Euler equations on a fast rotating sphere --- time-averages and zonal flows
EEULER EQUATIONS ON A FAST ROTATING SPHERE— TIME-AVERAGES AND ZONAL FLOWS
BIN CHENG AND ALEX MAHALOV
Abstract.
Motivated by recent studies in geophysical and planetary sciences, we investigate thePDE-analytical aspects of time-averages for barotropic, inviscid flows on a fast rotating sphere S .Of particular interests are the incompressible Euler equations. We prove that the finite-time-averageof the solution stays close to a subspace of longitude-independent zonal flows . The intial data canbe arbitrarily far away from this subspace. Meridional variation of the Coriolis parameter underliesthis phenomenon. Our proofs use Riemannian geometric tools, in particular the Hodge Theory. Introduction
Recent studies have seen increasing understandings of global characteristics of geophysical flowson Earth and giant planets in the Solar System. Simulations and observations have persistentlyshown that coherent anisotropy favoring zonal flows appears ubiquitously in planet scale circula-tions. For a partial list of computational results, we mention [5] for 3D models, [20, 13, 9, 18, 6]for 2D models, and references therein. These highly resolved, eddy-permitting simulations aremade possible by rapid developements of high performance computing. On the other hand, wehave observed zonal flow patterns (bands and jets) on giant planets for hundreds of years, whichhas attracted considerable interests recently thanks to spacecraft missions and the launch of theHubble Space Telescope (e.g. [7], [15]). Figure 1 shows a composite view of the banded strucure ofJovian atmosphere captured by the Cassini spacecraft ([12]). There are also observational data inthe oceans on Earth showing persistent zonal flow patterns (e.g. [16, 17, 11]).
Remark 1.1.
In the existing literature, a necessary process for the zonal flow pattern to emerge isaveraging of data and/or simulations over a period of time usually decades long. Also, a number ofheuristic arguments (e.g. [6] ) are made pointing to the north-south gradient of Coriolis parameteras the underlying machanism, even in the absence of temperature/density gradient and verticalvariability.
To this end, we study inviscid, barotropic geophysical flows on a unit sphere S centered at theorigin of R and fast rotating about the z -axis with constant angular velocity. Let vector field u ( t, q ), tangent to S at every point q ∈ S , denote the fluid velocity relative to this rotatingframe. Throughout this paper, we represent any point q ∈ S either by its relative-to-the-framecartesian coordinates ( x, y, z ) or its relative-to-the-frame spherical coordinates ( θ, φ ) with θ being Date : November 15, 2018.
Key words and phrases.
Rotating fluids, Euler equations, barotropic models on a rapidly rotating sphere, zonalflows, time-averages, PDE on surfaces. a r X i v : . [ m a t h . A P ] A ug igure 1.1. This true-color simulated view of Jupiter is composed of 4 images takenby NASA’s Cassini spacecraft on December 7, 2000. Credit: NASA/JPL/Universityof Arizona [12].the colatitude and φ the longitude . Also, let e φ be the unit vector in the zonal direction ofincreasing longitude and e θ be the unit vector in the meridional direction of increasing colatitude.In this study, we focus on a canonical PDE system: the incompressible Euler equations underthe Coriolis force ([1, 4, 14]), ∂ t u + ∇ u u + ∇ P = zε u ⊥ , div u = 0 (1.1)where constant ε , called the Rossby number, scales like the frequency of the frame’s rotation(usually 0 . ∼ . u ⊥ denotes a counterclockwise π/ u on S .Cartesian coordinate z indicates how the Coriolis parameter varies along the meridional direction.Note that the Coriolis force is not uniformly large and actually vanishes on the equator. It is thelarge gradient of the Coriolis parameter that drives the zonal flow patterns.Note that results have been established concerning solution regularity of the above systems andrelated ones in the fast rotating regime with ε (cid:28)
1. Please refer to [2, 3, 8] and references thereinfor further discussion. At z = 1, we fix θ = 0 but φ can be arbitrary. Such singularity issue does not occur for cartesian coordinates x, y, z ∈ C ∞ ( S ). ur theoretical investigation is then focused on the fast rotating regime with ε (cid:28) time-averages of u : u ( T, · ) := 1 T (cid:90) T u ( t, · ) dt (1.2)for positive times T .The main result is stated as following. Theorem 1.1.
Consider the incompressible Euler equations (1.1) on S with div-free initial data u (0 , · ) ∈ H k ( S ) for k ≥ . Define the time-averaged flow u as in (1.2) . Then, there exist ε -independent constants C , T s.t. for any given T ∈ [0 , T / (cid:107) u (cid:107) H k ] , there exist a function f ( · ) :[ − , (cid:55)→ R and a universal constant C s.t. (cid:13)(cid:13)(cid:13) u ( T, x, y, z ) − ∇ ⊥ f ( z ) (cid:13)(cid:13)(cid:13) H k − ( S ) ≤ C ε ( M T + M ) . (1.3) Here, M := (cid:107) u (cid:107) H k indicates the size of initial data. In spherical coordinates, the approximation ∇ ⊥ f ( z ) is ∇ ⊥ f ( z ) = − f (cid:48) (cos θ ) sin θ e φ . Our theoretical result proves computational and observational results in the literature mentionedat the beginning of this paper, especially Remark 1.1. Note that our result shows that the zonal-flow pattern becomes prominant with decreasing Rossby number ε (cid:38)
0. In other words, thetime-averaged flow u is only O ( ε ) away from a very restricted subspace consisting of longitude-independent zonal flows . The initial data, on the other hand, do not need any filtering and canbe arbitrarily far away from that subspace of zonal flows. Our proofs below will suggest that suchunique pattern is essentially due to the Coriolis parameter z/ε that varies meridionally from thestrongest at the poles to zero on the equator.Rossby number ε is typically at magnitude 0.01 ∼ ∼
100 Earth days according to Theorem 1.1. This suggests that zonal flowpatterns can occur at time scales far below those used in the literature. In fact, the Rossby numberis even smaller for giant planets, leading to the direct observability of banded structures.Our result for rotating incompressible Euler equations uses the abstract framework of the fol-lowing lemma.
