Yuri N. Skiba

Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Circuito de la Investigación Científica s/n, Ciudad Universitaria, 04510 México, D.F.
E-mail: skiba@unam.mx

Received: January 29, 2015; accepted: August 20, 2015

Resumen

Abstract

The nonlinear barotropic vorticity equation (BVE) describing the vortex dynamics of viscous incompressible and forced fluid on a rotating sphere is considered. The asymptotic behavior of solutions of nonstationary BVE as t → ∞ is studied. Particular forms of the external vorticity source are given that guarantee the existence of a bounded attractive set in the phase space of solutions. The asymptotic behavior of the BVE solutions is shown to depend on both the structure and the smoothness of external forcing. Three types of sufficient conditions for global asymptotic stability of smooth and weak BVE solutions are also given. Simple attractive sets of a viscous incompressible fluid on a sphere under quasi-periodic polynomial forcing are considered. Each attractive set represents a BVE quasi-periodic solution of the complex (2n + 1)-dimensional subspace H_n of homogeneous spherical polynomials of degree n. The Hausdorff dimension of its trajectory, being an open spiral densely wound around a 2n-dimensional torus in H_n, equals to 2n. As the generalized Grashof number G becomes small enough then the domain of attraction of such spiral solution is expanded from H_n to the entire BVE phase space. It is shown that for a given G, there exists an integer n_G such that each spiral solution generated by a forcing of H_n with n ≥ n_G is globally asymptotically stable. Thus we demonstrate the difference in the asymptotic behavior of solutions in the cases, then Grashof number G is fixed and bounded, but the forcing is stationary or non-stationary. Whereas the dimension of the fluid attractor under a stationary forcing is limited above by G, the dimension of the spiral attractive solution (equal to 2n) may become arbitrarily large as the degree n of the quasi-periodic polynomial forcing grows. Since the small-scale quasi-periodic functions, unlike the stationary ones, more adequately depict the forcing in the barotropic atmosphere, this result is of meteorological interest and shows that the dimension of attractive sets depends not only on the forcing amplitude, but also on its spatial and temporal structure. This example also shows that the search of a finite-dimensional global attractor in the barotropic atmosphere is not well justified.

Keywords: Incompressible viscous and forced fluid on a sphere, asymptotic behavior, global stability, attractor dimension.

1. Introduction

The nonlinear barotropic vorticity equation (BVE) describing the vortex dynamics of a viscous incompressible and forced fluid on a rotating sphere is considered, which takes into account the Rayleigh friction, the sphere rotation, the external vorticity source (forcing) F(t, x), and the turbulent viscosity term of common form ν(—∆)^s+1ψ where s ≥ 1 is an arbitrary real number. The case s = 1 corresponds to the classical form used in Navier-Stokes equations (Ladyzhenskaya, 1969; Szeptycki, 1973a, b; Temam, 1984; Tribbia, 1984; Ilyin and Filatov, 1988), while the case s = 2 was considered for example in Simmons et al. (1983), Dymnikov and Skiba (1987a, b), and Skiba (1989). The turbulent term of such form for natural numbers s is applied in Lions (1969) for studying the solvability of Navier-Stokes equations in a limited area by the artificial viscosity method. The unique solvability of nonstationary BVE for arbitrary real number s ≥ 1, as well as the existence of weak solution to the stationary BVE, was shown in Skiba (2012). A condition guaranteeing the uniqueness of such steady solution is given in the same work.

Many works have been devoted to the study of large-time behavior of 2D vorticity equation solutions (Temam, 1985; Marchioro, 1986; Ladyzhenskaya, 1987; Constantin et al., 1988; Giga and Kambe, 1988; Giga et al. 1988, 2010; Doering and Gibbon, 1991; Babin and Vishik, 1989; Ilyin, 1994; Skiba, 1994; Gibbon, 1996; Gallagher and Gallay, 2005; Gallay and Wayne, 2005, 2007; Yu, 2005). In particular, an extensive literature deals with the question whether or not the vorticity of unforced two dimensional flow on ² converges to a self-similar solution (Giga and Kambe, 1988; Giga et al., 1988, 2010; Carpio, 1994; Gibbon, 1996; Cao et al., 1999; Gallagher and Gallay, 2005; Gallay and Wayne, 2005, 2007). Both experimental and numerical studies of unforced viscous fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. It was proved by Gallay and Wayne (2005) that in two dimensions, localized regions of vorticity do evolve toward a vortex. More precisely they prove that any solution of the two-dimensional Navier-Stokes equation, whose initial vorticity distribution is integrable, converges to an explicit self-similar solution called Oseen's vortex. This implies that the Oseen vortices are dynamically stable for all values of Reynolds number, and these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Under slightly stronger assumptions on the vorticity distribution, they gave precise estimates on the rate of convergence toward the vortex. This result is applicable to the problem of the formation of the Burgers vortex in a three-dimensional flow (Gallay and Wayne, 2007), which is a very interesting topic in fluid mechanics.

In this work, particular forms of the external vorticity source have been found which guarantee the existence of a bounded set that eventually attracts all the BVE solutions. It is shown that the asymptotic behavior of solutions depends on both the structure and the smoothness of an external vorticity source. Sufficient conditions for the global asymptotic stability of both smooth and weak BVE solutions are also given.

