The influence of circulation on the stability of vortices to mode-one disturbances

The initial value problem for the two-dimensional inviscid vorticity equation, linearized about an azimuthal basic velocity field with monotonic angular velocity, is solved exactly for mode-one disturbances. The solution behaviour is investigated for large time using asymptotic methods. The circulation of the basic state is found to govern the ultimate fate of the disturbance: for basic state vorticity distributions with non-zero circulation, the perturbation tends to the steady solution first mentioned in Michalke & Timme (1967), while for zero circulation, the perturbation grows without bound. The latter case has potentially important implications for the stability of isolated eddies in geophysics.