## 1 Introduction

*S*= {

*x*∈ ℝ

^{3}: |

*x*| = 1} which are orthogonal to any constant, and by

*n*≥ 1, |

*m*| ≤

*n*) that form the orthonormal system in

*δ*is the Kronecker delta. Each spherical harmonic

_{mk}*[10] Let n be a natural, and*

*Then*

*n*≥ 1. Denote by

*n*+ 1)-dimensional eigensubspace of homogeneous spherical polynomials of degree

*n*, which corresponds to the eigenvalue (2) [8]. The set 2

*n*+ 1 of spherical harmonics

**H**

_{n}. The Hilbert space

**H**

_{n}is the closure of

_{n}(

*ψ*) the projection of function

**H**

_{n}. Thus,

*χ*=

_{n}*n*(

*n*+ 1). We introduce the derivative Λ

^{s}= (−Δ)

^{s/2}of real degree

*s*of

*ψ*(

*x*) as

*s*be a real. Denote by

The inner product in
*ψ*, *h*〉_{s} = 〈Λ^{s}*ψ*, Λ^{s}*h*〉. Thus,
*ψ*|| = ||Λ*ψ*|| [10].

*[11] Let r, s and t be real numbers, r < t*,

*and*

*Then*

## 2 Existence and uniqueness of the BVE solutions

*S*[11]:

*ψ*is the stream function, Δ

*ψ*is the relative vorticity, Δ

*ψ*+ 2

*μ*is the absolute vorticity,

*F*(

*t*,

*x*) is the forcing,

*σ*Δ

*ψ*is the Rayleigh friction in the planetary boundary layer,

*S*, while the nonlinear term

*J*(

*ψ*, Δ

*ψ*) and linear term

*J*(

*ψ*, 2

*μ*) = 2

*ψ*represent the advection and the rotation of sphere, respectively. The velocity vector

_{λ}*ν*(−Δ)

^{s+1}

*ψ*, where −Δ is the spherical Laplace operator and

*s*≥ 1 is a real number.

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 [11].

*[11] Let s*≥ 1,

*ν*>0

*and σ*≥ 0.

*Suppose that*

*at initial moment, and*

*Then the non-stationary BVE problem (4)–(5) has a unique weak solution*

*such that*

*and*

*holds for all t*∈ (0,

*T*)

*and*

*[11] Let s*≥ 1,

*ν*> 0

*and σ*≥ 0.

*Suppose that*

*Then there exists at least one weak solution*

*of the stationary equation*

*such that*

*holds for all*

*If additionally*

*then the problem solution is unique.*

Here *M* is the constant from the estimate |*J*(*ψ*, *h*)| ≤ *M* ||Δ*ψ*|| ||Δ*h*|| (see [11]). 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 [10]. 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**be a stationary forcing of equation (4), r* ≥ 1. *Then there is a limited set***B***in a space***X***that attracts all BVE solutions ψ*(*t*, *x*), *besides*,

*1**if r*≥ 0*then*$\text{X}={\mathbb{H}}_{0}^{2}$ *and*$\text{B}=\{\psi \in {\mathbb{H}}_{0}^{2}:{\Vert \psi \Vert}_{2}\le {C}_{1}(r,s){\Vert F\Vert}_{r}\hspace{0.17em}\hspace{0.17em}.$ *2**if r*∈ [−1, 0)*then*$\text{X}={\mathbb{H}}_{0}^{1}$ *and*$\text{B}=\{\psi \in {\mathbb{H}}_{0}^{1}:{\Vert \psi \Vert}_{1}\le {C}_{2}(r,s){\Vert F\Vert}_{r}$ *where*${C}_{1}(r,s)=\frac{{a}^{-r}}{\sigma +{2}^{s}\nu},\hspace{0.17em}\hspace{0.17em}{C}_{2}(r,s)=\frac{{a}^{-r-1}}{\sigma +{2}^{s}\nu},\hspace{0.17em}\hspace{0.17em}a=\sqrt{2}\hspace{0.17em}\hspace{0.17em}.$

*Part 1*. Let

*r*≥ 0 and

*ψ*and the use of relations

*Part 2.*Let

*r*∈ [−1, 0) and

*ψ*and the use of (9) give

*F*,

*ψ*〉 and

*ν*||Λ

^{s+1}

*ψ*||

^{2}leads to

*ρ*is defined by (11), or

*ψ*|| = ||

*ψ*||

_{1}, we obtain

Q.E.D.

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]. Theorem 3 is also valid if
*ω* is the period. In this case, one should only replace in (7) and (8) the norms ||*F*||* _{r}* by the norms

## 4 A functional for the stability study

*ψ*′(

*t*,

*λ*,

*μ*) of the solution

*s*≥ 1,

*ν*> 0 and

*σ*≥ 0. Taking the inner product (1) of equation (18) in series with

*ψ*′ and Δ

*ψ*′, then using (9), we obtain the integral equations

*ψ*′, respectively.

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*).

*ψ*′(

*t*,

*λ*,

*μ*) of a basic flow

*p*and

*q*are non-negative real numbers, not equal to zero simultaneously. Multiplying (19) and (20) by

*p*and

*q*, respectively, and combining the results, we obtain

*ρ*is defined by (11).

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.

*Let**(this solution exists if F*(*x*) ≡ 0*). Then**and**in (19) and (20). Therefore, in the non-dissipative case (σ* = *ν* = 0*), the zero solution is stable, since the perturbation energy and enstrophy are constant. In the dissipative case (σ* ≠ 0 *and/or ν* ≠ 0*), the zero solution is globally asymptotically stable, because the energy and enstrophy of any perturbation decrease exponentially with time*.

