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 $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ 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 $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, 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 $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, 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.
Highlights
The shallow water (SW) equations have been widely used to study basic mechanisms occurring in geophysical flows
We derived the SW analogue (2.25) of the ‘four-fifths’ law of Kolmogorov turbulence. Using this relation and straightforward statistical and geometrical arguments we developed a simple shock model predicting that the shock amplitude scales as ( d)1/3, where d is the mean distance between the shocks, and that the pth-order structure function above a certain minimum will scale as ( d)p/3r/d
The flow variables in each Fourier mode were found to evolve in accordance with linear wave dynamics, with equipartition between available potential energy (APE) and kinetic energy (KE) over a period
Summary
The shallow water (SW) equations have been widely used to study basic mechanisms occurring in geophysical flows (see for example Vallis 2006). Yuan & Hamilton (1994) showed that statistically stationary shallow water flows can be obtained by forcing quasi-geostrophic modes at large scales and with dissipation only at small scales They suggested that the k−5/3 energy spectrum observed at atmospheric mesoscales (Nastrom & Gage 1985; Li & Lindborg 2018) is generated by a downscale cascade of gravity waves which may be captured by the SW equations. In the case of inertia–gravity waves, Falkovich & Medvedev (1992) showed that the exact solutions of the kinetic equation corresponding to a constant downscale energy flux are associated with a spectrum scaling as E(k) ∼ k−8/3, while Zakharov & Sagdeev (1970) used the weak turbulence formalism to derive a spectrum of the form E(k) ∼ k−3/2 for three-dimensional acoustic turbulence. We present some theory and we report on a series of simulations of irrotational SW wave turbulence
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