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Wind Induced Water Circulation Of Lake Geneva
219
W I N D INDUCED WATER CIRCULATION OF LAKE GENEVA
S.W. BAUER and W.H.
GRAF
Laboratoire d'Hydraulique ( L H Y D R E P ) , Ecole Polytechnique FBdBrale, Lausanne Switzerland
(Received 1 7 August 1978; accepted 24 August 1978)
ABSTRACT A numerical modeling technique i s used t o simulate flow p a t t e r n s a t various depths i n t h e Lake of Geneva (Le LBman) f o r a homogeneous s i t u a t i o n encountered u s u a l l y during w i n t e r months. Subsequently, a v e r t i c a l l y i n t e g r a t e d flow p a t t e r n i s obtained. A measuring campaign i s under way which provides d a t a on t h e v e l o c i t y , d i r e c t i o n and temperature of t h e atmosphere and t h e water. Using t h e s e d a t a , two quasi-steady s t a t e s i t u a t i o n s a r e compared w i t h model simulations for winter months of 1977 and 1978.
DESCRIPTION OF MATHEMATICAL MODEL
A system of c u r r e n t s
i n a l a k e may be considered as water movement on a l a r g e
s c a l e . I t can be described by t h e t h r e e components of t h e momentum eauation and t h e c o n t i n u i t y equation f o r a homogeneous (non s t r a t i f i e d ) and incompressible f l u i d ( L i g g e t t , 1970) :
P[-
a U+ a 2 a (UW) a (UV) + aY aZ a t ax (U ) + -
av + aw = aU + -
ax
ay
aZ
- fV1
=
-
a aU a ap + -(v) ax aZ, ax aZ
4- -(E-)+
0
aU ax
a au ay ay
-(E-)
(1)
(4)
The boundary c o n d i t i o n s a p p l i c a b l e on s o l i d boundaries a r e u = v = w = o and a t the f r e e s u r f a c e with z = 0
(5)
220
The symbols i n equation 1-6 a r e as follows:
u , v and w a r e t h e v e l o c i t y components i n t h e x, y and z d i r e c t i o n s r e s p e c t i v e l y where x i s p o s i t i v e towards east, y i s p o s i t i v e towards n o r t h and z i s p o s i t i v e upwards w i t h zero a t t h e water s u r f a c e ,
t
is the t i m e ,
f
i s t h e C o r i o l i s parameter,
p
is the f l u i d density,
p
is the local pressure,
17 and T
&
a r e t h e v e r t i c a l and h o r i z o n t a l compnents of t h e eddy v i s c o s i t y ,
i s t h e a c c e l e r a t i o n of g r a v i t y
g X
and
and T
a r e t h e wind s h e a r s t r e s s e s on t h e water s u r f a c e i n t h e x and y d i r e c t i o n s Y respectively. Equation 3 expresses t h e h y d r o s t a t i c equilibrium, which i s a v a l i d assumption
f o r shallow l a k e s , i.e.: D/L << 1 where L and D a r e c h a r a c t e r i s t i c v e r t i c a l and h o r i z o n t a l dimensions ( f o r the LQman: D/L
2 0,03).
Equations 1-3 may be f u r t h e r s i m p l i f i e d according t o t h e following assumptions:
aU at --
a) s t a t i o n a r y flow:
0,
av at --
0;
such a s i t u a t i o n might be imagined f o r a wind which blows long enough t o establish a stationary circulation; b ) t h e i n e r t i a f o r c e s are small when compared with t h e C o r i o l i s f o r c e s , i . e . ,
the
Rossby number i s small. I n t h i s c a s e equations 1-3 may be l i n e a r i z e d :
( f o r t h e Ldman t h e Rossby number
2 0 , l )i
c ) h o r i z o n t a l d i f f u s i o n i s s m a l l compared t o v e r t i c a l d i f f u s i o n , which can be accepted f o r shallow l a k e s :
a E -)av a x ( ax
=
0,
a a~
-(&
( f o r t h e LQman: D/L
au -) aY
= 0, etc.
'J 0 , 0 3 ) ;
d) t h e v e r t i c a l component of t h e eddy v i s c o s i t y 17 which i s notknown i s assumed t o be c o n s t a n t over the e n t i r e lake: 2 a v -(n -) = 17, etc. a Z aZ 2
a
av
aZ
e ) t h e i n f l u e n c e of e x t e r n a l i n - and outflows i n t h e Leman on wind induced currents on t h e l a k e i s n e g l i g i b l e . Thus, i n t h e immediate v i c i n i t y of r i v e r mouths, t h e model i s not a p p l i c a b l e .
221 W i t h t h e above s i m p l i f i c a t i o n s and conditions t h e system of equations describing
wind induced c u r r e n t s i n a homogeneous shallow l a k e i s thus:
+pfu=
aU ax
ap
--+qaY
aw av + +ay aZ =
a2v a2
0
The numerical s o l u t i o n of t h e model described thus f a r i s obtained by Gallagher
et al.
(1973) and based on a mathematical formulation by L i g g e t t e t a l .
(1969).
ADAPTATION OF MATHEMATICAL MODEL FOR THE LEMAN
The mathematical model as described above has been coded and provided t o LHYDREP by J . A .
L i g g e t t of Cornell University. Subsequently a s u i t e cf r o u t i n e s
allowing g r a p h i c a l r e p r e s e n t a t i o n i n 2 and 3 dimensions of t h e numerical r e s u l t s
as well as of t h e geometrical r e p r e s e n t a t i o n i n f i n i t e elements
-
s e e Figure 1 -
of t h e l a k e geometry have been developed a t LHYDREP. A f t e r s e v e r a l t r i a l geometries 2 a f i n i t e element g r i d c o n s i s t i n g of 579 nodes (about one node per km 1 , 1025 triangular elements and 131 boundary p o i n t s
w a s found t o g i v e reasonable r e s u l t s
( s e e Bauer e t a l . , 1977).
GRILLE DES ELEMENTS FINIS F) TRBIS DIMENSlBNS RVEC UNE DISTBRTIBN DE X : Y : 2 = 1 : 2.00 : 0.020 LE LEMRN
t l
2 "Y
=m
mm
iii : w
'I:dl .I.-'
'*>.' '1 kil
'*,.I '&m
)n:-
'm'm
Fig. 1. "Three dimensional" Lake of Geneva ( F i n i t e element g r i d , nodes a and b showing p o s i t i o n of i n s t r u m e n t s ) .
I t should be noted t h a t compared t o i t s width,the Leman i s very shallow
(0,Ol < D/L < 0 , 0 3 ) . Due t o numerical reasons of t h e f i n i t e element approximation it i s necessary t h a t t h e minimum depth allowable is about 3 % of t h e maximum depth
222 Thus t h e r e a r e no zero depths a t t h e boundaries ( s e e Figure 1) which however i s n o t a severe l i m i t a t i o n f o r t h e geometrical r e p r e s e n t a t i o n . To execute t h e model a s e r i e s of parameters ( s e e equations 7-9) must be entered:
t h e d e n s i t y of water, p, the C o r i o l i s parameter, f , t h e a c c e l e r a t i o n of g r a v i t y , g, and t h e eddy v i s c o s i t y , q . Also, r e p l a c i n g i n euuation 6 t h e shear s t r e s s , T, by equation 11, v i z . T = c
f
where
.
.
'a
C
(11)
i s a wind shear s t r e s s c o e f f i c i e n t ,
f
i s t h e d e n s i t y of a i r
P
U
2 "wind
.
wind
and
i s t h e f r e e stream wind v e l o c i t y .
