Boundary Rossby wave

A boundary Rossby wave is supported by a buoyancy perturbation or potential-temperature anomaly on a rigid horizontal boundary. Its intrinsic propagation depends on the boundary orientation and basic buoyancy gradient; the upper and lower boundary waves of the Eady model propagate oppositely relative to their local background flows.

For a semi-infinite fluid above a rigid horizontal boundary, constant buoyancy frequency $N$, and thermal wind $U=\Lambda z$, a zero-interior-potential vorticity perturbation has vertical dependence $e^{-\mu z}$, where $\mu =NK/|f_0|$ and $K=(k^2+l^2{)}^{1/2}$. Material conservation of boundary buoyancy gives $c{\varphi }^{\prime }(0)+\Lambda \varphi (0)=0$. Thus its phase velocity relative to the boundary flow is $c=\Lambda /\mu$. The wave's time dependence resides in its boundary condition; the interior field follows by potential-vorticity inversion. MIT's Eady-edge-wave notes develop this boundary-wave interpretation.

With uniform thermal wind $U=\Lambda z$ above a plane boundary, write the zero-interior-potential vorticity response as ${\psi }^{\prime }=\mathrm{Re}\{A{e}^{-\mu z+i(kx+ly)}\}$, where $\mu =N\sqrt{k^2+l^2}/f$ and $f>0$. Boundary vertical velocity $ϵ\cos (kx+ly-{\omega }_{0}t)$ gives $C=N^2ϵ/(f\mu )$ and natural angular frequency $\omega =\Lambda k/\mu$. An initially zero response has $A=C(e^{-i{\omega }_{0}t}-e^{-i\omega t})/[i(\omega -{\omega }_0)]$. Its continuous resonant limit is $Ct{e}^{-i\omega t}$: forcing in phase with the boundary mode supplies energy every cycle, and linear undamped theory predicts secular growth.

For a semi-infinite stratified fluid with thermal wind $U=\Lambda z$ above a gently sloping boundary $z=\alpha y$, a zero-interior-potential vorticity perturbation decays as $e^{-\mu z}$, where $\mu =N|k|/|f_0|$. The boundary buoyancy perturbation obeys a single oscillatory equation with phase velocity $c=(\Lambda -\alpha N^2/f_0)/\mu$. The topographic and thermal-wind contributions cancel when the boundary follows a background isopycnal.

Two vertically unbounded regions with buoyancy frequencies $N_1,N_2$ and thermal wind shears ${\Lambda }_1,{\Lambda }_2$ support a localized wave if pressure and normal velocity are continuous at their common material interface. With decays ${\varphi }_1=A{e}^{{\mu }_{1}z}$ below and ${\varphi }_2=A{e}^{-{\mu }_{2}z}$ above, ${\mu }_j=N_{j}K/|f_0|$, matching givesWhen $N_1/N_2\to \infty$ with bounded shear, the lower layer becomes effectively rigid and the upper-layer Eady edge wave speed ${\Lambda }_2/{\mu }_2$ is recovered.