Pseudo-Lagrangian Dispersion Estimates from Current Meters

J. Great Lakes Res., October 1977. Internat. Assoc. Great Lakes Res. 3(1-2): 106-112.

PSEUDO- LAGRANGIAN DISPERSION ESTIMATES FROM CURRENT METERS

Merv D. Palmer Water Resources Branch Ontario Ministry of the Environment 135 St. Clair West Toronto, Ontario M4V 1P5 January 1977

ABSTRACT. A method for computing the Pseudo-Lagrangian auto-correlation function from three concurrent current meter records is discussed. The current meters were separated by approximately 400 ms parallel- to- the-bottom contours 2.4 ms above the bottom where the water depth is 11.6 ms. Integrating the Pseudo-Lagrangian auto-correlation function, the dispersion coefficients perpendicular to the bottom contours ranged from 3.7 x 10 3 cm 2 jsec at one hour to 1.0 x 10 5 cm 2jsec at 10 hours and parallel to the bottom contours from 5.2 x 104 cm 2 j sec at one hour to 1.2 x 10 5 cm j sec at 10 hours. Dispersion parallel to the bottom grows with time at a rate one- third of the growth rate perpendicular to the bottom contours. Integral space scales determined from the Pseudo-Lagrangian auto-correlation functions have a mean value of 185 and 150 meters parallel and perpendicular to the depth contours respectively. The ratio of the Lagrangian and Eulerian scales was found to be between 1.6 and 1.0.

INTRODUCTION Most of our present knowledge of dispersion estimates on lakes and oceans has been generated from experiments involVing the tracking of passive chemical components like chlorides or dye plumes or the tracking of drogue clusters. The dispersion processes in these environments are complex phenomena, consequently our understanding is based upon empirical analysis of the tracking experiments which have revealed a general relationship between dispersion coefficient and time and/ or space. Dispersion is a Lagrangian process so it is natural to follow a passive contaminant in the body of water to obtain an insight into the dispersion processes. However, these experiments generally require extensive planning and are expensive to execute. Therefore in anyone particular case one is faced with accepting the results of two or three experiments, normally conducted under ideal weather conditions, as being representative of the body of water. In highly variable bodies of water like the coastal regions we are aware from measurements with recording chemistry meters that a few determinations are inadequate for representing the water chemistry variations of the coastal waters

(Palmer 1972). Examinations of the records from current meters similarly demonstrate the inadequacies of a few spot measurements for representing water movement and the predominance of advection parallel to the bottom contours. These results suggest that one should attempt to estimate dispersion from recording devices at fixed locations for a statistical representation. We attempted to do this in earlier experiments for long periods of time (monthly) and using an equivalence between Eulerian and Lagrangian length scales for large Reynolds numbers 0 (1 OS) (Corrsin 1963; Palmer and Izatt 1970) and for the short-term in hours (Palmer & Izatt 1971). However, these experiments utilized the data from one recording current meter and were limited by the requirement of some approximation of the equivalence between the Eulerian and Lagrangian systems. The equivalence between Eulerian and Lagrangian systems is a reasonable approximation for large Reynolds numbers. Recently Murthy (1973) found an approximate equivalence for periods of less than 17 hours in the coastal regions of Lake Ontario. One is skeptical of the representativeness of a single point measurement for a body of water with 106

PSEUDO-LAGRANGIAN DISPERSION variable characteristics. The next step is to utilize data from many adjacent fixed recording locations to evolve estimates of the dispersion. This paper discusses the development of a method to determine dispersion estimates from the data from three current meters operated concurrently in the coastal regions of Lake Ontario.

DEVELOPMENT Let us derive a relationship between the dispersion coefficient based on the Lagrangian auto-correlation time function after Hinze (1959, pages 46 to 49). The Lagrangian correlation coefficient is:

where V2 (t) is the turbulent velocity (V2 (t) = V2 (t) - is an ensemble average. Rh(7) is symmetrical with respect to 7, is zero for large 7, and is 1 for 7 = O. It can be shown that the mean particle spread
For long periods of time, the dispersion coefficient €2 (t) is: €2(t)

=(rmsv2)2

.r;, dr Rh(7) ....

