Residual flow pattern in an estuary induced by the density difference between seawater and river water.
The classical theory of estuarine circulation was first developed by Hansen and Rattray in 1965 , after earlier investigations by D. Pritchard in the St. James river . Many other authors have subsequently contributed to further verification and refinement.
In the classical theory of estuarine circulation tidal motion does not affect the salinity distribution at the intratidal timescale. The tide moves the salinity field cyclically up and down the estuary without modification. The effect of tidal motion is therefore restricted to turbulence generation at the bed, which enhances momentum dissipation and mixing in and across the bottom boundary layer. Seawater intrusion is only driven by a longitudinal density gradient between water masses of different salinity. The corresponding residual flow is called (often without distinction) estuarine circulation, baroclinic flow or gravitational circulation.
In the following basic notions of the classical theory are presented. We consider a straight prismatic estuary (uniform depth and width ); the width is small enough for neglecting the influence of earth rotation (Rossby radius ). We use the following notations (see also figure 1):
- is the coordinate along the estuarine axis ( at the sea boundary, at the landward boundary of salt intrusion);
- is the velocity in the -direction, where is the velocity related to river runoff (river discharge divided by estuarine cross-section), is the estuarine circulation, is the tidal velocity amplitude and is the tidal radial frequency;
- is the ratio of the local salinity and the seawater salinity is related to the density by the relation where is the fresh water density and the expansion coefficient
- is the hydrostatic pressure; is the surface elevation relative to a horizontal plane and is the depth relative to the surface;
- is the eddy viscosity (vertical momentum diffusion coefficient ) and the eddy diffusivity (vertical salinity diffusion coefficient).
Depth-averaging is denoted by an overbar.
The following additional assumptions are made:
- Vertical exchanges of velocity and salinity are described as a diffusion process;
- The diffusion coefficients do not depend on the spatial and temporal coordinates ; 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 1);
- The river flow velocity and the tidal velocity amplitude do not depend on the spatial coordinates ;
- The river flow velocity is much smaller than the tidal velocity, ;
- The water surface variation is much smaller than depth, ; Stokes drift is neglected;
- The vertical flow velocity is neglected;
- Side wall effects are neglected, implying uniformity of all variables in lateral -direction;
- Longitudinal gradients in the velocity are neglected;
- The longitudinal salinity gradient is the same at all depths (
The above assumptions clearly do not hold near the upstream and downstream boundaries of the salinity distribution; they can be considered reasonable approximations only for the central part of the salinity distribution.
We consider the influence of constant river flow on the salinity distribution in a frame moving with the velocity ; we thus have In the moving frame the only driving force acting on the fluid is the pressure gradient
The fluid acceleration caused by this pressure gradient is compensated by momentum dissipation through turbulent diffusion:
This equation can be solved considering the boundary conditions:
- No slip at the bottom: ;
- No momentum diffusion through the surface: ;
- No net discharge:
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 has a gradient in -direction. The salt balance is restored by vertical diffusion of salinity:
This equation can be solved with the boundary conditions:
- No salt flux through bottom and surface at
The solution of equations Eqs. (1-3) reads (shown in figure 2):
A frequently used estimate of the eddy viscosity is where is the friction velocity. We then find that the estuarine circulation is related to the Simpson stratification number :
The motion in the moving frame is not exactly cyclic; after one tide the moving frame has moved seaward over a distance . In order to prevent flushing of the salinity a net landward salt transport should exist. Many seawater intrusion mechanisms can contribute to landward salt transport, as discussed in 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, , should be compensated by salt transport related to estuarine circulation:
Evaluation of the integral gives, using Eqs. (4,5),
is an estimate for the length of the seawater intrusion due to estuarine circulation; is the internal wave speed in a fresh water layer of depth overlying seawater. By approximating the maximum estuarine circulation velocity can be written as
From equation (4) it follows that estuarine circulation corresponds to a net landward flow in a near bottom layer of thickness and a net seaward flow of the same order in the upper layer, see figure 3. By calling the average salinity and average discharge in the upper layer and the average salinity and discharge in the bottom layer, we can determine the average salinity difference ("stratification") from the condition of net zero salt transport:
The "bulk" Richardson number is a measure for the degree of stratification in an estuary. For small Richardson numbers (typically smaller than 0.25) the diffusion coefficients are not strongly influenced by vertical density gradients. In this case one may approximate The friction coefficient is of the order of 0.002. Substitution in the expressions for (Eq. 8), (Eq. 9) and (Eq. 12) yields
The estuarine circulation and the stratification increase with increasing river runoff; they do not depend on the tidal velocity.