Two f u r t h e r parameters, C and p a r e t o be supplied. The following parameters f a have been taken a s c o n s t a n t s i n t h e model simulations: 3 d e n s i t y of a i r : Pa = 1 , 2 kg/m 3 d e n s i t y of water : P = 999,9 kg/m 2 a c c e l e r a t i o n of g r a v i t y : g = 9,80 m / s
C o r i o l i s parameter
:
f
= 0,000105 s
-1
( f o r a mean l a t i t u d e of t h e Leman of
46O 25 )
The parameters rl and C f , which a r e unknown a r e thus s u b j e c t t o determination by a model c a l i b r a t i o n . A t r i a l simulation f o r a s e t of f i x e d and chosen uarameters h a s been done and
i s reported by Bauer e t a l .
(1977).
MEASURING CAKZAIGNS
To allow f o r comparison between t h e c u r r e n t s occurring i n nature and t h e i r simulation by a model it i s necessary t o study ?he d i s t r i b u t i o n of t h e water v e l o c i t i e s and t h e winds which generate them. Therefore a measuring campaign has been s t a r t e d t o measure simultaneously and i n s i t u v e l o c i t i e s and d i r e c t i o n s of t h e wind and c u r r e n t s . Furthermore, temperature p r o f i l e s of atmosDhere and water were measured. I d e a l l y , measurements should be taken by a l a r g e number of recording instruments placed according t o some g r i d system over t h e e n t i r e domaine. Such measurements, h o r i z o n t a l l y and v e r t i c a l l y d i s t r i b u t e d , would allow t o o b t a i n a synoptic view of t h e phenomenon. Such a s o l u t i o n however i s a t p r e s e n t , for f i n a n c i a l and operational reasons, impossible and it was thus necessary t o a r r i v e a t a much more modest solution. Keeping t h e concept of a v e r t i c a l scheme, a s i n g l e s t a t i o n i s nlaced i n t h e l a k e . Measured and recorded a r e :
223 - v e l o c i t y o f t h e atmosphere a t t h r e e ( 3 ) d i f f e r e n t a l t i t u d e s as well a s i t s d i r e c t i o n s and t h e temperature;
- v e l o c i t y , d i r e c t i o n , temperature and p r e s s u r e of t h e c u r r e n t s a t f i v e ( 5 ) d i f f e r e n t depths. A d e s c r i p t i o n of this i n s t a l l a t i o n i s given by P r o s t e t a l .
(1977) and a schematic
view of t h e s t a t i o n i s shown i n Figure 2 . This s t a t i o n has been k e p t f o r a d u r a t i o n of weeks a t one place. Since f o r t h e s e r v i c i n g of t h e c u r r e n t meters it i s necessary t o l i f t t h e e n t i r e i n s t a l l a t i o n o u t of t h e w a t e r , t h i s o p e r a t i o n i s a l s o used t o change i t s p o s i t i o n i n t h e l a k e , thus allowing f o r a coverage of d i f f e r e n t a r e a s . During the winter months, t h e i n s t a l l a t i o n w a s l o c a t e d c l o s e t o t h e two nodes a and b a s i n d i c a t e d i n Figure 1. Measurements:
,
,
. .. .. .. ,,
I .
~
+...
Fig. 2 . Schematic view of measuring s t a t i o n For measurements i n t h e atmosphere, conventional cup anemometers of t h e type Aanderaa (WSS 219) a r e being used t o measure t h e v e l o c i t i e s . The number of r o t o r r e v o l u t i o n s i s counted e l e c t r o n i c a l l y and t r a n s m i t t e d i n d i g i t a l form t o a d a t a logger. Wind d i r e c t i o n s a r e determined with t h e a i d of a wind vane type Aanderaa (WDS 2053) and a magnetic compass measuring t h e o r i e n t a t i o n of t h e buoy with resp e c t t o n o r t h . The a i r temperature sensors used were Aanderaa (1289 A ) .
224 DATA
-
OBTAINED SINCE FEBRUARY 1977
The periods of t h e d a t a c o l l e c t e d thus f a r by LHYDREP i n t h e Lake o f Geneva
are summarized i n Table 1. TABLE 1
Sumary of d a t a p e r i o d s c o l l e c t e d by LHYDREP (Dates given by day, month and y e a r ) No
Period -
1 2 3 4 5
1/2/77
-
1/3/77
1/3/77 25/4/77 16/6/77 - 24/8/77 31/8/77 - 8/11/77 9/12/77 - 15/2/78
P o s i t i o n (km) X Y 528,28 531,24 538,39 542,02 542,02
149,18 147,84 148,68 147,48 147,48
Depth ( m ) 73,O 198,9 293,4 289,9 289.9
To allow i n s p e c t i o n of t h e s e d a t a s e v e r a l computer programs p e r m i t t i n g analog r e p r e s e n t a t i o n o f t h e d a t a were developed a t LHYDREP. A f t e r a l l d a t a weleplotted, v i s u a l i n s p e c t i o n showed t h a t t h e l a k e has been reasonably homogeneous f o r t h e e n t i r e p e r i o d s 1 and 2 and f o r period 5 a f t e r 14/1/78.
Using average temperatures
from 5,6; 10,O; 20,O; 35,O and 55,O m depth t h e o v e r a l l mean l a k e temperature was 5,725 OC w i t h a s t a n d a r d d e v i a t i o n of t h e means of 0,020 OC f o r p e r i o d 1. Similarly, f o r period 2 , using average temperatures from 5,8; 10,2; 20,2; 84,4 and 148,9 m depth, t h e o v e r a l l mean l a k e temperature was 6,143 OC with a standard d e v i a t i o n of 0 , 4 1 2
OC.
For period 5 a f t e r 14/1/78 t h e o v e r a l l mean temperature obtained from
3,9; 8 , 6 ; 18,5; 92,7 and 195,6 m depth was 5,968 OC with a s t a n d a r d d e v i a t i o n of 0,222 OC. Typical records of t h e d a t a a r e shown i n Figures 3 and 4 f o r t h e periods 2 and 5 r e s p e c t i v e l y . I n Figures 3 and 4, t h e t i m e a x i s is h o r i z o n t a l , whereby every 24 hours a v e r t i c a l l i n e , showing t h e d a t e i n y e a r s , months, days,hours and minutes
i n d i c a t e s t h e s t a r t of a new day. The d a t a represented i n t h e topmost band a r e the temperatures of t h e water i n 5 depths corresponding t o t h e p o s i t i o n s of each current meter. Proceeding downwards, t h e n e x t band shows the wind v e l o c i t i e s and then follow t h e v e l o c i t y observations of t h e 5 currentmeters. The subsequent 6 bands show d i r e c t i o n s , t h e f i r s t being t h e d i r e c t i o n of t h e wind followed by t h e directions of t h e 5 currentmeters. The s c a l e s - shown every f i v e days - have been s e l e c t e d
such t h a t t h e temperature band extends over a range of 5-10 OC, t h e wind speed band over 0-10 m / s , over 0-360°.
t h e water v e l o c i t y bands over 0-10 cm/s and the d i r e c t i o n bands
( I n c a s e t h e s e ranges a r e exceeded by t h e d a t a t o be p l o t t e d , t h e
225
t r a c e o f one band c o n t i n u e s o v e r n e i g h b o u r i n g bands b u t k e e p s i t s o r i g i n a l s c a l e . ) I n F i g u r e s 3 and 4 i t c a n b e s e e n t h a t r e c o r d s of wind were n o t always a v a i l a b l e ( F i g u r e 3: a f t e r 9 / 3 / 7 7 ,
F i g u r e 4 : between 3/1/78 and 2 3 / 1 / 7 8 ) . Thus, wind ob-
s e r v a t i o n s o f t h e S w i s s M e t e o r o l o g i c a l S e r v i c e r e c o r d e d a t 7 , 1 3 and 19 h o u r s a t Lausanne a r e i n s e r t e d i n t h e wind bands assuminq a s i x h o u r s d u r a t i o n ( s t r a i q h t l i n e s e g m e n t s ) f o r each o f t h e s e o b s e r v a t i o n s . Comparison o f t h e LHYDREP and "Lausanne" vind d a t a ( s e e F i g u r e 3, b e f o r e 9/3/77 a f t e r 23/1/78)
and F i q u r e 4 b e f o r e 3/1/78 and
show r e a s o n a b l e agreement. Also, c u r r e n t m e t e r 4 o f F i g u r e 3 r e -
c o r d e d o n l y c u r r e n t d i r e c t i o n s and t h u s no d a t a a r e shown on t h e v e l o c i t y band of c u r r e n t m e t e r 4 of t h i s F i g u r e .
