the Hamiltonian equations for a potential flow with a truncated non-linear term, or the kinetic equations resulting from weak turbulence theory at moderate spatial resolution [@dyachenko_weak_2004; @korotkevich_simultaneous_2008] (with the notable exception of [@yokoyama_statistics_2004]). Here, we solve the primitive equations, without truncating the non-linear terms, potentially allowing for the development of vortical motions and of strong interactions between waves, and with spatial resolutions up to $2048^2$ grid points.

The paper is organized as follows. In section \[equations\] we introduce both models, show the assumptions made in order to obtain them, derive their energy balance equations, and briefly discuss the predictions obtained in the framework of weak turbulence theory. In section \[simulations\] we describe the numerical methods employed and the simulations. Then, in section \[results\] we introduce several dimensionless numbers defined to characterize the flows, and present the numerical analysis and results. We present wavenumber energy spectra and fluxes, time resolved spectra (as a function of the wavenumber and frequency), frequency spectra, and probability density functions of the fluid free surface height. Finally, in section \[conclusion\] we present the conclusions. The most important results are: (a) As in previous experimental and observational studies [@smith_equilibrium_2003; @kaihatu_asymptotic_2007] we find now in simulations that different 