Mathematical Models in Environmental Problems (Studies in Mathematics and its Applications)

Mathematical Models in Environmental Problems

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 16

Editors: J. L. LIONS, Paris G. PAPANICOLAOU,New York H. FUJITA, Tokyo H. B. KELLER, Pasadena

NOKTFI-HOLLAND -AMSTERDAM

0

N E W YORK

OXFORD .TOKYO

IN ENVIRONMENTAL PROBLEMS G. I . MARCHUK Academician Head ofthe Computer Mathematics Departmcwt of the U.S. S. R. Academy of Scirnce Moscow, u.s.s.R.

I Y 8b NORTH-HOLLAND -AMSTERDAM

NEW YOKK

O X F O l i L ~* 7 ' O K Y O

1SBN:O 444 87965 x 7r~itrsl~iiioti OJ Matematicheskoye Modelirovanye v Prohlyem'e Okrazhayouschchey Sredy "Nauka. Moscow. 19x2 Pitblrshrrs:

F.I,SEVIEK SCIENCE PUBLISHERS B.V. P.O. BOX 190 I 1000 B Z AMSTERDAM T H E NETHERLANDS S d i , dr.strihrtior.s for the iJ. S . A . u i r d ('nnorlu.

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Library of Congress C,ltalogiiig-inPublicatioii h t a M a r c h u k , G . 1. ( C u r i ? I ~ ~ ~ n o v i c l 1i 9) 2, 5 M.i t hcma t i c 1 m o d c I s i n c n v i ironme n t J 1 p r o b I ems

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( S t u d i e s i n m a t h e m a t i c s ,rnd i t s , i p p l i c . i t i o n s ; v . 1 6 ) T r < i n s l , t t i o n of: M a t t y a t i c h e s k o e m o d r L i r o v , i n i e v p c o h I erne o k r LIZ h a ?u s h c tic- i s re, d y H i 0 Ii og r.ip1iy : p 1 I: n v i r n n m c i i I 1 p r n t cx c t 1on--Ma t 1 i c . n ~t i c ,i1 m o d e 1s . I. T i t l e . 11. Series. TD170.2.M3713 1986 361.7'00724 87-3 I I 6 0 ISliN 0 - 4 4 4 - 8 7 9 6 5 - X

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P l i l N IF I > I N I H t N F I H t l i l ANDS

3

LIST OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 . BASIC EQUATIONS OF TRANSPORT AND DIFFUSION . . . . 1.1. Equation Describing Pollutant Transport in Atmosphere . Uniqueness of the Solution . . . . . . . . . . . . . . . 1.2. Stationary Equation for Substance Propagation . . . . . . 1.3. Diffusion Approximation. Uniqueness of the Solution . . 1.4. Simple Diffusion Equation . . . . . . . . . . . . . . . 1.5. Transport and Diffusion of Heavy Aerosols . . . . . . .

.

. . . . . .

. . . . .

. . . . .

PREFACE

INTRODUCTION

1.6. The Structure and Simulation of Turbulent Motions in the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2

.

ADJOINT EQUATIONS OF TRANSPORT AND DIFFUSION

2.1. An Adjoint Equation f o r a Simple Diffusion Equation

. . . . . . .

. . . . .

5

7 10 10

13 18

26 30

. .

32

. . . .

36

2.2.A General Case of an Adjoint Problem for a Three-Dimensional Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 49

2.3. Uniqueness of the Solution for the Adjoint Problem

56

2.4. Adjoint Equation and Lagrange Identity

. . . . . . . . . . . . . . . . . . . .

59

. . . . .

62

Chapter 3 .

NUMERICAL SOLUTION OF BASIC AND ADJOINT EQUATIONS

. . . . . . . . . . 3.2. Splitting of a Problem According to Physical Processes . 3.3. Equation of Motion . . . . . . . . . . . . . . . . . . . . 3.4. Diffusion Equation . . . . . . . . . . . . . . . . . . . . 3.5. General Numerical Algorithm . . . . . . . . . . . . . . 3.6. Numerical Solution of Adjoint Problems . . . . . . . . . 3.1. Elements of General Splitting Theory

. . . . . . . . . . . . . . . . . .

3.7. Comments on Difference Approximation of Basic and Adjoint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 4.

. .

63 74 82 100

. .

. .

103 104 105

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

4.2. Adjoint Equations and Optimization Problem . . . . . . . . . . .

113

4.3. Multicriterional Optimization Problem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

4.4. Minimax Problem

121

OPTIMUM LOCATION OF INDUSTRIAL PLANTS

4.1. Statement of the Problem

4.5. A Generalized Optlmization Problem of Industrial Plant Location . . . . . . . . . . . . . . . . . . . . . . . . 4.6.

107

. . . .

123

Some General Remarks . . . . . . . . . . . . . . . . . . . . . .

12 1

List of Contents

4 Chapter 5 .

ECONOMIC CRITERIA OF PLANNING. PROTECTION AND RESTORATION 129 OFENVIRONMENT . . . . . . . . . . . . . . . . . . . .

5.1. Value of Biosphere Products Losses. due to Environmental Pollution with Industrial Emissions . . . . . . . . . . 5.2. Value of Losses Due to Atomospheric Pollution with Multicomponent Aerosol . . . . . . . . . . . . . . . .

. .

....

129 131

5.3. Economic Aspects of Natural Resources Depreciation when Ecological Conditions are Disturbed Due to Environmental Pollution . . . . . . . . . . . . . . . . . . . . . . . . .

133

5.4. General Economical Criterion

136

Chapter 6 .

. . . . . . . . . . . . . . . . .

MATHEMATICAL PROBLEMS OF OPTIMIZING EMISSIONS FROM OPERATING INDUSTRIAL PLANTS . . . . . . . . . . .

. . . . . . . . 6.2. Optimization by Basic Equations . . . . . 6.3. Optimization by an Adjoint Problem . . . 6.4. Perturbation Theory . . . . . . . . . . . . 6.1. Statement of the Problem

Chapter 7.

ACTIVE AEROSOL EMISSIONS

. . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Dimensional Approximation . . . . . . . . . . . . . . . .

145 145 146 148 152 154

7.1. Basic and Adjoint Equations

154

7.2. Influence Function

159

7.3.

160

Chapter 8 .

MODELLING THE LOCATION OF POLLUTION SOURCES IN WATER BODIES AND COASTAL SEAS . . . . . . . . . . . . . .

. .

164

8.1. Basic Equations

164

8.2.

. . . . . . . . . . . . . . . . . . . . . . Adjoint Equations . . . . . . . . . . . . . . . . . . . . . Finite-Difference Approximations . . . . . . . . . . . . . .

166

8.3.

.................... Appendix . MESOMETEOROLOGICAL AND MESOOCEANIC PROCESSES . . . . . . 1 . Mesometeorological Problem of Determining Local Atmospheric Circulations . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Quasihomogeneous Ocean Layer . . . . . . . . . . . . 8.4. Finite Element Method

REFERENCES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 172 184 184 206

215

5

PREFACE

In the past few years, environmental protection has become a challenging scientific task whose importance is highlighted by ever-increasing pace of technological progress throughout the world. The swift industrial development resulting in increased level of industrial pollution of the environment has already begun disturbing the ecological equilibrium in many regions of the globe. Meanwhile industry continues to develop at unprecedented rates, giving a powerful impetus to research associated with the location of new industrial plants and complexes exerting minimum impacts on the environment. The problem of environmental contamination by industrial plants whose maximum permissible level of safe pollution is still inconsistent with current requirements has become even more pressing. All this refers equally to the processes occurring on land and in the ocean. Environmental protection problems have been taken up in a series of investigations carried out in the Soviet Union, specifically at the Chief Geophysical Observatory of the State Hydrometeorological Committee, as well as abroad. Selective information on such investigations can be gained from the references at the end of this book. In the present monograph,special attention is paid to mathematical modelling of optimization problems associated with environmental protection. These problems were first posed by the author in 1970 at the International Environmental Protection Symposium, which was held in Czechoslovakia (RudohoGi). The author's talk at this Symposium served a starting point for his further research in the field, which was reported subsequently at international symposia in Italy (Rome, 1973), France (Nice, 1975), and FRG (Wurzburg, 1977). These findings provided the basis for the monograph. A substantial amount of research along these lines has been accomplished by staff of the Computer Center of the Siberian D i v i d m of the USSR Academy of Sciences. An active role in these efforts belongs to V.V.Penenko, N.N.Obraztsov, V.I.Kuzin, A.E.Aloyan, E.A.Tsvetova, and some other scientists. Much work has been done by the research probationer A.Yu.Sokolov, who computed examples illustrating the opportunities provided by the methods. Besides, the monograph draws on the calculations conducted by A.E.Aloyan, A.A.Kordzadze, V.I.Kuzin, N.N.Obraztsov, and V.A.Sukhorukov, to whom the author expresses his deep gratitude. In the present book, the author restricts himself to the study of direct environmental impact, leaving aside the problem of climatic fluctuations caused by manmade factors, which seems to be of interest in its own right. This latter issue is expected to be treated in a specific monograph which is being prepared at the Computer Center of the Siberian Division of the USSR Academy of Sciences.

6

Preface

Much of the monograph's text was edited by N.N.Obraztsov to whom the author pays special tribute. The book was primarily written by the author during his staying on vacation in the Turkmen Soviet Socialist Republic, where the environmental protection problems are being intensely studied in connection with major industrialization and irrigation projects for desert and arid regions of the republic. The author is grateful to M.G.Gaporov, Ch.S.Karriyev, and A.G.Babayev for many helpful discussions of these issues.

G.I.MARCHUK

7

INTRODUCTION

The rapid industrial development all over the world has posed an acute problem before the mankind striving to preserve the ecological systems that have formed historically in various regions of the globe. Local pollution caused by industrial emissions in many cities of the world have long surpassed the maximum permissible values

of safe standards. The gigantic scale of work associated with

the mining of coal, ferrous ores, non-ferrous metals and other mineral resources has resulted in erosion and contamination of vast expanses of land. Freon discharges from industrial and domestic refrigerators exert adverse effect on ozone layers of the atmosphere. Increased concentration of carbon dioxide as a result of burning a great amount of hydrocarbons

needed for large-scale power generation

has begun to tell on the heat balance of the globe. Volcanic eruptions change the optic and radiation properties of the

atmosphere. All this has led to the global

disturbance of the existing ecological systems. Therefore, an important task of contemporary science is to forecast changes in the ecological systems under the impact of natural and anthropogenic factors. First we should investigate the process of environmental pollution with industrial emissions and wastes and then assess the impact of noxious contaminations on the biological environment. The present monograph deals with an estimation of atmospheric pollution and contamination of the underlying surface with passive and active pollutants. A passive pollutant is taken to mean here the pollutant which underwent no changes up to its fall on the earth surface. Conversely, a pollutant will be called active if in the course of its spreading in the air it enters into chemical reactions with water vapors or other atmospheric components or passes from one chemical state to another. Industrial emissions spread in the air due to their advective transfer by air masses and diffusion caused by turbulent gusts of the air. If one watches a torch of smoke over a chimney, one may notice,firstly, that the smoke is being carried away by air and,secondly, that it grows larger and larger due t o minor turbulences asthc distance from the chimney is increasing. As a result the torch of smoke acquires the shape of an elongated cone expanding toward the direction of the air mass movement. Then under the effect of major turbulent fluctuations, the torch starts disintegrating into spaced whirl formations which are carried away far from its source.

8

Introduction If pollutantsin the atmospheric emissions consist of large particles, they

in the course of their spreading in the air begin to fall down by gravity with a constant velocity in compliance with Stock's law. Naturally, practically all the pollutants sooner o r later settle on the earth surface, heavy particles fall down mainly due to the gravitation field, whereas light ones are precipitated by diffusion. Gravitation flow of heavy particles by far surpasses that of diffusion, whereas for lightpollutants it is vertually insignificant. Since gaseous pollutants of oxide type pose a greater danger to the environment, it is the light compounds that the author treats at length in the monograph. Apart from minor diffusions dispersing torch pollutants much attention in the pollution dissipation theory is given to fluctuations of velocity and direction of wind over a long period of time, generally about one year. Over this period, the air masses carrying pollutants away from the source repeatedly change

their

direction and velocity. Statistically such many-year observations are generally described by a special diagram called the wind rose in which the size of the vector is proportional to the number of recurrent events connected with the movement of air masses toward a given direction. This means that the longer the vectors in the wind rose diagram the more often the air masses change their movement in a given direction. Thus, maxima in the wind rose diagram correspond to winds predominant in the given region. This information is used as a starting point for planning new industrial projects, but it proves insufficient in drafting plans for the location of industrial plants among a great number of ecologically important zones (human settlements, recriation zones, agricultural and forest lands, etc.) since each of them has a maximum permissible dose of pollution of its own. Maximum permissible doses should, of course, take into account pollution from already existing industrial plants in the given region, therefore, while planning the establishment of a new industrial unit, limitations by safe standards should be laid down with reference to actual pollution from operating plants. The monograph describes a method for preliminary calculations of regions for a possible location of industrial plants meeting safe pollution requirements for all ecologically important zones. To this end adjoint

equations of substance

transfer and diffusion have been introduced as a basis for solving the problem in question. The physical solution of adjoint

problems is a function of impacts

o r a function of sensitivity with regard to the main functional of the problem.

Specifically such a functional may be represented by the number of pollutants settled down inanecological zone during one year or permissible hazardness of pollutants in both settled and suspended states.These problems are considered in chapter 4 in which findings are summed up for cases when there is a wider set of functionals. Much attention in the book is given to problems of estimating expenditures for environment restoration, the gist of which lies in the following. If it is

9

Introduction n e c e s s a r y t o e s t a b l i s h a new i n d u s t r i a l p l a n t w i t h a s p e c i f i e d l e v e l of noxious a e r o s o l s e m i s s i o n , t h e f i r s t t h i n g t o do i s t o a s s e s s consequencies of e n v i r o n mental p o l l u t i o n : t h e s t a t e of a g r i c u l t u r a l l a n d s , f o r e s t a r e a s , w a t e r b o d i e s , a n i m a l s , b i r d s , i n s e c t s , s o i l , e t c . To do t h i s s i t i s c o n v e n i e n t t o u s e s t a t i s t i c -

a l e s t i m a t e s of d e t r i m e n t t o t h e environment and a n a t u r e r e s t o r a t i o n programmethe c o s t of which s h o u l d b e i n c l u d e d i n t h e c o s t of p r o d u c t s b e i n g manufactured and b e a l l o c a t e d d i r e c t l y f o r n a t u r e r e s t o r a t i o n p u r p o s e s . A l l t h i s has makes i t p o s s i b l e t o s o l v e , i n t h e f i n a l a n a l y s i s , t h e o p t i m i z a t i o n problem w i t h s e v e r a l l i m i t a t i o n s of a s a n i t a r y , e c o l o g i c a l and economic n a t u r e . Apart from t h e importance of t h e planned b u i l d i n g of new i n d u s t r i a l p r o j e c t s i t

i s p r a c t i c a l l y e q u a l l y important t o e l u c i d a t e requirements f o r i n d u s t r i a l emissi-

ons from a l r e a d y o p e r a t i n g p l a n t s so a s t o e n s u r e a minimum dose of environmental p o l l u t i o n of e c o l o g i c a l l y i m p o r t a n t z o n e s . T h i s problem i s t r e a t e d i n c h a p t e r 6 . To s o l v e i t , t h e a u t h o r chose a s t h e b a s i c c r i t e r i o n economic e x p e n d i t u r e s f o r t e c h n o l o g i c a l changes i n t h e i n d u s t r i a l p l a n t i n q u e s t i o n s o t h a t t o t a l o u t l a y s a t a s p e c i f i e d d e c r e a s e i n t h e p o l l u t i o n l e v e l would be minimal f o r a l l t h e e n t e r p r i s e s in t h e r e g i o n . A s a r e s u l t , t h e problem may be reduced t o a l i n e a r p r o g r a m i n g problem w i t h t h e u s e of s o l u t i o n s of main and

adjoint

equations f o r the

t r a n s f e r and d i f f u s i o n of a e r o s o l p o l l u t a n t s . C h a p t e r 7 i s p r i m a r i l y concerned w i t h e m i s s i o n s of a c t i v e a e r o s o l s which i n t h e c o u r s e of t h e i r i n t e r a c t i o n w i t h w a t e r vapor and o t h e r a i r components a r e t r a n s f e r r i n g from one compound

t o a n o t h e r w i t h s i m u l t a n e o u s change i n t h e n a t u r e

of i t s t o x i c i t y w i t h r e s p e c t t o t h e environment. T h i s problem may be b o i l e d down t o c o n s i d e r a t i o n of t h e o p t i m i z a t i o n problems which have been c o n s i d e r e d e a r l i e r

on t h e b a s i s of p e r t i n e n t

adjoint

equations.

The l a s t c h a p t e r d e a l s w i t h problems of m o d e l l i n g , l o c a t i n g p o l l u t i o n s o u r c e s i n w a t e r b o d i e s and c o a s t a l s e a s , and s t u d y i n g t h e p r o c e s s e s i n v o l v e d i n p o l l u t a n t s t r a n s f e r and d i f f u s i o n w i t h r e f e r e n c e t o mesometeorological and mesooceanic processes.

10

Chapter I . BASIC EQUA TIONS OF TRANSPORT A N D DIFFUSION P o l l u t i n g s u b s t a n c e s a r e t r a n s p o r t e d i n atmosphere by wind a i r s t r e a m s which c o n t a i n some s h o r t - r a n g e f l u c t u a t i o n s . The averaged f l u x of t h e s u b s t a n c e s c a r r i ed by a i r flows h a s , i n g e n e r a l , a d v e c t i v e and c o n v e c t i v e components, and t h e i r averaged f l u c t u a t i o n a l motions can b e c o n s i d e r e d a s d i f f u s i o n a g a i n s t t h e background of t h e main s t r e a m . The purpose of t h e p r e s e n t c h a p t e r i s t o c o n s i d e r v a r i o u s models f o r s u b s t a n c e t r a n s p o r t and d i f f u s i o n , t h e b a s i c e q u a t i o n s d e s c r i b i n g t h e s e p r o c e s s e s , and t h e domains of d e f i n i t i o n and p r o p e r t i e s of t h e solutions. 1.1. Equation D e s c r i b i n g P o l l u t a n t T r a n s p o r t i n Atmosphere. Uniqueness

of t h e S o l u t i o n

9

Let

( x , y , z , t ) b e t h e c o n c e n t r a t i o n of an a e r o s o l s u b s t a n c e t r a n s p o r t e d

with an a i r flow i n t h e atmosphere. We w i l l s t a t e t h e problem f o r a c y l i n d r i c a l domain

G

with t h e su r f a c e

1, t h e b a s e lo

S

c o n s i s t i n g of t h e l a t e r a l s u r f a c e of t h e c y l i n d e r

( a t z = O ) , and o t h e r c o v e r

1,

( a t z = H ) . We w r i t e t h e v e l o c i t y

v e c t o r of a i r p a r t i c l e s , which i s a f u n c t i o n of x , y , z, and t , a s + w& (where

L,i, k

Substance t r a n s p o r t

=

u i + vJ +

a r e u n i t v e c t o r s a l o n g t h e a x e s x , y , z, r e s p e c t i v e l y ) . a l o n g t h e t r a j e c t o r i e s of a i r p a r t i c l e s , when t h e p a r t i c l e

c o n c e n t r a t i o n i s c o n s e r v e d , i s d e s c r i b e d i n t h e s i m p l e s t way, namely, d@/dt = 0 The e x p l i c i t form of t h i s e q u a t i o n i s (1.1) Since t h e continuity equation

aa ux

+ aa vy + az w= O ,

(1.2)

h o l d s w i t h a r e a s o n a b l e a c c u r a c y i n t h e lower atmosphere, we have t h e e q u a t i o n

I n t h e f o l l o w i n g we s h a l l suppose t h a t d i v y = 0 , u n l e s s it i s s t a t e d o t h e r w i s e . B e s i d e s , we s h a l l assume t h a t w = O , a t z = O , z = H

(1.4)

I n t h e d e r i v a t i o n of E q . ( 1 . 3 ) we have used t h e i d e n t i t y (1.5)

Basic Equations which i s v a l i d i f t h e f u n c t i o n s

9

of Transport and Diffusion

and

u -

11

a r e d i f f e r e n t i a b l e . The second term

on t h e r i g h t - h a n d s i d e v a n i s h e s by v i r t u e of E q . ( 1 . 2 ) , and E q . ( 1 . 5 ) becomes (1.5') T h i s i s an i m p o r t a n t r e l a t i o n which w i l l f r e q u e n t l y be used i n t h e sequel Equation ( 1 . 3 ) s h o u l d b e supplemented w i t h t h e i n i t i a l d a t a

0

=

4,

and t h e boundary c o n d i t i o n s on t h e s u r f a c e

0 $o

where u

$s

and

=

9,

on

a r e given f u n c t i o n s and

o n t o t h e outward normal t o t h e s u r f a c e

i n t h e r e g i o n of

S

t

at

= 0,

(1.6)

S

of t h e domain

S

for u

u

G

< 0,

(1.7)

i s t h e p r o j e c t i o n of t h e v e c t o r

S . Condition ( 1 . 7 ) d e f i n e s t h e s o l u t i o n

where t h e a i r bulk c o n t a i n i n g t h e s u b s t a n c e i n q u e s t i o n i s

" i n j e c t e d " i n t o t h e domain

G . The e x a c t s o l u t i o n of t h e problem given by

E q . ( 1 . 3 ) i s p o s s i b l e i f t h e f u n c t i o n s u , v , and

w

a r e known throughout t h e

space and f o r every t i m e moment. I f i n f o r m a t i o n on t h e components of t h e v e l o c i t y v e c t o r i s i n s u f f i c i e n t , one h a s t o r e s o r t t o an a p p r o x i m a t i o n ; some of t h e s e r e l e v a n t methods a r e d i s c u s s e d below. Equation ( 1 . 3 ) can b e g e n e r a l i z e d . For i n s t a n c e , i f a f r a c t i o n of t h e subs t a n c e p a r t i c i p a t e s i n a chemical r e a c t i o n w i t h t h e e x t e r n a l medium, or i s decaying

d u r i n g t r a n s p o r t , t h e p r o c e s s can be t r e a t e d a s a b s o r p t i o n of t h e s u b s t a n c e .

I n t h i s c a s e , t h e e q u a t i o n i n c l u d e s an e x t r a term

a'30 t +

-

a 2 0

where

d i v LJ$

+ oQ

=

0,

i s a q u a n t i t y having t h e i n v e r s e t i m e dimension. The meaning of

t h i s q u a n t i t y i s e s p e c i a l l y c l e a r i f we p u t u = v = w = 0 e q u a t i o n i s j u s t a $ / a t + a@ = 0 , and i t s s o l u t i o n i s see that

0

(1.8)

i n E q . ( 1 . 8 ) . Now t h e

$ = $o exp ( - o t ) .

Hence,we

i s t h e r e c i p r o c a l time p e r i o d d u r i n g which t h e s u b s t a n c e c o n c e n t r a t i o n

f a l l s by a f a c t o r of

e

a s compared w i t h t h e i n i t i a l c o n c e n t r a t i o n 9

.

I f t h e domain of t h e s o l u t i o n c o n t a i n s s o u r c e s o f t h e p o l l u t i n g s u b s t a n c e d e s c r i b e d by a d i s t r i b u t i o n f u n c t i o n f ( x , y , z, t ) , E q . C l . 8 ) becomes inhomogeneous,

a+ + at

div

u$ + a$

=

f.

(1.9)

Now we t u r n t o i n v e s t i g a t i n g t h e problem s t a t e m e n t and c o n d i t i o n s r e l e v a n t t o Eq.(1.9).

time

t

from

L e t u s m u l t i p l y t h e e q u a t i o n by 0

$ and i n t e g r a t e it w i t h r e s p e c t t o

t o T , and o v e r t h e s p a c e domain

G . The r e s u l t i s t h e i d e n t i t y

Chapter 1

12

i

J

@2

dG

1 t=T

-

G

1 j T

j

dG

1 t=O

+

dt

O

G

I 1

T

$*

div

dG +

01

G

d t je2dG =

O

G

T =

dt

O

(1.10) f $ dG.

G

Applying t h e Ostrogradsky-Gauss

formula, one g e t s

(1.11) G

u

By v i r t u e of E q . ( 1 . 4 ) S

i n Eq.Cl.11)

surface

S

z = 0,

vanishes f o r

z

=

H, so t h a t i n t e g r a t i o n o v e r

c a n be r e p l a c e d by t h e i n t e g r a t i o n o v e r t h e l a t e r a l c y l i n d r i c a l

1. For

t h e s a k e of g e n e r a l i t y , however, w e r e t a i n h e r e t h e symbol

S,

having i n mind c o n d i t i o n ( 1 . 4 ) . Taking i n t o a c c o u n t t h e i n i t i a l and boundary conditions, a t t = 0,

$ = $o

(1.12) $ =

where $

[% k

and

$s

on

for u

S

t 0

$ s a r e g i v e n , w e o b t a i n from E q . ( l . l O )

1 1@ dt

0

1 1

T

T

dG +

dS + 0

dt

O

s

1

$’

dG =

G

1% G

T +

-

dt

0

j%

dS

t

s (1.13)

dt

f$

O where

T dG

dG,

G

u+ = [ u n , i f u > 0 , or 0 , i f u t 0 I ; u- = u - u+. n n n I d e n t i t y ( 1 . 1 3 ) i s fundamental i n i n v e s t i g a t i n g t h e u n i q u e n e s s of t h e s o l u -

t i o n s t o t h e problem s t a t e d i n E q s . ( l . 9 ) and ( 1 . 1 2 ) . I n d e e d , s u p p o s e w e h a v e two different solutions, say,

el

and $ 2 , s a t i s f y i n g E q . ( 1 . 9 )

e2,

is

+ d i v &w

+

-

The problem f o r t h e d i f f e r e n c e w = 0

a.

w = O

at

uw = 0,

and t h e c o n d i t i o n s ( 1 . 1 2 ) .

(1.14)

t = O , (1.15)

w

= O

on

S , i f

u

n

t O

13

Basic Equations of Transport and Diffusion Eq.Cl.13)

for t h e function

$

T

d t j

0

s

* d S + u [

f

dt

0

w # 0 , a l l t h e terms o n t h e l e f t - h a n d

expression vanishes only i f

,

+ 2

u w

d G + j

G

If

t a k e s t h e form

iil

w = 0, i . e .

6,

1

(1.16)

w2dC = 0

G

s i d e a r e p o s i t i v e , so t h a t t h i s

=

02 .

Thus, we have proved t h e

uniqueness of t h e s o l u t i o n . It goes without saying t h a t our deduction i s t r u e , provided a l l t h e proce-

d u r e s and t r a n s f o r m a t i o n s used i n t h e proof a r e c o r r e c t . I t i s n o t d i f f i c u l t t o

see t h a t t h i s i s t h e c a s e i f t h e s o l u t i o n

q? and t h e v e l o c i t y components

u , v, w

a r e d i f f e r e n t i a b l e f u n c t i o n s , and t h e i n t e g r a l s a p p e a r i n g i n E q . ( 1 . 1 3 ) do e x i s t . W e w i l l assume i n t h e s e q u e l t h a t a l l t h e smoothness c o n d i t i o n s e n s u r i n g t h e u n i q u e n e s s of t h e s o l u t i o n s a r e v a l i d . So w e have proved t h a t t h e problem

(1.17)

6

6

=

= b0

t

at

= 0,

(1.18)

6,

on

S,

if

< 0

u

has a unique s o l u t i o n . 1 . 2 . S t a t i o n a r v Eauation f o r Substance Prooaeation W e now d e s c r i b e a s t a t i o n a r y p r o c e s s of s u b s t a n c e p r o p a g a t i o n . u , v , ~ , f , +are ~ time-independent, Eqs.(l.l7)

I f input d a t a

t h e s t a t i o n a r y problem c o r r e s p o n d i n g t o

and ( 1 . 1 8 ) becomes q u i t e s i m p l e

+ uq?

d i v gq?

+

=

@son

S

= f ,

(2.1)

f o r un < 0

(2.2)

Evidently, t h e i d e n t i t y corresponding t o Eq.(1.13)

dS+o

S

J

G

+’

dG =

is

-

(2.3)

s

b

The method d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n can be used t o show t h a t t h e problem g i v e n by E q s . ( 2 . l ) ,

( 2 . 2 ) has a unique s o l u t i o n .

Thus t h e problem i n v i e w , E q s . ( 2 . 1 ) ,

( 2 . 2 ) , i s a p a r t i c u l a r c a s e where

s u b s t a n c e t r a n s p o r t p r o c e e d s w i t h i n v a r i a b l e i n p u t d a t a . However, t h e set of

14

Chapter 1

such p a r t i c u l a r s o l u t i o n s c o r r e s p o n d i n g t o v a r i o u s s t a t i o n a r y i n p u t f u n c t i o n s u -, f , $ s , i s a l s o u s e f u l i n t r e a t i n g more complicated p h y s i c a l s i t u a t i o n s , which t a k e p l a c e i n p r a c t i c e . To d e m o n s t r a t e t h i s f a c t , we suppose t h a t i n a r e g i o n under s t u d y t h e motions of a i r masses a r e s t a t i o n a r y d u r i n g c e r t a i n t i m e p e r i o d s s p e c i f i c f o r t h e e x i s t e n c e of any p a r t i c u l a r c o n f i g u r a t i o n of a t m o s p h e r i c f l o w s . Every such s t e a d y - s t a t e p e r i o d ends i n t h e rearrangement of t h e a i r motion, and a new s t a t i o n a r y c o n f i g u r a t i o n i s e s t a b l i s h e d . S i n c e t h e rearrangement t i m e i s much l e s s t h a n t h e time of t h e e x i s t e n c e of any p a r t i c u l a r c o n f i g u r a t i o n , i t can be assumed t h a t t h e s t a t e s a r e changed i n s t a n t l y . Suppose w e have a sequence of n

s t a t i o n a r y c o n f i g u r a t i o n s ; t h i s l e a d s t o a s e t of div u . @ . + -1

+i =

+is

1

S

on

UAi

oi

on t h e s u r f a c e

S

for u . in

and

uin

independent e q u a t i o n s ,

(2.4)

= f

The problem s t a t e d i n E q s . ( 2 . 4 ) , (2.5), where function

n

-

(2.5)

< 0, i = l , n

biS

i s t h e boundary v a l u e of t h e

i s t h e p r o j e c t i o n of t h e i - t y p e wind

stream upon t h e outward normal t o t h e boundary s u r f a c e , c o r r e s p o n d s t o t h e t i m e

t . < t < t i + l , t h e i n t e r v a l l e n g t h s Being At. = t . 1 i+1 - t i . Suppose E q s . ( 2 . 4 ) , ( 2 . 5 ) a r e s o l v e d f o r every i . Then t h e i m p u r i t y d i s t r i b u r? t i o n f u n c t i o n averaged o v e r t h e whole time i n t e r v a l T = )- @,ti i s t h e l i n e a r intervals

i=l

combination of t h e s o l u t i o n s

(2.5), (2.6) may b e c a l l e d t h e

The approach p r e s e n t e d i n E q s . ( 2 . 4 ) ,

s t a t i s t i c a l model. The s o l u t i o n of s t a t i o n a r y problems ( 2 . 1 ) , ( 2 . 2 ) and ( 2 . 4 ) ,

(2.5) i s s i m i l a r

t o d e t e r m i n i n g a t i m e averaged, o v e r a p e r i o d T , of t h e s u b s t a n c e d i s t r i b u t i o n , proceeding from s p e c i a l l y f o r m u l a t e d n o n s t a t i o n a r y problems. A c t u a l l y , w e can c o n s i d e r t h e f o l l o w i n g problem:

* at

+

=

6,

+

div

on

L &

S

+ IJ@= f , for

(2.7)

un < 0 ,

(2.8)

+

(r,T) = + ( r , O ) , ~~

r = (x,y,z)EG.

A s i n E q s . ( 2 . 1 ) , (2.2), we assume t h a t t h e f u n c t i o n s

u -

and

bS

a r e time-in-

dependent. The same method a s t h a t of s e c t i o n 1.1 i s used t o prove t h a t t h e problem s t a t e d i n E q s . ( 2 . 7 ) , (2.8) h a s a unique s o l u t i o n under some p r o p e r assumptions

15

Basic Equations o f Transport and Diffusion

on t h e s m o o t h n e s s o f t h e f u n c t i o n s i n v o l v e d . I n t e g r a t i n g E q . ( 2 . 7 ) o v e r t h e i n t e r v a l [O,Tl, w e o b t a i n t h e e q u a t i o n

div

I$

T

-

+ u$

-

f, $ =

=

T1

(2.9)

$ dt. 0

As t h e p r o b l e m s t a t e d i n E q s . ( 2 . 1 ) and ( 2 . 2 ) h a s a u n i q u e s o l u t i o n , w e s e e from E q . ( 2 . 9 ) t h a t t h e s o l u t i o n of E q s . ( 2 . 7 ) ,

(2.8) a v e r a e e d o v e r t h e p e r i o d

T , coin-

(2.2).

c i d e s w i t h t h e s o l u t i o n of E q s . ( Z . l ) ,

2

L e t u s c o n s i d e r a more c o m p l i c a t e d c a s e . Suppose w e h a v e a f u n c t i o n

i s s u f f i c i e n t l y smooth for t . + 1

Eq.

T

5t

i

ti+l ( i

= 0,

which

t E [O,T], d o e s n o t depend on t i m e f o r e v e r y i n t e r v a l

n-l),

and c o i n c i d e s o n t h e s e i n t e r v a l s w i t h u .

in

I

(2.4). The r e a r r a n g e m e n t t i m e f o r t h e s t r e a m c i r c u l a t i o n s ,

much l e s s t h a n t h e t i m e i n t e r v a l s

Ati,

by a s s u m p t i o n

i.e.

<<

T

1,is

(2.10)

At..

W e now d e a l w i t h t h e n o n - s t a t i o n a r y p r o b l e m ( 2 . 7 ) , (2.81, c o r r e s p o n d i n g t o t h e vector function

u -

d e f i n e d above

?$ at +

+ a+ = f ,

d i v ?$

(2.11)

div u = 0, @

where to

u.

-1

bS

is r e l a t e d t o

biS

=

$s

on

S

T

< 0

i n E q . ( 2 . 5 ) i n t h e same f a s h i o n a s

i n Eq.(2.4). Furthermore, suppose t h a t

o n s of a p e r i o d

(2.12)

for u

~

u

and

i.,

e q u a l t o o n e y e a r . The b o u n d a r y - v a l u e

is r e l a t e d

a r e periodic functiproblem ( 2 . 1 1 ) , ( 2 . 1 2 )

s h o u l d b e c o n s i d e r e d f o r t h e s e t of t i m e - p e r i o d i c f u n c t i o n s , t h a t i s t o s a y

4 (r,T) =

$(r,o),

Thus, s o l v i n g t h e problem ( 2 . 1 1 ) - ( 2 . 1 3 ) ,

=(x,Y,z)

(2.13)

we o b t a i n t h e annual average f o r t h e

substance d i s t r i b u t i o n

(2.14)

I t i s n o t d i f f i c u l t t o see t h a t t h e r e i s a n i n t i m a t e r e l a t i o n b e t w e e n t h e non-

s t a t i o n a r y p r o b l e m (2.11)-(2.13) a n d t h e p r o b l e m (2.4), ( 2 . 5 ) t r e a t e d a b o v e .

16

Chapter 1 Consider E q . ( 2 . 1 1 )

3 at +

d i v u . 4 + a4 = f

(2.15)

7

w i t h boundary c o n d i t i o n (2.12),

4 f o r t h e time i n t e r v a l [ t . +

4,

=

on

S

f o r uin

t = t. +

Let a t t h e moment

T ,

(2.16)

c 0,

T

t h e function

4 take the value

4 (ti

+ T) =

(2.17)

$?

Introduce t h e function =

0.

where

$i

4 - 4i,

(2.18)

i s t h e s o l u t i o n for t h e problem (2.4), ( 2 . 5 ) . T h i s i s t h e s o l u t i o n of

t h e boundary-value problem

aw.

at + wi

= 0

W.

for t h e i n t e r v a l

t€[ti +

T,

d i v U . W .+ 1

on

1

S

0,

=

Ow.

< 0,

f o r ui

0.

( t i + T) = 9'-

ti+l].

The n e x t s t e p i s t o m u l t i p l y d i f f e r e n t i a l Eq.(2.19)

11~ 11=

(

G

by w. and i n t e g r a t e

G . The r e s u l t i s

o v e r t h e domain

where

(2.19)

1

W2dG)1'2.

Hence, w e g e t t h e i n e q u a l i t y

IIwill 5

exp i-+(t

- ti -

T)III$~

Thus, t h e s o l u t i o n of t h e problem ( 2 . 1 1 ) - ( 2 . 1 3 )

-

"u.

(2.21)

for e v e r y t i m e i n t e r v a l

t . 5 t I t i + l , s t a r t i n g from a c e r t a i n t i m e moment, may d e v i a t e o n l y s l i g h t l y from t h e s o l u t i o n of t h e c o r r e s p o n d i n g problem s t a t e d i n E q s . ( 2 . 4 ) ,

(2.5). In general,

t h e e x t e n t of t h e d i f f e r e n c e between t h e s o l u t i o n s depends e s s e n t i a l l y on t h e

t i m e intervals

A t . . I t i s n a t u r a l f o r t h e c l a s s of t h e problems c o n s i d e r e d h e r e

-

a r e so chosen t h a t t h e

t h a t t h e l e n g t h s of t h e time i n t e r v a l s

At.

i n t e r v a l where t h e f u n c t i o n s

have a n e g l i g i b l e d i f f e r e n c e is s u h s t a n -

$ and $i

( i = 1,n)

t i a l l y g r e a t e r t h a n i t s complement t o A t . . With t h i s i n mind, w e i n t r o d u c e

a

17

Basic Equations o f Transport and Diffusion quantity

E

and regard two concentrations of the polluting substance

Jll

and

@'

as coinciding, if

- @211 Let

(2.22)

t

denote the time necessary for stabilization, in the sense of the defini-

T.

1

tion (2.22), of a steady-state propagation of the substance, given by the function

$ , during the time period

t. +

T~ = -

T

In

t 5

5

ti+l. Then we have

-

Il@O

5 at

(2.23)

i'

Proceeding along the same line, as it was shown above, we get by means of inequality (2.21) an estimate for the function t.

5 t

5 ti

+

9 in the time interval

T,

The annual average of the impurity concentration 9

is

(2.25)

t.+ T +Ti +'

; n

i=o

Hence, using relations (2.21)-(2.24),

[

$ dt.

i we get the inequality for the difference

between the solutions of the boundary-value problems (2.4), (2.5) and (2.11)-(2.13) -

averaged over the time interval [O,Tl: /I@ where

r

=

max

(T

+

T.),

5

E

+

(E

+ 311

f

/I

/a )r/At,

At = T/n. Taking into account condition (2.10) and

choosing the interval lengths At,

T.

according to the requirement

r/At 5

E

, we

obtain

[IS - SH

5

(1 - 31fIl/U)E

+

€2

(2.26)

Thus, under the assumptions adopted, the solution of the boundary-value problem for the substance distribution averaged over the time period

T , as obtained by

means of the statistical model, is sufficiently close to the solution of the nonstationary problem (2.11)-(2.13). Note in conclusion that solving problem (2.4), (2.5) or (2.11)-(2.13) and averaging the results by means of Eqs.(2.6)

or (2.14), respectively, we do take

into account substance diffusion induced by fluctuations of the input data.

Chapter 1

18

1.3. Diffusion Approximation. Uniqueness of the Solution The models f o r impurity propagation in the atmosphere from the air pollution sources considered in the preceding section describe the process in the essence; but they idealize, to a certain extent, the real processes, which have a more complicated and varied physical contents. Take, for instance, the case where the atmosphere is free from advective and convective motions, that is to say, u = v = w = 0. Then, according to the models considered above, the non-stationary problem for substance transport is (3.1) @

If f does not depend on

=dr0

att=O

t , the solution is

and it approaches the solution of the corresponding stationary problem, i.e.

9

=

f/o, as

t

O@

=

f,

goes to infinity.

It is known that the simplest model of this type does not describe the main properties of substance transport from a source

f. Actually, we do know

that in the atmosphere any impurity is as if dispersing, thereby forming a rather complicated aerosol distribution in a considerable neighbourhood of the exhaust source. No wonder, because even though the weather is calm there always exists turbulence in the atmosphere, where spontaneous short-range fluctuations (mostly, vortices) are permanently dissipated thereby giving rise to new short-range phenomena. The spectrum of such fluctuations has been investigated thoroughly enough, and we know that precisely these fluctuations are responsible for the dispersion of exhaust sources in the atmosphere. Since the effect in view is purely statistical, it is impossible to pre-calculate the fluctuations, or even detect them, in the real situation. If there were such a possibility, one could calculate the dispersion of an aerosol source and substance propagation, using the model of E q s . (2.4), (2.5), or the more accurate model of Eqs.(2.7),

(2.12),

(2.13). This

is inconceivable, however, so we should modify the modelsin such a way that they be appropriate to account for continuously generated atmospheric fluctuations. The physical nature of the fluctuational phenomena has been comprehended well enough, but until now their mathematical description has been based, for the most part, upon semi-empirical relations. The simplest approach is briefly considered below.

Basic Equations o f Transport and Diffusion Suppose w e d e a l w i t h a f u n c t i o n

19

which i s r e p r e s e n t e d a s t h e sum of i t s

a

-

averaged value

a

a ' , i.e.

and t h e f l u c t u a t i o n a l component

a = a + a ' , where

-

a'

((

a

(3.3)

i n o t h e r words, t h e f l u c t u a t i o n s o f t h e q u a n t i t y that the quantity

a

a r e s m a l l . Suppose f u r t h e r

is averaged over a s u f f i c i e n t l y l a r g e t i m e i n t e r v a l

a

T

t +T

a=r

1

j

adt,

(3.4)

t and on t h i s i n t e r v a l

(3.5)

t I f t h e p r o c e s s i n view s a t i s f i e s c o n d i t i o n s ( 3 . 3 ) - ( 3 . 5 ) ,

we c a n a p p l y t h e

f o l l o w i n g method o f c o n s t r u c t i n g t h e e q u a t i o n s d e s c r i b i n g s u b s t a n c e p r o p a g a t i o n s in various cases. C a l c u l a t e t h e i n t e g r a l of E q . ( 1 . 8 ) o v e r t h e i n t e r v a l

$ (t+T)

-

$(t)

+

div

T

-

Assuming $ = $ + $ '

and 2 =

tr

-

+ ?',

Q(t+T) - '(t) T

+

5 t

II

+ T,

t +T +

u$dt -

0

t

>

I

t

(3.6)

(idt = 0

t

a s i n E q . ( 3 . 4 ) , we g e t from E q . ( 3 . 6 ) . -

-

d i v >$ + d i v Fit + a$ = 0 ,

(3.7)

or e q u i v a l e n t l y -

-

'$(t

+

T) T

-

b(t)

+

divi<+ divmu; +

=

-

9'(t

+

-

T)

Next, w e do a r e s c a l i n g o f t h e f u n c t i o n s i n v o l v e d . W e w r i t e where a

((

0 and O '

$'(t)

(3.8)

T =

AZ,

$ ' = aO',

-

have t h e same o r d e r o f magnitude. By t h e a s s u m p t i o n , $ ' 5 $ , s o

A and w e have a/A = e < (

1. Then E q . ( 3 . 8 )

is reduced t o

where O(1) s t a n d s f o r a q u a n t i t y of t h e o r d e r o f @ ' . Thus, t h e r i g h t - h a n d

side

o f ( 3 . 9 ) is s m a l l b e c a u s e of t h e p a r a m e t e r E / T , and c a n b e n e g l e c t e d . The r e s u l t i n g e q u a t i o n is

20

Chapter 1 -

If the function replace

$ ( t ) h a s a s m a l l v a r i a t i o n on t h e t i m e i n t e r v a l

(o ( t + T ) - T(t))/T

a/at

by t h e t i m e d e r i v a t i v e

T , we can

t o obtain t h e equation

f o r t h e averaged component (3.11) which d i f f e r s from E q . ( 1 . 8 ) by t h e p r e s e n c e of t h e f l u c t u a t i o n a l moment d i v

u'lp'.

I t i s t h i s new term which i s r e s p o n s i b l e f o r t h e d i s p e r s i o n of a i r s t r e a m s c a r r y i n g p a r t i c l e s of p o l l u t i n g s u b s t a n c e s . I t h a s been found t h a t i n t h e c a s e of atmospheric p r o c e s s e s t h e components

__

of t h e v e c t o r

F ' $ ' can b e e x p r e s s e d s e m i - e m p i r i c a l l y i n t e r m s of t h e averaged

substance f i e l d s

Here

p

2 0 and

~i

2 0

a r e t h e h o r i z o n t a l and v e r t i c a l d i f f u s i o n c o e f f i c i e n t s ,

r e s p e c t i v e l y ; t h e y a r e t o b e determined e x p e r i m e n t a l l y . I f we s u b s t i t u t e r e l a t i o n s ( 3 . 1 2 ) i n t o E q . ( 3 . 1 1 ) , we g e t t h e d i f f u s i o n approximation f o r t h e e q u a t i o n of s u b s t a n c e p r o p a g a t i o n i n t h e atmosphere -

alp at +

div

where

-

D$

=

66 + uT =

DT,

-

-

(3.13)

-

a a -$ + a a a ax ay 2 ay + a z "3 az

ax

(3.14)

Of c o u r s e , E q . ( 3 . 1 3 ) s h o u l d b e completed w i t h t h e c o n t i n u i t y e q u a t i o n -

div 5 = 0

(3.15)

and t h e i n i t i a l d a t a .-

4

=

To

at

(3.16)

t = 0.

A s r e g a r d s t h e boundary c o n d i t i o n s , t h e y a r e d e r i v e d i n t h e i n v e s t i g a t i o n of t h e s o l u t i o n uniqueness f o r t h e problem. In t h e f o l l o w i n g w e s h a l l omit t h e b a r o v e r t h e f u n c t i o n , assuming t h a t we d e a l w i t h t h e averaged q u a n t i t y . Again, we m u l t i p l y Eq.(3.13) t h e r e s u l t over t h e time i n t e r v a l

0 5 t 5 T

by $

and t h e s p a c e domain:

and i n t e g r a t e

21

Basic Equations o f Transport and Diffusion rn

(3.17)

zH

z

0

Here $T = @ ( T ) , $o = $(O), the surface

z. R e c a l l

a@/an i s t h e d e r i v a t i v e a l o n g t h e outward normal t o

S

that

l a t e r a l cylindrical surface, a t t h e height

z = H,

L

i s t h e t o t a l s u r f a c e of t h e domain

CH

G,

).

is the

i s t h e h o r i z o n t a l c r o s s - s e c t i o n of t h e c y l i n d e r

is the cross-section

at

z = 0. Eq.(3.17)

is f u r t h e r

reduced t o

(3.18)

W e s e t t h e f o l l o w i n g boundary c o n d i t i o n s : $ =

a$&

Here

a

$s

Z

on

un < 0

for

1 for

u

a$/az = a $

on

1

a$/az

on

L

= 0

on

=

o

2

0

(3.19)

0’

H

.

2 0 i s a f u n c t i o n d e f i n i n g t h e i n t e r a c t i o n of t h e i m p u r i t i e s w i t h t h e

underlying s u r f a c e . Besides,

w = O

at

z = O ,

z = H .

(3.20)

Let u s show t h a t c o n d i t i o n s ( 3 . 1 9 ) , ( 3 . 2 0 ) t o g e t h e r w i t h i n i t i a l d a t a ( 3 . 1 6 ) e n s u r e t h e u n i q u e n e s s of t h e s o l u t i o n . A c t u a l l y , u s i n g E q . ( 3 . 1 9 ) , we o b t a i n t h e fundamental r e l a t i o n .

22

Chapter 1

G

J '

J

i:"

0

O

G

(3.21)

O

S

Here we have r e p l a c e d

o

G

z

Z, i n view of E q . ( 3 . 1 0 ) .

by

We now prove t h a t t h e s o l u t i o n i s u n i q u e . As i n s e c t i o n 1 . 2 , we assume t h a t t h e r e a r e two s o l u t i o n s ,

$1

and

$2, s a t i s f y i n g d i f f e r e n t i a l Eq.(3.13), i n i t i a l

d a t a ( 3 . 1 6 ) , boundary c o n d i t i o n s ( 3 . 1 9 ) . and s u b s i d i a r y c o n d i t i o n s ( 3 . 1 5 ) and

aw +

$2

we g e t t h e d i f f e r e n t i a l e q u a t i o n

+ u w = Do,

div g w

at

-

w = $

( 3 . 2 0 ) . Then f o r t h e d i f f e r e n c e

(3.22)

with t h e i n i t i a l d a t a

= o

0

t = 0,

at

(3.23)

and t h e boundary c o n d i t i o n s w

= 0

ao/an = 0

awaz

on X

for

u

< 0,

on I:

for

u

2 0

(3.24) =

C ~ W on

cOl

For t h i s problem s t a t e m e n t , i n t e g r a l r e l a t i o n ( 3 . 2 1 ) i s reduced t o

(3.25)

+

Since t h e f u n c t i o n s un,

1-1, u ,

0 ,

c1

i n Eq.

h o l d s o n l y i n t h e c a s e where w = 0 , i . e .

(3.25) a r e non-negative, t h e r e l a t i o n

$1 = $2. Thus, t h e s o l u t i o n is unique.

For t h e s a k e of s i m p l i c i t y w e have assumed h e r e t h a t

f = 0 . The e f f e c t of a

23

Basic Equations o f Transport and Diffusion s o u r c e s can b e accounted f o r a s i n s e c t i o n 1.1.

If one h a s t o i n c l u d e t h e s o u r c e s , t h e problem t o be a l s o c o n s i d e r e d h a s a unique s o l u t i o n i n t h e d i f f u s i o n a p p r o x i m a t i o n , provided t h e i n p u t d a t a a r e smooth. The problem i s

?$ at +

div

y$ + 00

=

D f$ + f ,

0

=

O0

at

t

= 0,

4

=

Qs

on

C

for

a$/an = 0

on

Z for u

2 4 / 3 2 = a$

on

a#/az

on

=

o

H

< 0,

u

0,

2

(3.26)

.

We a l s o assume h e r e t h a t div 5 = 0 ,

a t z = 0 , z = H.

w = 0

Note t h a t t h e r e i s a n o t h e r problem, b e s i d e t h a t of E q . ( 3 . 2 6 ) , which i s somet i m e s used i n c a l c u l a t i o n s and h a s a unique s o l u t i o n .

s d i v u$ at + -

$

= 0,

@

=

$s

+ 00

=

D$ + f ,

at

t = 0,

on

X,

on

. X0’

(3.27)

ae/a2

= CX$

I t goes w i t h o u t s a y i n g t h a t t h e r e a r e o t h e r s t a t e m e n t s of t h e problem e n s u r i n g a unique s o l u t i o n . To proceed f u r t h e r , we r e p r e s e n t t h e o p e r a t o r

D

d e f i n e d by E q . ( 3 . 1 4 ) a s

t h e sum of two o p e r a t o r s

To f a c i l i t a t e t h e f o l l o w i n g a n a l y s i s , we assume t h a t t h e d i f f u s i o n c o e f f i c i e n t p does n o t depend on t h e s p a c e c o o r d i n a t e s and t i m e . U n t i l now we have c o n s i d e r e d t h e g e n e r a l t h r e e - d i m e n s i o n a l problems, b u t t h e r e a r e many c a s e s where two-dimensional

(x,y)-approximations a r e adequate, t o

be s t a t e d on t h e b a s i s of E q s . ( 3 . 2 6 ) o r ( 3 . 2 7 ) . Take, f o r i n s t a n c e , E q s . ( 3 . 2 7 ) . I n t e g r a t i n g t h e d i f f u s i o n e q u a t i o n over t h e a l t i t u d e , we g e t

24

Chapter 1

4

+

(3.28) f dz.

0

Let u s t r a n s f o r m t h e second term on t h e l e f t - h a n d s i d e and t h e f i r s t t e r m on t h e r i g h t - h a n d s i d e t o a more e x p l i c i t form. Assuming t h a t t h e h o r i z o n t a l components

u

and

v

of t h e v e l o c i t y v e c t o r do n o t depend on t h e a l t i t u d e i n t h e

a c t i v e l a y e r of s u b s t a n c e t r a n s p o r t and d i f f u s i o n , we o b t a i n

7

div

$dz = a

J 0

1

H (u

ax

9

dz]

+

ay a

0

The l a s t term v a n i s h e s , s i n c e

1

z =H

H (v

$ dz]

+ WOI

w = 0

for

z = 0

(3.29) z=o

0

and

z

= H,

so t h a t

(3.30)

We now t u r n t o t h e r e l a t i o n

(3.31)

I t can be s i m p l i f i e d by v i r t u e of t h e boundary c o n d i t o n

84/32

=

a$ a t

z = 0

a s follows:

H

Assuming t h e approximate r e l a t i o n

one f i n a l l y g e t s

H

H (3.32)

25

Basic Equations o f Transport and Diffusion

I n t r o d u c e t h e i n t e g r a l i n t e n s i t i e s f o r t h e a e r o s o l d i s t r i b u t i o n and t h e source

P u t t i n g them i n t o Eq.(3.28), we o b t a i n

(3.33) -

u = u

where

+

crv/H.

Note t h a t u F

i s t h e amount of t h e a e r o s o l decaying i n t h e

p r o c e s s of i t s p r o p a g a t i o n i n t h e atmosphere, i n t e g r a t e d o v e r t h e a l t i t u d e , while -

(nv/H)

$ i s t h e amount of t h e a e r o s o l d e p o s i t e d a t t h e e a r t h ' s s u r f a c e . Omitting

t h e b a r o v e r t h e symbols

4 , f , and u , we a r r i v e a t t h e two-dimensional problem

statement

8 = Q 0

at

t = O

bS

on

L for

un < 0 ,

a$/an = 0

on

I: f o r

u

$ =

(3.34)

t 0.

Another s t a t e m e n t of t h e problem i s a l s o p o s s i b l e ,

$ = $o

at

(3.35)

t = 0,

I t i s worth n o t i n g t h a t a l l t h e arguments p r e s e n t e d i n s e c t i o n 1.2 a r e v a l i d f o r t h e d i f f u s i o n approximation a s w e l l . T h i s i s a l s o t r u e f o r t h e s t a t i o n a r y problem where t h e s o l u t i o n of t h e b a s i c problem i s o b t a i n e d , t o w i t h i n t h e t r a n s i e n t p r o c e s s e s , by a v e r a g i n g o v e r t h e ensemble of p a r t i c u l a r s t a t i o n a r y problems

of t h e f o l l o w i n g form: div u,$. -1

1

+ a$.

= Dqbi

+ f,

div u = 0 -i

(3.36)

w i t h t h e boundary conditicms $. =

+is

on L f o r

u. < 0, in (3.37)

&$./an = 0

on

Z

for

u. in

2

0,

Chapter 1

26

a@./az

= 0

CH.

on

averaged o v e r t h e t i m e p e r i o d

The s u b s t a n c e d i s t r i b u t i o n

T

is

n

1

$ = T

(3.38)

(li A t i ,

i=l

n where

T =

L A t i , and A t i i=l

is t h e t i m e d u r i n g which t h e r e e X i S t S a s t e a d y -

s t a t e c o n f i g u r a t i o n of t h e a i r s t r e a m s . 1 . 4 . Simple D i f f u s i o n E q u a t i o n

I t i s r e a s o n a b l e t o b e g i n an a n a l y s i s o f s u b s t a n c e t r a n s p o r t and d i f f u s i o n w i t h t h e s i m p l e s t examples of one-dimensional

problems, g r a d u a l l y complicating

t h e m a t h e m a t i c a l problem s t a t e m e n t . So, we a r e g o i n g t o s t a r t w i t h t h e p u r e d i f f u s i o n i n an i n f i n i t e o n e - d i m e n s i o n a l medium

where

-

< x <

m,

Q i s t h e c a p a c i t y of t h e s o u r c e e x h a u s t i n g t h e a e r o s o l i n t o

t h e atmosphere. The boundary c o n d i t i o n s i n t h i s c a s e are r e d u c e d t o t h e r e q u i r e ment t h a t t h e s o l u t i o n s h o u l d be f i n i t e t h r o u g h o u t t h e domain of d e f i n i t i o n . Note t h a t t h e source function

f

i n E q . ( 4 . 1 ) i s t a k e n i n a p a r t i c u l a r form c h a r a c -

t e r i s t i c of t h e problems of t h e c l a s s c o n s i d e r e d h e r e . I t i s c o n v e n i e n t t o s t a t e t h e problem i n an e q u i v a l e n t form w i t h o u t t h e t e g r a t e Eq.(4.1)

i n t h e v i c i n i t y of t h e p o i n t

6 f u n c t i o n . To t h i s e n d , w e i n x = x ; t h e r e s u l t is 0

d4 I -v dx / x - t / 2

As

E

+

t e n d s t o z e r o , we o b t a i n t h e important r e l a t i o n (4.2)

L e t us c o n s i d e r two r e g i o n s , s o l u t i o n s w i l l be d e n o t e d by

9-

-

< x

and

$+,

<

x

and x

C x <

w;

t h e corresponding

and t h e problems t o be s o l v e d a r e

(4.3) (p+

= 0

for x +

-7

;

27

Basic Equations of Transport and Diffusion

+-

for x

= 0

+

--.

