 2 ,’ \  J ¨ © 2

w 2 ’\ wt

· f ¸. ¹

(2.114)

The right-hand side vanishes on integration over a closed domain. Hence

³³ \ D

w 2 ’ \ dx dy wt

0.

(2.115)

Using the vector identity

\

w 2 ’\ wt

w\ · w ’\ .’\ § , ’ ˜ ¨\’ ¸ wt ¹ wt 2 ©

we obtain w

§

w\ ·

w ’\ ˜’\ dx dy . (2.116) 2

³³ \ wt ’ \ dx dy ³³ ’ ˜ ¨©\’ wt ¸¹ dx dy  ³³ wt D

2

D

D

As the first term on the right-hand side of (2.116) vanishes over a closed domain, we are left with

33

An Introduction to Finite Differencing 2

w  ’\ dx dy 2 w D t

³³

w

³³ \ wt ’ \ dx dy 2

0,

D

or

ª w «§ w\ ˆ w\ i ¨ wy wt «© wx ¬«

w (’\ ) 2 wt 2

ˆj ·¸ ¹

2

º 2» » »¼

2 ª º 2 w «§ w\ · § w\ · »  2 ¨ ¸ ¨ ¸ » wt «© wx ¹ © wy ¹ «¬ »¼

w u 2  v2 0. (2.117) wt 2 Thus the kinetic energy over a closed domain is time-invariant. In designing any simple atmospheric model, we generally try to write the finite differencing schemes so as to

satisfy conservation of kinetic energy (k ) , vorticity (] a ) , and the square of vorticity (] a2 ) over a closed domain.

2.13 The 5-Point Jacobian Let us first consider a 5-point Jacobian. Given the definition J ] ,\

w] w\ w] w\  wx wy wy wx

and starting with the five grid points as shown in Fig. 2.10, the Jacobian J(], \) can be expressed by ]  ] \ \ 4 ] 2  ] 4 \ 1 \ 3  J ] ,\ # 1 3 2 . 2'x 2'y 2'y 2 'x If 'x 'y ' , then 1 J ] ,\ (2.118) >(] 1  ] 3 ) (\ 2 \ 4 )  (] 2  ] 4 ) (\ 1 \ 3 )@  O  ' 2 . 4' 2

This is a second-order accurate Jacobian. It should be noted that this Jacobian does not satisfy all invariants.

2.14 Arakawa Jacobian We discuss here the design of a Jacobian which conserves kinetic energy and enstrophy (square of the vorticity) over a closed domain. This form of the Jacobian was first designed by Arakawa (1966), and is commonly known as the Arakawa Jacobian. We consider both the second- and fourth-order accurate Arakawa Jacobians. We can write J (] ,\ ) in the following three forms:

An Introduction to Global Spectral Modeling

34

Figure 2.11. A 9-point stencil for the Jacobian.

J ] ,\ J ] ,\

J ] ,\

w] w\ w] w\  , wx wy wy wx w § w\ · w § w\ ¨] ¸ ¨] wx © wy ¹ wy © wx

· ¸, ¹ w § w] · w § w] · ¨\ ¸. ¨\ ¸ wy © wx ¹ wx © wy ¹

(2.119) (2.120)

(2.121)

Analytically these three forms of J (] ,\ ) have the same value. However, in their finitedifference form they are not identical. For this consider the 9-point stencil with its different points labeled as in Fig. 2.11 with grid distance d. With respect to Fig. 211, the finite-difference form of the right-hand side of (2.119), (2.120), and (2.121) may be written as J 00 ] ,\

J 00u ] ,\

u J 00 ] ,\

1 ª] 10  ] 10 \ 01  \ 01 4d 2 ¬  ] 01  ] 01 \ 10  \ 10 º¼ ,

(2.122)

1 ª] 10 \ 11 \ 11  ] 10 \ 11 \ 11 4d 2 ¬ ] 01 \ 11 \ 11  ] 01 \ 11  \ 11 º¼ ,

(2.123)

1 ª\ 01 ] 11  ] 11 \ 01 ] 11  ] 11 4d 2 ¬ \ 10 ] 11  ] 11 \ 10 ] 11  ] 11 º¼ ,

which can also be written as u J 00 ] ,\

1 ª] 11 \ 01  \ 10  ] 11 \ 10  \ 01 4d 2 ¬ ] 11 \ 01  \ 10  ] 11 \ 10  \ 01 º¼ .

(2.124)

An Introduction to Finite Differencing

35

Superscripts u and + denote the location of ] and \ values (in that order) involved in the finite-difference approximation of the Jacobian. The + symbol indicates that values involved are located at points 10, -10, 01, and 0-1, while the x symbol indicates their location at points 11, -1-1, -11, and 1-1. The subscripts denote the point at which the Jacobian is centered. Let us now consider the conservation of mean-square vorticity, for which the condition is

] J (] ,\ ) 0 .

(2.125)

In finite-difference form this condition is satisfied if the products of ] J at various grid points in the finite-difference from of (2.125) cancel when added over the closed domain. Now from (2.122) we obtain nine equations, one equation for each grid point of our 9-point stencil 1 ª] 00] 10 \ 01  \ 01  !º¼ , ] 00 J 00 ] ,\ (2.126) 4d 2 ¬ 1 ª ] 10] 00 \ 11 \ 11  !¼º , (2.127) ] 10 J10 ] ,\ 4d 2 ¬

and so on. Also from (2.123) we get

] 00 J 00u ] ,\ ] 10 J10u ] ,\

1 ª] 00] 10 \ 11 \ 11  !º¼ , 4d 2 ¬ 1 ª ] 10] 00 \ 01  \ 01  !º¼ , 4d 2 ¬

(2.128) (2.129)

and so on. Similarly from (2.124) we have

] 00 J 00u ] ,\

] 11 J11u ] ,\

1 ª] 00] 11 \ 01 \ 10  !º¼ , 4d 2 ¬ 1 ª ] 11] 00 \ 01 \ 10  !º¼ , 4d 2 ¬

(2.130)

(2.131)

and so on. From (2.126) and (2.127) we see that the sum of the terms involving the product ] 00] 10 does not vanish. The various terms of ] J  (] ,\ ) therefore do not cancel when added over the whole closed domain. Therefore the domain integral of ] J  (] ,\ ) does not vanish. From (2.128) and (2.129) we find that the same is true for ] J u (] ,\ ) , that is, its domain integral also does not vanish. However, as the terms involving ] 00] 10 in (2.126) and (2.129) are equal and opposite in sign, as are those in (2.127) and (2.128), various terms of the sum ] J  (] ,\ )  ] J u (] ,\ ) cancel, that is,

]J



(] ,\ )  ] J  x (] ,\ ) 0 .

(2.132)

Also terms involving ] 00] 10 in (2.130) and (2.131) are equal and opposite. Therefore various terms of ] J u (] ,\ ) cancel on addition over a closed domain. Thus

An Introduction to Global Spectral Modeling

36

] J x  (] ,\ ) 0 .

(2.133)

Thus J  (] ,\ )  J u (] ,\ ) and J u (] ,\ ) individually conserve mean-square vorticity. Similarly it can be shown that

\J



(] ,\ ) \ J x  (] ,\ ) 0 ,

(2.134)

and

\ J  x (] ,\ ) 0 , 

u

(2.135)

u

so that J (] ,\ )  J (] ,\ ) and J (] ,\ ) individually conserve mean kinetic energy. From these we can take the three forms of the Jacobian, take a weighted average of the three, and add them up,

D J  9 ,\  E J u 9 ,\  J J u 9 ,\ 0 , 1 . Arakawa searched for values of D , E , and J that satisfied the E J 13 best satisfies the invariants. From this, we conclude that the finite difference Jacobian is

such that D  E  J

invariants and found that D

J1 ] ,\

1  ª J ] ,\  J u ] ,\  J u ] ,\ º¼ 3¬

(2.136)

as it conserves the square of vorticity and kinetic energy over a closed domain. To investigate the accuracy of the finite-difference scheme, we expand ] and \ in Taylor series: d 2 § w2 F · d 3 § w3 F · § wF · 4 Fr10 F00 r d ¨ ¸  ¨ 2 ¸ r ¨ 3 ¸ O d , © wx ¹00 2 © wx ¹00 6 © wx ¹00



F0r1 F1r1

§ wF · d 2 § w2 F · d 3 § w3 F · 4 F00 r d ¨ ¸  ¨ 2 ¸ r ¨ 3 ¸ O d , © wy ¹00 2 © wy ¹00 6 © wy ¹00 § wF wF · w2 F w2 F · d 2 § w2 F   F00  d ¨ ¸ ¸  ¨ 2 r2 wxwy wy 2 ¹00 © wx wy ¹00 2 © wx





F1r1

d 3 § w3 F w3 F w3 F w3 F · r 3 ¸  O d4 , ¨ 3 r3 2 3 2 6 © wx wx wy wxwy wy ¹00



§ wF wF · w2 F w2 F · d 2 § w2 F B  F00  d ¨ ¸ ¸  ¨ 2 B2 wxwy wy 2 ¹00 © wx wy ¹00 2 © wx w3 F w3 F w3 F · d 3 § w3 F  ¨ 3 B3 2 3 B 3 ¸  O d4 . 2 wx wy wxwy wy ¹00 6 © wx



In these expansions, F represents either ] or \. After substituting these expansions into (2.122), (2.123), and (2.124), we obtain

37

An Introduction to Finite Differencing

Figure 2.12. A 13-point stencil for the Jacobian.

J  ] ,\

J ] ,\ 

d 2 § w] w 3\ w] w 3\  ¨ 6 © wx wy 3 wy wx3

w 3] w\ w 3] w\ · 4  ¸O d , wx 3 wy wy 3 wx ¹ d 2 ª w] w 3\ w] w 3\ J  ] ,\   « 2 2 ¬ wx wx wy wy wxwy 2





J u ] ,\

§ w 2] w 2] ¨ 2  2 wy © wx

J u ] ,\

J  ] ,\ 

· w 2\ º 4 »O d , ¸ w w x y ¹ ¼



(2.137)

(2.138)

w\ w 3] d 2 ª w\ w 3]  « 2 ¬ wx wx 2 wy wy wxwy 2

§ w 2\ w 2\ ¨ 2  2 wy © wx

· w 2] º 4 »O d . ¸ w w x y ¹ ¼



(2.139)

Note that equations (2.138) and (2.139) contain all of equation (2.137). Therefore the second-order accurate Jacobian is J1 ] ,\

where

1  ª J ] ,\  J u ] ,\  J u º¼ 3¬ J ] ,\   d 2  O  d 4 ,

(2.140)

An Introduction to Global Spectral Modeling

38

6

w] w 3\ w] w 3\ w 3] w\ w 3] w\    wx wy 3 wy wx3 wx3 wy wy 3 wx w] w 3\ w] w 3\ § w 2] w 2] · w 2\  ¨  ¸ 2 wx wx wy wy wxwy 2 © wx 2 wy 2 ¹ wxwy § w 2] w 2\ · w 2] w\ w 3] w\ w 3]   ¨ 2  2 ¸ . 2 2 wx wx wy wy wxwy © wx wy ¹ wxwy



(2.141)

There are other finite-difference forms of the Jacobian which conserve square vorticity and kinetic energy. One such scheme can be designed with a 13-point stencil as shown in Fig. 2.12. This stencil can be obtained by rotating the stencil in Fig. 2.11 by 45q so that point 11, -11, -1-1, and 1-1 in Fig. 2.12 correspond to points 10, 01, -10, and 0-1 in Fig. 2.11 and points 20, 02, -20, and 0-2 in Fig. 2.12 correspond to points 1-1, 11, 11, and -1-1 in Fig. 2.11. Also we note that the distance between the points (i.e., between 00 and 11) is now 21/2d. As this stencil is obtained simply by a 45q rotation of the stencil in Fig. 2.11, J 2 ] ,\

1 uu ª J ] ,\  J u‡ ] ,\  J ‡u ] ,\ º¼ 3¬

(2.142)

will also conserve kinetic energy and the square of vorticity, where J 00xx ] ,\

u‡ J 00 ] ,\

‡u J 00 ] ,\

1 ª] 11  ] 11 \ 11  \ 11 8d 2 ¬  ] 11  ] 11 \ 11  \ 11 º¼ ,

(2.143)

1 ª] 11 \ 02  \ 20  ] 11 \ 20  \ 0 2 8d 2 ¬ ] 11 \ 02  \ 20  ] 11 \ 20  \ 0 2 º¼ ,

(2.144)

1 ª] 20 \ 11 \ 11  ] 20 \ 11  \ 11 8d 2 ¬ ] 02 \ 11  \ 11  ] 0 2 \ 11  \ 11 º¼ .

(2.145)

As J 2 ] ,\ also has centered differences, it is a second-order accurate Jacobian as well.

Using Taylor series we get an equation similar to (2.140), that is, J 2 ] ,\

1 uu ª J ] ,\  J u‡ ] ,\  J ‡u ] ,\ º¼ 3¬ J ] ,\  2  d 2  O  d 4 ,

(2.146)

where 2d2 is from (21/2d)2. From (2.140) and (2.146) we get J 3 ] ,\

2 J1 ] ,\  J 2 ] ,\

J ] ,\  O  d 4

(2.147)

as the fourth-order accurate Arakawa Jacobian, which conserves kinetic energy and mean-square vorticity.

An Introduction to Finite Differencing

39

Now that we have computed the fourth-order accurate Jacobian we ask what is the importance of finding it? If we were to make a barotropic forecast, which utilizes both the Laplacian and Jacobian, w 2 ’\ wt





 J \ , ’ 2\  E

w\ wx

and used a fourth-order accurate scheme for each the Laplacian, Jacobian, and w / wx , we could obtain a good forecast. In other words, understanding of fourth-order accurate schemes makes for better forecasts.

2.15 Exercises 2.1. Consider the function x u ( x)

0 0

S /6

S /4

S /2 12

1/2

1/(2 )

1.0

Evaluate du / dx at x S / 4 using (a) forward differencing, (b) centered differencing, (c) backward differencing, and (d) the second-order finite-difference analog. Analytically, u ( x) represents sin x . Which of the finite-difference formulas represents the best approximation to du / dx x

S /4

?

2.2. For the function of Exercise 2.1, evaluate u ( x) at x 7S /12 , given that the secondorder derivative of u ( x) at x S / 2 is exactly equal to its analytic value. 2.3. Consider Poisson’s equation ’ 2\ G , where \ \ ( x, y ) and G G ( x, y ) . Assume an even mesh in x and y and that 0 d x d 10 , 0 d y d 10 . Also take 'x 'y 1 and let

\  x, 0 \  x,10 0

0 d x d 10 ,

\  0, y \ 10, y 0 G  x, y

S  x, y 0

0 d y d 10 , 0 d x d 10 and 0 d y d 10 .

Find \ if S 0 , S 2( x 2  y 2 ) , S 2 exp( x  y ) / e10 , and S 2 cos x sin y . Plot the analytic solution as well as the solution obtained using the finite-difference method. 2.4. Let J

J M ,\

wM w\ wM w\  wx wy wy wx

represent the Jacobian. Show that ³³D J dx dy a d x d b and c d y d d .

0 over the closed domain D with

Chapter 3

Time-Differencing Schemes 3.1

Introduction

Atmospheric models generally require the solutions of partial differential equations. In spectral models, the governing partial differential equations reduce to a set of coupled ordinary nonlinear differential equations where the dependent variables contain derivatives with respect to time as well. To march forward in time in numerical weather prediction, one needs to use a time-differencing scheme. Although much sophistication has emerged for the spatial derivatives (i.e., second- and fourth-order differencing), the time derivative has remained constructed mostly around the first- and second-order accurate schemes. Higher-order schemes in time require the specification of more than a single initial state, which has been considered to be rather cumbersome. Therefore, following the current state of the art, we focus on the first- and second-order accurate schemes. However, higher-order schemes, especially for long-term integrations such as climate modeling, deserve examination. We start with the differential equation dF / dt G , where F F (t ) and G G (t ) . If the exact solution of the above equation can be expressed by trigonometric functions, then our problem would be to choose an appropriate time step in order to obtain a solution which behaves properly; that is, it remains bounded with time. This is illustrated in Fig. 3.1. We next show that: (1) if an improper time step is chosen, then the approximate finite difference solution may become unbounded, and (2) if a proper time step is chosen, then the finite difference solution will behave quite similar to the exact solution. The stability or instability of a numerical scheme will be discussed for a single Fourier wave. This would also be valid for a somewhat more general case, since the total solution is a linear combination of sine and cosine functions, which are all bounded. We next define an amplification factor O , the magnitude of which would determine whether

a scheme is stable or not.

3.2

Amplification Factor

Consider a linear wave equation such as

40

Time-Differencing Schemes

41

Figure 3.1. Stable and unstable time step 't. wu wu c wt wx

0

(3.1)

which contains the wave speed, c, and the advection term wu wx . What we learn from this equation is also applicable to the full spectral model. In other words, we can find the time-differencing scheme from this equation and then apply that scheme to the full spectral model. It happens that if the time-differencing scheme works with this equation then it seems to also work for the full model. Let the analytic solution of this equation be in the form of a single harmonic given by u ( x, t ) Re[U (t )eikx ] , where U (t ) is the wave amplitude and k is the wavenumber. Substituting this into (3.1), we obtain an equation for the amplitude function U (t ) as dU  ikcU 0 . (3.2) dt

Thus we have reduced the above partial differential equation to an ordinary differential equation whose exact solution is given by

U (t ) U (0)e  ikct .

(3.3)

Hence the desired harmonic solution is given by u ( x, t )





Re U (0)eik ( x ct ) .

(3.4)

An Introduction to Global Spectral Modeling

42

The exact solution is bounded and not amplifying in time so any finite difference representation that is made from this solution is also bounded in time. Therefore each wave component is advected at a constant velocity c along the xaxis with no change in amplitude. We now look for an analogous solution for the finitedifference equation, where we can substitute the form u nj Re(U n eikj'x ) , where U n is the

amplitude at the nth time level. We define the amplification factor O so that Un

OU n 1 ,

(3.5)

O U n 1 .

(3.6)

or Un

Substituting for | U n 1 | | O || U n  2 | and | U n  2 | | O || U n 3 | and following this procedure, we obtain | U n | | O |n | U 0 | . For the scheme to be stable, it is required that | U n | A , where A is some finite number. Therefore n

Un

O U0  A.

(3.7)

Taking the natural logarithm of both sides, we obtain ln U n

n ln O  ln U 0  ln A .

Simplifying, we obtain n ln O  ln

A . U0

Now let ln

t

A U0

A'.

Dividing the time t into n equal intervals, each of which is equal to 't , we obtain n't . Hence A' ln O  , (3.8) n A' ln O  't . (3.9) n

As we require the boundedness of the solution for a finite time t, we obtain ln O d O  't .

(3.10)

If we now define | O |{ 1  D , then ln 1  D D 

D2 2



D3 D4 3



4

!

for 1 d D d 1 .

The stability condition obtained is equivalent to D d O  't , so that

O d 1  O  't .

(3.11)

Time-Differencing Schemes

43

This is the Von Neumann necessary condition for stability. This condition allows for an exponential growth of the solution, and it is required when the true solution grows exponentially. However, when it is known that the true solution does not grow, it is customary to replace this condition by a sufficient condition

O d 1.

(3.12)

We apply all our schemes to the linear wave equation (3.1). If the amplification factor O is less than 1, then we obtain a stable solution. The reason why we apply our schemes to the above wave equation is that the solutions of wave equations are bounded trigonometric functions, which help us to examine the stability of the solution. The general stability criteria are:

3.3

Unstable scheme

if

O !1,

Neutral scheme

if

O

Stable scheme

if

O  1.

1,

Stability

In this section we discuss the computational stability of the following time-differencing schemes applied to the linear wave equation, namely the Euler, backward, trapezoidal, Matsuno and Heun predictor-corrector, leap-frog, Adams-Bashforth, and implicit schemes. 3.3.1

Euler, Backward, and Trapezoidal Schemes

Consider the linear wave equation (3.1) and assume u  x, t U  t eikx .

(3.13)

Let the value of U at time n't be known. We wish to predict the value of U at time (n  1)'t . Substituting the assumed solution, we obtain the marching equation for timedifferencing (3.2) dU f U , t U U t , (3.14) dt where f ikcU iZU and Z  kc . Integrating the above equation between time n't and (n  1)'t where n is the current time level and n  1 is the future time level, we obtain U n 1  U n

³

U n 1 U n 

³

( n 1) 't

n't

f U , t dt .

We can also write ( n 1) 't

n't

f U , t dt .

(3.15)

An Introduction to Global Spectral Modeling

44

We have two possible values for f (U , t ) , which we assume to be constant over the time interval [n't , (n  1)'t ] , namely, (a) f f n (U n , n't ) and (b) f f n 1[U n 1 , (n  1)'t ] . In the case of (b), if f f n 1 (i.e., f depends on U n 1 ), then the scheme is called implicit. If f f n (i.e., f depends on U n ), then the scheme is called explicit. We can also use a linear combination of f n and f n 1 to define the value of f (U , t ) in the time interval [n't , (n  1)'t ] , which we assume to be constant and defined as f (U , t ) D f n  E f n 1 , where D and E are two real constants satisfying D  E 1 . Thus (3.15) reduces to U n 1 U n 

³

( n 1) 't

n't

D f

n

 E f n 1 dt .

Alternatively, we can write the time-differencing scheme as U n 1 U n  't D f n  E f n 1 .

(3.16)

By assigning different values to D and E, we obtain the different time-differencing schemes. In particular, for

D 1 and E 0 D 0 and E 1 D E 0.5 .

Euler’s forward schemes, backward schemes, trapezoidal schemes, As f n

iZU n , f n 1

iZU n 1 , and U n 1 U n  't (D iZU n  E iZU n 1 ) , we can then write U n 1 U n  iDZ'tU n  i EZ'tU n 1 U n 1  i EZ'tU n 1 U n  iDZ'tU n U n 1 1  i EZ't U n 1  iDZ't U n 1 U n

1  iDZ't . 1  i EZ't

Comparing the above equation with U n 1

O

(3.17)

OU n , we observe that

1  iDZ't . 1  i EZ't

(3.18)

Let us define p Z't , so that O (1  iD p ) /(1  i E p ) , where Z is the frequency of the wave we are studying, which we have chosen, and 't is the time step we are assigning. After dividing and multiplying by the conjugate of (1  i E p ) , we obtain

O or, since D  E

1 , we can write

1  DE p 2  ip (D  E ) , 1  E 2 p2

Time-Differencing Schemes

45

O

1  DE p 2  ip . 1  E 2 p2

(3.19)

Now let us discuss the stability of the above-mentioned three schemes. Euler Forward Scheme. The Euler scheme is obtained by assigning D 1 and E 0 . Note that f is not centered in time in the Euler scheme. This now gives us O 1  ip , so knowing that if O a  ib then | O | (a 2  b 2 )1 2 we can also write

O

2 12

1  p

.

(3.20)

Since p Z't is real and p 2 ! 0 , it follows that | O |! 1 ; thus the Euler scheme is always unstable whatever the time step may be. It is a first-order-accurate time-differencing scheme.

Backward Scheme. The backward scheme is obtained by assigning D This gives 1  ip O , 1  p2 or (1  p 2 )1 2 1 O  1 , as p ! 0 . 2 1 p (1  p 2 )1 2

0 and E

1.

(3.21)

(3.22)

Hence the backward scheme is unconditionally stable for any time step. Furthermore, it is a damping scheme, and the amount of damping increases with p or as the frequency Z increases. During numerical integration, high-frequency modes often get excited and amplified unrealistically due to errors in the initial data. The damping property of the backward scheme is therefore desirable to reduce the amplitude of such high-frequency modes and to filter them out. It is also a first-order-accurate timedifferencing scheme. Trapezoidal Scheme. By assigning D this case,

O

E

0.5 , the trapezoidal scheme is obtained. In

4  p2  i4 p . 4  p2

Thus

O

(16  p 4  8 p 2  16 p 2 )1 2 4  p2

1.

(3.23)

Therefore | O | 1 for any time interval 't , and thus this scheme is always neutral. The amplitude of the numerical solution remains constant, just as in the exact solution. It is a second-order-accurate time-differencing scheme.

An Introduction to Global Spectral Modeling

46 3.3.2

Predictor-Corrector Schemes

These types of schemes work on the following basic principle. First an attempt is made to predict the value of U at the (n+1)th time step, which we denote by U ( n 1)* . Euler’s forward scheme is usually used to predict U ( n 1)* . This value is then used to find the value of f n 1 at the (n+1)th time step, and we denote this by f ( n 1)* . This value f ( n 1)* is used to correct the previous U ( n 1)* to get the final value of U ( n 1) . The step for finding U ( n 1)* is the predictor step, and the step for finding U ( n 1) using U ( n 1)* is the corrector step. The finite-difference equations for such a scheme are

U ( n 1)* U n  'tf n (predictor step)

(3.24)

and U ( n 1)

where D  E

U n  't D f n  E f ( n 1)*

(corrector step),

(3.25)

1 . As mentioned before, f ( n 1)* and f n are defined as f ( n 1)*

f ª¬U ( n 1)* , (n  1)'t º¼

(3.26)

and fn

f U n , n't .

(3.27)

To look into the stability of such a scheme, we again consider the linear wave equation (3.1). Recalling f n iZU n and f n 1 iZU n 1 the predictor step is described as

U ( n 1)* U n  iZ'tU n

(3.28)

and the corrector step is formulated as U ( n 1)

U n  iZ't ª¬DU n  E U ( n 1)* º¼ .

(3.29)

Substituting for U ( n 1)* from the predictor step into the corrector step, we obtain U ( n 1) U n  iZ't[DU n  E (U n  iZ'tU n )] , or we can write U n 1 U n  iZ'tDU n  iZ't E U n  i 2Z 2 't 2 E U n U ( n 1)

ª¬1  EZ 2 't 2  iZ (D  E )'t º¼ U n .

Thus O U ( n 1) / U n (1  EZ 2 't 2 )  iZ (D  E )'t . factor for this scheme is given by

O

Since D  E

(3.30) 1 , the amplification

12

ª¬(1  EZ 2 't 2 ) 2  Z 2 't 2 º¼ .

(3.31)

Examples of this kind of scheme are the Matsuno scheme and Heun’s scheme.

Matsuno Scheme. The Matsuno scheme is obtained by taking D

0 and E

1 such that

Time-Differencing Schemes

47

O

1  Z 't  Z 't 2 2

2

2

12

2

2 12

1  2Z 't  Z 't  Z 't 1  Z 't  Z 't . 2

2

If Z't d 1 , then Z 4 ('t ) 4 d Z 2 ('t ) 2 d 1 .

2

2

4

4

4

2

4 12

Thus if Z't d 1 , then O d 1 .

Hence the

Matsuno scheme is a stable scheme for Z't d 1 . This is also a first-order-accurate timedifferencing scheme. Heun’s Scheme. Heun’s scheme is obtained by assigning D write Z 2 ('t ) 2 O 1  iZ't . 2 Thus we obtain

E 1 2 . We can then

12

O

§ Z 4 ('t ) 4 ·  Z 2 ('t ) 2  Z 2 ('t ) 2 ¸ , ¨1  4 © ¹

or 12

O

§ Z 4 ('t ) 4 · ¨1  ¸ . 4 © ¹

(3.32)

As Z 4 ('t ) 4 / 4 ! 0 for any Z't , we observe that O ! 1 for all values of Z't . Therefore, Heun’s scheme is an unstable scheme. We next illustrate the leap-frog timedifferencing scheme. 3.3.3 Centered or Leap-Frog Scheme

We have discussed two-time-level differencing schemes so far. One of the most widely used time-integration schemes in numerical weather prediction is the centered, or leapfrog, time-integration scheme. This is a three-time-level scheme. The values of the function at time levels (n  1) and n are known (or predicted using one forward time step to start the process), and from these we predict the value of the function at time level (n  1) . We make a centered evaluation of the integral as U n 1 U n 1 

³

( n 1) 't

( n 1) 't

f U , t dt .

(3.33)

We take f = constant in the above integral. Let this constant be denoted as f n , the value of f at time t n't , so that (3.33) reduces to U n 1 U n 1  2'tf n .

(3.34)

An Introduction to Global Spectral Modeling

48

Thus we use the information at two previous time levels to generate the information at the next time level. The truncation error is of the order of ('t ) 2 . This scheme is called the centered or the leap-frog scheme. Consider the leap-frog scheme applied to the linear wave equation (3.1). Substituting u ( x, t ) Re[U (t )eikx ] as the solution, we obtain dU  iZU dt

where Z

0,

 kc .

(3.35)

Using a centered time-differencing scheme as in (3.34), we have U n 1 U n 1  i 2Z'tU n ,

where f n

(3.36)

iZU n . Furthermore, we have U n 1 n

OU n

O 2U n 1 ,

n 1

as U n

OU n 1

(3.37)

n 1

and we can replace all U and U with U and simplify equation (3.36). Since equation (3.36) is a second-order differential equation it has a quadratic equation for O . Substituting these into the above equations, we obtain

O 2U n 1 U n 1  i 2Z't OU n 1 O 2  i 2 pO  1 0 ,

(3.38)

where p Z't . The roots of this second-order equation are 12

O1

1  p

O2

 1  p2

2

 ip

(3.39)

and



12



 ip .

(3.40)

If U n 1 OU n is to represent an approximation to the true solution, then we must have O o 1 as 't o 0 . For the above roots, since p Z't o 0 as 't o 0 , we have O1 o 1 . However, at the same time, O2 o 1 . The solution associated with O1 is an approximation of the true solution and is called the physical mode. On the other hand, the solution associated with O2 is not the true solution and is called the computational mode. The complete solution of the centered time-differencing scheme is the sum of the solutions corresponding to the physical mode and the computational mode. Because O2n is positive for even values of n and negative for odd values of n, O2n alternatively takes positive and negative values as the integration proceeds. This results in an oscillation of the computational mode about the true solution. This is an undesirable feature of the scheme, and methods are available to suitably suppress the computational mode during the course of integration.

