Difference between revisions of "Estuarine circulation"
Dronkers J (talk  contribs) 
Dronkers J (talk  contribs) 

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<math>u_E=\large\frac{c^2 h^2}{48 K_M} \frac{\partial \overline S}{\partial x}\normalsize , \quad S_E=\large\frac{h^2 u_E}{60 K_S} \frac{\partial \overline S}{\partial x}\normalsize ,\qquad (5) </math>  <math>u_E=\large\frac{c^2 h^2}{48 K_M} \frac{\partial \overline S}{\partial x}\normalsize , \quad S_E=\large\frac{h^2 u_E}{60 K_S} \frac{\partial \overline S}{\partial x}\normalsize ,\qquad (5) </math>  
−  where <math> c=\sqrt{\beta s_0 gh} </math> is twice the maximum internal wave speed in a layered channel with density difference <math>\beta s_0</math> and total depth <math>h</math>.  +  where <math> c=\sqrt{\beta s_0 gh} </math> is twice the maximum internal wave speed in a layered channel with density difference <math>\beta s_0</math> and total depth <math>h</math>. 
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===Salt balance===  ===Salt balance===  
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This equation shows that <math>1/L</math> is a fair estimate for the salinity gradient <math>d \overline S / dx</math> due to estuarine circulation in the range <math>0 \le x \le L/2</math>. Substitution in Eq. (5) gives the maximum estuarine circulation velocity <math>u_E</math>:  This equation shows that <math>1/L</math> is a fair estimate for the salinity gradient <math>d \overline S / dx</math> due to estuarine circulation in the range <math>0 \le x \le L/2</math>. Substitution in Eq. (5) gives the maximum estuarine circulation velocity <math>u_E</math>:  
−  <math>u_E \approx 0.87(u_R c^2 \large\frac{K_S}{K_M})^{1/3}\normalsize. \qquad (9) </math>  +  <math>u_E \approx 0.87(u_R c^2 \large\frac{K_S}{K_M})^{1/3}\normalsize, \quad S_E \approx 0.6 \large (\frac{u_R}{c})^{2/3} (\frac{K_M}{K_S})^{1/3} \normalsize . \qquad (9) </math> 
−  +  Turbulent mixing over the water column is suppressed by the effect of density stratification, also in the case of continuously stratified (nonlayered) systems. This implies that the diffusion coefficients <math>K_M, K_S</math> depend not only on the turbulence generated by the tidal flow, but also on the damping by stratification. This dependence can be related to the socalled Richardson number,  
+  <math>Ri=  \large \frac{g}{\rho} \frac{\partial \rho / \partial z}{(\partial u /\partial z)^2} \normalsize</math>. Here we consider a rough approximation of the Richardson number,  
−  +  <math>Ri \approx (\large\frac{c}{U})^2 \normalsize \Delta S </math>, with <math>c^2=\beta s_0 gh </math> and <math>\Delta S = 9 S_E</math> (see figure A2).  
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−  <math>  +  For small Richardson numbers (typically much smaller than 0.25) the diffusion coefficients are not strongly influenced by vertical density gradients. In this case an estimate of the turbulent diffusion coefficients is given by <math>K_M \approx K_S \approx K = c_D hU</math>. The friction coefficient <math>c_D</math> is of the order of 0.002. Substitution in the expressions for <math>L</math> (Eq. 8), <math>u_E</math> (Eq. 9) and <math>\Delta S</math> (Eq. 12) yields 
−  +  <math> u_E \approx 0.87 (u_R c^2)^{1/3} , \; \Delta S \approx 5.4 (\large\frac{u_R}{c})^{2/3}\normalsize , \; L \approx \large\frac{0.024 h}{c_D}\normalsize (\large\frac{c}{u_R})^{1/3} \frac{c}{U}\normalsize . \qquad (13) </math>  
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−  <math> u_E \approx 0.87 (u_R c^2)^{1/3} , \; \Delta S \approx  
The estuarine circulation and the stratification increase with increasing river runoff <math>u_R</math>; they do not depend on the tidal velocity amplitude <math>U</math>. The seawater intrusion length <math>L</math> decreases with increasing river runoff and with increasing tidal velocity.  The estuarine circulation and the stratification increase with increasing river runoff <math>u_R</math>; they do not depend on the tidal velocity amplitude <math>U</math>. The seawater intrusion length <math>L</math> decreases with increasing river runoff and with increasing tidal velocity.  
