Stability of Relative Equilibria of Point Vortices on the Sphere
Fr\'ed\'eric Laurent-Polz
James Montaldi
Mark Roberts
We describe the linear and nonlinear stabilities of certain configurations of point vortices on the sphere forming relative equilibria. These configurations consist of up to two rings, with and without polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, enabling their eigenvalues to be calculated.
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\title{Stability of Relative Equilibria of Point Vortices on the
  Sphere} \author{Fr\'ed\'eric Laurent-Polz, James Montaldi, Mark
  Roberts} \date{\today}


\pagestyle{myheadings}
\markboth{\sl F.~Laurent-Polz, J.A.~Montaldi and R.M.~Roberts}%
{\sl Stability of Point Vortices on the Sphere \qquad(\today)}

\begin{document}

\maketitle

%\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
  We describe the linear and nonlinear stabilities of certain configurations
  of point vortices on the sphere forming relative equilibria.  These
  configurations consist of up to two rings, with and without polar
  vortices.  Such configurations have dihedral symmetry, and the
  symmetry is used to block diagonalize the relevant matrices,
  enabling their eigenvalues to be calculated.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Since the work of Helmholtz \cite{H} systems of point vortices on the
plane have been studied as finite-dimensional approximations to
vorticity dynamics in ideal fluids.  For a general survey of
patterns of point vortices see \cite{ANSTV03}. Point vortex
systems on the sphere, introduced by Bogomolov \cite{B77}, provide
simple models for the dynamics of concentrated regions of vorticity, such as
cyclones and hurricanes, in planetary atmospheres.  In this paper we consider
a non-rotating sphere, since the rotation of the sphere induces a non-uniform
background vorticity which makes the whole system infinite-dimensional.

As in the planar case, the equations governing the motion of $N$
point vortices on a sphere are Hamiltonian \cite{B77} and this
property has been used to study them from a number of different
viewpoints.  Phase space reduction shows that the three vortex
problem is completely integrable on both the plane and the sphere:
the motion of three vortices of arbitrary vorticity on a sphere is
studied in \cite{KN98}.  The stabilities of some of the relative
equilibria given in \cite{KN98} are computed in \cite{PM98} and
numerical simulations are presented in \cite{MPS99}.  The
existence of relative equilibria of $N$ vortices is treated in
\cite{LMR01}, and the nonlinear stability of a latitudinal ring of
$N$ identical vorticities is computed in \cite{BC03}, and
independently in the present paper. It follows that the linear
stability results obtained by \cite{PD93} are in fact Lyapunov
stability results. The stability of a ring of vortices on the
sphere together with a central polar vortex is studied in
\cite{CMS03}, and again independently in the present paper.  The
existence and nonlinear stability of relative equilibria of $N$
vortices of vorticity $+1$ together with $N$ vortices of vorticity
$-1$ are studied in \cite{LP02}. It has also been proved in
\cite{LP} that relative equilibria formed of latitudinal rings of
identical vortices for the non-rotating sphere persist to relative
equilibria when the sphere rotates. Finally, studies of periodic
orbits of point vortices on the sphere can be found in
\cite{ST,To,LPf}.

Our study of the stability of relative equilibria is based on the
symmetries of the system, and especially the isotropy subgroups of
the relative equilibria.  The Hamiltonian is invariant under
rotations and reflections of the sphere and permutations of
identical vortices.  However, some of these symmetries (eg
reflections) are not symmetries of the equations of motion: they
are \emph{time-reversing} symmetries.  From Noether's theorem, the
rotational symmetry provides three conserved quantities, the
components of the momentum map $\Phi: \p \to \R^3$ where $\p$ is
the phase space.


Relative equilibria are dynamical trajectories that are generated
by the action of a 1-parameter subgroup of the symmetry group.
More intuitively, they correspond here to motions of the point
vortices which are stationary in a steadily rotating frame.  In
other words, the motion of a relative equilibrium corresponds to a
rigid rotation of $N$ point vortices about some axis (which we
always take to be the $z$-axis). In the same way as equilibria are
critical points of the Hamiltonian $H$, relative equilibria are
critical points of the restrictions of $H$ to the level sets
$\Phi^{-1}(\mu)$. Section \ref{intro} is devoted to a description
of the system of point vortices on the sphere, and to an outline
of stability theory for relative equilibria.  The appropriate
concept of stability for relative equilibria of Hamiltonian system
is Lyapunov stability \emph{modulo a subgroup}.  The stability
study is realized using on one hand the \emph{energy momentum
method} \cite{Pa92,Or98} which consists of computing the
eigenvalues of a certain Hessian, and leads to \emph{nonlinear}
stability results, and on the other hand a linear study computing
the eigenvalues of the linearization of the equations of motion.
To this end, we block diagonalize these matrices using a suitable
basis, the \emph{symmetry adapted basis} (Section \ref{basis}),
which makes use of the specific dihedral isotropy subgroup of the
relative equilibrium.

The remaining five sections each treat one of five different types
of relative equilibria, consisting of rings of identical vortices
together with possible vortices at the poles, whose existence were
proved in \cite{LMR01}. The notation for the different
configurations is taken from the same source and is described at
the end of the introduction.  We now outline the main stability
results.

We begin in Section \ref{1ring-stability} by computing the
stability of the relative equilibria consisting of a single ring
of identical vortices, a configuration denoted $\CC_{nv}(R)$
(Figure \ref{cnvRevpolesch3c}(a)).  We show in Theorem
\ref{polvani} that for $n\geq 7$, they are unstable for all
co-latitudes of the ring, while for $n<7$ there exist ranges of
Lyapunov stability when the ring is near a pole. These results are
not new \cite{PD93,BC03}, but serve to demonstrate the method used
in later sections.

In Section \ref{Rp-stability}, we study the stability of the
relative equilibria $\CC_{nv}(R,p)$ (Figure
\ref{cnvRevpolesch3c}(b)) which are configurations formed of a
ring of $n$ identical vortices together with a polar vortex.  For
$n\geq 7$ they are all unstable if the vorticity $\kappa$ of the
polar vortex has opposite sign to that of the ring.  However if
the vorticities have the same sign then for each co-latitude of
the ring there exists a range of $\kappa$ for which the relative
equilibrium is Lyapunov stable. Adding polar vortices can
therefore \emph{stabilize} the unstable pure ring relative
equilibria. The detailed results are contained in Theorem
\ref{ring+pole/stab}, its corollary and the following discussion.
Our results are consistent with those of \cite{CMS03}, though our
methods give more regions of stability than theirs---see Remark
\ref{rmk:CMS}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\subfigure[$\CC_{4v}(R)$]{\includegraphics[width=4.6cm,height=5.8cm,angle=0]{cnvR.eps}}
\hspace{1.2cm}
\subfigure[$\CC_{4v}(R,p)$]{\includegraphics[width=4.6cm,height=5.8cm,angle=0]{cnvRp.eps}}
\caption{The $\CC_{nv}(R)$ and $\CC_{nv}(R,p)$ relative
equilibria.} ($\CC_{nv}(R,2p)$ has a vortex at the South pole as
well.) \label{cnvRevpolesch3c}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In Section \ref{R2p} we obtain analytic (in)stability criteria for
the relative equilibria $\CC_{nv}(R,2p)$ which are configurations
formed of a ring of $n$ identical vortices together with
\emph{two} polar vortices, but only with respect to certain modes
($\ell\geq2$). As in the case of a single polar vortex, the two
polar vortices play the role of control parameters for the
stability.  The details are contained in Theorem
\ref{ring+2poles/stab}.  A numerical investigation is needed for
the remaining ($\ell=1$) mode in order to provide stability
criteria; this is being pursued separately.



Finally, in Sections \ref{2R} and \ref{RR'} we investigate
configurations formed of two rings of arbitrary vorticities (each
ring, as always, consisting of identical vortices).  In
\cite{LMR01} it was shown that two rings of $n$ vortices can be
relative equilibria if and only if they are either aligned or
staggered.  These two arrangements are denoted $\CC_{nv}(2R)$ and
$\CC_{nv}(R,R')$ respectively (see Figure \ref{cnv2RRprimech3c}).
Here we show that for almost all pairs of ring latitudes there is
a unique ratio of the ring vorticities for which these
configurations are relative equilibria. Numerical computations of
their stabilities suggest that these relative equilibria can only
be stable if $n\leq 6$, and in the aligned case the two rings must
be close to opposite poles, and hence have opposite vorticities.
In some cases, staggered rings may also be stable when in the same
hemisphere.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\subfigure[$\CC_{4v}(2R)$]{\includegraphics[width=4.6cm,height=5.8cm,angle=0]{cnv2R.eps}} %\label{c2Rch3c}
\hspace{1.2cm}
\subfigure[$\CC_{4v}(R,R')$]{\includegraphics[width=4.6cm,height=5.8cm,angle=0]{cnvRRprim.eps}} %\label{cRRprimech3c}
\caption{Configurations of types $\CC_{nv}(2R)$ (2 aligned rings)
and $\CC_{nv}(R,R')$ (2 staggered rings).} \label{cnv2RRprimech3c}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\paragraph{Symmetry group notation}
All possible symmetry types of configurations of point vortices on
the sphere were classified in \cite{LMR01}.  The symmetry group of the
system is of
the form $\OO(3)\times S$, where $S$ is a group of permutations, and
a particular configuration with symmetry, or isotropy, subgroup
$\Sigma<\OO(3)\times S$ is
denoted $\Gamma(A)$, where $\Gamma$ is the projection of $\Sigma$ to
$\OO(3)$ and $A$ represents the way $\Sigma$ permutes the point
vortices.  The classical \emph{Sch\"{o}nflies-Eyring} notation for
subgroups of $\OO(3)$ is used.

