cathcart 1987 aquacultural-engineering

7/28/2019 Cathcart 1987 Aquacultural-Engineering

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Aquacultural Engineering 6 ( 19 8 7 ) 2 3 7 - 2 5 7

M odeling T emperature D istribution in FreshwaterPonds*T h o m a s P . C a t h c a r t a n d F r e d e r i c k W . W h e a t o n

D epa rtm en t of Agricultural Engineering, Un iversi ty of M aryland, College Park,M D 2 0 7 4 2, U S A

A B S T R A C TA m od el for predict ing the vertical temperature profi le o f turbid fresh-water po nd s is described. High resolution in space and t im e an d a reduc-tion in the num be r of meteorological inputs usually required by suc hm ode ls is achieved b y using po nd surface temperature as a mo de l input.A m eth od to est imate hourly surface temperature is prop osed a nddemonstrated.

N O M E N C L A T U R EABe l

EF( t )Hml?Q | O S SO,Omr

S ( z )

S u r f a c e a r e a ( m 2)F r a c t io n o f t h e s o l a r en e r g y s p e c t r u m h a v i n g w a v e l e n g t h s g r e a t e rt h an 7 0 0 n m .S l o p e o f r e g r e s s i o n l in eS p e c i fi c h e a t o f w a t e r ( k J ( k g C ) - 1)M o l e c u l a r d i f f u s iv i t y o f w a t e r ( m ~- h - ~)S u r f a c e t e m p e r a t u r e i n t e rp o l a t io n f u n c t io nH e a t c o n t e n t o f p o n d l a y e r ( J)N u m b e r o f m o d e l e l em e n t sE x t i n c t i o n c o e f f ic i e n t ( m - ~)S u r f a c e h e a t l o ss ( J)T o t a l h e a t fl u x a c r o s s t h e b o u n d a r i e s o f e l e m e n t 1 ( J)H e a t f lu x a c r o s s t h e lo w e r b o u n d a r y o f e l e m e n t 1 (J )F r a c t i o n o f t h e in c i d e n t s o l a r r a d i a t i o n r e f l e c t e d b y th e w a t e rs u r f a c eI n s o l a ti o n a t d e p t h z b e l o w t h e s u r f a c e (J m - 2 )

*Scien tif ic Ar t i c le N um ber A -4460 C on t r ibu t ion N um ber 7 452 o f t he Mary landAgricultural Ex perim ent S tat ion (D epa rtm ent of Agricultural Engineering).237Aquacultural Engineering 0 1 4 4 - 8 6 0 9 / 8 7 / S 0 3 .5 0 - - E l s ev i er A p p l i ed S ciencePub lishers Ltd, England, 1987. Printed in Grea t Bri tain


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238 T. P . Ca thcar t , F . W. W hea ton

S oT

ttLtHV(z)x ;Zz l/9

Insola t ion a t the w ater surface (J m -2)Tem pera tu re ( C)Da i ly low tem pera tu re (C)Da i ly h igh t empera tu re (C)T i m e (h)T im e of day o f TL (h)T i m e o f d a y of TH (h)Volume o f pond e lements (m3)H eat a dde d to e lem ent i due to wind mixing (J )Dep th , inc reas ing dow nw ard (m)D epth o f the bo t tom of the su r face e lement (m)W ater dens i ty (kg m-3 )

I N T R O D U C T I O NT he re i s current ly a la rge and growing body of li te ra ture assoc ia ted wi tha t tempts to predic t tempera ture d is t r ibut ion in aquat ic sys tems . Ingenera l, these e ffor ts have bee n fo cused on sa l t-s tra ti f ied (non-convect -ing) so lar ponds , cool ing ponds used to d iss ipa te was te hea t , and la rgewater b odies , such as lakes, reservoirs , and loca l ly def ined por t ions ofoceans (J irka e t a l . , 1978 ; De lnore , 1980 ; Har leman , 1982 ; Meyer ,1983; Sturm e t a l . , 1983; Pa t te rson e t a l . , 1984).The p red ic t ion o f wa te r co lumn tempera tu re i s a ma t t e r o f someimportance to aquacul tur is t s . For cold b looded organisms , tempera turedi rec t ly a ffec ts growth ra te , the onse t of reproduct ion , and the occur-rence of the rm al s tress. I t a lso m ay inf luence res is tance to d isease andbehav io r. Tem pera tu re g rad ien ts in f luence pond mix ing p roces sesthrough the concomitant c rea t ion of a ver t ica l dens i ty gradient . Thedens i ty gradien t tends to in hib i t ver t ica l m ixing and so m ay be animportant de terminant of d issolved oxygen and was te metabol i te d is t r i -bu t ion in the pond env i ronment . Because mos t aquacu l tu ra l ponds a rehighly turbid , tem pera tu re gradients (as grea t as 4C m -1) f requen t lyoccur dur ing the growing season. A re l iable tem pera ture d is t r ibut ionmode l wou ld enhance the capac i ty o f aquacu l tu r i s t s to moni to r pondcondi t ions , predic t p er iods of sub-opt im al growth, and m ake in te ll igentm anagem ent dec is ions .

