On Liapunov and Exponential Stability of Rossby–Haurwitz Waves in Invariant Sets of Perturbations

Stability of a class of solutions of the barotropic vorticity equation on a sphereequation on a sphere

“…of symmetric and positive definite spherical Laplace operator −Δ, corresponding to the eigenvalue χ n [22].…”

“…This equation also describes the main features of large-scale atmospheric dynamics [19]. The Liapunov and exponential stability of infinitely differentiable BVE solutions (Rossby-Haurwitz waves and Legendre polynomial flows) were analyzed in [22]. In this work, we study the stability of weak BVE solutions, namely, Verkley's modons [24,25,26] and antisymmetric Wu-Verkley (WV) waves [28].…”

“…Stationary modons and WV waves are considered in Section 7, and the conservation law for infinitesimal perturbations of these solutions and conditions for their normal mode stability are obtained in Sections 8 and 9, respectively. Unlike flows in the form of Legendre polynomials (LP-flows) and Rossby-Haurwitz waves (RH-waves) [22], these conditions depend not only on the degree of the spherical polynomial representing the basic solution, but also on the spectral distribution of the energy of the normal mode. The new conditions are used in Section 10 to estimate the maximum growth rate of unstable modes.…”

Asymptotic behavior of solutions to barotropic vorticity equation on a sphere

“…According to (12) and 17, if some solution ψ belongs to the set B at time t 0 then it will belong to B for all t > t 0 . Hence, all steady and periodic solutions (if they exist) belong to the set B. Evidently, the set B contains the maximal BVE attractor [14].…”

“…Let ψ(x) ∈ C ∞ 0 (S), and χ n = n (n + 1). We introduce the derivative Λ s = (−∆) s/2 of real degree s of ψ(x) as [11,12]. Let s be a real.…”