Effect of wind on the vertical circulation and stratification in the Volkerak Estuary

239

Netherlands Journal of Sea Research 23 (3): 239-253 (1989)

EFFECT OF WIND ON THE VERTICAL CIRCULATION AND STRATIFICATION IN T H E V O L K E R A K E S T U A R Y

J. VAN DE KREEKE 1 and K. ROBACZEWSKA 2 1 Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, Florida 33149, U.S.A. 2 Rijkswaterstaat Tidal Waters Division, P.O. Box 20907, 2500 EX The Hague, The Netherlands

ABSTRACT In the partially mixed part of the Volkerak Estuary current speed and salt concentration are measured in several stations for periods of 13 hours on August 16 and 17, 1977. The freshwater discharge on those days and during the preceding six months is kept constant at a rate of 50 m3.s-1. In spite of different wind conditions, the longitudinal distribution of the tidally- and crosssectionally averaged salt concentration is the same for both days. The vertical structure of circulation and salt concentration differ significantly. A similar set of measurements is carried out on March 18, 1980 with a constant freshwater discharge of 100 m3.s -1. The doubling of freshwater discharge leads to a relatively small increase in vertical circulation and stratification. In this study, the vertical circulation and stratification is examined in terms of the external forcings, freshwater discharge and wind, with special emphasis on wind. Tide conditions, for the three measurement days are similar. The estuary is schematized to a prismatic channel with a rectangular cross-section. In the schematized channel the tidally averaged salt concentration and velocity distribution are laterally uniform. To account for salt fluxes in the actual estuary associated with lateral and time-variations in current velocity and salt concentration, a diffusive salt flux is introduced. The effect of wind and longitudinal density gradients on the vertical circulation and stratification in the schematized channel is investigated using a simplified form of the two-dimensional conservation of momentum and salt equations. Using observed values along the axis of the estuary a scaling analysis shows that in the two-dimensional conservation of momentum equation the longitudinal pressure gradient and horizontal turbulent shear are the dominant terms, closely followed by the tidal stress. In the conservation of salt equation, the

dominant terms are those associated with advective fluxes resulting from the horizontal and vertical tidal mean velocities, a horizontal flux associated with tidal variations in the longitudinal velocity and salt concentration and the vertical turbulent flux. Retaining only first order terms, the equations are solved using the similarity solution of Hansen and Rattray. Comparing calculated and observed vertical circulation and stratification it is shown that wind mixing significantly increases the vertical exchange of momentum, thereby reducing the density-driven vertical circulation (=gravitational circulation) and stratification. At the same time, through the wind surface stress, wind generates its own vertical circulation and, therefore, stratification. An expected increase in vertical exchange of mass with increasing wind speed and accompanying reduction in stratification could not be confirmed by the analysis. It is concluded that on the days of the measurements wind is as important as the longitudinal salt concentration gradient in forcing vertical circulation and stratification. Care should be taken in generalizing this conclusion as on the days of the measurements the axial component of the wind velocity is always in the down estuary direction. 1. INTRODUCTION The Volkerak Estuary is located in the southwestern part of the Netherlands; see Fig. 1. The estuary connects to the Eastern Scheldt, a relatively large body of water with a fairly uniform salinity. At the head it is separated from a freshwater basin by a dam; a discharge sluice allows a controlled supply of freshwater. Typical dimensions of the estuary are length=35,000 m, width=1000 m and depth=10 m. For details on bathymetry see Fig. 1. For future reference, attention is drawn to the part of the estuary between Sections 4 and 6 which is characterized by a relatively uniform channel with a depth of 13 m. For

240

J. VAN DE KREEKE & K ROBACZEWSKA

VOLKERAK [~STUARY7 NORTH SEA

~ '

: ~

~ "

~

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: .....

.,,

~ _ 5 km

' _At FRESH WATER

:i!ii!ii!:!i:ii:il ii

iiii!iiitii:~:i:it:i:ii!!!it!i!iiiii!i

" '"*:s:ts!:iii::ili:ili::iiii!!iiis' "i{i

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Fig 1 Volkerak Estuary; bathymetry and location of measuring sections and stations average tide conditions, the amplitude of the damped co-oscillating tide ranges from 3.50 m at the mouth to 4 m at the head, The ratio of the spring and average tidal range is 1.15. Tidal current amplitudes range from 1.15 m.s -1 at the mouth to zero at the head. The tide is dominated by the M2 constituent. In connection with the construction of a storm surge barrier at the mouth of the Eastern Scheldt, the freshwater flow into the Volkerak Estuary was kept constant at a rate of 50 m3.s -1 during the period March 1977 through August 1977 and at a rate of 100 m3.s -1 during the period October 1979 through

March 1980. To document the resulting 'steady' velocity and salinity fields extensive velocity, conductivity and temperature measurements were carried out for periods of 13 hours on August 16 and 17, 1977 and on March 18 and 19, 1980. Inclement weather forced the measurements to be halted halfway in the tidal cycle on March 19. In this paper, using the observational data for the partially mixed part of the Volkerak Estuary, the vertical circulation and stratification for August 16 and 17 and March 18 are examined in terms of the external forcings Le. freshwater inflow, tide and wind stress. Values of

CIRCULATION AND STRATIFICATION IN AN ESTUARY

TABLE 1 External forcing * Tidal range at the head of the estuary at the day of the measurements ** Wind stress component in downestuary direction. Freshwater (m3.s - ' )

Tidal range * (m)

Wind stress * * (N.m -2)

