Barotropic Rossby wave

A barotropic Rossby wave has no vertical shear. For horizontal wavenumbers $k,l$, its quasi-geostrophic frequency is $\omega =-\beta k/(k^2+l^2)$.

For a stationary $n=2$, $l=0$ wave in a uniform zonal current, a perturbation $n>2$ has phase speed $U(1-4/n^2)$ and zonal group speed $U(1+4/n^2)$. Phase speed relative to the air is westward; packet propagation relative to the ground is faster downstream.

A wave packet can transfer energy downstream faster than individual phase features move, allowing successive downstream disturbances to develop while upstream ones weaken. In the simple uniform-current barotropic model with zero meridional wavenumber, the zonal group speed is $U+\beta /k^2$ while phase speed is $U-\beta /k^2$. Linear superposition and interference are not direct nonlinear eddy scattering. Real weather development also involves baroclinic energy conversion and wave–mean-flow feedback, which require additional dynamics.

In the uniform-current beta-plane model, $U=\beta /(k_2^2+l^2)$ with $k_2=2/(a\cos \varphi )$. This is a model-dependent balance between eastward advection and westward intrinsic propagation, not a unique speed set by zonal wavenumber alone.

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.