Conservation laws of perturbations of the Rossby-Haurwitz waves

Abstract

The work considers meteorologically important Rossby-Haurwitz (RH) waves, which are classical solutions of the nonlinear vorticity equation describing the motion of an unforced ideal fluid on a rotating sphere. These waves belong to subspaces H₁ ⊕ H_n (n ≥ 2), where H_k is the subspace of homogeneous spherical polynomials of degree k. The stability of waves and the dynamics of their perturbations are analyzed. A conservation law for arbitrary (and not just infinitesimal) perturbations is derived, which divides arbitrary perturbations of the RH wave into invariant sets Mⁿ_–, Mⁿ₀ and Mⁿ₊. The set Mⁿ₀ contains the invariant subspace H_n of neutral (stable) perturbations. A hyperbolic law between the energy and average spectral number of perturbations belonging to the sets Mⁿ_– and Mⁿ₊ is obtained. A geometric interpretation of the change in perturbation energy is given. Examples of the dynamics of RH wave perturbations are considered. A necessary condition for linear instability is obtained. For zonal solutions (Legendre polynomials and zonal RH waves), it complements the well-known Rayleigh-Kuo and Fjörtoft conditions. The bounds on the growth rate of unstable modes are also given, and the orthogonality of the amplitude of any unstable mode to the basic wave is demonstrated. The results clarify the spectral structure of unstable disturbances and are useful for testing computational algorithms in numerical studies of linear stability.