Point vortices on a rotating sphere

25 Pages·0.3 MB·English

by  Frederic Laurent-Polz

Point vortices on a rotating sphere
Fr´ed´eric Laurent-Polz
Institut Non Lin´eaire de Nice, Universit´e de Nice, France
3 [email protected]
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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
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6
AMS classification scheme number: 70E55,70H14, 70H33
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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׈ 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 momen-
tum 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 con-
tinuous 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 con-
served 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 rel-
ative 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” vor-
tices [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 con-
tinuous 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