Lemma 1.1.
Consider time-dependent equation u t = 1 ε L [ u ] + f ( t, q ) over certain spatial domain Ω . Here, ε > is a scaling constant, f ( t, q ) a source term and L alinear operator independent of time. Let operator Π null {L} denote (some) projection onto the null spaceof L . Assume a priori u , f ( t, q ) , L [ u ] , Π null {L} [ u ] have enough regularity as needed.Then, under the assumption (cid:107) u − Π null {L} u (cid:107) H k (Ω) ≤ C (cid:107)L [ u ] (cid:107) H k (Ω) (1.4) or some constant C , holds true the following estimate on the time-average of u , (cid:13)(cid:13)(cid:13) T (cid:90) T u dt − T (cid:90) T Π null {L} u dt (cid:13)(cid:13)(cid:13) H k ≤ εC (cid:18) MT + M (cid:48) (cid:19) where constants M := max t ∈ [0 ,T ] (cid:107) u ( t, · ) (cid:107) H k and M (cid:48) := max t ∈ [0 ,T ] (cid:107) f ( t, · ) (cid:107) H k . Remark 1.2.
The key hypothesis (1.4) is automatically true in a finite-dimensional space if u is avector in R n , L a linear transform R n (cid:55)→ R n and Π null {L} the l -projection onto null {L} . In such case,hypothesis (1.4) , with the norms understood as l norm on both sides, amounts to the boundednessof L − : image {L} (cid:55)→ R n / null {L} Proof of Lemma 1.1.
First, transform the original equation into u t = 1 ε L [ u − Π null {L} u ] + f and apply time-averaging T (cid:82) T · dt on both sides1 T ( u ( T, · ) − u (0 , · )) = 1 εT (cid:90) T L [ u − Π null {L} u ] dt + 1 T (cid:90) T f ( t, · ) dt Since all necessary regularities were assumed available and L was assumed to be linear and inde-pendent of time, we argue that (cid:82) T · dt and L [ · ] commute, so that the above equation becomes1 T ( u ( T, · ) − u (0 , · )) = 1 ε L (cid:104) T (cid:90) T u dt − T (cid:90) T Π null {L} u dt (cid:105) + 1 T (cid:90) T f ( t, · ) dt. Due to the factor ε in the first term on the RHS, we have (cid:13)(cid:13)(cid:13)(cid:13) L (cid:104) T (cid:90) T u dt − T (cid:90) T Π null {L} u dt (cid:105)(cid:13)(cid:13)(cid:13)(cid:13) H k ≤ ε (cid:18) MT + M (cid:48) (cid:19) . Finally, apply estimate (1.4) to arrive at the conclusion. (cid:3)
Note that, the constant M used in the above lemma depends on size of the solution up untiltime T and is not necessarily independent of ε . A priori estimates uniform in ε are therefore inorder. The proof requires considerations beyond the well established energy methods, which willbe explained in Appendix B. Proof of Main Theorem 1.1.
