Solvability of Vorticity Equation on a Sphere

The chapter is devoted to the questions of existence, uniqueness, and asymptotic stability of solutions to the barotropic vorticity equation (BVE) for a viscous and forced fluid on a rotating sphere. Unlike the other works devoted to the vorticity equation on a sphere, we generalize the theorems on the unique solvability and global asymptotic stability of the BVE solution to the case of a more general form of the turbulent viscosity term; to wit, instead of the classical Navier-Stokes form ν Δ 2 ψ $$\nu \Delta ^{2}\psi$$ (Dymnikov and Filatov, Mathematics of Climate Modeling, Birkhäuser, Boston (1997); Ilyin and Filatov, Mathematical Physics, pp. 128–145, Leningrad Pedagogical Institute, Leningrad (1987); Ilyin and Filatov, Dokl. Akad. Nauk SSSR 301(1), 18–22 (1988)), we consider the term ν ( − Δ ) s ψ $$\nu (-\Delta )^{s}\psi$$ where s ≥ 2 is a real number. Hyperdiffusion of this sort is used because ordinary linear diffusion is often too dissipative for many applications. The forcing is also considered as a function of fractional degree of smoothness on the sphere. In Sect. 3.1, a non-stationary problem for the nonlinear barotropic vorticity equation on the sphere is formulated. Some important properties of the nonlinear term (Jacobian) are established in Sect. 3.2. The existence and uniqueness of a weak solution of non-stationary problem is proved in Sect. 3.3. In the particular case when a weak solution is sufficiently smooth, it is a classical solution. In Sect. 3.4, the existence of weak stationary solutions for the barotropic vorticity equation on the sphere is proved, and a condition that guarantees the uniqueness of a solution of this problem is obtained. It is shown that the degree of smoothness of stationary solution depends on real degree s of the Laplace operator in the term of turbulent viscosity ν ( − Δ ) s + 1 ψ $$\nu (-\Delta )^{s+1}\psi$$ . The boundedness of attractive sets and the asymptotic behavior of solutions (as time tends to infinity) as well as sufficient conditions for the global asymptotic stability of classical and weak solutions are examined in Sect. 3.5. The role of the structure and smoothness of forcing is also analyzed. Sufficient conditions for the global asymptotic stability of smooth and weak solutions are obtained in Sect. 3.6. The dimension of the global vorticity equation attractor is evaluated in Sect. 3.7 provided that the external forcing is a quasi-periodic (in time) homogeneous spherical polynomial of degree n. It is known that in the case of a stationary forcing, the Hausdorff dimension of the global BVE attractor is limited by the Grashof number. We show that in the case of a non-stationary quasi-periodic forcing, the dimension of global BVE attractor depends not only on the generalized Grashof number, but also on the spatial and temporal structure of BVE forcing. Moreover, we show that the Hausdorff dimension of a globally attractive spiral solution may become arbitrarily large as the degree n of the quasi-periodic forcing grows, even if the generalized Grashof number is bounded. It should be noted that a quasi-periodic forcing, as compared with a time-invariant forcing, more adequately describes the effects of small-scale baroclinic processes which are not explicitly described by the barotropic vorticity equation. This result has a meteorological interest, since it shows that the search of a finite-dimensional global attractor in the barotropic atmosphere is not well justified.