Inviscid disturbance dynamics in barotropic shear flows

Abstract

The inviscid nature of disturbance evolution in shear flows is investigated as an initial-value problem within the framework of nondivergent vorticity dynamics. To ensure a basic understanding of physical processes, disturbance evolution is first considered in a rectilinear system of simple shear. Particular emphasis is placed on identifying how the disturbance evolution depends on the zonal wavenumber and on the meridional structure of the initial conditions. Insight acquired from the rectilinear problem is then applied to a bounded Rankine vortex. Here, the dependency of disturbance evolution on the azimuthal wavenumber is of special interest. Recent development of a low-frequency balance theory for rapidly rotating vortices has provided observational evidence that the low azimuthal wavenumber asymmetries, especially wavenumber one, are dominant in the near-vortex region. The results of this work provide further theoretical evidence of an inviscid wave number selection mechanism that preferentially damps the higher wavenumber asymmetries. The radial structure and location of the initial conditions are found to be critical factors in determining how rapidly a disturbance is compressed or elongated. This in turn controls the rate of disturbance growth or decay. For swirling flows, a definition of an effective shear that accounts for both the radial variations in the initial conditions as well as the radial variation in the angular velocity is proposed. Using the reciprocal of this effective shear, time scales for a disturbance to decay to half its initial energy, the half-life time, are calculated for initial conditions and symmetric wind profiles that are found in hurricanes. Simple shear flow and the bounded Rankine vortex do not admit discrete modal solutions since there is no mean state vorticity gradient to support them. The unbounded Rankine vortex is briefly considered in order to investigate how the presence of discrete neutral modes modifies the nonmodal solutions presented in this work.