Simulations of shallow water wave turbulence

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This dataset curates all the simulations used to reproduce the paper:

Shallow water wave turbulence
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

Abstract

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.