NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
MODELING A COMPLEX ENVIRONMENTAL SYSTEM:
THE LAKE BALATON STUDY
A u g u s t 1981 WP-81-108
W o r k i n g Papers a r e i n t e r i m r e p o r t s o n work o f t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s a n d h a v e r e c e i v e d o n l y l i m i t e d r e v i e w . V i e w s o r o p i n i o n s e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e - s e n t t h o s e o f t h e I n s t i t u t e o r o f i t s N a t i o n a l Member O r g a n i z a t i o n s .
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , A u s t r i a
THE AUTHOR
L2iszl6 ~ o m l y 6 d y i s t h e l e a d e r o f t h e B a l a t o n C a s e S t u d y , R e s o u r c e s a n d E n v i r o n m e n t A r e a , I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , L a x e n b u r g , A u s t r i a (on l e a v e f r o m t h e R e s e a r c h C e n t r e f o r Water R e s o u r c e s Development ( V I T U K I ) , B u d a p e s t , H u n g a r y ) .
PREFACE
One of the principal projects of the Task on Environmental Quality Control and Management in IIASA's Resources and Environ- ment Area is a case study of eutrophication management for Lake Balaton, Hungary. The case study is a collaborative project in- volving a number of scientists from several Hungarian institutions and IIASA (for details see WP-80-187).
This paper primarily considers the methodological framework of the Lake Balaton Case Study, using it as an example of how a system characterized by complexity and uncertainty can be
modelled. The suggested approach is supported by several examples and provides an account of both developments since the publication of WP-80-187, and also necessary research for the project's com- pletion.
CONTENTS
1 . INTRODUCTION 1
2 . MAJOR CHARACTERISTICS OF THE SYSTEM 3
3 . THE MODELING APPROACH 6
4 . ILLUSTRATION OF THE DIFFERENT STEPS OF THE APPROACH 1 4
4 . 1 W i n d I n d u c e d S e d i m e n t Water I n t e r a c t i o n ( S t r a t u m 5 ) 1 4 4 . 2 A p p l i c a t i o n of a ID H y d r o d y n a m i c Model ( S t r a t u m 4 ) 1 9 4 . 3 T h e N u t r i e n t L o a d u n d e r U n c e r t a i n t y a n d
S t o c h a s t i c i t y ( S t r a t u m 3 ) 2 1
4 . 4 T h e L a k e E u t r o p h i c a t i o n Model ( S t r a t u m 3 ) 2 3 4 . 5 T h e I n c o r p o r a t i o n of t h e L a k e Model i n t o
WQMM ( S t r a t u m 2 ) 2 5
5 . CONCLUDING REMARKS
INTRODUCTION
The e u t r o p h i c a t i o n o f l a k e s , a t y p i c a l , u n f a v o r a b l e manifestation o f t h e p a s t few d e c a d e s , i s a consequence o f t h e i n c r e a s e i n t h e amount o f n u t r i e n t s ( s u c h a s phosphorus and n i t r o g e n compounds) r e a c h i n g w a t e r b o d i e s . T h i s i n c r e a s e i s c l o s e l y r e l a t e d t o t h e g e n e r a l l y r a p i d development o f i n d u s t r y , a g r i c u l t u r e , and t o u r i s m w i t h i n t h e w a t e r s h e d o r i n s h o r t , t o a change i n t h e i n f r a s t r u c t u r e o f t h e r e g i o n . E u t r o p h i c a t i o n , a n i n - l a k e phenomenon, t h e o r i g i n of which l i e s o u t s i d e t h e l a k e , c a u s e s u n p l e a s a n t c o n s e q u e n c e s
( e . g . , a r i s e i n a l g a l biomass, w a t e r d i s c o l o r a t i o n , t a s t e and o d o r problems, b a c t e r i a l c o n t a m i n a t i o n , e t c . ) which c a n g r e a t l y l i m i t t h e u s e o f t h e l a k e ' s w a t e r f o r r e c r e a t i o n , w a t e r s u p p l y , e t c . , and l e a d t o a d r a s t i c change i n t h e ecosystem.
Thephenomenon o f e u t r o p h i c a t i o n and m a t h e m a t i c a l modeling o f it have been q u i t e w e l l e x p l o r e d f o r d e e p l a k e s , b u t o n l y t o a l e s s e r e x t e n t f o r s h a l l o w l a k e s . H e r e , d u e t o t h e low d e p t h and t h e g e n e r a l l y s t r o n g wind a c t i o n , s t r a t i f i c a t i o n r a r e l y oc- c u r s . The dynamics a r e more complex and a r e f a s t e r ; c o n s e q u e n t l y , s h a l l o w l a k e s a r e much more a f f e c t e d by c h a n g e s i n e n v i r o n m e n t a l f a c t o r s and show less c o n s i s t e n t p a t t e r n s from y e a r t o y e a r . The r e c o g n i t i o n o f t h i s g a p i n knowledge l e d IIASA's R e s o u r c e s and Environment Area t o i n i t i a t e r e s e a r c h on t h e e u t r o p h i c a t i o n o f s h a l l o w l a k e s . Lake B a l a t o n i n Hungary, t h e l a r g e s t l a k e i n C e n t r a l Europe, which h a s e x h i b i t e d t h e u n l i k e l y s i g n s o f e u t r o - p h i c a t i o n , was c h o s e n t o b e one o f t h e two c a s e s t u d i e s . The c a s e s t u d y i s b e i n g c a r r i e d o u t i n c o o p e r a t i o n w i t h t h e Hungarian Academy o f S c i e n c e s and w i t h t h e p a r t i c i p a t i o n o f s e v e r a l Hungarian
i n s t i t u t i o n s .
The g e n e r a l f e a t u r e s o f t h e problem from a m e t h o d o l o g i c a l s t a n d p o i n t a r e a s f o l l o w s : ( i ) The s y s t e m composed o f t h e w a t e r body and t h e w a t e r s h e d i s l a r g e i n a p h y s i c a l s e n s e ; t h e r e f o r e , u n d e r s t a n d i n g a n d m a n a g i n g it c a n n o t b e a c c o m p l i s h e d s o l e l y on
a n e x p e r i m e n t a l b a s i s . Modeling i s a b s o l u t e l y n e c e s s a r y and s h o u l d b e i n harmony and i n t e r a c t i o n w i t h i n s i t u and l a b o r a t o r y measure- ments and d a t a c o l l e c t i o n , r e s p e c t i v e l y . ( i i ) T h e r e a r e s t r o n g
i n t e r a c t i o n s among b i o l o g i c a l , c h e m i c a l , and h y d r o p h y s i c a l i n - l a k e p r o c e s s e s ; f u r t h e r m o r e , between t h e w a t e r s h e d and w a t e r body.
( i i i ) There a r e s e v e r a l s t o c h a s t i c i n f l u e n c e s ( e . g . , m e t e o r o l o g y , h y d r o l o g y ) . ( i v ) The d a t a a v a i l a b l e a r e o f t e n i n a d e q u a t e and s c a r c e . U n c e r t a i n t y c o n s e q u e n t l y p l a y s a n i m p o r t a n t r o l e .
( v ) The v i c i n i t y o f t h e l a k e i s t h e m a j o r t o u r i s t r e s o r t i n Hungary, s o t h e r e i s a s t r o n g economic i n t e r e s t i n a p r a c t i c a l s o l u t i o n t o t h e problem. I n o t h e r words, t h e management of t h e s y s t e m i s o f p r i m a r y i n t e r e s t .
From t h e s e c h a r a c t e r i s t i c s it f o l l o w s t h a t t h e l a k e and i t s r e g i o n form a complex e n v i r o n m e n t a l system; t h e problems r e l a t e d t o it r e q u i r e a s y s t e m s a n a l y t i c a l a p p r o a c h . The aim i s t o h a n d l e t h e p r o b l e m i n b o t h a r e s e a r c h and management c o n t e x t , t h a t i s , t o B e t t e r o u r u n d e r s t a n d i n g on t h e s c i e n t i f i c l e v e l and t h e n t o u t i l i z e t h i s knowledge f o r working o u t o p t i m a l c o n t r o l s t r a t e g i e s
f o r improving t h e w a t e r q u a l i t y o f t h e l a k e . The e l a b o r a t i o n o f such a model o r s e t o f models i s n o t a n e a s y t a s k and meets s e - r i o u s m e t h o d o l o g i c a l d i f f i c u l t i e s . I n t h i s r e s p e c t , t h e metho- d o l o g i e s d e v e l o p e d w i t h i n IIASA's R e s o u r c e s and Environment Area
( s e e Beck i n t h i s i s s u e [ 4 ] ) c o u l d p r o v i d e s u p p o r t , b u t a t t h e same t i m e , t h e c a s e s t u d y i t s e l f s h o u l d s e r v e t o answer i n a w i d e r c o n t e x t , t h e m e t h o d o l o g i c a l q u e s t i o n s a d d r e s s e d ; a n o t h e r
r e a s o n why t h e c a s e s t u d y was i n i t i a t e d .