There i s , a t present , no tempera ture d is t r ibut ion model spec i f ica l lydes igned for the types of ponds f requent ly encountered in aquacul ture .Th e app l i cab i li ty o f ex i st ing mode l s to the aquacu l tu ra l p ro b lem i s some-what l imi ted . Sa l t -s t ra t i f ied so lar ponds a re charac ter ized by very lowturbid i ty and an u nusu al (i .e . ' reversed ') temp era ture gradient. Mo dels of


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M odel ing temperature in f resh water po nd s 2 3 9l a rg e w a t e r b o d i e s a r e f r e q u e n t l y li m i t e d b y l o w r e s o l u t io n s i n s p a c e a n dt i m e ( o n t h e o r d e r o f m e t e r s a n d d a ys ). F o r a q u a c u l t u r is t s , re s o l u t i o n s i nh o u r s a n d f r a c t io n s o f m e t e r s a r e d e s i r a b le .O n e p o s s i b l e s t ra t e g y is to a d a p t a n a p p r o p r i a t e m o d e l t o t h e s p e ci a lc o n d i t i o n s o f a q u a c u l t u r a l p o n d s , b u t h e r e t o o t h e r e a r e p r o b l e m s . M o s tt e m p e r a t u r e d i s tr i b u ti o n m o d e l s r e q u i r e a n u m b e r o f m e t e o r o l o g ic a li n p u t s t o e s t i m a t e t h e h e a t f l u x a c r o s s t h e a i r - w a t e r i n t e r f a c e a n d t h eshea r s t r e s s p l aced on the wa te r su r f ace by the wind ( J i rka e t a l . , 1 9 7 8 ;S t u r m e t a l . , 1983) . Som e o f t he i npu t s , such a s i nc iden t so l a r r ad i a t i on ,t e n d t o r e m a i n r e la t iv e l y c o n s t a n t o v e r s u b s t a n t i a l g e o g r a p h i c a r e a s , a n ds o m a y b e d e r i v e d f r o m n e a r b y w e a t h e r s t a t i o n s ( J i r k a e t a l . , 1978) .Oth e r s , how eve r , such a s wind v e loc i ty and d i r ec t ion , a r e sub jec t t o con-s ide rab le l oca l va r i a t i on . I f u se o f empi r i ca l cons t an t s i s t o be avo ided ,m e a s u r e m e n t s m u s t b e t a k e n v e r y n e a r t o t h e s i te o f i n t e r e s t ( B lo s s a n dH a r l e m a n , 1 9 7 9 ; S t u r m e t a l . , 1 9 8 3 ). A c q u i s i t io n o f w e a t h e r m o n i t o r i n ge q u i p m e n t ( a n d t h e r e q u i r e m e n t s o f m a i n t e n a n c e ) is p r o b a b l y n o t ap r a c t i c a l o p t i o n f o r m o s t a q u a c u l t u r a l o p e r a t i o n s d u e t o c o s t . F o r amode l t o be use fu l i n t he p r iva t e s ec to r , some (pe rhaps subs t an t i a l )s i m p l i fi c a ti o n is r e q u i r e d o v e r m o s t c o n v e n t i o n a l t e m p e r a t u r e d i s tr i b u -t i o n m o d e l s ; a n d t h is s i m p l i f ic a t io n s h o u l d b e a c c o m p l i s h e d w i t h o u t ac o r r e s p o n d i n g d e c r e a s e i n a c c u r a cy !T he s t r a tegy ado p te d in th i s pa pe r i s r e l a t ive ly s imple : po nd su r f acet e m p e r a t u r e i s a s s u m e d k n o w n a n d i s u s e d a s a m o d e l i n p u t . T h i sa p p r o a c h o b v i a t e s th e n e e d t o e s t im a t e s u r f a c e h e a t f lu x a n d p r o v i d e s a na l t e r n a t e m e t h o d f o r c o m p u t i n g t h e e f f e c t o f w i n d o n t e m p e r a t u r e d i s -t r ib u t i o n . C le a r ly , t h is h a s c h a n g e d t h e n a t u r e o f t h e m o d e l . T h e p r o b l e mn o w i s t o e s t i m a t e n o t t h e t e m p e r a t u r e g r a d i e n t o f t h e e n t i r e w a t e rc o l u m n , b u t r a t h e r t h e g r a d i e n t b e n e a t h a s u r f ac e e l e m e n t o f s o m e p r e -s c r ib e d d e p t h . T h e r a t i o n a l e o f t h is a p p r o a c h is th a t , u n l i k e m a n y o t h e rt y p e s o f w a t e r b o d i e s , a q u a c u l t u r a l p o n d s r e c e i v e f r e q u e n t a t t e n t i o nd u r i n g t h e g r o w i n g s ea s o n . U s e o f th i s m o d e l r e q u i r e s t h a t s o m e m e t h o dbe u t i l i z ed to p rov ide hour ly e s t ima te s o f su r f ace wa te r t empera tu re .F o l l o w i n g a d e s c r i p t i o n o f t h e m o d e l , a m e t h o d t o p r o v i d e r e a s o n a b l eh o u r l y e s ti m a t e s is p r o p o s e d a n d d e m o n s t r a t e d .

M O D E L D E V E L O P M E N TT h e t e m p e r a t u r e d i s tr i b u ti o n m o d e l p r e s e n t e d h e r e c o n si st s o f th r e ep a r ts . T h e f ir s t p a r t a c c o u n t s f o r m o l e c u l a r d i f fu s i o n a n d i n t e r n a la b s o r p t i o n o f s o l a r r a d ia t i o n . T h e s e c o n d p o r t i o n i d e n t i fi e s a n d r e m o v e sb u o y a n t i n s ta b il it y f r o m t h e w a t e r c o l u m n a n d s o a p p r o x i m a t e s t h e r m a l


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24 0 T. P . Ca thcar t , F . W. W hea ton

convec t i on . T h e t h i rd pa r t e s t i m a t e s t he quan t i t y and d i s t r i bu t i on o f hea tm i x e d d o w n i n t o t h e w a t e r c o l u m n b y w i n d a c t i o n . T h e f i r s t t w o p a r t sc o r r e s p o n d r a t h e r c l o se l y t o o t h e r m o d e l s a n d a r e s u m m a r i z e d o n l yb r ie fl y. T h e a p p r o a c h u s e d t o e s ti m a t e t h e q u a n t i t y a n d d i s tr i b u ti o n o fw i n d m i x e d h e a t i s o ri g in a l t o t h is m o d e l a n d is tr e a t e d w i t h m o r e d e p t h .M olec ula r diffusion and solar absorptionFo l l ow i ng J i rka e t a / . (1978) , t he one -d i m ens i ona l h ea t equ a t i on ( eqn ( 1 ))w i t h a hea t sou rce i s u sed t o e s t i m a t e t he m o l ecu l a r d i f fu s ion o f hea t andt h e v a r i a b l e a b s o r p t i o n o f s o la r r a d i a t i o n w i th d e p t h .

