The mixing of saline and fresh water is a process of energy dissipation. The freshwater flow that enters an estuary from the river contains potential energy with respect to the saline ocean water. This potential energy is able to perform work. Looking from the ocean to the river, there is a gradual transition from saline to fresh water and an associated rise in the water level in accordance with the increase in potential energy. Alluvial estuaries are systems that are free to adjust dissipation processes to the energy sources that drive them, primarily the kinetic energy of the tide and the potential energy of the river flow and to a minor extent the energy in wind and waves. Mixing is the process that dissipates the potential energy of the fresh water. The maximum power (MP) concept assumes that this dissipation takes place at maximum power, whereby the different mixing mechanisms of the estuary jointly perform the work. In this paper, the power is maximized with respect to the dispersion coefficient that reflects the combined mixing processes. The resulting equation is an additional differential equation that can be solved in combination with the advection–dispersion equation, requiring only two boundary conditions for the salinity and the dispersion. The new equation has been confronted with 52 salinity distributions observed in 23 estuaries in different parts of the world and performs very well.

The mixing of fresh and saline water in estuaries is governed by the
dispersion–advection equation, which results from the combination of the
salt balance and the water balance under partial to well-mixed conditions
(see e.g. Savenije, 2005). The partially to well-mixed condition applies
when the increase in the salinity over the estuarine depth is gradual. The
salinity equation reads

System description of the salt and fresh water mixing in an estuary, with the seaside on the left and the riverside on the right. The water level (blue line) has a slope as a result of the salinity distribution (red line). In yellow are the hydrostatic pressure distributions on both sides. The black arrows show the fluxes. Subscript “0” represents the downstream boundary condition.

In the steady state, the flushing out of salt by the fresh river discharge
is balanced by the exchange of saline and fresh water resulting from a
combination of mixing processes, which causes an upriver flux of salt. The
sketch in Fig. 1 presents the system description with a typical
longitudinal salinity distribution (in red). It also shows the associated
water level (in blue), which has an upstream gradient due to the decreasing
salinity. Because of the density difference, the hydrostatic pressures on
both sides (in yellow) are not equal. The water level at the toe of the salt
intrusion curve is

The dispersion coefficient of Eq. (2) is generally determined by calibration
on observations of

The complication is that there are many different mixing processes at work. One can distinguish tidal shear, tidal pumping, tidal trapping, gravitation circulation (e.g. Fischer et al., 1979), and residual circulation due to the interaction between ebb and flood channels (Nguyen et al., 2008; Zhang and Savenije, 2017). And these different processes can be split up in many subcomponents. Park and James (1990), for instance, distinguished 66 components grouped into 11 terms. This reductionist approach, unfortunately, did not lead to more insight.

Here we take a system approach in which the assumption is that the different mechanisms are not independent but are jointly at work to reduce the salinity gradient that drives the exchange flows. We use the concept of maximum power as described by Kleidon (2016). Kleidon defines Earth system processes as dissipative systems that conserve mass and energy, but export entropy. These systems tend to function at maximum power, whereby the power of the system can be defined as the product of a process flux and the gradient driving the flux. The ability to maintain this power (i.e. work through time) in steady state results from the exchange fluxes at the system boundary, and when work is performed at the maximum possible rate within the system (“maximum power”), this equilibrium state reflects the conditions at the system boundary. The key parameter describing the process can then be found by maximizing the power.

From an energy perspective, we see that the freshwater flux, which has a lower density than saline water and without a counteracting process would float on top of the saline water, adds potential energy to the system, while the tide, which flows in and out of the estuary at a regular pace, creates turbulence, mixes the fresh and saline water, and hence works at reducing this potential energy. This is why dispersion predictors are generally linked to the estuarine Richardson number, which represents the ratio of the potential energy of the fresh water entering the estuary to the kinetic energy of the tidal flow.

In thermodynamic terms, the freshwater flux maintains a potential energy gradient, which triggers mixing processes that work at depleting this gradient. Because the strength of the mixing of fresh and saline water in turn depends on this gradient, there is an optimum at which the mixing process performs at maximum power. From a system point of view, it is not really relevant which particular mixing process is dominant or how these different processes jointly reduce the salinity gradient. What is relevant is how the optimum flux associated with this mixing process, yielding maximum power, depends on the dispersion.

