The rotating shallow water equations are
\begin{align*} u_t + \vec{u} \cdot\nabla u -fv &= -gh_x \\
v_t + \vec{u} \cdot\nabla v +fu &= -gh_y \\
h_t + \nabla \cdot (\vec{u}h) &= 0
\end{align*}
where \(\vec{u}=(u,v)\) is the horizontal velocity, \(h=h(x,y,t)\) is the depth of the fluid, \(g\) is the gravitational acceleration, and \(f\) is the Coriolis parameter.