a T E a ( v ( z ) a _ s ) 1 f f zc 3 t = V ( z---)) z A C pp ( S ( z ) A ) (1)T h e f u n c t io n S ( z ) c o r r e s p o n d s t o B e e r ' s L a w f o r t h e i n te r n a l a b s o r p t i o no f so l a r r ad i a t i on :

S(z ) = (1 - r)(1 - B ) S o - n z l (2)T h e t e r m 1 - B r e p r e s e n t s t h e p o r t i o n o f t h e s o la r e n e r g y s p e c tr u mb e t w e e n 3 0 0 a n d 7 0 0 r an . T h e v a l u e o f 1 - B a p p e a r s t o v a r y b e t w e e n0 .4 an d 0 .45 . J i rka e t a L (1978) a s sum e a va l ue fo r 1 - B o f 0 .45 fo r t he i rm o d e l . T h e s a m e a s s u m p t i o n is m a d e i n th e p r e s e n t m o d e l .The use o f e l em en t vo l um es (V (z ) ) i s r equ i red t o conse rve ene rgy i nt h e h e a t e q u a t i o n (1 ). E l e m e n t v o l u m e s a r e r e c a lc u l a te d f o r e v e r y c h a n g ei n p o n d e l ev a t io n . U s e o f t h e m o d e l p r e s u p p o s e s t h a t a n a c c u r a teb o t t o m c o n t o u r m a p o f t h e p o n d is a v a il a bl e.A n exp l ic i t f i n it e d i f f e rence num er i ca l m e t ho d i s u sed t o so l ve eqn (1)f o r e a c h t im e s t ep . T h e l o c a t io n s o f e l e m e n t n o d e s a r e r e d e t e r m i n e d f o re a c h c h a n g e i n p o n d e l e v a t i o n . T h e s u r f a c e b o u n d a r y c o n d i t i o n i s t h ek n o w n t e m p e r a t u r e o f t h e s u rf a c e e l e m e n t . It is a s s u m e d t h a t th e r e i s n oh e a t f l u x t h r o u g h t h e b o t t o m o r s i d e s o f t h e p o n d a n d t h a t t h e p o n dt e m p e r a t u r e s t r u c t u r e is h o r i z o n t a l l y h o m o g e n e o u s .Therm al convect ionW h e n t h e s u r fa c e h e a t l o ss e x c e e d s t h e a m o u n t o f ra d i a t io n e n t e r i n g th ep o n d f r o m a b o v e , t h e w a t e r c o l u m n b e c o m e s b u o y a n t l y u n s t a b l e (i.e .c o o l e r w a t e r o v e r l a y s w a r m e r ) . A d j a c e n t e l e m e n t s t h a t a r e b u o y a n t l yu n s t a b l e a r e l o c a t e d a n d m i x e d i n t h e p r e s e n t m o d e l . A s i n t h e c a s e o ft h e h e a t e q u a t i o n , k n o w l e d g e o f e l e m e n t v o l u m e s i s r e q u i r e d f o r e n e r g yt o b e c o n s e r v e d .


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M odeling temperature in freshwater ponds 241W i n d - m i x e d h e a t

I n o r d e r t o e s t i m a t e t h e a m o u n t o f e n e r g y m i x e d d o w n i n t o t h e w a t e rco l um n by t he w i nd , t he m ode l f i r s t ca l cu l a t e s an uppe r l i m i t , o r r e se r -v o ir , o f e n e r g y w h i c h c a n n o t o t h e r w i s e b e a c c o u n t e d f o r i n t h e p o n d h e a tba lanc e (eqn (3)) :O = S o A ( 1 - r ) - A H ~ - A H 2 - Q~oss (3 )

w h ere A H~ re fe r s t o t he h ea t ch ange i n t he su r face e l em en t du r i ng ag i ven ti m e s t ep , A H 2 i s t he he a t ch ange i n t he r em a i nd e r o f the w a t e rco lum n, an d Q~os, s t he he a t f lux across the a i r -w ate r in t e r face .

T h e h e a t c h a n g e i n t h e s u r fa c e e l e m e n t m a y b e w r i tt e n :A H I = S o A ( 1 - r ) ( 1 - B ) ( a - e - " ~ ) + S o A ( 1 - r ) B - Q ] (4)

t o a c c o u n t f o r t h e w a v e l e n g t h - d e p e n d e n t n a t u r e o f e n e r g y a b s o r p t i o n i nt he pond . T he f i rs t t e rm on t he r i gh t han d s i de (RH S) o f eq n (4) rep re -s e n ts s o l ar e n e r g y a b s o r p t i o n o f w a v e l e n g t h s i n t h e r a n g e 3 0 0 - 7 0 0 n m( R a b l a n d N i e l s o n , 1 9 7 5 ) . A b s o r p t i o n i n t h i s r e g i o n m a y b e a p p r o x i -m a t e d u s in g a B e e r ' s L a w f u n ct io n . T h e s e c o n d t e r m o n t h e R H S o f e q n( 4 ) r e p r e s e n t s a b s o r p t i o n a t w a v e l e n g t h s g r e a t e r t h a n 7 0 0 n m , w h i c hoccu r s w i t h i n a f ew cm o f t he su r face i n t u rb i d w a t e r s . The t h i rd t e rm ,Q , , r ep re sen t s t r anspor t ou t o f t he su r face l aye r , e i t he r ac ross t hea i r - w a t e r i n t e r f a c e o r d o w n i n t o t h e w a t e r c o lu m n .T h e h e a t c h a n g e i n th e r e m a i n d e r o f t h e w a t e r c o l u m n ( AH 2) m a y b ewri t t en :A H 2 = SoA ( 1 - r ) (1 - B)e'"ZJ + Qm (5 )

T h e f i r st t e r m o n t h e R H S r e p r e s e n t s a b s o r p t i o n o f so l a r r a d i a t i o n i n t h er a n g e o f 3 0 0 - 7 0 0 m n . T h e s e c o n d t e r m ( Q m ) a c c o u n t s f o r a d d i t i o n a lt r a n s p o r t o f h e a t f r o m t h e s u r f a c e la y e r.