Having Lemma 1.1 and ε -independent estimates in Theorem 6.1, itsuffices to study properties of properly defined operators L (c.f. Definition 2.1) and Π null {L} (c.f.Lemma 3.2), and to finally prove estimate (1.4) (c.f. Theorem 4.1). (cid:3) The rest of the paper is organized as following. We start Section 2 with describing a version ofthe Hodge decomposition in terms of differential operators on S . The definitions of these operatorsare given the Appendix. An elliptic operator L that plays the same role as the L in Lemma 1.1is defined by the end of Section 2. In Section 3, we characterize the null space of L , identifyingnull {L} as the space of longitude-independent zonal flows (c.f. Lemma 3.1). We also define theprojection operator Π null {L} and its complement. In Section 4, we obtain Sobolev-type estimates, in articual (1.4), regarding L and Π null {L} using the spherical coordinates and spherical harmonics. InAppendix A, we give the rigorous definitions of differential operators on surfaces such as S andprove related properties. It is necessary to adopt coordinate-independent differential geometrictools since any global coordinate system on S is bound to have singularity issues. On the otherhand, one can formally use spherical coordinates as well as cartesian coordinates for most of thearguments presented in this paper, knowing their validity is justified. In Appendix B, we prove an ε -independent estimates after carefully examining commutability properties of some differential-integral operators on a sphere. 2. Hodge Decomposition
The
Hodge decomposition theorem ([21], [19]) confirms that for any k -form ω on an orientedcompact Riemannian manifold, there exist a ( k − α , ( k + 1)-form β and a harmonic k -form γ , s.t. ω = dα + δβ + γ. In particular, if the manifold is a surface in the cohomology class of S (loosely speaking, thereis no “hole” or ”handle”), then there exist two scalar-valued functions Φ (called potential) and Ψ(called stream function) such that u = u irr + u inc where u irr := ∇ Φ and u inc := ∇ ⊥ Ψ . (2.1)Please refer to (5.14) in the Appendix and the discussion that leads to it.Moreover, the decomposition satisfiescurl u irr = div u inc = 0and u irr = ∇ ∆ − S div u , u inc = ∇ ⊥ ∆ − S curl u . (2.2)In other words, any (square-integrable) vector field on S can be written as superposition of anincompressible and an irrotational vector fields that are determined by (2.2). Such decompositionis unique because a harmonic scalar-valued function on a sphere (and any surface in the samecohomology class) is always constant and therefore ∆ − S is unique up to a constant.For simplicity, we will use ∆ for ∆ S from here on. Also, we assume that, unless specifiedotherwise, ∆ − f always has zero global mean over S . (2.3)We postpone the differential-geometric definitions and properties of ∇ , ∇ ⊥ , curl , div , ∆ on S till the Appendix.Now, rearrange (1.1) as zε u ⊥ − ∇ u u = ∂ t u + ∇ P. bserve that on the RHS, ∂ t u is incompressible and ∇ P is irrotational. Thus, the RHS is theunique Hodge decomposition of the LHS, which satisfies the elliptic PDEs (2.2). In particular, theincompressible part ∂ t u is uniquely determined by ∂ t u = ∇ ⊥ ∆ − curl (cid:16) zε u ⊥ − ∇ u u (cid:17) . (2.4)This is indeed an equivalent formulation of the original incompressible Euler’s equation (1.1).In the context of Lemma 1.1, we define the following operator Definition 2.1.
For any u , not necessarily div-free, define L [ u ] := ∇ ⊥ ∆ − curl ( z u ⊥ ) . (2.5) Here, ∆ − follows the convention (2.3) . Then, (2.4) can be reformulated as, ∂ t u + ∇ ⊥ ∆ − curl ( ∇ u u ) = 1 ε L [ u ] . (2.6)3. Null Space of L and associated L -orthogonal projection We observe that the definition (2.5) naturally implies a sufficient condition for L [ u ] = 0 iscurl ( z u ⊥ ) = 0 . Indeed this is also necessary by the virtue of (5.13), namely curl ∇ ⊥ = ∆, so thatcurl L [ u ] = curl ∇ ⊥ ∆ − curl ( z u ⊥ ) = curl ( z u ⊥ ) . Thus, for velocity field u , not necessarily div-free, L [ u ] = 0 ⇐⇒ curl ( z u ⊥ ) = 0 . (3.1)Further analysis reveals the following lemma. Lemma 3.1. (Characterization of null {L} ) For div-free u with sufficient regularity, L [ u ] = 0 ⇐⇒ u = ∇ ⊥ g ( z ) = − g (cid:48) (cos θ ) sin θ e φ (3.2) for some function g : [ − , (cid:55)→ R . Thus, we identify null {L} , when restricted to div-free velocityfields, with the space of longitude-independent zonal flows .Proof of Lemma 3.1. Apply the product rule (5.16) in the Appendix to curl ( z u ⊥ ),curl ( z u ⊥ ) = ( ∇ z ) · u + z div u = ( ∇ z ) · u (3.3)where the · product is given by the natural metric on S induced from R and we used the in-compressibility condition div u = 0. Therefore, by (3.1) and (3.3), the null space of L is identifiedas For any div-free u , L [ u ] = 0 ⇐⇒ ( ∇ z ) · u = 0 . (3.4) ince ∇ z is in the meridional direction, (3.4) implies any u in the null space of L flows in thezonal direction. But there is more than that. Hodge decomposition (2.1), (2.2) impliesdiv u = 0 ⇐⇒ u = ∇ ⊥ Ψ( x, y, z ) (3.5)with Ψ being a unique scalar function with zero