The o b j e c t i v e o f t h i s p a p e r i s t o i l l u s t r a t e w i t h t h e example o f Lake B a l a t o n , how s u c h a complex s y s t e m c a n b e u n d e r s t o o d and managed i n i t s e n t i r e t y , i . e . , t o g e t h e r w i t h t h e r e l a t e d metho-
d o l o g i c a l q u e s t i o n s . The p a p e r i s o r g a n i z e d a s f o l l o w s : f i r s t , t h e main c h a r a c t e r i s t i c s of t h e s y s t e m and t h e modeling a p p r o a c h a r e o u t l i n e d ( S e c t i o n s 2 and 3 ) . I n S e c t i o n 4 , t h e i n d i v i d u a l s t e p s o f t h e a n a l y s i s a r e i l l u s t r a t e d t h r o u g h examples. These w i l l be r e l a t e d t o t h e s e d i m e n t - w a t e r i n t e r a c t i o n , s p a t i a l mass exchange, n u t r i e n t l o a d i n g problem,and t h e l a k e e u t r o p h i c a t i o n model. I n most o f t h e examples t h e i n f l u e n c e o f u n c e r t a i n t i e s and s t o c h a s t i c e f f e c t s w i l l be a c c o u n t e d f o r . A t t h e end o f t h i s s e c t i o n , i t i s shown how t h e l a k e w a t e r q u a l i t y model can b e
i n c o r p o r a t e d i n t o t h e management framework.
I t h a s t o b e mentioned t h a t t h e s t u d y h a s n o t y e t been com- p l e t e d . T h e r e f o r e , i n some c a s e s , r e f e r e n c e w i l l b e made t o r e - s u l t s which a r e p r e l i m i n a r y o n l y .
2 . MaJOR CH?iRACTERISTICS O F THE SYSTEM
The l a k e and i t s w a t e r s h e d a r e i l l u s t r a t e d i n F i g u r e 1 . The l e n g t h of t h e l a k e i s 78 km, t h e a v e r a g e w i d t h around 8 km ( s u r f a c e a r e a n e a r l y 600 km 2 ) and t h e a v e r a g e d e p t h 3.1 m. The m a j o r i n - f l o w o f t h e l a k e i s t h e R i v e r Zala a t t h e s o u t h w e s t e r n end o f t h e l a k e which d r a i n s h a l f o f t h e t o t a l c a t c h m e n t a r e a ( " 5800 km 2 ) .
There is a s i n g l e o u t f l o w a t t h e o t h e r end o f t h e l a k e , s i g f o k , t h r o u g h a c o n t r o l g a t e . The mean r e s i d e n c e t i m e o f w a t e r i s a b o u t 2 y e a r s .
The f l u c t u a t i o n i n t h e w a t e r ' s t e m p e r a t u r e i s h i g h . There i s a r e l a t i v e l y l o n g i c e - c o v e r e d p e r i o d (around two i n o n t h s ) , w h i l e
t h e t e m p e r a t u r e i n summer may exceed 25O C. Concerning t h e c h e m i c a l c o m p o s i t i o n o f t h e w a t e r , t h e h i g h c a l c i u m c a r b o n a t e c o n t e n t and pH v a l u e ( 8 . 3 t o 8 . 7 ) s h o u l d b e mentioned. Wind a c t i o n i s i m p o r t a n t ( t h e r e a r e a b o u t 80 " s t o r m y " d a y s i n a y e a r ) r e s u l t i n g i n f a v o r a b l e oxygen c o n d i t i o n s and a permanent back and f o r t h motion ( s e i c h e ) a l o n g t h e l a k e and a c o m p l i c a t e d c i r - c u l a t i o n p a t t e r n . Wind s t r o n g l y i n f l u e n c e s s e d i m e n t a t i o n and re- l e a s e o f v a r i o u s m a t e r i a l s from t h e s e d i m e n t l a y e r ( i t s o r g a n i c m a t e r i a l c o n t e n t i s r e l a t i v e l y l o w ) .
I n r e c e n t y e a r s , r e m a r k a b l e c h a n g e s h a v e been o b s e r v e d i n t h e w a t e r q u a l i t y d u e t o t h e r a p i d i n c r e a s e i n t o u r i s m , sewage d i s c h a r g e s , f e r t i l i z e r u s e l a n d o t h e r f a c t o r s . The a l g a l biomass
( a l g a e i s t h e most i m p o r t a n t p r i m a r y p r o d u c e r i n t h i s c a s e ) i n - c r e a s e d by a f a c t o r o f 10 when compared w i t h t h e p a s t 15-20 y e a r s . The t r e n d i n p r i m a r y p r o d u c t i o n i s s i m i l a r and a t t h e most p o l l u t e d
3
w e s t e r n b a s i n , p e a k s o f up t o 13.6 g ~ / m " d were o b s e r v e d , a hyper- t r o p h i c v a l u e [ 1 2 ] . I n s h o r t , t h e a v e r a g e l a k e c o n d i t i o n s moved from m e s o t r o p h i c t o e u t r o p h i c , t h u s e n d a n g e r i n g t h e u s e o f t h e l a k e f o r r e c r e a t i o n a l p u r p o s e s , t h e prime w a t e r u s e i n t h i s c a s e .
Phosphorus p l a y s a dominant r o l e i n t h e e u t r o p h i c a t i o n o f t h e l a k e ( t h e l a k e i s p h o s p h o r u s - l i m i t e d most o f t h e t i m e ) . Thus, b o t h from t h e p o i n t o f view o f u n d e r s t a n d i n g and managing t h e s y s t e m , t r a c i n g t h e phosphorus compounds i n t h e l a k e and on t h e w a t e r s h e d i s o f p r i m a r y i n t e r e s t . The t o t a l p h o s p h o r u s l o a d o f t h e l a k e i s a r o u n d 1000 kg/d [ 1 5 ] , h a l f o f which i s a v a i l a b l e f o r a l g a l u p t a k e . The l o a d h a s many components: 33% i s d e r i v e d from sewage, 27% from d i f f u s e s o u r c e s , 2 2 % i s r e l a t e d t o r u n o f f
p r o c e s s e s i n t h e d i r c e t v i c i n i t y o f t h e l a k e , w h i l e t h e c o n t r i b u t i o n
o f a t m o s p h e r i c p o l l u t i o n i s 18%. The r a t i o o f sewage d i s c h a r g e s i n t h e a v a i l a b l e l o a d i s h i g h e r ; o n l y t h e sewage r e l e a s e d i n t h e r e c r e a t i o n a l a r e a ( F i g u r e 1 ) a c c o u n t s f o r 3 6 % o f t h e a v a i l a b l e l o a d . T h i s d i r e c t l o a d v a r i e s q u i t e a l o t , f o l l o w i n g t h e f l u c t u a - t i o n s i n p o p u l a t i o n due t o t o u r i s m , and h a s a 2-4 t i m e h i g h e r v a l u e i n summer t h a n d u r i n g t h e o f f - s e a s o n . The l o a d d i s t r i b u t i o n
a l o n g t h e l a k e i s a p p r o x i m a t e l y uniform; however t h e volume r e - l a t e d v a l u e i s twelve t i m e s h i g h e r a t t h e K e s z t h e l y Bay ( F i g u r e 1 ) t h a n a t t h e o t h e r end o f t h e l a k e , due t o d i f f e r e n c e s i n t h e
volume o f t h e f o u r main b a s i n s . T h i s f a c t i s a l s o r e f l e c t e d i n t h e pronounced l o n g i t u d i n a l g r a d i e n t o f v a r i o u s w a t e r q u a l i t y p a r a m e t e r s , e . g . , f o r C h l o r o p h y l l - a t h e r a t i o of t h e maximum and minimum v a l u e s r a n g e s between 4 and 2 0 [ 2 9 ] . The g r a d i e n t ob-
s e r v e d a t t h e same t i m e i n d i c a t e s t h a t t h e s t r o n g wind a c t i o n and t h e mixing a s s o c i a t e d w i t h it a r e s t i l l n o t s u f f i c i e n t f o r l e v e l i n g o u t t h e s p a t i a l n o n u n i f o r m i t i e s .
