Topographic Rossby-wave dispersion relation

In a one-layer quasi-geostrophic fluid with effective meridional PV gradient ${\beta }_{\mathrm{eff}}$,Bottom slope contributes to ${\beta }_{\mathrm{eff}}$ through vortex stretching.

In a square with impermeable walls and background topographic potential-vorticity gradient $\beta$, the linear shallow-water quasi-geostrophic potential vorticity equation is ${\partial }_t({\nabla }^2-L_D^{-2})\psi +\beta {\psi }_x=0$. For a nonzero-frequency normal mode, setting $\hat{\psi }=\psi e^{i\beta x/(2\omega )}$ gives a Helmholtz equation with homogeneous wall data. The eigenfunctions are $\hat{\psi }=\sin (m\pi x/L)\sin (n\pi y/L)$, with $m,n\ge 1$, and the displayed frequencies. The positive and negative frequency representations are complex conjugates for a real wave.

A depth discontinuity between two constant-depth rotating layers supports an exponentially trapped Rossby wave. Continuity of surface height and cross-step volume flux gives $\omega (H_{+}{\kappa }_{+}+H_{-}{\kappa }_{-})=fk(H_{+}-H_{-})$, where ${\kappa }_{\pm }^2=k^2+(f^2-{\omega }^2)/(g{H}_{\pm })$ and positive roots give decay. For $H_{\pm }=H_0\pm d$ with $|d|\ll H_0$, the low-frequency branch has $\omega \simeq (fd/H_0)k/\sqrt{k^2+R_D^{-2}}$. In the Northern Hemisphere it propagates with the shallow side on its right.

A long disturbance trapped on a depth step is sometimes called a double Kelvin wave. With $H_{+}>H_{-}$, $f>0$ and $c_{\pm }=\sqrt{g{H}_{\pm }}$, cross-step geostrophic balance and the along-step acceleration yield the transport equation shown for a background height jump ${\eta }_0\mathrm{sgn}(x)$. Its zero-initial-amplitude solution is $A={\eta }_0[\mathrm{sgn}(x)-\mathrm{sgn}(x-(c_{+}-c_{-})t)]$. The adjusted region diverts cross-step volume flux along the step; the propagating front still carries flux. This outer approximation does not resolve fast adjustment at the fronts.

The long step-trapped topographic Rossby wave has equal phase velocity and group velocity, $c_T=fd{R}_D/H_0$. For an outer surface profile $\eta ={\eta }_0\mathrm{sgn}(x)-A(x,t)e^{-|y|/R_D}$, cross-step matching gives $A_t+c_T{A}_x=2c_T{\eta }_0\delta (x)$. With $A(x,0)=0$, its weak solution is $A={\eta }_0[\mathrm{sgn}(x)-\mathrm{sgn}(x-c_{T}t)]$. This describes the slowly varying outer response; the fast adjustment region at the origin is unresolved.