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New article: Shallow water wave turbulence

Paper about how turbulence in shallow water equations evolves and gives rise to shock waves.

Just today another open access paper got published. This paper talks about some how turbulence in shallow water equations evolves and gives rise to shock waves. There are also some novel statistical results which might find use in other fields like acoustics.

Figure: Shocks becoming stronger and weaker with variations in Reynolds number and Froude numbers.

Find more about it in here.

## Abstract

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
(*ϵ**d*)^{1 ⁄ 3}, where *ϵ* 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 (*ϵ**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} = *ϵ*^{1 ⁄ 3} ⁄ *ck*^{1 ⁄ 3}_{f}, where *k*_{f} is the
forcing wave number 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
as *E* ~ √(*ϵ**c* ⁄ *k*_{f}), 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* ~ *F*^{1 ⁄ 2}_{f} ⁄ *k*_{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} → 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.

### About the author

Ashwin Vishnu Mohanan, Ph.D. in Fluid mechanics