The coupling between the solutions of the problems is given by the boundary condition

at x = x

0

d4dx

d@ dx

at

- ~ - + Q = o

~-r+

X = X

(4.5)

0

The second condition follows from the requirement that the solution must be continuous everywhere, including the point

x

=

at

$+ = $-

x

0’

x = x .

(4.6)

It is easy to see that the solutions for problems (4.3) and ( 4 . 4 ) are of the form @+

= C+

exp

~-G(x-x~ I ,)

+-

= C-

exp

1-vGZ(x~-x)1

(4.7)

Substitution of these expressions into Eqs.(4.5) and ( 4 . 6 ) yields a set of linear equations for the coefficients C+

c+

and C - ; the solution is

c-

=

=

L, 2

G

Thus, the solution of problem (4.1) is of the form exp

1- Jo/ll(x-x0)i

for

x 1

x

exp

1- Jo/ll(xo-x)i

for

x ‘z

x

$(x) = __ 2

The graph of the function

Q6 \

(4.8)

+(x) is given in figure 1.1. It shows that the diffusion

0

XO

FIGURE 1.1. process results in the distribution which falls exponentially and symmetrically in both directions from the source that

at

x = x

.

It is a simple matter to verify

28

Chapter 1

I

m

$ (x) dx =

Q,

4

We now turn to a more interesting case where the air stream velocity is non-zero. Suppose that it is constant and positive. Then we have the differential equation

x, -

for all

-

d@ + u dx < x <

-.

09

=

dx2

+ Q6

(X

- x

(4.9)

)

Just like the preceding case, Eq.(4.9) with the appro-

priate condition at infinity is reduced to a pair of problems

(4.10) $+ = 0

for x

+

-;

4-

for

+

- -.

(4.11) =

0

x

The solutions of these problems are coupled, as before, by the boundary condition at

x = x

0'

d@+ p

-

d@fl

+ Q

= 0,

0,

=

4-

at

x = x

(4.12)

The solutions of problems (4.10) and (4.11) are represented as

(4.13)

Substituting these expressions into Eq.(4.12),

we get

C+ = C-

= C,

and

The resulting solution is of the form

(4.14)

x 6 x0 '

29

Basic Equations o f Transport and Diffusion

FIGURE 1 . 2

FIGURE 1 . 3

The graph o f t h e s o l u t i o n i s g i v e n i n f i g u r e 1 . 2 . I t i s c l e a r l y seen t h a t for

u > 0

t h e l e f t - h a n d p a r t of t h e exponent ( w i t h r e s p e c t t o t h e p o i n t x = x )

g e t s n e a r e r t o t h e s o u r c e , w h i l e t h e r i g h t - h a n d p a r t i s p u l l e d from t h e s o u r c e and s p r e a d i n g ; t h e e f f e c t i s e v i d e n t l y caused by s u b s t a n c e d r i f t due t o t h e wind with simultaneous d i f f u s i o n .

A more c o m p l i c a t e d s i t u a t i o n t a k e s p l a c e when t h e wind i s blowing f o r a x (u,

long t i m e towards p o s i t i v e v a l u e s of

> 0)

and t h e n changes t h e d i r e c t i o n

< 0 ) . In t h i s c a s e , w e have two s o l u t i -

and i s blowing towards n e g a t i v e v a l u e s (u, on s

$1

=

Q\

___

JGq

exp

{-(m u'1) P

+

G

+

2P

x ' x

0'

(4.15)

(xo - x , ] , x 5 x ; 0

x > x 0

1

(4.16)

x s x . 0

days and t h e second p e r i o d (u < O), f o r Atl d a y s , t h e a v e r a g e d e n s i t y of t h e s u b s t a n c e i s g i v e n by t h e formula

If t h e f i r s t period endures f o r

At2

(4.17)

S o l u t i o n ( 4 . 1 7 ) i s shown s c h e m a t i c a l l y i n f i g u r e 1 . 3 . Note t h a t we have used t h e d i r e c t s i m u l a t i o n method when t h e t r a n s i e n t p r o c e s s e s a r e n o t t a k e n i n t o

30

Chapter 1

account. F i n a l l y , l e t us c o n s i d e r t h e s t a t i s t i c a l model where t h e wind i s a l s o desc r i b e d s t a t i s t i c a l l y . Let t h e wind v e l o c i t y b e

u ( S ) = UP ( S ) , where

5

(4.18)

i s a random q u a n t i t y i n t h e u n i t i n t e r v a l , 0 5

t h e p r o b a b i l i t y d e n s i t y normalized t o u n i t y , i . e .

{

p

(5)

5

5

dS

1,

and

p(S) is

= 1. I f t h e a i r

s t r e a m s f o l l o w t h e wind immediately, t h e n by analogy w i t h t h e p r e c e d i n g c a s e w e can w r i t e t h e s o l u t i o n of problem ( 4 . 9 ) ,

under c o n d i t i o n ( 4 . 1 8 ) , i n t h e form of

t h e i n t e g r a l with r e s p e c t t o t h e random v a r i a b l e

(4.19)

where

-

w(x

xo, u

(5))

=

r

For each f i x e d v a l u e of

x , t h e i n t e g r a l in E q . ( 4 . 1 9 ) i s c a l c u l a t e d by t h e Monte

C a r l o method. 1 . 5 . T r a n s p o r t and D i f f u s i o n of Heavy A e r o s o l s Heavy a e r o s o l s a r e of s p e c i a l i n t e r e s t for problems r e l e v a n t t o t h e l o c a l environmental p o l l u t i o n . P r o p a g a t i n g i n t h e atmosphere, heavy a e r o s o l s a r e d i f f u s i n g and s i n k i n g t o t h e s o i l under t h e a c t i o n of g r a v i t y . The s i n k i n g v e l o c i t y i s c a l c u l a t e d from t h e S t o k e s problem; i t is a c o n s t a n t v e c t o r d i r e c t e d down-

wards. So, i f we d e n o t e by

w -g

t o g r a v i t y , t h e new t e r m ,

w

t h e a b s o l u t e v a l u e o f t h e p a r t i c l e v e l o c i t y due a$/az,

appears i n t h e aerosol transport equations, g and t h e t r a n s p o r t and d i f f u s i o n problem of Eq.(3.27) t a k e s t h e form

+ +

= +o

at

t = 0,

o

on

1,

=

a$/az =

+

=

CY+

on

o

on

10, 1,.

(5.1)

Basic Equations of Transport and Diffusion

31

Let u s d e t e r m i n e t h e amount of a e r o s o l d e p o s i t e d d u r i n g t h e t i m e i n t e r v a l 0 5 t S T

lo

upon an a r e a

on t h e p l a n e

z

i n t e g r a t e Eq.(5.1) with r e sp e c t t o

z

For t h i s p u r p o s e , we

= 0.

over the interval

0 5 z i H . Using t h e

notation

1

H

[ P d z = $ ,

u

and assuming t h a t

fdz = F,

0

0

and

v

do n o t depend on

i n t h e “ a c t i v e “ zone, we g e t

z

t h e equation

where

$g = $ / z = o . In t h e d e r i v a t i o n of E q . ( 5 . 2 ) we have used t h e c o n d i t i o n s w = O

a t z = O

a$/az

=

a$

a n d z = H , z = 0,

at

a s w e l l a s a n o t h e r c o n d i t i o n , n a t u r a l f o r t h e c a s e i n view,

9

+

0,

as

z

-*

H.

I t i s s e e n from E q . ( 5 . 2 ) t h a t t h e amount of a e r o s o l i n t h e atmosphere above

-

-

t h e p o i n t (x, y ) d e c r e a s e s by (w + V a ) Qg p e r u n i t t i m e p e r i o d . Here w $ is g g g t h e f r a c t i o n of t h e a e r o s o l d e p o s i t due t o t h e p a r t i c l e f r e e f a l l under t h e a c t i o n of g r a v i t y , and

Va

$g

i s t h e d e p o s i t due t o t h e t u r b u l e n t exchange motion i n

-

the layer adjacent t o the e a r t h ‘ s surface.

If

-

term c o n t a i n i n g Va,

w

i n Eq.(5.1).

I f , however,

w

-g w

<<

VCI,

one can n e g l e c t t h e

i s comparable w i t h or exceeds

g g one s h o u l d c o n s i d e r problem ( 5 . 1 ) , i n s t e a d of problem g i v e n by E q . ( 3 . 2 7 ) .

Two f u n c t i o n a l s a r e , a s a r u l e , used i n t h e problems c o n s i d e r e d . F i r s t , t h e t o t a l amount of a e r o s o l i n a g i v e n domain

G.

1’

(5.3)

Second, t h e t o t a l amount of a e r o s o l d e p o s i t e d o n t o t h e a r e a of a c y l i n d r i c a l domain

li

a t t h e bottom

G, T

(5.4)

The c o n s t a n t

a

i s r e l a t e d t o t h e g r a v i t a t i o n a l and d i f f u s i o n mechanism of

a e r o s o l d e p o s i t i o n . In view of t h e above d i s c u s s i o n we have

32

Chapter 1 ~

a = w

g

+ wa.

(5.5)

Thus, w e have d e f i n e d t h e main f u n c t i o n a l s . I n t h e s e q u e l w e w i l l n o t draw s p e c i a l a

entio

0

he case o

eavy

a e r o s o l s ; one s h o u l d , however, b e a r i n mind t h a t i f t h e g r a v i t a t i o n a l f a l l i s a p p r e c i a b l e t h e problem s h o u l d be m o d i f i e d

t o include t h e r e s u l t s presented in

t h i s s e c t i o n . This t o p i c hardly deserves s p e c i a l c o n s i d e r a t i o n , f o r t h e necessary modification is t r i v i a l . 1 . 6 . The S t r u c t u r e and S i m u l a t i o n of T u r b u l e n t Motions i n t h e Atmosphere I t i s q u i t e i m p o r t a n t t o r e c o n c i l e t h e a d v e c t i v e and c o n v e c t i v e t r a n s p o r t p r o c e s s e s w i t h s u b s t a n c e d i f f u s i o n p r o c e s s e s , when i n v e s t i g a t i n g t h e p r o p a g a t i o n of p a s s i v e i m p u r i t i e s i n t h e atmosphere.

A c o r r e c t combined model t a k i n g i n t o

a c c o u n t b o t h phenomena would, p r o b a b l y , p r o v i d e u s w i t h an a d e q u a t e d e s c r i p t i o n of t h e p h y s i c a l p r o c e s s e s u n d e r l y i n g i m p u r i t y p r o p a g a t i o n i n t h e a t m o s p h e r e . An i n c o r r e c t or, t o be more p r e c i s e , an i n c o m p a t i b l e a c c o u n t f o r b o t h t y p e s o f t h e p r o c e s s e s may l e a d t o l a r g e e r r o r s . Now w e a r e g o i n g t o c o n c e n t r a t e on a c o n s i s t e n combination of t h e a d v e c t i v e - c o n v e c t i v e Suppose w e d e a l w i t h s h o r t - r a n g e

t r a n s p o r t and d i f f u s i o n .

t r a n s p o r t p r o c e s s e s of s p e c i f i c times o f a

few h o u r s . In t h i s c a s e , o n e c a n n e g l e c t t h e v a r i a b i l i t y of t h e m e t e o r o l o g i c a l c o n d i t i o n s , r e g a r d i n g them a s s t a b l e , i n t h e f i r s t a p p r o x i m a t i o n ; t h i s means t h a t u , v , and

w

a r e c o n s t a n t . I f t h e problem of s u b s t a n c e p r o p a g a t i o n from a s o u r c e

i s s o l v e d w i t h no a c c o u n t f o r d i f f u s i o n , w e o b t a i n a s i n g l e t r a j e c t o r y a l o n g which t h e i m p u r i t y i s t r a n s f e r r e d . T h i s approach i s , of c o u r s e , i n c o n t r a d i c t i o n w i t h t h e e s s e n c e of t h e p h y s i c a l p r o c e s s . A s t h e s u b s t a n c e e x p e r i e n c e s s p r e a d i n g a l l t h e way from i t s s o u r c e a l o n g t h e t r a j e c t o r y , w e do n o t d e a l , i n f a c t , w i t h a s i n g l e t r a j e c t o r y , b u t w i t h a whole r e g i o n where t h e c o n c e n t r a t i o n of t h e p o l l u t i n g a e r o s o l i s non-zero.

What a r e t h e f a c t o r s g o v e r n i n g t h e s m e a r i n g of

t h e d i s t r i b u t i o n ? F i r s t , i t i s short-range

f l u c t u a t i o n s of t h e wind v e l o c i t y

i n h e r e n t i n t h e s t o c h a s t i c n a t u r e o f t h e a t m o s p h e r i c m o t i o n s . Measuring t h e d i v e r g e n c e cone and u s i n g t h e s o l u t i o n s of model p r o b l e m s , one c a n d e t e r m i n e t h e d i f f u s i o n c o e f f i c i e n t s p e r t a i n i n g t o t h e m o t i o n s of t h e s p a c e - t i m e s c a l e cons i d e r e d . To make t h i s p o i n t c l e a r , w e s h a l l a n a l y s e t h e f o l l o w i n g example. Suppose s u b s t a n c e t r a n s p o r t i s a two-dimensional

p r o c e s s d e s c r i b e d by t h e

equation (6.1) where and

u IJ

and

v

a r e t h e v e l o c i t y components r e g a r d e d , f o r s i m p l i c i t y , c o n s t a n t ,

i s t h e d i f f u s i o n c o e f f i c i e n t t o b e d e t e r m i n e d . For an i n f i n i t e domain t h e

Basic Equations of Transport and Diffusion solution of Eq.(6.1)

33

is

where K (x) is the McDonald function

I

m

K (x)

=

e

-x ch y

dy,

x > 0

0

Perform the coordinate transformation, so that the origin is put at the source,

s,and the x-axis

is directed along the velocity vector 0. Then the solution

takes the form, (6.3) 1

where

=

( u z + vZ)'.

Now, let the divergence cone of the substance distribution have the width 2yl at a distance

x1

from the exhaust point (figure 1.4). Assuming that the

Y? ---------

FIGURE 1.4

distance

x1

FIGURE 1.5

is large enough to use the asymptotics of the function

K (x) and

that the accuracy of measuring the impurity density in the divergence cone is about

E,

we ohtain the equation for detendningthe coefficient

Q

(XI' Y1,

P) =

)1

E ,

or, explicitly,

The graphical solution of problem ( 6 . 5 ) is presented in figure 1.5.

(6.4)

Chapter 1

34

Next, we shall consider a transport process with a time scale of several days (an intermediate-range process). Using a similar procedure, averaging the weather factors, with due regard to their type, over the given time scale, and neglecting the diffusion processes, we obtain an averaged trajectory. Evidently, this solution does not describe the physical nature of impurity propagation, as the given mean interval consists of a set of ensembles of transport mesoprocesses with specific time periods of several hours, within which they can be considered as weakly variable. Meanwhile, the set of such mesoprocesses does determine their variations during several days. If we know the types of mesoprocesses and wind variations during several days, we can solve the corresponding mesoscale transport problems, determine the divergence cones for the polluting substance, and perform statistical averaging over the ensemble of mesoprocesses. A s a result, we obtain the substance distribution in the transport process with specific time scales of several days. The solution of this combined problem can be compared with the actual cloud shape, as observed experimentally. Performing the analysis, one gets an impression on how adequate is the description based on the set of typical mesoprocesses for understanding the physical nature of the phenomenon simulated. Clearly, in this case there is no need to introduce the macro-diffusion coefficient; it is sufficient to have information on the statistical nature of the mesoprocesses which constitute the overall transport process. The macrodiffusion approximation is, nevertheless, useful in some cases. Let us formally apply the diffusion equation to describe impurity propagation,

Assuming that the vector and considering the coefficient

is the result of averaging over several days M

as unknown, we can use the methods of per-

turbation theory to find the value of

p

which would lead to the best value, in

a certain sense, of some functional. It is possible, however, that even though the functional is obtained correctly with the chosen value of

)1.

the differential

field of solutions for the direct problem (6.6) may differ considerably from the actual one (or from the model field calculated using the set of the meso-problems). Hence, if we are interested in a single fixed functional, the model for substance transport with the chosen diffusion coefficient will be a suitable tool for the investigation. If we are interested in more subtle characteristics of the pollution field or in various functionals, it would be better to simulate the process by a set of

small-scale models. Therefore, while solving optimization problems analysed in the present book we shall proceed from the assumption that the model can be used

Basic Equations of Transport and Diffusion

35

i n t h e m a c r o - d i f f u s i o n p r o c e s s a p p r o x i m a t i o n . Meanwhile, t o h a n d l e t h e s u b j e c t i n g r e a t e r d e t a i l and t e s t t h e r e l i a b i l i t y of t h e r e s u l t s o b t a i n e d i t i s r e q u i r e d t o f i n d s o l u t i o n s f o r a s e t of problems c o r r e s p o n d i n g t o s h o r t - r a n g e

fluctuations.

The mathematical p r e s e n t a t i o n of t h e formalism i s a s f o l l o w s . We d e s c r i b e s u b s t a n c e t r a n s p o r t by t h e e l e m e n t a r y d i f f u s i o n e q u a t i o n

We assume h e r e t h a t

u, v,

and

w

a r e f u n c t i o n s g i v e n by

-

u = u + where

u ' , v ' , w'

-

v = v

U ' )

+

-

w = w + w';

v',

(6.8)

a r e t h e components of t h e v e l o c i t y v e c t o r v a r y i n g d u r i n g

s p e c i f i c t i m e s of a few h o u r s ; t h e s e components a r e d e v i a t i o n s from t h e main -

stream

-

-

u, v, w

and u'

a t a r a t e known a p r i o r i ; p'

are the diffusion coeffi-

c i e n t s c o r r e s p o n d i n g to t h e meso-pmcesses. The problem i s t r e a t e d a s p e r i o d i c w i t h a period

T

t y p i c a l of t h e p r o c e s s c h a r a c t e r i s t i c s c a l e . The boundary c o n d i t i o n s

are

a$/az

=

o

ag/az = a@ 4

+

as

0

at

z = H,

at

z = 0,

x, y

+

(6.9)

fm

A l l s t a t i s t i c a l f e a t u r e s of t h e meso-processes a r e t a k e n i n t o account by

s o l v i n g t h e s e t of problems s t a t e d i n t h i s way. Thus, we need t o s i m u l a t e t h e behaviour of t h e f u n c t i o n s u', v ' , and w ' . S i n c e t h e p r o c e s s i s l i n e a r , i n t h e s i m u l a t i o n of t h e s e q u a n t i t i e s c a r e s h o u l d be t a k e n of t h e i r r e c u r r e n c e c o r r e s p o n d i n g t o t h e i r a c t i o n d u r i n g t i m e i n t e r v a l s which a r e l o n g e r t h a n

T . The f i n a l r e s u l t must b e averaged o v e r t h e time i n t e r -

v a l [ O , TI. Thus, a l l t h e models c o n s i d e r e d i n t h e book can n a t u r a l l y be g e n e r a l i z e d t o account more a c c u r a t e l y f o r t h e s t o c h a s t i c s t r u c t u r e of t h e i n p u t d a t a . T h i s r e a s o n i n g can b e i l l u s t r a t e d by a s i m p l e example. Let s u b s t a n c e t r a n s p o r t b e d e s c r i b e d by E q . ( 6 . 1 ) , a s it was d i s c u s s e d above, and t h e s o l u t i o n i s

(6.2). Write t h e f u n c t i o n s

u

and

-

v

u = u + u'

as

(a), v

=

v + vu (a),

(6.10)

where t h e d e v i a t i o n s u ' and v ' from t h e main s t r e a m , g i v e n by t h e components

u

-

and v , depend on t h e parameter

CY and a r e t r e a t e d a s random f l u c t u a t i o n s of

a

q u a n t i t y r e s t r i c t e d a p r i o r i . S u b s t i t u t i n g e x p r e s s i o n s ( 6 . 1 0 ) i n t o E q . ( 6 . 2 ) we have

36

Chapter 2

The f u n c t i o n

$ ( x , y , a) i s a f a m i l y of p o s s i b l e r e a l i z a t i o n s of t h e a e r o s o l

c l o u d , depending on t h e s t a t i s t i c a l s t r u c t u r e of t h e f l u c t u a t i o n s . The i s o l i n e s f o r the function

$ , averaged o v e r t h e parameter

CL

under t h e assumption t h a t t h e

f l u c t u a t i o n d i s t r i b u t i o n i s normal, a r e p r e s e n t e d i n f i g u r e 1 . 6 (dashed l i n e s ) .

For comparison, t h e f i g u r e a l s o shows i s o l i n e s of t h e f u n c t i o n

$ f o r u' = v' = 0

(solid lines).

FIGURE 1.6 In t h e i n v e s t i g a t i o n of g l o b a l p r o c e s s e s w i t h c h a r a c t e r i s t i c t i m e s c a l e s of s e v e r a l weeks or even months t h e methods s u g g e s t e d above e n a b l e t h e c o n s t r u c t i o n of a h i e r a r c h y of t h e models which can b e i d e n t i f i e d , s a y , by means of p i c t u r e s made from a s p a c e c r a f t .

Chapter 2. ADJOINT EQUA TIONS OF TRANSPORT AND DIFFUSION The i n t e r e s t i n a d j o i n t e q u a t i o n s , a s a p p l i e d t o p a r t i c u l a r f u n c t i o n a l s of problems i n mathematical p h y s i c s , a r o s e long a g o . But o n l y on r a r e o c c a s i o n s a d j o i n t e q u a t i o n s have been i n s t r u m e n t a l i n s o l v i n g t h e s e problems. The c o n s t r u c t i o n of a d j o i n t problems was h i g h l y s t i m u l a t e d by p e r t u r b a t i o n t h e o r y , s i n c e i t s c o r r e c t f o r m u l a t i o n i n v o l v e s b o t h b a s i c and a d j o i n t problems. A s r e g a r d s p-oblems

Adjoint Equations o f Transport and Diffusion

37

on s e a r c h i n g f o r o n e l i n e a r f u n c t i o n a l or a n o t h e r , t h e a d j o i n t f o r m u l a t i o n , r e f l e c t i n g t h e d u a l i t y p r i n c i p l e , a l s o p r o v i d e s a l g o r i t h m s which p r o v e t o b e o p t i m a l , u n d e r s p e c i f i c c o n d i t i o n s , b o t h f o r a n a l y s i s of p r o b l e m s a n d f o r t h e i r implement a t i o n . Such a p p l i c a t i o n s o f a d j o i n t e q u a t i o n s w i l l b e e x e m p l i f i e d i n t h i s and subsequent c h a p t e r s . An A d j o i n t E q u a t i o n for a S i m p l e D i f f u s i o n E q u a t i o n

2.1.

Consider a simple d i f f u s i o n equation

(1.1) with t h e c o n d i t i o n

(Qo i s a g i v e n f u n c t i o n o f

x

(-

F i r s t , w e d e f i n e t h e s p a c e of f u n c t i o n s

6 , on which a s o l u t i o n f o r t h e t-

x - d i f f e r e n t i a b l e and have a g e n e r a l i z e d second d e r i v a t i v e w i t h r e s p e c t t o s h a l l assume t h e s e f u n c t i o n s t o b e f i n i t e i n t h e e n t i r e r a n g e - -' < x < x

L e t us w r i t e E q . ( l . l )

+

2

-,

W e w i l l consider t h a t

and

ri'

formally a s

L = -a + t 0 -

where

and

x. We

t h e r e b y e n s u r i n g a q u a d r a t i c summation, i . e .

(1.3)

= f ,

LQ

a

is

< x < ").

m

problem w i l l b e s o u g h t . Suppose i t i s t h e s p a c e of f u n c t i o n s which a r e

r a p i d l y d e c r e a s i n g when

4,

x ) and u n d e r t h e a s s u m p t i o n t h a t t h e s o l u t i o n

f i n i t e i n t h e e n t i r e range of

a2

f = Qd ( X

b;ixz'

-

x0).

Q is a G i l b e r t space with t h e s c a l a r product

1 1

T (g, h) =

m

dt

0

gh d x .

-m

We now p r o c e e d t o c o n s t r u c t i n g a n a d j o i n t p r o b l e m . T o t h i s e n d , we m u l t i p l y ( 1 . 1 ) by some f u n c t i o n Q * , whose p r o p e r t i e s w i l l b e c l a r i f i e d l a t e r o n , and i n t e g r a t e t h e r e s u l t o v e r t i m e and s p a c e T

j d t 0

] $ * [ $ + f l Q - b ~ , -m

1

dx = Q

I 0

dt

~ + * ~ \ ( x - x od)x . -_

(1.4)

38

Chapter 2

a p p e a r s under

Let u s t r a n s f o r m t h e l e f t - h a n d s i d e so t h a t t h e f u n c t i o n $

t h e i n t e g r a l o u t of t h e p a r e n t h e s i s , whereas t h e d i f f e r e n t i a l r e l a t i o n c o n t a i n i n g $*

a p p e a r s i n t h e p a r e n t h e s i s . For t h i s p u r p o s e , we i n t e g r a t e by p a r t s t o o b t a i n

(1.6)

S u b s t i t u t i o n of ( 1 . 5 ) and ( 1 . 6 ) i n t o ( 1 . 4 ) y i e l d s

Assume t h a t

b * = O

f o r x + + m

Then r e l a t i o n ( 1 . 7 ) i s s i m p l i f i e d a s follOWS T

i d t 0

1

1

$ ( - z + U $ * - p -

-a

= Q

1

dx

+

7

($T$;

- $o$g)

dx =

_m

(1.9)

$* (xo, t ) d t .

0

We now assume t h a t

$*

s a t i s f i e s t h e equation (1.10)

with t h e i n i t i a l d a ta $* = 0

and boundary c o n d i t i o n s ( 1 . 8 ) . H e r e

for p

(1.11)

t = T i s a f u n c t i o n Of

determined. T h i s problem w i l l b e r e f e r r e d t o a s an

x

and

t

t o be

a d j o i n t problem. With due

39

Adjoint Equations o f Transport and Diffusion account f o r (l.lO), we reduce ( 1 . 9 ) t o t h e form

T

0

T

m

dt

1 --

m

@*( x o ,

P@ dx = Q

O(x, 0) $ * ( x ,

t ) dt +

0

0) d x .

(1.12)

-m

Let

(1.13)

be a l i n e a r f u n c t i o n a l of b a s i c problem ( l . l ) , ( 1 . 2 ) .

@ which i s t o b e c a l c u l a t e d from t h e s o l u t i o n of I t f o l l o w s from ( 1 . 1 2 ) t h a t t h i s f u n c t i o n a l can a l s o ( l . l l ) , so t h a t

b e e v a l u a t e d by s o l v i n g a d j o i n t problem ( 1 . 1 0 ) .

(1.14) 0 T h i s is p r e c i s e l y t h e

-m

duality principle.

Consider t h e problem of f u n c t i o n a l s . F u n c t i o n a l ( 1 . 1 3 ) a d m i t s a v a r i e t y of p h y s i c a l meanings. L e t p ( x , t ) = 6(X

- 5)

6(t

-

(1.15)

T).

S u b s t i t u t i n g ( 1 . 1 5 ) i n t o (1.13), w e d e r i v e t h e f u n c t i o n a l

J =

(1.16)

$(L,T),

i . e . t h e v a l u e of t h e s o l u t i o n a t t h e p o i n t

x =

t h a t t h e same v a l u e w i l l b e o b t a i n e d by ( 1 . 1 4 ) i f

5,

t = p

T.

I t s h o u l d be n o t e d

i n a d j o i n t problem (l.lO),

(1.11) i s g i v e n by e x p r e s s i o n ( 1 . 1 5 ) . Consider a n o t h e r c a s e , where it is n e c e s s a r y t o f i n d t h e t o t a l amount of substance i n t h e i n t e r v a l

a 5 x 5 b . In t h i s case, t h e function

p ( x , t ) w i l l be

t a k e n i n t h e form

(1.17)

S u b s t i t u t i n g ( 1 . 1 7 ) i n t o ( 1 . 1 3 ) , we o b t a i n T J =

b

i i dt

O

@ dx.

a

The same f u n c t i o n a l can b e d e r i v e d u s i n g ( 1 . 1 4 )

(1. l a )

40

Chapter 2

Next, suppose t h a t we measure t h e t o t a l amount of s u b s t a n c e during t h e time i n t e r v a l strument depends on by t h e f u n c t i o n

x

IT1,

and

T

2

$ on [ a , b l

1 . Suppose a l s o t h a t t h e r e s o l u t i o n of t h e i n -

y , i . e . t h e instrumental c h a r a c t e r i s t i c is f i t t e d

x = V ( t ) ~ ( x ) .I t i s n e c e s s a r y t o compare t h e measured and

1

i n t h e form

c a l c u l a t e d f u n c t i o n a l s . Then, choosing t h e f u n c t i o n p ( x , t )

P(X, t )

V (x) ~ ( x ) x , E

[a,bl

=

0, x q l a , b l

t0

IT1,

Tzl,

[T

1’

T21,

(1.19)

tp

or

and

we a r r i v e a t t h e f u n c t i o n a l

1

b

‘2

J =

Tl

x

dt

( x ) V ( x ) $dx.

(1.20)

a

The s e t of p o s s i b l e f u n c t i o n a l s c o u l d b e e x t e n d e d . I t i s i m p o r t a n t t o n o t e t h a t i n t h i s way we can r e p r e s e n t any l i n e a r f u n c t i o n a l of a s o l u t i o n a n d , h e n c e , c o n s t r u c t Its own a d j o i n t problem. Thus, i f we c o n s i d e r some f u n c t i o n a l s of a s o l u t i o n

r a t h e r than t h e s o l u t i o n

i t s e l f , w e can f o r m u l a t e a p a r t i c u l a r a d j o i n t problem f o r each of them. I t seems a t f i r s t g l a n c e t h a t t h e most n a t u r a l way i s t o s o l v e t h e b a s i c problem a n d , u s i n g ( 1 . 1 3 ) , c a l c u l a t e any f u n c t i o n a l . Sometimes, t h i s approach i s most f a v o u r a b l e , indeed. Yet, a d j o i n t e q u a t i o n s a r e q u i t e i n d i s p e n s a b l e when we p l a n a e r o s o l - e m i t t i n g p l a n t s , or e s t i m a t e t h e f u n c t i o n a l s e n s i t i v i t y t o t h e ambient parameter v a r i a t i o n s or s o l v e o t h e r s i m i l a r problems. C o n s i d e r , f o r example, t h e f o l l o w i n g problem. Let t h e p r o c e s s of p o l l u t a n t p r o p a g a t i o n i n t h e r e g i o n c r i b e d by ( 1 . 1 ) .

I t i s n e c e s s a r y t o f i n d such a r e g i o n

of t h e form ( 1 . 1 6 )

t =

Q

T~

-

-

t h e amount of p o l l u t a n t a t t h e p o i n t C

does n o t exceed a given c o n s t a n t

i s assumed t o b e l o c a t e d a t t h e p o i n t

v a l u e of t h e f u n c t i o n

W

$

G = (-

m,

m)

be des-

G t h a t the functional x = 5,

a t t h e moment

( t h e p o l l u t i o n s o u r c e of c a p a c i t y

x E w ) . In t h i s problem t h e i n i t i a l

can b e t a k e n z e r o

$ = O

for

t = O

(1.21)

The problem can be s o l v e d i n two ways a t l e a s t . The f i r s t way i s to s o l v e r e p e a t e d l y E q . ( l . l ) f o r d i f f e r e n t v a l u e s of

xO€ G , t o e v a l u a t e f u n c t i o n a l ( 1 . 1 6 ) ,

and, hence, t o determine t h e region i n question

w . T h i s way, however, r e q u i r e s

t h e s o l u t i o n of a g r e a t number of problems of t h e t y p e ( 1 . 1 ) and i s n o t l i k e l y t o be p r a c t i c a b l e . The o t h e r approach i s based on a d u a l r e p r e s e n t a t i o n of f u n c t i o n a l ( 1 . 1 6 ) and t h e u s e of a s o l u t i o n of t h e a d j o i n t e q u a t i o n . For o u r problem, t h i s r e p r e s e n -

41

Adjoint Equations of Transport and Diffusion tation is (1.22)

where

is the solution for the equation

$*

- -”*

+

at

-

U$*

a2

LIT=

$*

6 (x

ax

- < 1)

6(t - T

(1.23)

)

1

with the initial data $* = 0

at

t = T.

(1.24)

Thus, proceeding from dual representation (1.22) of functional (1.16), we can

select a permissible pollution-source region wc

G

by solving the problem

adjoint to (1.1) only once. Let us find the function

$ * which fits (1.23),

(1.24).

Introducing a new

variable

we reduce problem (1.23), (1.24) to the problem

**

+ u$* -

6 (x

@ * = ax2

- 5 1) 6(T - t

-

T1)’

(1.26)

atl

$* = 0

tl = 0

at

It is noteworthy that the operator of problem (1.26) coincides formally with the operator of problem (1.1). The solution for problem (1.26) is given by

-

T)6(<

- C1)6(T

-

T

-

T1).

(1.27)

. dS where

$

(x, t) is the fundamental solution for the operator of problem (1.26)

+

?1

e

dT,

is the

(x, t) = e(t) exp 2 G

i

Heaviside f u n c t i o n

e

5 0,

(t) =

> 0. Referring again to the previous variables we obtain

Chapter 2

42

(1.28)

i

0

In accordance w i t h ( 1 . 2 7 ) , d u a l r e p r e s e n t a t i o n ( 1 . 2 2 ) of f u n c t i o n a l ( 1 . 1 6 )

is

(1.29)

dt

or

J

where

t . = jAt, A t = T /K. J 1 Consider f u n c t i o n a l (1.29) a s a f u n c t i o n of J ( x o ) f o r

t h i s function. A permissible pollution-source region equality t h e region

xoE

G

and p l o t

w i s found from t h e i n -

J ( x ) < C . The above example i s s o l v e d g r a p h i c a l l y i n f i g u r e 2 . 1 , where w

i s denoted by a double l i n e on t h e a b s c i s s a .

L

I

0

L

k

X FIGURE 2.1

P o l l u t a n t d i s t r i b u t i o n f o r any p a r t i c u l a r x E w s o l u t i o n of problem ( l . l ) , above. a s f o l l o w s

i s determined from t h e

(1.2), which i s w r i t t e n , w i t h due r e f e r e n c e t o t h e

Adjoint Equations o f Transport and Diffusion

43

The i s o l i n e s of t h e f u n c t i o n 4 (x, t ) a r e shown i n f i g u r e 2 . 2 .

FIGURE 2 . 2 Thus, we have c o n s i d e r e d a n o n - s t a t i o n a r y problem. I f t h e b a s i c problem

is stationary,

(1.32) $ = 0

x

when

+

f

m

t h e a d j o i n t problem i s a l s o s t a t i o n a r y

(1.33)

$*

when x

= 0

Clearly, the linear functional

-f

f

m

i n t h i s c a s e i s of t h e form

J

1

m

J =

P (x)8 d x

(1.34)

$* (xo).

(1.35)

-m

or J =

If t h e function

Q

p(x) i s taken a s

P(X) = 6 (x

- E),

t h e s o l u t i o n of a d j o i n t problem ( 1 . 2 2 ) , j u s t a s ( 4 . 8 ) of

(1.36) 1.4

is of t h e form

44

Chapter 2

(1.37)

for

I f we s e e k a s o l u t i o n

I

p ( x ) o t h e r t h a n (1.36), w e h a v e

m

@*(x) =

P ( X ) $* ( x ,

5)

(1.38)

d<,

_m

Here

$* (x,

function

5 ) i s t h e f u n d a m e n t a l s o l u t i o n (1.37). I n p a r t i c u l a r , i f t h e

p ( x ) i s t a k e n i n t h e form g i v e n by (1.17), w e o b t a i n

@*(x) =

]

@ *( x ,

5) dS

a S u b s t i t u t i o n o f (1.37) i n t o t h i s e x p r e s s i o n y i e l d s +*(x) = 1 (2 20

-

exp

1-

G(b-x)}-

-

exp

=(x-a)})

(1.39)

T h i s f u n c t i o n i s p l o t t e d i n f i g u r e 2.3.

a-8 2

0

FIGURE 2.3

W e now c o n s i d e r s u b s t a n c e t r a n s p o r t by d i f f u s i o n w i t h d u e a c c o u n t f o r a d v e c t i o n . I n t h i s c a s e , t h e b a s i c problem i s

& ! at

+

dx

+ 04 -

a z $ = &6(x

ax*

-

xo),

(1.40) $ = @ o

t = o

at

@ = O

as

x - t f m

S i m i l a r l y t o t h e f o r e g o i n g e x a m i n a t i o n , t h e a d j o i n t p r o b l e m i s o f t h e form

_ -aa @t * -

u

+* =

* 3X

0

+*

a2 + 0$'* - p = p,

for

dX2

t = T, x

(1.41) +

f

m.

Adjoint Equations o f Transport and Diffusion

45

The b a s i c f u n c t i o n a l of t h e p r o b l e m i s T J =

m

i i dx

0

(1.42)

P@ d x ,

-m

and i t s d u a l r e p r e s e n t a t i o n i s

(1.43)

C o n s i d e r a n o t h e r example r e l a t e d t o c o n s t r u c t i o n p l a n n i n g . L e t t h e p r o p a g a tion t o pollutant u = const > 0

i n t h e region

rp

and $

0

G

by d e s c r i b e d by p r o b l e m ( 1 . 4 0 ) for

= 0. I t is necessary t o determine a point

x

0

for locating

a p o l l u t i o n s o u r c e such t h a t

@(<,, T1) < c , lxo -

<,I

= min,

(1.44)

(1.45)

x o G~

where C 1, 'rl,

a r e given constants.

C

To s o l v e t h i s p r o b l e m , w e w i l l a g a i n employ t h e d u a l r e p r e s e n t a t i o n of f u n c t i o n a l (1.16)

J = Q

7

b*

(xo, t ) d t .

(1.46)

0

In t h i s c a s e ,

$*

i s t h e s o l u t i o n for E q . ( 1 . 4 1 ) w i t h t h e r i g h t - h a n d s i d e o f

t h e form g i v e n by ( 1 . 1 5 ) . A f u n d a m e n t a l s o l u t i o n f o r t h e o p e r a t o r of problem (1.41)

is o b t a i n e d by c a l c u l a t i n g t h e F o u r i e r t r a n s f o r m o f t h e e q u a t i o n (1.47)

Here

tl = T

- t.

Hence, w e a r r i v e a t (1.48)

where

The s o l u t i o n o f E q . ( 1 . 4 8 )

rr

is t h e function

46

Chapter 2 F[@] ([, t ) = B ( t ) exp ( - ( i u g + a + ~ 5 ' ) t l l . 1 1

Performing t h e i n v e r s e F o u r i e r t r a n s f o r m we o b t a i n t h e fundamental S o l u t i o n i n quest ion

(1.49)

Using (1.27) and t h e above v a r i a b l e s , we o b t a i n a s o l u t i o n for t h e problem, which is a d j o i n t t o ( 1 . 4 0 ) , w i t h t h e r i g h t - h a n d s i d e

p = 6(x

- 5 1) 6 ( t -

T

)

1

S u b s t i t u t i n g (1.50) i n t o ( 1 . 4 6 ) , we d e r i v e an e x p r e s s i o n f o r t h e f u n c t i o n a l studied

Here

t . = j A t , A t = T /K. J 1 S i m i l a r l y t o t h e above p r o c e d u r e , we s h a l l d e t e r m i n e t h e r e g i o n u c G i n

which i n e q u a l i t y ( 1 . 4 4 ) remains v a l i d . The p o i n t

x

i s found from c o n d i t i o n

(1.45). Figure 2 . 4 i s a g r a p h i c a l s o l u t i o n f o r t h i s problem. F i g u r e 2 . 5 g i v e s

FIGURE 2 . 4 t h e i s o l i n e s of t h e f u n c t i o n ( 1 . 4 0 ) , with

x

@ ( x , t ) , which a r e s o l u t i o n s f o r d i r e c t problem

found from t h e s o l u t i o n of t h e a d j o i n t problem.

Allowing f o r t h e formal c o i n c i d e n c e of t h e o p e r a t o r s of problems ( 1 . 4 0 ) and ( 1 . 4 7 ) , we can e x p r e s s t h e f u n c t i o n $ ( x , t ) a n a l y t i c a l l y a s

47

30 28 26 24

22 20

78 76 74 72 70 8 6 4 2

\ ,I

,,,,,,,,, I,,,,,,,,, FIGURE 2 . 5

(x-x - u ( t - t $(x,t) =

Qbt ~

2J;;; where

K-l

exp

t-

b t j )

1

.))2

; f i ( t - t . )J

+

+ O(&t),

Jt-t J

j=o

11 (1.52)

K = [t/At]. I f problem ( 1 . 4 0 ) i s s t a t i o n a r y : (1.53)

$ = 0

when

x

+

?

m,

i t w i l l a l s o g e n e r a t e a s t a t i o n a r y a d j o i n t problem

-

u

** ax

a 2 p

fi 7 + p ( x ) ,

+ u$* =

$* = 0

ax

for

The b a s i c problem ( 1 . 5 3 ) was s o l v e d i n

x

(1.53') + ?

m.

1.4 (see (4.14)).

Let us s o l v e t h e a d j o i n t problem. To t h i s e n d , we f i r s t f i n d a fundamental s o l u t i o n of ( 1 . 5 3 ' ) f o r

p(x) = 6(x

- 5).

S i m i l a r l y t o ( 4 . 1 4 ) , we o b t a i n

4fi2

2fi

(1.54)

48

Chapter 2

Any o t h e r s o l u t i o n of

( 1 . 5 3 ' ) can b e d e r i v e d from t h e fundamental s o l u t i o n m

i f p ( x ) i s t a k e n i n t h e form (1.171, t h e n we have

In p a r t i c u l a r ,

@*(x) =

]

9 * ( x , S ) dS.

(1.56)

a S u b s t i t u t i o n of

(1.54)

i n t o (1.56) y i e l d s

(1.57)

The f u n c t i o n

@*(x)

is p l o t t e d i n f i g u r e 2 . 6 .

FIGURE 2.6

I f we d e a l w i t h a s e t of b a s i c s t a t i o n a r y problems r e l a t e d t o v a r i o u s t y p e s of motion a t d i f f e r e n t a i r s t r e a m v e l o c i t i e s , w e s h o u l d c o n s i d e r a r e l e v a n t set of a d j o i n t problems.

In t h i s c a s e , t h e f u n c t i o n a l i n q u e s t i o n is o b t a i n e d e i t h e r

from t h e b a s i c e q u a t i o n s m

(1.58)

or from t h e a d j o i n t e q u a t i o n s

49

Adjoint Equations of Transport and Diffusion (1.59)

2.2.

A General Case of an Adjoint Problem f o r a Three-Dimensional

Region Consider a g e n e r a l t h r e e - d i m e n s i o n a l problem

~

a+ + at

d i v &$

+

a + u@ - a2

= O

%dn = o

a+

v - - PA$

= f ,

a2

on

1

for

u

< O

on

1

for

u

:O (2.1)

Let u s d e r i v e a d j o i n t e q u a t i o n s . Assume t h a t t h e s o l u t i o n f o r problem ( 2 . 1 ) belongs t o t h e H i l b e r t s p a c e

of s u f f i c i e n t l y smooth f u n c t i o n s , each element

0

of t h e space b e i n g q u a d r a t i c a l l y summable, i . e . T

dt O

1

-.

$* dG <

G

M u l t i p l y Eq.(2.1) by a f u n c t i o n

$*

and i n t e g r a t e t h e r e s u l t o v e r t h e

e n t i r e domain of t h e s o l u t i o n . Then we have T

1

O

I 1

T $*

dt

i

at a $ dG

G

+

dt

O

dt

5

G

O

T

-

1 I

T $ * d i v L$dG +

$ * -a V v d”$G

dt

az

-

dZ

1 i dt

p

G

O

T

@*A$

dG =

G

T

j

O

dt

dt D

I n t e g r a t i n g by p a r t s , u s i n g t h e Ostrogradsky-Gauss relation

,/

$*f dG.

G

theorem, and a l l o w i n g f o r t h e

div u = 0 , we can t r a n s f o r m some e x p r e s s i o n s of ( 2 . 2 )

1

G

1 I

T

+* d i v

K@dG=

dt

0

s

(2.2)

T

O

@ $ * dG

G

j 1

t o t h e form

T

un $ + * dS

-

dt

O

G

@ div

!+*

dG,

(2.4)

50

Chapter 2

(2.5)

S u b s t i t u t i n g ( 2 . 3 ) - ( 2 . 6 ) i n t o ( 2 . 2 ) , we o b t a i n T

dt

1

$

(- a$* -

+

d i v u $*

o$*

~

O

-

a az

Li

**az

p

)

A$*

dG =

G

I I

T

=

dt

O

$*f d t

1

1 $o$G

$T$; dG +

G

T

dG

-

G

(2.7)

T

- i d t i

0

-

G

un$@*dS+jdtj V ( $ * g - @ z )

s

O

$*

Let us assume t h a t

s a t i s f i e s t h e equation

t h e boundary c o n d i t i o n s f o r t h e f u n c t i o n

$

For t h i s p u r p o s e , we w i l l employ a s g i v e n by (2.1) and t h e c o n d i t i o n

i s p e r i o d i c i n t i m e . The f u n c t i o n

$

time-periodic with period

-

T . Then we have

$T8G dG + G

-

CH

and t r a n s f o r m t h e r i g h t - h a n d s i d e of ( 2 . 7 ) .

t h a t the function

dZ

i

G

$08G

dG = 0 .

8*

w i l l be a l s o assumed

51

Adjoint Equations o f Transport and Diffusion

S i n c e , by t h e a s s u m p t i o n , $ = 0

on

u

if

S,

< 0,

then T

)

dt

0

1

dt

o

s

Here i t i s a l s o t a k e n

! 1

T

un $$* dS =

W

= 0

:u

z = 0, z = H.

for

(2.10)

$ $ * dl:

c

Next, w e have

(2.11)

E x p r e s s i o n s ( 2 . 1 1 ) and ( 2 . 1 2 ) i n v o l v e t h e b o u n d a r y c o n d i t i o n s from ( 2 . 1 ) f o r

z = H and

a n d z = 0 , r e s p e c t i v e l y . And f i n a l l y , a s s u m i n g

a@/h

= 0

on

X for u

$ = 0

Z

for

u

< 0

2 0, we obtain

(2.13)

L

Here

X+

= {(x, y, z ) E

XI un 2 0 1 , C-=

{ ( x , y , z ) E Z 1 un < 0 )

With d u e r e g a r d t o (2.8)-(2.13) w e h a v e

52

Chapter 2 So f a r , n o boundary c o n d i t i o n s h a v e been f i x e d f o r t h e a d j o i n t problem

s o l u t i o n . Now

( i n c o n j u n c t i o n w i t h ( 2 . 8 ) ) w e assume t h a t

r$*

on

= 0

1

for u

<

o (2.15)

9*

(r, T)

= $*(G 0 )

In t h i s c a s e , t h e d u a l r e l a t i o n t a k e s t h e form

(2.16)

I f t h e b a s i c f u n c t i o n a l i s d e f i n e d by t h e e q u a l i t y

T J =

i

dt

O

p$ dG,

(2.17)

$ * f dG.

(2.18)

G

i t s i d e n t i c a l dual r e p r e s e n t a t i o n i s

1 1

T J =

dt

O Choosing d i f f e r e n t

p-functions,

G

w e c a n d e r i v e d i f f e r e n t f u n c t i o n a l s and

corresponding a d j o i n t e q u a t i o n s . I t i s noteworthy t h a t t h e above method c a n be u s e d t o d e r i v e a somewhat d i f f e r e n t a d j o i n t problem f o r t h e b a s i c e q u a t i o n s of t h e form ( 3 . 2 7 ) i n c h a p t e r 1, a d d i t i o n a l c o n d i t i o n s f o r t h e problem b e i n g

ar$* -

_ -a t

div

= 0 and w = 0 a t z = 0 , z = H.

div u $ * + or$* = Dr$* + p ,

r$* = 0

for

r$* = 0

on

t = T

1

53

Adjoint Equations o f Transport and Diffusion

We n o t e i n c o n c l u s i o n t h a t o u r p r i m a r y problem was homogeneous w i t h r e s p e c t t o t h e boundary c o n d i t i o n s a n d i n i t i a l d a t a . However, t h e above a n a l y s i s can e a s i l y b e e x t e n d e d t o inhomogeneous c o n d i t i o n s f o r t h e b a s i c problem.

In t h e l a t t e r

c a s e , o n l y t h e form o f f u n c t i o n a l ( 2 . 1 8 ) c h a n g e s t o i n c l u d e new terms r e l a t e d t o t h e inhomogeneity. Some f e a t u r e s u n d e r l y i n g t h e s t a t e m e n t of two-dimensional worth m e n t i o n i n g . S i n c e w e h a v e a l r e a d y d e f i n e d two-dimensional

problems a r e a l s o b a s i c problems

( s e e Eqs.(3.34) and (3.35) of t h e p r e v i o u s c h a p t e r ) , we c a n d e r i v e , u s i n g t h e above methods, t h e c o r r e s p o n d i n g a d j o i n t p r o b l e m s . Thus, t h e problem a d j o i n t t o

=

u A$*

=

T,

+ p,

(2.19) for u $'* = 0

on

1

for

u

<

~

0,

o

and t h a t t o (3.35),

$'* =

for

$;'

$'* = 0

on

t = T,

(2.20)

1

The c o r r e s p o n d i n g f u n c t i o n a l s a r e of t h e form

(2.21)

T J =

r J

O

1r

f $ * dG.

(2.22)

G

L e t u s c o n s i d e r t h e problem of d e t e r m i n i n g t h e amount o f t h e p o l l u t a n t which

i s e x p e c t e d t o f a l l down on t h e u n d e r l y i n g s u r f a c e . As was shown i n c h a p t e r 1, t h i s problem c a n b e s o l v e d i n t e r m s of a two-dimensional

t ion

model u n d e r t h e assump-

54

Chapter 2

(2.23)

(H

i s t h e h e i g h t of a c y l i n d r i c a l r e g i o n ) . The problem w i l l b e f o r m u l a t e d a s f o l l o w s . Let an amount

4

[ O , T I a t a point

Q

of t h e

It is a n e c e s s a r y t o d e t e r m i n e t h e t o t a l amount of t h e p o l l u t a n t t h a t w i l l f a l l down a t

pollutant

b e d i s c h a r g e d a t a moment T

E

r E G.

r E @ of t h e s u r f a c e a t t h e moment T E I T . T I . Basing on t h e r e s u l t s -1 1 J of t h e p r e c e d i n g c h a p t e r and a c c o r d i n g t o t h e f o r m u l a t i o n of t h e problem, we can

the point

w r i t e t h e b a s i c f u n c t i o n a l (2.21) i n t h e form -

w + av J = O L (El,Tl).

(2.24)

-

i s t h e a b s o l u t e v e l o c i t y of t h e p a r t i c l e f a l l d o w n due t o g r a v i t y . The g i d e n t i c a l d u a l r e p r e s e n t a t i o n of t h e f u n c t i o n a l i s

where

w

J =

QO* (&, To).

Suppose t h e boundary of t h e r e g i o n points

G , x1

(2.25)

i s s u f f i c i e n t l y f a r away from t h e

G

and t h e d i s c h a r g e e f f e c t n e a r t h e boundary i s n e g l i g i b l e . Then

we can employ t h e f o l l o w i n g model:

(2.26)

w

;=-

+ av H

The c o r r e s p o n d i n g a d j o i n t problem i s w r i t t e n a s

@ *=

0

for

@* o

for

+

(2.27)

t = T,

1'

+

m

S o l u t i o n s of problems (2.26) and (2.27) a r e o b t a i n e d from t h e fundamental s o l u t i o n s of t h e c o r r e s p o n d i n g o p e r a t o r s . The fundamental s o l u t i o n s f o r one-dimens i o n a l d i f f u s i o n e q u a t i o n s i n c l u d i n g t h e a d v e c t i v e t e r m s were d e r i v e d i n of t h i s c h a p t e r . The fundamental s o l u t i o n s

' L ' L

0, $ *

(2.26), (2.27) can be found i n a s i m i l a r way:

2.1

f o r t h e o p e r a t o r s of problems

Adjoint Equations o f Transport and Diffusion

where

-

tl = T

55

t . Formulas (2.28), (2.29) were d e r i v e d assuming t h a t u = c o n s t > O ,

v = const > 0. With t h e l a t t e r r e l a t i o n s i n mind, w e a r r i v e a t t h e f o l l o w i n g s o l u t i o n s f o r t h e d i r e c t and a d j o i n t problems:

(2.30)

(2.31)

where

F i g u r e s 2.7 and 2.8 show t h e i s o l i n e s f o r t h e f u n c t i o n s $ ( x . y , T ~ )and ( x , y , ‘to),

$

respectively.

Thus, t h e amount of t h e p o l l u t a n t f a l l i n g o u t a t a g i v e n p o i n t can be determined by s u b s t i t u t i n g e i t h e r (2.30) i n t o (2.24) or (2.31) i n t o (2.25).

I t s h o u l d b e s t r e s s e d h e r e t h a t a l t h o u g h t h e f i n a l r e s u l t does n o t depend on whether t h e b a s i c f u n c t i o n a l or i t s d u a l r e p r e s e n t a t i o n h a s been u s e d , s o l u t i o n s f o r t h e d i r e c t and a d j o i n t problems y i e l d q u a l i t a t i v e l y d i f f e r e n t d a t a . For example, s o l v i n g t h e d i r e c t problem, we g a i n i n f o r m a t i o n on t h e s p a t i a l and temporal d i s t r i b u t i o n s of t h e p o l l u t a n t f o r a e r o s o l d i s c h a r g e a t a f i x e d p o i n t and a t a g i v e n moment. On t h e o t h e r hand, s o l u t i o n of t h e a d j o i n t problem g i v e s i n f o r m a t i o n on how much of t h e p o l l u t a n t e m i t t e d by a p o l l u t i o n s o u r c e a t an a r b i t r a r y moment w i l l f a l l down a t a f i x e d p o i n t and a t a given moment. T h i s i m p l i e s , f o r i n s t a n c e , t h a t i f we do n o t know a p r i o r i t h e t i m e and t h e c o o r d i n a t e s of a s i n g l e e m i s s i o n , t h e problem f o r t h e p o i n t ( r l , u s i n g d u a l r e p r e s e n t a t i o n (2.25) of t h e f u n c t i o n a l .

T

1

)

should be solved

56

Chapter 2 Value 0 1 of l e v e l 5.7765

Value 20 of l e v e l 1 . 1 5 5 3

30

28 26 24 22 20 78 76

74 72 70

a

, , , , , , ~ , , , , ~ , , , , ~, ,, , ,, ,,, ~, , , 1

2

4

6

8

70

, / , , I/ ,

72 74 76

, ,

1 , , , , I/

78 20

/

,

I

1 1 I I I I I C ~ ~ I f i ~ ~ , I , , / ~

22 24

26 2 8

30

FIGURE 2 . 7

Value 01 of l e v e l 5.7765

Value 20 of l e v e l 1 . 1 5 5 3

24

FIGURE 2 . 8 2 . 3 . Uniqueness of t h e S o l u t i o n f o r t h e A d j o i n t Problem We now d e m o n s t r a t e t h a t t h e s o l u t i o n for a d j o i n t problem (2.15)

i s unique.

To t h i s e n d , we m u l t i p l y t h e e q u a t i o n of ( 2 . 1 5 ) by # * and i n t e g r a t e t h e r e s u l t over t h e e n t i r e domain of t h e s o l u t i o n . Then we have

57

Adjoint Equations of Transport and Diffusion

S i m i l a r l y t o t h e above c o n s i d e r a t i o n s , w e transform t h e i n t e g r a l s a s f 011 ows :

(3.2)

(3.3)

o

c

(3.4)

Using ( 3 . 2 ) - ( 3 . 4 ) , w e reduce ( 3 . 1 ) t o t h e form

(3.5) J

I:

cH

+

1

I 1

T

v$*

dC}

=

dt

O

p$* dG.

G

cO

S i n c e t h e problem is p e r i o d i c , w e have T

j d t j % d G = O . O

N e x t , by v i r t u e of t h e c o n d i t i o n

G

(3.6)

58

Chapter 2

b*

1

on

= 0

for

o

<

u

(3.7)

we have

(3.8) L

and by v i r t u e of t h e c o n d i t i o n

(3.9) we o b t a i n

i i

T

dt

u:$*.

dT:

+

x+

O

i i

T

dt

dZ

p$*

= 0.

(3.10)

x+

O

As f o r t h e l a s t b u t one terms on t h e l e f t - h a n d s i d e of (3.5), we have a c c o r d i n g t o t h e boundary c o n d i t i o n s on

for (3.7)-(3.11), r e l a t i o n (3.5) t a k e s t h e form

Thus, w i t h an allowance

(3.12)

1 i

T + p

dt

O

T

T

(V@*)z

dG + ctV

$**

dt

G

dX

=

i

dt

O

0

p$* dG. G

R e l a t i o n (3.12) w i l l b e b a s i c a l l y used t o p r o v e t h e u n i q u e n e s s of t h e s o l u t i o n f o r problem (2.15). Indeed, l e t t h i s problem have two e q u a l s o l u t i o n s , $;

and

9;.

