assume potential flow, and, more importantly, we do not truncate the non-linear term, thus retaining all high order non-linearities. Another difference is that we do not introduce an artificial dissipation term as it is usually done, but one based on physical grounds. The key motivation for these choices is to be able to compare with experiments in the future, where vortical motions can develop, and where dissipation also plays a non-negligible role. To achieve higher resolutions than the ones studied here becomes increasingly more expensive as the BQ model is dispersive.

All the simulations were started from the fluid at rest. An external mechanical forcing injected energy in the system, allowing it to reach for sufficiently long times an out-of-equilibrium turbulent steady state, after an initial transient. To excite waves, and prevent external injection of energy into vortical motions, the forcing had the following form $$\mathbf{F} = {\boldsymbol{\nabla}}f,$$ where $f$ is a randomly generated scalar function, with a time correlation of one time unit, amplitude $f_0$, and applied in a band of wavenumbers in Fourier space between $k_{f_1}$ and $k_{f_2}$ (see Tables \[table1\], \[table2\], and \[table3\]). Note that having a mechanical forcing in the momentum equation adds an extra term 