Re w r i t i ng eqn (3 ) w i t h eqns (4) and (5 ) and c l ea r i ng t e rm s g ives:Q] = Qm + Qio~s (6)

H ea t t r ansp or t ou t o f e l em e n t 1 (Q ~ ) m ay be d i v i ded i n t o tw o pa r t s :A H ] = ( S o A ( 1 - r)(a - B ) ( 1 - e - " ~ ) - Q 1 1 ) + ( S o A ( 1 - r ) B - Q 1 2 ) (7 )

w h ere Q ~ - -Q I~ + Q 12. W e t hen a s sum e t ha t a l l l ong -w ave ( > 70 0 nm )so l a r r ad i a t ion , ha v i ng be en abs o rbe d c l o se t o t he su r face , is a l m os ti m m e d i a t e l y r e t u r n e d t o th e a t m o s p h e r e t h r o u g h s u r f a c e lo s se s :Q I2 = S o A ( 1 - r )B (8 )


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242 T. P . Ca thcar t , F. W. W hea ton

so t ha tA H 1 = S o A ( 1 - r )( 1 - B ) ( 1 - e - n Z l ) - Q l l (9)

A p p r o x i m a t i n g o b s e r v e d h e a t c h a n g e s i n t h e s u r f a c e e l e m e n t w i t h al in e a r f u n c t i o n o f S o A yie lds :

Q n = S o A ( 1 - r ) ( 1 - B ) ( 1 - e - n Z l ) - C I S o A ( 1 - r ) ( I 0 )w h ere C1 i s t he s l ope o f t he r eg re s s i on l ine . T he qua n t i t y o f hea t ene rgy ,w h i ch is equa l t o

Q l l = Q m + {~o ss- Q12 (11)p r o v i d e s t h e u p p e r l i m i t o n t h e a m o u n t o f e n e r g y w h i c h m a y b e m i x e dd o w n i n t o th e w a t e r c o l u m n .A n u m b e r o f s e p a r a t e p r o c e s s e s , c a u s e d u l t im a t e l y b y t h e e f fe c ts o ft h e w i n d o n t h e w a t e r c o l u m n , m a y r e s u l t i n t h e d o w n w a r d t r a n s p o r t o fh e a t . T h e s e i n c l u d e t h e c r e a t i o n o f c u r r e n t s w h i c h l e a d t o t u r b u l e n c e ,t u rbu l enc e cause d b y t he b reak i ng o r b i l l ow i ng o f in t e rna l w aves , se i ch -i n g , a n d b o u n d a r y t u r b u l e n c e , t o n a m e a f e w ( F i s c h e r e t aL , 1979) .C l e a r ly , t h e t o p ic i s c o m p l e x . M a n y q u e s t i o n s r e m a i n a b o u t t h e m e c h a n -i sm s o f m i x i ng , and t he i r r e l a t ive i m p or t anc e , i n aqua t i c bod i e s . I t is no tw i t h in t h e s c o p e o f th e p r e s e n t m o d e l t o p r e d i c t th e m e c h a n i s m s w h i c hm a y b e a c t i v e i n a n a q u a c u l t u r a l p o n d . R a t h e r , t h e a t t e m p t h a s b e e nm ade t o u se t he s i ng l e p i ece o f i n fo rm a t i on ava i l ab l e to u s , t he de ns i t yg rad i en t , t o e s t i m a t e how t he r e se rvo i r o f ' w i nd -m i xed hea t ' m ay be d i s -t r ib u t e d i n th e w a t e r c o l u m n .T h e a p p r o a c h u s e d t o d i s t r i b u t e t h e w i n d - m i x e d h e a t i s s u m m a r i z e db y t h e s et o f m - 1 s i m u l t a n e o u s e q u a ti o n s :Fo r i--- 1 to m - 1 ,

mEx, Ev,J=i+ l J~ i+ l

m

E Ek = i k = i

e -p/z (12)

w h e r e m is t h e n u m b e r o f m o d e l e l e m e n t s a n d X i s t h e a m o u n t o f h e a ta d d e d t o e a c h e l e m e n t . U s i n g th i s a p p r o a c h , t h e a b s e n c e o f a d e n s it yg r a d i e n t ( O p / O z = 0 ) r e su l ts i n u n i f o r m d i s t r ib u t i o n o f w i n d - m i x e d h e a to n a p e r u n i t v o l u m e b a s is . U s e o f a n e x p o n e n t i a l d is t r ib u t i o n o f w i n d -m i xed hea t i n t he p re sence o f a dens i t y g rad i en t r e f l ec t s t he cos t i nk i ne t i c ene rgy o f t r ansp or t i ng h ea t a l ong a dens i t y g rad ien t . T he spec i fi cu s e o f a n e x p o n e n t i a l d is t r ib u t i o n is p u r e l y a m a t h e m a t i c a l c o n v e n i e n c e


7/21

M odeling temperature in fresh waterponds 24 3a n d is a f ir st a p p r o x i m a t i o n w i t h o u t i m p l y i n g a n y s p e c ia l in s i g h t i n t o t h ep h y s i c a l p r o b l e m .