9* -9*2 1

Then f o r t h e d i f f e r e n c e w* =

_ - aa t ~ -* d i v p

aw* a n

uw* + 5w* -

a

-

az

v

we have t h e homogeneous problem

** az

-

0 = 0,

p A

+ u w * = o

on

1

for

u

L O ,

o

on

1

for

u

<

n

w* =

o

(3.13)

59

Adjoint Equations o f Transport and Diffusion

&* az

= aw*

aZ

=

o

on

lo

on

1, (r , 0)

(r , T) = w *

w*

The r e l a t i o n of t h e form (3.12), c o r r e s p o n d i n g t o t h i s problem, i s

(3.14)

Since

a, v ,

u,

a 2 0 , t h i s r e l a t i o n h o l d s o n l y f o r w * = 0 , i . e . when $* =@* 1

2'

Thus, t h e u n i q u e n e s s of t h e problem s o l u t i o n i s proved. Of c o u r s e , i n t h e proof we have t a c i t l y assumed t h a t a l l t h e f u n c t i o n s i n v o l v e d a r e smooth, which e n s u r e d t h e v a l i d i t y of a l l t h e t r a n s f o r m a t i o n s made. S i m i l a r l y , we might prove t h e uniqueness theorem f o r a mixed problem a s w e l l , b u t i n t h a t c a s e we should have t o s e t $* = $+

for

t

=

T

and s o l v e t h e problem f o r d e c r e a s i n g

t . Precisely

under t h e s e c o n d i t i o n s t h e s o l u t i o n a l g o r i t h m g i v e s c o r r e c t r e s u l t s . I n c o n c l u s i o n , i t i s p e r t i n e n t t o p o i n t o u t t h a t b o t h t h e b a s i c and t h e

adjoint problems a l s o admit a f o r un < 0 , a@/an = 0 LI

unique s o l u t i o n when t h e c o n d i t i o n s

on 1 f o r u

a$*/an + u $* = 0 on I: f o r u

Q

=

0 on &

2 0 g i v e n by ( 2 . 1 ) . and t h e c o n d i t i o n s

2 0 , $* = 0 on Z f o r u

r e p l a c e d by t h e c o n d i t i o n s $ = 0 on C and

$ * = 0 on

< 0 g i v e n by ( 2 . 1 5 ) a r e

Z for a l l u . n

2 . 4 . A d j o i n t Equation and Lagrange I d e n t i t y

I n t h e p r e c e d i n g s e c t i o n s of t h i s c h a p t e r we s u g g e s t e d a method of d e r i v i n g a d j o i n t e q u a t i o n s for t h e e q u a t i o n s o f a e r o s o l t r a n s p o r t and d i f f u s i o n . These e q u a t i o n s can a l s o b e o b t a i n e d from g e n e r a l c o n s i d e r a t i o n s using t h e Lagrange i d e n t i t y . L e t u s analyse t h e procedure f o r d e r iv in g a d j o i n t equations f o r a g e n e r a l e v o l u t i o n e q u a t i o n . Thus, we s h a l l h a n d l e t h e problem

a$

+

@=$o where t h e l i n e a r o p e r a t o r functions

$€

A

A$ at

=

f,

(4.2)

t = O

i n t h e H i l b e r t s p a c e i s d e f i n e d on t h e s e t of

3 . Each element of t h i s o p e r a t o r meets t h e smoothness c o n d i t i o n s ,

some a d d i t i o n a l ( s a y , boundary) c o n d i t i o n s , and o t h e r r e q u i r e m e n t s i n h e r e n t

60

Chapter 2

i n t h e problem. T h i s p o i n t w i l l be i l l u s t r a t e d below by way o f s e v e r a l e x a m p l e s . L e t t h e s c a l a r p r o d u c t of two f u n c t i o n s , g

and

h , i n t h e H i l b e r t space

a

be d e f i n e d a s f o l l o w s :

I

T ( g , h) =

dt

O

gh dG,

(4.2)

G

where [ O , TI i s t h e r a n g e of t h e v a r i a b l e

t , and

G

i s t h e r a n g e of t h e s p a t i a l

v a r i a b l e s . Suppose, f o r c e r t a i n t y , t h a t t h e problem i s p e r i o d i c and p u t $ (I-, T) = =

Q ( r , 0). L e t E q . ( 4 . 1 ) be f o r m a l i z e d f u r t h e r by w r i t i n g LQ

a

where

L = -

at

operator

L*

+

A.

f ,

=

(4.3)

L

With t h e l i n e a r o p e r a t o r

we associated the adjoint

t h r o u g h t h e Lagrange i d e n t i t y ( 9 , Lh) = ( h , L*g),

where t h e o p e r a t o r Putting

L

and t h e f u n c t i o n s

g

(4.4)

and

h

a r e assumed t o be r e a l .

h = Q , g = $ * , w e have

LQ) = ( $ , I,*$*).

($*,

Since Eq.(4.3)

(4.5)

is v a l i d , t h e formal e x p r e s s i o n L*@* = p ,

(p

(4.6)

b e i n g s t i l l a n unknown f u n c t i o n ) l e a d s t o t h e f o l l o w i n g form o f ( 4 . 5 ) :

(Q*, f ) If

p

= ($, P ) .

i s a c h a r a c t e r i s t i c of measurement

(4.7) ( s a y , of an i n s t r u m e n t ) or of

a s y s t e m of measurements ( s a y , t h e sum o f measurements), J

= ($,

our basic functional is

P).

(4.8)

E x p r e s s i o n ( 4 . 7 ) i m p l i e s t h e d u a l formula J = ($*.

f).

(4.9)

W e now employ t h i s method t o s o l v e a p a r t i c u l a r problem.

2 + at

div

3

+ a$

+ = o

a

= -

v

on

1

aZ

az +

MA+

+ f, (4.10)

Adjoint Equations o f Transport and Diffusion

A s p r e v i o u s l y , w e assume t h a t t h e components

u, v,

61

and

satisfy the

w

conditions = 0

div

(4.11)

f o r z = O , z = H

w = O Consider t h e space

8

of q u a d r a t i c a l l y summable f u n c t i o n s which h a v e d e r i -

v a t i v e s with respect t o a l l t h e v a r i a b l e s , generalized second-order d e r i v a t i v e s z , x , a n d y , a n d which meet c o n d i t i o n s ( 4 . 1 0 ) . The s c a l a r p r o d u c t

with respect t o

i s d e f i n e d by

i 1

T (g, h) =

dt

O

gh dG.

G

W e w r i t e ( 4 . 1 0 ) for t h e f u n c t i o n s o f t h e H i l b e r t s p a c e

@

i n t h e form

= f ,

* + A *

at

(4.12)

U s i n g L a g r a n g e i d e n t i t y ( 4 . 4 ) and w r i t i n g o u t i t s l e f t - h a n d s i d e e x p l i c i t l y ,

we o b t a i n T ( g , Lh) = O

=

1

gThT dG

G

g

dt

(x+ ah

+ Oh

d i v uh -

-

a V az

ah dz

- UAh)

7

dG =

G

-

i

1 I

T goho dG +

dt

o

G

1

T ungh dC

+

z

(i

dt

O

j

gh dG

-

G

(4.13) T

T

T

- I d t [ ‘H

U g g d Z

+!dt! O

u g 2 d Z

zo

+ I d t l

‘H

U h Z d I

-

62

Chapter 3

T

dt

-

O

h d i v &g dG. G

I d e n t i t y ( 4 . 1 3 ) goes o v e r i n t o t h e e x p r e s s i o n T

The o p e r a t o r a c t i n g on t h e f u n c t i o n

g

i n t h e expression enclosed in

p a r e n t h e s e s w i l l be denoted by L* =

a

-at

a

a

+ o - aZ v az +

div (us)

-

UA.

(4.15)

Then, t h e r i g h t - h a n d s i d e of ( 4 . 1 4 ) i s a s c a l a r p r o d u c t of t h e form ( h , L * , g ) . Hence, w e n a t u r a l l y a r r i v e a t i d e n t i t y ( 4 . 4 ) and a d j o i n t o p e r a t o r (4.15) i n t h e s p a c e 9. Let u s d e f i n e f o r m a l l y t h e a d j o i n t problem a s L*$* = p ,

(4.16)

or i n e x p l i c i t form

(4.17)

$ * ( r .T)

=

$*(x,0 )

In t h a t c a s e , we o b t a i n any p a r t i c u l a r f u n c t i o n a l of

p . R e a l l y , i f we t a k e

h = $ , g = $*,

J , depending on t h e c h o i c e

t h e n , a c c o r d i n g t o ( 4 . 7 ) , we have

f u n c t i o n a l s ( 4 . 8 ) and ( 4 . 9 ) .

Chapter 3. NUMERICAL SOLUTION OF BASIC AND ADJOINT EQUATIONS T h i s c h a p t e r d i s c u s s e s t h e methods used t o s o l v e e v o l u t i o n problems a s a p p l i e d t o s u b s t a n c e t r a n s p o r t and d i f f u s i o n i n t h e atmosphere. I t w i l l he assumed t h a t t h e problem s o l u t i o n s a t i s f i e s homogeneous c o n d i t i o n s on t h e boundary of t h e

Numerical Solution of Basic and Adjoint Equations

63

r e g i o n . A s r e g a r d s t h e e q u a t i o n i t s e l f , i t i s n a t u r a l l y inhomogeneous. The s p l i t t i n g method i s t h e main a l g o r i t h m for s o l v i n g such problems. 3.1. Elements of G e n e r a l S p l i t t i n g Theory

A c o m p l i c a t e d problem e n c o u n t e r e d i n m a t h e m a t i c a l p h y s i c s c a n be reduced i n many c a s e s t o s i m p l e r problems s o l v a b l e s u c c e s s i v e l y and e f f e c t i v e l y by a computer. T h i s r e d u c t i o n p r o v e s f e a s i b l e i f t h e i n i t i a l p o s i t i v e s e m i d e f i n i t e o p e r a t o r c a n be r e p r e s e n t e d a s t h e sum of s i m p l e p o s i t i v e s e m i d i f i n i t e o p e r a t o r s . Such methods w i l l be r e f e r r e d t o a s s p l i t t i n g - u p methods. The methods f o r s p l i t t i n g n o n s t a t i o n a r y problems were o r i g i n a l l y developed by G.Douglas, D.Peaceman, and H.Rachford. S u b s e q u e n t l y , t h e y were g r e a t l y advanced by S o v i e t m a t h e m a t i c i a n s K.A.Bagrinovsky and S.K.Godunov, N.N.Yanenko, A.A.Samarsky, E.G.Dyakonov, V.K.Saulev, G.I.Marchuk.

A t f i r s t , t h e s p l i t t i n g - u p methods were f o r m u l a t e d and s u b s t a n t i a t e d t h e o r e t i c a l l y f o r s i m p l e problems w i t h commutative p o s i t i v e d e f i n i t e o p e r a t o r s . C l e a r l y , a s a p p l i e d t o problems of t h i s k i n d i n t h e s t u d i e s of v a r i o u s a u t h o r s , t h e s p l i t t i n g - u p methods proved e i t h e r a l m o s t s i m i l a r or e q u i v a l e n t b u t d i f f e r i n g o n l y i n t h e scheme of t h e i r i m p l e m e n t a t i o n . S i n c e t h e s e methods have become n e a r l y c l a s s i c a l t o d a y , w e w i l l n o t p e r s o n i f y them h e r e . Those w i s h i n g t o d e l v e more d e e p l y i n t o t h e d e t a i l s of t h e a l g o r i t h m s may s t a r t w i t h t h e r e f e r e n c e s t o o r i g i n a l l i t e r a t u r e . L a t e r o n , t h e r a n g e of n o n - t r i v i a l problems s o l v e d by t h e s p l i t t i n g methods w a s c o n s i d e r a b l y e x t e n d e d . C u r r e n t l y , t h e s p l i t t i n g - u p methods p r o v i d e p o w e r f u l mathematics u s e d t o s o l v e v e r y c o m p l i c a t e d problems of m a t h e m a t i c a l p h y s i c s . S i n c e t h e s p l i t t i n g - u p method t h e o r y h a s been e l a b o r a t e d most compreh e n s i v e l y f o r t h e c a s e where t h e i n i t i a l o p e r a t o r i s g i v e n a s t h e sum of two s i m p l e r o p e r a t o r s , w e w i l l s t a r t w i t h t h e e x p o s i t i o n of p r e c i s e l y t h i s c a s e . D e s p i t e t h e a v a i l a b i l i t y of multinumerous s p l i t t i n g - u p methods, t h e most u n i v e r s a l and g e n e r a l l y a p p l i e d i s , i n t h e o p i n i o n of t h e a u t h o r , component s p l i t t i n g . W e hope t h a t t h i s c i r c u m s t a n c e w i l l b e d u l y

c o n s i d e r e d by t h e r e a d e r

of t h i s c h a p t e r . As f o r t h e o t h e r methods o u t l i n e d below, t h e y a r e r a t h e r o f methodical

nature.

L e t u s c o n s i d e r t h e e v o l u t i o n problem % a t+ A $

= f ,

(1.1) $

with t h e o p e r a t o r

= g

in

at

t = O

A ? 0 r e p r e s e n t e d i n t h e form A = A

where

D

1

+ A

2

(1.2)

64

Chapter 3

A1 L O ,

(1.3)

A > O .

2

Assume t h a t t h e s o l u t i o n of problem (1.1) i s s u f f i c i e n t l y smooth. W e now c o n s i d e r t h e component s p l i t t i n g - u p methods u n d e r t h e assumption t h a t problem (1.1) h a s a l r e a d y been r e d u c e d t o a d i f f e r e n c e form a n d , h e n c e , t h e o p e r a t o r s A , A1,

and A2 a r e m a t r i c e s . L e t A 1 ( t ) 2 0 , A 2 ( t ) L 0 . Examine t h e a p p r o x i m a t i o n s of t h e s e m a t r i c e s on t h e i n t e r v a l t . 5 t <= t J j+l

assuming t h a t t h e e l e m e n t s of A1

and A2 a r e s u f f i c i e n t l y smooth. W e w r i t e t h e

d i f f e r e n c e system of e q u a t i o n s s u g g e s t e d by N.N.Yanenko, which c o n s i s t s of s u c c e s s i v e s o l u t i o n s of s i m p l e Cranck-Nickolson schemes-

@J+1/2

- $J

+ *j

4j+1/2

1

2

T

E l i m i n a t i n g t h e a u x i l i a r y f u n c t i o n $ j+1'2

(1.4)

from t h e system o f d i f f e r e n c e E q s . ( l . 4 ) ,

we c a n r e d u c e i t t o a s i n g l e e q u a t i o n

where

F i r s t , w e examine t h e a p p r o x i m a t i o n problem. Expand t h e o p e r a t o r powers of

assuming t h a t

7 ,

'1 I

1

+

%

2

T. = E - TAJ J If t h e operators

:A

u A l.l

T . in the J

5 1. Simple t r a n s f o r m a t i o n y i e l d s

2

'

+ 2AJ 2 A J1 + (Ai)z)-

((Af)2

a r e commutative, i . e . i f A1A2

= A2

...

(1.7)

A1, e x p a n s i o n ( 1 . 7 ) c a n

be w r i t t e n a s '

Tj = E

- ,AJ

2

+

'

2 (AJ)2

- ...

(1.8)

Thus, i f A ( t ) 2 0 , A ( t ) 2 0 , t h e n d i f f e r e n c e scheme ( 1 . 4 ) i s a b s o l u t e l y 1 2 s t a b l e f o r s u f f i c i e n t l y smooth m a t r i x e l e m e n t s and s o l u t i o n 4 of problem (1.1)-(1.3)

( t h i s f o l l o w s from t h e i n e q u a l i t y

1 I T J. 1 I

< 1 which i s v a l i d , a c c o r d -

* T h e o r e t i c a l s u b s t a n t i a t i o n and m o d i f i c a t i o n s o f t h e scheme were r e p o r t e d by t h e a u t h o r a t t h e Symposium on Numerical methods f o r s o l v i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s , which was h e l d i n t h e USA i n 1970.

65

Numerical Solution of Basic and Adjoint Equations i n g t o t h e K e l l o g lemma). The scheme a p p r o x i m a t e s t h e i n i t i a l E q , ( l . l ) t o t h e

:A

second o r d e r i n T , i f A;,

a r e commutative, and t o t h e f i r s t o r d e r i n

I ,

if

they a r e n o t commutative. Suppose now t h a t t h e o p e r a t o r s A ( t ) and A ( t ) a r e approximated i n t h e 1 2 interval t . 6 t 5 t ( r a t h e r than i n t h e i n t e r v a l t . 2 t t. a s was t h e J-1 j+l J J +I c a s e w i t h (1.4) by A J = A a ( t j ) . C o n s i d e r t h e f o l l o w i n g two s y s t e m s of d i f f e r e n c e e q u a t i o n s :

$J

-

+J-1/2

+

A j $J

T

+

2

$ j-1/2 = 0;

2

(1.10) + $J+lI2 @J+l - $ J + l l 2 + A J @ J+l 2 1

= o

Computation p r o c e d u r e i s p r e c i s e l y t h e s u c c e s s i v e u s e of d i f f e r e n c e schemes ( 1 . 9 ) and ( 1 . 1 0 ) . S i m i l a r l y t o t h e above c o n s i d e r a t i o n s , i t i s e a s y t o show t h a t a c o m p l e t e c o m p u t a t i o n p r o c e d u r e i n v o l v i n g ( 1 . 9 ) and ( 1 . 1 0 ) g i v e s

where

Comparison of t h e o p e r a t o r Tj w i t h t h e s t e p o p e r a t o r i n t h e Cranck-Nickolson scheme $J+1

2T

+J-1

+

AJ

@

2

= 0,

(1.13)

shows t h a t i n a t w o - c y c l e s p l i t t i n g scheme t h e f o r m e r o p e r a t o r c o i n c i d e s , up t o T*, with t h e l a t t e r o p e r a t o r a s a p p l i e d t o a double t i m e i n t e r v a l ,

whether t h e o p e r a t o r s

Aa

no m a t t e r

a r e commutative or n o t . T h u s , t h i s method r e l a x e s

s i g n i f i c a n t l y t h e r e q u i r e m e n t f o r o p e r a t o r commutation. L e t u s now d i s c u s s c o m p u t a t i o n s t a b i l i t y p r o v i d e d by t h e method. For t h i s

p u r p o s e , w e e s t i m a t e r e l a t i o n ( 1 . 5 ) i n t e r m s of t h e e n e r g y norm

66

Chapter 3 5 1 f o r Aa

Since

2 0, the estimate is

\ITj\!

(1.14) whence

(1.15) I f w e d e a l w i t h a t w o - c y c l e method, e a c h s t e p of t h e c y c l e i n v o l v e s e s t i m a t i o n s of t h e t y p e ( 1 . 1 4 ) . T h i s means t h a t t h e t w o - c y c l e method i s a b s o l u t e l y s t a b l e , t o o . I t s h o u l d b e n o t e d t h a t a s i m i l a r s y m m e t r i z a t i o n t e c h n i q u e was

J. S t r a n g who c o n s i d e r e d t h e method of v a r i a b l e d i -

s u g g e s t e d i n d e p e n d e n t l y by rect ions.

C o n s e q u e n t l y , i f A ( t ) 2 0 , A ( t ) ? 0 , s y s t e m s of d i f f e r e n c e E q s . ( 1 . 9 ) and 1 2 ( 1 . 1 0 ) a r e a b s o l u t e l y s t a b l e and scheme ( 1 . 1 1 ) a p p r o x i m a t e s i n i t i a l E q . ( l . l ) u p t o T 2 , provided t h e s o l u t i o n

4 of p r o b l e m (1.1)-(1.3) and t h e e l e m e n t s of A l ( t )

and A ( t ) a r e s u f f i c i e n t l y smooth. 2 W e now t u r n t o t h e inhomogeneous problem and s e e k i t s s o l u t i o n by a c o m p l e t e t w o - c y c l e s p l i t t i n g . To t h i s e n d , w e a n a l y s e a s y s t e m of d i f f e r e n c e e q u a t i o n s o f t h e form (1.9), (1.10) e x p r e s s e d more c o n v e n i e n t l y a s

(1.16)

where

f J = f ( t . ) . S o l v i n g t h e s e e q u a t i o n s for J $J+1 = T j $ j - l

+

+J+',

2TTJTJfj, 1 2

we obtain

(1..17)

where

Expanding ( 1 . 1 7 ) i n t h e powers o f t h e s m a l l p a r a m e t e r T , w e r e d u c e t h i s expression t o

67

Numerical Solution o f Basic and Adjoint Equations

and e l i m i n a t e 4 j - l .

To t h i s e n d , we expand t h e s o l u t i o n i n a T a y l o r s e r i e s i n

t h e neighborhood of t h e p o i n t t j - l . We h a v e , up t o

The d e r i v a t i v e

T2,

a $ / a t i s e l i m i n a t e d by t h e e q u a l i t y hJ$J-' + f j

+

(1.23)

o(T).

S u b s t i t u t i n g ( 1 . 2 3 ) i n t o ( 1 . 2 2 ) , we o b t a i n $J = (E

-

TAj)$J-'

+

TfJ + 0 ( 1 2 ) ,

-

TfJ

whence

(E - TA')$J-'

= $J

+

(1.24)

O(T2).

S u b s t i t u t i n g now (1.24) i n t o ( 1 . 2 1 ) , we f i n d

E v i d e n t l y , (1.25) approximates i n i t i a l E q . ( l . l ) i n t h e i n t e r v a l t .

J-1

t o t h e second o r d e r i n

S t l t . J+1

t . Thus, u s i n g a two-cycle method, we have accomplished

a second-order d i f f e r e n c e approximation of an inhomogeneous e v o l u t i o n e q u a t i o n .

I t i s a s i m p l e m a t t e r t o prove t h e s t a b i l i t y of t h e method i n terms of t h e j+l energy norm. I n d e e d , l e t u s e s t i m a t e ,$ from ( 1 . 1 7 ) w i t h r e s p e c t t o t h e norm (1.26)

\\+JII5 where \If

(1

= max

llg

11

+

T j ~ \ f l l ~

(1.28)

( ( f J ( (R . e l a t i o n ( 1 . 2 8 ) i m p l i e s t h e c o m p u t a t i o n a l s t a b i l i t y of

t h e scheme i n any f i n i t e t i m e i n t e r v a l .

68

Chapter 3 The system of E q s . ( l . l 6 ) can a l s o be e x p r e s s e d i n t h e f o l l o w i n g e q u i v a l e n t

form:

(1.29)

E l i m i n a t i n g t h e unknown q u a n t i t i e s w i t h f r a c t i o n a l i n d i c e s , we a r r i v e a t a resolved equation @J+'

= T T T T

4 J - I + 2rT T f J ,

1 2 2 1

(1.30)

1 2

which c o i n c i d e s w i t h (1.17). Sometimes, i t i s more p r e f e r a b l e t o w r i t e t h e e q u a t i o n s i n t h e form ( 1 . 2 9 ) r a t h e r t h a n i n t h e form ( 1 . 1 6 ) . Thus, i f t h e m a t r i c e s A ( t ) , A ( t ) Z 0 , and i f t h e s o l u t i o n $,, the function 1 2 f ( t ) , and t h e e l e m e n t s of A l ( t ) , A ( t ) a r e s u f f i c i e n t l y smooth, t h e system of 2

d i f f e r e n c e E q s . ( 1 . 1 6 ) i s a b s o l u t e l y s t a b l e i n t h e i n t e r v a l 0 5 t 5 T and i t approx i m a t e s t h e i n i t i a l e q u a t i o n up t o t h e second o r d e r i n We have assumed so f a r t h a t t h e i n i t i a l o p e r a t o r

?.

A

can be r e p r e s e n t e d a s

t h e sum of two s i m p l e r o p e r a t o r s . When t a c k l i n g c o m p l i c a t e d problems of mathemat i c a l p h y s i c s , one h a s f r e q u e n t l y t o s p l i t o p e r a t o r s i n t o a l a r g e number of comp o n e n t s . G e n e r a l l y , w e have n

1

A =

(1.31)

a=1 with

A

2 0. S i n c e t h e c a s e of

n

=

2 was t r e a t e d i n d e t a i l e a r l i e r , w e s h a l l

c o n c e n t r a t e h e r e o n l y on t h e s i t u a t i o n where n > 2 . F i r s t of a l l , one i s r e a d i l y convinced t h a t s t r a i g h t forward e x t e n s i o n of above s p l i t t i n g - u p methods ( f o r n = 2 ) i s , g e n e r a l l y , i m p o s s i b l e . That i s why we s h a l l t r y t o e x t e n d t h e component s p l i t t i n g a l g o r i t h m s t o t h e c a s e n > 2 under t h e assumptions t h a t make such an e x t e n s i o n p o s s i b l e . A n a t t e m p t w i l l be undertaken t o develop a d i f f e r e n c e analogue t o t h e

problem of t h e second-order approximation w i t h r e s p e c t t o

7 ,

which i s a b s o l u t e l y

s t a b l e i n t i m e . According t o t h e assumption of a multicomponent s p l i t t i n g , w e write

69

Numerical Solution of Basic and Adjoint Equations

(1.32) a=1

where a l l

A:

a r e positive semidefinite operators and, therefore, AJ

i 0.

Consider t h e system of e q u a t i o n s (1.33) w = 1

When

, 2,

...,

n

Z 0 , t h e o p e r a t o r s Aa a r e commutative and e i t h e r A’(1

or

=

A’ = (A’+’ + A ’ ) , scheme ( 1 . 3 3 ) i s a b s o l u t e l y s t a b l e and i s a second-order a 2 a p p r o x i m a t i o n . T h i s can b e demonstrated i n an e l e m e n t a r y way u s i n g t h e F o u r i e r t r a n s f o r m a t i o n method. E v i d e n t l y , f o r non-commutative o p e r a t o r s

is, g e n e r a l l y , of t h e f i r s t o r d e r i n t i o n t h a n t h e f o l l o w i n g second-order

T

A J scheme ( 1 . 3 3 ) a

and, t h e r e f o r e , l e s s a t t r a c t i v e f o r applica-

scheme s u g g e s t e d by E.G.Dyakonov:

(1.34)

a = n + l , n + 2 ,

...

,2n.

I n what f o l l o w s we s h a l l t r y t o g i v e a s p e c i f i c d e s c r i p t i o n of t h e complete s p l i t t i n g method based on ( 1 . 3 3 ) which a d m i t s a s o l u t i o n t o t h e Cauchy problem f o r p o s i t i v e semidef i n i t e and non-commutative o p e r a t o r s A J and which p r o v i d e s a second-order a p p r o x i m a t i o n . T h i s would a c t u a l l y b e a d e f i n i t i v e s o l u t i o n t o t h e s p l i t t i n g problem. Note t h a t s y s t e m ( 1 . 3 3 ) i s reduced t o a s i n g l e e q u a t i o n

n (1.35) ol=l

which can b e used t o o b t a i n an e s t i m a t e w i t h r e s p e c t t o t h e norm

From t h e K e l l o g lemma we have (1.37) I f t h e o p e r a t o r i s a n t i s y m m e t r i c , t h e n we have

70

Chapter 3

Thus, we have proved t h a t t h e scheme i s a b s o l u t e l y s t a b l e . To d e t e r m i n e t h e o r d e r of a p p r o x i m a t i o n , we expand t h e e x p r e s s i o n n

- 1 A'),

1 AJ)-'(E

TJ = n ( E +

2

a

2

(1.38)

a

a=1

i n t h e powers of t h e s m a l l parameter T (assuming t h a t . n S i n c e T J = II T i , we f i r s t expand t h e o p e r a t o r T J : u=1 T J = E - 'rAJ

'

a

2

+

'

2

f

))AalI(

< 1).

...;

a

(1.39)

We have

If t h e o p e r a t o r s A J a r e commutative, t h e e x p r e s s i o n under t h e double sum v a n i s h e s and we o b t a i n TJ = E -

TAJ

+

7' (AJ)~+ 2

(1.41)

o(T3).

Comparison of ( 1 . 4 1 ) w i t h t h e s i m i l a r expansion f o r t h e Cranck-Nickolson scheme i n d i c a t e s t h a t i n t h i s p a r t i c u l a r c a s e scheme ( 1 . 3 3 ) i s a second-order approximation i n

T.

I f t h e o p e r a t o r s AJ a r e n o t commutative, t h e s p l i t t i n g scheme

g i v e s o n l y a f i r s t - o r d e r approximation i n T

T.

To d e v e l o p a second-order

scheme i n

f o r t h e non-commutative c a s e , one s h o u l d modify scheme ( 1 . 3 3 ) a s f o l l o w s :

(1.42)

From t h e a l g o r i t h m i c s t a n d p o i n t , t h i s means t h a t we f i r s t s o l v e system ( 1 . 3 3 ) in the interval t .

5 t 5 t

f o r a = 1,2,

...,

n

and t h e n s o l v e t h e same system

J-1 j in t h e i n t e r v a l t . 5 t 5 t j + l , but i n t h e r e v e r s e o r d e r (a = n , n J

a = 1, 2 ,

...

-

1,

...,

1):

, n (1.43)

a = n , n - 1 ,

...,

1.

71

Numerical Solution of Basic and Adjoint Equations E v i d e n t l y , a complete c y c l e of c o m p u t a t i o n s i n (1.43) r e q u i r e s t h a t @ J + 1= T J $J-l

where

a=1

a=n

Thus, scheme (1.43) g i v e s a second-order approximation i n

T

on t h e i n t e r v a l

p r o v i d e d f o r A J we t a k e one of t h e a n a l o g u e s d e f i n e d b y ( 1 . 2 2 ) t . C t 5 t . J-1 J+1' -(1.24). T h i s f o l l o w s from t h e comparison of (1.43) and (1.33) f o r t h e double ~

time i n t e r v a l . In c o n c l u s i o n , we s h o u l d l i k e t o n o t e t h a t d i f f e r e n c e scheme (1.43) i s a b s o l u t e l y s t a b l e f o r A J 2 0 . Hence, we have o b t a i n e d an optimum a l g o r i t h m of multicomponent s p l i t t i n g . For t h e non-homogeneous problem = f ,

*+A$

at

$ = g

(1.44) at

t = O ,

n where A ( t ) 2 0 , A =

Acl,

A ( t ) 2 0 t h e f o l l o w i n g s p l i t t i n g scheme h o l d s i n

the interval t . 5 t 5 t . J-1 j+l'

. . . . . . . . . . . . . . . . . . (E + 1 A J ) ( $ J 2

n

-

T f j ) = (E

-

' 2

'

AJ)$ n

j-l/n

,

(1.45)

. . . . . . . . . . .

Here A' = A ( t , ) . One can e a s i l y s e e t h a t t h e scheme g i v e s a second-order approa a J ximation i n T and i s a b s o l u t e l y s t a b l e under t h e assumption t h a t ,+ i s s u f f i c i e n t l y smooth. S i m i l a r l y t o t h e c a s e of c1 = 2 , system (1.45) can be w r i t t e n i n an e q u i v a l e n t form

72

Chapter 3

(1.46)

a

=

2 , 3,

...,

n + 1

Let u s now t u r n t o t h e s p l i t t i n g - u p method f o r i m p l i c i t d i f f e r e n c e approximat i o n s . To t h i s e n d , we h a n d l e t h e problem

*+

A$ = 0 ,

at

(1.47) at

$ = g

t = O .

Suppose A = C A a , A 2 0 , and A a=l s p l i t t i n g a l g o r i t h m i s of t h e form

do n o t depend on t i m e . Then t h e

. . . . . . . . . .

(1.48)

I t w i l l be shown t h a t t h i s a l g o r i t h m i s a b s o l u t e l y s t a b l e . Consider t h e equation

$j+a/n

-

+j+(a-l)/n +

and m u l t i p l y it s c a l a r l y by 4'

(+j+a/n

-+

j+(a-l)/n

+a/n

A $j+a/n =

a

T

o;

(1.49)

:

$j+a/n)

+

r(Aa+ j+Wn , + j + u / n ) =

o.

S i n c e t h e o p e r a t o r s Aa a r e p o s i t i v e s e m i d e f i n i t e , we o b t a i n * ($ j + a / n

j+(a-l)/n

j+a/n

) 5 0,

or

On t h e o t h e r hand,

whence

*For methodic r e a s o n s , o n l y homogeneous boundary c o n d i t i o n s a r e c o n s i d e r e d .

73

Numerical Solution of Basic and Adjoint Equations

By v i r t u e of t h i s r e c u r r e n c e i n e q u a l i t y we have

T h i s means t h a t under t h e above a s s u m p t i o n s , computation by s p l i t t i n g scheme (1.48) i s a b s o l u t e l y s t a b l e . One can r e a d i l y s e e t h a t ( 1 . 4 8 ) approximates t h e i n i t i a l problem t o

the f i r s t order in

7.

We now c o n s i d e r t h e non-homogeneous problem

*+,a

at

$ = g

= f

(1.51) at

t = O

The s p l i t t i n g scheme f o r t h i s problem i s d e f i n e d a s

. . . . . . . . . . . . .

(1.52)

T h i s scheme approximates t h e i n i t i a l non-homogeneous e q u a t i o n t o w i t h i n t h e first-order

i n T.

The s t a b i l i t y of t h e scheme can b e proved a s f o l l o w s . M u l t i p l y each e q u a t i o n s c a l a r l y by $J+'ln,

...,

$J+',

r e s p e c t i v e l y . S i m i l a r l y t o t h e above,

we have

Examine t h e

Since A

l a s t e q u a t i o n of ( 1 . 5 2 ) more c l o s e l y . We have

2 0 , we o b t a i n

By v i r t u e of t h e Cauchy-Bunyakovsky i n e q u a l i t y , we have

Consequently,

C a n c e l l a t i o n of

Ilb j + l I Il$J+' I\

SIl$

j+(n-l)/n

leads t o the inequality

74

Chapter 3

11

Ij@j+l

ll@j+(n-l)/n

Eliminating s o l u t i o n s with f r a c t i o n a l i n d i c e s , w e g e t

\ $'J'l Allowing f o r

Il$"I( = 11 g 11

5

II$J )I

I(f(1 = max

11 f J j l .

1).

(1.54)

and e l i m i n a t i n g i n t e r m e d i a t e s o l u t i o n s , we o b t a i n

IIg (I+ T j I(f)I,

1(9J+111 5 where

+ TIlfJ

(1.55)

T h i s i m p l i e s a b s o l u t e s t a b i l i t y of t h e d i f f e r e n c e

scheme for any moment w i t h i n t h e i n t e r v a l

0 6 t 2 T. j T h i s s p l i t t i n g a l g o r i t h m can b e extended t o t h e c a s e of a time-dependent

o p e r a t o r A . Here, i n t h e computation c y c l e performed by t h e s p l i t t i n g scheme,

A

should be r e p l a c e d by a s u i t a b l e d i f f e r e n c e approximation of t h i s o p e r a t o r i n each i n t e r v a l

t. 6 t 5 t J

j+l'

3.2. S p l i t t i n g of a Problem According t o P h y s i c a l P r o c e s s e s We w i l l assume t h a t each element of t h e W i l b e r t s p a c e 0 , where t h e b a s i c problem s o l u t i o n i s s o u g h t , i s s u f f i c i e n t l y smooth and meets some r e q u i r e m e n t s r e l a t e d t o t h e boundary c o n d i t i o n s on

1, 1

0'

and

1H .

Then t h e s o l u t i o n of t h e

b a s i c problem s a t i s f i e s t h e e v o l u t i o n problem

+ = g

at

t = O .

As p r e v i o u s l y , we assume t h a t f o r u , v , and

w

t h e following r e l a t i o n s a r e

valid:

w = O

at

z = O,

z=H.

G e n e r a l l y s p e a k i n g , problem ( 2 . 1 ) d e s c r i b e s two e s s e n t i a l l y d i f f e r e n t p r o c e s s e s . One i s s u b s t a n c e t r a n s p o r t ( w i t h c o n s e r v a t i o n ) a l o n g a t r a j e c t o r y . T h i s p r o c e s s

is d e s c r i b e d by t h e e q u a t i o n

$ = g

4 = o

on

at

1

(2.3)

t = O , for

u

< 0.

The o t h e r p h y s i c a l p r o c e s s i s r e l a t e d t o s u b s t a n c e d i f f u s i o n and a b s o r p t i o n i n t h e c o u r s e of p r o p a g a t i o n . T h i s p r o c e s s i s f i t t e d by t h e problem

75

Numerical Solution of Basic and Adjoint Equations

a$

-a +t

a$

=

a

v 3+ dz at

$ = g

at

$ = o

3

32

t = O ,

1

on

= u$ on

32

=

(2.4)

10

on

0

+ f,

IH.

Here w e h a v e c h o s e n t h e s i m p l e s t boundary c o n d i t i o n s . They c o u l d b e r e p a c e

( a s p o i n t e d o u t i n c h a p t e r 1 ) by t h e f o l l o w i n g c o n d i t i o n s :

@ =

3 an

=

o

on

1

for

u

< 0,

o

on

I

for

u

:0 ,

’9 = u~ az

on

lo

?a z! k z 0

on

IH

From t h e c o m p u t a t i o n a l v i e w p o i n t , h o w e v e r , t h e p r o b l e m w i t h s u c h boundary c o n d i t i o n s d o e s n o t d i f f e r , i n p r i n c i p l e , from t h e p r o b l e m f o r m u l a t e d a b o v e . T h e s e p r o c e s s e s are two l i m i t i n g cases o f p r o b l e m ( 2 . 1 ) . I n d e e d , p u t t i n g v = o, p = 0,

v = 0, w

= 0,

a = 0 i n (2.1), w e a r r i v e a t p r o b l e m (2.3), w h e r e a s s e t t i n g u

= 0,

w e come t o p r o b l e m ( 2 . 4 ) . T h e r e f o r e , i n l a t e r s e c t i o n s of t h e book

we s h a l l examine s o l u t i o n s o f t h e s e p h y s i c a l l y e l e m e n t a r y p r o b l e m s a t g r e a t e r length.

Now w e w i l l t r y t o r e l a t e p r o b l e m s (2.3) a n d (2.4). I t t u r n s o u t t h a t t h i s c a n b e d o n e a p p r o x i m a t e l y d u e t o l o c a l a d d i t i v i t y of t h e p r o c e s s e s . Choose a sufficiently small interval

t . 5 t 5 t 3 j+l

a n d s o l v e i n t h i s i n t e r v a l t h e problem

76

Chapter 3

+:

o

#1 =

t = t

for

$1 =

1

on

if

j 5 0,

u

where $' i s d e f i n e d b e l o w . A s a r e s u l t , we f i n d $'+I = 2 1 N e x t , we s o l v e a n o t h e r problem i n t h e same i n t e r v a l

b2

=

+2 =

o

$l(r,t j + l ) .

j+l

b1

1

on

(2.6)

= (12(', t . ) i s a n a p p r o x i m a t e s o l u t i o n for p r o b l e m ( 2 . 1 ) . To demonJ +I s t r a t e t h i s , we i n t e g r a t e ( 2 . 5 ) w i t h i n t h e l i m i t s ( t ., t )

Then,

J

(11 =

$1 - j

d i v K $ ~d t .

(2.7)

t. J H e r e w e employed t h e c o n d i t i o n

=

$1

t

at

=

i n t e g r a l s i g n i n ( 2 . 7 ) ( u s i n g t h e same f o r m u l a )

$l =

$1

t

..

t

- (t

- t J. ) d i v

~ $ ji +

E l i m i n a t e b1

J

under t h e

t div(u

t.

j

div

uQ1 d t )

dt,

(2.8)

t. J

J

or $1 =

where

T

=

t .

J+1

- t . J

+J2

- (t

- t

.) d i v

J

~$1

+ O(T~),

(2.9)

Using (2.9), w e o b t a i n $J+' 1

=

:6 -

T d i v g$j + O ( T 2 ) .

(2.10)

L e t u s now c o n s i d e r problem ( 2 . 6 ) . I n t e g r a t e t h e e q u a t i o n o v e r t h e i n t e r v a l t . J

1-

t 2 t .

J+1

w i t h t h e i n i t i a l c o n d i t i o n s 4 J = $J+'. 2 1

The r e s u l t i s

I7

Numerical Solution of Basic and Adjoint Equations

(2.11) t . J Using a s i m p l e e x p l i c i t a p p r o x i m a t i o n , we o b t a i n

(2.12)

D i v i d i n g ( 2 . 1 3 ) by for

T

and c a l c u l a t i n g t h e l i m i t of t h e e x p r e s s i o n t h u s d e r i v e d

0 , we a r r i v e a t t h e e q u a t i o n

T

Hence, i t f o l l o w s t h a t $ 2

-f

$ , when T

+

0 . T h u s , w e h a v e shown t h a t a c o m b i n a t i o n

of p r o b l e m s ( 2 . 5 ) and ( 2 . 6 ) d o e s y i e l d a n a p p r o x i m a t e s o l u t i o n f o r problem ( 2 . 1 ) i n a small t i m e i n t e r v a l

T ,

provided a l l t h e functions involved a r e s u f f i c i e n t l y

smooth. I t s h o u l d b e n o t e d t h a t our e s t i m a t e s a r e f a i r l y c r u d e . Hence, t h e a b o v e a n a l y s i s c a n b e r e g a r d e d a s a p r o o f of a f u n d a m e n t a l r e s u l t p e r t a i n i n g t o a p p r o x i m a t i o n . A c t u a l l y , t h e a p p r o x i m a t i o n of t h e i n i t i a l p r o b l e m by t h e s p l i t problem p r o v i d e s much b e t t e r r e s u l t s . To make s u r e o f i t , w e c o n s i d e r a model s p a t i a l t w o - d i m e n s i o n a l p r o b l e m c o r r e s p o n d i n g t o ( 2 . 1 ) f o r c o n s t a n t v a l u e s of

u , v . In

t h i s c a s e , w e have

@ = g

at

t = O .

L e t g ( x , y ) b e d e f i n e d i n t h e e n t i r e p l a n e ( x , y ) and d e c r e a s e r a t h e r r a p i d l y a t i n f i n i t y , so t h a t t h e s o l u t i o n c a n b e g i v e n a s t h e ' F o u r i e r i n t e g r a l

(2.15)

I n a s i m i l a r way,

78

Chapter 3 m

m

n ) eimx+iny _m

dm d n .

M u l t i p l y i n g t h e e q u a t i o n and t h e i n i t i a l d a t a of and i n t e g r a t i n g t h e

(2.16)

_m

r e s u l t with respect t o

x

and

1 ( 2 . 1 4 ) by 7 4 n eimx-iny y , w e o b t a i n t h e problem

f o r the Fourier c o e f f i c i e n t s

a@

+ at

[-imu - i n v + o + p(mz + n Z ) 1 0 @ = A

at

= 0,

(2.17)

t = O ,

The s o l u t i o n of t h i s problem i s o f t h e form 0 = A e x p “imu

+ inv -

ii

-

w(m*

+ n2)]tl.

(2.18)

S u b s t i t u t i o n of ( 2 . 1 8 ) i n t o ( 2 . 1 5 ) y i e l d s t h e e x a c t s o l u t i o n

4,

=

15 _m

A ( m , n ) e x p {im(x + u t ) + i n ( y + v t ) - [ a + p(mz + n 2 ) ] t } *

(2.19)

-m

‘dm dn. take a t i m e interval 0 5 t i T

a n d w r i t e on i t two p r o b l e m s :

w1

a+l

Q1 u -+ v - = dX aY

-+

at

b,=g

@

=

at

0,

t = O

@ ( X , Y, T )

L e t t h e f u n c t i o n g ( x , y ) b e o f t h e form ( 2 . 1 6 ) . C o n s i d e r s i m i l a r F o u r i e r

i n t e g r a l s f o r t h e f u n c t i o n s $1 a n d $ . T h e n , j u s t a s i n t h e a b o v e c o n s i d e r a t i o n , we come t o two p r o b l e m s : do 1

(imu + i n v )

- dt 0

1

f o r the Fourier coefficients

= A

a1

at

a1

= 0,

(2.22)

t = O ;

of p r o b l e m ( 2 . 2 0 )

(2.23)

0 = O1(~)

at

t

= T

79

Numerical Solution of Basic and Adjoint Equations @ of problem ( 2 . 2 1 ) .

f o r the Fourier c o e f f i c i e n t s

The s o l u t i o n f o r (2.22) i s

(2.24)

Q ( t ) = A exp { ( i m u + i n v ) t l ; 1

(2.25) Putting t =

i n (2.24) and s u b s t i t u t i n g

T

Finally, putting t =

exp {(imu + i n v ) r - [n +

= A

O(T) T

O1(T)

i n t o (2.25), we have

v(m2

+ n2)]tt.

(2.26)

i n (2.26), we o b t a i n

O(T) = A exp {[imu

+ inv -

-

13

p(m2 + n * ) l r i ,

and, hence,

A(m, -m

n ) exp i i m ( x + u ? ) + i n ( y + vT) -

(2.27)

_m

The v a l u e of Let t =

T

T

[U

+ u(m2 + n n ) ] - r 1 dm dn.

h a s n o t so f a r been f i x e d . Suppose i t i s chosen a r b i t r a r i l y .

i n (2.19). Then, (2.19) and (2.27) a r e j u s t i d e n t i c a l ( f o r any

I t i s e v i d e n t , however, t h a t t h e s e s o l u t i o n s c o i n c i d e o n l y when t = endeavored t o f i n d a s o l u t i o n w i t h i n t h e i n t e r v a l 0 Z t

2-

T

T .

I).

If w e

u s i n g a s p l i t problem,

t h e r e s u l t would b e o n l y an a p p r o x i m a t e e s t i m a t e . S o , i f we s e e k a s o l u t i o n f o r a d i s c r e t e set of p o i n t s t = t . , we s h o u l d s u c c e s s i v e l y s o l v e t h e s p l i t p r o b l e m s , J i . e . t h e problems

2 + u -a + 1 41

a+l

v -= aY

ax

at

+1 =

9’

for

t

0,

(2.28) =

t . ; J

(2.29)

T h i s r e m a r k a b l e f a c t u n d e r l i e s t h e s p l i t t i n g of problems a c c o r d i n g t o physical processes. Since i n actual s i t u a t i o n s a s a r u l e , time-dependent exact solution f o r

t = t

the coefficients

u

and

v

are,

q u a n t i t i e s , t h e s p l i t t i n g a l g o r i t h m d o e s n o t y i e l d an j

(j

=

1, 2 ,

. . .) .

To o b t a i n more a c c u r a t e r e s u l t s , t h e

t i m e i n t e r v a l s should be taken s u f f i c i e n t l y s m a l l , thereby ensuring a b e t t e r

80

Chapter 3

a p p r o x i m a t i o n u n d e r s u b s t a n t i a l v a r i a t i o n of t h e c o e f f i c i e n t s

u , v . W e have

a n a l y z e d a two-dimensional c a s e , b u t t h e Same c a l c u l a t i o n s a p p l y t o a g e n e r a l three-dimensional

c a s e as w e l l .

L e t u s now examine problem ( 2 . 2 8 ) , p r o v i d e d u = c o n s t , v = c o n s t . T h i s

problem w i l l be s o l v e d i n two s t a g e s . F i r s t , w e s o l v e t h e problem "11 a$lI a t + u - = O ,

ax

(2.30)

and t h e n , t h e problem

(2.31)

S i m i l a r l y t o t h e f o r e g o i n g d i s c u s s i o n , we c a n show, u s i n g t h e F o u r i e r a n a l y s i s , t h a t t h e s o l u t i o n s f o r t h e i n i t i a l problem ( 2 . 2 8 )

t. 5 t 5 t J

j+l

i

+;,+I = + j + l = 1

_m

Here 0 ;

in the interval

and f o r s p l i t problem ( 2 . 3 0 ) . (2.31) c o i n c i d e a t t = t j + l :

$ i ( r a , n ) exp {im(x + uT) + i n ( y + vT)}

dm dn.

(2.32)

_m

i s t h e F o u r i e r t r a n s f o r m of t h e f u n c t i o n $ J . 1

Now, l e t u s s o l v e problem (2.29) by t h e component s p l i t t i n g method. W e s p l i t t h i s problem i n t o two p a r t s : f o r the variable

x

(2.33)

42, and f o r t h e v a r i a b l e

j+l = $12

at

t

=

t.; J

y

(2.34)

S o l v i n g t h e s e problems by t h e F o u r i e r t r a n s f o r m a t i o n method, w e o b t a i n t h e expression

81

Numerical Solution of Basic and Adjoint Equations

(2.35)

- [u +

~ ( m z + n Z ) ] T } dm d n ,

which i s an e x a c t s o l u t i o n t o t h e problem i n t h e i n t e r v a l t .

1 t tJ+l. J And f i n a l l y , we c o n s i d e r a more g e n e r a l c a s e of s p l i t t i n g when problem ( 2 . 1 4 )

i s s p l i t i n t o t h e f o l l o w i n g components: t h e f i r s t problem

(2.36)

t h e s e c o n d problem

(2.37)

F o u r i e r t r a n s f o r m a t i o n y i e l d s a s o l u t i o n f o r problems (2.36). (2.37) which

i s j u s t i d e n t i c a l t o t h e s o l u t i o n of problem (2.14) t h e i n t e r v a l t . i t J m

2

+ =

m

ii -m

tj+l:

'

cbJ(m, n ) exp Iim(x + UT) + i n ( y + VT)

-

_m

(2.38)

- [a +

p(m2

+ n Z ) ] ~ dlm dn.

A s c a n be s e e n , t h e s p l i t t i n g of problem (2.38) f a l l s u n d e r a v a r i e t y of p o s s i b i l i t i e s and i n e a c h c a s e t h e r e e x i s t s an e x a c t s o l u t i o n a t a g i v e n moment

t . . An u n c e r t a i n t y t h a t may a r i s e u n d e r a c t u a l c o n d i t i o n s i s o n l y due t o v a r i a J t i o n s i n t h e v a l u e s o f u . v . To minimize a p o s s i b l e s p l i t t i n g error, t h e t i m e i n t e r v a l should be taken r a t h e r s m a l l . P r e c i s e l y f o r t h i s purpose, it i s reasona b l e t o s p l i t g e n e r a l problem (2.14) on e a c h i n t e r v a l t . 5 t J

i

tj+l,

T

=

t

j+l

- t.

i n t o two problems r e l a t e d t o d i f f e r e n t p h y s i c a l p r o c e s s e s , s i n c e each o f t h e proc e s s e s g e n e r a l l y i n v o l v e s p a r t i c u l a r p h y s i c a l b a l a n c e r e l a t i o n s which must n o t be v i o l a t e d even i n t h e s t a g e of t h e development of d i f f e r e n c e a p p r o x i m a t i o n s .

A s r e g a r d s t h e d i f f e r e n c e a n a l o g u e s of s p l i t s y s t e m s , t h e y s h o u l d be s o l v e d by t h e multicomponent c y c l i c r e d u c t i o n o u t l i n e d i n t h e p r e v i o u s s e c t i o n . Using t h e s p l i t t i n g method, w e c a n a l s o o b t a i n s o l u t i o n s i d e n t i c a l t o t h e s o l u t i o n of problems l i k e

J

82

Chapter 3 *+A$

at

(2.39)

= 0 ,

n A i , A . a r e pairwise-commutative where A = , X 1=1 1 c a s e , t h e s p l i t t i n g scheme i s a s f o l l o w s :

simple-shape m a t r i c e s .

In t h i s

(2.40)

(2.41)

at

t = t.; J

. . .

. .

. .

(2.42) at

t = t. J

I t i s c l e a r from t h e above a n a l y s i s why t h e s p l i t t i n g method e n s u r e s s u c h remarkably e x a c t r e s u l t s i n s o l v i n g problems r e l a t e d t o s u b s t a n c e t r a n s p o r t a n d d i f f u s i o n , t h e o r y of c l i m a t e , e t c . N a t u r a l l y , a l l t h i s r e f e r s e q u a l l y t o a t h r e e dimensional c a s e , t o o . 3 . 3 . E q u a t i o n of Motion

The e q u a t i o n of s u b s t a n c e t r a n s p o r t a l o n g t r a j e c t o r i e s , which w i l l b e h e r e i n a f t e r c a l l e d , f o r t h e s a k e of b r e v i t y , an

equation

Of

motion, i s one of

t h e s i m p l e s t problems i n m a t h e m a t i c a l p h y s i c s . I t s n u m e r i c a l s o l u t i o n , however,

i s h a r d l y a s i m p l e t a s k and r e q u i r e s c a r e f u l c h o i c e of t h e s o l u t i o n a l g o r i t h m . L e t u s c o n s i d e r t h e e q u a t i o n of motion a s a c o n s t i t u e n t of a more g e n e r a l

problem (2.14). W e have

where

d - 2 + u a a+ va - + w - , dt

- at

ax

ay

az

T h i s e q u a t i o n w i l l be s o l v e d u n d e r an a d d i t i o n a l c o n d i t i o n

The i n i t i a l d a t a w i l l be t a k e n as

83

Numerical Solution of Basic and Adjoint Equations $ = g

at

t = O .

The s i m p l e s t problem of t h i s t y p e i s a one-dimensional substance

t r a n s p o r t of a

$

*+u*=o,

ax

at

$ = g(x) where

u

(3.1) t = 0

at

i s t h e v e l o c i t y , g ( x ) i s t h e i n i t i a l d i s t r i b u t i o n . The f u n c t i o n s g ( x , y )

and g ( x ) a r e assumed t o b e

ZIT-periodic i n

x . I f u = c o n s t , problem ( 3 . 1 ) h a s

an o b v i o u s s o l u t i o n $(x, t) = g(x

-

ut)

(3.2)

p r o v i d e d g ( x ) i s a d i f f e r e n t i a b l e f u n c t i o n . S o l u t i o n ( 3 . 2 ) d e s c r i b e s t h e propagat i o n of t h e i n i t i a l p e r t u r b a t i o n a l o n g t h e c h a r a c t e r i s t i c c u r v e

on any s t r a i g h t l i n e

T h i s means t h a t $ ( x , y ) = c o n s t

x

-

x - u t = const.

u t = c o n s t . Thus, when

u > 0 , problem ( 3 . 1 ) d e f i n e s p e r t u r b a t i o n p r o p a g a t i o n t o w a r d s i n c r e a s i n g v a l u e s of

x. These well-known c o n c e p t s s h o u l d b e b o r n e i n mind w h i l e e l a b o r a t i n g d i f f e r -

e n c e a n a l o g u e s o f problem ( 3 . 1 ) . I f t h e v e l o c i t y u = u ( x , t ) i s v a r i a b l e , problem (3.1) cannot, i n g e n e r a l , be solved a n a l y t i c a l l y . P r e c i s e l y i n t h e s e c a s e s , n u m e r i c a l methods b a s e d on d i f f e r e n c e a p p r o x i m a t i o n s a p p e a r t o b e v e r y u s e f u l . W e now c o n s i d e r s i m p l e d i f f e r e n c e schemes w i t h c = c o n s t . L e t u s assume, f o r c e r t a i n t y , t h a t u > 0. Then w e h a v e t h e e x p l i c i t scheme

(3.3) and t h e i m p l i c i t scheme

T

where $:

+ u

Ax

= 0,

(3.4)

.

= ( x k , t .) J One c a n see t h a t scheme ( 3 . 3 ) becomes a b s o l u t e l y s t a b l e i f it i s m o d i f i e d

as

where n = k - e n t i e r

(r/Ax).

But i f t h e d i f f e r e n c e e x p r e s s i o n s f o r form

ua$/ax i n ( 3 . 3 ) a r e t a k e n i n t h e

84

Chapter 3

i n s t e a d of

Lhe r e s u l t a n t scheme i s u n s t a b l e f o r any r e l a t i o n between t h e s t e p s . The two schemes a r e approximate t o t h e f i r s t o r d e r i n Ax and

T.

Indeed,

suppose t h e i n i t i a l v a l u e s of g ( x ) and $ ( x , t ) a r e s u f f i c i e n t l y smooth f u n c t i o n s . We expand t h e s o l u t i o n of Eq.(3.1) point

x =

Xk,

i n a T a y l o r s e r i e s i n t h e neighborhood of t h e

t = t.: J

S u b s t i t u t i n g (3.5) i n t o (3.3), we o b t a i n

a24 2

a t 2

I

(3.6)

t h e d i s c a r d e d terms b e i n g of a h i g h e r o r d e r of s m a l l n e s s . I t f o l l o w s from (3.1) that (3.7)

Then (3.6) t a k e s t h e form

T h i s a n a l y s i s of d i f f e r e n c e schemes was s u g g e s t e d by A.I.Zhukov. When ur/Ax < 1, r e l a t i o n (3.8) can be i n t e r p r e t e d a s an e q u a t i o n w i t h t h e s o l u t i o n range xk-1

j+l' Assuming t h a t t h e d i s c a r d e d terms i n (3.8) a r e s m a l l , we a r r i v e a t t h e

< X ' X k , t . J' t ' t

h e a t conduction e q u a t i o n

where p = (uAx

-

u Z r ) / 2 i s t h e s o - c a l l e d c o e f f i c i e n t of a r t i f i c i a l or computation

v i s c o s i t y . I t should be n o t e d t h a t i f

UT/AX

= 1, t h e n fl = 0 and a l l t h e o t h e r

d i s c a r d e d t e r m s a r e z e r o , i . e . e x p l i c i t scheme (3.3) becomes a scheme of an i n f i n i t e - o r d e r approximation i n

x and

T.