50 50 100

4.3 4.5 4.8

0.14 0.05 0.23

Aug. 16 Aug. 17 March 18

these parameters are listed in Table 1. The wind stress is calculated from .rw= Q C D Wx/W/ with C o = 3.2.10 -6, DRONKERS (1964), ~'w is wind stress, O is density of water, W is wind speed at sealevel + 10 m, Wx is wind velocity component along the axis of the estuary. On all 3 days, the angle between the wind direction and the longitudinal axis in the study area was 30 ° . Values of wind speeds used in the calculations are daily averages, i.e., on August 16, 7 m.s -1, on August 17, 4 m.s -1 and on March 18, 9 m.s -1. The precipitation-evaporation surplus in August 1977 amounted to a withdrawal of freshwater at a rate of 1 m3.s -1 and in March 1980 amounted to an addition of 1 m3.s - 1. These fluxes are negligible compared to the freshwater discharge at the head. Acknowledgement.--The measurements referred to in this paper were carried out by the survey department Zierikzee of the Rijkswaterstaat. We would like to thank M. Bubbert, M. Geurtz and G.J. Kolld for their help in data reduction and plotting.

£

241

2. OBSERVATIONS 2.1. MEASUREMENTS Simultaneous measurements of current speed and direction, conductivity and temperature were carried out in the cross-sections numbered 2, 4, 6, 8, 13 and 15 in Fig. 1. The number of stations in the measuring sections ranged from 5 to 6, depending on the width of the section. Each station was occupied by a survey vessel. Depending on the water depth, measurements were carried out at 4 to 7 different depths every 15 to 30 minutes. Current speed and direction were measured by an EImar current meter. Beckman salinometers were used to measure conductivity and temperature. The meters were lowered to the desired depth on a cable. Readings were averages over a one-minute interval. For a number of observations the chlorosity calculated from the conductivity and temperature were compared to the result of titrations. If neccessary a correction was made to the chlorosity. Here chlorosity rather than salinity is used as a measure for the salt concentration to facilitate comparison with the study on salt transport mechanisms in the Volkerak Estuary presented in DRONKERS & VAN DE KREEKE (1986). To convert chlorosity expressed in g.dm -3 CI- to salinity in ppt, multiply chlorosity by 1.80. 2.2. THE LONGITUDINAL SALT CONCENTRATION DISTRIBUTION The longitudinal distributions of the tidally and crosssectionally averaged salt concentrations on the days of the measurements are presented in Fig. 2. Salt concentrations are normalized using the concentration at the mouth of the estuary as the norm. As expected, salt concentrations are lowest and the longitudinal salt concentration gradients are largest

O

[]

O

O

A A

A [3 0

A

w

~

A 35

2'8

21 14 DISTANCE TO DAM [ km]

7

0

A

~

9 A

PARTIALLY

mrxED

2'a

Fig. 2. Longitudinal salt concentration distribution for different freshwater discharges Qr Salt concentrations are tidally and cross-sectionally averaged values and normalized with respect to salt concentration at the mouth <~>m" [] 16-08-77, Qf=50 m3.s-1, <~>m = 16.7 g.dm -3 CI-; O 17-08-77, Qf=50 m3.s-l; <~_>rn--16.7 g.dm -3 CI-; & 18-03-80, Qf=100 m3.s-1, <~>m=15.1 g.dm -3 CI-.

A 0

;1 ~aI DISTANCE TO DAM [km] I SECTION 6

l~ SECTION4

Fig. 3. Differences in salt concentration between bottom and surface along the axis of the estuary for different freshwater discharges Qf. Salt concentrations are tidally averaged values. [] 16-08-77, Qr= 50 m3.s- 1; O 17-08-77, Qf=50 m3.s-1; A 18-03-80, Qf= 100 m3.s-1.

242

J. VAN DE KREEKE & K. ROBACZEWSKA

TABLE 2 Circulation and stratification characteristics for the partially mixed part of the estuary * See VAN DE KREEKE& WANG (1984).

Aug. 16 Aug. 17 March 18 Standard error*

<~> o

d < ~ >/dx

(g.dm -3 CI-)

(g.dm -3 CI-)

12.4 12.5 8.9 0.02

2.1.10 -4 2.1-10 -4 2.7.10 -4 0.04.10 -4

for the freshwater inflow of 100 m3.s -1. Differences in tidally averaged salt concentrations between bottom and surface along the axis of the estuary are plotted in Fig. 3. A distinct stratification exists only between km 0 and km 15. Therefore, for purposes of this study, the region between the measuring sections 4 and 6 is defined as the partially mixed part of the estuary. The region between the dam at the head of the estuary and measuring section 4 is discarded because here the circulation and stratification are affected by the localized discharge of the freshwater. Seaward of section 6 the stratification is too small to be of interest. In the partially mixed part, the longitudinal gradient of the tidally- and crosssectionally averaged salt concentrations, d < ~ >/dx, is practically independent of x; values of d < ~ > / d x together with the values of < ~ > o for the different measurement campaigns are listed in Table 2. Here c is the salt concentration expressed in terms of chlorosity in g.dm -3 C l - , x=longitudinal coordinate, positive seaward, x = 0 is taken in the centre of the partially mixed part of the estuary, double overbars denote averaging over the cross-section, the angle brackets denote averaging over the tidal period and the subscript refers to values at x = 0 . Also, along the centreline to a good approximation d < c > / d x is independent of the vertical coordinate z

Stratification I(- )l (g.dm -3 CI-)

Circulation I(-)l (cm.s- ~)

