Seawater intrusion and mixing in estuaries

We describe in this article the physical processes involved in seawater intrusion and mixing in estuaries under the influence of tidal motion and explain some simple methods for deriving quantitative estimates. Several of the dispersion mechanisms discussed in this article are illustrated by dye experiments shown in Estuarine dispersion: dye experiments in the Eastern Scheldt scale model. This article is largely based on experience from large-scale experiments to unravel tidal dispersion processes in the Eastern Scheldt by Dronkers and van de Kreeke (1986 ) and on the book Mixing in Inland and Coastal Waters by Fischer et al. (1979).

Introduction

Estuaries are generally defined as semi-enclosed transition zones between river and sea. The intrusion of seawater in estuaries determines the salinity distribution. Salinity is usually expressed as the weight in grams of all dissolved salts per kilogram of water or alternatively according to the Practical Salinity Scale (see Salinity). The seawater-intrusion limit must be defined by a chosen salinity threshold, for example a salinity increase of 0.1 above the river-water salinity. Seawater intrusion is mainly due to tides and to the influence of the density difference between seawater and river water on currents and turbulence (this influence is often termed 'buoyancy effects', see the article Estuarine circulation for further details). It is counteracted by fresh water inflow from rivers, groundwater, and other sources. Seawater intrusion in estuaries is an important phenomenon to man and nature: it limits fresh water availability for human and agricultural use, and it determines the type of habitats and species that can develop in an estuarine environment. Besides, density-driven currents and salinity play a role in estuarine turbidity and sedimentation processes.

The net effect of estuarine mixing processes during a tidal cycle on along-channel seawater intrusion or on net along-channel spreading (diffusion) of dissolved substances is generally called longitudinal dispersion. This net effect is described by a longitudinal dispersion coefficient [math]K^{(x)}[/math]. The longitudinal-dispersion approach is mainly applicable after salinity or tracer patches have become sufficiently mixed over the cross-section. In wide multichannel estuaries, strongly stratified reaches, and short estuaries where cross-sectional mixing is slow relative to flushing, a single local dispersion coefficient may not be meaningful.

The different mixing processes contributing to longitudinal dispersion are termed dispersion mechanisms. In well-mixed and partially mixed estuaries these mixing processes are dominantly induced by tidal motion; the net effect on seawater intrusion is called tidal longitudinal dispersion or simply tidal dispersion. Longitudinal dispersion and the flushing time of an estuary are related concepts. The combined effect of all tidal dispersion processes on seawater intrusion is sometimes called 'tidal pumping'. Here we use the term 'tidal pumping' only for the fraction of outflowing estuarine water that is replaced by 'new' seawater at the estuarine mouth.

Seawater intrusion has a strong variability. At a fixed station, salinity generally increases during flood and decreases during ebb. The maximum and minimum salinities usually occur around high and low water, respectively, but their phases depend on tidal propagation, advection, stratification and local geometry. Besides, seawater intrusion is small with high river discharge and strong with low river discharge. Change in seawater intrusion with river discharge is more gradual than the variability due to flood and ebb. If the river discharge varies on a time scale greater than the flushing time of the estuary, the tide-averaged salinity distribution tends towards an equilibrium distribution. The longitudinal dispersion coefficient can be determined by measuring the salinity distribution under equilibrium conditions. This coefficient generally varies along the estuary and depends on the river discharge.

The dispersion processes that are responsible for seawater intrusion determine the flushing time as well as the residence time of dissolved substances in an estuary. This holds similarly for the residence time of micro-organisms that are carried along passively (not swimming, floating or settling) with the water.

Definitions and assumptions

For the purpose of this article, the estuarine mouth is defined as the transition between zones of channeled tidal flow (tidal discharge mainly following the channel axis) and non-channeled flow (discharge deviated along the open coast). The estuarine head is defined here as the location where the seawater influence on the salinity is (almost) nil; this location depends on the river discharge. We are interested in the net effect of tidal mixing processes during a tidal cycle on seawater intrusion along the estuarine axis, defined as the [math]x[/math]-axis ([math]x=0[/math] at the estuarine mouth). A one-dimensional longitudinal description is most useful when the cross-sectionally averaged longitudinal salinity variation is the dominant large-scale variation, although vertical (along [math]z[/math]-axis) and lateral (along [math]y[/math]-axis) salinity differences may still contribute importantly to the dispersive salt flux. Stratified estuaries where salinity differences between bottom and surface are comparable to the salinity differences between estuarine mouth and estuarine head are dealt with in the article Salt wedge estuaries. The estuaries considered in this article are often designated as well-mixed or partially mixed estuaries.

Successive tidal cycles are not identical. Determining the net effect of tidally induced mixing processes requires integration over a large number of tidal cycles. To circumvent this complication, we assume that successive tidal cycles are sufficiently comparable (no extreme spring-neap variation) to enable the definition of a representative cyclic tide. This assumption will be used throughout this article.

Random walk

Under certain conditions, explained below, estuarine mixing processes can be described in terms of a random walk. For simplicity we consider a tidal basin with small river inflow (river discharge much smaller than tidal discharges). Situations where river flow exerts a substantial influence on estuarine water motion are dealt with in the article Estuarine circulation. Water motion is described as the average motion of a very large number of individual water parcels. If individual water parcels follow identical paths during flood (inflow) and ebb (outflow) all seawater parcels entering the basin during flood will be expelled during ebb: no seawater intrusion will occur. In reality this will not happen; mixing processes ensure that individual water parcels follow different paths during flood and ebb. Water parcels move back and forth in a tidal basin for some time before finally leaving the estuary and being evacuated offshore. We call [math]\tau_x [/math] the characteristic longitudinal dispersion time scale of water parcels in the tidal basin. It has the same order of magnitude as the flushing time [math]\tau_f[/math] (the ratio of fresh water volume in the tidal basin to fresh water discharge ) or the transit time [math]\tau_t[/math] (the average time it takes for a fluid parcel to travel from the upstream boundary of the estuary to the sea). During their transit, water parcels also move in lateral and vertical directions, due to flow circulations and turbulent eddies. The characteristic time scale for vertical mixing, [math]\tau_z [/math], and the characteristic time scale for lateral mixing, [math]\tau_y [/math], are related to the vertical and lateral turbulent diffusion coefficients, [math]K^{(z)} [/math] and [math]K^{(y)} [/math], by the relationships [math]\tau_z=h^2/ K^{(z)}[/math] and [math]\tau_y=b^2/ K^{(y)}[/math], respectively, where [math]h[/math] and [math]b[/math] are representative mean values of the depth and the width of the tidal basin. Numerical values depend on geometry and boundary conditions. The time scales [math]\tau_y [/math] and [math]\tau_z [/math] indicate the time during which the motions of individual water parcels remain correlated; after a longer time they have lost the memory of their initial position [math](y_0,z_0)[/math] in the estuarine cross-section. The probability that after this time they will be in a cross-sectional position [math](y,z)[/math] from where their net displacement during the next tidal cycle will be in landward direction is then equal to the probability that they will be in a position where their net displacement will be in seaward direction (we assume that the influence of river discharge is small and can be ignored). The cross-sectional decorrelation time or mixing time [math]\tau[/math] is approximately equal to the lateral mixing time [math]\tau_y[/math], if we assume that lateral mixing takes more time than vertical mixing; this is the case for most well-mixed or partially mixed wide estuaries. If [math]\tau[/math] is much smaller than the longitudinal flushing time [math]\tau_x[/math] of water parcels in the estuary, the path in longitudinal direction of a water parcel then follows a random walk. This condition only holds for estuaries of sufficient length; it may not be the case for wide, multichannel estuaries. The longitudinal displacements [math]X (1), X (2), … [/math] in successive time intervals [math]\tau[/math] are uncorrelated, by choosing [math]\tau=nT[/math] equal to an integer number of cyclic tidal periods such that [math]\tau \ge T[/math]. After a time [math]t_N[/math] comprising a large number [math]N[/math] of time intervals [math]\tau[/math] the average net displacement of all water parcels initially situated in a given cross-section of the tidal basin is approximately zero in the absence of river discharge:

[math]E[\sum_{i=1}^N X (i)] \; \approx \; 0 , \qquad(1)[/math]

where [math]E[..][/math] is the ensemble average over all water parcels initially situated in a given cross-section of the tidal basin. Here we have assumed [math]t_N \le \tau_x[/math]. Because [math]X(i)[/math] and [math]X(j)[/math] are uncorrelated for [math]i \ne j[/math], we have [math]E[X(i)X(j)] \approx 0 \;[/math] for [math]i \ne j[/math] and [math]E[X(i)^2] \approx \overline{X^2} \;[/math] is the average square displacement over a time [math]\tau[/math]. The average square displacement over the time interval [math]t_N =N \tau[/math] is therefore given by

[math]E[(\sum_{i=1}^N X (i))^2] \; = \; E[\sum_{i=1}^N X (i) \; \sum_{j=1}^N X (j)] \; \approx \; \sum_{i=1}^N E[X (i)^2] \; \approx \; t_N \; \overline{X^2} / \tau . \qquad(2)[/math]

Figure 1: Longitudinal dispersion coefficients at different locations in the Eastern Scheldt and Ems-Dollard estuaries.