Of s p e c i a l i n t e r e s t i s t h e c a s e of U T / A X

>

1, where t h e e q u a t i o n

Numerical Solution of Basic and Adjoint Equations

85

holds t r u e . One c a n e a s i l y see t h a t E q . ( 3 . 9 ) w i t h t h e i n i t i a l c o n d i t i o n

+ l e a d s t o an i l l - d e f i n e d

= +O(x)

t =

at

o

(3.10)

Hadamard's problem whose s o l u t i o n i s u n s t a b l e u n d e r

small v a r i a t i o n s of t h e i n i t i a l d a t a . In d e r i v i n g d i f f e r e n c e e q u a t i o n s f o r problem ( 3 . 1 ) , w e s h o u l d t a k e i n t o a c c o u n t t h e c o n d i t i o n LI = ur/Ax 1 1 t h a t t h e problem i s w e l l - d e f i n e d . L e t u s now examine t h e c o m p u t a t i o n s t a b i l i t y of scheme ( 3 . 3 ) . F i r s t , we c o n s i d e r t h e s p e c t r a l problem

i n an i n f i n i t e - d i f f e r e n c e which i s bounded i n D h ,

-

interval D = h

< xk <

-.

The s o l u t i o n of

(3.11),

is ikpAx

(3.12)

Wk = where

p

i s an a r b i t r a r y i n t e g e r . S u b s t i t u t i n g (3.12) i n t o (3.11), w e a r r i v e

a t t h e formula f o r t h e eigenvalue

+ i s i n PAX].

(3.13)

Write ( 3 . 3 ) i n t h e o p e r a t o r form

(3.14) and s e e k a s o l u t i o n i n t h e form m

+j =

1

$;

elkpax

(3.15)

p=-m

where + J i s t h e F o u r i e r c o e f f i c i e n t of t h e f u n c t i o n + J . For t h e c o e f f i c i e n t s $J P P we o b t a i n t h e e q u a t i o n

(3.16) Hence,

where T = 1 P

-

rh

P

is t h e t r a n s i t i o n f a c t o r f o r the Fourier coefficients.

86

Chapter 3 The c o n d i t i o n t h a t none of t h e components $ J i n c r e a s e i n modulus i s P 11

-

Th

P

1

1.

5

(3.17)

t h e i n e q u a l i t y t a k i n g p l a c e i f u-r/Ax 5 1. I n d e e d ,

> 1 , then [ I - T X

If UT/AX

of v a r i a t i o n i n 11 -

-rx 1

(El'

+

2 '

AX

s i n 2 PAX =

1

> 1. T h e r e f o r e , it remains t o c o n s i d e r t h e r a n g e P f o r 0 < uT/Ax C 1 ( r e c a l l t h a t we assume u > 0 ) . In t h e

P above i n t e r v a l , t h e f u n c t i o n

AX

uT/Ax = 1 / 2 . In t h i s c a s e , 11

(1

- TX

-

=)

t a k e s a maximum v a l u e e q u a l t o 1/4 f o r

1

= 1

-

AX

sin'

2

=

cos'

9.

For o t h e r v a l u e s

i n t h e i n t e r v a l 0 5 u-r/Ax 5 1, t h i s p o s i t i v e f u n c t i o n d e c r e a s e s and t e n d s

of U T / A X

t o z e r o a t t h e extreme p o i n t s ur/Ax = 0 and U T / A X

= 1. T h i s means t h a t we have

the inequality

(F) (1 - =) Ax 5

1

which proves t h e s t a t e m e n t . Thus, we have e s t a b l i s h e d t h e c o n d i t i o n of computation s t a b i l i t y of a d i f f e r e n c e scheme. E v i d e n t l y , t h e s t a b i l i t y c r i t e r i o n c o i n c i d e s i n our c a s e with t h e c o n d i t i o n t h a t E q . ( 3 . 8 ) i s w e l l - d e f i n e d . Using t h e above method, we can r e a d i l y show t h a t scheme ( 3 . 4 ) a l s o p r o v i d e s t h e f i r s t - o r d e r approximation i n Ax and x = x

k

T.

Expansion i n t o a T a y l o r s e r i e s f o r

and t = t . y i e l d s t h e e q u a t i o n s i m i l a r t o ( 3 . 8 ) : J

g + u aa+= x

where II = (uAx

-

a''

I.-+ ax*

...,

(3.18)

u2r)/2.

Even now we can s e e a fundamental d i f f e r e n c e between ( 3 . 8 ) and ( 3 . 1 8 ) . The c o e f f i c i e n t of computation v i s c o s i t y i s always p o s i t i v e i n t h e l a t t e r e q u a t i o n . Consequently, E q . ( 3 . 1 8 ) i s always w e l l - d e f i n e d f o r t h e r e l e v a n t s u f f i c i e n t l y smooth i n i t i a l d a t a . I t i s a s i m p l e m a t t e r t o show t h a t d i f f e r e n c e Eq.(3.4)

i s s t a b l e f o r any r e l a t i o n between t h e s t e p s , i . e . it i s a b s o l u t e l y

s t a b l e s i n c e t h e t r a n s i t i o n f a c t o r f o r each F o u r i e r c o e f f i c i e n t i s T This implies t h a t

IT

1

P

=

1/(1 + TX ).

P

5 1.

Among t h e most i n t e r e s t i n g and commonly used d i f f e r e n c e schemes, o t h e r than ( 3 . 3 ) and ( 3 . 4 ) , we can p o i n t o u t t h e f o l l o w i n g two:

(3.19)

Numerical Solution of Basic and Adjoint Equations

+k ++ll2 l J

+ u

T

Here

+kj + W =

t(+;+l

- *k-1

87

J

2 Ax

=

0;

(3.20)

,$;),

+

One c a n e a s i l y see t h a t scheme ( 3 . 1 9 ) g i v e s a f i r s t - o r d e r

approximation

i n T and a s e c o n d - o r d e r a p p r o x i m a t i o n i n Ax. The d i f f e r e n t i a l e q u a t i o n c o r r e s p o n d i n g t o t h i s scheme i s ( 3 . 1 8 ) where p = u z ~ / 2 . The scheme s t a b i l i t y i s d e t e r m i n e d by t h e t r a n s i t i o n o p e r a t o r s f o r F o u r i e r c o e f f i c i e n t s . W e a r r i v e , t h e r e f o r e , a t t h e s p e c t r a l problem h

(A w

) =~ u

W k + l - W k - l = hw 2 Ax k

(3.21)

I t s s o l u t i o n w i l l b e s o u g h t i n t h e form ( 3 . 1 2 ) . Thus, we h a v e A

2 sin

= i

P

Ax

(3.22)

PAX,

h e n c e w e immediately o b t a i n t h e e q u a t i o n f o r t h e F o u r i e r c o e f f i c i e n t s

+ P J +-l

+,:

+

A

P P

= 0,

where T p

= L l+Th

With due r e g a r d t o ( 3 . 2 2 ) , we h a v e T

(J 1 +

lTpl =

(3.23)

P

P

= 1/(1 + i

( ~ 1 's i n 2

-1

PAX)

u1 sin Ax

PAX) a n d , h e n c e ,

5 1

T h i s i n d i c a t e s a n a b s o l u t e c o m p u t a t i o n s t a b i l i t y of scheme ( 3 . 1 9 ) . Of p a r t i c u l a r i n t e r e s t t o u s , i n view of a p p l i c a t i o n s , i s t h e CranckNickolson scheme ( 3 . 2 0 ) . One i s r e a d i l y c o n v i n c e d t h a t t h i s scheme g i v e s a secondo r d e r approximation i n

T

and Ax and i s n o t d i s s i p a t i v e . T h i s means t h a t i n d i f f e r -

p = 0 and t h e d i s c a r d e d terms a r e of

e n t i a l Eq.(3.18)

t h e o r d e r s o f TAX, T z , and

Ax2. As r e g a r d s t h e c o m p u t a t i o n s t a b i l i t y , w e h a v e i n t h i s c a s e T

P

= (1

-

i EL

2 Ax

s i n pax)/

(1 + i KL s i n p ~ x ) , 2 Ax

1

IT = 1. T h u s , t h i s scheme i s a b s o l u t e l y s t a b l e . P I n c o n c l u s i o n , w e c o n s i d e r a n o t h e r a t t r a c t i v e method f o r n u m e r i c a l s o l u t i o n

and, hence,

of problem ( 3 . 1 ) b a s e d on t h e s o - c a l l e d

running cornputation schcmc s u g g e s t e d by

88

Chapter 3 N.N.Neiman, and 1.M.Khalatnikov. The scheme i s of t h e form

L.D.Landau,

J

One can e a s i l y s e e t h a t t h i s scheme i s approximate t o t h e second o r d e r i n t o the f i r s t order in

T

and

x . I t i s d e f i n e d by t h e r e c u r r e n c e r e l a t i o n

(3.25) I t can be r e a d i l y demonstrated by t h e F o u r i e r a n a l y s i s of s t a b i l i t y , i n t h e s e n s e of J.von Neumann, t h a t scheme ( 3 . 2 4 ) i s a b s o l u t e l y s t a b l e . S i m i l a r l y , we can e l a b o r a t e an a b s o l u t e l y s t a b l e running computation scheme f o r a multidimensional problem of motion and prove t h a t i t i s a b s o l u t e l y s t a b l e f o r an e q u a t i o n w i t h c o n s t a n t c o e f f i c i e n t s .

We have always assumed above t h a t t h e f u n c t i o n positive. If

u

is negative,

then replacing x

u

i s t i m e - c o n s t a n t and

by - x , we a r r i v e a t t h e above

e q u a t i o n . Of p a r t i c u l a r i n t e r e s t for a p p l i c a t i o n s , however, i s t h e c a s e o f u = u ( x , t ) . Even a s i m p l e a n a l y s i s r e v e a l s t h a t i n t h i s s i t u a t i o n computation s t a b i l i t y can f a i l , d e s p i t e t h e u s e of i m p l i c i t d i s s i p a t i v e schemes. T h i s f a c t shows up e s p e c i a l l y i n n o n - l i n e a r

problems. The main p o i n t h e r e i s a s f o l l o w s .

I f t h e s o l u t i o n of a d i f f e r e n c e problem and t h e c o e f f i c i e n t u ( x k , t . ) a r e expanded J i n a F o u r i e r s e r i e s , t h e product of t h e F o u r i e r s e r i e s g i v e s r i s e t o harmonics which a r e e i t h e r l o n g e r or s h o r t e r t h a n t h e i n t e r a c t i n g harmonics. As a r e s u l t , t h e energy r e l a t e d t o rounding-off

e r r o r s may sometimes be t r a n s f e r r e d from

l o n g e r t o s h o r t e r waves, t h e r e b y making t h e computation p r o c e s s u n s t a b l e , a l t h o u g h t h i s d i f f e r e n c e scheme w i t h a c o n s t a n t c o e f f i c i e n t i s c o m p u t a t i o n - s t a b l e . k i n d of i n s t a b i l i t y i s g e n e r a l l y termed non-tinear

instability.

This

Sometimes, it

a l s o a r i s e s i n s o l v i n g l i n e a r problems w i t h v a r i a b l e c o e f f i c i e n t s .

It is, there-

f o r e , of c u r r e n t i n t e r e s t t o e l a b o r a t e p e r t u r b a t i o n - s t a b l e d i f f e r e n c e schemes f o r n o n - l i n e a r e q u a t i o n s o r for e q u a t i o n s w i t h v a r i a b l e c o e f f i c i e n t s . In most c a s e s , computation i n s t a b i l i t y can

b e e l i m i n a t e d by u s i n g d i s s i p a t i v e d i f f e r e n c e

schemes w i t h a s u i t a b l e c h o i c e of t h e computation v i s c o s i t y c o e f f i c i e n t IJ. But such schemes g i v e , a s a r u l e , a f i r s t - o r d e r approximate e i t h e r i n

or b o t h i n

T

T ,

or i n hx,

and i n ax.

Of p a r t i c u l a r i n t e r e s t t o a p p l i c a t i o n s a r e e q u a t i o n s of t h e form

(3.26) where u = u ( x , t ) . The d i f f e r e n c e schemes f o r t h i s t y p e of e q u a t i o n s , which a r e a b s o l u t e l y s t a b l e and p r o v i d e a f i r s t - o r d e r or even a second-order approximation

Numerical Solution of Basic and Adjoint Equations

89

for some c l a s s e s of c o e f f i c i e n t s , w i l l be s t u d i e d f o r m u l t i d i m e n s i o n a l e q u a t i o n s

of t h e form (3.26). Let u s a n a l y s e t h e motion of a p a r t i c l e ensemble a l o n g given t r a j e c t o r i e s i n t h e (x, y ) p l a n e . Within t h e framework of t h e mechanics of c o n t i n u o u s media, we a r r i v e a t t h e problem

(3.27)

9

(x, Y , 0 ) = g ,

where u = u ( x , y , t ) , v = v ( x , y , t ) . Assume t h a t t h e components of t h e v e l o c i t y vector, u

and

v , a t each moment s a t i s f y t h e c o n t i n u i t y e q u a t i o n

au

av

-ax + -a =y o

(3.28)

'

Let a s o l u t i o n b e s o u g h t i n t h e r e c t a n g u l a r domain D , { O

~

x ' a,

0 5 y 5 b } , and l e t t h e s o l u t i o n of problem (3.27) t o g e t h e r w i t h t h e c o e f f i c i e n t s

u

and

v

be p e r i o d i c and t a k e t h e same v a l u e s on t h e opposing b o u n d a r i e s of t h e

rectangle. W r i t e (3.27) i n t h e o p e r a t o r form *+A$

= 0 ,

at

(3.29)

$ = g

a ax

where A = u - + v

a

ay

at

t = O

.

E v i d e n t l y , f o r o u r problem t h e o p e r a t o r (A$,

A

s a t i s f i e s the condition

4) = 0 . I n d e e d , i n t r o d u c i n g a s c a l a r p r o d u c t , we w r i t e a

b

(3.30)

With due r e g a r d t o (3.28), t h e i n t e g r a n d i s t r a n s f o r m e d t o

whence i t f o l l o w s t h a t

(3.31)

Hence, t a k i n g i n t o account t h a t t h e s o l u t i o n i s p e r i o d i c on t h e b o u n d a r i e s , we o b t a i n (A$,

$) = 0 .

(3.32)

90

Chapter 3 Thus, the operator

A

can be reduced to an antisymmetric form, thereby

permitting an elaboration of absolutely stable difference schemes.

A

Now we shall try to split the operator the elementary operators

A

a

(a

=

in such a manner that each of

1 , 2 ) also satisfies the condition (Aa$,

4)

=

0.

(3.33)

In this case, the difference scheme of component splitting provides an absolutely stable scheme with a second-order approximation.

A formal decomposition of the operator A

into constituents

is not compatible with condition (3.33). One can easily see that the following relations are valid:

This means that we cannot use A1 and A2 as elementary operators in the elaboration of the successive splitting scheme. Now let us choose the operators A1 and A2 in a more complicated form

+$aV 2 ay'

(3.35)

It is evident that each of these operators satisfies condition (3.33) and their sum is exactly equal to A (A1 + A ) @ = u 2

3 ax +

ay

+

* (+ ax ?)! 2

+

=

Here we have employed the fact that the coefficients u

ax

+

and

?#! ay = v

satisfy Eq.(3.28)

So, all the necessary conditions for the splitting method applicability are now valid, and we come to the splitting scheme in the interval t . J-1

' '

j +1

(3.36)

Numerical Solution

If t h e functions

u

and

v

91

Basic and Adjoint Equations

of

as w e l l a s the solution

9

are sufficiently

smooth f o r a l l t h e v a r i a b l e s , scheme ( 3 . 3 6 ) i s a p p r o x i m a t e t o t h e second o r d e r and i s a b s o l u t e l y s t a b l e

(1 $ J + l 11 = p - 1 I( =

...

=

(1 61).

(3.37)

T h i s i s an i n s t r u c t i v e example t o show t h a t a f o r m a l d e c o m p o s i t i o n i n t o o p e r a t o r s ( 3 . 3 4 ) may compromise t h e v e r y i d e a of s p l i t t i n g and t h a t o n l y some a d d i t i o n a l c o n s i d e r a t i o n s may l e a d t o schemes which a r e proved t h e o r e t i c a l l y and e f f e c t i v e i n a p p l i c a t i o n . F o l l o w i n g t h i s p r e l i m i n a r y i n t r o d u c t i o n , w e w i l l now e l a b o r a t e d i f f e r e n c e schemes f o r s o l v i n g problem ( 3 . 2 7 )

i n s p a c e v a r i a b l e s and i n t i m e . To t h i s e n d ,

w e f i r s t c o n s i d e r t h e o p t i m a l methods of a p p r o x i m a t i n g t h e o p e r a t o r A w i t h r e s p e c t t o space v a r i a b l e s

x

and

y . A s was a l r e a d y n o t e d , a c o n v e n i e n t method o f

a p p r o x i m a t i n g problems i n m a t h e m a t i c a l p h y s i c s which p r e s e r v e s t h e a d d i t i v e p r o p e r t i e s and q u a l i t a t i v e f e a t u r e s of t h e o p e r a t o r , i s a c o o r d i n a t e approximat i o n . P r e c i s e l y t h i s method w i l l b e u s e d t o e l a b o r a t e d i f f e r e n c e schemes. Suppose t h e c o e f f i c i e n t s Eq.(3.27)

u

and

v

are s u f f i c i e n t l y smooth and c o n s i d e r

i n t h e d i v e r g e n t form

(3.38) $ = g

in

D for

t = 0.

To c o n s t r u c t t h e d i f f e r e n c e scheme, w e s h a l l u s e t h e o p e r a t o r d e f i n e d a s (3.39) and c o n s i d e r t h e f o l l o w i n g d i f f e r e n c e a n a l o g u e of t h i s r e l a t i o n :

- -'k!L

["k+l,!L

2

-

Uk-l,P. 2Ax

Evidently, d i f f e r e n c e expression

Vk,L+l

(3.40)

.k,L-l]

~ A Y

( 3 . 4 0 ) a p p r o x i m a t e s ( 3 . 3 9 ) t o t h e second

o r d e r i n Ax a n d Ay f o r s u f f i c i e n t l y smooth f u n c t i o n s s i o n has a s u b s t a n t i a l shortcoming,

-

u , v , and

4 . But t h i s e x p r e s -

s i n c e i n t h i s form t h e o p e r a t o r A is no l o n g e r

92

Chapter 3

a n t i s y m m e t r i c , i .e . (Ah$,

9)

# 0.

(3.41)

T h i s i m p l i e s t h a t t h e c o n v e n t i o n a l a p p r o x i m a t i o n does n o t f i t t h e form of t h e computation a l g o r i t h m used t o s o l v e problem ( 3 . 2 7 ) .

W e now show t h a t t h e a p p r o x i m a t i o n o f ( 3 . 3 9 ) i n t h e form

(3.42)

- v k,k-1/2 4 k , k - 1

Vk,P.+l/2'k,!.,+l

2AY s a t i s f i e s the basic relation h

(A

$,

(3.43)

$1 = 0

and i s of t h e second o r d e r i n bx and by. T o t h i s e n d , w e s h a l l a p p r o x i m a t e t h e coefficients a s follows:

(3.44) 'k-1/2,!.,

+

= uk - 1 , e

UkV.

-

Uk-l,l

2

S u b s t i t u t i n g (3.44) i n t o ( ( 3 . 4 2 ) , we o b t a i n a f t e r simple t r a n s f o r m a t i o n s

'i

@ Vk,l+l'k,Y.+l

vk,l-l k , l - 1

2AY

- -'kk 2

~ k + l , k u k - l , V . Vk,E+l-vk,l-l' J 2by 2Ax +

-

where

f o r Ax -> 0 and Ay

+

0.

N o w l e t u s assume t h a t t h e c o e f f i c i e n t s u k l , vk9, s a t i s f y t h e d i f f e r e n c e

a n a l o g u e of t h e

c o n t i n u i t y equation

93

Numerical Solution of Basic and Adjoint Equations Uk+l,P - Uk-l,k 2Ax If the coefficients

u , v,

d e r i v a t i v e s with r e s p e c t t o expression

x

,

- Vk,k-l = 0 ( h 2 ) . 2AY

'k,k+l

T

(3.46)

and t h e s o l u t i o n $) h a v e f i n i t e s e c o n d - o r d e r and y

and i f c o n d i t i o n (3.46) 1s s a t i s f i e d ,

(3.45) d i f f e r s from (3.40) i n t h e same o r d e r a s (3.40) d i f f e r s from

(3.39). T h u s , we h a v e d e m o n s t r a t e d t h a t e x p r e s s i o n (3.42) a p p r o x i m a t e s (3.39) t o t h e s e c o n d o r d e r i n Ax and Ay. We now d e m o n s t r a t e t h a t t h e o p e r a t o r t h u s c o n s t r u c t e d meets c o n d i t i o n (3.43) a n d , what i s more, e a c h of t h e o p e r a t o r s A1

and

Az

d e f i n e d by

(3.47)

A'$= 2

V k , L+l/Z"k, k + l - V k ,Q-l/Z'k, 1-1 2AY

also s a t i s f i e s the condition

(Ae@, 9)

=

0.

(3.48)

For t h i s p u r p o s e , w e d e f i n e a s c a l a r p r o d u c t f o r v e c t o r q u a n t i t i e s

5,

as

(3.49)

C o l l e c t i n g t e r m s i n (3.49), w e o b t a i n Eqs.(3.48) which d i r e c t l y imply c o n d i t i o n (3.43). T h u s , w e have a c c o m p l i s h e d n e c e s s a r y s p a c e a p p r o x i m a t i o n s . Now, our aim i s t o r e d u c e i n t i m e t h e system of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s

'

h

h

- = g , where

A = A1

+

A2,

9. i s

(3.50)

t h e v e c t o r f u n c t i o n w i t h t h e components ,$ ,,

and

s a t i s f i e s c o n d i t i o n (3.48). T h i s means t h a t problem (3.50) c a n be s o l v e d u s i n g the splitting

method. O m i t t i n g t h e s u p e r s c r i p t

f u n c t i o n s and o p e r a t o r s w e o b t a i n t h e s y s t e m

h (as inessential) in the

94

Chapter 3

(3.51)

t . 5 t 2 t j+l' J-1 Thus, problem ( 3 . 2 7 ) h a s been reduced t o t h e system of s i m p l e one-dimension-

on t h e i n t e r v a l

a 1 d i f f e r e n c e e q u a t i o n s which can b e s o l v e d by f a c t o r i z i n g t h r e e - p o i n t

difference

e q u a t i o n s . S i m i l a r l y , we can a l s o s o l v e t h e e q u a t i o n of motion i n three-dimensiona 1 s p a c e , when A = A

+

A2 + A s .

I t i s noteworthy t h a t t h e s o l u t i o n of t h e e q u a t i o n of motion by c o o r d i n a t e s p l i t t i n g , w i t h t h e space o p e r a t o r approximated by f i n i t e - d i f f e r e n c e o p e r a t o r s i n t h e form ( 3 . 4 0 ) , i s n o t a unique method a d m i t s a q u a d r a t i c i n v a r i a n t i n t h e d i f f e r e n c e form. We now g i v e a n o t h e r example concerned w i t h t h e s o l u t i o n of a ( t h r e e - d i m e n s i o n a l ) t r a n s p o r t e q u a t i o n based on s e p a r a t i o n of t h e b a r o t r o p i c component of t h e v e l o c i t y . In t h i s c a s e , t h e problem i s reduced t o a s e r i e s of p l a n e problems for domain s e c t i o n s , which e n a b l e s u s t o e f f e c t i v e l y employ t h e f i n i t e - e l e m e n t method t o make t h e d i f f e r e n t i a l o p e r a t o r s d i s c r e t e .

S o , we c o n s i d e r t h e e q u a t i o n (3.52) $ = g i n t h e domain (0, T ) G , where base

lo,

upper b a s e

In, and

G

at

t = O

is t h e three-dimensional c y l i n d e r with t h e

l a t e r a l surface

1. A s

previously, the coefficients

a r e assumed t o obey t h e c o n t i n u i t y e q u a t i o n

aa ux + -da Y+v - =aw aoZ

(3.53)

B e s i d e s , i t is r e q u i r e d t h a t

I t i s w e l l known t h a t p r o c e s s e s t a k i n g p l a c e i n t h e atmosphere or i n t h e ocean a r e q u a s i - h o r i z o n t a l , i . e . t h e h o r i z o n t a l s c a l e s of motion and v e l o c i t y markedly exceed,on t h e a v e r a g e , t h e v e r t i c a l s c a l e s . Hence, h o r i z o n t a l motions

Numerical Solution of Basic and Adjoint Equations

p l a y a dominant r o l e i n p l a n e t a r y - s c a l e

95

geographical processes. Nevertheless,

v e r t i c a l p r o c e s s e s , though of a r e l a t i v e l y s m a l l s c a l e , s h o u l d a l s o be t a k e n into consideration

t o e n s u r e a c o r r e c t d e s c r i p t i o n of t h e v e r t i c a l d i s t r i b u t i o n

of t h e c h a r a c t e r i s t i c s i n t h e medium. By s e p a r a t i n g t h e b a r o t r o p i c component i n t h i s c a s e w e c a n d i s t i n g u i s h between t h e m o t i o n s of d i f f e r e n t q u a n t i t i e s and s c a l e s and s i m p l i f y c o n s i d e r a b l y t h e s o l u t i o n of t h e problem. T h u s , w e s h a l l r e p r e s e n t t h e v e l o c i t y components a s

(3.55) where

0

- _

0

Then, 5 , v , u ' , v ' w i l l s a t i s f y t h e c o n t i n u i t y e q u a t i o n s of t h e form

(3.56) au'

+

ax Using Eq.(3.56), w e c a n d e f i n e a

ay

+

;iw aZ

o.

=

(3.57)

Y' by t h e r e l a t i o n s

StYWm fUnCtiOn

(3.58)

or -

AY

= rot u, ~

y17. = O .

W e s h a l l employ (3.57) t o f i n d t h e v e r t i c a l v e l o c i t y component i n g (3.57) a l o n g t h e v e r t i c a l between w

H

= 0, we

z

H

and

(3.59) w. Integrat-

and a l l o w i n g f o r t h e f a c t t h a t

obtain H

w =

a ax

u'dz +

-

51

z

Introducing t h e n o t a t i o n

I

H

&

=

Z

v' dz.

(3.60)

Z

H u' dz,

c=

v ' dz

(3.61)

Z

and s u b s t i t u t i n g (3.56), (3.58), (3.61) i n t o (3.52), w e a r r i v e a t t h e e q u a t i o n

96

Chapter 3

Consider t h e s p a c e o p e r a t o r of t h i s e q u a t i o n .

I t c a n be r e p r e s e n t e d i n t h e

form

+ Axz + A

A = A XY

YZ'

where

(3.63)

a A a

a A a

YZ

Analysing t h e f u n c t i o n a l of e n e r g y for Ax y , A x z ,

Thus, t h e o p e r a t o r

A

we can prove t h a t

Ayz,

i s a g a i n r e p r e s e n t e d a s t h e sum of a n t i h e r m i t i a n

o p e r a t o r s . Hence, w e c a n s o l v e ( 3 . 6 2 ) f o r t i m e , u s i n g t h e above two-cycle

splitt-

i n g scheme. Then, f o r e a c h i n t e r v a l w e o b t a i n a set of problems t o be s o l v e d successively +J-2/3

-

'+'j-1

T

$j-l/3

$j-2/3 T

+

AJ

'+ J-2/3

+

2

XY

+

j-1/3 AJ

XZ

+

$'

6J-I =

0,

+j-2/3 = 0,

2

(3.65)

+J+2/3

-

+J+1/3 T

+

$J+2/3 A'

XZ

+

$J+1/3 = 0,

2

Since conditions (3.64) a r e s a t i s f i e d f o r t h e o p e r a t o r s A

Axz,

Ayz,

scheme

XY'

(3.65) is absolutely s t a b l e .

L e t u s now c o n s i d e r t h e method of s p a t i a l d i s c r e t i z a t i o n of two-dimensional o p e r a t o r s Ax y , A x z ,

and A YZ

.

In t h i s c a s e , a p p r o x i m a t i o n s on s e c t i o n s p a r a l l e l

t o t h e c o o r d i n a t e p l a n e s a r e c a r r i e d o u t b e s t of a l l by t h e f i n i t e element method, which y i e l d s d i f f e r e n c e o p e r a t o r s o b e y i n g a n a n a l o g u e o f t h e c o n s e r v a t i o n l a w .

97

Numerical Solution of Basic and Adjoint Equations L e t u s i l l u s t r a t e t h e a p p l i c a t i o n of t h i s method f o r a two-dimensional

equation

a s a s p a c e o p e r a t o r d e f i n e d by t h e f i r s t of r e l a t i o n s (3.63)

containing A XY

a4

+

Axy@ = 0 , (3.66)

+ = g i n a r e c t a n g u l a r domain G = { ( x , y )

at

1

t = O

0

~

x

a, 0

~

y

<

b l w i t h t h e boundary

1.

A s w e know, t h e f i n i t e element method i s t h e G a l e r k i n method for c o o r d i n a t e f u n c t i o n s w i t h a f i n i t e s u p p o r t . To d e f i n e t h e c o o r d i n a t e f u n c t i o n s , w e i n t r o d u c e in

G

a network domain Gh w i t h s t e p s Ax and Ay a l o n g t h e c o o r d i n a t e d i r e c t i o n s Gh = { ( x i , y j )

1

xi = iAx, y . = j A y , i = J

M = a/Ax, GM, j = GN,

N = b/Ay)

The c e l l s o f t h e define in

G

network

a r e t r i a n g u l a t e d by p o s i t i v e d i a g o n a l s . L e t u s

G

continuous functions w

mn

( x , y ) which a r e l i n e a r i n each t r i a n g l e

and s u c h t h a t

0 ( x , y) have a hexagon a s a c a r r i e r ( f i g u r e 3.1), and t h e mn f u n c t i o n s t h e m s e l v e s a r e shown i n f i g u r e 3.2.

The f u n c t i o n s

FIGURE 3.1 According t o

FIGURE 3.2

t h e G a l e r k i n method, an a p p r o x i m a t e s o l u t i o n i s s o u g h t a s

a l i n e a r c o m b i n a t i o n of t h e c o o r d i n a t e f u n c t i o n s M

N

(3.67) m=O n=O Then, u s i n g a q u a d r a t u r e f o r m u l a t o c a l c u l a t e t h e e v o l u t i o n t e r m , w e c a n w r i t e

98

Chapter 3

t h e e q u a t i o n s f o r t h e c o e f f i c i e n t s 9*(t)

(3.68)

Qmn ( 0 )

=

g

S u b s t i t u t i n g (3.67) i n t o (3.68), we o b t a i n

(3.69)

a r e l i n e a r , t h e i r d e r i v a t i v e s on K = i w dx d y . S i n c e t h e f u n c t i o n s w mn D Y mn the triangles T t a k e t h e v a l u e s p r e s e n t e d i n t a b l e 3.1. mn

where

TABLE 3.1

E v a l u a t i n g t h e i n t e g r a l s i n (3.69) and c o l l e c t i n g terms, w e f i n a l l y a r r i v e a t a problem which c a n be e x p r e s s e d i n t h e m a t r i x form K

a@h at +

ib = g Here [ Oh]mn = $mn

t h e form

I

iiimn dx

dy

I?Qh

= 0,

(3.70) at

t = 0.

and I? i s t h e d i f f e r e n c e o p e r a t o r which i s of

D

(3.71)

99

Numerical Solution of Basic and Adjoint Equations

y(i)

=

mn

I

Y dx dy.

Ti n D mn at the point (xm, yn). To verify how problem (3.70) approximates the initial problem (3.52), we substitute a smooth function into (3.52) and expand this function as well as the coefficients of (3.70) into a Taylor series in the neighborhood of the point (xm, y ) . Performing all calculations, we find that problem (3.70) approximates (3.52) up to the terns of the order Axz

,0

If each Eq.(3.70) is multiplied by

and sunned over all

m = 0, M,

~

n = 0, N , we get the relation +h)h

-

= 0,

at

2

which is a difference representation of the quadratic conservation law for (3.70). This follows from the fact that (3.72)

(Adh, +h)h= 0.

Let us prove the validity of this relation. Examining (3.71), we see that the coefficients of the operator satisfy the following relations: (ly(2

mn

-(6)

(*mn ($1)

mn

)

-

$3)

-

$1))

-

mn

) =

-(6

-( $ 5 )

m,n+l - 'n,n+l)'

= -($3)

mn

m+l,n

p(2) ) = -(+)

-

m+l,n+l

mn

$4)

m+l,n)'

(3.73)

- $5)

m+l,n+l).

Rearranging the terms of (3.72) one can easily demonstrate that

Consider the structure of the operator A. It is evident from (3.71) that this operator can be represented as the sum of one-dimensional operators A = A where

1

+ A

2

+ A

3'

100

Chapter 3

(3.74)

A s f o r the operator A , the relations

a r e a l s o valid for the operators

A 1, A 2 ,

and A3.

T h i s means t h a t t o s o l v e ( 3 . 7 0 )

f o r t i m e , w e c a n employ t h e s p l i t t i n g method, which w i l l be a b s o l u t e l y s t a b l e .

In e a c h s t e p , t h e o p e r a t o r i s r e v e r s e d by t h e f a c t o r i z a t i o n method. In t h e i n t e r v a l t .

J-1

2 t S t

j +1

t h e scheme i s

(3.76)

$j+2/3 K

-

#J+1/3

h

#j,+2/3

+ A

2

+

= o

2

S i m i l a r l y , we can approximate t h e o p e r a t o r s A

on t h e v e r t i c a l

and A XZ

YZ

s e c t i o n s of t h e domain G . T h u s , problem ( 3 . 5 2 ) i s r e d u c e d t o a s u c c e s s i v e s o l u t i o n o f a s e t of one-dimensional 3.4.

difference equations.

Diffusion Equation

In a c c o r d a n c e w i t h t h e s p l i t t i n g method, w e r e d u c e t h e b a s i c problem of substance d i f f u s i o n i n t h e i n t e r v a l t . 5 t 5 t j+l J

101

Numerical Solution of Basic and Adjoint Equations + = + j

at

+ = o

t = O

on

1 (4.1)

t o t h r e e problems. The f i r s t d e s c r i b e s d i f f u s i o n a l o n g t h e z - a x i s

+1 =

+J

at

t

o

=

(4.2)

t h e second problem

-

along th e x-axis

j+l

+2 = +1

and t h e t h i r d problem

-

at

t = t

(4.3)

j

along t h e y-axis

(4.4)

i s sought e i t h e r i n Assume t h a t t h e s o l u t i o n f o r problem ( 4 . 1 ) w i t h t = t . J+1 -m < x < m , -m < y < m , 0 5 z 5 H or i n t h e p a r a l l e l e p i p e d -a 5 x a,

the layer

-b 5 y 5 b , 0 5 z 5 H , where

a

and

b

c a n be i n f i n i t e i n a p a r t i c u l a r c a s e .

Then, f o r t h e s t r i p - s h a p e d r e g i o n t h e s o l u t i o n can be r e p r e s e n t e d a s a F o u r i e r i n t e g r a l with r e s p e c t t o

x,

y

z; f o r t h e p a r a l l e l e p i p -

and a F o u r i e r s e r i e s i n

ed t h e s o l u t i o n is represented a s a Fourier s e r i e s with r e s p e c t t o a l l t h e t h r e e v a r i a b l e s ( x , y , z). In b o t h c a s e s , f o r y i e l d s an e x a c t s o l u t i o n of ( 4 . 1 ) ,

i.e.

f = 0 and t = t .

+J+' 3

J+1

= +J+';

t h e s p l i t t i n g method

i n t h e remaining c a s e s , t h e

Chapter 3

102

s o l u t i o n i s approximate. We have assumed so f a r t h a t we d e a l w i t h a d i f f e r e n t i a l s t a t e m e n t of t h e problem. In a c t u a l s i t u a t i o n s , h o w e v e r , w e u s u a l l y r e p l a c e d i f f e r e n t i a l e q u a t i o n s by f i n i t e - d i f f e r e n c e a n a l o g u e s . T h i s means t h a t i n s t e a d of (4.1) we c o n s i d e r t h e problem

+

2'

(Al

+

A2 + A ) $ = f , 3

(4.5)

where

~

k = 1, n ,

-

Y. = 1, p , m

= 0 , s ; Azm =

zm

- z ~ - Azo ~ ,

=

A Z ~ += ~0 .

For s i m p l i c i t y , we assume h e r e t h a t t h e x-y network i s uniform w i t h s t e p s Ax and Ay, r e s p e c t i v e l y . For c o n v e n i e n c e , w e have a l s o o m i t t e d t h e i n d i c e s which a r e n o t e s s e n t i a l f o r t h e o p e r a t o r s ha i n ( 4 . 5 ) and ( 4 . 6 ) . Note t h a t i n t h e neighborhood of t h e boundary p o i n t s e x p r e s s i o n s ( 4 . 6 ) a r e somewhat modified i n accordance w i t h t h e boundary c o n d i t i o n s

4 k = 0

at

k = O , k = n + l ,

@,=O

at

L = O , I ! = p + l ,

'm

AZ

'm-1

- 'm

'm-1 AZ

-

at

m = 0,

m

= 0

at

m

= 3.

m+l

One can r e a d i l y s e e t h a t d i f f e r e n t o p e r a t o r s A 1, A 2 ,

and A 3 on t h e f u n c t i o n s

Qkem s a t i s f y i n g boundary c o n d i t i o n s ( 4 . 7 ) a r e p o s i t e v e l y d e f i n i t e . I n d e e d , m u l t i p l y

r e s p e c t i v e E q s . ( 4 . 6 ) by $k Ax, $,A$,

and $m(Azm + A Z ~ + ~ and ) / ~sum up e v e r y r e s u l t

o v e r nodes of t h e network. Taking i n t o account t h e boundary c o n d i t i o n s , we d e r i v e

(4.8)

103

Numerical Solution of Basic and Adjoint Equations

A s p r e v i o u s l y , h e r e i n d i c e s a r e used o n l y f o r e s s e n t i a l v a r i a b l e s . Thus, we have approximated ( 4 . 1 ) i n t h e s p a c e v a r i a b l e s . I t remains t o approximate ( 4 . 5 ) w i t h r e s p e c t t o t i m e . To t h i s e n d , we employ t h e s p l i t t i n g method:

(4.9)

+ J +-l $J+5/6 7

+

+ J ++l + J + 5 / 6 Al 2

= 0.

A s was shown e a r l i e r , s p l i t t i n g i n t h e form ( 4 . 9 ) e n s u r e s a second-order approx i m a t i o n i n T , whereas t h e o r d e r of approximation i n

x , y , z, which was adopted

p r e v i o u s l y , p r o v i d e s a c c u r a c y t o t h e second o r d e r i n Az, Ax, and Ay. 3 . 5 . General Numerical Algorithm A s was a l r e a d y p o i n t e d o u t , i t i s sometimes r e a s o n a b l e t o s o l v e problem

( 2 . 1 ) by t h e

component splitting-up method, which comes about a s f o l l o w s . Problem

(2.1) is written a s

where

Using d i f f e r e n c e a p p r o x i m a t i o n s o u t l i n e d i n t h e f o r e g o i n g d i s c u s s i o n , we reduce problem ( 5 . 1 ) t o t h e system of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s

104

Chapter 3

>2+ (Al + at

A2 + A ) + 3

f ,

=

(5.3)

where A 1, h 2 , /I3, a r e t h e d i f f e r e n c e o p e r a t o r s approximating A 1, A 2 ,

A3,

res-

p e c t i v e l y . E v o l u t i o n problem ( 5 . 3 ) i s reduced by component s p l i t t i n g t o s u c c e s s i v e s o l u t i o n s of a number of problems +J+l/S

$j+1/6 T

-

$j+2/6

1

$j+1/6

$j+3/6 - +j+2/6 7

#J+4/6

-'

j+5/6

-

$j+4/6

j+2/6

A

fj+1/2 -__

2

+j+3/6

+

+j+5/6

+ A

T

= 0,

2 #j+I

+

'

#J+4/6 = 0,

2

1

1'5 . 4 )

fj+1/2 2

+

'

-~

2

$j+4/6 T

$J+1/6

2

3

@J+ - l@ j + 5 / 6 +

A s shown i n

qJ

+

3

j+3/6 + A

9

+

2

,+j+3/6 + A

#j

= 0,

+j+2/6

+ A2

7

+

2

@J+5/6 = 0. 2

3 . 1 , system ( 5 . 4 ) g i v e s a second-order approximation i n T.

3 . 6 . Numerical S o l u t i o n of A d j o i n t Problems A l g o r i t h m i c a l l y , t h e s o l u t i o n of

adjoint problems does n o t d i f f e r from t h e

s o l u t i o n of b a s i c problems i f t h e p r o c e d u r e i s performed i n t h e r e v e r s e t i m e d i r e c t i o n , i . e . t h e problem

- -"*

at

-

a v d i v y$* = az

*az

+

qJ* = g*

at

t = T

+* = o

on

7,

MA$*

+

p,

(6.1)

s h o u l d be s o l v e d , b e g i n n i n g a t t = T , towards d e c r e a s i n g v a l u e s of t . In t h i s c a s e , t h e numerical a l g o r i t h m i s w e l l - d e f i n e d .

Problem ( 6 . 1 ) c a n be reduced t o

t h e form t y p i c a l of b a s i c e q u a t i o n s by s u b s t i t u t i n g t ' = T variable

t

and t h e f u n c t i o n

at'

s' =

- t

f o r t h e independent

-u- f o r y. Then it goes o v e r t o t h e problem

+ div u'+* = .-

a

-

az

v

*a z

+ MA+* + p ,

Numerical Solution of Basic and Adjoint Equations

+* = +* =

g*

at

t' =

0

On

1

105

o

C l e a r l y , t h e o p e r a t o r of a d j o i n t problem ( 6 . 2 ) i n t h e new v a r i a b l e s f o r m a l l y c o i n c i d e s w i t h t h e o p e r a t o r of t h e d i r e c t problem. F o r t h i s r e a s c n , a l l t h e numerical a l g o r i t h m s used t o s o l v e t h e d i r e c t problem apply a u t o m a t i c a l l y t o t h e s o l u t i o n of t h e a d j o i n t problem.

3.7. Comments on D i f f e r e n c e Approximation of Basic and A d j o i n t Problems The i d e a s and methods used by up-to-date

c o m p u t a t i o n a l mathematics make i t

p o s s i b l e t o e l a b o r a t e many q u a l i t a t i v e d i f f e r e n c e schemes and a l g o r i t h m s b o t h f o r d i r e c t ( 2 . 1 ) and a d j o i n t (6.1), ( 6 . 2 ) e q u a t i o n s . But t h e u s e of a d j o i n t e q u a t i o n s f o r n u m e r i c a l s o l u t i o n of t h i s c l a s s of problems imposes c e r t a i n cons t r a i n t s on t h e c h o i c e of o p e r a t o r s approximating an a d j o i n t problem. T h i s s e c t i o n d e a l s w i t h t h e methods of e l a b o r a t i n g a d i f f e r e n c e analogue of an a d j o i n t problem w i t h due a c c o u n t of t h e r e q u i r e m e n t t h a t t h e f u n c t i o n a l s i n v o l v e d admit d u a l r e p r e s e n t a t i o n . Thus, suppose w i t h t h e problem

a+

+ A+ = f ,

(7.1)

+ where

A

= +O

at

t

=

0,

i s a l i n e a r o p e r a t o r i n t h e H i l b e r t s p a c e d e f i n e d on t h e s e t of s u f f i -

c i e n t l y smooth f u n c t i o n s , t h e r e

associates the difference analope

(7.2)

Here

&'

= ($;,

.. .

+ i ) T i s t h e v e c t o r f u n c t i o n d e f i n e d a t p o i n t s ( w i t h numbers

~

k

=

~

1, n ) of some r e g i o n

G

and a t p o i n t s ( w i t h numbers j = 1, r ) of t h e time

a x i s ; Ah i s t h e m a t r i x o p e r a t o r d e f i n e d i n t h e s p a c e of network f u n c t i o n s

(7.3) the superscript

T

refers t o transposition.

L e t u s i n t r o d u c e t h e s p a c e of network f u n c t i o n s

106

Chapter 3

o2

=

I-+

=

{+JI,

j =

-

GI

(7.4)

and d e f i n e t h e s c a l a r product i n t h e s p a c e s O1 and Q2 by

(7.5)

Here D = d i a g

(k

=

[%I

i s t h e p o s i t i v e d e f i n i t e m a t r i x . In t h e s i m p l e s t c a s e ,

4,

=

Ax

1 , ) . With r e s p e c t t o t h e f u n c t i o n space LhL

a2, =

E q . ( 7 . 2 ) c a n be f o r m a l i z e d a s f o l l o w s :

z-,

(7.7)

where t h e s t r u c t u r e s of t h e o p e r a t o r Lh and v e c t o r f depend on t h e form of ( 7 . 2 ) . 21

M u l t i p l y i n g ( 7 . 7 ) s c a l a r l y by t h e f u n c t i o n $ , we o b t a i n

Considering t h a t t h e o p e r a t o r a d j o i n t t o a r e a l - e l e m e n t m a t r i x i s a t r a n s p o s e d m a t r i x and a l l o w i n g f o r t h e form of s c a l a r p r o d u c t (7.5), we can t r a n s f o r m ( 7 . 8 ) to

Let t h e a d j o i n t f u n c t i o n J;-

(j =

T

(z, $J)l]

= 0.

r,) satisfy

t h e equation

(7.10)

+ = "r+l ? =o.

% -

Then, r e l a t i o n (7.9) t a k e s t h e form (7.11) Eq.(7.10)

i s a d j o i n t t o (7.2) i n m e t r i c (7.6) and i t s s o l u t i o n y i e l d s a d u a l

r e p r e s e n t a t i o n of t h e f u n c t i o n a l

Adjoint e q u a t i o n s f o r any o t h e r approximation i n t h e v a r i a b l e

t , a s well

a s f o r t h e s p l i t t i n g can b e d e r i v e d i n a s i m i l a r way scheme. I t s h o u l d be s t r e s s e d

Optimum Location of Industrial Plants

107

t h a t t h e e q u a t i o n a d j o i n t t o t h e d i f f e r e n c e analogue of problem ( 2 . 1 ) must be made c o n s i s t e n t w i t h t h e approximation of t h e d i r e c t problem and w i t h t h e m e t r i c of t h e network f u n c t i o n s p a c e , u s i n g t h e method

which p e r m i t s d u a l r e p r e s e n t a t i o n

of t h e f u n c t i o n a l s .

Chapter 4. OPTIMUM LOCATION OF INDUSTRIAL PLANTS The present-day

r a t e of economic development demands c o n s t r u c t i o n of more

and more e f f i c i e n t i n d u s t r i a l i n s t a l l a t i o n s and complexes. Such i n d u s t r i a l proj e c t s a r e g e n e r a l l y b u i l t i n or c l o s e t o t h i c k l y p o p u l a t e d a r e a s c a p a b l e of meeting r e q u i r e m e n t s

f o r man power. T h i s imposes p a r t i c u l a r l i m i t a t i o n s on t h e l o c a -

t i o n of works or p l a n t s t h a t d i s c h a r g e i n t o t h e a i r a e r o s o l s harmful t o man and d i s t u r b t h e e c o l o g i c a l system i n t h e r e g i o n . Optimal l o c a t i o n of i n d u s t r i a l p l a n t s

i s a m u l t i - a s p e c t and a l g o r i t h m i c a l l y r a t h e r complex problem. To s o l v e t h i s problem, t h e a u t h o r had t o develop a body of mathematics f o r a d j o i n t problems whose s o l u t i o n s a r e t y p i c a l f u n c t i o n s of e f f e c t s caused by a e r o s o l p o l l u t i o n of t h e environment. In t h i s c h a p t e r , t h e r e s u l t s of i n v e s t i g a t i o n s and 2

given i n c h a p t e r s 1

have been a p p l i e d t o a d e f i n i t e o b j e c t f o r studying optimum l o c a t i o n of

p l a n t s ; d i v e r s e mathematical models of most t y p i c a l s i t u a t i o n s have heen examined. The c h a p t e r a l s o c o n t a i n s methods of s o l v i n g o p t i m i z a t i o n problems and an i n t e r p r e t a t i o n of r e s u l t s . The i d e a s and methods d e s c r i b e d h e r e w i t h w i l l be f u r t h e r developed i n subsequent c h a p t e r s . 4 . 1 . S t a t e m e n t of t h e Problem Suppose t h a t a new i n d u s t r i a l p l a n t h a s t o be c o n s t r u c t e d c l o s e t o populated a r e a s , r e c r e a t i o n z o n e s , and o t h e r e c o l o g i c a l l y important zones so t h a t t h e i r t o t a l annual p o l l u t i o n w i t h noxious i n d u s t r i a l s u b s t a n c e s w i l l n o t exceed t h e v a l u e s s t i p u l a t e d by s a n i t a r y r e q u i r e m e n t s and t h e t o t a l e c o l o g i c a l burden on t h e whole r e g i o n

1

due t o p o l l u t i o n w i l l b e minimum o r w i t h i n t h e range s p e c i f i e d

i n g l o b a l s a n i t a r y requirements. Suppose t h a t an i n d u s t r i a l p l a n t d i s c h a r g e s p e r t i m e u n i t and with i n t e n s i t y Q noxious a e r o s o l a t a h e i g h t z = h , which i s t h e n c a r r i e d away by a i r mass and

s p r e a d s under t h e a c t i o n of low-scale t u r b u l e n c e . Assume t h a t t h e a e r o s o l s o u r c e

i s l o c a t e d i n a neighborhood of p o i n t r

-il

by f u n c t i o n

and we a r r i v e a t t h e e q u a t i o n

= (xo, yo.

h ) . Then it can b e d e s c r i b e d

Chapter 4

108

A s boundary c o n d i t i o n s w e t a k e

+ = o

on

1

W e s h a l l f i n d a s o l u t i o n t o problem (1.2), ( 1 . 3 ) on a s e t o f s u f f i c i e n t l y smooth p e r i o d i c f u n c t i o n s w i t h p e r i o d $

( r , T)

The t a s k i s t o select a zone w

G

T =

i n variable

9(r,

t: (1.4)

0 )

C G where t h e p l a n t c a n be l o c a t e d

SO

that

t h e g l o b a l and l o c a l s a n i t a r y r e q u i r e m e n t s f o r p o l l u t i o n w i l l be o b s e r v e d b o t h f o r t h e whole r e g i o n

lo and

t h e c h o s e n zone R k .

I n s o l v i n g a problem c o n c e r n i n g

t r a n s f e r and d i f f u s i o n o f p o l l u t a n t s t h e components o f t h e v e l o c i t y v e c t o r i n t h e p l a n e t a r y boundary l a y e r o f t h e atmosphere i n t h e g i v e n r e g i o n are computed by t h e methods used i n mesometeorology ( s e e Appendix). Having g a t h e r e d n e c e s s a r y i n f o r m a t i o n on t h e wind f i e l d , d i r e c t m o d e l l i n g methods a r e employed f o r s o l v i n g problems of s p r e a d i n g of i n d u s t r i a l a e r o s o l e m i s s i o n s a t t h e g i v e n p o i n t r E G. --o

For t h i s p u r p o s e w e t a k e a v e r a g e weakly c l i m a t i c d a t a on wind v e l o c i t y components o b t a i n e d by mesometeorological

methods. T h e r e a f t e r , problem ( 1 . 2 ) - ( 1 . 4 )

F i g u r e s 4.1 and 4.4 show how problem (1.2)-(1.4) i s Value o f 0 1 l e v e l 2.2755

Value of 20 l e v e l 11.332

30 28

26 24 22

20 78

76 74 72 70 8

6 4

2

FIGURE 4.1

is solved.

s o l v e d on t h e b a s i s of

Optimum Location of Industrial Plants

109

s t a t i s t i c a l model d i s c u s s e d i n c h a p t e r 1. These f i g u r e s d e p i c t t h e i s o l i n e s of t h e function -

4 (x, Y)

T

H

0

0

1

j $ ( x , Y.

= -TH

f o r d i f f e r e n t t y p e s of a i r mass motion.

2,

t ) dt

F i g u r e 4 . 1 conforms t o t h e c a s e

u = v = w = 0 , and f i g u r e 4 . 2 , t o t h e c a s e u = 5 m/s, v = w = 0 . F i g u r e 4 . 3 Value of 0 1 l e v e l 0 . 1 9 4 7 3

Value of 20 l e v e l 3.4992

30 28 26 24

22 20 78 76 74 72

70 8

6 4 2

FIGURE 4 . 2

Value of 01 l e v e l 0 , 3 8 9 4 5

Value of 20 l e v e l 6 . 9 9 8 3

FIGURE 4.3

110

Chapter 4

i l l u s t r a t e s a c a s e when t h e to u = 5 m/s,

components of t h e wind v e l o c i t y v e c t o r a r e e q u a l

v = w = 0 i n [ O , T/21, and u = 0 , v = 5 m / s ,

w

= 0

i n [ T/2,

TI.

F i g u r e 4 . 4 was o b t a i n e d f o r t h e f o l l o w i n g components of t h e wind v e l o c i t y v e c t o r : 5,

t C

lo,

0,

t G

[T/3, 2T/3],

-5,

t G

[2T/3, T I ;

Value of 0 1 l e v e l 0.77289

T/3),

Value of 20 l e v e l 1 1 , 0 3 7

2 b , l , , , , I , , , , , , , , , , , , , , , , , , , , , , I , , , , ~ , , , , ) , , ,, ~ , , , , ~ , ,1 , ,, , , 1 , , , , I , , , , 2 4 6 8 70 72 74 76 78 20 22 24 26 28 30 FIGURE 4 . 4 The s o l u t i o n o b t a i n e d i s i n t e g r a t e d w i t h i n t h e annual i n t e r v a l l i m i t s 0 5 t 5 T . Then e i t h e r t h e a v e r a g e , f o r p e r i o d T , amount of a e r o s o l i n a u n i t

c y l i n d e r over an e c o l o g i c a l l y s i g n i f i c a n t zone

ClkC

1:

T JB = b I d t !

k

$dG,

(1.5)

Gk

or t h e t o t a l amount of a e r o s o l t h a t has s e t t l e d on t h e E a r t h ’ s s u r f a c e i n t h a t very zone

ilkC

lo

i s estimated:

111

Optimum Location of Industrial Plants

T

Here b = 1/T; c o n s t a n t

a

t a k e s i n t o a c c o u n t t h e f r a c t i o n of t h e a e r o s o l t h a t

g e t s i n t o t h e s o i l ( f i r s t o f a l l , t h e s e a r e heavy a e r o s o l s which f a l l on Earth under g r a v i t y and a l s o a f r a c t i o n of l i g h t p a r t i c l e s t h a t f a l l on E a r t h due t o v e r t i c a l d i f f u s i o n ) . A s shown i n s e c t i o n 1 . 5 , i n t h i s c a s e

Note t h a t b o t h f u n c t i o n a l s a r e i m p o r t a n t f o r e s t i m a t i n g p o l l u t i o n and i t s impact on t h e e c o l o g i c a l c o n d i t i o n s o f zone i l k . T h u s , f u n c t i o n a l ( 1 . 5 ) i s s i g n i f i c a n t f o r s t a t i s t i c a l e s t i m a t i o n of t h e d i r e c t e f f e c t on a n i m a t e n a t u r e consum-

-

i n g oxygen, and f u n c t i o n a l ( 1 . 6 )

f o r judging about t h e

p o l l u t i o n of s o i l cover

and w a t e r b o d i e s whose e f f e c t on t h e e n v i r o n m e n t a l e c o l o g y may a l s o be a p p r e c i a b l e , though o f i n d i r e c t n a t u r e w i t h i n t h e c o r r e s p o n d i n g l i n e s of b i o c e n o s i s . A s noted e a r l i e r , functionals(l.5)

and ( 1 . 6 ) a r e p a r t i c u l a r c a s e s o f a more g e n e r a l

functional

1 1

T J

P

=

dt

O

pQ dG,

(1.8)

G

whose u s e i s e x p e d i e n t t o estimate p o l l u t i o n i n d i f f e r e n t c a s e s w i t h known v a l u e s of p . Assuming t h a t

Gk

b,

'E

0 ,

L E Gk

we a r r i v e a t f u n c t i o n a l ( 1 . 5 ) . Assuming t h a t

r

w e a r r i v e a t f u n c t i o n a l ( 1 . 6 ) . Recall t h a t Qk

1,

i s t h e b a s i s of t h e c y l i n d r i c -

a l r a n g e Gk on p l a n e z = 0 . The c a l c u l a t i o n h a s t o b e r e p e a t e d when t h e i n d u s t r i a l p l a n t i s l o c a t e d a t a d i f f e r e n t point r

-1

€

G . T h i s i m p l i e s t h a t a l a r g e number of v a r i a n t s have

t o be examined f o r f i n d i n g a n optimum l o c a t i o n of a p l a n t ; s u b s e q u e n t comparison o f f u n c t i o n s JA and JE or t h e i r l i n e a r c o m b i n a t i o n s w i t h d e f i n i t e c o n s t a n t s a k and b i s a l s o n e c e s s a r y A B J k = Jk + Jk (1.9) Keeping i n view t h a t p o l l u t i o n e s t i m a t e i s , i n t h e f i n a l a n a l y s i s , a s s o c i a t A B e d w i t h t h e c o m b i n a t i o n o f f u n c t i o n a l s Jk and J k , w e i n t r o d u c e a g e n e r a l i z e d

112

Chapter 4

functional 1

Jk

r J

=

(1.10)

dt

0

which c a n b e w r i t t e n 8 s f u n c t i o n a l ( 1 . 8 ) p r o v i d e d

+ a6

b

in G

(2)

0 outside G

k

k

T h e r e f o r e , i n o u r f u r t h e r c o n s i d e r a t i o n w e s h a l l d e a l w i t h f u n c t i o n a l (1.10) o n l y . We i n t r o d u c e a l s o an i m p o r t a n t f u n c t i o n a l J

pk:

T

m

r

I

(1.11)

Here w e h a v e assumed t h a t c o n s t a n t s a k and bk c a n b e d i f f e r e n t f o r d i f f e r e n t (non-intersecting)

zones Gk and may depend, f o r example, on t h e n a t u r e of t h e

u n d e r l y i n g s u r f a c e . Then, f u n c t i o n a l ( l . 1 1 ) c a n be r e w r i t t e n a s

1 1

T

J

P

=

dt

O

(1.12)

p9 dG.