2. Spherical harmonics, projectors and fractional derivatives

Let S = {X∈³:|X|=1} be a unit sphere in the 3D Euclidean space and let be the set of infinitely differentiable functions on S. We denote by

the inner product and norm in , respectively. Here x = (λ, μ) is a point of the sphere, dS = dλdμ, is an element of sphere surface, μ= sinφ; μ∈[—1,1], φ is the latitude, λ∈[0,2π) is the longitude and ḡ is the complex conjugate of g. It is well known that for each integer n ≥ 0, the 2n + 1 spherical harmonics (λ, μ) with |m|≤ n are orthogonal eigenfunctions of the spectral problem —∆= X_n, (|m|≤ n, X_n(n+1)) for spherical Laplace operator —∆, which form a generalized (2n + 1)-dimensional eigensubspace

of homogeneous spherical polynomials of degree n (Richtmyer, 1981).

Orthogonal projector Y_n: ↦ H_n is introduced by means of the convolution with Legendre polynomial P_n(x) (Skiba, 1989, 2004):

Note that each function ψ(x)∈ is represented by its own Fourier-Laplace series and .

Let s > 0 and ψ(x)∈. The derivative Λ^s = (—Δ)^s/2 of real order s is defined as a multiplier operator, defined by infinite set of multiplicators :

Besides,

Obviously, operator Λ^s may be defined on functions from = {ψ∈ : Y₀(ψ)=0} by means of (4) for any real degree s. In particular, Λ²ⁿ = (—Δ)ⁿ for a natural n, and operator Λ can be interpreted as the square root of nonnegative and symmetric Laplace operator. Unlike the local derivatives δⁿ/δλⁿ and δⁿ/δμⁿ, the derivatives Λ^s and projectors Y_n are invariant with respect to any element of the group SO(3) of sphere rotations (Skiba, 2012).

3. Hilbert spaces ^s

We denote the completion of in norm (1) as the Hilbert space ⁰ = of functions on S. For any real s, we introduce the inner product and norm ||•||_s in as

We denote the Hilbert space obtained by closing the space in norm (10) as ^s. We will keep the symbols and ||•||_s for the inner product and norm in ⁰. Let 0 < s < r. Then the imbeddings are continuous, and the dual space (^s)* coincides with ^-s (Agranovich, 1965).

Let s and r be real numbers. Operator Λ^{r =} ↦ is symmetric, , and hence, closable, and extended to ^s. Namely, an element z ∈ ^s is called the rth derivative Λ^rψ of a function ψ ∈ ^s if 〈z, h〉_s = 〈z^r, Λh〉_s holds for all h ∈ (Skiba, 1989).

Lemma 1 (Skiba, 1989). Let s and r be real numbers, r > 0, and ψ ∈ ^s+r. Then

Due to Lemma 1, the mapping Λ^r : ^s+r ↦ ^s is isometric and isomorphic for any real s and r. In particular, at r = -2s, the operator Λ^-2s : ^-s ↦ ^s is an isometric isomorphism.

Lemma 2 (Skiba, 1989). Let r, s and t be real numbers, r< t, α and ψ ∈ ^s+t. Then

4. Vorticity equation

The dynamics of viscous and forced nondivergent barotropic fluid on the sphere S is described by the nonlinear barotropic vorticity equation

written in the geographical coordinate system (λ, μ) whose pole N is on the axis of rotation of the sphere (Skiba, 1969). Here ψ is the streamfunction, ∆ψ(t, x) is the relative vorticity, ∆ψ + 2μ is the absolute vorticity, F(t, x) is the forcing, σ∆ψ describes the Rayleigh friction in the planetary boundary layer (σ ≥ 0),

is the Jacobian, J(ψ,2μ)=2ψ_λ describes the rotation of sphere, and is the unit outward normal at a point of the sphere. The fluid velocity is solenoidal: . The turbulent viscosity term has the form ν(—Δ)^s+1ψ, where v > 0 and s ≥ 1 is arbitrary real number (Skiba, 2012). As it was mentioned before, the value s = 1 corresponds to the classical viscosity term in Navier-Stokes equations (Szeptycki,1973a, b; Temam, 1984, 1985; Ladyzhenskaya, 1987; Ilyin and Filatov, 1988), s = 2 was considered in (Simmons et al., 1983; Dymnikov and Skiba, 1987a, b; Skiba, 1994), while natural numbers of s were used in (Lions, 1969) for proving the solvability of Navier-Stokes equations in a limited area by means of the method of artificial viscosity. Note that (14) is considered in the classes of functions, orthogonal to a constant on S, thus, Y₀(ψ) = 0 and Y₀(F) = 0.

We now briefly consider the main properties of Jacobian (15). Let all functions be complex-valued. It is clear that

Let n be a natural, and r be a real. Since Λ^s Yⁿ(ψ) = Yⁿ(ψ) we get J(ψ, Λ^sψ)= 0 for any ψ ∈ H_n. Obviously, J(ψ, h)= 0 for any ψ(μ) and h(μ). Also note that

and

holds for sufficiently smooth complex-valued functions ψ, g and h on S (Skiba, 1989).

Let be the set of complex numbers, ψ ∈ , ψ : S → , and let G(ψ) = G ° ψ be a superposition of two functions. Then 〈J(ψ, h),〉 = 0.