From a n a n a l y s i s o f t h e d a t a it i s c l e a r t h a t t h e r e i s n o t o n l y a c r i t i c a l s t a t e o f t h e w a t e r q u a l i t y a t K e s z t h e l y Bay, b u t a l s o a s p r e a d i n g d e t e r i o r a t i o n p r o c e s s which e x t e n d s t o w a r d s o t h e r a r e a s of t h e l a k e where t h e w a t e r q u a l i t y i s s t i l l good.
Thus a c t i o n i s u r g e n t l y r e q u i r e d from t h e view o f t h e e n t i r e l a k e .
Based on h y d r o l o g i c and w a t e r q u a l i t y c o n s i d e r a t i o n s , t h e l a k e was d i v i d e d i n t o f o u r b a s i n s , a s i n d i c a t e d i n F i g u r e 1 . The a p p l i c a t i o n o f t h e p r i n c i p l e o f s e g m e n t a t i o n proved t o b e a u s e f u l t o o l f o r m o d e l i n g , d a t a c o l l e c t i o n , and d a t a h a n d l i n g .
Concerning d a t a , e x t e n s i v e r e c o r d s a r e a v a i l a b l e on h y d r o l o g y and m e t e o r o l o g y . R e g u l a r w a t e r q u a l i t y m o n i t o r i n g s t a r t e d t e n
y e a r s a g o , i n two network s y s t e m s c o n s i s t i n g o f 9 and 16 s p a t i a l s a m p l i n g p o i n t s , r e s p e c t i v e l y (10-20 measurements p e r y e a r ) , b u t i r r e g u l a r d a t a a r e a l s o a v a i l a b l e back t o t h e e a r l y s i x t i e s .
S e v e r a l o t h e r i n s i t u and l a b o r a t o r y measurements w e r e a l s o t a k e n ( p r i m a r y p r o d u c t i o n , e x t i n c t i o n , s e d i m e n t - w a t e r i n t e r a c t i o n ,
v e l o c i t y , e t c . ) . A s u r v e y was done on t h e n u t r i e n t l o a d between 1975-79, which i n v o l v e d 2 0 t r i b u t a r i e s and 27 sewage d i s c h a r g e p o i n t s [ I S ] ( i n d i c a t e d i n F i g u r e 1 ) . On t h e m a j o r t r i b u t a r y , d a l l y o b s e r v a t i o n s were made d u r i n g t h i s p e r i o d [ 1 6 1 .
On t h e b a s i s o f d a t a c o l l e c t e d by Hungarian i n s t i t u t i o n s r e c e n t l y , more o f t e n i n i n t e r a c t i o n w i t h modeling--the "Lake B a l a t o n Data Bank" was c r e a t e d a t IIASA, s e r v i n g a s a s t a r t i n g p o i n t f o r t h e modeling work.
F o r f u r t h e r d e t a i l s on t h e c a s e s t u d y , t h e r e a d e r i s r e - f e r r e d t o [ 2 9 , 301.
3. THE MODELING APPROACH
3.1 The p r e v i o u s s e c t i o n s d e m o n s t r a t e d t h e c o m p l e x i t y o f t h e
-
w a t e r q u a l i t y problem. Here, m e t h o d o l o g i c a l q u e s t i o n s w i l l be d i s c u s s e d . For t h e p u r p o s e o f i l l u s t r a t i o n t h e i n - l a k e p r o c e s s e s w i l l b e c o n s i d e r e d . From t h i s example, c o n c l u s i o n s w i l l b e drawn, l e a d i n g t o t h e modeling a p p r o a c h t o b e a d o p t e d .
The w a t e r q u a l i t y o f a l a k e i s t h e r e s u l t o f s e v e r a l p h y s i c a l , c h e m i c a l , and b i o l o g i c a l p r o c e s s e s . The development o f a model i s g e n e r a l l y b a s e d on t h e a p p r o p r i a t e c o m b i n a t i o n o f knowledge g a i n e d from t h e o r y and measurements, r e s p e c t i v e l y ( t h e s o - c a l l e d t h e o r e t i c a l and measurement knowledge [8]). Depending on t h e s o l i d i t y o f t h e t h e o r y o f v a r i o u s p r o c e s s e s , q u i t e d i f f e r e n t
a p p r o a c h e s may b e employed. For example, i n most o f t h e hydrodynamic
applications ("hard" science, see [3] ) the model structure is basically determined by the partial differential equations (PDEs) of continuity and momentum (exceptions may be caused by not well defined or unknown boundary conditions, see Section 4.1). With- in the domain of water quality, the theoretical background of biology and chemistry ("soft" sciences) is less satisfactorily established and model development should be supported more exten- sively by the measurement knowledge. The development is generally based on testing hypotheses, and uncertainty plays an important role [9, 14, 301. Under such conditions, steps such as model structure identification, parameter estimation and the analysis of error propagation [2] are inevitably required--techniques which are available mostly for ordinary differential equations
(ODES) only.
In the majority of the water quality models, the description of processes of both of the categories mentioned is necessitated;
the dilemma then is [26] : how should the modeling procedure of biology and chemistry (ODE structure with interactive use of data) be combined with the "precise" treatment of hydrodynamics, the PDE structure of which a priori excludes the application of most of the techniques required for the cognition of phenomena of the other group and consequently the exploration of the entire process?
The answer seems to be relatively easy, namely one should start elaborating detailed models for both groups, and keeping in mind the main features of the other group (time and length scales [Ill, etc.), simplify them through a sensitivity analysis (aggre- gation on the base level). Then the same procedure should be repeated after coupling the models (aggregation on the higher
level) which should then establish the relative importance of various processes and harmony among t h e m e t h o d o l o g i e s employed.
From this conclusion, an o f f - l i n e modeling approach follows, which avoids the direct coupling of all the detailed submodels.
The procedure starts with a resonable d e c o m p o s i t i o n of the com- plex structure into smaller, more tractable units which are accessible for separate and detailed studies. This is followed by two different kinds of a g g r e g a t i o n s , the aim of which is to preserve and integrate only essentials, but ruling out the un- necessary details. In contrast to the paper by Beck [4], the
procedure of decomposition and subdivision has a different meaning here. In the course of submodel development, experiments of
"isolated" character are certainly required, but then the coupled, aggregated ("smaller") model is validated against its detailed
("larger") version and data on the higher level (a tedious task which is rarely documented in the literature). It is also stressed
that for such a complex system one "small" model is as unrealistic as one "large" model (accordingly, the solution cannot be looked for in this contrast, i.e., "larger" or "smaller" model--the question rasied by Beck [4]). Thus, a procedure which is re- quired progresses through the subsequent development of d e t a i l e d and s i m p l i f i e d m o d e l s by maintaining only the aggregated knowledge on the higher level.
Each modeling step is associated with model and data u n c e r -
t a i n t i e s ; consequently, the aggregation is required in this re-
spect as well: another guideline in our approach.
The tactics of decomposition are especially important if the objective of the study involves such different levels as the scientific understanding of a system and the decision making
r e l a t e d t o t h e same problem: a s i t u a t i o n t h a t we f a c e h e r e .
3.2 The a p p l i c a t i o n o f t h i s approach f o r t h e Lake B a l a t o n problem
-
i s e x p l a i n e d w i t h t h e h e l p o f F i g u r e 2 , which shows t h e frame- work o f t h e r e s e a r c h [ 3 0 ] . The f i r s t d e c o m p o s i t i o n t h a t d i r e c t l y comes t o mind i s t h e d i s t i n c t i o n between l a k e and w a t e r s h e d , s i n c e a s mentioned b e f o r e , t h e w a t e r q u a l i t y problem l i e s i n t h e l a k e , b u t t h e c a u s e s , and p r a c t i c a l l y a l l c o n t r o l p o s s i b i l i t i e s a r e t o be found i n t h e w a t e r s h e d . Next, t h e v a r i o u s u n i t s s h o u l d be
s e p a r a t e d and t h e e s s e n t i a l r e s u l t s p u t t o g e t h e r on a s u c c e s s i v e l y h i g h e r l e v e l o f i n t e g r a t i o n . The p r o c e d u r e i n v o l v i n g f i v e s t r a t a w i l l be d i s c u s s e d i n g r e a t e r d e t a i l f o r t h e Lake E u t r o p h i c a t i o n Model ( L E M ) , w i t h r e f e r e n c e t o models now b e i n g e l a b o r a t e d . The p a r a l l e l s i n t h e N u t r i e n t Loading Model (NLM) c a n b e found t h r o u g h F i g u r e 1.
