According to

Estuaries are water bodies in which rivers with fresh water meet the open sea. The longitudinal salinity difference causes a water level gradient along the estuary. As a result, the water level at the limit of salt intrusion is set up several centimeters above sea level (about 0.012 times the estuary depth). Therefore, the hydrostatic forces from the seaside and riverside have different lines of action (a third of the setup apart). Since the hydrostatic forces at the seaside and the salinity limit are equal but opposed, this difference in the lines of action triggers an angular moment (a torque) that drives the gravitational circulation, whereby fresh water near the surface flows to the sea and saline water near the bottom moves upstream

In contrast to the earlier work by

Traditionally, the empirical Van der Burgh (VDB) method has worked very well to describe the mixing in alluvial estuaries, leading to predictive equations to describe the salinity intrusion in alluvial estuaries

In an estuary, the cross-sectional average hydrostatic forces have equal values along the estuary axis. Over a segment, the forces are opposed but working on different lines of action due to the density gradient in the upstream and downstream directions. As a result, they exert an angular moment (torque)

The friction force during the dispersive circulation is expressed as

Because the velocity of the dispersive gravitational circulation is small, the mixing flow is assumed to be laminar. The shear stress is typically a function of flow velocity (

Power is defined by the product of a force and its velocity. The power of torque (angular moment) is defined as the product of the moment and its angular velocity. Hence, the power is defined as

Sketch of the sensitivity of the exchange flow velocity

Here, the accelerating force (

The accelerating moment has an arm

Subsequently, the accelerating moment due to the density gradient driving gravitational circulation over a tidal cycle can be described as

In steady state, the one-dimensional advection–dispersion equation averaged over the cross section and over a tidal cycle reads

Accordingly, with

Assuming that the steady state over a tidal cycle is driven mainly by the accelerating moment, especially in the upstream part where tidal mixing is relatively small and this gravitational circulation (

This equation indicates that the dimensionless dispersion coefficient is proportional to the root of the estuarine Richardson number

Equations derived from the maximum power concept are obtained along the estuary axis, and hence they can be used at any segment along the estuary. Then, Eq. (

The following equations are used for the tidal excursion and width in alluvial estuaries:

At the boundary, Eq. (

Substitution of Eqs. (

Differentiating

The cross-sectional area

Substituting Eq. (

At the salinity intrusion limit (

The solution for the longitudinal salinity distribution yields

This solution is comparable to other research. It is similar to

With these new analytical equations, the dispersion and salinity distribution can be obtained using the boundary conditions (

The boundary condition is often taken at the geometric inflection point (

Semilogarithmic presentation of estuary geometry, comparing simulations (lines) to the observations (symbols), including cross-sectional area (blue diamonds), width (red dots), and depth (green triangles). Vertical lines display the inflection point.

Subsequently, Eq. (

Summary of application results using two methods.

It can be seen that the simulated curves by the new MP method do not perform well in the wider part of the estuary (particularly upstream from the inflection point) where tidal mixing is dominant. However, the salinity observations can be very well simulated landward from the inflection point in most estuaries. In the Bernam, the Pangani, the Rembau Linggi, and the Incomati estuaries, the central part, where

However, in the Thames (no. 8), the Delaware (no. 20), the Scheldt (no. 21), and the Pungwe (no. 22), the new approach seems not to work for both the salinity and dispersion profiles. In these estuaries tide-driven mixing is dominant. Figure

Comparison of the geometry to the Van der Burgh coefficient. Numbers show the labels of the estuaries.

Overall, the maximum power approach in open systems is a useful tool to understand the mixing processes in most estuaries. In the upstream part where the effect of the tide is small, gravitational circulation plays the main role. There, the MP approach yields good results. At the same time, the predictions upstream are more relevant for water users. Where the salinity is high, it is less relevant since the water is already too saline for domestic or agricultural use.

This study provides an approach to define the dispersion coefficient due to gravitational circulation, which is proportional to the product of the dispersive velocity of the gravitational circulation and the tidal excursion length (which is the longitudinal mixing length over which one particle travels during a tidal cycle). The dispersive velocity actually represents the strength of the density-driven mechanism. Based on the maximum power method (Eq.

Hence, the dispersive flow due to gravitational circulation strengthens with larger freshwater discharge

Using the calibrated dispersion coefficient at the inflection point,

Comparison of calibrated and predicted

Finally, there is uncertainty about the timescale of reaching this optimum. If this timescale is longer than the tidal period, then such an optimum is not reached. In a low-flow situation, however, which is the critical circumstance for salt intrusion, the variation of the river discharge is slow (following an exponential recession). If the timescale of flow recession is large compared to the timescale of salinity intrusion then it is reasonable to assume that the maximum power optimum is approached.

The fact that the MP method works well for density-driven mixing but not for tide-driven mixing, whereas the VDB method works well for the combination of the two, offers an excellent opportunity for the combination of the two methods. The VDB method requires two parameters: the

This combined approach also allowed for more accurate predictive equations as derived before. The correlation between

An estuary is an open system that has a maximum power limit when the accelerating source is stable. This study has described a moment balance approach to nonthermal systems, yielding a new Eq. (

As can be seen in Appendix

This study is a further development of the paper by

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

Notations for symbols used in this study.

This Appendix represents the application in 23 estuaries around the world of the maximum power method for determining the dispersion coefficient and the salinity distribution using Eqs. (

Left: Application of the analytical solution from the maximum power method (solid lines) to observations (symbols) for high water slack (HWS, in red) and low water slack (LWS, in blue). The green line shows the tidal average (TA) condition. Dash dot lines reflect applications of the Van der Burgh method. Vertical dash lines display the inflection point. Right: Simulated dispersion coefficient using different methods.

HS conceptualized and supervised the study. ZZ executed the research and prepared the article.

The authors declare that they have no conflict of interest.

This article is part of the special issue“Thermodynamics and optimality in the Earth system and its subsystems (ESD/HESS inter-journal SI)”. It is not associated with a conference.

The first author is financially supported for her PhD research by the China Scholarship Council.

This paper was edited by Stefan Hergarten and reviewed by Axel Kleidon and one anonymous referee.