SELECTION OF PERIODS USED FOR SIMULATION
I t was s t a t e d above t h a t t h e m a t h e m a t i c a l model i s c o n c e i v e d f o r t h e f o l l o w i n s
limiting conditions:
(1) homogeneity of t h e w a t e r body, i . e . ,
t h e l a k e must h a v e a l m o s t no t e m n e r a t u r e
stratification; ( 2 ) s t a t i o n a r i t y of t h e v e l o c i t y and d i r e c t i o n of t h e wind and t h e w a t e r . The manner of s a t i s f y i n r j t h e above c o n d i t i o n s i.s d i s c u s s e d i n t h e f o l l o w i n g . S i n c e t h e " s t a t i o n a r i t y " i s more r e s t r i c t i v e we s h a l l s t a r t w i t h i t . ( a d 2 ) A c l i m a t o l o g i c a l p a r t i c u l a r i t y of t h e Leman b a s i n i s t h a t t h e r e e x i s t two i m p o r t a n t and s t r o n g w i n d s : one n o r t h - e a s t e r l y - t h e b i s e
- and one s o u t h -
w e s t e r l y - t h e v e n t ( P r i m a u l t , 1 9 7 2 ) . These winds o f t e n blow f o r u e r i o d s o f days a n d , w i t h t h e p o s s i b l e e x c e p t i o n of t h e e a s t e r n end of t h e l a k e , p r o d u c e p r o b a b l y a c o n s t a n t i n t e n s i t y and d i r e c t i o n of s h e a r on t h e l a k e s u r f a c e . S e a r c h i n g f o r s u c h s i t u a t i o n s i n p e r i o d s 2 and 5 o f T a b l e 1 ( s e e F i q u r e s 3 and 4 ) , we h a v e s e l e c t e d two t i m e p e r i o d s of d a t a , s u b s e q u e n t l y t o b e u s e d f o r a model s i m u l a t i o n . These p e r i o d s a r e : bise : vent
:
29/3/77, 2/2/78,
12h00 - 30/3/77,
24h00
OOh00 -
12h00
3/2/78,
I f ' o n e r e g a r d s t h e d i r e c t i o n of c u r r e n t s a t t h e s e l a y e r s , one f i n d s them t o b e r e a s o n a b l y c o n s t a n t ; however, t h e v e l o c i t y o f c u r r e n t s a t t h e d i f f e r e n t l a y e r s i s n o t a t a l l c o n s t a n t ( I ) . V e c t o r i a l l y a v e r a g e d v a l u e s f o r t h e v e l o c i t i e s of wind and c u r r e n t s w e r e c a l c u l a t e d as i n d i c a t e d
- F i g u r e s 3 and 4 w i t h b o l d l i n e s
and a r e summarized i n T a b l e s 2 (BISE) and 3 (VENT). ( a d 1) For t h e p e r i o d s s e l e c t e d a b o v e , i . e . b i s e and v e n t , d e n s i t y v a r i a t i o n s were c a l c u l a t e d and found t o be weak; r e s u l t s a r e g i v e n i n T a b l e s 2 and 3.
-
227 TABLE 2
BISE: Mean (temporal) v e l o c i t i e s and temperatures of c u r r e n t s a t x = 531,24 km and y = 147,84 km between 29/3/77, 1 2 hours and 31/3/77, zero hours 3,5 m/s Average wind v e l o c i t y a t Lausannel) : ('wind'x - (U . ) = - 8 , 4 m / s UWlnd = 9 , l m / s = 32,8 km/h wind Depth ( m )
5,6 10,2 20,2 84,4 148,9 Weighted mean with 200 m max. depth
u (cm/s)
v (cm/s)
- 14,77 - 12,84
- 1,60
-
(cm/s)
t (°C)2)
p (g/ml)
1,16 0,96
14,86 12,89 9,61
6,287 6,293 6,320 5,962
0,999931014 0,999930803 0,999929846 0,999941645
-
9,56
-
1,l
0,02
1,lO
5,664
0,999950001
- 3,64
0.09
3,64
-
0,999942124
-
-
-
&/Po 2,1101221 9r5665536 -1,1799582 -8,3563836
-
. lo;; .
*
'-5
. 10
Po =
Velocity components of wind and c u r r e n t s : u i s p o s i t i v e f o r flow towards e a s t v i s p o s i t i v e f o r flow towards north 2, S u b j e c t t o accuracy of 5 0,Ol OC according t o instrument s p e c i f i c a t i o n
')
TABLE 3
VENT:
Mean (temporal) v e l o c i t i e s and temperatures of c u r r e n t s a t x = 542,02 km and y = 1 4 7 , 4 8 km between 2/2/78 zero hours and 3/2/70, 1 2 hours Average wind v e l o c i t y a t station') : ( U . ) = 4 , 7 m / s
(;wind)y wind x = 4 , 3 m / s = 6 , 4 m / s = 22,9 km/h wind
-
-
-
3,92 - 1,29 - 0,60
4,OO 1,16 1,08 2,66 2,22
0,21
2,20
0 ro
3,9 8r6 18,5 92,7 195,6 Weighted mean with 305 m max. depth
8,40 7,92
See Table 2 2 , See Table 2
~~
9,30 8,OO
6,055 5,930 5,911
0,999938764 0,999942607 0,999943190
4,07 2,96 2,30
5,911 5,978 5,527
0,999943170 0,999941159 0,999953397
2,21
-
Po = 0,999947942
-
-3,8424000
. 10:;
-5r6352933
*
0
2,0113047 -1,2234736
. .
lo
-6 10-5
228 CALIBRATION O F MODEL
A s has been s t a t e d above, t h e two model parameters s u b j e c t t o model c a l i b r a t i o n
( i n this study) a r e t h e wind s h e a r c o e f f i c i e n t , C f ,
and t h e eddy v i s c o s i t y ,
n.
Let u s look f i r s t a t t h e i n f l u e n c e of t h e eddy v i s c o s i t y upon t h e simulated v e l o c i t i e s , keeping t h e s h e a r stress c o e f f i c i e n t constant. This is done i n Figure 5, 2 T l = 1o00, 500 and 100 cm /s f o r t h r e e ( 3 ) p o i n t s
where t h e v e l o c i t y v e c t o r s f o r a r e shown every 2,s m.
I t can be seen t h a t i n each c a s e t h e r e i s a d e v i a t i o n on
-
-
/= :
DIRECTION DU VENT I
Fig. 5. Ekman s p i r a l s a t nodes a , b and c
229
t h e water s u r f a c e of about 45O t o t h e r i g h t between t h e wind d i r e c t i o n and t h e c u r r e n t d i r e c t i o n . Then, proceeding downwards, t h e v e c t o r s diminish and t u r n t o t h e r i g h t forming the &man s p i r a l . A t a c e r t a i n depth, D , t h e vectors have turned 180° r e l a t i v e t o t h e v e c t o r s on t h e w a t e r s u r f a c e . For an i d e a l i z e d ocean of in-
f i n i t e dimensions t h i s depth, Dp,
i s given by ( D i e t r i c h e t a l . , 1975) (12)
Thus, t h e depth, D
, is
p r o p o r t i o n a l t o t h e square r o o t of rl a s seen i n Figure 5.