The mean-square longitudinal displacement of water parcels in the tidal basin therefore increases approximately linearly with time. This is precisely the characteristic of a diffusion process of a dissolved substance with concentration [math]s(x,t)[/math]. This diffusion process is described by the equation

[math]\Large \frac{\partial s}{\partial t} \normalsize = K^{(x)}\Large \frac{\partial^2 s}{\partial x^2} \normalsize , \quad [/math] with diffusion coefficient [math]\quad K^{(x)}= \Large \frac{\overline{X^2} }{ 2 \tau} \normalsize . \qquad(3)[/math]

The random walk assumption [math] \tau_x \gt \gt \tau[/math], expressing the condition that particle paths become uncorrelated for time intervals larger than the cross-sectional mixing time [math]\tau[/math] but much smaller than the longitudinal mixing time [math]\tau_x[/math], thus implies that salt transport by seawater intrusion processes, [math]Q_{disp}[/math], can be represented by a gradient-type transport formula,

[math]Q_{disp} = - A_0 K^{(x)}\Large \frac{ds_0}{dx} \normalsize , \qquad(4) [/math]

where [math]s_0[/math] is the tidally averaged salinity concentration, [math]s_0(x)=\lt s(x,t)\gt [/math], [math] K^{(x)}[/math] is the longitudinal dispersion coefficient and [math]A_0=\lt A\gt [/math] (the brackets <> represent the average over a tidal cycle). The dispersion coefficient [math] K^{(x)}[/math] has the important property that it does not depend explicitly on the salinity distribution in the estuary , but only on the flow characteristics during the tidal cycle (which may be influenced by the salinity distribution, by the way). The magnitude of the random displacements depends on the location [math]x [/math] in the estuary; the longitudinal dispersion coefficient [math] K^{(x)}[/math] is thus a function of [math]x [/math]. This is illustrated in Fig. 1 for the Eastern Scheldt and Ems-Dollard estuaries.

Longitudinal dispersion mechanisms

Mixing processes, such as those described by the random walk model, cause seawater to intrude further and further into the estuary. However, seawater is also expelled from the estuary due to the inflow of river water at the estuarine head. Seawater intrusion is stabilised when the net upstream salt transport by mixing processes equals the net downstream salt transport by the fluvial discharge. This can be represented by the following formulas.

The fresh water discharge [math]Q_R[/math] is given by

[math]Q_R=- \langle A\overline{\overline{u}} \rangle \equiv - \large \frac{1}{T}\int_0^T \int_{-b/2}^{b/2} \int_{-D}^0 \normalsize u(x,y,z,t) dzdy dt \qquad(5) [/math]

and the net total salt flux (by mixing processes and river discharge) is given by

[math]Q_S = \langle A\overline{\overline{us}} \rangle \equiv \large \frac{1}{T} \int_0^T \int_{-b/2}^{b/2} \int_{-D}^0 \normalsize u(x,y,z,t)s(x,y,z,t) dzdydt , \qquad(6) [/math]

where [math]D(x,y,t)=h(x,y)+\eta(x,y,t)[/math] is the instantaneous local water depth and [math]b(x,t)[/math] the estuarine width, [math]A(x,t)[/math] is the estuarine cross-section (approximated by [math]\int_{-b/2}^{b/2} \int_{-D}^0 dy dz[/math]), [math]u[/math] the longitudinal current velocity and [math]s[/math] the salinity. The brackets stand for averaging over the tidal period [math]T[/math] (assuming a cyclic tide) and the overbars stand for averaging over the depth and the width. The coordinate [math]x[/math] follows the upstream positive longitudinal direction (along the thalweg), the coordinate [math]y[/math] the lateral direction and the coordinate [math]z[/math] the upward positive vertical direction.

We call [math] s_0 \equiv \langle \overline{\overline{s}} \rangle [/math] the salinity averaged over the estuarine cross-section [math]A[/math] and the tidal period. We may then decompose

[math]Q_S=Q_{disp} - Q_R s_0, \quad Q_{disp} = \langle A\overline{\overline{u(s-s_0)}} \rangle \qquad(7) [/math] .

This decomposition singles out the fresh water discharge (the term [math] Q_R s_0[/math] ) as a mechanism for flushing seawater out of the estuary, while the term [math] Q_{disp}[/math] represents the sum of all processes contributing to seawater intrusion. These processes are:

• Horizontal circulations in the estuary (mainly the net relative displacement of water masses circulating between ebb and flood channels and the net relative displacements due to geometry-induced eddies, followed by lateral mixing of these water masses);
• Horizontal tidal straining (lateral mixing between water masses which are advected at different speeds, due to lateral gradients in the longitudinal velocity);
• Vertical circulation in the estuary, also called estuarine circulation (seawater intrusion induced by the density-driven net displacement of near-surface water relative to near-bottom water, followed by mixing over the vertical);
• Vertical tidal straining (vertical mixing between water masses advected at different speeds due to vertical gradients in the longitudinal velocity);
• Lateral mixing of water masses captured in "dead zones" with the main flow;
• Chaotic dispersion, related to the chaotic character of particle trajectories when travelling through a complex field of tide-generated eddies;
• Tidal pumping at the inlet, which is defined here as the partial replacement of the ebb tidal prism with ‘new’ seawater flowing in from the nearshore zone during flood.

These mechanisms are not independent; for example, density stratification modifies tidal shear, and tidal mixing controls the strength of density-driven exchange flow. In a fixed Eulerian cross-section, the dispersive salt flux can be separated into a spatial or shear contribution, arising from correlations between cross-sectionally varying velocity and salinity, and a tidal oscillatory contribution, arising from correlations between the cross-sectionally averaged tidal velocity and salinity. In a reference plane moving with the tide, the latter can be interpreted as nonlocal shear dispersion generated at other locations within a tidal excursion.

The different dispersion processes will be discussed in the following.

Analytical expressions for the longitudinal dispersion coefficient

Under strongly simplifying conditions it is possible to derive analytical expressions for the longitudinal dispersion coefficient. These assumptions are:

• the estuarine geometry does not vary strongly in [math]x[/math]-direction over distances comparable to or larger than the tidal excursion;
• the cross-section of the estuarine main channel can be represented approximately by a rectangular shape.

We also have the condition [math]\tau\lt \lt \tau_x[/math], cross-sectional mixing takes place on a time scale much shorter than the estuarine flushing time. In the following we consider different seawater intrusion processes under these conditions and present an approximate analytical expression for the dispersive transport produced by each process.

Dispersion by residual circulation

First we consider seawater intrusion caused by estuarine circulation: the up-estuary near-bottom flow caused by the higher density of seawater relative to estuarine water. The estuarine circulation is represented by the velocity component

[math]u_0 (z)= \langle u \rangle - \overline{ \langle u \rangle }^z, \qquad(8) [/math]

where the brackets [math]\langle … \rangle[/math] stand for averaging over the tidal period and [math]\overline{u}^z [/math] for averaging over the vertical. Here we consider idealized longitudinally uniform estuaries, but in practice it can be advantageous to remain in the same salinity range by performing the averaging in a frame moving with the cross-sectional mean velocity. The longitudinal dispersive transport can be estimated by a procedure outlined by G.I. Taylor . The result is

[math] K^{(x)}= f_0^{(z)} \tau_z \overline{(u_0)^2}^z , \qquad(9) [/math]

with [math] f_0^{(z)} [/math] a coefficient of the order of 0.1 . The formula (9) shows that the dispersion coefficient increases with increasing mixing time [math]\tau_z[/math]. However, there is a limit, because of the assumption [math]\tau_z \lt \lt \tau_x[/math].

For the dispersion coefficient related to lateral horizontal residual circulation a similar formula can be derived, replacing in the expression for [math] K^{(x)}[/math] everywhere [math]z[/math] by [math]y[/math]. Approximate semi-empirical expressions for the vertical mixing time [math]\tau_z[/math] and the lateral mixing time [math]\tau_y[/math] are given in the section #Time scales for vertical and lateral mixing.

Figure 2: Circulation cells in the Western Scheldt related to ebb- and flood-dominated channels.

Estuarine circulation is an important seawater intrusion mechanism in estuaries with a single deep (dredged) channel and small to moderate tide. Lateral circulations are important in wide natural estuaries with a complex geometry (meandering main channel, secondary channels, channel bars and tidal flats) and strong tides. The possible dominance of lateral circulations for seawater intrusion relative to vertical circulations appears in the analytical expression of [math] K^{(x)}[/math] through the much larger lateral mixing time [math]\tau_y [/math] compared to the vertical mixing time [math]\tau_z [/math]. The presence of distinct ebb and flood channels is a major cause of lateral circulation in wide estuaries, see for example Fig. 2. However, density gradients related to seawater intrusion also produce lateral circulations (see Estuarine circulation), which contribute often even more to longitudinal dispersion than the vertical density-induced circulation .

Dispersion by tidal straining

If residual circulations are weak, dispersion is mainly caused by tidal straining, the relative displacement of water masses due to vertical and horizontal gradients in the tidal current velocity. In river flow, the usual term for this mixing mechanism is shear dispersion. Seawater intrusion in narrow deep estuaries is primarily caused by vertical tidal velocity gradients, whereas lateral tidal velocity gradients are important in wide estuaries. We present formulas for vertical tidal straining; the expressions for lateral tidal straining are similar. The process of longitudinal dispersion through tidal straining is explained in Fig. 3.

