Navier-Stokes equations on a rapidly rotating sphere

We extend our earlier \beta-plane results [al-Jaboori and Wirosoetisno, 2011, DCDS-B 16:687--701] to a rotating sphere. Specifically, we show that the solution of the Navier--Stokes equations on a sphere rotating with angular velocity 1/\epsilon\ becomes zonal in the long time limit, in the sense that the non-zonal component of the energy becomes bounded by \epsilon M. We also show that the global attractor reduces to a single stable steady state when the rotation is fast enough.


Introduction
On a sphere of unit radius rotating with angular velocity 1/ε about the z-axis, the non-dimensional Navier-Stokes equations read where ∇ v is the covariant derivative and v ⊥ is the velocity vector v rotated by +π/2. Here θ ∈ [0, π] is the (co)latitude and φ ∈ [0, 2π) is the longitude. Observations of flows on rotating planetary atmospheres and many numerical studies (cf., e.g., [10] and references herein) strongly suggest that as the rotation rate increases, the flow will become more zonal, that is, the velocity will become more aligned with the planetary rotation. The main aim of this article is to prove that the solution of the 2d Navier-Stokes equations (1.1) does indeed become more zonal, in a sense to be precised below, as the planetary rotation rate ε −1 → ∞. As a corollary, we show that the global attractor reduces to a point for small enough ε. We note a related work for the Euler equations [3]. It is more convenient to work with the vorticity ω := curl v, which by definition has zero integral over the sphere. We thus have Poincaré inequality (1.2) c 0 |ω| 2 L 2 ≤ |∇ω| 2 L 2 := (−∆ω, ω) L 2 ; here one can show that c 0 = 2, but we shall write c 0 to make its origin clear. The evolution equation for ω is where here and henceforth ∆ −1 is uniquely defined by requiring that the result has zero integral over the sphere. In polar coordinates (θ, φ), the Jacobian takes the and satisfies the so-called abc identity: for any a, b and c ∈ H 1 (S 2 ), We shall also make use of the functional form of (1.3), The following properties are readily verified: with (·, ·) and | · | denoting the L 2 inner product and norm, for all ω and ω * whenever the expressions make sense. Furthermore, , and they commute, For functions of vanishing integral on S 2 (all that is relevant in this paper), we define the Sobolev spaceḢ s by (1.9) |u| 2 H s := ((−∆) s u, u) L 2 for positive integers s. As usual, spaces of non-integral order are then defined by interpolation.
As with periodic boundary conditions, one can show that with f ∈ L ∞ (R + , H s−1 ) the solution ω of the NSE (1.3) belongs to L ∞ ((1, ∞), H s ) ∩ L 2 loc ((1, ∞), H s+1 ), regardless of the initial data ω(0) (assumed to be in L 2 ). Moreover, for each s ∈ {0, 1, · · · }, there exist N s (f ; µ) and τ s (ω(0), f ; µ) such that, for all t ≥ τ s , The proof is essentially identical to that in the continuous case [1] (see also [6] who obtained a closely related result for s = 1), so we shall not repeat it here. Regularity in Gevrey spaces has also been established [2], although in this case the result is slightly weaker than in the doubly-periodic case [5].
The planetary rotation, represented by the antisymmetric operator L in (1.6), breaks the symmetry of the sphere and defines a special direction. We therefore introduce the averaging operatorP, (1.12) 2. L 2 Estimate for the Non-zonal Part We do the initial stage of the computation here in order to motivate the crucial "non-resonance" Lemma 1. We start by multiplying (1.6) byω in L 2 . Using (1.7) and (1.12), we find Using (1.7b) and (1.12b), we rewrite the nonlinear term as giving us Let ν := µc 0 . Using Poincaré inequality (1.2) on half of µ |∇ω| 2 in (2.3) and multiplying by e νt , we find d dt Integrating, this gives As will be shown below, the integrand on the right-hand side is rapidly oscillating, so the integral will be of order ε, giving an order-ε bound on |ω(t)| 2 for large t.
We briefly recall the properties of spherical harmonics. We denote by k := (k,k) a wavevector , with k ∈ {0, 1, · · · } andk ∈ {−k, −k + 1, · · · , k}. Writing the Laplacian ∆ has eigenvalues −|k| 2 , with each eigenspace having dimension 2k + 1 and spanned by the spherical harmonics Y k (θ, φ), The spherical harmonics are orthonormal, where δ jk = 1 when j = k and =k, and δ jk = 0 otherwise. Their explicit expressions (including phase and normalisation) can be found in [9, §14.30], whose convention we follow; we note that [8] used the same convention for spherical harmonics and 3j-symbols. For our purpose here, we only note that where C k is real and the Legendre polynomial Pk k (·) has real coefficients and is of degree (exactly) k. Now it is clear from (2.9) that Y k is also an eigenfunction of L, and we use this fact to define the frequency Ω k (ignoring the case k = 0), Assuming sufficient regularity, we Fourier-expand the vorticity as where the factor e −iΩ k t/ε has been included since we expectω to oscillate rapidly. Consistent with our definitions ofω andω, we writeω k = ω k whenk = 0 andω k = 0 whenk = 0, andω k = ω k whenk = 0 andω k = 0 otherwise. Similar notations are understood forf andf . Here and henceforth, the sum over wavevector is understood to be although the k = 0 term is often zero. Similarly, we expand the forcing f as without the rapidly oscillating exponential since later we will demand that f vary slowly in time. Writing the nonlinear term in (2.5) is (Ω j +Ω k )/ε whereω l denotes the complex conjugate ofω l (here and elsewhere, small overbar denotes φ-average and full overline denotes complex conjugate). As will be seen below, our main difficulty is the resonances in (2.15), which occur when Ω j +Ω k = 0.
Let J jkl denote the coupling coefficients of the Jacobian, viz., We handle the resonances with the following: Lemma 1. For all wavevectors j, k and l withk = 0 andl = 0, We note that the right-hand side is symmetric in j and k as expected: J kjl = −J jkl whilel = 0 implies thatk = −. Deferring the proof of the Lemma to the Appendix, we state our main result: Let the forcing f in (1.3) be bounded as for all t ≥ T and some T ≥ 0. Then there exist T 0 (f, ω(0); µ) and M 0 (K 0 ; µ) such that, for t ≥ T 0 , Proof. Except for the handling of resonances in the nonlinear term (the main difficulty), the proof essentially follows that in [1].
Here and in what follows, x ≤ c y means x ≤ cy for some positive absolute constant c which may assume different values in different inequalities. We note that the hypothesis (2.18) and (1.10) imply that our solution ω is bounded uniformly in We bound the integral on the right-hand side in (2.5). Starting with the last term (omitting the e −νt factor for now), we integrate it by parts in time to bring out a factor of ε: Here and henceforth, the prime on the sum indicates that the resonant terms (i.e. those with Ω k = 0) are omitted. Taking note of (2.10), we rewrite (2.20) as In the integral, we bound the first term as above and the middle term as This is the worst term (in the sense of requiring the most regularity on f ) in the entire L 2 estimate. For the nonlinear term in (2.25), we write ψ := ∆ −1 ω, extend ψ and ω to functions independent of the radius r in a thin shell in R 3 around the unit sphere, and write the Jacobian as where the gradients and cross product are in R 3 and e r is the unit vector in the radial direction. Working in R 3 , we obtain after a little computation ∆∂(ψ, ω) = ∂(∆ψ, ω) + ∂(ψ, ∆ω) + 2 ∂(∇ψ, ∇ω) Noting that ∇e r is smooth on the unit sphere, we then restrict back to the sphere and estimate (with "l.o.t." denoting lower-order terms majorisable by those already present) where all norms are L 2 on the last line. We conclude that Turning to the nonlinear term in (2.5), in Fourier components it reads where the prime on the sum again indicates that resonant terms, i.e. those with Ω j + Ω k = 0, are omitted (since then B jkl + B kjl = 0 by Lemma 1). As in (2.20), we integrate this by parts to bring out a factor of ε. Defining the symmetric bilinear operator B Ω (·, ·) by As before, we bound each term in the last integral. Now Lemma 1 implies (2.34) (B Ω (ω,ω),ω) = 1 4 (∂(∂ −1 φω ,ω),ω), so we can bound the first term as (2.35) (B Ω (ω,ω),ω) ≤ c |∇ω| 2 L 2 |ω| L ∞ ≤ c |∇ω| 2 |∇ω|.
Taking t 0 = τ 3 , this and (1.10) bound the integral on right-hand side of (2.45). The theorem follows by taking t − t 0 sufficiently large.