In this paper we single out configurations consisting of concentric rings of
identical vortices, with the same number of vortices in each ring, and with possible
polar vortices. These configurations have cyclic symmetry (in the ``horizontal
plane''), and the Sch\"{o}nflies-Eyring notation for this subgroup of $\OO(3)$ is
$C_n<\SO(3)$.  In fact we only consider the cases where the rings are either aligned
(the vortices lie on the same longitudes) or staggered (they lie on intermediate
longitudes, out of phase by $\pi/n$). In this case the symmetry group is the larger
\emph{dihedral} group $C_{nv}$ ($n$ being the number of vortices in each ring, and
$v$ denoting the fact that there are vertical planes of reflection). For such
configurations, we write $C_{nv}(k_1R,k_2R',k_pp)$ to mean that there are
$k_r=k_1+k_2$ rings and $k_p$ polar vortices. The difference between $R$ and $R'$ is
that the $k_1$ rings R are aligned and the $k_2$ rings $R'$ are staggered with
respect to the first (and so aligned with each other). Of course $k_p=0,1$ or 2.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Point vortices on the sphere and stability theory}
\label{intro}

In this section we briefly recall that the system of point
vortices on a sphere is a $n$-body Hamiltonian system with
symmetry and we review the stability theory for relative
equilibria.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Point vortices}

Consider $n$ point vortices $x_1,\dots,x_n\in S^2$ with
vorticities $\kappa_1,\dots,\kappa_n\in\R$.  Let $\theta_i,\phi_i$
be respectively the co-latitude and the longitude of the vortex
$x_i$. The dynamical system is Hamiltonian with Hamiltonian given
by
$$
H \ =\  -\sum_{i<j}\kappa_i\kappa_j\ln (
1-\cos\theta_i\cos\theta_j-\sin\theta_i\sin\theta_j\cos(\phi_i-\phi_j)
)
$$
and conjugate variables given by
$q_i=\sqrt{\vert\kappa_i\vert}\cos\theta_i$ and
$p_i=\mathrm{sign}(\kappa_i)\sqrt{\vert\kappa_i\vert}\phi_i$.

The phase space is $\p = \lbrace (x_1,\dots,x_{n})\in
S^2\times\cdots\times S^2\mid x_i\neq x_j\ \mbox{if}\ i \neq j
\rbrace$ endowed with the symplectic form $\omega=\sum_{i}
\kappa_i \sin\theta_i\ d\theta_i\land d\phi_i$.  The Hamiltonian
vector field $X_H$ satisfies $\omega(\ \cdot\ ,X_H(x))=dH_x$.  If
we consider $S^2$ as a subset of $\R^3$, so the vortices $x_j\in
\R^3$, then we obtain
$$
\dot {x_i} \ =\  X_H(x)_i\ =\ \sum_{j,j\neq i} \kappa_j\,
\frac{x_{j}\times x_{i}}{1-x_{i}\cdot x_{j}}\ ,\ i=1,\dots,N,
$$
$$
H \ =\  - \sum_{i<j}\kappa_i\kappa_j\ln(\|x_i-x_j\|^2/2).
$$
It follows that $H$ is invariant under the action of $\OO(3)$. The
symplectic form is $\SO(3)$ invariant and so $X_H$ is $\SO(3)$-equivariant.
The reflections in $\OO(3)$ reverse the sign of the symplectic form
and so are time-reversing symmetries of $X_H$.
Moreover $H$, $\omega$ and $X_H$ are all invariant or equivariant
with respect to permutations of vortices with equal vorticity.

The rotational symmetry implies the existence of a momentum map
$\Phi:\p\to\so^*\simeq\R^3$:
$$
\Phi(x)\ =\ \sum_{j=1}^{N}\kappa_j x_j\qquad (x_j\in
S^2\subset\R^3)
$$
which is conserved under the dynamics. In other words each of the
three components of $\Phi(x)$ is a conserved quantity.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Relative equilibria}

A point $x_e\in\p$ is a relative equilibrium if and only if there
exists $\xi\in\so\simeq\R^3$ (the angular velocity) such that
$x_e$ is a critical point of the function $H_{\xi}(x)=H(x)-\left<
\Phi(x),\xi\right>$, where the pairing $\left<\ ,\ \right>$
between $\R^3$ and its dual is identified with the canonical
scalar product on $\R^3$.  Equivalently, relative equilibria are
critical points of the restriction of $H$ to $\Phi^{-1}(\mu)$,
since the level set $\Phi^{-1}(\mu)$ are always non-singular for
point vortex systems of more than two vortices.  The function
$H_\xi$ is called the \emph{augmented Hamiltonian}.

Since the momentum is conserved, we can choose a frame for $\R^3$ such
that $\Phi$ is parallel to the $z$-axis (provided the momentum is
non-zero).  It follows from the symmetry that the angular velocity
$\xi\in\R^3$ is also parallel to the $z$-axis.  We can therefore
identify $\xi$ and $\Phi$ with their $z$-components and the
augmented Hamiltonian becomes simply $H_{\xi}(x)=H(x)-\xi\Phi(x)$.

Let $f:\p\to\R$ be a $K$-invariant function with $K$ a compact group.
Recall that $\Fix(K)=\lbrace x\in\mathcal{P} \mid g\cdot x = x,
\ \forall g\in K \rbrace$.  The \emph{Principle of Symmetric
  Criticality} \cite{P79} states that a critical point of the
restriction of a $K$-invariant function $f$ to $\Fix(K)$ is a
critical point of $f$.  As a corollary, if the Hamiltonian is
invariant under $K$ and $x_e$ is an isolated point in
$\Fix(K)\cap\Phi^{-1}(\mu)$, then $x_e$ is a relative equilibrium.
It follows in particular that all configurations of type
$\CC_{nv}(R)$, $\CC_{nv}(R,p)$ (Figure \ref{cnvRevpolesch3c}), and
$\CC_{nv}(R,2p)$ are relative equilibria: take $K$ such that
$\pi(K)=\CC_{nv}$ where $\pi:\OO(3)\times S_n\to \OO(3)$ is the
cartesian projection.

Finally, one can show that if $x_e$ is a relative equilibrium with
angular velocity $\xi$, then $H_\xi$ is a $G_{x_e}$-invariant
function.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Stability theory}

Stability is determined by the energy momentum method together
with an isotypic decomposition of the symplectic slice. We recall the
main points of the method.

Let $x_e\in\p$ be a relative equilibrium, $\mu=\Phi(x_e)$, and $\xi$
its angular velocity.  The energy momentum method consists of
determining the symplectic slice
$$
\G\ =\ (\so_{\mu}\cdot x_e)^\bot \cap \Ker D\Phi(x_e)
$$
transversal to $\so_{\mu}\cdot x_e$, where
$$
\SO(3)_{\mu}\ =\ \lbrace g\in \SO(3)\mid\CoAd_g\cdot \mu=\mu \rbrace
$$
and then examining the definiteness of the restriction
$d^2H_{\xi}|_{\G}(x_e)$ of the Hessian $d^2H_\xi(x_e)$ to $\G$.  If
$K$ is a group acting on the phase space a relative equilibrium $x_e$
is said to be \emph{Lyapunov stable modulo K} if for all
$K$-invariant open neighbourhoods $V$ of $K\cdot x_e$ there is an open
neighbourhood $U\subseteq V$ of $x_e$ which is invariant under the
Hamiltonian dynamics.  The \emph{energy momentum theorem} of Patrick
\cite{Pa92} holds since $\SO(3)$ is compact, and so we have:
%
\begin{center} \textit{If $d^2H_{\xi}|_{\G}(x_e)$ is definite, then
  $x_e$ is Lyapunov stable modulo $\SO(3)_{\mu}$.}
%
\end{center}
For $\mu\neq 0$, $\SO(3)_{\mu}$ is the set of rotations with axis
$\<\mu\m$, and so isomorphic to $\SO(2)$, while for $\mu = 0$,
$\SO(3)_{\mu} = \SO(3)$.  If $\mu\neq 0$ Lyapunov stability modulo
$\SO(3)_{\mu}$ of a relative equilibrium with non-zero angular
velocity corresponds to the ordinary stability of the
corresponding periodic orbit.

The second step consists of performing an isotypic decomposition of
the symplectic slice $\G$ in order to block diagonalize
$d^2H_{\xi}|_{\G}(x_e)$.  Let $V$ be a finite dimensional
representation of a compact Lie group $K$.  Recall that a $K$
invariant subspace $W\subset V$ of $K$ is said to be \emph{irreducible} if
$W$ has no proper $K$ invariant subspaces. Since $K$ is compact, $V$
can be expressed as a direct sum of irreducible representations:
$V=W_1\oplus\cdots\oplus W_n$. In general this is not unique.  There
are a finite number of isomorphism classes of irreducible
representations of $K$ in $V$, say $U_1,\dots,U_\ell$.  Let $V_k$
($k=1,\dots,\ell$) be the sum of all irreducible representations
$W_j\subset V$ such that $W_j$ is isomorphic to $U_k$.  Then
$V=V_1\oplus\cdots\oplus V_\ell$.  This decomposition of $V$ is unique
and is called the \emph{$K$-isotypic decomposition} of $V$
\cite{Se78}.  By Schur's Lemma, the matrix of a $K$-equivariant linear
map $f:V\to V$ block diagonalizes with respect to a basis
$\mathcal{B}=\{\mathcal{B}_1,\dots,\mathcal{B}_l \}$ where
$\mathcal{B}_k$ is a basis of $V_k$, each block corresponding to a
subspace $V_k$.  The basis $\mathcal{B}$ is called a \emph{symmetry
adapted basis}.

