Linearization about rest replaces the <material derivative> by $\partial_t$, so the <quasi-geostrophic potential-vorticity equation> becomes
$$
\partial_t\left(
\psi_{xx}+\psi_{yy}+\frac{f_0^2}{N^2}\psi_{zz}
\right)+\beta\psi_x=0.
$$
Insert the <separation of variables> $\psi=\phi(x,y,t)P(z)$ and impose
$$
P''=-\frac{N^2}{c^2}P.
$$
The horizontal amplitude then satisfies
$$
\boxed{
\partial_t\left(
\nabla_h^2\phi-\frac{f_0^2}{c^2}\phi
\right)+\beta\phi_x=0
}.
$$
The <Linear Rossby-wave equation> therefore governs each vertical normal mode, whose <Rossby deformation radius> is $c/f_0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 3 / ii / Solution

{c}

For a mode with streamfunction $\phi$, deformation radius $L_D$, and constant planetary-vorticity gradient $\beta$, the linear Rossby-wave equation is
$$
\partial_t(\nabla_h^2\phi-L_D^{-2}\phi)+\beta\phi_x=F.
$$
In the long-wave limit it becomes a first-order westward transport equation.


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 Incoming links: Linear Rossby-wave equation