Lemma 3 (Skiba, 2012). Let r be a real number, and ψ, h ∈ . Then

Lemma 4 (Skiba, 1989). Let ψ, h ∈ ². Then J(ψ, h) belongs to ⁰ and

5. Attractive set of BVE solutions

The existence and uniqueness of the weak solution to nonstationary BVE (14), as well as the existence of a weak solution of steady BVE were proved in Skiba (2012). A condition for the uniqueness of the steady solution was also given in Skiba (2012). Besides self-interest, the analysis of classes of functions, in which there exist BVE solutions is particularly important in the stability study of solutions.

We now consider particular forms of the forcing guaranteeing the existence in a phase space X of a bounded set B.

Theorem 1 (Skiba, 2013). Let s ≥ 1 in (14) and let F(x)∈^r be a steady forcing where r ≥ —1. Then every solution ψ(t, x) of BVE (14) will eventually be attracted by a bounded set B⊂X. Moreover,

I. If r ≥ 0 then X = ² and

Here α = is the constant from Lemma 2.

Remark 1. All the steady and periodic solutions (if they exist) belong to the set B. Obviously, the set B contains the maximal attractor of the BVE (Temam, 1985).

Remark 2. If F(t, x) is a periodic forcing of space C(0, ω;^r) where ω is the time period, then Theorem 1 is also valid with the obvious change of the norm ||F||_r by the norm .

We now show that under certain conditions on the forcing and dissipation, the maximal attractor of BVE (9) coincides with the zero solution.

Theorem 2. If forcing F(t, x) is such that the integral converges then ||ψ(t)||_x → 0 as t → ∞. Besides, X = ² if r ≥ 0 and X = ¹ if —1 ≤ r < 0.

Proof. Consider only the case F(t, x) ∈ ^r where r ∈[—1,0) is proved similarly. Taking the inner product of Eq. (9) with ∆ψ and using Lemma 3 we get

With lemmas 1 and 2, the terms 〈F, ∆ψ〉 and ν||Λ^s+2 ψ||² can be estimated as

where α = , and

Then (19) implies

Integrating (20) with respect to t from τ to t we obtain

where ||ψ||₂ = ||∆ψ|| (see [6] and [7]). Multiplying (20) by ||∆ψ|| and integrating the result with respect to t from τ to t, we obtain

Estimating the norm ||ψ(t')||₂ in the r.h.s. of the last inequality with the help of (21) we get

Here we have 0 ≤ τ ≤ t ≤ T. Hence, if integral is finite then integral is also finite. Therefore, there exists a subsequence t_k→ ∞ such that ||ψ(t_k)||₂ → 0. Using (21) in the case when τ = t_k → 0 we obtain that ||ψ(t)||₂ → 0 as → ∞. The theorem is proven.

6. Positive functional for the stability study

We now introduce a positive functional related with kinetic energy and enstrophy of perturbations and derive an equation describing its behavior in time. Examples are given for the meteorologically important flows having the form of a super-rotation flow, a homogeneous spherical polynomial of degree n or a Rossby-Haurwitz wave.

Let (t, λ, μ) be a solution to BVE (9) under consideration, and let (t, λ, μ) be another solution of (9). Then

where s ≥ 1, v > 0, and σ ≥ 0, holds for the perturbation ψ(t, λ, μ)= (t, λ, μ) — (t, λ, μ) of .Taking the inner product (1) of Eq. (22) successively with ψ and ∆ψ and using the relations (14) and Lemma 3, we obtain two integral equations:

for the kinetic energy K(t) = ||∇ψ||² and enstrophy η(t) = ||∆ψ||² of the perturbation, respectively.

It follows from (23) and (24) that the Jacobian J(ψ, ∆) in (22) can change the perturbation enstrophy η(t) but does not affect the perturbation energy K(t). On the contrary, the other Jacobian J(, ∆ψ) in (22) has no effect on n(t) but can change K(t). As for the last two terms in the l.h.s. of (22) (the super-rotation term and the non-linear term), they both have no influence on the behavior of K(t) and η(t).

Example 1. If = 0 (such a solution exists if F(x)≡0) then 〈J(ψ, ∆ψ),)〉 = 0 and 〈J(ψ, ∆ψ),∆)〉 in (23) and (24). Therefore, in the non-dissipative case (σ, μ = 0), the zero solution is stable, since the perturbation energy and enstrophy will be constant. In a dissipative case (σ ≠ 0 and/or μ ≠ 0), the zero solution is globally asymptotically stable, since the perturbation energy and enstrophy will exponentially decrease in time.

Let now be a solution of (9), and let p and q be non-negative numbers, not equal to zero simultaneously. We will measure the magnitude of perturbation with the functional

Multiplying (23) and (24) byp and q, respectively, and combining the results, we obtain

where

Lemma 2 leads to —||Λ^s+1ψ||² ≤ —2^s||∇ψ||², and —||Λ^s+2ψ||² ≤ —2^s||∆ψ||², and hence (26) can be estimated as

Example 2. Super-rotation basic flow. Let = (μ) ≡ Cμ where C is a constant. Then R(t) = 0 due to (15) and Q(p, q, ψ, t) is the Liapunov function. Thus the super-rotation flow is Liapunov stable if σ = μ = 0, and is the global attractor (asymptotically Liapunov stable) if p > 0. The same is true for any flow from subspace H₁ (Skiba, 1989).

