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Point vortices on a rotating sphere Frédéric Laurent-Polz Institut Non Linéaire de Nice, Université de Nice, France 3 [email protected] 0 0 2 n Abstract a J We study the dynamics of N point vortices ona rotating sphere. The Hamilto- 0 nian system becomes infinite dimensional due to the non-uniform background 3 vorticity coming from the Coriolis force. We prove that a relative equilibrium ] formed of latitudinal rings of identical vortices for the non-rotating sphere per- S siststo be a relativeequilibriumwhenthe sphererotates. We study the nonlin- D earstabilityofapolygonofplanarpointvorticesonarotatingplaneinorderto . approximatethecorrespondingrelativeequilibriumontherotatingspherewhen h t the ring is close to the pole. We then perform the same study for geostrophic a vortices. To end, we compare our results to the observations on the southern m hemisphere atmospheric circulation. [ 1 Keywords: point vortices, rotating sphere, relative equilibria, nonlinear sta- v bility, planar vortices, geostrophic vortices, Southern Hemisphere Circulation 0 6 AMS classification scheme number: 70E55,70H14, 70H33 3 1 0 PACS classification scheme number: 45.20.Jj, 45.50.Jf, 47.20.Ky,47.32.Cc 3 0 / 1 Introduction h t a The interest of studying point vortices on a rotating sphere is clearly geophys- m ical. This may permit also to understand the motion of concentrated regions v: of vorticity on the surface of planets such as Jupiter [DL93]. The literature is i now numerous on point vortices on a non-rotating sphere [B77, KN98, KN99, X KN00, N00, PM98, LMR01, BC01, LP02] but only few papers consider a ro- r tating sphere [F75, B77, B85, KR89, DP98]. In [F75], the interaction of three a identical point vortices equally spaced on the same latitude is investigated via the β-plane approximation. It is shown — under some additional assumptions —thatthisconfigurationisanequilibrium,andthatitslinearstabilitydepends onthe strength of the vortices: linearly stable for a negative or a strongly posi- tive strength, linearly unstable otherwise. In [B77], the equations of motion on a rotating sphere are given, while in [B85] the motion of a single point vortex is given. It appears that a single vortex moves westward and northward as a hurricane does. In [KR89], the approach is completely different from ours and [B77];theyprovedtheexistenceofrelativeequilibriaformedoftwoorthreevor- ticesofpossiblynon-identicalvorticities. In[DP98],theymodelthebackground 1 vorticity coming from the rotation by latitudinal strips of constant vorticities and they study the motion of a vortex pair (point-vortex pair as well as patch- vortex pair). A vortex pair is a solution formed of two vortices with opposite sign vorticities rotating around the North pole. A vortex pair moving eastward or strongly westward is stable, unless the vortex sizes are too large. Different types of instabilities are described for weak westward pairs. In this paper, we first recall some basics notions of geometric mechanics in Section 2. In particular, the notion of relative equilibrium is defined. In the cases of that paper, a relative equilibrium corresponds to a rigid rotation of N point vortices about some axis. We then give the equations of motion for point vortices on a rotating sphere [B77] in Section 3. The Hamiltonian system is infinite dimensional due the background vorticity coming from the rotation of the sphere. We prove that a relative equilibrium formed of latitudinal rings of the non- rotatingsystempersistswhenthesphererotates. From[LMR01]and[LP02],we know that the following arrangementsof latitudinal ringsare relative equilibria of the non-rotating system: C (R,k p), D (2R,k p), D (R,R,k p), and Nv p Nh p Nd ′ p D (R ,k p) (k is the number of polar vortices). See Figures 1, 2, and 3. 