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

Abstract

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.