Example 3. Flow in the form of a homogeneous spherical polynomial. Let (t,x) ∈ H_n for some n ≥ 2, that is,

In particular, it can be a zonal Legendre-polynomial flow: (μ) = CP_n (μ). Then J(ψ, ∆ψ) = 0 for any perturbation of H_n, and R(t) ≡ 0. By (22), any perturbation of H_n will never leave H_n, that is, H_n is invariant set of perturbations to flow (29); besides, due to (28), Q(p, q, ψ, t) ≤ Q(p, q, ψ,0) exp(—2pt). Thus any initial perturbation of H_n will exponentially tend to zero with time not leaving H_n, that is, set H_n belongs to the domain of attraction of solution (29).

Example 4. The basic flow is a linear combination of the flows considered in examples 2 and 3. In particular, (t, λ, μ) can be a Rossby-Haurwitz wave (Skiba, 1989, 2004).Then the result obtained in example 3 is also valid in this case.

7. Evolution of functional Q(p, q, ψ, t)

Eq. (22) for a perturbation (t, x) of solution (t, x) can be written as

where

is a linear operator that has a compact resolvent if v > 0 (Skiba, 1989, 1998). Using the Lagrange identity for defining the adjoint operators, Eq. (26) can be written as

where

is the symmetric operator in the space ⁰.

Let ω_n and G_n(x) be eigenvalues and orthonormal eigenfunctions of the spectral problem

Note that functions G_n(x) represent the orthonormal basis in the real space ⁰, and operator B : ⁰ → ⁰ also has a compact resolvent (v > 0), and ω_n are real isolated eigenvalues of geometrical multiplicity one. The only possible limit point of the spectrum is ω — ∞. Thus the number of positive eigenvalues of B is finite.

Let us now renumerate the eigenvalues {ω_n} of operator B in such a way that their values are decreasing with increasing values of n. Besides, assume that the first N eigenvalues ω₁,..., ω_n are positive. The streamfunction of a perturbation ψ(t,x) can be represented by its Fourier series

As a result we obtain

Assume that an initial perturbation ψ(t₀, x) has the form of a single eigenfunction G_n(x): Ψ(t₀, x) = α_n(t₀)G_n(x), then

Therefore, if ω_n > 0 (or ω_n< 0) the functional Q(t) of such initial perturbation will increase (decrease). Note that the growth (decay) rate of Q(t) at the moment t₀ is determined by the modulus of eigenvalue ω_n and by the amplitude α_n(t₀) of perturbation. In particular, if p = 1, q = 0 and all the eigenvalues are ordered so that ω_n+1 ≤ ω_n, then perturbation of the form of eigenfunction G₁(x) will cause the fastest growth of the perturbation energy K(t).

Let us consider a sequence {α_n} of the Fourier coefficients of a perturbation (35) as a point in the phase space of perturbations to BVE solution (t, x). Then due to (36), the condition

defines a subset in the coordinate space of points {α_n}. Any perturbation y, whose Fourier coefficients {α_n} belong to , generates the growth of Q(t).

It is interesting to study a structure of the set . Obviously, set is unbounded because it includes the N-dimensional Euclidean space 𝔼_N of vectors {α₁, α₂,..., α_N} except for its origin {0, 0,..., 0}. The set is of infinite dimension and is not invariant with respect to applying the nonlinear operator Lψ —J(ψ, ∆ψ) of Eq. (22), that is, the trajectory of perturbation ψ(t, x) can enter and leave the set . Obviously, among all the points {α_n} belonging to a surface the maximum of Q(t) is achived when α₁ = and α_n =0 for all n ≥ 2.

It follows from (38) that if is bounded then is also bounded. Since |ω_n| → ∞ to as n → ∞ (recall that the operator B has a compact resolvent), the inequality < C defines a compact set in the coordinate space of sequences , which is orthogonal to the N-dimensional Euclidean space 𝔼_N.

Thus, the nonlinear evolution process for perturbations can be described by the following way. Assume that at an initial moment t₀ the perturbation Ψ(t₀, x) is such that α_n(t₀) = 0 for all n > N, i.e. the point {α_n(t₀)}∈𝔼_N. Then functional Q(t) will grow. Since 𝔼_N is not invariant, nonzero coefficients α_n(t₀) for n > N will appear. Their growth will render the inequality (38) invalid, and the point {α_n(t)} will leave the set . From this moment the functional Q(t) will decrease. Note that the larger the number of nonzero coefficients α_n(t) with n > N, the higher the possibility for the point {α_n(t₀)} to leave the set .

Example 5. Super-rotation basic flow. Let ≡ (μ) = Cμ where C is a constant. Then

Since Δ and are commutative, the operator [2 (C—1) — CΔ] is skew symmetric, and hence,

Thus, the spherical harmonics (x) (n = 1,2,3,...; |m| ≤ n) are the eigenfunctions G_n(x) of operator B corresponding to the eigenvalues ω_n = —(σ + ν)( + PX_n) where X_n = n(n +1). Since ω_n< 0 for all n, the set defined by (38) is empty and Q(t) → 0 in time for any perturbation of super-rotation flow (μ)= Cμ (see Example 2).

