On Liapunov and Exponential Stability of Rossby–Haurwitz Waves in Invariant Sets of Perturbations

Autor: Yuri N. Skiba
Rok vydání: 2018
Předmět:
Zdroj: Journal of Mathematical Fluid Mechanics. 20:1137-1154
ISSN: 1422-6952
1422-6928
Popis: In this work, the stability of the Rossby–Haurwitz (RH) waves from the subspace $$\mathbf {H}_{1}\oplus \mathbf {H}_{n}$$ is considered ( $$n\ge 2$$ ) where $$\mathbf {H}_{k}$$ is the subspace of the homogeneous spherical polynomials of degree k. A conservation law for arbitrary perturbations of the RH wave is derived, and all perturbations are divided into three invariant sets $$\mathbf {M}_{-}^{n}$$ , $$\mathbf {M}_{0}^{n}$$ and $$\mathbf {M} _{+}^{n}$$ in which the mean spectral number $$\chi (\psi ^{\prime })$$ of any perturbation $$\psi ^{\prime }$$ is less than, equal to or greater than $$ n(n+1) $$ , respectively. In turn, the set $$\mathbf {M}_{0}^{n}$$ is divided into the invariant subsets $$\mathbf {H}_{n}$$ and $$\mathbf {M}_{0}^{n}{\setminus } \mathbf {H}_{n}$$ . Quotient spaces and norms of the perturbations are introduced, a hyperbolic law for the perturbations belonging to the sets $$\mathbf {M}_{-}^{n}$$ and $$\mathbf {M}_{+}^{n}$$ is derived, and a geometric interpretation of variations in the kinetic energy of perturbations is given. It is proved that any non-zonal RH wave from $$\mathbf {H}_{1}\oplus \mathbf {H} _{n}$$ ( $$n\ge 2$$ ) is Liapunov unstable in the invariant set $$\mathbf {M} _{-}^{n}$$ . Also, it is shown that a stationary RH wave from $$\mathbf {H} _{1}\oplus \mathbf {H}_{n}$$ may be exponentially unstable only in the invariant set $$\mathbf {M}_{0}^{n}{\setminus } \mathbf {H}_{n}$$ , while any perturbation of the invariant set $$\mathbf {H}_{n}$$ conserves its form with time and hence is neutral. Since a Legendre polynomial flow $$aP_{n}(\mu )$$ and zonal RH wave $$-\,\omega \mu +aP_{n}(\mu )$$ are particular cases of the RH waves of $$\mathbf {H} _{1}\oplus \mathbf {H}_{n}$$ , the major part of the stability results obtained here is also true for them.
Databáze: OpenAIRE
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