−  +  For situations with strong (but continuous) stratification the Richardson number is larger than 1. In this case we use expressions for <math>K_M, K_S</math> established experimentally by MunkAnderson <ref>Munk, W.H. and Anderson, E.R. (1948). Notes on the theory of the thermocline. J. Marine Res. 7: 276295</ref>,  
<math>K_M = K(1+10Ri)^{1/2} \approx 0.316 K Ri^{1/2} , \; K_S = K (1+3.3Ri)^{3/2} \approx 0.167 K Ri^{3/2} , \; K=c_D h U . \qquad (14) </math>  <math>K_M = K(1+10Ri)^{1/2} \approx 0.316 K Ri^{1/2} , \; K_S = K (1+3.3Ri)^{3/2} \approx 0.167 K Ri^{3/2} , \; K=c_D h U . \qquad (14) </math>  
−  +  With these approximations we find  
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−  <math>u_E \approx 0.  +  <math>u_E \approx 0.27 U , \; \Delta S \approx 17 \large\frac{u_R}{U}\normalsize , \; L \approx \large\frac{h}{c_D}\normalsize (\large\frac{c}{U})^3 (\frac{u_R}{U})^{1/2}\normalsize . \qquad (15) </math> 
With an increase of runoff <math>u_R</math> the estuarine circulation <math>u_E</math> remains unchanged, whereas the intrusion length <math>L</math> increases. This counterintuitive result can be explained by the fact that the river runoff loses its 'grip' on the nearbottom seawater intrusion with increasing stratification.  With an increase of runoff <math>u_R</math> the estuarine circulation <math>u_E</math> remains unchanged, whereas the intrusion length <math>L</math> increases. This counterintuitive result can be explained by the fact that the river runoff loses its 'grip' on the nearbottom seawater intrusion with increasing stratification.  
===Numerical example===  ===Numerical example===  
−  Consider an estuary characterized by: <math>U</math>=1 m/s, <math>h</math>=10 m, <math>u_R</math>=0.05 m/s, <math>c_D</math>=0.002.  +  Consider an estuary characterized by: <math>U</math>=1 m/s, <math>h</math>=10 m, <math>u_R</math>=0.05 m/s, <math>c_D</math>=0.002. Using the formulas (15) we find <math>u_E\approx 0.27 m/s, \; \Delta S \approx 0.85, \; Ri \approx 1.7</math>. An improved estimate can be obtained by using the nonapproximated form of Eqs. (14); this yields similar values. 
−  The salt intrusion length is given by <math>3L/2</math> according to Eq. (8).  +  The salt intrusion length is given by <math>3L/2</math> according to Eq. (8). This gives a seawater intrusion length of about 5 km. Taking other dispersion processes into account, in addition to estuarine circulation, the salt intrusion length will increase by a certain factor. In this case the estuarine circulation <math>u_E</math> will decrease by the same factor, according to equation (5). Field observations show that the estuarine circulation is generally larger than 0.1 m/s, even when longitudinal and vertical salinity gradients are small. 
Revision as of 12:34, 17 May 2020
Definition of Estuarine circulation:
Residual flow pattern in an estuary induced by the density difference between seawater and river water.
This is the common definition for Estuarine circulation, other definitions can be discussed in the article

Contents
Introduction
Seawater has a higher density than river water, due to its high salinity. The density difference is of the order of 3%. In estuaries where seawater and river water meet, seawater will tend to intrude beneath the river outflow. Part of the intruding seawater will mix with river water, whereby seawater is returned to the sea by estuarine outflow in the upper water layer. This allows 'new' seawater to enter the estuary via the lower water layers. In this way, a circulation of seawater is created, which is directed inland along the bottom and seaward along the surface, see figure 1. This circulation is called estuarine circulation or gravitational circulation. It is driven by the longitudinal density gradient along the estuary: higherdensity seawater water in the downstream part of the estuary 'diving' under lowerdensity mixed water in the inner estuary. The circulation is kept going by mixing of seawater with river water. Mixing is the result of turbulence, which in estuaries is mainly caused by strong tidal flows. Most of the mixing takes place vertically in the water column due to turbulence generated by tidal flow along the estuarine bottom^{[1]}. The different processes that drive estuarine circulation are discussed in this article. In estuaries with little tide there is hardly any mixing of seawater with river water. In that case a more or less stationary salt wedge forms along the bottom; this case is discussed in the article Salt wedge estuaries.
