The Charney Stability Problem with a Lower Ekman Layer

Abstract

A linear, continuous, infinite-depth beta-plane model is used to study the effects of Ekman pumping on the stability of a zonal flow similar to that considered by Charney (1947). The linear, quasi-geostrophic, inviscid potential vorticity equation is used in the interior, and the effects of friction at the earth?s surface are modeled by prescribing a vertical velocity at the lower boundary which is consistent with homogeneous Ekman layer dynamics. That equation is solved analytically and the solution substituted into the lower boundary condition to yield a dispersion relation which is then solved numerically. The results indicate that Ekman pumping at the lower boundary leads to a reduction in instability for disturbances of all wavelengths. The dissipation causes the formation of a short-wave cutoff to instability for all shear values, and in cases of small shear a long-wave cutoff is also produced.