On the structure and growth rate of unstable modes to the Rossby-Haurwitz wave

Authors

  • Yuri N. Skiba,

    Corresponding author
    1. Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito Exterior, CU, C.P. 04510, México, D.F., México
    • Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito Exterior, CU, CP 04510, México, DF, México
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  • Ismael Pérez-García

    1. Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito Exterior, CU, C.P. 04510, México, D.F., México
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Abstract

The normal mode instability study of a steady Rossby-Haurwitz wave is considered both theoretically and numerically. This wave is exact solution of the nonlinear barotropic vorticity equation describing the dynamics of an ideal fluid on a rotating sphere, as well as the large-scale barotropic dynamics of the atmosphere. In this connection, the stability of the Rossby-Haurwitz wave is of considerable mathematical and meteorological interest. The structure of the spectrum of the linearized operator in case of an ideal fluid is studied. A conservation law for perturbations to the Rossby-Haurwitz wave is obtained and used to get a necessary condition for its exponential instability. The maximum growth rate of unstable modes is estimated. The orthogonality of the amplitude of a non-neutral or non-stationary mode to the Rossby-Haurwitz wave is shown in two different inner products. The analytical results obtained are used to test and discuss the accuracy of a numerical spectral method used for the normal mode stability study of arbitrary flow on a sphere. The comparison of the numerical and theoretical results shows that the numerical instability study method works well in case of such smooth solutions as the zonal flows and Rossby-Haurwitz waves. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

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