Time-Differencing Schemes

49

3.3.3.1 Centered in Time and Space One method used to suppress the computational mode is to start the integration with one forward time step and then compute 49 centered difference time steps. After these 50 time steps, we want to break the process and repeat it again for as many time steps that are needed. This is done so that the computational factor (O2 ) does not blow up. In the discussion on the centered time-differencing scheme, we have used a centered approximation for the time differential wu / wt and an analytical value for the space differential wu / wx in the wave equation. We now examine the scheme using centered differencing both in time and space. With x m'x and t n't , we write u ( x, t ) U (t )eikx as

u  m'x, n't U  n't eikm'x

U n eikm'x

(3.41)

which is the finite difference representation of the trial solution. Substituting (3.41) into the linear wave equation (3.1) gives U n 1  U n 1 2't

(U n eik 'x  U n e  ik 'x ) 2'x c't n ik 'x ik 'x  U e e , 'x c



U n 1  U n 1



or

U n 1 U n 1 

Futhermore, using U n 1

c't n ik 'x ik 'x U e e . 'x





(3.42)

O 2U n 1 , we obtain

OU n

O 2U n 1 U n 1 

c't OU n 1 eik 'x  e  ik 'x 'x



c't O eik 'x  e ik 'x  1 0 . 'x





sin k 'x

eik 'x  e  ik 'x 2i

O2 



Using the identity

we obtain

O 2  i 2c

't sin k 'xO  1 0 'x

(3.43)

which is the quadratic representation for space-time differencing. The roots of (3.43) are 1/ 2

O1

ª § c't · 2 2 º «1  ¨ ¸ sin k 'x » «¬ © 'x ¹ »¼

O2

ª § c't · 2 2 º «1  ¨ ¸ sin k 'x » x ' © ¹ ¬« ¼»

 ic

't sin k 'x , 'x

 ic

't sin k 'x . 'x

1/ 2

An Introduction to Global Spectral Modeling

50 The absolute values of O1 and O2 are

1/ 2

O1

2 ª § c't · 2 2 º § c't · 2 «1  ¨ ¸ sin k 'x  ¨ ¸ sin k 'x » © 'x ¹ «¬ © 'x ¹ »¼

O2

Thus we can write O1

O2

1.

(3.44)

1.

When 'x o 0 and 't o 0 , then the numerical solution should converge to the true solution. This turns out to be true with O O1 , since in this case U n 1 O1U n U n . However, if we instead take O2 as the solution, then U n 1 O2U n U n . Hence O2 corresponds to the oscillating computational mode of the equation and O1 corresponds to the physical mode. 3.3.3.2 Alternative Method for Space-time Differencing ˆ iQ n't then substituting for Consider again (3.42). If we assume U n to be of the form ue n 1 n 1 n U , U , and U into this equation gives

c't iQ n't ik 'x  ik 'x ˆ ue e e 'x c't iQ n't ik 'x  ik 'x ˆ iQ n't eiQ't  ue ˆ iQ n't e  iQ't  ˆ ue ue e  e 'x c't ik 'x  ik 'x eiQ't  e  iQ't  e e , 'x ˆ iQ ( n 1) 't  ue ˆ iQ ( n 1) 't ue











or

sin Q't



c't sin k 'x . 'x

(3.45)

Because k is a real wavenumber, sin k 'x d 1 so that (3.45) is satisfied for a real value of v if c't / 'x d 1 . However, if c't / 'x ! 1 , v must have an imaginary component. In that case, the finite-difference solution becomes unstable. The condition c't / 'x d 1 is a necessary condition for the stability of an explicit time-differencing scheme. It was first discovered by Courant, Friedrichs, and Levy (1928) and is commonly known as the CFL criterion for the stability of a time-differencing scheme. According to this, in fine-mesh models where 'x is small, we need small time steps 't to ensure computational stability. Also 't should be small for fast-moving waves such as gravity waves, because c is small. Some examples of models and their respective time steps and grid spacing are listed below: Model Barotropic model Primitive equation model MM5 Micrometeorology

'x 100 km 100 km 20 km 10 m

't 1 hour 6 minutes 1 minute Few seconds

Time-Differencing Schemes

51

3.3.4 Adams-Bashforth scheme This is a three-time-level scheme in which the value of the space derivative is approximated from a linear extrapolation of f in the center to time levels n and (n  1) from its value at time levels n and (n  1) . With such an approximation, the finitedifference analog of (3.1) takes the form

1 §3 · U n 1 U n  't ¨ f n  f n 1 ¸ 2 ©2 ¹

(3.46)

which is a second-order differential equation. After applying this scheme to the linear wave equation, we obtain 1 §3 · U n 1 U n  ikc't ¨ U n  U n 1 ¸ . 2 ©2 ¹

Substituting U n 1

OU n and U n

(3.47)

OU n 1 , we obtain

O2

§3 ©

1·

O  ick 't ¨ O  ¸ , 2 2 ¹

or § ©

3 2

· ¹

1 2

O 2  ¨1  i Z't ¸ O  i Z't 0 , where Z

(3.48)

kc . The roots of the above equation are 1/ 2

O1

O2

1ª 3 § 9 2 · 2 «1  i Z't  ¨1  Z ('t )  iZ't ¸ 2 «¬ 2 © 4 ¹

º », »¼ 1/ 2 1ª 3 § 9 2 · º 2 «1  i Z't  ¨1  Z ('t )  iZ't ¸ » . 2 «¬ 2 © 4 ¹ »¼

(3.49)

(3.50)

We note that if 't o 0 , then O1 1 and O2 o 0 . Thus O1 denotes the physical mode and O2 denotes the computational mode. As the time step 't is reduced, the Adams-Bashforth method approaches the true solution. On power series expansion of (3.49) and (3.50), it can be shown that | O1 |! 1 while | O2 | 1 . Thus the computational mode in this scheme tends to dampen, which is a beneficial property. However, the physical mode tends to amplify. The rate of amplification is small if 't is small, and the scheme is weakly unstable. The AdamsBashforth scheme is therefore suitable for short periods of integration with a small time step such as in a squall or flow past a mountain.

52

An Introduction to Global Spectral Modeling

3.3.5 Implicit Schemes There are two types of implicit schemes, one is the fully implicit scheme discussed below and the other is a semi-implicit scheme discussed in Sec. 3.4. In the time-differencing schemes discussed so far, the time step should satisfy the CFL condition. The implicit time-integration scheme permits longer time steps than specified by the CFL condition and is therefore more economical than explicit time-differencing schemes. In the implicit scheme, the space derivative at time level n is obtained as the mean of the space derivatives at time levels (n) and (n  1) . Since the future value of the function is not known explicitly, the scheme is called implicit. If m is the space index and n is the time index, the fully implicit finite-difference analog of the linear wave equation takes the form

umn 1  umn 't

c § u n 1  u n 1 u n  u n ·  ¨ m 1 m 1  m 1 m 1 ¸ 2 © 2'x 2'x ¹

(3.51)

which is a first-order non-separable difference equation. In this equation, the unknown function u at time level (n  1) appears at the three space points (m  1) , m, and (m  1) . Such equations are associated with each space grid point; this system of equations, in principle, can be solved by inverting a matrix with proper boundary conditions. However, for a large number of grid points, the procedure becomes difficult. In such cases, relaxation or spectral methods are more convenient. Assuming a solution of the form umn U n eikm'x , we obtain

U n 1eikm'x  U n eikm'x 't

c ª U n 1eik ( m 1) 'x  U n 1eik ( m1) 'x  « 2¬ 2'x n ik ( m 1) 'x  U n eik ( m 1) 'x º Ue  » 2'x ¼

U n 1eikm'x  U n eikm'x 't

c ª U n 1eikm'x eik 'x  U n 1eikm'x e ik 'x  « 2¬ 2'x n ikm'x ik 'x Ue e  U n eikm'x e  ik 'x º  » 2'x ¼

U n 1  U n 't

c ª U n 1eik 'x  U n 1e  ik 'x U n eik 'x  U n e ik 'x º  «  » 2¬ 2'x 2'x ¼

U n 1  U n 't











n 1 n ik 'x  ik 'x eik 'x  e  ik 'x cU c U e e  . 2 2'x 2 2'x

Rearranging equation (3.52) and using Euler’s relation, i.e., sin x obtain n 1 n ik 'x  ik 'x eik 'x  e  ik 'x U n 1 c U Un c U e  e   't 2 2'x 't 2 2'x









(3.52)

(eix  e  ix ) / 2i , we

Time-Differencing Schemes

53

U n 1 ic n 1 sin k 'x  U 't 2 2 'x

U n ic n sin k 'x  U 't 2 2'x

't 't § · § · U n 1 ¨1  ic sin k 'x ¸ U n ¨1  ic sin k 'x ¸ . x x ' ' 2 2 © ¹ © ¹

(3.53)

Hence 't sin k 'x 2'x . 't 1  ic sin k 'x 2'x

1  ic

U n 1 Un

(3.54)

This is the ratio of the amplitude at time level (n  1) to that at time level n at a particular spatial point m. Furthermore, multiplying (3.54) by the complex conjugate of the denominator and taking the absolute value gives | O | | U n 1 / U n | 1 , which implies stability regardless of the value of c't / 'x . The future values of the function u will be of the same order as the previous values. Hence this scheme is nonamplifying. We next show another way of demonstrating the stability of the implicit scheme, which is called the trigonometric method. Starting from the finite-difference analog of the linear wave equation, that is, umn 1  umn



c't ª n 1 um 1  umn 1  umn 11  umn 1 º¼ , 4'x ¬





(3.55)

we can write umn 11

ˆ ik [( m 1) 'x  c ( n 1) 't ] ue

ˆ ik ( m'x  cn't ) eik ( 'x  c't ) ue where u

n m

ˆ ue

ik ( m'x  nc't )

umn eik ( 'x  c't ) ,

(3.56)

. Similarly, ˆ ik [( m 1) 'x  cn't ] ue

umn 1

ˆ ik ( m'x  cn't ) eik 'x ue

umn eik 'x ,

ˆ ik [( m 1) 'x  cn't ] ue

umn 1

ˆ ik ( m'x  cn't ) e  ik 'x ue umn 11

umn e  ik 'x ,

ˆ ik [( m 1) 'x  c ( n 1) 't ] ue ˆ ik ( m'x  cn't ) eik ( 'x  c't ) ue

umn 1

umn eik ( 'x  c't ) ,

ˆ ik [ m'x  c ( n 1) 't ] ue ˆ ik ( m'x  cn't ) eikc't ue

umn eikc't .

Then substituting these equations into (3.55), we obtain





umn eikc't  1



c't ik ( 'x  c't ) ik 'x ik ( 'x  c't ) ik ( 'x ) n e e e e um . 4'x



From this we can write eikc't  1 

c't ik 'x ikc't ª e (e  1)  e  ik 'x (eikc't  1) º¼ , 4'x ¬



(3.57)

An Introduction to Global Spectral Modeling

54 or

c't ª(eik 'x  e  ik 'x ) (eikc't  1) º¼ . 4'x ¬

eikc't  1 

(3.58)

We then obtain eikc't  1 eikc't  1

c't ik 'x  ik 'x e e , 4'x





or

tan

kc't 2



c't sin k 'x . 2'x

(3.59)

The sine of an angle is bounded between –1 and 1 while the tangent of an angle varies from f to f . Hence, any value of 't can satisfy this equation, i.e. (3.59) is satisfied for all values of (c't ) / 'x . Thus the stability of this scheme does not depend on the value of (c't ) / 'x and is therefore unconditionally stable. In the following section we illustrate the application of the semi-implicit time-differencing scheme to the shallowwater model equations.

3.4

Shallow-Water Model

To demonstrate the semi-implicit time-differencing scheme the shallow water equations are used instead of the linear wave equation. The linear wave equation is a one-wave solution equation, whereas the shallow water equations are, in essence, a stripped down version of the full model equations, i.e., they have some properties of the atmosphere. In the semi-implicit scheme the nonlinear part of an equation is handled explicitly while the linear part is handled implicitly. In the shallow-water model we have a layer of fluid with constant density. Under the hydrostatic assumption, there is no vertical variation of the pressure gradient force if density is constant. Therefore, if initially the horizontal velocities are independent of height, they will remain so during all times. As wu / wz w / wz 0 , the vertical advection terms do not appear in the shallowwater equations. These equations may be written in the Cartesian coordinate system as the momentum equations:

wu wu wu wI u  v  fv  wt wx wy wx wv wv wv wI  u  v  fu  wt wx wy wy

0,

(3.60)

0,

(3.61)

and the mass continuity equation :

wu wv ww   wx wy wz

0.

(3.62)

Here u is the x-component of the wind vector, v is the y-component of the wind vector, f is the Coriolis parameter, I is the geopotential, and w is the vertical component of the wind vector.

Time-Differencing Schemes

55

The mass continuity equation can be integrated in the vertical to obtain

§ wu wv · wtop  wbottom  ¨  ¸ h © wx wy ¹

0,

(3.63)

since u and v are independent of height. Let H be the mean depth of the fluid, which is invariant with time, and h' be the perturbation height, so that h H  h ' . Furthermore, with a flat surface at the bottom we assume wbottom 0 . We then have wtop

dh dt

§ wu wv · § wu wv · H ¨  ¸  h '¨  ¸ . © wx wy ¹ © wx wy ¹

This equation can be written as

dh dt

wh wh wh u v wt wx wy

§ wu wv · § wu wv · H ¨  ¸  h '¨  ¸ . x y w w © ¹ © wx wy ¹

(3.64)

After multiplying by g and defining the geopotential (I ) as I gh , we obtain the continuity equation: § wu wv · wI ' w w  (uI ')  (vI ')  I ¨  ¸ 0 . (3.65) wt wx wy © wx wy ¹

Here, I gH and I ' I  I . Equation (3.65) describes the variation in the height of the free surface. Equations (3.60), (3.61) and (3.65) form the shallow-water model. A linearized shallow-water system has a frequency equation which is cubic. Two of the solutions are gravitational modes, and the third is a Rossby wave. To look into the nature of the gravity wave solution of the shallow-water equations, we examine their linearized form (hence, no nonlinear dynamics) on a nonrotating frame (i.e., f 0 ) and assume no basic flow (i.e., u 0, v 0 ): wu wI '  0, wt wx wv wI '  0, wt wy § wu wv · wI ' I ¨  ¸ 0. wt © wx wy ¹

(3.66)

(3.67)

(3.68)

Differentiating (3.68) with respect to t and making use of (3.66) and (3.67), we can write w 2I '  I’ 2I ' 0 . wt 2

(3.69)

Assuming that the perturbation is only in the x-direction, the above equation further simplifies to w 2I ' w 2I ' I 2 0 . (3.70) 2 wt wx

An Introduction to Global Spectral Modeling

56

Assuming a solution to be of the form I ' eik ( x ct ) and solving for phase speed c, we obtain 12 c 2 I gH , or c r  gH .

This is the gravity wave phase speed. This disturbance is forced by gravity and moves both in the positive and negative directions. For an isothermal atmosphere, taking the mean height as H = 9 km and g = 9.8 ms-1, we obtain c # 300 ms-1, which is very close to the speed of sound waves. We next look at the nature of the Rossby wave solution. Consider the horizontal motion of the fluid under the influence of the Coriolis force and assume no basic flow. Furthermore, assume the flow to be nondivergent, so that u



w\ ,v wy

wu wv w\  , and wx wy wx

0.

(3.71)

The linearized equations of motion are then wu  fv wt wv  fu wt

wI c , wx wI c .  wy



(3.72) (3.73)

Differentiating (3.73) with respect to x and (3.72) with respect to y and subtracting, we get the linearized form of the vorticity equation. That is,

w] / wt

where E w f / wy and ]

Ev ,

(3.74)

’ 2\ . In other words, we can write

wv / wx  wu / wy w 2 ’\ wt

E

w\ . wx

Let us assume a solution of the form \ 2 Q (k  l 2 ) k E , k E Q . k2  l2

(3.75)

\ˆ ei ( kx ly Q t ) . Then we have (3.76)

Using this, we obtain the phase speed in the x-direction

cx

v k

E . k  l2 2

(3.77)

This is the phase speed of Rossby waves caused by the variation in the Coriolis force (the

E effect). The phase speed is directly proportional to E and inversely proportional to the square of the wavenumber k 2  l 2 . Therefore, we observe that the shallow-water equations contain both slow-moving Rossby waves and high-frequency grravity waves. Handling of high-frequency gravity waves will require shorter time steps as demanded by the CFL criterion, which is computationally expensive.

Time-Differencing Schemes

57

To overcome this, it is economical to apply the implicit time integration scheme, which permits relatively longer time steps. Because the nonlinear part of the equations cannot be dealt with implicitly, we calculate it explicitly. The linear part of the equations is treated implicitly. Such a scheme is known as a semi-implicit time-integration scheme. Based on these considerations, we write the fully nonlinear shallow-water equations as § wu · wu wI ' wu   ¨ u  v  fv ¸ N u , (3.78) wt wx wy © wx ¹

wv wI '  wt wy

§ wv · wv  ¨ u  v  fu ¸ N v , wy © wx ¹ § wu wv · § wuI ' wvI ' · wI ' I ¨  ¸ ¨  ¸ Nh , wt wy ¹ © wx wy ¹ © wx

(3.79)

(3.80)

where the left-hand sides of the equations are the terms that excite gravity waves, the right-hand sides of the equations are the terms for the Rossby waves, and N u , N v , and N h are the nonlinear terms. We assume nonlinear dynamics do not by themselves excite gravity waves. For simplicity, the prime (') is dropped from the I ' term in the following discussion. The finite-difference analog of (3.78) can be written as

u n 1  u n 1 1 § wI n 1 wI n 1 ·  ¨  ¸ 2't 2 © wx wx ¹

 Nu

n

,

(3.81)

where n denotes the time level. The term 1/ 2(wI n 1 / wx  wI n 1 / wx) denotes the mean value for wI / wx at time levels (n  1) and (n  1) . Furthermore, the nonlinear term

 Nu

n

is calculated at time level n. Thus we can write u n 1

v n 1

§ wI n 1 wI n 1 · n  u n 1  't ¨ ¸  2 't  N u , w w x x © ¹ § wI n 1 wI n 1 · n n 1  v  't ¨ ¸  2 't  N v . w w y y © ¹

(3.82)

(3.83)

In addition, the mass continuity equation can be written as ª§ wu n 1

I n 1 I n 1  I 't Ǭ

© wx



wv n 1 · § wu n 1 wv n 1 · º n  ¸¨ ¸ »  2 't  N h . wy ¹ © wx wy ¹ ¼

(3.84)

At this point there are three unknowns from equations (3.82), (3.83), and (3.84): u n 1 , v n 1 , and I n 1 . From these equations we can obtain a single equation for the geopotential. This can be accomplished by first taking w / wx of (3.82) and w / wy of (3.83) and substituting into the continuity equation as

58

An Introduction to Global Spectral Modeling

w n 1 u wx

w n 1 w § w w w n · u  't ¨ I n 1  I n 1 ¸  2't  N u wx wx © wx wx wx ¹

w n 1 v wy

· w n 1 w § w w w n v  't ¨ I n 1  I n 1 ¸  2't  N v . wy wy © wy wx w y ¹

and

Next, substituting these two equations into equation (3.84) we obtain w § w w w n ª w n 1 · u  't ¨ I n 1  I n 1 ¸  2't  N u wx © wx wx wx ¬ wx ¹ § w · º w § w w w w w n  ¨ v n 1  't ¨ I n 1  I n 1 ¸  2't  N v  u n 1  v n 1 » , w w w w w w w y y y y y x y © © ¹ ¼



I n 1 I n 1  I't «



ª§ w n 1 w n 1 · § w2 · w2 w2 w2 u  v ¸  't ¨ 2 I n 1  2 I n 1  2 I n 1  2 I n 1 ¸ w w w w w w x y x y x y ¹ © ¹ ¬©

I n 1 I n 1  I't Ǭ

§ w ·º w w n n· § w n 2't ¨  N u   N v ¸  ¨ u n 1  v n 1 ¸ »  2't  N h , wy wy © wx ¹ © wx ¹¼ 2

2

I n 1 I n 1  I  't ’ 2I n 1  I  't ’ 2I n 1 G w 2§ w n n· I't ’ ˜ V n 1  2I  't ¨  N u   N v ¸ wy © wx ¹ G n 1 n I't ’ ˜ V  2 't  N h ,

 



G where ’ ˜ V n 1 wu n 1 / wx  wv n 1 / wy . We place all terms at time level (n  1) on the lefthand side and the rest of the terms on the right-hand side. Then we write G I ('t ) 2 ’ 2I n 1  I n 1  ª¬I n 1  I ('t ) 2 ’ 2I n 1  2I 't (’ ˜ V n 1 º¼ w 2 ª w n nº n 2I  't «  N u   N v »  2't  N h . w w x y ¬ ¼ This equation can be written as

’ 2I n 1  where F n 1 Gn

I n 1 I ('t ) 2

F n 1  G n , I ('t ) 2

(3.85)

G  ª¬I n 1  I ('t ) 2 ’ 2I n 1  2I 't (’ ˜ V n 1 º¼ , n w( Nv )n º 2 ª w( Nu ) n 2I  't «  »  2't  N h . wy ¼ ¬ wx

This is a Helmholtz equation for the variable I n 1 , i.e. one equation and one unknown. It can be solved using relaxation techniques or casting it into a tridiagonal matrix. Once a solution I n 1 is obtained, it can be substituted into (3.82) and (3.83) to

Time-Differencing Schemes

59

obtain future values of the velocity components u n 1 and v n 1 . Therefore, in principle, one can march forward in time and obtain future values of I, u, and v. This technique can be used to forecast an event such as the propagation of an African wave and will be good for at most one day since it does not include convection.

Chapter 4

What Is a Spectral Model? 4.1

Introduction

If one takes a closed system of the basic meteorological equations and introduces within this system a finite expansion of the dependent variables using functions such as double Fourier or Fourier-Legendre functions in space, then the use of the orthogonality properties of these spatial functions enables one to obtain a set of coupled nonlinear ordinary differential equations for the coefficients of these functions. These coefficients are functions of time and the vertical coordinate, since the horizontal spatial dependence has been removed by taking a Fourier or a Fourier-Legendre transform of the equations. The coupled nonlinear ordinary differential equations for the coefficients are usually solved by simple time-differencing and vertical finite-differencing schemes. The mapping of the solution requires the multiplication of the coefficients with the spatial functions summed over a set of chosen finite spatial basis functions. This is what defines spectral modeling. Meteorological application of the spectral method was initiated by Silberman (1954), who studied the nondivergent barotropic vorticity equation in the spherical coordinate system using the spectral technique. In its earlier days, the spectral method was particularly suitable for low-resolution simple models. The equations of these simple models involved nonlinear terms evaluated at each time step. Evaluation of the nonlinear terms was performed using the interaction coefficient method and thus required large memory allocations, which was an undesirable proposition. However, with the introduction of the transform method, developed in dependently by Eliasen et al. (1970) and Orszag (1970), the method for evaluation of these nonlinear terms changed completely. This transform method also made it feasible to include nonadiabatic effects in the model equations. For the past couple of decades, the spectral method has become an increasingly popular technique for studies of general circulation and numerical weather prediction at the operational and research centers.

4.2

The Galerkin Method

This method forms the basis for spectral modeling, and it is easy to understand if the reader has some background in linear algebra. We have a set of linearly independent functions T i ( x) , which are called the basis functions. The dependent variables of the 60

What Is a Spectral Model?

61

problem are represented by a finite sum of these basis functions. To illustrate the application of the Galerkin method, let us consider the following problem:

g  x ,

L u

D dxdE ,

(4.1)

where L is a differential operator, u is the unknown dependent variable, g is the forcing function, and x is the independent variable. The bounds of x are indicated by the real numbers D and E. Our objective is to solve (4.1) given appropriate boundary conditions. The first step in obtaining a solution to (4.1) is to approximate u ( x) by a finite sum of basis functions as N

u  x #

¦ u T  x .

(4.2)

i i

i 1

Here ui represents the coefficient of the ith basis function T i ( x) and N is some prescribed integer. When one approximates u ( x ) by (4.2), the error involved in satisfying (4.1) is given by § N · (4.3)  ERROR N L ¨ uiT i  x ¸  g  x . ©i1 ¹

¦

Our objective is to determine ui for i 1, 2, 3, ! , N . This is done by imposing the condition that the error given by (4.3) is orthogonal to each and every basis function T i ( x) in the interval D d x d E . Mathematically, this condition can be written as

³

E

D

 ERROR NTi  x dx

for i 1, 2, 3, ! , N .

0

(4.4)

Then from (4.4) we can write

³

E

D

§ L¨ ¨ ©

·

N

¦u T j

j

j 1

E

 x ¸¸T i  x dx  ³D g  x T i  x dx

0,

(4.5)

¹

for i 1, 2, 3, ! , N . Equation (4.5) represents N algebraic equations for N unknowns u1 , u2 , u3 , ! , u N . Therefore, in principle, one can solve for these unknowns. To illustrate the Galerkin method, let us consider the following example. We want to solve the system given by d 2u dx 2

where 0 d x d S and du / dx x

0

du / dx x

F  x S

(4.6)

0 are the boundary conditions.

By

inspection, the following basis functions are suitable for the above problem, since they satisfy the prescribed boundary conditions:

T n  x cos nx, In this example, L

d 2 / dx 2 and

n 1, 2, 3, ! , N .

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An Introduction to Global Spectral Modeling

d2 un  x dx 2

L ª¬un  x º¼

d2 cos nx dx 2

 n 2 cos nx .

Thus

§ L¨ ©

N

N

·

¦ u T ( x) ¸¹ ¦ i u T  x . 2

i i

i i

i 1

i 1

The error in satisfying (4.6) is given by

 ERROR N

§ L¨ ©

N

·

¦ u T ( x) ¸¹  F  x , i i

i 1

which can also be written as N

 ERROR N

¦ i u T  x  F  x . 2

i i

(4.7)

i 1

Imposing the condition that  ERROR N is orthogonal to each and every basis function for the range 0 d x d S , we obtain N

¦ i u ³ 2

i

i 1

S

0

T i  x T j  x dx

³

S

0

T j  x F  x dx .

(4.8)

It should be noted that in this example,

³

S

0

T i  x T j  x dx

³

S

cos ix cos jx dx .

0

Furthermore, we know from the orthogonality property of the Fourier function that

³

S

cos ix cos jx dx

0

S 2

G ij ,

(4.9)

where G is the Kronecker delta function with the values

G ij

­1 ® ¯0

if i

j

if i z j

.

(4.10)

With this, (4.8) leads to  j 2u j

S 2

³

S

0

T j  x F  x dx,

j 1, 2, 3, ! , N ,

from which we can write uj

2 j 2S

³

S

0

T j  x F  x dx,

j =1, 2, 3, ! , N ,

(4.11)

If F ( x) can also be represented using a finite sum of the basis functions cos nx , then the solution would be exact. Furthermore,

What Is a Spectral Model?

63

³

S

0

T n  x F  x dx

represents a kind of transform using the basis functions. Let us consider the specific case when F ( x) cos x  cos 2 x  cos 3 x . Then using (4.11) we have 2 S u1 2 cos x  cos x  cos 2 x  cos 3 x dx 1S 0 2 S (4.12) cos 2 x dx 1 .

³

³

S

Similarly we get u2

0

1/ 4 and u3 u4

Thus, u1 1 , u2 solution of

where du / dx x

0

u5

1/ 4 and u3

d 2u dx 2 du / dx x

1/ 9 . Furthermore, uN

0.

1/ 9 are the desired spectral coefficients. Hence the

0d x dS ,

cos x  cos 2 x  cos 3 x,

(4.13)

0 , is given by

S N

u

!

u6

1 1  cos x  cos 2 x  cos 3 x . 4 9

¦ u T ( x) i i

i 1

(4.14)

In this case, we have an exact solution since F ( x) , the forcing function, can be represented exactly as the sum of our basis functions. Two of the most useful Galerkin methods are the spectral method and the finite element method. In the above example, if we introduce time as one of the independent variables, then the ordinary differential equation has to be recast as a partial differential equation and the spectral coefficients would depend on time.

4.3

A Meteorological Application

Suppose we want to represent the observed temperature field at 850 mb around a latitude circle using wave-like functions. We use a sum of sine and cosine functions to do this. This gives us the finite discrete Fourier series as n

Ti

A0 

¦ A cos k

k 1

ikS  n

n 1

¦ B sin k

k 1

ikS , n

(4.15)

where n is the total number of wave components used to describe the temperature field. The points at which the temperature is known are represented by i 0, 1, 2, 3, ! , 2n  1 . If T is defined at 100 points, then we will have a total of 50 waves. The integer k is called the zonal wavenumber, which denotes the number of waves along a latitude

64

An Introduction to Global Spectral Modeling

circle. Thus k 10 implies a wavelength of L 360q /10 36q . The coefficients Ak and Bk are obtained using the following formulas, i.e., the discrete Fourier transform:

Ak

1 2 n 1 ikS Ti cos , ni0 n

¦

Bk

1 2 n 1 ikS Ti sin . n i0 n

¦

(4.16)

The temperature data are used to obtain Ak and Bk . Furthermore, (4.15) can be used along with the above equation in order to reconstruct the original temperature field if we know the coefficients. The functions used in most global atmospheric spectral models as the basis functions are the spherical harmonics, a combination of sine and cosine functions that represent the zonal structure and associated Legendre functions that represent the meridional structure. Moreover, we work with the spherical domain. This is unlike the grid-point finite-difference models, where we generally work with a limited area in the Cartesian coordinate system. Using the above basis functions, a dependent variable, say A(O , I ) , can be represented as j

A  O ,I

j m

¦¦A e

m imO m n n

P

 sin I ,

(4.17)

m jn m

where m is the zonal wavenumber, n is the two-dimensional (total) wavenumber, j is the maximum wavenumber resolved, O is the longitude, I is the latitude, Anm is the spectral coefficient, and Pnm is an associated Legendre function of the first kind. A more detailed discussion of these functions is presented in Chapter 6.

