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Every forced propagating mode must have the imposed zonal frequencyFor $C>0$, only the equatorial Kelvin wave has the correct sign. Upward group velocity in the fluid $z>0$ selectsWith $K(y)=\exp [-\beta y^2/(2C)]$, its pressure field can be writtenHere $v_K=0$, $u_K={\varphi }_K/C$, and the complex vertical-velocity amplitude isThe lower boundary is matched only ifafter choosing the phase of $a_K$ to make the prescribed cosine amplitude real.

For $C<0$, the upward-radiating equatorial Rossby waves haveChoose each pair $({\hat{v}}_n,{\hat{\varphi }}_n)$ as in part c for this $m_n$. The propagating sum isThe other fields follow mode by mode from part c and ${\hat{w}}_n=-kC{m}_n{\hat{\varphi }}_n/N^2$. Thus the boundary matching condition is

For $C>0$, the space of upward-radiating wave profiles is only the one-dimensional Gaussian Kelvin profile, so a generic $W(y)$ cannot be matched by propagating waves. Its Kelvin projection radiates upward; the remaining forcing produces a balanced, vertically evanescent response trapped near the lower boundary. This is the equatorial analogue of the fact that quasi-geostrophic Rossby waves have westward rather than eastward phase propagation.

A quasi-geostrophic mountain wave is a stationary Rossby wave forced when a geostrophic basic flow crosses topography. Its vertical wavenumber is real only when the intrinsic Rossby-wave phase speed permits vertical propagation.

A stationary boundary corrugation can resonate with a Rossby wave whose intrinsic propagation cancels the background current. For constant $N$, rigid upper height $H$, and $m=\pi /H$, define ${\kappa }^2=k^2+{\ell }^2+f_0^2{m}^2/N^2$. When $U=\beta /{\kappa }^2$, a lower corrugation proportional to $e^{ikx}\sin (\ell y)$ admits a disturbance proportional to $e^{ikx}\sin (\ell y)[Bt\cos (mz)+C(z-H)\sin (mz)]$, with $B=2ik{f}_0{U}^2/\beta$ and $C=N^2/(f_{0}m)$. Its wave activity grows quadratically with time while the upward Eliassen–Palm flux grows linearly. The linear prediction applies only while the disturbance remains small.

The wavevectors of a nonzero-frequency Rossby wave in a resting shallow layer satisfyThis follows by completing the square in the shallow-water Rossby-wave dispersion relation. A real circle requires $|\omega |\le |\beta |/(2|a|)$. Its two intersections with a given tangential wavenumber provide the incident and reflected normal wavenumbers at a meridional wall.

In zonally averaged barotropic quasi-geostrophic approximation flow with zero mean meridional transport, the Taylor identity for quasi-geostrophic flux relates the meridional potential vorticity flux to the convergence of Reynolds stress. A local downgradient closure $\overline{v^{\prime }{q}^{\prime }}=-\kappa {\overline{Q}}_y$, with $\kappa \ge 0$, therefore gives westward mean acceleration where the background gradient is positive. Rossby wave breaking can stir PV into fine filaments; coarse-graining or weak diffusion produces the inferred mixing. Globally the momentum-flux convergence integrates to boundary stresses, so localized westward changes need compensating transport or momentum changes elsewhere in an isolated system.

For a resting midlatitude layer, the slow Rossby wave equation is $({\nabla }_h^2-a^2){\eta }_t=-\beta {\eta }_x$, where $a=f_0/\sqrt{gH}$. With the plane wave convention $e^{i(kx+ly-\omega t)}$,The zonal phase velocity is westward for $\beta >0$, but the zonal group velocity changes sign at $k^2=l^2+a^2$.

For a uniform eastward current $U>0$ on a beta plane with $\beta >0$, stationary nondivergent Rossby waves have $k^2+l^2=\beta /U$. Their ground-frame group velocity is $2U(k^2,kl)/(k^2+l^2)$, tracing a circle centered at $(U,0)$ with radius $U$. Packets emitted by a localized source throughout $0<s<t$ occupy the disk shown in geometrical ray theory. This envelope is distinct from the full balanced near field. A westward current has no nontrivial stationary propagating branch in this model.

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.

Thermal damping makes the vertical wavenumber of a stationary topographic Rossby wave complex. A vertically propagating wave then decays while oscillating, whereas an inviscidly evanescent disturbance acquires both decay and phase tilt; both can exert wave drag.