E q u a t i o n ( 1 2) r e su l ts i n m - 1 s i m u l t a n e o u s e q u a t io n s . S i n c e t h e r e a r em u n k n o w n s ( X ~ , . . . , X m ) , a n m t h e q u a t i o n is r e q u ir e d . T h i s i s p r o v i d e db y t h e s u m m a t io n :Z X ~ = Qm+ Q~oss- Q12 (13)i=~

T h e e q u a t i o n s a r e t h e n s o l v e d u s in g G a u s s i a n e l im i n a t i o n .

Est imat ion o f hour ly sur face temp eratureT h e m o d e l , a s h e r e p r e s e n t e d , r e q u i r e s t h a t h o u r l y s u r f a c e w a t e r t e m -p e r a t u r e s b e a v a i la b l e a s m o d e l i n p u t s . I t is p r o b a b l y n o t r e a s o n a b l e t oa s s u m e t h a t s in g le c h a n n e l t e m p e r a t u r e r e c o r d e r s w i ll b e u s e d t o p r o v i d eac tua l measu remen t s . I t i s neces sa ry , t hen , t o subs t i t u t e a me thod fo re s t i m a t i n g h o u r l y t e m p e r a t u r e s o n t h e b a s i s o f s o m e c o n v e n i e n t l ym e a s u r e d p h y s i c a l p a r a m e t e r . A p o s s i b le s o l u t i o n t o t h is p r o b l e m m a yb e t h e u s e o f d ai ly m a x i m u m - m i n i m u m t h e rm o m e t e rs . S u c h a t h e r m o -me te r , if he ld s l igh t ly be low the su r f ace (on the o rd e r o f 1 0 -1 5 cm) andread on a da i ly bas i s, w i ll p rov ide va lues fo r da i ly h igh and low su r f acewa te r t empera tu re s . I f t hese va lues cou ld be r e l i ab ly a s soc i a t ed wi thhou r s o f the da y ( i.e . ave rage hou r s o f l ow and h igh t empe ra tu re ) t hen abas is wou ld ex i s t fo r i n t e rpo la t ing su r f ace tem pera tu re fo r the r em a in ing2 2 h o u r s . T h e p r e s e n t m o d e l e s t i m a t e s h o u r l y t e m p e r a t u r e s b e t w e e nd a i ly h i g h a n d l o w w a t e r t e m p e r a t u r e s u s in g a f u n c t i o n F ( t ) h a v i n g t h eder iva t ive :

d Fd t K ( t - - t L ) ( t - - t H ) (14)wh ere K i s an e qu a t ion con s t an t , t r ep re sen t s t he t ime o f day (h ), and tLand tH a re t he t imes o f day w hen the t em pera tu re i s a t it s da i ly m in im umor m ax im um va lue , r e spec tive ly . Th en , u s ing the cond i t i ons :

F( /L ) - - rL( 1 5 )

F ( t . ) = r .E q n ( 14 ) c a n b e i n t e g r a t e d a n d t h e v al u e s o f K a n d t h e i n t e g r a ti o n c o n -s t an t a sce r t a ined . Th i s y i e lds a cu b ic func t io n hav ing fi r st de r iva t ives o f


8/21


9/21

M odeling temperature in freshwater pond s 245

(a)

(b)

+ + +

Fig. 2. Sc hem atic epresentation o f the pond (L-- 78 m, W -- 22 m) used to validate thepond temperature model. (a): Contour map of pond bottom. Th e ou ter line is at depth 0,the seco nd lin e is at depth 0.15 m. Rem aining lines correspond to depth increments of0.3 m. (b): C ross es ma rk the approxim ate locations of stakes used to support therm o-couples.

T h e p e n e t r a t i o n o f l ig h t i n to t h e w a t e r c o l u m n w a s m e a s u r e d u s i n g aL y c o r L I - 1 8 8 B l i g h t m e t e r w i t h a s u b m e r s i b l e s e n s o r . M e a s u r e m e n t sw e r e m a d e a p p r o x i m a t e l y t w i c e a w e e k a n d w e r e u s e d t o c a l c u la t ee x t i n c t i o n c o e f f i c i e n t s . O b s e . r y a t i o n s o f p o n d e l e v a t i o n w e r e m a d e ,a p p r o x i m a t e l y t w i c e a w e e k , u s i n g a d e p t h m a r k e r m o u n t e d n e a r t h ee d g e o f t h e p o n d . E x t i n c t i o n c o e f f i c ie n t s a n d p o n d e l e v a t i o n s w e r e e st i-m a t e d b y l i n e a r i n t e r p o l a t i o n o n d a y s w h e n m e a s u r e m e n t s w e r e n o tt a k e n . H o u r l y v a l u e s o f i n c i d e n t s o l a r r a d i a t i o n w e r e t a k e n a t a si tea p p r o x i m a t e l y 6 5 k m f r o m t h e W y e In s ti tu t e.O ne d im e ns io na li tyU s e o f e q n ( 1 ) i n t h e p r e s e n t m o d e l r e q u i r e s t h a t t h e t e m p e r a t u r e d i st ri -b u t i o n i n t h e p o n d b e h o r i z o n t a l l y h o m o g e n e o u s . T h e r e i s s o m e a g r e e -m e n t a m o n g m o d e l e r s (e .g . J i rk a e t a l . , 1 9 7 8 ; P a t t e r s o n e t a l . , 1 9 8 4 ) t h a tt h e v o l u m e f l o w r a t e i n t o a n d o u t o f t h e p o n d m a y b e u s e d t o a s s e s s t h e