4.4

Exercises

4.1 Consider the heat equation, wu wt

k

w 2u , wx 2

0d xd L,

with the boundary conditions u (0, t ) 0 and u ( L, t ) 0 and the initial condition u ( x, 0) f ( x) . Find the required solution u ( x, t ) of the above problem. (Hint: Use separation of variables). 4.2. In Exercise 4.1 assume (a) f ( x) 12  8cos nS x / L and (b) f ( x) Obtain a solution for Exercise 4.1.

8sin S x / L .

Chapter 5

Lower-Order Spectral Model 5.1

Introduction

This system was first developed by Lorenz (1960b). It is an elegant system that provides an introduction to the concepts of spectral modeling. It is based on the premise that the dynamic equations governing the atmosphere can be simplified to the greatest extent possible and still be realistic enough to describe certain desired features. The extent to which we can simplify the equations depends on the particular phenomena we are studying. This simplification can aid in our understanding of a certain phenomena and help to form reasonable hypotheses which we can then test through a more advanced system of equations. The system is based on the use of double Fourier series representations of the basic equations in a doubly periodic domain. That is, the equations are expanded into a series of eigenfunctions, some of whose coefficients are retained as the new dependent variables. These new dependent variables correspond to features of the largest scale. Here we examine the barotropic vorticity equation. We start with the equation governing the conservation of vorticity of a parcel for two-dimensional, homogeneous, incompressible, and inviscid fluid flow on an f-plane given by G w 2 w 2 or ’ \  J \ , ’ 2\ , ’ \  k ˜’\ u ’ ’ 2\ (5.1) wt wt









where \ is the streamfunction and J is the Jacobian. Since we are working on an f-plane, the E term does not appear in this equation. We let \ be doubly periodic and state the periodicity property by the relation 2S 2S · § \  x, y , t \ ¨ x  ,y ,t ¸ , k l ¹ ©

where k and l are the wavenumbers and are specified constants. The area is now finite but unbounded. We seek a solution to (5.1) in a closed horizontal domain. For this, we expand \ following Lorenz (1960b), that is, f

\

f

¦¦ m k 2

m 0 n n0

1 ª Amn cos  mkx  nly  Bmn sin  mkx  nly º¼ .  n 2l 2 ¬

2

65

(5.2)

An Introduction to Global Spectral Modeling

66

This is the double Fourier representation of the function \. The coefficients Amn and Bmn are functions of time, where m and n are integers representing east-west and northsouth wavenumbers, respectively. Note that A00 0 . In addition, the lower limit n0 of n is specified as ­ f if m ! 0 . n0 ® if m 0 ¯0 The series will now be truncated and we consider only those terms for which m equals 0 and +1 and n equals -1, 0, and +1; in other words, we include only one wave in both directions. Equation (5.2) for the streamfunction \ then reduces to

\

A10 A A cos kx  201 cos ly  2 11 2 cos  kx  ly 2 k l k l B10 B A11  2 2 cos  kx  ly  2 sin kx  201 sin ly k l k l B11 B11  2 2 sin  kx  ly  2 2 sin  kx  ly , k l k l



(5.3)

which is the time dependent solution for (5.1) for one wave. The corresponding relative vorticity is given by ’ 2\

A10 cos kx  A01 cos ly  A11 cos  kx  ly

 A1,1 cos  kx  ly  B10 sin kx  B01 sin ly  B11 sin  kx  ly  B1,1 sin  kx  ly .

(5.4)

Substituting the Fourier expansion of \ and ’ 2\ into (5.1) and taking the Fourier transform of both sides of the resulting equation, we get the prediction equations for the amplitude of the different wave components. In all, we have eight equations providing time tendencies for each of the eight amplitudes.

5.2

Maximum Simplification

After substituting for \ and ’ 2\ from (5.3) and (5.4) into (5.1) and equating the coefficients of the various Fourier functions on both sides of the resulting equation, we get a set of differential equations for the coefficients A10 , A01 , A11 , A11 , B10 , B01 , B11 , and B11 . Following Lorenz (1960b), we assume: (a) If B10 , B01 , B11 , and B11 , vanish initially, then they will remain zero for all time since their tendencies are always equal to zero, that is, dB10 dB01 dB11 dB11 0. dt dt dt dt

We thus obtain B10 B01 B11 B11 0 . (b) If A11  A11 initially, then A11 will remain equal to  A11 for all time. Furthermore, let A01 A , A10 F , and  A11 G . With this, (5.3) reduces to

Lower-Order Spectral Model

\



67

A F G cos ly  2 cos kx  2 2 2 sin kx sin ly . l2 k k l

(5.5)

A, F, and G are functions of time. The term ( A / l 2 ) cos ly describes the basic zonal current, that is, it has no x dependence. The term ( F / k 2 ) cos kx  [2G /(k 2  l 2 )]sin kx sin ly describes the eddies. The corresponding relative vorticity is given by

’ 2\

A cos ly  F cos kx  2G sin kx sin ly .

(5.6)

Next we obtain an expression for



J \ , ’ 2\



w\ w’ 2\ w\ w’ 2\  . wx wy wy wx

(5.7)

Now,

w\ F 2Gk sin kx  2 2 cos kx sin ly , wx k k l w\ A 2Gl sin ly  2 2 sin kx cos ly , wy l k l w 2 ’ \  Fk sin kx  2Gk cos kx sin ly , wx w 2 ’ \  Al sin ly  2Gl sin kx cos ly . wy

(5.8) (5.9) (5.10) (5.11)

Hence



J \ , ’ 2\

2Gk §F · ¨ sin kx  2 2 sin ly cos kx ¸ k k l  © ¹ u   Al sin ly  2Gl sin kx cos ly



2Gl §A ·  ¨ sin ly  2 2 cos ly sin kx ¸ k l ©l ¹ u   Fk sin kx  2G cos kx sin ly .

After simplifying, we obtain



J \ , ’ 2\



§1 1 · ¨ 2  2 ¸ AFkl sin kx sin ly k ¹ ©l 1 · § 1  ¨ 2  2 2 ¸ 2 FGkl sin 2 kx cos ly k l ¹ ©k 1 · § 1  ¨  2  2 2 ¸ 2 AGkl sin 2 ly cos kx . l ¹ l k ©

(5.12)

Differentiating (5.6) with respect to t, we get w 2 ’\ wt

dA dF dG cos ly  cos kx  2 sin kx sin ly . dt dt dt

From the barotropic vorticity equation (5.1) along with (5.12) and (5.13), we get

(5.13)

An Introduction to Global Spectral Modeling

68

dA dF dG cos ly  cos kx  2 sin kx sin ly dt dt dt ª§ 1 1 ·  «¨ 2  2 ¸ AFkl sin kx sin ly k ¹ ¬© l 1 1 · §  ¨ 2  2 2 ¸ 2 FGkl sin 2 kx cos ly k l ¹ ©k º 1 · § 1  ¨  2  2 2 ¸ 2 AGkl sin 2 ly cos kx » . k l ¹ © l ¼

(5.14)

If we multiply (5.14) by cos ly and integrate both sides over the entire doubly periodic fundamental domain, then using the orthogonality properties of the Fourier functions we obtain dA dt

2S

³ ³ 0

2S

cos 2ly dx dy

0

1 · § 1  ¨ 2  2 2 ¸ 2klFG k k l ¹ ©

u

2S

³ ³ 0

2S

sin 2 kx cos 2 ly dx dy .

(5.15)

0

Integrating, we obtain 2

dA 2 S dt

1 · § 1 2S 2 klFG ¨ 2  2 2 ¸ , k l ¹ ©k

or dA dt

1 · § 1  ¨ 2  2 2 ¸ klFG . k l ¹ ©k

(5.16)

Similarly, if we multiply (5.14) by cos kx and sin kx sin ly and integrate over the domain, we get dF § 1 1 · (5.17) ¨  ¸ klAG , dt © l 2 k 2  l 2 ¹ dG 1§ 1 1 ·  ¨ 2  2 ¸ klAF . (5.18) dt k ¹ 2©l

Equations (5.16), (5.17), and (5.18) are a system of three coupled nonlinear firstorder ordinary differential equations that constitute the barotropic vorticity equation in the three unknowns A, F, and G. If their initial values are known, then their future values can be obtained using numerical integration. The above system has exact solutions which can be expressed by elliptic functions (or circular functions) in time.

5.3

Conservation of Mean-Square Vorticity and Mean Kinetic Energy

In this section we will attempt to answer the following question: Does this simplified system conserve mean-square vorticity and mean kinetic energy? Consider the expression for the total-square vorticity, which is written as

Lower-Order Spectral Model

69 2S

2S

2

³ ³  ’ \ dx dy . 0

2

0

On substituting the Fourier expansion for ’ 2\ , we obtain 2S

2S

2S

2

³ ³  ’ \ dx dy ³ ³ 0

2

0

0

2S

0

2

 A cos ly  F cos kx  2G sin kx sin ly dx dy .

This can be written as 2S

2S

2S

2

2S

³ ³  ’ \ dx dy ³ ³  A 0

2

0

0

2

0 2

cos 2 ly  F 2 cos 2 kx

4G sin 2 kx sin 2 ly  2 AF cos ly cos kx 4 AG sin kx sin ly cos ly  4 FG sin kx sin ly cos kx dx dy .

(5.19)

The integrals involving terms AF, AG, and FG turn out to be zero using the orthogonality rules, and we are left with 2S

2S

2S

2

2S

³ ³  ’ \ dx dy ³ ³  A 0

2

0

0

2

0

cos 2 ly  F 2 cos 2 kx



 4G 2 sin 2 kx sin 2 ly dx dy .

(5.20)

This gives 2S

2S

2

³ ³  ’ \ dx dy 0

2

2S 2 A2  2S 2 F 2  4S 2G 2

0





2S 2 A2  F 2  2G 2 .

(5.21)

Thus the total-square vorticity over the domain is 2S

2S

2

³ ³  ’ \ dx dy 0

2



2S

2S

Because the domain area is given by ³0 ³0 dx dy

’ \ 2

2



2S 2 A2  F 2  2G 2 .

0

(5.22)

4S 2 , the mean-square vorticity is

1 2 A  F 2  2G 2 . 2





(5.23)

Next we show that the mean-square vorticity of this system is conserved. From d §1 2· dA (5.23), using the relation the time rate of change of mean-square ¨ A ¸ A dt © 2 ¹ dt vorticity (’ 2\ ) 2 is given by .

d ’ 2\ dt





2

d ª1 2 º ( A  F 2  2G 2 ) » dt «¬ 2 ¼

A

dA dF dG F  2G . dt dt dt

(5.24)

Substituting for dA / dt , dF / dt , and dG / dt from (5.16), (5.17), and (5.18), we get d ’ 2\ dt





2

ª AFG « kl ¬

1 · 1 · § 1 §1 ¨ 2  2 2 ¸  kl ¨ 2  2 2 ¸  l ¹ k k l l k © ¹ ©

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70

§ 1 1 ·º kl ¨ 2  2 ¸» k ¹¼ ©l

0.

(5.25)

Thus, the mean-square vorticity of this three-component low-order system is conserved. We next show that the mean kinetic energy is also conserved by the low-order system. Following a similar procedure,

Total K.E.

1 2 1 2

2S

³ ³

2S

0

0

2S

2S

³ ³ 0

0

ª§ w\ · 2 § w\ · 2 º «¨ ¸ » dx dy ¸ ¨ «¬© wx ¹ © wy ¹ »¼ ª§ F 2Gk · «¨ k sin kx  k 2  l 2 cos kx sin ly ¸ ¹ ¬©

2

2 2Gl §A · º  ¨ sin ly  2 2 sin kx cos ly ¸ » dx dy k l ©l ¹ »¼ 2 2 2 2 2 S S ª§ F 1 4k G sin 2 kx  2 2 2 cos 2 kx sin 2 ly «¨ 2 0 0 ¬© k 2 (k  l ) 4 FG ·  2 2 sin kx sin ly cos kx ¸ k l ¹ 2 2 2 §A 4l G  ¨ 2 sin 2 ly  2 2 2 cos 2 ly sin 2 kx (k  l ) ©l

³ ³



(5.26)

4 AG ·º sin kx sin ly cos ly ¸» dx dy k2  l2 ¹¼

1 ªF2 2 4k 2G 2 4l 2G 2 A2 2 2 2º « 2 2S  2 2 2 S  2 2S  2 2 2 S » . 2¬k (k  l ) l (k  l ) ¼ Hence

§ F 2 A2 2G 2 · Total K.E. S 2 ¨ 2  2  2 2 ¸ . l k l ¹ ©k

(5.27)

Note that the domain area equals 4S 2 . Thus the mean kinetic energy is given by K.E.

1 § F 2 A2 2G 2 · ¨ 2  2  2 2¸ . 4© k l k l ¹

Now we show that d / dt K.E. 0 ,

d K.E. dt

d ª 1 § F 2 A2 2G 2 · º « ¨ 2  2  2 2 ¸» dt ¬ 4 © k l k  l ¹¼ 1 ª A dA F dF 2G dG º   2 «¬ l 2 dt k 2 dt k 2  l 2 dt »¼

(5.28)

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71

1ª 1§ 1 1 · 1 §1 1 ·  ¨  ¸ ¨  ¸ 2 ¬« l 2 © k 2 k 2  l 2 ¹ k 2 © l 2 k 2  l 2 ¹



1 § 1 1 ·º ¨  ¸» AFGkl k  l 2 © l 2 k 2 ¹¼ 2

1ª 1 1 1    2 «¬ l 2 k 2 l 2 (k 2  l 2 ) k 2 (k 2  l 2 ) º 1 1 1   AFGkl k 2l 2 l 2 (k 2  l 2 ) k 2 (k 2  l 2 ) »¼ 0.



(5.29)

Thus d / dt K.E. 0 . Hence the mean kinetic energy is conserved with time.

5.4

Energy Transformations

In the previous section, we saw that the mean kinetic energy of the low-order system is conserved with time. However, the zonal kinetic energy and the eddy kinetic energy individually are not conserved, as we show below. Thus, continuous exchanges of kinetic energy take place between the zonal flow and the eddies. At times, the zonal may gain kinetic energy at the expense of the eddies while at other times the eddies may gain kinetic energy at the expense of the zonal flow. This occurs in such a way that the total kinetic energy is invariant. In our low-order system, the zonal flow is represented by

\Z



A cos ly , l2

(5.30)

while the eddy flow is given by

\E



F 2G cos kx  2 2 sin kx sin ly . 2 k k l

(5.31)

The mean zonal and mean eddy kinetic energies therefore are given by

KZ

KE

1

2S

³ ³

4S 2 0 1 A2 , 4 l2 1 2S 4S 2 0

0

4S

2S

2

2S

0

³ ³ 0

1 2  ’\ Z dx dy 2

1 4S 2

2S

³ ³ 0

2S

0

1 A2 sin 2 ly dx dy 2 l2

(5.32)

³ ³

1

2S

2S

0

1 2  ’\ E dx dy 2 2 1 ª§ F 2Gk · «¨ sin kx  2 2 cos kx sin ly ¸ 2 «¬© k k l ¹

2 § 2Gl · º  ¨  2 2 sin kx cos ly ¸ » dx dy © k l ¹ »¼

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72

ª 1 «F2 4G 2 k 2 sin 2 kx  cos 2 kx sin 2 ly 2 2 2 2 4S 0 0 2 « k k l  ¬ 2 2 º 4G l 4GF  2 2 2 sin 2 kx cos 2 ly  2 2 sin kx cos kx sin ly » dx dy (k  l ) k l ¼ 1

2S

2

³ ³

2S





1 § F2 2G 2 · ¨ 2  2 2 ¸. 4© k k l ¹

(5.33)

Thus we have dK Z dt

1 d A2 2 dt l 2

A dA . 2l 2 dt

(5.34)

This, after substituting the value of dA / dt from (5.16), gives dK Z dt

 AFG

kl § 1 1 · ¨  ¸ 2l 2 © k 2 k 2  l 2 ¹

 AFG

k § 1 1 · ¨  ¸. 2l © k 2 k 2  l 2 ¹

(5.35)

Similarly, dK E dt

1 dF 1 dG F  2 2G 2 dt k  l dt 2k AFGkl ª 1 § 1 1 · 1 § 1 1 ·º ¨  ¸ ¨  ¸ 2 «¬ k 2 © l 2 k 2  l 2 ¹ k 2  l 2 © l 2 k 2 ¹ »¼ AFGk § 1 1 · ¨  ¸. 2l © k 2 k 2  l 2 ¹

(5.36)

We denote wK Z / wt as  K E , K Z ! , which is the change in K Z due to energy transfer from the eddy to the zonal flow. Also,  K Z , K E ! denotes wK E / wt and represents the change in K E due to transfer from the zonal to the eddy flow. Note that  K E , K Z ! and  K Z , K E ! are equal in magnitude but opposite in sign. Thus in this low-order system, the gain of kinetic energy by the zonal flow is equal to the loss of kinetic energy by the eddy flow or vice versa, so that the total kinetic energy is invariant with time. Such barotropic energy exchange is an important property of a nondivergent flow.

5.5

Mapping the Solution

In the expression for the streamfunction of a low-order system, that is,

\



A F G cos ly  2 cos kx  2 2 2 sin kx sin ly , l2 k k l

(5.37)

the latitudes S /(2l ), 3S /(2l ), 5S /(2l ), ! correspond to the zonal wind maxima and are fixed. However, the intensity of the zonal flow given by A / l may vary. Thus the variable A denotes the zonal index. The last two terms represent disturbances superimposed on the zonal flow. Both of these combined describe a wave of a single

Lower-Order Spectral Model

73

wavenumber, but with a variable shape and phase. This simple model simulates the interaction of the zonal flow with the superimposed disturbance. This problem is easy to solve on any simple calculator or a personal computer. The student is strongly advised to try this as an exercise. The initial values of A, F, G and the wavenumbers k and l need to be assigned. For numerical integration in time, one forward time step may be followed by centered time steps. For that purpose, one uses (5.16), (5.17), and (5.18). At the end of each day of integration, the predicted values of A, F and G can be substituted in (5.37) to map the forecast.

5.6

An Example of Chaos

In order to illustrate the sensitivity of the low-order model to small changes of parameters, we examine the following sixth-order system:

VD 1 3 D 2 A  V (1  D 2 ) A  D  D BC 0 , 1D 2 2 1D 2 3 B  V B  D AC 0 , 4 VD 1 D3  C  V (4  D 2 ) C  F AB 0 , 2 4 D 2 4 D 2 1 D  (1  D 2 ) D  RD A  D AE  D BF 0 , 2 1  E  4 E  D AD 0 , 2 1 F  (4  D 2 ) F  RD C  D BD 0 . 2

(5.38) (5.39)

(5.40) (5.41) (5.42)

(5.43)

This is a coupled nonlinear system of six equations for the six dependent variables A, B, C, D, E, and F. The independent variable here is time. Furthermore, an overdot (.) represents a derivative with respect to time, D is the wavenumber, and R is the Rayleigh number, which is expressed as R g E'Td 3 /(NQ ) . Here g is the acceleration due to gravity, E is the thermal expansion coefficient, N is the thermal diffusivity, Q is the kinematic viscosity, 'T is the temperature excess of the bottom boundary over the top boundary, and d is the distance between these two boundaries. This problem is solved numerically (see Fig. 5.1) in the same manner as the simple three-component system illustrated above. In fact, with B C F 0 , this sixcomponent system reduces to a three-component Lorenz system. A time-differencing scheme is needed to solve this coupled system numerically. In the coding of this problem, we have used an Adams-Bashforth time-differencing scheme, which was described in Chapter 3. The sensitivity of the solution to small changes in the Rayleigh number R is illustrated in Fig. 5.1. Here we show a periodic solution for R = 50.2299 and a chaotic solution for R = 50.2300. The abscissa denotes the value of the coefficient B, and the ordinate denotes the value of coefficient C. The time evolution of the solution for R =

An Introduction to Global Spectral Modeling

74

Figure 5.1. Periodic and chaotic solutions of a six-component system.

50.2299 repeats itself, whereas for R = 50.2300 the solution is chaotic. It should be noted that the same sort of multidimensional behavior (periodic or chaotic) is found for other pairs of variable choices as well. Hence, we see that there are huge initial uncertainties in non-linear systems.

5.7.

Exercises

5.1. In the equations

dA dt dF dt dG dt

1 · § 1  ¨ 2  2 2 ¸ klFG , k k l ¹ © 1 · §1 ¨ 2  2 2 ¸ klAG , k l ¹ ©l 1§ 1 1 ·  ¨ 2  2 ¸ klAF , k ¹ 2©l

set 2S / l 5000 and 2S / k 2500 , so that D k / l 2 . For initial conditions, let A(0) 0.06 units, F (0) 0.12 units, and G (0) 0 . Obtain the future values of A(t ) , F (t ) , and G (t ) using 't 6 hours.

Lower-Order Spectral Model

75

5.2. For A(t ) , F (t ) , and G (t ) in Exercise 5.1, map the resulting streamfunction at time t 0 and time t 24 hours.

Chapter 6

Mathematical Aspects of Spectral Models 6.1

Introduction

In this chapter we provide an introduction to the topic of spherical harmonics as basis functions for a global spectral model. The spherical harmonics are made up of trigonometric functions along the zonal direction and associated Legendre functions in the meridional direction. A number of properties of these functions need to be understood for the formulation of a spectral model. This chapter describes some useful properties that will be used to illustrate the procedure for the representation of data sets over a sphere with spherical harmonics as basis functions. The calculations of Fourier and Legendre transforms and their inverse transforms are an important part of global spectral modeling, and these are covered in some detail in this chapter. Finally, this chapter addresses the formulation of two simple spectral models. One of these is a single-level barotropic model, and the other is a shallow-water model. Consider the equation

’ 2u

1 w 2u , c 2 wt 2

which is satisfied by the velocity potential of a compressible fluid. Here u represents the velocity potential and c is the speed of gravity waves in the fluid. If the fluid is in a steady state, this equation reduces to Laplace’s equation, which is ’ 2u

w 2u w 2u w 2u   wx 2 wy 2 wz 2

0.

(6.1)

A similar equation describes the steady-state diffusive process or the heat conduction process in a medium. Laplace’s equation and its solutions, which are harmonic functions, are of fundamental importance in the study of fluid dynamics. The solutions of Laplace’s equation in a spherical coordinate system are the spherical harmonics, and are obtained by the method of separation of variables. That is, the solutions can be broken up into factors, each factor being the function of a single coordinate. The transformation between Cartesian and spherical coordinates is given by x

r cosO sinT ,

y

r sinO cosT ,

76

z

r cosT ,

(6.2)

Mathematical Aspects of Spectral Models

77

where r is the radius, O is longitude, and T is co-latitude. Substituting x, y, and z from (6.2) into (6.1), we obtain Laplace’s equation on a sphere:

w 2u 2 wu 1 w 2u cot T wu 1 w 2u     wr 2 r wr r 2 wT 2 r 2 wT r 2 sin 2 T wO 2

0,

wu · 1 ª w § 2 wu · 1 w § 1 w 2u º T   sin r ¨ ¸ ¨ ¸ » wT ¹ sin 2 T wO 2 ¼ r 2 «¬ wr © wr ¹ sin T wT ©

0.

’ 2u

(6.3)

or

In the case of atmospheric models, latitude is usually used instead of co-latitude as one of the coordinates. Hereafter, T denotes the latitude. Laplace’s equation (6.3) in the (r, O, T) coordinate system (where T is latitude and remembering sin(colat) = cos(lat)) then takes the form

wu · 1 ª w § 2 wu · 1 w § 1 w 2u º ¨r ¸ ¨ cos T ¸ » 2 « r ¬ wr © wr ¹ cos T wT © wT ¹ cos 2 T wO 2 ¼

0,

or

1 w § 1 w 2u º w § 2 wu · wu · ¨r ¸ ¨ cos T ¸ » wr © wr ¹ cos T wT © wT ¹ cos 2 T wO 2 ¼

0.

(6.4)

We use the method of separation of variables to solve (6.4). For this, assume a solution of the form u

R (r ) L(O ) P(T )

(6.5)

where R is a function of r, L is a function of O, and P is a function of T. Substituting (6.5) into (6.4) we obtain LP

d § 2 dR · RL d § dP · RP d 2 L ¨r ¸ ¨ cos T ¸ dr © dr ¹ cos T dT © dT ¹ cos 2 T d O 2

0,

or, on dividing by RLP, we get

1 d § dP · 1 d 2L ¨ cos T ¸ 2 dT ¹ L cos T d O 2 P cos T dT ©



1 d § 2 dR · ¨r ¸. R dr © dr ¹

(6.6)

The left-hand side of (6.6) is a function of O and T, while the right-hand side is a function of r only. Therefore, both sides should be equal to a constant, say k. If we now consider the right-hand side of (6.6), we obtain 1 d § 2 dR · ¨r ¸ k, R dr © dr ¹ 1 d § 2 dR · k ¨r ¸ 0, R dr © dr ¹ d § 2 dR · ¨r ¸  kR 0 . dr © dr ¹



(6.7)

An Introduction to Global Spectral Modeling

78

Let the solution of (6.7) be of the form R = rn, where n is an integer. If n is positive, then on substitution of R = rn into (6.7) we obtain

d § 2 dr n · n ¨r ¸r k 0, dr © dr ¹ d 2 n 1  r nr  r n k 0 , dr d n r n 1  r n k 0 , dr n(n  1)r n  r n k 0 ,

> n(n  1)  k @ R

0 , or

n(n  1) .

k

We see that the value of k is not altered if we replace n by  n  1 . Therefore r  n 1 is also a solution of (6.7), for which k  n(n  1) . Thus k is of the form  n(n  1) , n being a positive integer including zero. Equation (6.6) thus reduces to

1 d § dP · 1 d 2L ¨ cos T ¸ P cos T dT © dT ¹ L cos 2 T d O 2

 n(n  1) ,

or multiplying by cos 2 T and rearranging we obtain cos T d § dP · 2 ¨ cos T ¸  n(n  1) cos T P dT © dT ¹



1 d 2L . L dO 2

(6.8)

Since the left-hand side of (6.8) is a function of T and the right-hand side is a function of O, both sides must be equal to a constant, say m2. Considering the right-hand side of (6.8) gives 1 d 2L m2 , L dO2 d 2L  m2 L 0 , dO2 

(6.9)

which has a solution of the form

L

e r imO , m 0, 1, 2, 3,!

Hence after multiplying (6.8) by P cos 2 T and remembering both sides of the equation are equal to m2 the equation reduces to

1 d § dP · m2 P , ¨ cos T ¸  n(n  1) P cos T dT © cos 2 T dT ¹ 1 d § dP · § m2 · ¸P ¨ cos T ¸  ¨ n  n  1  cos T dT © dT ¹ © cos 2 T ¹

0.

(6.10)

79

Mathematical Aspects of Spectral Models

If P = sinT and

d d sin T d P dT

d dP d P dT

d dT

cos T

d , then (6.10) can be written as dP

d § m2 · 2 dP · §     1 P n n 1   ¨ ¸P ¨ ¸ dP © dP ¹ © 1 P 2 ¹

0,

(6.11)

remembering 1  sin 2 T cos 2 T . Since T varies from -S/2 to S/2, P varies from -1 to 1. Equations (6.10) and (6.11) are both known as associated Legendre equations, as they apply to all wavenumbers, m.

6.2

Legendre Equation and Associated Legendre Equation

If m = 0 in equation (6.11), we obtain d § 2 dP · ¨ 1  P ¸  n(n  1) P dP © dP ¹

0,

(6.12)

which is called the Legendre equation. We next discuss the solutions of the Legendre equation and the associated Legendre equation. Furthermore, we discuss the properties of their solutions without rigorous mathematical proofs. Solutions of the Legendre equation are known as Legendre polynomials and are denoted by Pn ( P ) . For a given n, Pn ( P ) is a polynomial of degree n and is given by M

¦  1

Pn ( P )

 2n  2r ! P n2r , 2n r ! n  r ! n - 2r !

r

r 0

(6.13)

where M n / 2 if n is even and M (n  1) / 2 if n is odd. A more convenient form of Pn ( P ) is given by Rodrigues’ formula, namely

Pn ( P )

n 1 dn P 2  1 , n n  2 n! d P n

0, 1, 2, 3,! , P d 1 .

In particular, P0 ( P ) 1 , 1 d P1 ( P )  P 2  1 2 dP

P,

P2 ( P )

2 1 d2 P 2  1 2  8 dP

P3 ( P )

3 1 d3  P 2  1 48 d P 3

P4 ( P )

4 1 d4  P 2  1 384 d P 4

P5 ( P )

5 1 d5 P 2  1 5  3840 d P

1  3P 2  1 , 2

1  5P 2  3P , 2

1  35P 4  30P 2  3 , 8 1  63P 5  70P 3  15P . 8

(6.14)

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80

Figure 6.1. Representation of Legendre polynomials P0 ( P ) to P5 ( P ) . The number of zero crossings is the same as the degree of the polynomial, n. If n is even, Pn ( P ) is symmetric about the equator. If n is odd, Pn ( P ) is antisymmetric about the equator.

Figure 6.1 shows the graphs of P0 ( P ) to P5 ( P ) for -1 d P d 1. Three useful properties of Pn ( P ) are as follows: (1) Pn ( P 1) 1 . (2) If n is even, Pn ( P ) has only even powers of P and is symmetric with respect to the equator (P = 0). (3) If n is odd, Pn ( P ) has only odd powers of P and is antisymmetric with respect to the equator. In other words, for odd n, the graph on the negative side of the P-axis is a mirror image of the graph on the positive side of the P-axis, as shown in Fig. 6.1. We now consider the associated Legendre equation (6.11). Solutions of this equation involve two parameters, m and n, and are denoted by Pnm  P . Pnm  P are called associated Legendre functions of the first kind of order m and degree n. Here m is any integer and n is a non-negative integer such that n t m . One can obtain Pnm  P by using Rodrigues’ formula, Pnm  P

n (1  P 2 ) m / 2 d n  m  P 2  1 , 2n n ! d P n  m

|P| d 1.

