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.

E I S E V I E R SCIENCE PUBI~ISHINGCOMPANY, INC 52 VAN D E RB ILT AV EN U E NEW YORK. N.Y. 10017 U.S.A.

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

.

( 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

.

.

.

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.


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


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


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

;

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

--

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.

This Page Intentionally Left Blank