(3)

and the integral space scale is: AL

_

-

L

(rms V2) f o d7 R22 (7) .... 00

(4)

If we can measure the Lagrangian correlation function it would be possible to determine the dispersion coefficient. A similar approach can be used to determine the particle spread and dispersion in the Xl direction. In the following method section we will attempt to evaluate a Pseudo-Lagrangian correlation function using Eulerian data histories from three locations which are spaced out along the predominant current direction at the same depth. Monin and Yaglom (1971) discuss the relationship of the Lagrangian velocity correlation function and Eulerian statistical Characteristics in Section 9.5. In general the theoretical relationships are very complex. However, for a one-point and one-time probability distribution and homogeneous turbulence for an incompressible fluid, the probability distributions for Eulerian and Lagrangian velocities will coincide with each other at all times. For

107

homogeneous and stationary turbulence the Lagrangian correlation function is greater than the Eulerian function and it follows that the Lagrangian integral and microscales are greater than the corresponding Eulerian scales. The assumption of near equivalence between the Lagrangian and Eulerian system is justified as an approximation in this case as the Reynolds numbers are large (Corrsin 1963; Palmer and Izatt, 1970) ranging from (0.7 to 5.11) X 10 5 • Franz (1974) shows that there is a relationship between Eulerian and Lagrangian correIation functions and Hay and Pasquill (1959, p. 142) show from measurements that the Eulerian and Lagrangian scales are related A L = I'A E where 1.3 < I' < 1.6. Although no concurrent Lagrangian data are available it is possible to validate the method. The Pseudo-Lagrangian correlation coefficients determined from three simultaneous Eulerian current histories will be compared to the Eulerian correlation coefficient demonstrating the differences. Then mean particle spreads and dispersion coefficients computed from the PseudoLagrangian correlation coefficients will be compared to Lagrangian dye experiments. Method Three submersible self-recording current meters (A, B and C) were separated by 357 and 419 meters respectively parallel to the bottom contours in a water depth of 11.6 meters at 2.4 meters from the bottom (Figure I). The separation of the meters was of the order of 4 integral space scales. The current meters were Geodyne Models 920 and 850 attached to fixed bottom-resting towers. Velocity was measured with Savonius rotors with a threshold of 2 em/sec and accuracy of approximately 2.5 em/sec. These meters were set to sample current speed and direction for 40 seconds every 20 minutes. Proper functioning of the meters was checked by independently measuring current speed (agreement within 1.5 em/sec) and direction (agreement within 12°) in the vicinity of each meter every two or three weeks. We now have three simultaneous time histories of Eulerian currents at the same depth. Let us designate these histories UA(t), UB(t) and UC(t). For ease of computation we define the direction connecting A, Band C as Xi positive B to C and the orthogonal direction as X 2 positive to the south (see Figure I). For every turbulent velocity UB(t) value we find a UB(t + 7) from the UA(t + 7) or UC(t + 7) records. Every UB(t) consists of a UBI (t) and UB 2 (t) component. If UBI (t) is positive, 7 = distance BC/UB i (t) or if

M. D. PALMER

108

.'

'e'

METER

.....

79°35'30" W 43°29'52" N

..

METER 'B'

0°

..'

'

•••••••

79°35'41" W 43°29'42" N

... \8········

'A'

METER

t' .

.. '

.... '.

79°35'51"W 43°29'34" N

'"

.~

'.

..

• • 0'

GULF OIL ./ .:....

43°30'

,.:~. (.':-

1m1Q::

,'. ,<::..~~~ . ,

:

:..

:

.0'

~:

•0° •• 0

.0° .0"

:., 1<)' : ~

~:

,":--:'

:

.0°

:..,

X2

." 00

.'

~

.,'

..-

................. 0°

:.

•••••

~.

:

.

: ....

,'.

0°

! ....,' ,'.,:

:

XI

..,

,'.,
ONTARIO

•• -

0" •••• 0" ."

....

o

I Km.

DEPTH IN METERS

FIG. 1. Current Meter Locations. negative 7 = distance BA/UB 1 (t). Once 7 is known UB 2 (t + 7) = UC 2 (t + 7) or UA 2 (t + 7) provided UBI (t) > UB 2 (t). For all records UB(t) > 2UB 2 (t)

