An effective-β vector for linear planetary waves on a weak mean flow

An effective-β vector and an accompanying mean flow advection vector are computed for linear planetary waves on a weak, horizontally uniform mean flow, for arbitrary stratification and vertical shear profiles. These vectors arise from the projections of mean flow vertical shear and horizontal advection terms onto vertical structure functions related to the rest-state linear vertical mode. As such, these vectors are independent of depth, and determine the propagation direction and speed for the linear planetary waves on the weak mean flow. The resulting dispersion relation gives the modified wave frequency as the sum of a Doppler shift term and a dispersion function that has the same form as the rest-state planetary wave dispersion function, but relative to the effective-β vector. The propagation of long-waves remains non-dispersive. A modal decomposition of the mean flow illustrates the non-Doppler effect. For two examples of climatological mean flow and stratification profiles, the theory is shown to reproduce accurately the modified long-wave zonal phase speeds from the full linear long-wave solution on a purely zonal mean flow.