In particular, for n = 5, P50  P

P51  P P52  P

1  63P 5  70P 3  15P , 8 12 15 1  P 2  21P 4  14P 2  1 , 8 105 1  P 2  3P 3  P , 2

(6.15)

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81

Figure 6.2. Representation of associated Legendre functions P50  P to P55  P . Each Pnm  P is normalized by multiplying by  n  m !/  n  m ! . If n  m is even, Pnm  P is

symmetric about the equator and is antisymmetric if n  m is odd.

P53  P

32 105 1  P 2  9 P 2  1 ,  2 2

P54  P 945 1  P 2 P , 52

P55  P 945 1  P 2 .

Figure 6.2 shows the graphical representation of the above associated Legendre functions. Since Pnm  P is a polynomial of degree n, it has n roots given by the equation Pnm  P 0 . It is clear from (6.15) that m of these roots are at the poles (P = r1), while

n  m roots are between the poles. The n  m roots between the poles are called the zeroes of the associated Legendre function Pnm  P . If n  m is even, then Pnm  P is symmetric with respect to the equator. If n  m is odd, then Pnm  P is antisymmetric with respect to the equator. Three useful properties of Pnm  P are as follows: Pnm  P 0

if n  m , m

Pn m  P

 1

Pnm   P

 1

(n  m)! m Pn  P , (n  m)!

nm

Pnm  P .

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82

6.3

Laplace’s Equation

So far we have been working with only one independent variable. We next consider two independent variables. Laplace’s equation can be written as

d § 1 w 2Y 2 dY ·  n(n  1)Y ¨ (1  P ) ¸ dP © d P ¹ 1  P 2 wO 2

0,

(6.16)

where O is longitude, P = sin T, with T being latitude. Equation (6.16) is of the form ’ 2Y  n(n  1)Y 0 , where ’2 is the twodimensional Laplacian on a sphere. Let the solution of (6.16) be of the form Y ( P , O ) P  P L  O . Substituting this into (6.16), we obtain L

P w2 L  n(n  1) PL 1  P 2 wO 2

0,

(6.17)

After dividing by PL and multiplying by 1  P 2 , (6.17) may be written as

1 P 2 d § 2 dP · 2 ¨ (1  P ) ¸  n  n  1 1  P dP ¹ P dP ©

1 d 2 L . L dO 2

(6.18)

The left-hand side of the above equation is a function of P, while the right-hand side is a function of O . This implies that both sides must be equal to some constant. Let this constant be m2, so that from the right-hand side of (6.18) we obtain 1 d 2L m2 , L dO2 d2  Lm 2 dO2 d 2L  m2 L 0 . dO2



The solution of (6.19) is given by L (6.18) we obtain

(6.19)

e r imO . Furthermore, from the left-hand side of

1 P 2 d § 2 dP · 2 ¨ (1  P ) ¸  n  n  1 1  P P dP © dP ¹

m2 ,

or upon multiplication by P 1  P 2 d § m2 · 2 dP · § ¸P ¨ (1  P ) ¸  ¨ n(n  1)  dP © dP ¹ © 1 P 2 ¹

0,

(6.20)

which is an associated Legendre equation. As mentioned in Section 6.2, this has a solution of the form Pnm  P .

Mathematical Aspects of Spectral Models

83

Thus given the solution of equation (6.16), Y ( P , O ) L

e

r imO

P  P L  O , we found that

using equation (6.19) and that P  P has the form Pnm  P using equation

(6.20). Hence, the solution of Laplace’s equation on a sphere is of the form Ynm  P , O Pnm  P eimO . Ynm  P , O is a spherical harmonic of order m and degree n. The factor eimO describes the east-west variation, and the factor Pnm  P describes the northsouth variation of the spherical harmonic wave Ynm  P , O . Some useful mathematical properties of Ynm  P , O are Ynm  P , O 0 m* n

Y

 P, O

Yn m  P , O

for n  m , m n

P

 P eimO ,

Pn m  P e  imO

 1

m

(n  m)! m Pn  P e  imO , (n  m)!

n(n  1) m Yn , a2

’ 2Ynm

where Ynm*  P , O is the complex conjugate of Ynm  P , O and

’2

1 a cos T 2

§ 1 w2 w w · cos T  ¨ ¸, 2 T O T T¹ cos w w w ©

and a is the radius of the sphere.

6.4

Orthogonality Properties

The spectral equations that we work with contain a system of nonlinear differential equations that are functions of time only. In order to derive such a set of equations, one needs to remove the spatial dependence. Spherical harmonics (containing trigonometric functions in the zonal direction and associated Legendre functions in the meridional direction) describe this spatial dependence. These are removed by invoking orthogonality properties of the trigonometric and Legendre functions using what are called Fourier and Legendre transforms. In this section, we provide a theoretical background for the desired orthogonality properties. Legendre polynomials satisfy the orthogonality condition

³

1

1

Pm  P Pn  P d P

if m z n ­0 ° . ® 2 °¯ 2n  1 if m n

(6.21)

To prove this, we first evaluate the integral

³

1

1

P m Pn  P d P ,

mdn.

(6.22)

Using Rodrigues’ formula and integrating the above equation by parts m times, we have

An Introduction to Global Spectral Modeling

84

³

1

1

P m Pn  P d P

³

1

1

1 d n ( P 2  1) n dP 2 n! dPn

Pm

n

1 d n 1 ( P 2  1) n Pm n d P n 1 2 n!

 1

m

0

1

 1 n

m 2n n !

m  P 2  1 d P 2n n ! 1 if m  n .

³

1

³

1

1

P m 1

d n 1 ( P 2  1) n dP d P n 1

if m = n, (6.23)

Hence

³

1

1

­ n 1 °(  ) n 2 ® °¯ 0

P m Pn  P d P

 1

Let us next evaluate

n 1 1

³

³

1

1

( P 2  1) n d P

if m

n

.

(6.24)

if m  n

P

2

n

 1 d P .

Writing P

sin T

and after

integrating by parts, we obtain

 1

n

1

³ P

2

1

n

 1 d P

³

S /2

S / 2

cos 2 n 1 T dT

cos 2 n T sin T 2n

³

S /2

S / 2

S /2 S / 2

 2n

³

S /2

S / 2

cos 2 n 1 T sin 2 T dT

cos 2 n 1 T dT 1  cos 2 T dT ,

or

³

S /2

S / 2

cos 2 n 1 T dT

2n S / 2 cos 2 n 1 T dT 2 n  1 S / 2 2n 2n  2 2 S / 2 ... cos T dT 2 n  1 2 n  1 3 S / 2

³

(6.25) 2(2n ) 2 (n !) 2 . (2n  1)!

³

Thus

³

1

1

P Pn  P d P m

­ 2n 1 (n !) 2 ° ® (2n  1)! ° ¯0

if m

n

.

(6.26)

if m  n

Now we prove the orthogonality condition given by (6.21). For this, without loss of generality, assume that m d n in (6.21). Writing Pm  P in polynomial form using Rodrigues’ formula, we have

³

1

1

Pm  P Pn  P d P

§ (2m)! m P ¨ m © 2 m !m ! (2m  2)! P m  2  m  ! Pn  P d P . 2 1!(m  1)!(m  2)!

³

1

1

(6.27)

Mathematical Aspects of Spectral Models

85

From (6.26), if m z n (i.e., m < n), then each term in the brackets, when multiplied by Pn  P and integrated over the interval -1 to 1, has a value of zero. If m = n, then only the term with the highest degree n contributes toward the integral, so that

³

1

1

2

Pn  P d P

(2n)! 2n (n !) 2

³

1

1

P n Pn  P d P

(2n)! 2n 1 (n !) 2 2n (n !) 2 (2n  1)!

2 . 2n  1

(6.28)

This proves the orthogonality conditions given by (6.21). For associated Legendre functions, a similar orthogonality condition is

³

1

1

­ (n  m)! 2 ° (n  m)! 2n  1 ° ® ° 0 °¯

Pnm1 1  P Pnm2 2  P d P

if m1 m2 m and if n1 n2 n if m1 z m2 and/or

.

if n1 z n2

To prove this, first let m1 z m2 and/or n1 z n2 and write the associated Legendre equation with indices m1, n1 and m2, n2 as

§ m12 · m1 ¨ n1 (n1  1)  ¸ Pn  P 1 P 2 ¹ 1 ©

 ª¬1  P 2 Pnm1 1 '  P º¼

'

(6.29)

and ' ' § m22 · m2 P P  ª1  P 2 Pnm1 2  P º , ¨ n2 (n2  1)  2 ¸ n2  ¬ ¼  1 P © ¹

(6.30)

where the prime denotes a derivative with respect to P. Multiplying the first equation by Pnm2 2 , the second by Pnm1 1 , and subtracting the second from the first, we obtain

§ m12  m22 · m1 P P P m2 P ¨ n1 (n1  1)  n2 (n2  1)  2 ¸ n1  n2   1 P © ¹

^

'

(6.31) '

 ª¬(1  P 2 ) Pnm1 1 '  P º¼ Pnm2 2  P  ª¬1  P 2 Pnm2 2 '  P º¼ Pnm1 1  P `

^

'

m '  ª¬(1  P 2 ) Pnm1 1 '  P Pnm2 2  P º¼  ª¬1  P 2 Pn2 2  P Pnm1 1  P º¼ ' ` .

Because the right-hand side of (6.31) vanishes on integrating between the limits -1 and 1, we have § m2  m2 · n1 (n1  1)  n2 (n2  1)  1 2 2 ¸Pnm1 1  P Pnm2 2  P d P ¨ 1 1 P ¹ ©

³

1

0.

(6.32)

As m1 z m2 and/or n1 z n2, the term within brackets does not vanish. Therefore, the 1 integral will vanish only if ³1 Pnm1 1  P Pnm2 2  P d P 0 for m1 z m2 and/or n1 z n2.

86

An Introduction to Global Spectral Modeling

We next consider the case when m1 = m2 = m and n1 = n2 = n. Then using

Pn m  P

 1

m

(n  m)! m Pn  P , (n  m)!

we have

³

1

1

2

Pnm  P d P

³

1

1

Pnm Pnm d  P

 1

m

 1

m

(n  m)! 1  m Pn  P Pnm  P d P (n  m)! 1 1 d n  m ( P 2  1) n d n  m ( P 2  1) n (n  m)! 1 u dP . n 2 2 1 (n  m)! (2 ) (n !) d P nm d P nm

³

³

(6.33)

Let

K mn

 1

m

(n  m)! 1 , (n  m)! (2n ) 2 (n !) 2

and integrating by parts we obtain

³

1

1

2

Pnm  P d  P

ª d n  m ( P 2  1) n d n  m 1 ( P 2  1) n K mn « d P nm d P n  m 1 ¬

1

1

º d n  m 1 ( P 2  1) n d n  m 1 ( P 2  1) n dP» .  1 d P n  m 1 d P n  m 1 ¼

³

1

Then we can write

³

1

1

2

Pnm  P d  P

 K mn

d n  m 1 ( P 2  1) n d n  m 1 ( P 2  1) n dP . 1 d P n  m 1 d P n  m 1

³

1

Continuing the process m times, we obtain

³

1

1

2

Pnm  P d  P

 1

m

K mn

d n ( P 2  1) n d n ( P 2  1) n dP 1 dPn dPn

³

1

2

§ 1 d n ( P 2  1) n · ¨ n ¸ dP 1 2 n ! dPn © ¹ (n  m)! 1 (n  m)! 2 2 Pn  P d P . (n  m)! 1 (n  m)! 2n  1

(n  m)! (n  m)!

³

1

(6.34)

³

In practice, for spectral modeling it is more convenient to use normalized Legendre polynomials and associated Legendre functions. We may normalize a 12

Legendre polynomial Pn  P by multiplying it by ¬ª 2n  1 / 2 ¼º . Denoting a normalized polynomial by P  P , we then write the orthogonality relation as n

³

1

1

Pm ( P ) Pn ( P )

­0 if m z n . ® ¯1 if m n

Likewise, a normalized associated Legendre function Pnm  P is given by

(6.35)

Mathematical Aspects of Spectral Models

87 1/ 2

§ (n  m)! · Pnm ( P ) ¨ ¸ © (n  m)! ¹

1/ 2

§ (2n  1) · ¨ ¸ © 2 ¹

Pnm ( P ) ,

and the corresponding orthogonality relation becomes

³

1

1

­ 0 if m1 z m2 and/or n1 z n2 ® ¯1 if m1 m2 m and n1 n2

Pnm1 1 ( P ) Pnm2 2 d ( P )

Note that with this normalization, Pn m  P

 1

m

n

.

(6.36)

Pnm .

Throughout the remainder of the book, we deal with normalized Legendre polynomials and normalized associated Legendre functions. We will, however, drop the tilde (a) so that from now on Pn ( P ) and Pnm ( P ) represent normalized Legendre polynomial and normalized associated Legendre function, respectively, unless otherwise stated.

6.5

Recurrence Relations

Taking a derivative in the east-west direction of a spherical harmonic given by Ynm Pnm eimO we obtain w m w m imO Yn Pn e wO wO w m Yn imPnm eimO . wO However, taking a derivative in the north-south direction of the same harmonic is more complicated. For this we need recurrence relations. There are a number of relations relating the associated Legendre functions of different orders and degrees as well as their derivatives. These are useful for calculating associated Legendre functions and their derivatives needed for spectral modeling. For our purpose, the following four recurrence relations for the normalized associated Legendre functions are most useful: (1) P Pnm ( P ) mn 1 Pnm1 ( P ) mn Pnm1  P , or

sin T Pnm (sin T ) mn 1 Pnm1 (sin T ) nm Pnm1 (sin T ) ,

(6.37)

where 1/ 2

§ n2  m2 · mn ¨ 2 ¸ © 4n  1 ¹

(2) 1  P 2 dPnm  P / d P cos T

.

n mn 1 Pnm1  P  (n  1) mn Pnm1  P , or

dPnm (sin T ) dT

n mn 1 Pnm1  sin T  (n  1) mn Pnm1  sin T .

Eliminating Pnm1 between (6.37) and (6.38), we obtain the following relations:

(6.38)

An Introduction to Global Spectral Modeling

88 (3) 1  P 2 dPnm  P / d P

cosT 12

 2n  1 nm Pnm1  P  nP Pnm  P , or

dPnm (sin T ) dT

(4) 1  P 2 Pnm  P

(2n  1) nm Pnm1  sin T  n sin T Pnm  sin T .

(6.39)

g nm Pnm11 ( P )  hnm Pnm11 ( P ) , or

cos T Pnm  sin T

g nm Pnm11 (sin T )  hnm Pnm11 (sin T ) ,

(6.40)

where 1/ 2

g nm

§ (n  m  1) (n  m  2) · ¨ ¸ © (2n  1) (2n  3) ¹

hnm

§ (n  m  1) (n  m) · ¨ ¸ . © (2n  1) (2n  1) ¹

and 1/ 2

Using (6.37) to (6.40), we can calculate Pnm  P and their derivatives for any given m and n. Starting with the value of P00  P , the recurrence relations (1) and (4) given by (6.37) and (6.40) can generate the values of associated Legendre functions of any given order m and degree n. The value of normalized P00  P for the global domain (P = -1 to P = 1) is 1/(21/2), and for the hemispheric domain (P = -1 to P = 0 or P = 0 to P = 1) it is 1. With m = 0 and n = 0, recurrence relation (4) becomes 2 12

1  P

g 00 P11  P

P00  P .

(6.41)

From the above equality, if we know P00  P , then we can determine P11  P . If we know P11  P , then the same recurrence relation can be used to obtain P22  P . m m

proceeding recursively, we can find the value of P m m

know P m m

P

P ,

P ,

P

Thus

for any given m. Likewise if we

then recurrence relation (1) determines the value of Pmm1  P . From

and Pmm1  P , the same relation gives value of Pmm 2  P . Thus proceeding

recursively, we can calculate the value of Pnm  P for any given degree n. Recurrence relation (2) can be used to calculate the differential of any Pmm  P from the values of Pnm1  P and Pnm1  P .

Recurrence relation (3) serves a similar

 P and P  P to calculate differentials of Pnm  P . To calculate the derivative of Pnm  P of the highest degree n using recurrence relation (2), we need the value of Pnm1  P , which is beyond the usual truncation of the series. m n 1

purpose, but we need P

m n

Recurrence relation (3) uses the values of associated Legendre functions within the truncation limit only. For calculation of differentials of Pnm  P , it is therefore preferable to use recurrence relation (3) rather relation (2).

89

Mathematical Aspects of Spectral Models

6.6

Gaussian Quadrature

To obtain the coefficients of the truncated spherical harmonics in the spectral model we have to perform a Fourier transform as well as evaluate the Legendre transform. This requires the evaluation of numerical integrals. The Gaussian quadrature is used for the Legendre transform of data in the north-south direction. We show that the Gaussian quadrature is an efficient numerical quadrature for this purpose. Furthermore, we describe the procedure to design it. b Consider the integral ³a f ( x)dx , where f ( x) is an integrable function on a d x d b. We can write this as

³

n

b

f ( x)dx

a

¦ w f (x ) , i

(6.42)

i

i 1

where the expression on the right-hand side is the numerical equivalent of the integral on the left-hand side. The expression on the right-hand side is known as a numerical integral quadrature. Our aim is to select wi and xi such that the summation on the righthand side is exactly equal to the integral on the left-hand side. If we choose xi equally spaced within the interval of integration, then we shall have n values of wi as a function of the location of these points. It is possible to make the quadrature ¦ in 1 wi f ( xi ) exact for f ( x) of degree d n 1 by suitable selection of wi. However, we show that if we can choose both wi and xi suitably, it is possible to make the numerical quadrature ¦ in 1 wi f ( xi ) exactly equal to the integral on the left-hand side for f ( x) of degree d 2n - 1, which is the highest possible accuracy attainable from 2n degrees of freedom (n for xi and n for wi). We call wi the Gaussian weights and xi the Gaussian ordinates, and the numerical quadrature is called the Gaussian quadrature. For convenience, we transform the interval of integration from (a, b) to (-1, 1). This can be done by defining a new variable, that is,

z

2 x  ( a  b) ; ba

x

(b  a) z  a  b , 2

(6.43)

so that

³

b

a

f ( x)dx

ba 2

§ (b  a) z  a  b · f¨ ¸ 1 2 © ¹

³

1

³

1

1

F ( z )dz ,

(6.44)

where F ( x)

b  a § (b  a) z  a  b · f¨ ¸. 2 2 © ¹

For Gaussian quadrature, one needs to find n values of xi and wi such that 1 n ³1 F ( x)dx ¦ i 1 wi F ( xi ) , which is exact for F ( x) of degree d 2n -1. Consider first the case when xi is arbitrarily predefined, and also suppose that they are equidistant with x1 = -1 and xn = 1. In addition, F (x1), F(x2),. . ., F(xi), . . ., F(xn) represent the values of F ( x ) at x1, x2,. . ., xi,. . ., xn, respectively. Then using Lagrange’s

An Introduction to Global Spectral Modeling

90

interpolation formula, we can find a unique polynomial Gn 1 ( x) of n-1 degree passing through these n values. This is given by n

¦ L( x  x ) F ( x ) ,

Gn 1 ( x)

i

(6.45)

i

i 1

where ( x  x1 )( x  x2 )...( x  xi 1 )( x  xi 1 )...( x  xn ) . ( xi  x1 ) ( xi  x2 )...( xi  xi 1 )( xi  xi 1 )...( xi  xn )

L( x  xi )

We can write this as

S n  x  xi , ( x  xi )S n ( xi )

L( x  xi )

(6.46)

where

S n ( x  xi ) ( x  x1 )( x  x2 )! ( x  xn ) and

S n ' ( xi  xi ) ( xi  x1 )( xi  x2 )! ( xi  xi 1 )( xi  xi 1 )! ( xi  xn ) . Because

°­1 ®0 °¯

L( x  xi )

if x x i , if x z x i

Gn 1 ( x) has values F ( xi ) at xi, i = 1, 2, 3,. . . , n. Thus Gn 1 ( x) is the desired polynomial passing through the given n values. Integrating (6.45) over the limit (-1, 1), we obtain

³

1

1

1

n

³ ¦ L( x  x ) F ( x )dx .

Gn 1 ( x)dx

i

1 i 1

(6.47)

i

Interchanging the order of summation and integration operations on the right-hand side, we obtain

³

1

1

1

Gn 1 ( x)dx

n

n

³ ¦ L( x  x ) F ( x )dx ¦ w F ( x ) , i

1 i 1

i

i

i

(6.48)

i 1

where wi

³

1

1

L( x  xi )dx .

(6.49)

Thus for such values of wi, the summation ¦ in 1 wi F ( xi ) is exactly equal to the integral 1

³1 Gn 1 ( x)dx , where Gn 1 ( x) is a curve of degree d n -1 passing through the n ordinates F ( x1 ), F ( x2 ),! , F ( xn ) . Suppose next that we have F ( x) as an arbitrary polynomial of degree 2n-1 having values at each xi, i = 1, 2, . . . , n. By Lagrange’s interpolation formula, we can again find a polynomial Gn 1 ( x) passing through F ( x) at the given points, so that

Gn 1 ( xi )

F ( xi ) .

We write this arbitrary polynomial F ( x) of degree 2n - 1 in the form

(6.50)

Mathematical Aspects of Spectral Models

91 n

– (x  x ) ,

F ( x) Gn 1 ( x)  In 1 ( x)

(6.51)

i

i 1

where Gn 1 ( x) is a polynomial of degree n -1 passing through the n ordinates F ( xi ) ,

In 1 ( x) is an arbitrary polynomial of degree n - 1, and – in 1 ( x  xi ) is a polynomial of degree n. Then the polynomial on the right-hand side of (6.51) is equal to F ( x) and is of degree 2n - 1. If we integrate (6.51) from (-1, 1), we obtain

³

1

1

F ( x)dx

³

1

1

Gn 1 ( x)dx 

³

n

1

1

In 1 ( x)– ( x  xi )dx , i 1

or

³

1

1

n

¦ w F (x )  ³

F ( x)dx

i

i

1

1

i 1

n

In 1 ( x)– ( x  xi )dx .

(6.52)

i 1

1 If ¦ in 1 wi F ( xi ) is to be exactly equal to ³1 F ( x)dx for F ( x) of degree 2n - 1, then the second term on the right-hand side of (6.52) must vanish. This can be achieved by suitable positioning of points x1, x2, x3, . . . , xn so that

³

1

1

n

In 1 ( x)– ( x  xi )dx 0 .

(6.53)

i 1

Because In 1 ( x) is an arbitrary polynomial, the integral on the left-hand side of (6.53) must vanish for every polynomial In 1 ( x)– in 1 ( x  xi ) .

This is possible if

n i 1

In 1 ( x)– ( x  xi ) 0 . Since In 1 ( x) is an arbitrary polynomial (which may not necessarily equal zero), we have n

– (x  x )

0.

i

(6.54)

i 1

Thus xi are the roots of – in 1 ( x  xi ) 0 . We may represent In 1 ( x) and – in 1 ( x  xi ) in terms of Legendre polynomials as n 1

In 1 ( x)

n

n

¦ a P ( x) and – ( x  x ) ¦ b P ( x) . k

k 0

i

k

i 1

j

j

(6.55)

j 0

Note that because In 1 ( x) and – in 1 ( x  xi ) are polynomials of degree n-1 and n, respectively, the coefficients of Pn 1 in the summation for In 1 and for Pn in the summation for – in 1 ( x  xi ) must not vanish; in other words, an 1 z 0 and bn z 0 . Substituting (6.55) into (6.53) and using the orthogonality property of Legendre polynomials, we obtain

An Introduction to Global Spectral Modeling

92

³

1

1

1 n 1

n

n

³ ¦ a P ( x)¦ b P ( x)dx

In 1 ( x)– ( x  xi )dx

1

i 1

k

k

j

k 0

j

j 0

1 n 1

³ ¦ a b P ( x)dx . 1

k k

2 k

(6.56)

k 0

Since In 1 ( x) is an arbitrary polynomial, ak for k = 0, 1 , 2, 3, . . . . , n - 1 in general do not vanish. Therefore, the above integral will vanish only if bk 0 for k = 0, 1, 2, 3, . . . . , n - 1. Thus the integral (6.53) vanishes for all values of In 1 ( x) only if n

– (x  x )

bn Pn ( x) .

i

(6.57)

i 1

From (6.54) and (6.57) we note that xi are the roots of Pn ( x) (i.e., of the Legendre polynomial of degree n). Also, d dx

S n ' ( xi )

n

d bn Pn ( x) dx x

– (x  x ) i

i 1

x xi

bn P 'n ( xi ) .

(6.58)

xi

The weights wi in (6.49) may now be expressed as wi

³

1

1

L( x  xi )dx

S n ( x  xi ) ³1 ( x  xi )S n ' ( xi ) dx 1

³

1

1

bn Pn ( x) dx , ( x  xi )bn P 'n ( xi )

or wi

1 P 'n ( xi )

³

1

1

Pn ( x) dx . x  xi

(6.59)

1

To evaluate ³1 Pn ( x) /( x  xi )dx , we make use of recurrence relation (1) given by (6.37), that is, H l01 Pl 1 ( x) xPl ( x)  H l0 Pl 1 ( x) . (6.60) For x

xi , (6.60) may be written as

H l01 Pl 1 ( xi ) xi Pl ( xi )  H l0 Pl 1 ( xi ) .

(6.61)

If we multiply (6.60) by Pl ( xi ) and (6.61) by Pl ( x) and then subtract the resulting equations, we obtain

H l01 > Pl 1 ( x) Pl ( xi )  Pl 1 ( xi ) Pl ( x)@ Pl ( x) Pl ( xi )( x  xi )  H l0 > Pl 1 ( x) Pl ( xi )  Pl 1 ( xi ) Pl ( x)@ . (6.62)

After summation of the two sides of (6.62) over the range l = 1 to l = n, we obtain

Mathematical Aspects of Spectral Models

93

n

¦H >P 0 l 1

l 1

( x) Pl ( xi )  Pl 1 ( xi ) Pl ( x) @

l 0

n

¦ ^P ( x) P ( x )( x  x )  H > P l

l

i

0 l

i

l 1

( x) Pl ( xi )  Pl 1 ( xi ) Pl ( x) @` .

l 0

After some simplification we obtain n

¦ P ( x) P ( x )( x  x ) l

l

i

i

H n01 > Pn 1 ( x) Pn ( xi )  Pn 1 ( xi ) Pn ( x) @

l 0

H10 > P0 ( x) P1 ( xi )  P0 ( xi ) Pn1 ( x) @ .

(6.63)

Noting that for the spherical domain, 1 1 , , P0 ( xi ) 21/ 2 21/ 2 1/ 2 1/ 2 § 3· § 3· P1 ( x) ¨ ¸ x , P1 ( xi ) ¨ ¸ xi , H10 © 2¹ © 2¹

P0 ( x)

1 , 31/ 2

and Pn ( xi ) 0 , since xi are the roots of the Legendre polynomial Pn ( x) . We finally obtain n x  xi Pl ( x)Pl ( xi )( x  xi ) H n01 Pn 1 ( xi ) Pn ( x)  (6.64) 2 l 1

¦

H n01 Pn 1 ( xi ) Pn ( x)  P0 ( xi ) P0 ( x)( x  xi ) . Transferring the second term from the right-hand side to the left-hand side, we obtain n

¦ P ( x)P ( x )( x  x ) l

l

i

i

H n01 Pn 1 ( xi ) Pn ( x) ,

l 0

or



n

0n 1 Pn 1 ( xi ) Pn ( x) x  xi

¦ P ( x)P ( x ) . l

l

i

(6.65)

l 0

Integrating (6.65) from -1 to 1, we obtain  0n 1 Pn 1 ( xi )

n

Pn ( x) dx 1 x  x i

³ ¦ P ( x) P ( x )dx ,

Pn ( x) dx 1 x  x i

¦ P ( x )³

³

1

1

1

l

l

i

l 0

or  0n 1 Pn 1 ( xi ) 1

Because ³1 Pl ( x) dx

³

1

0 for all l except l

 0n 1 Pn 1 ( xi )

n

l

l 0

1

1

Pl ( x)dx .

(6.66)

0 , (6.66) reduces to

Pn ( x) dx 1 x  x i

³

i

1

P0 ( xi )

³

1

1

P0 ( x)dx 1 .

(6.67)

An Introduction to Global Spectral Modeling

94 Thus



³

1

1

Pn ( x) 1 dx . 0n 1 Pn 1 ( xi ) x  xi

(6.68)

Substituting into (6.59), we obtain wi

0 n 1



1 . Pn 1 ( xi ) Pn' ( xi )

(6.69)

These are the corresponding weights assigned to the various xi. The expression for wi given by (6.69), though proper, is not very convenient, because it needs the values of Pn 1 ( xi ) and the derivatives P '( xi ) . We express wi in a more convenient form below. Using recurrence relation (2) given by (6.39) at x = xi, we obtain

1  x P ( x ) 2 i

' n

(2n  1) 0n Pn 1 ( xi )  0 .

i

Pn ( xi ) 0 since xi are the roots of Pn ( x) , or

Pn' ( xi )

(2n  1)0n Pn 1 ( xi ) . 1  xi2

(6.70)

Also, using recurrence relation (1) given by (6.37) at x = xi and noting that Pn ( xi ) 0 , we obtain 0n 1 Pn 1 ( xi )

 0n Pn 1 ( xi ) ,

or

Pn 1 ( xi )



0n Pn 1 ( xi ) . 0n 1

(6.71)

Substituting (6.70) and (6.71) into (6.69), we obtain 1  xi2 , (2n  1) (0n ) 2 Pn 1 ( xi ) 2

wi or wi

2(1  xi2 ) (n  12 ) , n 2 Pn 1 ( xi ) 2

(6.72)

which is a more convenient expression for the Gaussian weights wi. Thus if a polynomial F ( x) of degree 2n  1 has values F ( xi ) available at n points xi, i 1, 2, 3, ! , n , as the roots (zeroes) of the Legendre polynomial Pn, then the 1

integral ³1 F ( x)dx is evaluated exactly by the Gaussian quadrature ¦ in 1 wi F ( xi ) where xi are the zeroes of the Legendre polynomial of degree n and wi are the weights given by (6.72). The efficiency of the Gaussian quadrature lies in the fact that it can evaluate the integral of a function of degree 2n  1 from its values given at n points. For designing a Gaussian quadrature, we need to take the following steps:

Mathematical Aspects of Spectral Models

95

1.

