The inner product in
2 Existence and uniqueness of the BVE solutions
Note that if ψ is a BVE solution then ψ + const is also the solution. We ignore this constant by searching a solution in spaces of functions orthogonal to a constant on the sphere. Spaces of functions in which a solution exists are important in many applications, and, in particular, in studying the stability of solutions.
In this section, we formulate two theorems proved in .
Here M is the constant from the estimate |J(ψ, h)| ≤ M ||Δψ|| ||Δh|| (see ). The case s = 1 and σ = 0 was proved in [4, 5], whilst the cases s = 1 and s = 2 (σ ≠ 0) were proved in . Theorem 2 considers the general case when s ≥ 1 is a real number.
3 Existence of a limited attractive set
Let us study the asymptotic behavior of the BVE solutions as t → ∞.
Let s ≥ 1, and let
- 1if r ≥ 0 then
- 2if r ∈ [−1, 0) then
According to (12) and (17), if some solution ψ belongs to the set B at time t0 then it will belong to B for all t > t0. Hence, all steady and periodic solutions (if they exist) belong to the set B. Evidently, the set B contains the maximal BVE attractor . Theorem 3 is also valid if
4 A functional for the stability study
One can see from (19) and (20) that the first Jacobian in (18) does not affect the behavior of the perturbation energy K(t), while the second Jacobian in (18) does not affect the perturbation enstrophy η(t). Moreover, the sphere rotation and nonlinear term (the last two terms in the LHS of (18)) do not affect the behavior of K(t) and η(t).
Let us consider three examples when the basic solution is zero or represents meteorologically important flows, such as super-rotation, or a homogeneous spherical polynomial. Each basic solution is assumed to be supported by appropriate forcing.
The basic flow is a super-rotation:
Then J(ψ′, Δψ′) = 0 for any initial perturbation ψ′ from the subspaceHn, and R (t) ≡ 0. Besides, such a perturbation will never leaveHn, i.e., the subspaceHn is the invariant set of perturbations to the polynomial flow (24). Moreover, due to (23), Q(t) ≤ Q(0) exp(−2ρt), and therefore any initial perturbation ψ ′ (0, λ, μ) fromHn will exponentially tend to zero with time, without leavingHn. In other words, the invariant setHn belongs to the domain of attraction of solution (24).
5 Global Asymptotic Stability of BVE solutions
Let us obtain sufficient conditions for the BVE solution to be a global attractor.
Note that in a limited domain on the plane, the condition for the global asymptotic stability of a smooth BVE solution was earlier obtained in  under the condition that rotation and linear drag are not taken into account (σ = 0) and s = 1. Theorem 4 extends this result to smooth flows on a rotating sphere when s ≥ 1, and the linear drag is also taken into account (σ ≠ 0).
It should be emphasized that in both assertions, the basic solution
then Q(t) is the Lyapunov function, and the solution
In contrast to Theorem 4, Theorem 5 requires a non-zero viscosity coefficient ν. According to conditions (28), the basic solution must have continuous derivatives only up to the second order. Therefore, (31) can be applied to a wider class of BVE solutions. For example, the main solution to problem (4)–(5) can be one of the non-stationary modons [7, 15] supported, despite the dissipation, by the corresponding external forcing. As it is known,
6 Dimension of global spiral BVE attractor
Besides, it was shown in Example 3 that the subspace Hn is the domain of attraction of this solution.
Since the frequencies ωm are rationally independent, the solution
Thus, we prove the following assertion:
Let s ≥ 1, ν > 0, σ ≥ 0, and let F(t, x) ∈ Hnbe a quasiperiodic forcing (34) – (35) of the BVE equation (4). Then solution (37) from the subspaceHn is a global attractor provided that condition (43) is satisfied.
It follows from (44) that for a fixed finite value of the generalized Grashof number G, it is always possible to determine such an integer n(G) that the spiral solution generated by any quasiperiodic forcing (34)–(35) from subspace Hn with n ≥ n(G) is a global BVE attractor. For example, if we take
The result obtained is not unexpected. Indeed, for a fixed coefficient v(s), the number G(s) is fixed if the L2-norm (36) of the forcing is a constant independent of n. Let the amplitudes |fm| of oscillations of forcing be nonzero for all m. Then they must decrease as n grows, and for a sufficiently large number n (or for sufficiently small amplitudes |fm|), the viscosity v(s) can become sufficient to satisfy condition (27) for the global asymptotic stability of the quasiperiodic solution (37).
Thus, unlike the case of stationary forcing when the Hausdorff dimension of the global BVE attractor is limited above by the generalized Grashof number G , in the case of the quasiperiodic forcing (34), the Hausdorff dimension 2n of the global spiral attractor (37) is not limited by the generalized Grashof number G and can become arbitrarily large as the degree n of the BVE forcing increases.
This result is of particular meteorological interest, since it shows that the dimension of the global attractor in the barotropic atmosphere can be unlimited, even if the generalized Grashof number (33) is bounded. Thus, the dimension of the global attractor crucially depends not only on the generalized Grashof number, but also on the time-space structure of the BVE forcing. This also shows that the search for a global attractor of small dimension in the barotropic atmosphere  is theoretically unjustified due to the fact that forcing usually has a very complex structure with a huge number of degrees of freedom.
This work was partially supported by the National System of Researchers (SNI, CONACYT, Mexico) through grant 14539.
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