The practical value of the surprisingly simple Van der Burgh equation in
predicting saline water intrusion in alluvial estuaries is well documented,
but the physical foundation of the equation is still weak. In this paper we
provide a connection between the empirical equation and the theoretical
literature, leading to a theoretical range of Van der Burgh's coefficient of

Estuaries play an essential role in the human–earth system, affecting fresh water resources, the mixing between ocean and river water, and the health of aquatic ecosystems. This makes the functioning of estuarine systems an important field of research. A crucial element of estuarine dynamics is the interaction between saline and fresh water. The river discharges fresh water into estuaries, flushing out the salt, while saline water penetrates landward as a result of density gradients. The temporal and spatial distribution of salinity in an estuary is determined by the competition between fresh water flushing and penetration of saline water by gravity.

Dispersion is the mathematical reflection of the spreading of a substance
(e.g., salinity

The dispersion resulting from density gradients is closely connected to the
stratification number

Traditionally, researchers focused on vertical/longitudinal dispersion in
prismatic estuaries

Although the processes of mixing and saline water intrusion are clearly complex and three-dimensional, it is remarkable that a very simple, empirical and one-dimensional approach, such as Van der Burgh's relationship, has yielded such surprisingly good results. This paper tries to bridge the gap between the theoretical approaches developed in the literature and the empirical results obtained with Van der Burgh's relationship, considering the complex interaction between tide, geometry, salinity and fresh water that govern dispersion in alluvial estuaries. In addition, we present a one-dimensional general dispersion equation for convergent estuaries that includes lateral exchange through preferential ebb and flood channels, using local tidal and geometrical parameters. This equation was validated on a broad database of salinity distributions in alluvial estuaries.

The one-dimensional mass-conservation equation averaged over the cross
section and over a tidal cycle can be written as

At steady state, where

Combining Eqs. (

Interestingly, using Eqs. (

After division of all terms by the salinity gradient, it becomes an equation
for the dispersion coefficient

Based on Eq. (

Hence

Comparison between the factors in the Taylor series expansion of

Considering only the density-dependent terms in Eqs. (

According to Eq. (10),

Overall, there are three results for the estimation of Van der Burgh's
coefficient: (1) by comparison with traditional studies (

In the theory about mixing in estuaries, several authors have distinguished
between tide-driven and density-driven dispersion

Figure

Conceptual sketch for lateral and longitudinal mixing.
Longitudinal and lateral mixing lengths are

The balance equation then becomes

The assumption used is that the lateral exchange is proportional to the
longitudinal

As a result, longitudinal and lateral processes can be combined into one
single one-dimensional equation:

Comparing Eq. (

Subsequently, the longitudinal exchange flow

The reason why the exchange flow is a function of the stratification number
to the power of

We then obtain a simple dimensionless expression for the dispersion
coefficient, simulating to the one by

In almost all estuaries, the ratio of width to excursion length is quite
small, particularly upstream where salinity intrusion happens. So for further
analytical solutions we can focus on the first part of Eq. (

The traditional approach by

The following equations are used for the tidal velocity amplitude, width and
tidal excursion:

At the inflection point, the predicted equation is given by

Substitution of Eqs. (

Differentiating

Combining the result with the time-averaged salt balance, Eq. (

For a prismatic channel (

The cross-sectional area

Substitution of Eq. (

In analogy with

The maximum salinity intrusion length is obtained from Eq. (

This is the same equation as in

Using Eq. (

This solution is similar to the solution by

So with these new analytical equations, the local dispersion and salinity can
be obtained, using the boundary condition at the inflection point. This
method is limited since it only works when

Eighteen estuaries with quite
different characteristics, covering a diversity of sizes, shapes and
locations, have been selected from the database of

Semi-logarithmic presentation of estuary geometry, comparing simulated (lines) to observations (symbols), including cross-sectional area (green), width (red) and depth (blue).

Summary of the geometry of the estuaries.

In Table

Summary of salinity measurement.

Dispersion parameters using

Tables

Comparison between simulated and observed salinity at
high water slack (thin lines) and low water slack (thick lines),
scaled by the salinity

Through the use of

To check the sensitivity to

The poor fit in the downstream parts of the Lalang and Chao Phraya, in which measured salinities are lower than simulated, can be explained by a complex downstream boundary. The Lalang estuary has a pronounced riverine character and is a tributary to the complex estuary system of the Banyuasin, sharing its outfall with the large Musi River. So the salinity near its mouth is largely affected by the Musi. Also, pockets of fresh water can decrease the salinity near the confluence. The Chao Phraya opens to the Gulf of Thailand where the salinity is influenced by historical discharges rather than ocean salinity, remaining relatively fresh. Other measurement uncertainties may cause outliers as well.

The physical meaning of Van der Burgh's coefficient has been analyzed,
linking it to traditional theoretical research. Equation (

A

Comparison between predicted and calibrated

Overall, the single one-dimensional salinity intrusion model including
residual circulation appears to work well in natural estuaries with a
diversity of geometric and tidal characteristics, by both analytical and
numerical computation. The new equation is a simple and useful tool for
analyzing local dispersion and salinity directly on the basis of local
hydraulic variables. In a calibration mode,

The addition of the factor

Van der Burgh's coefficient determines the way dispersion relates to the
stratification number by a power function. Two approaches, theoretical
derivation from the traditional literature and empirical validation based on
observations in a large set of estuaries, provided similar estimates of Van
der Burgh's coefficient. Under MacCready's assumptions, there are three ways
to estimate

A previous analytical salinity intrusion model was developed by

An important consequence of this research is that

In some particular cases, the simulated salinity with

About the data, all observations are available on the
website at

The same as Fig. 3.

The same as Fig. 4.

The authors declare that they have no conflict of interest.

The first author is financially supported for her PhD research by the China Scholarship Council. Edited by: Insa Neuweiler Reviewed by: two anonymous referees