If N is the highest degree of the polynomial to be integrated, then the minimum number of points n needed for the Gaussian quadrature is given by 2n  1 N , or n ( N  1) / 2 .

2.

The Gaussian points xi, i 1, 2, ! , n , are determined as the zeroes of the Legendre polynomials of degree n, that is, as the roots of Pn ( x) 0 . The roots of Pn ( x) are not equally spaced, but are nearly so. They can be determined iteratively using the Newton-Raphson method starting with the first guess of xi as n equally spaced points between -1 and 1.

3.

Once xi are found, the weights wi can be obtained from (6.72).

In the case of a Legendre polynomial of even degree, its zeroes are located symmetrically, in other words, at r xi . In the case of a Legendre polynomial of odd degree, x 0 is one of the zeroes and the rest of the zeroes would be located symmetrically at r xi . The weight wi has the same value for r xi . It is therefore sufficient to calculate the zeroes of the Legendre polynomials and the corresponding weights for x t 0 to complete the Gaussian quadrature. In a spectral model, the use of a suitable fast Fourier transform (FFT) for the Fourier transform and Gaussian quadrature for the Legendre transform achieves the optimally fast and accurate calculations.

6.7

Spectral Representation of Physical Fields

Any smooth function over a sphere can be expressed as a sum of spherical harmonics. However, the convergence of term-by-term derivatives of the function is assured for much less liberal conditions, that is, the absolute convergence of the series. With most meteorological quantities (scalars), this is generally not a problem. However, sharp discontinuities across clouds, rain areas, fronts, and so on, are not easy to represent. There is also a problem with the horizontal wind components u and v, which have a singular behavior at the poles. While u and v can be expressed to any desired accuracy by a series of spherical harmonics, there is no guarantee that the derivative of such a series will converge properly (this is the pole problem). The cross-polar flow cannot be expressed by a series. At one side, the flow may appear to be northerly, while at the other side of the pole the flow might appear southerly. This poses a singularity at the pole. In the past u and v were set equal to zero at the poles for all times. However, due to this singularity at the poles, the velocity components are carried in terms of the following pseudoscalar fields (called Robert functions): U

u cos T a

and

V

v cos T , a

(6.73)

where T is the latitude and a is the radius of the earth. At the poles, T r S 2 , so U V 0 , but we can find U and V very close to 90°N and 90°S. The Robert functions U and V are therefore well defined at the poles. It is these functions rather than u and v that we work with.

An Introduction to Global Spectral Modeling

96

One can obtain a simple kinematic relationship between the flow-field variables in the spectral domain using the formula discussed in earlier sections. We deal with variables such as \ , F , ] , D, U and V, where \ is the stream function, F is the velocity potential, ] is the relative vorticity, D is the divergence, and U and V are the Robert functions defined in (6.73). The relationships between U, V and \ , F (or ] , D) are given by 1 § wF w\ ·  cos T ¨ ¸, a 2 © wO wT ¹ 1 § w\ wF ·  cos T V ¸, 2 ¨ a © wO wT ¹ 1 § wV wU ·  cos T ] ¨ ¸, cos 2 T © wO wT ¹ 1 § wU wV · D  cos T ¨ ¸, cos 2 T © wO wT ¹ U

(6.74) (6.75)

(6.76)

(6.77)

where T is the latitude, O is the longitude, and a is the radius of the earth. If \ and F are both well-behaved, which generally is the case, then for a given truncation we can expand \ and F by the relations

a2

\

¦¦\

m m n n

O, P

(6.78)

¦¦ F

m m n n

O, P .

(6.79)

m

Y

n

and a2

F

m

Y

n

For the same truncation, remembering the mathematical property ’ 2Ynm



n(n  1) m Yn a2

given in Sec. 6.3, we can express the vorticity and divergence as

]

’ 2\

¦¦ n(n  1)\ m

m m n n

Y

(6.80)

m m n n

(6.81)

n

and D ’2 F

¦¦ n(n  1) F m

Using the spectral expansions ]

Y .

n

m

m

¦ ¦ ] n Yn

and D

m n

m m ¦ ¦ Dn Yn we get the spectral m n

form of ] and D for one harmonic so that

] nm  n(n  1)\ nm

(6.82)

Dnm

(6.83)

and  n(n  1) F nm .

Mathematical Aspects of Spectral Models

97

By substituting the spectral representations of \ (6.78) and F (6.79) into (6.74) and (6.75) and making use of the zonal meridional differentials of Ynm  O , P given by Pnm  P eimO ,

Ynm (O , P ) wYnm (O , P ) wO wYnm (O , P ) wO

imPnm  P eimO ,

imYnm  O , P

(6.84)

and using the recurrence relation (6.38) where eimO has been multiplied through m n

1  P wY w(PO , P ) 2

 n mn 1 Ynm1  O , P

(n  1) mn Ynm1  O , P ,

(6.85)

we obtain the spectral expansions of both U and V. That is,

U

¦¦U m

Remembering Ynm

1 § 2 ¨a a2 ©

m m n n

Y

n

Pnm eimO ,

w wT

m

Using the identity cos 2 T

m

m m n n

Y  cos T

m

w

¦¦ wT \ m

n

· Y ¸. ¹

m m n n

sin T and using (6.84), we obtain

w

¦¦ cosT wP\ m

m m n n

Y .

n

1  P 2 , we obtain

¦¦ imF

U

Y  cos T a 2

w , and P wP

n

1  sin 2 T

m m n n

n

cos T

¦¦ imF

U

w

¦¦ wO F

Y  1  P 2

m m n n

n

¦¦\ m

n

m n

wYnm . wP

Finally, using (6.85) we obtain U

¦¦U Y ¦¦ imF ¦¦\ ª¬ n  Y m m n n

m

n

m

m n

m

m m n n

Y

n

m n 1

m n 1

n

 (n  1) mn Ynm1 º¼ .

(6.86)

Similarly for V we obtain V

¦¦V Y ¦¦ im\  ¦¦ F ª¬  n  Y m m n n

m

n

m

m n

m

n

m m n n

Y

n

m n 1

m n 1

 (n  1) mn Ynm1 º¼ .

Equating the coefficients of Ynm on both sides of the above equations, we obtain

(6.87)

An Introduction to Global Spectral Modeling

98

U nm

imF nm  (n  1) nm \ nm1  (n  2) mn 1 \ nm1 ,

(6.88)

m n

im\ nm  (n  1) mn F nm1  (n  2) mn 1 F nm1 .

(6.89)

V

Since ] nm

 n(n  1)\ nm and Dnm  (n  1) F nm , we may also write n(n  1)U nm m n

n(n  1)V

imDnm  (n  1) nm ] nm1  n mn 1 ] nm1 , m n

m n

m n 1

im]  (n  1)  D m n

For m = 0 and n = 0 and remembering Y U 00

O, P

m n 1

 n

m n 1

D

(6.90)

.

(6.91)

0 for n  m ,

2 10 \ 10 10 ] 10 ,

0 0

0 1

2  F

V

0 1

0 1

(6.92)

0 1

 D .

(6.93)

At this point, let us ask the following questions: How does kinematics work, given \ nm and F nm ? If we are given the spectral coefficients of the stream function and the velocity potential (i.e., given \ nm and F nm ), can we obtain ] nm , Dnm , U nm , and Vnm ? Since we know \ nm and F nm for all desired values of n and m, we can use (6.88) and (6.89) to obtain U nm and Vnm , respectively. Using (6.82) and (6.83), one can obtain ] nm and Dnm , respectively. Furthermore, one can obtain the true velocity components u and v using the inverse Robert functions. That is, equations (6.82), (6.83), and (6.88)-(6.91) are a series of equations that enable one to solve for \, F, ], D, U and V given two spectral coefficients. This can be called the triangle problem. Without worrying about details, the orthogonality properties of Legendre polynomials and associated Legendre functions are used for determining the expansion coefficients. This leads to 2S

³ ³

1

1

0

AYnm*d P d O

2S

¦¦ A ³ ³ m n

m

n

0

1

1

YnmYnm* d P d O ,

(6.94)

where Ynm* is the complex conjugate of Ynm . We have used the orthogonality of the spherical harmonics, that is, 1 2S

2S

³ ³ 0

1

1

YnmY jk *d P d O G mk G nj

­0 ® ¯1

if m z j and/or n z j if m

k and n

j

.

Transformation from grid space to spectral space and vice versa is important in spectral modeling. Grid space is represented by A(O , P ) and is the representation on a weather map. Spectral space is represented by Anm and are the amplitudes of the spherical harmonics of A(O , P ) . It is important to learn to switch between grid ( A(O , P ) ) and spectral space ( Anm ), i.e. to get weather maps from the spectral coefficients. To do this we must utilize the Fourier transform, Legendre transform, inverse Fourier transform, and inverse Legendre transform. We can write a function A as

Mathematical Aspects of Spectral Models

99

¦¦ A Y

m m n n

A

m

where A

,

n

A(O , P ) and

1 2S

Anm

2S

³ ³

1

1

0

AYnm*d P d O .

(6.95)

It is necessary to truncate the series at some wavenumber. There are two common ways of truncating the series: (1) Rhomboidal truncation, which has the form m N

N

A O, P

¦ ¦AY

m m n n

O, P ,

(6.96)

O, P .

(6.97)

m N n m

and (2) triangular truncation, which has the form N

AO, P

N

¦ ¦A Y

m m n n

m N n m

We discuss these truncations in detail in Chapter 7. If the set of spectral coefficients Anm is known, then by using either (6.96) or (6.97) the function A(O , P ) can be defined everywhere over the globe. In practice, it is necessary to evaluate the function at a finite set of grid points. Likewise, the evaluation of the integrals in the Fourier-Legendre transform is based on data at only a finite number of points. 6.7.1 Grid to Spectral Space In practice, the evaluation of Anm from A(O , P ) is carried out in two steps. Step 1: Perform the Fourier transform of the space field along latitudes, Am  P

1 2S

³

2S

0

A  O , P e  imO d O .

(6.98)

Step 2: Perform the Legendre transform of the Fourier components, Anm

³

1

1

Am  P Pnm  P d P .

(6.99)

The Fourier transform is evaluate using the trapezoidal quadrature formula, that is, Am  P k

where O j

[(2S ) /(2M )] j

1 2M

2 M 1

¦ AO , P e j

k

 imO j

,

(6.100)

j 0

(S / M ) j . This integration is exact for any function which

may be represented by a truncated Fourier series with wavenumbers d 2 M  1 . This calculation is very efficiently done by the FFT. The Legendre transform is evaluated using the Gaussian quadrature formula, that is,

An Introduction to Global Spectral Modeling

100 K

Anm

¦W  P

k

Am  P k Pnm  P k ,

(6.101)

k 1

where P k are the Gaussian latitudes and W ( P k ) are the Gaussian weights, as discussed

in Section 6.6. This formula is exact for any polynomial [here Am ( P ) Pnm ( P ) ] of degree d 2K 1 . 6.7.2 Spectral to Grid Space The transformation from spectral to grid space is also achieved similarly in two steps. Step 1: Perform the reverse Legendre transform, Am  P k

¦I A

m m n n

P

 Pk .

(6.102)

n

Step 2: Perform the reverse Fourier transform, A  O j , Pk

Anm  P k e

imO j

.

(6.103)

Thus one proceeds from spherical harmonic components Anm to Fourier components Am and then to grid-point values A(O j , P k ) . It should be noted that the use of Gaussian latitudes and weights enables one to calculate the Legendre transform exactly.

6.8

Barotropic Spectral Model on a Sphere

In Chapter 2 we discussed the finite-difference barotropic model where we were concerned with the proper formulation of the space-differencing schemes, the Jacobian, the Laplacian, and the solution of Poisson’s equation. In this section, we consider a spectral model on a sphere, the integration of which will be through a coupled system of nonlinear ordinary differential equations. This system of equations is obtained by transforming the model equations from the space to the spectral domain. The basis functions for this transformation will be surface spherical harmonics, some important properties of which have already been discussed. If O represents the longitude and T the latitude, then the barotropic vorticity equation on a sphere is given by

w] wt

where P

· 1 § w\ w w\ w (]  f )  (]  f ) ¸ , ¨ a 2 © wP wO wO wP ¹

sin T . Noting that wf / wO

w] wt

0 and wf / wP

(6.104)

2: , we obtain

1 § w\ w] w\ w] · 2: w\  , ¨ ¸ a 2 © wP wO wO wP ¹ a 2 wO

or

w] wt

F O, P 

2: w\ . a 2 wO

(6.105)

Mathematical Aspects of Spectral Models

101

In the above equation, the nonlinear advective term

1 § w\ w] w\ w] ·  ¨ ¸ a 2 © wP wO wO wP ¹ is written as F (O , P ) . The second term on the right-hand side of (6.105) is the earth’s rotation term and it is linear. To transform (6.105) into its spectral form, we take the Fourier-Legendre expansion of variables ] , \ , and F

] O, P , t \ O, P , t F O, P , t

¦¦] ¦¦\ ¦¦ F m

n

m

n

m

n

m n

 t Ynm  O , P ,

(6.106)

m n

 t Ynm  O , P ,

(6.107)

m n

 t Ynm  O , P .

(6.108)

Substituting into (6.105) we obtain

¦¦ m

n

d] nm (t ) m Yn  O , P dt 

Using ]

¦¦  F

m n

m

(t )Ynm (O , P )

n

2: m w · \ n (t ) Ynm (O , P ) ¸ . wO a2 ¹

(6.109)

’ 2\ and looking at the left-hand side of equation (6.105) only, we get d] nm (t ) m Yn  O , P dt

remembering ’ 2Ynm



n(n  1) d\ nm (t ) m Yn  O , P , a2 dt

(6.110)

n(n  1) m Yn . Also, a2 w m Yn  O , P imYnm  O , P . wO

(6.111)

Substituting (6.110) and (6.111) into (6.109) and then equating the coefficients of Ynm  O , P on both sides of the above equation, we obtain n  n  1 d\ nm  t a

2

dt

Ynm  O , P

d\ nm dt

Fnm  t Ynm  O , P 

2: m \ n  t imYnm  O , P a2

a2 2:im m Fnm \n  n(n  1) n(n  1)

(6.112)

as the spectral form of the barotropic vorticity equation (6.105). The first term on the right-hand side of (6.112) is the beta term (the acceleration due to the advection of the earth’s vorticity), while the second term is the advection of relative vorticity.

An Introduction to Global Spectral Modeling

102

The spectral form of the barotropic vorticity equation consists of a set of ordinary nonlinear differential equations in spectral space. It should be noted that Fnm is a nonlinear term. We use the transform method to evaluate such nonlinear terms. The transform method involves calculating the terms w\ / wO , w\ / wP , w] / wO , and w] / wP on the grid points by projecting the spectral coefficients onto the space domain. These are then multiplied to get values of the nonlinear terms F (O , P ) on the grid points. The spectral analysis of F (O , P ) then involves the Fourier analysis of the F (O , P ) along latitude circles, followed by a Legendre transform of the resulting Fourier coefficients to obtain the spectrum Fnm . This procedure is called the FourierLegendre transform. We can write

F (O , P )

1 § w\ w] w\ w] ·  ¨ ¸ a 2 © wP wO wO wP ¹ § 1 w\ w] · 2 w\ w] (1  P 2 )  ¨ (1  P ) ¸. a 2 (1  P 2 ) © wP wO wO wP ¹

(6.113)

Let

\ O, P

¦¦\ m

m n

(t )Ynm  O , P ,

(6.114)

n

so that upon taking the derivative of (6.114) with respect to P , multiplying by (1  P 2 ) , and remembering that Ynm (O , P ) Pnm ( P )eimO we obtain the north-south derivative of \ (1  P 2 )

w\ wP

¦¦\ m

m n

(t )eimO (1  P 2 )

n

w m Pn ( P ) , wP

(6.115)

where by the recurrence relation (6.38) we have (1  P 2 )

d m Pn ( P ) dP

n mn 1 Pnm1 ( P )  (n  1) nm Pnm1 ( P )

and 1/ 2

§ n2  m2 · mn ¨ 2 ¸ . © 4n  1 ¹

Hence using the recurrence relation in equation (6.115) we obtain (1  P 2 )

w\ wP

¦¦\ > n  m n

m

m n 1

Pnm1 ( P )

n

  n  1 mn Pnm1 ( P ) º¼ eimO

(6.116)

where the term in brackets is always known. If we differentiate (6.114) with respect to O , we obtain the east-west derivative of \

Mathematical Aspects of Spectral Models w\ wO

¦¦ im\ m

103 m n

(t ) Pnm ( P )eimO .

(6.117)

n

Using (6.116) and (6.117) we can obtain the grid-point values of w\ / wO and (1  P 2 )w\ / wP . Similarly, the grid-point values of (1  P 2 )w] / wP and w] / wO are

calculated from the spectral coefficients of vorticity ] nm .

Having obtained

(1  P 2 )w\ / wP , w\ / wO , (1  P 2 )w] / wP , and w] / wO in grid space, we obtain F (O , P ) from (6.113) in grid space. The Fourier-Legendre transform is then applied to F (O , P ) to obtain Fnm .

Knowing the spectral coefficients \ nm and Fnm , one can calculate the term on the right-hand side of the spectral form of vorticity equation (6.112). This gives us d\ nm / dt , the time tendency of \ nm , which, along with the value of \ nm at the previous time step, can be used to obtain \ nm at a future time step. In practice, the following steps are needed to integrate the barotropic vorticity equation: Step 1: From the coefficients \ nm and ] nm , obtain the grid-point values of (1  P 2 )w\ / wP and (1  P 2 )w] / wP along a latitude circle using (6.116). Step 2: Similarly, using (6.117), obtain grid-point values of w\ / wO and w] / wO along the latitude circle. Step 3: Multiply (1  P 2 )w\ / wP , w] / wO , w\ / wO and (1  P 2 )w] / wP to calculate

F (O , P )

1 § w\ w] w\ w] ·  ¨ ¸ a 2 © wP wO wO wP ¹ § 1 w\ w] · 2 w\ w] (1  P 2 )  ¨ (1  P ) ¸ wP wO wO wP ¹ a 2 (1  P 2 ) ©

on the grid points (O , P ) . Step 4: Perform the Fourier transform of F (O , P ) along the latitude circle to obtain the Fourier components F m ( P ) . This is done by using the FFT. Step 5: Perform the Legendre transform of F m ( P ) at the various latitude to obtain Fnm , the spherical harmonic amplitudes of F (O , P ) . This is done by using Gaussian quadrature. Step 6: From\ nm and Fnm , calculate the right-hand side of the spectral vorticity equation

and obtain d\ nm / dt . Step 7: From\ nm (t  't ) and d\ nm / dt , obtain \ nm (t  't ) at the next time step as

\ nm (t  't ) \ nm (t  't )  2't

d\ nm . dt

An Introduction to Global Spectral Modeling

104

6.9

Shallow-Water Spectral Model

Earlier we discussed the integration of the shallow-water equations using finite differences in both time and space. We now structure the shallow-water system to a similar time-differencing scheme, but the space differentials will be calculated spectrally. In this system, we use the momentum and the mass continuity equations. In a shallow-water model, we consider a homogeneous incompressible fluid with a rigid lower boundary and an upper free surface. The horizontal velocity is taken to be invariant with height. The continuity equation for an incompressible fluid is

wu wv ww   wx wy wz

0.

(6.118)

Integrating (6.118) vertically with a lower boundary condition w0 at z0 § wu

ww

0 , we obtain

wv ·

³ wz dz ³ ¨© wx  wy ¸¹dz , § wu wv · z ¨  ¸ , © wx wy ¹ § wu wv · z ¨  ¸ , © wx wy ¹

wz  w0

wz

(6.119)

where z is the height of the free surface and wz is the vertical velocity of the free surface. Also, dz wz G wz  VH ˜’z . (6.120) dt wt From (6.119) and (6.120) we obtain § wu wv · wz G  VH ˜’z  z ¨  ¸ , wt © wx wy ¹ G G wz VH ˜’z  z’ ˜ VH , wt

which on multiplication by g gives wI wt

G G VH ˜’I  I’ ˜ VH ,

(6.121)

where I is the geopotential at the free surface. Writing I I  I ' as the sum of the mean and its deviation from the mean, where I is the time-variant area mean, we obtain wI ' wt

G G VH ˜’(I  I ')  (I  I ')’ ˜ VH

G G G VH ˜’I '  I ' ’ ˜ VH  I’ ˜ VH ,

Mathematical Aspects of Spectral Models or

105

G G ’ ˜ I 'VH  I ˜’VH .

wI ' wt





Equation (6.122) now represents the continuity equation. The horizontal momentum equation for the shallow-water model is G G G G G wVH  VH ˜’ VH  f k u VH  ’I . wt





(6.122)

(6.123)

Using the vector identity G G VH ˜’ VH





G G G G § VH ˜ VH · ’¨ ¸  ] k u VH , 2 © ¹

the horizontal momentum equation may be written as G G G G G § VH ˜ VH · wVH (]  f )k u VH  ’ ¨ I  ¸, wt 2 ¹ © or G G G G G § VH ˜ VH · wVH (]  f )k u VH  ’ ¨ I ' ¸, wt 2 ¹ ©

(6.124)

because ’I 0 . We express the momentum equation (6.124) in terms of the vorticity and divergence equations, which are more suitable for a spectral model. The vorticity equation is obtained by operating on (6.124) with the vector G operator k ˜’ u . Thus G G G G w k ˜’ u VH G G G G § VH ˜ VH · ª º  k ˜’ u ¬(]  f )k u VH ¼  k ˜’ u ’ ¨ I ' ¸. 2 ¹ wt ©





After simplifying, we obtain w] wt

G ’ ˜ (]  f )VH

(6.125)

as the desired vorticity equation. Similarly, the divergence equation is obtained by applying the vector operator ’ ˜ to (6.124). Thus G G G w ’ ˜ VH G G § VH ˜ VH · ª º ’ ˜ ¬(]  f )k u VH ¼  ’ ˜’ ¨ I ' ¸. wt 2 ¹ ©





After further simplification we obtain the divergence equation as G G VH ˜ VH · wD wv(]  f ) wu cos T (]  f ) 2§   ’ ¨I ' + ¸. wt a cosT wO a cos T wT 2 ¹ ©

(6.126)

An Introduction to Global Spectral Modeling

106

The vorticity, divergence, and continuity equations given by (6.125), (6.126), and (6.122) form the shallow-water model on a sphere. To remove the discontinuity in the wind field at the poles, we replace u and v by the Robert functions defined by (6.73). Using these functions, the vorticity, divergence, and continuity equations take the following form in spherical coordinates: w] wt wD wt

1 § wU (]  f ) wV (]  f ) ·  cos T ¨ ¸, cos 2 T © wO wT ¹ 1 § wV (]  f ) wU (]  f ) ·  cos T ¨ ¸ cos 2 T © wO wT ¹



§ U 2 V 2 · ’ 2 ¨  I '¸ , 2 T 2 cos © ¹ wI ' 1 § w (U I ') w (V I ') ·   cos T ¨ ¸ ID . wt cos 2 T © wO wT ¹

(6.127)

(6.128)

(6.129)

These can be written in terms of \ , F , and I ' as w’ 2\ wt

w’ 2 F wt

wI ' wt

1 § wU (’ 2\  f ) wV (’ 2\  f ) ·  cos T ¨ ¸, 2 wO wT cos T © ¹ wU (’ 2\  f ) · 1 § wV (’ 2\  f )  cos T ¨ ¸ cos 2 T © wO wT ¹



§ U 2 V 2 · ’ 2 ¨  I '¸ , 2 © 2 cos T ¹ 1 § w (U I ') w (V I ') ·   cos T ¨ ¸ ID . cos 2 T © wO wT ¹

(6.130)

(6.131)

(6.132)

It can be shown that the above set of vorticity, divergence, and continuity equations ensures the conservation of vorticity, kinetic energy, and potential energy during the course of computations. Because the first term on the right-hand side in these equations is of the same form, the equations can be calculated by an identical computation algorithm. Our aim is to solve (6.130), (6.131), and (6.132) spectrally using a fully explicit method. This is achieved by expanding \, F , I ' , U, and V spectrally, that is,

\

a2

¦¦\ m

m m n n

Y ,

(6.133)

n

and similarly for F, I‘, U, and V. In (6.130) to (6.132), \, F , and I ' are the prognostic variables. In addition to these, we need U and V to calculate the right-hand side of these equations. To obtain U mn and V mn , the spectral components of U and V, from the spectral components of \ and F, we make use of (6.88) and (6.89). Substituting the spectral expansions of \, F, I ' , U, and V into (6.130) to (6.132), the spectral form of the shallow-water equations takes the following form:

Mathematical Aspects of Spectral Models

n  n  1 n  n  1

wF nm wt

w\ nm wt

107

D nm ( A, B) ,

(6.134) m

§ U 2 V 2 ·  I '¸ , 2 © 2 cos T ¹n

D nm ( B,  A)  ’ 2 ¨

wI 'mn wt

 D nm (U I ,V I )  I Dnm ,

(6.135)

(6.136)

G G G G where A U (’ 2\  f ) , B V (’ 2\  f ) , D ’ ˜ V , V Ui  Vj , and the operator D is defined as

D  A, B

1 cos 2 T

wB · § wA  cos T . ¨ w w O T ¸¹ ©

The spectral amplitudes of the nonlinear terms on the right-hand side of (6.134) to (6.136) are calculated by first projecting the values of \, F, I ' , U, and V from the spectral domain onto the space domain on a Gaussian grid, and then multiplying them in the space domain to obtain the grid-point values of the nonlinear terms. The FourierLegendre transform is then used to convert these grid-point values to the spectral amplitudes of the nonlinear terms. The sum of the nonlinear terms and linear terms forms the tendencies of the various spectral amplitudes in the shallow-water model. These tendencies, along with the values of \ nm , F nm , and I 'mn at the previous time step, are used to obtain the future values of these functions using the following finitedifference analog:

§ w\ nm · ¸ 2 't , © wt ¹

(6.137)

§ wF nm · ¸ 2 't , © wt ¹

(6.138)

§ wI 'mn · ¸ 2 't . © wt ¹

(6.139)

\ nm  t  't \ nm  t  't  ¨ F nm  t  't F nm  t  't  ¨

I 'mn  t  't I 'mn  t  't  ¨

To march forward in time, we make use of centered time differencing except for the first time step, where forward time differencing is applied. Marching forward in time over the forecast period, we obtain the final forecast values of \ nm , F nm , and I 'mn . These amplitudes are then projected onto the space domain to obtain the forecast fields of stream function (or vorticity), velocity potential (or divergence), and the geopotential (or height) at the free surface. This shallow water spectral model has gravity waves and Rossby waves. Gravity waves require small computational time steps while Rossby waves can take larger computational time steps. If we are using centered time differencing we need to use small time steps for this model which is computationally expensive. For this reason, the fully explicit method is rarely used and the semi-implicit shallow water model is employed instead.

An Introduction to Global Spectral Modeling

108

6.10

Semi-implicit Shallow-Water Spectral Model

We next address the formulation of a shallow-water model based on the semi-implicit time-integration scheme. In this formulation, the linear terms are integrated implicitly, while the nonlinear terms are integrated explicitly. As the vorticity equation has only nonlinear terms on its right-hand side which do not excite gravity waves, it is integrated explicitly. The divergence equation and the continuity equation have both linear and nonlinear terms on their right-hand sides and therefore are integrated implicitly. Separating the linear and nonlinear terms, the divergence and continuity equations can be written as

w D wt

F1 \  ’ 2I '

(6.140)

w I' wt

F2 I '  I D ,

(6.141)

and

where F1 \

§ U 2 V 2 · w 1 § w · 2 U (’ 2\  f ) ¸  ’ 2 ¨ ¸ ¨ V (’ \  f )  cos T 2 2 cos T © wO wT ¹ © 2 cos T ¹

and F2 I '



1 § w w · (V I ') ¸ ¨ (U I ')  cos T cos 2 T © wO wT ¹

are the nonlinear terms. The reason for doing such a partitioning is that we want to treat the gravitational modes differently from the Rossby waves. We integrate the linear terms responsible for the fast-moving gravitational modes implicitly and the nonlinear terms explicitly. Let us define a time-average operator ( )t as

Dt

D t 't  D t 't , 2

(6.142)

noting that from the centered time differencing scheme we have wD wt

D t 't  D t 't . 2't

Solving equation (6.142) for D t 't and substituting it into the above equation we can rewrite (6.140) using the above averaged value as D t 't  D t 't 2't

D t  D t 't 't

Similarly, equation (6.141) can be rewritten as

F1t (\ )  ’ 2 I 't .

(6.143)

Mathematical Aspects of Spectral Models

109

I 't 't  I 't 't

I 't  I 't 't

2't

't

F2t (I ')  I D t .

(6.144)

It should be noted that the values of I ' and D in the linear terms on the right-hand side of (6.143) and (6.144) are averages of their values at time levels t  't and t  't , while the nonlinear terms are calculated at time level t. Equations (6.143) and (6.144) may be written as D t 't  F1t (\ )'t  ’ 2 I 't 't ,

Dt

(6.145)

I 't I 't 't  F2t (I ')'t  I D t 't .