8. Conditions for global asymptotic stability

In a limited domain on the plane, in the absence of linear drag, a condition for global asymptotic stability of a BVE solution was derived by Sundstrom (1969), provided that the solution has continuous derivatives up to the third order. In this section we give three sufficient conditions for the global asymptotic stability of smooth and weak BVE solutions (theorems 3-5). These conditions differ by the smoothness of basic solution. The first two conditions were proved in Skiba (2013). The first condition (Theorem 3) generalizes Sundstrom's result to a flow on a rotating sphere when the linear drag is also taken into account and s > 1 in the turbulent term. However, in the general case, the solvability theorems (Skiba, 2012) do not guarantee the existence of the solution whose third or higher derivatives are continuous. Therefore in the second condition (Theorem 4), the restriction on the smoothness of basic solution is weakened (only continuous derivatives up to the second order are required) and is in full accordance with the solvability theorems. The third condition (Theorem 5) is proved in this section for a weak BVE solution. Examples are given for a homogeneous spherical polynomial of subspace H_n and a pure dipole modon.

First, consider a rather smooth basic solution (t, x) of BVE (9) that has continuous derivatives up to the third order, so that

are both finite. Let us estimate the inner product (27) by means of functional (25) with p and q defined by (39):

Then substitution of (40) in (28) leads to

Theorem 3 (Skiba, 2013). Let s > 1, v > 0 and σ ≥ 0. If a solution (t, x) of Eq. (9) is such that the numbers p and q defined by (39) are finite, and

then Q(p, q, y, t) is the Liapunov function, besides any perturbation of (t, x) will exponentially decrease in time.

In the particular case when s = 1 and σ = 0, Theorem 3 is analogous to the assertion (Sundstrom, 1969; Yu, 2005) for the flows in a limited domain on the plane. Note that both results demand the uniform boundedness of |∇Δ(t, x)| and |∇(t, x)|. However, the existence of BVE solutions is proved only in the classes of twice continuously differentiable stream-functions (Skiba, 2012). It was shown in Skiba (2013) that the restriction on the smoothness of solution could be weakened so as to be in accordance with the requirements of the solvability theorems.

Theorem 4 (Skiba, 2013). Let s ≥ 1, v > 0, σ ≥ 0, and let (t, x) be a solution of Eq. (9) such that numbers

defined by (42) are finite. If

then Q(p, q, y, t) is the Liapunov function, besides any perturbation of (t, x) will exponentially decrease in time.

Note that in contrast to Theorem 3, Theorem 4 requires a non-zero viscosity coefficient v.

Unlike (39), condition (43) can be applied to a wider class of BVE solutions. For example, if (t, x) is a modon by Tribbia (1984), Verkley (1987) or Neven (1992), subjected to the linear drag and viscosity and supported by a certain forcing, then condition (43) is applicable, whereas (41) cannot be used because ∇Δ(t, x) is discontinuous on the boundary between the inner and outer regions of the modon.

Example 6. Let σ = 0 and s = 1 in Eq. (9), and let (x) be a stationary BVE solution from H_n (n ≥ 2):

This solution is supported by a steady forcing F(x) whose Fourier coefficients are equal to = (— + i2m) where i= and X_n = n(n+1).Due to Example 3, subspace H_n is the basin of attraction of solution (x) for any v. Besides, (42) leads to , and by Theorem 4, solution (44) is the global attractor if ν ≥ q. Thus, the larger the velocity and degree n of flow (50), the larger must be the viscosity coefficient v to provide its global asymptotic stability.

Example 7. Pure dipole modon. Let us now specify condition (43) for the pure dipole stationary modon by Verkley (1987) provided that σ = 0 and s = 1 in Eq. (9). Such a modon has the form

in the local coordinate system (λ', μ'), besides,

in the inner region S_i = {(λ', μ') ∈ S : μ' > α}, and

in the outer region S_o = {(λ', μ') ∈ S : μ' > α} of the modon (|α| < 1). It can be shown that

for the stationary modon (43). In the relations (45)-(47), μ is the sine of latitude in the geographical system (λ, μ) whose pole μ= 1 is on the axis of rotation of sphere, and X_α and X_σ are the eigenvalues of the spherical harmonics used in constructing the modon in regions S_i and S_o, besides, X_α > 0 and X_σ is a real. Note that p ≤ max {X_α, |X_σ|}q +2, where q is defined by (42). As a result, condition (43) for the global asymptotic stability of modon explicitly depends on X_α, |X_σ| and q: ν² ≥ q max {X_α, |X_σ|} + 1.

We now derive a condition for the global asymptotic stability of a steady weak solution (x) ∈ ^s+2(s ≥ 1) whose existence is proved in (Skiba, 2012).

Theorem 5. Let s ≥ 1, ν > 0 and σ ≥ 0. A steady weak solution (x) ∈ ^s+2 of BVE (9) is globally asymptotically stable if

where p = C₀||Δ||_𝕃⁴(S)= (∫_S|Δ|⁴dS)^1/4 and q= C₀||||_𝕃⁴(S) and C₀ is the constant from the estimate

(see lemmas 6 and 7 from Leray, 1933).

