Nonlinear Waves and Coherent Vortex Structures in Barotropic β-plane Jets

Abstract

We examine the nonlinear evolution of barotropic ?-plane jets on a periodic domain with a pseudospectral. A calculation of the linear growth rate yields an infected U-shaped curve on the ? versus k0 plane which separates regions of stability and instability. This curve aids in clarifying the morphology of the nonlinear structures which evolve from monochromatic small-amplitude perturbations of wavenumber k0. At very small or zero ?, we recover and further quantify previously obtained results, including formation of: dipolar vortex structures or bound pools of opposite-signed vortex regions at small k0; staggered streets isolated vortex pools at intermediate k0; and ?cat-eyes? or staggered connected pools of vorticity at large but still unstable k0. As ? is increased, the jet exhibits quite different evolutionary patterns. At low k0, where the laminar jet may be stable, we find a multistage instability. First, neutrally stable long-wavelength modes of small amplitude interact nonlinearly to produce harmonics in the linear unstable band. These grow at an exponential rate until a near-steady wake appears. However, the wake is unstable to the initial long wavelength modes and a rapid merger (i.e., backward energy cascade) occurs. At an intermediate k0, the presence of ? causes a ?reversal? of vortex pools in the meridional direction of the near-steady vortex street. That is for a west-to-east flowing jet the Positive pools of vorticity are south of the negative pools causing a decrease in the near-steady velocity of the jet. Retrograde Rossby radiation is observed and weak ?shingles? or cast-away vortex pools are observed. The meander amplitude pulsates, pumping Rossby radiation into the far field. The merger and binding processes also occur on a jet excited by many harmonics, with an ensuing chaos that is very sensitive to initial conditions.