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.