Equilibrium statistical mechanics of waves in inhomogeneous moving media

Abstract

We adapt the microcanonical framework of equilibrium statistical mechanics to predict the statistics of short waves in inhomogeneous moving media. For steady inhomogeneities and background flow, we compute the wave spectrum at any location in the domain based on an ergodic prescription for the action density in phase space, constrained by conservation of absolute frequency. We illustrate the method for shallow-water waves subject to a background flow or to topographic inhomogeneities, and for deep-water surface capillary waves over a background flow, validating the predicted maps of rms surface elevation and interfacial slope against numerical simulations.

Figures (3)

• Figure 1: Steady background flow (a) and background topography (b) employed in the various numerical simulations. Panels (c-e) display snapshots of the surface elevation normalized by its maximum initial value $\eta_0$ for a simulation of shallow-water waves above the topography in panel (b). The initial rightward-propagating wavepacket (c, $t=0$) gets scattered at early time (d, $t=0.8L/\sqrt{gH_0}$), eventually reaching a statistically steady state at longer time (e, $t=4.9L/\sqrt{gH_0}$).
• Figure 2: Comparison of the ergodic predictions for $\overline{\eta^2}$ and $\overline{|\boldsymbol{\nabla} \eta|^2}$ with the numerical simulations for shallow-water waves. Panels (a–d) show the case with topography, while panels (e–h) correspond to the case with a background flow. The top row displays the theoretical predictions \ref{['eq:eta2SW']} and \ref{['eq:gradeta2SW']}, and the bottom row shows the corresponding numerical simulations.
• Figure 3: Comparison of the ergodic predictions with numerical simulations for capillary waves over a background flow ${\bf U}({\bf x})$. Panels (a,b) show the predicted and observed $\overline{\eta^2}$, respectively, while panels (c,d) show the predicted and observed $\overline{|\boldsymbol{\nabla} \eta|^2}$.