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The Short-Wave Limit of Linear Equatorial Kelvin Waves in a Shear Flow

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  • 1 University of Michigan, Ann Arbor, Michigan
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Abstract

The effects of latitudinal shear on equatorial Kelvin waves in the one-and-one-half-layer model are examined through a mixture of perturbation theory and numerical solutions. For waves proportional to exp(ikx), where k is the zonal wavenumber and x is longitude, earlier perturbation theories predicted arbitrarily large distortions in the limit k → ∞. In reality, the distortions are always finite but are very different depending on the sign of the equatorial jet. When the mean jet is westward, the Kelvin wave becomes very, very narrow. When the mean jet flows eastward, the Kelvin wave splits in two with peaks well off the equator and exponentially small amplitude at the equator itself. The phase speed is always a bounded function of k, asymptotically approaching a constant. This condition has important implications for the nonlinear behavior of Kelvin waves. Strong nonlinearity cannot be balanced by contracting longitudinal scale, as in the author’s earlier Korteweg–deVries theory for equatorial solitons: for sufficiently large amplitude, the Kelvin wave must always evolve to a front.

Corresponding author address: John P. Boyd, Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward Ave., Ann Arbor, MI 48109-2143. Email: jpboyd@umich.edu

Abstract

The effects of latitudinal shear on equatorial Kelvin waves in the one-and-one-half-layer model are examined through a mixture of perturbation theory and numerical solutions. For waves proportional to exp(ikx), where k is the zonal wavenumber and x is longitude, earlier perturbation theories predicted arbitrarily large distortions in the limit k → ∞. In reality, the distortions are always finite but are very different depending on the sign of the equatorial jet. When the mean jet is westward, the Kelvin wave becomes very, very narrow. When the mean jet flows eastward, the Kelvin wave splits in two with peaks well off the equator and exponentially small amplitude at the equator itself. The phase speed is always a bounded function of k, asymptotically approaching a constant. This condition has important implications for the nonlinear behavior of Kelvin waves. Strong nonlinearity cannot be balanced by contracting longitudinal scale, as in the author’s earlier Korteweg–deVries theory for equatorial solitons: for sufficiently large amplitude, the Kelvin wave must always evolve to a front.

Corresponding author address: John P. Boyd, Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward Ave., Ann Arbor, MI 48109-2143. Email: jpboyd@umich.edu

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