In our case, the power derived from the potential energy of the freshwater flux is described by the product of the upstream dispersive water flux and the gradient in geopotential height driving this flux or, alternatively, the product of the dispersive exchange flux and the water level gradient. The optimum situation is achieved when the system is in an equilibrium state.

The water level gradient follows from the balance between the hydrostatic
pressures of fresh and saline water (see e.g. Savenije, 2005), resulting in

One can express the density of saline water as a function of the salinity:

Geometry of the Maputo Estuary, showing the cross-sectional area

We introduce three length scales:

Geometry of the Limpopo Estuary, showing the cross-sectional area

What the maximum power equation has contributed is that it provides an
additional equation. In the past, a solution could only be found if an
empirical equation was added describing

The two Eqs. (2) and (12) together can be solved numerically by using a
simple linear integration scheme. As a boundary condition it requires values
for

The downstream part of estuaries with an inflection point has a much shorter
convergence length, giving the estuary a typical trumped shape. This wider
part is generally not longer than about 10 km, which is the distance over
which ocean waves dissipate their energy. Beyond the inflection point, the
shape is determined by the combination of the kinetic energy of the tide and the
potential energy of the river flow. If the tidal energy is dominant over the
potential energy of the river, then the convergence is short, leading to a
pronounced funnel shape; if the potential energy of the river is large due
to regular and substantial flood flows, then the convergence is large, which
is typical for deltas. Hence, the topography can be described by two branches:

Subsequently we have integrated Eqs. (2) and (12) conjunctively by using
a simple explicit numerical scheme in a spreadsheet and confronted the
solution with observations. The solutions are fitted to the data by
selecting values for

Application of the numerical solution to observations in the Maputo Estuary for high water slack (HWS) and low water slack (LWS). The green line shows the tidal average (TA) condition. The red diamonds reflect the observations at HWS and the blue dots the observations at LWS on 29 May 1984.

Application of the numerical solution to observations in the Limpopo Estuary for high water slack (HWS) and low water slack (LWS). The green line shows the tidal average (TA) condition. The red diamonds reflect the observations at HWS and the blue dots the observations at LWS on 10 August 1994.

By making use of the maximum power (MP) concept, it was possible to derive an additional equation to describe the mixing of salt and fresh water in estuaries. Together with the salt balance equation, these two first-order and linear differential equations only require two boundary conditions (the salinity and the dispersion at some well-chosen boundary) to be solved. If the estuary has an inflection point in the geometry, then the preferred boundary condition lies there; otherwise the boundary condition is chosen at the ocean boundary.

This new equation can replace previous empirical equations, such as the Van der Burgh equation, and does not require any calibration coefficients (besides the boundary conditions). The new equation appears to fit very well to observations, which adds credibility to the correctness of applying the MP concept to fresh and salt water mixing.

The method presented here is based on a system perspective, which is holistic rather than reductionist. Reductionist theoretical methods have tried to break down the total dispersion in a myriad of smaller mixing processes, some of which are difficult to identify or to connect to conditions that make them more or less prominent. The idea here is that in a freely adjustable system, such as an alluvial estuary, individual mixing processes are not independent of each other, but rather influence each other and jointly work at reducing the salinity gradient at maximum dissipation. The resulting level of maximum power and dissipation is set by the boundary conditions of the system. It is then less important which mechanism is dominant, as long as the combined performance is correct. The maximum power limit is a way to derive this joint performance of mixing processes. The fact that the relationship derived from maximum power works so well in a wide range of estuaries is an indication that natural systems evolve towards maximum power, much like a machine that approaches the maximum performance of the Carnot limit.

Regarding the data, all observations are available on the web at

Notation.

The supplement related to this article is available online at:

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 authors would like to thank the two reviewers for their valuable comments and two colleagues, Xin Tian and Sha Lu, for specifying the mathematic concepts. The first author is financially supported for her PhD research by the China Scholarship Council. Edited by: Valerio Lucarini Reviewed by: Zhengbing Wang and Axel Kleidon