Linearized shallow water equations

For a homogeneous layer of mean depth $H_0$ on an f-plane, linearization about rest gives

On a beta plane with $f=f_0+\beta y$, set ${\mathcal{F}}_y={\partial }_t^2+f(y{)}^2$. Exact elimination of the depth-integrated shallow-water transports givesThe essential commutator is ${\partial }_y({\mathcal{F}}_{y}V)={\mathcal{F}}_y{V}_y+2f\beta V$. Freezing undifferentiated $f$ factors to $f_0$ is a local approximation. The slow-wave limit gives the shallow-water Rossby-wave dispersion relation with a negative zonal phase velocity for $\beta >0$.

For a depth-independent horizontal velocity in a layer of reference depth $H$, the linear transports are $U=Hu$, $V=Hv$. They have dimensions of area per time and satisfy the surface continuity equation ${\eta }_t+U_x+V_y=0$. Their momentum equations are obtained by multiplying the depth-independent velocity equations by $H$.

In a homogeneous layer under the hydrostatic approximation, the horizontal pressure gradient is independent of depth. The depth derivatives of velocity obey $(u_z{)}_t-f{v}_z=0$, $(v_z{)}_t+f{u}_z=0$. If the motion starts without shear, these derivatives remain zero. Hydrostatic pressure alone does not eliminate a pre-existing freely oscillating shear.

For linear rotating shallow water equations with fixed depth $H(y)$ and constant Coriolis parameter $f$, elimination of the horizontal velocity givesAt a depth step this is interpreted distributionally, with continuity of surface elevation and normal volume flux. Because velocity elimination raises the time order, initial height alone is insufficient; the original velocity and continuity equations determine the remaining initial derivatives.