S t r a t u m 5
F i r s t , t h o s e segments of t h e l a k e s h o u l d b e i s o l a t e d which can b e c o n s i d e r e d a p p r o x i m a t e l y u n i f o r m from t h e v i e w p o i n t o f w a t e r q u a l i t y ( c o m p l e t e mixing i n s i d e e a c h u n i t ) and from t h e f a c t o r s i n f l u e n c i n g them. The o b j e c t i v e o f t h e models on t h i s s t r a t u m i s t o d e s c r i b e t h e a l g a l dynamics and n u t r i e n t c y c l i n g f o r a l l t h e s e g m e n t s , i n v o l v i n g b o t h t h e w a t e r body and t h e s e d i - ment, s i n c e t h e l a t t e r i s a s i n k and s o u r c e o f v a r i o u s m a t e r i a l s and t h e i r i n t e r a c t i o n p l a y s a n i m p o r t a n t r o l e i n s h a l l o w l a k e s
( a l s o from t h e p o i n t o f view o f management, a f t e r a r e d u c t i o n i n l o a d , t h e new e q u i l i b r i u m o f t h e l a k e w i l l be d e t e r m i n e d by t h e n u t r i e n t r e l e a s e o f t h e s e d i m e n t ) . These k i n d s o f models b a s e d on t h e mass c o n s e r v a t i o n p r i n c i p l e and f o r m u l a t e d t h r o u g h a s e t o f n o n l i n e a r ODES a r e well-known i n t h e l i t e r a t u r e [ 2 1 ] . I n t h e frame o f t h e p r e s e n t s t u d y , t h r e e submodels, BEM, BALSECT, and
SIMBAL were developed (see [13, 19, 311, with respect to their comparison [30]) which differs basically in the number of state variables (between 4 and 7) and essential parameters (10-171, as well as in the mathematical formulation of various processes and in the parameter estimation technique adopted. It is noted here that some of the parameters can be derived from further
isolation up to a lower level with appropriately designed experi- ments. As examples, the estimation of algal growth parameters
from vertical primary production measurements [321 and the study of wind induced sediment-water interaction (see Section 4.1) may be mentioned.
To end the discussion on stratum 5, it is stressed that several steps of the analysis are based on intuition (starting from the segmentation to the formulation of various biochemical processes), thus the inclusion of data with their uncertainties is imperative.
Stratum 4
On the next level the segment-oriented biochemical and sedi- ment models are coupled by involving mass in- and outflows at
the boundaries of the units. For this purpose, a hydrodynamic- transport model can be used. In light of the experiences gained from the study of the Great Lakes [7], it was decided not to use a coupled more-dimensional hydrodynamic-transport model incor- porating the submodels of the lower stratum: the gain in infor- mation is not proportional to the increase in complexity; further- more it causes methodological difficulties as explained in Section
3.1. Here again, an off-line technique is applied. The basic assumption is that it is sufficient to subdivide the lake in a
l o n g i t u d i n a l d i r e c t i o n o n l y . T h i s i s s u p p o r t e d by t h e r i v e r i n e s h a p e o f t h e l a k e and t h e presumably e x t e n s i v e t r a n s v e r s a l m i x i n g , s i n c e t h e p r e v a i l i n g wind d i r e c t i o n i s n e a r l y p e r p e n d i c u l a r t o t h e l a k e ' s a x i s ( t h e d e s c r i p t i o n o f t h e s h o r e l i n e e f f e c t s i s n o t t h e o b j e c t i v e h e r e ) . C o n s e q u e n t l y t h e p a r a l l e l d e v e l o p m e n t o f a n u n s t e a d y t h r e e - d i m e n s i o n a l (3D) and o n e - d i m e n s i o n a l ( I D ) hydro- dynamic model was d e c i d e d on. The r e s u l t s g a i n e d ( f o r d e t a i l s see [ 2 2 ] and [ 2 7 ] showed t h a t t h e two models c o u l d b e e q u a l l y
c a l i b r a t e d a g a i n s t dynamic w a t e r l e v e l d a t a and s u g g e s t e d t h a t t h e cross-sectionallyaverageddischarge, Q ( t ) , c a n b e p r o p e r l y d e r i v e d from t h e much s i m p l e r 1D model which a l s o a l l o w e d a n u n c e r t a i n t y a n a l y s i s o n t h e wind d a t a , a m e t h o d o l o g i c a l l y r e m a r k a b l e a s p e c t
( S e c t i o n 4 . 2 ) . The 1D v e r s i o n c a p t u r e s o n l y t h e r a t h e r r a p i d
s e i c h e m o t i o n o n t h e l a k e ( c o n v e c t i o n ) b u t n o t t h e d i f f e r e n t k i n d s o f b a c k f l o w s and c i r c u l a t i o n s : t h e i r m i x i n g i n f l u e n c e s h o u l d b e d e r i v e d from t h e 3D model. I n f a c t , a l o n g i t u d i n a l d i s p e r s i o n c o e f f i c i e n t , DL, c a n b e a p p r o x i m a t e l y c a l c u l a t e d from t h e v e l o c i t y f i e l d by t h e method s i m i l a r t o t h a t employed f o r r i v e r s [ l o ] .
Based on t h e i n i t i a l e x p e r i e n c e s [ 2 3 ] i t seems t o b e s u f f i c i e n t t o p e r f o r m t h e c o n l p u t a t i o n s f o r some t y p i c a l s t o r m y e v e n t s and a f t e r w a r d t o c o r r e l a t e D t o wind p a r a m e t e r s . The r e l a t i o n s h i p
L
D L [ W ( t ) ] g a i n e d , w i l l a l l o w r e p l a c e m e n t o f t h e c o u p l e d 3D hydro- d y n a m i c - t r a n s p o r t model by a ID v e r s i o n . Thus t h e submodels o f
s t r a t u m 5 w i l l be i n c o r p o r a t e d i n t o a s e t o f l o n g i t u d i n a l d i s - p e r s i o n e q u a t i o n s on s t r a t u m 4 . T h i s a g g r e g a t i o n i s a c h i e v e d by a s e n s i t i v i t y a n a l y s i s s o l e l y on t h e hydrodynamics. The s e c o n d a g g r e g a t i o n (see S e c t i o n 3 . 1 ) c a n be a r r i v e d a t w i t h t h e u s e o f t h e c o u p l e d d i s p e r s i o n b i o c h e m i c a l model c u r r e n t l y b e i n g
e l a b o r a t e d , which a l l o w s a s e n s i t i v i t y s t u d y o f a l l t h e p r o c e s s e s i n v o l v e d .
I t s h o u l d b e mentioned h e r e ( s e e S e c t i o n 2 ) t h a t p r o v i s i o n a l l y , i n a l l t h r e e b i o c h e m i c a l models f o u r segments ( s e e F i g u r e 1 ) a r e assumed; t h e i r c o u p l i n g i s b a s e d on h y d r o l o g i c t h r o u g h f l o w and a wind i n f l u e n c e d mass exchange p r o c e s s d e s c r i b e d g l o b a l l y . S i n c e t h e model s t r u c t u r e b a s e d on ODES h a s many a d v a n t a g e s , o n e o f t h e o b j e c t i v e s o f t h e s t u d y on t h e ID c o u p l e d model i s w h e t h e r t h e f o u r b a s i n s c o n c e p t c a n be m a i n t a i n e d o r n o t .
S t r a t u m 3
The i n v o l v e m e n t o f mass exchange among segments a s d e s c r i b e d b e f o r e w i l l r e s u l t i n t h e Lake E u t r o p h i c a t i o n Model (LEM) ( F i g u r e 2 ) which h a s s e v e r a l f o r c i n g f u n c t i o n s , s u c h a s s o l a r r a d i a t i o n , w a t e r t e m p e r a t u r e , wind, e t c . ( n a t u r a l o r u n c o n t r o l l a b l e f a c t o r s ) and t h e n u t r i e n t l o a d . S i n c e t h e l a t t e r i s t h e o n l y f a c t o r t o be c o n t r o l l e d , i t p l a y s a d i s t i n g u i s h e d r o l e ; however, l e s s modeling work was d o n e ' o n i t i n t h e frame o f t h e c a s e s t u d y . T h i s c a n be e x p l a i n e d by t h e r e l a t i v e l y h i g h c o n t r i b u t i o n o f t h e sewage l o a d (modeling i s b a s i c a l l y n o t needed h e r e b e c a u s e o f t h e n a t u r e o f t h e problem) and t h e l i m i t e d amount o f w a t e r s h e d d a t a a v a i l a b l e f o r n o n - p o i n t s o u r c e modeling. A t h o r o u g h d a t a c o l l e c t i o n and t h e d e r i v a t i o n o f a n u t r i e n t b a l a n c e f o r t h e whole l a k e w e r e p r e f e r r e d ( f o r d e t a i l s see [ I 51 )
,
t h e r e s u l t s o f which w e r e a l - r e a d y summarized i n S e c t i o n 2 . T h i s s t u d y a l s o i n v o l v e d an u n c e r t a i n t y a n a l y s i s i n r e l a t i o n t o t h e unobserved c o n t r i b u t i o n o f f l o o d s t o t h e l o a d ( S e c t i o n 4 . 3 ) . The r e s e a r c h a l l o w e d t h e d e r i v a t i o n of t h e t e m p o r a l and s p a t i a l p a t t e r n o f t h e l o a d com- p o n e n t s i n a d e s c r i p t i v e f a s h i o n , b o t h f o r LEM and t h e WaterQ u a l i t y Management Model (WQMM) on s t r a t u m 2 . I n f a c t , a t t h e f i r s t s t a g e o f l a k e model development t h e modeling o f any o f t h e d r i v i n g f u n c t i o n s i s n o t n e c e s s a r i l y r e q u i r e d ; b o t h f o r c a l i b r a t i o n and v a l i d a t i o n , h i s t o r i c a l d a t a c a n b e employed. For p l a n n i n g
p u r p o s e s t h e s i t u a t i o n i s d i f f e r e n t , t h e r e f o r e t h e s t o c h a s t i c e f f e c t s o f b o t h t h e l o a d and m e t e o r o l o g y were i n v o l v e d t h r o u g h Monte-Carlo s i m u l a t i o n ( S e c t i o n 4 . 4 ) .
