Abstract :
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.
