Simulations of shallow water wave turbulence

<strong>About</strong> This dataset curates all the simulations used to reproduce the paper: <em>Shallow water wave turbulence</em><br> DOI: 10.1017/jfm.2019.375 The source code and scripts necessary to generate the manuscript are archived at: https://github.com/ashwinvis/augieretal_jfm_2019_shallow_water See the README in the repository above to generate the manuscript <strong>Abstract</strong> The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as \((\epsilon d)^{1/3}\), where \( \epsilon\) is the mean dissipation rate and \(d\) the mean distance between the shocks, and that the \(p\)^th-order displacement and velocity structure functions scale as \((\epsilon d)^{p/3} r/d\), where \(r\) is the separation. Then we carry out a series of forced simulations with resolutions up to 7680², varying the Froude number,\(F_{f} = (\epsilon L_f)^{1/3}/ c \), where \(L_f\) is the forcing length scale and \(c\) is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as \(E \sim \sqrt{\epsilon L_f c}\), and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualizations. The mean distance between the shocks is found to scale as \( d \sim F_f^{1/2} L_f\). Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, \(F_f \rightarrow 0 \), shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given \(F_f\), no matter how small, shocks will prevail if the Reynolds number is sufficiently large.