S t r a t u m 2
The o b j e c t i v e o f WQMM i s t o g e n e r a t e a l t e r n a t i v e management o p t i o n s and s t r a t e g i e s ( t h e e f f e c t o f t h e s e b e i n g e x p r e s s e d t h r o u g h NLM which s h o u l d be u s e d h e r e i n a p l a n n i n g mode) and t o s e l e c t from among t h e s e a l t e r n a t i v e s , on t h e b a s i s o f o n e o r more ob- j e c t i v e s . Both w a t e r q u a l i t y and e x p e n s e s c a n be u s e d a s ob- j e c t i v e f u n c t i o n s o r c o n s t r a i n t s , and q u i t e o f t e n t h e i r w e i g h t i n g i s r e q u i r e d . F r e q u e n t l y t h e l o a d c a n r e p l a c e t h e l a k e ' s w a t e r q u a l i t y i n t h e o p t i m i z a t i o n i n which c a s e LEM i s u s e d m e r e l y t o check t h e r e a c t i o n o f t h e l a k e and WQMM may have a s i m p l e r s t r u c - t u r e . A d m i t t e d l y however, t h e f i r s t v e r s i o n i s more o b v i o u s be- c a u s e o f t h e n a t u r e o f t h e problem. T h i s f o r m u l a t i o n however l e a d s t o t h e dilemma: how s h o u l d a complex dynamic model b e in- c o r p o r a t e d i n t o t h e o p t i m i z a t i o n framework [ 2 8 ] ?
A t t h i s s t e p a g g r e g a t i o n i s a l s o needed. T h i s s t a r t s w i t h t h e s e l e c t i o n o f c e r t a i n w a t e r q u a l i t y i n d i c a t o r s c h a r a c t e r i z i n g t h e l a r g e scale and long-term b e h a v i o r o f t h e s y s t e m s e r v i n g as a b a s i s f o r d e c i s i o n making. D i f f e r e n t p a r a m e t e r s ( y e a r l y peak, d i f f e r e n t a v e r a g e s , d u r a t i o n o f c r i t i c a l c o n c e n t r a t i o n s , f r e q u e n c y d i s t r i b u t i o n s , e t c . ) o f t y p i c a l w a t e r q u a l i t y components ( p r i m a r y p r o d u c t i o n , a l g a l biomass, C h l o r o p h y l l - a , e t c
.
) c a n b e u s e d asi n d i c a t o r s . S u b s e q u e n t l y t h e dynamic model LEM c a n be u s e d i n t e r m s o f i n d i c a t o r s e s t a b l i s h e d , I , under r e d u c e d l o a d i n g con- d i t i o n s o r i n a n o t h e r way under s e v e r a l l o a d i n g s c e n a r i o s , L . S i n c e t h e d e f i n i t i o n o f i n d i c a t o r s i n t r o d u c e s t e m p o r a l a v e r a g i n g , it i s e x p e c t e d t h a t t h e l a k e ' s r e s p o n s e w i l l b e l e s s complex com- p a r e d t o t h e dynamic s i m u l a t i o n and a s i m p l e , d i r e c t I ( L ) t y p e r e l a t i o n s h i p c a n be found. I f s u c h a s o l u t i o n h a s a l r e a d y been a t t a i n e d , LEM c o u l d b e r e p l a c e d by I ( L ) i n WQMM; an e s s e n t i a l a g g r e g a t i o n (see S e c t i o n 4 . 5 )
.
Among t h e management a l t e r n a t i v e s , o n l y t h e two most impor- t a n t o p t i o n s a r e mentioned h e r e : ( i ) t e r t i a r y t r e a t m e n t ( p o i n t s o u r c e l o a d r e d u c t i o n )
,
( i i ) e s t a b l i s h i n g r e s e r v o i r s ( c o n s i s t i n g o f two segments s e r v i n g f o r t h e removal o f b o t h p a r t i c u l a t e and d k s s o l v e d n u t r i e n t f o r m s , r e s p e c t i v e l y [ 2 9 ] ) a t t h e mouth o f r i v e r s which a r e t h e r e c i p i e n t s of p o i n t and n o n - p o i n t s o u r c e p o l l u t a n t s . The o p t i m i z a t i o n s h o u l d t h e n b e b a s e d on t h e t r a d e - o f f between t h e two b a s i c a l t e r n a t i v e s , w i t h r e s p e c t t o t h e i r l o - c a t i o n s ( e . g . , r e g i o n a l v e r s u s l o c a l t r e a t m e n t ) and t h e s p a t i a l v a r i a t i o n o f t h e l a k e ' s w a t e r q u a l i t y .S t r a t u m 1
For t h e s a k e o f c o m p l e t e n e s s it h a s t o be mentioned t h a t
WQMM c o u l d be t h o u g h t o f a s b e i n g a p a r t o f a r e g i o n a l development p o l i c y model f o r m i n g t h e t o p o f t h e pyramid, a f i e l d which i s beyond t h e s c o p e o f t h i s s t u d y .
4 . ILLUSTRATION OF THE DIFFERENT STEPS OF THE APPROACH 4.1 Wind Induced Sediment Water I n t e r a c t i o n ( S t r a t u m 5 )
F o r s t u d y i n g t h e s e d i m e n t - w a t e r i n t e r a c t i o n i n l a k e s , s e v e r a l a p p r o a c h e s a r e p o s s i b l e (see, f o r example [ 2 4 ] )
.
I n t h i s s t u d y ,y e t a n o t h e r method w a s chosen [ 2 5 ] , i n r e c o g n i t i o n t h a t when e u t r o p h i c a t i o n i s c o n s i d e r e d , more t h a n j u s t t h e p h y s i c a l pro- c e s s e s s h o u l d b e examined. D a i l y measurements were t a k e n f o r 6 months, a t t h e m i d - p o i n t ( d e p t h H = 4.3 m ) o f t h e Szemes b a s i n
( B a s i n 2 , F i g u r e 1 ) . The measurements i n v o l v e d S e c c h i d e p t h , t e m p e r a t u r e , suspended s o l i d s ( S S ) , C h l o r o p h y l l - a , and phosphorus f r a c t i o n s a t d i f f e r e n t v e r t i c a l l o c a t i o n s . Wind v e l o c i t y and d i - r e c t i o n were r e c o r d e d c o n t i n u o u s l y , from which h o u r l y a v e r a g e s were c a l c u l a t e d . The o b j e c t i v e o f t h e f i r s t p a r t o f t h e a n a l y s i s was t o d e s c r i b e t h e dynamics o f t h e suspended s o l i d s a s a f u n c t i o n o f wind. T h i s t h e n a l l o w e d f o r a c h a r a c t e r i z a t i o n o f t h e t e m p o r a l c h a n g e s i n t h e l i g h t c o n d i t i o n s , t h e d e p o s i t i o n , and r e s u s p e n s i o n o f o t h e r p a r t i c u l a t e m a t e r i a l and t o some e x t e n t , a l s o t h e r e - l e a s e s o f d i s s o l v e d components. Here o n l y t h e b e h a v i o r o f SS w i l l be r e p o r t e d .