The velocity component [math]u_1 (z,t) [/math] responsible for vertical tidal straining is

[math]u_1= u - \overline{u}^z , \qquad(10) [/math]

where [math]\overline{u}^z[/math] is the depth-average current velocity. By a procedure outlined by Aris (1956), the following first-order estimate of the longitudinal dispersion coefficient is obtained:

[math] K^{(x)}\approx f_1^{(z)} \Large \frac{\tau_z \langle \overline{ u_1^2}^z \rangle }{1+( f_2^{(z)} \tau_z / T)^2 } \normalsize . \qquad(11) [/math]

[math]T[/math] is the tidal period. The coefficients [math] f_1^{(z)}, f_2^{(z)} [/math] depend on the velocity profile; order-of-magnitude estimates are [math] f_1^{(z)} \sim 0.1[/math] and [math] f_2^{(z)} \sim 0.5-1 [/math].

Longitudinal dispersion produced by lateral tidal straining can be expressed by a similar formula as for vertical tidal straining. Dispersion by tidal straining is largest if the time scale for vertical or lateral mixing is on the order of [math]T/ f_2[/math]. The time scale for vertical mixing is generally smaller than the tidal period and the time scale for lateral mixing is generally larger. Dye experiments illustrating dispersion by lateral tidal straining are shown in Estuarine dispersion: dye experiments in the Eastern Scheldt scale model.

A derivation of the expressions (9) and (11) is given in Appendix A.

It should be realised that longitudinal dispersion is not simply the sum of transport processes related to circulation and straining. Circulation causes not only net relative displacements of water masses in the estuary, but it also influences tidal straining.

Figure 3: Schematic representation of longitudinal dispersion by a vertical gradient in the tidal current [math]u(z,t)[/math]. A patch of dye is introduced homogeneously over the vertical when flood starts at low-water slack tide ([math]t=0[/math]). The patch of dye is stretched by the tidal current while being mixed vertically by turbulent diffusion. The time scale for vertical mixing [math]\tau_z[/math] is assumed of the same order as the tidal period [math]T[/math]. The tidal cycle is represented by four discrete time steps, illustrating the combined action of turbulent vertical mixing and shear of the current in the vertical. The stretching of the dye patch combined with vertical diffusion results in a strong longitudinal spreading after a tidal cycle.

Time scales for vertical and lateral mixing

A difficulty for the practical use of the preceding expressions for longitudinal dispersion arises from the uncertainty associated with estimating the vertical and (especially) lateral mixing times, [math]\tau_z=h^2/ K^{(z)}[/math] and [math]\tau_y=b^2/K^{(y)}[/math]. In the case of a logarithmic velocity profile (in the absence of density stratification), the vertical diffusion coefficient is given by [math] K^{(z)}=-0.4z(1+z/h)u_* \, , \; -h\lt z\lt 0[/math], where [math]u_* \approx 0.05 U[/math] is the friction velocity and [math]U[/math] is the flow velocity. This yields a longitudinal dispersion coefficient [math] K^{(x)}\approx 6u_*h[/math], for steady flow . However, in the case of buoyant flows, vertical diffusion can be much slower (smaller [math] K^{(z)}[/math]), leading to stronger longitudinal dispersion. Lateral diffusion is strongly dependent on the geometry of the estuary. The lateral diffusion coefficient is generally expressed as [math] K^{(y)}\approx \alpha u_* b[/math]. An empirical estimate for moderately meandering channels is [math]\alpha \approx 0.6 [/math] and a model estimate for a meandering channel system is [math] \alpha \approx 150 (b/ R)^2 [/math] , where [math]b[/math] is the channel width and [math]R[/math] is a characteristic channel bend radius.

Dispersion by "dead zones"

Fig. 4. The Krabbenkreek, a large creek that contributes to longitudinal dispersion in the Eastern Scheldt estuary (Netherlands)

Estuaries usually have an irregular shape with areas along the main channel where the water flow has only a weak or even no longitudinal component. Such areas may correspond to tidal creeks. An example is the Krabbenkreek in the Eastern Scheldt estuary, shown in Fig. 4. Longitudinal flow is also weak on large tidal flats along the channel. These areas are sometimes called "dead zones". These dead zones should not be confused with the anoxic zones, called "dead zone" in ecology. Water that has flooded dead zones along the main estuarine channel often empties during falling tide with a different salinity than the water passing at that time in the channel. This can be caused by a phase difference [math]\phi[/math] between the flow on and off the dead zone relative to the phase of the channel flow. It can also be caused by mixing of the water inside the dead zone. The result after each tidal cycle is in both cases a net longitudinal dispersion of estuarine waters. The processes underlying longitudinal estuarine dispersion can be considered as special cases of tidal straining. Mathematical derivations are given in Appendix B for a few strongly idealized situations. Based on the analytical expressions, estimates can be derived for the contribution of the dead zones to the estuarine longitudinal dispersion. For illustration, we use as an example the following order of magnitudes: [math]A \sim 10^4 m^2[/math] is the channel cross-section area, [math]U \sim 1 m/s[/math] is the maximum tidal current velocity in the channel, [math]\phi \sim 45° , \, \sin \phi \sim 0.7[/math] is the phase delay of the tidal current velocity in the channel relative to lateral exchange flows. Other notations used are: [math]T[/math] is the tidal period, [math]|x|[/math] is the along-channel seaward distance from the creek entrance and [math]L=UT/\pi[/math] is the tidal excursion.

The idealized situations considered in Appendix B are:

Dispersion by a shallow tidal creek
We consider a tidal creek with bed level around low water (LW) or below. The creek tidal volume is [math]V^0_S[/math] with an order of magnitude of 10⁶ m³ in the numerical examples.

First case: The creek filling and emptying rate leads the channel velocity by [math]\phi[/math] radians. This generates a net salt transport, [math]M(x) = - K^{(x)} A T \dfrac{d s_0}{dx}[/math], through a plane situated at distance [math]-x[/math] seaward from the creek. In the idealized model without mixing it is given by (Eq. B6)

[math]M(x) = \, V^0_S \dfrac{ds_0}{dx} \, f(x) \, , \quad f(x,\phi) = x \sin \phi \sqrt{1 – \big(\cos \phi + \dfrac{2 x}{L} \big)^2} \, . \qquad (12)[/math]

This net salt transport is restricted to the range [math] -\frac{1}{2} L (1+ \cos \phi) \le x \le \frac{1}{2} L (1- \cos \phi) \, .[/math].

In the numerical example, the maximum net salt transport occurs at about 9 km seaward from the creek and yields a corresponding dispersion coefficient [math]K^{(x)}_{max}[/math] of about 13 m²/s.

Second case: the water entering the creek is completely and instantaneously mixed before outflow; the phase [math]\phi=0[/math]. Mixing in the tidal creek generates net landward salt transport through a plane situated at distance [math]-x[/math] seaward from the creek. In the idealized model it is given by (Eq. B19)

[math]M(x) = \frac{1}{2} V^0_S \dfrac{ds_0}{dx} \, x \big(1+\dfrac{x}{L}\big) \, . \qquad (13)[/math]

The corresponding maximum longitudinal dispersion [math]K^{(x)}_{max}[/math] is of the order of 4 m²/s and occurs at [math]x=-L/2[/math], about 7 km seaward of the creek.

The influence of a tidal creek on dye dispersion has been simulated in a hydraulic scale model of the Eastern Scheldt estuary, the Netherlands. Snapshots of the released dye at different tidal phases are shown in Estuarine dispersion: dye experiments in the Eastern Scheldt scale model. Experiment 3, Fig. 5, illustrates the influence of a tidal phase difference between creek filling/emptying and the main channel flow. Besides the phase difference, lateral mixing and other dispersion mechanisms due to the intricate estuarine morphology also play a role.

Dispersion by an along-channel tidal flat We consider a tidal flat that borders the tidal channel over its entire length. In the numerical examples, the ratio [math]r[/math] of the tidal flat cross-section to the adjacent channel cross-section at high water is chosen to be [math]r \approx[/math] 0.1.

First case: the tidal flat in- and outflow advances the channel flow by [math]\phi[/math] radians. The dispersion in the estuary is substantially enhanced when mixing on the tidal flat is insignificant. The idealized model gives a dispersion coefficient (Eq. B10)

[math]\quad K^{(x)} = \dfrac{1}{16 \pi} r U^2 T \sin(2\phi) \, .\qquad (14) [/math]

In the numerical example, the contribution to estuarine longitudinal dispersion is of the order of 100 m²/s.

Second case: the tidal flat in- and outflow is in phase with the channel flow, [math]\phi=0[/math]. The water entering the tidal flat during rising tide is completely and instantaneously mixed across the tidal flat before flowing back to the channel. The idealized model predicts a contribution to the estuarine dispersion given by (Eq. B22)

[math] K^{(x)} = \dfrac{r}{12 \pi^2} U^2 T \, . \qquad (15)[/math]

In the numerical example, the contribution to the estuarine longitudinal dispersion is about 40 m²/s. The expressions obtained from the idealized morphological schematizations should be considered order-of-magnitude estimates, not realistic predictions.