Furthermore, s i n c e t h e m a s s t r a n s p o r t over t h e depth, D eddy v i s c o s i t y ( D i e t r i c h e t a l . , 1975)
-
i.e.
De
1
i s independent of t h e e' vdz = c o n s t a n t - a change of D
produces a change i n t h e v e r t i c a l d i s t r i b u t i o n of t h e v e l o c i t i e s . This i s a l s o e v i d e n t i n Figure 5 . values on the simulation, it i s t o be noted f t h a t t h i s f a c t o r acts merely l i k e a s c a l e f a c t o r on the v e l o c i t y s c a l e s . A s f o r the i n f l u e n c e of t h e C
Concluding from t h e above remarks t h e following procedure i s proposed f o r c a l i b r a t i o n of t h e model:
(1) t r y i n g d i f f e r e n t values of Q ; s e l e c t Q such t h a t reasonable d i r e c t i o n a l agreement between observations and model s i m u l a t i o n i s obtained ( g r e a t e r importance must be a t t a c h e d t o c l o s e agreement i n the t o p l a y e r s ) ; ( 2 ) check t h e thus obtained rl value a g a i n s t values found i n t h e l i t e r a t u r e ;
( 3 ) f o r the chosen value of rl a d j u s t t h e wind shear s t r e s s c o e f f i c i e n t , C f , such t h a t the magnitudes o f simulated and observed v e l o c i t i e s agree reasonably w e l l ( a g a i n , g r e a t e r importance must be a t t a c h e d t o c l o s e agreement i n t h e top layers); ( 4 ) check i f t h e s e l e c t e d value o f C
agrees with t h e ones f observations o r found i n t h e l i t e r a t u r e .
c a l c u l a t e d from d i r e c t
Following t h e above procedure, t h e b i s e ( f o r d e s c r i p t i o n s e e T a b l e 2 ) w a s inv e s t i g a t e d f i r s t and is discussed herewith ( t h e same w a s a l s o done f o r t h e v e n t ) : (ad 1) Trying v a r i o u s values of 50 cm'/s
5 rl 5
2000 cm"/s,
it was found t h a t
2
rl = 500 cm / s gave d i r e c t i o n s of v e c t o r s t h a t agreed w e l l w i t h t h e observations, as it i s t o be seen i n Figure 5 (ad 2) I f t h e rlvalue i s derived from a well-accepted equation (Neumann and Pierson, 1966) : Q
= 0,1825
where
. 10-4
U
5/2
.
wind
/ p
Uwind
i s i n cm/s,
p
i s i n g/cm3 and 2 i s i n cm / s
T)
2 with a wind v e l o c i t y of 32,7 h / h , an eddy v i s c o s i t y of 460 cm / s i s obtained.
230
F i g . 6 . S i m u l a t i o n of c u r r e n t v e c t o r s f o r a B i s e , and o b s e r v a t i o n s .
Fig. 7 . Simulation of v e r t i c a l l y i n t e g r a t e d c u r r e n t v e c t o r s f o r a Bise, and observations.
231
Fig. 8. Simulation of c u r r e n t v e c t o r s f o r a Vent, and observations RXE X l K M l
"ISCOSIlE OE TURBULENCE: 187.0 C " Z / S PRRRHETRE OE CORIOLIS: 0.000105 RROIRNSIS
RXE X l K M l
Fig. 9 . Simulation of v e r t i c a l l y i n t e g r a t e d c u r r e n t v e c t o r s f o r a Vent, and observations.
232 2 2 This i s thought t o be reasonably c l o s e t o 500 cm / s , and a value of 460 cm / s was adopted f o r f u r t h e r c a l c u l a t i o n s . (ad 3) I n o r d e r t o o b t a i n agreement of t h e magnitudes of t h e v e c t o r s , t h e wind s h e a r stress c o e f f i c i e n t , C f , w a s taken as C
f
= 0,004.
v a l u e s have been c a l c u l a t e d a t LHYDREP by P r o s t (personal communication, f 1978) f o r our measuring campaign and do indeed corroborate with our chosen values. (ad 4) C
The r e s u l t s o f this model simulation ( c a l c u l a t i o n ) are shown f o r t h e b i s e i n Figure 6 and f o r t h e v e n t i n Figure 8. Drawn a r e t h e computer c a l c u l a t e d c i r c u l a t i o n p a t t e r n s a t d i f f e r e n t l a y e r s and t h e i n s i t u measured v e l o c i t y v e c t o r s a t r e s p e c t i v e depths. Considering t h e assumptions made i n t h e mathematical simulation and t h e d i f f i c u l t i e s encountered i n an i n s i t u measuring campaign, t h e agreement i s cons i d e r e d t o b e reasonably good. Furthermore, i n Figure 7 ( b i s e ) and Figure 9 ( v e n t ) we show a comparison of depth-average v e l o c i t i e s c a l c u l a t e d and measured. Agreement i s extremely encouraging f o r t h e b i s e , b u t c e r t a i n l y less good f o r t h e vent.
(We de n o t exclude t h a t f u r t h e r
c a l i b r a t i o n could b e a p p l i e d t o reach b e t t e r agreement f o r t h e v e n t a s w e l l ) . I n t e r e s t i n g l y enough t h e model c i r c u l a t i o n p a t t e r n l e a d s u s t o agree with a conc l u s i o n r e c e n t l y drawn by Hamblin (1976):"Currents demonstrate t h e g e n e r a l tendency t o follow the wind i n t h e nearshore region, whereas t h e r e t u r n c u r r e n t opposed t o t h e wind d i r e c t i o n i s s i t u a t e d i n t h e c e n t r a l p o r t i o n of t h e l a k e " .
ACKNOWLEDGEMENT
This work was p a r t i a l l y sponsored by t h e Swiss National Science Foundation (FNSFS) under i t s s p e c i a l program "Fundamental problems of t h e water c y c l e i n Switzerland"
REFERENCES
Bauer, S.W., Graf, W.H. and T i s c h e r E . , 1977. L e s courants dans l e L6man en s a i s o n f r o i d e . Une simulation mathematique. B u l l . Techn. S u i s s e Romande, 103 :239-243. D i e t r i c h , G . , K a l l e , K . , Krauss, W . and S i e d l e r , G . , 1975. Allgemeine Meereskunde, Gebnider Borntrager, B e r l i n - S t u t t g a r t . Gallagher, R.H., L i g g e t t , J . A . and Chan, S.K.T., 1973. F i n i t e Element Shallow Lake C i r c u l a t i o n . Proc. Am. SOC. Civ. Engs., Vol. 99, NO HY7 Hamblin, P.F., 1976. Seiches, c i r c u l a t i o n , and storm surges of an i c e - f r e e Lake Winnipeg. J . Fish. R e s . Board Can., 33:2377-2391. L i g g e t t , J . A . , 1970. C e l l Method f o r computing l a k e c i r c u l a t i o n . Proc. Am. SOC. C i v . Engs., V o l . 96, No HY3. L i g g e t t , J . A . and Hadjitheodorou C . , 1969. C i r c u l a t i o n i n shallow homoqeneous l a k e s . Proc. Am. SOC. C i v . Engs., V o l . 95, No H Y 2 .
233 Neumann, G. a n d P i e r s o n , W.H. J r . , 1966. P r i n c i p l e s of p h y s i c a l oceanogranhv. P r e n t i c e H a l l , Englewood C l i f f s , N.J. P r i m a u l t , B . , 1972. E t u d e m d s o c l i m a t i a u e du Canton de Vaud, C a h i e r d e l'amenagement r d g i o n a l 1 4 , O f f i c e c a n t o n a l v a u d o i s d e l ' u r b a n i s m e , Lausanne. P r o s t , J . - P . , B a u e r , S.W., G r a f , W.H. and G i r o d , H., 1977. Campagne d e mesure des c o u r a n t s d a n s l e L h a n . B u l l . Techn. S u i s s e Romande, 103:243-249.
