and their associated frequencies, $T_{0,1,2,3}$ is the coupling coefficient between the four modes, and $\textrm{d} \mathbf{k}_{123} = \textrm{d}
\mathbf{k}_{1} \textrm{d} \mathbf{k}_{2} \textrm{d} \mathbf{k}_{3}$. From this equation, dimensional analysis yields the following expression for the energy spectrum $$\label{predicc_wt}
    E(k) \sim k^{-4/3} .$$ From this spectrum and using dimensional analysis, it is easy to show that in the presence of dissipation, the dissipation wavenumber in such a flow is $k_\eta \sim [\epsilon/(h^2_0 \nu^3)]^{1/5}$, where $\epsilon$ is the mean energy injection rate.

A scaling compatible with a $\sim k^{-4/3}$ spectrum was observed in laboratory and field datasets [@smith_equilibrium_2003; @kaihatu_asymptotic_2007], where a spectrum compatible with $\sim k^{-2}$ was also found in shallower regions of the fluid.

The prediction in Eq.  applies to the BQ model, when dispersion is not negligible. Before proceeding, we should comment on some peculiarities of the SW model regarding wave turbulence. First, an inspection of its dispersion relation, Eq. , indicates that three wave interactions are possible in this model, and as a result the arguments above for four-wave interactions do not apply. Weak turbulence theory can be used in systems with three-waves interactions (with the case of deep water flows being a paradigmatic one, but see also the case 