Fig. 5. Schematic of flow pattern at a channel constriction.

Numerical results of Garcia and Geyer (2023) show that constrictions, bends and the estuarine mouth can act as localized dispersion hotspots. Dispersion generated at these locations can contribute nonlocally to the salt balance at sections situated elsewhere within a tidal excursion. At a channel constriction, the flood flow initially concentrates in a jet, so the downstream lateral sides of the channel ("dead zones") are only later filled by the flood flow via recirculating currents. However, most of this lateral water is absorbed into the ebb current almost without delay, resulting in a displacement of water parcels stored in the dead zone relative to water parcels following the center of the channel. A similar relative displacement is caused on the seaward side of the channel constriction by the ebb current jet. The longitudinal dispersion caused by a channel constriction strongly depends on the local bathymetry. Therefore, a simple analytical expression for this dispersion mechanism cannot be given. The same holds for flow separation zones that occur in channel bends.

Dye experiments illustrating longitudinal dispersion by dead zones are shown in Estuarine dispersion: dye experiments in the Eastern Scheldt scale model.

Dispersion by deterministic chaos

Figure 6: Stretching of an initially rectangular patch of dye in a few tidal cycles, according to a 2D numerical hydrodynamic model of the central part of the Western Wadden Sea .

Dye experiments show that dispersion in wide estuaries with complex geometry generally proceeds in an irregular way, by advection of fluid parcels through a field of geometry-induced tidal eddies. The result is very different from turbulent diffusion (i.e. diffusion by a cascade of turbulent eddies of progressively decreasing size). Parts of the dye can be trapped within gyres without any significant turbulent diffusion, while other dye patches can be highly stretched in the flow direction. Strong stretching occurs in particular in the interface zones between tidal eddies, see Fig. 6. Zimmerman (1986) described the dispersion process in such systems as the result of Lagrangian chaos produced by 'a tidal whirlpool'. Fluid parcels can be dispersed over the entire length of the estuary before lateral mixing has taken place. In this situation the simple random-walk model based on rapid cross-sectional mixing and independent tidal-cycle displacements is not valid.

However, Zimmerman showed that longitudinal dispersion can still be described as a random process, even if turbulent mixing is completely absent. He called this random process “deterministic chaos” . In his model, fluid parcels are dispersed by moving along chaotic orbits through a lattice of tidal eddies. Most dispersion is generated by eddies with a size [math]\lambda[/math] comparable to the tidal excursion length [math]L[/math] . This suggests that the longitudinal dispersion coefficient for chaotic mixing should scale as

[math] K^{(x)}\propto UL \, , \qquad(16) [/math].

where [math]U[/math] represents the magnitude of the tidal current velocity. The size of the eddies also depends on the basin width [math]b[/math]. A field study in Willapa Bay (US Pacific coast) suggests that chaotic dispersion could be described by a dispersion coefficient [math] K^{(x)}= 0.06 Ub[/math] . If the lateral mixing time [math]\tau_y[/math] is comparable to or larger than the flushing time [math]\tau_x [/math], the representation of the dispersive transport [math] Q_{disp}[/math] by the product of a dispersion coefficient and the local salinity gradient is no longer valid. Nevertheless, deterministic chaotic advection can produce apparently random parcel trajectories and, after sufficiently long times, an approximately diffusive growth of mean-square displacement.

Dispersion by tidal pumping

The term 'tidal pumping' is often used for all the estuarine dispersion processes that are not due to density-driven estuarine circulation. A different convention is used here, in which salt intrusion by tidal pumping is related to the fraction of estuarine outflow water that is replaced by seawater on the next flood. In this definition, tidal pumping refers to the rate of renewal of estuarine water in the outflow region and therefore on tidal inflow-outflow hydrodynamics. In general, the seawater entering the estuary during flood has a higher salinity compared to the average salinity of estuarine water flowing out into the sea during ebb. Tidal exchange through the inlet therefore constitutes a major mechanism of seawater intrusion, importing seawater into the estuary as far as the tidal flood incursion.

Outflow of estuarine water often has a jet-like character, whereas flood water enters the estuary more evenly distributed from different directions, depending on the tidal phase, see Fig. 7. The inflowing flood water therefore contains ‘new’ seawater from the sides of the ebb channel. Replacement of outflowing estuarine water by 'new' seawater is enhanced when river discharge is high, because the ebb jet is concentrated in a surface layer (salinity stratification), whereas seawater inflow is more evenly distributed over the vertical.

Figure 7: Schematic representation of seawater intrusion by tidal pumping.

We call [math]\alpha[/math] the replacement rate of outflowing estuarine water by seawater during a tidal cycle. For small fresh water discharge, the corresponding dispersion coefficient at the estuarine mouth is given by

[math]K^{mouth} = \alpha \Large \frac{L^2 }{2T} , \normalsize \qquad(17) [/math]

where [math]L=\int_{flood} u(t)dt [/math] is the tidal excursion at the estuarine mouth. Savenije derived the following empirical expression for [math]\alpha[/math], based on observations suggesting that the renewal of estuarine water in the mouth area depends on the buoyancy of the ebb jet:

[math]\alpha = \text{min} \Bigg[ 1, \dfrac{2800 \pi h}{l_A} \sqrt{Ri_E} \Bigg] \, . \qquad(18) [/math]

In this expression [math]h[/math] is the depth and [math]l_A[/math] the convergence length of the estuary in the outflow zone (assuming exponential convergence of the cross-section in the outflow zone, [math] A(x) \sim exp(-x/l_A)[/math]);

[math]Ri_E[/math] is the Richardson estuary number [math]Ri_E=g \Delta \rho Q_R T^3 / ( \pi^2 \rho b L^3)[/math];

[math] \Delta \rho / \rho[/math] is the relative density difference seawater-fresh water and [math]b[/math] the estuarine width at the mouth. The value of [math]\alpha[/math] should not exceed 1. This expression is empirical and should mainly be used for order-of-magnitude estimates in estuaries comparable to those on which it was based. Because of its dependency on buoyancy, tidal pumping is a major mechanism of seawater intrusion in estuaries during periods of high river flow.

However, tidal pumping at the estuarine sea inlet depends not only on the buoyancy of the estuarine outflow jet. Outflowing water can be directed away from the estuarine inlet if the alongshore tidal flow is not in phase with the tidal flow through the inlet. This is in some way similar to the earlier described 'dead zone' dispersion mechanism for a tidal creak.

The fraction of estuarine water that re-enters the estuary on the next flood can also be influenced by a net alongshore current in the coastal zone. Strong alongshore currents can carry estuarine outflow water away from the inlet and prevent it from returning on the next flood. Alongshore currents can also transport the water outflowing from adjacent estuaries to the inlet. Wind is generally the main driving agent for alongshore currents through Ekman transport and geostrophic flow. Besides wind forcing, density gradients and remotely forced shelf circulation can be important drivers of alongshore currents.

For estuaries located on narrow shelves, the salinity of the seawater near the inlet can be influenced by upwelling or downwelling currents (see Ekman transport). Upwelling currents bring high salinity water to the estuary entrance, while downwelling currents can prevent the ebb current plume from spreading out to sea and thus increase the fraction of estuary water that returns to the estuary on the next flood.

Residence time scale

The residence time [math]\tau_r(x)[/math] is defined as the ensemble-averaged time required for a water parcel, initially located at a distance [math]x[/math] from the seaward boundary, to leave the estuary. With the terminology adopted here, residence time and transit time are equivalent, although these terms are sometimes defined differently in the literature.

Under statistically stationary conditions, the mean residence time of water parcels released at the location where freshwater enters the estuary, is equal to the freshwater flushing time [math]\tau_f[/math] (see Appendix C):

[math]\tau_f = \dfrac{V_{fresh}}{Q_R} \, , \qquad (19)[/math]

where [math]V_{fresh}[/math] is the freshwater volume contained in the estuary and [math]Q_R[/math] is the freshwater inflow. The averaging includes parcels released at different times and therefore, in a tidal estuary, at different tidal phases.

If longitudinal dispersion can be represented by a one-dimensional random-walk model with a uniform and constant dispersion coefficient [math]K^{(x)}[/math], and if the stationary freshwater inflow is sufficiently small not to affect the dispersion process, it can be shown that the flushing time is equal to the characteristic longitudinal dispersion time scale (see Eq. C2):

[math]\tau_f = \tau_x = \dfrac{l^2}{2 K^{(x)}} \, , \qquad (20)[/math]

where [math]l[/math] is the estuarine length.

In complex estuaries, residence times can differ greatly, even between nearby release locations. A single residence-time scale is useful for first estimates, but ecological exposure times and pollutant retention often require particle-tracking calculations or age and tracer simulations.

