Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations |
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Authors: | Yu. N. Skiba |
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Affiliation: | Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México, DF 04510, Mexico |
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Abstract: | Stability of the Rossby–Haurwitz (RH) wave of subspace H1Hn in an ideal incompressible fluid on a rotating sphere is analytically studied (Hn is the subspace of homogeneous spherical polynomials of degree n). It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets M−n, M+n, Hn and M0n−Hn depending on the value of their mean spectral number χ(t)=η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M−n and M+n due to a hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced using the invariant subspace Hn of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to Hn, the factor norm controls the perturbation part orthogonal to Hn. It is shown that in the set M−n (χ(t)<n(n+1)), any nonzonal RH wave of subspace H1Hn (n2) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M0n−Hn. A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n+1), where n is the RH-wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave is shown. The instability in the invariant set M+n of small-scale perturbations (χ(t)>n(n+1)) is still open problem. |
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Keywords: | Ideal incompressible flow Liapunov and exponential instability Rossby– Haurwitz wave |
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