The classical theory
The classical theory of estuarine circulation was first developed by Hansen and Rattray in 1965 ^{[2]}, after earlier investigations by D. Pritchard in the St. James river ^{[3]}. Many other authors have subsequently contributed to further verification and refinement^{[4]}^{[5]}.
In the classical theory of estuarine circulation, advection of seawater up and down the estuary by the tidal motion and associated mixing processes are not considered explicitly. This theory considers a stationary salinity field in a frame that moves cyclically up and down the estuary while vertical mixing processes are represented by a tideaveraged diffusivity. The role of tidal motion boils down to the generation of turbulence at the channel bed, which produces turbulent eddies that exchange of mass and momentum across the water column.
Nowadays, the intrusion if seawater in an estuary can be simulated in numerical models that incorporate detailed descriptions of turbulent exchange processes and complex bathymetries ^{[6]}^{[7]}. However, insight in the seawater intrusion process and some rough estimates of the contribution of estuarine circulation can already be obtained from the classical analytical approach. It shows the important role of density stratification for the suppression of turbulent vertical mixing. In the case of weak stratification and strong vertical mixing the river discharge opposes the intrusion of seawater. The strength of the estuarine circulation increases with increasing river discharge whereas the seawater intrusion length decreases. In the case of strong stratification and suppression of turbulent mixing the strength of the estuarine circulation depends almost solely on the tidal velocity. The seawater intrusion length strongly decreases with increasing tidal velocity, but increases with increasing river discharge (increase relative to tidal velocity). This counterintuitive result can be explained by tilting of the pycnoclines due to the increase of stratification and decrease of internal friction.
Formulas for the estuarine circulation and seawater intrusion, as well as their derivation are presented in the appendix, together with a numerical example. Observed salt intrusion lengths are typically several times larger than the estimate from the analytical model. A plausible explanation is the neglect of other processes that also contribute to seawater intrusion (see Seawater intrusion and mixing in estuaries).
The analytical model is a strong simplification of reality. The prismatic channel approximation and the tidal averaging preclude important seawater intrusion mechanisms, such as:
 dispersion by vertical tidal straining (see next paragraph);
 dispersion by horizontal tidal shear effects and by horizontal residual circulations;
 dispersion by horizontal residual circulations related to the flow structure in channel bends;
 dispersion related to estuarine geometrical characteristics such as secondary channels and tidal flats ('dead zones');
 dispersion related to the chaotic nature of water parcel trajectories.
Seawater intrusion produced by these dispersion mechanisms is discussed in the article Seawater intrusion and mixing in estuaries.
Asymmetric Tidal Mixing
In 1986, Van Aken ^{[8]} explained the onset of seasonal stratification in the North Sea by the process of tidal advection of surface water with lower density, related to the presence of strong salinity gradients. A few years later, Simpson et al. ^{[9]} observed and modelled tidal variation of stratification in Liverpool Bay by this differential advection mechanism, which they called "tidal straining". Observations by Stacey and Ralston (2005)^{[10]} and numerical model studies by Jay and Musiak ^{[6]} and Scott ^{[7]} revealed that tidal straining can strongly influence estuarine circulation. When during flood high salinity seawater enters the estuary, the less dense estuarine surface water is repulsed, leading to more upright pycnoclines. Density stratification is often further reduced by convective overturning (seawater being advected by tidal flow on top of less dense estuarine water ^{[11]}). Except in the case of strong stratification, the surface flow exchanges momentum with the nearbottom flow, which is accelerated in upstream direction. During ebb, the tidal flow restores stratification, by advecting less dense estuarine waters seaward over the denser nearbottom waters. The momentum exchange between surface and nearbottom water is now reduced; seaward tidal flow is concentrated near the surface, see figure 2. The net effect is an enhancement of residual landward nearbottom flow, as shown by numerical experiments ^{[12]}.
Enhancement of estuarine circulation by asymmetric tidal mixing only occurs in situations of weak stratification. In case of strong stratification (weak tides, high river runoff) momentum exchange between surface and nearbottom water hardly occurs (see Salt wedge estuaries). The major effect of weak tides in the latter case is an increase of eddy viscosity in the bottom layer, which reduces gravitational circulation ^{[13]}. Enhancement of estuarine circulation by asymmetric tidal mixing is strongest for small Richardson numbers (typically [math]Ri \lt 1[/math], see appendix) and periodic or weak stratification.
