Estuarine and Coastal ikfarine Science (1974) 2, 273-281.

The Salt Balance in Stratified Estuaries

K. R. Dyer Institute of Oceanographic Sciences, Crossway, Taunton, Somerset, U.K. Received 17 November I973

There are several methods by which the dispersion of salt in estuaries is analysed using cross-sectional aa well as time averaging. These are compared using data from three estuaries. The results show that in a highly stratified estuary the lateral contribution to the dispersion is small. In two partially mixed estuaries the lateral contribution is of the same order of magnitude as the vertical.

The distribution of salinity within an estuary is maintained by a balance between advection and diffusion. At a point cube this can be described by the Fickian equation of salt continuity using a coefficient of diffusion. As soon as the volume becomes large enough for there to be non-uniform advection through the faces of the cube then the balance can be thought of as maintained by the mean flow through the cube face and a dispersive flux of salt which can be represented by a dispersion coefficient. The contribution of diffusion to dispersion is generally small and most of the observed dispersion is the result of inhomogeneities in the advective fluxes of salt. I f the volume is a complete cross-section of the estuary enclosed by the water surface and the estuary bed and sides, the dispersion is the result of gravitational circulation and diffusion and balances the cross-sectional mean flow, which, in the case of a standing tidal wave and steady state conditions, corresponds to the river discharge. In partially mixed estuaries the gravitational circulation involves vertical and lateral variations in current velocity and salinity leading to an upstream mean salt flux on the bottom and a downstream mean flux on the surface. Lateral effects can be caused by Coriolis Force and by centrifugal accelerations due to the estuarine topography.

In the absence of more detailed knowledge the dispersion coefficient is a useful means of comparing estuaries and of attempting prediction of effluent distributions. However, the values obtained depend not only on the adequacy of the data in describing the cross-sectional variations, but also on the decomposition of the fluxes into their tidal cycle average and fluctuating parts. There is some uncertainty as to the relative magnitudes of vertical and lateral contributions to the dispersion coefficient in different types of estuaries. This report examines these contributions from observations in a highly stratified and two partially strati- fied estuaries.

The shear effect

Taylor (1954) examined the flow and mixing of dye in pipes. Because of shear on the pipe walls, dye in the centre of the pipe travelled faster than that near the walls. Vertical diffusion then mixed the faster layers downwards and the slower layers upwards giving an increase in

273

a74 K. R. Dyer

the length of the dye patch downstream. It was found that a dispersion coefficient describing this effect can be defined as:

D = Io-IaU, (1)

where a is the pipe radius and U, the friction velocity. This analysis was extended by Elder (1959) for flow in a channel and, assuming laterally homogeneous conditions, he obtained:

D = S-ghU, (2)

where h is the water depth. By use of a friction coefficient, Equations (I) and (2) can be obtained relative to the mean flow rather than the friction velocity.

Bowden (1965) used more complicated velocity profiles and calculated larger dispersion coefficients, but the values were still much smaller than those observed in rivers. This was explained by Fischer (1967) who obtained realistic values for dispersion coefficients by consideration of transverse shear. In estuaries, however, the flow is oscillatory and stratifica- tion is present. With certain assumptions, including lateral homogeneity, Bowden (1965) found that for an alternating flow the dispersion coefficient is about half that for unidirectional flow. For a tidal current of amplitude U, the dispersion coefficient is:

D = 0.15 U,h (Bowden, 1963) (3)

Dispersion in estuaries can also be thought of in terms of the one-dimensional steady-state transport equation :

iii=DEandifii=R/A,D=E Ad”

dx

(4)

Where R is the river flow and A the cross-sectional area. Values calculated using this equation are an order of magnitude greater than calculations using Equation (3), but the relative effects of cross-sectional variations in mean flow and of cross correlations between tidally fluctuating values cannot be distinguished.

Bowden (1963) has considered the salt flux provided by changes in the vertical profiles of velocity and salinity during a tidal cycle. The instantaneous rate of transport of salt through a unit width of a section perpendicular to the mean flow is given by:

F = <us>h (5)

where the angled brackets denote depth mean values. At any depth let u = u,+u’ and s = s,,+s’, where us and ss are the observed velocity and salinity averaged over a number of minutes and zi and s’ are the turbulent variations. Also Y,, = <u> fu, and ss = <s> +si, where u1 and si are the deviations of the observed values from the depth means. Because of tidal fluctuations <u> = ii+ U and <s> = 3+S, where P and s are the mean values over a tidal cycle.