global mean. Combining (3.5) it with (3.4), wehave, for any incompressible velocity field u , u = ∇ ⊥ Ψ ∈ null {L} ⇐⇒ ( ∇ z ) (cid:107) ( ∇ Ψ) . The condition ( ∇ z ) (cid:107) ( ∇ Ψ) implies Ψ is a function of z only. Thus, we arrive at the conclusion.The very last term in (3.2) is due to the fact that, in spherical coodinates, ∇ ⊥ z = − sin θ e φ . (cid:3) It is then easy to show the following characterization of Π null {L} , the L -orthogonal-projectionoperator onto null {L} . Lemma 3.2. (Characterization of Π null {L} ) For any div-free vector field u ∈ L ( S ) , its L -orthogonal-projection onto null {L} satisfies Π null {L} u = (cid:16)(cid:72) C ( θ ) u · e φ (cid:17) e φ (cid:72) C ( θ ) e φ · e φ = 12 π sin θ (cid:32)(cid:73) C ( θ ) u · e φ (cid:33) e φ (3.6) where (cid:72) C ( θ ) is the line integral along the circle C ( θ ) at a fixed colatitude θ . Several remarks are in order. First, among all possible projection operators, we chose onethat nullifies the L ( S ) orthogonal complement of null {L} . Secondly, intuitively, Π null {L} u at agiven latitude is a uniform zonal flow equal to the mean circulation of u at that latitude; thus,(id − Π null {L} ) u is of zero circulation along the circle at a fixed latitude. Such intuition is consistentwith the orthogonality condition (cid:90) S ( Π null {L} u ) · (id − Π null {L} ) u (cid:48) = 0 . Thirdly, even though (3.6) runs into singularity at the poles θ = 0 and θ = π , such singularity isremovable. In fact, apply the Stokes’ theorem on the RHS of (3.6) so that, for θ ∈ (0 , π ),Π null {L} u = 12 π sin θ (cid:32)(cid:90)(cid:90) interior of C ( θ ) curl u (cid:33) e φ and by taking the limit as θ →
0+ and θ → π − , we obtainΠ null {L} u (cid:12)(cid:12)(cid:12) poles = . n terms of the stream function, for any div-free velocity field u = ∇ ⊥ Ψ which amounts to u = ( ∂ θ Ψ) e θ ⊥ + (cid:16) ∂ φ Ψsin θ (cid:17) e φ ⊥ with removable singularity at the poles,Π null {L} ( ∇ ⊥ Ψ) = 12 π (cid:18)(cid:90) π ∂ θ Ψ dφ (cid:19) e φ = ∇ ⊥ (cid:18) π (cid:90) π Ψ dφ (cid:19) . (3.7)In other words, Π null {L} maps the stream function to its zonal means.4. Key Estimates
This section is dedictated to proving an estimate similar to (1.4). A convinient tool in studyingSobolev norms of functions on S is the spherical harmonics.To this end, introduce the spherical harmonc of degree l and order m , Y ml ( φ, θ ) = N ml e imφ Q ml (cos θ ) , for l = 0 , , , ... and m = − l, − l + 1 , ..., , ..., l − , l where the normalizing constant N ml = (cid:113) (2 l +1)4 π ( l − m )!( l + m )! so that (cid:107) Y ml (cid:107) L ( S ) = 1. It satisfies theeigenvalue problem ∆ Y ml = − l ( l + 1) Y ml (4.1)and orthonomal condition (cid:104) Y ml , Y m (cid:48) l (cid:48) (cid:105) L ( S ) = δ ll (cid:48) δ mm (cid:48) . Here and below, (cid:104) f, g (cid:105) L ( S ) := (cid:90) S f g d Ωwith d Ω being the area element of S and locally equals sin θdθdφ . The assoicated Legendrepolynomial Q ml ( z ) satisfies the general Legendre equation ddz (cid:18)(cid:112) − z ddz Q ml ( z ) (cid:19) + (cid:18) l ( l + 1) − m − z (cid:19) Q ml = 0and can be expressed via the Rodrigues’s formula, Q ml ( z ) = 12 l l ! (1 − z ) m d m + l dz m + l (1 − z ) l . In order to estimate the Sobolve norms (esp. H k norms) of Y ml , we take the L ( S ) inner productof (4.1) with Y (omitting indices for simplicity), invoke Green’s identity (5.15) to calculate l ( l + 1) = l ( l + 1) (cid:104) Y, Y (cid:105) L ( S ) = − (cid:104) Y, ∆ Y (cid:105) L ( S ) = (cid:104)∇ Y, ∇ Y (cid:105) L ( S ) which implies Y ∈ H ( S ) so inductively Y ∈ H k ( S ) , k ≥ . As a matter of fact, a little more rigor is needed in defining Sobolev norms on a manifold, but wewill skip the technical details and only use the fact that C (cid:48) k (1 + l ) k ≤ (cid:107) Y ml (cid:107) H k ( S ) ≤ C k (1 + l ) k , for l ≥ , k ≥ his estimate allows us to use the following definition, among many other equivalent definitions([19]), of the H k norm of a scalar-valued function Ψ defined on S . Definition 4.1.
For a scalar function Ψ with (cid:82) S Ψ = 0 and series expansion
Ψ = ∞ (cid:88) l =1 l (cid:88) m = − l ψ ml Y ml , where ψ ml = (cid:104) Ψ , Y ml (cid:105) L ( S ) , (4.2) define it H k norms, among other equivalent versions, as (cid:107) Ψ (cid:107) H k := (cid:118)(cid:117)(cid:117)(cid:116) ∞ (cid:88) l =1 l (cid:88) m = − l (1 + l ) k | ψ ml | . (4.3) Remark 4.1.
Here and below, we always start series and sums with l = 1 and assume ψ = (cid:82) S Ψ = 0 . Consequently, we define H k norms for u . Definition 4.2.
For a vector field u with Hodge Decomposition u = ∇ Φ + ∇ ⊥ Ψ with (cid:82) S Φ = (cid:82) S Ψ = 0 , we define its H k norm, among other equivalent versions, as (cid:107) u (cid:107) H k := (cid:113) (cid:107) Φ (cid:107) H k +1 + (cid:107) Ψ (cid:107) H k +1 . (4.4) In particular, if u is div-free with u = ∇ ⊥ Ψ and (cid:82) S Ψ = 0 , then (cid:13)(cid:13)(cid:13) ∇ ⊥ Ψ (cid:13)(cid:13)(cid:13) H k = (cid:118)(cid:117)(cid:117)(cid:116) ∞ (cid:88) l =1 l (cid:88) m = − l (1 + l ) k +1) | ψ ml | . (4.5) Remark 4.2.