Let $G$ denote the group of all symmetries of $H$ and $X_H$ and $G^\chi$
the subgroup consisting of time-preserving symmetries.  In the case of
$N$ identical vortices we have $G=\OO(3)\times S_N$ and
$G^\chi=\SO(3)\times S_N$.  Since $H_\xi$ is
a $G_{x_e}$-invariant function, $d^2H_{\xi}|_{\NN}(x_e)$ is
$G_{x_e}$-equivariant as a matrix.  Moreover the symplectic slice
$\NN$ is a $G_{x_e}$-invariant subspace and so we can implement a
$G_{x_e}$-isotypic decomposition of $\NN$ to block diagonalize
$d^2H_{\xi}|_{\NN}(x_e)$. This block diagonalization of
$d^2H_{\xi}|_{\NN}(x_e)$ simplifies the computation of its
eigenvalues, and hence its definiteness. If it is definite
then the relative equilibrium is Lyapunov stability modulo $\SO(3)_{\mu}$.

If $d^2H_{\xi}|_{\NN}(x_e)$ is {\it not} definite then we
study the spectral stability of $x_e$. In particular we examine the
eigenvalues of $L_\NN$, the matrix of the linearized system on the
symplectic slice, that is
$L_\NN=\J_{\NN}d^2H_{\xi}|_{\NN}(x_e)$,
where $\J_{\NN}^{-1}$ is the matrix of $\omega|_{\NN}$.  The matrix $L_\NN$ is
$G_{x_e}^\chi$-equivariant and so we perform a $G_{x_e}^\chi$-isotypic
decomposition of $\NN$ to obtain a block diagonalization of $L_\NN$,
and so to determine the spectral stability of $x_e$. In particular,
if $L_\NN$ has eigenvalues with non-zero real part, then $x_e$ is
linearly unstable.  Note that the block diagonalization of
$d^2H_{\xi}|_\NN(x_e)$ refines that of $L_\NN$ since
$G_{x_e}^\chi\subset G_{x_e}$.

Throughout this paper, \emph{Lyapunov stable} will mean \emph{Lyapunov
  stable modulo $\SO(2)$} unless specified otherwise, and
$\mu=\Phi(x_e)$ will denote the momentum of the configuration
$x_e$. We will assume in most of the results that the relative
equilibrium has a non-zero momentum.  It is straightforward to
check that almost all points of type $\CC_{nv}(R,k_p p)$ have a
non-zero momentum, so this is not a strong assumption for that
case.  This statement is also true for $\CC_{nv}(2R)$ and
$\CC_{nv}(R,R')$ relative equilibria, as we shall see.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Symmetry adapted bases for rings and poles}
\label{basis}


In this section we give the ingredients needed to determine the
symmetry adapted bases for the symplectic slice at the
configurations described above, that is those of type
$\CC_{nv}(k_1R,k_2R', k_pp)$.  In the first subsection we give a
general symmetry adapted basis for the tangent space $T_{x_e}\p$
to the phase space at such a configuration, and express the
derivative of the momentum map and tangent space to the group
orbit in this basis. In the following two subsections we describe
the isotypic decomposition of $T_{x_e}\p$, first for a single ring
and then in general. Recall that the isotropy subgroup is always a
dihedral group $\CC_{nv}$ and that the irreducible representations
of this group are of dimension $1$ or $2$. The actual symmetry
adapted bases of the symplectic slices will be given case-by-case
in the following sections. We do not give the proof of the results
in the first subsection, since they can be easily deduced from the
proofs of Propositions 4.1, 4.2, 4.3, 4.4 in \cite{LP02}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Description of the symplectic slice}

Let $x_e$ be a $\CC_{nv}(k_1R,k_2R', k_pp)$ configuration.  The
total number of vortices is $N = n(k_1+k_2) + k_p$. We suppose the
vorticities in the $k_r=k_1+k_2$ rings are $\kappa_j$ for $j =
1,\ldots,k_r$ while the vorticities of the possible polar vortices
are $\kappa_N$ for the North pole and $\kappa_S$ for the South
pole.

For each ring $j=1,\dots,k_r$ let $s=1,\dots,n$ label the vortices in
the ring in cyclic order, and define tangent vectors in $T_{x_e}\p$ by
\begin{equation} \label{eqn:alphas,betas}
\begin{array}{rcl}
  \alpha^{(\ell)}_{j,\theta} +i\beta^{(\ell)}_{j,\theta}
  &=&
  \sum_{s=1}^n\exp(2i\pi\ell s/n + i\ell\phi_j^0)\,\delta\theta_{j,s}
  \\[4pt]
  \alpha^{(\ell)}_{j,\phi} +i\beta^{(\ell)}_{j,\phi}
  &=&
  \sum_{s=1}^n\exp(2i\pi\ell s/n + i\ell\phi_j^0)\,\delta\phi_{j,s}
\end{array}
\end{equation}
where $\ell=0,\dots,n-1$, $i=\sqrt{-1}$ and $\phi_j^0=0$ or
$\pi/n$ depending on whether the $j^{\rm th}$ ring of vortices is of type
$R$ or $R'$.  Note that
$\beta^{(\ell)}_{j,\theta},\beta^{(\ell)}_{j,\phi}$ vanish for
$\ell=0$ and $n/2$ (for $n$ even).  For each pole
$j=1,\dots,k_p$ we also have tangent vectors $\delta x_j$ and $\delta
y_j$.

The tangent vectors defined in the last paragraph are almost
canonical, in the sense that
$$\begin{array}{rcl} \omega\left(\alpha^{(\ell)}_{j,\theta},
    \alpha^{(\ell)}_{j,\phi}\right) &=&
  \cases{n\sin\theta_j\,\kappa_j & if $\ell=0, n/2$ \\ \frac{n\sin\theta_j}{2}\,\kappa_j & otherwise} \\[8pt]
  \omega\left(\beta^{(\ell)}_{j,\theta}, \beta^{(\ell)}_{j,\phi}\right) &=& \frac{n\sin\theta_j}{2}\,\kappa_j \\[6pt]
  \omega\left(\delta x_j,\delta y_j \right)&=&{\rm
    sign}(z_j)\,\kappa_j,
\end{array}$$
while the other pairings vanish.

In order to compute a specific basis for the symplectic slice it
is necessary to have expressions for the derivative of the
momentum map and the tangent space to the group orbit at a
$\CC_{nv}(k_1R,k_2R',k_pp)$ configuration.  These expressions are
given in the next two propositions.  Since $\CC_{nv}$ refers to
fixed vertical reflection planes, the values of $\phi_j^0$ in
(\ref{eqn:alphas,betas}) above can be taken to be:
$$
\phi_j^0 = \cases{0 & if $j=1\dots k_1$\cr \pi/n & if
$j=(k_1+1)\dots k_r$}
$$
where $k_r=k_1+k_2$ is the total number of rings.

\begin{prop}
\label{kernel}
At a $\CC_{nv}(k_1R,k_2R',k_pp)$ configuration, the differential of
the momentum map is given by,
\begin{eqnarray*}
\lefteqn{d\Phi(\delta x,\delta y,\alpha,\beta)} \\ &=&
\sum_{j\,\mbox{polar}}\kappa_j(\delta x_j + i\delta y_j) + \sum_{j=1}^{k_r}
\kappa_j\cos\theta_j(\alpha^{(1)}_{j,\theta}+i\beta^{(1)}_{j,\theta})
+i\sum_{j=1}^{k_r} \kappa_j\sin\theta_j(\alpha^{(1)}_{j,\phi}+i\beta^{(1)}_{j,\phi})
\\
&&\quad \bigoplus -\sum_{j=1}^{k_r} \kappa_j\sin\theta_j\alpha^{(0)}_{j,\theta}
\end{eqnarray*}
where the direct sum corresponds to the $\CC_{nv}$-invariant
decomposition of $\so^*$ as a direct sum of a plane and the line
$\Fix(\CC_{nv},\so^*)$ (the ``$z$-axis'').
\end{prop}


\begin{prop}
\label{tangentspace}
Let $x_e$ be a $\CC_{nv}(k_1R,k_2R',k_pp)$ configuration, and
$\mu=\Phi(x_e)$.
%
If $\mu\neq 0$, then the tangent space to the orbit $\so_{\mu}\cdot
x_e$ is generated by the vector
$$\sum_{j=1}^{k_r} \alpha^{(0)}_{j,\phi}.$$ %
If $\mu = 0$, then $\so_{\mu}\cdot x_e = \so\cdot x_e$ and is
generated by the three vectors:
$$\begin{array}{c} \displaystyle
\sum_{j=1}^{k_r} i\left(\alpha^{(1)}_{j,\theta} +
    i\beta^{(1)}_{j,\theta}\right)
  -\cos\theta_j\sin\theta_j\left(\alpha^{(1)}_{j,\phi}
    + i\beta^{(1)}_{j,\phi}\right)\;+\; i \sum_{j\,\mbox{polar}} {\rm sign}(z_j)(\delta x_j + i\delta y_j)\\[6pt]
  \displaystyle\sum_{j=1}^{k_r} \alpha^{(0)}_{j,\phi}
\end{array}
$$
where one must take the real and imaginary parts of the first line.
\end{prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{A single ring}
\label{singlering}

Let $x_e$ be a $\CC_{nv}(R,k_pp)$ configuration, that is a single ring
together with $k_p$ polar vortices where, of course, $k_p=0,1$ or $2$.
Since there is only one ring, we write
$\alpha^{(\ell)}_{\theta},\beta^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi},\beta^{(\ell)}_{\phi}$
instead of
$\alpha^{(\ell)}_{1,\theta},\beta^{(\ell)}_{1,\theta},\alpha^{(\ell)}_{1,\phi},\beta^{(\ell)}_{1,\phi}$
in order to lighten the formulae.
\bigskip

