  We study wave turbulence in shallow water flows in numerical simulations
using two different approximations: the shallow water model, and the Boussinesq
model with weak dispersion. The equations for both models were solved using
periodic grids with up to $2048^2$ points. In all simulations, the Froude
number varies between $0.015$ and $0.05$, while the Reynolds number and level
of dispersion are varied in a broader range to span different regimes. In all
cases, most of the energy in the system remains in the waves, even after
integrating the system for very long times. For shallow flows, non-linear waves
are non-dispersive and the spectrum of potential energy is compatible with
$\sim k^{-2}$ scaling. For deeper (Boussinesq) flows, the non-linear dispersion
relation as directly measured from the wave and frequency spectrum (calculated
independently) shows signatures of dispersion, and the spectrum of potential
energy is compatible with predictions of weak turbulence theory, $\sim
k^{-4/3}$. In this latter case, the non-linear dispersion relation differs from
the linear one and has two branches, which we explain with a simple qualitative
argument. Finally, we study probability density functions of the surface height
and find that in all cases the distributions are asymmetric. The probability
density function can be approximated by a skewed normal distribution as well as
by a Tayfun distribution.