Figure 3 illustrates asymmetric tidal mixing in Rotterdam Waterway. The velocity and salinity profiles were measured on 9 September 1967 at an estimated net discharge of about 1000 m3/s (runoff velocity [math]u_R \approx 0.2 m/s[/math]). Characteristic values of the Richardson and Simpson numbers are [math]Ri \approx 1.8, Si \approx 0.5 .[/math] The sections at maximum ebb and maximum flood flow display asymmetry between nearbottom flow velocities; nearbottom velocities are substantially higher during flood than during ebb (as schematically depicted in figure 2). The sections at H.S.W. and L.S.W. show asymmetry in stratification: at the end of the flood the water column is less stratified than at the end of the ebb. Another survey in Rotterdam Waterway was conducted by De Nijs et al. in 2006 at higher river discharge (stronger stratification) ^{[14]}. These authors show that in Rotterdam Waterway asymmetry in tidal mixing is primarily related to asymmetry of the tide at the mouth (faster tidal rise than tidal fall, which enhances flood currents over ebb currents), more than to differential advection. This is consistent with the fast thickening of the bottom boundary layer during flood, caused by strong bedgenerated turbulence. A similar behaviour was observed in the Hudson estuary ^{[15]}.
Lateral estuarine circulation
In natural estuaries, the depth decreases towards the lateral boundaries (in a meander the depth decreases from the outer bend towards the inner channel bend). Tidal velocities are lower near the channel banks than at the channel centre. The salinity front is therefore not advected at the same speed at different lateral locations. This results in lateral salinity gradients. During flood (especially shortly before the end of the flood phase) the salinity is higher at the channel centre than near the channel banks. During ebb the situation is reversed; near the surface, the salinity at the channel centre is lower than near the channel banks. The lateral salinity gradients related to this differential tidal salinity advection, induce a baroclinic (densitydriven) lateral current. This current has a vertical structure; during flood it is directed near the bottom toward the channel banks and toward the channel centre near the surface, see figure 4. The resulting flow convergence at the channel centre is often visible in the field by a scum line following the channel axis ^{[16]}. During ebb the lateral exchange flow is divergent at the surface and convergent near the bottom. This lateral baroclinic exchange flow has important consequences for longitudinal dispersion in natural estuaries, as first pointed out by R. Smith ^{[17]} and explained in the article Seawater intrusion and mixing in estuaries.
Lateral exchange flows also influence estuarine circulation. Downwelling flow at the channel axis during flood transports momentum down the vertical; surface flow is decelerated and near bottom flow accelerated in upestuary direction. During ebb, vertical momentum transport reduces turbulent momentum dissipation by reducing the vertical shear of the tidal current. This also favours the baroclinic flow component ^{[18]}. Lateral baroclinic exchange flow therefore enhances estuarine circulation. It also induces an a net downstream flow component in the shallower parts of the crosssection (lateral inflow of downestuary momentum from the channel centre during ebb, outflow of upestuary momentum toward the channel centre during flood, see figure 4); this flow component enhances the downestuary flow component related to river runoff ^{[19]}.
Other lateral circulation patterns
Several mechanisms can alter the symmetric flow pattern of figure 5. In wide estuaries the symmetry is broken by earth's rotation. Currents are deviated to the right on the northern hemisphere and to the left on the southern hemisphere (see Coriolis acceleration). Looking in upestuary direction, flood currents are concentrated at the right side of a straight tidal channel and ebb currents at the left side (northern hemisphere). Estuarine circulation is influenced by earth's rotation in a similar way ^{[20]}: the residual upestuary flow is shifted to the right bank and the downestuary flow to the left bank, see figure 5. The mean crosssectional salinity distribution is tilted accordingly.
Channel meandering induces transverse circulation, directed to the inner channel bend in the lower part of the vertical and to the outer bend near the surface. The spiraling tidal flow modifies the salinity distribution and the densitydriven longitudinal residual flow in a similar way as earth's rotation when the tidal flow follows a channel meander with the inner channel bend at the right bank ^{[21]}. For an inner channel bend at the left bank the picture is mirrored; in this case the effects of channel meandering and earth's rotation are opposite. Channel bends and other topographic structures (bars, shoals) generate longitudinal circulating flow patterns in estuaries also in the absence of density differences. These horizontal topographically induced flow circulations are often stronger than horizontal estuarine circulation generated by density gradients. However, residual flow circulation in the longitudinalvertical plane is mainly due to the density difference between seawater and river water.
