Asymptotic behavior of solutions to barotropic vorticity equation on a sphere

Yuri N. Skiba 1
  • 1 Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Av. Universidad 3000, CU/UNAM, Coyoacán Mexico City, Mexico
Yuri N. Skiba
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  • Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Av. Universidad 3000, CU/UNAM, Coyoacán Mexico City, C.P. 04510, Mexico
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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.

1 Introduction

Let us denote by 0(S) the space of infinitely differentiable functions on the unit sphere S = {x ∈ ℝ3 : |x| = 1} which are orthogonal to any constant, and by
f,g=Sf(x)g(x)¯dSandf=f,f1/2
the inner product and the norm of functions of 0(S) , respectively. Denote by Ynm(λ,μ) the spherical harmonics (n ≥ 1, |m| ≤ n) that form the orthonormal system in 0(S):Ynm,Ylk=δmkδnl where δmk is the Kronecker delta. Each spherical harmonic Ynm is the eigenfunction of the eigenvalue problem
ΔYnm=χnYnm,|m|n
for symmetric and positive definite spherical Laplace operator, where
χn=n(n+1)
Lemma 1
[10] Let n be a natural, andKn=[2n+14π]1/2 . Then
m=nn|Ynm(x)|2=Kn2andm=nn|Ynm(x)|2=χnKn2.
Let n ≥ 1. Denote by
Hn={ψ:Δψ=χnψ}
the (2n + 1)-dimensional eigensubspace of homogeneous spherical polynomials of degree n, which corresponds to the eigenvalue (2) [8]. The set 2n + 1 of spherical harmonics Ynm(λ,μ)(n<mn) form the orthonormal basis in Hn. The Hilbert space 𝕃02(S)=n=1Hn being the direct orthogonal sum of the subspaces Hn is the closure of 0(S) in the norm (1). We denote by Yn (ψ) the projection of function ψ𝕃02(S) onto Hn. Thus, ψ=n=1Yn(ψ) for every function ψ(x)𝕃02(S) .
Let ψ(x)0(S) , and χn = n(n + 1). We introduce the derivative Λs = (−Δ)s/2 of real degree s of ψ(x) as
Λsψ(x)=n=1χns/2Yn(ψ(x))
[11, 12].
Let s be a real. Denote by 0s the Hilbert space obtained by closing the space 0(S) in the norm
ψs={n=1χnsYn(ψ)2}1/2.

The inner product in 0s is defined as 〈ψ, hs = 〈Λsψ, Λsh〉. Thus, ψs=ψ,ψs1/2 . It can be shown that ||∇ψ|| = ||Λψ|| [10].

Lemma 2
[11] Let r, s and t be real numbers, r < t, a=2 , andψ0s+t . Then
ΛrψsartΛtψsandψs+t=Λtψs.

2 Existence and uniqueness of the BVE solutions

Let us consider the non-stationary nonlinear BVE problem on S [11]:
Δψt+J(ψ,Δψ+2μ)+σΔψν(Δ)s+1ψ=F
Δψ(0,x)=Δψ0(x).
Here ψ is the stream function, Δψ is the relative vorticity, Δψ + 2μ is the absolute vorticity, F(t, x) is the forcing, σΔψ is the Rayleigh friction in the planetary boundary layer,
J(ψ,h)=(n×ψ)h
is the Jacobian determinant, n is the outward unit normal vector to the surface of the sphere S, while the nonlinear term J(ψ, Δψ) and linear term J (ψ, 2μ) = 2ψλ represent the advection and the rotation of sphere, respectively. The velocity vector v=n×ψ is nondivergent: v=0 . The viscosity is modeled by the term ν(−Δ)s+1ψ, 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].

Theorem 1
[11] Let s ≥ 1, ν >0 and σ ≥ 0. Suppose thatΔψ0𝕃02(S)at initial moment, andF(t,x)𝕃2(0,T;0s) . Then the non-stationary BVE problem (4)(5) has a unique weak solutionψ(t,x)𝕃(0,T;02)such that
ψ(t,x)𝕃(0,T;0)𝕃2(0,T;0s),Δψt𝕃2(0,T;0s),Δψ(0,x)=Δψ0(x)
and
0t[Δψt,hJ(ψ,h),Δψ+2μ+σΔψ,h]dtν0tΛs+2ψ,Λshdt=0tF,hdt
holds for all t ∈ (0, T) andh𝕃2(0,T;0s) .
Theorem 2
[11] Let s ≥ 1, ν > 0 and σ ≥ 0. Suppose thatF(x)0s . Then there exists at least one weak solutionψ(x)0s+2of the stationary equation
J(ψ,Δψ+2μ)+σΔψν(Δ)s+1ψ=F(x)
such that
νΛs+2ψ,ΛshσΔψ,h+J(ψ,h),Δψ+2μ=F,h
holds for allh0s . If additionally
ν2>21sMF(x)s
then the problem solution is unique.