The a n a l y s i s s t a r t e d from a s i m p l i f i e d t r a n s p o r t e q u a t i o n f o r d e s c r i b i n g t h e t e m p o r a l and v e r t i c a l c h a n g e s o f t h e a v e r a g e SS c o n c e n t r a t i o n i n t h e b a s i n , n e g l e c t i n g i n f l o w and o u t f l o w . I t w a s r e c o g n i z e d however, t h a t t h e problem had a n u n d e f i n e d boundary c o n d i t i o n a t t h e bottom, z = H , [ 2 5 ]
where c i s c o n c e n t r a t i o n , w i s s e t t l i n g v e l o c i t y , E i s v e r t i c a l eddy v i s c o s i t y , and @d and @ e a r e t h e f l u x e s o f d e p o s i t i o n and r e s u s p e n s i o n , r e s p e c t i v e l y . I n f a c t , o n e o f t h e o b j e c t i v e s o f t h e measurements w a s t o f o r m u l a t e t h e boundary c o n d i t i o n . From t h e o b s e r v a t i o n s made, i t a p p e a r e d t h a t t h e t e m p o r a l c h a n g e s governed t h e s y s t e m (see F i g u r e 3a f o r t h e d e p t h i n t e g r a t e d v a l u e s and
wind s p e e d ) . The c ( z ) v e r t i c a l p r o f i l e s w e r e q u i t e u n i f o r m , ex- c e p t c l o s e t o t h e b o t t o m where t h e e x p e c t e d , b u t sudden i n c r e a s e c o u l d b e o b s e r v e d . A c c o r d i n g l y , i t was d e c i d e d n o t t o d e t e r m i n e t h e unknown b o u n d a r y c o n d i t i o n from t h e PDE f o r m u l a t i o n ( a r a t h e r t e d i o u s p o r c e d u r e ) , b u t t o i n t e g r a t e t h e t u r b u l e n t d i f f u s i o n
e q u a t i o n a l o n g t h e d e p t h , a n d u s e t h e ODE d e r i v e d , which t h u s d i r e c t l y i n v o l v e s t h e b o u n d a r y c o n d i t i o n i t s e l f .
I n o r d e r t o c a r r y o u t t h i s s t e p , h y p o t h e s e s w e r e n e e d e d f o r t h e f l u x e s Qd and Qe. The d e p o s i t i o n was c h a r a c t e r i z e d by i t s . P p r o b a b i l i t y ( e x p r e s s i n g which p o r t i o n o f t h e p a r t i c l e s r e a c h i n g t h e i n t e r f a c e would r e m a i n t h e r e ) :
( h e r e , t h e wavy l i n e i n d i c a t e s d e p t h a v e r a g e d v a l u e ) , w h i l e
O e
by a n e m p i r i c a l r e l a t i o n s h i p [I81
where p s a n d pw a r e s e d i m e n t and w a t e r d e n s i t i e s , and w i s e n - e
t r a i n m e n t v e l o c i t y . To f i n d w t h e c o n c e p t o f e n e r g y t r a n s f o r - e
m u l a t i o n between p o t e n t i a l a n d t u r b u l e n t k i n e t i c e n e r g i e s o f t e n employed f o r s t r a t i f i e d l a k e s was a d o p t e d [ 6 ] . A c c o r d i n g l y , u n d e r s i m p l i f i e d c o n d i t i o n s :
where t h e power d e p e n d s on t h e R i c h a r d s o n number.
Using t h e s e h y p o t h e s e s , t h e d e p t h i n t e g r a t e d t r a n s p o r t equa- t i o n t a k e s t h e f o l l o w i n g form [ 2 5 ] :
where K1 and K 2 c o m p r i s e on t h e one hand p a r a m e t e r s l i s t e d i n p a r t b e f o r e , b e i n g a p p r o x i m a t e l y c o n s t a n t f 0 r . a g i v e n s i t u a t i o n , and unknown c o e f f i c i e n t s on t h e o t h e r hand, d e r i v e d from t h e h y p o t h e s e s
( E q u a t i o n s 2 t o 4 ) . C o n s e q u e n t l y , t h e s t r u c t u r e o f t h e model s h o u l d b e i d e n t i f i e d and t h e p a r a m e t e r v a l u e s ,
Kit
K 2 , and n , e s t i m a t e dfrom measurements. The f e a s i b i l i t y o f E q u a t i o n ( 5 ) c a n b e a p p r e - c i a t e d from F i g u r e 3 a , which c l e a r l y shows t h e i n f l u e n c e o f t h e wind v e l o c i t y on t h e c o n c e n t r a t i o n . However, a s i m p l e r e g r e s s i o n between t h e W and SS i s n o t p r e c i s e enough; t h e i n v o l v e m e n t of SS i n a t i m e series f a s h i o n improves i t , t h u s s u g g e s t i n g t h e i n - f l u e n c e o f s e t t l i n g and d e p o s i t i o n .
F i r s t a d e t e r m i n i s t i c e s t i m a t i o n t e c h n i q u e was a d o p t e d t o d e r 3 v e t h e unknown c o e f f i c i e n t s which r e s u l t e d i n r e a l i s t i c v a l u e s b u t w3thout p r o v i n g t h e c o r r e c t n e s s o f t h e h y p o t h e s e s ( a p o s t e r i o r i model s t r u c t u r e i d e n t i f i c a t i o n , s e e Beck i n t h i s i s s u e [ 4 ] ) .
For t h i s p u r p o s e , a s a second s t e p , t h e Extended Kalman F i l t e r (EKF) method was a p p l i e d [ 2 , 51
.
F o r t h e power n a v a l u e n e a r t o 1 was d e r i v e d which c o r r e s p o n d e d t o t h e s m a l l R i c h a r d s o n number [ 2 6 ] . S u b s e q u e n t l y n was f i x e d t o 1 s i n c e i n t h i s c a s e t h e p h y s i c a l i n t e r p r e t a t i o n o f t h e r e s u l t s i s more o b v i o u s . The r e c u r s i v e e s t i m a t i o n s t a r t e d from t h e e s t i m a t e s o f t h e d e t e r m i n - i s t i c t e c h n i q u e . The r e s u l t s a r e i l l u s t r a t e d i n F i g u r e 3a. A s i s a p p a r e n t , t h e a g r e e m e n t between o b s e r v a t i o n s and model c a l c u - l a t i o n i s r e a s o n a b l y good, and t h e p a r a m e t e r s become a p p r o x i m a t e l yc o n s t a n t a f t e r t h e f i r s t 40-50 d a y s ( F i g u r e 3 b ) , p r o v i n g t h a t t h e mode2 s t r u c t u r e i s c o r r e c t [ 2 , 4 1 . Some s l i g h t p a r a m e t e r c h a n g e s
c a n b e o b s e r v e d a t t h e end o f t h e p e r i o d ; t h i s may b e c a u s e d , e . g . , by t h e e x c l u s i o n o f i n f l o w - o u t f l o w p r o c e s s e s ( o r by o t h e r phenomena s u c h a s a l g a l b l o o m s ) . T h i s s u g g e s t s t h a t t h e i s o l a t i o n o f sub-
p r o c e s s e s i s g e n e r a l l y n o t c o m p l e t e . From t h e a n a l y s i s , a r e a l i s t i c o r d e r o f m a g n i t u d e s f o l l o w s f o r a l l t h e e s s e n t i a l p h y s i c a l quan-
t i t i e s ; i n t h i s c o n n e c t i o n see [ 2 6 ] .
A s c a n b e o b s e r v e d i n F i g u r e 3 , f o r o n e month i n t h e m i d d l e o f t h e t o t a l p e r i o d , no measurements w e r e a v a i l a b l e , s o t h e model was u s e d f o r p r e d i c t i o n . The a p p r o p r i a t e n e s s o f t h e model i s a l s o i l l u s t r a t e d by t h e f a c t t h a t a f t e r g e t t i n g new d a t a , t h e p a r a m e t e r v a l u e s d i d n o t c h a n g e . T h i s s e c o n d p e r i o d s e r v e d f o r v a l i d a t i o n , f o l l o w i n g t h e i d e n t i f i c a t i o n a n d c a l i b r a t i o n p r o c e d u r e .
The s t u d y , w h i c h u n d e r l i n e s t h e d e f i n i t e n e e d t o combine b o t h t h e o r e t i c a l a n d measurement knowledge, r e s u l t e d i n two b a s i c a c h i e v e m e n t s : ( i ) t h e e s t i m a t i o n o f t h e unknown b o u n d a r y c o n d i t i o n o f a t r a n s p o r t problem (which was t h e n a l s o s o l v e d by u s i n g a n i m - p l i c i t f i n i t e d i f f e r e n c e method b u t t h e "submodel" was n o t main- t a i n e d f o r t h e c o m p l e x s t u d y a s t h e v e r t i c a l c h a n g e s a r e n o t e s s e n - t i a l i n t h i s c a s e from t h e p o i n t o f v i e w o f e u t r o p h i c a t i o n )
,
and( i i ) t h e d e s c r i p t i o n o f t h e p r o c e s s e s o f d e p o s i t i o n a n d r e s u s p e n - s i o n t h r o u g h a n ODE which c a n b e e a s i l y i n c o r p o r a t e d i n t o t h e b i o - c h e m i c a l submodel w i t h a s i m i l a r ODE s t r u c t u r e .