5.5 5 6.5 0.5

0.35 0,3 0.4 0.02

(measured from the surface downward). 2.3. STRATIFICATION AND VERTICAL CIRCULATION Salt concentrations and velocities measured in stations 17, 18 and 19 of section 4 and in stations 27, 28 and 29 of section 6 are averaged over the M 2 period. Time averaging is carried out for constant values of z/h where h is the mean depth at the station. For each station the salt concentration deficit -<~>, (=stratification) and the velocity deficit < u > - < ~ > (=vertical circulation) are calculated. The overbar denotes averaging over the vertical. Characteristic profiles of stratification and vertical circulation for the study area are obtained by averaging the deficit values of the six stations. The results for each of the experiments are presented in Fig. 4 for the salt concentration and in Fig. 5 for the vertical circulation. Variations in the circulation and stratification in the study area are small as witnessed by fig. 18 in DRONKERS & VAN DE KREEKE (1986). To facilitate comparison, the parameters I < c > - < ~ > I and I < u > - < 5 > I are taken as measures of respectively the stratification and vertical circulation. The values of these parameters are listed in Table 2. For future reference, the freshwater velocities on

- 0

.5 0

o .11

z

.5

I

~

! Am

0 O

AgO Am I

i

.5-

O 0

• O i o

o

:nA n •

•

•

Fig. 4. Characteristic salt concentration deficit, - , for each of the experiments. • 16-08-77, Qf= 50 m&s-1; G 17-08-77, Qf=50 m3.s-1; • 18-03-80, Qf= 100 m3.s-1.

CIRCULATION AND STRATIFICATION IN AN ESTUARY

243

:~ -<0> -2 0

-.1 ~

.1

0 t

I

.-

Z H

.2 ,,.

,,,.

•

no• la =& me• °

.5 AJ •

•

•

•

•

;o om

•go

Fig. 5. Characteristic velocity deficit, < u > - < ~ > , for each of the experiments. • 16-08-77, Qf= 50 m3.s-1; G 17-08-77, Qf=50 m3.s-1; • 18-03-80, Qf=100 m3.s-1. August 16, August 17 and March 18 are respectively 0.004 m.s -1, 0.004 m.s -1 and 0.008 m.s -1. The tidal current amplitude in the partially mixed part of the estuary is approximately 0.25 m.s-1. 3. OBJECTIVES AND APPROACH As shown in Table 2, stratification and vertical circulation increase with increasing wind stress and freshwater inflow. In particular, the results for August 16 and 17 pertaining to the same freshwater inflow suggest that the response of the partially mixed part of the estuary to short-term wind events manifests itself primarily in a change in vertical circulation and stratification and hardly affects the longitudinal salt concentration distribution. This is not surprising as the response time of the longitudinal salinity distribution to changes in the forcing is in the order of the residence time of freshwater and is measured in weeks. The appropriate time scale for the adjustment of the vertical circulation and stratification to changes in wind forcing is the time to transport the momentum imparted by the wind to the bottom layers (i.e., h2/Nz , see section 4 for definitions) and is in the order of hours. On March 18, the doubling of the freshwater inflow leads to a relatively small increase in stratification and circulation. To explain the observed variations in vertical cirTABLE 3 Salt fluxes at x=0. Salt fluxes in kg CI- .m-1.s-1. i., is ratio of diffusive and advective salt flux. The standard error in the fluxes is estimated at 10% Advective Flux due to Diffusive Flux verticalcirculation flux

Aug. 16 Aug. 17 March 18

0.65 0.65 0.93

0.33 0.23 0.44

0.32 0.42 0.49

0.49 0.65 0.52

culation and stratification, the partially mixed part of the estuary is schematized to a prismatic channel with a rectangular cross-section having a width b and a depth h. In the schematized channel/estuary salt concentrations and velocities are laterally homogeneous. The stratification, < c > - < c > , and the vertical circulation, < u > - < ~ > , are assumed to be independent of longitudinal position and equal to the values presented in respectively Figs 4 and 5. The tidally- and depth-averaged salt concentration, < ~ > , varies linearly with x. The values of < 5 > 0 and d < E > / d x are taken equal to the corresponding values of respectively < ~ > o and d < ~ > / d x in Table 2. The tidally- and depth-averaged velocity < ~ > is set equal to the freshwater velocity uf. Assuming the tidal variations in water level to be small compared to the mean depth, the tidally averaged salt flux in the schematized channel is h bh uf<~'c> + b f ( < u > -uf) ( < c > - < ' c > ) d z + o

d + bhKx.-dx

=0

(1)

The last term is a diffusive flux representing the salt flux in the actual channel associated with horizontal circulation and horizontal- and vertical oscillatory shear. For each of the experiments, the values of the different fluxes in Eq. (1) are presented in Table 3. The advective flux is calculated from the known values of h, uf and < ~ > . The flux associated with the vertical circulation is calculated using the stratification and circulation profiles in respectively Figs 4 and 5. The diffuse flux is calculated from Eq. (1).

244

J. VAN DE KREEKE & K. ROBACZEWSKA

To analyse the effect of wind, tide and longitudinal density gradients on the vertical circulation and stratification, use is made of a simplified form of the two-dimensional equations for conservation of momentum and salt and the similarity solution proposed by HANSEN & RATTRAY (1965). Using observations in the centre of the actual channel, a scaling analysis is carried out to determine whether the use of the reduced equations is appropriate. 4. CONSERVATION OF M O M E N T U M AND SALT EQUATIONS The tidally-averaged two-dimensional equations for the conservation of x-momentum is, according to PRITCHARD (1956) and RIGTER (1971):

+

--

u = ef (1 + k c)

~

ax

5z

10 -6

10 -6

1 c~U2

a2 + :~x

ax

where e is density and ef is density of freshwater. D. v~ and (~ are amplitudes and c~, f3 and -~ are phase angles. The amplitudes and phase angles are associated with the following decomposition of velocities and salt concentration.