(6.146)

In order to solve this system of equations, we eliminate one of the unknown variables ( D t or I 't ) from these equations in order to obtain an equation in a single unknown variable. We eliminate I 't to get an equation in the single variable D t . This is achieved by applying the ’ 2 operator to (6.146) and multiplying the resultant equation by 't 2

2

't’ 2I 't 't   't ’ 2 F2t I '   't I’ 2 D t ,

’ 2 I 't 't

then substituting for ’ 2 I 't 't into (6.145) to obtain Dt

2

2

D t 't  F1t \ 't  't’ 2I 't 't   't ’ 2 F2t I '   't I’ 2 D t ,

or, on rearranging, we obtain a Helmholtz equation, that is, 2  't I’ 2 Dt  Dt

F3  D, I ',\

t ,t 't

,

(6.147)

where F3  D, I ',\

t ,t 't

 't

2

’ 2 F2t I '  'tF1t \   't ’ 2I 't 't  D t 't .

We can write (6.147) in spectral form as 2

  't I

t n(n  1) mt Dn  Dnm a2

F3mn ,

or t

Dnm



F3mn . § n(n  1) 2 · I 1 ¨ ( ) ' t ¸ 2 © a ¹

(6.148)

t

Hence we can obtain the unknown variable Dnm using the above equation. The value of Dnm at time level t  't is then obtained from

Dnm

t 't

t

t 't

2 Dnm  Dnm

.

(6.149)

An Introduction to Global Spectral Modeling

110 t

By substituting Dnm into (6.146), one also obtains the spectrum of I 't and hence I 't 't .

The spectrum of ] t 't is obtained explicitly from the spectral form of the vorticity equation (6.134). This completes a one-time-step forecast of the shallow-water model using a semi-implicit time-integration scheme.

6.11 Inclusion of Bottom Topography It is relatively easy to cast a spectral shallow-water model that includes the effects of mountains. The only changes that are involved in the basic equations are in the pressure gradient force, where wI wO

is replaced by

w I  gh wO

wI wT

is replaced by

w I  gh . wT

and

The mass continuity equation remains unaltered. Here h is the height of the mountain (i.e., the bottom topography of the shallow-water model) and I  gh defines the geopotential of the free surface. The derivation of the Helmholtz equations for the shallow-water equations with bottom topography is left as an exercise for the student.

6.12 Exercises 6.1. In Section 6.2, Rodrigues’ formula, Pn  P

n 1 dn  P 2  1 , 2 n! d P n n

0, 1, 2, 3, ! ,

n

P d1,

was described. Using (6.13), try to prove the above formula. Hint: Apply a binomial expansion of (1  P 2 ) n . 6.2. We define the generating function of the ordinary Legendre polynomial as follows: f

1

¦ P P x , n

1/ 2

1  2P x  x 2

n

P  1,

x  1.

h 0

Prove the above equality. Hint: Use the binomial expansion of 1/(1  y )1 2 , set y 2 P x  x 2 , and multiply the powers of 2 P x  x 2 out and collect terms involving xn .

6.3. Prove the recurrence relation

 n  1 Pn1  P  2n  1 P Pn  P  nPn1  P ,

n 1, 2, 3, !

Mathematical Aspects of Spectral Models

111

Hint: Differentiate the generating function with respect to x. The above equation is called Bonnet’s recursion. 6.4. Represent the following polynomials in terms of Legendre polynomials: 1, P , P 2 , and P 3 ; i.e., find the matrix A such that

ª P0 ( P ) º « P (P ) » » A« 1 « P2 ( P ) » « » ¬ P3 ( P ) ¼

ª1º «P» « ». «P 2 » « 3» ¬P ¼

Describe the most salient feature of matrix A. What does this feature imply? 6.5. Show that Pn  P d 1 for 1 d P d 1 .

Chapter 7

Multilevel Global Spectral Model 7.1

Introduction

Since the 1970s, the spectral method has become an increasingly popular technique for global numerical weather prediction. Global numerical models formulated using the spectral technique are used worldwide for both research and operational purposes. The success of the spectral technique can be attributed to the spectral transform technique developed independently by Eliasen et al. (1970) and Orszag (1970), and later refined by Bourke (1972). Prior to the introduction of the transform technique, the nonlinear terms were computed using a very tedious process called the interaction coefficients method. This method required large amounts of computer resources as well as enormous bookkeeping. The transform technique facilitates the computation of the nonlinear terms, as discussed later in Section 7.4. Furthermore, the Galerkin method discussed in Chapter 4 is widely used in most spectral models and provides us with alias-free computation of the nonlinear terms. The transform technique enables the current spectral models to be competitive in terms of computational overhead with respect to their grid-point counterparts. The transform technique is also well suited for incorporating the terms dealing with physics in the prediction scheme. There are a number of advantages to using the spectral technique over the conventional grid-point method. However, we will not get into this discussion here. It should be noted that the model truncation limit specifies the scale of the shortest wavelength that can be resolved by the model. In the following section, we discuss the two most widely used truncations in a spectral model.

7.2

Truncation in a Spectral Model

As we saw earlier, any variable field (say ], \, etc.) can be represented by a series of spherical harmonics as ] O, P ] nmYnm  O , P , (7.1)

\ O, P

¦¦ ¦¦\ m

n

m

n

112

m m n n

Y

O, P ,

(7.2)

Multilevel Global Spectral Model

113

where the double summation is infinite in general, in other words, f d m d f and | m |d n d f . However, in practice, representation in terms of an infinite series is not possible and one needs to truncate the series at some point. It can be shown that once we decide on the maximum number of spectral components to be present in the double summation, then we can achieve a consistent energy, momentum, and vorticity conservation for the spectrally truncated equations. In doing so, we neglect waves outside the spectrum (i.e., beyond the designated highest wavenumber). In practice, one truncates smaller-scale waves in the spectral truncation. In a global spectral model, two types of truncation schemes are generally used. They are called the triangular truncation and the rhomboidal truncation. In the case of triangular truncation, the highest degree of the various spherical harmonics Ynm (m denotes the order of a spherical harmonic and n the degree) is fixed and is set equal to the highest order of these waves. Thus, this truncation is represented mathematically by N

\ O, P

N

¦ ¦\

m m n n

Y

O, P ,

(7.3)

m  N n | m|

and is schematically represented in Fig. 7.1. Under this truncation, the number of zeroes n  m for different wavenumbers m vary, and thus different wavenumbers m have different degrees of freedom along a latitude. The two-dimensional wave index has the same maximum value for all waves, and all waves are truncated at the same two-dimensional scale. The rhomboidal truncation has a fixed value of the maximum number of zeroes for every wavenumber m, thus giving equal degrees of freedom along a latitude for all waves. This truncation is represented as N

\ O, P

|m| J

¦ ¦\

m m n n

Y .

(7.4)

m  N n |m|

If J N , the truncation is called rhomboidal. When J z N , it may be called parallelogramic truncation. Schematically, this can be represented as in Fig. 7.2. The triangular truncation at wavenumber N is often abbreviated as T  N ; likewise, rhomboidal truncation at wavenumber N is abbreviated as R  N .

Figure 7.1. Triangular truncation.

114

An Introduction to Global Spectral Modeling

Figure 7.2. Rhomboidal truncation.

There has been considerable discussion as to which of the two truncations is better suited for different problems. Both of these representations have their advantages and disadvantages. The triangular truncation assumes that the variance contained in waves shorter than a particular two-dimensional wave is zero. However, the representation in the meridional direction does not have the same number of degrees of freedom for all waves. In general, it is seen that for the same number of wave components, triangular truncation contains more variance than the rhomboidal one. The total number of Pnm for a triangular truncation that is truncated at wavenumber N is equal to ( N  1)( N  2) / 2 (see Fig. 7.1). This can be obtained as follows: In a triangular truncation with maximum wavenumber N, there are N  1 Legendre functions with P00 , P10 , P20 , …, PN0 zonal wavenumbers. Likewise, the number of Legendre functions containing wavenumbers 1, 2, 3, …, N is N, N  1 , N  2 , …, 1, respectively. Thus on average there are ( N  2) / 2 Legendre functions per zonal wavenumber. The total number of zonal wavenumbers, including wavenumber zero, equals N  1 . This means that the total number of Legendre functions, or spectral components, in a triangular truncation truncated at wavenumber N is set at ( N  1)( N  2) / 2 . Therefore a T  42 spectral model (truncated at wavenumber 42) contains (43 u 44) / 2 946 spectral components. Likewise, T  106 and T  170 models would have 5778 and 14706 spectral components, respectively. The total number of Pnm for a rhomboidal truncation R  N is equal to ( N  1)( N  1) . This can be obtained as follows: In a rhomboidal truncation, there are N  1 Legendre functions ( Pnm , Pnm1 , Pnm 2 , …, Pnm J ) for any zonal wavenumber m. The total number of zonal wavenumbers, including wavenumber zero, is N  1 . Therefore, the total number of Legendre functions or wave components in this truncation is ( N  1)( N  1) . Thus R  42 truncation has 43 u 43 1849 Pnm .

Multilevel Global Spectral Model

7.3

115

Aliasing

Given some data at 2 N  1 points along a line, we can analyze them into a maximum of N-Fourier (N-sine and N-cosine) components. If one tries to calculate more than N components, a false representation, or aliasing, of lower wavenumbers into higher wavenumbers occurs. To show this, let M

y  x

¦a e

imx

m

,

(7.5)

m 0

where y has values y1 , y2 , ! , y2 N 1 at 2 N  1 equally spaced points x1 , x 2 , ! , x 2 N 1 , with x1 0 and x2 N 1 2S . Multiplying both sides of (7.5) by e  imx and summing over all possible grid points j, we obtain am

1 2 N 1 imx j y je 2N 1 j 1

¦

If m ! N , then we can write m

1 2 N 1  im (S / N )( j 1) y je . 2N 1 j 1

¦

(7.6)

2 N  m ' , where m '  N ,

2 N 1

am

¦ye

 i (2 N  m ')(S / N )( j 1)

j

j 1

2 N 1

¦ye

i m '(S / N )( j 1)

j

,

because e  i 2S ( j 1) 1

am

am ' .

j 1

am ' ,

or

(7.7)

Thus we see that the amplitude am is actually reflected as am ' , which is the amplitude of the lower wavenumber m', where m '  N . Thus if one tries to analyze a data set at 2 N  1 equally spaced points into wavenumbers greater than N, one runs into aliasing problems. The minimum number of points needed to analyze a field into NFourier components is 2 N  1 . When we analyze the product terms, we need still more points for alias-free calculating, as will be seen in the next section.

7.4

Transform Method

The nonlinear terms in a spectral model truncated at the model’s spectral resolution can be calculated by one of the following methods: (a) the interaction coefficients method or (b) the transform method. The interaction coefficients method consists of obtaining the spectral amplitudes of the nonlinear products directly from the spectral amplitudes of the individual members of the product terms using relevant trigonometric or Legendre function formulas. This procedure is exact and free from any computational errors. If the number of spectral components in the model is small, say four to six, as is the case in many theoretical studies, this method is very efficient and elegant. However, when dealing with a large number of spectral components, the number of interaction coefficients becomes too large and their storage , bookkeeping, and off and on retrieval

An Introduction to Global Spectral Modeling

116

becomes very cumbersome. For a large number of wave components, the transform method is simpler and more efficient. In the transform method, the nonlinear terms are calculated in the space domain and then transformed to the spectral domain. The nonlinear terms in an atmospheric model are generally quadratic products of the form uwu / wO , vwu / wP , etc., where u and v are the spectrally truncated variables. We are required to spectrally analyze these products truncated at the model’s resolution. To spectrally analyze the product of variables A(O , P ) and B(O , P ) , the transform method consists of the following three steps, which are repeated at every time step during model integration:

1. Perform a spectral to grid-point transform of the model variables. (a) This involves the calculation of the Fourier amplitudes from the spherical harmonic amplitudes at each latitude using the relations N

Am  P

¦a

m m n n

P

P ,

m m n n

P ,

n m N

Bm  P

¦b

P

n m

where anm and bnm are the spherical harmonic amplitudes of the two variables involved in the nonlinear term, while Am ( P ) and B m ( P ) are their Fourier amplitudes at a particular latitude P. (b) Calculate the grid-point values from the Fourier components as M

AO, P

¦

Am ( P )eimO ,

m M M

B O, P

¦B

m

( P )eimO .

m M

Processes 1(a) and 1(b) are the inverse Legendre and inverse Fourier transforms, respectively. 2. Perform a calculation of the nonlinear products on grid points. The components A  O , P and B  O , P at each grid point are multiplied to get the nonlinear product

C  O , P , that is, C O, P

A O, P B O, P .

Here, A and B may be the grid point values of variables like u, v, wu / wO , wv / wP , and so on. 3. Perform a spectral transform of the nonlinear products at grid points. This process is again achieved in two transform steps:

Multilevel Global Spectral Model

117

(a) A Fourier transform of grid-point products on latitude circles. (b) A Legendre transform of the Fourier components obtained in step 3(a). Now we consider the requirements for alias-free calculations. For this, let A  O , P and B  O , P be the space values of the two variables spectrally truncated at

order (zonal wavenumber) M and degree N of the spherical harmonics series. Then the product C  O , P may be written as M

N

M

¦ ¦a

C O, P

Y

m1  M n1 m1

M

N

¦ ¦b

O, P

m1 m1 n1 n1

m2 m2 n2 n 2

m2  M n2 m2

M

N

Y

O, P

N

¦ ¦ ¦ ¦a

b Pnm1 1 Pnm2 2 ei ( m1  m2 ) O .

m1 m2 n1 n2

(7.8)

m1  M m2  M n1 m1 n2 m2

The two-dimensional spectral amplitude Cnm of a spherical harmonic wave is given by 1 4S 1 4S

Cnm

1

³ ³

2S

C (O , P )Ynm*  O , P d O d P

1 0 1

§

³ ¨© ³ 1

2S

C (O , P )e  imO d O ·¸ Pnm  P d P . ¹

0

Substituting for C  O , P from (7.8), we get Cnm

1 2

³

1

1

§ 1 ¨¨ © 2S

M

2S

M

N

0

m1  M m2  M n1 |m1 | n2 |m2 |

m1 m2 i ( m1  m2 ) O  imO n1 n2

m1 m2 n1 n2

a b P

N

³ ¦ ¦ ¦ ¦ P e e dO P  P d P . m n

(7.9)

The integral with respect to O on the right-hand side of (7.9) is nonzero only for cases when m m1  m2 . Thus the zonal wavenumber m of the nonlinear product results from the interaction of those wavenumbers m1 and m2 of the individual components of

m1  m2 . The integration with respect to O results in the

the product for which m

Fourier analysis of C  O , P along latitude circles. Also, if C m is the Fourier amplitude of wavenumber m, then

Cnm

1 2

³

1

1

C m ( P ) Pnm ( P )d P .

Comparing (7.9) and (7.10), we have

Cnm

1 2S

2S

M

M

N

N

³ ¦ ¦ ¦ ¦a 0

m1 m2 n1 n2

b Pnm1 1 ( P ) Pnm2 2 ( P )

m1  M m2  M n1 |m1 | n2 |m2 |

u ei ( m1  m2  m ) O d O

(7.10)

An Introduction to Global Spectral Modeling

118 M

M

N

N

¦ ¦ ¦ ¦a

m1 m2 n1 n2

b Pnm1 1 Pnm2 2 ,

(7.11)

m1  M m2  M n1 |m1 | n2 |m2 |

where m1  m2

m . Thus M

Cm

M

N

N

1 m1 m2 an1 bn2 |m2 | 2

¦ ¦ ¦ ¦

m1  M m2  M n1 |m1 | n2

³

1

1

Pnm1 1 Pnm2 2 Pnm d P .

This calculation of spectral components Cnm of the product C 1 m1 m2 n2 1 n1

(7.12) AB involves the

m n

evaluation of an integral of the form ³ P P P d P , where m1  m2 m . During the Legendre transform, it is necessary that this integral be evaluated exactly. We examine the requirements of numerical quadrature to achieve such exact evaluation of the integral. We may write a normalized Legendre function, 1/ 2

Pnm ( P )

1 § (n  m)! · ¨ ¸ 2 n ! © (n  m)! ¹ n

1  P 2

m/2

1/ 2

§ 2n  1 · ¨ ¸ © 2 ¹

(1  P 2 ) m / 2

d n  m ( P 2  1) n d P n m

pn  m ,

(7.13)

where pn  m is a polynomial of degree n  m . With this, we can write

Pnm1 1 ( P ) Pnm2 2 ( P ) Pnm ( P ) (1  P 2 ) m1 / 2 pn1  m1 u(1  P 2 ) m2 / 2 pn2  m2 (1  P 2 ) m / 2 pn  m

(1  P 2 )( m2  m2  m ) / 2 pn1  m1 pn2  m2 pn  m

p2m pn1  m1 pn2  m2 pn  m . Because m

(7.14)

m1  m2 ,

(1  P 2 )( m1  m2  m ) / 2  (1  P 2 ) m

p2 m

is a polynomial of degree 2m. Furthermore, pn1  m1 , pn2  m2 , and pn  m , are polynomials in

P of degree n1  m1 , n2  m2 , and n  m , respectively. Consider first the rhomboidal or the parallelogramic truncation of the form N

|m| J

¦ ¦.

m  N n |m|

The highest degree of p2 m is 2N and that of pn1  m1 , pn2  m2 , and pn  m is J. Therefore, Pnm1 1 ( P ) Pnm2 2 ( P ) Pnm ( P ) has the highest degree 2 N  3 J . By using a Gaussian quadrature, we can integrate a polynomial of highest degree d 2k  1 from its values available at k Gaussian quadrature points. In this case, the highest degree of the triple-product Pnm1 1 Pnm2 2 Pnm is 2 N  3 J . Therefore

³

1

1

Pnm1 1 ( P ) Pnm2 2 ( P ) Pnm ( P )d P

(7.15)

Multilevel Global Spectral Model

119

can be evaluated exactly if 2k  1 2 N  3 J or if k (2 N  3 J  1) / 2 . In a rhomboidal truncation J N , therefore k (5 N  1) / 2 is the minimum number of Gaussian latitudes for the Legendre transform. Consider next the triangular truncation, which has the form N

N

¦ ¦.

m N n m

For triangular truncation, the product Pnm1 1 ( P ) Pnm2 2 ( P ) Pnm ( P ) can be represented by a

polynomial of the form (7.14). However, in this case the highest degree of Pnm1 1 ( P ) , Pnm2 2 ( P ) , and Pnm ( P ) is fixed at N. The highest degree of the product Pnm1 1 ( P ) Pnm2 2 ( P ) Pnm ( P ) in this case may therefore be represented by the polynomial (1  P 2 )

m1  m2  m 2

pN  m1 pN  m2 pN  m

p2 m pN  m1 pN  m2 pn  m

p3N ,

since m

(7.16)

m1  m2 . The highest degree of the triple product Pnm1 1 ( P ) Pnm2 2 ( P ) Pnm ( P ) in this

case is 3N, and it can be integrated exactly using Gaussian quadrature with a minimum of k Gaussian latitudes, where 2k  1 3 N or k (3N  1) / 2 . For the Fourier transform in an R  42 resolution model, the minimum number of grid points in the zonal direction is 3 u 42  1 127 . Because the FFT needs these points to be multiples of 2, 3, or 5, the minimum number of points is taken as 128. The same number of points in the zonal direction are required for T  42 truncation. However, the minimum number of Gaussian latitudes for an R  42 resolution is 106 [t (5 u 42  1) / 2] , and for T  42 it is 64 [t (3 u 42  1) / 2] .

7.5

The x-y-V Coordinate System

In most applications of the hydrodynamic equations, the horizontal coordinate system is either the Cartesian coordinate system or the spherical coordinate system. However, the vertical coordinate system has more possible variations. The most common vertical coordinates are height (z) and pressure (p). One of the disadvantages of using z or p as a vertical coordinate is that the lower coordinate surfaces are frequently below the ground in regions of high elevation (Andes, Himalayas, Rockies), which may cause difficulties when incorporating the effect of surface topography into the model equations. To avoid this, use is made of the sigma (V) coordinate (Phillips 1957), in which the lowest V surface follows the earth’s surface. The V vertical coordinate is defined as

V where ps is the surface pressure. Then

p , ps

(7.17)

An Introduction to Global Spectral Modeling

120

Figure 7.3. Location of points on pressure and sigma surfaces.

V

­ 1 at the earth's surface ® ¯0 at the top of the atmosphere.

In a quasi-static system, V varies monotonically with height and has the desired properties of a vertical coordinate. The transform from an x-y-p coordinate system to an x-y-V coordinate will be presented here. As shown in Fig. 7.3 let Q1 and Q2 be the values of variable Q on a V surface at two points a zonal distance 'x apart from each other. Let Q3 be the value at the projection of Q2 on the p surface so that the zonal distance between Q1 and Q3 is also 'x . Let the vertical distance between Q2 and Q3 in the V coordinate be 'V . With this, we may write Q3  Q1 Q2  Q1 Q3  Q2 'V  . (7.18) 'x 'x 'V 'x

Taking the limits 'x o 0 and 'V o 0 , we obtain wQ wx

p

wQ wx

 V

wQ wV wV wx

.

(7.19)

p

Similarly, one can write an expression for the derivatives with respects to y and t. However, along vertical we have

wQ wp

wQ wV . wV wp

(7.20)

If we assume that V is a single-valued function of pressure, then the above expression is justified. In a non-hydrostatic, vigorous, small-scale weather system such as a tornado, pressure may not decrease monotonically with height. Therefore (7.20) will not be applicable in such cases. The following transformation laws describe the conversion from the p to the V coordinate system:

Multilevel Global Spectral Model

wQ wx wQ wy

wQ wt

121

p

wQ wx

p

wQ wy

p

wQ wt wQ wp

 V

 V

 V

wQ wV wV wx

p

wQ wV wV wy

p

wQ wV wV wt

p

,

(7.21)

,

(7.22)

,

(7.23)

wQ wV . wV wp

(7.24)

Now that we have the template for conversion from the p to the V coordinate system, in order to form a closed system of equations for the full spectral model, we first need to transform the substantial derivative, the pressure gradient term, the hydrostatic equation, and the mass continuity equation from pressure coordinates to sigma coordinates. This is done in the following sections. 7.5.1

Substantial Time Derivative (d / dt ) in V Coordinates

The substantial derivative (i.e., dQ / dt ), in the (x, y, p) system is given by

wQ wt

dQ dt

u p

wQ wx

v p

wQ wy

Z p

wQ . wp

(7.25)

On making use of (7.21) to (7.24), this can be written as dQ dt

wQ wt

u V

wQ wx

v V

wQ wy

V

§ wV ¨ ¨ wt ©

u p

wV wx

v p

wV wy

+Z p

wV wp

· wQ . ¸¸ ¹ wV

Thus we obtain

dQ dt

wQ wt

dQ dt

wQ wt

u V

wQ wx

v V

wQ wy

 V

wQ dV , wV dt

or

where V

u V

wQ wx

v V

wQ wy

 V V

wQ , wV

(7.26)

dV / dt , the vertical velocity in V coordinates.

7.5.2. Pressure Gradient Term in V Coordinates We start from the terms  g wz / wx p and  g wz / wy p in the pressure coordinate system.

Using transformation rule (7.21), we can write

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An Introduction to Global Spectral Modeling

g

wz wx

g p

wz wV

g

wV wx

g V

wz . wV

p

 g ( p / RT ) , we can rearrange g wz wV

g U

Using the hydrostatic equation, wp / wz from the above equation to obtain

g

wz wx

wz wp wp wV

RT wp . p wV



Hence g

If we let Q

wz wx

g p

wz wx

 V

p in (7.21) and noting that wp / wx p 0

wp wx

 V

RT wV p wx

p

wp . wV

(7.27)

0 , we obtain

wp wV wV wx

.

(7.28)

p

From (7.27) and (7.28) we obtain

g

wz wx

g p

wz wx

 V

RT wp p wx

V

RT wp p wy

V

.

(7.29)

.

(7.30)

Similarly, g

wz wy

g p

wz wy

 V

Equations (7.29) and (7.30) give the conversion of the horizontal pressure gradient terms from p to V coordinates. It is often convenient to introduce the Exner function

S

§ p· ¨ ¸ © p0 ¹

R / cp

,

(7.31)

where p0 1000 mb . From the above definition, by taking the natural logarithm of both sides of (7.31) and differentiating with respect to x it follows that R

ln S

ln S w ln S wx

1 wS S wx

V

§ p · cp ln ¨ ¸ , © p0 ¹ R ln p  ln p0 , cp · w §R ¨¨ ln p  ln p0 ¸¸ , wx © c p ¹

R w  ln p  ln p0 , c p wx V

123

Multilevel Global Spectral Model

1 wS S wx

 V

R wp pc p wx

.

(7.32)

V

Now (7.29) may be written as g Furthermore, using T

wz wx

g p

wz wx

 cp V

T wS S wx

. V

TS , we obtain g

wz wx

p

wz wy

p

g

wz wx

 c pT V

wS wx

V

wS wy

V

.

(7.33)

.

(7.34)

Similarly, g

g

wz wy

 c pT V

It should be noted that the individual terms on the right-hand side of (7.29), (7.30), (7.33), and (7.34), like  g wz / wx V and ( RT / p )wp / wx V , are very large in

magnitude but have opposite signs. Small errors in the calculation of these terms can lead to larger errors in the estimation of the pressure gradient force, which is a small sum of these two large terms with opposite signs. This problem becomes particularly serious over regions covered by steep mountains. 7.5.3

Hydrostatic Equation in V Coordinates

The hydrostatic equation may be expressed as wI wp



1



U

RT , p

or

wI wV wV wp Using the relation V



RT . p

p / ps and noting that

wV wp

w § p· ¨ ¸ wp © ps ¹

1 wp ps wp

p wI ps wV

 RT ,

1 , ps

we get

or

V

wI wV

 RT .

(7.35)

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124

This is the hydrostatic equation in V coordinates. An alternative form of the hydrostatic equation can be obtained from the Exner function (7.31). We take the natural logarithm of both sides of (7.31) and differentiate with respect to V to obtain

1 wS S wV

R wp . c p p wV

g U

Using the hydrostatic equation, wp / wz

1 wS S wV



(7.36)

 g ( p / RT ) , we obtain

1 wz g , c pT wV

or g

wz wV

c pT

wS , wV

T

as

TS .

(7.37)

This is also an hydrostatic equation in (x, y, V) coordinates. 7.5.4

Mass Continuity Equation in V Coordinates

The mass continuity equation in the (x, y, p) system is wu wx

 p

wv wy

wZ wp

 p

0,

(7.38)

where Z dp / dt is the vertical velocity in the pressure coordinate system. Making use of (7.21) and (7.22), we may write

wu wx

wv wy

p

wu wx

p

wv wy

 V

 V

wu wV wV wx

p

wv wV wV wy

p

,

(7.39)

.

(7.40)

The term wZ / wp may also be written as wZ wp

w § dp · w § dp · wV ¨ ¸ ¨ ¸ wp © dt ¹ wV © dt ¹ wp w § wp wp wp wp · wV ¨  u  v  V ¸ wV © wt wx wy wV ¹ wp ª§ w w w w · wp «¨  u  v  V V ¸¹ wV w w w w t x y ¬©



or

wu wp wV wx

 V

wv wp wV wy

 V

wV wp º wV , wV wV »¼ wp

(7.41)

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125

wV d § wp · § wu wp ¨ ¸¨ wp dt © wV ¹ © wV wx

wZ wp

Letting Q

 V

wv wp wV wy

 V

wV wp · wV . ¸ wV wV ¹ wp

(7.42)

p in (7.21) and (7.22), we obtain

wp wx

wp wx

p

wp wV wV wx

 V

p

and

wp wy Since wp / wx

p

wp / wy

p

wp wy

p

wp wV wV wy

 V

. p

0, we obtain

wp wx

 V

wp wV wV wx

p

and

wp wy Substituting for wp / wx wZ wp

V

and wp / wy

wV d § wp ¨ wp dt © wV

V

 V

wp wV wV wy

.

(7.43)

p

into (7.42), we obtain

· wp ¸ ¹ wV

§ wV ¨ ¨ © wx

p

wu wV  wV wy

p

wv wV · wV  . ¸ wV wV ¹¸ wp

(7.44)

From (7.39), (7.40) and (7.44), the continuity equation (7.38) may be written in the (x, y, V) coordinate system as

wu wx

 V

wu wV wV wx

 p

wp § wV  ¨ wV ¨© wx

wv wy

p

 V

wv wV wV wy

wu wV  wV wy

p

 p

wV d § wp · ¨ ¸ wp dt © wV ¹

wv wV · wV  ¸ wV wV ¸¹ wp

0,

or

wu wx As V

p / ps and wp / wV

 V

wv wy

 V

(7.45)

ps , (7.45) can be written as 1 dps wu  ps dt wx

or

wV wV d § wp ·  ¨ ¸ 0. wV wp dt © wV ¹

 V

wv wy

 V

wV wV

0,

(7.46)

An Introduction to Global Spectral Modeling

126 d ln ps wu  wx dt

 V

wv wy

 V

wV wV

0.

(7.47)

This is the mass continuity equation in the (x, y, V) coordinate system. Often it is useful to write the continuity equation in flux form. This can be done by multiplying (7.46) by ps to obtain § wu dps  ps ¨ dt © wx

 V

wv wy

 V

wV · ¸ 0, wV ¹

or

wps wp u s wt wx

v V

wps wy

 V V

wps wu  ps wV wx

 ps V

wv wy

 ps V

wV wV

0,

or w w w w ps  (ups )  (vps )  (V ps ) 0 . wt wx wy wV

(7.48)

This is the flux form of the continuity equation in the (x, y, V) system.

7.6

A Closed System of Equations in V Coordinates on a Sphere

In Section 7.5 we transformed the substantial time derivative, the pressure gradient term, and the mass continuity equation from the (x, y, p) to the (x, y, V) coordinate system. We make use of these to write a closed system of equations for a global model with V as the vertical coordinate. The appropriate coordinate system for a global domain is the spherical curvilinear coordinates. The position of any point in this coordinate system is given by (O, T, r), where O is the longitude, T is the latitude, and r is the distance from the center of the earth. It is convenient to regard the earth as a sphere with radius a, so that r a  z , where z is the vertical height of the point from the earth’s surface. With this, we obtain

G x r cos TGO ,

G y rGT ,

and G z G r .

