to study waves propagating in only one direction. The Boussinesq equations for surface waves (see, e.g., [@whitham_linear_1974; @choi_nonlinear_1995]) provide a model to study weakly dispersive waves propagating in any direction. This model not only broadens the range of physical phenomena encompassed by the SW model, but adding dispersive effects also makes it more enticing to weak turbulence theory, for which dispersion effects are of the utmost importance.

Let us take a look at the first terms in the Taylor expansion of the dispersion relation in Eq. , $$\label{taylor_exp}
    \omega^2 = c^2_0 k^2 - \frac{1}{3} c^2_0 h^2_0 k^4 + \ldots ,$$ where the first term is the non-dispersive shallow water term. The idea is to add terms to Eqs.  and such that the linear dispersion relation of the new system coincides, up to the fourth order, with the expansion in Eq. . This can be done by adding the term $h^2_0 {\nabla^2}\partial_t \mathbf{u}/3$ to Eq. , resulting in the following system, $$\begin{aligned}
    \frac{\partial \mathbf{u}}{\partial t} =& -(\mathbf{u} \cdot {\boldsymbol{\nabla}}) \mathbf{u}  -g
    {\boldsymbol{\nabla}}h + \frac{1}{3} h^2_0 {\nabla^2}\frac{\partial \mathbf{u}}{\partial t} \nonumber \\
{}& + \frac{\nu}{h} {\boldsymbol{\nabla}}\cdot ( h {\boldsymbol{\nabla}}\mathbf{u}) + \mathbf{F}, \label{bq_u} \\
    \label{bq_h}
    \frac{\partial h}{\partial t} =& -{\boldsymbol{\nabla}}\cdot (h \mathbf{u}) .\end{aligned}$$ We 