W I N D INDUCED WATER CIRCULATION OF LAKE GENEVA
S.W. BAUER and W.H.
GRAF
Laboratoire d'Hydraulique ( L H Y D R E P ) , Ecole Polytechnique FBdBrale, Lausanne Switzerland
(Received 1 7 August 1978; accepted 24 August 1978)
ABSTRACT A numerical modeling technique i s used t o simulate flow p a t t e r n s a t various depths i n t h e Lake of Geneva (Le LBman) f o r a homogeneous s i t u a t i o n encountered u s u a l l y during w i n t e r months. Subsequently, a v e r t i c a l l y i n t e g r a t e d flow p a t t e r n i s obtained. A measuring campaign i s under way which provides d a t a on t h e v e l o c i t y , d i r e c t i o n and temperature of t h e atmosphere and t h e water. Using t h e s e d a t a , two quasi-steady s t a t e s i t u a t i o n s a r e compared w i t h model simulations for winter months of 1977 and 1978.
DESCRIPTION OF MATHEMATICAL MODEL
A system of c u r r e n t s
i n a l a k e may be considered as water movement on a l a r g e
s c a l e . I t can be described by t h e t h r e e components of t h e momentum eauation and t h e c o n t i n u i t y equation f o r a homogeneous (non s t r a t i f i e d ) and incompressible f l u i d ( L i g g e t t , 1970) :
P[-
a U+ a 2 a (UW) a (UV) + aY aZ a t ax (U ) + -
av + aw = aU + -
ax
ay
aZ
- fV1
=
-
a aU a ap + -(v) ax aZ, ax aZ
4- -(E-)+
0
aU ax
a au ay ay
-(E-)
(1)
(4)
The boundary c o n d i t i o n s a p p l i c a b l e on s o l i d boundaries a r e u = v = w = o and a t the f r e e s u r f a c e with z = 0
(5)
220
The symbols i n equation 1-6 a r e as follows:
u , v and w a r e t h e v e l o c i t y components i n t h e x, y and z d i r e c t i o n s r e s p e c t i v e l y where x i s p o s i t i v e towards east, y i s p o s i t i v e towards n o r t h and z i s p o s i t i v e upwards w i t h zero a t t h e water s u r f a c e ,
t
is the t i m e ,
f
i s t h e C o r i o l i s parameter,
p
is the f l u i d density,
p
is the local pressure,
17 and T
&
a r e t h e v e r t i c a l and h o r i z o n t a l compnents of t h e eddy v i s c o s i t y ,
i s t h e a c c e l e r a t i o n of g r a v i t y
g X
and
and T
a r e t h e wind s h e a r s t r e s s e s on t h e water s u r f a c e i n t h e x and y d i r e c t i o n s Y respectively. Equation 3 expresses t h e h y d r o s t a t i c equilibrium, which i s a v a l i d assumption
f o r shallow l a k e s , i.e.: D/L << 1 where L and D a r e c h a r a c t e r i s t i c v e r t i c a l and h o r i z o n t a l dimensions ( f o r the LQman: D/L
2 0,03).
Equations 1-3 may be f u r t h e r s i m p l i f i e d according t o t h e following assumptions:
aU at --
a) s t a t i o n a r y flow:
0,
av at --
0;
such a s i t u a t i o n might be imagined f o r a wind which blows long enough t o establish a stationary circulation; b ) t h e i n e r t i a f o r c e s are small when compared with t h e C o r i o l i s f o r c e s , i . e . ,
the
Rossby number i s small. I n t h i s c a s e equations 1-3 may be l i n e a r i z e d :
( f o r t h e Ldman t h e Rossby number
2 0 , l )i
c ) h o r i z o n t a l d i f f u s i o n i s s m a l l compared t o v e r t i c a l d i f f u s i o n , which can be accepted f o r shallow l a k e s :
a E -)av a x ( ax
=
0,
a a~
-(&
( f o r t h e LQman: D/L
au -) aY
= 0, etc.
'J 0 , 0 3 ) ;
d) t h e v e r t i c a l component of t h e eddy v i s c o s i t y 17 which i s notknown i s assumed t o be c o n s t a n t over the e n t i r e lake: 2 a v -(n -) = 17, etc. a Z aZ 2
a
av
aZ
e ) t h e i n f l u e n c e of e x t e r n a l i n - and outflows i n t h e Leman on wind induced currents on t h e l a k e i s n e g l i g i b l e . Thus, i n t h e immediate v i c i n i t y of r i v e r mouths, t h e model i s not a p p l i c a b l e .
221 W i t h t h e above s i m p l i f i c a t i o n s and conditions t h e system of equations describing
wind induced c u r r e n t s i n a homogeneous shallow l a k e i s thus:
+pfu=
aU ax
ap
--+qaY
aw av + +ay aZ =
a2v a2
0
The numerical s o l u t i o n of t h e model described thus f a r i s obtained by Gallagher
et al.
(1973) and based on a mathematical formulation by L i g g e t t e t a l .
(1969).
ADAPTATION OF MATHEMATICAL MODEL FOR THE LEMAN
The mathematical model as described above has been coded and provided t o LHYDREP by J . A .
L i g g e t t of Cornell University. Subsequently a s u i t e cf r o u t i n e s
allowing g r a p h i c a l r e p r e s e n t a t i o n i n 2 and 3 dimensions of t h e numerical r e s u l t s
as well as of t h e geometrical r e p r e s e n t a t i o n i n f i n i t e elements
-
s e e Figure 1 -
of t h e l a k e geometry have been developed a t LHYDREP. A f t e r s e v e r a l t r i a l geometries 2 a f i n i t e element g r i d c o n s i s t i n g of 579 nodes (about one node per km 1 , 1025 triangular elements and 131 boundary p o i n t s
w a s found t o g i v e reasonable r e s u l t s
( s e e Bauer e t a l . , 1977).
GRILLE DES ELEMENTS FINIS F) TRBIS DIMENSlBNS RVEC UNE DISTBRTIBN DE X : Y : 2 = 1 : 2.00 : 0.020 LE LEMRN
t l
2 "Y
=m
mm
iii : w
'I:dl .I.-'
'*>.' '1 kil
'*,.I '&m
)n:-
'm'm
Fig. 1. "Three dimensional" Lake of Geneva ( F i n i t e element g r i d , nodes a and b showing p o s i t i o n of i n s t r u m e n t s ) .
I t should be noted t h a t compared t o i t s width,the Leman i s very shallow
(0,Ol < D/L < 0 , 0 3 ) . Due t o numerical reasons of t h e f i n i t e element approximation it i s necessary t h a t t h e minimum depth allowable is about 3 % of t h e maximum depth
222 Thus t h e r e a r e no zero depths a t t h e boundaries ( s e e Figure 1) which however i s n o t a severe l i m i t a t i o n f o r t h e geometrical r e p r e s e n t a t i o n . To execute t h e model a s e r i e s of parameters ( s e e equations 7-9) must be entered:
t h e d e n s i t y of water, p, the C o r i o l i s parameter, f , t h e a c c e l e r a t i o n of g r a v i t y , g, and t h e eddy v i s c o s i t y , q . Also, r e p l a c i n g i n euuation 6 t h e shear s t r e s s , T, by equation 11, v i z . T = c
f
where
.
.
'a
C
(11)
i s a wind shear s t r e s s c o e f f i c i e n t ,
f
i s t h e d e n s i t y of a i r
P
U
2 "wind
.
wind
and
i s t h e f r e e stream wind v e l o c i t y .