Here M is the constant from the estimate |J(ψ, h)| ≤ M ||Δψ|| ||Δh|| (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 → ∞.

Theorem 3

Let s ≥ 1, and letF(x)0rbe a stationary forcing of equation (4), r ≥ 1. Then there is a limited setBin a spaceXthat attracts all BVE solutions ψ(t, x), besides,

  1. 1if r ≥ 0 thenX=02and
    B={ψ02:ψ2C1(r,s)Fr.
  2. 2if r ∈ [−1, 0) thenX=01and
    B={ψ01:ψ1C2(r,s)Fr
    where
    C1(r,s)=arσ+2sν,C2(r,s)=ar1σ+2sν,a=2.

Proof
Part 1. Let r ≥ 0 and F(x)0r . The inner product (1) of equation (4) with Δψ and the use of relations
J(g,μ),Λrg=0andJ(g,h),g=0
valid for real functions due to relations
J(ψ,g),h=J(g,h),ψ=J(ψ,h),g
[12], imply
Δψt,Δψ=σΔψ,Δψ+ν(Δ)s+1ψ,Δψ+F,Δψ=σΔψ2νΛs+2ψ2+F,Δψ.
Due to Lemma 2,
|F,Δψ|FΔψarFrΔψandνΛs+2ψ22sνΔψ2.
We denote
ρ=σ+2sν.
The use of the obtained inequalities in (10) leads to
tΔψρΔψ+arFr.
The last inequality gives
Δψ(t)Δψ(0)exp(ρt)+arρFr[1exp(ρt)]
and hence,
ψ(t)2C1(r,s)Frast.
Part 2. Let r ∈ [−1, 0) and F(x)0r . The inner product (1) of equation (4) with ψ and the use of (9) give
Λψt,Λψ=σΛψ2νΛs+1ψ2F,ψ.
Applying Lemma 2 to the terms 〈F, ψ〉 and ν ||Λs+1ψ||2 leads to
|F,ψ|=|ΛrF,Λrψ|ΛrFΛrψar1FrΛψ
and
νΛs+1ψ22sνΛψ2.
Then (13) implies
tΛψρΛψ+ar1Fr
where ρ is defined by (11), or
Λψ(t)Λψ(0)exp(ρt)+ar1ρFr[1exp(ρt)].
Since ||Λψ|| = ||ψ||1, we obtain
ψ(t)1C2(r,s)Frast.

Q.E.D.

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 [14]. Theorem 3 is also valid if F(t,x)C(0,ω;0r) where ω is the period. In this case, one should only replace in (7) and (8) the norms ||F||r by the norms maxt0,ω]Fr .

4 A functional for the stability study

We now introduce a positive functional for estimating arbitrary perturbations of the BVE solution. Let ψ˜(t,λ,μ) be a real BVE solution. Then a perturbation ψ′(t, λ, μ) of the solution ψ˜ satisfies the equation
tΔψ'+J(ψ',Δψ˜)+J(ψ˜,Δψ')+2ψ'λ+J(ψ',Δψ')=[σ+νΛ2s]Δψ'
where s ≥ 1, ν > 0 and σ ≥ 0. Taking the inner product (1) of equation (18) in series with ψ′ and Δψ′, then using (9), we obtain the integral equations
ddtK+J(ψ',Δψ'),ψ˜+2σK+νΛs+1ψ'2=0
and
ddtηJ(ψ',Δψ'),Δψ˜+2ση+νΛs+2ψ'2=0
for the kinetic energy K(t)=12ψ'(t,x)2=12Λψ'(t,x)2 and enstrophy η(t)=12Δψ'(t,x)2 of perturbation 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).

