Thermodynamic solutions for equilibrium beach profiles

Abstract

Solutions are developed for beach profiles using equilibrium principles of thermodynamics applied to simple representations of the nearshore fluid dynamics. Equilibrium beaches are posed as isothermal shorezone systems of constant volume that dissipate external work by incident waves into heat given up to the surroundings. By the maximum entropy production formulation of the second law of thermodynamics (the law of entropy increase), the shorezone system achieves equilibrium with profile shapes that maximize the rate of dissipative work performed by wave‐induced shear stresses. Dissipative work is assigned to two different shear stress mechanisms prevailing in separate regions of the shorezone system, an outer solution referred to as the shorerise and a bar‐berm inner solution. The equilibrium shorerise solution extends from closure depth (zero profile change) to the breakpoint, and maximizes dissipation due to the rate of working by bottom friction. In contrast, the equilibrium bar‐berm solution between the breakpoint and the berm crest maximizes dissipation due to work by internal stresses of a turbulent surf zone. Both shorerise and bar‐berm equilibria were found to have an exact general solution belonging to the class of elliptic cycloids. The elliptic cycloid allows all significant features of the equilibrium profile to be characterized by the eccentricity and the size of the ellipse axes. These basic ellipse parameters are evaluated by process‐based algorithms and empirically validated parameters for which an extensive literature already exists. The elliptic cycloid solutions displayed wave height, period and grain size dependence and demonstrate generally good predictive skill in point‐by‐point comparisons with measured profiles.