*The basic flow is a super-rotation:**where C* = *const. Then R*(*t*) = 0 *due to (9), while Q*(*t*) *is the Lyapunov function. Thus, the super-rotation flow is Lyapunov stable if σ* = *ν* = 0, *and is the global BVE attractor (asymptotically Lyapunov stable) if ρ* > 0. *It is easy to prove that the same is true for any flow from subspace***H**_{1}, *since it represents a super-rotation flow about some axis of a sphere [10]*.

*The basic flow is a homogeneous spherical polynomial:*

*(n*: ≥ 2

*):*

*Then J*(*ψ*′, Δ*ψ*′) = 0 *for any initial perturbation ψ′ from the subspace***H*** _{n}, and R* (

*t*) ≡ 0.

*Besides, such a perturbation will never leave*

**H**

_{n}, i.e., the subspace**H**

*(*

_{n}is the invariant set of perturbations to the polynomial flow (24). Moreover, due to (23), Q*t*) ≤

*Q*(0) exp(−2

*ρt*),

*and therefore any initial perturbation ψ*′ (0,

*λ*,

*μ*)

*from*

**H**

_{n}will exponentially tend to zero with time, without leaving**H**

_{n}. In other words, the invariant set**H**

*.*

_{n}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.

*J*(

*ψ*,

*h*)| ≤ |∇

*ψ*| · |∇

*h*| we get

The use of inequality (26) in (23) leads to

*Let s*≥ 1,

*ν*> 0

*and σ*≥ 0.

*If*

*where p and q are defined by (25), then Q*(

*t*)

*is the Lyapunov function, and solution*

*is a global BVE attractor, besides, any its perturbation ψ′ will exponentially decrease with time.*

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 [13] 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

*ɛ*-inequality one can estimate

*R*(

*t*) as:

*ɛ*

^{2}=

*pa*

^{s−1}/

*ν*in order to eliminate the two terms containing

*Let s*≥ 1,

*ν*> 0

*and σ*≥ 0.

*And let*

*be such a solution of equation (4) that the numbers p and q defined by (28) are limited. If*

*then Q*(*t*) *is the Lyapunov function, and the solution**is a global BVE attractor, besides, any its perturbation will exponentially decrease with time.*

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,

*Let σ*= 0

*and s*= 1

*in equation (4), and let*

*be a stationary solution from the subspace*

**H**

*≥ 2):*

_{n}of homogeneous spherical polynomials (n*The solution (32) is supported by a steady forcing whose Fourier coefficients*

*are*

*where*

*and χ*=

_{n}*n*(

*n*+ 1).

*According to Example 3, subspace*

**H**

*≠ 0.*

_{n}is the domain of attraction of solution (32) for any ν*Moreover, it follows from (28) that p*=

*χ*

_{n}q, and due to Theorem 5, solution (32) is the global attractor of equation (4) if*Thus, in order for basic flow (32) to be a global BVE attractor, the viscosity coefficient ν must increase with increasing velocity (q) and degree n of the flow (32).*

## 6 Dimension of global spiral BVE attractor

*s*= 1 and

*s*= 2 is limited by the generalized Grashof number

*χ*

_{1}= 2 is the smallest positive eigenvalue of the spherical Laplace operator (see (2)).

*n*in space:

*f*is a constant, and the frequencies

_{m}*ω*are incommensurate. Thus,

_{m}*F*(

*t*,

*x*) ∈

**H**

_{n}. Note that

*n*or/and amplitudes

*f*.

_{m}*J*(

*ψ*, Δ

*ψ*) = 0 for any function

*ψ*∈

**H**

_{n}, there exists an exact BVE solution

**H**

_{n}defined by the Fourier coefficients

Besides, it was shown in Example 3 that the subspace **H**_{n} is the domain of attraction of this solution.

Since the frequencies *ω _{m}* are rationally independent, the solution

*n*-dimensional torus in the (2

*n*+ 1)-dimensional complex space

**H**

_{n}. According to Theorem 3 in [9], the closure of this trajectory coincides with the torus. Hence, the Hausdorff dimension of the attractive set, that is solution (37), coincides with that of the torus and equals 2

*n*.

*p*=

*χ*=

_{n}q*n*(

*n*+ 1)

*q*. Therefore, condition (27) for the global asymptotic stability of solution (37) accepts the form

*F*|| of forcing (34), (35) is time-independent (see (36)). Due to (39)–(41),

*χ*

_{1}= 2, the last inequality can be written in terms of the generalized Grashof number:

Thus, we prove the following assertion:

*Let s* ≥ 1, *ν* > 0, *σ* ≥ 0, *and let F*(*t*, *x*) ∈ **H**_{n}*be a quasiperiodic forcing (34) – (35) of the BVE equation (4). Then solution (37) from the subspace***H*** _{n} is a global attractor provided that condition (43) is satisfied*.

*s*= 2 and

*s*= 1, solution (37) is the global BVE attractor if

*s*= 1 corresponds to the Navier-Stokes equations.

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 **H**_{n} with *n* ≥ *n*(*G*) is a global BVE attractor. For example, if we take
*n* is equal to or greater than 8.

The result obtained is not unexpected. Indeed, for a fixed coefficient *v*(*s*), the number *G*(*s*) is fixed if the *L*_{2}-norm (36) of the forcing is a constant independent of *n*. Let the amplitudes |*f _{m}*| 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 |

*f*|), the viscosity

_{m}*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* [3], in the case of the quasiperiodic forcing (34), the Hausdorff dimension 2*n* 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 [1] 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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