regimes arise depending on the fluid depth and the degree of nonlinearity of the system. (b) We obtain a power spectrum of the surface height compatible (within statistical uncertainties) with $\sim k^{-2}$ in the shallow water (non-dispersive) case, and one compatible with a $\sim k^{-4/3}$ spectrum as the fluid depth is increased using the Boussinesq (weakly dispersive) model. The latter spectrum is also compatible with predictions of weak turbulence theory [@onorato_four-wave_2008]. (c) Dispersion in the Boussinesq model results in more prominent small scale features and the development of rapidly varying waves. (d) In the weakly dispersive regime, the non-linear dispersion relation obtained from the simulations has two branches in a range of wavenumbers, one branch corresponding to non-dispersive waves, and another corresponding to dispersive waves. We interpret this as the result of short wavelength waves seeing an effectively deeper flow resulting from the interaction with waves with very long wavelength. (e) The probability density function of surface height can be approximated by both skewed normal and Tayfun distributions. In the latter case, the parameters of the distribution are compatible with those previously found in observations and experiments [@falcon_observation_2011].

The shallow water and Boussinesq equations {#equations}
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Let us consider a 