G

where on Gk, k = 1,

bk + ak 6 ( z )

5

outside

0

...,

m,

Gk.

k= 1

Functional J has t h e following physical i n t e r p r e t a t i o n : t h e f u n c t i o n a l P y i e l d s o v e r a l l e c o l o g i c a l l y s i g n i f i c a n t r e g i o n s Gk an i n t e g r a l e f f e c t of t h e environmental p o l l u t i o n , provided t h e i n d u s t r i a l emission source i s i n a p o i n t r Q G . C l e a r l y , t h i s i s a g l o b a l f u n c t i o n a l , i n a c e r t a i n s e n s e , for e v e r y r a n g e

4

G

and a l l r e g i o n s G

k' P a r a l l e l w i t h (1.11) we c o n s i d e r a n o t h e r g l o b a l f u n c t i o n a l

(1.13)

i n which

Bk + Ak 6(z), 0

outside

~e

Gk,

;I

k=l

k = 1,

Gk.

... ,

m,

Optimum Location

of

Industrial Plants

113

Here Ak and Bk s t a n d f o r v a l u e s r e l a t e d t o t h e s a n i t a r y ( p h y s i o l o g i c a l ) i m p a c t of i n d u s t r i a l e m i s s i o n s o v e r t h e w h o l e d i s t r i c t G C G . They c a n b e s e l e c t e d by k d i f f e r e n t m e t h o d s . F o r e x a m p l e , c o n s t a n t s Bk c a n e x p r e s s c o r r e l a t i o n between t h e amount o f a e r o s o l s p r e s e n t i n Gk a n d e i t h e r t h e i r s a n i t a r y h a z a r d o r t h e i r d i r e c t a c t i o n on v a r i o u s o b j e c t s ( i n c l u d i n g l i v i n g o n e s ) i n r e g i o n Gk. T h i s i s a l s o t r u e of c o n s t a n t s

%.

Such f i n d i n g s a r e o b t a i n e d on t h e b a s i s of e x p e r i m e n t a l d a t a

collected for several years. F i n a l l y w e g i v e o n e more g e n e r a l i z a t i o n . Suppose t h a t we have e v e r y n e c e s s a r y i n f o r m a t i o n a b o u t p h y s i o l o g i c a l i m p a c t s n o t o n l y i n t h e most i m p o r t a n t e c o l o g i c a l zones

nk,

but a l s o i n a l l other points

r E -

1,.

Then t h e v a l u e s

and

A

i n t o f u n c t i o n s : A = A ( x , y ) , B = B ( x , y ) . L e t u s compare f u n c t i o n a l Y

P

B

turn

with t h e

value pc = B + A 6 ( z ) W e s h a l l use f u n c t i o n a l s (1.13),

in G

( 1 . 1 4 ) t o e s t i m a t e e c o l o g i c a l b u r d e n on a

r e g i o n d u e t o p o l l u t i o n . The t a s k i s t o f i n d a p o i n t Y

(1.14)

P

r E G such t h a t

4

= min

(1.15) r E G

-0

A s m e n t i o n e d e a r l i e r , t h i s p r o b l e m is s o l v e d w i t h t h e h e l p of b a s i c p r o b l e m

(1.2)-(1.4)

by e x h a u s t i o n o f r

i n G . Such a n e x h a u s t i o n demands l a r g e c o m p u t a t i -

-0

o n s and i s d i f f i c u l t l y r e a l i z e d u s i n g even modern c o m p u t e r s . T h i s i s t h e r e a s o n why i n a r e a l - l i f e s i t u a t i o n p u r p o s e f u l e x h a u s t i o n i s u s e d t a k i n g i n t o a c c o u n t

a

wind r o s e a n d o t h e r c o n s i d e r a t i o n s o f s t a t i s t i c a l n a t u r e . However, a s w e w i l l show

l a t e r , p r o b l e m ( 1 . 1 5 ) c a n b e s o l v e d u n i q u e l y by o n l y o n e v a r i a n t of computing a n a d j o i n t problem. T h i s p r o v e s t o

b e p o s s i b l e due t o o n e o f t h e most r e m a r k a b l e

p r o p e r t i e s o f d u a l i t y . L e t u s now go o v e r t o t h e c o n s i d e r a t i o n of t h i s a l g o r i t h m . 4 . 2 . A d j o i n t E q u a t i o n s and O p t i m i z a t i o n Problem Inasmuch a s we h a v e t a k e n t h e main f u n c t i o n a l of t h e p r o b l e m i n t h e form

T Y

P

p,

dt

=

pcd) dG,

O

G

= B

+ A6

(2)

in G

t h e f o l l o w i n g w i l l b e a n a d j o i n t p r o b l e m w i t h r e s p e c t t o t h e main o n e i n a c c o r d w i t h t h e r e s u l t s of c h a p t e r 2

114

Chapter 4

_ - ”*3 t

-

d

div u+* + u$* = - v

az

$*(z,T )

aQ* az +

(z,

= $*

4,

M u l t i p l y i n g ( 1 . 2 ) by $* and ( 2 . 2 ) by

,,A#*

+

C’

0).

i n t e g r a t i n g t h e r e s u l t s over t h e

whole time i n t e r v a l and r a n g e G , s u b t r a c t i n g one from t h e o t h e r , and u s i n g boundary v a l u e s and i n i t i a l d a t a of (1.3), ( 1 . 4 ) and ( 2 . 2 ) , w e , i n v i r t u e of t h e problem conjugacy ( s e e c h a p t e r 2), o b t a i n t h e d u a l form of f u n c t i o n Y ( 2 . 1 7 ) and (2.18) o f c h a p t e r 2) a f t e r t r a n s f o r m a t i o n s and s i m p l i f i c a t i o n s

j

(see P

T Y

P

=

dt

pc$ dG,

O

(2.3)

G

(2.4)

Let u s d e n o t e Y by Y ( r ) s i n c e t h i s f u n c t i o n a l i s p a r a m e t r i c a l l y dependent P P-0 on l o c a t i o n r E G of t h e i n d u s t r i a l p l a n t . -0

Assume t h a t t h e a d j o i n t problem ( 2 . 2 ) i s s o l v e d and a f u n c t i o n $ * ( r , t )

is

f o u n d . S u b s t i t u t i n g it i n ( 2 . 4 ) , w e o b t a i n

T

Y

P

( r )=

Q

J -

0

(2.5)

$*(r,t) dt.

Now w e u s e t h e a u x i l i a r y f u n c t i o n Y ( r ) f o r f i n d i n g r P-

Y ( r ) = min P -

from t h e c o n d i t i o n

(2.6)

r E G Here, t h e p o i n t minimizing Y ( r ) w i l l b e r . P-0 F u r t h e r , o p e r a t i o n s a r e e v i d e n t . I t is necessary t o c o n s t r u c t a f i e l d of

f u n c t i o n Y ( x , y , h ) where h s t a n d s f o r e m i s s i o n h e i g h t which i s l i m i t e d by P c o n s t r u c t i o n requirements. A s a r e s u l t , we o b t a i n a f i e l d of i s o l i n e s

the

Y p ( x , y , h ) = c o n s t on t h e ( x , y ) p l a n e . F i g u r e s 4.5-4.8

r e p r e s e n t f i e l d s of i s o l i n e s o f f u n c t i o n Y ( x , y ) ( 2 . 5 ) f o r P

t h e t y p e s of a i r mass m o t i o n s shown i n f i g u r e s 4 . 1 - 4 . 4 .

Optimum Location of Industrial Plants Value of 0 1 l e v e l 2.2755

115

Value of 20 l e v e l 11.332

FIGURE 4.5 Value of 01 l e v e l 0.19473

.

.

.

.

Value of 20 l e v e l 3.4992

42 2

4

6

8

70

72

74

76

78

2 0 22

24

26

28

30

FIGURE 4.6 However, i n most c a s e s i t i s n o t r e q u i r e d t o have a unique s o l u t i o n of t h e o p t i m i z a t i o n problem because t h e f i n a l s o l u t i o n r e q u i r e s t h a t s e v e r a l l i m i t a t i o n s be met t a k i n g i n t o account t h e r e g i o n geology, p r o x i m i t y of manpower r e s o u r c e s , w a t e r s u p p l y and v a r i o u s communication f a c i l i t i e s . T h e r e f o r e , it is n e c e s s a r y t o choose a whole range v a l u e s s a t i s f y i n g s a n i t a r y r e q u i r e m e n t s . Let u s d e n o t e by uG t h e r e g i o n where t h e c o n d i t i o n Yc

2

c

(2.7)

116

Chapter 4

Value of 0 1 l e v e l 0.38945

Value o f 20 l e v e l 6 . 9 9 8 3

30

28

FIGURE 4 . 7 Value of 20 l e v e l 11.037

Value of 0 1 l e v e l 0.77289

2

4

6

8

70

72

74

76

78

20

22

24

26 28 i

FIGURE 4 . 8 c a n be f u l f i l l e d , and which w i l l be t h e s o l u t i o n t o o u r problem. Now w e go back t o t h e c l a s s i c a l s o l u t i o n of t h e o p t i m i z a t i o n problem. Suppose that r

-0

i s f o u n d . Then f o r t h i s f i x e d p o i n t it i s n e c e s s a r y t o s o l v e t h e b a s i c

problem ( 1 . 2 ) - ( 1 . 3 )

( s e e f i g u r e s 4.1-4.4)

and o b t a i n c o m p l e t e i n f o r m a t i o n on

p o l l u t i o n f i e l d s n o t on t h e g l o b a l b u t l o c a l s c a l e , t h a t i s , t o o b t a i n i n f o r m a t i on on p o l l u t i o n on d i f f e r e n t z o n e s . I f t h e p o l l u t i o n l e v e l i n a l l zones d o e s n o t exceed t h e maximum p e r m i s s i b l e d o s e s , t h e n t h e problem i s s o l v e d . On t h e c o n t r a r y ,

w e s h a l l have t o s o l v e a more complex and more i n f o r m a t i v e m u l t i c r i t e r i o n a l

Optimum Location of Industrial Plants

117

problem which w i l l b e c o n s i d e r e d i n t h e f o l l o w i n g s e c t i o n .

4.3. M u l t i c r i t e r i o n a l O p t i m i z a t i o n Problem I n g l o b a l e s t i m a t i o n o f s a f e p o l l u t i o n of t h e whole r e g i o n

1,

with noxious

i n d u s t r i a l e m i s s i o n s c e r t a i n e c o l o g i c a l l y i m p o r t a n t zones SIk may be p o l l u t e d beyond t h e p e r m i s s i b l e l e v e l . To a v o i d t h i s , w e s o l v e t h e f o l l o w i n g m u l t i c r i t e r i o n a l problem. L e t Rk

(k = 1 , 2 ,

...,

m) be s e v e r a l e c o l o g i c a l l y i m p o r t a n t zones (human

s e t t l e m e n t s , r e c r e a t i o n z o n e s , d r i n k i n g w a t e r s o u r c e s , e t c . ) p i c k e d up on p l a n e z = 0 in

10 . ,F i n d

a n a r e a where a new i n d u s t r i a l p l a n t c a n be l o c a t e d such t h a t

the pollution i n a l l

m

zones of Rk w i l l n o t e x c e e d t h e maximum p e r m i s s i b l e

v a l u e s ( i f s u c h an a r e a e x i s t s i n

lo,

10

i t i s i m p o s s i b l e to f i n d t h i s a r e a i n

). If

t h e n c e r t a i n l i m i t a t i o n s c a n be imposed on t h e e m i s s i o n r a t e

t h e p l a n t l o c a t i o n a r e a w i l l appear i n

2:

Q

such t h a t

.

F i r s t w e s h a l l c o n s i d e r a s i m p l e problem when t h e r e i s o n l y one a r e a RkC

Io. A

p r i o r i w e demand t h a t s a f e p o l l u t i o n i n t h i s a r e a s h o u l d be less t h a n

t h e maximum p e r m i s s i b l e v a l u e C k , t h a t i s , T

Y pk =

pck$ dG > Ck '

dt

( 3 1)

Gk

where

(3 2)

U n l i k e t h e c a s e c o n s i d e r e d i n s e c t i o n 4 . 2 , i n t e g r a t i o n i n ( 3 . 1 ) i s performed n o t o v e r t h e whole r e g i o n G , b u t o v e r p a r t of G

k

.

Then w e have t h e f o l l o w i n g

problem i n s t e a d of ( 2 . 2 ) :

(3.3)

Q;<:,

T) =

Q;(z,O ) ,

where pck i s a f u n c t i o n of t y p e (3.2). S i n c e t h e p r i n c i p l e of d u a l i t y h o l d s

118

Chapter 4 T

I

f

(3.4)

t h e n along w i t h (3.1) an e q u i v a l e n t c o n d i t i o n i s a l s o v a l i d

Y

pk

1

(r ) = Q -0

(3.5

$$(ro,t ) d t 5 C

k'

0

I t i s t h i s c o r r e l a t i o n t h a t we u s e t o d e f i n e t h e a r e a f o r a p o s s i b l e p l a n t l o c a t i o n . Indeed, we assume t h a t problem (3.3) i s s o l v e d and we have + * ( r , t ) . Let u s kf i n d Y ( r ) by formula pk -

(r) = Q

Ypk

and draw i s o l i n e s Y

(r) = const. Pk C a l l t h e unknown a r e a u k C

1

0'

T 0

(3.6)

$*(r,t) dt

Thus, we c l e a r l y know t h e a r e a wk where it

i s p o s s i b l e t o l o c a t e an i n d u s t r i a l p l a n t . F u r t h e r t h e e c o l o g i c a l and o t h e r

c r i t e r i a of s e l e c t i n g t h e most s u i t a b l e p l a c e f o r c o n s t r u c t i o n come i n t o e f f e c t .

lo,

10

I f w i s found t o l i e o u t s i d e t h e n we can always b r i n g uk i n t o by dek c r e a s i n g Q . T h i s , of c o u r s e , imposes d e f i n i t e l i m i t a t i o n s on e m i s s i o n s a n d , p o s s i b l y , on t h e manufacturing technology of t h e p l a n t . We a l s o assume t h a t t h e r e a r e s e v e r a l e c o l o g i c a l l y important zones have t o s o l v e

m

nk

(k = 1,2,

...,

m).

In t h i s case,we

a d j o i n t problems of t h e t y p e (3.3) and o b t a i n $,;

Using t h e s e s o l u t i o n s we f o r m u l a t e T

Ypk(zo) = Q

[

dt

m

I

$,;

...,

.$;

functionals

$$(zo,t) dz,

k = 1,2,

. .. ,

m,

(3.7)

'k and o b t a i n r e s p e c t i v e l y

m

limitations Y p k ( ~ )5 C k ,

k = 1,2,

... ,

m.

(3.7')

we f i n d w k . I n t e r s e c t i o n of t h e s e r e g i o n s w k k m) i s a r e g i o n f o r c o n s t r u c t i n g a s a n i t a r y s a f e p l a n t f o r a l l

F u r t h e r , f o r each zone (k = 1,2, zones fl

k

.

...,

T h i s s i t u a t i o n i s r e p r e s e n t e d i n f i g u r e s 4.9 through 4.12 where t h e

f i e l d s of i s o l i n e s a r e shown. These f i g u r e s , s i m i l a r l y t o f i g u r e s 4.5 t h r o u g h

Optimum Location of Industrial Plants Value of 0 1 l e v e l 2.3874

Value o f 0 1 l e v e l 0.33878

119

Value o f 10 l e v e l 10.866

Value of 10 l e v e l 3.3396

4.8., h a v e been drawn f o r t h e c a s e when t h e r e a r e s e v e r a l p r o t e c t i o n z o n e s . Thus, t h e a r e a

0

of p o s s i b l e l o c a t i o n of an i n d u s t r i a l p l a n t i s f o u n d . I f

t h i s a r e a does n o t e x i s t f o r a given

Q , i t c a n b e o b t a i n e d by d e c r e a s i n g G .

T a k i n g a c c o u n t of what h a s been d e s c r i b e d ,

it i s q u i t e r e a l a t p r e s e n t t o

set f o r e v e r y e c o l o g i c a l r e g i o n a program of l o c a t i n g i n d u s t r i a l p l a n t s t h a t d i s c h a r g e some of t h e i r w a s t e s i n t o atmosphere i n t h e form of a e r o s o l s . For e v e r y region

& it

i s n e c e s s a r y t o have c l i m a t i c maps showing wind f i e l d s w i t h due

c o n s i d e r a t i o n of t h e c h a r a c t e r i s t i c f e a t u r e s of l o c a l t o p o g r a p h y . On t h e b a s i s

Chapter 4

120 Value of 0 1 l e v e l 0.67755

Value 10 l e v e l 6 . 6 7 9 3

FIGURE 4 . 1 1 Value of 0 1 l e v e l 1.1526

Value of 1 0 l e v e l 10.535

FIGURE 4.12 of t h e s e d a t a we s o l v e t h e a d j o i n t problems and f i n d f u n c t i o n s The l a t t e r a r e used for o b t a i n i n g f u n c t i o n a l s Y

( I which )

+*, +;, . . . , +;.

1 d e f i n e t h e a r e a of

Pk p o s s i b l e c o n s t r u c t i o n s i t e s and d e t e r m i n e t h e p e r m i s s i b l e e m i s s i o n l e v e l . T h i s j o b , f i r s t of a l l , should be accomplished i n p l a n n i n g c o n s t r u c t i o n s i t e s i n econimical r e g i o n s which need t o b e developed and where d e c i s i o n s s a t i s f y i n g environment p r o t e c t i o n r e q u i r e m e n t s can be t a k e n . In f u t u r e t h i s w i l l become a priority criterion.

Optimum Location of Industrial Plants

121

4 . 4 . Minimax Problem

The p r i n c i p l e s f o r m u l a t e d i n s e c t i o n 4 . 3 e n a b l e u s t o a r r i v e a t a s o l u t i o n t o t h e minimax problem which w e f o r m u l a t e a s f o l l o w s . Let of a s p a c e w i t h boundary

S =

c u 1 u 1".

Inside

G

on

1

G

be a c l o s e d r e g i o n

m ecologically

important zones needing s p e c i a l p r o t e c t i o n a g a i n s t i n d u s t r i a l p o l l u t i o n a r e l o c a t e d . L e t u s d e n o t e them by $Il, ( I 2 ,

... ,

12

.

7

FIGURE 4 . 1 3 Value of 0 1 l e v e l 0.25707

Value o f 2 0 l e v e l 3.5023

122

Chapter 4

FIGURE 4 . 1 5 Value of 01 l e v e l 1.0881

Value o f 20 l e v e l 11.053

FIGURE 4 . 1 6 For every zone we c o n s t r u c t f u n c t i o n a l s Ypl,

...,

Ypm u s i n g s o l u t i o n s of a d j o i n t

problems ( 3 . 3 ) . F u r t h e r , w e c o n s i d e r t h e f u n c t i o n a l for every p o i n t M ( r ) E G - -0

Of

p o s s i b l e l o c a t i o n of an i n d u s t r i a l p l a n t which d i s c h a r g e s a e r o s o l a t a r a t e Q

and h e i g h t h ,

(4.1)

123

Optimum Location of Industrial Plants Then, p o l l u t i o n i n a l l zones Rk w i l l be minimum when M ( r )

i s chosen from t h e

~

condition max Ypk(z) = min, (4.2)

(r) a t a l l points pk of w , we f i n d a p o i n t M ( r ) where c o n d i t i o n ( 4 . 2 ) i s f u l f i l l e d . T h i s i s t h a t

Whence, by d i r e c t e x h a u s t i o n of v a l u e s of t h e f u n c t i o n a l max Y k

0

~

4

very r a r e c a s e when, based on s i n g l e e x h a u s t i o n of f u n c t i o n a l s , a minimax problem f o r a complex problem of mathematical p h y s i c s i s s o l v e d i n an e x p l i c i t f o r m with an a d j o i n t problem. The i s o l i n e s of t h e f u n c t i o n a l Y ( x , y ) which c h a r a c t e r i z e P t h e q u a l i t a t i v e a s p e c t of t h e problem, g i v e n t h r e e e c o l o g i c a l l y important zones, a r e shown i n f i g u r e s 4 . 1 3 through 4 . 1 6 . Here, t h e t y p e s of wind c i r c u l a t i o n s a r e s i m i l a r t o t h o s e i n f i g u r e s 4 . 1 through 4 . 4 . 4 . 5 . A G e n e r a l i z e d O p t i m i z a t i o n Problem o f I n d u s t r i a l P l a n t Location Thus, we have a r r i v e d a t two s t a t e m e n t s of t h e problem of clrosing a r e g i o n where a p l a n t can p o s s i b l e be l o c a t e d w i t h due c o n s i d e r a t i o n of i t s i n d u s t r i a l e m i s s i o n s . Global e s t i m a t i o n of p o l l u t i o n i n an o p t i m i z a t i o n problem i s made for t h e whole r e g i o n

lo, b u t

i t may n o t meet c e r t a i n s p e c i f i c c o n d i t i o n s f o r a l l of i t s

e c o l o g i c a l l y i m p o r t a n t zones f l k . The m u l t i c r i t e r i o n a l problem i s s o l v e d i n r e s p e c t of a l l e c o l o g i c a l l y i m p o r t a n t z o n e s , b u t i t does n o t f u l l y t a k e account of p o l l u t i o n h a z a r d i n o t h e r a r e a s of t h i s r e g i o n , though, i n p r i n c i p l e , t h e e n t i r e r e -

I0,

1,

gion c a n b e c o v e r e d w i t h R zones such t h a t Unk = and a d j o i n t problems k can be s o l v e d f o r a l l t h e z o n e s , T h i s i s a p o s s i b l e b u t a d i f f i c u l t way because such zones may be nunerous. Meanwhile a combination o f b o t h problems can meet

us w i t h s u c c e s s . Indeed, f i r s t a m u l t i c r i t e r i o n a l problem is s o l v e d and a r e g i o n uk of p o s s i b l e l o c a t i o n of an i n d u s t r i a l p l a n t is found provided t h a t s a n i t a r y r e q u i r e m e n t s a r e observed f o r a l l g l o b a l e s t i m a t i o n of p o l l u t i o n of

flk

zones.

T h e r e a f t e r , t h e problem

of

t h e e n t i r e r e g i o n i s s o l v e d . A s a r e s u l t , we

o b t a i n an a r e a f o r p o s s i b l e l o c a t i o n of i n d u s t r y i n view of l i m i t i n g e c o l o g i c a l burden from environmental p o l l u t i o n of t h e r e g i o n

1,.

Let i t b e an a r e a (wG.

Then

t h e i n t e r s e c t i o n of uk and uG w i l l y i e l d an a r e a f o r which b o t h c o n d i t i o n s a r e s a t i s f i e d . Denote i t w

.

G F i g u r e 4 . 1 7 i l l u s t r a t e s t h i s p r o c e d u r e . T h e r e i n a r e shown t h e i s o l i n e s of

f u n c t i o n a l s o b t a i n e d upon s o l v i n g a d j o i n t problems w i t h t h e r i g h t - h a n d s i d e of t h e form p1 = 1

when

(x,y)

pp = 1

when

( x , y ) ~G1,

E G

124

Chapter 4

-

Value of 0 1 l e v e l 0 , 2 7 1 8 solid lines

Value of 10 l e v e l 1 . 8 3 7 0 solid lines

-

FIGURE 4 . 1 7 where G 1 i s t h e p r o t e c t e d zone i n t h e m i d d l e of t h e f i g u r e . S o l i d l i n e s c o r r e s p o n d t o t h e l o c a l c r i t e r i o n ; dashed l i n e s

-

t h e g l o b a l c r i t e r i o n (1 l e v e l 1 . 0 6 0 2 , 10

l e v e l 2 , 0 5 8 4 ) . The v e l o c i t y v e c t o r components were t a k e n t o b e : u = 1 0 m / s ,

v = 0.

In c o n c l u s i o n i t may be mentioned t h a t w e s h a l l r e p e a t e d l y r e t u r n t o t h e o p t i m i z a t i o n problem by making t h e l i m i t a t i o n s more r i g i d a t t h e e x p e n s e of f a c t o r s a s s o c i a t e d w i t h t h e r e s t o r a t i o n of e c o l o g y o f p o l l u t e d environment and

i t s d e p r e c i a t i o n . A l l of t h e s e problems a r e o f economic n a t u r e and w i l l be c o n s i d e r e d i n t h e following c h a p t e r . 4.6.

Some G e n e r a l Remarks Up t o now t h e a e r o s o l s

r e g i o n G (on

1).

d e n s i t y was assumed t o b e z e r o a t t h e boundary o f

In most c a s e s , t h i s c o n d i t i o n i s n o t j u s t i f i e d b e c a u s e a e r o s o l s

of one r e g i o n c a n be d r i f t e d t o a n o t h e r t h r o u g h t h e boundary

1. T h e r e f o r e ,

account

must b e t a k e n of t h i s e f f e c t of i n t e r n a l p o l l u t i o n c a u s e d by i n d u s t r i a l p l a n t s located i n neighboring regions.

Industrial plants discharging similar aerosols

i n t o atmosphere c a n now o p e r a t e i n t h e g i v e n r e g i o n . The e f f e c t o f t h i s f a c t o r c a n be a l l o w e d f o r w i t h o u t d i s t u r b i n g t h e b a s i c s t r a t e g y of s o l v i n g o p t i m i z a t i o n problems d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s of t h i s c h a p t e r . L e t u s c o n s i d e r a problem

$ 8

= fs

on

1,

Optimum Location of Industrial Plants

(z, T)

9'

125

(r , O),

= $'

where q ( x , y , z ) i s an a e r o s o l s o u r c e of t h e p l a n t s i n o p e r a t i o n and f

denotes

t h e d e n s i t y o f a e r o s o l s t h a t a r e d r i f t e d t h r o u g h boundary G from t h e r e g i o n s neighboring with

1.

I n v i r t u e o f l i n e a r i t y of t h e problem t h e s o l u t i o n of

can be r e p r e s e n t e d a s a sum

$+

9'

= *o

+ 9, where Oo s a t i s f i e s t h e problem

d i v u +o + -

(6.1)

=

04j0

5v a 2

3 q5O

-+ 32

~ . ' i 4 3 ~+ q ,

(6.2)

4' and

9

(z, T)

(r, - 0)

4'

=

s a t i s f i e s t h e a l r e a d y known problem

(6.3)

+

(T, T) = $

(5,0).

I t is now n e c e s s a r y t o r e l a t e t h e f u n c t i o n a l s of a l S of

(6.2),

( 6 . 1 ) and s i m i l a r f u n c t i o n -

(6.3). Let us consider a functional

T Y'

Pk

=

dt

1

pck$'

dG,

(6.4)

Gk where

1 pck =

b

I

k

+ a 6 k

(z)

on G

k'

outside G O

k = 1,2,

...,

m.

k'

Suppose t h a t w e have t h e f o l l o w i n g l i m i t a t i o n s f o r f u n c t i o n a l s Y' Pk

Chapter 4 5

Y' Pk

Inserting

+

Ck

(6.5)

i n ( 6 . 4 ) i n s t e a d of $ ' g i v e s

where

(6.7)

Keeping i n mind (6.6), c o n d i t i o n ( 6 . 5 ) can be r e w r i t t e n a s

+ Ypk 5 Ck

YOpk

5 Ck - Yo Having denoted Ck pk Pk' l i m i t a t i o n for problem ( 6 . 3 ) :

or Y

-

(6.8) -

YOpk = Ck

we a r r i v e a t b o t h t h e d e s i r e d

and t h e problem we have c o n s i d e r e d though w i t h new l i m i t a t i o n s . F u n c t i o n a l s Yo a r e found by s o l v i n g problem ( 6 . 2 ) . The f o l l o w i n g remark r e l a t e s t o l o c a l i n time f u n c t i o n a l s . Everywhere t h e b a s i c f u n c t i o n a l s were assumed t o be i n t e g r a l v a l u e s o v e r t h e e n t i r e time i n t e r v a l

0 5 t 5 T, t h a t i s , we proceeded from t h e assumption t h a t t h e o p t i m i z i n g f u n c t i o n a l was connected w i t h t h e t o t a l annual dose of a e r o s o l s , b o t h i n s e t t l e d and suspended s t a t e o v e r zone

1, .

However, i n t h i s assumption no a l l o w a n c e s were made

f o r p o s s i b l e s h o r t - t e r m b u t very i n t e n s e a e r o s o l e m i s s i o n s due t o sudden changes i n t h e d i r e c t i o n and v e l o c i t y of a i r mass motion. T h i s can a p p r e c i a b l y a f f e c t t h e p o l l u t i o n l e v e l i n t h e zones of

lo,

though t h e a v e r a g e annual s a f e dose of p o l l u -

t i o n w i t h such a e r o s o l s w i l l n o t exceed t h e s t i p u l a t e d v a l u e . T h i s i m p l i e s t h a t l o c a l i n time f u n c t i o n a l s ( f o r t h e p e r i o d s when v a r i a t i o n s i n m e t e o r o l o g i c a l cond i t i o n s a r e very l i k e l y ) may a l s o b e c o n s i d e r e d a l o n g w i t h t h e s t u d i e d f u n c t i o n a l s of t o t a l annual v a l u e . Suppose t h a t i n a given r e g i o n , jo t y p i c a l v a r i a t i o n s i n m e t e o r o l o g i c a l cond i t i o n s o c c u r and t o t a l annual d u r a t i o n of every c o n d i t i o n i s e q u a l t o t . . Summing J up over all j = 1 , 2 , . . . , jo we o b t a i n JO

1

A t . = T.

j=1

J

N e g l e c t i n g t r a n s i e n t p r o c e s s e s , and assuming a d d i t i v i t y , i t i s p o s s i b l e t o c o n s i d e r t h e g e n e r a l p r o c e s s t o be c o n t i n u o u s w i t h a r b i t r a r y a l t e r n a t i o n

127

Optimum Location of Industrial Plants

of d i f f e r e n t t y p e s of m e t e o r o l o g i c a l c o n d i t i o n s and t o s o l v e j

different basic

problems ( c h a p t e r 1):

(6.10)

Note t h a t i f t r a n s i e n t e f f e c t s a r e d i s r e g a r d e d , t h e n t h e s o l u t i o n t o ( 6 . 1 0 ) can b e o b t a i n e d by t h e f o l l o w i n g n o n s t a t i o n a r y , b u t p e r i o d i c problem:

*

u$ + a$ a t + div -

a w ?!! az az

= -

$ = 0

3 aZ =

a$

+

MA$ + Q6 (-r - r ) , -0

on

1,

on

lo,

(6.11)

where t j , t j + l a r e t h e l i m i t s o f t h e g e n e r a l i z e d t i m e i n t e r v a l A t . i n which a J

t y p i c a l m e t e o r o l o g i c a l s i t u a t i o n d e v e l o p s . Then

t.

J+1

1

1 $. = J At. J t. J

$ dt.

(6.12)

Along w i t h t h e b a s i c problem ( 6 . 1 0 ) w e s h a l l c o n s i d e r t h e f o l l o w i n g a d j o i n t problem:

(6.13)

where

128

Chapter 4

[

bk + ak6

on G

(2)

k’

The s o l u t i o n t o problem ( 6 . 1 3 ) can be found by t h e n o n s t a t i o n a r y problem:

a +;

-- at -

d i v u+* k ~

+ u+*k

a

241;

= -V dz

+ dz

MA+*

k

+ pc k ’

(6.14)

Then

J

Let u s c o n s i d e r a l s o t h e f u n c t i o n a l s

Ypjk

=

1

P

Y$ .

~ dG,~

PJk

= ~

Q+* (r ), Jk 7 (6.15)

Gk j = 1,2,

...,

jo;

...,

k = 1,2,

m.

Having a s e t of a d j o i n t problems, we i n t r o d u c e a s b e f o r e t h e i n e q u a l i t y

Y .

PJk

C

G.

Jk’

where C . a r e t h e maximum p e r m i s s i b l e d o s e s of p o l l u t i o n . As a r e s u l t , t h e problem Jk is b o i l e d down t o f i n d i n g r e g i o n s w where an i n d u s t r i a l p l a n t can be c o n s t r u c t e d kj

without v i o l a t i n g s a n i t a r y r e q u i r e m e n t s f o r zone R k . t i o n of r e g i o n s w k j

(j = 1,2,

...,

Let u s d e n o t e t h e i n t e r s e c -

-

j ) by w k .

I t is t h i s region t h a t w i l l s a t i s f y

a l l of t h e r e q u i r e m e n t s with c o n s i d e r a t i o n f o r a l l t y p e s of m e t e o r o l o g i c a l p r o c e s s e s involved i n t h e t r a n s f e r and d i f f u s i o n of a e r o s o l s . I t f o l l o w s t h a t t h e i n t e r s e c t i o n of a l l

ik(k

=

1,2,

..., m)

yields a region

~

loc

where an i n d u s t r i a l

u n i t can s a f e l y be l o c a t e d from t h e viewpoint of s a n i t a r y

r e q u i r e m e n t s and w i t h c o n s i d e r a t i o n f o r s t a n d a r d i z a t i o n of m e t e o r o l o g i c a l processes.

129

Chapter 5. ECONOMIC CRITERIA OF PLANNING, PROTECTION A N D RESTORATION OF ENVIRONMENT I n t h e p r e v i o u s c h a p t e r we have d i s c u s s e d t h e problem of l o c a t i n g i n d u s t r i a l u n i t s w i t h c o n s i d e r a t i o n f o r maximum p e r m i s s i b l e d o s e s o f p o l l u t i o n f o r a l l ecol o g i c a l l y i m p o r t a n t zones i n a g i v e n r e g i o n . H e r e , w e s h a l l i n t r o d u c e a new f u n c t i o n a l c o n n e c t e d w i t h economic e x p e n d i t u r e on r e s t o r a t i o n of t h e environment p o l l u t e d with i n d u s t r i a l e m i s s i o n s . Together with t h e f a c t s considered

before,

t h i s f u n c t i o n a l g i v e s a s u f f i c i e n t l y c o m p l e t e image o f p o s s i b l e consequences of b i o s p h e r e p o l l u t i o n and t h e economic e x p e n d i t u r e on e c o l o g i c a l r e s t o r a t i o n of t h e environment. 5 . 1 . Value of B i o s p h e r e P r o d u c t s L o s s e s , due t o Environmental P o l l u t i o n with I n d i s t r u a l Emissions S i n c e , w i t h a r a r e e x c e p t i o n , i n d u s t r i a l e m i s s i o n s t e n d t o o p p r e s s t h e animal and v e g e t a b l e kingdom, i n c l u d i n g b i r d s , f i s h , m o l l u s c , i n s e c t s , u s e f u l b a c t e r i a , and o t h e r components o f t h e b i o s p h e r e , it i s q u i t e n a t u r a l t o g i v e some i n t e g r a l , over t h e e n t i r e region

lo, e s t i m a t e

of l o s s d u e t o i n d u s t r i a l e m i s s i o n s .

With t h i s aim w e s h a l l c o n s i d e r d i f f e r e n t i a l c h a r a c t e r i s t i c s t h a t d e s c r i b e t h e amount of biomass o f a g i v e n component P. l o s t due t o a e r o s o l p o l l u t i o n

j

r e f e r r e d t o a u n i t a r e a p e r t i m e u n i t of u n i t a e r o s o l c o n c e n t r a t i o n . Let u s d e n o t e

i t by n Q b j Q (j = 1,2,

...,

1-th population i n

and b . . i s l o s s i n t h e p o p u l a t i o n biomass c a l c u l a t e d p e r

10

m;

!. =

...,

1,

s ) , where n , ( x , y )

i s t h e d e n s i t y of t h e

1J

u n i t d e n s i t y . Then t h e t o t a l a n n u a l l o s s o f t h e biomass component !. due t o

10

aerosol pollution a t concentration $ . in J

T

Rkj

dt

=

Uk

'

nlbjk+jdC

I0

O

Let

1

i s d e f i n e d by f3rmula

be t h e v a l u e of a u n i t component of t h e b i o m a s s . The l o s s due t o p o l l u t i o n

i n such a c a s e w i l l be e q u a l t o

ckj =

7

(1.2)

dt

J

0

Summing up c e j o v e r a l l components 1 we o b t a i n

1

T c. = J

O

dt

pi

I0

9j

dZ,

1

(1.3)

130

Chapter 5

where

The v a l u e s b . J

showing t h e l e v e l of p h y s i o l o g i c a l o p p r e s s i o n of b i o s p h e r e com-

p o n e n t s by a e r o s o l s of a g i v e n t y p e a r e o b t a i n e d on t h e b a s i s of e x p e r i m e n t a l s t u d i e s . I t may be n o t e d t h a t a t l a r g e c o n c e n t r a t i o n s of p o l l u t a n t s ,

bjt

t o b e l i n e a r f u n c t i o n s . Now i f w e sum up t h e r e s u l t s o v e r a l l components of t h e a e r o s o l s d i s c h a r g e d by i n d u s t r i a l p l a n t s , we o b t a i n t o t a l l o s s o f b i o s p h e r e i n t h e region:

I I

T

c =

dt

j=l

o

p:

(1.5)

6 j dC.

c

L e t u s now f o r m u l a t e t h e o p t i m i z a t i o n problem. To t h i s e n d , w e c o n s i d e r

m

problems c o r r e s p o n d i n g t o b a s i c components of e m i s s i o n s :

(1.6)

and

m

a d j o i n t problems:

(1.7)

$!

(5, T)

=

+:

( r , O),

j = 1, 2 ,

...,

m.

W e s h a l l assume t h a t problems ( 1 . 6 ) and ( 1 . 7 ) a r e s o l v e d . L e t u s now c o n s i d e r the functional

131

Economic Criteria of Planning

J

J

0

I t s d u a l form i s w r i t t e n by s o l v i n g an a d j o i n t e q u a t i o n and t a k e s t h e form

1

@; I. = Q . J J 0

(zo,t )

dt.

(1.9)

I t i s i m p o r t a n t t o n o t e t h a t i n t h e c a s e of t h e f u n c t i o n a l i n q u e s t i o n prob-

lem ( 1 . 7 ) i s s o l v e d o n l y once f o r a f i x e d a e r o s o l . F u r t h e r we compute f u n c t i o n

$*(F. t ) and f i n d a r e g i o n wB where t h e l o s s of biomass due t o p o l l u t e d e n v i r o n ment w i l l be minimum. T h u s , a l o n g w i t h t h e r e g i o n oc i n t r o d u c e d i n t h e p r e v i o u s c h a p t e r and e n s u r i n g s a t i s f a c t i o n of s a n i t a r y r e q u i r e m e n t s f o r e c o l o g i c a l l y i m p o r t a n t o b j e c t s , t h e r e a p p e a r s a r e g i o n wB where c o n d i t i o n s of p e r m i s s i b l e environmental l o s s e s f o r t h e r e g i o n

1

a r e f u l f i l l e d . The i n t e r s e c t i o n of t h e s e

r e g i o n s g i v e s t h e most s u i t a b l e a r e a f o r l o c a t i n g a new i n d u s t r i a l p l a n t (figure 5.1).

FIGURE 5 . 1 5 . 2 . Value of Losses Due t o Atmospheric P o l l u t i o n w i t h Multicomponent Aerosol Let u s c o n s i d e r a multicomponent m i x t u r e of a p a s s i v e a e r o s o l which underg o e s no chemical changes upon s c a t t e r i n g and t h e r e f o r e

U,

J

= 0 . Assume t h a t t h e

a e r o s o l i s l i g h t and c o n s i s t s of o x i d e s of d i f f e r e n t compounds. T h i s i m p l i e s t h a t t h e e n t i r e m i x t u r e i s s c a t t e r e d a c c o r d i n g t o one law. In t h i s c a s e , problems ( 1 . 6 ) and ( 1 . 7 ) may immediately be f o r m u l a t e d f o r t h e m i x t u r e d e n s i t y .

Suppose t h a t t h e d i s c h a r g e of s o u r c e s Q . of given a e r o s o l components i s J known and Q./Q=

J

S i m i l a r l y we assume t h a t

y. J'

j = 1,

...,

m.

(2.1)

132

Chapter 5

where

t

Y j = 1.

(2.3)

j=1

Taking i n t o account (2.3) i t f o l l o w s from (2.1) and (2.2) t h a t

(2.4)

j

Summing up every r e l a t i o n of ( 1 . 6 ) o v e r

w i t h t h e u s e of

(2.4) we o b t a i n

an a e r o s o l m i x t u r e s c a t t e r i n g problem:

(2.5)

*a z= o

IH>

on

@(c,T)

+ ( r ,0).

=

Analogously we a r r i v e a t an a d j o i n t problem for t h e m i x t u r e :

- ?!? -

at

a

az

az +

0

on

I,

a$*

on

lo,

on

IH>

div u $ * = - LI

+* =

* az

=

_ a$* -

az

- 0

$ * ( z ,T)

= $*

wA$*

+

p 6(z),

(T,O),

m =

where p O

I

p:.

j=1

Thus, t h e f u n c t i o n a l under s t u d y

-

v a l u e of b i o s p h e r e l o s s e s i n r e g i o n

1

t a k e s t h e form

1

T I =

0

or

dt

po+

xo

dZ

(2.7)

-

133

Economic Criteria of Planning

Here

A f t e r s o l v i n g problem ( 2 . 6 ) and d e t e r m i n i n g t h e l o c a t i o n of t h e planned i n s t a l l a t i o n , w e f i n d a s o l u t i o n $ t o problem ( 2 . 5 ) . F u r t h e r , u s i n g (2.2), r e -

.

c a l c u l a t i o n i s made i n t e r m of component-wise d e n s i t y # . = y . $ ( j = 1, . . , m). J J Together w i t h t h i s t h e d i f f e r e n t i a t e d d i s t r i b u t i o n of a e r o s o l d e n s i t i e s i n r e g i o n

1,

i s a s c e r t a i n e d . I f t h e p r o p a g a t i o n of a e r o s o l s $ . f o l l o w s d i f f e r e n t l a w s , then

m

problems w i l l have t o b e s o l v e d f o r f i n d i n g t h e t o t a l dose C .

J

In c o n c l u s i o n i t may be n o t e d t h a t i n such c a l c u l a t i o n s t h e d e t e r m i n a t i o n of c o e f f i c i e n t s b . . i s t h e most s i g n i f i c a n t element of i n i t i a l problems. T h i s i s 1.l

a very n o n t r i v i a l problem a s i t r e q u i r e s e i t h e r s i m u l t a n e o u s m u l t i - y e a r c o n t r o l o v e r a l l b i o s p h e r e components found i n t h e p o l l u t e d zone, or c o n s t r u c t i o n of complex b i o c o e n o s i s mathematical models t h a t would t a k e account of b i o s p h e r e comp o n e n t s i n t e r a c t i o n a s much a s p o s s i b l e . Let u s i l l u s t r a t e i t . Suppose t h a t an a e r o s o l $ . proves f a t a l f o r i n s e c t s . This w i l l e n t a i l a sharp decrease i n t h e J abundance of i n s e c t i v o r o u s b i r d s and t h i s , i n i t s t u r n , w i l l have an e f f e c t on birds-of-prey

f e e d i n g on i n s e c t i v o r o u s b i r d s and r o d e n t s . A s a r e s u l t , rodent

p o p u l a t i o n w i l l i n c r e a s e and t h e l o s s of c e r e a l s i n f i e l d s w i l l grow, and t h e l i k e . T h e r e f o r e , much a t t e n t i o n s h o u l d be p a i d t o t h e d e t e r m i n a t i o n of c o e f f i c i e n t s b . . s i n c e it i s of prime importance i n p l a n n i n g expansion of i n d u s t r y and 1J

f o r e c a s t i n g i t s e c o l o g i c a l e f f e c t on t h e environment. 5.3. Economic Aspects of N a t u r a l Resources D e p r e c i a t i o n when E c o l o g i c a l C o n d i t i o n s a r e D i s t u r b e d Due t o Environmental P o l l u t i o n E a r l i e r we have touched upon v a l u e s of l o s s e s i n v a r i o u s b i o s p h e r e components due t o environmental p o l l u t i o n . T h i s i s o n l y one way of l o o k i n g a t t h i s problem. The o t h e r way r e s i d e s i n r e s t o r a t i o n j o b s aimed a t improving or, a t l e a s t , maint a i n i n g e c o l o g i c a l c o n d i t i o n s of t h e r e g i o n s . I n f a c t , i f f o r example, i n p o l l u t e d w a t e r b o d i e s t h e f i s h b r e e d i n g d e c r e a s e s due t o t h e o p p r e s s i o n of spawning, then f i s h h a t c h a r i n g farms a r e e s t a b l i s h e d t o r e s t o r e f i s h p o p u l a t i o n i n a g i v e n w a t e r body up t o t h e o p t i m a l l e v e l which i s determined by a v a i l a b l e f i s h - f e e d r e s o u r c e s . A d e c r e a s e d c r o p y i e l d s h o u l d be augmented by improving farming p r a c t i c e s , applyi n g o r g a n i c and m i n e r a l f e r t i l i z e r s , and l a n d r e c l a m a t i o n . When abundance of b e a s t s d e c r e a s e s , measures a r e t a k e n t o f e e d and look a f t e r them; h u n t i n g q u o t a s a r e s e t , and s o o n .

134

Chapter 5

Thus, t h e damage t o n a t u r e , p a r t i c u l a r l y t o a n i m a t e n a t u r e , s h o u l d b e comp e n s a t e d by p l a n n e d a d d i t i o n a l d e d u c t i o n s made from t h e o p e r a t i n g p r o f i t s of t h e e n t e r p r i s e s . Such d e d u c t i o n s s h o u l d b e made compulsary l i k e d e p r e c i a t i o n deduct ions. L e t u s f o r m u l a t e a m a t h e m a t i c a l model for e s t i m a t i n g s u c h compulsary ex-

penses. L e t p

t

b e t h e c o s t of measures aimed a t r e s t o r i n g a u n i t mass of t h e 2 - t h

b i o s p h e r e component, o p p r e s s e d by n o x i o u s i n d u s t r i a l e m i s s i o n s , t o i t s i n i t i a l l e v e l . L e t u s now c a l c u l a t e t h e l o s s of t h e j - t h b i o s p h e r e component:

T r

i

(3.1)

Then t h e c o s t of r e s t o r a t i o n measures w i l l e q u a l t o

Adding (3.2) o v e r B and j w e o b t a i n

(3.3)

W e now w r i t e t h i s e x p r e s s i o n a s

(3.4)

where

If w e assume t h a t n o x i o u s i n d u s t r i a l e m i s s i o n is

then

0 .

J

= 0 (j = 1, 2 ,

...,

l i g h t and

undecomposable,

m). Then, w i t h a l l o w a n c e made f o r (2.2), w e w r i t e

1

T R

g

=

O where

Lb

dt

Lo

dC,

(3.6)

135

Economic Criteria of Planning Now we can f o r m u l a t e an a d j o i n t problem w i t h r e s p e c t t o f u n c t i o n a l R : g

(3.8)

Then on t h e b a s i s of t h e g e n e r a l t h e o r y we have two e q u a l i t i e s :

To s o l v e

$ * ( rt,)

we c o n s t r u c t a graph of f u n c t i o n R ( r ) , which i s shown i n --o

figure 5.2.

FIGURE 5 . 2

FIGURE 5 . 3

With t h e a l g o r i t h m g i v e n we f i n d a domain uR where o p e r a t i o n of an i n d u s t r i a l p l a n t would r e q u i r e l e s s e r d e p r e c i a t i o n e x p e n d i t u r e s f o r r e s t o r a t i o n of p o l l u t e d environment t h a n t h o s e p e r m i t t e d by s a n i t a r y r e q u i r e m e n t s : R

g

< BR.

Thus, i n

p l a n n i n g t h e l o c a t i o n of a n i n d u s t r i a l p l a n t t h a t d i s c h a r g e s noxious a e r o s o l s into the a i r , we arrive a t three c r i t e r i a . P o l l u t i o n of e c o l o g i c a l l y most i m p o r t a n t zones must s a t i s f y maximum p e r m i s s i b l e v a l u e s s t i p u l a t e d by s a n i t a r y r e q u i r e m e n t s . A s a r e s u l t , we f i n d a region w c

c

10.

The l o s s e s of b i o l o g i c a l r e s o u r c e s , when p o l l u t e d , s h o u l d be minimum. In

I

BC o. D e p r e c i a t i o n e x p e n d i t u r e s f o r r e s t o r a t i o n of b i o l o g i c a l r e s o u r c e s a b a t e d

t h i s c a s e , we f i n d a r e g i o n

0

due t o environmental p o l l u t i o n . T h i s g i v e s us a r e g i o n w

R

lo.

Chapter 5 The i n t e r s e c t i o n of t h e s e r e g i o n s y i e l d s a most s u i t a b l e a r e a f o r a c o n s t r u c t i o n s i t e . In f i g u r e 5 . 3 t h i s a r e a i s h a t c h e d . I f t h e i n t e r s e c t i o n of r e g i o n s does n o t y i e l d any a r e a , t h e n t h e l i m i t i n g economical c r i t e r i a s h o u l d b e changed. A c a s e when e v e r y component of t h e a e r o s o l behaves a s t h e e n t i r e m i x t u r e was a l s o c o n s i d e r e d . I f some a e r o s o l components s c a t t e r d i f f e r e n t l y , t h e n a l l of t h e problems must b e s o l v e d f o r e v e r y component j . A s a r e s u l t , we a r r i v e a t a more complex, b u t a s i m p l e r problem of i n t e r s e c t i o n of 3j d i f f e r e n t r e g i o n s . 5 . 4 . General Economical C r i t e r i o n

I t can happen t h a t t h e i n t e r s e c t i o n of t h e r e g i o n s f o r which a l l o p t i m i z a -

t i o n c r i t e r i a a r e f u l f i l l e d i n t h e given region

lo,

i s empty. I t w i l l t h e n b e

n e c e s s a r y t o r e l a x t h e s e or o t h e r l i m i t a t i o n s on t h e f u n c t i o n a l s b e i n g s t u d i e d ,

or t o i n c r e a s e t h e dimensions of t h e r e g i o n i n q u e s t i o n . Such changes i n t h e o p t i m i z a t i o n problem a r e h e u r i s t i c , i n p r i n c i p l e , and do n o t r e p r e s e n t a s t r i c t l y deterministic process.

On t h e o t h e r hand, t h e o p t i m i z a t i o n c r i t e r i a f o r m u l a t e d i n t h e p r e v i o u s c h a p t e r s e n a b l e u s , s t r i c t l y s p e a k i n g , t o u n i q u e l y i s o l a t e zones where g e o g r a p h i c a l d i s t r i b u t i o n of i n d u s t r y i s i n a d m i s s i b l e . A d d i t i o n a l i n f o r m a t i o n i s needed t o l o c a l i z e t h e r e g i o n s where t e r r i t o r i a l d i s t r i b u t i o n of i n d u s t r y i s economically justified.

In t h i s s e c t i o n we s h a l l t h e r e f o r e c o n s i d e r a common economical c r i -

t e r i o n of t o t a l e x p e n d i t u r e s on r e s t o r a t i o n of environment p o l l u t e d w i t h i n d u s t r i a l w a s t e s ; we s h a l l do t h i s w i t h due r e g a r d f o r mutual optimum l o c a t i o n of i n d u s t r i a l p l a n t s and f o r e c o l o g i c a l l y i m p o r t a n t z o n e s . The t a s k of mutual optimum l o c a t i o n of i n d u s t r i a l p l a n t s i n c l u d e s c o n s i d e r a t i o n of a l a r g e number of f a c t o r s r e l a t e d t o economic e x p e n d i t u r e s on c o n s t r u c t i o n i n t h e g i v e n s i t e , c o n s t r u c t i o n and o p e r a t i o n c o s t of s e r v i c e l i n e s ( r a i l ways, motor r o a d s , w a t e r p i p e l i n e s , t e l e p h o n e and t e l e g r a p h s e r v i c e , e t c . ) under d e f i n i t e c o n d i t i o n s , and f u t u r e development of t h e r e g i o n a s a whole. Methodolog i c a l l y t h i s problem c a n b e s t a n d a r d i z e d a s f o l l o w s : - by p l a n n i n g t h e s i t i n g o f p l a n t s

i n t h e region i n accord with t h e e s t a b l i s h -

ed e c o l o g i c a l s t r u c t u r e ( f o r example, c o n s t r u c t i o n of a l a r g e i n t e g r a t e d p l a n t i n a suburb a r e a ) ;

-

by forming an e c o l o g i c a l s t r u c t u r e around i n d u s t r i a l i n s t a l l a t i o n s ( f o r

example, by b u i l d i n g l i v i n g accomodations a c c o r d i n g t o t h e proposed c o n s t r u c t i o n of i n d u s t r i a l u n i t s e x p l o i t i n g m i n e r a l r e s o u r c e s a n d , t h e r e f o r e , "connected" w i t h one or a n o t h e r a r e a ) ;

-

by s i m u l t a n e o u s p l a n n i n g of a new i n d u s t r i a l l o c a t i o n and l o c a t i o n of

e c o l o g i c a l l y important zones. Let u s now f o r m u l a t e t h e g e n e r a l economic c r i t e r i o n w i t h due c o n s i d e r a t i o n f o r a s i t u a t i o n of t h e f i r s t t y p e . We t h u s d i r e c t o u r a t t e n t i o n t o e x p e n d i t u r e s

137

Economic Criteria of Planning

on r e s t o r a t i o n of e n v i r o n m e n t , which i n c l u d e : e x p e n d i t u r e s on m e d i c a l p r o t e c t i o n of l o c a l p o p u l a t i o n , on e x t r a n o u r i s h m e n t a n d on improving i t s q u a l i t y , on d i s -

p e n s a r i e s , rest h o u s e s , h o l i d a y h o t e l s , and h o s p i t a l s . L e t u s d e n o t e by a . t h e e x p e n d i t u r e s on r e s t o r a t i o n of h e a l t h of t h e Jk p e o p l e l i v i n g i n a n e c o l o g i c a l l y i m p o r t a n t r e g i o n , marked w i t h k , c a l c u l a t e d p e r p e r s o n p e r y e a r and r e f e r r e d t o u n i t c o n c e n t r a t i o n of a e r o s o l j . Suppose t h a t N i s t h e t o t a l p o p u l a t i o n of t h e g i v e n r e g i o n . Then t h e amount s p e n t on r e s t o r a k t i o n of h e a l t h undermined by p o l l u t i o n w i l l e q u a l t o

T dt

c. = Jk

j

a . N $ . dZ Jk k J

(4.1)

'k Summing up t h i s e x p r e s s i o n o v e r a l l e c o l o g i c a l l y i m p o r t a n t r e g i o n s , we obtain T

(4.2)

or

(4.3)

where

k=l P 0

o u t s i d e t h e domain

The s e c o n d p a r t of t h e e x p e n d i t u r e s i s r e l a t e d t o t h e d e c r e a s e i n t h e biomass of a l l e n v i r o n m e n t a l components ( a n i m a l kingdom, v e g e t a t i o n c o v e r , e t c . ) due t o t h e d e c l i n e i n s o i l p r o d u c t i v e a r e a s . Denote them by R b j .

In c o m p l i a n c e

w i t h t h e r e s u l t s of s e c t i o n 5 . 1 w e w r i t e T

(4.4)

where PJ =

3

e=1

nQBQ b j k .

