Nonlinear and linear instability of the Rossby–Haurwitz wave

The dynamics of perturbations to the Rossby–Haurwitz (RH) wave is analytically analyzed. These waves, being of great meteorological importance, are exact solutions to the nonlinear vorticity equation describing the motion of an ideal incompressible fluid on a rotating sphere. Each RH wave belongs to a space $H_1\oplus H_n$, where $H_n$ 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 $\eta(t)$ 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 $\chi(t)=\eta(t)/K(t)$. 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 $\chi(t)$. 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 $M_{-}^n$ ($\chi(t)<n(n+1)$), any nonzonal RH wave of subspace $H_1\oplus H_n$ ($n\ge 2$) 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 $\chi(t)$ 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 are shown in two Hilbert spaces. The instability in the invariant set $M_{+}^n$ of small-scale perturbations ($\chi(t)>n(n+1)$) is still an open problem.