Nonlinear shallow water equation

The name of the equation pretty much gives the quantities that matter in this scenario: the nonlinearity and shallowness. For the standard input of the water wave equation with velocity potential as the unknown, we have:

$\nabla^2 \phi=0$ as governing equation;

$\eta_t+\phi_x\eta_x+\phi_y\eta_y-\phi_z=0$ as kinematic boundary condition (KBC) at free surface;

$\phi_t+g\eta+\frac{1}{2}(\nabla \phi)^2+\frac{p}{\rho}=0$ as dynamics boundary condition (DBC) at free surface.

In order to quantify the effect of nonlinearity and shallowness, non-dimensional form of the equation is introduced by using normalized variables:

$(x, y)\leftarrow (x, y)/l,\quad z\leftarrow z/h, \quad t\leftarrow ct/l, \quad \eta\leftarrow \eta/a,\quad \phi\leftarrow h\phi/(alc)$

where $l$ is characteristic length scale, $a$ is wave amplitude, $h$ is water depth, $c$ is shallow water celerity by linear theory $c=\sqrt{gh}$. The quantity that evaluates nonlinearity is defined as

$\epsilon=a/h$

and that for shallowness is

$\delta=(h/l)^2$

Now the equations become

$\Delta(\phi_{xx}+\phi_{yy})+\phi_{zz}=0$

$\phi_t+\frac{\epsilon}{2}(\phi_x^2+\phi_y^2)+\frac{\epsilon}{2\delta}\phi_z^2+\eta$, for $z=1+\epsilon\eta$    (DBC)

$\delta[\eta_t+\epsilon(\phi_x\eta_x+\phi_y\eta_y)]-\phi_z=0$, for $z=1+\epsilon\eta$   (KBC)

If the unknown $\phi$ is expanded with $\delta$:

$\phi=\phi_0+\delta\phi_1+\delta^2\phi_2+\dots$

One can find for zero order term $\phi_0$, governing equations gives:

$\phi_{0zz}=0$

combining homogeneous Neumann boundary condition at the bottom, this means $\phi_{0z}=0$ all over the region, and $\phi_0$ is independent of vertical location. If denote

$u(x,y,t)=\phi_{0x}, \quad v(x,y,t)=\phi_{0y}$

we have for $\phi_1$:

$\phi_1=-z^2/2(u_x+v_y),\quad \phi_2=z^4/24[(\nabla^2u)_x+(\nabla^2v)_y]$

All above results come from the governing equation and bottom boundary condition, and have nothing to do with free surface boundary conditions, i.e. they are concluded by the fact of flow being incompressible, irrotational, and having immiscible bottom. Now, if we put the expansion of $\phi$ in free surface boundary condition equations, and retain only up to 2nd order of $\delta$, $\epsilon$ in KBC, and up to 1st order of $\delta$, $\epsilon$ in DBC, combing the definition notation of $(u,v)$, we have:

$u_t+\epsilon(uu_x+vv_x)+\eta_x-1/2\delta(u_{txx}+v_{txy})=0$

$v_t+\epsilon(uu_y+vv_y)+\eta_y-1/2\delta(u_{txy}+v_{tyy})=0$

$\eta_t+[u(1+\epsilon\eta)]_x+[v(1+\epsilon\eta)]_y=\delta/6[(\nabla^2u)_x+(\nabla^2v)y]$

In above equations if we take $\delta\rightarrow 0$ and $\epsilon\rightarrow 0$, i.e., assuming linearity and shallowness, we have the linear wave equation for $\eta$:

$\eta_{tt}-c^2(\eta_{xx}+\eta_{yy})=0$

On the other hand, the 1D version of above equations is so called Boussinesq equations:

$u_t+\epsilon uu_x+\eta_x-1/2\delta u_{txx}=0$

$\eta_t+[u(1+\epsilon\eta)]_x=\delta/6 u_{xxx}$