I n a d d i t i o n t o t h e wind i n d u c e d i n t e r a c t i o n d i s c u s s e d h e r e , t h e s e d i m e n t b i o c h e m i s t r y i s a l s o o f m a j o r i m p o r t a n c e , a f i e l d where f u r t h e r r e s e a r c h i s r e q u i r e d .
4.2 Application of a ID Hydrodynamic Model (Stratum 4)
The objective of the model has already been explained. The complete one-dimensional equation of momentum and continuity was solved by using a conventional implicit finite difference scheme
[27]. For the matrix inversion, an effective decomposition tech- nique was developed, resulting in economic computations [27].
Dynamic input is the longitudinal component of the wind force, while the output is the water level and streamflow rate at each cross section (Ax = 2000 m). The two parameters of the model
(drag coefficient and bottom friction) were calibrated on the ba- sis of the work of Muszkalay [201. From the data of nearly ten years of observations, he derived empirical relationships between some typical wind parameters of a storm and the corresponding maximum denivellation of water surface and velocity at the Tihany peninsula (Figure I), respectively. For validation, more than ten historical events were selected. The results of one example are shown in Figure 4. This event can be characterized by a long- lasting longitudinal wind followed by smaller shocks of different directions (Figure 4a). It is apparent that the agreement between measured and simulated water levels is satisfactory (Figure 4b).
The discharge shows a striking oscillation between-2000 and 3000 m /s (Figure 4c: associated with the seiche phenomenon) which is 3
higher by two orders of magnitude than the hydrologic throughflow and may cause considerable fluctuation in the volume of the basins
(Figure 1). The seiche phenomenon alone will certainly not cause mixing and its influence can be judged only by the method described
in Section 3.2. Still, its effect can be illustrated from another angle: at a given location the rapid seiche motion will cause an
o s c i l l a t i o n o f v a r i o u s c o n s t i t u e n t s w i t h i n a d a y , which s t r o n g l y d e p e n d s a l s o o n t h e l o n g i t u d i n a l g r a d i e n t . T h i s may r e s u l t i n q u i t e a c r i t i c a l e r r o r i n t h e c o n c e n t r a t i o n d e t e r m i n e d t h r o u g h i n - s t a n t a n e o u s s a m p l i n g , a r e c o g n i t i o n which w i l l a l l o w d e f i n i t i o n t h r o u g h t h e model o f a n u n c e r t a i n t y r a n g e o f h i s t o r i c a l m e a s u r e - m e n t s . To f i n d a more s a t i s f a c t o r y a g r e e m e n t t h a n i n t h e p r e v i o u s example i s o f t e n i m p o s s i b l e . The r e a s o n i s q u i t e s i m p l e : a s m a l l e r r o r i n t h e wind d i r e c t i o n may c a u s e a d r a s t i c c h a n g e i n t h e wind f o r c e , i f t h e d i r e c t i o n i s f a r f r o m t h e l o n g i t u d i n a l o n e . I n f a c t , t h e r e a r e many k i n d s o f u n c e r t a i n t i e s i n t h e wind d i r e c t i o n , s u c h a s random f l u c t u a t i o n ( t u r b u l e n c e ) , t h e i n f l u e n c e o f h i l l s o n t h e n o r t h e r n s i d e o f t h e l a k e , which c a u s e n o n u n i f o r m i t i e s i n t h e wind f i e l d , m e a s u r e m e n t e r r o r s , e t c . F i g u r e 5 i l l u s t r a t e s t h e c a s e
( t r a n s v e r s a l wind c o n d i t i o n s ) . A d e t e r m i n i s t i c s i m u l a t i o n d i d n o t p r o v e a c c e p t a b l e . B e a r i n g i n mind t h e p o s s i b l e r o l e o f un- c e r t a i n t i e s , a random component w a s s u b s e q u e n t l y a d d e d t o t h e wind d i r e c t i o n ( G a u s s i a n d i s t r i b u t i o n , z e r o mean, 12.5O s t a n - d a r d d e v i a t i o n : a m o d e s t v a l u e ) and a Monte C a r l o s i m u l a t i o n was p e r f o r m e d . F i g u r e 5, w h i c h shows t h e r e s u l t s o f a l l t h e
100 r u n s , d o e s n o t r e q u i r e d e t a i l e d d i s c u s s i o n : i t shows t h e ex- t r e m e s e n s 2 t i v i t y t o i n p u t d a t a u n c e r t a i n t y (compared t o t h i s t h e p a r a m e t e r s e n s i t i v i t y i s n e g l i g i b l e ) a n d i l l u s t r a t e s how d i f f i - c u l t it i s t o v a l i d a t e a d e t e r m i n i s t i c model ( t o a lesser e x t e n t t h i s i s a l s o t r u e f o r a m o r e - d i m e n s i o n a l m o d e l ) . T h i s b e h a v i o r a l s o s u g g e s t s t h a t some t i m e a v e r a g i n g s h o u l d b e i n t r o d u c e d f o r t h e t r a n s p o r t model p a r t . A c c o r d i n g t o o u r a n a l y s i s , t h e mean and v a r i a n c e o f t h e f l o w r a t e t i m e s e r i e s o n a d a i l y b a s i s shows l i m i t e d s e n s i t i v i t y o n l y . S i n c e g e n e r a l l y o n e i s n o t i n t e r e s t e d
i n s h o r t - t e r m c o n c e n t r a t i o n c h a n g e s ( t h e c h a r a c t e r o f t h e s a m p l i n g problem mentioned b e f o r e i s d i f f e r e n t ) t h i s a l l o w s u s e o f t h e
mean v a l u e f o r c o n v e c t i o n i n t h e d i s p e r s i o n model, w h i l e t h e e f f e c t b e h i n d t h e v a r i a n c e c a n b e i n c o r p o r a t e d t o t h e d i s p e r s i o n c o e f f i c i e n t .
The example s u g g e s t s t h a t a l t h o u g h a d e t e r m i n i s t i c hydro- dynamic model c a n h a r d l y b e v e r i f i e d i n a s t r i c t s e n s e , t h e same c a n b e done f o r a t r a n s p o r t model i n a w a t e r q u a l i t y s t u d y by f i l t e r i n g o u t t h e i n f l u e n c e o f u n c e r t a i n t i e s .
4.3 The N u t r i e n t Load under U n c e r t a i n t y and S t o c h a s t i c i t y ( S t r a t u m 3 ) I n a c c o r d a n c e w i t h t h e n u t r i e n t l o a d i n g e s t i m a t e done [ 1 5 ] ,
more t h a n 4 0 % o f t h e t o t a l phosphorus ( T P ) l o a d r e a c h e s t h e l a k e t h r o u g h t r i b u t a r i e s . A s i s w e l l known, f l o o d s p l a y a d e c i s i v e r o l e i n t h e y e a r l y t o t a l t r a n s p o r t , t h e i r c o n t r i b u t i o n r a n g i n g gener- a l l y between 70-90%. T h i s f a c t i s i n most o f t h e c a s e s n o t r e - f l e c t e d i n t h e m o n i t o r i n g s t r a t e g y ; g e n e r a l l y o n e o r two o b s e r v a - t i o n s made monthly a t t h e mouth o f t h e r i v e r a r e a v a i l a b l e . Thus t h e i n f l u e n c e o f f l o o d s r e m a i n s unobserved. The i n f r e q u e n t d a t a c o l l e c t i o n i s c h a r a c t e r i s t i c f o r 19 of t h e 2 0 t r i b u t a r i e s o f Lake B a l a t o n , w h i l e f o r t h e m a j o r p o l l u t i o n s o u r c e , t h e R i v e r Zala which a c c o u n t s f o r more t h a n h a l f o f t h e t r i b u t a r y l o a d , d a i l y measurements w e r e performed ( 1975-79, [ I 61 )
.