Experimental determination of the longitudinal dispersion coefficient

The dispersion coefficient [math] K^{(x)}[/math] can be determined experimentally in situations where the fresh water discharge [math]Q_R[/math] is constant over a period longer than [math]\tau_x[/math]. In that case the tidally averaged salinity distribution [math]\lt s(x,t)\gt [/math] is close to equilibrium, [math] \lt s(x,t)\gt \approx s_{eq}(x) [/math]. The total residual salt flux [math]Q_S[/math] approximately equals zero. We thus have (for seaward positive [math]x-[/math]direction and [math]Q_R \gt 0[/math])

[math]A K^{(x)}\Large \frac{ds_{eq}}{dx} \normalsize = Q_R s_{eq} . \qquad (22) [/math]

Values of the dispersion coefficient can be derived from measurements of the residual discharge [math]Q_R[/math] and the salinity distribution [math]s_0(x)[/math]. Estimates of longitudinal dispersion coefficients determined in this way are shown in Table 1. More detailed analyses show that the dispersion coefficient [math] K^{(x)}[/math] is a function of [math]x[/math] and [math]Q_R[/math]. The dependence of [math] K^{(x)}[/math] on [math]Q_R[/math] has two causes. It is related in the first place to the influence of the salinity distribution on the velocity flow field [math]u(x,y,z,t)[/math]; such an influence is due to salinity-induced density gradients (see Estuarine circulation). In the second place, it is related to the location of the fresh-water-seawater transition zone. If this zone is situated near the estuarine mouth, the dispersion coefficient is strongly influenced by tidal pumping. This explains the high longitudinal dispersion coefficients for Rotterdam Waterway, Seine and Loire in Table 1, which are determined for situations with important river flow [math]Q_R/hb[/math]. The same holds, to a somewhat lesser degree, for the Elbe, Weser, Mekong and Sinnamary.

The influence of salinity-induced density gradients on the longitudinal dispersion coefficient is generally small in estuaries with complex topography and small river inflow ([math]Q_R/hb[/math] several orders of magnitude smaller than the tidal velocity). This is the case for many tidal lagoons, such as the Wadden Sea.

If a non-buoyant dissolved substance is introduced in the estuary, it will be mixed over the cross-section after a time [math]\tau[/math] . From that time, the longitudinal dispersion of the substance is similar to salinity dispersion (same dispersion coefficient). For a permanent discharge of a non-buoyant dissolved substance, the same dispersion coefficient applies in the estuarine zones where the substance is mixed over the estuarine cross-section.

Longitudinal dispersion coefficients derived from observed salinity distributions at constant river discharge. The figures for tidal range, depth and width are representative values for the zone with the strongest salinity gradient. The values are effective coefficients inferred with one-dimensional equilibrium models and are not strictly comparable local material properties. Data from , , , , , , .

estuary	tidal range [math]2a[/math] [m]	depth [math]h[/math] [m]	width [math]b[/math] [km]	discharge [math]Q_R[/math] [m3/s]	dispersion coefficient [math] K^{(x)}[/math] [m2/s]
Bay of Fundy (Canada)	10	20	20	150	300
Bristol Channel (UK)	8	30	20	480	60
Chao Phraya (Thailand)	2.5	7.2	0.6	30	330
Corantijn (Surinam)	4.3	6.5	3	500	230
Delaware (US)	1.5	6.6	7	300	300
Eastern Scheldt (Netherlands)	3	12.5	2	60	200
Elbe (Germany)	3.3	12	2.5	750	700
Ems-Dollard (Netherlands)	3	9	4	100	275
Gambia (The Gambia)	1.2	8.7	4	2	200
Hudson (US)	1.6	11.6	2.2	100	110
Humber (UK)	5.5	12	3	250	300
Incomati (Mozambique)	5.5	2.9	0.6	1	10
Limpopo (Mozambique)	2.6	7	0.2	5	150
Loire (France)	4.5	9	0.9	825	900
Mae Klong (Thailand)	2	5.2	0.2	30	200
Maputo (Mozambique)	6.7	3.6	1.3	10	100
Mekong-Co Chien+Cung Hau (Vietnam)	2.1	7	4	2125	570
Mekong-Tran De+Dinh Anh (Vietnam)	2.8	8	3	2250	530
Mersey (UK)	7.5	20	1	80	400
Potomac (US)	1.4	8.4	9	110	70
Pungue (Mozambique)	5	11.5	1.8	20	140
Rotterdam Waterway (Netherlands)	0.9	15	0.6	1000	1000
St. Lawrence (Canada)	3	74	48	8500	200
Seine (France)	5.5	8	0.8	440	800
Sinnamary (Guiana)	2.3	3.8	0.3	100	560
Solo (Indonesia)	1.1	9.2	0.17	10	240
Tha Chin (Thailand)	2.9	5.3	0.2	10	270
Thames (UK)	4.5	12	3	60	100
Weser (Germany)	3.8	9	2	324	1000
Western Scheldt (Netherlands)	3.8	16	3.5	100	200

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Appendix A: Longitudinal dispersion by a shearing current

In this appendix we first reproduce the analysis of G.I. Taylor (1954) for the derivation of an analytical expression of the longitudinal dispersion coefficient related to vertical (turbulent) diffusion in a mean velocity field with a stationary circulation current. It may apply, for example, to the longitudinal dispersion of salinity produced by a density-driven net upstream flow near the bottom and a net downstream flow near the surface. The analytical model is an extreme simplification of real estuarine velocity fields. Major simplifications relate to the assumptions that the mean velocity field [math]u[/math] and the depth [math]h[/math] are uniform in longitudinal and lateral directions over the full tidal excursion and that vertical diffusion can be characterized by a uniform and constant eddy diffusion coefficient [math]K^{(z)}[/math]. The longitudinal and vertical coordinates [math]x, z[/math] refer to a reference frame moving with the cross-sectionally averaged tidal flow.

The mean velocity field with stationary vertical circulation in the moving reference frame is represented by

[math]u(z) = - U \cos(\large\frac{\pi z}{h}\normalsize) \, , \quad -h \lt z \lt 0 \, , \quad U^2 = 2 \, \overline{u^2} \, , \qquad (A1)[/math]

The salinity concentration is denoted [math]s(x,z,t)[/math]. The salt balance equation in the moving frame (salinity change by longitudinal advection and vertical diffusion) can be written

[math]\Large\frac{\partial s(x,z,t)}{\partial t}\normalsize + u(z) \Large\frac{\partial s(x,z,t)}{\partial x}\normalsize - \Large\frac{\partial }{\partial z}\normalsize \Big( K^{(z)} \Large\frac{\partial s(x,z,t)}{\partial z}\normalsize \Big) =0 \, . \qquad (A2)[/math]

We consider a stationary situation, [math]\Large\frac{\partial s(x,z,t)}{\partial t}\normalsize =0[/math]. The longitudinal salinity gradient [math]\Large\frac{\partial s(x,z,t)}{\partial x}\normalsize = \Large\frac{ds}{dx}\normalsize [/math] can be considered constant and uniform over the depth. The salt transport by longitudinal dispersion is then given by

[math] h K^{(x)} \Large\frac{ds}{dx}\normalsize = - \int_{-h}^0 u(z) s(x,z) dz \, . \qquad (A3)[/math]

The salinity [math]s(x,z)[/math] can be found by integrating Eq. (A2) twice over the depth, considering that [math]K^{(z)} \Large\frac{\partial s}{\partial z}\normalsize =0[/math] at [math]z=0,-h[/math] (no salt transport through surface and bottom) and [math]\int_{-h}^0 u(z) dz =0[/math]. The [math]z[/math]-dependent component of [math]s(x,z)[/math] is

[math]s(x,z) = \Large\frac{1}{K^{(z)}}\frac{ds}{dx}\normalsize \int_{-h}^z dz' \int_{-h}^{z'} u(z'') dz'' \, . \qquad (A4)[/math]

Substitution in Eq. (A3) and using Eq. (A1) gives

[math]K^{(x)} = \Large\frac{U^2}{h K^{(z)}}\normalsize \int_{-h}^0 dz \cos(\large\frac{\pi z}{h}\normalsize) \int_{-h}^{z} dz' \int_{-h}^{z'} \cos(\large\frac{\pi z''}{h}\normalsize) dz'' \, . \qquad (A5)[/math]

The result is

[math]K^{(x)} = \Large\frac{h^2 U^2}{2 \pi^2 K^{(z)}}\normalsize = \Large\frac{\overline{u^2} \, \tau_z}{\pi^2}\normalsize \, , \qquad (A6)[/math]

where the vertical mixing time scale is given by

[math]\tau_z = \Large\frac{h^2}{K^{(z)}}\normalsize \, . \qquad (A7)[/math]

Shear dispersion can also be estimated by considering diffusion as a random walk process. In this approach, the longitudinal and vertical position of a fluid parcel are given by the stochastic variables [math](X(t), Z(t))[/math]. The mean velocity field [math]u(x,z,t)[/math] is averaged over the turbulent eddy time scale and is defined in a reference frame moving with the cross-sectionally averaged tidal flow.

We consider again a mean velocity field with a vertical circulation which is stationary in the moving reference frame, Eq. (A1). We thus may write

[math]X(t) = \int_0^{t} u(Z(s)) ds \, . \qquad (A8)[/math]

We also have [math]E[u(Z(t))]=0[/math], where [math]E[ .. ][/math] is the ensemble average over all possible parcel trajectories. According to the random walk model Eq. (3), the longitudinal dispersion coefficient is related to the net squared parcel displacement due to vertical diffusion:

[math]K^{(x)} = \lim_{\tau \to \infty} \dfrac{E\big[X^2(\tau)\big]}{2\tau} = \int_0^\infty E\big[u(Z(0))u(Z(t))\big] dt = U^2 \int_0^\infty E\big[\cos(\dfrac{\pi Z(0)}{h})\cos(\dfrac{\pi Z(t)}{h})\big] dt \, , \qquad (A9)[/math]

The expression (A9) is the Taylor–Kubo relation for dispersion, in which the effective diffusivity is obtained as the time integral of the Lagrangian velocity autocorrelation with initially uniform tidal phase.

