the height of the fluid column, namely $$\label{eq_h}
    \frac{\partial h}{\partial t} + \frac{\partial }{\partial x} (h v_x)
    + \frac{\partial }{\partial y}(h v_y) = 0 .$$ Note that we do not have to assume irrotationality to derive neither Eq.  nor Eq. .

If we linearize these equations, we find again the dispersion relation given by Eq. , as expected. In the presence of external forcing $\mathbf{F}$, and viscosity $\nu$, the equations can be written as $$\begin{aligned}
 \label{sw_u} \frac{\partial \mathbf{u}}{\partial t} &=
    -(\mathbf{u} \cdot {\boldsymbol{\nabla}}) \mathbf{u}  -g {\boldsymbol{\nabla}}h + \frac{\nu}{h} {\boldsymbol{\nabla}}\cdot (
    h {\boldsymbol{\nabla}}\mathbf{u}) + \mathbf{F}, \\ \label{sw_h} \frac{\partial h}{\partial
        t} &= -{\boldsymbol{\nabla}}\cdot (h \mathbf{u}) .  \end{aligned}$$ We will refer to this set of equations as the shallow water model, or SW model. In these equations the viscosity $\nu$ was added to the horizontal velocity field $\mathbf{u}$, which behaves as a compressible flow (i.e., it has non-zero divergence, see [@marche_derivation_2007]). This choice of the viscous term ensures conservation of the momentum $h \mathbf{u}$, and also that energy dissipation is always negative, as in Sec. \[energybalancesec\].

Boussinesq model
----------------

As the depth of the fluid increases, dispersion becomes important. There are several models that introduce weak dispersive effects perturbatively, but many are built 