Dynamic Instability of Two-Dimensional Nondivergent Flow in a Barotropic Atmosphere

In discussing wave motions along a strong jet of westerlies, we can no longer neglect the variation of west wind with latitude, and therefore must consider the possibility of having a phase velocity whose value is between the maximum and minimum west-wind velocities. The study of wave motions is thus extended from the very long and slowly moving or retrograding waves into the realm of ordinary waves and cyclone waves and it is found that, for nondivergent barotropic motion, the condition for the presence of neutral and amplified waves with a phase velocity whose value is between the maximum and minimum wind velocity in the belt is the existence of critical points where the absolute vorticity has an extreme value. If no such point exists, then all perturbations must be damped. When this condition is satisfied, both amplified (unstable) and neutral waves can be expected. The wave moving with a velocity equal to the current velocity at the critical point is neutral while those with a velocity less than this value but greater than the minimum wind velocity will be amplified. The amplification will be greatest when the phase velocity is intermediate between the latter two values; therefore, both fast and slowly moving waves will have little amplification. The degree of instability will also depend upon the sharpness of the velocity profile. When the wave is unstable, the trough line will be directed from southeast toward northwest to the south of the point of minimum absolute vorticity and from southwest toward northeast to the north of the point of maximum vorticity.