In case of strong (but continuous) stratification the Richardson number is larger than 1. In this case the diffusion coefficients depend on and are strongly reduced following the relations of Munk-Anderson 
Substitution of these approximations yields
With an increase of runoff the estuarine circulation remains unchanged, whereas the intrusion length increases. This counterintuitive result is due to the strong increase of stratification with increasing runoff.
The assumption that seawater intrusion is due only to gravitational circulation strongly limits the salt intrusion length The previous formulas indicate for realistic values an intrusion length Observed salt intrusion lengths are typically several times larger (see Seawater intrusion and mixing in estuaries). This reduces the value of the estuarine circulation by a similar factor, according to the classical theory. However, field observations indicate that residual circulation can be considerable (order 10 cm/s), even in the case of small longitudinal and vertical salinity gradients. Reasons will be discussed is the following paragraph.
Asymmetric Tidal Mixing
In 1986, Van Aken  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.  observed and modeled tidal variation of stratification in Liverpool Bay by this differential advection mechanism, which they called "tidal straining". Numerical model studies by Jay and Musiak  and Scott  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. Vertical salinity stratification is therefore reduced, a process further strengthened in some cases by convective overturning . For sufficiently weak stratification, the surface flow exchanges momentum with the near-bottom flow, which is accelerated in upstream direction. During ebb, the tidal flow restores stratification, by advecting less dense estuarine waters seaward over the denser near-bottom waters. The momentum exchange between surface and near-bottom water is now reduced; seaward tidal flow is concentrated near the surface, see figure 4. The net effect is an enhancement of residual landward near-bottom flow, as shown by numerical experiments .
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 near-bottom 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 . Enhancement of estuarine circulation by asymmetric tidal mixing is strongest for small Simpson numbers (typically ) and periodic or weak stratification.
Figure 5 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 ). Characteristic values of the Richardson and Simpson numbers are The sections at maximum ebb and maximum flood flow display asymmetry between near-bottom flow velocities; near-bottom velocities are substantially higher during flood than during ebb (as schematically depicted in figure 4). 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) . 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 bed-generated turbulence. A similar behaviour was observed in the Hudson estuary .
Lateral estuarine circulation
In straight natural estuaries the depth decreases towards the lateral boundaries (in a meander the depth decreases 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 differential tidal salinity advection, induce a baroclinic (density-driven) 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 6. The resulting flow convergence at the channel centre is often visible in the field by a scum line following the channel axis . 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 , see also 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 up-estuary 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 . Lateral baroclinic exchange flow therefore enhances estuarine circulation. It also induces an a net downstream flow component in the shallower parts of the cross-section (lateral inflow of down-estuary momentum from the channel centre during ebb, outflow of up-estuary momentum toward the channel centre during flood, see figure 6); this flow component enhances the down-estuary flow component related to river runoff .
Other lateral circulation patterns
Several mechanisms can alter the symmetric flow pattern of figure 6. In wide estuaries the symmetry is broken by earth rotation. Currents are deviated to the right on the northern hemisphere and to the left on the southern hemisphere (see Coriolis acceleration). Looking in up-estuary 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 rotation in a similar way : the residual up-estuary flow is shifted to the right bank and the down-estuary flow to the left bank, see figure 7. The mean cross-sectional 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 density-driven longitudinal residual flow in a similar way as earth rotation when the tidal flow follows a channel meander with the inner channel bend at the right bank . For an inner channel bend at the left bank the picture is mirrored; in this case the effects of channel meandering and earth 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 longitudinal-vertical 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 up-estuary near-bottom 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 up-estuary near-bottom 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 oxygen-rich surface water is inhibited by stratification. However, estuarine circulation is not the only cause of up-estuary sediment transport. Many field and model investigations show that tidal asymmetry due to non-linear tidal propagation in shallow water (leading to stronger flood currents than ebb currents and longer high-water slack periods than low-water slack periods) is often a more important mechanism.
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