Wave turbulence in shallow water models

TL;DR

This study investigates wave turbulence in shallow water models through numerical simulations, revealing different spectral scalings and dispersion characteristics depending on the depth and regime, with implications for understanding wave energy distribution.

Contribution

It compares shallow water and Boussinesq models, demonstrating how dispersion and turbulence spectra vary with depth and non-linearity, advancing understanding of wave turbulence regimes.

Findings

Shallow flows exhibit a $k^{-2}$ energy spectrum.

Deeper Boussinesq flows show a $k^{-4/3}$ spectrum consistent with weak turbulence theory.

Probability density functions of surface height are asymmetric and can be modeled by skewed normal or Tayfun distributions.

Abstract

We study wave turbulence in shallow water flows in numerical simulations using two different approximations: the shallow water model, and the Boussinesq model with weak dispersion. The equations for both models were solved using periodic grids with up to $2048^2$ points. In all simulations, the Froude number varies between $0.015$ and $0.05$, while the Reynolds number and level of dispersion are varied in a broader range to span different regimes. In all cases, most of the energy in the system remains in the waves, even after integrating the system for very long times. For shallow flows, non-linear waves are non-dispersive and the spectrum of potential energy is compatible with $\sim k^{-2}$ scaling. For deeper (Boussinesq) flows, the non-linear dispersion relation as directly measured from the wave and frequency spectrum (calculated independently) shows signatures of dispersion, and the spectrum of potential energy is compatible with predictions of weak turbulence theory, $\sim k^{-4/3}$. In this latter case, the non-linear dispersion relation differs from the linear one and has two branches, which we explain with a simple qualitative argument. Finally, we study probability density functions of the surface height and find that in all cases the distributions are asymmetric. The probability density function can be approximated by a skewed normal distribution as well as by a Tayfun distribution.