\noindent
\emph{The irreducible representations (subspaces) of dimension $1$ of
the action of $G_{x_e}$ on $T_{x_e}\p$ are:
  $$
  \< \alpha^{(0)}_{\theta} \m ,\; \< \alpha^{(0)}_{\phi} \m\
  \quad \mbox{if $n$ is odd,}
  $$
  while they are
  $$
  \< \alpha^{(0)}_{\theta} \m ,\; \< \alpha^{(0)}_{\phi} \m ,\; \<
  \alpha^{(n/2)}_{\theta} \m ,\; \< \alpha^{(n/2)}_{\phi} \m \
  \quad \mbox{if $n$ is even.}
  $$
  Of these, only $\< \alpha^{(n/2)}_{\theta}\m$ and $\<
  \alpha^{(n/2)}_{\phi}\m$ lie in the symplectic slice.  Moreover, $\<
  \alpha^{(n/2)}_{\theta}\m$ and $\< \alpha^{(n/2)}_{\phi}\m$ are
  $G_{x_e}^\chi$-isomorphic representations, but not $G_{x_e}$-isomorphic representations.}\\
\bigskip

\noindent\emph{The following spaces are irreducible representations of dimension
$2$ of the action of $G_{x_e}$ on $T_{x_e}\p$:
  $$
  \< \alpha^{(\ell)}_{\theta},\beta^{(\ell)}_{\theta} \m ,\; \<
  \alpha^{(\ell)}_{\phi},\beta^{(\ell)}_{\phi} \m , 1\leq\ell\leq n-1
  , \ell\neq n/2 ,\ \< \delta x_j , \delta y_j \m , j=1,\dots,k_p.
  $$
  The representations $\<
  \alpha^{(1)}_{\theta},\beta^{(1)}_{\theta}\m,\; \<
  \alpha^{(1)}_{\phi},\beta^{(1)}_{\phi} \m,\; \< \delta x_j , \delta
  y_j \m$ do not lie in the symplectic slice, while the others do.
  Moreover, $\<\alpha^{(1)}_{\theta},\beta^{(1)}_{\theta}\m$,
  $\<\alpha^{(1)}_{\phi},\beta^{(1)}_{\phi}\m$, and $\< \delta x_j ,
  \delta y_j \m$ are $G_{x_e}^\chi$-isomorphic representations.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{General case}
\label{tworings}

In the general case where $x_e$ is a $\CC_{nv}(k_1R,k_2R',k_pp)$
configuration ($k_p=0,1$ or 2), we have the following
decomposition.\\

\noindent\emph{The subspaces
  $\<\alpha^{(0)}_{j,\theta}\m,\; \<\alpha^{(0)}_{j,\phi}\m,\;
  j=1,\dots,k_r$ are $G_{x_e}$-irreducible representations of
  dimension $1$ and are $G_{x_e}^\chi$-isomorphic representations.  If
  $n$ is even, then we have $2k_r$ additional $G_{x_e}$-irreducible
  representations
  $$\<\alpha^{(n/2)}_{j,\theta}\m,\; \<\alpha^{(n/2)}_{j,\phi}\m,\;
  j=1,\dots,k_r$$
  which are $G_{x_e}^\chi$-isomorphic.}
\bigskip

\noindent\emph{The subspaces
  $\< \alpha^{(\ell)}_{j,\theta},\beta^{(\ell)}_{j,\theta} \m ,\; \<
  \alpha^{(\ell)}_{j,\phi},\beta^{(\ell)}_{j,\phi}\m,\;
  j=1,\dots,k_r,\; 1\leq\ell\leq n-1 , \ell\neq n/2$ and $\< \delta
  x_r , \delta y_r \m,$ $r=1,\dots,k_p$ are $G_{x_e}$-irreducible
  representations of dimension $2$.  Moreover, the subspaces
  $$
  \<\alpha^{(1)}_{j,\theta},\beta^{(1)}_{j,\theta} \m ,\;
  \<\alpha^{(1)}_{j,\phi},\beta^{(1)}_{j,\phi}\m,\;
  \< \delta x_r , \delta y_r \m,
  $$
  ($j=1,\dots,k_r$, \ $r=1,\dots,k_p$)
  are $G_{x_e}^\chi$-isomorphic representations.}\\

\noindent The difficulty in the case of several rings is that
usually the subspaces listed above do not lie in the symplectic
slice.  One needs therefore to find linear combinations of the
above vectors  that do lie in the symplectic slice, and that the
\emph{irreducible} and \emph{isomorphism} properties of the
representations are preserved.  We will give the symmetry adapted
bases for the cases of two rings without polar vortices, that is
$\CC_{nv}(2R)$ and $\CC_{nv}(R,R')$ configurations, in the last
two sections.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A ring of identical vortices: $\CC_{nv}(R)$}
\label{1ring-stability}

The linear stability of $\CC_{nv}(R)$ relative equilibria was determined by Polvani
and Dritschel in \cite{PD93}. Recently, Boatto and Cabral \cite{BC03} studied their
Lyapunov stability and found that the two types of stability coincide: whenever the
relative equilibrium fails to be Lyapunov stable the linearization of $X_H$ has real
eigenvalues.  In this section, we give another proof using the geometric method of
this paper.

For $n$ vortices of unit vorticity the Hamiltonian is
$$
H(\theta_j,\phi_j) \ =\ -\sum_{j<k}
\ln\left(1-\sin\theta_j\sin\theta_k\cos(\phi_j-\phi_k) -
  \cos\theta_j\cos\theta_k\right)
$$
and the augmented Hamiltonian is $H_\xi=H-\xi\sum_j\cos\theta_j$.

Let $x_e$ be a $\CC_{nv}(R)$ relative equilibrium and $\theta_0$
the co-latitude of the ring.  The angular velocity of $x_e$ is
$$\xi \ =\  \frac{(n-1)\cos\theta_0}{\sin^2\theta_0},$$
since $H_{\xi}$
has a critical point there and $\frac{\partial
  H_\xi}{\partial\theta_j}(x_e) =
\frac{(n-1)\cos\theta_0-\xi\sin^2\theta_0}{\sin\theta_0}$.
\bigskip

The second derivatives of $H$ at the relative equilibrium are:
$$\left.\begin{array}{rclcrcl} \frac{\partial^2H}{\partial\theta_j^2}
    &=& -\frac{(n-1)(n-5)}{6\sin^2\theta_0} &\qquad&
    \frac{\partial^2H}{\partial\theta_j\partial\theta_k} &=&
    \frac{1}{2\sin^2\theta_0\sin^2(\pi(j-k)/n)} \\[6pt]
    \frac{\partial^2H}{\partial\theta_j\partial\phi_j} &=& 0 &&
    \frac{\partial^2H}{\partial\theta_j\partial\phi_k} &=& 0 \\[6pt]
    \frac{\partial^2H}{\partial\phi_j^2} &=&
    \sum_{r=1}^{n-1}\frac{1}{2\sin^2(r\pi/n)} &&
    \frac{\partial^2H}{\partial\phi_j\partial\phi_k} &=&
    -\frac{1}{2\sin^2(\pi(j-k)/n)}.
\end{array}\right.
$$
We note that $\sum_{r=1}^{n-1}1/\sin^2(\pi r/n) =
\frac{1}{3}(n^2-1)$ \cite{H75}. %??


\paragraph{Notation.}
In order to harmonize the statements of the results between $n$ even
and $n$ odd, we introduce the following notation: let
$\eta^{(\ell)}_1,\eta^{(\ell)}_2,\eta^{(\ell)}_3,\eta^{(\ell)}_4$ be
objects defined for all $2\leq \ell\leq \left[\frac{n-1}{2}\right]$,
where $[m]$ is the integer part of $m\in\N$, and only
$\eta^{(\ell)}_1,\eta^{(\ell)}_2$ for $\ell=n/2$ when $n$ is even.
Then define $\left\lbrace
  \eta^{(\ell)}_1,\eta^{(\ell)}_2,\eta^{(\ell)}_3,\eta^{(\ell)}_4\mid
  2\leq\ell\leq [n/2]\right\rbrace^*$ to be
\begin{equation}\label{eq:set*}
  \begin{array}{rcl}
\mbox{for even $n$} &:&
\left\lbrace
  \eta^{(\ell)}_1,\eta^{(\ell)}_2,\eta^{(\ell)}_3,\eta^{(\ell)}_4\mid
  2\leq\ell\leq \frac{n}{2}-1\right\rbrace\cup\left\lbrace
  \eta^{(n/2)}_1,\eta^{(n/2)}_2\right\rbrace   \\
\mbox{for odd $n$} &:& \lbrace
\eta^{(\ell)}_1,\eta^{(\ell)}_2,\eta^{(\ell)}_3,\eta^{(\ell)}_4\mid
2\leq\ell\leq [n/2]\rbrace.
  \end{array}
\end{equation}

\noindent
The following proposition gives the symmetry adapted basis for a
$\CC_{nv}(R)$ configuration.

\begin{prop}
\label{diagonering}
Assume $\mu\neq 0$.  With respect to the following basis for the
symplectic slice $\NN$ at $x_e$,
$$
\left( e_1,\ e_2,\ \left\lbrace
    \alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi},\beta^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi}
    \mid 2\leq \ell\leq [n/2] \right\rbrace^* \right)
$$
where
$$
\begin{array}{lll}
e_1&=&\sin\theta_0\;\alpha^{(1)}_{\theta}+\cos\theta_0\;\beta^{(1)}_{\phi}\\
e_2&=&\sin\theta_0\;\beta^{(1)}_{\theta}-\cos\theta_0\;\alpha^{(1)}_{\phi}
\end{array}
$$
the Hessian $d^2H_\xi|_\NN(x_e)$ block diagonalizes in $1\times 1$
blocks, and $L_\NN$ block diagonalizes in $2\times 2$ blocks.
\end{prop}

\begin{proof}
  It is straightforward to check that the vectors above do form
  a basis for the symplectic slice at $x_e$ thanks to Propositions
  \ref{kernel} and \ref{tangentspace}.