10/21

246 T . P . Ca thca r t , F . IV. W h ea ton

o n e - d i m e n s i o n a l a ss u m p t i o n . A h i g h fl o w r a te , r e l at iv e t o p o n d v o l u m e ,m a y r e s u lt i n m i x in g a n d / o r t h e c r e a t i o n o f c u r re n t s w h i c h c o u l d l e ad t ohor i zon ta l t empera tu re va r i a t i ons . Excep t du r ing pe r iods o f r a in , t h i sw a s p r o b a b l y n o t a f a c t o r i n th e p o n d u s e d f o r m o d e l v a l i d a ti o n , I n f lo wto the po nd was , in gene ra l, l es s than the ev apo ra t ion r a t e a s ev iden cedb y t h e s l o w b u t s t e a d y d e c r e a s e i n p o n d e l e v a t i o n b e t w e e n p e r i o d s o fp rec ip i t a t i on . Du r ing an d im m edia t e ly a f t e r s to rms , fl ows in to and o u t o ft h e p o n d w e r e o b s e r v e d t o d r a m a t i ca l l y i n c re a s e . T h e r e w a s, h o w e v e r,n o w a y t o q u a n t i t a ti v e l y m e a s u r e t h e s e fl ow s a n d t h e d e g r e e a n d d u r a -t i o n o f p e r t u r b a t i o n c o u l d n o t b e c a l cu l a te d .A q u a l it a ti v e e x a m i n a t i o n o f th e p o n d t e m p e r a t u r e r e c o r d s ug g es tst h a t t h e o n e - d i m e n s i o n a l a s s u m p t i o n w a s v a l id m o s t o f t h e t i m e . D u r i n gdry pe r iods , ho r i zon ta l va r i a t i ons appea r t o be l e s s t han 0 . 5C . Dur ings to rm ac tiv ity , va r i a t i ons a r e , fo r t he m os t pa r t , l es s t han 0 .5C a l thou gho c c a s i o n a l v a r i a ti o n s o f I C a n d m o r e w e r e o b s e r v e d . G a p s i n t h e t e m -p e r a t u r e r e c o r d , w h i c h f r e q u e n t l y o c c u r r e d d u r i n g p e r i o d s o f s t o r m ywea the r , l imi t the c om ple t eness o f such obse rva t ions . A s t a ti s ti ca l com -p a r i s o n o f h o r i z o n t a l t e m p e r a t u r e s h a s n o t , a s y e t, b e e n u n d e r t a k e n .Surface losse sT h e p o n d t e m p e r a t u r e m o d e l a s s u m e s t h a t a l l o f t h e a b s o r b e d s o l a rr a d i a t i o n h a v i n g a w a v e l e n g t h g r e a t e r t h a n 7 0 0 n m i s a l m o s t i m m e -d i a te l y l o s t t o t h e a t m o s p h e r e t h r o u g h t h e a i r - w a t e r i n te r fa c e . F i g u r e 3r e p r e s e n t s d a i l y a b s o r p t i o n o f s o la r r a d i a t i o n in t h e p o n d a s a f r a c ti o n o ft h e t o t a l in c i d e n t r a d i a t i o n o n t h e p o n d s u rf ac e . T h e q u a n t i t y o f r a d ia -t i o n a b s o r b e d is b a s e d o n t h e o b s e r v e d t e m p e r a t u r e c h a n g e s i n t h e p o n dd u r i n g p e r i o d s o f i n s o l at io n . T h e m e a n f r a c t io n o f s o l a r r a d i a t io n t h a twas ab so rb ed was 0 . 29 w i th m os t o f the va lues l es s t han 0 . 40 . Th e f r ac -t i o n o f t h e s o l a r e n e r g y s p e c t r u m h a v i n g w a v e l e n g th s g r e a te r t h a n 7 0 0nm is , i n gene ra l , 0 . 55 -0 . 6 0 , sugges ting tha t su r face lo s ses du r ing in so l a -t i o n a r e i n d e e d g r e a t e r i n m a g n i t u d e t h a n t h e e n e r g y c o n t a i n e d i n t h el o n g -w a v e p o r t i o n o f t h e s o l a r s p e c t ru m .Tem perature predictionD u r i n g t h e p e r i o d w h e n t e m p e r a t u r e s w e r e r e c o r d e d ( 1 5 A u g u s t t o 3 1O c t o b e r ) t h e w a t e r c o l u m n e x p e r i e n c e d a l a r g e s e a s o n a l d e c r e a s e i nt e m p e r a t u r e . F i g u r e 4 r e p r e s e n t s o b s e r v e d a n d p r e d i c t e d t e m p e r a t u r e sp lo t t ed aga ins t t ime fo r t h i s pe r iod . Fo r i l lu s t r at i ve pu rpo ses , t he p lo t isl im i t e d to t e m p e r a t u r e s a t 4 p . m . f r o m t h e f o u r t h e l e m e n t ( a p p r o x i m a t e l y1 m in dep th ) . A r igo rous eva lua t ion o f t he mode l r equ i r e s t ha t t heseasonal e f fec t be l imi ted in so fa r as i t i s poss ib le to do so . This was


11/21

M odefing temperature in freshwa ter ponds~ . 5

247

OH / SR

Fig . 3 .

~ . 0

0.5 e _ e. 0 . 4 0- . . . . . . . . . . . _- . . . . . . . . .

oo ol ~l ~ a O . 2 9o % o o o + o oo o ~ o o

o oc~

0 . 0 I J I = I = I = I220 240 260 280 300 320

T i m e ( J u l i a n D a y s )

C h a n g e o f p o n d h e a t e n e r g y d u r i n g i n s o l a t io n a s a fr a c t io n o f i n c i d e n t s o l a re n e r g y .

F ig . 4 .