Consequence of estuarine circulation
Estuarine circulation is highly relevant for transport of fine sediments and sedimentation (see Dynamics of mud transport). The upestuary nearbottom flow captures a substantial part of the downstream flux of fluvial sediment, and moves this sediment back toward the tip of the salt intrusion zone. The upestuary nearbottom flow contributes also to import of marine sediment into the estuary. This sediment settles on the bed during periods of neap tide and slack water. It is resuspended in periods of high tidal currents, leading to a strong increase in turbidity. Decomposition of organic material in the high turbidity zone can cause oxygen depletion, especially in the lower part of the vertical where mixing with oxygenrich surface water is inhibited by stratification. However, estuarine circulation is not the only cause of upestuary sediment transport. Many field and model investigations show that tidal asymmetry due to nonlinear tidal propagation in shallow water (leading to stronger flood currents than ebb currents and longer highwater slack periods than lowwater slack periods) is often a more important mechanism, see the articles Estuarine turbidity maximum and Tidal asymmetry and tidal basin morphodynamics.
Appendix
In this appendix the classical analytical model for estuarine circulation is presented. The model applies to a straight prismatic estuary (uniform depth [math]h[/math] and width [math]b[/math] ); the width is small enough for neglecting the influence of earth's rotation (Rossby radius [math]U/fb \lt \lt 1[/math] ).
The following notations are used (see also figures 1 and A1):
 [math]x[/math] is the coordinate along the estuarine axis ([math]x=0[/math] at the sea boundary, [math]x=L[/math] at the landward boundary of salt intrusion);
 [math]u[/math] is the velocity in the [math]x[/math]direction, [math]u(z) = u_R + u_C(z) + U \cos \omega t,[/math] where [math]u_R[/math] is the velocity related to river runoff (river discharge divided by estuarine crosssection), [math]u_C(z)[/math] is the estuarine circulation, [math]U[/math] is the depthaveraged tidal velocity amplitude and [math]\omega = 2 \pi / T[/math] is the tidal radial frequency;
 [math]S=s/s_0[/math] is the ratio of the local salinity [math]s[/math] and the seawater salinity [math]s_0 :[/math] [math]S[/math] is related to the density [math]\rho[/math] by the relation [math]\rho=\rho_0(1+\beta s_0 S),[/math] where [math]\rho_0[/math] is the fresh water density and the expansion coefficient [math]\beta=7.7 \times 10^{4};[/math]
 [math] p(z) = g \int_z^\eta \rho dz[/math] is the hydrostatic pressure; [math]\eta[/math] is the surface elevation relative to a horizontal plane and [math](z)[/math] is the depth relative to the surface;
 [math]K_M[/math] is the eddy viscosity (vertical momentum diffusion coefficient [math][m^2/s] [/math] ) and [math]K_S[/math] the eddy diffusivity (vertical salinity diffusion coefficient).
Depthaveraging is denoted by an overbar.
