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

Abstract

Stability of the Rossby–Haurwitz (RH) wave of subspace H 1⊕ H n in an ideal incompressible fluid on a rotating sphere is analytically studied ( H n 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 , H n and M 0 n − H n 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 H n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H n , the factor norm controls the perturbation part orthogonal to H n . It is shown that in the set M − n ( χ( t)< n( n+1)), any nonzonal RH wave of subspace H 1⊕ H n ( 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 M 0 n − H n . 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.