S i n c e from t h i s d a t a s e t t h e " a c c u r a t e " l o a d f o r d i f f e r e n t a v e r a g i n g p e r i o d s ( s u c h a s a month o r y e a r ) c a n b e d e r i v e d , i t a l l o w s o n e t o s t u d y t h e e r r o r c a u s e d by s c a r c e o b s e r v a t i o n s . The p r o c e d u r e i s a s t r a i g h t f o r w a r d Monte C a r l o t y p e t e c h n i q u e which s t a r t s w i t h a random s e l e c t i o n on t h e d e t a i l e d d a t a s e t
f o l l o w i n g t h e s a m p l i n g s t r a t e g y o f t h e o t h e r t r i b u t a r i e s and c a l - c u l a t e s t h e l o a d o f t h e p e r i o d i n q u e s t i o n . A f t e r making a s u f - f i c i e n t number o f random s e l e c t i o n s t h e s t a t i s t i c a l p a r a m e t e r s o f
t h e l o a d c a n be d e t e r m i n e d . The r e s u l t s f o r t h e l o n g - t e r m monthly a v e r a g e l o a d (on t h e b a s i s o f a f o u r y e a r l o n g o b s e r v a t i o n p e r i o d and 2 0 0 d a t a s e l e c t i o n s ) a r e i l l u s t r a t e d i n F i g u r e 6. The c h o i c e o f a month was made f o r two r e a s o n s : ( i ) b e i n g i n p o s s e s s i o n o f two monthly o b s e r v a t i o n s f o r a p e r i o d o f s e v e r a l y e a r s o n l y t h e monthly a v e r a g e l o a d c a n b e e s t i m a t e d a t b e s t ; ( i i ) from a s e n s i -
t i v i t y s t u d y on LEN [ 2 8 ] , it t u r n e d o u t t h a t it i s s u f f i c i e n t t o u s e t h i s l o a d t y p e a s a f o r c i n g f u n c t i o n . A s c a n b e s e e n from F i g u r e 6, which shows t h e mean and e x t r e m e v a l u e s , . a s w e l l a s t h e domain o f
+
s t a n d a r d d e v i a t i o n , t h e e r r o r i s q u i t e h i g h and i t s f l u c t u a t i o n f o l l o w s t h e change i n t h e mean v a l u e . On t h e b a s i s o f t h i s s t u d y and a s i m i l a r a n a l y s i s f o r t h e y e a r l y a v e r a g e s , t h e a n n u a l l o a d o f o t h e r t r i b u t a r i e s was c o r r e c t e d ( h e r e t h e s i m i l a r i - t i e s o f m a j o r s u b w a t e r s h e d s were a l s o examined, b u t t h e c o r r e c t i o n c e r t a i n l y h a s a n e x t r a p o l a t i v e c h a r a c t e r ) and a random component was added t o t h e monthly a v e r a g e l o a d component [ 2 8 ] .To i n c o r p o r a t e t h e s t o c h a s t i c i n f Z u e n c e o f t h e h y d r o l o g i c regime a r e g r e s s i o n a n a l y s i s w a s done on t h e d a t a s e t o f R i v e r Z a l a . I t was found t h a t t h e monthly a v e r a g e TP l o a d c o r r e l a t e d s a t i s f a c t o r i l y w i t h t h e c o r r e s p o n d i n g s t r e a m f l o w r a t e . A c c e p t i n g t h e s t a t i s t i c s o f t h e monthly a v e r a g e s t r e a m f l o w from l o n g - t e r m o b s e r v a t i o n s [ I ] t h e l o a d c a n be c a l c u l a t e d i n a s t o c h a s t i c f a s h i o n . F i g u r e 7 shows t h e c h a r a c t e r i s t i c s o f t h e l o a d p a t t e r n f o r 1976-79 and t h e 90% c o n f i d e n c e l e v e l s d e r i v e d . For i l l u s t r a t i n g t h e i n - f l u e n c e o f t h e h y d r o l o g i c regime a n e v e n t o f low p r o b a b i l i t y i n J u l y 1975, i s l i k e w i s e i n d i c a t e d .
A s a f i n a l o u t p u t o f t h e r e s e a r c h o u t l i n e d i n t h i s s e c t i o n a l o a d s c e n a r i o g e n e r a t o r was d e v e l o p e d f o r t h e whole l a k e , which
a c c o u n t e d f o r b o t h u n c e r t a i n t y and s t o c h a s t i c i t y , d i s c u s s e d above.
For f u r t h e r d e t a i l s s e e [ 2 8 ] .
I t Is n o t e d h e r e t h a t u s i n g h i s t o r i c a l d a t a , a s i m i l a r a n a l y - s i s was made on c l i m a t i c ( u n c o n t r o l l a b l e ) f a c t o r s , which a l l o w e d t h e w a t e r t e m p e r a t u r e and d a i l y r a d i a t i o n t o be g e n e r a t e d i n h a r - mony w i t h e a c h o t h e r , i n a random f a s h i o n [ 2 8 ] . Thus, f u t u r e
s c e n a r i o s c a n be g e n e r a t e d f o r a l l t h e e s s e n t i a l f o r c i n g f u n c t i o n s o f t h e l a k e model--an e s s e n t i a l t o o l f o r p l a n n i n g p u r p o s e s .
4 . 4 The Lake E u t r o p h i c a t i o n Model ( S t r a t u m 3 )
The p r e l i m i n a r y r e s u l t s g a i n e d w i t h t h e s i m p l e s t model, SIMBAL [ 3 1 ] , d e v e l o p e d f o r Lake B a l a t o n a r e g i v e n below. The
model i s a phosphorus c y c l e model, t h a t i s , a l l t h e s t a t e v a r i a b l e s ( t h e e s s e n t i a l s a r e two a l g a l g r o u p s , d e t r i t u s , and d i s s o l v e d
i n o r g a n i c p h o s p h o r u s ) a r e e x p r e s s e d i n t e r m s o f p h o s p h o r u s , f o r t h e f o u r b a s i n s i n d i c a t e d i n F i g u r e l ( s e e S e c t i o n 3 . 2 ) . A Monte C a r l o s i m u l a t i o n i s i n c o r p o r a t e d i n t o t h e model t o f i n d a r e a s i n p a r a m e t e r s p a c e where t h e model p r o d u c e s r e s u l t s f u l l y w i t h i n
s p e c i f i e d b o u n d a r i e s drawn a r o u n d t h e d a t a t o a c c o u n t f o r d a t a u n c e r t a i n t y (see, e . g . , S e c t i o n 3.2) and t h u s e a s i l y a p p l i c a b l e t o t e s t v a r i o u s h y p o t h e s e s ( [31] and a l s o [9, 1 4 1 )
.
Among t h e c a l i b r a t i o n r u n s , r e s u l t s f o r , t h e p h y t o p l a n k t o n p h o s p h o r u s , PPP, f o r t h e f o u r b a s i n s a r e g i v e n i n F i g u r e 8 ( a s
1977 f o r c i n g s d a t a was u s e d ) t o g e t h e r w i t h t h e c o r r e s p o n d i n g o b s e r v a t i o n v a r i a b l e , C h l o r o p h y l l - a ( b a s i n a v e r a g e v a l u e s ) . I t
2s p o i n t e d o u t t h a t C h l o r o p h y l l - a and PPP c a n n o t be d i r e c t l y com- p a r e d t o e a c h o t h e r ; however, s i n c e a more o r l e s s l i n e a r measure- ment e q u a t i o n Is e x p e c t e d among them, PPP s h o u l d f o l l o w t h e p a t t e r n o f C h l o r o p h y l l - a : a t r e n d which c a n be g e n e r a l l y o b s e r v e d . For
i l l u s t r a t i o n t h e s t a n d a r d d e v i a t i o n a r o u n d t h e t r a j e c t o r y f o r Basin 2 e s t i m a t e d t h r o u g h t h e Monte C a r l o s i m u l a t i o n i s a l s o i n - d i c a t e d ( p a r a m e t e r u n c e r t a i n t y ) . F u r t h e r d i s c u s s i o n on t h e c a l i - b r a t i o n and model improvement r e q u i r e d c a n be found i n [ 3 1 ] .
For management p u r p o s e s h i s t o r i c a l d a t a c a n n o t be u s e d .
E i t h e r some c r i t i c a l , u n f a v o r a b l e e n v i r o n m e n t a l c o n d i t i o n s s h o u l d be i n t r o d u c e d o r t h e model s h o u l d b e c o n s i d e r e d s t o c h a s t i c t h r o u g h
i n p u t d a t a which a r e b a s i c a l l y random v a r i a b l e s when f u t u r e p l a n - n i n g i s i n q u e s t i o n . H e r e t h e l a t t e r a p p r o a c h was a d o p t e d and t h e g e n e r a t o r s o u t l i n e d i n t h e p r e v i o u s s e c t i o n c o u p l e d t o t h e l a k e model. Two e s s e n t i a l r e s u l t s f o r B a s i n 1 a r e p r e s e n t e d i n F i g u r e s 9 and 10.
I n t h e f i r s t c a s e , u n c e r t a i n t i e s c a u s e d by n a t u r a l f a c t o r s were c o n s i d e r e d and t h e 1977 l o a d was m a i n t a i n e d . The summary o f
100 Monte C a r l o r u n s (mean,
