Maximum power of saline and fresh water mixing in estuaries

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 density-driven 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.

Maximum power of saline and fresh water mixing in estuaries

Natural systems evolve towards a state of maximum power (Kleidon, 2016), leading to higher levels of entropy production by different mechanisms, including the gravitational circulation in alluvial estuaries. Gravitational circulation is driven by the potential energy of the fresh water. Due to the density difference between seawater and riverwater, the water level on the river side is higher. The hydrostatic forces on both sides are equal, but have different working lines. This triggers an (accelerating) angular moment, providing rotational kinetic energy into the system, part of which drives mixing by gravitational circulation mixing; the remainder is transferred into dissipated energy by friction while mixing. With a constant discharge over a tidal cycle, the density-driven gravitational circulation in the estuarine system performs work at maximum power, lifting up saline water and bringing down fresh water against gravity. The rotational flow causes the spread of salinity, which is mathematically represented by the dispersion coefficient. Accordingly, a new equation for the dispersion coefficient due to the density-driven mechanism has been derived. Together with the steady state advection-dispersion equation, this resulted in a new analytical model for gravitational 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 eighteen out of twenty-three estuaries, with relatively large convergence length. Finally, a predictive equation is presented for the dispersion coefficient at the boundary. Overall, the maximum power concept has provided an alternative for describing the dispersion coefficient due to gravitational circulation in alluvial estuaries.

Frictional interactions between tidal constituents in tide-dominated estuaries

When different tidal constituents propagate along an estuary, they interact because of the presence of nonlinear terms in the hydrodynamic equations. In particular, due to the quadratic velocity in the friction term, the effective friction experienced by both the predominant and the minor tidal constituents is enhanced. We explore the underlying mechanism with a simple conceptual model by utilizing Chebyshev polynomials, enabling the effect of the velocities of the tidal constituents to be summed in the friction term and, hence, the linearized hydrodynamic equations to be solved analytically in a closed form. An analytical model is adopted for each single tidal constituent with a correction factor to adjust the linearized friction term, accounting for the mutual interactions between the different tidal constituents by means of an iterative procedure. The proposed method is applied to the Guadiana (southern Portugal–Spain border) and Guadalquivir (Spain) estuaries for different tidal constituents (M2, S2, N2, O1, K1) imposed independently at the estuary mouth. The analytical results appear to agree very well with the observed tidal amplitudes and phases of the different tidal constituents. The proposed method could be applicable to other alluvial estuaries with a small tidal amplitude-to-depth ratio and negligible river discharge.

Thermodynamics of saline and fresh water mixing in 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.

Thermodynamics of Saline and Fresh Water Mixing in Estuaries

Mixing of saline and fresh water is a process of energy dissipation. The fresh water 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 of the water level in accordance with the increase of 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 performed very well, even better than the well-tested empirical Van der Burgh equation that required a calibration parameter, which with this equation is no longer needed.

The physics behind Van der Burgh's empirical equation, providing a new predictive equation for salinity intrusion in estuaries

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 1∕2 < K < 2∕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.

The physics behind Van der Burgh's empirical equation, providing a new predictive equation for salinity intrusion in estuaries

The practical value of the surprisingly simple Van der Burgh's equation to predict 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 for the Van der Burgh's coefficient of 1/2