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

G. R. FlierlMassachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences, Center for Meteorology and Physical Oceanography, Cambridge, MA 02139;

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P. Malanotte-RizzoliMassachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences, Center for Meteorology and Physical Oceanography, Cambridge, MA 02139;

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N. J. ZabuskyUniversity of Pittsburgh, Department of Mathematics and Statistics, Pittsburgh, Pittsburgh, PA 15261

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Abstract

We examine the nonlinear evolution of barotropic β-plane jets on a periodic domain with a pseudospectral. A calculation of the linear growth rate yields an infected U-shaped curve on the β 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 β, 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 β 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 β 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.

Abstract

We examine the nonlinear evolution of barotropic β-plane jets on a periodic domain with a pseudospectral. A calculation of the linear growth rate yields an infected U-shaped curve on the β 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 β, 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 β 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 β 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.

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