Proof. The functions (x) and Δ(x) belong to 𝕃⁴(S) due to lemma 4 from Leray (1933), and applying Hölder's inequality to (27) we get

Integrating the last inequality and applying (49) we obtain

Setting p = C₀||Δ||_𝕃⁴(S) and q= C₀||||_𝕃⁴(S) and using ε-inequality, we get

with p = σ + 2^sν. Applying (50) in (51) we obtain

The last term in this inequality is eliminated by setting ε² = p/(2^s+1ν). Thus (x) is globally asymptotically stable if p— pε² > 0. The theorem is proved.

Remark 3. The restriction on the coefficient v weakens when the order s in the turbulent term of Eq. (9) increases. Also note that although the number q is absent in (48), it is present implicitly through the definition of functional Q(p, q, ψ, t).

9. Hausdorff dimension of spiral attractor for a quasi-periodic forcing of H„

9.1 Hausdorff dimension of BVE attractor subjected to a stationary forcing

Estimates of the Hausdorff dimension of attractor in a viscous incompressible 2D fluid subjected in the periodic hypercube to a stationary forcing were made in Babin and Vishik (1989) and, in the more general case, in Constantin et al. (1988), Doering and Gibbon (1991), and Gibbon (1996). They state that the attractor dimension is limited by the nondimensional generalized Grashof number (Temam, 1984, 1985). In the case of sphere, the Hausdorff dimension of attractor of vorticity equation

was estimated in Ilyin (1994) as

where ε_G(s) → 0 G(s) → ∞ and G(s) is the generalized Grashof number

is the L₂-norm of the stationary forcing, and is its Fourier coefficient with respect to the orthonormal set of spherical harmonics (x) of the zonal number m and degree n ≥ 1. In particular, the inequality (52), as applied to large-scale barotropic processes of the atmosphere for s = 2 and G(2) = 1500, yields the upper limit of the barotropic atmosphere attractor dimension (Ilyin, 1994):

Despite the fact that the results (52) and (55) are of considerable theoretical interest in hydrodynamics, their practical application to the barotropic atmosphere raises doubts. Indeed, the forcing of the BVE (9) describes the influence of nonstationary small-scale baroclinic processes with rather complicated spatial and temporal behavior. So it is natural to expect that in contrast to stationary functions, small-scale quasi-periodic functions more adequately represent the BVE forcing.

9.2 Quasi-periodic spiral solution

where

i is the imaginary unit, ƒ_m is a constant amplitude, and the numbers ω_m are some incommensurate fundamental frequencies. Obviously, the geometric scale of forcing decreases as n increases. Note that the spherical variant of the well-known example by Marchioro (1986) corresponds to the time-independent forcing (56) of degree n = 1. Also note that the norm

of forcing (56), (57) is time-independent, and we will use the same generalized Grashof number (53) as in the case of stationary forcing. Obviously, there is a host of quasi-periodic forcings (56) that have the same norm (58) (or the same Grashof number [53]), but differ in their degrees n and amplitudes ƒ_m.

Due to (2) and (11), J(ψ, ∆ψ) = 0 for any ψ ∈ H_n and hence, H_n is the invariant set of BVE solutions: any solution of (9) that starts in H_n will never leave H_n. Moreover, it follows from (27) that R(t) ≡ 0, and by (26), the subspace H_n is the attraction domain for the solution _m(t, x)∈ H_n defined by its Fourier coefficients (Skiba, 2013):

Since the frequencies ω_m are rationally independent, the attractive solution (t, x) is quasi-periodic, and its path is an open (endless) spiral densely wound around a 2n-dimensional torus in the (2n + 1)-dimensional complex subspace H_n. According to theorem 3 by Samoilenko (1991), the closure of this trajectory coincides with the torus. Hence, the Hausdorff dimension of attractive solution (59) equals 2n.

Sufficient condition for the global asymptotic stability of solution (59) is given by Theorem 3. In our case (t, x)∈ H_n, and (39) leads to = n(n+1)q. As a result, the condition (41) for global asymptotic stability of solution (59) accepts the form

where

It is easy to show (see Skiba, 1994, Appendix B) that

According to (59) we have ||(t,x)||≤[σ + ν]^—1x||F|| where ||F|| is the time-independent norm (58) of forcing (56). Using the last estimates in (61) we get

where b_n[σ + ν]^—1 and hence, condition (60) is satisfied if σ + 2^Sν > b_n||F|| and even more so if

Using (53), condition (62) can be written in terms of generalized Grashof number: 2^2—s> G(s), and hence, it is satisfied if

Thus, we have proved the following assertion:

Theorem 6. Let s ≥ ν > 1, ν > 0, σ ≥ 0, F(t, x) ∈ H_n be a quasi-periodic BVE forcing (56), and let G(s) be the generalized Grashof number (53) of this equation. Then solution (59) of H_n is globally asymptotically stable provided that inequality (63) is fulfilled. Moreover, for a fixed finite Grashof number G(s), there is an integer n_G so that for any n ≥ n_G the spiral solution generated by any forcing (56) of H_n with Grashof number G is globally asymptotically stable.

