The interaction of a baroclinic mean flow with long Rossby waves

The effect of a baroclinic mean flow on long oceanic Rossby waves is studied using a combination of analytical and numerical solutions of the eigenvalue problem. The effect is summarized by the value of the nondimensional number R-2 = beta N(2)h/f(2)U(z) when the mean flow shear keeps a constant sign throughout the water column. Because previous studies have shown that no interaction occurs if the mean flow has the shape of the first unperturbed mode (the non-Doppler shift effect), an implicit assumption in the application of the present work to the real ocean is that the relative projections of the mean flow on the second and higher modes remain approximately constant. Because R-2 is large at low latitudes between 10 degrees and 30 degrees (the southern branches of subtropical gyres or the regions of surface westward shear), the phase speed of the first mode is very slightly decreased from the no mean flow standard theory case. Between 30 degrees and 40 degrees latitudes (the northern branches of the subtropical gyres or the regions of surface eastward shear), R-2 is O(10) and the westward phase speed can increase significantly (up to a factor of 2). At still higher latitudes when R-2 is O(1) a critical transition occurs below which no discrete Rossby waves are found that might explain the absence of observations of zonal propagations at latitudes higher than 50 degrees. Our case studies, chosen to represent the top-trapped and constant-sign shear profiles of observed mean flows, all show the importance of three main effects on the value of the first baroclinic mode Rossby wave speed: 1) the meridional gradient of the quantity N-2/f (where N is the buoyancy frequency) rather than that of the potential vorticity fN(2); 2) the curvature of the mean flow in the vertical direction, which appears particularly important to predict the sign of the phase speed correction to the no-mean-flow standard theory case: increase (decrease) of the westward phase speed when the surface-intensified mean flow is eastward (westward); and 3) a weighted vertical average of the mean flow velocity, acting as a Doppler-shift term, which is small in general but important to determine the precise value of the phase speed.