The Eady problem for a basic state with zero PV gradient but beta does not equal 0

Abstract

The classic Eady problem is modified to include beta does not equal 0, but with the basic distributions of temperature and zonal flow adjusted to preserve zero meridional gradients of basic-state potential vorticity in the fluid interior. Much of the mathematical simplicity of the classic problem is retained; however, the results differ in important ways. Specifically, the instability now has a long-wave cutoff in addition to the traditional short-wave cutoff. The former is associated with the fact that the phase speeds of the edge waves begin to differ so much as wavenumber is reduced that the two edge waves can no longer interact in order to form unstable modes. For the unstable modes, this manifests itself in that the steering level for unstable modes is always below the middle of the fluid and approaches the lower boundary near the long-wave cutoff. Relatedly, the amplitude of the unstable geopotential perturbations is larger at the upper boundary than at the lower boundary. Finally, below the long-wave cutoff, one of the neutral waves has a phase speed that becomes increasingly easterly as wavenumber decreases. This allows a resonant response to planetary-scale stationary forcing.