(Table 1). We are now in a position to infer a Lagrangian correlation function from three concurrent Eulerian records which we will call a Pseudo-Lagrangian correlation function. To evalu-

ate the Pseudo-Lagrangian correlation function of equation (1) this process must be repeated many times storing the UB 2 (t) UB 2 (t + 7) values in column vectors. Let us operate on the UB 2 (t) and detennine UB 2 (t + 7) by using the records of UA 2 (t) and UC 2 (t) so that we can detennine a Pseudo-Lagrangian correlation function for each 7 to obtain a statistical result. For each 7 column a mean value < UB 2 (t) UB 2 (t + 7) > is determined as well as the variance of UB 2 (t). The variances of UB 2 (t) are weighted by the number of readings in each column vector. We now have a series of Rt2 (7) correlation coefficients which must be numerically smoothed. In this case the coefficients were smoothed using a moving average technique with binominal smoothing coefficients over three coefficients. Does this Pseudo- Lagrangian detennination approximation compare with other results of Lagrangian experiments? If the determined Pseudo-Lagrangian correlation function is valid it should be a continuous function with a value near 1.0 at zero time lag and zero at some large time lag. It should also contain similar periodicities as the Eulerian auto-correlation time function for location B. For the X2 direction one expects the Pseudo-Lagrangian correlation function to be greater than the Eulerian similar to the findings of Hay and Pasquill (1959) in an atmospheric boundary layer. The validity of the Pseudo-Lagrangian correlation coefficient can be checked by comparing the integral scales for the Eulerian and Pseudo-Lagrangian. Hay and Pasquill (1959) found the Lagrangian integral scale to be 1.3 to 1.6 times the Eulerian scale. The· time lag determination in the Pseudo-Lagrangian method is based on a velocity measured at location B

TABLE 1. Sample Current Frequency Table for Location A, June 1974.

Direction in Degrees (0° North) Speed (Cm/Sec) 0.0 to 0.30 0.31 to 1.99 2.00 to 3.99 4.00 to 5.99 6.00 to 7.99 8.00 to 9.99 10.00 to 37.99 Column Sums

157.5 202.5 0.0 1.48 3.98 139 1.20 0.79 0.14 8.98

202.5· 247.5 0.0 1.99 6.11 9.72 8.15 7.13 10.42 43.52

247.5· 292.5 0.0 1.06 5.65 4.54 2.22 0.69 0.51 14.68

292.5· 337.5 0.0 0.74 4.44 2.50 0.69 0.0 0.0 8.38

337.522.5 0.0 0.83 2.82 2.08 0.60 1.06 1.62 9.03

22.567.5 0.0 0.60 1.76 4.21 1.34 0.69 0.09

67.5112.5 0.0 0.51 1.67 0.74 0.51 0.05 0.0

8.70

3.47

Resultant Current is 3.20 Cm/Sec at 59 degrees. Mean current is 5.86 Cm/Sec. Maximum current is 23.33 Total number of readings = 2160. Persistence is 0.55. Reading taken every 20.00 minutes.

112.5 . 157.5 0.0 0.65 1.53 0.97 0.09 0.0 0.0 3.24

Row Sums 0.0 7.87 27.46 16.16 14.81 10.42 12.78 100.00

109

PSEUDO-LAGRANGIAN DISPERSION

persisting on the average for the period of the lag, e.g., for 2 and 25 hour lag an average of 5.4[71(419 +357)m/2 hours] and 0.43 cm/sec respectively are assumed. The shorter time lags are a better measure of single episodes while longer time lags will contain more current direction reversals associated with periodic motion like lake sieches with a fundamental period of 5.1 hours (Rao and Schwab 1974) and inertial periodicity of 17.4 hours (Verber 1966). The method assumes a general current field in one predominant direction for the lag period, consequently the applicability of the assumptions of the method are weaker for longer lag times. Measurements for the longer lag times are also weakened by the accuracy of the measuring device. The author suggests that lag times greater than 22 hours (component of mean velocity in the Xl direction of 0.5 cm/sec) are crude and a lag time of 12 hours or less the most reliable for the existing meter spacing. The dispersion coefficient e(t) and the variance of the particles spread
Two months of current meter data were collected at locations A, Band C. A sample record for location A in June 1974 appears in Table 2. From these records the Pseudo-Lagrangian correlation functions in the X2 direction for June and July TABLE 2. Summary of Current Statistics. Locations A, B and Cfor June and July 1974. Current Meter Locations

A Jun Mean Speed «U cm/sec 5.9 Maximum Speed 23 cm/sec Percent of record along shore 70 Percent of record offshore 30

»

B

•

C

Jill

Jun

Jill

Jun

5.0

3.1

2.5

3.9

Jill 3.0

21

21

13

21

15

58

67

55

62

54

42

33

45

38

46

1.0

~~

0.8

IL

0.6

~

0.4

z

~

g

8

02

0.0

-0.2

r;:---,---.,---,-----,---.,---,----, '.;:;;.