Now let [math]p(z,t|z_0)dz[/math] be the probability that a random fluid parcel initially at [math]z_0[/math] is found after a time [math]t[/math] between [math]z[/math] and [math]z+dz[/math]. The probability density function [math]p(z,t|z_0)[/math] must satisfy the diffusion equation [math]\; K^{(z)}\dfrac{\partial^2 p}{\partial z^2} = \dfrac{\partial p}{\partial t} \;[/math], where [math]K^{(z)}=\dfrac{h^2}{\tau_z}[/math]. Further conditions are: [math]p(z,0|z_0) = \delta(z-z_0) \;[/math], [math]\; \int_{-h}^0 p(z,t|z_0) dz= 1 \;[/math] and reflecting boundaries, [math]\dfrac{\partial p}{\partial z} = 0 \, , \; z=-h,0 \, .[/math] These conditions are satisfied by the function

[math]p(z,t|z_0) = \dfrac{1}{h} + \dfrac{2}{h} \sum_{n=1}^\infty \cos(\dfrac{n \pi z}{h}) \cos(\dfrac{n \pi z_0}{h}) \exp(-\dfrac{n^2 \pi^2 t}{\tau_z} ) \, . \qquad (A10)[/math]

Substitution in Eq.(A9) gives

[math]K^{(x)} = U^2 \int_0^\infty dt \int_{-h}^0 dz_0 \, p_0(z_0) \cos(\dfrac{\pi z_0}{h}) \int_{-h}^0 dz \, p(z,t|z_0) \cos(\dfrac{\pi z}{h}) \, .\qquad (A11) [/math]

Orthogonality of the cosine modes implies

[math]\int_{-h}^0 dz \, p(z,t|z_0) \cos(\dfrac{\pi z}{h}) = \cos(\dfrac{\pi z_0}{h}) \exp(-\dfrac{\pi^2 t}{\tau_z}) \, .\qquad (A12) [/math]

Substitution in Eq. (A11) gives

[math]K^{(x)} = U^2 \int_0^\infty dt \int_{-h}^0 dz_0 \, p_0(z_0) \cos^2(\dfrac{\pi z_0}{h}) \exp(-\dfrac{\pi^2 t}{\tau_z}) \, .\qquad (A13) [/math]

The initial position [math]z_0[/math] of the fluid parcel is random within the interval [math][-h,0][/math], therefore [math]p_0(z_0)=\dfrac{1}{h}[/math].

The final result is

[math]K^{(x)} = \dfrac{U^2 \, \tau_z}{2 \pi^2} = \dfrac{\overline{u^2} \, \tau_z}{\pi^2} \, . \qquad (A14)[/math]

This gives the same scaling as Eq. (A6), with the same numerical coefficient. Comparing with Eq. (9) we find [math]f_0^{(z)}=1/\pi^2[/math].

The same procedure applies for the case of a sheared tidal current,

[math]u(z,t)=-U\cos(\dfrac{\pi z}{h})\cos(\dfrac{2\pi t}{T}) \, , \quad U^2=4 \overline{\langle u^2 \rangle } \, ,\qquad (A15)[/math].

where [math]\langle …\rangle[/math] stands for averaging over the tidal period. Now [math]p(z,t|z_0, t_0)dz[/math] is the probability that a random fluid parcel at [math]z_0[/math] at time [math]t_0[/math] is found after a time [math]t_0 + t[/math] between [math]z[/math] and [math]z+dz[/math]. The longitudinal dispersion is given by

[math]K^{(x)} = \lim_{\tau \to \infty} \dfrac{E\big[X^2(\tau)\big]}{2\tau} = \int_0^\infty E\big[u(Z(0),t_0)u(Z(t), t_0+t)\big] dt = U^2 \int_0^\infty E\big[\cos(\dfrac{\pi Z(0)}{h})\cos(\dfrac{\pi Z(t)}{h})\big] \cos(\dfrac{2 \pi t_0}{T}) \cos(\dfrac{2 \pi (t_0+t)}{T}) dt \, . \qquad (A16)[/math]

The expression of the probability Eq. (A10) and the result (A12) remain valid for the sheared tidal current case.

Eq. (A13) now becomes

[math]K^{(x)} = U^2 \int_0^\infty dt \int_{-h}^0 dz_0 \, p_0(z_0) \cos^2(\dfrac{\pi z_0}{h}) \exp(-\dfrac{\pi^2 t}{\tau_z}) \cos(\dfrac{2 \pi t_0}{T}) \cos(\dfrac{2 \pi (t_0+t)}{T}) \, .\qquad (A17) [/math]

Averaging over the tidal phase [math]t_0[/math] gives [math] \Big\langle \cos(\dfrac{2 \pi t_0}{T}) \cos(\dfrac{2 \pi (t_0+t)}{T}) \Big\rangle = \dfrac{1}{2} \cos(\dfrac{2 \pi t}{T}) [/math]

Substitution gives

[math]K^{(x)} = \dfrac{U^2}{2} \int_0^\infty dt \int_{-h}^0 dz_0 \, p_0(z_0) \cos^2(\dfrac{\pi z_0}{h}) \exp(-\dfrac{\pi^2 t}{\tau_z}) \cos(\dfrac{2 \pi t}{T}) \, .\qquad (A18) [/math]

Substituting [math]p_0(z_0)=\dfrac{1}{h}[/math] and integrating gives the final result

[math]K^{(x)} = \dfrac{U^2 \tau_z}{4 \pi^2} \dfrac{1}{1+ \big(\large\frac{2 \tau_z}{\pi T}\normalsize \big)^2} = \dfrac{\overline{\langle u^2 \rangle} \tau_z}{ \pi^2} \dfrac{1}{1+ \big(\large\frac{2 \tau_z}{\pi T}\normalsize \big)^2} \, . \qquad (A19)[/math]

This is the same expression as Eq. (11) with the values [math]f_1^{(z)}=1/\pi^2\, , \; f_2^{(z)} = 2 /\pi[/math].

If the vertical mixing time is larger than the tidal period, [math]\tau_z \gt \gt \pi T /2 [/math], then [math] K^{(x)} = \, \dfrac{T^2 \overline{\langle u^2 \rangle} }{4 \tau_z} \, . \qquad (A20)[/math]

If the vertical mixing time is much smaller, [math]\tau_z \lt \lt \pi T /2[/math], then [math] K^{(x)}\approx \dfrac{\tau_z \overline{\langle u^2 \rangle} }{\pi^2}\normalsize \, . \qquad (A21)[/math]

Longitudinal dispersion by tidal straining is greatest when the vertical mixing time scale is a factor of about [math]\pi/2[/math] longer than the tidal period. The longitudinal dispersion coefficient then becomes [math] K^{(x)} = \dfrac{ T \overline{\langle u^2 \rangle} }{4 \pi}\normalsize \, . \qquad (A22)[/math]

The physical interpretation is as follows. If the vertical mixing time scale [math]\tau_z[/math] is very short, a solute patch remains almost uniformly distributed over the depth. The vertical shear therefore produces little relative displacement between different parts of the patch, and the shear-dispersion contribution is small. An equivalent statement is that the net displacement of fluid parcels is almost uncorrelated with their instantaneous vertical position in the estuary. This holds for both the cases of residual circulation and tidal shear. If, on the other hand, the mixing time scale is very long, a patch of solute will be stretched over a large longitudinal distance by a residual circulation before it is vertically mixed, resulting in a large longitudinal dispersion. The net longitudinal displacement of fluid parcels will thus be strongly correlated with their instantaneous vertical position in the estuary. However, in the case of a vertically sheared tidal current, there will be little longitudinal dispersion if the time scale of vertical mixing is very long compared with the tidal period. This is because the stretching of a solute patch during flood is largely reversed during ebb before vertical diffusion produces irreversible exchange between the layers (see Figure 3).

For a horizontally sheared current, the analogous statement uses lateral position instead of vertical position.

Appendix B: Dispersion by dead zones

A longitudinally uniform estuary with a prismatic channel (mean cross-sectional area [math]A[/math]) is bordered by a tidal flat of width [math]b_S[/math]. The tidal flat level corresponds to the low tidal level (level at [math]t=0[/math]). The longitudinal velocity on the tidal flat is nil; the tidal flat constitutes a "dead zone". Within the estuary we consider a long straight section where the tide level is represented by [math]\eta(t)= - a \cos \omega t[/math] and the tidal velocity in the channel by [math]u=U \sin (\omega t -\phi)=\frac{1}{2} \omega L \sin (\omega t -\phi)[/math]. Here [math]L=2U/ \omega=UT/\pi[/math] is the tidal excursion. Channel flow is assumed homogeneous over the cross-section and the influence of the dead zone filling and emptying on the channel flow is neglected.

