Seawater intrusion and mixing in estuaries

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Estuaries are generally defined as semi-enclosed transition zones between river and sea. The intrusion of seawater in estuaries is mainly due to tides and buoyancy (related to the density difference between seawater and river water, see Estuarine circulation). 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.

We describe in this article the physical processes involved in seawater intrusion and mixing in estuaries 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.

Seawater intrusion mechanisms

The fresh water discharge Q_R and the salt flux Q_S in an estuary are given by

Q_R=-<A\overline{\overline{u}}> \equiv - \frac{1}{T}\int_0^T \int_{-B/2}^{B/2} \int_0^D u(x,y,z,t) dzdy dt,

Q_S=<A\overline{\overline{us}}> \equiv \frac{1}{T} \int_0^T \int_{-B/2}^{B/2} \int_0^D u(x,y,z,t)s(x,y,z,t) dzdydt,

where A(x,t) is the estuarine cross-section, D(x,y,t) the instantaneous local water depth and B(x,t) the estuarine width, u the longitudinal current velocity and s the salinity. The brackets stand for averaging over the tidal period T (assuming a cyclic tide) and the overbars stand for averaging over the depth and the width. The coordinate x follows the upstream positive longitudinal direction (along the thalweg), the coordinate y the lateral direction and the coordinate z the upward positive vertical direction.

We call  s_0 \equiv <\overline{\overline{s}}> the salinity averaged over the estuarine cross-section A and the tidal period. We may then decompose

Q_S=Q_{disp}- Q_R s_0, \quad Q_{disp} = <A\overline{\overline{u(s-s_0)}}> .

This decomposition singles out the fresh water discharge (the term  Q_R s_0 ) as a mechanism for flushing seawater out of the estuary, while the term  Q_{disp} 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 (partial replacement of the ebb tidal prism with ‘new’ seawater flowing in from the nearshore zone during flood).

Random walk

Water parcels move some time forth and back in an estuary before they are evacuated offshore. We call T_x the flushing time of water parcels in the estuary (the average residence time fluid parcels entering the estuary at the upstream boundary). During this time, water parcels also move in lateral and vertical directions, due to flow circulations and turbulent eddies. The time scale for vertical mixing, T_z , and the time scale for lateral mixing, T_y , are related to the vertical and lateral turbulent diffusion coefficients, K_z and K_y , by the relationships T_z=D^2/K_z and T_y=B^2/K_y , respectively. If the vertical and lateral mixing time scales are both much smaller than the flushing time T_x of water parcels in the estuary, the longitudinal path of a water parcel follows a random walk. The longitudinal displacements X_n in successive time intervals T_n are uncorrelated, by choosing T_n=T equal to an integer number of cyclic tidal periods such that T_n > T_A. The cross-sectional mixing time T_A = T_y, assuming that lateral mixing takes more time than vertical mixing, as is the case for most estuaries. In estuaries satisfying the condition  T_x > T_A, salt transport by seawater intrusion processes, Q_{disp}, can be represented by a gradient-type transport formula,

Q_{disp} = <A\overline{\overline{u(s-s_0)}}> = - A_0 K_x \frac{ds_0}{dx},

where K_x is the longitudinal dispersion coefficient and A_0=<A>. The dispersion coefficient K_x has the important property that it does not depend explicitly on the salinity distribution in the estuary [1], but only on the flow characteristics during the tidal cycle (which may be influenced by the salinity distribution, by the way). According to random walk theory [2],

K_x =\frac{\overline{X^2}}{2 T_n},

where \overline{X^2} is the average of the squared successive random displacements,

\overline{X^2}= \frac{1}{N} \sum_{n=1}^N X_n^2 , \quad  N>>1 .

The magnitude of the random displacements depends on the location x in the estuary; the longitudinal dispersion coefficient K_x is thus a function of x . This is illustrated in figure 1 for the Eastern Scheldt and Ems-Dollard estuaries.

