Abstract
Abstract The behavior of a viscous incompressible fluid on a rotating sphere is described by the nonlinear barotropic vorticity equation (BVE). Conditions for the existence of a bounded set that attracts all BVE solutions are given. In addition, sufficient conditions are obtained for a BVE solution to be a global attractor. It is shown that, in contrast to the stationary forcing, the dimension of the global BVE attractor under quasiperiodic forcing is not limited from above by the generalized Grashof number.
Highlights
Let us denote by C∞0 (S) the space of infinitely differentiable functions on the unit sphere S = {x ∈ R3 : |x| = 1} which are orthogonal to any constant, and byf, g = f (x)g(x)dS and f = f, f 1/2 (1)S the inner product and the norm of functions of C∞0 (S), respectively
Denote by Ynm (λ, μ) the spherical harmonics (n ≥ 1, |m| ≤ n) that form the orthonormal system in C∞0 (S): Ynm,Ylk = δmkδnl where δmk is the Kronecker delta
Each spherical harmonic Ynm is the eigenfunction of the eigenvalue problem
Summary
Denote by Ynm (λ , μ) the spherical harmonics (n ≥ 1, |m| ≤ n) that form the orthonormal system in C∞0 (S): Ynm,Ylk = δmkδnl where δmk is the Kronecker delta. Each spherical harmonic Ynm is the eigenfunction of the eigenvalue problem. The set 2n + 1 of spherical harmonics Ynm (λ , μ) (−n < m ≤ n) form the orthonormal basis in Hn. The Hilbert space L20(S) = ⊕∞n=1Hn being the direct orthogonal sum of the subspaces Hn is the closure of C∞0 (S) in the norm (1). [11] Let r, s and t be real numbers, r < t , a = 2, and ψ ∈ Hs0+t.
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