2Nh e p p µ µ λ +1 +1 +1 +1 +1 +1 +1 +1 (a)C4v(R) (b)C4v(R,p) Figure 1: The C (R,k p) relative equilibria with k =0,1. nv p p InSection4,wegivethestabilityofrelativeequilibriaC (R)andC (R,p) Nv Nv forthreedifferentapproximationsorlimitingcases: pointvorticesonarotating plane,geostrophicvortices,andpointvorticesonanon-rotatingsphere. Indeed in that particular cases the system becomes finite dimensional and we can use the techniquesofSection2toobtainbothnonlinearandlinearstability results. The stability is determined with a block diagonalizationversionof the Energy- 2 + + + + + + + + (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) PSfrag repla ements PSfrag repla ements (cid:0) (cid:0) (cid:0) D4h(2R) D4d(R;R0) D4h(2R) D4d(R;R0) Figure 2: Relative equilibria D (2R) and D (R,R). 4h 4d ′ (cid:0) + (cid:0) + (cid:0) + + (cid:0) + (cid:0) + (cid:0) PSfrag repla ements Figure 3: Equatorial( )ring D (R ). 2Nh e ± 3 momentummethod[OR99,LP02]andtheLyapunovstabilityresultsaremodulo SO(2). In particular,we compute the (nonlinear)stability of a polygonformed of N identical point vortices in the plane together with a central vortex of arbitrary vorticity in Appendix B. Our results differ from those of [CS99] but agree with the linear study of [MS71]. We also improve some stability results ongeostrophicvorticesof[MS71]provingthatsomelinearstableconfigurations are actually Lyapunov stable. The paper ends with a discussion on the Southern Hemisphere Circulation and its relationship with vortices. 2 Geometric mechanics We recall thereafter some basics notions of geometric mechanics. We refer to [MR94, Or98] for further details. Let G be a connected Lie group acting smoothly on a symplectic manifold ( ,̟). ConsideranHamiltoniandynamicalsystem( ,̟,H)withmomentum P P mapJ : g suchthatthe HamiltonianvectorfieldX andthe momentum ∗ H P → map are G-equivariant. A point x is called a relative equilibrium if for all e ∈P t there exists g G such that x (t)=g x , where x (t) is the dynamic orbit t e t e e ∈ · of X with x (0) = x . In other words, the trajectory is contained in a single H e e grouporbit. Arelativeequilibriumx isacriticalpointoftheaugmented e ∈P Hamiltonian: H (x)=H(x) J(x) µ,ξ ξ −h − i for some ξ g. The vector ξ is unique if the action of G is locally free, and is ∈ called the angular velocity of x . e In the case of point vortices on a rotating sphere, we will have G= SO(2). Hence relativeequilibria arerigidrotationswith angularvelocityξ so(2) R ∈ ≃ around the axis of rotation of the sphere. The Hamiltonian is G-invariant, but may have additional symmetries. De- notebyGˆ thegroupofsymmetriesoftheHamiltonian. Forexample,inthecase ofN identicalpointvorticesonanon-rotatingsphere,onehasG=SO(3) S N andGˆ =O(3) S [LMR01]. Let the fixed point set of a subgroupK of G×ˆ be: N × Fix(K, )= x g x=x, g K . P { ∈P | · ∀ ∈ } The following theorem permits to determine relative equilibria, it is a corollary of the Principle of Symmetric Criticality of Palais [P79]. Theorem 2.1 LetK beasubgroupofGˆ,x Fix(K, )andµ=J(x). Assume that Gˆ is compact. If x is an isolated point∈in Fix(KP, ) J 1(µ), then x is a − P ∩ relative equilibrium. Note that this result depends only on the symmetries, the phase space and the momentum map, and not depends on the form of the Hamiltonian. A relative equilibrium obtained with that result is said to be a large symmetry relative equilibrium since its isotropy subgroup must be large. 4 Tocomputethe stabilityofrelativeequilibria,weusetheEnergy momentum method: let x be a relative equilibrium, µ = J(x ) g and ξ be its e e ∗ ∈ angular velocity. The method consists first to determine the symplectic slice =T (G x ) KerDJ(x ) N xe µ· e ⊥∩ e where G = g G Coad µ=µ . µ g { ∈ | · } The second step consists to examine the definitness of d2H (x ), and apply ξ e |N the following result [Pa92, OR99] which holds in particular if G is compact: If d2H (x ) is definite, then x is Lyapunov stable modulo G . ξ e e µ |N In Section 4.1, we will consider vortices in the plane (point vortices and geostrophic vortices), the symmetry group G is SE(2) or SO(2) depending on whether the plane is rotating. However we will forget translational symmetries since SE(2) is not compact, hence we take G=SO(2) and the previous result holds. Moreover we have G = SO(2) for all µ so(2) R. In Section µ ∗ ∈ ≃ 4.2, the sphere is non-rotating, thus the symmetry group is G =SO(3) [LP02, LMR]. We have SO(3) = SO(2) for µ = 0, and SO(3) = SO(3). Since µ µ=0 6 we will consider only relative equilibria with a non-zero momentum µ for that section, Lyapunov stable will mean Lyapunov stable modulo SO(2) throughout that paper. The linear stability is investigated calculating the eigenvalues of the lin- earization in the symplectic slice, that is of L = Ω♭−1d2Hξ (xe) where Ω♭ is the matrix of ̟ . N N |N N |N Thesymplecticslice isaG -invariantsubspace. Hencewecanperforman N xe G -isotypicdecompositionof ,thispermitstoblockdiagonalizethematrices xe N d2H (x ) and L , their eigenvalues are then easier to compute and we can ξ e |N N conclude about both Lyapunov and linear stability. A basis of the symplectic slice in which these matrices block diagonalize is called a symmetry adapted basis. The symmetry adapted bases do not depend on the particular form of the system, they depend only on the symmetries of the system. The different steps of the method are widely detailed in [LP02] which is a study on point vortices on a non-rotating sphere. A relative equilibrium x is said to be elliptic if it is spectrally stable with e d2H (x ) not definite. An elliptic relative equilibrium may be Lyapunov ξ e |N stable, but this can not be proved via the Energy-momentum method (but KAMtheorymaywork). Notealsothatanellipticrelativeequilibriumbecomes linearly unstable when some dissipation is added to the system [DR02]. 3 Equations of motion and relative equilibria In this section we consider N point vortices on a unit sphere rotating with a constant angular velocity Ω around the axis (Oz). Hence the SO(3) symmetry of the non-rotating system breaks to SO(2). The Coriolis force induces a continuous vorticity ω on the sphere: ω = ω +ω where ω = λ δ(x x ), Ω 0 Ω 0 i i − i P 5 λ is the vorticity of the vortex x , and ω (t = 0) = 2Ωcosθ. The continuous i i Ω vorticity is not uniform and thus interacts a priori with the singular vorticity i.e. the vortices. This interaction makes ω a function of time. For example, Ω without any vortices ω =ω =2Ωcosθ is a steady solution. Ω The stream function ψ satisfies ∆ψ = ω. Hence ψ = G ω where G is the ∗ Greenfunction onthe sphereG(x,x)=1/(4π)ln(1 x x), andwe obtain the ′ ′ − · following expression for the stream function: N 1 ψ = λ ln(1 x x )+G ω . i i Ω 4π − · ∗ i=1 X The equations governing the motion of the vortices are therefore [B77]: N sinθ sin(φ φ ) 1 ∂G ω θ˙ = λ j i− j + ∗ Ω(θ ,φ ) i j i i 1 x x sinθ ∂φ i j i i j=1,j=i − · X6 N sinθ cosθ sinθ cosθ cos(φ φ ) ∂G ω sinθ φ˙ = λ i j − j i i− j ∗ Ω(θ ,φ ) i i j i i − 1 x x − ∂θ i j i j=1,j=i − · X6 foralli=1,...,N. Moreover,thevorticitysatisfies∂ ω+(u )ω =0(Euler),u t ·∇ beingthe velocity,andthis equationcanbe writtenequivalently withaPoisson bracket: ω˙ = ω,ψ { } where ∂f ∂g ∂f ∂g f,g = { } ∂cosθ∂φ − ∂φ∂cosθ for two smooth functions f,g on the sphere. The full dynamical system is therefore: N sinθ sin(φ φ ) 1 ∂G ω θ˙ = λ j i− j + ∗ Ω(θ ,φ ) i j i i 1 x x sinθ ∂φ i j i i j=1,j=i − · X6 N sinθ cosθ sinθ cosθ cos(φ φ ) ∂G ω sinθ φ˙ = λ i j − j i i− j ∗ Ω(θ ,φ ) i i j i i − 1 x x − ∂θ i j i j=1,j=i − · X6 ω˙ = ω ,ψ Ω Ω { } since the strengths of the vortices are constant. Remark. The vorticity must satisfy ω dS = 0 from Stoke’s theorem. It should be noted that when the sum Λ = λ is non-zero, then the stream i R function remains as before taking ω = Λ/(4π)+ λ δ(x x ). Since ω 0 − P i i − i 0 satisfies ω dS =0, it follows that ω dS =0 when ω =2Ωcosθ. 0 Ω P R R 6 It is easy to verify that the following quantity (total kinetic energy) is conserved and serves as a Hamiltonian: 1 1 H = ω ψ dS = ω G ω dS 2ZS2 · 2ZS2 · ∗ N 1 1 = λ λ ln(1 x x )+ λ G ω (θ ,φ ) i j i j k Ω k k 4π − · 2 ∗ i<j k=1 X X 1 + ω (θ,φ)ω (θ ,φ)G(θ,φ,θ ,φ)dSdS . Ω Ω ′ ′ ′ ′ ′ 2ZS2ZS2 Let SO(2) be the group of rotations with as axis the axis of rotation of the sphere,andconsiderthe diagonalactionofSO(2)onthe productofN spheres. Clearly the continuous vorticity satisfies ω (g θ,g φ,g θ ,g φ ,...,g θ ,g φ )=ω (θ,φ,θ ,φ ,...,θ ,φ ) Ω 1 1 N N Ω 1 1 N N · · · · · · for all g SO(2). It follows that H is SO(2)-invariant and the dynamical ∈ system is SO(2)-equivariant. Due to the continuous vorticity, the phase space becomes infinite dimen- P sional: = (x ,...,x ) S2 S2 x =x if i=j (S2) 1 N i j P { ∈ ×···× | 6 6 }×F where (S2) is the set of smooth functions on the sphere. F The following vector (momentum vector) is conserved and thus provides three conserved quantities [B77]: M~ = ω ~x dS ZS2 · N S2ωΩ(θ,φ)sinθcosφ dS = λ x + ω (θ,φ)sinθsinφ dS . j j  RS2 Ω  Xj=1 RS2ωΩ(θ,φ)cosθ dS   R The term λ x corresponds to the moment map for point vortices on a non- j j rotating sphere [LMR01]. Indeed in the case of the non-rotating sphere, the P dynamicalsystemis SO(3)-equivariant,andthisleadstothreeconservedquan- tities since SO(3) is of dimension three. In the case of the rotating sphere the system is only SO(2)-equivariant, hence the symmetries provide only one con- servedquantity(whichisthez-componentofM~). Actually,thethreeconserved quantitiescomefromageneralpropertyof2D incompressibleandinviscidfluid flows on compact and simply connected surfaces which states that the vector ω ~xdSisconserved,nomatterthesymmetriesofthesurfacewehave[B]. That · vector and the momentum map simply coincide in the case of a non-rotating R sphere. We would like to know if the relative equilibria found on the non-rotating sphere persist when the sphere rotates. We call an N-ring a latitudinal regular 7 polygon formed of N identical vortices. The following theorem show that relative equilibria formed of N-rings (together with possibly some polar vortices) persist. Theorem 3.1 Let x be a relative equilibrium of the non-rotating system, with e angular velocity ξ , formed of m N -rings (N 2) together with possibly 0 m m ≥ some polar vortices. Then (x ,ω (t)=2Ωcosθ) is a relative equilibrium of the e Ω rotating system with angular velocity ξ =ξ +Ω. 0 Corollary 3.2 The relative equilibria C (R,k p), D (2R,k p), D (R,R,k p), D (R ,k p) Nv p Nh p Nd ′ p 2Nh e p persist when the sphere rotates. Proof. Let (x ,ξ ) be a relative equilibrium of the non-rotating system e 0 formed of k N -rings, a polar vortex being considered as a 1-ring. Hence r k 1 ∂G ω θ˙ = ∗ Ω(θ ,φ ) i i i sinθ ∂φ i i 1 ∂G ω φ˙ = ξ ∗ Ω(θ ,φ ) i 0 i i − sinθ ∂θ i i for all i=1,...,N. Let the continuous vorticity be ω (t) = 2Ωcosθ for all time t. We have Ω ∂G ωΩ =0 since G(θ,,φ,θ ,φ )dφ does not depend on φ , and ∂φ∗i i i i ∂G ω ΩR π 2π cosθsinθ sinθcosθ cos(φ φ ) Ω i i i ∗ = sin2θ − − dφdθ ∂θ 4π 1 cosθcosθ sinθsinθ cos(φ φ ) i Zθ=0 Zφ=0 − i− i − i Ω π cosθsinθ cosθ 1 cosθcosθ i i i = sin2θ + 1 − dθ. 2 cosθ cosθ sinθ − cosθ cosθ Zθ=0 (cid:20)| − i| i (cid:18) | − i|(cid:19)(cid:21) And we obtain after some calculus that ∂G ωΩ = Ωsinθ . ∂∗θi − i Thus we provedthat θ˙ =0 and φ˙ =ξ +Ω for all i=1,...,N. It remains i i 0 then to prove that ω˙ = 0. One has ω˙ = ω ,ψ = 2Ω∂ψ. Since ∂G ωΩ = 0, Ω Ω { Ω } ∂φ ∂∗φ it follows that ∂ψ kr Nk sinθsinθ sin(φ φ ) k j,k = λ − k ∂φ 1 cosθcosθ sinθsinθ cos(φ φ ) k k j,k k=1j=1 − − − XX where λ and θ are respectively the vorticity and the co-latitude of the ring k k k, and φ = ǫ +2π(j 1)/N. It can be shown that this sum vanishes (see j,k k − Appendix A), hence ω˙ =0 and (x ,ω (t)=2Ωcosθ) is a relative equilibrium Ω e Ω of the rotating system with angular velocity ξ =ξ +Ω. 2 0 Remark. In order to prove existence of relative equilibria, we may think to use the Principle of Symmetric Criticality of Section 2. The Principle holds since SO(2) is compact, however the fixed point sets Fix(C , ) are infinite N P dimensional, it is therefore quite a task to find relative equilibria with that method. 8 4 Stability of relative equilibria on the rotating sphere In this section, we unfortunately do not compute the stability of the relative equilibria determined in the previous section. Indeed the Hamiltonian system is here infinite dimensional and the method described in Section 2 work only for finite dimensional Hamiltonian systems. Hence we will give in this section thestabilityofdifferentapproximationsorlimitingcasesofthe“rotatingsphere problem” which lead to a finite dimensional Hamiltonian system. 4.1 Vortices on a rotating plane We consider point vortices on a rotating plane in order to approximate the C (R) and C (R,p) relative equilibria on a rotating sphere when the ring Nv Nv is close to the pole. This approximation can be done in two different manners: the “classical” point vortices on a rotating plane, and the “geostrophic” vortices [S43]. The equivalent of the C (R,p) arrangement on the plane is the Nv arrangementC (R,p): aregularpolygonformedofN vorticesofstrength1to- N getherwith a centralvortexof strengthλ. Whenthere is no centralvortex,the configuration is denoted C (R) and corresponds to the C (R) arrangement N Nv on the sphere (see Figure 4). We assume that N 3 since the cases N =2 are ≥ degenerate due the colinearity of the arrangements. +1 +1 λ +1 +1 +1 +1 +1 +1 (a)C4(R) (b)C4(R,p) Figure 4: The C (R,k p) planar relative equilibria. N p Point vortices on a rotating plane. Considerasystemofpointvorticesonaplane rotatingwith aconstantangular 9 velocity Ω around its normal axis. Contrary to the rotating sphere, the continuous vorticity induced by the rotation is here uniform, thus the continuous vorticity does not interact with the point vortices. The dynamics of point vor- ticesontherotatingplaneisthereforesimilartothatforthenon-rotatingplane as the following proposition shows. Denote a relative equilibrium x =(z ,...,z ) CN with angular velocity e 1 N ∈ ξ by (x ,ξ) and set the origin of the plane to be the centre of the rotation. e Recall that Λ is the sum of the vorticities, that is Λ= λ . i i Theorem 4.1 If (x ,ξ) is a relative equilibrium of thePnon-rotating plane such e that λ z = 0 and Λ = 0, then (x ,ξ +Ω) is a relative equilibrium of the j j e 6 rotating plane. Moreover (x ,ξ+Ω) satisfies the stability properties of (x ,ξ) e e P (as elements of /SO(2)). P Proof. The continuous vorticity induced by the rotation of the plane is ω (x,y) = 2Ω, thus the distribution of vorticity for N point vortices z = Ω j ρ exp(iφ ), j =1,...,N is given by ω =2Ω+ λ δ(z z ). It follows that j j j j − j the Hamiltonian is Ω P H = λ ρ2+H , 2 j j 0 j X where H is the Hamiltonian for the non-rotating plane (see Appendix B). 0 Since λ ρ˙ = 1 ∂H and λ φ˙ = 1 ∂H, the dynamical system is given by j j −ρj ∂φj j j ρj ∂ρj ρ˙j = 0 + −λj1ρj ∂∂Hφj0 . φ˙ Ω 1 ∂H0 (cid:18) j (cid:19) (cid:18) (cid:19) λjρj ∂ρj ! Since λ z =0 and Λ=0, one can show that the relative equilibrium of the j j 6 non-rotating plane (x ,ξ) satisfies e P ∂H 1 ∂H 0 0 =0, =ξ ∂φ λ ρ ∂ρ j j j j forallj(Λ=0isrequiredtoinsuretheexistenceofthe“barycentre” λ z /Λ). 6 j j j Hence(x˜ ,ξ˜)withx˜ =x andξ˜=Ω+ξisarelativeequilibriumoftherotating e e e P plane. Ifavortexz oftherelativeequilibriumx isattheorigin(acentralvortex), j e this discussion is not valid due to the degeneracy of polar coordinates at the origin, we then use cartesian coordinates and find the same conclusions. We are interested in stability in /SO(2) (stability modulo SO(2)) where P SO(2) is the group of rotations around the origin. The “rotating” dynamical system is SO(2)-equivariant. The momentum map coming from the SO(2) symmetryisJ = λ ρ2/2 R so(2)⋆ (seeAppendixB),thusH =H +ΩJ. j j j ∈ ≃ 0 Hence P d2H (x˜ )=d2H(x ) (Ω+ξ)d2J(x )=d2H (x ) ξd2J(x )=d2H (x ). ξ˜ e e − e 0 e − e 0ξ e 10