to the right hand side of Eq. , $$\label{balance_full}
    \frac{\mathrm{d} E}{\mathrm{d} t} = - 2 \nu Z + \epsilon,$$ where the mean energy injection rate can be computed as $$\label{eps_balance}
    \epsilon = \frac{1}{A} \iint\limits_A h \mathbf{u}\cdot \mathbf{f} \mathrm{d}x
    \mathrm{d}y .$$

Under the procedure described above, the typical evolution of the energy in a numerical simulation is shown in Fig. \[eng\_cons\]. The energy starts from the value corresponding to the fluid at rest (i.e., all the energy is the potential energy associated with the equilibrium height $h_0$). The total energy then grows under the action of the external mechanical forcing, and after $t \approx 80$ the system reaches a turbulent steady state in which the energy fluctuates around a mean value, and in which the energy injection and dissipation are equilibrated on the average. Even though pseudospectral methods are known to introduce no numerical dissipation, in the inset of Fig. \[eng\_cons\] we also show explicitly that the energy balance (Eq. ) is satisfied with an error of order $10^{-7}$, which remains stable and does not grow even after integrating for very long times.

To ensure that the flow in the simulations remained shallow for all excited wavenumbers, we enforced the following 