5. 10 -6

cos(~-b)

1 ~#~

2

c3X2

(4)

+

,,d d<~-> + g ( z + < C>)K --+gdx dx 10-5 10-5

2

The number below each term represents the magnitude of the term. In Eqs (2) and (3), u and w are the current velocities in respectively the x and z direction averaged over the turbulence time scale, c is the salt concentration averaged over the turbulence time scale, ~- is the water level measured with respect to z=0, g is gravity acceleration. Here the turbulence time scale is 15 min., the sampling interval of the velocity measurements; see section 2. and ~ are effective turbulent exchange coefficients for respectively momentum and salt representing mean mixing conditions during a tidal period; they are assumed to be independent of z. The coefficient k=1.4.10 -3 relates density and chlorosity through the eqation of state.

5.10 -8

u(x,z,t)= < u > (x,z)+ u'(x,z,t)

(5)

w(x,z,t)= < w > (x,z)+ w'(x,z,t)

(6)

c(x,z,t)= < c > (x,z)+ c'(x,z,t)

(7)

az <2.5.10 -6

~2

+ '~z-

(2)

az2 10--5

Ires-2]

The tidally-averaged equation for the conservation of salt is, according to PRITCHARD (1954) and RIGTER 1971: < c>

-----

+
~t< c >

ax

~z

10-5

5.10-6

~2 + Ex clX2

5.10-8

1 ~D~ c o s ( ~ - -~) 2

1 01,~/Cc o s ( ~ - " f )

2

az <5.10 -4

+

< u > and < w > are tidally averaged velocities and < c > is the tidally averaged salt concentration. In particular, < u > contains the freshwater velocity ut= < ~ > and the vertical circulation < u > - u f . The time-dependent components u', w' and c' are approximated by a single harmonic component

u'(x,z,t)= O(x,z) sin (~;t + ,,, (x,z))

(8)

w'(x,z,t) = d/(x,z) sin (~t + i~ (x,z))

(9)

ax c'(x,z,t)= #(x,z) sin (~,t + "r (x,z))

<5.10-6 a2
~"~z

az 2

(3)

10-5 [kg.m-3.s -1]

(10)

in which ~., is the tidal frequency. In deriving Eq. (2) use is made of the earlier observation that d < c > / d x is independent of z. Here, time averaging is carried out for constant z. This requires that for a given z, values of u, w and c are defined at all times. Consequently, the domain of the variable z must be time invariant. For this reason, in this analysis it will be assumed that .~-< < h, where h is the depth.

CIRCULATION AND STRATIFICATION IN AN ESTUARY

The order of magnitude values for the terms on the left-hand side of Eqs (2) and (3) directly follow from the values of the velocities, salt concentrations and their gradients, presented in Appendix I. The values in Appendix I are derived from the measurements in the longitudinal axis of the estuary. Estimating the magnitude of the terms on the right-hand side of the equations requires additional input. This is discussed in more detail below. The first and third term on the right-hand side of Eq. (2)--These terms represent momentum fluxes associated with tidal fluctuations in the velocity. The magnitude of the first term follows from the values of O and a~lax in Appendix I. The magnitude of the third term in addition to the amplitudes ~ and ~, depends on the phase angles c~and ft. Insufficient information is available on these phase angles. In view of this, only an upperbound for the third term can be estimated. Taking cos ( e - / ~ ) = l and making use of the values for ~ and ~, in Appendix I, it follows 9(1/2 LI I~ cos(c~-~))laz<2.5 10 -6 m-s -2. The second and fourth term on the right-hand side of Eq. (2)--Both terms are associated with turbulent fluctuations in the velocity. Observations in the Volkerak Estuary on turbulent fluctuations in the horizontal velocity are presented in DRONKERS & VAN DE KREEKE (1986). It follows from their fig. 10 that the magnitude of the turbulent velocities is 3 cm.s-1 and the dominant period is approximately 10-15 min. Most likely these variations in current speed can be attributed to internal waves. Taking the distance between the measurement stations and the dam at the head of the estuary (=15 km) as the appropriate length scale it follows that 0 (~xO2/Ox2)=5.10 -8 •m.s-2. To estimate the magnitude of the fourth term use is made of values of ~z reported in the literature. For the Mersey Estuary, BOWDEN & GILLIGAN (1971) estimated Xz= 100 cm2.s -1. The stratification for the Volkerak Estuary and the Mersey Estuary is approximately the same. However, tidal currents in the Volkerak Estuary are considerably lower. In view of this, for the Volkerak Estuary the values of the vertical turbulent exchange coefficient is estimated as ~z=30 cm2.s -1. With the order of magnitude value for o32/Oz 2 listed in Appendix I, it follows 0 (~zO2laz2)=lO -5 m.s -2. The first and third term on the right-hand side of Eq. (3)--These terms represent salt fluxes associated with tidal fluctuations in the velocity and the salt concentration. Because insufficient information is available on the phase angles ~, fl and % only an upperbound for the values of these terms can be estimated. As before, taking cos (o~-.y)=cos (fl-..f)= 1 and making use of the values of 0, ~, and # in Appendix I, it follows that a(1/2D # cos (o~-.y))lOx <5.10 -6 g.dm -3 C l - . s -1, 0(1/2WCCOS (~--'y))/OZ <5.10 -4 g.dm -3 C l - . s -~. In estimating the upperbound for