In particular, solution (59) is globally asymptotically stable if 2^1/27/2> G(2) and 2^1/23/2 > G(1). The last case corresponds to the Navier-Stokes equations. For example, the condition 2^1/27/2 > G(2) is satisfied for any forcing (56) of degree n ≥ 8 if G(2) = 1500.

The result obtained is not unexpected. Indeed, for a fixed coefficient v(s), the number G(s) is fixed if the L₂-norm (58) of the forcing is a constant independent of n. The amplitudes |ƒ_m| of forcing must decrease as n grows, and for n large enough (or for the amplitudes |ƒ_m| small enough) the viscosity v(s) is sufficient to make the quasi-periodic solution (59) globally asymptotically stable.

Thus, whereas the Hausdorff dimension of the BVE attractor subjected to a stationary forcing is limited above by the generalized Grashof number G (Ilyin, 1994), the Hausdorff dimension 2n of globally attractive spiral solution (59) may become arbitrarily large as the degree n of quasi-periodic forcing (56) grows. This result has a meteorological interest, since it shows that the dimension of the global attractor in the barotropic atmosphere crucially depends on the spatial and temporal structure of BVE forcing and it can be unlimited even if the Grashof number (53) is bounded.

Remark 4. For a steady forcing F(t, x) ∈ H_n, condition (63) guarantees the global asymptotic stability of the steady BVE solution (t, x) ∈ H_n as well.

The nonlinear barotropic vorticity equation (BVE) describing the vortex dynamics of viscous incompressible and forced fluid on a rotating sphere is considered. It takes into account the Rayleigh friction, the rotation of sphere, the external vorticity source (forcing) F(t,x), and the turbulent viscosity term of common form v(—Δ)^S+1ψ, where Δ is the spherical Laplace operator and s ≥ 1 is a real number.

The large-time behavior of its solutions is studied. Specific forms of steady BVE forcing have been found which guarantee the existence of a limited set that eventually attracts all the BVE solutions (Theorem 1). Additionally, theorem 2 shows that under certain conditions on the nonstationary forcing, the maximal BVE attractor coincides with the zero solution. Thus, the asymptotic behavior of BVE solutions depends on both the structure and the smoothness of forcing. Three sufficient conditions for the global asymptotic stability of BVE solutions of different degree of smoothness are also given (theorems 3-5).

Simple attractive sets of the BVE (9) subjected to a quasi-periodic forcing of subspace H_nof homogeneous spherical polynomials of degree n are analyzed. Each such set is a spiral quasi-periodic BVE solution densely wound on a 2n-dimensional torus in H_n. The Hausdorff dimension of its trajectory equals to 2n. As the generalized Grashof number G becomes small enough then the basin of attraction of such spiral solution is expanded from H_n to the entire BVE phase space. It is shown that for a given G, there exists an integer n_G such that each spiral solution generated by a forcing of H_n with n ≥ n_G (and Grashof number G) is globally asymptotically stable (Theorem 6). Thus, whereas the Hausdorff dimension of global attractor of the BVE subjected on a sphere to a stationary forcing is limited by Grashof number G, the Hausdorff dimension of globally attractive spiral solution (59) may become arbitrarily large as the degree n of quasi-periodic forcing (56) grows. Unlike a steady forcing, a quasi-periodic forcing more adequately describes the effects of small-scale baroclinic processes in the BVE, and therefore the last result is of meteorological interest, showing that the dimension of the global BVE attractor can be unlimited even if the generalized Grashof number is limited, and hence, the global attractor dimension crucially depends not only on the magnitude but also on the spatial and temporal structure of forcing. This also shows that the search of a finite-dimensional global attractor in the barotropic atmosphere is not well justified.

Acknowledgments

This work was supported by the grant 14539 of National System of Investigators (CONACyT, SNI, Mexico).

References

Agranovich M. S., 1965. Elliptic singular integro-differential operators (Russian). Russ. Math. Surv. 20, 1-121.         [ Links ]

Babin A. and M. Vishik, 1989. Attractors of evolution equations. Nauka, Moscow (English transl., North-Holland, Amsterdam, 1992).         [ Links ]

Carpio A., 1994. Asymptotic behavior for the vorticity equations in dimensions two and three. Commun. Part. Diff. Eq. 19, 827-872.         [ Links ]

Cao C., M. A. Rammaha and E. S. Titi, 1999. The Navier-Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom. Z. Angew. Math. Phys. 50 341-360.         [ Links ]

Constantin P., C. Foias, R. Temam, 1988. On the dimension of the attractors in two dimensional turbulence. Physica D 30, 284-296.         [ Links ]

Doering C. R., J. D. Gibbon, 1991. Note on the Constantin-Foias-Temam attractor dimension estimate for two dimensional turbulence. Physica D 48, 471-480.         [ Links ]

Dymnikov V. P. and Y. N. Skiba, 1987a. Spectral criteria for stability of atmospheric barotropic flows. Izv. Atmos. Ocean. Phy.+ 23, 263-274.         [ Links ]

Dymnikov V. P. and Y. N. Skiba, 1987b. Barotropic instability of zonally asymmetric atmospheric flows over topography. Russ. J. Numer. Anal. M. 2, 83-98.         [ Links ]

Gallagher I. and T. Gallay, 2005. Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity. Math. Ann. 332, 287-327.         [ Links ]