Two f u r t h e r parameters, C and p a r e t o be supplied. The following parameters f a have been taken a s c o n s t a n t s i n t h e model simulations: 3 d e n s i t y of a i r : Pa = 1 , 2 kg/m 3 d e n s i t y of water : P = 999,9 kg/m 2 a c c e l e r a t i o n of g r a v i t y : g = 9,80 m / s
C o r i o l i s parameter
:
f
= 0,000105 s
-1
( f o r a mean l a t i t u d e of t h e Leman of
46O 25 )
The parameters rl and C f , which a r e unknown a r e thus s u b j e c t t o determination by a model c a l i b r a t i o n . A t r i a l simulation f o r a s e t of f i x e d and chosen uarameters h a s been done and
i s reported by Bauer e t a l .
(1977).
MEASURING CAKZAIGNS
To allow f o r comparison between t h e c u r r e n t s occurring i n nature and t h e i r simulation by a model it i s necessary t o study ?he d i s t r i b u t i o n of t h e water v e l o c i t i e s and t h e winds which generate them. Therefore a measuring campaign has been s t a r t e d t o measure simultaneously and i n s i t u v e l o c i t i e s and d i r e c t i o n s of t h e wind and c u r r e n t s . Furthermore, temperature p r o f i l e s of atmosDhere and water were measured. I d e a l l y , measurements should be taken by a l a r g e number of recording instruments placed according t o some g r i d system over t h e e n t i r e domaine. Such measurements, h o r i z o n t a l l y and v e r t i c a l l y d i s t r i b u t e d , would allow t o o b t a i n a synoptic view of t h e phenomenon. Such a s o l u t i o n however i s a t p r e s e n t , for f i n a n c i a l and operational reasons, impossible and it was thus necessary t o a r r i v e a t a much more modest solution. Keeping t h e concept of a v e r t i c a l scheme, a s i n g l e s t a t i o n i s nlaced i n t h e l a k e . Measured and recorded a r e :
223 - v e l o c i t y o f t h e atmosphere a t t h r e e ( 3 ) d i f f e r e n t a l t i t u d e s as well a s i t s d i r e c t i o n s and t h e temperature;
- v e l o c i t y , d i r e c t i o n , temperature and p r e s s u r e of t h e c u r r e n t s a t f i v e ( 5 ) d i f f e r e n t depths. A d e s c r i p t i o n of this i n s t a l l a t i o n i s given by P r o s t e t a l .
(1977) and a schematic
view of t h e s t a t i o n i s shown i n Figure 2 . This s t a t i o n has been k e p t f o r a d u r a t i o n of weeks a t one place. Since f o r t h e s e r v i c i n g of t h e c u r r e n t meters it i s necessary t o l i f t t h e e n t i r e i n s t a l l a t i o n o u t of t h e w a t e r , t h i s o p e r a t i o n i s a l s o used t o change i t s p o s i t i o n i n t h e l a k e , thus allowing f o r a coverage of d i f f e r e n t a r e a s . During the winter months, t h e i n s t a l l a t i o n w a s l o c a t e d c l o s e t o t h e two nodes a and b a s i n d i c a t e d i n Figure 1. Measurements:
,
,
. .. .. .. ,,
I .
~
+...
Fig. 2 . Schematic view of measuring s t a t i o n For measurements i n t h e atmosphere, conventional cup anemometers of t h e type Aanderaa (WSS 219) a r e being used t o measure t h e v e l o c i t i e s . The number of r o t o r r e v o l u t i o n s i s counted e l e c t r o n i c a l l y and t r a n s m i t t e d i n d i g i t a l form t o a d a t a logger. Wind d i r e c t i o n s a r e determined with t h e a i d of a wind vane type Aanderaa (WDS 2053) and a magnetic compass measuring t h e o r i e n t a t i o n of t h e buoy with resp e c t t o n o r t h . The a i r temperature sensors used were Aanderaa (1289 A ) .
224 DATA
-
OBTAINED SINCE FEBRUARY 1977
The periods of t h e d a t a c o l l e c t e d thus f a r by LHYDREP i n t h e Lake o f Geneva
are summarized i n Table 1. TABLE 1
Sumary of d a t a p e r i o d s c o l l e c t e d by LHYDREP (Dates given by day, month and y e a r ) No
Period -
1 2 3 4 5
1/2/77
-
1/3/77
1/3/77 25/4/77 16/6/77 - 24/8/77 31/8/77 - 8/11/77 9/12/77 - 15/2/78
P o s i t i o n (km) X Y 528,28 531,24 538,39 542,02 542,02
149,18 147,84 148,68 147,48 147,48
Depth ( m ) 73,O 198,9 293,4 289,9 289.9
To allow i n s p e c t i o n of t h e s e d a t a s e v e r a l computer programs p e r m i t t i n g analog r e p r e s e n t a t i o n o f t h e d a t a were developed a t LHYDREP. A f t e r a l l d a t a weleplotted, v i s u a l i n s p e c t i o n showed t h a t t h e l a k e has been reasonably homogeneous f o r t h e e n t i r e p e r i o d s 1 and 2 and f o r period 5 a f t e r 14/1/78.
Using average temperatures
from 5,6; 10,O; 20,O; 35,O and 55,O m depth t h e o v e r a l l mean l a k e temperature was 5,725 OC w i t h a s t a n d a r d d e v i a t i o n of t h e means of 0,020 OC f o r p e r i o d 1. Similarly, f o r period 2 , using average temperatures from 5,8; 10,2; 20,2; 84,4 and 148,9 m depth, t h e o v e r a l l mean l a k e temperature was 6,143 OC with a standard d e v i a t i o n of 0 , 4 1 2
OC.
For period 5 a f t e r 14/1/78 t h e o v e r a l l mean temperature obtained from
3,9; 8 , 6 ; 18,5; 92,7 and 195,6 m depth was 5,968 OC with a s t a n d a r d d e v i a t i o n of 0,222 OC. Typical records of t h e d a t a a r e shown i n Figures 3 and 4 f o r t h e periods 2 and 5 r e s p e c t i v e l y . I n Figures 3 and 4, t h e t i m e a x i s is h o r i z o n t a l , whereby every 24 hours a v e r t i c a l l i n e , showing t h e d a t e i n y e a r s , months, days,hours and minutes
i n d i c a t e s t h e s t a r t of a new day. The d a t a represented i n t h e topmost band a r e the temperatures of t h e water i n 5 depths corresponding t o t h e p o s i t i o n s of each current meter. Proceeding downwards, t h e n e x t band shows the wind v e l o c i t i e s and then follow t h e v e l o c i t y observations of t h e 5 currentmeters. The subsequent 6 bands show d i r e c t i o n s , t h e f i r s t being t h e d i r e c t i o n of t h e wind followed by t h e directions of t h e 5 currentmeters. The s c a l e s - shown every f i v e days - have been s e l e c t e d
such t h a t t h e temperature band extends over a range of 5-10 OC, t h e wind speed band over 0-10 m / s , over 0-360°.
t h e water v e l o c i t y bands over 0-10 cm/s and the d i r e c t i o n bands
( I n c a s e t h e s e ranges a r e exceeded by t h e d a t a t o be p l o t t e d , t h e
225
t r a c e o f one band c o n t i n u e s o v e r n e i g h b o u r i n g bands b u t k e e p s i t s o r i g i n a l s c a l e . ) I n F i g u r e s 3 and 4 i t c a n b e s e e n t h a t r e c o r d s of wind were n o t always a v a i l a b l e ( F i g u r e 3: a f t e r 9 / 3 / 7 7 ,
F i g u r e 4 : between 3/1/78 and 2 3 / 1 / 7 8 ) . Thus, wind ob-
s e r v a t i o n s o f t h e S w i s s M e t e o r o l o g i c a l S e r v i c e r e c o r d e d a t 7 , 1 3 and 19 h o u r s a t Lausanne a r e i n s e r t e d i n t h e wind bands assuminq a s i x h o u r s d u r a t i o n ( s t r a i q h t l i n e s e g m e n t s ) f o r each o f t h e s e o b s e r v a t i o n s . Comparison o f t h e LHYDREP and "Lausanne" vind d a t a ( s e e F i g u r e 3, b e f o r e 9/3/77 a f t e r 23/1/78)
and F i q u r e 4 b e f o r e 3/1/78 and
show r e a s o n a b l e agreement. Also, c u r r e n t m e t e r 4 o f F i g u r e 3 r e -
c o r d e d o n l y c u r r e n t d i r e c t i o n s and t h u s no d a t a a r e shown on t h e v e l o c i t y band of c u r r e n t m e t e r 4 of t h i s F i g u r e .