138

Chapter 5 The t h i r d p a r t i s s p e n t on m a i n t a i n i n g t h e p r o d u c t i v i t y of b i o r e s o u r c e s a t

a given c o n s t a n t l e v e l : T R

.

gJ

dt

=

O . where P J =

P i l j dz,

(4.5)

zo

s

1

penebje. ?“=1 g Then f o r m d i f f e r e n t components of t o x i c i m p u r i t i e s we o b t a i n

m

functi-

onals (4.6)

A f t e r adding we g e t m R =

1

R..

j=1

J

(4.7)

Let u s now d e n o t e by c . ( r ) t h e c o n s t r u c t i o n c o s t of an i n d u s t r i a l p l a n t , 1 -

lo;

of by c . ( r ) w e d e n o t e t h e c o n s t r u c t i o n and ik o p e r a t i o n c o s t of s e r v i c e l i n e s , c a l c u l a t e d p e r u n i t of t h e s h o r t e s t d i s t a n c e marked w i t h i , a t a p o i n t

between p o i n t

r

-

r -

of t h e i - t h u n i t and t h e k - t h p r o t e c t e d zone ( i n c l u d i n g a v e r a g e

c o s t of planned t r a n s p o r t a t i o n of goods and p a s s e n g e r s ) . Let r . ( r ) be t h e 1k t o t a l i t y of such d i s t a n c e s . Now we c o n s i d e r a f u n c t i o n a l E . (r) = c . ( r ) + c i k ( ~ ) r i k ( z ) , Ik1 -

(4.8)

c h a r a c t e r i z i n g t h e c o n s t r u c t i o n c o s t of t h e i - t h u n i t and of n e c e s s a r y s e r v i c e l i n k s w i t h t h e k-th e c o l o g i c a l zone. Summing up ( 4 . 8 ) o v e r a l l

k

and t a k i n g

account of f u n c t i o n a l ( 4 . 7 ) , we f i n d a common f u n c t i o n a l

whose l e v e l c u r v e s , I ( r ) = c o n s t , g i v e u s a l o c a l i z e d domain most s u i t a b l e f o r plants siting. Let u s now c o n s i d e r t h e f o l l o w i n g model s i t u a t i o n . Two s e t t l e m e n t s a r e l o c a t e d a t p o i n t s ( x l , yl)

and ( x z , y 2 ) , t h e f i r s t b e i n g w i t h double p o p u l a t i o n

of t h e s e c o n d . Wind blowing p a r a l l e l t o t h e x - a x i s predominates i n t h e c o n s i d e r e d region over a time c y c le [0, T I . A point with c o o rdi nat es (x2, y ) i s a place 1 where t h e c o n s t r u c t i o n c o s t of t h e i n d u s t r i a l p l a n t i s minimum. As w e go f a r t h e r and f a r t h e r away from t h i s p o i n t t h e c o s t i n c r e a s e s p r o p o r t i o n a l t o t h e f u n c t i o n of t y p e Y(x, y) = 2-exp

I-

a[(x

-

x 2 ) 2 + (y

-

Y , ) ~ ] } . And, f i n a l l y , i n t h e a r e a

x > ( x l + x ) / 2 t h e c o s t of s e r v i c e l i n e s exceeds two-fold t h e c o s t of communi2

c a t i o n s i n [O, (x, + x 2 ) / 2 1 .

139

Economic Criteria of Planning Value o f 01 l e v e l 10,0000

Value of 20 l e v e l 7 . 5 4 8 5

FIGURE 5 . 4

Value of 01 l e v e l 4 . 5 1 5 1

2

4

6

8

70

12

Value of 20 l e v e l 1 . 5 5 7 2

74

76

78

2 0 22

24

26

28 30

FIGURE 5 . 5 F i g u r e 5 . 4 shows i s o l i n e s of a f u n c t i o n a l t h a t d e s c r i b e d o n l y t h e c o s t of medical s e r v i c e s rendered t o t h e p o p u l a t i o n . The f u n c t i o n a l shown i n f i g u r e 5 . 5 d e s c r i b e s t h e c o s t of communications. Figure 5 . 6 g i v e s t h e c o n s t r u c t i o n c o s t o f a f a c t o r proper.

140

Chapter 5 Value of 0 1 l e v e l 1 , 0 4 0 5

Value of 20 l e v e l 1.8102

FIGURE 5 . 6 I s o l i n e s of a f u n c t i o n a l of t h e t y p e ( 4 . 9 ) a r e shown i n f i g u r e s 5 . 7 through 5 . 1 0 . D i f f e r e n t v a r i a n t s conform t o p r e v a l e n c e of v a r i o u s t y p e s of t h e c o n s i d e r ed c o s t s . For example, f i g u r e 5 . 7 a shows predominance of t h e c o s t of l a y i n g communications, t h a t i s ,

whereas f i g u r e 5 . 7 b i l l u s t r a t e s t h e s u r f a c e formed by t h i s f u n c t i o n a l . F i g u r e 5 . 8 conforms t o a l a r g e r s h a r e of e r e c t i o n c o s t of a p l a n t , t h e c o s t of l a y i n g communications being comparable w i t h t h e e r e c t i o n c o s t , t h a t i s ,

The s u r f a c e c o r r e s p o n d i n g t o t h i s f u n c t i o n a l i s r e p r e s e n t e d i n f i g u r e 5 . 8 b . F i g u r e 5.9a i l l u s t r a t e s a s i t u a t i o n when t h e c o s t of communications i s almost e q u a l t o o t h e r c o s t s , t h a t i s ,

Economic Criteria of Planning Value of 0 1 l e v e l 1 . 5 3 2 8

Value of 20 l e v e l 4 . 1 7 7 4

b)

FIGURE 5 . 7

14

142

Chapter 5 Value of 01 level 6.0643

Value of 20 level 1.2102

b)

FIGURE 5.8

Economic Criteria Value of 01 l e v e l 1 . 7 3 4 3

of

Planning

Value of 20 l e v e l 3 . 1 8 8 1

b)

FIGURE 5 . 9

143

144

Chapter 5 Value of 0 1 l e v e l 2 . 5 9 5 9

Value of 20 l e v e l 9.2086

30

28

26

24

22 20 18 16

74

rz ru 8 6 4

2 2

4

6

E

I0

12

14

16

78

20 22 24

26 28

30

b) FIGURE 5 . 1 0 A c o r r e s p o n d i n g s u r f a c e i s shown i n f i g u r e 5 . 9 b . F i n a l l y , f i g u r e 5 . 1 0 a i l l u s t r a t e s

a c a s e when a l l t y p e s of c o s t s a r e e q u a l , and f i g u r e 5 . 1 0 b shows a s u r f a c e f o r t h i s c a s e . Areas of i n d u s t r i a l p l a n t s l o c a t i o n , where t h e t o t a l e x p e n d i t u r e s w i l l be minimum w i t h r e g a r d t o e a r l i e r l i s t e d f a c t o r s , a r e marked i n t h e f i g u r e s ; t h e r e i n t h e t o t a l e x p e n d i t u r e s i n t h e chosen u n i t s of measurements a r e shown f o r t h i s point.

145

Chapter 6. MA THEMA TICAL PROBLEMS OF OPTIMIZING EMISSIONS FROM OPERATING INDUSTRIAL PLANTS P r o t e c t i o n of environment a g a i n s t i n d u s t r i a l p o l l u t i o n i s becoming a t o p i c a l problem of s c i e n c e and e n g i n e e r i n g . In t h e p r e v i o u s c h a p t e r s w e have c o n s i d e r e d an a s p e c t of t h i s p r o b l e m , which i s r e l a t e d t o t h e l o c a t i o n of new i n d u s t r i a l u n i t s ( e m i t t i n g n o x i o u s a e r o s o l s i n t o t h e a i r ) w i t h r e g a r d t o minimum p o l l u t i o n of t h e n e i g h b o r i n g s e t t l e m e n t s , r e c r e a t i o n z o n e s , a g r i c u l t u r a l l a n d s , and o t h e r e c o l o g i c a l l y i m p o r t a n t z o n e s . I n t h i s c h a p t e r we s h a l l c o n s i d e r a n o t h e r a s p e c t of t h i s problem. W e s h a l l assume a l l i n d u s t r i a l u n i t s i n t h i s r e g i o n t o be o p e r a t i n g and d i s c h a r g i n g n o x i o u s a e r o s o l s i n t o t h e a i r . The t a s k r e s i d e s i n d e t e r m l n i n g f o r e v e r y u n i t a p e r m i s s i b l e amount of a e r o s o l s t o be d l s c h a r g e d such t h a t t h e i r sum w i l l n o t e x c e e d p e r m i s s i b l e v a l u e s s t i p u l a t e d by s a n i t a r y r e q u i r e m e n t s . A t t h e same t i m e , t o t a l e m i s s i o n s c a n n o t b e s i g n i f i c a n t l y d e c r e a s e d s i n c e t h i s w i l l result

i n t h e l o w e r i n g of economic i n d i c e s of t h e i n d u s t r i a l u n i t s . Thus,

w e s h a l l t a l k a b o u t such l i m i t a t i o n s on e m i s s i o n s which w i l l , n e v e r t h e l e s s , e n s u r e maximum economic e f f e c t a t t h e g i v e n l i m i t a t i o n s . 6 . 1 . S t a t e m e n t of t h e Problem Suppose t h a t a t r . (i = 1, 2 , -1

boundary

S n

...,

n ) p o i n t s of a g i v e n r e g i o n

G

with

i n d u s t r i a l u n i t s A . a r e l o c a t e d and t h a t t h e y d i s c h a r g e e v e r y 1

second Q . ( i = 1, 2 ,

...,

n ) a e r o s o l s . For s i m p l i c i t y , we s h a l l assume t h e com-

p o s i t i o n of t h e s e a e r o s o l s t o b e u n i f o r m . e c o l o g i c a l zones Gk (k = 1, 2 ,

...,

In r e g i o n

we s h a l l i s o l a t e

G

m

m); t h e l i m i t i n g p e r m i s s i b l e c o n c e n t r a t i o n

of a e r o s o l s e m i t t e d i n a t i m e i n t e r v a l [ O , TI, a r e g i v e n f o r e a c h of t h e s e z o n e s . A s a r e s u l t , w e a r r i v e a t t h e f o l l o w i n g m a t h e m a t i c a l s t a t e m e n t of t h e problem. L e t a n e q u a t i o r , f o r d i f f u s i o n of s u b s t a n c e s from

n

i n d u s t r i a l u n i t s be

the following:

at

+ div u$ -

+ a# =

a aZ

v

-

a# az

-

+ PA$

+

i=l

Qi6(r - ri)

(1.1)

provided

(1.2)

_ '4 - o az

on

I".

Assuming t h e problem ( l . l ) , ( 1 . 2 ) t o be c l i m a t i c a l l y p e r i o d i c ( w i t h a p e r i o d equal t o 1 y e a r ) , w e o b t a i n t h e i n i t i a l d a t a

146

Chapter 6

L e t us consider t h e functional

(1.4)

which c h a r a c t e r i z e s t h e s a f e dose of t h e a e r o s o l t h a t h a s s e t t l e d on E a r t h ' s s u r f a c e ( z = 0) w i t h i n t h e e c o l o g i c a l zone G k . The problem r e s i d e s i n f i n d i n g a t o t a l i t y of p l a n n e d e m i s s i o n of a e r o s o l s Q . which w i l l e n s u r e a v e r a g e - a n n u a l J' maximum p e r m i s s i b l e d o s e s o f a e r o s o l p o l l u t i o n

Y

k

5 ck,

k = 1, 2 ,

...,

..., m

(1.5)

w i t h minimum o u t l a y for t e c h n o l o g i c a l r e c o n s t r u c t i o n of p l a n t s ; which w i l l a s s u r e a s p e c i f i e d volume of o u t p u t a t t h e g i v e n d e c r e a s e i n t h e e m i s s i o n l e v e l . Of c o u r s e , i n t h i s problem it i s n e c e s s a r y t o c o n s i d e r a minimizing f u n c t i o n a l along with t h e l i m i t a t i o n s ( 1 . 5 ) . A s such a f u n c t i o n a l w e t a k e

(1.6) -

i n which Q . and Q . a r e t h e i n i t i a l and t h e p l a n n e d e m i s s i o n r a t e s , r e s p e c t i v e l y ; c o e f f i c i t n t < . determines c a p i t a l o u t l a y f o r technology t h a t w i l l e n s u r e t h e same volume of o u t p u t on d e c r e a s i n g t h e e m i s s i o n l e v e l ( c a l c u l a t e d f o r a u n i t e m i s s i o n r a t e ) . Then t h e f u n c t i o n a l

I

r e p r e s e n t s t o t a l expenses necessary f o r

improving t h e p r o d u c t i o n p r o c e s s e s a t a l l t h e u n i t s A . a s w e p a s s from

8.

to Q..

A s a r e s u l t , w e a r r i v e a t a problem of f i n d i n g e m i s s i o n s Q . i n ( 1 . 1 ) - ( 1 . 3 )

such

t h a t t h e following conditions a r e f u l f i l l e d

Y

Problem ( 1 . 1 ) - ( 1 . 3 ) ,

k

Cck,

k = l , 2,

...,

m.

( 1 . 7 ) c a n be r e d u c e d t o a l i n e a r programming problem

T h i s c a n be done i n two d i f f e r e n t ways: w i t h t h e u s e of b a s i c e q u a t i o n s and by a d j o i n t problems.

6.2. O p t i m i z a t i o n by B a s i c E q u a t i o n s L e t u s r e p r e s e n t t h e s o l u t i o n t o problem ( 1 . 1 ) - ( 1 . 3 )

s o l u t i o n s of e l e m e n t a r y p r o b l e m s . Suppose t h a t

as s u p e r p o s i t i o n o f

147

Mathematical Problems of Optimizing Emissions n

1

@ =

i=l

Q.Q.(r, t ) 1 1

+

@s,

(2.1)

-

where $ i ( x , t ) i s t h e s o l u t i o n of t h e problem

agi

-+ u

at

a$.

2+

ax

v

v .+ 4 dY

wi

a$i a w - + u + . = - u - + PA$. + 6 ( r - r . )

az

aZ

,jz

1

-

-1

(2.2)

w i t h boundary c o n d i t i o n s

(2.3)

and t h e p e r i o d i c i t y c o n d i t i o n

Along w i t h t h e problems ( 2 . 2 ) - ( 2 . 4 ) , for i = 1, 2 ,

...,

n , we s h a l l c o n s i d e r

one more problem for d e t e r m i n i n g t h e background of a e r o s o l s t h a t come i n t o r e g i o n G through boundary

S:

(2.5)

given t h a t

az =

on

lo,

Assume t h a t each of t h e problems ( 2 . 2 ) - ( 2 . 4 ) , f o r i = 1, 2 ,

(2.6)

...,

n , and

a l s o t h e problem ( 2 . 5 ) - ( 2 . 7 ) a r e s o l v e d . Then t h e r e p r e s e n t a t i o n of ( 2 . 1 ) i s j u s t i f i e d . Now

we

can make u s e of

(2.1) for computing f u n c t i o n a l s Y k .

s u b s t i t u t i n g (2.1) i n t o ( 1 . 4 ) g i v e s

Indeed,

148

Chapter 6

(2.8)

where

...,

i = 1, 2 ,

Now a i k , b

n;

...,

k = 1, 2 ,

m.

k a r e t h e known c o n s t a n t s . Combining ( 1 . 7 ) and ( 2 . 8 ) w e a r r i v e a t t h e

problem n

1

ti(Pi

-

o i ) = min,

i=l

-

From Q . w e c a n c o n v e n i e n t l y change o v e r t o q . = Q . 1

1

-

Q . 2 0. Then w e a r r i v e a t 1

a l i n e a r programming problem i n f i n d i n g a n optimum s e t q . on t h e s t r e n g t h of t h e s o l u t i o n of t h e problem

i

ciqi

= min,

i=l

n

(2.10)

q i t O , n where R

=

1

a.

a. + bk

i=l i k

1

- c

k

i = l , 2,

...,

n,

.

I t i s n a t u r a l t h a t t h e number of l i m i t a t i o n s c a n b e i n c r e a s e d a t t h e e x p e n s e of s o c i a l and economic r e q u i r e m e n t s r e s u l t i n g from one o r a n o t h e r p r i o r i t y considerat ions.

6.3. O p t i m i z a t i o n by a n A d j o i n t Problem An a d j o i n t problem is g e n e r a t e d by t h e L a g r a n g e ' s i d e n t i t y . shall consider the operator

In f a c t , w e

149

Mathematical Problems of Optimizing Emissions

L =

% + 3t

a

a

- aZ V

d i v (u')

j z

-

+ 0

(3.1)

on a s e t of f u n c t i o n s $ E 9 c o n t i n u o u s t o g e t h e r with t h e i r f i r s t o r d e r ( w i t h respect t o

t ) and second o r d e r ( w i t h r e s p e c t t o x , y) d e r i v a t i v e s . As t o t h e

d e r i v a t i v e with r e s pe c t t o

z, t h e f l u x v a $ / a z

i s assumed t o be c o n t i n u o u s .

F u r t h e r , we suppose t h a t e v e r y element of t h i s s e t s a t i s f i e s t h e c o n d i t i o n s

o

on

3 az = o

on

$ =

1, (3.2)

Let u s i n t r o d u c e a s c a l a r product

T (g, h) =

j

I d t J1 gh

O

dG.

(3.3)

G

In t h e e a r l i e r a c c e p t e d n o t a t i o n s , t h e b a s i c problem ( 2 . 2 ) - ( 2 . 4 ) can now be r e p r e s e n t e d i n t h e o p e r a t o r form L$. = 6 (r

-

r ).

(3.4)

4

Let u s now c o n s i d e r an a d j o i n t o p e r a t o r L* w i t h t h e u s e of t h e Lagrange's identity ($*,

L$) = ( $ , L * + * ) ,

(3.5)

Here, $ and $* a r e assumed t o be r e a l . Using a s t a n d a r d t e c h n i q u e , we a r r i v e at the

d e f i n i t i o n of t h e a d j o i n t o p e r a t o r :

Every element $ * E @ , on which o p e r a t o r L* a c t s , s h o u l d be a c o n t i n u o u s f u n c t i o n having f i r s t d e r i v a t i v e w i t h r e s p e c t t o a continuous a d j o i n t f l u x va$*/az.

t , second, with respect t o

Besides,

@ *s h o u l d

x , y , and

s a t i s f y t h e f o l l o w i n g con-

d i t ions :

+* = o =

u$*

on

1,

on

lo,

(3.7)

150

Chapter 6

$*(z, T)

$ * (-r , 0).

=

L e t u s c o n s i d e r an a d j o i n t problem w i t h r e s p e c t t o (3.4) =

L*$C

undefined f u n c t i o n . Multiplying ( i n scalar product terms)

where pk i s s t i l l an

chi,

(3.8) by

(3.4) by $,:

(3.8)

'k

and s u b t r a c t i n g one r e s u l t from t h e o t h e r , w e o b t a i n

1

T

(a:,

-

Lai)

( 4 ~ ~L*$c) , = Qi

@;(zi,

t)dt

-

($i,

p,).

(3.9)

0 Since $ . €

1 , 4*E 1 ,

k

s u b j e c t t o (3.5) t h e l e f t - h a n d s i d e of (3.9) v a n i s h e s .

Then

(3.10)

A s pk w e t a k e b + a6 (z)

on

Gk,

PkQ) =

( A s a r e s u l t , for

t h e f u n c t i o n a l Yik

outside G . k

O

we have two e q u i v a l e n t forms g e n e r a t e d by

e q u a l i t y (3.10) and e l e m e n t a r y problems w i t h a u n i t s o u r c e Q . :

I 1

T

T

Y i. k =

dt

O T h u s , Yik

pk9i

dx,

R

Yik

=

Qi

[

@;(yi,

t ) dt.

(3.11)

0

c a n b e found by s o l v i n g e i t h e r a b a s i c or an a d j o i n t problem.

S i n c e o p t i m i z a t i o n w i t h b a s i c e q u a t i o n s h a s been c o n s i d e r e d e a r l i e r , w e s h a l l f o c u s our a t t e n t i o n on t h e a d j o i n t problem (3.8). L e t u s w r i t e i t a s

_-

a 9;: at

-

d i v u @ * + u$* = - k k

a 26; u - + ~ a $ *+ az az k

p (r), k -

(3.12)

(3.13)

(3.14)

Mathematical Problems of Optimizing Emissions

151

I f t h e problem ( 3 . 1 2 ) - ( 3 . 1 4 ) i s s o l v e d , t h e n we f i n d t h e f u n c t i o n a l Y . i k by s i m p l e i n t e g r a t i o n . F u r t h e r s t e p s needed t o f o r m u l a t e an o p t i m i z a t i o n problem a r e

now e v i d e n t . Let u s i n t r o d u c e a f u n c t i o n a l

and d e n o t e

(3.16)

Then ( 3 . 1 5 ) i s w r i t t e n a s

Yk =

i

(3.17)

afkQi + b*

k'

i=l

Thus, s i m i l a r t o t h e c a s e of a b a s i c problem, we a r r i v e a t an o p t i m i z a t i o n problem f o r t h e a d j o i n t e q u a t i o n s :

(3.18)

i=l -

or, i n t r o d u c i n g q . = Q . 1

1

-

afkQi + b* k 5 ck,

Q. 2 0 1

k = 1, 2 ,

. .. ,

m.

we transform (3.18) i n t o t h e following: n

1

tiqi

= min,

(3.19)

i=l

n

1

i=l

where R* = problem.

-

a . Q + b* ik k i=l

-

a*q.tR;, ik 1

k = l , 2 , 9 . 2 0 ,

c k . Thus, we have :gain

...,

m,

i = l , 2,

...,

n,

a r r i v e d a t a l i n e a r programming

In d i f f e r e n t c a s e s i t i s c o n v e n i e n t t o f o r m u l a t e an o p t i m i z a t i o n problem by s o l v i n g e i t h e r b a s i c e q u a t i o n s or a d j o i n t problems. I f t h e number of i n d u s t r i a l plants emitting aerosol is

s m a l l and t h e

number of e c o l o g i c a l l y i m p o r t a n t zones

i s l a r g e , t h e n i t i s c o n v e n i e n t t o u s e b a s i c e q u a t i o n s . On t h e c o n t r a r y , u s e i s made of a d j o i n t problems.

152

Chapter 6

6 . 4 . P e r t u r b a t i o n Theory

The i m p o r t a n c e of an a d j o i n t problem i s by f a r n o t e x h a u s t e d by t h e p o s s i b i l i t y of i n d e p e n d e n t l y s t a t i n g an o p t i m i z a t i o n problem.

I n many c a s e s , w i t h o u t

s o l v i n g t h e problem a t l a r g e , one c a n o b t a i n v e r y v a l u a b l e i n f o r m a t i o n a b o u t t h e s e n s i t i v i t y of f u n c t i o n a l s of Yk t y p e t o t h e d e v i a t i o n of c e r t a i n problem p a r a L e t u s c o n s i d e r t h e s i m p l e s t c a s e of t h e p e r t u r b a t i o n

meters from t h e "normal".

t h e o r y by u s i n g v a r i a t i o n s of a e r o s o l e m i s s i o n s . Suppose t h a t an i n d u s t r i a l p l a n t d i s c h a r g e s Q I = Qi

+6Qi

a e r o s o l s i n s t e a d of 9 . . Then LO;

where

# I1

=

9. + 64.. Let u s 1

= &;A("

-xi),

(4.1)

combine an a d j o i n t problem L*#*

=

k

p

(4.2)

k'

w i t h t h i s problem. M u l t i p l y i n g ( 4 . 1 ) ( i n s c a l a r p r o d u c t t e r m s ) by by

9;,

and

+* and k

(4.2)

s u b t r a c t i n g one r e s u l t from t h e o t h e r , w e o b t a i n T ($;,

L$i)

-

-

$cdt

0 The l e f t - h a n d

1 1

T

L*$c) = Q;

($;,

dt

p k $ i dG.

(4.3)

Gk

s i d e of t h i s e q u a l i t y v a n i s h e s i n v i r t u e of t h e L a g r a n g e ' s r e l a t i o n

We h a v e

I

+

SQi)

1 1

T

T

(Qi

-

Ocdt

dt

0

pk(Qi + 64,)

dG = 0.

(4.4)

Gk

Since T

Pi

i

1 1

T

4;

dt =

dt

0

pkOi dG = Y . ik

(4.5)

Gk

and

T

u s i n g ( 4 . 4 ) , w e a r r i v e a t a f o r m u l a of t h e p e r t u r b a t i o n t h e o r y

T GYik

= bQi

j Q;

0

dt,

i = 1, 2 ,

...,

n;

k = 1, 2 ,

.. .,

m.

(4.6)

153

Mathematical Problems of Optimizing Emissions This formula can be obtained also in a simple manner by using (3.15). T After computing $; dt and constructing isolines J +* dt = const, we can 0 O k determine the regions where aerosol pollution will be maximum. The factories located in these regions make maximum contribution to aerosol pollution, and therefore the maximum permissible emissions f o r them should be computed in the first place. This, of course, does not solve the entire optimization problem though it somewhat clarifies its main points. Finally, we shall consider another important question. Up to now we had assumed the input parameters of basic and adjoint problems to be constant. However, in solving optimization problems the concentration of aerosols over a region G

varies and this causes certain changes in the local circulation of

the atmosphere. This means that the components of vector

u:

2'

=

y + b y , the

coefficients of exchange turbulence: v ' = v + 6 v , p ' = p + 6 p , as well as the parameters: a' =

a + 6a, B ' = B + 613 can vary. As a result of perturbations the

basic problem takes the form (L + 6L)

$I = (Q +

6Q)6(r - ~

~

1

.

(4.7)

To this equation we add an unperturbed adjoint equation L*$* = p

k

(4.8)

k'

Multiplying (4.7) (in scalar product terms) by $ * , ( 4 . 8 ) by $;, subtracting one k result from the other, and taking account of the Lagrange's identity, we obtain

Hence,

Assuming the perturbations 6$. and

6 L to be small, we arrive at a formula of

the theory of minor perturbations, accurate to small terms of second order,

6Yik = 6Qi

i

$;

dt - ( $ { ,

6L$i).

(4.11)

0

Having inserted scalar product in this formula, we get the possibility of estimating the "feedback" of atmospheric processes, which is generated by a change in the aerosol background in

G.

The optimization problem considered in this chapter can be extended to other functionals as it was done in chapter 4.

Chapter 7 I n c o n c l u s i o n we s h a l l s a y a few words about numerical r e a l i z a t i o n of

a l g o r i t h m s . S i n c e a b a s i c and an a d j o i n t problems a r e l i n e a r and p e r i o d i c i n t i m e , t h e y c a n be s o l v e d by t h e p e r i o d i c i t y method ( s t a r t i n g from some i n i t i a l d a t a and c o n t i n u i n g u n t i l a p e r i o d i c i t y o c c u r s ) . U s u a l l y , two- o r t h r e e - y e a r c y c l e s of c a l c u l a t i o n s a r e s u f f i c i e n t for t h i s . I t i s i m p o r t a n t t o n o t e t h a t an a d j o i n t problem s h o u l d b e s o l v e d i n t h e o p p o s i t e d i r e c t i o n t o t i m e , b e c a u s e , a s we have a l r e a d y mentioned, i n t h i s c a s e t h e c o r r e c t n e s s of t h e problem w i l l b e observed i n c o u n t i n g . A s t o l i n e a r programming problems, t h e y a r e s o l v e d by s t a n d a r d p r o c e d u r e s . S i n c e q . 2 0 and a l l t h e c o e f f i c i e n t s a i k ,

ark are also

p o s i t i v e , t h e s o l u t i o n t o t h e problem i s found on t h e f a c e s of p o l y h e d r o n s , which a r e formed i n c o n s t r u c t i n g l i m i t a t i o n s a r e a s .

Chapter 7. ACTIVE AEROSOL EMISSIONS The problem of optimum l o c a t i o n of i n d u s t r i a l p l a n t s i n t h e e c o l o g i c a c t i v i t y zones has been c o n s i d e r e d i n c h a p t e r 6 w i t h r e g a r d t o t h e i r minimum p o l l u t i o n l e v e l . In doing so, t h e a e r o s o l was assumed t o b e p a s s i v e , t h a t i s , a e r o s o l does n o t change d u r i n g i t s t r a n s f e r and d i f f u s i o n i n t o o t h e r , may b e , more t o x i c forms. I n t h e same manner,we c o n s i d e r e d t h e o p t i m i z a t i o n problem f o r a e r o s o l emission l e v e l s of o p e r a t i n g p l a n t , which e n s u r e minimum p o l l u t i o n w i t h i n t h e given i n d u s t r i a l e f f i c i e n c y l i m i t s .

In t h e p r e s e n t c h a p t e r we s h a l l examine a more g e n e r a l s t a t e m e n t of t h e problem: i n d u s t r i a l e m i s s i o n s c o n s i s t o f s e v e r a l components of a e r o s o l s and p a r t of them forms a c h a i n of c o n s e c u t i v e l y changing c h e m i c a l s of d i f f e r e n t t o x i c i t y under t h e impact of atmospheric w a t e r v a p o r s , oxygen, n i t r o g e n and o t h e r compounds 7 . 1 . Basic and A d j o i n t E q u a t i o n s Suppose t h a t an i n d u s t r i a l p l a n t e m i t t i n g v a r i o u s k i n d s of a e r o s o l components

i s l o c a t e d a t a p o i n t A(x, y ) i n r e g i o n G . C a l l them a 1' a 2 ' They s c a t t e r o v e r t h e g i v e n r e g i o n p a r t i a l l y s e t t l i n g on E a r t h ' s s u r f a c e

a t a height z = h

... .

and p o l l u t i n g t h e environment. During t r a n s f e r and d i f f u s i o n , p a r t of such a e r o s o l compounds changes i n t o o t h e r forms under t h e a c t i o n of chemical r e a c t i o n s t h a t t a k e p l a c e i n t h e atmosphere. Thus, t h e c h a i n of t r a n s f o r m a t i o n s can b e r e p r e s e n t ed a s : al

+

all

+

a12

+

...

and r e s p e c t i v e l y

a2

+

aZ1

+

aZ2

+

...

Of course,

t h e number of a e r o s o l components and t h e t r a n s f o r m a t i o n c h a i n can b e l a r g e enough, b u t t h i s does n o t change t h e e s s e n c e of t h e problem i n p r i n c i p l e . The t a s k r e s i d e s i n f i n d i n g an optimum l o c a t i o n of t h e i n d u s t r i a l p l a n t i n G , t a k i n g i n t o account t h e minimum p o l l u t i o n of a l l of t h e e c o l o g i c a l l y important zones and t h e g i v e n maximum p e r m i s s i b l e l e v e l of p o l l u t i o n .

155

Active Aerosol Emissions

For t h e s a k e of c o n v e n i e n c e , we c o n s i d e r f o r m a l i z e d o p e r a t o r n o t a t i o n s . With t h i s i n view we s h a l l examine a problem on t r a n s f e r and d i f f u s i o n of some a e r o s o l substance

4 . (j J

__

= 1, n ) :

a+j

a$,

w.

a$.

- + u J + v J + W J + at ax aY

J

1,

= aj+j on

lo,

o

a+. J=

aZ

$j

J

on

u. = J

a$ 2

a

o =

0. J

32

pA4'. = f . (1.1)

J

dZ

J

(1.2)

IH:

on

(z, T)

a*.

a.+.--u+

dZ

(z, 0).

Let u s now i n t r o d u c e a l i n e a r o p e r a t o r

a at +

A = -

d i v (u-)

a a - UA - az u az

and a s p a c e of f u n c t i o n s 0 , whose e l e m e n t s s a t i s f y c e r t a i n smoothness c o n d i t i o n s , a s w e l l a s t h e boundary c o n d i t i o n s ( 1 . 2 ) and t h e i n i t i a l d a t a ( 1 . 3 ) . Problem ( 1 . 1 ) - ( 1 . 3 ) can now be f o r m a l l y r e p r e s e n t e d a s

A$jo + a . $ . JO

JO

= f.. J

(1.6)

A s mentioned e a r l i e r , $ , i s n o t a p a s s i v e s u b s t a n c e . On t h e c o n t r a r y , a f t e r J going i n t o t h e atmosphere it undergoes t h e f o l l o w i n g changes: j. = 4 j. o -- ' j l

+

+$x2

+

...

This implies t h a t

m

+

new e q u a t i o n s s h o u l d b e added t o ( 1 . 6 ) :

(1.7)

For s i m p l i c i t y , h e r e and f u r t h e r we have dropped index

j

at

m . A term of t h e

form 0 4 i s a b s e n t i n t h e l a s t e q u a t i o n , because t h e c h a i n of t r a n s f o r m a t i o n s of s u b s t a n c e @ i s assumed t o b e

f i n i t e , and, t h e r e f o r e , 4 .

Jm

i s an undercomposed

mixture. To s o l v e ( 1 . 6 ) ,

( 1 . 7 ) i t i s n e c e s s a r y t h a t t h e smoothness c o n d i t i o n s and

r e l a t i o n s h i p s , s i m i l a r t o (1.2), ( 1 . 3 ) , should b e f u l f i l l e d :

156

Chapter 7

~

S j L ( ~ ,T) = $je(z, O), V. = 0 , m . Let us consider a space of vectors

and the matrix

A + a.

0

JO.

L. J

'.

- u

=

0

A+a.

J ,m-l

-0.

.

For the migration and transformation of

J ,m-l

1 A]

9 . aerosols into new components, problem

-J

(1.61, (1.7) can now be represented in the vector-matrix form: Lj

lj

=

fj

(1.9)

Let us now introduce a Hilbert space of vector-function with a scalar product

and consider an adjoint problem L*4* = p. J--J

-J

(1.10)

'

'*

-J

(1.11)

=

where p. is still an undefined vector-function. We find operator L* by the J

-J

Lagrange's identity ( $ ? , L.*.) = ( * . , L ; y . -J J-J -J

(1.12)

Active Aerosol Emissions

157

Using t h e d e f i n i t i o n of s c a l a r p r o d u c t and i n t e g r a t i n g t h e l e f t - h a n d r e l a t i o n s h i p by p a r t s , we a r r i v e a t L ? : J

A*

+

!

0 .

J1

L? = J

‘A*+n.

j,m-l

-0

j,m-1

1

where o p e r a t o r A * a p p e a r s a s A* =

a - at

div (us)

-

-

d

az

@*c o n t a i n s

The space of a d j o i n t f u n c t i o n s

a

-W -

az

-

~

b

.

e l e m e n t s of n e c e s s a r y smoothness which

s a t i s f y t h e f o l l o w i n g c o n d i t i o n s for t h e i r components: $*

J8

on

= 0

1,

(1.13)

F u r t h e r , we w r i t e ( 1 . 1 0 ) i n t h e componentwise

form

(1.14)

. . . . . . . . . . . . . . .

Now w e go o v e r t h e d e t e r m i n a t i o n of v e c t o r - f u n c t i o n p . . To t h i s e n d , we -3 f i r s t assume t h a t t h e a e r o s o l s o u r c e w i t h t h e j - t h component f . i s a d e l t a J f u n c t i o n of t h e c o o r d i n a t e s , i . e . f . = Q . 6(r J J -

-

(1.15)

r )

-0

where Q . i s t h e r a t e a t which a e r o s o l s a r e e m i t t e d i n t o a i r . M u l t i p l y i n g ( 1 . 9 ) J i n s c a l a r product terms by $ ? , ( 1 . 1 0 ) by $ . , and s u b t r a c t i n g one r e s u l t from t h e -J -J other, gives ($*, -J

L.@.) - ( $ . , Ltdrt) = ( 4 ~ . f . ) - ( P . $ . ) . J-J

-J

J-J

-J’-J

-J’-J

(1.16)

158

Chapter 7

By t h e Lagrange's i d e n t i t y t h e l e f t - h a n d s i d e v a n i s h e s and we g e t

(+*, -J

f.) = ( P . ,

-J

-J

+

-3

(1.17)

.)

Now we choose p . i n t h e f o l l o w i n g f a s h i o n . Suppose t h a t s a f e p o l l u t i o n w i t h -J a e r o s o l component $ . - s e t t l e d on a r e a G o v e r a y e a r i s e q u a l t o JJ. k T

where

5.

Jk

i s a c o e f f i c i e n t which c h a r a c t e r i z e s t o x i c i t y of t h i s component. T o t a l j

t o x i c i t y of a l l t h e components marked w i t h

w i l l then equal

(1.18)

Y. J

Let u s t a k e t h i s f u n c t i o n a l a s i n i t i a l f u n c t i o n a l and demand ( p .

$.)

-J' -J

with Y . . J

t o agree

For t h i s i t s u f f i c e s t o assume t h a t I

Si,,

if

0 For t h e

P E

outside

Gk,

G

k

.

chosen v a l u e of p . t h e r i g h t - h a n d s i d e i n ( 1 . 1 7 ) w i l l e q u a l t o Y .

-J

and t h e l e f t - h a n d s i d e w i l l ,

J

t h e r e f o r e , be equal t o

or, more s i m p l y ,

1 @30(s,

T

Y. J

=

Q. J

t) d t

(1.19)

0

Thus, t o d e t e r m i n e t h e e x t e n t t o which r e g i o n Gk is p o l l u t e d w i t h a l l components of t h e c h a i n @Jo .

+

$jl

+

@jz +

...,

i t i s necessary t o solve t h e a d j o i n t

problem ( 1 . 4 ) i n t h e r e v e r s e o r d e r :

. . . . . . . . . . . .

(1.20)

159

Active Aerosol Emissions

For t h i s we u s e boundary c o n d i t i o n s and i n i t i a l d a t a i n t h e form ( 1 . 1 3 ) d u r i n g c o u n t down. Now we go o v e r t o complete d e t e r m i n a t i o n of p o l l u t i o n due t o a l l a e r o s o l ~

components j = 1, m.

I n t h i s c a s e t h e f u n c t i o n a l s (1.18) and ( 1 . 1 9 ) w i l l change

t o the following: T

(1.21)

(1.22)

H e r e , Q d e n o t e s t h e emission r a t e of a l l a e r o s o l s and ~

of e v e r y component (j = 1, n ) 7.2.

.

TI.

J

represents a fraction

Influence Function Formula ( 1 . 2 2 ) h a s p a r t i c u l a r s i g n i f i c a n c e i n choosing t h e c o n s t r u c t i o n s i t e

for a new p l a n t w i t h t h e g i v e n s t r u c t u r e of a e r o s o l s and t h e t o t a l emission r a t e Q . A s a m a t t e r of f a c t , we can

a p r i o r i compute t h e f u n c t i o n Y * ( r ) k -

T (2.1)

J

O

a f t e r s o l v i n g an a d j o i n t problem i n r e l a t i o n t o t h e p o l l u t i o n of r e g i o n Gk. Note t h a t t h e r i g h t - h a n d s i d e of ( 2 . 1 ) depends on Gk because t h e v a l u e s of p .

Ja. were t a k e n t o be d i f f e r e n t from z e r o f o r t h i s e c o l o g i c a l l y important r e g i o n . Using t h e

n o t a t i o n s of (1.21), we w r i t e t h e f u n c t i o n a l r e p r e s e n t i n g t o t a l dose of p o l l u t i o n i n t h e region

C

k

as

We c a l l Y * t h e influence

k

function of an i n d u s t r i a l p l a n t e m i t t i n g a e r o s o l a t

a

r E G , t h a t i s t h e function a f f e c t i n g t h e average yearly s a f e dose f o r a t h e region G k' I f we have s e v e r a l p r o t e c t i o n zones, s a y G k , t h e a d j o i n t problem w i l l then

point

have t o h e s o l v e d f o r a s many t i m e s ; e v e r y t i m e a new f u n c t i o n Y * {k = 1, 2 , . . . ) k i s o b t a i n e d . T h e r e a f t e r , t h e problem of c h o o s i n g t h e b e s t s i t e f o r an i n d u s t r i a l p l a n t i s s o l v e d by c l e a r e x h a u s t i o n of f u n c t i o n s Y* f o r every p o i n t of C . In k consequence, we can f i n d a p o i n t where t h e minimax c o n d i t i o n i s f u l f i l l e d max

Y;(l)

= min

(2.3)

160

Chapter 7

7 . 3 . Two-dimensional Approximation We s h a l l now c o n s i d e r i n d e t a i l t h e problem f o r m u l a t e d i n s e c t i o n 7 . 1 . ; t h e domain of t h e f u n c t i o n $ i s assumed t o be two-dimensional.

For d e f i n i t e n e s s assume

t h a t t h e c h a i n of a e r o s o l $ t r a n s f o r m a t i o n s c o n s i s t s of t h r e e l i n k s . Thus, we a r r i v e a t t h e f o l l o w i n g problem:

wl

wl

ax

aY

-+ u - + v - +

2t

w2

w2

wz

at

ax

aY

ul$l -

- + u - + v - + u2$'2

a$3

a$3

at

ax

-+ u -+

a+3

--

v

aY

= f ,

- ol$l -

u

~

- $ MA$,~

= 0,

(3.1)

= 0;

The s e t of E q s . ( 3 . 1 ) d e s c r i b e s a very p a r t i c u l a r c a s e of s c a t t e r i n g a mixture t h a t d o e s n o t i n t e r f e r e w i t h a t m o s p h e r i c components. Summing up t h e s e t

+

of e q u a t i o n s and c a l l i n g $ = $

$2

+

$3

we i n f a c t a r r i v e a t t h e f o l l o w i n g

equation (3.4)

which c o r r e s p o n d s t o t h e c a s e of a p a s s i v e s u b s t a n c e . Suppose t h a t t h e i n i t i a l s u b s t a n c e r e a c t s d u r i n g i t s t r a n s f o r m a t i o n w i t h t h e ambient a i r and t h e change i n mass p e r u n i t of t i m e f o r e v e r y subsequent component of t h e c h a i n i s p r o p o r t i o n a l t o t h e amount of s u b s t a n c e t h a t h a s changed i n t o t h e new s t a t e . In a d d i t i o n , we s h a l l assume t h e s u b s t a n c e $3 t o b e decomposable. T h i s i m p l i e s t h a t t h e f i r s t t h r e e s u b s t a n c e s (more i m p o r t a n t i n one or o t h e r r e s p e c t ) , s a y t h e s u b s t a n c e s may prove hazardous f o r humane h e a l t h , a r e chosen from t h e c h a i n of t r a n s f o r m a t i ons $

+

$12

-f

...

-f

$ n . The s e t of E q s . ( 3 . 1 ) i s t h e n w r i t t e n a s

w1 a @ l 3 + u + v - + ul$l at ax aY -

a$2 at

w2

a$z

ax

ay

at

ax

- + u - + v - + a $

z

-

V A $ ~= f ,

2

-

where

0. = 1

u , + S . and S . i s t h e q u a n t i t a t i v e c h a r a c t e r i s t i c of v a r i a t i o n i n t h e 1

1

s u b s t a n c e mass a s a r e s u l t of i t s i n t e r a c t i o n w i t h atmospheric components.

161

Actiue Aerosol Emissions For d e t e r m i n i n g a f u n c t i o n a l of t h e t y p e ( 1 . 2 1 ) from t h e s o l u t i o n of t h e problem ( 3 . 5 ) ,

( 3 . 2 ) , ( 3 . 3 ) we s h a l l u s e m a t h e m a t i c a l methods d e s c r i b e d i n

c h a p t e r 1, and f o r m u l a t e a s t a t i s t i c a l model f o r t h e p r o c e s s e s i n q u e s t i o n . For t h i s , w e s h a l l assume t h e a d a p t a t i o n t i m e T f o r t h e t y p e s of c i r c u l a t i o n s t o be much less t h a n t h e c h a r a c t e r i s t i c s t i m e A t f o r a c i r c u l a t i o n of t h e g i v e n t y p e . Then f o r e v e r y i n t e r v a l A t w e o b t a i n a set of s t a t i o n a r y e q u a t i o n s

u

a$

ax

a+l

+ v -+ u

- $N

~

aY

A~@ ~= Q S ( j

-

zo), (3.6)

Let us introduce a notation L = u

a +

ax

v

aay - FA

(3.7)

and w r i t e ( 3 . 6 ) i n t h e f o r m a l i z e d form

(3.8)

where f = Q6(r

G).

When no u . a r e e q u a l among t h e m s e l v e s ,

( 3 . 8 ) c a n be r e d u c e d t o a sequence

of e q u a t i o n s f o r m a l l y n o t l i n k e d w i t h e a c h o t h e r . I n f a c t , m u l t i p l y i n g t h e f i r s t two e q u a t i o n s by i n d e t e r m i n a t e f a c t o r s a and B and summing up t h e s e e q u a t i o n s yields

W e now p i c k a r b i t r a r y f a c t o r s a and

B such t h a t t h e e q u a l i t y

is f u l f i l l e d

T h i s r e q u i r e m e n t i s s a t i s f i e d , f o r example, by

a = u1’

B = o

1

- a .

(3.11)

2

Then ( 3 . 9 ) t a k e s t h e form L(0lbl + 13@2)

+

u2(a$l

+

W2)

= af.

(3.12)

162

Chapter 7

M u l t i p l y i n g t h e s e t of e q u a t i o n s by a r b i t r a r y f a c t o r s a , L(a$l

+

w2 +

YO,)

a3(a,$1 + M2

+

-

where a = ul, I 3 = u

u

_.

= a2,

Y =

a2

- u

1

+

6, and y g i v e s

Y,$3)

=

__

-

- u3, y

- u3)(u2 - a3)/u2 o r a

= (ul

(3.13)

af,

= a 1a 2/ ( a 1

-

a3)9

3' We i n t r o d u c e t h e f o l l o w i n g n o t a t i o n s : '3

= $ 1'

o2

= UlOl

-

U3)O2

(Dl

-

'3 3 = o191

+

(a1

-

+ (al

-

a2)$2,

- a3)

U3)(U2

-

+

93'

O2

Qs(5- L ) ,

F1 =

(3.14)

-

F2 = OlQ6(L

F3 = a Q 6 ( r 1 -

-

-51,

r ).

--o

The s e t of E q s . ( 3 . 6 ) now t a k e s t h e form

LO1 + alO1 = F1, LO2 + u 2 @ 2 = F2' La3

+

(3.15)

= F3.

U3Q3

A l l t h e e q u a t i o n s have a s i m i l a r s t r u c t u r e and can be s o l v e d i n s u c c e s s i o n by

one and t h e same method. Having determined @ 1, 0 2 , and

m3,

functions $

1'

$21

$3

a r e found by formulae ( 3 . 1 4 ) . Let u s now c o n s i d e r a very i m p o r t a n t c a s e when wind of c o n s t a n t d i r e c t i o n predominates i n e v e r y t y p e of c i r c u l a t i o n , t h a t is u = c o n s t , v = c o n s t . We s h a l l f i n d s o l u t i o n t o every e q u a t i o n of t h e s y s t e m ( 3 . 1 5 )

am ax

am ay

u - + v - + um

-

wA0 = Q6(r - r ) --o

(3.16)

- :o)/(2~)},

(3.17)

i n t h e f o l l o w i n g form @

= ,$

exp

{(y,z

where (E, L ) = ux + vy. S u b s t i t u t i n g ( 3 . 1 7 ) i n t o ( 3 . 1 6 ) we g e t an e q u a t i o n i n ,$:

- wA9 + B @ where (i = a

+

(u*

+

= Q6(?

-5 1 ,

v 2 ) / ( 4 p ) . The s o l u t i o n t o Eq.(3.18)

(3.18)

i n p l a n e (x, y ) i s

o b t a i n e d by (3.19)

163

Active Aerosol Emissions where K

i s t h e McDonald f u n c t i o n of t h e form m

K (x) =

i

e

-x c h y

x > 0

dy,

(3.20)

0 With c o n s i d e r a t i o n f o r (3.19) we o b t a i n t h e s o l u t i o n t o (3.16):

Let u s now e s t i m a t e t h e c h a r a c t e r i s t i c dimensions of t h e r e g i o n G , l o r which

(3.21) g i v e s an approximate s o l u t i o n t o a boundary problem. With t h i s i n mind we s h a l l introduce f o r every f u n c tio n 0 , a value

E.

and assume t h a t c o n d i t i o n of t h e

t y p e (3.2) i s f u l f i l l e d , p r o v i d e d

ai The s e l e c t i o n of

E.

5

(3.22)

E,

i s d i c t a t e d by t h e i n s t r u m e n t a l

e r r o r s of measuring t h e l e v e l

of a e r o s o l c o n c e n t r a t i o n , i t s background c o n c e n t r a t i o n , and a l s o by t h a t minimum l e v e l f o r which t h e e f f e c t of a g i v e n t y p e of m i x t u r e can be n e g l e c t e d . Taking a p r i o r i t h e v a l u e s

and u s i n g t h e i n e q u a l i t y

(u, r -

1r-

r --o

r ) S a

1

such t h a t t h e a s y m p t o t i c formula h o l d s

lu/ Ir -

,

we f i n d t h a t

Hence, we a r r i v e a t t h e f o l l o w i n g r e l a t i o n

(3.24)

F u l f i l m e n t of t h i s r e l a t i o n for t h e chosen domain of d e f i n i t i o n G g u a r a n t e e s a s o l u t i o n t o t h e problem (3.6), (3.2) w i t h a g i v e n d e g r e e of a c c u r a c y . Formulae (3.21) and ( 3 . 1 4 ) y i e l d a s o l u t i o n t o t h e problem (3.1)-(3.3) from t h e s t a t i s t i c a l model of

(3.6), (3.2):

(3.25)

Chapter 8

164

where

8. 1

=

u . + (uz + v 2 ) / ( 4 u ) ( i = 1, 2 , 3).

When n circulations

1

-

t h e number of c o n s i d e r e d t i m e i n t e r v a l s w i t h d i f f e r e n t t y p e s of

- is

more t h a n or e q u a l t o two, t h e s o l u t i o n t o t h e problem i s found

by formula of c h a p t e r 1:

(3.26)

Here

k

i s t h e c i r c u l a t i o n t y p e number.

In c o n c l u s i o n , i t may b e n o t e d t h a t t h e s o l u t i o n of e v e r y e q u a t i o n of ( 3 . 1 5 ) , when u = u ( x , y ) , v = v ( x , y ) , i s found n u m e r i c a l l y by t h e methods d e s c r i b e d i n t h e previous c h a p t e r s.

Chapter 8. MODELLING THE LOCATION OF POLLUTION SOURCES IN WATER BODIES AND COASTAL SEAS The r a p i d development of i n d u s t r y c a l l s f o r t h e d e t e r m i n a t i o n of optimum c o n d i t i o n s of l o c a t i n g new i n d u s t r i a l p l a n t s and f o r t e c h n o l o g i c a l r e s t r i c t i o n s on r u n - o f f s t h a t p o l l u t e water b o d i e s ( s e a s , l a k e s , b a y s , e t c . ) so t h a t t h e p o l l u t i o n of water a r e a s , i n c l u d i n g t h e chosen c o a s t a l a r e a s , w i l l b e minimum. Mathem a t i c a l l y t h i s i s reduced t o a minimax problem. I n t h i s chapter,we have examined d i f f e r e n t ways of s o l v i n g t h e - p r o b l e m of d e t e r m i n i n g t h e l o c a t i o n of a hydrosol p o l l u t i o n s o u r c e . For t h i s , t h e i n f o r m a t i on on f l o w v e l o c i t y f i e l d and t u r b u l e n t d i f f u s i o n c o e f f i c i e n t s i s assumed t o be known. Meteorological and p r a c t i c a l a s p e c t s of u s i n g t h i s i n f o r m a t i o n w i l l be considered i n t h e following c h a p te r s. 8.1. Basic Equations

Consider a l i m i t e d r e g i o n assume t h i s r e g i o n t o and c o n s t a n t depth

G

i n some w a t e r body. For s i m p l i c i t y , w e s h a l l -.

be c y l i n d r i c a l w i t h a l a t e r a l s u r f a c e

1, b a-s e s lo, lH, 1 i s a combina-

H. W e s h a l l suppose t h a t t h e l a t e r a l s u r f a c e -

t i o n of t h e s o l i d ( c o a s t a l c o n t o u r )

1, and

-

liquid

12

boundaries:

Let u s t a k e t h e h y d r o s o l d i f f u s i o n e q u a t i o n i n t h e form +

at

? ! + i

ax

3+ aY

3+ az

o+

- a v a+ az az

M A + = f,

(1.2)

165

Modelling the Location of Pollution Sources where @ d e n o t e s t h e c o n c e n t r a t i o n of t h e p o l l u t i n g h y d r o s o l , f = Q o (r

r

--o

= ( x o , y o , z ) a r e t h e c o o r d i n a t e s of t h e c o n j e c t u r a l r u n - o f f ;

t h e components of t h e f l o w v e l o c i t y v e c t o r

u -

-

r ), *

u, v, w

are

which s a t i s f i e s t h e c o n t i n u i t y

equation (1.3)

As t o t h e flow v e l o c i t y v e c t o r , we s h a l l assume i n a d d i t i o n t h a t

i s t h e p r o j e c t i o n of v e c t o r

where u -

u

on t h e o u t e r normal t o t h e s u r f a c e

The boundary and i n i t i a l c o n d i t i o n s , which have d i f f e r e n t p h y s i c a l meanings and l e a d t o c o r r e c t s t a t e m e n t of t h e problem, have been d i s c u s s e d i n t h e p r e v i o u s c h a p t e r s . Here we s h a l l c o n s i d e r a mixed problem (1.5)

where

n

i s t h e normal t o t h e boundary of t h e r e g i o n

G.

T o s o l v e t h e problem of l o c a t i n g t h e s o u r c e s of i n d u s t r i a l r u n - o f f s i n t h e

w a t e r body a r e a , we s h a l l c o n s i d e r a f u n c t i o n a l

T Jk =

d t 1 p k (:)@(I',

O where

pk")

=

t ) dG,

(1.7)

G

I-

when

r =Gk,

TSk outside

G

k

.

i s a measure of t h e r e g i o n Gk which h a s t o b e p r o t e c t e d a g a i n s t p o l l u t i o n k (for example, a measure of t h e r e g i o n i n t h e immediate neighborhood of a s e t t l e -

Here, S

ment, r e c r e a t i o n z o n e , e t c . ) . F u n c t i o n a l (1.7) r e p r e s e n t s t h e a v e r a g e c o n c e n t r a t i o n of t h e p o l l u t i n g i m p u r i t y i n t i m e T . In c a s e of a c c u m u l a t a b l e i m p u r i t i e s t h e c u r r e n t v a l u e of t h e functional

166

Chapter 8

f o r a s t a t i o n a r y s o u r c e may exceed t h e a v e r a g e v a l u e . To t a k e a c o r r e c t d e c i s i o n on t h e p o s s i b l e l o c a t i n g a p o l l u t i o n s o u r c e i n a p a r t i c u l a r r e g i o n of w a t e r a r e a

it is n e c e s s a r y t o s t u d y a f u n c t i o n a l o f t h e t y p e

(1.8)

Let u s c o n s i d e r l i m i t a t i o n s imposed by s a n i t a r y r e q u i r e m e n t s , t h a t i s , we s h a l l demand t h a t t h e p o i n t r

-(I

l i e s i n t h e r e g i o n w k and t h e f o l l o w i n g c o n d i t i o n

i s f u l f i l l e d f o r a l l p o i n t s of t h e r e g i o n

J k (-r ) 5 C k ,

(1.9)

where Ck i s a c o n s t a n t r e l a t e d t o s a n i t a r y r e q u i r e m e n t s f o r t h e e c o l o g i c a l region G k .

Thus, t h e problem i s reduced t o i n t e g r a t i n g E q . ( 1 . 2 ) w i t h c o n d i t i o n s

( 1 . 5 1 , ( 1 . 6 ) and a r e s t r i c t i o n of t h e t y p e ( 1 . 9 ) . 8 . 2 . Adjoint Equations We s h a l l suppose t h a t t h e e a r l i e r f o r m u l a t e d problem p e r m i t s s u f f i c i e n t l y smooth s o l u t i o n s t o be o b t a i n e d from t h e s p a c e $ with a s c a l a r p r o d u c t of t h e type

Taking t h e e a r l i e r d e s c r i b e d procedure of s o l v i n g problems of t h e given type a s t h e b a s e , t h e problem of d e t e r m i n i n g t h e p o l l u t i n g h y d r o s o l d i s p o s a l s i t e with r e g a r d t o r e s t r i c t i o n ( 1 . 9 ) can be s o l v e d by dual r e p r e s e n t a t i o n of t h e functional (1.7):

T

$*(%,

Jk = Q

t ) dt.

0

Here

@*r e p r e s e n t s

t h e s o l u t i o n of t h e a d j o i n t problem

L*$* = p

k

(2.3)

where L* i s an o p e r a t o r a d j o i n t from t h e viewpoint of t h e L a g r a n g e ' s i d e n t i t y t o t h e o p e r a t o r of problem ( 1 . 2 ) ,

(1.5), ( l . S ) , and

$*(T) = 0 and t h e problem (2.3), ( 2 . 4 ) i s s o l v e d i n t h e d e c r e a s i n g d i r e c t i o n of form of o p e r a t o r L* i s o b t a i n e d from t h e r e l a t i o n

(2.4)

t . Definite

Modelling the Location

of

167

Pollution Sources

T

T (2.5)

where

In deducing ( 2 . 5 ) we used t h e c o n t i n u i t y e q u a t i o n

and c o n d i t i o n s (1.5), (1.6), (2.4). Now we t r a n s f o r m ( 2 . 5 ) u s i n g t h e i d e n t i t i e s d i v au uVa + cldiv u = u Q a, -= -

Vfi

agVB = a&

t

(2.6)

- fi uV(afi),

d i v a@

(2.7)

where uvw. = u

-

aa ax +

v

aa

ay +

w

aa

-,a 2

which h o l d t r u e when t h e c o n t i n u i t y e q u a t i o n i s s a t i s f i e d and fi :0 . Choosing functions

@ *from

t h e s u b s e t of non-negative f u n c t i o n s of t h e s e t 4 , we o b t a i n

1

T (L*$*

-

- Pk,$)

=

[-

dt

O ( $ G ) y V G + MV$V$*

G

+V*

t

@

*

aZ az

*?V($*)

(2.8)

-

$p ) dG = 0 .

k

Note t h a t a t $ = c o n s t a n t ( 2 . 8 ) changes i n t o t h e b a l a n c e r e l a t i o n w i t h o u t any additional transformations rn

Now i m p a r t i n g t h e meaning of t r i a l f u n c t i o n t o @ i n ( 2 . 8 ) , we s h a l l t a k e t h e g i v e n i d e n t i t y a s d e f i n i t i o n of a g e n e r a l i z e d s o l u t i o n t o t h e a d j o i n t problem.