  The Hessian $d^2H_\xi|_\NN(x_e)$ and the linearization $L_\NN$ are
  both $G_{x_e}^\chi$-invariant.  Assume $n$ odd.  It follows from
  Section \ref{singlering} and Schur's Lemma (see the Introduction) that
  $d^2H_\xi|_\NN(x_e)$ and $L_\NN$ both block diagonalize into $4\times 4$
  blocks and one $2\times 2$ block corresponding to the subspaces
  $V_\ell=\<\alpha^{(\ell)}_{\theta},\beta^{(\ell)}_{\theta},
  \alpha^{(\ell)}_{\phi},\beta^{(\ell)}_{\phi}\m$ and  $\<e_1,e_2 \m$,
  respectively. See the proof of Theorem 4.5 of \cite{LP02} for a
  detailed proof of a similar assertion.

  Now fix $\ell$ and denote by $s$ an anti-symplectic (time-reversing)
  element of ${G_{x_e}}$.  For example $s$ could be the reflection
  $y\mapsto -y$ together with an order two permutation of $S_n$.
  The restriction of $H_\xi$ to $V_\ell$ is $\Z_2[s]$-invariant. Moreover
  $\<\alpha^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi}\m$ and
  $\<\beta^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi}\m$ are
  non-isomorphic irreducible representation of $\Z_2[s]$ on $V_\ell$.
  Hence $d^2H_\xi|_\NN(x_e)$ block diagonalizes into $2\times 2$ blocks
  which correspond to subspaces
  $\<\alpha^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi}\m$,
  $\<\beta^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi}\m$, and $\<
  e_1,e_2 \m$.  This result does not depend on the details of the
  Hamiltonian, only its symmetries.  However taking in account its particular
  form, one can improve the block diagonalization. Indeed one has
  $$d^2H_\xi(x_e)\cdot
  (\alpha^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi})\ =\
  d^2H_\xi(x_e)\cdot
  (\beta^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi})\ =\ 0$$
  and
  $d^2H_\xi(x_e)\cdot(e_1,e_2)=0$ which gives the desired
  diagonalization of the Hessian.

  The particular form of the symplectic form also enables us to improve
  the diagonalization of $L_\NN$. Among the basis vectors of
  $V_\ell$, only
  $\omega(\alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi})$ and
  $\omega(\beta^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi})$ do not
  vanish, and so the restriction of $\omega$ to $V_\ell$ block
  diagonalizes into two $2\times 2$ blocks which correspond to the
  subspaces $\<\alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi}\m$ and
  $\<\beta^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi}\m$.  The block
  diagonalization of $L_\NN$ then follows from
  $L_\NN=\J_{\NN}d^2H_{\xi}|_{\NN}(x_e)$.

  The case $n$ even is very similar, except that there is an additional
  $2\times 2$ block in the $G_{x_e}^\chi$-isotypic decomposition, and
  leads to the same result.
\end{proof}

The block diagonalization of $d^2H_\xi|_\NN(x_e)$ and $L_\NN$ enable
us to find formulae for their eigenvalues, and thus to conclude
criteria for both Lyapunov and linear stability.

\begin{theo}\label{polvani}
  The stability of a ring of $n$ identical vortices depends on $n$ and
  the co-latitude $\theta_0$ as follows:
  \begin{description}
  \item[n=2] is Lyapunov stable at all latitudes;
  \item[n=3] is Lyapunov stable at all latitudes;
  \item[n=4] is Lyapunov stable if $\cos^2\theta_0 > 1/3$, and
    linearly unstable if the inequality is reversed;
  \item[n=5] is Lyapunov stable if $\cos^2\theta_0 > 1/2$, and
    linearly unstable if the inequality is reversed;
  \item[n=6] is Lyapunov stable if $\cos^2\theta_0 > 4/5$, and
    linearly unstable if the inequality is reversed;
  \item[n$\geq$7] is always (linearly) unstable.
  \end{description}
\end{theo}

\begin{proof}
  Any arrangement of two vortices is a relative equilibrium
  \cite{KN98}. When perturbing such a relative equilibrium, we obtain
  a new relative equilibrium close to the first. Thus any relative equilibrium of
  two vortices is Lyapunov stable modulo $\SO(2)$ (modulo $\SO(3)$ if
  $\mu=0$).

  Hence assume $n\geq 3$.  We first study Lyapunov stability.
  Suppose further that $\mu\neq 0$ and so the ring is not
  equatorial.  A simple calculation shows that $d^2H_\xi(x_e)\cdot
  (\beta^{(\ell)}_{\theta},\beta^{(\ell)}_{\theta})=d^2H_\xi(x_e)\cdot
  (\alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\theta})$ and
  $d^2H_\xi(x_e)\cdot
  (\beta^{(\ell)}_{\phi},\beta^{(\ell)}_{\phi})=d^2H_\xi(x_e)\cdot
  (\alpha^{(\ell)}_{\phi},\alpha^{(\ell)}_{\phi})$. Hence it follows
  from Proposition \ref{diagonering} that
  $$
  d^2H_\xi|_\NN(x_e)\ =\ \diag\left(\lambda_1,\lambda_1,
    \lbrace\lambda^{(\ell)}_\tet,\lambda^{(\ell)}_\phi,\lambda^{(\ell)}_\tet,\lambda^{(\ell)}_\phi
    \mid 2\leq \ell\leq [n/2]\rbrace^* \right)
  $$
  (recall notation from (\ref{eq:set*})) where
  $$
\begin{array}{lll}
\lambda_1&=&\sin^2\theta_0\lambda^{(1)}_\tet
+\cos^2\theta_0\lambda^{(1)}_\phi,\\[6pt]
\lambda^{(\ell)}_\tet&=&d^2H_\xi(x_e)\cdot
(\alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\theta}),\\[6pt]
\lambda^{(\ell)}_\phi&=&d^2H_\xi(x_e)\cdot (\alpha^{(\ell)}_{\phi},\alpha^{(\ell)}_{\phi}).
\end{array}
$$
Thanks to the following formula \cite{H75} (p.271)
$$
\sum_{j=1}^{n-1} \frac{\cos(2\pi\ell j/n)}{\sin^2(\pi
  j/n)}\ =\ \frac{1}{3}(n^2-1)-2\ell(n-\ell),
$$
we find after some computations that
$\lambda^{(\ell)}_\phi=n\ell(n-\ell)/2$ and
$$
\lambda^{(\ell)}_\tet\ =\ \frac{n}{2\sin^2\tet_0}\left[
  -(\ell-1)(n-\ell-1)+(n-1)\cos^2\tet_0 \right].
$$
The eigenvalues $\lambda^{(\ell)}_\phi$ are all positive and
$\lambda_1=n(n-1)\cos^2\tet_0>0$, thus the relative equilibrium is
Lyapunov stable (modulo $\SO(2)$) if
$(n-1)\cos^2\tet_0 > (\ell-1)(n-\ell-1)$ for all
$\ell=2,\dots,[n/2]$, that is if $\cos^2\tet_0 >
([n/2]-1)(n-[n/2]-1)/(n-1)$. This gives the desired values.

We now turn to linear stability.  It follows from Proposition
\ref{diagonering} and the block diagonalization of
$d^2H_\xi|_\NN(x_e)$ that
$$
L_\NN=\diag\left( \left( \begin{array}{cc}
      0&-\lambda_1\\
      \lambda_1&0
\end{array}\right),
\left\lbrace \left( \begin{array}{cc}
      0&-\lambda^{(\ell)}_\phi\\
      \lambda^{(\ell)}_\tet&0
\end{array}\right),
\left( \begin{array}{cc}
    0&-\lambda^{(\ell)}_\phi\\
    \lambda^{(\ell)}_\tet&0
\end{array}\right)
\mid 2\leq \ell\leq [n/2]\right\rbrace^* \right)
$$
where the blocks are given up to a strictly positive scalar factor.
The eigenvalues of $L_\NN$ are therefore
$$
\pm i \lambda_1,
\left\lbrace\pm i\sqrt{\lambda^{(\ell)}_\tet\lambda^{(\ell)}_\phi}\mid
  2\leq \ell\leq [n/2]\right\rbrace,
$$
(up to a positive factor) and so the relative equilibrium is
linearly unstable if $\lambda^{(\ell)}_\tet>0$ for some $\ell$,
that is if
$$
\cos^2\tet_0 <
\frac{1}{n-1}\left(\left[\frac{n}{2}\right]-1\right)
\left(\left[\frac{n+1}{2}\right]-1\right).
$$
In particular this inequality is satisfied if $\tet_0=\pi/2$ and
$n>3$.