JUNE

.\,.,.,\

"\.

1974

i\

\

.V\' \'!,/ \ \_-:// /, ",.

v\/

\,'/

•

•

~'

.>:

_._

.~._-----.--f9-s-

% CONF IDENCE INTERVAL

-0.4 L -__--'-_ _-'---_ _L-.--_--'-_ _-'---_ _l.----l o 15 30 45 60 75 90 TIME (HOURS)

LEGEND - - - AUTOCORRELATION (EULERIAN) LOCATION B - - - - - - PSEUDO-LAGRANG I AN 1.0 I-

z w -

O.B

'=

0.6

IL IL

W

0 <..>

0.4

z 0 -

0.2

I-

«

...J W

0.0

a:: a::

8 -0.2

T I ME (HOURS)

FIG. 2. Pseudo-Lagrangion Correlation and Eulerion Autocorrelation Function X 2 Direction at Location B June and July 1974.

were computed and plotted in Figure 2. The Eulerian auto- correlation time function for location B also appears in Figure 2 with the 95% confidence intervals (Jenkins and Watts 1969, p. 188). Both functions approach zero for long lag times within the 95% confidence interval. It is observed that the Pseudo-Lagrangian functions are well behaved, greater than the Eulerian functions and exhibit similar periodicities as the auto-correlation time functions for lag times less than 28 hours. As the Reynolds numbers are large (0.7 to 5.1) X 10 5 (using water depth for the length scale) one expects some similarity between the Eulerian and Lagrangian correlation functions (Corrsin 1963; Franz 1974) which appears to be true. The variances < y~(t) > determined from the Pseudo-Lagrangian functions are plotted in Figure 3 where the slope is observed to be 2.4 and 2.5 for July and June respectively which is very close to the 2.3 value shown by Okubo (1971) in his summary of oceanographical results and by Murthy (1976). This indicates that eddy diffusity grows faster than a Fickian model where < Yht) > ex: t and slower than in the "Inertial Subrange" where < y~ (t) > ex: t 3

110

M. D. PALMER

.u.

N.

-... N.

1010

E

u

E

u /'..

,.,

NN

(1970)

'-/

W

-

U

Z

U

«

tr

lL lL W

« >

w ...J u

o

109

u

z o

-

I-

tr

«

ll.

LEGEND JUNE 'SLOPE 1.42 X JULY 'SLOPE 1.59

@

LEGEND Gl JUNE ' SLOPE 2.43 X JULY , SLOPE 2.5

JUNE LESS ONE STANDARD ERROR JULY PLUS ONE STANDARD ERROR

JUNE LESS ONE STANDARD ERROR JULY PLUS ONE STANDARD ERROR 10

10 3

FIG. 4. Dispersion Coefficients X 2 Direction vs. Time June and July 1974.

(Murthy 1976). Plots of the dispersion coefficients (~(t) versus time are presented in Figure 4. The (rms velocities) used are listed in Table 3. For a quantitative comparison, the effective dispersion coefficients given by Murthy (1970) for Lake Ontario 13 km offshore at a depth of 2.5 meters are 4.1 X 10 4 , 5.0 X' 10 4 and 1.4 X 105 cm 2 /sec for 10.3, 14.8 and 30 hours respectively (Figure 4). The Pseudo-Lagrangian correlation functions in the Xl direction for June and July are plotted in Figure 5 with the Eulerian autocorrelation functions for location B. These generated PseudoLagrangian functions are similar in behavior to the TABLE 3. Summary of (rms velocities) used in dispersion coefficient and integral scale computation. (rrns velocity) cm/sec Xl X2 Jun Jul Jun Jul

3.3

2.3 3.0

LJ 100

TIME (HOURS)

100

FIG. 3. Particle Spread X 2 Direction at Location B vs. Time June and July 1974.