H s Estimates and Dimension of the Global Attractor
As in [1], with f (t) ∈ H s+2 , one can obtain similar bounds forω in H s . Since the proof is similar to that in [1], we only sketch briefly here the case s = 1.
Bounding the terms as Moving the first term to the l.h.s. and using (2.19) along with (2.46) to obtain an O(ε) bound for the integral on the second term, we conclude that there exist T 1 (f, ω(0); µ) and M 1 (|∇ 3 f | 2 ; µ), such that for all t ≥ T 1 , Proceeding along similar lines (see [1] for more details), for s = 2, 3, · · · , we have T s (f, ω(0); µ) and M s (|∇ s+2 f | 2 When the forcing is independent of time, ∂ t f = 0, the regularity results of the NSE allow one to conclude that there exists a global attractor A whose Hausdorff dimension is bounded as [4,7] (3.7) dim H A ≤ c S G 2/3 (1 + log G) 1/3 where G := |∇ −1 f | L 2 /µ 2 is the Grashof number and c S is an absolute constant. It has long been known (and easily shown by some computation) that dim H A = 0 for G ≤ G 0 , that is, A reduces to a single stable steady state for sufficiently small Grashof number. Using (3.6) for s = 3 and proceeding as in [1], one can show that there exists ε * (|∇ 2 f |; µ) such that (3.8) The proof is essentially identical to that in [1] and we shall not repeat it here. We remark that (A.9) is valid whether or not the 3j symbols vanish: with j + k + l odd, one can verify that the combined triangle conditions (A.5) for (j, k, l) and (j − 1, k, l) are equivalent to those for (j, k, l) and (j, k − 1, l). Although this proof is based on direct computation of the coefficients, an alternate group-theoretic approach (cf., e.g., [11]) may be possible.