Shallow water wave turbulence

Authors: Augier, P., Mohanan, A. V., Lindborg, E.
Document Type: Article
Pubstate: Submitted
Journal: Journal of Fluid Mechanics
Year: 2019


The dynamics of irrotational shallow water wave turbulence forced in 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 $7680^2$, 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 visualisations. 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.