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

Analytical expressions for the longitudinal dispersion coefficient

Under certain 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 x-direction over distances comparable to or larger than the tidal excursion;
  • the cross-section of the estuarine main channel has approximately a rectangular shape.

We also have the condition T_A<<T_x. 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

Fist 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

u_0 (z)= <u>-<\overline{u}^z>,

where the brackets <u> stand for averaging over the tidal period (in fact, the averaging is done in a frame moving with the cross-sectional mean velocity), and \overline{u}^z for averaging over the vertical. The longitudinal dispersive transport can be estimated by a procedure outlined by G.I.Taylor [3]. The result is

K_x =  f_0^{(z)} T_z \overline{(u_0)^2}^z ,

with  f_0^{(z)} \approx 0.1 [4].

For the dispersion coefficient related to lateral horizontal residual circulation a similar formula can be derived, replacing in the expression for K_x everywhere z by y.

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 dominance of lateral circulations for seawater intrusion relative to vertical circulations appears in the analytical expression of K_x through the much larger lateral mixing time T_y compared to the vertical mixing time T_z . The presence of distinct ebb and flood channels is a major cause of lateral circulation in wide estuaries, see for example figure 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 [5].

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

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 is shear dispersion). Seawater intrusion is primarily caused by vertical tidal velocity gradients in narrow deep estuaries, 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 figure 3.

The velocity component u_1 (z,t) responsible for vertical tidal straining is

u_1= u-\overline{u}^z ,

where \overline{u}^z is the depth-average current velocity. By a procedure outlined by Holley, Harleman and Fischer [6], the following first-order estimate of the longitudinal dispersion coefficient is obtained:

K_x  \approx  f_1^{(z)} \frac{T_z <\overline{ u_1^2}^z> }{1+( f_2^{(z)} T_z / T)^2 } ,

The coefficients  f_1^{(z)}, f_2^{(z)} depend on the velocity profile; order-of-magnitude estimates are  f_1^{(z)} \sim 0.1-0.2 and  f_2^{(z)} \sim  0.5-1 .

Longitudinal dispersion produced by lateral tidal straining can be expressed by a formula similar 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 T/ f_2. 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.

It should be realised that longitudinal dispersion is not simply the sum of transport processes related to circulation and straining. Circulation causes not only a 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 u(z,t). A patch of dye is introduced homogeneously over the vertical when flood starts at low water slack tide (t=0). The patch of dye is stretched by the tidal current while being mixed vertically by turbulent diffusion. The time scale for vertical mixing T_z is assumed equal to the tidal period T. The tidal cycle is represented by four discrete time steps, illustrating the longitudinal spread of the patch of dye.

Dispersion by “dead zones”

The formula for lateral tidal straining includes the influence of "dead zones", if they are considered part of the estuarine cross-section and if they are distributed regularly along the estuary. Dead zones are areas along the main estuarine channel where water is not transported in longitudinal direction, for instance, tidal flats or lateral creeks. The longitudinal dispersion coefficient is given by an expression of the type [7]


where U is the maximum tidal velociy and r is the HW storage cross-section of the dead zones relative to the channel cross-section. The coefficient f_{dz} depends on the mixing rate within the dead zone; in case of complete mixing during the tidal cycle f_{dz}=1/12 \pi^2, assuming that filling of the dead zones starts at low water (LW) [8].

Even without any mixing, storage areas along the channel contribute to longitudinal dispersion, because of a non-zero phase shift \phi that generally exists between horizontal and vertical tidal motion (between u(t) and d\eta /dt, respectively, where \eta(t) is the tidal level). The process is illustrated in figure 4. We assume dead zones with bed level at low water (or below), which are distributed evenly along the estuarine main channel. Filling and emptying of the dead zones during the tidal period then produces a net transport through a plane at x=0 given by

