the Phillips’ spectrum [@phillips_equilibrium_1958], derived using dimensional arguments from strong turbulence and considering the coupling between waves, was long considered to be correct. However, different observational and experimental data [@toba_local_1973; @donelan_directional_1985], as well as numerical simulations [@badulin_weakly_2007], suggest that the actual spectrum is closer to that of weak turbulence. In fact, Phillips himself suggested that his spectrum may not be valid in the ocean [@phillips_spectral_1985]. Nonetheless, a scaling compatible with Phillips’ spectrum is still observed in numerical simulations [@korotkevich_simultaneous_2008] when the forcing is strong. This suggests that while weak turbulence provides an elegant theoretical way to study wave turbulence in the ocean, more considerations are necessary to appropriately describe the diversity of regimes found in these flows [@newell_wave_2011].

Most of the work done in ocean surface waves under the weak turbulence approximation concerns deep water flows. But the theory can also be applied to the shallow water case, i.e., for gravity waves whose wavelengths are large compared to the height of the fluid column at rest (see [@zakharov_statistical_1999; @onorato_four-wave_2008]). In this case, the theory leads to the prediction that the energy spectrum follows a $\sim
k^{-4/3}$ behavior. Behavior compatible with this prediction was found both experimentally and observationally [@smith_equilibrium_2003; @kaihatu_asymptotic_2007; 