Here and below, we always choose Φ , Ψ with zero global mean such that the abovedefinition is consistent with (cid:107) (cid:107) H k = 0 . Remark 4.3.
Apparently, under above definition, div-free and curl-free vector fields are H k -orthogonal. We now characterize operator L using the spherical harmonics. Let incompressible velocity field u = ∇ ⊥ Ψ . Then, by definition (2.5) and identity (3.3) L [ ∇ ⊥ Ψ] = ∇ ⊥ ∆ − (cid:16) ∇ z · ∇ ⊥ Ψ (cid:17) It is easy to verify that, in spherical coordinates, ∇ z = − sin θ e θ and ∇ ⊥ Ψ = ( ∂ θ Ψ) e θ ⊥ + (cid:18) ∂ φ Ψsin θ (cid:19) e φ ⊥ . Thus, combining the three equalities above, we obtain L [ ∇ ⊥ Ψ] = ∇ ⊥ ∆ − ∂ φ Ψ . (4.6) emma 4.1. (Spherical-harmonic representation of L .) For a scalar function Ψ with a seriesexpansion (4.2) , the identity (4.6) leads to L [ ∇ ⊥ Ψ] = ∇ ⊥ ∞ (cid:88) l =1 l (cid:88) m = − lm (cid:54) =0 − iml ( l + 1) ψ ml Y ml . (4.7)Here, we used the fact that ∂ φ Y ml = imY ml and ∆ − Y ml = − l ( l +1) Y ml for l ≥
1. We exclude l = 0 from the series due to ψ = (cid:82) S Ψ = 0. We also exclude m = 0 since it doesn’t contribute to(4.7) anyway.We now use spherical harmonics to characterize the projection operator Π null {L} given in (3.7). Itfollows from (4.7) that u ∈ null {L} iff u = ∇ ⊥ ∞ (cid:88) l =1 (cid:88) m =0 ψ ml Y ml , which is consistent with Lemma 3.1 since Y l is a function of θ only.Thus, the only modes that survive Π null {L} are those with m = 0 since Π null {L} is an L -orthogonalprojection and (cid:90) S ∇ ⊥ Y ml · ∇ ⊥ Y m (cid:48) l (cid:48) = − (cid:90) S ∆ Y ml Y m (cid:48) l (cid:48) = l ( l + 1) δ ll (cid:48) δ mm (cid:48) . Lemma 4.2. (Spherical-harmonic representation of Π null {L} .) For a scalar function Ψ with a seriesexpansion (4.2) , Π null {L} ( ∇ ⊥ Ψ) = ∇ ⊥ (cid:32) ∞ (cid:88) l =1 (cid:88) m =0 ψ ml Y ml (cid:33) (4.8)(id − Π null {L} )( ∇ ⊥ Ψ) = ∇ ⊥ ∞ (cid:88) l =1 l (cid:88) m = − lm (cid:54) =0 ψ ml Y ml (4.9)Note that the above 2 equations can also be derived from (3.7) together with the fact that12 π (cid:90) π Y ml ( φ, θ ) dφ = δ m . Combining (4.7) with (4.9) and using the absence of m = 0 modes from both series, we inducethat, when L is restricted to the image of (id − Π null {L} ), its null space is trivial and its inverse is“bounded” (as noted in Remark 1.2, this is automatically true for linear transform L : R n (cid:55)→ R n ).More precisely, Theorem 4.1.
For any div-free vector field u ∈ H k ( S ) and k ≥ , (cid:13)(cid:13)(cid:13) u − Π null {L} u (cid:13)(cid:13)(cid:13) H k ≤ (cid:13)(cid:13)(cid:13) L (cid:2) u − Π null {L} u (cid:3)(cid:13)(cid:13)(cid:13) H k +2 = (cid:107)L [ u ] (cid:107) H k +2 roof. Consider the stream function Ψ so that u = ∇ ⊥ Ψ. Combining (4.5) and (4.9), we obtain (cid:13)(cid:13)(cid:13) u − Π null {L} u (cid:13)(cid:13)(cid:13) H k = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) ∞ (cid:88) l =1 l (cid:88) m = − lm (cid:54) =0 (1 + l ) k +1) | ψ ml | . Combining (4.5) and (4.7), we obtain (cid:107)L [ u ] (cid:107) H k +2 = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) ∞ (cid:88) l =1 l (cid:88) m = − lm (cid:54) =0 (1 + l ) k +3) (cid:12)(cid:12)(cid:12)(cid:12) mψ ml l ( l + 1) (cid:12)(cid:12)(cid:12)(cid:12) . The key observation here is that m = 0 modes are absent in both series; thus, by a simple inequality(1 + l ) k +1) ≤ (1 + l ) k +3) (cid:12)(cid:12)(cid:12)(cid:12) ml ( l + 1) (cid:12)(cid:12)(cid:12)(cid:12) for l ≥ , | m | ≥ , we arrive at the conclusion! (cid:3) Appendix A: Preparation in Differential Geometry
Let M denote a 2-dimensional, compact, Riemmanian manifold without boundary, typically theunit sphere M = S − endowed with metric g induced from the embedding Euclidean space R .Let p ∈ M denote a point with local coordinates ( p , p ). Any vector field u in the tagent bundle T M is identified with a field of directional differential operator which is written in local coordinatesas u = (cid:88) i a i ∂∂p i . We use the notation ∇ u f := (cid:88) i a i ∂f∂p i to denote the directional derivative of a scalar-valued function f in the direction of u . Using theorthogonal projection Proj T R → T M induced by the Euclidean metric of R , we define the covariantderivative of a vector field v ∈ T M along another vector field u ∈ T M , ∇ u v := Proj T R → T M (cid:88) i =1 ( ∇ u v i ) e i . (5.1)Here, v is expressed in an orthonomal basis of R as v = v e + v e + v e .The metric g is identified with a (0 ,