Temperature(C )

3 0

2 5

2 0

t 5

10 ' ' ~ ' 5001 , A + , l O 00 1 , , , , 150 01 , , ~ , 20100Time ( h o u r s )

O b s e r v e d a n d p r e d i c te d t e m p e r a t u r e s s h o w i n g s e a s o n a l c o o li n g . T e m p e r a t u r e sa r e fo r 4 p . m . a t a d e p t h o f a p p r o x i m a t e l y 1 m .

a c c o m p l i s h e d b y f i t t i n g a l i n e a r f u n c t i o n t o t h e o b s e r v e d t e m p e r a t u r e sf r o m e a c h o f t h e 6 s u b s u r f a c e e l e m e n t s . T h e r e g r e s s i o n w a s t h e n s u b -t r a c t e d f r o m b o t h o b s e r v e d a n d p r e d i c t e d t e m p e r a t u r e s . F i g u r e 5 i l l u s -t r a t e s t h e d a t a o f F i g . 4 a f t e r s e a s o n a l a d j u s t m e n t . B e c a u s e o f t h e l a r g e


12/21

2 4 8 T . P . Cath cart , F . W. W heaton

F i g . 5 .

T (C)

6

4

r2 x x

l

0

~ g

i i i I i

5 0 0

-2

- 4 0

LEGEND

: OBSERVED- x PREDICTED

i i i ~ 0 0 0 [ i i i i ~ 5 [ 0 0 I i i

Time (Mours)' 2 0 0 0

Observed and predicted seasonal ly adjusted temperatures (T*) . The data int h i s g r a p h c o r r e s p o n d s t o t h a t o f F i g . 4.

P r e O L c t e OT w

(C]

F i g . 6 .

5 e e e ~ . . "

0 - "

"

0 " L E G E N D

. ~ ~ - - R E G ~ E B B I O N L I N E I,IT LINE ]@

- ~ = I l I i I , I , I i I- 8 - 4 2 O 2 4 6

ODserved T' (C}

Predicted vs observed seasonal ly adjusted temperatures (T*) for element 2(depth approxim ately 0 .45 m) at 4 p .m. ( r= 0 .97).

n u m b e r o f d a t a p o i n t s , i t i s n o t p o s s i b l e t o r e p r e s e n t p r e d i c t e d v e r s u so b s e r v e d t e m p e r a t u r e s f o r a ll h o u r s a n d e l e m e n t s .

F i g u r e s 6 - 1 1 o f f e r a s a m p l e o f t y p i c a l c u r v e s . C o r r e l a t io n c o e f f i c i e n t ss h o w n o n t h e f ig u r e s a r e t h o s e f o r a ll o f t h e d a t a p o i n t s f o r th a t e le m e n t .


13/21

M odel in g empera tu re n resh wa ter on ds 249

F i g . 7 .

PredictedT K

(C )- 2

- 4

LEGEND [ o , 7 7 - - //~$- - REGRESSION LINE J ~ . "IDEAL FIT LINE J ~ "

. ~ 0 0

;

IE i I Ii 1 4 1 2 l I t I- 6 - - 0 2 4 6

O b s e r v e d T ~ (C )P r e d i c te d v s o b s e r v e d s e a s o n a l ly a d j u s t e d te m p e r a t u r e s ( T * ) f o r e l e m e n t 3

( d e p t h a p p r o x i m a t e l y 0 . 7 5 m ) a t 4 p . m . ( r = 0 - 9 7 ) .

Pre0lcted

[C )

F i g . 8 .

6

2 4 0 @ " @ " " 0 @ e@ ~ . .@ @

o ~ . 2 = ~ I L EG E .O JI J I - :. ~ I - - R E G .E G G IO N L : N E I

- d J I , I , i I , 1- 4 - 2 0 2 4 6

P r e d i c te d v s o b s e r v e d s e a s o n a l ly a d j u s t e d t e m p e r a t u r e s ( T * ) f o r e le m e n t 4( d e p t h a p p r o x i m a t e l y 1 . 0 5 m ) a t 4 p . m . ( r -- 0 - 9 6 ) .

T h e m o d e l e x h i b i te d a s y st em a t i c te n d e n c y t o u n d e r p r e d i c t p o n d t e m -p e r a t u r es i n m o s t e l e m e n t s . T h i s p a t t e r n i s e s p e c i a l l y p r o m i n e n t i n F i g s7 - 9 a n d is fo r m a l i z e d i n t h e c o m p a r i s o n o f t h e m e a n o b s e r v e d a n d p r e-d i c t e d t e m p e r a t u r e s f o r e a c h e l e m e n t . V a r i a n c e s f o r e a c h e l e m e n t w e r e


14/21

2 5 0 T. P. Cathcar t , F . W. Wh eaton

F i g . 9 .

P r e d i c t e dT"(c }

LEGENDe DATA

2 IDEAL FIT LINE [- - REGRESSION LINE|

0 Q ;

o

- 4 , 1 2 ~ I- 4 m 0

O b ser ved T " ( e l

~c . ~ ~ 0 " 0

~ - O ~ O " ~ O 0 00O

// -// /// o

i I i I2 4

P r e d i c t e d v s o bs e r v e d s e a s o na l l y a d j us t e d t e m pe r a t ur e s ( T * ) f o r e l e m e nt 5( de pt h a p pr o x i m a t e l y 1 . 3 5 m ) a t 4 p . m . (r - - 0 -9 5 ).

Predicted

T ~(e l

F i g . 1 0 .

- 2

+LEGEND / //

/ .DATA /- - R E G R E S S I O N L I N E ~ ~ "

IDEAL FIT LINE ~/ ~ . "

- - - - . y o -0

- 4 4 i J * , _ l3 . . . . _ 12 . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i- i 0 ~ 2 3 4

Observed T (C)

P r e d i c t e d v s o bs e r v e d s e a s o na l l y a d j us t e d t e m pe r a t ur e s ( T * ) f o r e l e m e nt 6(dep th approx imate ly 1 .65 m ) a t 4 p .m. (r= 0"93) .

n o t s i g n i f i c a n t l y d i f f e r e n t ( P > 0 .0 5 ) . P r e d i c t e d a n d o b s e r v e d m e a n s o fe l e m e n t s 3 , 4 , 5 , a n d 7 w e r e s i g n i f ic a n t l y d i f f e r e n t f r o m o n e a n o t h e r( P < 0 - 0 5 ).