Consequently Equation (5) averaged over a tidal cycle becomes :

F = %i.T + hUS + h<u,s,> + h<u’s’> (6) I 2 3 4

Term I IS the flux associated with the river discharge. Term 2 will give a finite contribution if the tidal response of the estuary has a progressive component which is correlated with the tidal salinity variation. Term 3 is the shear effect and Term 4 the diffusive contribution due to the turbulent fluctuations. Comparison between Equations (6) and (4) shows that Term 2

Salt balance in stratified estuaries 275

in Equation (6) would be regarded as part of the dispersive flux even though it is caused by a component of the non-tidal drift. For results from stations in the Mersey Narrows, Bowden (1963) considers the contribution of Term 4 to be small and the dispersion coefficients obtained using Term 3 alone were comparable with values using Equation (4). The flux of salt due to Term z was directed upstream and during two of the three periods of observation its value was about three times that of Term 3. During all of the measurements there was a total upstream salt flux. During a subsequent investigation (Bowden & Sharaf el Din, 1966) three stations on a section were occupied and the results analysed in the same way. The values of Term 3 were of the same sign and magnitude as calculated previously, with only small differences between the stations. The values of Term 2, however, were large and positive. The observed mean velocities were an order larger than R/A, but were negative on one side of the estuary and positive on the other. The cross-sectional mean velocity was probably equivalent to the river discharge.

Hansen (1965) has presented an analysis of salinity and velocity data from a cross-section at the mouth of the Columbia River. Instantaneous salt fluxes are considered as a cross- sectional average and deviations therefrom. Thus:

F = 4uAsA + 04~) (7)

where the subscript A denotes averaging over the cross section. Both the cross-sectional averages, the deviations from them and the cross-sectional area can be considered as the sum of a tidal mean, a tidal fluctuation and a turbulent fluctuation. Thus uA = ti*+U,+u’,,

SA = S,+SA+S’A, (u&A = (~dsd)A+(udSd)A+(~d~d)‘A and a = A+A+A’- Putting these values into Equation (7) and averaging over a tidal cycle gives:

-- - -- - F= iiiAIA+A~,SA+A~, ii. A+AuAsA+AuAsA+A(UdSd)AfA(udsd)A. (8)

I 2 3 4 5 6 7

There are additional turbulent terms in Equation (8), but their magnitudes cannot be inde- pendently assessed. The first two terms on the righthand side are associated with the ‘non- tidal drift’, the first being the result of the river flow and the second being a compensation flow for the inward transport on the partially progressive tidal wave. Hansen (1965) found that about 70% of the non-tidal drift in the Columbia River estuary was due to the river discharge and consequently 30% of the downstream salt flux on the mean flow was a com- pensation current. Term 3 is due to correlation of tidal period variation of tidal height and salinity, Term 4 the correlation of tidal period variations of salinity and current. Term 5 is the third order correlation of tidal period variations in salinity, velocity and cross-sectional area, Term 6 is the mean shear effect and Term 7 the covariance of the shear effect and cross sectional area.

Evaluation of the magnitudes of the terms in Equation (8) showed that, of the seaward salt flux on the mean flow, 45% was balanced by the shear effect (Term 6) and 40% by Term 4. The other terms were comparable in magnitude to the possible errors and the remaining 15% was taken as being the result of short period turbulent fluctuations.

The shear effect was separated into lateral and vertical components by Fischer (1972).

He has considered separately the deviations of velocity and salinity from the cross-sectional mean. Thus ud = ii,+U, and s, = ?,+S,. Considering three dimensional profiles both z& and U, can be separated into variations in the vertical transverse directions. Thus ii, = iid,+& and U, = U,+U,, where iiht is the deviation of the depth mean at any position from the cross sectional mean, and tid, is the deviation of the mean value at any depth from the depth mean value Udt, i.e. <ti,> = riA+id = iiA+tidt and ii, = UA+z&,+ridV. A

276 K. R. Dyer

similar treatment can be carried out for salinity. Fischer (1972) has neglected tidal fluctua- tions in cross-sectional area and the mean salt flux over a tidal cycle through a unit area of the cross section is then :

- - k% = uAiA + uAsA + &&t)A + (&,i.dA + (u,‘%)A + (“vsv)A~ (9)

I 2 3 4 5 6

Term 2 is equivalent to term 4 in Equation (8) and the last four terms should be equivalent to term 6 in Equation (8). Term 3 is the contribution of the net transverse circulation, Term 4 the net vertical circulation, Term 5 the transverse oscillatory shear and Term 6 the vertical oscillatory shear.

If we include a tidally fluctuating cross-sectional area then the mean flux of salt through a section over a tidal cycle becomes:

I 2 3 4 5 6 7

8 9 IO II

Terms I to 5 are equivalent to those in Equation (8). Terms 6 to 9 correspond to the last four terms in Equation (9) and the last two in Equation (8). Terms IO and II represent the covariance of tidal fluctuations in cross-sectional area and the transverse and vertical oscil- latory shear.

Fischer (1972) assessed the relative magnitudes of the last four terms in Equation (9) from theoretical considerations, based on data from the Mersey. He concluded that the net transverse gravitational circulation was an order greater than the net vertical gravitational circulation and the vertical oscillatory shear.

An assessment of the relative magnitudes of the various terms in Equations (6)-(10) using the same sets of data has not so far been done.

Data and results

The Vellar estuary has been described by Dyer & Ramamoorthy (1969). On four occasions during a period of decreasing river flow measurements of current velocity and direction and of salinity were observed over a complete tidal cycle at three stations on each of two sections in the lower estuary. Measurements were taken at 0.5 m intervals from the surface to the bottom. On the first three occasions the estuary was of a salt wedge type, though with upstream mean flow on the bottom, the flow regime being dominated by river discharge. During the final period at the lower section the estuary was of a partially mixed type with significant differences in mean velocity across the estuary.

The hydrography of Southampton Water has been described by Dyer (1973). The estuary is a typical partially mixed estuary and has a large lateral gradient of salinity and lateral variations in the mean flow caused mainly by the estuarine topography. The data used here were obtained at four stations on each of two sections in the lower estuary. Measurements were taken at 1-5 m intervals from the surface to the bottom, One section was observed on one day and the other section the following day.

The Mersey has been described by Hughes (1958) and Bowden & Sharaf el Din (1966). It is a partially mixed estuary with lateral variations of salinity and velocity. The data used here is that described by Bowden & Sharaf el Din for their section C. The observations were

Salt balance in stratified estuaries 277

at three stations across the estuary, the stations being occupied for consecutive as-hour periods.

The data was analysed treating the results at each station as being a true representation of the mean in a prismatic segment of the cross section. The possible errors in the results were assessed by re-running the computer programmes with reasonable error bands on the data. The velocities were assumed to have errors of -foe05 m s-1, salinity in the Vellar &~*a%, and in Southampton Water and the Mersey fo*03%~. Areas are taken to be accurate to 15 %. The values for each term in the equations were calculated for the original data and also when increased and decreased by the above amounts. Also the data for the surface layer on one side of each estuary was increased and for the bottom layer on the other side decreased by the error values. The last calculation was then repeated using the opposite sides of the estuaries. The errors quoted in the Tables 2 and 3 are the mean deviations of these four values from those calculated from the original data. They are likely to be the outside limits as these calculations are maximising consistent errors. Random errors would lead to smaller deviations.

The results shown in Table I were calculated using Equation (6). The salt flux of Term I

calculated using the observed mean velocities are much larger than the other two terms but, having positive and negative values, the cross sectional average could produce a negligible total salt flux. If the mean flow is considered as C = R/A, large downstream salt fluxes appear that are difficult to explain without a decreasing salt content in the upper part of the estuary. Similar results are apparent in the Mersey (Bowden & Sharaf el Din, 1966).

TABLE I. Salt flux in Southampton Water calculated using Equation (6)

Station

Al 2

3 4

BI 2 3 4

<3> <ii> (%.A (cm s-l)

32’72 4.28 32’53 -0.18

31.84 - 8.50 31.58 - 10’10

32.28 - 4’15 31.98 - 1.52 30’51 4.10 29’94 4’07

12’1 11’1

5.8 4.6

12.3 8.8 3’7 3’7

ii?”

5.46 5’44

5’32 5.28

5.39 5’34 5’10 4’99

I(EE, GGJ h 7;

(IO-~ g cm-a 9-l)

- 0.83 I ‘07 - 1.18 0.23

1’30 - 0’35 1.92 0’04

- 2.07 - 1'32 - 0.85 - 2.68

5’50 - 1.16 3.10 - 0.86

“In this column ii = R/A.

Data for the three estuaries analysed using Equation (8) are shown in Table 2. The sum of Terms I and 2, which represents the downstream flux of salt on the non-tidal drift, shows variable results, mainly because of probable errors in riA. Consequently the relative magni- tudes of the other terms should be considered rather than their absolute values. In most cases the salt balance is maintained by a combination of Terms 4 and 6, though the contribution of Term 3 appears to be relatively smaller in Southampton Water than in the Vellar. The values of Terms 4 and 5 are within the probable error range and Term 7 and a further term -- (A . (U,S,),) were both negligible. These results are in agreement with those of Hansen (1965) for the Columbia River.

On one section in Southampton Water Term 6 gave a downstream salt flux. This may be caused by observational errors, or by a transient flow situation caused by a change in the proportion of river discharge issuing from the Itchen (the proportion changed by 10% two

TABL

E 2.

Sa

lt flu

x in

Ve

llar,

Sout

ham

pton

W

ater

an

d th

e M

erse

y ca

lcula

ted

usin

g Eq

uatio

n (8

)

Stat

ion

Aii

ASA

A?,

. SA

AxA.

iiA

A.

lJA

SA

-- A

UASA

A

. (w

a)~

(1)

(2)

(3)

(4)

(5)

(6)

%O

ma

s-l

Velia

r 20

.1.6

7

5-7

8-10

27.1

.67

iZ0

9.2.

67 5-7

8-10

15.2.

67

5-7

8-10

Sout

ham

pton

W

ater

Al

-4

BI-4

M

erse

y Na

rrows

Cl

-3

351.

4fIO

2.8

-7.5

dco.

3

66.4

sc25

.3

0*4+

0~0

87.8h

245.2

74.9

fIO8.

7

179.

8kI3

4.9

- 52

'3f2.

1

- 12

.6~b

0.7

214'2

ztI59

'1 -4

6*9*

1.5

-35'3

fI.I

76.6

k245

.6

68.7

ztI33

.9

58.1

h2

.5

45'2k

I.4

-261

8.4f10

279'0

-296

I.Of9

128.

8 21

04.8f

57*3

- 84

'0&32

*3

2774

'7It1

3539

.7

- 10

045.5

h316

.7

10.4

I-t3.

8

6.3f

2.1

2.6io

.3

15.2

fo.4

2.11

7.0

4.3f

6.3

- 31

'1fO.

9

- 20

.6*0

.9

3.81

3.8

-76.

1i1.9

4'3zk

3.2

-56.

131.

5

2.9i9

.1

2.4f

7.4

- 1.8

57'1

- 1.

7f4.

8

- 32

.2k1

56.9

-42'0

zk1.

4

- 24

'3fo.

8

- 26

.315

.4

2.1-

12

'1

2188

.6155

.3 -

0’0

ko.0

1.9f

o.1

3'7Yt

O.I

1.8fo.

1

- 2'2

io.

1 -

1.2f

O.I

- 5'9

*oQ

- 6.

5 k0

.2

- 6.

3 10

.2

- 2.

5 f0

.1

136.

155.

9

- 27

0.112

2.7

57.6

zt3.7

- 21

7'51t2

5.4

-70'7

*13'9

- 16

5.1&x

5.9

- 17

5'1&1

4'0

- 66

.8j,

19'7

- 34

'3k7.

7

288.5

FI5.1

-461.4

h28.4

-1x41

.6&32

.3

Salt balance in stratified estuaries 279

days before the survey) and an increase in mean sea level of about IO cm due to barometric fluctuations on the survey day. To reproduce downstream salt tlux due to the river flow an error in I, of about I cm s-l is necessary in Term I. Using u = R/A gives a value for Term I of the order 450%~ m2 s-i.

Table 3 shows the additional terms of Equation (IO) evaluated for the same sets of data. Terms IO and I I were negligible. It is gratifying to note that in all cases the sum of Terms 6 to 9 in Equation (IO) equals the value of Term 6 in Equation (8), in spite of the different procedures involved in the analysis. In the Vellar the largest term is the mean vertical gravitational convection. The vertical oscillatory contribution is smaller but increases relative to the mean vertical term as river flow decreases. The lateral contributions are smaller. The upstream section, where depth is more constant across the section has a smaller lateral contribution than the lower section which spans a scour hole. On the last occasion at the lower section, where the characteristics of a partially mixed estuary were becoming apparent, the mean lateral circulation contained a larger proportion of the salt flux.

TABLE 3. Calculated additional terms in Equation (IO)

Station -GdtGdA &Ud”S,“) A

-- -- A(U,St)* A(

(6) (7) (8) (9)

(%, n-2 s-Y

Vellar 20.1.67 5-7 -3'7rt2.8

8-10 -0.7fo.3

27.1.67 5-7 - 6.1 f3.1 8-10 -0.3+0.6

9.2.67 s-7 - 13'9k5.4 8-10 -0’1+0.3

152.67 5-7 - 15.9f8.7 8-10 o'3*5'0

Southampton Water AI-~ 258.31kS7.1 BI-4 - 223.8*86.3 Mersey Narrows CI-3 -287.9f41.0

- 256.94~19.2 - 37'2f2.8

- 195.8f23.1 -66.ri13.4

- IO5'7i7'4 - 14I.OiI3'1

-32'0+13'2 -25.x&-8.6

102.5i-48.9 - 198.oi67.7

-344.2*47'S

0.6~ko.o - 3’5 fO.1

3'7io.2 -007*0.1 - 4'0&0'2

0.2ho.o -2.2*0.1 - 0.6&0,1

- 20.8io.7 -21.1 io.9

- 222.41t9.5

- 7'250.4 - 16.9fo.5 - 20.0&0.7

- 4.6io.3

-41'9io.9

- 33'9hO.9 - 16.9fo.7

- 9'5fO.2

- 51’9f4’1 - 12.410.6

- 282.3f9.3

In Southampton Water the transverse mean circulation is larger, being of the same order as the vertical. The oscillatory contributions are also of similar magnitudes, but are about an order smaller than the mean terms.

In the Mersey the lateral and vertical terms are again of the same magnitude. The oscilla- tory terms are, however, much larger than in Southampton Water, perhaps due to the greater tidal amplitude and higher oscillatory currents. These results are of different importance to that anticipated by Fischer (1972).

Dispersion coefficients can be calculated from these results using terms 6 to 9 of Equation

(IO)-

D= XTerms6to9

A d?/dx (11)

The results range between 20-320 ma s-1 and are shown in Table 4 together with values derived from the measured river flow using Equation (4). The high values obtained in the Vellar on the 9 February 1967 are the result of a low salinity gradient between the two cross-sections. This may have been caused by a discharge of water from an irrigation channel

280 K. R. Dyer

above the upper section at this time. The results can be compared with values of 5000 m2 s-l for the Columbia River (Hansen & Rattray, 1965)~ 24 m2 s-l for the James River (Hansen & Rattray 1965), 161-360 m2 s-l for the Mersey (Hughes, 1958), 54-535 m2 s-i for the Severn, 158 m2 s-i for Southampton Water (Dyer, 1973), 50-300 m2 s-r for the Tay (Williams & West, 1973) and 53-338 m2 s-i for the Thames (Bowden, 1963). Most were obtained with the one-dimensional diffusion equation using the observed river flow [Equation (4)]. It appears that more reasonable values of dispersion coefficients would be obtained using the river flow rather than the mean flow observed in the estuary. The causes for the negative dispersion coefficient for one section of Southampton Water cannot definitely be stated.

TABLE 4. Dispersion coefficients calculated using Equations (I I) and (4) (m” s-l)

Vellar Equation (I I) Equation (4)

ao.x .67 5-7 8-10

27.1.67 5-7 8-10

9.2.67 5-7 8-10

152.67 s-7 8-10

Southampton Water Al-4 BI-4 Mersey Narrows CI-3

61.4 19’7 62.5 32’4

231’5 263.0

37’3 20.9

- 40’0 119 63.2

315’0 247

conclusion

It appears that the proportion of the salt balance effected by the lateral circulation is greater in partially mixed than in salt wedge estuaries. This is consistent with the greater observed effect of centrifugal and Coriolis forces in producing lateral inhomogenity. With further decreasing stratification and the development of a vertical homogeneous estuary, lateral effects should predominate. This gradational sequence suggests that errors can be expected in consideration of partially mixed and vertically homogeneous estuaries if lateral homogeneity is assumed.

The width to depth ratios in the Vellar and Southampton Water are the same, so that the relative magnitudes of lateral and vertical circulations may be better considered as being governed by a flow ratio, densimetric Froude number or estuarine Richardson number. Further data is necessary to determine which parameter will provide the best indicator of lateral effects.

Acknowledgements

I would like to thank Professor K. F. Bowden for the data on the Mersey and for useful comments on the manuscript. I am grateful to Mrs L. Cummings for computer program- ming and assistance with the data analysis.

Salt balance in stratified estuaries 281

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