Because z  a , r | a , so that we may write

G x a cosTGO ,

G y aGT ,

and G z G r .

(7.49)

If instead of height we take pressure as the vertical coordinate, then the position of any point is given by (O, T, p) and the distance increments are

G x a cos TGO ,

G y aGT ,

and G z



Gp . gU

(7.50)

The basic equations governing the atmosphere are the three momentum equations, the mass continuity equation, and the thermodynamic equation. The large-scale atmospheric flow is quasi-horizontal with very small vertical acceleration. For such an

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127

atmosphere, the vertical momentum equation is approximated by the hydrostatic assumption: wp g U , wz where p is the pressure, z is the height, g is the acceleration due to gravity, and U is the density of air. With this, the equations for a global model in (O, T, p) coordinates are as follows: x The horizontal momentum equation G G G G G G G wV wV  V ˜’ V  Z  fk u V  ’I  F . wt wp





(7.51)

x The hydrostatic equation wp wI

U .

(7.52)

G § wT RT · Q  V ˜’T  Z ¨   . ¨ wp c p p ¸¸ c p © ¹

(7.53)

x The thermodynamic equation wT wt

x The mass continuity equation

G wZ ’ ˜V  wp

0.

(7.54)

x The moisture equation wr wt G Here V

u

wr wr wr  v Z EP. wx wy wp

(7.55)

G G ui  vj is the horizontal wind vector, t is time, I

gz is the geopotential, f is G the Coriolis parameter, T is the temperature, Z is the vertical velocity dp / dt , F is the frictional force vector, Q represents all the diabatic sources and sinks of heat, r is the specific humidity, E is the evaporation, and P is the precipitation. If we take the vector equation (7.51) as two scalar equations, (7.51) to (7.55) are the six equations in the six dependent variables u, v, Z, I , T, and r (x, O, T, and p are independent variables) that form a closed system. We now write the above set of equations with V as the vertical coordinate. With V as the vertical coordinate, we have V 0 at the top ( p 0) and V 1 at the bottom ( p ps ) of the atmosphere. Therefore, V dV / dt 0 at V 0 and V 1 . These are the top and bottom boundary conditions for V . From (7.29) and (7.30), we have

’ pI

’V I 

RT ’V p p

’V I  RT ’V ln p ,

(7.56)

An Introduction to Global Spectral Modeling

128

where ’ is a two-dimensional operator. With the fact that p V ps and that V is constant on V surfaces, (7.56) may be written as ’ pI

’V I  RT ’V ln ps .

(7.57)

Thus the momentum equation in the V coordinate system becomes G G G G G G G wV wV  V ˜’ V  V  fk u V  (’I  RT ’ ln ps )  F . wt wV





Furthermore, making use of the identity G G G G G G § V ˜V · V ˜’ V ’ ¨ ¸  (’ u V ) u V 2 © ¹





G G G G § V ˜V · ’¨ ¸  ] k uV , 2 © ¹

the momentum equation takes the form G G G G G G G § V ˜V · wV wV  (]  f )k uV  V  ’ ¨I  ¸  RT ’ ln ps  F . wt wV 2 © ¹

(7.58)

The hydrostatic equation takes the form

V

wI wV

 RT .

(7.59)

G G wV V ˜’ ln ps  ’ ˜V  . wV

(7.60)

The continuity equation (7.47) is then w ln ps wt

It should be noted that since ln ps is a function of O and T only, then w ln ps / wV 0 . Thus the vertical advection term V w ln ps / wV does not appear in (7.47). The thermodynamic equation, or the first law of thermodynamics, is cp

dT 1 dp  dt U dt

Q .

(7.61)

From the definition of V, we obtain dp / dt V ps  V p s . Then (7.61) takes the form wT wt

G wT RT V ˜’T  V  (V ps  p sV )  H T wV c p p G § wT RT · RT dps V ˜’T  V ¨    HT ¨ wV c pV ¸¸ c p ps dt © ¹ G § wT RT · RT d ln ps V ˜’T  V ¨    HT . ¨ wV c pV ¸¸ c p dt © ¹

From (7.62) and (7.60) we get

(7.62)

Multilevel Global Spectral Model

wT wt where J

G RT   V ˜’T  VJ cp

129

G wV · § ¨ ’ ˜V  ¸  HT , wV ¹ ©

RT / c pV  wT / wV is the static stability and H T

heat sources and sinks. The moisture equation is G wr wr V ˜’r  V  E  P. wt wV

(7.63)

Q / c p represents the diabatic

(7.64)

Thus (7.58), (7.59), (7.60), (7.63), and (7.64) represent a closed system of equations in (O, T, V) coordinates. For a global spectral model, it is convenient to write the horizontal momentum G equation (7.58) as vorticity and divergence equations. If we operate on (7.58) by k ˜’ u , we obtain the vorticity equation. Operating on the same equation by ’ ˜ results in the divergence equation. Furthermore, it is advantageous to write these equations in flux form. With this, the desired closed system of equations takes the following form as given by Daley et al. (1976): G G G § w] wV G · ’ ˜ (]  f ) V  k ˜’u ¨ RT ’q  V F¸, (7.65) wt wV © ¹ G wD G k ˜’u (]  f ) V wt G G G § § V ˜V · wV G · ’ ˜ ¨ RT ’q  V  F ¸  ’2 ¨I  (7.66) ¸, wV 2 ¹ © ¹ © G wT RT § wV ·   ’ ˜ TV  TD  VJ (7.67) ¨D ¸  HT , wt cp © wV ¹ G wq wV V ˜’ q  D  , (7.68) wt wV wI  RT , V (7.69) wV G wr wr ’ ˜ rV  rD  V  E  P. (7.70) wt wV





A summary of the basic equations can be found in Appendix B. In the above equations, ] is the vorticity, D is the divergence, f is the Coriolis G G G parameter, V is the horizontal wind vector ui  vj , T is the temperature, q ln ps , r is G the specific humidity, J RT / c pV  wT / wV is the static stability parameter, F is the frictional force, H T is the diabatic heat sources and sinks, E is the evaporation, and P is the precipitation. There are different ways to treat the moisture variables in a model. Most of the models treat the moisture in terms of specific humidity, as in (7.70). In the Florida State University Global Spectral Model, however, it is the dew-point depression T  Td that is

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130

used as the moisture variable. We present below a prediction equation for the dew-point depression also. The dew-point temperature Td is a function of pressure p and specific humidity r, so that dTd § wTd · dp § wTd · dr . (7.71)  ¨ ¸ dt © wp ¹ r dt ¨© wr ¸¹ p dt

The first term on the right-hand side gives the change in Td due to changes in pressure, keeping specific humidity constant. This change occurs due to atmospheric dynamics. The second term represents the change in Td due to changes in specific humidity at constant pressure. This results from external moisture sources or sinks (evaporation and precipitation, etc.) and may be denoted by H M . To determine a suitable expression for wTd / wp r , consider the Clausius-Clapeyron

equation, 1 wes es wp

 L(Td ) 1 wTd , R Td 2 wp

(7.72)

where es is the saturation vapor pressure,  0.622 is the ratio of the molecular weight of water vapor to the molecular weight of dry air, and R is the gas constant for dry air, and L is the latent heat of phase change and is a function of Td . The specific humidity is defined as  es r . (7.73) p Taking the natural logarithm of (7.73) and differentiating the resulting equation with respect to p, we obtain ln r

§e · ln ¨ s ¸  ln  es  ln p  ln   ln es  ln p , © p ¹ § w · w w w ln r ¨ ln   ln es  ln p ¸ , p p p wp w w w © ¹ w ln r 1 wes 1  . wp es wp p

(7.74)

From (7.72) and (7.74) we get w ln r wp

 L(Td ) wTd 1  . R Td 2 wp p

For constant r, w ln r wp

Therefore

0. r constant

(7.75)

Multilevel Global Spectral Model

131

 L(Td ) wTd 1 1  R wp Td 2 p

0,

or wTd wp

r constant

RTd 2 .  pL(Td )

(7.76)

Thus from (7.71) RTd 2 Z  HM .  pL(Td )

dTd dt

(7.77)

Vertical velocity Z is given by

§ V

Z V ps  V p s V ps ¨



p s · ¸ ps ¹

©V wV · § V p¨  D  ¸. wV ¹ ©V

§ V d ln ps · p¨  dt ¸¹ ©V

(7.78)

With this, (7.77) takes the form

wTd wt

G wT RTd 2 § wV V · ’ ˜ TdV  Td D  V d   ¸  HM . ¨D wV  L(Td ) © wV V ¹



(7.79)

This is a prediction equation for dew-point temperature, Td . Subtracting (7.79) from (7.67), we obtain a prediction equation for dew-point depression as G wS wS ’ ˜ VS  SD  V wt wV § RT RTd 2 · § wV V · (7.80) D ¨    HT  H M , ¨ c p  L(Td ) ¸¸ ©¨ V V ¹¸ w © ¹



T  Td is the dew-point depression. We have used V , the vertical velocity in V coordinates, in the model equations, but have not discussed the procedure to calculate it. We now derive a diagnostic equation to obtain V . We define the vertical integral operators ( š ) and ( š V ) as

where S

Fˆ

1

³ FdV

Fˆ V

and

0

Integrating the continuity equation (7.68) from V 1 wq 1 G dV  V ˜’q dV  0 wt 0

³

As q

³

1

V

FdV .

(7.81)

0 to V 1 , we obtain G 1 wV ’ ˜ V dV  dV . 0 0 wV

³

1

ln ps is not a function V, and V vanishes at V wq wt

³

³

0 and V

Gˆ Gˆ V ˜’q  ’ ˜ V ,

1 , we obtain

(7.82)

An Introduction to Global Spectral Modeling

132

Gˆ where V obtain

G 1 ³0 VdV . If we integrate the continuity equation from V

³

1

V

wq dV  wt

1

G

G

1

³ V ˜’q dV  ³ ’ ˜V dV  ³ V

V

1

V

V to V

1 , we

wV dV , wV

or

1  V

wq wt

Gˆ V Gˆ V V ˜’q  ’ ˜ V  V .

(7.83)

Eliminating wq / wt from (7.82) and (7.83), we obtain

Gˆ Gˆ Gˆ V Gˆ V § · (V  1) ’ ˜ V  V ˜’q  ¨ ’ ˜ V  V ˜’q ¸ , © ¹



V



or Gˆ Gˆ V § · (V  1) Dˆ  V ˜’q  ¨ Dˆ V  V ˜’q ¸ . (7.84) © ¹ G This is the diagnostic equation for V , where D ’ ˜ V . Before we transform the global model equations into their spectral form, we separate their linear and nonlinear parts. This is necessary in order to integrate the model using a semi-implicit time-integration scheme. The pressure gradient term is divided into linear and nonlinear parts assuming T T *  T ' , where T * is the horizontal mean at a level, which is a function of V only, and T' is the deviation from T * . Likewise, J J *  J ' . To remove the wind singularity at the poles, we define the Robert functions as

V



U

u cos T a



and

V

v cos T . a

(7.85)

We also define an operator D as

D  A, B

1 cos 2 T

wB · § wA  cos T . ¨ O T ¸¹ w w ©

(7.86)

We now make use of this operator D, the Robert functions, and the definition of T * to recast the model equations. If we now consider the terms on the right-hand side of the vorticity equation (7.65), we have G 1 § w (]  f )U ’ ˜ ]  f V  ¨ cos 2 T © wO w (]  f )V ·  cos T (7.87) ¸, wT ¹ Gˆ Gˆ Gˆ  k ˜’ u  RT ’q  k ˜’ u  RT ' ’q  k ˜’ u  RT *’q Gˆ k ˜’ u  RT ' ’q

133

Multilevel Global Spectral Model

ª w § RT ' wq · « wO ¨ a 2 cos T wT ¸ © ¹ ¬ w § RT ' wq · º  cos T ¨ ¸» , wT © a 2 wO ¹ ¼ 

G Gˆ § wV · k ˜’ u ¨ V ¸ © wV ¹

Gˆ G k ˜’ u F

1 cos 2 T

1 ª w § wV · ¨ V ¸ cos 2 T «¬ wO © wV ¹ w § wU · º  cos T ¨ V ¸» , wT © wV ¹ ¼ 1 w FT w FO  . cos T wO a wT a

(7.88)



(7.89) (7.90)

Thus using (7.87), (7.88), (7.89) and (7.90), the vorticity equation takes the form

w] wt

ªw § wV « wO ¨ ]  f U +V wV ¬ © F · RT ' wq w  2 cos T  cos T T ¸  cosT  (]  f )V a a ¹ wT wT F ·º wU RT ' wq  V  2  cos T O ¸» wV a wO a ¹¼ 1 § wA wB ·   cos T ¨ ¸. cos 2 T © wO wT ¹



1 cos 2 T

(7.91)

In a more compact form, we may write the vorticity equation as w] wt

D  A, B .

(7.92)

Similarly, the divergence equation may be written as wD  ’ 2 I  RT *q wt





D  B,  A  a 2’ 2 E .

(7.93)

The thermodynamic equation may be expressed as wT RT * wq  J *V  wt c p wt

D UT ', VT '  BT .

(7.94)

The continuity equation can be written as wq ˆ ˆ G D wt

0.

(7.95)

The hydrostatic equation is given as

V

wI wV

 RT .

(7.96)

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134

Equations (7.92) to (7.96) along with the equation of state form a closed set of equations for the dry atmosphere. Finally, the moisture (dew-point depression) equation takes the form wS D US ,VS  BS . (7.97) wt

In the above set of equations,

F wV RT ' wq  2 cos T  cos T T , wV a wT a FO wU RT ' wq B ]  f V  V  2  cos T , wV a wO a 1 § wq wq · G  V cos T ¨U ¸, cos 2 T © wO wT ¹

A

E

V

]  f U  V

U 2 V 2 , 2 cos 2 T (V  1) Gˆ  Dˆ  Gˆ V  Dˆ V ,





BT

T ' D  J 'V 

BS

SD  V

RT ' ˆ ˆ RT GD  G  H T  J * Gˆ V  Gˆ , cp cp









RTd2 · § V wS § RT · ¨  ¸ ¨  G  Gˆ  Dˆ ¸  H T  H M . wV ¨© c p  L(Td ) ¸¹ © V ¹

We next define a pseudopressure function P as

P I  RT *q .

(7.98)

This function enables the pressure gradient term to be split into linear and nonlinear parts. Differentiating (7.98) by V, we get

V

wP wV

V

wI wT * V R q. wV wV

Making use of the hydrostatic equation (VwI / wV

V

wP wV

 RT  V R

 RT ) , we can write

wT * q. wV

(7.99)

If we take w / wt of the above equation and substitute for wT / wt from (7.94), we obtain w § wP · ¨V ¸ wt © wV ¹

§ · RT * wq  R ¨ J *V   D (UT ', VT ')  BT ¸ ¨ ¸ c p wt © ¹ * wT wq wV wT * w wT *  RV R q  RV q . wV wt wt wV wt wV

(7.100)

Since T * is independent of time, the term w / wt (wT * / wV ) is zero. Furthermore, the local time rate of change of V on a V surface is zero. Thus

Multilevel Global Spectral Model w § wP · ¨V ¸ wt © wV ¹

135

§ · RT * wq  R ¨ J *V   D (UT ',VT ')  BT ¸ ¨ ¸ c p wt © ¹ * wT wq .  RV wV wt

(7.101)

From (7.95) and (7.101), we get w § wP · ¨V ¸ wt © wV ¹

 R ª¬J *V  D (UT ', VT ')  BT º¼ § wT * RT * · R ¨  V Gˆ  Dˆ . ¨ wV c pV ¸¸ © ¹





(7.102)

We next introduce a pseudo-vertical velocity function W by the expression  W V  V (Gˆ  Dˆ ) , where Gˆ is the vertically integrated advection of surface pressure and Dˆ is the vertically integrated divergence. W may also be expressed as W

that J

*

wq . wt

W (at V

The surface value of W is given by Ws *

V  V

(7.103)

1)

(Gˆ  Dˆ ) . With this and noting

*

( RT ) / c pV  wT / wV , (7.102) becomes

w § wP · * ¨V ¸  RJ W wt © wV ¹

RD (UT ', VT ')  RBT .

(7.104)

The divergence equation takes the form wD  ’V 2 P D ( B,  A)  a 2’V 2 E . wt

(7.105)

Furthermore, the vertically integrated continuity equation (7.95) can be expressed as wq wt



 Gˆ  Dˆ



Ws .

(7.106)

Relating the four variables P, W, D, and q, we have three equations. Thus, we need an additional equation to complete this system. For this purpose, we write the continuity equation in the form wq wt

Since W



 (D  G) 

wV wV

at any level.

(7.107)

wV wq  . wV wt

(7.108)



V  V Gˆ  Dˆ , we obtain wW wV

Substituting for wq / wt , we obtain

wV  (Gˆ  Dˆ ) wV

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136

wW D wV

BW ,

(7.109)

where BW G . We have four equations for the four unknowns P, W, D, and q. A summary of the multi-level spectral model equations can be found in Appendix B. We will be using a semi-implicit time-differencing scheme for integrating (7.104) to (7.106) and (7.109). It is customary to treat the vorticity and moisture equations explicitly. Along with the other model equations, we transform the above four equations to spectral form, and then we combine them to form a single spectral equation for semi-implicit integration.

7.7

Spectral Form of the Primitive Equations

The model equations are transformed into their spectral form similar to Daley et al. (1976) by (a) writing each of the model variables as a truncated series of spherical harmonics and (b) multiplying both sides of the equations by the complex conjugate of the spherical harmonics and integrating, making use of the orthogonality relationship 1

³ ³

2S

1 0

­1 if l1 l2 and m1 m2 Yl1m1Yl2m2 *d O du ® . ¯0 if l1 z l2 and/or m1 z m2

This enables us to obtain the spectral form of the equations for each of the spherical harmonic amplitudes. The spectral amplitudes of the nonlinear parts of the equations are, however, obtained by the transform method. The spectral amplitude of the horizontal component of wind in terms of Robert functions is obtained from the relations U lm

1 ªimF lm  (l  1) lm \ lm1  (l  2) lm1 \ lm1 º¼ a2 ¬

(7.110)

Vl m

1 ªim\ lm  (l  1) lm F lm1  (l  2) lm1 F lm1 º¼ , a2 ¬

(7.111)

and

where 1/ 2

§ l 2  m2 · lm ¨ 2 ¸ . © 4l  1 ¹

There are also relationships between U lm , Vl m , ] lm , and Dlm , where ] lm and Dlm are coefficients of ] and D, respectively. Namely,

l  l  1 U lm

imDlm   l  1 lm ] lm1  l lm1 ] lm1 ,

(7.112)

l  l  1 Vl

 l  1 

(7.113)

m

m l

m l 1

m l 1

D l

m l 1

m l

D  im] .

Following the above procedure, we get the spectral form of the equations: Vorticity equation: w] lm m  >D ( A, B ) @l . (7.114) wt

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137

Divergence equation:

wDlm m  a 2l (l  1) Pl m >D ( B,  A) @l  l (l  1) Elm . wt Thermodynamic equation: w 2 Pl m  RJ *Wl m R[D (UT ', VT ')  BT ]lm . V wt wV Hydrostatic equation: wI m V l  RTl m . wV Pseudo-vertical velocity equation: wWl m m  Dlm  BW l . wV Mass continuity equation: wqlm m Ws l . wt Moisture (dew-point depression) equation: m wSlm ª¬ D US ,VS  BS º¼ l . wt

(7.115)

(7.116)

(7.117)

(7.118)

(7.119)

(7.120)

The left-hand side of the above equations contain the linear parts, while the nonlinear parts are put on the right-hand side. This is to facilitate their integration using the semiimplicit time-integration scheme. Note that in (7.116), Pl m is a spectral component of

P I  RT *q . It should not be confused with the Legendre polynomial Pnm ( P ) . Semi-implicit Time-Differencing Scheme. The primitive equation model has both slow-moving atmospheric waves and fast-moving gravity waves as its solution. Gravity waves have phase speeds of the order of 300 ms-1, which is more than an order of magnitude higher than the phase speeds of atmospheric waves. A centered timedifferencing scheme used to accommodate the gravity waves will need very small time steps to avoid violating the CFL criterion. This makes the integration of a primitive equation model computationally very expensive, particularly for a high-resolution model. To overcome this problem, the gravity wave part of the primitive equation solution is integrated using an implicit time-integration scheme, which is an unconditionally stable scheme at time steps much larger than those permitted by the CFL criterion. The atmospheric wave solution is integrated using an explicit time-differencing scheme. Implicit integration is used only for the linear parts of the equations. The nonlinear parts are integrated explicitly. This time-differencing scheme, where part of the equations are integrated explicitly and part implicitly, is called a semi-implicit timedifferencing scheme. The time step needed for this scheme will be that needed by the fastest-moving atmospheric waves to satisfy the CFL criterion. The gravity waves are excited in the divergent part of flow by the force of gravity. The divergence equation, the thermodynamic equation, and the mass continuity equation contain the gravity wave solution as well as the atmospheric wave solution. They will be

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integrated using the semi-implicit time-integration scheme. The vorticity equation and the moisture equation are integrated explicitly. To develop semi-implicit time differencing, we define F

F (t  't )  F (t  't ) , 2

t

(7.121)

so that wF wt

F (t  't )  F (t  't ) 2't

t

F  F (t  't ) . 't

(7.122)

The terms in the model equations that are treated explicitly are calculated at time t, while those that are treated implicitly are calculated as means at time t  't and t  't . With this, the spectral forms of the divergence equation may be written as t

t Dlm  Dlm (t  't )  a 2l (l  1) Pl m 't

[D ( B,  A)]lm  l (l  1) Elm ,

or t

Dlm  a 2 'tl (l  1) Pl m

t

(7.123) m l

m l

m l

't{[D ( B,  A)]  l (l  1) E }  D (t  't ) .

The thermodynamic equation is written as t

V

w Pl m wP m (t  't ) V l t wV wV  RJ *Wl m 't

R[D (UT ', VT ')  BT ]lm ,

or t

t w Pm V l  'tRJ *Wl m wV

'tR[D (UT ', VT ')  BT ]lm ,

V

wPl m (t  't ) . wV

(7.124)

The equation for pseudo-vertical velocity is written as t

t wWl m  Dlm wV

m

 BW l

,

(7.125)

and the mass continuity equation becomes t

qlm  qlm (t  't ) 't

m

Ws l

,

or qlm

t

m

't Ws l  qlm (t  't ) .

(7.126)

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139

t

After eliminating Dlm between the divergence equation (7.123) and the pseudovertical velocity equation (7.125), we get t

t wW m  l  a 2 'tl (l  1) Pl m wV

(7.127)

^

`

m

m

't ª¬D  B,  A º¼ l  l  l  1 Elm   BW l  Dlm  t  't . t

Now eliminating Wl m from the above equation and the thermodynamic equation (7.124), we get t w § V w m t · 2 Pl ¸  a R't 2l (l  1) Pl m ¨ wV © J * wV ¹ w § 'tR V wPl m (t  't ) · m ¨ * >D (UT ', VT ')  BT @l  * ¸ wV © J wV J ¹

^

m

 R't 't ¬ªD  B,  A  l  l  1 E ¼º l

  BW l  Dlm  t  't ` , m

which may be written as w wV

t § V w m t · 2 Pl ¸  a R't 2l (l  1) Pl m ¨ * © J wV ¹

w wV

m

§ CT · m ¨ * ¸  (CD )l , J © ¹l

(7.128)

where m

 CT l

m

R't ª¬D UT ', VT '  BT º¼ l  V

wPl m (t  't ) wV

and m

 C D l

^

m

m

`

R't 't ª¬D  B,  A  l  l  1 E º¼ l   BW l  Dlm  t  't . t

Equation (7.128) is a Helmholtz-type second-order differential equation in Pl m . The term on the right-hand side of this equation is a function of time t or t  't and can be calculated via the transform method. One can solve this equation using a finite-difference analog. For this, consider the nth level of an N-level model (i.e., 1 d n d N ). Let V n be the sigma value at the layer interface and let V n be the sigma value in the center of the layer, where V n Also, the logarithmic spacing between two adjacent levels is defined as

dn

ln V n 1  ln V n

§V · ln ¨ n 1 ¸ , © Vn ¹

(V nV n 1 )1 2 .

(7.129)

and the vertical spacing between adjacent V levels is Gn V n 1  V n for 1 d n d N  2 . Considering the top of the model as the top of the atmosphere (where V 0 ) and the

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140

Figure 7.4. Vertical structure of the model. bottom of the model as the earth’s surface (where V 1 ), we get G0 GN 1 1  V N 1 . Figure 7.4 shows the vertical structure of the model. One can then write the finite-difference analog of (7.128) as t t t t t 1 § P n 1  P n P n  P n 1 · R't 2 ¨ ¸  2 l (l  1) P n  * *  ¨ ¸   J n 1d n 1 ¹ a G n 1 © J n d n · 1 § 1  1 ¨ (CT ) n  * (CT ) n 1 ¸  (CD ) n , Jn 1 Gn 1 © Jn* ¹

V1 and

(7.130)

where the tilde ( a ) indicates the value in the middle of the layer. For simplicity, we have dropped the wavenumber index m, l in the above equation. t

Equation (7.130) leads to N  2 algebraic equations for the N unknowns P n . Two equations can be obtained by using the boundary conditions; thus this leads to a closed set of N algebraic equations for the N unknowns. At the top boundary ( V 0 ), V 0 and thus W 0 . Then from (7.125) we get t

t W1  0 D G

 BW 1 ,

0

or t

t W1 D G

 BW 1

at

V

V1 .

0

t

Substituting the above value of W 1 into the finite-difference analog of (7.124), we get

141

Multilevel Global Spectral Model t t t 1 § P 2  P1 · ¨ * ¸  R't ( BW )1  D1 G0 ¨© J1 d1 ¸¹





1 (CT )1

G0 J1*

.

(7.131)

t

Eliminating D1 between (7.131) and (7.123) at level one gives t t t 1 § P 2  P1 · R't 2 ¨ * ¸ l (l  1) P1 G0 ¨© J1 d1 ¸¹ a 2

(CT )1  (CD )1 . G0J1*

(7.132)

At the lower boundary, the finite-difference analog of (7.125) becomes t

t

t P N 1  P N  R't JN* W N dN t

As P N 1

t

Ps

 C T

N

.

Is  RTs*q t , the above equation may be written as t



t P N Is RTs* t   q  R't JN* W N dN dN dN

 C T

N

.

(7.133)

Assuming that

JN* 1 JN*

1 § RTN* TN* 1  TN* ·  ¨ ¸¸ , dN V N ¨© c p ¹

Ts*

§R · d N ¨ TN*  V N JN* 1 ¸ . ¨ cp ¸ © ¹

we get TN* 1

(7.134)

Also, from the definition of pseudo-vertical velocity in layers N  1 and N, we get t

t W N 1 V N 1  V N 1W s

(7.135)

and t

W N Combining (7.135) and (7.136) gives t

W N

t

V N  V N W s .

t

(7.136) t

hW N 1  (hV N 1  V N )W s ,

(7.137)

where h V N / V N 1 . t

Substituting for Ts* from (7.133) and W N from (7.136) into (7.132) and with some further simplification, we get the finite-difference analog of the thermodynamic equation at level V N as t CT t t P N R'T N d N  * h JN* 1W N 1  R'TW s , (7.138) *   O O O*



N

where

N

N

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142

ON* Here 0 d h d 1 and V N

RTN* TN*   hJN* 1V N 1 . cp dN

hV N 1 . t

t

t

A relation can be obtained between W s , W N 1 , and P N using finite-difference analogs of the divergence equation and the pseudo-vertical-velocity equation applied to t t t the V level. Furthermore, a relation between P N 1 , P N , and W N 1 can be obtained by N

the finite-difference analog of the thermodynamic equation at the V N 1 level. t

t t t Thus we have four unknowns, namely P N 1 , P N , W N 1 , and W s , and three t t equations. These three equations can be simplified to eliminate W N 1 and W s to obtain a t

t

single equation involving P N 1 and P N , that is, t t t t 1 §  P N P N  P N 1 · R't 2l (l  1) P N ¨ * ¸  * a2 GN 1 ¨© ON d N ON 1d N 1 ¸¹ 1 § 1  1 CT  * CT ¨  N G N 1 © ON* ON 1





(7.139)

N 1

· ¸  C D ¹



N

,

where

ON* 1

JN* 1ON* . O  hJN* 1 * N

Finally, (7.130), (7.132), and (7.139) define a system of N equations in N unknowns. The coefficient matrix is an N u N tridiagonal matrix and the unknowns are t t t t t t P n for 1 d n d N . Once P n is obtained, one can find D n , W n , W N 1 , and W s . t

Furthermore, q can be obtained from W s using the mass continuity equation. summary of the spectral equations can be found in Appendix B.

7.8

A

Examples

We show here some examples of the quality of forecasts produced by the multilevel spectral model. Overall performance of the model is determined by anomaly correlation. Also we demonstrate the prediction of hurricanes by such models. Anomaly Correlation. We start with an anomaly which is defined as ’Q(t ) Q(t )  Qn , where Q(t ) is a variable and Qn is its climatological mean value. The anomaly correlation, as shown in Figure 7.5, is defined by

A

1 N

N

§ · ¨ ¸ ª 'Q p (t )  'Q p (t ) º ª 'Qo (t )  'Qo (t ) º ¬ ¼¬ ¼ ¨ ¸. 1/ 2 1/ 2 ¨ ¸ 2½ 2½ ­ 1 ­ ¨¨ ® ¬ª 'Q p (t )  'Q p (t ) ¼º ¾ ® ¬ª 'Qo (t )  'Qo (t ) ¼º ¾ ¸¸ ¿ ¯ ¿ ¹ ©¯

¦ n

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143

Figure 7.5. (a) Geopotential height anomaly correlation at (a) 1000 mb and (b) 500 mb and root-mean-square error at (c) 1000 mb and (d) 500 mb for the Northern Hemisphere at model resolutions T-213 and T-106. Source: ECMWF.