2) tensor, simply put, an 2 × g ij ) × in localcoordinates. Thus, the vector inner product follows g ( ∂∂p i , ∂∂p j ) = g ij for 1 ≤ i, j ≤ . .1. Hodge Theory. ([21, 19])The Hodge *-operator, defined in an orthonormal basis ∂∂p , ∂∂p , satisfies ∗ dp = dp , ∗ dp = − dp , ∗ dp ∧ dp , ∗ ( dp ∧ dp ) = 1 . Using the Hodge star operator, we define the co-differential for any k -forms α in an n -dimensionalmanifold, codifferential : δα := ( − k ∗ − d ∗ α = ( − n ( k +1)+1 ∗ d ∗ α, and in particular, for n = 2, δα = − ∗ d ∗ α. In the case when the Riemannian manifold M has no boundary, the codifferential is the adjoint ofexterior differential w.r.t. L ( M ) inner product induced by the given metric g , (cid:104) dα, β (cid:105) L ( M,g ) = (cid:104) α, δβ (cid:105) L ( M,g ) . (5.2)The Hodge Laplacian (a.k.a. Laplace-Beltrami operator and Laplace-de Rham operator) is thendefined by ∆ H := dδ + δd. (5.3)In particular, for a scalar-valued function f in a local basis (cid:110) ∂∂p i (cid:111) with metric g, it is identified as∆ H f = − (cid:112) | g | (cid:88) i,j ∂ i ( (cid:112) | g | g ij ∂ j f )where ( g ij ) is the matrix inverse of ( g ij ). Thus, on a surface M , the Hodge Laplacian ∆ H definedin(5.3) amounts to -1 times the surface Laplacian ∆ M . In particular, if M is a two-dimensionalsurface, then for scalar function f, ∆ M f = − δdf = ∗ d ∗ df (5.4)since δf = 0 on a two-dimensional manifold.For now on, we will use ∆ for ∆ M .The Hodge decomposition theorem in its most general form states that for any k -form ω on an oriented compact Riemannian manifold, there exist a ( k − α , ( k + 1)-form β and aharmonic k -form γ satisfying ∆ H γ = 0, s.t. ω = dα + δβ + γ. In particular, for any 1-form ω on a 2-dimensional manifold with the 1st Betti number 0 (looselyspeaking, there is no “holes”), there exist two scalar-valued functions Φ , Ψ such that ω = d Φ + δ ( ∗ Ψ) = d Φ − ∗ d Ψ . (5.5)Here, we used the Hodge theory to equate the dimension of the space of harmonic k -forms on M with the k -th Betti number of M . For the cohomology class containing the unit sphere S − , the0th, 1st and 2nd Betti numbers are respectively 1 , , The existence of such basis is guaranteed by the Gram-Schmidt orthogonalization process. .2. In Connection With Vector Fields.
Let (cid:91) and (cid:93) denote the musical isomorphism between
T M and T ∗ M induced by the given metric g , i.e. for vector field a = (cid:80) i a i ∂ i and covector field(1-form) A = (cid:80) i A i dp i index-lowering, ( (cid:88) i a i ∂ i ) (cid:91) = (cid:88) i,j g ij a i dp j index-raising, ( (cid:88) i A i dp i ) (cid:93) = (cid:88) i,j g ij A i ∂ j In connection with the metric g , for vector fields u , v , ∗ ( u · v ) = u (cid:91) ∧ ( ∗ v (cid:91) )so that consistently, (cid:104) u , v (cid:105) L ( M,g ) = (cid:104) u (cid:91) , v (cid:91) (cid:105) L ( M,g ) . (5.6)In a 2-dimensional Riemannin manifold, the divergence and curl of a vector field u ∈ T M arethen defined as scalar-valued functions on M , divergence , div u := − δ ( u (cid:91) ) = ∗ d ∗ ( u (cid:91) ) (5.7) curl , curl u := − ∗ d ( u (cid:91) ) (5.8)For a scalar field f , we define gradient and its π/ gradient , ∇ f := ( df ) (cid:93) (5.9) rotated gradient , ∇ ⊥ f := ( δ ( ∗ f )) (cid:93) = − ( ∗ df ) (cid:93) (5.10)We also define the counterclockwise π/ ⊥ acting on a vector field as u ⊥ := − ( ∗ u (cid:91) ) (cid:93) (5.11)so that, consistantly, ∇ ⊥ f = ( ∇ f ) ⊥ , and div u = curl u ⊥ . It is then easy to use these definitions to verify the following properties, for scalar function f ona surface, curl ∇ f = div ∇ ⊥ f = 0 (5.12)due to dd = 0 and δδ = 0; and div ∇ f = curl ∇ ⊥ f = − δd f (5.13)with − δd being the classical surface Laplacian ∆ (c.f. (5.4)).To this end, the vector-field version of Hodge decomposition (5.5) becomes u = ∇ Φ + ∇ ⊥ Ψ . (5.14) Here and below, 0-forms are identified with scalar-valued functions. e note that, by the virtue of (5.13), the decomposition satisfiesdiv u = ∆Φ , curl u = ∆Ψ . We finally establish a version of the Green’s identity and a version of