We will estimate the behavior of perturbations ψ′(t, λ, μ) of a basic flow ψ˜(t,λ,μ) using the functional
Q(t)=pK(t)+qη(t)=12(pψ'2+qΔψ'2)
where 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
ddtQ(t)=2σQ(t)R(t)νpΛs+1ψ'2νqΛs+2ψ'2
where
R(t)=J(ψ',Δψ'),pψ˜qΔψ˜.
Applying Lemma 2 we get
Λs+1ψ'22sψ'2,Λs+2ψ'22sΔψ'2.
Therefore, (21) leads to
ddtQ(t)2ρQ(t)R(t)
where 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.

Example 1

Letψ˜=0(this solution exists if F(x) ≡ 0). ThenJ(ψ',Δψ'),ψ˜=0andJ(ψ',Δψ'),Δψ˜=0in (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.

Example 2

The basic flow is a super-rotation:ψ˜ψ˜(μ)=Cμ , 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 subspaceH1, since it represents a super-rotation flow about some axis of a sphere [10].

Example 3
The basic flow is a homogeneous spherical polynomial:ψ˜Hn(n : ≥ 2):
ψ˜(t,λ,μ)=m=nnψ˜nm(t)Ynm(λ,μ).

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.

First, assume that the basic solution ψ˜(t,λ,μ) of equation (4) is rather smooth, such that
p=supt0max(λ,μ)S|Δψ˜(t,λ,μ)|andq=supt0max(λ,μ)S|ψ˜(t,λ,μ)|
are bounded. Then using the inequality |J(ψ, h)| ≤ |∇ψ| · |∇h| we get
|R(t)|=|J(pψ˜qΔψ˜,ψ'),Δψ'|2pqψ'Δψ'2pqQ(t).

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

Theorem 4
Let s ≥ 1, ν > 0 and σ ≥ 0. If
σ+2sν>pq
where p and q are defined by (25), then Q(t) is the Lyapunov function, and solutionψ˜(t,λ,μ)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 ψ˜(t,λ,μ) must have continuous derivatives up to the third order. But Theorems 1 and 2 guarantee the existence of a solution that has continuous derivatives only up to the second order. Theorem 5 of this section gives new global asymptotic stability conditions for smooth solution of equation (4), in which the requirement on the smoothness is weakened. These instability conditions are in full accordance with Theorem 1.

We now show that the restriction (25) on the smoothness of the basic solution can be weakened to be consistent with the solvability theorem (Theorem 1). To this end, let us consider a smooth solution ψ˜(t,λ,μ) such that
p=suptmax(λ,μ)S|Δψ˜(t,λ,μ)|andq=suptmax(λ,μ)S|ψ˜(t,λ,μ)|
are bounded. Using the ɛ-inequality one can estimate R(t) as:
|R(t)|2pqψ'Δψ'=2pqψ'Λ3ψ'2pqψ'ψ'3=(pqψ'1)(2pqψ'3)2qɛ2Q(t)+pqɛ2ψ'32.
In addition, the use of Lemma 2 in (21) leads to
ddtQ(t)2ρQ(t)R(t)νpψ's+12νqψ's+222ρQ(t)R(t)νqψ's+222ρQ(t)R(t)νqa1sψ'32
where a=2 . Combining (29) with (30) and setting ɛ2 = pas−1/ν in order to eliminate the two terms containing ψ'32 , we prove the following assertion:
Theorem 5
Let s ≥ 1, ν > 0 and σ ≥ 0. And letψ˜(t,λ,μ)be such a solution of equation (4) that the numbers p and q defined by (28) are limited. If
ν(σ+2sν)>2(s1)/2pq

then Q(t) is the Lyapunov function, and the solutionψ˜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, Δψ˜(t,λ,μ) is discontinuous at the boundary between the inner and outer regions of the modon. Therefore, condition (31) is applicable, while condition (27) is not.

Example 4
Let σ = 0 and s = 1 in equation (4), and letψ˜(λ,μ)be a stationary solution from the subspaceHn of homogeneous spherical polynomials (n ≥ 2):
ψ˜(λ,μ)=m=nnψ˜nmYnm(λ,μ).
The solution (32) is supported by a steady forcing whose Fourier coefficientsFnmare
Fnm=(νχn2+i2m)ψ˜nm
wherei=1and χn = n(n + 1). According to Example 3, subspaceHn is the domain of attraction of solution (32) for any ν ≠ 0. Moreover, it follows from (28) that p = χnq, and due to Theorem 5, solution (32) is the global attractor of equation (4) if
νqn(n+1)2.