168

Chapter 8

For c o n s t r u c t i n g an e f f e c t i v e c o m p u t a t i o n a l a l g o r i t h m o f t h e s o l u t i o n of

(2.8)

w e s h a l l u s e t h e i d e a of t h e method of imaginary r e g i o n s . With t h i s i n view w e s h a l l introduce a rectangular region D = [X

where X

1

=

inf (x)

,

1

5 x 5 X2,

X2 = s u p (x) , Y1 = i n f

Y

1

(y)

Y21,

f y 5

(2.10) ~~

,

Y2 = s u p ( y ) , x , Y E

G I and t h e

region

G

= D * [O,

HI.

(2.11)

A s a b o v e , we s h a l l assume t h a t t h e boundary

1,

and l i q u i d

12

1 of

r e g i o n G c o n s i s t s of s o l i d

b o u n d a r i e s . We s h a l l e x t e n d f u n c t i o n s $ , $ * ,

2,

and p t o t h e

whole r e g i o n G a s f o l l o w s :

rGc

Cu.

u

=(-'

-

(2.12)

Then t h e i n t e g r a l i d e n t i t y ( 2 . 8 ) t a k e s t h e form

T (L*'$*

-

P,

0)

= j d t j O

-

($*)

E V G

+

( - %a'$* T + LjTLy($*)

pV$V$*

+ v

* az

- p$

I

For s i m p l i c i t y , h e r e and f u r t h e r w e h a v e dropped i n d e x 1.e.

-

G

(2.13) dG = 0 . k

a t function

p. This,

( 2 . 1 3 ) , i s an i n i t i a l i d e n t i t y for c o n s t r u c t i n g a n u m e r i c a l a l g o r i t h m f o r

t h e problem s o l u t i o n . 8.3. F i n i t e - D i f f e r e n c e Approximations L e t u s now p r o c e e d t o t h e c o n s t r u c t i o n of f i n i t e - d i f f e r e n c e Of

(2.3).

In range

G

r e g u l a r one-dimensional

approximations

w e s h a l l d e t e r m i n e a network r e g i o n a s d i r e c t p r o d u c t of

nets: (3.1)

where G h = r1 x i €

IX,,

x 11x. 2

1

=

x1

+ i6x,

i =

O,,6x

= x2 ~

- x1 N

''

169

Modelling the Location of Pollution Sources

[yl,

G~ = { y j =

Y

G:

=

{z

Y ~ I I J~ =.

Y2 - Y

__ j = 0 , M,

yl + i s y ,

k [~O , HI1 zk = k 6 2 ,

k =

6y =

1

___ M t '

H = ~ 1 . FK, 62

h We s h a l l now c o n s i d e r a space of network f u n c t i o n s d e f i n e d on t h e n e t G :

Qh = { + = { * . -

. ] I + . ijk .

ijk

= b(t,

Y . J'

Xi'

Zk)'

(3.2)

~

i =-,

j

=-,

k = 0 , K}.

S c a l a r product i s d e t e r m i n e d by t h e r e l a t i o n

(3.3)

where 6x/2, &Xi

i = 0, N,

=

6x,

i = 1, N - 1

6y. =

1

M,

6y/2,

. i=

6y,

j = 1, M - 1

6z/2,

k = 0, K,

62,

k = 1, K

-

0,

1.

Using i d e n t i t y ( 2 . 1 3 ) a s t h e b a s e , we s h a l l c o n s t r u c t f i n i t e - d i f f e r e n c e schemes t o s o l v e ( 2 . 3 ) . Suppose t h a t t h e d i f f e r e n c e o p e r a t o r s

Ax ,

y l A2

are

g i v e n . They approximate d i f f e r e n t i a l o p e r a t o r s a / a x , a / a y , 2 / 2 2 a t t h e nodes h of t h e network r e g i o n G w i t h due r e g a r d t o t h e r e l a t i o n ?$*/an = 0 , -

rE -

1u l o u 1,

which i s n a t u r a l f o r t h e f u n c t i o n a l ( 2 . 5 ) . C o n s i d e r i n g t h e f a c t

t h a t t h e a d j o i n t problem i s s o l v e d i n t h e d e c r e a s i n g d i r e c t i o n of

t

we i n t r o d u c e

a network on [ O , T I a s f o l l o w s : Gh =

t

it,"

[0, T ] l t Q = T

-

Tg,,

L =

c,

T = T/Ll.

(3.4)

Let u s c o n s i d e r an e x p r e s s i o n of t h e t y p e ( 2 . 1 3 ) i n an elementary time i n t e r v a l [t,, t,+l].

Using t h e d i f f e r e n c e a n a l o g i e s of d i f f e r e n t i a l o p e r a t o r s

and c o n s i d e r i n g t h e form of t h e s c a l a r p r o d u c t ( 3 . 3 ) , we o b t a i n

170

Chapter 8

(3.5)

where U , V , W , A ,

A, P ,

CLe+r

located the values uijk,

a r e block-diagonal

vijk,

wijk,

uijk,

m a t r i c e s on whose d i a g o n a l s a r e L+r pijk, uijk, with respect t o t h e

Uijk,

o r d e r d e f i n e d by t h e s t r u c t u r e of v e c t o r s from t h e s p a c e

Qh;

e

is a u n i t v e c t o r ;

a r e t h e o p e r a t o r s a d j o i n t towards A block-diagonal R = d i a g { 1 / 2 , 1,

...,

1, 1/21; index

T

X’

m a t r i x of t h e t y p e

denotes t h e transposition operation.

We t a k e t h e r e l a t i o n

(L*&*

-

Pe,

8) -

h

(3.6)

= 0

a s a d e f i n i t i o n of t h e g e n e r a l i z e d s o l u t i o n t o t h e d i f f e r e n c e a n a l o g of problem ( 2 . 3 , ( 2 . 4 ) . For $=

e,

( 3 . 6 ) c h a n g e s i n t o t h e f o l l o w i n g e q u a l i t y w i t h o u t any a d d i -

tional transformations

T h i s e q u a l i t y i s t h e d i f f e r e n c e a n a l o g of b a l a n c e of f i r s t moments of t h e d i f f e r e n t i a l problem. The f i n i t e - d i f f e r e n c e scheme f o r t h e numerical i s determined by choosing

{&‘+‘I

s o l u t i o n t o problem ( 2 . 3 )

f u n c t i o n s i n t h e form

a = l , 3 , 5 , . . . , 13. S u b s t i t u t i n g ( 3 . 8 ) i n t o ( 3 . 5 ) we o b t a i n a d i f f e r e n c e scheme $‘+ll7 -

= $‘ -

+

TPe,

-.

(3.9a) (3.9b)

171

(E

(3.9d)

+

(3.9e)

(3.9f)

(3.9g) where

__

k = 0 , K.

For a b b r e v i a t i o n , h e r e and f u r t h e r we have d r o p d a s t e r i s k a t f u n c t i o n

9.

Note t h a t b i c y c l i c s p l i t t i n g schemes c o n s i d e r e d i n c h a p t e r 2 a r e o b t a i n e d upon symmetric approximation of t h e f u n c t i o n a l ( 2 . 1 3 ) i n r e s p e c t of p o i n t tQ+1,2. Such schemes l e a d , a s mentioned e a r l i e r , t o approximation of t h e second o r d e r t i m e v a r i a b l e . For p o s i t i v e s e m i - d e f i n i t e n e s s P , t h e scheme ( 3 . 9 ) d o e s n o t t a k e function

L out

of t h e c l a s s of t h e p o s i t i v e f u n c t i o n s . I n t h i s c a s e , ( 3 . 7 ) i s a

t e s t i m o n y t o t h e s t a b i l i t y of t h e d i f f e r e n c e scheme ( 3 . 9 ) . The o p e r a t o r s A x ,

AZ

A

Y' approximating t h e d i f f e r e n t i a l o p e r a t o r s i n r e g i o n G a r e chosen very a r b i t r a r i -

l y . We d e t e r m i n e them so a s t o e n s u r e s i m p l e and e f f e c t i v e r e a l i z a t i o n of t h e network ( 3 . 9 ) . For ( 3 . 9 b ) - ( 3 . 9 d )

we choose t h e s e o p e r a t o r s from a c l a s s conform-

i n g t o approximation on a t h r e e - p o i n t p a t t e r n :

and f o r ( 3 . 9 e ) - ( 3 . 9 g ) ,

from a c l a s s conforming t o a two-point

pattern:

Such a s e l e c t i o n e n s u r e s a second o r d e r approximation i n s p a c e v a r i a b l e s f o r ( 2 . 3 ) . In t h i s c a s e , t h e o p e r a t o r s i n ( 3 . 9 ) r e p r e s e n t b l o c k - t h r e e - d i a g o n a l and a r e r e v e r t e d by t h e run method. The s t e p

T

matrices

i s chosen i n such a way a s t o

e n s u r e t h e s u f f i c i e n t c o n d i t i o n f o r run s t a b i l i t y .

Chapter 8 In c o n c l u s i o n , i t may be mentioned t h a t t h e g e n e r a l form of t h e scheme and t h e s t r u c t u r e of t h e a l g o r i t h m remain u n a l t e r e d when any o t h e r network r e g i o n i n G

i s chosen.

8 . 4 . F i n i t e Element Method Here we s h a l l d e s c r i b e a n o t h e r method of s o l v i n g ( 1 . 2 ) which i s a b a s i c e q u a t i o n f o r f i n d i n g optimum l o c a t i o n of a run-off

s o u r c e i n t h e w a t e r body. In

t h i s s e c t i o n we have a p p l i e d t h e i d e a s used i n c h a p t e r 3 f o r s o l v i n g motion e q u a t i o n s by t h e method of f i n i t e e l e m e n t s i n combination w i t h t h e s p l i t t i n g method.

In doing s o , we succeed i n combining approximation advantages of t h e

f i n i t e e l e m e n t s method w i t h s i m p l e r e a l i z a t i o n of t h e s p l i t t i n g method. F i r s t , a s b e f o r e , we s h a l l r e d u c e t h e a d v e c t i v e terms of

(1.2) t o t h e a n t i -

symmetric form by i s o l a t i n g t h e b a r o t r o p i c component of s p e e d . Suppose t h a t

-

-

u = u + u',

v = v + v',

where -

u

=

1

u dz,

1

=

0

Then t h e v e l o c i t y components

w = w'

(4.1)

v dz.

0

u, v,

u'

,

v'

,

w i l l s a t i s f y the continuity

w'

equations

?X+Y=o

(4.2)

'

aY aax u * + - + - =aw o avi

ay

(4.3)

az

Equation ( 4 . 2 ) e n a b l e s a f l o w f u n c t i o n t o be i n t r o d u c e d by t h e r e l a t i o n s (4.4) For a c l o s e d b a s i n t h e flow f u n c t i o n c a be found from t h e s o l u t i o n of t h e boundary problem A$

= rot

i = o I n t e g r a t i n g over v e r t i c a l from find

w

z

to

u, o n r

H

and u s i n g t h e f a c t t h a t wH = 0 , we

from ( 4 . 3 )

(4.6) Z

Z

173

Modelling the Location of Pollution Sources Introducing notations

=

7

I

H

$=

u dz,

v dz

(4.7)

2

Z

and s u b s t i t u t i n g ( 4 . 6 ) , ( 4 . 4 ) , ( 4 . 7 ) i n t o (1.2), we o b t a i n

In f u r t h e r d i s c u s s i o n we s h a l l assume t h e b a s i n t o b e c l o s e d . Then t h e following r e l a t i o n s hold f o r t h e c o e f f i c i e n t s t h a t appear in (4.8):

jilr = o ;

(4.9)

Also we s h a l l r e c a l l t h a t t h e f o l l o w i n g e x p r e s s i o n s were c o n s i d e r e d a s boundary c o n d i t i o n s f o r ( 1 . 2 ) :

(4.13)

If t h e conditions (4.9)-(4.11) (4.12)-(4.14)

a r e s a t i s f i e d , t h e boundary c o n d i t i o n s

a r e n a t u r a l i n t h e v a r i a t i o n s e n s e f o r t h e space o p e r a t o r a p p e a r i n g

i n ( 4 . 8 ) , and t h i s e n a b l e s u s t o c o n s t r u c t t h e p r o j e c t i o n - d i f f e r e n c e

approxima-

t i o n s of t h e e q u a t i o n . Let u s now c o n s i d e r t h e s p a c e o p e r a t o r t h a t a p p e a r s i n ( 4 . 8 ) . As i n t h e c a s e of motion e q u a t i o n s , it may be p r e s e n t e d a s a sum of t h r e e p l a n e o p e r a t o r s : A = A

xy

+ A

xz

+ A

yz

(4.15)

where

(4.16)

Chapter 8

174

Let u s now study t h e p r o p e r t i e s of t h e o p e r a t o r s A -

XY

Ayz,

'

Axz,

and prove

t h a t they a r e p o s i t i v e l y d e f i n e d i n G . W e s h a l l c o n s i d e r , f o r example, o p e r a t o r and c o n s t r u c t an energy f u n c t i o n a l ( A

A XY

(A XY

it,@)

=

1 [-'&$g

- -aa x

@,@ ) f o r it.

Then

XY'

$

a+ + a $ ait ay ax ay

7

dy

2

ay

G

(4.17)

where ( n , x ) d e n o t e s t h e a n g l e between x - a x i s and t h e d i r e c t i o n of t h e o u t e r normal. Taking account of t h e c o n d i t i o n

$1- c

= 0

i n t h e boundary i n t e g r a l , we

a r r i v e a t t h e following expression

(4.18)

Considering c o n d i t i o n ( 4 . 1 2 ) we have

(4.19) G

T h i s proves t h e p o s i t i v e d e f i n i t e n e s s of o p e r a t o r A . When

is positively semi-definite. o p e r a t o r s Ax z , Ayz

0

= 0 the operator A

In a s i m i l a r manner t h e p o s i t i v e d e f i n i t e n e s s of

XY

i s proved. T h i s p r o p e r t y h a s a dominant r o l e i n c o n s t r u c t i n g

t h e s p l i t t i n g network and i n approximation by t h e f i n i t e e l e m e n t s method. Dividi n g t h e i n t e g r a t i o n i n t e r v a l 10, TI: t approximation scheme on tE-l 5 t S

on t h e i n t e r v a l t

9,-1

5 t 5 t tR+l- t R= T , we w r i t e a weak R R+l' f o r Eq.(1.2):

2 t 5 t : R

(4.20)

175

Modelling the Location of Pollution Sources on t h e i n t e r v a l t Q5 t 5 t

'

k+l'

(4.20')

a' 6 at

+

Axy96

=

f,

Q6(te = )4 5 ( t Q + l ) .

A t every s t e p of s p l i t t i n g t h e boundary c o n d i t i o n s f o r o p e r a t o r s Ax y ' Ay z '

A

YZ

a r e s e l e c t e d so t h a t they a r e n a t u r a l i n t h e v a r i a t i o n s e n s e f o r a correspond-

i n g o p e r a t o r . The boundary c o n d i t i o n s i n t h i s c a s e a r e so s p l i t t e d t h a t t h e i r sum w i l l y i e l d i n i t i a l boundary c o n d i t i o n s . F u r t h e r we s h a l l c o n s i d e r t h e s e c t i o n s

Denote t h e b o u n d a r i e s of t h e s e c t i o n s

I., I., 1, 1 J

Ii, I.,1k J

a r e p a r a l l e l t o t h e c o o r d i n a t e p l a n e s and d i v i d e

G

of t h e region G :

by

r 1' . r J' . rk.

The s e c t i o n s

w i t h s t e p s 6x,6 y , 6z i n t h e

Ii, I., 1,

d i r e c t i o n of t h e c o r r e s p o n d i n g c o o r d i n a t e . In e v e r y s e c t i o n we can J h h h h d e t e r m i n e network r e g i o n s which c o n s i s t of G p o i n t s belonging t o J h t h e s e s e c t i o n s . C a l l t h e network b o u n d a r i e s Ti, r:, . : - i In t h e n e x t s t e p of con-

ci, I,,Ik

s t r u c t i o n Eqs. s e t (N

+

( 4 . 2 0 ) w i l l be c o n s i d e r e d o n l y on t h e s e s e c t i o n s . Thus, we g e t a

1 ) (M + 1 ) ( K

+

1) of p l a n e p a r a b o l i c e q u a t i o n s on t h e s e c t i o n s of G .

Now i f we approximate e v e r y problem w i t h network e q u a t i o n s on c o r r e s p o n d i n g n e t h h i n such a manner t h a t t h e p r o p e r t y of p o s i t i v e s e m i - d e f i n i t e n e s s works

Ii, ij, 1;

of s p a c e network o p e r a t o r s is r e t a i n e d i n t h e s e a p p r o x i m a t i o n s , t h e n t h e scheme o b t a i n e d from (4.20) w i l l be c o n v e r g e n t . Let u s denote by Axyi, t h e o p e r a t o r s Ax y , A x z ,

Ayz

Axzj,

Axzk

t h e network o p e r a t o r s which approximate

on t h e s e c t i o n s

li, I., lk,r e s p e c t i v e l y . J

Equations

( 4 . 2 0 ) , a f t e r t h e i r q u a n t i z a t i o n by s p a c e v a r i a b l e s w i l l t a k e t h e form: on t h e i n t e r v a l

2 t S t 2. :

(4.21)

176

Chapter 8

on the interval t Q 5 t C tQ+l:

(4.21) a%k

+

7

xyk $k

= Fk’

$kcti)

=

$kCtQ+l)

On

1;’

Now we shall determine the type of approximation of operators appearing in (4.21) and the method of solving every problem. As a quantization method we shall use the method of finite elements described in section 3.3 of chapter 3 for SOlVing equations of motion. Let u s now consider in detail an approximation and the method of solving the Eqs.(4.20). Let it be an equation containing operator A

only one of

XY‘

All

other splitting steps are realized similarly, as the type of operators and their properties are completely analogous. Thus, we shall describe an approximation of plane equations of the type (4.20) by the finite elements method, as it has been done for equations of motion in chapter 3. In this case, the equations contain diffusion terms demanding formulation of boundary conditions. The equations take the form *+A$=f, at

A = - p ” 2- a $ a + a ai i , - - M + u .axz ax ay ay ax ay2 a2

(4.22)

We choose the boundary and initial conditions as follows: (4.23)

@

1 t=O

(4.24)

= 0.

In the integration range D we now introduce the network Dh formed by the intersection of straightlines x . = X1 + iSx, y . = t.&y. As before, we triangulate J

J

&

the network cells with positive direction diagonals. Suppose that region D with boundary

2.

r

has the least union of triangles, which is contained in D. At every

point (xi, y.) of the network Dh we determine the functions w , .(x, y) continuous in D, linear on every triangle, and satisfying the relations 1, i = m U . .(Xm’

1J

Yn)

1J

and j = n ,

=

0 , i f m or

j+n.

We shall find an approximate solution in the form of a linear combination of functions

w. .:

177

Modelling the Location of Pollution Sources Then t h e e q u a t i o n s f o r f i n d i n g c o e f f i c i e n t s $ . . ( t ) , when a q u a d r a t u r e formula

i s used t o approximate an e v o l u t i o n a r y

=J

t e r m , can b e w r i t t e n a s

(4.26)

The mathematical a s p e c t s of o b t a i n i n g E q s . ( 4 . 2 6 )

a r e not d e t a i l e d s i n c e they

a r e d i s c u s s e d i n t h e monographs on t h e method of f i n i t e e l e m e n t s . Computing ( 4 . 2 6 ) w i t h t h e u s e of t a b l e 3 . 1 ( s e c t i o n 3.3) and g r o u p i n g l i k e terms, we f i n d a system of d i f f e r e n t i a l e q u a t i o n s

(4.27)

where

K

i s t h e diagonal matrix o p e r a to r :

A i s t h e m a t r i x o p e r a t o r which can b e r e p r e s e n t e d a s

(4.28)

178

Chapter 8

where

y i.j . 1I

= O ~ ( L ! .

J-1

o..dD,

i,j-1

= u j f d .

yfj. 1 , J+1

ij

$k. . =

dD,

=

(4.29)

i dD.

13

TZjn D

ij

D

D

BFj

w..dD,

i,j+l

TZjn D

The set of Eqs.(4.27) a p p r o x i m a t e s t h e second o r d e r d i f f e r e n t i a l problem

(4.22) w i t h r e s p e c t t o t h e s t e p of t h e n e t w o r k . On t h e s t r e n g t h of

(4.28),

a p p e a r i n g i n (4.27) c a n be r e p r e s e n t e d , as i n t h e c a s e of network

operator

o p e r a t o r i n s e c t i o n 3 . 3 , as a sum of one-dimensional

operators:

A = A l + A Z + A 3’

(4.30)

where

..

..

..

.. +

Y iij , j + l + i , j+l

..

+

(Yi, i jj+l

+

Y;;j-l),$ij

ij

+

Yi ,J-1 . ,$. . + 1. J-1

179

Modelling the Location of Pollution Sources

(4.31)

Analysing t h e t y p e of c o e f f i c i e n t s u ,

6, y g i v e n by r e l a t i o n s

( 4 . 2 9 ) one

may n o t e t h a t t h e f o l l o w i n g r e l a t i o n s a r e s a t i s f i e d f o r them:

..

,

.. a'J

Bi. f. 1 , j ,

B i. j .

i+l,j

= a , .

a?i f 1 , j

=

-

i,jfl.

= a , .

i,jfl

1,J f 1

13

13

=

(4.32)

'

- Bf , j + l 13

(4.33)

if1,j

ij

Yi+l,j

=

Yij

Yi,j+l

..

Using ( 4 . 3 2 ) - ( 4 . 3 4 ) s i t i v e semi-definiteness

(A,$,

&)

i ,j f l

ij I

Yij

=

,

ifl, j f l

(4.34)

it can be shown t h a t t h e o p e r a t o r s h l , h a , A 3

have po-

i n t h e s p a c e of network f u n c t i o n s , t h a t i s , 2 0,

(Az&,

9) 2

0,

(Ad,9) 2

0.

(4.35)

T h i s e n a b l e s t h e s p l i t t i n g method t o be u s e d f o r s o l v i n g ( 4 . 2 7 ) , which ( t h e s p l i t t i n g method) w i l l be a b s o l u t e l y s t a b l e . Making u s e of t h e b i c y c l i c s p l i t t i n g p l a n , we f i n a l l y o b t a i n

Chapter 8

180

(4.36)

T h i s i s r e a l i z e d by performing o p e r a t i o n s in t h r e e d i r e c t i o n s , i n c l u d i n g t h e d i a g o n a l d i r e c t i o n of t h e network t r i a n g u l a t i o n . F i g u r e s 8 . 1 through 8 . 4 r e p r e s e n t i s o l i n e s of d i f f e r e n c e a n a l o g u e s of f u n c t i o n a l s (1.7), ( 1 . 8 ) p a r a m e t r i c a l l y dependent on t h e c o o r d i n a t e s r

-0

.

F i g u r e 8 . 1 shows t h e f i e l d of i s o l i n e s of t h e f u n c t i o n a l (1.7), c a l c u l a t e d by ( 3 . 9 ) , i n t h e p l a n e 11-type "channel" w i t h t h e i n f l o w i n i t s r i g h t branch For t h e s e v e r y c o n d i t i o n s t h e i s o l i n e s Of t h e

being constant i n t i m e .

f u n c t i o n a l ( 1 . 8 ) a r e shown i n f i g u r e 8 . 2 . F i g u r e s 8.3 and 8 . 4 i l l u s t r a t e t h e c a s e of a c l o s e d two-dimensional

r e s e r v o i r i n which t h e l i q u i d c i r c u l a t e s i n

t h e c l o c k w i s e d i r e c t i o n w i t h a c o n s t a n t i n time v e l o c i t y . For t h i s c a s e , t h e i s o l i n e s of f u n c t i o n a l s ( 1 . 7 ) and ( 1 . 8 ) a r e shown i n f i g u r e s 8.3 and 8 . 4 , resp e c t i v e l y . The l a s t two f i g u r e s have been o b t a i n e d by t h e method of f i n i t e e l e m e n t s . The t r i a n g u l a t i o n f o r t h i s method i s shown i n f i g u r e 8 . 5 .

FIGURE 8.1

Modelling the Location of Pollution Sources

FIGURE 8 . 2

FIGURE 8 . 3

181

182

Chapter 8

FIGURE 8 . 4

FIGURE 8.5

183

Modelling the Location of Pollution Sources

L i n e s show e q u a l v a l u e s of t h e f u n c t i o n a l J ( r ) because t h e p o l l u t i o n k-o s o u r c e s l o c a t e d a t t h e p o i n t s of one and t h e same l i n e J ( r ) = c o n s t have k a s i m i l a r e f f e c t on t h e r e g i o n G k . As t h e i n d u s t r i a l waste s o u r c e s a r e moved f a r t h e r away from t h e r e g i o n G k . t h e i r p o l l u t i o n e f f e c t d e c r e a s e s . Suppose t h a t i n t h e range of d e t e r m i n a t i o n of s o l u t i o n we have found a s u r f a c e on which Jk = c o n s t = I t i m p l i e s t h a t w i t h i n t h e r a n g e J ( r ) > C k , and o u t s i d e t h e range boundk -< ed by t h e g i v e n l i n e , J ( r ) 5 C k . Hence, t h e s a n i t a r y r e q u i r e m e n t s f o r e m i s s i o n s k w i l l be observed w i t h r e g a r d t o p o l l u t i o n of t h e w a t e r body a r e a G . k __ Now we suppose t h a t t h e r e a r e n e c o l o g i c a l l y p r o t e c t e d zones D (k = 1 , n ) , k e a c h having i t s own s a n i t a r y r e q u i r e m e n t s . We s h a l l s o l v e n a d j o i n t problems

= Ck.

of t h e t y p e (2.3), ( 2 . 4 ) and o b t a i n

n

f u n c t i o n a l s J k . For e v e r y f u n c t i o n a l we

s h a l l have a s i t u a t i o n s i m i l a r t o t h a t shown i n f i g u r e s 8 . 1 through 8 . 4 . The i n t e r s e c t i o n of a l l t h e r e g i o n s u k w i l l y i e l d t h e d e s i r e d r e g i o n ( i f it e x i s t s )

1 , )

f o r which a l l of t h e c o n d i t i o n s J ( r ) 5 Ck(k = a r e s a t i s f i e d . T h i s means k-O i n d u s t r i a l w a s t e s o u r c e s i n such a zone w i l l s i m u l -

t h a t t h e l o c a t i o n of t h e

t a n e o u s l y p r o v i d e a normal e c o l o g i c a l s i t u a t i o n i n a l l t h e w a t e r a r e a s G G

.

1’ ’ . ‘ I I f such a zone d o e s n o t e x i s t , p a r t i c u l a r r e q u i r e m e n t s must be p l a c e d on t h e

p r o d u c t i o n p r o c e s s and t h e amount of e m i s s i o n s

Q

s h o u l d be d e c r e a s e d t o a

l e v e l u n t i l t h e r e g i o n becomes non-empty. Let u s now c o n s i d e r a problem of f i n d i n g such a p o i n t r

--o

f o r hydrosols d i s -

c h a r g e a t which t h e maximum v a l u e of Jk f o r a l l e c o l o g i c a l zones Gk w i l l be minimum. For t h i s w e i n t r o d u c e normalized f u n c t i o n a l s

f o r which t h e r e s t r i c t i o n h o l d s

Exact e q u a l i t i e s B ( r ) = l k v w i l l h o l d t r u e and s a n i t a r y r e q u i r e m e n t s f o r t h e s u r f a c e be met.

F u r t h e r , we s h a l l compute B ( r ) a t a l l t h e p o i n t s of t h e domain of p e r k -0 m i s s i b l e v a l u e s meeting s a n i t a r y r e q u i r e m e n t s f o r a l l D and f i n d a maximum k v a l u e f o r each of them, t h a t i s , w e s h a l l perform e x p l i c i t e x h a u s t i o n of a l l

) . Going o v e r from one p o i n t --o r t o a n o t h e r w e s h a l l perform e x p l i c i t k w exhausion a t a l l t h e p o i n t s . As a r e s u l t , we s h a l l f i n d a p o i n t M ( 5 ) a t which

B (r

t h e maximum component B (r ) h a s a minimum v a l u e , and t h i x w i l l be t h e s o l u t i o n k -0 of t h e minimax problem, t h a t i s

Appendix

Note t h a t h e r e t h e minimax problem i s s o l v e d by t h e e x p l i c i t e x h a u s t i o n method. In t h i s r e s i d e s t h e remarkable p r o p e r t y of a d j o i n t problems which permit of t h e most s i m p l e r e a l i z a t i o n of t h e minimax problem. I f w e had n o t made u s e of t h e a d j o i n t problems, t h e n f o r s o l v i n g a minimax problem we had t o s o l v e a v a s t number of problems w i t h d i f f e r e n t l o c a t i o n s of t h e i n d u s t r i a l waste s o u r c e s . Such a methodology would have h a r d l y provided t h e r e q u i r e d a c c u r a c y even w i t h t h e u s e of most advanced computers.

Appendix. MESOME TEOROLOGICAL AND MESOOCEANIC PROCESSES 1. Mesometeorological problem of d e t e r m i n i n g l o c a l a t m o s p h e r i c circulations With t h e growing s c a l e of man's economic a c t i v i t i e s energy p r o d u c t i o n i n c r e a s e s , and so do t h e amount of h e a t and i m p u r i t i e s d i s c h a r g e d i n t o atmosphere. Discharged i m p u r i t i e s undergo v a r i o u s t r a n s f o r m a t i o n s and s p r e a d o v e r l a r g e d i s t a n c e s , p o l l u t i n g environment. Environmental p o l l u t i o n w i t h harmful s u b s t a n c e s depends n o t only on t h e t e c h n o l o g i c a l p a r a m e t e r s b u t a l s o on such m e t e o r o l o g i c a l f a c t o r s a s wind v e l o c i t y , a t m o s p h e r i c s t r a t i f i c a t i o n , t e r r a i n o r o g r a p h y , c h a r a c t e r i s t i c s of t h e u n d e r l y i n g s u r f a c e , t u r b u l e n c e f i e l d and t h e l i k e . A c c o r d i n g l y , g r e a t e r emphasis s h o u l d be p l a c e d on t h e s t u d y of m e t e o r o l o g i c a l f a c t o r s of environmental p o l l u t i o n and on t h e development of a p p r o p r i a t e mathematical models. In most c a s e s t h e main b u l k of c o n t a m i n a n t s i s d i s c h a r g e d i n t o t h e lower atmospheric l a y e r s . Then, under t h e i n f l u e n c e of l o c a l c i r c u l a t i o n s t r q h t about by l a r g e - s c a l e

movement due t o t h e r m a l and o r o g r a p h i c inhomogeneity of u n d e r l y i n g

s u r f a c e , i m p u r i t i e s a r e l i f t e d t o t h e boundary l a y e r . Hence, c o n s t r u c t i o n of mathematical models r e q u i r e s s i m u l t a n e o u s s o l u t i o n of atmosphere dynamics and i m p u r i t y t r a n s f e r problems. Here we g i v e some examples on t h e s o l u t i o n of problems of atmosphere dynamics.

We begin w i t h a model f o r t h e dynamics of t h e boundary l a y e r . Accordingly, we c o n s i d e r t h e system of e q u a t i o n s used i n mesometeorological problems. The e q u a t i o n s a r e w r i t t e n i n l e f t - h a n d C a r t e s i a n system of c o o r d i n a t e s x , y , z ( x - a x i s

is d i r e c t e d eastward, y

-

northward, z

-

v e r t i c a l l y upward). For t h e i n i t i a l d a t a

we t a k e t h e e q u a t i o n of motion:

(1.1)

185

Mesome teorological and Mesooceanic Processes

dw--ldp-g+2w; dt -

p

az

t h e e q u a t i o n of c o n t i n u i t y

-3u

av aw 2y az

+

+

Ox

+

Id0 =

0;

p dt

t h e e q u a t i o n of s t a t e p = pRT; and t h e e q u a t i o n of h e a t i n f l u x

* dt

=

LW ~

c

tt,

+ Q, + > e ,

(1.4)

P

where

2

The o p e r a t o r s

d@/dt

and

Ap a r e :

(1.6)

Here, t along p

is time; u , v, w

a r e t h e components of t h e wind v e l o c i t y v e c t o r

x , y, z , respectively; T

is pressure; p is density; R

is t e m p e r a t u r e ;

0

is p o t e n t i a l temperature;

is t h e u n i v e r s a l gas c o n s t a n t ; L

h e a t of e v a p o r a t i o n ; 0 i s t h e r a t e of l i q u i d p h a s e f o r m a t i o n ; c h e a t c a p a c i t y of a i r a t c o n s t a n t p r e s s u r e ; Q

P

is the latent

is specific

i s t h e r a d i a t i v e component of h e a t

f l o w ; 1-1, v a r e t h e h o r i z o n t a l and v e r t i c a l t u r b u l e n t exchange c o e f f i c i e n t s , respectively; A

i s t h e r m a l e q u i v a l e n c e of work; g

is t h e a c c e l e r a t i o n due t o

gravity. W e do n o t u s e E q s . ( l . l ) - ( l . 7 )

i n s o l v i n g m e s o m e t e o r o l o g i c a l problems a s ,

b e s i d e s t h e l a t t e r , t h i s system a l s o d e s c r i b e s l a r g e - s c a l e

meteorological pro-

c e s s e s , a c o u s t i c waves, e t c . On t h e o t h e r hand, a l l terms i n t h e e q u a t i o n s a r e n o t of t h e same o r d e r of m a g n i t u d e ; t h e r e f o r e , i n s o l v i n g s p e c i f i c mesometeorolog i c a l problems t h e s y s t e m c a n be s i m p l i f i e d by n e g l e c t i n g s m a l l terms. By way of 1

example, c o n s i d e r l i n e a r i z a t i o n of n o n l i n e a r terms of t h e t y p e - g r a d

p . For

t h i s p u r p o s e , i n t r o d u c e a new f u n c t i o n

where

0

i s t h e mean p o t e n t i a l t e m p e r a t u r e . Using t h e e q u a t i o n of s t a t e , d e f i n i -

186

Appendix

t i o n s of p o t e n t i a l t e m p e r a t u r e and f u n c t i o n

- -1

T,

we o b t a i n

e

- - grad

grad p =

(1.9)

T.

OO

S i n c e we a r e i n t e r e s t e d i n mesoprocesses i n which h o r i z o n t a l s c a l e s c o n s i d e r a b l y exceed t h e G e r t i c a l o n e s , t h e terms c o n t a i n i n g

w

i n t h e t h i r d equation

of motion (1.1) is s m a l l compared w i t h t h e o t h e r two t e r m s . I n t h i s c a s e , t h e t h i r d e q u a t i o n i n (1.1) i s c o n s i d e r a b l y s i m p l i f i e d and w e o b t a i n t h e e q u a t i o n of statics

A f t e r t h e s e s i m p l i f i c a t i o n s and s i n c e i n ( 1 . 9 ) 8 / O 0 = ( 0 + 8 ' ) / o o :1 a s 070

< < 1, t h e system (1.1) can be e x p r e s s e d a s

_ d' _ dt

an ax

dv _

an

ay -

dt

Assuming t h e

+ Pv + l u , ?I

(1.11)

PU + A v ,

space-time v a r i a t i o n s of p t o b e s m a l l we w r i t e t h e e q u a t i o n

of c o n t i n u i t y ( 1 . 2 ) a s

a u + -a +v a w = o . ax d y az

(1.12)

For t h e s a k e of s i m p l i c i t y , we can a l s o n e g l e c t t h e i n f l u e n c e of r a d i a t i v e h e a t flow i n t o

t h e atmospheric boundary l a y e r .

I n n a t u r e meso- and l a r g e - s c a l e p r o c e s s e s always i n t e r a c t . But t h e i n f l u e n c e of l a r g e - s c a l e c i r c u l a t i o n on mesoscale one is t h e most i m p o r t a n t i n t h e mesometeorological theory.

To o b t a i n a c o n s i s t e n t system of boundary l a y e r e q u a t i o n s , we r e p r e s e n t t h e meteorological f i e l d s a s

u =

u +

u*, T

= =

v +

ll +

v*,

T I ,

w =

p = P

w + +

w',

e

=

o + el, (1.13)

p',

where c a p i t a l l e t t e r s denote background l a r g e - s c a l e components of m e t e o r o l o g i c a l f i e l d s , and l e t t e r s w i t h prime d e n o t e d e v i a t i o n s . S u b s t i t u t i n g (1.13) i n t o (1.10)-(1.12),

and n e g l e c t i n g s m a l l q u a n t i t i e s which a r i s e on t h e assumption t h a t

<< P,

8'

<<

o,

TI

< < I[,

jo - o I

<<

o0'

(1.14)

and t a k i n g i n t o account a p p r o p r i a t e e q u a t i o n s for background f i e l d s , we o b t a i n

187

Mesometeorological and Mesooceanic Processes t h e r e q u i r e d system of mesometeorology e q u a t i o n s . There a r e d i f f e r e n t ways of d e f i n i n g background f i e l d s . To c o n s t r u c t t h e p e r t u r b a t i o n e q u a t i o n s , i t i s nec e s s a r y t h a t t h e background f i e l d s s a t i s f y t h e i n i t i a l system to s m a l l t e r m s .

Let u s now c o n s i d e r a s p a t i a l n o n s t a t i o n a r y numerical model of an atmospheric boundary l a y e r o v e r o r o g r a p h i c a l l y and t h e r m a l l y inhomogeneous s u r f a c e based on a system of p e r t u r b a t i o n e q u a t i o n s . We f o r m a l l y d i v i d e t h e atmosphere i n t o two l a y e r s : t h e c o n s t a n t - f l u x l a y e r and upper l a y e r . The c o n s t a n t - f l u x l a y e r i s e x p r e s s e d i n a p a r a m e t r i z e d form. We supplement t h e i n i t i a l system with an equation f o r s p e c i f i c humidity:

aei

set

+ div

+

Sw' = - U ' ( S ~+~ BX)

-

v'(s6

Y

+

B ) + Y (1.18)

*' ax + div gq? = -

at

2

+

av' ay

+

(w' + u ' 6

u +

o

u',

= 0,

az

+ u =

"'

+ V'6y)

-

Aq' v =

v +

(1.19)

q = Q + q';

-

-

UQ,

vQY +

a

aqq

v0

+

(1.20)

a; v',

w =

w +

w!, (1.21)

=

o

+ 81,

=

II +

T I ;

here

x, y , z z = z.

a r e c u r v i l i n e a r coordinates mutually orthogonal along t h e r e l i e f ,

-

6 ( x , y)zi

is t h e h e i g h t above t h e s e a l e v e l ; t h e e q u a t i o n i s t h e wind v e c t o r ; h , S

describes the r e l i e f ; c a t i o n parameters;

u1, v2

ex,

'a',

y)

a r e c o n v e c t i o n and s t r a t i f i -

a r e horizontal turbulence coefficients;

v e r t i c a l t u r b u l e n c e c o e f f i c i e n t s f o r momentum and h e a t ; q d i t y ; u', v ' , w ' ,

z . = 6(x,

Vu,

lie

are

i s t h e s p e c i f i c humi-

q ' , n ' a r e d e v i a t i o n s from t h e r e s p e c t i v e background v a l u e s ;

By and Qx, Qy a r e h o r i z o n t a l g r a d i e n t s of t h e background p o t e n t i a l tempera-

188

Appendix

t u r e and t h e background s p e c i f i c h u m i d i t y . To d e s c r i b e t h e c o n s t a n t - f l u x

layer

s t r u c t u r e , w e employ t h e Monin-Obukhov s i m i l a r i t y t h e o r y a n d Businger e m p i r i c a l f u n c t i o n s . To approximate t h e e l e v a t i o n p r o f i l e s o f m e t e o r o l o g i c a l l a y e r under s t r o n g i n s t a b i l i t y , w e use t h e

t h e constant-flux str-ng s t a b i l i t y

-

l i n e a r l a w . Then, t h e c o n s t a n t - f l u x

'IL

-

fields in l a w " and u n d e r

3 l a y e r model c a n be e x p r e s s

ed by t h e f o l l o w i n g e q u a t i o n s :

(1.24)

z,-I!.

fu(F',

Su) =

I

(1.26)

Z

ze __ dS,

@u(5)

fO(C,

5,)

=

1 ~ =1

rate;

e,,

__ d c ;

(1.27)

50

F'U

here,

/ "l"

1

(uz + v Z ) ' i s t h e modulus of t h e v e l o c i t y v e c t o r ; u + i s f r i c t i o n

9, a r e t h e s c a l e s of p o t e n t i a l t e m p e r a t u r e and s p e c i f i c h u m i d i t y ; h i s

the constant-flux layer height;

K

is t h e KaMnan c o n s t a n t ; z u , z v a r e r o u g h n e s s

p a r a m e t e r s f o r wind and t e m p e r a t u r e ( i n d i c e s 0 and f i e l d s a t z = 0 and z = h , r e s p e c t i v e l y ) ; H on and h e a t t r a n s f e r c o e f f i c i e n t s ,

h

denote meteorological

i s t h e h e a t f l u x ; c u , c8 a r e f r i c t i -

respectively;

",

f . a r e continuous u n i v e r s a l

functions. N u m e r i c a l l y , t h e problem (1.15)-(1.21) i s s o l v e d a s f o l l o w s . The c o n s t a n t f l u x l a y e r i s i n c l u d e d i n t o t h e model i n s u c h a manner t h a t w e h a v e o n l y t o v a r y t h e way of a s s i g n i n g t h e c o n d i t i o n s for z = h ,

l e a v i n g t h e a l g o r i t h m unchanged.For

t h i s p u r p o s e , w e t r a n s f o r m t h e Eqs.(1.22)-(1.27)

so t h a t t h e y y i e l d t h e r e s p e c -

t i v e conditions f o r z = h. F i n a l l y , t h e boundary c o n d i t i o n s f o r t h e s y s t e m ( 1 . 1 5 ) - ( 1 . 2 1 )

take the

Mesometeorological and Mesooceanic Processes

189

f o l l o w i n g form:

au' = o ,

av*

3' = aY

dV' ay

ax

0,

ax

= 0,

= 0,

a09

aov ay

= 0,

= 0,

aq' ~

ax

= 0

3' = o 3Y

for x = 2 X;

(1.29)

for y = 2 Y

(1.30)

u=

here ( u , v ) ; i.. = +i(Ch)/fi(C 5 ) , i = u , 8 ; H i s t h e conventional height h' o of a t m o s p h e r i c boundary l a y e r ; X , Y a r e l a t e r a l b o u n d a r i e s o f t h e domain under consideration. C o n d i t i o n s (1.29), ( 1 . 3 0 ) p r o v i d e a way f o r c o m p l e t i n g t h e problem. They e x p r e s s t h e r e q u i r e m e n t for t h e d i s t u r b a n c e t o b e smooth i n t h e v i c i n i t y of t h e domain boundary t o a g r e a t e r e x t e n t , t h a n t h e p h y s i c a l meaning of t h e s i m u l a t e d p r o c e s s e s . T h e r e f o r e , a l o n g w i t h (1.29), ( 1 . 3 0 ) w e c o n s i d e r t h e c o n d i t i o n s t h a t a l l o w .,le p r o c e s s e s o c c u r r i n g w i t h i n t h e domain D t o t e n d t o t h e background f l o w mode. These c o n d i t i o n s c a n be w r i t t e n a s f o l l o w s :

(1.33) q = Q,, Over w a t e r , t h e f u n c t i o n s B O ,

where E

and q

y = f Y.

x = f X,

a r e supposed t o be g i v e n

i s t h e r e s i l i e n c e o f w a t e r vapor s a t u r a t i o n a t t e m p e r a t u r e 8 ; p

is

a t m o s p h e r i c p r e s s u r e ; f (t) i s w a t e r s u r f a c e t e m p e r a t u r e assumed t o be a known f u n c t i o n of t h e h o r i z o n t a l c o o r d i n a t e s and t i m e . Land t e m p e r a t u r e is found from t h e h e a t b a l a n c e e q u a t i o n , and t h e f u n c t i o n q

i s computed by formula

(1.35)

qo = 0,622r70E 0(Q0)/p, where 11

i s r e l a t i v e h u m i d i t y assumed t o b e a known f u n c t i o n

heat balance condition a t t h e atmosphere-soil

where GS = As

of x , y , t . The

i n t e r f a c e is of t h e form

( a T / a z ) s i s h e a t t r a n s f e r t h r o u g h s o i l s u r f a c e ( s u b s c r i p t S shows

t h a t t h e v a l u e s a r e t a k e n f o r z = 0 ) ; AS = c P K ; p s , c s , KS, T a r e d e n s i t y ,

s s s

s p e c i f i c h e a t , t e m p e r a t u r e c o n d u c t i v i t y c o e f f i c i e n t and a b s o l u t e s o i l t e m p e r a t u r e ;

190

p

Appendix

i s atmospheric d e n s i t y ; I

i s s o l a r short-wave r a d i a t i o n , A

S

i s t h e subsurface

a l b e d o ; FS i s t h e e f f e c t i v e long-wave r a d i a t i o n ; a p i s a d i m e n s i o n l e s s c o e f f i c i e n t which a c c o u n t s f o r d i f f e r e n t amount of Ilent s p e n t on e v a p o r a t i o n ( c o n d e n s a t i o n ) a t d i f f e r e n t p o i n t s of t h e u n d e r l y i n g s u r f a c e due t o s u r f a c e n o n - u n i f o r m i t y ; I

S

(x, y , t ) i s a f u n c t i o n which d e s c r i b e s t h e flow of h e a t r e l e a s e d d u r i n g energy p r o d u c t i o n and consumption. The models of h y d r o m e t e o r o l o g i c a l regime over i n d u s t r i a l r e g i o n s do n o t t a k e account of a r t i f i c i a l h e a t f l o w s . T h i s i s s t i l l an open q u e s t i o n . Heat c a n be r e l e a s e d i n t o atmosphere e i t h e r i n r e a l or l a t e n t form, hence t h e r e a r e d i f f e r e n t ways of p a r a m e t r i z a t i o n of h e a t r e l e a s e . In p a r t i c u l a r , a r t i f i c i a l h e a t flow can be accounted f o r by adding i t t o

t h e r a d i a t i v e heat flow.

Under calm c o n d i t i o n s , t h e t e m p e r a t u r e d r o p between t h e l e v e l s z = 0 and

z

may be q u i t e h i g h . Hence, i n s o l v i n g ( 1 . 3 6 ) ,

= zo

semi-empirical

parametriza-

t i o n i s used f o r v i s c o u s s u b l a y e r : (1.37) where

V

is t h e c l i m a t i c v i s c o s i t y f a c t o r .

The s o i l t e m p e r a t u r e d i s t r i b u t i o n i s d e s c r i b e d by t h e e q u a t i o n aT

a

aT

(1.38)

where KS = K S ( x , y , z ) i s t h e s o i l t e m p e r a t u r e c o n d u c t i v i t y c o e f f i c i e n t which i s c o n s i d e r e d t o be g i v e n . Equation ( 1 . 3 6 ) i s t h e boundary c o n d i t i o n f o r ( 1 . 3 8 ) f o r z = 0 . We g i v e t h e second boundary c o n d i t i o n f o r ( 1 . 3 8 ) a t a s m a l l d e p t h HTI

and assume t h a t d i u r n a l

temperature f l u c t u a t i o n d e c a y , i . e . T = T n

at

z = - H

n'

(1.39)

To d e t e r m i n e t h e h e a t f l o w i n t o s o i l a s a f u n c t i o n of TS, we approximate G

S

as follows:

where AL,

is t h e g r i d s t e p with r e s p e c t t o depth, T

1

is s o i l temperature a t t h e

f i r s t level. To s o l v e ( 1 . 3 8 ) , l e t u s c o n s t r u c t i t s f i n i t e - d i f f e r e n c e analogue and employ t h e t r i a n g l e f a c t o r i z a t i o n t e c h n i q u e . Omitting t h e u n t e r m e d i a t e f o r m u l a s , we r e p r e s e n t s o l u t i o n i n t h e form T1 = BITS

where G1

+ z 1'

(1.41)

and z1 a r e p a r a m e t e r s t r i a n g l e f a c t o r i z a t i o n t e c h n i q u e . S u b s t i t u t i n g

191

Mesome teorological and Mesooceanic Processes ( 1 . 4 1 ) i n t o (1.40) we o b t a i n

A S (l-B1)TS-hSZ1 GS =

By v i r t u e of

(1.42)

A5

(1.37), (1.42) w e c a n w r i t e t h e s o l u t i o n of (1.36) a s

-

e

e

=

+ cFOh

O

0

(1.43)

1 t c F '

where

Knowing O0 from (1.37), w e can d e t e r m i n e BS. To c a l c u l a t e t h e near-ground char a c t e r i s t i c s t h e following empirical formulas a r e used. Albrecht formula:

I. I

0

t 0; sin h

bod s i n h

= (t

= 2 cal/(sq'cm'min);

b

C'

cos a c o s $ cos

n;

(1.44)

- 12) n/12;

i s sun's z e n i t h a n g l e ; @ i s t h e l a t i t u d e ;

declination; a

-

= sin @ sin $

n h

-

= a sin h

n i s sun's

hour a n g l e ; $ is s u n ' s

= 0.3 c a l / ( s q *cm'min);

( t i s reckoned

by l o c a l time i n h o u r s , s t a r t i n g from m i d n i g h t ) ; Brent formula : FS = afcTo(ae where

U

-

be

i s t h e Stefan-Boltzmann c o n s t a n t ; f

constants; e

&),

is s o i l albedo; a e , b

(1.45) a r e empirical

i s w a t e r vapor r e s i l i e n c e ; and t h e Cbarnok formula f o r d e t e r m i n i n g

roughness p a r a m e t e r o v e r w a t e r : z

= 0,035 u:/g.

(1.46)

192

Appendix

formula i s s u i t a b l e f o r c a l c u l a t i n g s o l a r

I t must be n o t e d t h a t t h e AZbrecht

r a d i a t i o n f l u x f o r a f l a t t e r r a i n . I t i s w e l l known t h a t t h e ground t e m p e r a t u r e and a g g r e g a t e e v a p o r a t i o n of wet s u r f a c e depend on t h e a c t i v e s u r f a c e i n s o l a t i o n . Consequently, t h e d i f f e r e n c e s i n i n s o l a t i o n of s l o p e s , depending on t h e i r e x p o s i tion, may,under rough t e r r a i n , l e a d t o s i g n i f i c a n t mesometeorological c o n t r a s t s . T h e r e f o r e , we w r i t e t h e formula f o r c a l c u l a t i n g s o l a r r a d i a t i o n from s l o p e s u r f a c e as

sh where c o s

=

01

cos a

sin h

=

so

+ cos

c o s a,

(sin

(1.47)

4 c o s Ji c o s

Q

-

s i n $ cos

Ji

cos $)

s i n Cl s i n a ; S i s t h e s o l a r c o n s t a n t ; a i s a n g l e of i n r o c i d e n c e of sun r a y s on s u r f a c e ; i i s azimuth of t h e p r o j e c t i o n of normal on sin a

+ s i n $ cos

1)

i s p o s i t i v e i n reckoning

h o r i z o n t a l s u r f a c e reckoned from t h e m e r i d i a n p l a n e from t h e s o u t h c l o c k w i s e ) . The f u n c t i o n s a

and $a a r e c a l c u l a t e d a s f o l l o w s :

r

I

0 1 ~=

a r c t g [(6;

The v a l u e of t h e p a r a m e t e r

k

+ 62)21,

$a = a r c t g ( 6 x / 6

Y

Y

) + kn.

(1.48)

v a r i e s depending on t h e o r i e n t a t i o n of t h e s l o p e .

By analogy w i t h ( 1 . 4 4 ) , we w r i t e a formula f o r S-the

t o t a l r a d i a t i o n flow

t o the slope: Sr = a

cos

01

-

b

G.

(1.49)

I t can e a s i l y be s e e n t h a t f o r a, = 0 , formulas (1.44) and ( 1 . 4 9 ) c o i n c i d e . Exact c a l c u l a t i o n of d i f f u s e r a d i a t i o n i s more d i f f i c u l t due t o a n i s o t r o p y . D i f f u s e r a d i a t i o n f l u x from d i f f e r e n t s e c t i o n s of t h e h e a v e n l y dome depends on t h e Sun p o s i t i o n . F i g u r e A . l i l l u s t r a t e s d i u r n a l v a r i a t i o n of S r , c a l c u l a t e d by ( 1 . 4 9 ) and t h a t of t h e t o t a l h e a t Sc r e c e i v e d by t h e s l o p e s i n c e s u n r i s e . The f o l l o w i n g v a l u e s a r e used f o r p a r a m e t e r s : $r

=

42.5O,

$ = 1l0, a

r

=

loo,

a

= 2

cal/(sq.cm*min),

bo = 0.3 c a l / ( s q * c m * m i n ) . I t i s c l e a r from t h e f i g u r e t h a t t h e s o u t h e r n s l o p e r e c e i v e s much more h e a t t h a n t h e n o r t h e r n one. For ar = 0 , t h e c u r v e s f u s e i n t o one c u r v e . For a s p e c i f i c r e g i o n with complex t e r r a i n , t h e v a l u e s of a The systems ( 1 . 1 5 ) - ( 1 . 2 0 )

and b

must b e found e x p e r i m e n t a l l y .

and (1.28)-(1.32) a r e n o t c l o s e d a s t h e v e r t i c a l

t u r b u l e n c e exchange c o e f f i c i e n t s

Vh

s i d e r t h e t u r b u l e n t energy e q u a t i o n

and

Vo

a r e unknown. To d e t e r m i n e them, con-

193

Mesome teorological and Mesooceanic Processes

t

A-si 4-

3-

t

*

FIGURE A.l

(1.50)

where aT = vB/vU = @ u ( S ) / $ o ( S ) ; a b and

a r e dimensionless u n i v e r s a l c o n s t a n t s .

c

To d e t e r m i n e t h e t u r b u l e n c e c h a r a c t e r i s t i c s , we u s e t h e Kolmogorov approximate

s i m i l a r i t y h y p o t h e s i s which i m p l i e s t h a t t h e t u r b u l e n t exchange c o e f f i c i e n t s

v

and t h e rate of t u r b u l e n t energy d i s s i p a t i o n i n t o h e a t c a r e u n i q u e l y e x p r e s s -

e d i n t e r m s of t u r b u l e n c e k i n e t i c energy

b

1

V

= c!,b2,

E

and t u r b u l e n c e s c a l e E: = cb2/v

0 ’

(1.51)

194

Appendix

The t u r b u l e n c e s c a l e i s computed by t h e Karmn g e n e r a l i z e d formula f o r d i s p l a c e ment p a t h : (1.52) Let u s now f o r m u l a t e t h e boundary c o n d i t i o n s f o r t h e system ( 1 . 5 0 ) - ( 1 . 5 2 ) . A s i s known, m e t e o r o l o g i c a l f i e l d s n e a r t h e u n d e r l y i n g s u r f a c e can have s t e e p

g r a d i e n t s , and t o d e s c r i b e them t h e n u m e r i c a l models need h i g h v e r t i c a l r e s o l u t i -

on. Hence, d i r e c t i n c l u s i o n of ( 1 . 5 0 ) - ( 1 . 5 2 )

i n t o s p a t i a l models i t r a t h e r d i f f i -

c u l t due t o l i m i t e d computer c a p a b i l i t i e s . T h e r e f o r e , we use ( 1 . 2 2 ) - ( 1 . 2 7 ) ( 1 . 5 1 1 , (1.52) t o near-ground

d e r i v e p a r a m e t r i z a t i o n formula f o r t h e f u n c t i o n

b

and

in t h e

atmospheric l a y e r .

F i r s t , c o n s i d e r p a r a m e t r i z a t i o n of t h e mean k i n e t i c energy of t u r b u l e n t

_ _ _ _ _ _

f l u c t u a t i o n s b2 = ( u ' u '

+ v ' v ' + w'w')/2

( u ' , v ' , w ' - f l u c t u a t i o n s of wind v e l o -

c i t y components; b a r above d e n o t e s a v e r a g i n g ; b

=

- &)

i n t h e near-ground atmo-

s p h e r i c l a y e r . The v a l u e of b2 for z = h can s e r v e a s a boundary c o n d i t i o n f o r t h e e q u a t i o n of t u r b u l e n c e energy i n t h e domain above t h e ground l a y e r . E x p r e s s t h e system ( 1 . 5 1 ) , ( 1 . 5 2 ) t o t h e form

a bU-

f ub u + g ub u2 = 0 ,

x

(1.53)

where (1.54)

If f

and g

a r e known f u n c t i o n s of h e i g h t , t h e n w e f i n d b

from ( 1 . 5 3 ) , t a k i n g

no energy flow a t roughness l e v e l a s t h e boundary c o n d i t i o n , i . e .

