Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations

Stability of the Rossby–Haurwitz (RH) wave of subspace H1 ⊕ Hn 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 Mn− , Mn+ , Hn and Mn 0 −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 Mn− and Mn+ 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 Mn− (χ(t ) < n(n + 1)), any nonzonal RH wave of subspace H1 ⊕Hn (n 2) 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 Mn 0 − 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 Mn+ of small-scale perturbations (χ(t ) > n(n+ 1)) is still open problem.  2003 Elsevier Inc. All rights reserved.