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

It was shown in [3] that in the case of a stationary forcing, the Hausdorff dimension of the global attractor of the barotropic vorticity.equation (4) on a sphere for s = 1 and s = 2 is limited by the generalized Grashof number
G(s)=F(x)/χ12s1ν2(s)
where χ1 = 2 is the smallest positive eigenvalue of the spherical Laplace operator (see (2)).
This is an important hydrodynamic result. However doubts arise about its applicability to the dynamics of large-scale barotropic processes in the atmosphere. Indeed, the BVE forcing represents the influence of small-scale baroclinic processes (convection, etc.) on the large-scale dynamics of the barotropic atmosphere, and therefore is essentially nonstationary with rather complex spatio-temporal behavior. In order to show that the dimension of the global BVE attractor crucially depends on the space-time structure of the forcing, we now estimate the dimension of the global BVE attractor provided that the forcing of the BVE model is a quasiperiodic function in time and a homogeneous spherical polynomial of degree n in space:
F(t,x)=m=nnFm(t)Ynm(x)
where
Fm(t)=fmexp{imωmt},|m|n,
each fm is a constant, and the frequencies ωm are incommensurate. Thus, F(t, x) ∈ Hn. Note that
F(t,x)=(m=nn|fm|2)1/2Const
i.e., the norm (36) of forcing and the generalized Grashof number (33) are time-independent constants. Obviously, there are many quasiperiodic forcings of the form (34) which have the same norm (36), or the same Grashof number (33), but differ in degrees n or/and amplitudes fm.
Since J(ψ, Δψ) = 0 for any function ψHn, there exists an exact BVE solution ψ˜(t,x)=m=nnψ˜m(t)Ynm(x) from the subspace Hn defined by the Fourier coefficients
ψ˜m(t)=AmFm(t){[σ+νχns]χn+im(χnωm2)}1Fm(t),|m|n.

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 ψ˜(t,x) is quasiperiodic, and its path is an open endless spiral densely wound around a 2n-dimensional torus in the (2n + 1)-dimensional complex space Hn. 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 2n.

Sufficient conditions for the global asymptotic stability of smooth solution (37) are given by Theorem 4. In our case ψ˜(t,x)Hn , and (25) gives p = χnq = n(n + 1)q. Therefore, condition (27) for the global asymptotic stability of solution (37) accepts the form
σ+2sν>qχn
where
q=supt0max(λ,μ)S|ψ˜(t,λ,μ)|.
Due to Lemma 1,
|ψ˜(t,x)|2=|m=nnψ˜nm(t)Ynm(x)|2ψ˜(t,x)2{m=nn|Ynm(x)|2}=χnKn2ψ˜(t,x)2.
Also, according to (37),
ψ˜(t,x)1(σ+νχns)χnF
where the norm ||F|| of forcing (34), (35) is time-independent (see (36)). Due to (39)(41),
qχnKnσ+νχnsF
and hence condition (38) is satisfied if
σ+2sν>Kn[σ+νχns]F
and also if
2sν>KnνχnsF.
Using the definition (33) where χ1 = 2, the last inequality can be written in terms of the generalized Grashof number:
22sχnsπ2n+1>G(s).
Since
χns12n+1=ns(n+1)s12n+112n2s1/2
condition (42) holds if
23/2sπn2s1/2>G(s).

Thus, we prove the following assertion:

Theorem 6

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.

In particular, for s = 2 and s = 1, solution (37) is the global BVE attractor if
21/2πn7/2>G(2)and21/2πn3/2>G(1),
respectively. Note that 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 Hn with nn(G) is a global BVE attractor. For example, if we take G(2)=F(x)/χ13ν2(2)=1500 for the barotropic atmosphere (this value was used in [3]), then condition (44) is satisfied for any forcing (34)(35) whose degree 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 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 [3], 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 [1] is theoretically unjustified due to the fact that forcing usually has a very complex structure with a huge number of degrees of freedom.

Acknowledgements

This work was partially supported by the National System of Researchers (SNI, CONACYT, Mexico) through grant 14539.

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    Grassberger, P. (1986). Do Climatic Attractors Exist?. Nature, 323, 609–612.

    • Crossref
    • Export Citation
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    Helgason, S. (1984) Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions. Academic Press, Orlando.

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    Ilyin, A.A. (1993). On the dimension of attractors for Navier-Stokes equations on two-dimensional compact manifolds. Diff. Integr. Equations, 6(1), 183–214.

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