Q_{disp} = -\frac{1}{T} \int_0^{T/2} \frac{d A_s}{dt} dt \int_{\xi(t)}^{\xi(T-t)} [s(x,t)-s(0,0)] dx ,

where A_s(t) = b_s \eta (t) is the dead zone volume at time t per unit estuarine length,  s(x,t)-s(0,0)\approx  (x-\xi(t)) ds_0/dx and \xi(t) is the net distance travelled by fluid parcels from the time of low water (LW). Other symbols are shown in figure 4. The integral evaluates mass transport due to water parcels passing through the plane x=0 during ebb but not during flood (because of a net seaward displacement related to storage in the dead zone, represented by the first term in square brackets), taking into account an equivalent landward water flow in the channel (second term in square brackets). By evaluating the integral we find for the coefficient f_{dz} the expression f_{dz} = \sin^2 \phi / (3 \pi^2).

A usual order of magnitude for the phase shift T\phi/2 \pi is 30-60 minutes, yielding f_{dz} \approx 0.005 . In estuaries with large tidal flats dead zones can significantly enhance longitudinal dispersion.

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

Figure 4: Dispersion by an intertidal "dead zone" along a tidal channel. The figure illustrates the case of a phase shift \phi between the vertical tidal motion (d\eta /dt) and the horizontal tidal motion (u(t)). The paths of two fluid parcels are shown, both starting from the same location at low water (t=0). The first parcel (the square) follows the tidal motion in the channel, while the second parcel (the circle) is stored in the dead zone around high water in a time interval [t, T-t]. Without mixing, the latter parcel returns to the channel before the former parcel has arrived at the same location. If particle paths in the main channel are given by X(x,t)=x+\xi(t), \; \xi=(L/2)[\cos \phi - \cos (\omega t - \phi)], then the relative seaward displacement of particles leaving the dead zone at time T-t is given by \Delta X(t)=\xi(T-t)-\xi(t) . The result is longitudinal dispersion: a net relative displacement of the two fluid parcels during the tidal cycle.

Time scales for vertical and lateral mixing

A difficulty for practical use of the previous expressions for longitudinal dispersion, results from the uncertainty related to estimating the vertical and (especially) lateral mixing times, T_z=D^2/K_z and T_y=B^2/K_y. In case of a logarithmic velocity profile, the vertical diffusion coefficient is given by K_z=0.4z(1-z/D)u_*, where u_*\approx 0.05 U is the friction velocity and U the flow velocity. This yields a longitudinal dispersion coefficient K_x \approx 6u_*D, for steady flow [9]. However, in case of buoyant flows, vertical diffusion can be much slower (smaller K_z), leading to stronger longitudinal dispersion. Lateral diffusion depends strongly on the geometry of the estuary. The lateral diffusion coefficient is generally expressed as K_y \approx \alpha u_* D. An empirical estimate for moderately meandering channels is \alpha \approx 0.6 [10] and a model estimate is  \alpha \approx 150 (B/ R)^2 [11], where R is a characteristic channel bend radius.

Dispersion by deterministic chaos

Dye experiments show that dispersion in wide estuaries with complex geometry generally proceeds in an irregular way, by advection through a field of geometry-induced tidal eddies. The result is very different from diffusion by a cascade of turbulent eddies of progressively decreasing size. Parts of the dye can be trapped within gyres with almost no 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. Zimmerman described the dispersion process in such systems as the result of Lagrangian chaos produced by the tidal whirlpool [12]. Fluid parcels can be dispersed over the entire length of the estuary before lateral mixing has taken place. In this case, the random walk description of tidal dispersion is no longer valid. 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” [13]. 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 \lambda comparable to the tidal excursion length L [14]. This suggests that the longitudinal dispersion coefficient for chaotic mixing should be proportional to

K_x \propto UL.

The size of the eddies also depends on the basin with B. A field study in Willapa Bay (US Pacific coast) suggests that chaotic dispersion could be described by a dispersion coefficient K_x = 0.06 UB [15]. If the lateral mixing time T_y is comparable to or larger than the flushing time T_x , the representation of the dispersive transport  Q_{disp} by the product of a dispersion coefficient and the local salinity gradient is no longer valid.

