@falcon_observation_2011]. It was also found that an inertial range with a $\sim k^{-2}$ dependency can develop in the shallower regions of the fluid. The comparison between different shallow water models, with different degree of shallowness (and of dispersion) is therefore of interest, e.g., for the study of waves in basins with inhomogeneous depth.

In the shallow water regime there are several models that can be considered to describe the ocean surface. There is the linear theory (see, e.g., [@landau_fluid_2004]) which can predict the dispersion relation of small amplitude waves, but which is insufficient to study turbulence. There are also non-linear theories, such as the shallow water model [@pedlosky_geophysical_1987] for non-dispersive waves, as well as the Boussinesq model [@whitham_linear_1974] for weakly dispersive waves which is the one used in some of the most recent works on the subject [@onorato_four-wave_2008]. While the former non-linear model is valid in the strict shallow water limit, the latter can be used in cases in which the wavelengths are closer to (albeit still larger than) the depth of the basin.

In the present work, we study turbulent solutions of the shallow water model and of the Boussinesq equations using direct numerical simulations. Previous numerical studies considered 