the product rule onRiemannian manifolds. First, the duality relation (5.2) together with (5.4) and (5.6) implies (cid:104) f, ∆ g (cid:105) L ( M ) = −(cid:104)∇ f, ∇ g (cid:105) L ( M,g ) . (5.15)Secondly, as a consequence of the product rule for differential d acting on wedge product, for scalarfunction z and vector field u , we havecurl ( z u ⊥ ) = div ( z u ) = ∇ z · u + z div u . (5.16)5.3. Local Expression in Terms of Spherical Coordinates for M = S . Let φ denote thelogitude and θ the colatitude of a point on a sphere. Let e φ , e θ denote the unit tangent vectors inthe increasing directions of φ and θ . Then, at point p that is away from the poles, ∂ φ = sin θ e φ , ∂ θ = e θ , namely, 1sin θ ∂ φ and ∂ θ form an orthonromal basis of T M p (5.17)Therefore, the musical isomorphisms, in φ, θ coordinates, satisfy( 1sin θ ∂ φ ) (cid:91) = sin θdφ and ( ∂ θ ) (cid:91) = dθ form an orthonomal basis of T ∗ M p . In this context, the Hodge *-operator satisfiesfor 1-forms, ∗ ( A dφ + A dθ ) = A sin θ dθ − A sin θdφ for 0-forms and 2-forms, ∗ ( Adφ ∧ dθ ) = A sin θ , ∗ A = A sin θdφ ∧ dθ The differential operators defined in (5.7) — (5.10) then become,for vector field u = u e φ + u e θ div u = 1sin θ ( ∂ φ u + ∂ θ ( u sin θ ))curl u = 1sin θ ( ∂ φ u − ∂ θ ( u sin θ ))and for scalar field f ∇ f = 1sin θ ∂ φ f e φ + ∂ θ f e θ ∇ ⊥ f = ∂ θ f e φ − θ ∂ φ f e θ ∆ f = 1sin θ ( ∂ φ f + sin θ∂ θ (sin θ∂ θ f )) . Appendix B: Uniform Estimates Independent of ε In this section, we use energy methods to prove local-in-time existence of classical solutions forthe incompressible Euler equations independent of the Rossby number ε . Rewrite the equation asin (2.6), ∂ t u + ∇ ⊥ ∆ − curl ( ∇ u u ) = 1 ε L [ u ] , (6.1)where operator L , as in (2.5), is defined by L [ u ] := ∇ ⊥ ∆ − curl ( z u ⊥ ) . (6.2)The main challenge rises from the nontrivial geometry of S : only a selective set of differential-integral operators on S commute with each other. Although this is not a problem regardingwell-posedness with fixed ε , it causes difficulties in obtaining ε -independent estimates. The factthat our L has variable coefficients adds another layer of difficulties. In proving the followingtheorem, we will address these commutability issues specifically. Theorem 6.1.
Consider the incompressible Euler equations (6.1) , (6.2) on a rotating sphere S with div-free initial data u . Given any integer k > , assume u ∈ H k ( S ) . Then, there existsuniversal constants C , T independent of ε so that (cid:107) u ( t, · ) (cid:107) H k ≤ C (cid:107) u (cid:107) H k for any t ∈ (cid:20) , T (cid:107) u (cid:107) H k (cid:21) . Proof.
For simplicity, we only prove the case when k is even.First, we show that for any u ∈ L ( S ), (cid:90) S u · L [ u ] = 0 if div u = 0 . (6.3)Indeed, by definition (6.2) and Hodge decompositon (2.1), (2.2), we have z u ⊥ − L [ u ] = (id − ∇ ⊥ ∆ − curl )[ z u ⊥ ] = ∇ ∆ div [ z u ⊥ ]which is curl-free and therefore L -orthogonal to a div-free flow u , namely, (cid:90) S u · ( z u ⊥ − L [ u ]) = 0 . Thus, (cid:90) S u · L [ u ] = (cid:90) S u · ( z u ⊥ ) = 0 . Secondly, we show that ∆ and L commute. Indeed, for any incompressible flow u = ∇ ⊥ Ψ∆ L [ u ] = ∆ ∇ ⊥ ∆ − ∂ φ Ψ by (4.6)= ∇ ⊥ ∆∆ − ∂ φ Ψ by (5.4), (5.10)= ∇ ⊥ ∆ − ∆ ∂ φ Ψ he key step remaining is to show that ∆ and ∂ φ commute. This can be done using the fact thatspherical harmonics Y ml are eigenfunctions for both ∆ and ∂ φ . More specifically, for any sphericalharmonic Y ml , ∂ φ Y ml = ∂ φ e imφ Q ml (cos θ ) = imY ml and therefore ∆ ∂ φ Y ml = ∆( imY ml ) = − iml ( l + 1) Y ml = ∂ φ ∆ Y ml . Lastly, once the commutability of ∆ and L are established as above, we easily obtain for eveninteger k ≥ (cid:90) S ∆ k/ u · ∆ k/ L [ u ] = (cid:90) S ∆ k/ u · L [∆ k/ u ] = 0where the second equality is due to (6.3) and ∆ k/ u also being incompressible (note: ∆ and divcommute). Now, take the L inner product of ∆ k u with both sides of (6.1), knowing that the RHSshould vanish, (cid:90) S ∆ k u · ∂ t u + ∆ k u · ∇ ⊥ ∆ − curl ( ∇ u u ) = 0and invoke the standard energy methods (e.g. [10]) to arrive at conclusion, which is clearly ε -independent. (cid:3) Acknowledgments
The work of the second author is sponsored in part by AFOSR contract FA9550-08-1-0055.