Gallay T. and C. E. Wayne, 2005. Global Stability of vortex solutions of the two dimensional Navier-Stokes equation. Commun. Math. Phys. 255, 97-129.         [ Links ]

Gallay T. and C. E. Wayne, 2007. Existence and stability of asymmetric Burgers vortices. J. Math. Fluid. Mech. 9, 243-261.         [ Links ]

Gibbon J. D., 1996. A voyage around the Navier-Stokes equations. Physica D 92, 133-139.         [ Links ]

Giga Y. and T. Kambe, 1988. Large time behavior of the vorticity of two dimensional viscous flow and its application to vortex formation. Commun. Math. Phys. 117, 549-568.         [ Links ]

Giga Y., T. Miyakawa and H. Osada, 1988. Two-dimensional Navier-Stokes flow with measures as initial vorticity. Arch. Rational Mech. Anal. 104, 223-250.         [ Links ]

Giga M.-H., Y. Giga and J. Saal, 2010. Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions, Birkhäuser, Boston, 128 pp. (Progress in Nonlinear Differential Equations and Their Applications, 79).         [ Links ]

Ilyin A. A. and A. N. Filatov, 1988. On unique solvability of Navier-Stokes equations on two-dimensional sphere. Dokl. Akad. Nauk SSSR+ 301, 18-22.         [ Links ]

Ilyin A. A., 1994. Navier-Stokes equations on the rotating sphere. A simple proof of the attractor dimension estimate. Nonlinearity 7, 31-39.         [ Links ]

Ladyzhenskaya O. A., 1969. The mathematical theory of viscous incomprehensible flow. Gordon and Breach, New York, 198 pp.         [ Links ]

Ladyzhenskaya O. A., 1987. Minimal global B-attractors of semigroups and of initial-boundary value problems for nonlinear partial differential equations (Russian). Dokl. Akad. Nauk SSSR 29 , 33-37.         [ Links ]

Leray J., 1933. Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique. J. Math. Pures. Appl. 12, 1-82.         [ Links ]

Lions J.-L., 1969. Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Gauthier-Villars, Paris, 554 pp.         [ Links ]

Marchioro C., 1986. An example of absence of turbulence for the Reynolds number. Commun. Math. Phys. 105, 99-106.         [ Links ]

Neven E. C., 1992.Quadrupole modons on a sphere. Geophys. Astro. Fluid 65, 105-126.         [ Links ]

Richtmyer R. D., 1981. Principles of advanced mathematical physics. Springer-Verlag, New York, Vol. 2, 322 pp. (Texts and Monographs in Physics, 2).         [ Links ]

Samoilenko A. M., 1991. Elements of the mathematical theory of multi-frequency oscillations. Kluwer Academic Publishers, Dordrecht, 332. pp.         [ Links ]

Simmons A. J., J. M. Wallace and G. W. Branstator, 1983. Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci. 40, 1363-1392.         [ Links ]

Skiba Y. N., 1989. Mathematical problems of the dynamics of viscous barotropic fluid on a rotating sphere (Russian). VINITI, Moscow (English translation: Indian Institute of Tropical Meteorology, Pune, 1990).         [ Links ]

Skiba Y. N., 1994. On the long-time behavior of solutions to the barotropic atmosphere model. Geophys. Astro. Fluid. Dynamics 78, 143-167.         [ Links ]

Skiba Y. N., 1998. spectral approximation in the numerical stability study of non-divergent viscous flows on a sphere. Numer. Meth. Part. D. E. 14, 143-157.         [ Links ]

Skiba Y. N., 2004. Instability of the Rossby-Haurwitz wave in invariant sets of perturbations. J. Math. Anal. Appl. 290, 686-701.         [ Links ]

Skiba Y. N., 2012. On the existence and uniqueness of solution to problems of fluid dynamics on a sphere. J. Math. Anal. Appl. 388, 627-644.         [ Links ]

Skiba Y. N., 2013. Asymptotic behavior and stability of solutions to barotropic vorticity equation on a sphere. Commun. Math. Anal. 14, 143-162.         [ Links ]

Sundström A., 1969. Stability theorems for the barotropic vorticity equations. Mon. Weather Rev. 97, 340-345.         [ Links ]

Szeptycki P., 1973a. Equations of hydrodynamics on compact Riemannian manifolds. Bull. Acad. Pol. Sci., Series Sci. Math. Astr. Phys. 21, 335-339.         [ Links ]

Szeptycki P., 1973b. Equations of hydrodynamics on manifold diffeomorfic to the sphere. Bull. Acad. Pol. Sci., Series Sci. Math. Astr. Phys. 21, 341-344.         [ Links ]

Temam R., 1984. Navier-Stokes equations. Theory and numerical analysis. North-Holland, Amsterdam, 500 pp.         [ Links ] Temam R., 1985. Attractors for Navier-Stokes equations. Res. Not. in Mat. 122, 272-292.         [ Links ]

Tribbia J. J., 1984. Modons in spherical geometry. Geophys. Astro. Fluid. Dynamics 30, 131-168.         [ Links ]

Verkley W. T. M., 1987. Stationary barotropic modons in westerly background flows. J. Atmos. Sci. 144,2383-2398.         [ Links ]

Yu N., 2005. The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Commun. Math. Phys. 255, 161-181.         [ Links ]