When the ring is equatorial, one has $\tet_0=\pi/2$ and $\mu=0$.  In
particular $\lambda_1=0$.  This is because the symplectic slice is
smaller ($G_{\mu=0}=\SO(3)$): it follows from Proposition
\ref{tangentspace} that we have to remove the vectors $e_1$, $e_2$
from the basis for $\mu\neq 0$ (that is to remove $\lambda_1$ from the
previous eigenvalue study).  It follows that the $\CC_{nv}$ equatorial
relative equilibria are linearly unstable for $n>3$, and Lyapunov
stable (modulo $\SO(3)$) for $n=3$.
\end{proof}

The proof shows that the `critical mode' for stability is $\ell =
[n/2]$. For $n \geq 7$ a ring is always unstable to this mode,
while for $4 \leq n \leq 6$ the ring first loses stability to this
mode as it moves closer to the equator. This loss of stability is
accompanied by a pitchfork bifurcation to a pair of staggered
rings when $n = 4$ or $6$ (i.e.\ to types $\CC_{2v}(R,R')$ and
$\CC_{3v}(R,R')$ respectively, in the notation of \cite{LMR01}).
In the case $n=5$ the bifurcation is transcritical to  an
``equatorial'' vortex and two pairs that are reflections of each
other in that equator (ie type $C_h(2R,E)$ in \cite{LMR01}).  The
bifurcations for $n=4$ and $5$ are illustrated in Figures 7 and 8
of \cite{LMR01} respectively.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A ring and a polar vortex: $\CC_{nv}(R,p)$}
\label{Rp-stability}

We assume that the polar vortex lies at the North pole and its
vorticity is $\kappa$, while the other vortices are all identical
with vortex strength 1. The relative equilibrium is of symmetry
type $\CC_{nv}(R,p)$ and denoted $x_e$. In this case, the
Hamiltonian is given by
$$H \ =\  H_r + H_p$$
where $H_r$ is the ring Hamiltonian given in the
previous section and
$$
H_p(x,y,\theta_i,\phi_i) = -\kappa\sum_{j=1}^n \ln\left(1 -
  x\sin\theta_j\cos\phi_j-y\sin\theta_j\sin\phi_j -
  \sqrt{1-x^2-y^2}\cos\theta_j\right),
$$
is the Hamiltonian responsible for the interaction of the pole and
the ring.

In this case
$$
H_\xi \ =\  H - \xi\left(\sum_j\cos\theta_j +
  \kappa\sqrt{1-x^2-y^2}\right)
$$
and the relative equilibrium at $x=y=0,\ \theta_j=\theta_0,\
\phi_j=2\pi j/n$ has angular velocity
$$
\xi \ =\ \frac{(n-1)\cos\theta_0 +
  \kappa(1+\cos\theta_0)}{\sin^2\theta_0}
$$
since $\frac{\partial H_\xi}{\partial\theta_j}(x_e) =
-\frac{(n-1)\cos\theta_0+\kappa(1+\cos\theta_0)-\xi\sin^2\theta_0}{\sin\theta_0}$
must vanish.
\bigskip

The second derivatives at the relative equilibrium of $H$ can be
derived from those for $H_r$ given in the previous section, together
with:
$$
\left.
  \begin{array}{rclcrcl}
    \frac{\partial^2H_p}{\partial\theta_j^2} &=& \frac{\kappa}{1-\cos\theta_0}
    &\quad&
    \frac{\partial^2H_p}{\partial x^2} &=& \frac{n\kappa}{2} \;=\;
    \frac{\partial^2H_p}{\partial y^2}  \\[6pt]
    \frac{\partial^2H_p}{\partial x\partial \theta_j} &=&
    -\frac{\kappa\cos(2\pi j/n)}{1-\cos\theta_0} &&
    \frac{\partial^2H_p}{\partial y\partial \theta_j} &=&
    -\frac{\kappa\sin(2\pi j/n)}{1-\cos\theta_0}\\[6pt]
    \frac{\partial^2H_p}{\partial x \partial\phi_j} &=&
    -\frac{\kappa\sin\theta_0\sin(2\pi j/n)}{1-\cos\theta_0} &&
    \frac{\partial^2H_p}{\partial y \partial\phi_j} &=&
    \frac{\kappa\sin\theta_0\cos(2\pi j/n)}{1-\cos\theta_0},
  \end{array}\right.
$$
while the other second derivatives all vanish.  Here we have used
that $\sum\cos^2(2\pi j/n) = n/2$ for $n>2$, but for $n=2$ the sum is
$2$.  Thus for $n=2$, one obtains
$$
\textstyle \frac{\partial^2H_p}{\partial x^2} =
\frac{2\kappa}{1-\cos\theta_0} \qquad\qquad
\frac{\partial^2H_p}{\partial y^2} =
-\frac{2\kappa\cos\theta_0}{1-\cos\theta_0}.
$$


The following proposition gives the symmetry adapted basis for
$\CC_{nv}(R,p)$ relative equilibria.


\begin{prop}
\label{diag}
Let $n\geq 3$ and $\mu\neq 0$.  With respect to the following basis for the
symplectic slice:
$$
\left( e_1,e_2,e_3,e_4,\left\lbrace
    \alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi},\beta^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi}
    \mid 2\leq \ell\leq [n/2] \right\rbrace^* \right)
$$
where
$$
\begin{array}{lll}
e_1&=&\cos\theta_0\;\beta^{(1)}_{\theta}-\sin\theta_0\;\alpha^{(1)}_{\phi}-n\cos(2\theta_0)/(2\kappa)\;\delta y_n\\
e_2&=&\sin\theta_0\;\alpha^{(1)}_{\theta}+\cos\theta_0\;\beta^{(1)}_{\phi}\\
e_3&=&\cos\theta_0\;\alpha^{(1)}_{\theta}+\sin\theta_0\;\beta^{(1)}_{\phi}-n\cos(2\theta_0)/(2\kappa)\;\delta x_n\\
e_4&=&\sin\theta_0\;\beta^{(1)}_{\theta}-\cos\theta_0\;\alpha^{(1)}_{\phi}
\end{array}
$$
the Hessian $d^2H_\xi|_\NN(x_e)$ block diagonalizes into $1\times 1$
blocks and two $2\times 2$ blocks, and $L_\NN$ block diagonalizes into
$2\times 2$ blocks and one $4\times 4$ block.
\end{prop}

\begin{proof}
  The proof is similar to the proof of Proposition \ref{diagonering}.
\end{proof}


These block diagonalizations enable us to prove the following stability
theorem for $n \geq 4$, illustrated by Figures \ref{CRp1} and \ref{CRp2}.
The cases $n=2$ and $3$ are treated afterwards.

\begin{theo}
\label{ring+pole/stab}
A $\CC_{nv}(R,p)$ relative equilibrium with $n\geq 4$ and $\mu\neq 0$

\noindent (i) is spectrally unstable if and only if
$$
\kappa < \kappa_0\ \ \mbox{or}\ \ 8a\kappa >
(n\sin^2\theta_0+4(n-1)\cos\theta_0)^2,
$$
\noindent (ii) is Lyapunov stable if
$$
\kappa > \kappa_0\ \ \mbox{and}\ \ a\kappa
(\kappa+n\cos\theta_0)(\kappa-\kappa_1) < 0,
$$
where
$$
\begin{array}{lll}
a&=&(n\cos\theta_0-n+2)(1+\cos\theta_0)^2\\
\kappa_1&=& (n-1)\cos\theta_0\; (n\sin^2\theta_0+2(n-1)\cos\theta_0)/a\\
\kappa_0&=&(c_n - (n-1)(1+\cos^2\theta_0))/(1+\cos\theta_0)^2\\
c_n&=&\cases{n^2/4 & if $n$ is even, \cr
              (n^2-1)/4 & if $n$ is odd.}
\end{array}
$$
\end{theo}

\begin{proof}
  We first study the Lyapunov stability.  Following the beginning of
  the proof of Theorem \ref{polvani}, we obtain from Proposition
  \ref{diag} that
  $$
  d^2H_\xi|_\NN(x_e)=\diag(A,A,D)
  $$
  where
  $D=\diag(\lbrace\lambda^{(\ell)}_\tet,\lambda^{(\ell)}_\phi,
  \lambda^{(\ell)}_\tet,\lambda^{(\ell)}_\phi \mid 2\leq \ell\leq
  [n/2]\rbrace^* )$,
  $$
  A=\left( \begin{array}{cc}
      q_{11}&q_{12}\\
      q_{12}&q_{22}
\end{array}\right)
$$
and
$$
\begin{array}{lll}
\lambda^{(\ell)}_\tet&=&d^2H_\xi(x_e)\cdot (\alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\theta})\\
\lambda^{(\ell)}_\phi&=&d^2H_\xi(x_e)\cdot (\alpha^{(\ell)}_{\phi},\alpha^{(\ell)}_{\phi})\\
q_{11}&=&d^2H_\xi(x_e)\cdot (e_1,e_1)\\
q_{12}&=&d^2H_\xi(x_e)\cdot (e_1,e_2)\\
q_{22}&=&d^2H_\xi(x_e)\cdot (e_2,e_2).
\end{array}
$$
Note that $D$ exists only for $n\geq 4$.  From the previous section
one has $\lambda^{(\ell)}_\phi=n\ell(n-\ell)/2$ and some additional
computations give
$$
\lambda^{(\ell)}_\tet=\frac{n}{2\sin^2\tet_0}\left[
  -(\ell-1)(n-\ell-1)+(n-1)\cos^2\tet_0+\kappa(1+\cos\tet_0)^2
\right].
$$
The eigenvalues $\lambda^{(\ell)}_\phi$ are all positive, thus $D$
is definite if $-(\ell-1)(n-\ell-1) + (n-1)\cos^2\tet_0 +
\kappa(1+\cos\tet_0)^2 > 0$ for all $\ell=2,\dots,[n/2]$, that is
if $\kappa >
(([n/2]-1)(n-[n/2]-1)-(n-1)\cos^2\tet_0)/(1+\cos\theta_0)^2$ which
corresponds to $\kappa > \kappa_0$.

The relative equilibrium is therefore Lyapunov stable if $A$ is
positive definite, that is if $q_{11}q_{22}-q_{12}^2 >0$ and
$q_{11}+q_{22} > 0$.  Some lengthy computations give
$$
\begin{array}{rcl}
  q_{11}q_{22}-q_{12}^2&=& -\frac{n^2\cos^22\tet_0}{\kappa^2\sin^2\theta_0}
  a\kappa (\kappa+n\cos\theta_0)(\kappa-\kappa_1)\\
  q_{22}&=& \frac{n}{2(1+\cos\theta_0)^2}(\kappa-\kappa_2)
\end{array}
$$
where $a,\kappa_1$ are given in the theorem and $\kappa_2 =
-2(n-1)\cos^2\theta_0/(1+\cos\theta_0)^2$. Now we show that if
$q_{11}q_{22}-q_{12}^2 >0$ and $\kappa > \kappa_0$, then
$q_{11}+q_{22} > 0$: we have $q_{22}(q_{11}+q_{22})>0$ since
$q_{11}q_{22}-q_{12}^2 >0$, and $q_{22}>0$ since $\kappa >
\kappa_0>\kappa_2$, hence $q_{11}+q_{22} >0$. We proved therefore
that $\CC_{nv}(R,p)$ is Lyapunov stable if $\kappa > \kappa_0$ and
$a\kappa (\kappa+n\cos\theta_0)(\kappa-\kappa_1)<0$.