References [1] Arnold, Vladimir I.; Khesin, Boris A.
Topological methods in hydrodynamics.
Applied Mathematical Sciences,125. Springer-Verlag, New York, 1998.[2] Babin, A.; Mahalov, A.; Nicolaenko, B.
Global splitting and regularity of rotating shallow-water equations.
Eu-ropean J. Mech. B Fluids, 16 (1997), no. 5, 725–754.[3] Babin, A.; Mahalov, A.; Nicolaenko, B.
Global splitting, integrability and regularity of 3D Euler and Navier-Stokesequations for uniformly rotating fluids.
European J. Mech. B Fluids, 15 (1996), no. 3, 291–300.[4] Chorin, Alexandre J.; Marsden, Jerrold E.
A mathematical introduction to fluid mechanics.
Third edition. Textsin Applied Mathematics, 4. Springer-Verlag, New York, 1993.[5] Galperin, B.; H. Nakano; H. Huang; S. Sukoriansky.
The ubiquitous zonal jets in the atmospheres of giant planetsand Earths oceans . Geophys. Res. Lett., 31 (2004), L13303, doi:10.1029/2004GL019691.[6] Galperin, B.; S. Sukoriansky; N. Dikovskaya; P. L. Read; Y. H. Yamazaki; R. Wordsworth.
Anisotropic turbulenceand zonal jets in rotating flows with a β -effect. Nonlinear Processes in Geophysics, 13, 1 (2006), 83–98.[7] Garc´ya-Melendo, E.; S´anchez-Lavega, A.
A study of the stability of Jovian zonal winds from HST images: 1995–2000 , Icarus, 152 (2001), 316–330.[8] Goncharov, Yevgeny.
On existence and uniqueness of classical solutions to Euler equations in a rotating cylinder.
Eur. J. Mech. B Fluids, 25 (2006), no. 3, 267–278.[9] Huang, H.-P.; B. Galperin; S. Sukoriansky.
Anisotropic spectra in two-dimensional turbulence on the surface ofa rotating sphere , Phys. Fluids, 13 (2001), 225–240.[10] Majda, Andrew J.; Bertozzi, Andrea L.
Vorticity and incompressible flow.
Cambridge Texts in Applied Mathe-matics, 27. Cambridge University Press, Cambridge, 2002. xii+545 pp.
11] Maximenko, N. A.; B. Bang; H. Sasaki.
Observational evidence of alternating jets in the World Ocean . Geophys.Res. Lett., 32 (2005), L12607, doi:10.1029/2005GL022728.[12] NASA/JPL/University of Arizona. http://photojournal.jpl.nasa.gov/catalog/PIA02873 [13] Nozawa, T.; S. Yoden.
Formation of zonal band structure in forced two-dimensional turbulence on a rotatingsphere , Phys. Fluids, 9 (1997), 2081–2093.[14] Pedlosky, J.
Geophysical fluid dynamics.
Springer-Verlag, Berlin, 1992.[15] Porco, C., et al.
Cassini imaging of Jupiters atmosphere, satellites and rings , Science, 299 (2003), 1541–1547.[16] Roden, G.
Upper ocean thermohaline, oxygen, nutrients, and flow structure near the date line in the summer of1993 , J. Geophys. Res., 103 (1998), 12,919 – 12,939.[17] Roden, G.
Flow and water property structures between the Bering Sea and Fiji in the summer of 1993 , J. Geophys.Res., 105 (2000), 28,595–28,612.[18] Sukoriansky, S.; B. Galperin; N. Dikovskaya.
Universal spectrum of two-dimensional turbulence on a rotatingsphere and some basic features of atmospheric circulation on giant planets , Phys. Rev. Lett., 89 (2002), 124501.[19] Taylor, Michael E.
Partial differential equations. I. Basic theory.
Applied Mathematical Sciences, 115. Springer-Verlag, New York, 1996.[20] Vallis, G.; M. Maltrud.
Generation of mean flows and jets on a beta plane and over topography , J. Phys.Oceanogr., 23 (1993), 1346–1362.[21] Warner, Frank W.
Foundations of differentiable manifolds and Lie groups.
Corrected reprint of the 1971 edition.Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
School of Mathematical and Statistical SciencesArizona State University, Wexler Hall (PSA)Tempe, Arizona 85287-1804 USA
E-mail address , Bin Cheng: [email protected]
E-mail address , Alex Mahalov: