The Charney Stability Problem with a Lower Ekman Layer

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  • 1 Department of Meteorology, and Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee 32306
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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.

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.

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