The model
The model is based on the following assumptions:
 The diffusion coefficients [math]K_M, K_S[/math] do not depend on the spatial and temporal coordinates [math]x,z,t[/math] ; this means that the classical model is realistic only for continuously stratified estuaries (no layered salinity structure; such estuaries are often called "partially mixed", see figure A1);
 The river flow velocity [math]u_R[/math] and the tidal velocity amplitude [math]U[/math] do not depend on the spatial coordinates [math]x,z[/math] ;
 The river flow velocity is much smaller than the tidal velocity, [math]u_R \lt \lt U[/math] ;
 The water surface variation is much smaller than depth, [math]\eta \lt \lt h[/math] ; Stokes drift is neglected;
 The vertical flow velocity [math] w[/math] is neglected;
 Side wall effects are neglected, implying uniformity of all variables in lateral [math]y[/math]direction;
 Longitudinal gradients in the velocity [math]u[/math] are neglected;
 The longitudinal salinity gradient is the same at all depths ([math]\large\frac{\partial S}{\partial x}\normalsize=\large\frac{\partial \overline S}{\partial x}\normalsize ).[/math]
We consider the influence of a constant river flow [math]u_R[/math] on the salinity distribution in a frame moving with the velocity [math]u_R + U \cos \omega t[/math]. In the moving frame the velocity [math]u[/math] is timeindependent and depends only on the vertical coordinate [math]z[/math]; we thus have [math]u=u_C(z)[/math]. The only driving force acting on the fluid in the moving frame is the pressure gradient [math]\partial p / \partial x [/math], which is a linearly increasing function of depth ([math]z[/math]) below mean water level:
[math]\large\frac{1}{g \rho} \frac{\partial p}{\partial x}\normalsize = \large\frac{\partial \eta}{\partial x}\normalsize + (\eta z) \beta s_0 \large\frac{\partial S}{\partial x}\normalsize . \qquad (1) [/math]
The fluid acceleration caused by this pressure gradient is balanced by momentum dissipation through turbulent diffusion:
[math]\large\frac{\partial p}{\rho \partial x}\normalsize = \large\frac{\partial }{\partial z}\normalsize ( K_M \large\frac{\partial u_C}{\partial z}\normalsize) . \qquad (2) [/math]
This equation can be solved considering the boundary conditions:
No slip at the bottom: [math]u_C(h)=0[/math] , no momentum diffusion through the surface: [math]du_C / dz =0[/math] at [math]z=\eta \approx 0[/math] and no net discharge: [math]\int_{h}^0 u_C dz =0[/math].
Because the river runoff is constant and the tide cyclic, the salinity distribution should not change with time in the moving frame. However, the estuarine circulation modifies the salinity distribution, because salt advection [math]u_C(z) S(x,z)[/math] has a gradient in [math]x[/math]direction. According to the classical theory, the salt balance is restored by vertical diffusion of salinity:
[math]u_C \large\frac{\partial \overline S}{\partial x}\normalsize = \large\frac{\partial }{\partial z}\normalsize ( K_S \large\frac{\partial S}{\partial z}\normalsize) . \qquad (3) [/math]
This equation can be solved with the boundary conditions: no salt flux through bottom and surface, [math]\partial S / \partial z =0[/math] at [math] z=h, z=0 .[/math]
The solution of equations Eqs. (13) is given by (shown in figure A2):
[math] u_C(z)=u_E (1+9 \zeta^2+8 \zeta^3) , \; S=\overline S + s_C(z), \; s_C(z)=S_E (5+30 \zeta^245 \zeta^424 \zeta^5), \; \zeta=z/h , \qquad (4) [/math]
[math]u_E=\large\frac{c^2 h^2}{48 K_M} \frac{\partial \overline S}{\partial x}\normalsize , \quad S_E=\large\frac{h^2 u_E}{60 K_S} \frac{\partial \overline S}{\partial x}\normalsize ,\qquad (5) [/math]
where [math] c=\sqrt{\beta s_0 gh} [/math] is twice the maximum internal wave speed in a layered channel with density difference [math]\beta s_0[/math] and total depth [math]h[/math].
Salt balance
The motion in the moving frame is not exactly cyclic; after one tide the moving frame has moved seaward over a distance [math]T u_R[/math] . In order to prevent complete flushing of the salinity a net landward salt transport should exist. Many seawater intrusion mechanisms can contribute to landward salt transport, as discussed in the article Seawater intrusion and mixing in estuaries. In this case, we assume that estuarine circulation is the primary mechanism for seawater intrusion. This means that the loss of salt content from the estuary due to the net shift of the moving frame (moving from any start position in the estuary), [math]hu_rS[/math], should be compensated by salt transport related to estuarine circulation:
[math]h u_R \overline S = \int_{h}^0 u_C(z)S(z) dz. \qquad (7) [/math]
This balance equation holds for all crosssections [math]x[/math] landward of the estuarine mouth. Using Eqs. (4, 5) this equation can be rewritten as a nonlinear differential equation for the depthaveraged salinity
[math]\overline S(x) = ( L \large\frac{d \overline S}{dx} )^3 \normalsize , \quad [/math] with solution [math]\overline S(x) = (1  \large\frac{2x}{3L})^{3/2}\normalsize , \qquad L \approx 0.024 h^2 c (\large\frac{c}{u_R K_S K_M^2})^{1/3}\normalsize . \qquad (8) [/math]
This equation shows that [math]1/L[/math] is a fair estimate for the salinity gradient [math]d \overline S / dx[/math] due to estuarine circulation in the range [math]0 \le x \le L/2[/math]. Substitution in Eq. (5) gives the maximum estuarine circulation velocity [math]u_E[/math]:
[math]u_E \approx 0.87(u_R c^2 \large\frac{K_S}{K_M})^{1/3}\normalsize, \quad S_E \approx 0.6 \large (\frac{u_R}{c})^{2/3} (\frac{K_M}{K_S})^{1/3} \normalsize . \qquad (9) [/math]
Turbulent mixing over the water column is suppressed by the effect of density stratification, also in the case of continuously stratified (nonlayered) systems. This implies that the diffusion coefficients [math]K_M, K_S[/math] depend not only on the turbulence generated by the tidal flow, but also on the damping by stratification. This dependence can be related to the socalled Richardson number, [math]Ri=  \large \frac{g}{\rho} \frac{\partial \rho / \partial z}{(\partial u /\partial z)^2} \normalsize[/math]. Here we consider a rough approximation of the Richardson number,
[math]Ri \approx (\large\frac{c}{U})^2 \normalsize \Delta S [/math], with [math]c^2=\beta s_0 gh [/math] and [math]\Delta S = 9 S_E[/math] (see figure A2).