We now study the spectral stability of the relative equilibrium.  It
follows from Proposition \ref{diag} and the block diagonalization of
$d^2H_\xi|_\NN(x_e)$ that
$$
L_\NN=\diag\left(A_L, \left\lbrace \left( \begin{array}{cc}
        0&-\lambda^{(\ell)}_\phi\\
        \lambda^{(\ell)}_\tet&0
\end{array}\right),
\left( \begin{array}{cc}
    0&-\lambda^{(\ell)}_\phi\\
    \lambda^{(\ell)}_\tet&0
\end{array}\right)
\mid 2\leq \ell\leq [n/2]\right\rbrace^* \right)
$$
where the blocks are given up to a positive scalar factor and
$$
A_L=\left( \begin{array}{cccc}
    0 & 0 & a & b\\
    0 & 0 & c & d\\
    -a & -b & 0 & 0\\
    -c & -d & 0 & 0
\end{array}\right),\ \ \left\lbrace \begin{array}{ccc}
a&=&\beta q_{11}-\gamma q_{12}\\
b&=&\beta q_{12}-\gamma q_{22}\\
c&=&\alpha q_{12}-\gamma q_{11}\\
d&=&\alpha q_{22}-\gamma q_{12}
\end{array}\right.
$$
and
$$
\begin{array}{lll}
\alpha&=&\omega\left( e_1,e_3 \right) = n\cos\theta_0\sin^2\theta_0-n^2\cos^2(2\theta_0)/(4\kappa)\\
\beta&=&\omega\left( e_2,e_4 \right) = n\cos\theta_0\sin^2\theta_0\\
\gamma&=&\omega\left( e_1,e_4 \right) = \omega\left( e_2,e_3
\right)=n\sin\theta_0/2.
\end{array}
$$
The eigenvalues (up to a positive factor) of $L_\NN$ are therefore
$$
\pm\frac{1}{\sqrt{2}}\sqrt{\pm\sqrt{\nu}+\sigma} ,\left\lbrace\pm
  i\sqrt{\lambda^{(\ell)}_\tet\lambda^{(\ell)}_\phi}\mid 2\leq
  \ell\leq [n/2]\right\rbrace^*
$$
where $\nu=a^4+4a^2bc-2a^2d^2+4bcd^2+d^4+8adbc$ and
$\sigma=-a^2-2bc-d^2$. The eigenvalues $\pm
i\sqrt{\lambda^{(\ell)}_\tet\lambda^{(\ell)}_\phi}$ are all purely
imaginary if and only if $\kappa > \kappa_0$. After some lengthy
but straightforward computations we obtain that
$$
\begin{array}{lll}
\nu&=&\frac{n^{10}}{2^{12}\kappa^4}(1+u)^2(2u^2-1)^8(\kappa+nu)^4(1-u)^2\\
 & &\quad\times\left[ -8(1+u)^2(nu+2-n)\kappa+(nu^2-4(n-1)u-n)^2      \right]\\[6pt]
\sigma&=&-\frac{n^4}{2^6\kappa^2}(2u^2-1)^4(\kappa+nu)^2\\
 & &\quad\times\left[ -4(1+u)^2(nu+2-n)\kappa+n^2u^4-4n(n-1)u^3+\right.\\
   & & \qquad\quad \left.+2(3n^2-8n+4)u^2+ 4n(n-1)u+n^2 \right]
\end{array}
$$
where $u=\cos\theta_0$. One can check that if $\nu\geq0$, then
$\sqrt{\nu}+\sigma\leq0$ and the eigenvalues are purely imaginary.
If $\nu<0$, then the eigenvalues have a non-zero real part. Thus
the eigenvalues $\pm\sqrt{\pm\sqrt{\nu}+\sigma}$ are purely
imaginary if and only if $\nu\geq0$ which is equivalent to
$8a\kappa \leq (n\sin^2\theta_0+4(n-1)\cos\theta_0)^2$.
\end{proof}


A spectrally stable relative equilibrium for which the Hessian
$d^2H_\xi|_\NN(x_e)$ is not definite is said to be
\emph{elliptic}. Note that in principle an elliptic relative
equilibrium may be Lyapunov stable, but if there are more than 4
vortices then it is expected to be unstable as a result of Arnold
diffusion. Moreover an elliptic relative equilibrium typically
becomes linearly unstable when some dissipation is added to the
system \cite{DR02}.

\begin{coro}
  A $\CC_{nv}(R,p)$ relative equilibrium with $n\geq 4$ and $\mu\neq
  0$ is elliptic if and only if $$
  \kappa \geq \kappa_0\mbox{,}\ \
  a\kappa (\kappa+n\cos\theta_0)(\kappa-\kappa_1) \geq 0\ \
  \mbox{and}\ \ 8a\kappa \leq (n\sin^2\theta_0+4(n-1)\cos\theta_0)^2.
  $$
\end{coro}

\paragraph{Discussion of results for $n \geq 4$} \quad
See Figure \ref{CRp2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
  \begin{center}
    \includegraphics[width=4.7in,height=3.4in,angle=0]{real5p.eps}
                                %\vspace{1cm}
    \includegraphics[width=4.7in,height=3.4in,angle=0]{real8p.eps}
    \caption{Bifurcation diagrams for  $\CC_{nv}(R,p)$ relative equilibria.
      The bifurcation diagrams for $n\geq7$ are similar to that for
      $n=8$, while those for $n=4,6$ are similar to that for $n=5$.
      The circles represent the eigenvalues of the mode $1$, while
      the cross represent those of the mode $[n/2]$.  A relative equilibrium can be
      Lyapunov stable (S), linearly unstable (U) or elliptic (E).}
    \label{CRp2}
  \end{center}
\end{figure}
%%%%%%%%%%%

\begin{itemize}
\item If the sign of the vorticity of the polar vortex is opposite
to that of the ring then there are stable configurations with
$\kappa<0$ only for $n\leq 6$. Conversely configurations with
$n\leq 6$ and $\theta_0$ close to $\pi$, ie with the ring close to
the opposite pole, are Lyapunov stable for all $\kappa<0$.

\item The region of Lyapunov stability is larger when the
vorticities of the pole and the ring have the same sign
($\kappa>0$). The stability frontiers in the upper-left corners of
Figure \ref{CRp2} go the infinity when $\theta_0$ goes to
$\arccos(1-2/n)$. It follows that for $n\geq 4$ and
$\theta_0>\arccos(1-2/n)$, the relative equilibria are Lyapunov
stable for all sufficiently large $\kappa$. Thus, a ring of
vortices is stabilized by a polar vortex with a sufficiently large
vorticity of the same sign as the vortices in the ring. Note that
for $4 \leq n \leq 6$ and $\kappa$ positive, but sufficiently
small, a ring near the opposite pole is only elliptic and may not
be Lyapunov stable.

\item The limiting stability results for $\theta_0=0$, ie when the
ring is close to the polar vortex, coincide with the stability of
a planar $n$-ring plus a central vortex, see \cite{CS99} and
\cite{LP}. This is also true for $n = 2$ and $n =3$.

\item One of the main stability boundaries corresponds to the mode
$[n/2]$ and is analogous to the stability boundary for a single
ring. When $n$ is even stability is probably lost through a
pitchfork bifurcation to a relative equilibrium of type
$\CC_{\frac{n}{2}v}(R,R',p)$ consisting of two staggered
$\frac{n}{2}$-rings and a pole as $(\kappa, \theta_0)$ passes
through this boundary. When $n$ is odd there is an analogous
transcritical bifurcation to relative equilibria with only a
single reflectional symmetry which fixes two vortices and permutes
the others. These are denoted by $\CC_{h}(\frac{n-1}{2}R,2E)$ in
\cite{LMR01}.  A nice illustration in the case $n=3$ can be found
in Figure 8 of \cite{CMS03}.

\item The other stability boundary corresponds to the mode $1$.
Stability is lost through a Hamiltonian-Hopf bifurcation: two
pairs of imaginary eigenvalues `collide' and leave the imaginary
axis. For a detailed description of the bifurcations that can be
expected in this case see \cite{vM85}.

\item Note that it also happens that pairs of $\ell = 1$
eigenvalues pass through zero without leaving the imaginary axis.
In this case the relative equilibrium changes from being Lyapunov
stable to elliptic or vice versa and these stability changes are
accompanied by bifurcations.

