mean kinetic energy, and $$\label{ec_pot}
    V = \frac{1}{A} \iint g \frac{h^2}{2} \mathrm{d} x \mathrm{d} y$$ as the mean potential energy, such that the sum of both gives the mean total energy $E$.

The dispersive term present in the BQ model changes the balance given by Eq. . However, since the extra term is of order $(h_0/L)^2$, as long as we are in a sufficiently shallow flow it will be very small, and therefore, negligible for the conservation of energy. We verified this is the case in our numerical simulations.

\[sec:weak\]Weak Turbulence prediction
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We briefly present some results obtained in the framework of weak turbulence theory for the BQ model (as the derivation is a bit cumbersome, only a general outline will be given here; please see [@onorato_four-wave_2008] for details). Weak turbulence is studied in the BQ model assuming the fluid is inviscid and irrotational, so that the velocity can be written in terms of a velocity potential. To obtain a statistical description of the wave field, it is also assumed that it is homogeneous and that the free modes are uncorrelated.

At first sight, the quadratic nonlinear terms in Eqs.  and indicate modes interact in triads, with the wave 