Assuming t h a t b

# 0

for z = z

and u s i n g ( 1 . 5 3 ) , we f i n a l l y o b t a i n

bU = f U / g U I t is a p p a r e n t from ( 1 . 2 4 ) and ( 1 . 5 3 ) , b

for

z = z

(1.55)

(1.54) t h a t t h e f u n c t i o n s f u I gu and

p a r a m e t r i c a l l y depend on u* and L . To make f u r t h e r p r e s e n t a t i o n c o n v e n i e n t ,

we p a s s t o d i m e n s i o n l e s s v a r i a b l e s

(1.56)

195

Mesometeorological and Mesooceanic Processes

and r e w r i t e ( 1 . 5 3 ) - ( 1 . 5 5 )

as (1.57)

where b = f / g f o r

5

=

Lo (C0

=

On s u b s t i t u t i n g y = :-'into

zo/L). (1.57),

t h e l a t t e r i s reduced t o a f i r s t - o r d e r

l i n e a r d i f f e r e n t i a l e q u a t i o n whose s o l u t i o n i s

(1.58)

S u b s t i t u t i n g ( 1 . 5 6 ) i n t o (1.58) and p u t t i n g z = h , we o b t a i n

(1.59)

where (1.60)

5h = and

E; = 5 /H, H = h/z . o h In d e r i v i n g ( 1 . 5 9 ) we have t a k e n t h a t t h e f u n c t i o n 4 i s always p o s i t i v e

h/L,

5 changes i t s s i g n , depending on s t r a t i f i c a t i o n . Note t h a t

F,

i s no l o n g e r

an independent v a r i a b l e , h e n c e , t h e f u n c t i o n s a c t u a l l y depend on 5,

and H,

though t o i s s t i l l r e t a i n e d . Specific expressions f o r laws of

+(5),

found from e x p e r i m e n t a l d a t a and a s y m p t o t i c

t h e s i m i l a r i t y t h e o r y , a s w e l l a s t h e f u n c t i o n s O(5

( 1 . 6 0 ) a r e l i s t e d i n t a b l e A . l where

and

V,

C',

F",

6

a r e empirical constants

' F> o) determined from

196

Appendix

TABLE A . l

S i n c e Ch v a r i e s i n t h e i n t e r v a l H E ' 5 follows t h a t

Q ( c o ) and

5'

5

6(C0).

C0

5

5

h

5

c",

t h e n from F,

= Sh/H i t

5". T h e r e f o r e , t h e e x p r e s s i o n f o r @ ( 5 , 5 ) c o n t a i n s o n l y

Substituting

@ ( 5 ) and @(C,

o b t a i n t h e e x p r e s s i o n s for b(E % h,

C0)

stratification in the interval 0 C

5

) from t a b l e A . l

i n d i f f e r e n t r a n g e s of

5h

E",

5

i n t o ( 1 . 5 9 ) we

5,.

Thus, f o r s t a b l e

we have

where A(x) = ln[JBx + & ( x ) ] . I t i s e a s y t o d e m o n s t r a t e t h a t

Then from t a b l e A . 1 and ( 1 . 2 2 ) - ( 1 . 2 4 ) , t h a t b = 1 (b

= u,/c),

(1.56), (1.57), (1.62), a s

5,

+

0 it follows

1 = n z , corresponding t o n e u t r a l s t r a t i f i c a t i o n .

In a s i m i l a r manner, t h e s t r o n g s t a b i l i t y range 5" 5

En

5

5" we o b t a i n

(1.63)

which, f o r

5,

=

5" c o i n c i d e s w i t h ( 1 . 6 2 ) .

Under s t a b l e s t r a t i f i c a t i o n , s o l u t i o n in t h e i n t e r v a l

5 ' <= 5, 5

0 can be

represented a s

(1.64) where

197

Mesome teorological and Mesooceanic Processes

The i n t e g r a l s I ( x ) a r e c a l c u l a t e d a p p r o x i m a t e l y .

In t h e s t r o n g i n s t a b i l i t y r a n g e , we have

I.

I

(1.65)

%

where c = 1 ( - 5 ' )

3.955.

Thus, f o r d i f f e r e n t s t r a t i f i c a t i o n s we o b t a i n boundary c o n d i t i o n s f o r t h e functions b

a t t h e upper boundary of t h e near-ground l a y e r (z = h ) . T h i s approach

i s u s e f u l i n s o l v i n g t h e t u r b u l e n t energy e q u a t i o n i n a s p a c e model w i t h g r i d s of r a t h e r rough v e r t i c a l r e s o l u t i o n . Using ( 1 . 6 2 ) - ( 1 . 6 5 ) , a r y c o n d i t i o n f o r t h e problem ( 1 . 5 0 ) - ( 1 . 5 2 ) b = 0

for

2,

h = aZ(5 5 ) u h' o

*a z

=

o

we can w r i t e t h e bound-

as

t = 0;

(1.66)

for

z = h;

(1.67)

for

z

= H;

(1.68)

$=o

f o r z = k ~ ;

(1.69)

-ab= 0

f o r y = + Y .

(1.70)

aY

Note t h a t boundary c o n d i t i o n (1.68) h o l d s f o r a s t a b l e s t r a t i f i e d atmos p h e r e . For u n s t a b l e s t r a t i f i c a t i o n t h i s c o n d i t i o n t a k e s a d i f f e r e n t form. We s o l v e system ( 1 . 1 5 ) - ( 1 . 2 1 ) ,

( 1 . 2 8 ) - ( 1 . 3 2 ) by s p l i t t i n g w i t h r e s p e c t t o

p h y s i c a l p r o c e s s e s . We s h a l l r e p r e s e n t t h e s o l u t i o n on t i m e s t e p t

i

t '. t

j+l

a s a sequence of two more s i m p l e problems: 1. T r a n s p o r t of l e s s of h u m i d i t y ) :

m a t t e r a l o n g t r a j e c t o r i e s and t u r b u l e n t exchange ( r e g a r d -

198

Appendix

at

a

+ div ue' - A e ' -

+ 8x) + v ' ( S 6

u'(S6

Y

+ @

Y

) = 0. ( 1 . 7 3 )

i s s o l v e d under t h e f o l l o w i n g boundary c o n d i t i o n s :

System ( 1 . 7 1 ) - ( 1 . 7 3 )

_auu_

aei ax -

av' = 0 ,

ax

-

ay -

aY

'3

= 0,

au

e'

v* = 0 ,

x;

=

y;

w' = 0 ,

P =

h V a z = a vv

- O1

o,

=

a

h -a = z au,

x = ?

'7

(1.74)

aei -

av' = 0 ,

aui _ -

uq

ae- +

v8

-

h a ea Z- = a (e@

1

(1.75)

H;

(1.76) z = h . (1.77)

2 . Matching of mesometeorological f i e l d s :

x - a,. at

an* - _.

=

(1.78)

ax '

(1.79)

ae+ + at

SW' = 0 ;

(1.80)

_ an' - he';

(1.81)

az

aui

+

ax

ay

az

+

=

(1.82)

under t h e boundary c o n d i t i o n s

au* ax = 0,

i~ ax

=

o

for

x

f X;

(1.83)

K = , , - a-y -

0

for

y = ? Y;

(1.84)

avi

aY

w = o e l

=

o,

for

(1.78)-(1.86)

for t = t

(1.85)

z = h;

f o r z = H

W ' = O

S o l u t i o n of t h e problem ( 1 . 7 1 ) - ( 1 . 7 7 ) f o r t h e problem

=

for t = t .

J+1

,. J

is the i n i t i a l condition

Omitting i n t e r m e d i a t e t r a n s f o r m a t i o n s , w r i t e t h e f i n i t e - d i f f e r e n c e of system ( 1 . 7 1 ) - ( 1 . 7 3 )

analogue

i n t h e form

2 + (A: where

(1.86)

+

A:

+

A:)+

=

F,

(1.87)

Mesometeorological and Mesooceanic Processes A2

A3h

1

=

0

A3

0

1;

1 0

0

A

'

U. i+1/2,jkY i+l,jk .

(AIY)ijk

[

-1 (A y )

,

2

iJk

O

0

A2

31

- u. i-l/Z,jkY i-1,jk .

-

Y.

Y

Y . .

l+l,jk 1Jk Axi+l "I, i-1/2

'1, i+1/2

= V 1, . j+l/2,kyi,j+l,k

ijk

-

1

'i-1, jk Axi J'

- v.1,j-l/2,kYi,j-l,k -

2r.

J

Y . .

- -1 r J.

(Uz,j+l/z

Y . . -Y. , i,j+l,k - y ijk 1Jk l,J-l,k AY. - "2,j-1/2 AY. J+1 J

w 1~ . .,k+l/ZYij,k+l - wij ,k-l/Z'ij (A3Y)ijk

A2

26i

=

6i

0

0

0

199

1.

,k-1

=

2dk

[:.

'ij,k+l - 'ijk -1 , A2 dk 1J,k+1/2 k+l ""

v = vu ,

P = Ve,

AYj + A y .

J+1

r. = J

B

1

=

diag I 6 (xi)},

-

;,

. ij,k-1/2

6. =

, dk

+

Azk + b z

B2 = diag {6 (yi)>;

Y

Az k

Axi+l 2

=

- "ij,k-l ~.

'ijk

k+l

1,

,

,

2

+

= ( u , v, O ' , q ' ) .

To approximate the problem (1.87) with respect to time, we use the scheme of component-wise splitting

+n+i/3

- +n+(i-1)/3 A t /2

+ A;

n+i/3

+

+n+(i-1)/3 = 0,

2

(1.88)

i = 1, 2 , 3, $n+i/3

- +n+(i-1)/3 At/2

n+i/3

+ Ah7-i

@n+(i-1)/3

L-2+

= 0,

(1.89)

200

Appendix i = 4 , 5, 6,

where A t i s t i m e s t e p . The s t a b i l i t y of n u m e r i c a l scheme i s g u a r a n t e e d by t h e condition h 'A d, 9)2

9E

where

0,

a = 1I 2, 3,

(1.90)

.-

(D

i s an a r b i t r a r y v e c t o r - f u n c t i o n i n t h e domain of d e f i n i t i o n of s a t i s f y i n g t h e homogeneous boundary c o n d i t i o n s .

L e t u s now c o n s i d e r t h e problem ( 1 . 7 8 ) - ( 1 . 8 6 ) .

t i o n w e have chosen t h e

By v i r t u e of t h e approxima-

e n e r g y b a l a n c e of i t s d i s c r e t e a n a l o g u e s i s g u a r a n t e e d

and t h e y a r e c o n s i s t e n t w i t h t h e a p p r o x i m a t i o n of c o n t i n u i t y e q u a t i o n s u s e d a t the transport step

J

.+1 j+l w?1 j , k + 1 / 2 - w 1. J. , k - 1 / 2

=

(1.91)

o.

dk S i n c e S # 0 , from ( 1 . 8 0 ) w e f i n d w = - -1E

s at

(1.92)

.

Regarding t h e d i f f e r e n c e a n a l o g u e s (1.78), ( 1 , 7 9 ) a s a s y s t e m o f l i n e a r a l g e b r a i c e q u a t i o n s i n u J + l and v

j+l

, we obtain

Here, j u s t l i k e i n ( 1 . 8 7 ) , i t i s more c o n v e n i e n t t o u s e d i f f e r e n t i a l e x p r e s s i o n s i n s t e a d of f i n i t e - d i f f e r e n c e o n e s f o r s p a c e v a r i a b l e s . S u b s t i t u t i n g ( 1 . 9 2 ) , and (1.94)

(1.93)

i n t o c o n t i n u i t y e q u a t i o n ( 1 . 9 1 ) , we o b t a i n a n e q u a t i o n for t h e

f u n c t i o n F' : (L1 + L2 + L 3) 2' = where

2,

(1.95)

Mesometeorological and Mesooceanic Processes

a=The v e c t o r

F

-

depends on

1 l+(RAt)Z ’

=

vJ?”~, 1Jk

u?:”~, 1Jk

~

1 SAAt

OJt1/2 1Jk

201

’

and n o n u n i f o r m i t y of boundary

conditions. D i r e c t v e r i f i c a t i o n shows t h a t

(La?’, T ‘ )

2 0,

(Lait,

5)

= 0,

(F, e)

= 0,

01

= 1,2,3, (1.96)

where t h e s c a l a r p r o d u c t i s d e f i n e d i n t h e g r i d f u n c t i o n s p a c e by t h e e x p r e s s i o n

and

2- i s a u n i t v e c t o r . To s o l v e (1.95) w e u s e s e p a r a t i o n of v a r i a b l e s i n t h e d i s c r e t e c a s e . For

t h e a l g o r i t h m t o be e f f i c i e n t w i t h r e s p e c t t o t h e number o f c o m p u t a t i o n a l s t e p s , t h e s p e c t r a l p r o b l e m s , a r i s i n g i n t h e c o u r s e of v a r i a b l e s e p a r a t i o n , must be s o l v e d for two m a t r i c e s of l o w e r o r d e r . I f t h e o p e r a t o r of m a t r i c e s L2 and L

3

s a t i s f y t h i s c o n d i t i o n , t h e n t h e f o l l o w i n g s p e c t r a l problems a r e s o l v e d : L w = xw; 2 L rl = 3

kl.

A s t h e o p e r a t o r s L2 and L3 a r e s e l f - c o n j u g a t e

(1.98)

(1.99) i n t h e f u n c t i o n space with

t h e s c a l a r p r o d u c t ( 1 . 9 7 ) , t h e s e problems d e t e r m i n e t h e c o m p l e t e o r t h o n o r m a l s y s t e m s of e i g e n f u n c t i o n s

z‘

{i

q

1

and

1

and t h e s e q u e n c e s of nownegative e i g e n -

F),

= r e s p e c t i v e l y . Representing vectors 1 and 1 ( q = -, q i n ( 1 . 9 5 ) a s F o u r i e r s e r i e s i n {o 1 and 10 1 : 4

values { A and

{Id

(1.100)

where F .

wrl

and n ! a r e F o u r i e r c o e f f i c i e n t s , w e o b t a i n t h e s y s t e m of e q u a t i o n s 1qn

(1.102)

202

Appendix

T h i s system f o r each p a i r q and

n i s s o l v e d by t h e f i n i t e - d i f f e r e n c e

t e c h n i q u e i n x . A f t e r d e t e r m i n i n g IT', we f i n d u j + l and vJ+' (1.94),

e7J+'

from t h e e q u a t i o n of s t a t i c s , and w

j+l

-

from (1.93) and

from t h e c o n t i n u i t y

equation. Let u s g i v e an example of computation i n t h i s model. Computation i n v o l v e s f i v e s t e p s and i s so c a r r i e d t h a t s t a r t i n g from a r e l a t i v e l y s i m p l e problem, a d d i t i o n a l f a c t o r s a r e g r a d u a l l y i n t r o d u c e d i n t h e model. T h i s p r o c e d u r e r e v e a l s t h e i n f l u e n c e of d i f f e r e n t f a c t o r s on p r o c e s s e s t a k i n g p l a c e i n t h e boundary l a y e r .

In t h e c a l c u l a t i o n s we have t a k e n t h e f o l l o w i n g v a l u e s f o r t h e i n p u t p a r a m e t e r s : X = Y = 8 5 km, h = 50 m , H = 2050 m, Ax = Ay = 10 km IAz = 100 m i f

z 5 300 m ; Az = 150 m i f 300 m 9 z 9 750 m ; Az = 200 m i f 750 m < z 9 2150 m, 4 -3 h = 0.035 m/(sec g r a d ) , k = 10-4s-1, p1 = p2 = 10 s q m / s , S = 3 10 deg/m, z

= 0.01 m ,

K

=

0 . 3 5 , p = 1300 g/cu m , P = 1000 mb, C

= 0.24 c a l / ( g d e g ) , P c a l / ( c u m d e g ) , kS = 3 10 s q m/s,g =

Lo = 530 c a l / g , A = 0 . 3 , P c = 44 l o 4 s s 2 , f c = 0 . 9 , a = 0 . 5 6 , be = 0.08, Is = 0 ( i n v e r s i o n 2 , a

= 9.8 m/s

b

= 0 9 m $ = 42.5O,

$J

= 0.2,

= 1l0,

f(t) =

300

+

cos

(s (

t

-

22)

)

.

The t u r b u l e n c e c o e f f i c i e n t s i n t h e range h 5 z 5 H were t a k e n a s l i n e a r f u n c t i o n s d e c r e a s i n g w i t h h e i g h t from ( v ~ t )o ~z e r o . The r e l a t i v e humidity ( i n p e r c e n t ) on t h e s o i l s u r f a c e was g i v e n i n t h e form rl

a

-

3 c o s 2a

-

4 s i n a, a = a ( t

-

-

11 s i n

-

* ( - ~ z , ) ,Q, = Q = 0 , = Y In c a l c u l a t i o n s , i t was assumed t h a t

=

8 = I3 + Sz

-

0 = @ = 0 , Q = Q exp * x Y BQ where B = 300 K, Q = 12 g/km, B = 6.10-5 m.

s p e c i f i c humidity were t a k e n a s follows:

awaz

= 59-22 c o s a

4 ) / 1 2 ; background f i e l d s of t e m p e r a t u r e and

Bo, Q

1 :

= 0, W = 0 , @ = 0.

Let us c o n s i d e r t h e mesometeorological p r o c e s s d e v e l o p i n g i n t h e absence of background wind over a c i r c u l a r i s l a n d of 30 kin r a d i u s , r e g a r d l e s s of humidity p r o c e s s e s i n atmosphere and i n s o i l . From s u n r i s e , t h e i s l a n d s t a r t s t o warm u p , and a t e m p e r a t u r e g r a d i e n t a p p e a r s between l a n d and s e a promoting b r e e z e c i r c u l a t i o n which r e a c h e s i t s peak a t 14 h r s (15 h r s 40 min)*. Maximum wind speed modulus, 9 m / s

(7 m / s )

is reached a t an a l t i t u d e of 100 m a t a d i s t a n c e of 20 km

away from t h e c e n t e r of t h e i s l a n d . Ascending c u r r e n t s , which a r e of an o r d e r of magnitude h i g h e r t h a n t h e descending o n e s , reach t h e i r maximum 29 cm/s

(20 cm/s)

over t h e c e n t e r of t h e i s l a n d a t an a l t i t u d e of 600 m. The power of t h e s e a b r e e z e i s 1400 m (1200 m ) ; speed of 2 . 6 m / s

(2 m / s )

a t h i g h e r a l t i t u d e s a n t i b r e e z e o c c u r s w i t h a maximum

a t an a l t i t u d e of 1700 m (1500 m ) .

I n t h e evening t h e

s e a b r e e z e s u b s i d e s and l a n d b r e e z e d e v e l o p s . I t i s weaker t h a n t h e s e a b r e e z e . The power of t h e

l a n d b r e e z e is 600 m , a t a h i g h e r p o s i t i o n a s t r o n g a n t i b r e e z e

t a k e s p l a c e with a speed of 0 . 5 m / s . *Brackets c o n t a i n t h e v a l u e s f o r t h e second v e r s i o n .

Mesometeorological and Mesooceanic Processes

203

In c o n t r a s t t o t h e f i r s t , t h e second v e r s i o n t a k e s account of humidity which markedly changes b r e e z e c i r c u l a t i o n development o v e r t h e i s l a n d . Changes o c c u r due t o d i u r n a l ground t e m p e r a t u r e v a r i a t i o n d u r i n g t h e s e mesoprocesses. F i g u r e A.2 i l l u s t r a t e s t h e c a l c u l a t e d time v a r i a t i o n of components of h e a t b a l a n c e and ground t e m p e r a t u r e w i t h ( f i g u r e A.2a) and w i t h o u t ( f i g u r e A.2b) humidity a t t h e c e n t e r of t h e

i s l a n d . During d a y t i m e , when a major p a r t of h e a t i s spend on e v a p o r a t i o n ,

wet s o i l warms up s l o w e r t h a n d r y s o i l . A s a consequence, compared t o t h e f i r s t v e r s i o n , t u r b u l e n t h e a t f l o w i n t o atmosphere d e c r e a s e s and a l o c a l c i r c u l a t i o n develops l e s s i n t e n s i v e l y . A t n i g h t , t h e r e a r e no t u r b u l e n t flows of h e a t and m o i s t u r e , o n l y two com-

ponents G

and

F

remain i n t h e h e a t b a l a n c e . S i n c e a 3 and b3 i n t h e Brent

formula were so s e l e c t e d t h a t t h e r a d i a t i o n b a l a n c e s i n v e r s i o n s 1 and 2 were approximately t h e same, i t i s c l e a r why a t n i g h t t h e p r o c e s s e s w i t h and without humidity a r e p r a c t i c a l l y i d e n t i c a l .

t

40

30 20

c

70

FIGURE A . 2

204

Appendix

FIGURE A . 3 F i g u r e A . 3 shows r e s u l t s c a l c u l a t e d f o r 16 h r s ( l e f t h a l f ) and midnight ( r i g h t h a l f ) a t z = 100 m and y = 90 km. Wind f i e l d s a r e r e p r e s e n t e d b y arrows (arrows ending w i t h s q u a r e s a r e low speed w i n d ) . The o u t l i n e of t h e i s l a n d i s drawn i n t h i c k l i n e ; i s o l i n e s of v e r t i c a l speeds ( c m / s ) of t e m p e r a t u r e ( d e g r e e s , C e n t i g r a d e )

-

-

i n bold l i n e ; i s o l i n e s

i n broken l i n e .

A s compared t o v e r s i o n 2, i n t h e t h i r d v e r s i o n t h e outward

wind depends

on t i m e ; i t s speed i n t h e range 3 h r s 5 t 5 15 h r s i s g i v e n b y e x p r e s s i o n

while a t o t h e r moments i t

is considered constant { U(t) = 3 m/s,

V(t) = 0 ) .

During daytime t h e l o c a l c i r c u l a t i o n and outward flow a r e u n i d i r e c t i o n a l o v e r t h e w e s t e r n p a r t of t h e i s l a n d u p t o an a l t i t u d e of 1200 m . T h i s r e s u l t s i n s t r o n g e r wind. But o v e r t h e e a s t e r n p a r t t h e wind g e t s weaker. Near t h e e a s t e r n s h o r e , a zone of convergence is formed by 16 h r s i n which v e r t i c a l c u r r e n t s and t e m p e r a t u r e

205

Mesometeorological and Mesooceanic Processes

t a k e on e x t r e m e v a l u e s . A t n i g h t , l o c a l c i r c u l a t i o n i s o p p o s i t e t o t h a t i n day-

t i m e . For t h i s r e a s o n , t h e f l o w i n l o w e r l a y e r s , w h i l e a p p r o a c h i n g t h e i s l a n d , s l o w s down as i f f l o w i n g round two s i d e s of t h e i s l a n d . F i g u r e A.4 shows r e s u l t s c a l c u l a t e d as i n f i g u r e A . 3 .

I50fl

--


-60 -30

0

30

X,HM

FIGURE A.4 Now l e t u s c o n s i d e r t h e i n f l u e n c e of e x t e r n a l wind on d i u r n a l v a r i a t i o n of components o f h e a t b a l a n c e ( f i g u r e A.2c) and ground t e m p e r a t u r e ( f i g u r e A.2d) i n t h e c e n t e r of t h e i s l a n d . I n d a y t i m e , when wind e n h a n c e s t u r b u l e n t m i x i n g , t h e maximum t e m p e r a t u r e on t h e e a r t h ' s s u r f a c e i s 4OC l o w e r t h a n i n t h e second v e r s i o n . A t n i g h t , a l s o due t o e x t e r n a l wind, t u r b u l e n t f l o w s of h e a t and m o i s t u r e d o o c c u r b u t t a k e on s m a l l v a l u e s , b u t d u r i n g t h e whole n i g h t t h e y d o n o t change d i r e c t i o n , t h e r e f o r e t h e minimum v

i s 3°C h i g h e r t h a n w i t h o u t wind.

The f o u r t h v e r s i o n t a k e s a c c o u n t of t h e i n f l u e n c e of l a n d r e l i e f on t h e development of l o c a l c i r c u l a t i o n . On t h e

i s l a n d , a s compared t o t h e f i r s t v e r -

s i o n , t h e r e is a n e x i s y m m e t r i c 500 m h i g h mountain whose s u r f a c e is d e s z r i b e d by t h e e x p r e s s i o n 6 ( r ) = ( 2 1 3 + 300 c o s ( n r / 4 0 ) ) where

r

i s t h e d i s t a n c e from

t h e c e n t e r of t h e i s l a n d . The t i m e v a r i a t i o n p r o c e s s i n t e n s i t y i s a p p r o x i m a t e l y

206

Appendix

t h e same a s i n t h e second v e r s i o n , t h e d i f f e r e n c e s b e i n g c o m p l e t e l y r e l a t e d t o t h e r e l i e f . In daytime, c i r c u l a t i o n i n t e n s i t y h a r d l y v a r i e s . Wind by a mere 1 m / s .

speed i n c r e a s e s

Ascending c u r r e n t s , whose range g e t s narrower w i t h a l t i t u d e ,

reach 23 cm/s o v e r t h e c e n t e r of t h e mountain. A t n i g h t , t h e mountain i n f l u e n c e

i s much g r e a t e r . The l a n d b r e e z e i s almost t w i c e a s reaches 4 m / s .

s t r o n g and a t midnight i t

8 cm/s.

The descending c u r r e n t s reach

F i g u r e A.5 shows t h e

i s o l i n e s of v e r t i c a l speeds and t e m p e r a t u r e a t 16 h r s ( f i g u r e A.5a) and a t 0 h r s ( f i g u r e A.5b) a t y = 90 km. N o t a t i o n i s t h e same a s i n f i g u r e A.3.

i

_ - - - - -74---

----__ ?I------2o---

-60 -30

U

3U

60

X,XM

-60--30

(a)

U

3U

60

X,XM

(b) FIGURE A.5

The f i f t h v e r s i o n r e f e r s t o t h e s o l u t i o n of t h e (1.14)-(1.18)

problem ( 1 . 1 ) - ( 1 . 7 ) ,

j o i n t l y with t h e t u r b u l e n c e energy E q s . ( 1 . 3 6 ) - ( 1 . 3 8 ) ,

(1.51)-(1.55).

A l l i n p u t parameters a r e t h e same a s i n v e r s i o n 3 , o n l y U = -5 m / s ,

V = 0 , V = 0,

AX = Ay = 4 km. F i g u r e A.6a i l l u s t r a t e s t h e i s o l i n e s of t u r b u l e n c e energy

b

and v e r t i c a l t u r b u l e n c e exchange c o e f f i c i e n t v i n ( x , z) p l a n e a t y = 32 km a t t i m e , t = 15 h r s . F i g u r e A.6b shows t h e v e r t i c a l p r o f i l e s of t h e f u n c t i o n s b , V

and

E

a t d i f f e r e n t p o i n t s f o r t = 1 5 h o u r s and f i g u r e A.6c f o r t = 12 h o u r s .

2 . Quasihomogeneous ocean l a y e r

World Ocean p o l l u t i o n became an a c u t e problem i n t h e e a r l y 1 9 5 0 ' s due t o r a d i o a c t i v e f a l l o u t from n u c l e a r t e s t s i n atmosphere. A t p r e s e n t , o i l p o l l u t i o n of oceans i s i n c r e a s i n g and t h e f e a s i b i l i t y of burying decay r a d i o a c t i v e p r o d u c t s a t t h e ocean bed is being examined. A l l t h e s e f a c t o r s s t i m u l a t e t h e development of mathematical t e c h n i q u e s for f o r e c a s t i n g i m p u r i t y p r o p a g a t i o n i n t o t h e ocean. The e v o l u t i o n of c o n s e r v a t i v e i m p u r i t y c o n c e n t r a t i o n f i e l d i n

t h e Eulerian

r e p r e s e n t a t i o n i s d e s c r i b e d by averaged e q u a t i o n of molecular d i f f u s i o n :

207

Mesometeorological and Mesooceanic Processes

0

72

24

36

48

60 X,MM 0

24

36

x )6(t

-

72

48

60 s , ~ n

4

FIGURE A . 6 * + u

at

where

a

" ax

- = MA$

a -u''' -ax

ci

+

Q6(x -

-

-0

to),

(2.1)

4 is t h e impurity c o n c e n t r a t i o n , M i s t h e molecular d i f f u s i o n c o e f f i c i e n t

of i m p u r i t y i n t o t h e o c e a n , Q i s t h e power of i n s t a n t a n e o u s p o i n t p o l l u t i o n source. Impurity t u r b u l e n t f l o w v e c t o r u ' 9 '

is g e n e r a l l y determined by a g r a d i e n t

t y p e ( B u s s i n e s q ) approximation

The t u r b u l e n t d i f f u s i o n t e n s o r k

UB

does n o t depend on p a s s i v e i m p u r i t y concen-

t r a t i o n and is t h e major c h a r a c t e r i s t i c of t h e t u r b u l e n t f i e l d .

208

Append k

I n g e n e r a l , i n s o l v i n g t h e problem ( 2 . 1 ) ,

( 2 . 2 ) t h e v e l o c i t y f i e l d and

t u r b u l e n t d i f f u s i o n c o e f f i c i e n t s a r e assumed t o b e known. To s t u d y t h e g l o b a l i m p u r i t y p r o p a g a t i o n i n t h e World Ocean, i t i s n e c e s s a r y t o h a v e a l a r g e - s c a l e v e l o c i t y f i e l d and t e n s o r f i e l d of t u r b u l e n t d i f f u s i o n c o e f f i c i e n t s o v e r t h e e n t i r e o c e a n . Models of t h i s k i n d , b a s e d on n u m e r i c a l i n t e g r a t i o n o f g e o p h y s i c a l e q u a t i o n s , a r e b e i n g d e v e l o p e d b o t h i n t h e USSR and a b r o a d . For t h e i m p u r i t y d e p o s i t e d on t h e ocean s u r f a c e , o f s p e c i a l s i g n i f i c a n c e i s t h e f o r e c a s t o f i t s p r o p a g a t i o n i n t h e u p p e r boundary l a y e r of t h e o c e a n . I n t h i s c a s e , s t r a t i f i c a t i o n being s t a b l e o r

n e u t r a l d e t e r m i n a t i o n of h o r i z o n t a l v e l o c i t y f i e l d and

v e r t i c a l t u r b u l e n t exchange i s c o n s i d e r a b l y s i m p l i f i e d . W e now t a k e up t h i s problem. C o n s i d e r t h e e q u a t i o n s of an Ekman o c e a n i c boundary l a y e r w i t h r e g a r d t o v e r t i c a l t u r b u l e n t exchange: (2.3)

aatv + -

a k &u = az

-aT = - -

at

a& at

- = 1,38

k

a , v aZ

(2.4)

aQ

aZ

(2.5)

I

[au(z+] (83'1 + & k

- 1,4

2

' b

E

'

- 1,4 b

g d ,

k = 0,OS b2/E,

(2.7)

(2.8)

under t h e boundary and i n i t i a l c o n d i t i o n s

z

=

0: k

a = 2

az

-u2 + I

k

az = 0, Q

= WIT' = Q

0

2 0;

(2.9)

(2.10) z = H ( H > h ) :

t = 0: u

=

u = v = O ,

is t u r b u l e n t energy;

E

(2.11)

u o , v = vo, T = To, b = b o , E = s o , ( 2 . 1 2 )

Here, a l l q u a n t i t i e s i n ( 2 . 1 ) a r e a v e r a g e d , k b

b = E = O ;

is t u r b u l e n t v i s c o s i t y c o e f f i c i e n t ,

is t u r b u l e n t d i s s i p a t i o n v e l o c i t y ;

c o e f f i c i e n t , g i s a c c e l e r a t i o n due t o g r a v i t y , h

is the Coriolis

i s t h e d e p t h of t h e u p p e r q u a s i -

homogeneous l a y e r (UQL) of t h e o c e a n , a i s t h e r m a l e x p a n s i o n c o e f f i c i e n t of t h e medium. From t h e v i e w p o i n t of n u m e r i c a l s o l u t i o n , problem ( 2 . 3 ) - ( 2 . 1 2 ) simple. T h i s formulation w a s used

is q u i t e

i n n u m e r i c a l i n v e s t i g a t i o n of s t o r m d e v e l o p -

209

Mesometeorological and Mesooceanic Processes

ment. The s o l u t i o n r e v e a l e d a new phenomenon: s t o r m b r i n g s about a l o c a l boundary l a y e r a t t h e r m a l shock. A n a l y s i s of n u m e r i c a l s o l u t i o n s d e m o n s t r a t e t h a t a n a l y t i c a l s o l u t i o n s of (2.3)-(2.12),

which a r e more i l l u s t r a t i v e t h a n numerical o n e s ,

a r e extremely i m p o r t a n t f o r v e r i f i c a t i o n

of d i f f e r e n c e schemes. Let u s c o n s i d e r

some o f them. Assume t h a t a s t a t i o n a r y t u r b u l e n t s u r f a c e l a y e r i s formed i n t h e ocean under d r i f t motion w i t h a c o n s t a n t c o e f f i c i e n t k . The a n a l y t i c a l s o l u t i o n of t h e e q u a t i o n of motion ( 2 . 3 ) , by V.1.Ekman i n 1905 (H

(2.4) f o r a constant coefficient

k

was f i r s t o b t a i n e d

+ m):

u, v = f(2kk)

-1

(cos 42 f s i n 9z),

u:e-B2

(2.13)

1

where B = ( L / ( 2 k ) ” .

In (2.6)-(2.7) d e f i n e t h e f u n c t i o n

G

a s a s o u r c e of t u r -

bulence displacement (2.14) According t o dimension t h e o r y , t h e d e p t h of t h e upper quasihomogeneous l a y e r can be determined a s h = yX, where

X

= u,/f

(2.15)

(A i s Ekman s c a l e of l e n g t h ) , y i s a f u n c t i o n t h a t d e f i n e s dimen-

s i o n l e s s p a r a m e t e r s i n o u r problem. Equating t h e d e p t h , a t which t h e t u r b u l e n c e displacement s o u r c e ( 2 . 1 4 ) d e c r e a s e s by a f a c t o r of e n , i . e .

(2.16)

t o t h e d e p t h of t h e upper quasihomogeneous l a y e r (2.15), we o b t a i n :

(2.17) K i n e t i c energy flow from atmosphere i n t o ocean i s

(2.18)

Assume t h a t t h e d i f f u s i o n term

For a known

k, the quantities

b

a and

k E

i n ( 2 . 6 ) i s s m a l l compared t o o t h e r s . a r e r e a d i l y found i n t h i s c a s e from

( 2 . 6 ) , (2.8). Equation (2.7) t u r n s t o be s u p e r f l u o u s :

210

Appendix

The d i m e n s i o n l e s s f u n c t i o n y i s found from t h e c o n d i t i o n t h a t t h e r e i s no turbulence a t z 2 h , i . e .

E

= 0 a t z = h:

y

=

(n/fi) e

-n/2 M

-*

(2.20)

I

where p = X/L i s a d i m e n s i o n l e s s s t r a t i f i c a t i o n p a r a m e t e r , L = u:/(gaQ)

is

Monin-Obukhov l e n g t h s c a l e . S u b s t i t u t i n g y from (2.20) i n t o (2.15) w e o b t a i n

(2.21) -

.

h a s h = yL; i n t h i s c a s e y = J Both d e f i n i t i o n s fi e* a r e e q u i v a l e n t i n c a s e of s t a b l e s t r a t i f i c a t i o n ( f o r Q > 0 ) . For n e u t r a l

We can a l s o d e f i n e of

h

s t r a t i f i c a t i o n (Q = 0 ) t h e r e a r e no parameter p and s c a l e L , t h e r e f o r e t h e most g e n e r a l d e f i n i t i o n of

h

is (2.15).

S t a t i o n a r y s o l u t i o n s can be used f o r e v a l u a t i n g n o n s t a t i o n a r y p r o c e s s e s . R e l a x a t i o n time A t of dynamic and t h e r m a l p e r t u r b a t i o n s i n t h e boundary l a y e r and c h a r a c t e r i s t i c t i m e T i n which t u r b u l e n c e a d a p t s t o hydrodynamic environment are : At =

k

=

2lI (-),

(2.22)

4 9 .

(2.23)

These s o l u t i o n s d e r i v e d a r e determined a s f u n c t i o n s of two d i m e n s i o n l e s s parameters:

I.I

and z / h . The a n a l y s i s i s based on t h e c l a s s i c a l Ekman s o l u t i o n of

e q u a t i o n s of motion, t h e o n l y d i f f e r e n c e b e i n g t h a t t h e v e r t i c a l t u r b u l e n t exchange c o e f f i c i e n t i s c o n s i s t e n t w i t h t h e e x t e r n a l p a r a m e t e r s of t h e problem. This a n a l y s i s i s v a l i d , provided t h e s u r f a c e t u r b u l e n t l a y e r i s t h i c k e r t h a n t h e quasilaminar sublayer : h > 10 k /u*,

y

> 10 k v u :

= 10 ko/(u*X),

i f no account is t a k e n of t h e wave p r o c e s s e s on t h e ocean s u r f a c e . Here, c o e f f i cient k

t a k e t h e minimal v a l u e found from t h e c o n d i t i o n Ret = k/V 2 10, under

which t u r b u l e n c e e q u a t i o n s ( 2 . 6 ) - ( 2 . 8 ) h o l d , hence k

= 1Ov. The c o n d i t i o n

h > 10 ko/u+ i s chosen by analogy w i t h t h e l a m i n a r s u b l a y e r t h i c k n e s s e q u a l t o 10 V/u,

from Laufer experiment

v i s c o s i t y . On t h e

(1951), v i s a k i n e m a t i c f a c t o r of m o l e c u l a r

o t h e r hand, f o r y > 0 . 4

t h e medium can be c o n s i d e r e d n e u t r a l l y

s t r a t i f i e d and we can t a k e y = 0 . 4 . Let u s f o r m u l a t e t h e a n a l y s i s a p p l i c a b i l i t y condition :

102 V/(U,A)

< y < 0,4.

(2.24)

21I

Mesome teorological and Mesooceanic Processes The q u a l i t a t i v e s o l u t i o n i s based on t h e assumption t h a t Numerical s o l u t i o n s of ( 2 . 3 ) - ( 2 . 1 2 ) Suppose t h a t

k

k

indicate that

k

is c o n s t a n t .

diminishes with depth.

i s d e f i n e d by a power f u n c t i o n :

x = (1 - z/H).

k = klx2, In t h i s c a s e , t h e s o l u t i o n s of

(2.25)

(2.4) under t h e boundary c o n d i t i o n s ( 2 . 9 ) ,

(2.3),

(2.12) can be w r i t t e n a s

m

uH : U

,

"

=

?

-

-

[cos(q

L

kl ( m 2 + n 2 ) u 2

n I n x ) , s i n ( q - n In x)],

Flow of k i n e t i c e n e r g y from atmosphere i n t o t h e ocean i s

m H 4 u u 2 =-0 m2 + n2 k u*' 1

(2.27)

*

Define

E.

b as

4

u*

= -x kl

E

2m

-

guQ,

(2.28)

t

b = - (5u

(2.29)

To s o l v e t h e problem,we have t o f i n d t h e d e p t h of t h e upper quasihomogeneous layer h, k

1

and t h e d e p t h H . Beside t h e c o n d i t i o n s h = yX,

Go/Gh

= e',

E

(z=h =

(2.30)

we impose t h e c o n s t r a i n t on k a t t h e boundary of t u r b u l e n c e domain:

k = k

1

(1

:I2

-

= kO.

(2.31)

S o l u t i o n s of ( 2 . 3 0 ) , ( 2 . 3 1 ) a r e k

1

= e

-71

-T/2

H

=

v5

e ( X U

f

u,L,

A(Ret),

(2.32)

(2.33)

(2.34)

(2.35)

212

Appendix

where Re

= k /k

t

1

0

= u,Le-'/ko,

The c h a r a c t e r i s t i c t i m e of hydrodynamic and t u r b u l e n t r e l a x a t i o n i s of t h e f orm

At

H3/

Q

1

kdz =

4

A2

(Ret)/&,

(2.36)

-

(2.37)

0 T

Q

b/E = ( 5 / 6 ) e-'x(xZm

e-T)-'/(pg).

In a d d i t i o n t o ( 2 . 2 4 ) t h e c o n d i t i o n for a p p l i c a b i l i t y of o u r a n a l y s i s i s k 1 > i . e . Re > 1; h e n c e , i t i s n a t u r a l t h a t kl > min k. Moreover, t h e i n e q u a l i t i e s H > 0 , v > 0 , 0 < h < H , m > 0 , n > 0 always h o l d .

S u b s t i t u t i n g (2.32)-(2.35)

w e o b t a i n t h e s o l u t i o n which

i n t o (2.26)-(2.29),

L , R e t and p. For y > 0 . 4 t h e

depends on t h r e e d i m e n s i o n l e s s p a r a m e t e r s : z / f i

medium i s n e u t r a l l y s t r a t i f i e d , p = 0 . Then, kl, H a r e g i v e n by e q u a t i o n s kl = k o ( l

-2

-

h/H)

,

h = 0,4X,

{ [ 2 ( 1 + 2 / l n Ret)*

H = (k1/(4&))'

-

(2.38)

1 1 2 - 1}1'4

(2.39)

and t h e s o l u t i o n depends on two d i m e n s i o n l e s s p a r a m e t e r s :

L e t us a n a l y s e s o l u t i o n s of v e r s i o n s A : k = c o n s t , a n d B: k = k 1x 2 . For

t h e s a k e of c o n v e n i e n c e , w r i t e them a s v e r s i o n u , v = +(eTi/2)'

G = e'pu3/(~e

-2

version B: k = e

p

-1

u , v = ?(eTp/4)'

-

p'u*e-Bz(cosfiz

02 ) ,

E

sin ~z),

-282

= e'pu:/[A(e

~ b = ( g / ~ ) u : ( e - ~-~ e-')', -T

A : k = e-'p-lu*X;

- e -71 ) I ,

y = n/(2e*p)-';

u*Xx2; A(m2

+

n2)-'

u,xm ' [ c o s ( q

G = eTlpuZ/(xx2m ) ,

E

-

nlnx), sin(q

-

= e T p u 3 / [ ~ ( x Z m- e-')],

b = (5/fi) u:(xZm

y = A(Re )[1 t

-

-

e-')'

(Ret)-'I/(2eTp)'.

x,

nlnx)],

213

Mesometeorological and Mesooceanic Processes The i n f l u e n c e of e x t e r n a l p a r a m e t e r s i n b o t h v e r s i o n s i s i d e n t i c a l ; t h e d i f f e r e n c e l i e s i n t h e p r e s e n c e of an a d d i t i o n a l p a r a m e t e r R e t

i n t h e second

v e r s i o n and i n f u n c t i o n a l dependence on d e p t h : e x p o n e n t i a l and power o n e s . A s f l o w Qo i n c r e a s e s (L d i m i n i s h e s , p , A , m i n c r e a s e ) t h e medium s t a b i l i t y grows, h e n c e l a r g e scale t u r b u l e n t f l u c t u a t i o n s a r e s u p p r e s s e d . A t t h e same t i m e , t h e v a l u e s of c o e f f i c i e n t

k

a n d t h i c k n e s s d i m i n i s h , and t u r b u l e n c e d i s s i p a t i o n r a t e

and s u r f a c e f l o w v e l o c i t y i n c r e a s e . S u r f a c e t u r b u l e n c e e n e r g y d o e s n o t depend on

Q

.

Growth of Q

l e a d s t o t h e growth of

B and m; a s a r e s u l t , t h e p r o f i l e s of b o t h

s o l u t i o n s become s t e e p e r . A n a l y t i c a l s o l u t i o n s c o r r e c t l y r e f l e c t p a r t i c u l a r i t i e s of s o l u t i o n of t h e s y s t e m (2.3)-(2.12) d e r i v e d n u m e r i c a l l y , and i n v e r s i o n diffusion coefficient k

B

t h e i n f l u e n c e of

i s t h e same a s i n n u m e r i c a l s o l u t i o n : t h e growth o f k

l e a d s t o growth of UQL t h i c k n e s s : dh/dko i l l u s t r a t e s t h e b a l a n c e of

> 0 , dY/dko < 0 , dH/dko > 0 . F i g u r e A.7

terms o f t h e e n e r g y t u r b u l e n c e e q u a t i o n . D i f f u s i o n

term becomes s i g n i f i c a n t o n l y i n t h e v i c i n i t y of t h e lower UQL boundary. A n a l y t i c a l s o l u t i o n s p r o v i d e a s i m i l a r e s t i m a t e . For example, i n c a s e

A

the relation

i s small a t z = 0 and i n c r e a s e s w i t h d e p t h .

II

FIGURE A.7 L e t u s c o n s i d e r n o n s t a t i o n a r y development of s u r f a c e t u r b u l e n t l a y e r . Assume t h a t t h e s o l u t i o n s a r e u n i v e r s a l f u n c t i o n s i n n o n s t a t i o n a r y c a s e , t o o . Multiplyi n g (2.3), ( 2 . 4 ) by t o h ( t ) , we o b t a i n

u

and v , r e s p e c t i v e l y , a d d i n g them and i n t e g r a t i n g from

0

214

Appendix

uz + vz

at

d

2

uz

-

au + az

dh

P vz / z = h

= ko [ u -

avl

v -

a Z j z=h

+

0

Substituting

u, v, k

from s o l u t i o n

i n t o ( 2 . 4 0 ) and i n t e g r a t i n g , we

B

o b t a i n t h e f o r e c a s t e q u a t i o n f o r UQL d e p t h :

(2.41)

In d e r i v i n g ( 2 . 4 1 ) , H ( t ) h a s been e l i m i n a t e d through t h e u s e of r e l a t i o n (2.35).

The q u a n t i t i e s u , ( t ) ,

a r e e x t e r n a l p a r a m e t e r s in ( 2 . 4 1 ) .

Q(h, t ) , f , k

The q u a n t i t y Q ( h , t ) i s found from h e a t E q . ( 2 . 5 ) . I f Q i n ( 2 . 5 ) i s d e f i n e d a s Q = -k x2aT/az w i t h a r b i t r a r y boundary c o n d i 1 t i o n s for x = 0 : T = T ( t ) , Q = Q ( t ) , t h e n Q ( h , t ) i s found by i n t e g r a t i n g (2.5) with respect t o

h . In t h i s c a s e , w e a r r i v e a t t h e assumptions u n d e r l y i n g t h e

t r a d i t i o n a l " i n t e g r a l " models of UQL.

I

h (24)

I

0

n/f

1

t p n

FIGURE A . 8 F i g u r e A . 8 shows t h e n u m e r i c a l s o l u t i o n s of ( 2 . 4 1 ) f o r t h e f o l l o w i n g v a l u e s

of e x t e r n a l p a r a m e t e r s : Q =

K cm/s,

u*

=[

.t =

s-',

0.5 cm/s,

t = 0,

1 cm/s,

t > 0;

ko = 1 sq.crn/s:

References

u* =

{

21 5

1 cm/s,

t = 0,

0.5 cm/s,

t > 0.

Stationary solutions (2.35) are employed as initial values of Solution of (2.41) as t

+ m

h

at t = 0.

asymptotically approaches the stationary solution

(2.35), and in the inertia period 2n&-'

the depth

h

tends to stationary solution.

I n concluding the dimensional analysis, let us note that instead of dynamic

velocity u+, as the external parameter, one can also take c 0 , the mean turbulent dissipation rate per mass unit which is the energy flux from atmosphere into the ocean: H (2.42)

From c 0 , ,.P

Q

one can proceed to the parameters u*, E , Q

by substituting their

values obtained from the above solutions into the energy relation (2.42). In conclusion we may note that this idealization yields analytical solution which clearly demonstrates the effect of external parameters on the development of surface turbulent layer and is consistent with solutions of the system of equations (2.3)-(2.12).

REFERENCES 1) M.E. Berlyand, Modern problems of atmospheric diffusion and atmospheric pollution, Leningrad, Gidrometeoizdat, (1975) (Russian ed.). 2) N.L. Byzova, Impurity diffusion into atmospheric boundary layer, Leningrad,

Gidrometeoizdat, (1974) (in Russian). 3) V . S . Vladimirov, Equations of mathematical physics, Moscow, Nauka, (1981)

(in Russian). 4) N.L. Karol and G.S. Malakhov, Problems of nuclear meteorology, Moscow,

Gosatomizdat, (1962) (in Russian). 5) A.N.Kolmogorov, Equations of turbulent motion of incompressible liquid, Izv. AN SSSR, Fizika, (1942), 6, No.1-2, pp.56-58. 6) K.Ya. Kondratyev, Radiant energy of the Sun, Leningrad, Gidrometeoizdat, (1956) (in Russian). 7 ) Yu.A. Kuznetsov, On the convergence of splitting method for a system with

positively semidefinite matrices, in: Differential and Integral Equations,

216

References N o v o s i b i r s k , Computer C e n t e r of t h e S i b e r i a n D i v i s i o n o f t h e USSR Academy o f S c i e n c e s , ( 1 9 7 7 ) , pp.112-119 ( i n R u s s i a n ) .

8) O . A . Ladyzhenskaya, Mathematical problems i n t h e dynamics of i n c o m p r e s s i b l e l i q u i d , Moscow, Nauka, (1970) ( i n R u s s i a n ) .

9 ) G . I . Marchuk, Methods of n u c l e a r r e a c t o r d e s i g n , Moscow, A t o m i z d a t , (1961) ( i n Russian). 10) G.I.Marchuk, On c e r t a i n i n v e r s e p r o b l e m s , Doklady AN SSSR, ( 1 9 6 4 ) , 1 5 6 , No.3, pp.503-506.

11) G . I . Marchuk, E l u a t i o n f o r t h e s i g n i f i c a n c e o f i n f o r m a t i o n from m e t e o r o l o g i c a l s a t e l l i t e s and f o r m u l a t i o n of i n v e r s e p r o b l e m s , Kosmicheskiye i s s l e d o v a n i y a , ( 1 9 6 4 ) , 2 , No.3, pp.462-477. 12) G.I.Marchuk, On t h e e q u a t i o n s of b a r o c l i n i c ocean dynamics, Doklady AN SSSR, ( 1 9 6 7 ) , 1 7 3 , No.6. 13) G . I . Marchuk, Numerical methods i n w e a t h e r f o r e c a s t , L e n i n g r a d , Gidrometeoi z d a t , (1967) ( i n R u s s i a n ) . 14) G.I. Marchuk, Numerical s o l u t i o n of atmosphere and ocean dynamics problem, L e n i n g r a d , G i d r o m e t e o i z d a t , (1974) ( i n R u s s i a n ) . 15) G . I . Marchuk, Environment a n d some o p t i m i z a t i o n p r o b l e m s , N o v o s i b i r s k , ( 1 9 7 5 ) , P r e p r i n t o f t h e Computer C e n t e r of S i b e r i a n D i v i s i o n of t h e USSR Academy of Sciences. 1 6 ) G . I . Marchuk, Environment and o p t i m i z a t i o n problems of e n t e r p r i s e l o c a t i o n , Doklady AN SSSR, ( 1 9 7 6 ) , 2 2 7 , No.5, pp.1056-1059. 17) G.I.Marchuk, Some problems of e n v i r o n m e n t a l p r o t e c t i o n , i n : Comprehensive A n a l y s i s and i t s A p p l i c a t i o n , Moscow, Nauka, ( 1 9 7 8 ) , pp.375-381 ( i n R u s s i a n ) . 18) G . I . Marchuk, Methods o f c o m p u t a t i o n a l m a t h e m a t i c s , Moscow, Nauka, (1980) ( i n Russian). 19) (3.1. Marchuk, A p p l i c a t i o n of a d j o i n t e q u a t i o n s t o s o l u t i o n o f m a t h e m a t i c a l p h y s i c s p r o b l e m s , Uspekhi mekhaniki, ( 1 9 8 1 ) , No.1. 20) G . I . Marchuk, V.P. Dymnikov e t a l . , Hydrodynamic model of common c i r c u l a t i o n of atmosphere and o c e a n , I n f o r m a t i o n B u l l . , N o v o s i b i r s k , Computer C e n t e r of S i b e r i a n D i v i s i o n of t h e USSR Academy of S c i e n c e s , ( 1 9 7 5 ) . 21) G.I. Marchuk, V . B . Zalesny and V . I . Kuzin, On methods of f i n i t e d i f f e r e n c e s and f i n i t e e l e m e n t s i n problem of g l o b a l wind c i r c u l a t i o n of o c e a n , I z v . AN SSSR, F i z i k a a t m o s f e r y i o k e a n a , ( 1 9 7 5 ) , 11, No.12. 22) G . I . Marchuk, V.P. K o c h e r g i n , V . I . Klimok and V . A . Sukhorukov, A n a l y t i c a l s o l u t i o n s f o r t h e Ekman t u r b u l e n t l a y e r of t h e o c e a n , Doklady AN SSSR, ( 1 9 7 9 ) , 247, No.1. 23) G . I . Marchuk, V.P. K o c h e r g i n , A.S. S a r k i s y a n e t a l . , Mathematical models of c i r c u l a t i o n i n t h e o c e a n , N o v o s i b i r s k , Nauka, (1980) ( i n R u s s i a n ) . 24) G . I . Marchuk and V . V . O r l o v , To t h e o r y of c o n j u g a t e f u n c t i o n s , i n : Neutron physics,Moscow, G o s a t o m i z d a t , ( 1 9 6 1 ) , pp.30-45 ( i n R u s s i a n ) .

217

References

25) G . I . Marchuk, V . V . Penenko, A . E . Aloyan and G . L . L a z r i y e v , Numerical s i m u l a t i o n of c i t y m i c r o c l i m a t e , M e t e o r o l o g i y a i g i d r o l o g i y a , ( 1 9 7 9 ) , No.8, pp.5-15. 26) G . I . Marchuk, Environment and some o p t i m i z a t i o n p r o b l e m s , L e c t . Notes Comp. S c i . , (19761, 4 0 , No.1, pp.13-30. 27) G . I . Marchuk,V.P. K o c h e r g i n , V . I . Klimok and V . A . Sukhorukov, On t h e dynamics of t h e ocean s u r f a c e mixed l a y e r , J. P h y s . O c e a n o g r . , ( 1 9 7 7 ) , 7 , pp.865-875. 28) G.I. Marchuk, V . I . Kuzin and N . N . O h r a z t s o v , Numerical m o d e l l i n g of t h e d i s t r i b u t i o n s o u r c e s i n a w a t e r b a s i n , N o v o s i b i r s k , ( 1 9 7 9 ) , Academy of S c i e n c e s of t h e USSR, Computing C e n t e r , ( P r e p r i n t ) . 29) G . I . Marchuk, and V . V . Penenko, A p p l i c a t i o n of o p t i m i z a t i o n methods t o t h e problem of m a t h e m a t i c a l s i m u l a t i o n of a t m o s p h e r i c p r o c e s s e s and e n v i r o n m e n t , P r o c . of t h e I F I P , B . , ( 1 9 7 9 ) , pp.240-252. 30) A.S. Monin and A . M . Obukhov, B a s i c l a w s of t u r b u l e n t mixing i n t h e ground a t m o s p h e r i c l a y e r , Trudy I n s t i t u t a G e o f i z i k i , t h e USSR Academy of S c i e n c e s , ( 1 9 5 4 ) , N0.24 ( 1 5 1 ) , pp.163-187. 31) A.S. Monin and A . M . Vol.1 ( i n Russian).

Yaglom. S t a t i s t i c a l h y d r o m e c h a n i c s , Moscow, Nauka, ( 1 9 6 5 ) ,

32) R . V . Ozmidov, On t u r b u l e n t exchange i n a s t e a d i l y s t r a t i f i e d o c e a n , I z v . AN SSSR, F i z i k a a t m o s f e r y i o k e a n a , ( 1 9 6 5 ) , 1, No.8, pp.853-860. 33) B.L. Rozhdestvensky and N . N . Yanenko, Systems of q u a s i l i n e a r e q u a t i o n s and t h e i r a p p l i c a t i o n t o g a s dynamics, Moscow, Nauka, (1968) ( i n R u s s i a n ) . 34) G . S t r e n g and J. F i c s , Theory of f i n i t e e l e m e n t s method, Moscow, M i r , (Russian e d . )

.

(1977)

35) A.S. S a r k i s y a n , Numerical a n a l y s i s and s e a c u r r e n t f o r e c a s t , L e n i n g r a d , G i d r o m e t e o i z d a t , (1977) ( i n R u s s i a n ) . 36) A . M . Yaglom, On t u r b u l e n t d i f f u s i o n i n t h e a t m o s p h e r i c ground l a y e r , I z v AN SSSR, F i z i k a a t m o s f e r y i o k e a n a , ( 1 9 7 2 ) , 9 , No.6. 37) R . G . Yanovsky, Environmental p r o t e c t i o n i s a p u b l i c d u t y of s c i e n t i s t s , i n : M e t h o d o l o g i c a l Problems of t h e Modern S c i e n c e , Moscow, P o l i t i z d a t , ( 1 9 7 9 ) , pp. 196-211 ( i n R u s s i a n ) . 38) V . W . Ekman, On t h e i n f l u e n c e of t h e e a r t h ' s r o t a t i o n on o c e a n - c u r r e n t s , Arkhiv M a t . , A s t r o n o c h . f i z i k , ( 1 9 0 5 ) , 2 , No.11.

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