SELECTION OF PERIODS USED FOR SIMULATION
I t was s t a t e d above t h a t t h e m a t h e m a t i c a l model i s c o n c e i v e d f o r t h e f o l l o w i n s
limiting conditions:
(1) homogeneity of t h e w a t e r body, i . e . ,
t h e l a k e must h a v e a l m o s t no t e m n e r a t u r e
stratification; ( 2 ) s t a t i o n a r i t y of t h e v e l o c i t y and d i r e c t i o n of t h e wind and t h e w a t e r . The manner of s a t i s f y i n r j t h e above c o n d i t i o n s i.s d i s c u s s e d i n t h e f o l l o w i n g . S i n c e t h e " s t a t i o n a r i t y " i s more r e s t r i c t i v e we s h a l l s t a r t w i t h i t . ( a d 2 ) A c l i m a t o l o g i c a l p a r t i c u l a r i t y of t h e Leman b a s i n i s t h a t t h e r e e x i s t two i m p o r t a n t and s t r o n g w i n d s : one n o r t h - e a s t e r l y - t h e b i s e
- and one s o u t h -
w e s t e r l y - t h e v e n t ( P r i m a u l t , 1 9 7 2 ) . These winds o f t e n blow f o r u e r i o d s o f days a n d , w i t h t h e p o s s i b l e e x c e p t i o n of t h e e a s t e r n end of t h e l a k e , p r o d u c e p r o b a b l y a c o n s t a n t i n t e n s i t y and d i r e c t i o n of s h e a r on t h e l a k e s u r f a c e . S e a r c h i n g f o r s u c h s i t u a t i o n s i n p e r i o d s 2 and 5 o f T a b l e 1 ( s e e F i q u r e s 3 and 4 ) , we h a v e s e l e c t e d two t i m e p e r i o d s of d a t a , s u b s e q u e n t l y t o b e u s e d f o r a model s i m u l a t i o n . These p e r i o d s a r e : bise : vent
:
29/3/77, 2/2/78,
12h00 - 30/3/77,
24h00
OOh00 -
12h00
3/2/78,
I f ' o n e r e g a r d s t h e d i r e c t i o n of c u r r e n t s a t t h e s e l a y e r s , one f i n d s them t o b e r e a s o n a b l y c o n s t a n t ; however, t h e v e l o c i t y o f c u r r e n t s a t t h e d i f f e r e n t l a y e r s i s n o t a t a l l c o n s t a n t ( I ) . V e c t o r i a l l y a v e r a g e d v a l u e s f o r t h e v e l o c i t i e s of wind and c u r r e n t s w e r e c a l c u l a t e d as i n d i c a t e d
- F i g u r e s 3 and 4 w i t h b o l d l i n e s
and a r e summarized i n T a b l e s 2 (BISE) and 3 (VENT). ( a d 1) For t h e p e r i o d s s e l e c t e d a b o v e , i . e . b i s e and v e n t , d e n s i t y v a r i a t i o n s were c a l c u l a t e d and found t o be weak; r e s u l t s a r e g i v e n i n T a b l e s 2 and 3.
-
227 TABLE 2
BISE: Mean (temporal) v e l o c i t i e s and temperatures of c u r r e n t s a t x = 531,24 km and y = 147,84 km between 29/3/77, 1 2 hours and 31/3/77, zero hours 3,5 m/s Average wind v e l o c i t y a t Lausannel) : ('wind'x - (U . ) = - 8 , 4 m / s UWlnd = 9 , l m / s = 32,8 km/h wind Depth ( m )
5,6 10,2 20,2 84,4 148,9 Weighted mean with 200 m max. depth
u (cm/s)
v (cm/s)
- 14,77 - 12,84
- 1,60
-
(cm/s)
t (°C)2)
p (g/ml)
1,16 0,96
14,86 12,89 9,61
6,287 6,293 6,320 5,962
0,999931014 0,999930803 0,999929846 0,999941645
-
9,56
-
1,l
0,02
1,lO
5,664
0,999950001
- 3,64
0.09
3,64
-
0,999942124
-
-
-
&/Po 2,1101221 9r5665536 -1,1799582 -8,3563836
-
. lo;; .
*
'-5
. 10
Po =
Velocity components of wind and c u r r e n t s : u i s p o s i t i v e f o r flow towards e a s t v i s p o s i t i v e f o r flow towards north 2, S u b j e c t t o accuracy of 5 0,Ol OC according t o instrument s p e c i f i c a t i o n
')
TABLE 3
VENT:
Mean (temporal) v e l o c i t i e s and temperatures of c u r r e n t s a t x = 542,02 km and y = 1 4 7 , 4 8 km between 2/2/78 zero hours and 3/2/70, 1 2 hours Average wind v e l o c i t y a t station') : ( U . ) = 4 , 7 m / s
(;wind)y wind x = 4 , 3 m / s = 6 , 4 m / s = 22,9 km/h wind
-
-
-
3,92 - 1,29 - 0,60
4,OO 1,16 1,08 2,66 2,22
0,21
2,20
0 ro
3,9 8r6 18,5 92,7 195,6 Weighted mean with 305 m max. depth
8,40 7,92
See Table 2 2 , See Table 2
~~
9,30 8,OO
6,055 5,930 5,911
0,999938764 0,999942607 0,999943190
4,07 2,96 2,30
5,911 5,978 5,527
0,999943170 0,999941159 0,999953397
2,21
-
Po = 0,999947942
-
-3,8424000
. 10:;
-5r6352933
*
0
2,0113047 -1,2234736
. .
lo
-6 10-5
228 CALIBRATION O F MODEL
A s has been s t a t e d above, t h e two model parameters s u b j e c t t o model c a l i b r a t i o n
( i n this study) a r e t h e wind s h e a r c o e f f i c i e n t , C f ,
and t h e eddy v i s c o s i t y ,
n.
Let u s look f i r s t a t t h e i n f l u e n c e of t h e eddy v i s c o s i t y upon t h e simulated v e l o c i t i e s , keeping t h e s h e a r stress c o e f f i c i e n t constant. This is done i n Figure 5, 2 T l = 1o00, 500 and 100 cm /s f o r t h r e e ( 3 ) p o i n t s
where t h e v e l o c i t y v e c t o r s f o r a r e shown every 2,s m.
I t can be seen t h a t i n each c a s e t h e r e i s a d e v i a t i o n on
-
-
/= :
DIRECTION DU VENT I
Fig. 5. Ekman s p i r a l s a t nodes a , b and c
229
t h e water s u r f a c e of about 45O t o t h e r i g h t between t h e wind d i r e c t i o n and t h e c u r r e n t d i r e c t i o n . Then, proceeding downwards, t h e v e c t o r s diminish and t u r n t o t h e r i g h t forming the &man s p i r a l . A t a c e r t a i n depth, D , t h e vectors have turned 180° r e l a t i v e t o t h e v e c t o r s on t h e w a t e r s u r f a c e . For an i d e a l i z e d ocean of in-
f i n i t e dimensions t h i s depth, Dp,
i s given by ( D i e t r i c h e t a l . , 1975) (12)
Thus, t h e depth, D
, is
p r o p o r t i o n a l t o t h e square r o o t of rl a s seen i n Figure 5.