245

the first term the appropriate horizontal length scale is taken as the distance between the centre of the partially mixed part of the estuary (x =0) and the dam at the head of the estuary. This distance is approximately 10 km. The second and fourth term on the right-hand side of Eq. (3)--Both terms are associated with turbulent fluctuations in the velocity and the salt concentration. From observations in the Volkerak Estuary (DRONKERS & VAN DE KREEKE, 1986) referred to earlier it follows that the order of magnitude of the horizontal turbulent salt flux is 5.10 -4 g-din -3 Cl•m.s -1. Taking the same horizontal length scale used to evaluate the second term in Eq. (2), i.e. 15 km, it follows 0 (~xO2/Ox2)=5.10 -8 g.dm -3 CI-"S -1. To estimate the magnitude of the fourth term use is made of values of ez reported for the Mersey Estuary. Taking into account the differences in tidal currents between the Mersey Estuary and the Volkerak Estuary it is estimated that for the Volkerak Estuary cz =10 cm2.s -1. With the order of magnitude values for o32/~3z2 listed in Appendix I, it follows 0 (Cz~2/Oz2)--lO -5 g.dm -3 CI-.s -1. In the final equations the fluxes associated with the tidal and turbulent fluctuations in velocity and salt concentration are combined and new exchange coefficients N x, N z, K x, and K z are defined by the following equations

_ ZI~I 2 + Xx ~q< u >

2 10 -2

_ N x O_< u >

0x

(11)

ax

10 -4

[m2.s -2]

1^ . oq - - - u w cos (o~-/~+ x z - - = N 2 #z < 2-10 -5

1 ^ .

10-4

0

- - - u c cos (o~-',/)+ ~x 2 o~x 1.5.10 -2

1

0

10-4

(12) [m2.s -2]

=k,. o ~ < C >

"'x

~x

10 -4

- - ~ # cos (/3- "y)+% - - 2 az <10-4

o~ z #z

(13)

[kg.m -2.s-1]

0

- Kz

(14)

~z

[kg.m-2.s-1]

246

J. VAN DE KREEKE & K. ROBACZEWSKA

In view of the magnitude of the terms in Eqs (11)-(14) it follows that in the two-dimensional equations the horizontal exchange coefficients N x and K x are largely determined by tidal exchanges and the vertical exchange coefficients N z and K z are largely determined by turbulent exchanges. Therefore, values of N z and K z are reasonable measures for the values of the coefficients for vertical turbulent exchange, respectively "z and {z, whereas the values of N x and K x are orders of magnitude larger than the values of the coefficients for horizontal turbulent exchange respectively, ~x and ~x. Retaining only terms of 0 (10-5) and making use of Eqs (11) and (12) it follows from Eq. (2) el gzk

d

+gdO N x

dx

dx

O~X-+NzO2 Ox

c)Z2

(15) Here it is assumed that N z is independent of z. It follows from Eq. (15) that to a first approximation the momentum balance is formed by the gradients of the horizontal pressure and the tidal stress ( = 1 (2~2 ~ e N x O < U > ) 2 #x

and the horizontal turbulent shear. Retaining all terms in Eq. (3) and making use of Eqs (13) and (14) it follows,

< u > - ~ < c >- - + ~x



cl - ~ K x c)z

clx Ox

4- K ~ - 2 < c > z •z 2

(16) Here it is assumed that K z is independent of z. The horizontal and vertical advective fluxes of salt are balanced by a vertical diffusive flux that is a result of turbulent fluctuations of the vertical velocity and salt concentrations and a horizontal diffusive flux. Integrating Eq. (16) over the vertical and making use of the zero flux conditions at surface and bottom and the condition that the integral salt flux is zero, yields Eq. (1). Finally, it is pointed out that the preceding scaling analysis is carried out based on the assumption of steady state. Although this was not strictly the case, it follows from the values of the circulation for August 16 and 17 in Table 2 that 0 ( c 3 < u > l a t ) = l O - 7 m.s -2, which is negligible compared to the order of magnitude values of most other terms in the conser-

vation of momentum equation. Similarly, comparing the salt concentrations for August 16 and 17, Table 2, it follows that the difference in salt concentration from one day to the other is at best equal to the accuracy of the salt concentration measurements, i.e., 0.1 g.dm -3 CI-, and therefore at best, 0 (a/et)=10 -6 g.dm -3 C l - ' s -1. This again is considerably smaller than the order of magnitude values of most other terms in the conservation of salt equation. In many studies dealing with estuarine circulation and stratification, the two-dimensional equations of conservation of momentum and salt as derived by PRITCHARD (1954, 1956) for the James River Estuary are used. Based on an extensive set of velocity and salinity measurements, PRITCHARD (1956) determined that in the conservation of momentum equation the longitudinal pressure gradient and the vertical exchange of momentum were the dominant terms, closely followed by the gradient of the tidal stress. The magnitude of the tidal stress term was about half the magnitude of the pressure gradient and momentum exchange terms. The foregoing analysis shows that the dynamics of the partially mixed part of the schematized Volkerak Estuary is governed by the same balance of forces. Among other things this implies that, when studying first order processes, neglecting the tidal stress term is only marginally justified. With regard to the two-dimensional salt balance of the James River Estuary, PRITCHARD (1954) concluded that the horizontal advection and the diffusive vertical flux are the dominant transport modes closely followed by vertical advection. These processes are also important in the Volkerak Estuary. However, in addition the horizontal diffusive flux plays an important role in the salt balance. The horizontal diffusive flux is associated with tidal rather than turbulent fluctuations in the horizontal velocity and the salt concentration. The Eqs (15) and (16) are derived assuming a twodimensional field of velocities and concentrations. When defining < u > . ~.~w>, < c > and <~'> as laterally averaged values the equations are also valid for channels with lateral variations in these variables. In that case, the coefficients K x and N x include exchange of respectively mass and momentum by horizontal circulation and horizontal oscillatory shear and the coefficients K z and N z include exchange of respectively mass and momentum by transverse (in the cross-section) circulation and vertical shear. 5. HANSEN-RATTRAY SOLUTION An approximate analytical solution for < u > and < c > in Eqs (15) and (t6) is presented by HANSEN & RATTRAY (1965). The solution can be formally written as