The tidal amplitude is [math]a[/math] and the tidal radial frequency is [math]\omega = 2 \pi /T[/math]. The phase [math]\phi[/math] is the delay of maximum flood velocity relative to the time of maximum tidal-level rise. Thus [math]\phi=0[/math] corresponds to exact quadrature between tidal elevation and velocity. The path [math]X(t)[/math] followed by fluid particles in the channel is given by

[math]X(t)=X_0 + \frac{1}{2} L [\cos \phi - \cos(\omega t - \phi)] \, , \qquad (B1) [/math]

where [math]X_0[/math] is the position at low water [math]t=0[/math] and the landward direction is taken positive.

It is assumed that the salinity is instantaneously mixed over the whole channel cross-section. This cross-sectionally homogeneous salt concentration [math]s(x,t)[/math] is advected through the channel with the tidal velocity without changing the gradient [math]ds/dx=ds_0/dx[/math]. The gradient [math]ds_0/dx[/math] is assumed to be constant throughout the estuary (or the considered uniform estuarine section).

No mixing, non-zero phase shift [math]\phi[/math]

Our first assumption is that salt is transported only by tidal advection. No mixing occurs when the tidal flat is flooded between low water (LW, [math]t=0[/math]) and high water (HW, [math]t=T/2[/math]) and when the tidal flat is emptied in the channel during falling tide ([math]T/2\lt t\lt T[/math]). In this case, water parcels that enter the tidal flat at time [math]t[/math] experience a net seaward displacement

[math]\Delta X(t)= - L \sin \phi \sin \omega t \qquad (B2)[/math]

when flowing back into the channel compared to water parcels that travel all the time in the channel, see Fig. B1. This relative displacement generates a residual longitudinal salt transport and, on the seaward side of the creek, a dispersive transport.

Fig. B1. Dispersion by an intertidal “dead zone” along a tidal channel. The figure illustrates the case of a phase shift [math]\phi[/math] between the dead-zone filling and emptying rate, proportional to [math]d\eta/dt[/math], and the horizontal tidal velocity [math]u(t)[/math] in the channel. The paths of two fluid parcels are shown, both starting from the same location at low water, [math]t=0[/math]. The first parcel, shown by the square, follows the tidal motion in the channel. The second parcel, shown by the circle, enters the dead zone at time [math]T/2-t[/math], is stored there around high water, and returns to the channel at time [math]T/2+t[/math]. Without mixing in the dead zone, this parcel returns to the channel at the same along-channel position [math]X_1(t)= \frac{1}{2}L\left[\cos\phi-\cos(\pi-\omega t-\phi)\right][/math] where it entered the dead zone. At the return time [math]T/2+t[/math], the parcel that remained in the channel has reached [math]X_2(t)= \frac{1}{2}L\left[\cos\phi-\cos(\pi+\omega t-\phi)\right].[/math] The temporarily stored parcel is therefore displaced seaward relative to the channel parcel by [math]\Delta X(t)=X_1(t)-X_2(t)= - L \sin\phi\,\sin\omega t .[/math] This relative displacement produces longitudinal dispersion: during a tidal cycle, parcels that remain in the channel and parcels that are temporarily stored in the dead zone do not return to the same along-channel position.

Dispersion by a creek

Fig. B2. Schematic of a tidal creek. Left: Cross-sectional view. Right: Plan view.

The longitudinal dispersive salt transport is evaluated by considering the net transport of salt during a tidal cycle through a fixed cross-sectional plane located at [math]x[/math] in the estuary, seaward of the creek.

We represent the creek by a narrow storage element whose plan area is [math]b_S \, dx[/math], [math]b_S[/math] is therefore an effective storage width, not necessarily the physical cross-channel width of a rectangular creek. A sketch of the creek is shown in Fig. B2. The location of the creek in the estuary is taken as [math]x=0[/math]. The water volume in the creek is [math]\; V_S= b_S \, dx \, (a + \eta(t)) \;[/math] ; the high-water volume [math]\; V^0_S= 2 \, a\, b_S \, dx \;[/math]. The amount of salt entering the creek in the interval [math][t, t+dt][/math] is given by [math]\; dM_S = \dfrac{d V_S}{dt} s(0,t) \, dt = \frac{1}{2} V^0_S \omega \sin \omega t \, s(0,t) \, dt \;[/math].

We now consider a water parcel in the channel starting from [math]X_0[/math] at low water ([math]t=0[/math]), thus following the trajectory Eq. (B1). The initial position [math]X_0[/math] is assumed landward from the plane [math]x[/math], thus satisfying the condition [math]X_0 -x \gt 0[/math]. This water parcel may enter the creek at [math]x=0[/math] at time [math]t[/math] so that [math]X(t)=0[/math]. This implies [math]X(t)=X_0+\frac{1}{2}L [\cos \phi - \cos(\omega t-\phi)] =0[/math], giving [math]X_0=\frac{1}{2}L[\cos(\omega t - \phi) - \cos \phi ][/math]. When leaving the creek at [math]T-t[/math], it has experienced a relative seaward shift of [math]\; L \sin \phi \sin \omega t \;[/math] with respect to the water parcels that have remained in the channel (see Fig B1). The parcel will thus end its journey at the next low water at [math]X_2=X_0-L \sin \phi \sin \omega t = \frac{1}{2}L[\cos(\omega t +\phi) - \cos \phi ][/math]. Now consider water parcels that start at low water at a position [math]X_0 \gt x[/math] landward of the plane [math]x[/math] and that enter the creek some time afterwards. Part of these water parcels experience a relative seaward displacement such that [math]X_2 \lt x[/math], thus crossing the plane [math]x[/math] by the ebb flow. These water parcels therefore create a net seaward transport of water and salt across the plane [math]x[/math] during a tidal cycle. The net seaward salt transport through the plane [math]x[/math] is given by the integral

[math]M_{out} = - \int_0^{T/2} dt \dfrac{d V_S}{dt} s(0,t) \, \Theta(X_0-x)\Theta(x-X_2) \, , \qquad (B3)[/math]

where [math]\Theta(x)[/math] is the Heaviside function ([math]\Theta(x)=1, \, x\gt 0; \; \Theta(x)=0, \, x\lt 0[/math]). The integral (B3) can be rewritten by substituting the expressions of [math]X_0[/math] and [math]X_2[/math], giving

[math]M_{out} = - \frac{1}{2} V^0_S \int_0^{T/2} \omega dt \sin \omega t \, s(0,t) \, \Theta\big(\frac{1}{2} L [\cos(\omega t -\phi)-\cos \phi] – x \big) \, \Theta \big(x+\frac{1}{2} L [\cos \phi -\cos(\omega t +\phi)] \big) \, . \qquad (B4)[/math]

To evaluate the salt transport associated with a zero net transferred water volume, the selected seaward parcel transport is balanced by an equal landward water volume carrying the local salinity [math]s(x,t)[/math]. Subtracting both integrals and replacing [math]s(x,t)-s(0,t)=x \, ds_0/dx[/math], the net landward salt transport through the plane [math]x[/math] is given by

[math]M(x) = \frac{1}{2} V^0_S \dfrac{ds_0}{dx} \, x \int_0^{T/2} \omega dt \sin \omega t \, \Theta\big(\frac{1}{2} L [\cos(\omega t -\phi)-\cos \phi] – x \big) \, \Theta \big(x+\frac{1}{2} L [\cos \phi -\cos(\omega t +\phi)] \big) \, . \qquad (B5)[/math]

Fig. B3. The maximum dispersion by a tidal creek due to a phase difference between filling/emptying and the channel current. The corresponding scaled dispersion function [math]f_{max}/L[/math] is shown as a function of the phase [math]\phi[/math]. The dispersion is maximum for [math]\phi \approx 50°[/math] and occurs at a distance [math]x \sim 0.6 L[/math] seaward from the creek at low water.

Evaluation of the integral gives

[math]M(x) = \, V^0_S \dfrac{ds_0}{dx} \, f(x) \, , \quad f(x,\phi) = x \sin \phi \sqrt{1 – \big(\cos \phi + \dfrac{2 x}{L} \big)^2} \, . \qquad (B6)[/math]

There is no net salt transport at distances from the creek outside the range

[math] -\frac{1}{2} L (1+ \cos \phi) \le x \le \frac{1}{2} L (1- \cos \phi) \, . \qquad (B7)[/math].

For values of [math]x[/math] in the range [math] 0 \le x \le \frac{1}{2} L (1- \cos \phi)[/math] the residual transport is seaward and directed against the Fickian dispersive direction. Water stored in the creek originated landward of the plane and therefore has a higher salinity than the local tidal-mean salinity at the plane. This transport results from coherent tidal advection rather than down-gradient dispersion; it is therefore better described as residual advective transport.

For [math]x[/math] seaward of the creek, a dispersion coefficient [math] K^{(x)} (x)[/math] can be defined by the usual formula [math]M(x) = - K^{(x)} \,A \, T\, \dfrac{ds_0}{dx}[/math]. The dispersion coefficient is given by

[math] K^{(x)} (x) = \dfrac{ V^0_S}{A T} |f(x,\phi)| \, . \qquad (B8) [/math]

The maximum value occurs at [math] x /L= - \frac{1}{8} ( 3 \cos \phi + \sqrt{ \cos^2 \phi + 8})[/math] and is to a good approximation given by [math]\dfrac{f_{max}}{L} \approx 0.25 \sin \phi (1 + 1.55 \cos \phi) [/math], see Fig. B3. This gives

[math]\quad K^{(x)}_{max} \approx \dfrac{1}{4 \pi} \sin \phi (1 + 1.55 \cos \phi) \, \dfrac{V^0_S U}{A} \, . \qquad (B9)[/math]

Because distances refer to the low-water salinity distribution, the location of the maximum dispersion is shifted landward when considering a tidally averaged salinity distribution. The maximum dispersion then occurs closer to the creek.