Furthermore, s i n c e t h e m a s s t r a n s p o r t over t h e depth, D eddy v i s c o s i t y ( D i e t r i c h e t a l . , 1975)
-
i.e.
De
1
i s independent of t h e e' vdz = c o n s t a n t - a change of D
produces a change i n t h e v e r t i c a l d i s t r i b u t i o n of t h e v e l o c i t i e s . This i s a l s o e v i d e n t i n Figure 5 . values on the simulation, it i s t o be noted f t h a t t h i s f a c t o r acts merely l i k e a s c a l e f a c t o r on the v e l o c i t y s c a l e s . A s f o r the i n f l u e n c e of t h e C
Concluding from t h e above remarks t h e following procedure i s proposed f o r c a l i b r a t i o n of t h e model:
(1) t r y i n g d i f f e r e n t values of Q ; s e l e c t Q such t h a t reasonable d i r e c t i o n a l agreement between observations and model s i m u l a t i o n i s obtained ( g r e a t e r importance must be a t t a c h e d t o c l o s e agreement i n the t o p l a y e r s ) ; ( 2 ) check t h e thus obtained rl value a g a i n s t values found i n t h e l i t e r a t u r e ;
( 3 ) f o r the chosen value of rl a d j u s t t h e wind shear s t r e s s c o e f f i c i e n t , C f , such t h a t the magnitudes o f simulated and observed v e l o c i t i e s agree reasonably w e l l ( a g a i n , g r e a t e r importance must be a t t a c h e d t o c l o s e agreement i n t h e top layers); ( 4 ) check i f t h e s e l e c t e d value o f C
agrees with t h e ones f observations o r found i n t h e l i t e r a t u r e .
c a l c u l a t e d from d i r e c t
Following t h e above procedure, t h e b i s e ( f o r d e s c r i p t i o n s e e T a b l e 2 ) w a s inv e s t i g a t e d f i r s t and is discussed herewith ( t h e same w a s a l s o done f o r t h e v e n t ) : (ad 1) Trying v a r i o u s values of 50 cm'/s
5 rl 5
2000 cm"/s,
it was found t h a t
2
rl = 500 cm / s gave d i r e c t i o n s of v e c t o r s t h a t agreed w e l l w i t h t h e observations, as it i s t o be seen i n Figure 5 (ad 2) I f t h e rlvalue i s derived from a well-accepted equation (Neumann and Pierson, 1966) : Q
= 0,1825
where
. 10-4
U
5/2
.
wind
/ p
Uwind
i s i n cm/s,
p
i s i n g/cm3 and 2 i s i n cm / s
T)
2 with a wind v e l o c i t y of 32,7 h / h , an eddy v i s c o s i t y of 460 cm / s i s obtained.
230
F i g . 6 . S i m u l a t i o n of c u r r e n t v e c t o r s f o r a B i s e , and o b s e r v a t i o n s .
Fig. 7 . Simulation of v e r t i c a l l y i n t e g r a t e d c u r r e n t v e c t o r s f o r a Bise, and observations.
231
Fig. 8. Simulation of c u r r e n t v e c t o r s f o r a Vent, and observations RXE X l K M l
"ISCOSIlE OE TURBULENCE: 187.0 C " Z / S PRRRHETRE OE CORIOLIS: 0.000105 RROIRNSIS
RXE X l K M l
Fig. 9 . Simulation of v e r t i c a l l y i n t e g r a t e d c u r r e n t v e c t o r s f o r a Vent, and observations.
232 2 2 This i s thought t o be reasonably c l o s e t o 500 cm / s , and a value of 460 cm / s was adopted f o r f u r t h e r c a l c u l a t i o n s . (ad 3) I n o r d e r t o o b t a i n agreement of t h e magnitudes of t h e v e c t o r s , t h e wind s h e a r stress c o e f f i c i e n t , C f , w a s taken as C
f
= 0,004.
v a l u e s have been c a l c u l a t e d a t LHYDREP by P r o s t (personal communication, f 1978) f o r our measuring campaign and do indeed corroborate with our chosen values. (ad 4) C
The r e s u l t s o f this model simulation ( c a l c u l a t i o n ) are shown f o r t h e b i s e i n Figure 6 and f o r t h e v e n t i n Figure 8. Drawn a r e t h e computer c a l c u l a t e d c i r c u l a t i o n p a t t e r n s a t d i f f e r e n t l a y e r s and t h e i n s i t u measured v e l o c i t y v e c t o r s a t r e s p e c t i v e depths. Considering t h e assumptions made i n t h e mathematical simulation and t h e d i f f i c u l t i e s encountered i n an i n s i t u measuring campaign, t h e agreement i s cons i d e r e d t o b e reasonably good. Furthermore, i n Figure 7 ( b i s e ) and Figure 9 ( v e n t ) we show a comparison of depth-average v e l o c i t i e s c a l c u l a t e d and measured. Agreement i s extremely encouraging f o r t h e b i s e , b u t c e r t a i n l y less good f o r t h e vent.
(We de n o t exclude t h a t f u r t h e r
c a l i b r a t i o n could b e a p p l i e d t o reach b e t t e r agreement f o r t h e v e n t a s w e l l ) . I n t e r e s t i n g l y enough t h e model c i r c u l a t i o n p a t t e r n l e a d s u s t o agree with a conc l u s i o n r e c e n t l y drawn by Hamblin (1976):"Currents demonstrate t h e g e n e r a l tendency t o follow the wind i n t h e nearshore region, whereas t h e r e t u r n c u r r e n t opposed t o t h e wind d i r e c t i o n i s s i t u a t e d i n t h e c e n t r a l p o r t i o n of t h e l a k e " .
ACKNOWLEDGEMENT
This work was p a r t i a l l y sponsored by t h e Swiss National Science Foundation (FNSFS) under i t s s p e c i a l program "Fundamental problems of t h e water c y c l e i n Switzerland"
REFERENCES
Bauer, S.W., Graf, W.H. and T i s c h e r E . , 1977. L e s courants dans l e L6man en s a i s o n f r o i d e . Une simulation mathematique. B u l l . Techn. S u i s s e Romande, 103 :239-243. D i e t r i c h , G . , K a l l e , K . , Krauss, W . and S i e d l e r , G . , 1975. Allgemeine Meereskunde, Gebnider Borntrager, B e r l i n - S t u t t g a r t . Gallagher, R.H., L i g g e t t , J . A . and Chan, S.K.T., 1973. F i n i t e Element Shallow Lake C i r c u l a t i o n . Proc. Am. SOC. Civ. Engs., Vol. 99, NO HY7 Hamblin, P.F., 1976. Seiches, c i r c u l a t i o n , and storm surges of an i c e - f r e e Lake Winnipeg. J . Fish. R e s . Board Can., 33:2377-2391. L i g g e t t , J . A . , 1970. C e l l Method f o r computing l a k e c i r c u l a t i o n . Proc. Am. SOC. C i v . Engs., V o l . 96, No HY3. L i g g e t t , J . A . and Hadjitheodorou C . , 1969. C i r c u l a t i o n i n shallow homoqeneous l a k e s . Proc. Am. SOC. C i v . Engs., V o l . 95, No H Y 2 .
233 Neumann, G. a n d P i e r s o n , W.H. J r . , 1966. P r i n c i p l e s of p h y s i c a l oceanogranhv. P r e n t i c e H a l l , Englewood C l i f f s , N.J. P r i m a u l t , B . , 1972. E t u d e m d s o c l i m a t i a u e du Canton de Vaud, C a h i e r d e l'amenagement r d g i o n a l 1 4 , O f f i c e c a n t o n a l v a u d o i s d e l ' u r b a n i s m e , Lausanne. P r o s t , J . - P . , B a u e r , S.W., G r a f , W.H. and G i r o d , H., 1977. Campagne d e mesure des c o u r a n t s d a n s l e L h a n . B u l l . Techn. S u i s s e Romande, 103:243-249.