Variable-Coriolis shallow-water height equation (${\mathcal{F}}_y={\partial }_t^2+f(y{)}^2$)

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 $\beta >0$.