M e a n t e m p e r a t u r e s f o r e a c h e l e m e n t a r e c o m p a r e d i n F i g . 1 2 . D e s p i t e


15/21

M odel i ng tempera tu r e i n reshw a ter pon ds 2 51

Predicted

T *(C)

F i g . 11 .

4 1L E G E N D ]

@ DATA- - REGRESSIONLINE]

2 -- IDEAL FIT LINE[_j e ~ "

0 (~ 0@@ O f " ~ @ @@ @ -O . " @

. "@ @ @- " @

-2 .e"

- 4 . . . . I . . . . [ . . . . I . . . . I . . . . I . . . . I- 3 - 2 -~ 0 i 2 3

Observed T * (C)

P r e d i c t e d v s ob s e r ve d s e as on a l l y ad ju s te d t e mp e r a t u r e s ( T * ) f or e l e me n t 7( d e p t h a p p r ox i m at e l y 2 . 00 m) a t 4 p . m . (r - - 0" 90 ).

M e a n

T

F i g . 12 .

0 . 2 0 F

0 . ~.0~

0 . 0 0 ! ~ ,

-c.lohrr

- o . 2 o I- 0 , 3 0 !

- 0 . 4 0 I2

LEGENDI:" - o sE.vEO- P R E D I C T E D

4D

@

i , , ~ , I , , i3 4Element

5 6 7A c o m p ar i s on o f m e an an d p r e d i c te d ob s e r ve d s e as on a l l y ad j u st ed t e mp e r a -t u r e s f or t h e 6 s u b s u r f ac e e l e me n t s o f t h e p on d .

t h e s y s t e m a t i c u n d e r p r e d i c t i o n , th e m a g n i t u d e o f t h e d i f f er e n c e i s 0 .5 Cor l e s s .

F i g u r e 1 3 r e p r e s en t s a m o r e q u a l i ta t iv e v i e w o f t h e m o d e l . U s i n g 1 Ca s a b e n c h m a r k , F i g. 1 3 s h o w s t h e p e r c e n t a g e o f p r e d i c t io n s w h i c h fa ll


16/21

252 T. P . Ca thcar t , F . W. W hea ton

P e r c e n t

W i t h i n

I C

40

20

Fig. 13.

0H o u r : I 2 4 1 2 4 ~ 2 4 I 2 4 1 2 4 ! 2 4

2 3 4 5 6 7E l e m e n t

Percentage of pred ictions wh ich deviate from observed temperatures by ICor less.

w i t h i n ' r e a s o n a b l e ' l im i t s o f th e o b s e r v e d t e m p e r a t u r e f o r e a c h e l e m e n ta t e a c h h o u r o f t h e d a y . T h e m o d e l a p p e a r s t o p e r f o r m w o r s t f o r e l e -m e n t s 2 a n d 4 , w i th a g r e e m e n t f a ll in g b e l o w 8 0 % f o r c e r t a in h o u r s o f th ed a y. F o r o t h e r h o u r s a n d e l e m e n t s , p r e d i c t io n s a r e w i t h in I C 8 0 - 1 0 0 %of the t ime.W i n d m i x in gT h e i m p o r t a n c e o f th e w i n d m i x i n g p o r t i o n o f th e m o d e l is h i g h li g h te db y F i g s 1 4 a n d 1 5. F i g u r e 1 4 c o m p a r e s t h e m e a n p r e d i c t e d t e m p e r a t u r e sf o r e a c h e l e m e n t , w i t h a n d w i th o u t w i n d m i x in g , t o t h e o b s e r v e d m e a nt e m p e r a tu r e s . T h e t e n d e n c y to u n d e r p r e d i c t p o n d t e m p e r a t u re is m u c he x a c e r b a t e d i n t h e a b s e n c e o f t h e w i n d m i x in g p r o c e d u r e . F i g u r e 1 5s h o w s t h a t t h e im p r o v e d a g r e e m e n t o f t h e p r e d ic t e d t e m p e r a t u r e s w i t hw i nd m i x i ng is r e l a ti ve l y un i fo rm fo r a l l hou r s o f t he day .E s t i m a t i ng s ur fa c e t e m p e r a tur e sT h e u s e f u ln e s s o f t h e p o n d t e m p e r a t u r e m o d e l i s, t o a d e g r e e , p r e d i c a t e do n t h e a v a i la b i li ty o f h o u r l y s u r fa c e w a t e r t e m p e r a t u r e s . T h e m a x i -m u m - m i n i m u m p r o c e d u r e o u t li n e d in th e p r e c e d in g s e c ti o n w as u se d t oe s t im a t e h o u r l y te m p e r a t u r e s . I t w a s a s s u m e d t h a t d a i ly m i n i m u m a n dm a x i m u m w a t e r t e m p e r a t u r e s o c c u r r e d a t 8 a .m . a n d 4 p . m . r e s p e c ti v e ly


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( d a y li g h t s a v i n g s t im e ) . T h e e s t i m a t e d t e m p e r a t u r e s w e r e t h e n a d j u s t e dt o e l i m i n a t e th e s e a s o n a l b i a s a n d c o m p a r e d t o o b s e r v e d s u r fa c e t em -p e r a t u r e s. T h e m e a n s a n d v a r i a n c e s o f t h e t w o d a t a se t s w e r e n o t s ig n i fi -c a nt ly d if fe r e nt ( P > 0 ' 0 5 ) . T h e e s t i m a t ed m e a n t em p e r a t u r e w a s


18/21

2 5 4 T . P . Cathcar t , F. W. W heatona p p r o x i m a t e l y 0 . 1 7 C g r e a te r t h a n th e o b s e r v e d m e a n . T h e c o r r e la t i o nc o e f f i c i e n t w a s 0 . 9 9 .

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cathcart 1987 aquacultural-engineering

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