Saltwater intrusion is a crucial issue in estuaries. The spread of salinity is described by the dispersion coefficient. A purely empirical equation which links the effective tidal average dispersion to the freshwater discharge was developed by Van der Burgh [1972]. Combining it with the salt balance equation, Savenije [1986] derived a one-dimensional model for salinity intrusion in estuaries. This Van der Burgh model has performed surprisingly well around the world. However, the physical basis of the empirical Van der Burgh coefficient (퐾) is still weak. This study provides a theoretical basis for the Van der Burgh method and presents alternative equations. MacCready [2004] presented a theoretical expression for the dispersion coefficient following a reductionist approach. Comparing the density-related parts of the equations of the dispersion coefficient developed by Savenije and MacCready, a predictive equation is obtained for the coefficient 퐾 using physical parameters. In addition, a new box-model has been developed considering the longitudinal densitydriven gravitational circulation and the lateral tide-driven horizontal circulation. The coefficient 퐾 (closely related to the Van der Burgh’s coefficient) is used as an index of the density-driven mixing mechanism while the tide-driven part is included by assuming that it is proportional to the longitudinal dispersion. This model is validated in sixteen alluvial estuaries worldwide by using calibrated 퐾 values (and the boundary conditions). These calibrated values correspond well with the predicted values from the theoretical derivation, revealing that 퐾 has smaller values when the tide is stronger. From a system perspective, alluvial estuaries are free to adjust dissipation processes to the energy sources that drive them. The potential energy of the river flow drives mixing by gravitational circulation. The maximum power concept assumes that the mixing takes place at the maximum power limit. To describe the complex mixing processes in estuaries holistically, different assumptions had to be made. The maximum power concept did not work satisfactorily when estuaries were assumed as isolated systems. However, by including the accelerating moment provided by the freshwater discharge, the open estuary system could be solved in analogy with Kleidon [2016] applying the maximum power concept. A new expression for the dispersion coefficient due to gravitational circulation has been derived and solved in combination with the advection-dispersion equation. This maximum power model works well in eighteen estuaries with a large convergence length, providing an alternative equation for the dispersion. These estuaries also have larger calibrated 퐾 values by the Van der Burgh method, revealing a relation between the empirical coefficient 퐾 and the geometry. All these models: the Van der Burgh model, the box-model, and the maximum power model, can describe the longitudinal salinity profiles. The comparison between these models implies that the empirical Van der Burgh coefficient is associated with the geometry and stratification conditions. Finally, new predictive equations have been obtained by regression with physical-based parameters which make the Van der Burgh salinity intrusion method predictive with a solid theoretical basis.

According to Kleidon (2016), natural systems evolve towards a state of maximum power, leading to higher levels of entropy production by different mechanisms, including gravitational circulation in alluvial estuaries. Gravitational circulation is driven by the potential energy of fresh water. Due to the density difference between seawater and river water, the water level on the riverside is higher. The hydrostatic forces on both sides are equal but have different lines of action. This triggers an angular moment, providing rotational kinetic energy to the system, part of which drives mixing by gravitational circulation, lifting up heavier saline water from the bottom and pushing down relatively fresh water from the surface against gravity; the remainder is dissipated by friction while mixing.With a constant freshwater discharge over a tidal cycle, it is assumed that the gravitational circulation in the estuarine system performs work at maximum power. This rotational flow causes the spread of salinity inland, which is mathematically represented by the dispersion coefficient. In this paper, a new equation is derived for the dispersion coefficient related to density-driven mixing, also called gravitational circulation. Together with the steady-state advection-dispersion equation, this results in a new analytical model for densitydriven salinity intrusion. The simulated longitudinal salinity profiles have been confronted with observations in a myriad of estuaries worldwide. It shows that the performance is promising in 18 out of 23 estuaries that have relatively large convergence length. Finally, a predictive equation is presented to estimate the dispersion coefficient at the downstream boundary. Overall, the maximum power concept has provided a new physically based alternative for existing empirical descriptions of the dispersion coefficient for gravitational circulation in alluvial estuaries.

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 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 1ĝ•2 < K < 2ĝ•3 for density-driven mixing which falls within the feasible range of 0 < K < 1. In addition, we developed a one-dimensional predictive equation for the dispersion of salinity as a function of local hydraulic parameters that can vary along the estuary axis, including mixing due to tide-driven residual circulation. This type of mixing is relevant in the wider part of alluvial estuaries where preferential ebb and flood channels appear. Subsequently, this dispersion equation is combined with the salt balance equation to obtain a new predictive analytical equation for the longitudinal salinity distribution. Finally, the new equation was tested and applied to a large database of observations in alluvial estuaries, whereby the calibrated K values appeared to correspond well to the theoretical range.

Salt intrusion in alluvial estuaries is affected by the interaction between fresh water and saline water. When the river discharge is practically stable, the degree to which salt water from the ocean penetrates landward depends on the tide. The 1-D longitudinal dispersion is the parameter that describes the mixing between fresh and saline water. The dimension of this parameter is [L2/T], representing the spreading of a substance (e.g., salinity) per unit of time. The dimensionless dispersion is a function of the stratification, described by the estuarine Richardson number. But which physical parameters should be used to make the dispersion dimensionless? Basically, it should be scaled by a spatial distance (the mixing length) and a measure for the spreading velocity. However, the questions are: 1) which mixing length to use: the depth of the estuary (as a measure for vertical gravitational circulation), the tidal excursion (as a measure for the exchange with longitudinal salinity gradient and trapped pockets on the banks), or a mixture of the two? and 2) which velocity to use: the tidal amplitude (as a measure for the flow velocity), or the shear velocity (as a measure for the turbulence). Using the depth instead of the tidal excursion implies that the stronger the tide (e.g., spring tide), the smaller the stratification and the shorter the intrusion length, while using the tidal excursion implies that the weaker the tide (e.g., neap tide), the shorter the intrusion length. If we use the tidal velocity amplitude instead of the shear velocity, the effect of bottom shear is not taken into account explicitly. Most observational data in real estuaries are available during spring tide, when the estuaries are better mixed and when salinity is supposed to intrude furthest inland. On top of this, it is questionable if the neap-tide variations lead to approximate steady state salt intrusion at the extremes. Hence, the field data so far can’t provide unequivocal answers to these questions. Hence, the main question of this research is what the effects of neap and spring variations are on salt intrusion based on field observations.

The Van der Burgh’s equation for longitudinal effective dispersion is a purely empirical method with practical implications. Its application to the effective tidal average dispersion under equilibrium conditions appears to have excellent performance in a wide range of alluvial estuaries. In this research, we try to find out the physical meaning of Van der Burgh’s coefficient. Researchers like MacCready, Fischer, Kuijper, Hansen and Rattray have tried to split up dispersion into its constituents which did not do much to explain overall behaviour. In addition, traditional literature on dispersion is mostly related to flumes with constant cross-section. This research is about understanding the Van der Burgh’s coefficient facing the fact that natural estuaries have exponentially varying cross-section. The objective is to derive a simple 1-D model considering both longitudinal and lateral mixing processes based on field observations (theoretical derivation). To that effect, we connect dispersion with salinity using the salt balance equation. Then we calculate the salinity along the longitudinal direction and compare it to the observed salinity. Calibrated dispersion coefficients in a range of estuaries are then compared with new expressions for the Van der Burgh’s coefficient K and it is analysed if K varies from estuary to estuary. The set of reliable data used will be from estuaries: Kurau, Perak, Bernam, Selangor, Muar, Endau, Maputo, Thames, Corantijn, Sinnamary, Mae Klong, Lalang, Limpopo, Tha Chin, Chao Phraya, Edisto and Elbe.