5.2

~___l

10

TIME (HOURS)

Eulerian Pseudo- Lagrangian

L....L

1.9

23

1.9 4.2

Eulerian auto-correlation functions for location B computed in the X2 directions except that the Lagrangian functions are less than the Eulerian. !he computed particle spread < yt > (Figure 6) mcreases at a 1.5 power of time which is significantly slower than the 2.4 power for the X2 direction. Similarly the computed dispersion coefficients €1 (Figure 7) increase at a 0.5 power of time compared to the 1.5 power for the X 2 direction. These predictions show that the dispersion coefficients in a direction parallel (Xl) to the bottom contours (along shore) are 10 and 2 times greater than dispersion in a direction perpendicular to the bottom contours (X 2 ) at 1 and 10 hours respectively similar to the findings of Murthy (1976). But as the growth rate of the dispersion coefficient with time in a direction perpendicular to the bottom contours is 3.0 times greater than ~hat parallel to the bottom contours, the dispersion 1S approximately equal in both directions at 16 hours. This result is different than Murthy's (1976) and may be a result of the method used here or the influences of lake sieche and/or inertial perio-

111

PSEUDO-LAGRANGIAN DISPERSION 1.0

.-.\

0.6 0.4

z

~

0.2

I-

""

--' 0.0 a: a: 0-0.2

'"

<.>

JUNE 1974

\ .- ........

0.8

"

\Jv-.~::::_-

,, ,, "

\

\ '.

,

\/

'.

......

/\ /

95 % CONFIDENCE INTERVAL

//0/

--:::-::-::---. --- -.

~

:$

-0.4 ' - - - - - - - - - ' - - - - - - ' - - - - ' - - - - - - ' - - - - - - ' - - - - - - ' - - - - - - ' o I5 30 45 60 75 90 T I ME (HOURS)

LEGEND - - - AUTOCORRELAT I ON (EULER I AN) LOCAT ION 8 - - - - - - PSEUDO-LAGRANG I AN 1.0

N. /'"-,

~I

,--------.--~--~--~--___,_---,-----.

10 10

'"z<.> ""a:

.. >

'"--'<.>

..

I-

a:

0..

JULY 1974 LEGEND o x

I 10 8

JUNE 'SLOPE JULY 'SLOPE

1.5 I 1.54

JUNE PLUS ONE STANDARD ERROR JULY LESS ONE STANDARD ERROR

~-------L-

10

:-'-...J

100

T I ME (HOURS) -0.4 '----'----------'-------'------'------'------'-------" o 15 30 45 60 75 90 T I ME (HOURS)

FIG. 6. Particle Spread Xl Direction at Location B vs. Time June and July 1974.

FIG. 5. Pseudo-Lagrangian Correlation Function and Eulerian Auto-correlation Xl Direction Function at Location B.

dicities in the current meter records. The method appears to produce reasonable results; however, the spacing of the current meters is obviously important. The Pseudo-Lagrangian correlation is well behaved and the particle variances and dispersion coefficients compare favourably with other Lagrangian experiments. The Lagrangian integral space scales computed from Equation (4) have a maximum value at approximately seven hours lag. The first zero values for the PseudoLagrangian correlation function occurred at 48 and 28 hours perpendicular to bottom contours in June and July respectively and at 13 hours parallel to the contours in both June and July. The reason for this is that the variance of the mean velocity at longer lag times decreases as there are fewer records at the longer lags to obtain a variance estimate and the accuracy of method decreases with longer lags as discussed previously. Other distances between the meters would permit different estimates to be made. Applying Equation (4) with the limits of the integral set at seven hours and using a weighted mean velocity variance, the integral space scales parallel to the bottom depth contours are 170

~

"-

N.

~

8

"'

10

I-

z

'"<.> u. u.

'"o<.> z

o U)

a:

~

10 5

U)

o

x

o

x

I 104

~

JUNE ' SLOPE 0.5 I JULY' SLOPE 0.54 JUNE PLUS ONE STANDARD ERROR JULY LESS ONE STANDARD ERROR

----:-L-

--'---J

10

100

T I ME (HOURS)

FIG. 7. Dispersion Coefficients Xl Direction vs. Time June and July 1974.