CIRCULATION AND STRATIFICATION IN AN ESTUARY

< u > _f [7; ~,Ra, T]

(17)

uf

-

- M f[ff; ~,Ra, T]

<6>

o

(18)

P

with

.v=f[v Ra, M, T]

(19)

P

In Eqs (17)-(19), r/=z/h, Ra and M are dimensionless mixing numbers and T is a dimensionless wind stress number. The functions f are different for each of the equations, and together with the expressions for the parameters Ra, M and T are presented in Appendix II. In deriving the analytical solution Hansen & Rattray assumed that --the longitudinal gradient of the salt concentration is independent of x and z; this condition is reasonably satisfied for the Volkerak Estuary. --velocity profiles and salt concentration deficit, ( < c > - < ~ > ) , profiles are similar. - - N z and K z are constant The similarity assumption for the velocity profiles implies that 0 < u > l O x = 0 meaning that the tidal stress term in Eq. (15) is neglected. It follows from the magnitude of the tidal stress terms, see Eq. (2), that this is only partly justified. Furthermore, the assumptions introduced by HANSEN & RATTRAY (1965), lead to the condition that the horizontal diffusion (dispersion) coefficient, K x, must increase seaward at a rate equal to the freshwater velocity (20)

dKx - uf dx

This condition has often been criticized as being physically unrealistic. Among other things, Eq. (20) implies that the upestuary salt flux by vertical circulation is independent of x. For this multiply both sides of Eq. (20) by d <-c > / d x and integrate with respect to x. Realizing that d < E > / d x is assumed to be independent of x, this yields TABLE 4 Values of parameters in Hansen-Rattraysolution. The standard error in the parameters is estimated at 20%.

Aug. 16 Aug. 17 March 18

vRa

M/u

T

u

430 640 200

40 45 15

120 65 75

0.47 0.63 0.54

Kx d < - ~ > - - U f < ' C > dx

247

+ constant

(21)

Comparison with Eq. (1) shows that the constant in Eq. (21) represents the salt flux by vertical circulation. The uniformity of this salt flux is consistent with the assumption of similar velocity and salt concentration profiles and is in agreement with observations by DRONKERS & VAN DE KREEKE (1986); see fig. 20 in their paper. The exact form of the expression of the circulation, Eq. (17), is presented as Eq. (1) in Appendix I1. With the definitions of ~,, Ra and M, this equation can be rewritten as

<'u.~>_>_3 (1 -~72).+ Uf

h "rw

2

4Uf#

-+ gkh 3 d < c > / d x 48 uf

1.( 1 _4~7+3,r/2)+ Nz

1 (1 -9r/2+8r/3)

(22)

Nz

In Eq. (22) the coefficients h~-wl4UfQ and gkh 3 d < c > /dx

48 uf are known from the measurements, see Tables 1 and 2. Making use of the observed velocity profiles, Fig. 5, and applying a least square technique, values of N z are determined from Eq. (22). With the known values of N z, values of T and ~,Ra are calculated using Eqs (4), (6) and (7) in Appendix I1. Similary using a least square technique, values of Mh, are determined from the observed salt concentration profiles in Fig. 5 and Eq. (2) in Appendix I1. With the known values of ~,Ra, T and M/v values of v, Kxo and Kz are then calculated from respectively Eqs (3), (7) and (5) in Appendix I1. The results are presented in Table 4 and 5. Note that the values of K z and N z in Table 5 are in agreement with the values of these parameters used in the scaling analysis. For comparison the theoretical velocity profiles corresponding to the values of ~,Ra and T in Table 4 TABLE 5 Values of exchange coefficients Kxo in m2.s-1; Nz and Kz in cm2.s -1. ( )* calculated from diffusive salt fluxes in Table 3 and longitudinal salt concentration gradients in Table 2. The standard error in the parameters is estimated at 20%.

Aug. 16 Aug. 17 March 18

K~o

N~

K~

110 (123)* 150 (161) 145 (138)

36 25 100

4.5 5 6.5

248

J. VAN DE KREEKE & K. ROBACZEWSKA

g[

10

-20 o

0

10

20

30

50

4(;

Z

h -'<~'-

J

T : !)(]

5

0

.....

/

..'b.o : 21

i/

'\ ,, \

J!{<~:430

~ k

-,..~_ xX.x

~R(

bL.q

If -

-10

2O

ot

I

.

.

.

.

0

10

--L ........

~

20 ........

J.

.

.

.

.

30

40

J_

L

50 ] ........

Z

t

T

// \~

/

"/

: 6.~)

.......

vRq=O

.....

,'Ra : 3 2 0 -

•

\

./Ro : 6 4 0

\

.....

/Ro : 960

__->

' -U ) ]J{

-20 0 _

-10 _ _ _ - .

o

I

10

__$- .........

i

20 .......~ . . . . .

30 J~

....

4(] i. . . . .

50 J_ . . . . .

h

Z

.......
h

::-< .....

i i

j;<~, /

/"-?/"'

~'

'JRa

,RG

I

'. "~ \

k~

"-". ". "~-eL. \"

---~

....... .....

= () :

200

.,Ra = z,00 JRa

: 600

Fig. 6. Comparison of observed and theoretical velocity profiles. From top to bottom, 16-08-77, 17-08-77 and 18-03-80. are presented in Fig. 6. The corresponding standard deviation for the values of /uf on Aug. 16, Aug. 17 and March 18 are respectively, 1.97, 2.74, and 1.55. The theoretical salt concentration profiles corresponding to the values of M/~, in Table 4 are presented in Fig. 7. The standard deviations for the values of

(- )o

on Aug. 16, Aug. 17 and March 18 are respectively, 4.5.10 - 3 , 5.1.10 - 3 and 7.3.10 - 3 . The agreement between the observed and theoretical velocity and saltconcentration profiles is acceptable. Only for the velocity profiles is there a systematic differerence near the bottom. This most likely is the result of the no-slip bottom boundary condition used by HANSEN & RATTRAY (1965). In spite of the many assumptions, the Hansen-Rattray solution appears to reasonably account for the important physical processes. This is

CIRCULATION AND STRATIFICATION IN AN ESTUARY

M -3

-2

I

I

-1 I

249

- <~.>

0

1

I

I

o

2 I

Z

h T = 120

.5-

-

-

"~"%~Lt'~ ~/Ra

:

-

"qRa= 215

.....

"qRa= 430

-

.

-

.

.

.

"~.,~.~.

0

"~-~:-. X

/ ',,

\'','\,

.

)D

-3

-2

l

-1 I

I

M

--

<(;> -

• -

0

1

I

I

o

2 I

Z

h

M/9 = 45

"'-C2Z..

- - - - qRa = 0 .... "qRa: 320

"'~.~.~-~*"C.."--.... "

.....

"VRa= 640

~

X/Ra= 960

~

\ .'.

'

~)

0

-3

-2

-1

M

- -

.

')

0

"\.

~", " "!

- _

1

_

<~>o

2

K

'\~'.'k,t k

Z

h

T

.5-

T ----'qRa

"c'c<" "" =0

~..>.

....

~R~ = 20o

.....

qRa = 400

~

~\'---:...

.....

VRa:600

~

.

\ . ' , ,"-,,

!.i

\'

Fig. 7. Comparison of observed and theoretical stratification. From top to bottom, 16-08-77, 17-08-77 and 18-03-80. further confirmed by the close agreement between the values of v and Kxo determined from the Hansen-Rattray solution and the values of these parameters calculated directly from the measurements. See Tables 3 and 4 for ~ and Table 5 for Kxo. 6. DISCUSSION Because tide conditions and freshwater discharge

are approximately the same for August 16 and 17, the variability in vertical circulation and stratification can be attributed to local wind. The wind contribution to the circulation /ufwould be contained entirely in the second term on the right-hand side of Eq. (1) in Appendix II were it not that the coefficient for the vertical exchange of momentum N z, present in both T and vRa, is affected by wind and stratification. In general, the value of Nz increases with increasing wind stress and decreases with increasing stratifica-

250

J. VAN DE KREEKE & K. ROBACZEWSKA

-20 L

0

0

-10 .......

~

......

L

10 . . . . . . . . .

~

30

20

. . . . . . . . .

uf

L.

~ ..................

.

40 .

.

.

.

.

.

Z

h

1' t

I I

I] Fig. 8. Wind-induced circulation on 16-08-77 ( tion. With regard to the wind stress this is clearly demonstrated by the values of N z in Table 5. Therefore, wind affects the circulation in two ways. The wind surface stress generates a wind-driven vertical circulation and wind-generated waves increase the value of N z thereby decreasing the value of ~,Ra and the density-induced circulation (= gravitational circulation). This explains the relatively small increase in vertical circulation on March 18 in spite of a doubling of the freshwater discharge. For August 16 and 17 the contribution of the wind-induced circulation (second term on the right-hand side of Eq. (1) in Appendix I) to the total circulation is shown in Fig. 8 and similarly for the gravitational circulation (third term on the right-hand side of Eq. (1) in Appendix II) in Fig. 9. Even though the longitudinal salt concentration gradient is the same for both days, the gravitational circulation is considerably smaller on August 16, the day with the largest windstress. Note that even for the smaller windstress values, the winddriven circulation is of the same order of magnitude as the gravitational circulation. This agrees with

-20

-) and 17-08-77 ( . . . . ).

observations in the Providence River by WEINBERG & STURGES (1976) and for the Potomac Estuary by ELLIOT (1978). Using the result of long-term current observations they showed that local meteorological forcing could lead to variations in the vertical circulation that are much larger than the mean value. The contribution of the wind-driven circulation and gravitational circulation to the stratification is represented by the second and third term in square brackets of Eq. (2) in Appendix II, multiplied by the factor viM. For August 16 and 17, the contribution of each of these terms is depicted in Figs 10 and 11. The larger windstress on August 16 causes an increase in wind-driven circulation (Fig. 8) and a reduction in gravitational circulation (Fig. 9). As expected, the increase in wind-driven circulation and decrease in gravitational circulation correspond to respectively an increase and decrease in stratification; see Figs 10 and 11. Furthermore, the larger windstress on August 16 should be accompanied by a larger value of the vertical mixing coefficient, Kz, everything else being equal. This in turn would lead to a decrease in

-10

0

uf 20

10 f

Z

h

.5-

/ j'1

~

( Fig. 9. Gravitational circulation on 16-08-77 (

- ) and 17-08-77 ( . . . . . . )

CIRCULATION AND STRATIFICATION IN AN ESTUARY

M -3

-2

0

251

-

-1

0

1

2

I

I

I

I

I

I



o

%%% %'%~"

%

.5-

Fig. 10. Wind-induced stratification on 16-08-77 ( the value of ~,IM and therefore a reduction in stratification. However, contrary to N z, an increase in K z with increasing windspeed could not be detected from the measurements. In view of the limited accuracy of the analysis, the small variations in the values of K z (see Table 5) are deemed insignificant. The increase in vertical circulation and stratification on August 16 relative to the corresponding values on August 17 is accompanied by an increase in the upestuary salt flux. Because of the steady state conditions this increase is compensated for by a decrease in the diffusive flux, see Table 3, and therefore a decrease in the value of Kxo, see Table 4. The observations do not provide sufficient information on the physical processes responsible for the decrease in the value of Kxo. Finally, it is pointed out that the sequence of two 13-hour measurements on August 16 and 17 is too short to include possible effects on the circulation of the fortnightly and monthly components of the tide.

) and 17-08-77 ( - - -). 7. CONCLUSIONS

From observations it follows that during periods of constant freshwater discharge, but different wind conditions, the longitudinal salt concentration distribution shows little variation whereas variations in vertical circulation and stratification are considerable. Furthermore, doubling the freshwater discharge leads to a small increase in the vertical circulation and stratification. To explain the foregoing observations, the partially mixed part of the estuary is schematized to a prismatic channel with a rectangular cross-section. Applying the similarity solution of HANSEN & RATTRAY (1965), it is shown that the observed variations in vertical circulation and stratification can be explained by variations in the local wind conditions. On the days of the measurements the axial component of the wind speed is always in the downestuary direction. As a result, the wind surface stress forces a vertical circulation similar to the gravitational

M -<~> -2 0

i

-1

0

i

I

1

2

I

I

!

i 'k \

.5-

%% |

Fig. 11. Density-induced stratification on 16-08-77 (

) and 17-08-77 ( - - -).

3

252

J. VAN DE KREEKE & K. ROBACZEWSKA

circulation, Also, through wave action, wind intensifies the vertical exchange of turbulent m o m e n t u m which lessens the gravitational circulation. For the partially mixed part of the Volkerak Estuary windand gravitational circulation are of equal magnitude. An increase in vertical circulation due to wind is accompanied by an increase in stratification that is partly offset by a decrease in gravitational circulation and a postulated intensification of vertical mixing. The latter could not be confirmed by the field measurements and the subsequent analysis using the similarity solution. Whereas values of the coefficient for vertical m o m e n t u m exchange N z clearly increased with increasing values of the windspeed, variations in K z were too small to be d e e m e d significant. 8. REFERENCES BOWDEN, K.F. & R.M. GILLIGAN, 1971. Characteristic

features of estuarine circulation as represented in the Mersey Estuary.--Limnol. Oceanogr. 16: 490-502. DRONKERS, J.J., 1964. Tidal Computations in Rivers and Coastal Waters. North Holland, Amsterdam: pp. 518.

DRONKERS, J. & J. VAN DE KREEKE. 1986. Experimental

determination of salt intrusion mechanisms in the Volkerak Estuary.--Neth. J. Sea Res. 20: 1-19. ELLIOTT, A., 1978. Observations of the meterologically induced circulation in the Potomac Estuary.--Estuar coast, mar. Sci. 6: 285-299. HANSEN, D.V. & M. RATTRAY, 1965. Gravitational circulation in straits and estuaries.--J, mar. Res. 23: 104-122. KREEKE, J. VAN DE & J.D WANG, 1984. An analysis of the salt transport mechanisms in the Volkerak Estuary, The Netherlands. RSMAS-Univ. Miami Tech Rep TR84-3:48 pp. PRITCHARD, g.w., 1954. A study el the salt balance in a coastal plain estuary --J. mar. Res. 13: 133-144. --, 1956. The dynamic structure of a coastal plain estuary.--J, mar. Res 15:33-42 RIGTER, B.P., 1971. Reproduction of salinity distribution m tidal rivers.--Delft Hydraulics Lab. Rep M896-111 (Dutch Text). WEINBERG, R.H & W. STURGES, 1976. Velocity observations in the west passage of Narragansett Bay: A partially mixed estuary.--J. Phys. Oceanogr. 6: 345-354. (received 21 June 1988; revised 28 February 1989)

APPENDIX I Order of magnitude of velocities, salt concentrations, and their gradients for the partially mixed part of the Volkerak Estuary. =0.1

m.s -~

L~ = 0 . 2 5

~/ax=

10-5s -I

~lbz=2.10-2

m.s-!

~Olc~z= lO- 2 s

at~/~x = 2.10 - 5 s - 1

s -1

1

, ? , 2 < - u > / a z 2 = 5 . 1 0 - 3 m - 1 s-1



= 1 0 -4 m.s -1

d<~l>/dx=2.10

=10 d/dx

w = 2 . 1 0 - 4 m.s -1

aTvlaz=2.10- S s

~,

-4

g.dm - 3 C I = 2-10 -4. g.dm - 3 CI

# = 0 . 5 g.dm - 3 CI.m - 1

~#/~,X=3.10-5 g.dm - 3 C l - . m - 1 ( - ) 2 < c > / a z 2 = 1.5.10 - 2 g.dm - 3 C I - . m - 2

~laz=5.10

-2 g.dm - 3 C l - - m

8#/~z= 5 . t 0 - 2 g.dm - 3 C l - . m - ~

CIRCULATION AND STRATIFICATION IN AN ESTUARY

253

APPENDIX II Hansen-Rattray solution < u > - 23 (1 - 7"/2)+ 74 (1 - 4r/+ 3~/2)+ ~ 8 a (1 - 9r/2 + 87"/3)

(1)

uf < C > - - <'C>

:(M)_ 1 [(_ 7 + 1 / 2 _ l q 4 ) + _T (___1 + 1 / 2 _ 2,r/3 + 1 / 4 ) + z,Ra

<-C > o

120

v

4

8

4

20

2

3

4

48

(- ~ +±,72- 3,7.+2,7s)] 12

2

4

5 (2)

vRa 152 vRa 2 32+107+ T2 + (76 + 14T)~-~- + ~(-~--~-)

v=l

(3) 1680

Ra - g k < ~ > o h3

Nz K~o M=

KxoKz u2f h 2

T= h r ~ NzUfe d<-~ > Kxodx Uf<'C > o

with ~=z/h

-M-

p (4)

(5) (6)

(7)