of $h$.

The two remaining dimensionless numbers are the Reynolds number, $$\label{reynolds}
    \textrm{Re} =\frac{U_0 L_f}{\nu},$$ where $L_f$ is the forcing scale (defined as $2 \pi/k_{f_0}$), and what we will call the dispersivity, $D_s$, defined as $$\label{dispersivity}
    D_s = h_0 k_\textrm{max} = \frac{2\pi
        h_0}{\lambda_\textrm{min}} = \frac{N h_0}{6 \pi}$$ following Eq. . This last number, only relevant for the Boussinesq model, measures how strong the dispersion is at the smallest scales, and for sufficiently small $D_s$ we can expect the solutions of the Boussinesq model to converge to the solutions of the shallow water model. In fact, it is easy to show from the weak turbulence spectrum in Eq.  that when the maximum resolved wavenumber $k_{\textrm{max}}$ is associated with the dissipation wavenumber $k_\eta$, then $$\textrm{Re} \sim \frac{U_0 L_f}{h_0 \epsilon^{1/3}} D^{5/3}_s .
    \label{reycurve}$$ Decreasing $D_s$ below the value given by this relation should result in negligible dispersion at all resolved wavenumbers. Note that the level of dispersion in a given Boussinesq run depends on the wavenumber, and $D_s$ actually quantifies the strongest possible dispersion at the smallest scales in the flow.

By qualitatively assessing each run, we can classify them into three sets, $A$, $B$, and $C$. In tables \[table1\], \[table2\], \[table3\] the 