M. D. PALMER

112

meters in June and 200 meters in July, and the scales perpendicular to the bottom contours (the offshore direction) are 101 meters in June and 205 meters in July. These integral scales are 80 times larger than those found by Palmer (1973) using data from one current meter supplemented by hot film data and 10 to 20 larger than those found from Lagrangian drogue trackings by Okubo and Farlow (1967) and Jones and Kenney (1974). However, these results are compatible for the coastal regions of the oceans with scales 10 to 100 meters (Stommel 1949). It is not known why the integral scales computed from the Pseudo- Lagrangian technique are larger than those determined from Lagrangian experiments when both the particle spread and dispersion coefficient estimates computed by the Pseudo-Lagrangian technique compare so well with other Lagrangian experiments. The author suggests that integral scales of the order of 100 meters are probably realistic for the lakeshore region particularly at depth. A better insight into the method can probably be obtained by applying the method to data from other meters at smaller separations of 100 to 200 meters or approximately an integral space scale. The characteristic time and space scales parallel to the shore are 38 hours and 185 meters and perpendicular to the shore (offshore) 11 hours and 150 meters. The relationship between the Eulerian and Lagrangian scales using the relationship proposed by Hay and Pasquill (1959) A L ={3A E was evaluated from the data and for the Xl direction {3 = 1.6 and 1.3 respectively for June and July and for the X 2 direction {3 = 1.6 and 1.0. These values are within the range published by Pasquill (1962, p. 142) but on the low side. Dispersion characteristics could be obtained in other directions by employing an array of meters instead of meters in a line.

ACKNOWLEDGEMENTS I would like to thank B. S. Kohli who ably carried out the computer processing and Drs. C. R. Murthy and G. K. Sato who reviewed the manuscript.

REFERENCES Corrsin, S. 1963. Estimates of the relation between Eulerian and Lagrangian scales in large Reynolds number turbulence, J. of the A tmos. Sc. 20: 11 5-119. Franz, H. W. 1974. On Lagrangian and Eulerian correlations. In Symposium Pour L 'exploration de la Mer, pp. 125128. Arhus: Conseil Inti. Hay, J. S., and Pasquill, F. 1959. Diffusion from a continuous source in relation to the spectrum and scale of turbulence in Atmospheric Diffusion and Air Pollution,

ed. F. N. Frenkiel and P. A. Shep'pard, Advances in Geophysics, Academic Press. 6:345. Hinze, J. O. 1959. Turbulence: An Introduction to Its Mechanisms and Theory. New York: McGraw-Hill Book. Jenkins, G. M., and Watts, D. G. 1969. Spectral Analysis and Its Applications. San Francisco: Holden & Day. Jones, I. S. F., and Kenney, B. C. 1974. Turbulence in Lake Huron, Water Res. 5 :765-776. Monin, A. S., and Yaglom, A. M. 1971. Statistical Fluid Mechanics of Turbulence, Vol. 1. Cambridge: The MIT Press. Murthy, C. R. 1970. An experimental study of horizontal diffusion in Lake Ontario, Proc. 13th Can! Great Lakes Res: 477-489. Murthy, C. R. 1973. A comparison of Lagrangian and Eulerian current measurements in coastal waters of Lake Ontario, Proc. 16th Can! Great Lakes Res: 1034-1037. Murthy, C. R. 1976. Horizontal Diffusion in Lake Ontario, J. Phys. Oceanogr.6:76-84. Okubo, A., and Farlow, J. 1967. Analysis of some Great Lakes droque studies, Proc. 4th Can! Great Lakes Res: 299-308. Okubo, A. 1971. Oceanic diffusion diagrams, Deep-SeaRes. 18:789-802. Palmer, M. D. 1972. Some chemical and physical relationships on Lake Ontario, Water Res. 6:843-858. Palmer, M. D. 1973. Some kinetic energy spectra in a nearshore region of Lake Ontario,J. Geophys. Res. 78:35853595. Palmer, M.D., and Izatt, J. B. 1970. Dispersion prediction from current meters, Proc. Am. Soc. Civ. Eng., Hyd. 96 (HY8): 1667-1680. Palmer, M. D., and Izatt, J. B. 1971. Lake hourly dispersion estimates from a recording current meter, J. Geophys. Res. 76:688-693. Pasquill, F. 1962. Atmospheric Diffusion. London: D. Van Nostrand Co. Ltd. Rao, D. B., and Schwab, D. J. 1974. Two-dimensional normal modes in arbitrary enclosed basins on a rotating Earth: Application to Lakes Ontario and Superior, Center for Grea~ Lakes Studies, University of WisconsinMilwaukee, Spec. Rep. No. 19. Stommel H. 1949. Horizontal diffusion due to oceanic turbulence. J. Mar. Res. 8: 199- 225. Verber, J. L. 1966. Inertial currents in the the Great Lakes, Proc. 9th Can! Great Lakes Res: 374-379.