Dispersion by an along-channel tidal flat
As a second case we consider a "dead zone" consisting of a tidal flat running along the channel with constant width [math]b_S[/math] and bed level at tidal low water. This tidal flat contributes to net tidal salt transport in a similar way as the tidal creek case. Instead of considering a narrow dead zone with length [math]dx[/math], we have to integrate the time integral (B6) over [math]x[/math], using [math]\; V^0_S= 2 \, a\, b_S \, dx [/math]. The integral gives

[math]\quad K^{(x)} = \dfrac{\pi}{8} \dfrac{a b_S}{A} \dfrac{L^2}{T} \sin(2\phi) = \dfrac{1}{16 \pi} r U^2 T \sin(2\phi) \, .\qquad (B10) [/math]

where [math]r=\dfrac{2 a b_S}{A}[/math] is the relative tidal flat volume. The integral represents the net result of both the down-gradient contribution from dead zones landward of the plane and the opposite residual-advective contribution from dead zones seaward of it. The same result was found by MacVean and Stacey (2011) using the Aris momentum method.

Complete mixing, no phase shift

Dispersion by a creek
Our second assumption is that complete mixing takes place in the creek, which is again situated at [math]x=0[/math]. We evaluate the longitudinal dispersive salt transport by considering the net transport of salt during a tidal cycle through a fixed cross-sectional plane located seaward of the creek at [math]x[/math] in the estuary ([math]-L \le x \le 0[/math]).

Consider a fluid parcel that starting at low water from a position [math]X_0[/math], is carried landward by the flood flow. Its trajectory in the channel is given by [math]\; X(t) = X_0 +\frac{1}{2}L(1-\cos \omega t) \; [/math], where [math]L=UT/\pi[/math] is the tidal excursion. Water parcels travelling along these trajectories can enter the creek located at [math]x=0[/math] at time [math]t[/math] if [math]X(t)=0=X_0+\frac{1}{2}L(1-\cos \omega t)[/math]. Their start position at [math]t=0[/math] is thus given by [math]X_0=\frac{1}{2}L(\cos \omega t -1)[/math].

Now consider the amount of salt that enters the creek during flood without crossing the plane [math]x[/math]. This salt content is given by

[math]M_{in} = \int_{t_0}^{T/2} dt \dfrac{dV_S}{dt} s(0,t) \, , \qquad (B11)[/math]

where [math]t_0[/math] is the time where [math]X_0=x[/math],

[math]\omega t_0 = \arccos(\dfrac{2 x}{L} + 1) \, . \qquad (B12)[/math]

During ebb, [math]X_0=x[/math] occurs at time [math]T-t_0[/math].

After HW, the creek empties into the channel with the average concentration [math]\overline{s}(0)[/math] of the stored salt content. This average concentration is given by

[math]\overline{s}(0) = \dfrac{1}{V^0_S} \int_0^{T/2} dt \dfrac{dV_S}{dt} s(0,t) = \frac{1}{2} \int_0^{T/2} dt \, \omega \, \sin \omega t \, s(0,t) \, . \qquad (B13)[/math]

Water parcels released by the creek cross the plane [math]x[/math] during ebb for [math]t \gt T-t_0[/math]. The associated salt content is

[math]M_{out} = \int_{T/2}^{T-t_0} dt \dfrac{dV_S}{dt} \overline{s}(0) = \frac{1}{2} \int_{T/2}^{T-t_0} dt \dfrac{dV_S}{dt} \int_0^{T/2} dt' \, \omega \, \sin \omega t' \, s(0,t') \, . \qquad (B14)[/math]

Now we use the property that the salt concentration is conserved along a channel-water trajectory,

[math]\dfrac{ds}{dt}(X(t),t) = \dfrac{\partial s}{\partial t} + \dfrac{dX}{dt}\dfrac{ds_0}{dx} =0 \, .[/math]

Integrating this equation gives, for any [math]x[/math],

[math]s(x,t_1)-s(x,t_2)= - [X(t_1)-X(t_2)]\dfrac{ds_0}{dx}=\frac{1}{2} L (\cos \omega t_1 - \cos \omega t_2) \dfrac{ds_0}{dx} \, . \qquad (B15)[/math]

Therefore,

[math]M_{out} = \frac{1}{2} \int_{T/2}^{T-t_0} dt \dfrac{dV_S}{dt} \int_0^{T/2} dt' \, \omega \, \sin \omega t' \, \Big[ s(0,t) +\frac{1}{2} L (\cos \omega t' - \cos \omega t)\dfrac{ds_0}{dx} \Big] \, . \qquad (B16)[/math]

Evaluating the last integral gives

[math]M_{out} = \int_{T/2}^{T-t_0} dt \dfrac{dV_S}{dt} \Big[ s(0,t) -\frac{1}{2}L \cos \omega t \dfrac{ds_0}{dx} \Big] \, . \qquad (B17) [/math]

The net salt transport through the plane [math]x[/math] is thus given by

[math]M(x) = M_{in}+M_{out} = \int_{t_0}^{T-t_0} dt \dfrac{dV_S}{dt} s(0,t) - \dfrac{1}{4} L V^0_S \dfrac{ds_0}{dx} \int_{T/2}^{T-t_0} dt \, \omega \, \sin \omega t \cos \omega t \, . \qquad (B18) [/math]

The first integral is zero because [math]\dfrac{dV_S}{dt}[/math] is antisymmetric about [math]T/2[/math], whereas [math]s(0,t)[/math] is symmetric. The second integral can be evaluated by changing variables [math]y=\cos \omega t[/math] and using the definition (B12). The result is

[math]M(x) = \dfrac{1}{4} L V^0_S \dfrac{ds_0}{dx} \int_{-1}^{\frac{2x}{L}+1} y \, dy = \frac{1}{2} V^0_S \dfrac{ds_0}{dx} \, x \big(1+\dfrac{x}{L}\big) \, . \qquad (B19)[/math]

Because [math]x\lt 0[/math], the longitudinal dispersion coefficient is given by

[math] K^{(x)} = \frac{1}{2} \dfrac{V^0_S}{A} \, \dfrac{L}{T} \, \dfrac{|x|}{L} \big(1- \dfrac{|x|}{L} \big) \, . \qquad (B20)[/math]

Its maximum occurs at [math]x=-L/2[/math] and equals

[math]K^{(x)}_{max}= \dfrac{1}{8} \dfrac{V^0_S}{A} \dfrac{L}{T} = \dfrac{1}{8 \pi} \dfrac{V^0_S U}{A} \, . \qquad (B21)[/math]

Dispersion by an along-channel tidal flat
The dispersion by complete mixing on a tidal flat that extends uniformly along the tidal channel is directly found by integrating (B19). Therefore we write [math]\; V^0_S = 2 \, a \, b_S \, dx . \;[/math] and contributions are summed from all infinitesimal flat elements situated up to one tidal excursion landward of the selected transport plane. The dispersive transport in the estuary produced by mixing on the tidal flat is

[math] K^{(x)} = \frac{1}{6} \dfrac{a b_S}{A} \dfrac{L^2}{T} = \dfrac{r}{12 \pi^2} U^2 T . \qquad (B22)[/math]

Other analytical estimates of estuarine longitudinal dispersion by dead zones have been published by Okubo (1967), Dronkers (1978) and MacVean and Stacey (2011). These estimates, which are similar to the ones derived above, are based on different approaches and different approximations.

Appendix C: Flushing time scale

Using Eq. (22), the saltwater-equivalent volume and freshwater volume of the estuary are given by

[math]V_{fresh} = V - V_{salt} \, , \quad V = \int_0^l A\, dx \, , \quad V_{salt} = \int_0^l A \exp big(\dfrac{Q_R}{ K^{(x)} A} (x-l) \big) dx \, . \qquad (C1) [/math]

We assume [math]A[/math] and [math] K^{(x)}[/math] uniform along the estuary,

[math]V_{fresh}= A\, l - \dfrac{K^{(x)} A^2}{Q_R} \Big[ 1 – \exp \big(- \dfrac{Q_R \, l}{K^{(x)} A} \big) \Big] \, . [/math]

For small fresh water inflow [math]\; \dfrac{Q_R \, l}{K^{(x)} A} \rightarrow 0[/math],

[math] V_{fresh} \approx Q_R \, \dfrac{l^2}{2 \, K^{(x)}} \, . \qquad (C2) [/math]

This yields Eq. (21)

References

The main author of this article is Job Dronkers Please note that others may also have edited the contents of this article. Citation: Job Dronkers (2026): Seawater intrusion and mixing in estuaries. Available from http://www.coastalwiki.org/wiki/Seawater_intrusion_and_mixing_in_estuaries [accessed on 28-07-2026] For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Dronkers J