For small Richardson numbers (typically much smaller than 0.25) the diffusion coefficients are not strongly influenced by vertical density gradients. In this case an estimate of the turbulent diffusion coefficients is given by [math]K_M \approx K_S \approx K = c_D hU[/math]. The friction coefficient [math]c_D[/math] is of the order of 0.002. Substitution in the expressions for [math]L[/math] (Eq. 8), [math]u_E[/math] (Eq. 9) and [math]\Delta S[/math] (Eq. 12) yields
[math] u_E \approx 0.87 (u_R c^2)^{1/3} , \; \Delta S \approx 5.4 (\large\frac{u_R}{c})^{2/3}\normalsize , \; L \approx \large\frac{0.024 h}{c_D}\normalsize (\large\frac{c}{u_R})^{1/3} \frac{c}{U}\normalsize . \qquad (13) [/math]
The estuarine circulation and the stratification increase with increasing river runoff [math]u_R[/math]; they do not depend on the tidal velocity amplitude [math]U[/math]. The seawater intrusion length [math]L[/math] decreases with increasing river runoff and with increasing tidal velocity.
For situations with strong (but continuous) stratification the Richardson number is larger than 1. In this case we use expressions for [math]K_M, K_S[/math] established experimentally by MunkAnderson ^{[22]},
[math]K_M = K(1+10Ri)^{1/2} \approx 0.316 K Ri^{1/2} , \; K_S = K (1+3.3Ri)^{3/2} \approx 0.167 K Ri^{3/2} , \; K=c_D h U . \qquad (14) [/math]
With these approximations we find
[math]u_E \approx 0.27 U , \; \Delta S \approx 17 \large\frac{u_R}{U}\normalsize , \; L \approx \large\frac{h}{c_D}\normalsize (\large\frac{c}{U})^3 (\frac{u_R}{U})^{1/2}\normalsize . \qquad (15) [/math]
With an increase of runoff [math]u_R[/math] the estuarine circulation [math]u_E[/math] remains unchanged, whereas the intrusion length [math]L[/math] increases. This counterintuitive result can be explained by the fact that the river runoff loses its 'grip' on the nearbottom seawater intrusion with increasing stratification.
Numerical example
Consider an estuary characterized by: [math]U[/math]=1 m/s, [math]h[/math]=10 m, [math]u_R[/math]=0.05 m/s, [math]c_D[/math]=0.002. Using the formulas (15) we find [math]u_E\approx 0.27 m/s, \; \Delta S \approx 0.85, \; Ri \approx 1.7[/math]. An improved estimate can be obtained by using the nonapproximated form of Eqs. (14); this yields similar values. The salt intrusion length is given by [math]3L/2[/math] according to Eq. (8). This gives a seawater intrusion length of about 5 km. Taking other dispersion processes into account, in addition to estuarine circulation, the salt intrusion length will increase by a certain factor. In this case the estuarine circulation [math]u_E[/math] will decrease by the same factor, according to equation (5). Field observations show that the estuarine circulation is generally larger than 0.1 m/s, even when longitudinal and vertical salinity gradients are small.
Related articles
 Salt wedge estuaries
 Seawater intrusion and mixing in estuaries
 Coriolis acceleration
 Dynamics of mud transport
References
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