Numerical Study of the Unstable Modes of a Hyperbolic-Tangent Barotropic Shear Flow

Abstract

An unbounded hyperbolic-tangent barotropic shear flow is assumed. The complex phase speeds and eigenfunction structure of unstable modes are studied numerically. A reversal of the planetary and shear vorticity gradient is required for these modes to exist. For sufficiently small wavenumber, the unstable modes, which can be associated with the modes of a discontinuous shear layer, are found on both sides of the neutral solution curve in wavenumber-shear parameter space. For sufficiently large wavenumbers, the unstable modes are contiguous to the neutral solution. For an intermediate range of wavenumbers, there are two sets of unstable modes (one contiguous to the neutral solution and one not); one set merges with the zero β unstable mode and one set vanishes for sufficiently small β (or large shear). Above some critical value of wavenumber, the set of modes at fixed wavenumber merging with the neutral solution is contiguous to the neutral boundary, whereas below the critical wavenumber value this set of modes extends beyond the neutral boundary, while the set of modes contiguous to the neutral boundary vanishes for sufficiently large shear. For shears comparable to or less than the neutral boundary shear and for values of wavenumber at least as small as those for which two modes are found, one set of unstable modes is essentially wavelike on the relatively westerly side of the shear zone. These waves can extract momentum from great distances beyond the region of shear. More generally, latitudinally propagating Rossby waves incident on a region of negative vorticity gradient with phase speed matching mean flow speed within this region should be over-reflected, with largest over-reflection occurring for phase speed near that of an unstable mode.