Figure 7.6. Predicted positions of a monsoon depression on day five of the forecast are shown by dots. Each dot is the forecast at a particular horizontal resolution ( T  21 , T  31 , T  42 , T  63 , T  106 , and T  170 ) of the global spectral model. The flow field shows the verification chart on day five, based on ECMWF analysis.

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An Introduction to Global Spectral Modeling

Here N denotes the number of cases, the subscript p denotes the predicted state, the subscript o denotes the observed state, and the overbar denotes an ensemble mean of all cases. Figure 7.5 illustrates the anomaly correlation and root-mean-square errors of typical forecasts at the European Centre for Medium-Range Weather Forecasts (ECMWF) in England. Panels 7.5a and 7.5c show the 1000-mb errors, whereas panels 7.5b and 7.5d show the 500-mb errors. Resolution. The impact of resolution on spectral modeling is illustrated here in the context of monsoon forecasts (see Fig. 7.6). Here we show the ECMWF analysis of a tropical depression over northeastern India. This is the flow field at 850 mb for 7 July 1979 at 1200 UTC. The dots shown are in the vicinity of the forecast positions of the center of this storm (assessed from minimum geopotential height at the 850-mb surface). We find that as the resolution was increased from T  21 successively (i.e., T  31 , T  42 , T  63 , T  106 , and T  170 ), the predicted position of the depression at day five of the forecast improved. This illustrates the sensitivity of tropical forecasts to resolution. Extratropical Cyclone. An example of a successful forecast of an extratropical storm is shown in Fig. 7.7. Here the top-left panel (Fig. 7.7a) illustrates the observed sealevel pressure for an extratropical cyclone located southwest of England. Panels 7.7b to 7.7f illustrate forecasts (valid at the same time as panel 7.7a) at the end of day two, three, four, five, and six. These forecasts were experimental and based on a new advancement in the analysis procedure called four-dimensional assimilation (Rabier et al. 1993). It is interesting to note that all of these forecasts were successful in predicting the intense cyclone over southwestern Europe through day six of the forecasts. Hurricane Forecast. An example of hurricane formation is illustrated in Fig. 7.8. This relates to hurricane Frederic of 1979, which formed over the Atlantic Ocean. Here we illustrate the growth of the speed field during a 72-hour forecast. This wind speed at 850 mb is around 6 ms-1 at the start of the forecast. The intensity grows to around 38 ms1 as the hurricane forms by hour 72 of the forecast. This is one of several such forecasts that have been made by research and operational groups. This kind of study requires very high-resolution global modeling (Krishnamurti et al. 1994a).

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Figure 7.7. An example of the prediction of an extratropical cyclone. (a) Observed field of surface pressure (mb). Days two (b), three (c), four (d), five (e), and six (f) of an ECMWF model prediction are valid at the same time as (a).

Figure 7.8. Forecasts of the wind speed (ms-1) at 850 mb made with a global spectral model for hurricane Frederic of 1979 over the Atlantic Ocean. The three-hourly fields cover the period between hours 3 and 72 of a high-resolution ( T  170 ) forecast.

Chapter 8

Physical Processes 8.1

Introduction

In this chapter we present some of the physical processes that are used in numerical weather prediction modeling. Grid-point models, based on finite differences, and spectral models both generally treat the physical processes in the same manner. The vertical columns above the horizontal grid points (the transform grid for the spectral models) are the ones along which estimates of the effects of the physical processes are made. In this chapter we present a treatment of the planetary boundary layer, including a discussion on the surface similarity theory. Also covered is the cumulus parameterization problem in terms of the Kuo scheme and the Arakawa Schubert scheme. Large-scale condensation and radiative transfer in clear and cloudy skies are the final topics reviewed.

8.2

The Planetary Boundary Layer

There are at least three types of fluxes that one deals with, namely momentum, sensible heat, and moisture. Furthermore, one needs to examine separately the land and ocean regions. In this section we present the so-called bulk aerodynamic methods as well as the similarity analysis approach for the estimation of the surface fluxes. The radiation code in a numerical weather prediction model is usually coupled to the calculation of the surface energy balance. This will be covered later in Section 8.5.6. This surface energy balance is usually carried out over land areas, where one balances the net radiation against the surface fluxes of heat and moisture for the determination of soil temperature. Over oceans, the sea-surface temperatures are prescribed where the surface energy balance is implicit. Thus it is quite apparent that what one does in the parameterization of the planetary boundary layer has to be integrated with the radiative parameterization in a consistent manner. 8.2.1

Bulk Aerodynamic Calculations Over Oceans and Land

Here the surface fluxes of sensible heat, water vapor, and momentum are expressed by relations of the type FH U C p CH Va Tw  Ta sensible heat,

146

Physical Processes

147

Fq

U Lq Cq Va  qw  qa

latent heat,

FM

U CD Va Va

momentum.

(8.1)

Va , Ta , and qa are the wind speed, the air temperature, and the specific humidity, respectively, at the anemometer level. Tw and qw denote the surface temperature and the saturation specific humidity at the surface. The currently accepted values of the dimensionless exchange coefficients are

CH

1.4 u103 ,

Cq

1.6 u 103 ,

and CD

1.1u 103 .

These were determined experimentally during the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE). The choice of units is quite important for the calculation of these fluxes. For most meteorological purposes it is desirable to express FH and Fq in units of W m-2, while FM is usually expressed in the familiar units of dynes cm-2. For these units, one can simplify the bulk formulas (using U 1.23 u103 g cm-3) to read FH

1.72 Va Tw  Ta ,

Fq

4.9 u 103 Va  qw  qa ,

FM

2

(8.2)

2

1.35 u 10 Va .

Here the wind speed is measured in m s-1, temperature in K or °C and specific humidity in g kg-1. In situations of strong wind speed, the ocean waves exert a large drag on the air, and one finds it desirable to allow for a variation of drag coefficient as a function of wind speed. The variation of drag as a function of wind speed has been expressed by CD

1.1u 103

for

V  5.8 m s-1,

CDO  0.74  0.046V

for

5.8 d V d 16.8 m s-1,

CDO  0.94  0.034V

for

CDO

V ! 16.8 m s-1.

In dealing with tropical storms and hurricanes, a variation of drag as a function of wind speed is generally invoked. The variation of the surface drag coefficient as a function of wind speed is also formally described in the section on the roughness parameter, z0 , described below. 8.2.2. The Roughness Parameter z0

Although z0 varies in space, it should be regarded as a function of time as well. Over oceans under conditions of strong winds, the wave drag can be quite large. Given a domain of calculation 0 d x d L and 0 d y d M , one needs a tabulation of a land-ocean matrix IL and a tabulation of orographic height h. Let us suppose IL 0 over oceans and IL 1 over land. The tabulation of h can be obtained from data centers that provide such information.

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148

Over Oceans. The well-known Charnock formula (Charnock 1955) is an accepted method for defining z0 over oceans, that is,

z0

M

u *2 , g

(8.3)

where M is a constant and has a value around 0.04. One cannot calculate z0 from this formula until u * , the friction velocity, is known. That, as we will show, appears in the similarity analysis and requires a knowledge of z0 . Thus an iteration procedure is required to solve for z0 and u * successively. Note that u *2

u ' w '

W0 , U0

(8.4)

where W 0 is the surface stress and U0 is the density of air. A first-guess field of u * can be obtained from the bulk aerodynamic representation, noting that u ' w ' represents the vertical eddy flux of momentum. Given a first-guess field of z0 , one next goes to the similarity approach (described below) to determine the surface fluxes which define u * , and in turn define the final value of z0 , the roughness parameter.

Over Land. Over land, an empirical method described by Manobianco (1989) allows for the variation of the roughness parameter as a function of the elevation based on the mesoscale variance of the mountain heights. In its simplest form, it is expressed by the relation z0 15   473.6  0.0368h u106 , (8.5)

where units of the grid-scale mountain height h and of the roughness parameter z0 are cm. For numerical weather prediction, it is desirable to restrict the upper limit of z0 to 4000 cm. 8.2.3

Surface Similarity Theory

The basis of the similarity analysis presented here follows planetary boundary-layer observations, e.g., Businger et al. (1971). According to these observations, nondimensionalized vertical gradients of large-scale quantities such as wind, potential temperature, and specific humidity can be expressed as universal functions of a nondimensional height z / L , where z is the height above the earth’s surface and L is the Monin-Obukhov length, which is defined by L



u *2

NET *

.

(8.6)

Physical Processes

149

Here E g / T 0 , where T 0 is a reference temperature, u * is the friction velocity, T * is a characteristic temperature, and N is the von Karman constant. These are usually expressed by the following relations: Nondimensional shear: N z wu Im  z / L . (8.7) u * wz

Nondimensional vertical gradient of potential temperature:

N z wT T * wz

Ih  z / L .

(8.8)

Nondimensional vertical gradient of specific humidity:

N z wq q* wz

Iq  z / L ,

(8.9)

where q* is a characteristic specific humidity. Here u * , T * , and q* are related to the surface fluxes by the expressions:

u *2

u ' w ' |0 ,

(8.10)

* *

uT

T ' w ' |0 ,

(8.11)

u *q*

q ' w ' |0 .

(8.12)

The right-hand sides of the above equations are eddy correlations estimated over a period of time. This defines the variables u * , T * , and q* . The empirical fits of the boundary-layer observations are usually separated in terms of stability. Stability is usually expressed as a function of the sign of the MoninObukhov length L, or of the bulk Richardson number Ri B . The definition of stability is based on surface heat flux. Unstable (heat flux up), stable (heat flux down), or neutral (heat flux = 0) correspond to L  0 , L ! 0 , and L 0 , respectively. Note that u * is a velocity and is always positive definite. Hence, this implies that for an unstable case ( L  0 ) T * is always positive and vice versa for the stable case. We next express this in terms of the bulk Richardson number, which is the ratio of stability to the square of wind shear. Here

Ri B

dT E dz 2 . § du · ¨ ¸ © dz ¹

(8.13)

For a stable surface layer, the heat flux is downward, in other words, L ! 0 or RiB ! 0 . For an unstable surface layer, the heat flux is upward, that is, L  0 or RiB  0 . For a neutral surface layer, there is no vertical heat flux, that is L 0 or RiB 0 . Since a priori the Monin-Obukhov length is an unknown quantity, stability is assessed from the

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150

sign of the bulk Richardson number Ri B . Empirical fits of the boundary-layer observations (Businger et al. 1971) for the unstable and stable cases may be expressed following Chang (1978) by nondimensional relations such as

N z wu

1/ 4

u wz

z· § ¨1  15 ¸ L © ¹

N z wT T * wz

z· § 0.74 ¨1  9 ¸ L © ¹

N z wq

z· § 0.74 ¨1  9 ¸ L © ¹

*

*

q wz

,

(8.14) 1/ 2

,

(8.15)

1/ 2

(8.16)

for the unstable case and

N z wu u * wz N z wT T * wz N z wq q* wz

z , L z 0.74  4.7 , L z 0.74  4.7 L

1.0  4.7

(8.17)

(8.18)

(8.19)

for the stable case. The above six relations were obtained from a least-squares fit of observations. For both the stable and unstable cases, the definition of the Monin-Obukhov length yields a fourth equation, z zNET *  *2 . (8.20) L u The four equations (for the stable or the unstable case) need to be solved for the four variables u * , T * , q* , and the Monin–Obukhov length L. The surface fluxes of momentum, heat, and moisture are defined by FM u *2 { u ' w ' |0 ,

FH Fq

u *T * { T ' w ' |0 ,

(8.21)

* *

u q { q ' w ' |0 .

The solution procedures followed by different investigators for the aforementioned equations vary somewhat. We illustrate here a method developed in our studies with a global spectral model (Krishnamurti et al. 1983, 1984). A two-level surface-layer representation is convenient for the evaluation of surface fluxes. Let z1 and z2 denote two levels, where z1 z0 (the roughness length), z2 is the top of the surface layer, and 'z z2  z1 is roughly 10 to 50 meters, which is considered the depth of the constant flux layer. At these two levels, the respective winds, potential temperature, and specific humidity may be denote by u1 , v1 , u2 , v2 , T1 ,T 2 , and



 q , q . These are known quantities. 1

2







151

Physical Processes

We next write the bulk Richardson number in finite-difference form, that is,

E

Ri B

T



 T1 'z

2

2

u2  v2 2

.

(8.22)

Note that u1 v1 0 . The solution for the unstable case is more complicated because of the fractional negative powers in (8.14) to (8.16). Following Chang (1978), we can express the two-level representation of these flux relations for the unstable case by

z2 L

· ln  z2 / z1 \ 1 z2 § ¨ ¸, 'z ¨© 0.74 ª¬ln  z2 / z1 \ 2 º¼ ¸¹

Ri B

(8.23)

where \ 1 and \ 2 are functions of L and are defined as

ª§ 1  · 2 1  2 º 2 2 »  2 tan 1 2 2 tan 1 1 , ¸ 2 «¬© 1 1 ¹ 1 1 »¼ 1 J 2 \ 2  z2 / L, z1 / L 2 ln . 1 J1

\ 1  z2 / L, z1 / L ln Ǭ

Here 1/ 4

§ z /L· 1 ¨1  15 2 ¸ , z 2 / z1 ¹ © 1/ 4

z · § 2 ¨1  15 2 ¸ L¹ ©

, 1/ 2

§

J 1 ¨1  9 ©

z2 / L · ¸ , z2 / z1 ¹

and 1/ 2

z2 · § ¨1  9 ¸ L¹ ©

J2

.

Integrating (8.14) from z1 to z2 , we obtain

³

u2

u1

N u

*

du

³

z2

z1

1 § 15 z · ¨1  ¸ z© L ¹

1/ 4

dz ,

or

N u

*

(u2  u1 )

z2

³

z1

³

z2

z1

ª 1 1 1 § 15 z · 1/ 4 º «   ¨1  ¸ » dz L ¹ ¼» ¬« z z z ©

1 dz  z

³

z2

z1

1 ª § 15 z · «1  ¨1  ¸ z ¬« © L ¹

1/ 4

º » dz ¼»

(8.24) (8.25)

152

An Introduction to Global Spectral Modeling

§z · §z z · ln ¨ 2 ¸  \ 1 ¨ 2 , 2 ¸ . z © 1¹ © L z1 ¹

(8.26)

Similar vertical integration of (8.15) and (8.16) results in





N T 2  T1 0.74T *



§z · §z z · ln ¨ 2 ¸  \ 2 ¨ 2 , 2 ¸ , z © 1¹ © L z1 ¹

(8.27)

§z · §z z · ln ¨ 2 ¸ \ 2 ¨ 2 , 2 ¸ . z © 1¹ © L z1 ¹

(8.28)



N q2  q1 0.74q* Here

\2

³

\1

³

1 ª § 9z · «1  ¨1  ¸ z ¬« © L¹

z2

z1

z2

z1

1/ 2

1 ª § 15 z · «1  ¨1  ¸ z ¬« © L ¹

º » dz , ¼»

1/ 4

º » dz . ¼»

Equations (8.14), (8.15), (8.16), and (8.20) are solved for the variables L, u * , T * , and q rather simply. The variation of the Monin-Obukhov length is monotonic with respect to u * and T * (Businger et al. 1971). A simple linear incremental search of L in (8.20) provides a rapid solution to the desired degree of accuracy. After substitution for L on the right-hand side of (8.14), (8.15), and (8.16), one obtains the corresponding solution for u * , T * , and q* . The solution for the stable case is relatively straightforward and requires a sequential solution of four linear algebraic equations. One of these is *

z2  z1 L

§ z · 9.4Ri B  0.74  (4.89Ri B  0.55)1/ 2 ln ¨ 2 ¸ . 9.4  44.18Ri B © z1 ¹

(8.29)

Equation (8.29) is obtained from eliminating u * and T * from vertically integrated forms of (8.17), (8.18), and (8.20) within the constant flux layer. The finite-difference forms of these three vertically integrated equations are



N u2  u1 u



N T 2  T1 T

*

*





§z · z z ln ¨ 2 ¸  4.7 2 1 , L © z1 ¹

§z · z z 0.47 ln ¨ 2 ¸  4.7 2 1 , L © z1 ¹ * z2  z1 ( z2  z1 )NET . L u *2

(8.30)

(8.31) (8.32)

The lower level z1 is identified with the roughness length z0 . At these two levels, one needs to define wind, potential temperature, and specific humidity in order to carry

Physical Processes

153

out the desired computations. At the lower level z1 we set all wind ( u1 , v1 ) to zero. The temperature T1 at the lower level is set to the sea-surface temperature. Over land surfaces, T1 is determined from surface energy balance, which is described below.

Furthermore one sets T1

T1 , since the surface pressure is close to 1000 mb. The specific

humidity q1 at the lower level over the ocean is set to the saturation value at temperature T1 . The upper-level height z2 is set to a value of 10 meters above the surface. The wind components u2 and v2 are usually interpolated using a log-linear profile, while the temperature T2 and the specific humidity q2 are linearly interpolated between 1000 and 850 mb from the analyzed data. The numerical model algorithm requires information at the top of the constant flux layer, which is usually obtained by interpolation of large-scale atmospheric information between 1000 and 850 mb. The von Karman constant N has a value of 0.35. E stands for g / T 0 where T 0 is a constant reference potential temperature. Since Ri B is know from large-scale data sets, this sequence of calculations yields L, u * , T * , and q* . A rigorous comparison of the aforementioned method was made with respect to direct observations of fluxes obtained from the so-called eddy correlation method. This was done using GATE observations, and results were in close agreement. The virtue of this method over the bulk method lies in the dependence of the implicit bulk coefficients on stability and shear. These are expressed by the following equivalent bulk coefficients: CD CT

Cq

N2

> ln ( z2 / z1 )  (4.7'z ) / L@

2

,

N 2 1 >ln ( z2 / z1 )  (4.7'z ) / L @ >0.74 ln ( z2 / z1 )  (4.7'z ) / L@

for the stable case and CD CT

Cq

N2 2

>ln ( z2 / z1 ) \ 1 @

,

N 2 0.74 > ln ( z2 / z1 )  \ 2 @ [ln ( z2 / z1 )  \ 1 ]

for the unstable case. The neutral cases are usually calculated in the same manner as the stable situations. 8.2.4

Vertical Disposition of Surface Fluxes

The vertical disposition of eddy fluxes of momentum, heat, and moisture above the surface layer is based on K-theory. This theory uses an eddy diffusion coefficient K that depends on the mixing length l, the vertical wind shear, and the stability of the atmosphere as determined by the bulk Richardson number Ri B . The eddy diffusion coefficient for heat and moisture is

An Introduction to Global Spectral Modeling

154

KH

l2

KQ

w|V | Fh  Ri B , wz

(8.33)

and for momentum it is given by l2

KM

w|V | Fm  Ri B . wz

(8.34)

Here, l is the mixing length based on a formula given by Blackadar (1962), i.e., l

Nz , 1 N z / O

where N is the von Karman constant, taken as 0.35, and O is the asymptotic mixing length, which is set to 450 m for heat and moisture exchange and 150 m for momentum exchange. Following Louis (1979), Fh and Fm are given as Fh

Fm

1 , (1  5Ri B ) 2

Ri B t 0

(8.35)

for the stable case and Fh

1  1.286| Ri B |1/ 2 8Ri B 1  1.286| Ri B |1/ 2

Ri B  0 ,

(8.36)

Fm

1  1.746| Ri B |1/ 2 8Ri B 1  1.746| Ri B |1/ 2

Ri B  0

(8.37)

for the unstable case. Here, the bulk Richardson number over an atmospheric layer is given by wT g wz Ri B . (8.38) T w |V | 2 wz It can be shown that Ri B d 0.212 is a physically valid value. The time tendency due to diffusive fluxes at any level is given by wW wt

where W

1 w §

U wz ¨©

UK

wW · ¸, wz ¹

(8.39)

u , v, T , or q , and U is density of air. In V coordinates, (8.39) takes the form

wW wt

g 2 w § 2 wW · ¨U K ¸. ps 2 wV © wV ¹

(8.40)

The upper and lower boundary conditions for (8.40) are wW / wV 0 at the top and wW / wV the boundary-layer fluxes at the top of the constant flux layer at the bottom. Using these boundary conditions, (8.40) is solved implicitly, where it takes the finite-difference form

Physical Processes

155

W (t  't )  W * (t  't ) 2't

g2 w § wW (t  't ) · ¨ UK ¸, ps2 wV © wV ¹

(8.41)

where W * (t  't ) is the value of u , v, T , or q before vertical disposition of the fluxes at time (t  't ) , and W (t  't ) is the value after vertical disposition of the fluxes. The solution of (8.41) is obtained by writing the equation in the form A W (t  't ) W * (t  't ) ,

where A is a tridiagonal coefficient matrix and W (t  't ) and W * (t  't ) are vectors

defining the values of W (t  't ) and W * (t  't ) at different levels.

8.3

Cumulus Parameterization

The scale of cumulus clouds is much smaller than the scale of the grid squares (or the smallest resolvable scale) of a numerical weather prediction model. The individual cumulus cloud has a scale of a few kilometers, whereas the model grid square is more like a few hundred kilometers. Within larger synoptic-scale disturbances often reside organized mesoscale convective systems with scales of the order of a few hundred kilometers. These are the mesoscales over which the clouds show organization. Thus the interaction of cumulus clouds with the broad synoptic scale appears to be an important multiscale problem. Cumulus parameterization addresses the effects of the cumulus scale on the resolvable scale, given the information on the latter scales. Minimally, one needs to derive three parameters from a cumulus parameterization scheme. These are the vertical distribution of both heating and moistening, and the rainfall rates. We first outline a version of Kuo’s scheme which has been modified in recent years by Krishnamurti et al. (1983) to provide successful forecasts of the life cycle of tropical cyclones. We follow that with a scheme based on the work of Arakawa and Schubert (1974), which has also been used successfully by several research scientists and numerical weather prediction centers. 8.3.1

Kuo’s Scheme

According to Kuo, organized cumulus convection requires the presence of conditional instability and a net supply of moisture. In the earlier formulation of Kuo (1965), in those regions where the two conditions were met the following form of the moisture equation was used, i.e., q q wq , (8.42) a s wt 'W where 'W is a cloud time-scale parameter and a denotes the fraction area of the grid box that would be covered by the newly formed convective clouds. The parameter a is defined by the relation

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156

G w § · ¨ ’ ˜ qV  wp qZ ¸ dp © ¹ . 1 pB § c p (Ts  T ) qs  q · dp  ¨ ¸ g pT © L'W 'W ¹ 

a

1 g

³

pB

pT

(8.43)

³

In the following we use the symbol Q for the denominator of (8.43). In this relation the denominator may be interpreted as the amount of moisture supply needed to cover the entire grid-square area by a model cloud (a local moist adiabat Ts , qs ). The numerator, on the other hand, is a measure of the available moisture supply. The total convective rainfall rate is expressed by

1 g

PT

³

pB

ac p (Ts  T ) L'W

pT

dp .

(8.44)

It should be noted that the definition of a as given by (8.43) and the parameterization of the moisture in (8.42) are consistent with the principle of conservation of moisture. This can be shown by considering the moisture conservation law in the form G w wq (8.45) ’ ˜ qV  qw  E  P . wt wp

After integration of (8.45) from pT to pB with the assumption that evaporation E occurs only at the air-sea interface (i.e., below pB ), we obtain 1 g

³

pB pT

wq dp wt

I  PT ,

(8.46)

where PT is the total precipitation rate for the vertical column and I is the moisture convergence rate at the grid point. Noting that I

aQ

§1 T T 1 qs  q · a¨ cp s dp  dp ¸ L'W g 'W ©g ¹

³

³

(8.47)

and substituting for PT from (8.44) into (8.46), we obtain 1 g

³

pB pT

wq dp wt

1 a g

³

pB pT

qs  q dp . 'W

(8.48)

Hence the use of (8.43) is consistent with the moisture conservation law (8.45). We furthermore note that part of the moisture convergence I is being used for raising the level of moisture, and part of it goes into condensational heating. This partitioning may be expressed by the relation I

I q  IT .

(8.49)

We note that in general I q is larger than IT initially. However, as q tends to qs , Q becomes small; as a consequence, a becomes very large. From then on, IT starts to

Physical Processes

157

become very large and I q tends to zero. This leads to a slight computational instability at a few grid points due to the very large heat release and very strong resulting vertical motions. This defect is primarily related to a disproportionate partitioning of I into I q and IT . Kuo applied this early formulation of the cumulus parameterization scheme to hurricanes for which it worked well. However, it was found that it did not succeed for modeling other large-scale disturbances. This scheme produces too much moistening, too little rain, and too little heating. In a high-resolution model, such as used to model a hurricane, there is a great deal of moisture convergence and hence more rain and heating. However, in a coarser resolution model, there is less moisture convergence and therefore less heating and rain. Now we consider the parameterization of deep, moist convection with a modified Kuo scheme. The large-scale convergence of the flux of moisture is expressed by the relation G wZ q CM ’ ˜ V q  , wp or in advective form, G wq . (8.50) CM V ˜’q  Z wp

In the Kuo type of cumulus parameterization schemes, the supply of moisture is usually defined from a vertical integral of the above expression. The supply is then used to define the moist adiabatic cloud elements in various versions of Kuo’s scheme. Krishnamurti et al. (1983) and Anthes (1977) have noted that the supply of moisture for the definition of clouds may be expressed simply by the second term in (8.50). The first term, namely the horizontal advection, is used for a direct moistening of the air as a large-scale advective term. Thus we define the supply of moisture by

IL

1 g

³

pB pT

Z

wq dp . wp

(8.51)

Here pT and pB denote the cloud top and the cloud base, respectively. They are defined in terms of the vanishing buoyancy level and the lifting condensation level, respectively. From our experience (Krishnamurti et al. 1980, 1983), we have noted that the above definition for the large-scale supply is a close measure of the rainfall rate, and thus sufficient supply is not available to account for the observed moistening of the vertical columns. These statements are based on semiprognostic studies made with GATE observations. This research leads us to propose a mesoscale convergence parameter K and a moistening parameter b I I L 1  K , (8.52)

where I LK denotes the net mesoscale moisture supply. The total moisture supply is partitioned into a precipitation part and a moistening part via the following relations:

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158

R

I 1  b

I L 1  K 1  b ,

M

Ib

I L 1  K b .

(8.53)

Following Krishnamurti et al. (1983), we define the total supply of moisture required to produce a grid-scale moist-adiabatic sounding by Q

1 g

³

pB pT

qs  q 1 dp  'W g

³

pB pT

c pT wT · § c pT (T s  T ) Z ¨ ¸ dp . L T ' W LT wp ¹ ©

Here 'W denotes a cloud time-scale approximately equal to 30 minutes. respective terms are denoted by Qq and QT , so that Q Qq  QT .

(8.54) The two (8.55)

Note that equation (8.54) differs from the definition of Q in the classic Kuo scheme. The last term on the right-hand side has been added and it is the amount of energy needed to overcome adiabatic cooling. In the classic Kuo scheme when saturation was reached Q o 0 so that a I / Q would blow up. The new term was added so that if saturation is reached Q would tend toward the value of this term. The total supply I may likewise be split into moistening and heating parts by Iq

Ib

I Lb 1  K

(8.56)

I L 1  b 1  K .

(8.57)

and IT

I 1  b

The temperature and moisture equations are expressed by wT G wT  V ˜’T  Z wt wp wq G  V ˜’q wt

§ T T wT · Z aT ¨ s ¸, wp ¹ © 'W q q aq s , 'W

(8.58)

(8.59)

where aT , the fractional area of a cloud that releases latent heat, and aq , the fractional area of a cloud that moistens, are defined by aT

IT QT aq

I (1  b) QT Iq Ib Qq

Qq

I L (1  K ) (1  b) , QT

(8.60)

I L b(1  K ) . Qq

(8.61)

The parameterization is closed if b and K are somehow determined. Then aT and aq may be evaluated from (8.60) and (8.61). Note that QT and Qq are known quantities. Krishnamurti et al. (1983) proposed a closure for b and K based on a screening multiregression analysis of GATE observations. Here they regressed normalized heating R / I L and moistening M / I L against a number of large-scale variables. From GATE

Physical Processes

159

observations they noted significant correlations for the heating and moistening from the following relations. M a1]  b1Z  c1 , (8.62) IL R a2]  b2Z  c2 . (8.63) IL

Here ] is the relative vorticity of the lower troposphere, Z is a vertically averaged vertical velocity, and a1 , b1 , c1 , a2 , b2 , and c2 are regression constants whose magnitudes may be found in Krishnamurti et al. (1983). Thus in numerical weather prediction, ] and Z determine M / I L and R / I L since M IL

b (1  K )

(8.64)

(1  K ) (1  b) .

(8.65)

and R IL

These two relations determine b and K. We can find aT and aq from I L (1  b) (1  K ) QT

aT

R QT

(8.66)

and I L b(1  K ) Qq

aq

M . Qq

(8.67)

It is also of interest to note that the apparent heat source Q1 and the apparent moisture sink Q2 for this formulation are expressed by Q1

§ T Ts T T wT · T  Zcp  c  HR  Hs , aT ¨ c p T wp ¸¹ p T © T 'W §q q wq · Q2  Laq ¨ s Z ¸, wp ¹ © 'W

(8.68)

(8.69)

where H R is the total radiative potential temperature rate of change and H s is the vertical sensible heat flux by subgrid-scale motions. The total rainfall is given by P 8.3.2

1 g

³

pB pT

aT

cp T LT

§ Ts T wT · Z ¨ ¸ dp . wp ¹ © 'W

(8.70)

The Arakawa–Schubert Cumulus Parameterization

This scheme is considered to be the best scheme i