Dispersion by tidal pumping

Salt intrusion by tidal pumping is related to the higher average salinity of seawater entering the estuary during flood compared to the average salinity of estuarine water flowing out into the sea during ebb. It depends on the rate of renewal of estuarine water in the outflow region and therefore on tidal inflow-outflow hydrodynamics. Outflow of estuarine water often has a jet-like character, whereas flood water enters the estuary more distributed from different directions, depending on the tidal phase, see figure 5. 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 5: Schematic representation of seawater intrusing by tidal pumping.

We call \alpha 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

K_{mouth} = \alpha \frac{L^2 }{2T} ,

where L=\int_{flood} u(t)dt is the tidal excursion at the estuarine mouth. Savenije [16] derived the following empirical expression for \alpha:

\alpha=\frac{2800 \pi D}{l_A} \sqrt{Ri_E} ,

where D is the depth and l_A the convergence length of the estuary in the outflow zone (assuming exponential convergence of the cross-section in the outflow zone,  A(x) \sim exp(-x/l_A));

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

 \Delta \rho / \rho is the relative density difference seawater-fresh water and B the estuarine width at the mouth. The value of \alpha should not exceed 1. Tidal pumping is a major mechanism of seawater intrusion in estuaries during periods of high river flow.

Residence time scale

The residence time T_r is defined as the average time a water parcel located at a distance x from the sea boundary, will take to leave the estuary. If the fresh water discharge is zero or very small, and if the dispersion coefficient K_x is assumed constant along the estuary, the residence time is given by random walk theory [17]:

T_r = \frac{x^2}{2K_x}.

The flushing time T_f=T_x is the average time it takes for a fresh water parcel to move through the estuarine zone to the sea. According to the random walk model, for small discharge Q_R,

T_f = \frac{l^2}{2K_x} ,

where l is the estuarine length. This is equivalent to T_f = V_f / Q_R , where V_f is the fresh water volume in the estuary.

Experimental determination of the longitudinal dispersion coefficient

The dispersion coefficient K_x can be determined experimentally in situations where the freshwater discharge Q_R is constant over a period longer than T_x. In that case the salinity distribution s(x,t) is close to equilibrium ( s(x,t)\approx s_0(x) ). The total residual salt flux Q_S approximately equals zero. We thus have (with the sign convention for Q_R)

AK_x \frac{ds_0}{dx}+Q_R s_0=0.

Values of the dispersion coefficient can be derived from measurement of the residual discharge Q_R and the salinity distribution s_0(x). Examples of longitudinal dispersion coefficients determined in this way are shown in table 1. It should be noticed that the dispersion coefficient K_x is a function of x and Q_R. The dependence of K_x on Q_R has two causes. It is related in the first place to the influence of the salinity distribution on the velocity flow field u(x,u,z,t); 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 freshwater-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 Q_R/hb. The same holds, to a somewhat lesser degree, for the Elbe, Weser, Mekong and Sinnamary.

In estuaries with a complex geometry and river flow Q_R/hb several orders of magnitude smaller than the tidal velocity, the influence of salinity-induced density gradients on the longitudinal dispersion coefficient is generally small. This is often the case for tidal lagoons with small river inflow.

If a non-buoyant dissolved substances is introduced in the estuary, it will be mixed over the cross-section after a time T_A . 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. Data from [18], [19], [20], [21], [22], [23], [24].
estuary tidal range 2a [m] depth D [m] width B [km] discharge Q_R [m3/s] dispersion coefficient K_x [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

Related articles

Estuarine dispersion: dye experiments in the Eastern Scheldt scale model

Estuarine circulation

Salt wedge estuaries

Physical processes and morphology of synchronous estuaries


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The main author of this article is Job Dronkers
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Citation: Job Dronkers (2016): Seawater intrusion and mixing in estuaries. Available from [accessed on 19-01-2018]