\item Finally we note that when $\kappa$ crosses zero eigenvalues
change sign without crossing zero due to the fact that the
symplectic form becomes degenerate for $\kappa=0$.
\end{itemize}


\paragraph{Discussion of the case $n=3$}
\quad See Figure \ref{CRp1}.
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
  \begin{center}
    \includegraphics[width=4.7in,height=3.4in,angle=0]{real2p.eps}
                                %\vspace{1cm}
    \includegraphics[width=4.7in,height=3.4in,angle=0]{real3p.eps}
    \caption{Bifurcation diagrams for  $\CC_{2v}(R,p)$ and $\CC_{3v}(R,p)$
      relative equilibria.  The circles represent the eigenvalues of
      the mode $1$ (some are filled in order to trace them).  A relative
      equilibrium can be Lyapunov stable (S), linearly unstable (U) or elliptic
      (E).}
    \label{CRp1}
  \end{center}
\end{figure}
%%%%%%%%%%%

\noindent We assume in this discussion that the relative equilibria have non-zero
momentum.
%
For $n=3$, by the proof of Proposition \ref{ring+pole/stab} we
have $d^2H_\xi|_\NN(x_e)=\diag(A,A)$ and $L_N=A_L$. Hence
$\CC_{3v}(R,p)$ is Lyapunov stable if $a\kappa
(\kappa+3\cos\theta_0)(\kappa-\kappa_1) < 0$, and spectrally
unstable if and only if
$$
8a\kappa > (3\sin^2\theta_0+8\cos\theta_0)^2,
$$
where $a=(3\cos\theta_0-1)(1+\cos\theta_0)^2$ as in Theorem \ref{ring+pole/stab}.
These results are illustrated in Figure \ref{CRp1}. Notice that a polar vortex
\emph{destabilizes} a 3-ring if either the polar vortex is in the same hemisphere as
the ring and has a sufficiently strong vorticity of the same sign as the ring, or
the polar vortex has the opposite sign vorticity and the ring lies in an interval
containing $\theta_0 = 2\pi/3$ that grows as the magnitude of the polar vorticity
increases. Outside these regions there is a patchwork of regimes in which the
relative equilibrium is either Lyapunov stable or elliptic.


\paragraph{Discussion of the case $n=2$}
\quad See Figure \ref{CRp1}.
\medskip

\noindent The $\CC_{2v}(R,p)$ relative equilibria are isosceles
triangles lying on a great circle, and for $\theta_0 = 2\pi/3$ the
triangle becomes equilateral. We again discuss the stability of
those with non-zero momenta. Indeed, any 3-vortex configuration
with zero momentum is a Lyapunov stable (relative to $\SO(3)$)
relative equilibrium since the reduced space is just a point
\cite{Pa92}.
%
For $n=2$ the symmetry adapted basis is
$(\kappa\,\delta\tet_1-\kappa\,\delta\tet_2-2\cos\theta_0\,\delta
x,\kappa\,\delta\phi_1-\kappa\,\delta\phi_2-2\sin\theta_0\,\delta
y)$. Following the proof of Theorem \ref{ring+pole/stab} we obtain
after some straightforward computations that $\CC_{2v}(R,p)$ is
Lyapunov stable if
$$
(1+2\cos\theta_0)[(1+\cos\theta_0)^2\kappa+\cos\theta_0(2+3\cos\theta_0)]<0,
$$
and linearly unstable if the inequality is reversed. See Figure \ref{CRp1}.
%
\begin{itemize}
\item
  There are two stable regions.
  For $\theta_0 < 2\pi/3$  the relative equilibria are stable provided the polar
  vorticity is less than a certain $\theta_0$ dependent critical
  value, while for $\theta_0 > 2\pi/3$ they are stable for all
  polar vorticities greater than a critical value. As
  $\theta_0 \to\pi$ this value goes to $-\infty$.

\item
  For $\theta_0=\pi/2$, where the $2$-ring is equatorial and the
  isosceles triangle is right-angled, they are stable if and only if
  $\kappa<0$. This is in agreement with \cite[Theorem III.3]{PM98},
  with $\Gamma_1=\Gamma_2=1$, and $\Gamma_3=\kappa$.

\item \emph{The restricted three vortex problem} \quad The range
of stability when $\kappa=0$ does not coincide with the range of
stability for a single ring. Indeed the $\CC_{2v}(R)$ relative
equilibria are Lyapunov stable for all co-latitudes (see Theorem
\ref{polvani}) while $\CC_{2v}(R,p)$ is unstable for $\kappa=0$
and $\theta_0\in (0,\pi/2)$. This means that if we place a
\emph{passive tracer} (or \emph{ghost vortex}) at the North pole
and a ring of two vortices in the Northern hemisphere, then the
passive tracer will be unstable.
\end{itemize}



\begin{remark}\label{rmk:CMS}
 The stability of $\CC_{nv}(R,p)$ relative equilibria
has also been studied in \cite{CMS03}. However our method differs
significantly from theirs in that we consider the definiteness of
the Hessian $d^2H_\xi|_\NN(x_e)$ on the $2n$ dimensional
symplectic slice, while in \cite{CMS03} the authors determine
conditions for the Hessian to be definite on the whole $2(n+1)$
dimensional tangent space. The result is that we prove the
relative equilibria to be Lyapunov stable in a larger region of
the parameter space. Notice in particular that for $n \leq 6$ our
results say that a positive vorticity $n$-ring near the south pole
is Lyapunov stable if the north pole has either negative or
sufficiently positive vorticity. However in \cite{CMS03} only the
case of negative north polar vorticity is shown to be Lyapunov
stable. In this paper we  also give criteria for when the relative
equilibria are {\it unstable} by considering the eigenvalues of
the linearisation $L_\NN$.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Stability of a ring and two polar vortices: $\CC_{nv}(R,2p)$}
\label{R2p}

In this section we consider a relative equilibrium $x_e$ of
symmetry type $\CC_{nv}(R,2p)$, that is configurations formed of a
ring of $n$ vortices of strength $1$, together with two polar
vortices $p_N$, $p_S$ of strengths $\kappa_N$, $\kappa_S$
respectively at the North and South poles. We assume without loss
of generality that the ring lies in the Northern hemisphere.

We obtain analytic (in)stability criteria for the relative
equilibria with respect to the $\ell\geq2$ modes, which of course
give sufficient conditions for genuine instability.  A numerical
investigation is needed for the $\ell=1$ mode and hence to provide
stability criteria; this is being pursued separately.

The Hamiltonian is given by
$$H = H_r + H_{p_N} + H_{p_S} + H_{NS}$$
where $H_r$ is given in
Section \ref{1ring-stability} and
$$
\begin{array}{lll}
  H_{p_N}&=&-\kappa_N\sum_{i=1}^{n} \kappa_i \ln (1-\sin\theta_i \cos\phi_i\; x_N -\sin\theta_i \sin\phi_i\; y_N-\sqrt{1-x_N^2-y_N^2}\cos\theta_i)\\
  H_{p_S}&=&-\kappa_S\sum_{i=1}^{n} \kappa_i \ln (1-\sin\theta_i \cos\phi_i\; x_S -\sin\theta_i \sin\phi_i\; y_S+\sqrt{1-x_S^2-y_S^2}\cos\theta_i)\\
  H_{NS}&=&-\kappa_N\kappa_S\ln (1-x_N x_S - y_N y_S+\sqrt{1-x_N^2-y_N^2}\sqrt{1-x_S^2-y_S^2}),
\end{array}
$$
and the augmented Hamiltonian is:
$$
H_{\xi}=H-\xi\left(\sum_{i=1}^{n}\kappa_i\cos\theta_i
  +\kappa_N\sqrt{1-x_N^2-y_N^2}-\kappa_S\sqrt{1-x_S^2-y_S^2}\right).
$$

The angular velocity of the relative equilibrium at
$x_N=y_N=x_S=y_S=0,\ \theta_j=\theta_0,\ \phi_j=2\pi j/n$ has
angular velocity
$$
\xi = \frac{(n-1)\cos\theta_0 +
  \kappa_N(1+\cos\theta_0)-\kappa_S(1-\cos\theta_0)}{\sin^2\theta_0}.
$$

The second derivatives of $H$ at the relative equilibrium can be
derived from those for $H_r$ (Section \ref{1ring-stability}), those
for $H_p$ (Section \ref{Rp-stability}), together with:
$$
\frac{\partial^2 H_{NS}}{\partial x_N^2}=\frac{\partial^2
  H_{NS}}{\partial y_N^2}= \frac{\partial^2 H_{NS}}{\partial
  x_S^2}=\frac{\partial^2 H_{NS}}{\partial y_S^2}= \frac{\partial^2
  H_{NS}}{\partial x_N\partial x_S}=\frac{\partial^2 H_{NS}}{\partial
  y_N\partial y_S}= \kappa_N\kappa_S/2
$$
while the other second derivatives of $H_{NS}$ vanish.


As in the previous sections, we can choose a symmetry adapted basis of
the symplectic slice such that the matrices $d^2H_\xi|_\NN(x_e)$ and
$L_\NN$ block diagonalize.

\begin{prop}
\label{diag2p}
Let $n\geq 3$ and $\mu\neq 0$.  In the following basis for the
symplectic slice,
$$
\left( e_1,e_2,e_3,e_4,e_5,e_6,\left\lbrace
    \alpha^{(\ell)}_{\theta},\alpha^{(\ell)}_{\phi},\beta^{(\ell)}_{\theta},\beta^{(\ell)}_{\phi}
    \mid 2\leq \ell\leq [n/2] \right\rbrace^* \right)
$$
where
$$
\begin{array}{lll}
e_1&=&\cos\theta_0\;\beta^{(1)}_{\theta}-\sin\theta_0\;\alpha^{(1)}_{\phi}-n\cos(2\theta_0)/(2\kappa_N)\;\delta y_N\\
e_2&=&\cos\theta_0\;\beta^{(1)}_{\theta}-\sin\theta_0\;\alpha^{(1)}_{\phi}-n\cos(2\theta_0)/(2\kappa_S)\;\delta y_S\\
e_3&=&\sin\theta_0\;\alpha^{(1)}_{\theta}+\cos\theta_0\;\beta^{(1)}_{\phi}\\
e_4&=&\cos\theta_0\;\alpha^{(1)}_{\theta}+\sin\theta_0\;\beta^{(1)}_{\phi}-n\cos(2\theta_0)/(2\kappa_N)\;\delta x_N\\
e_5&=&\cos\theta_0\;\alpha^{(1)}_{\theta}+\sin\theta_0\;\beta^{(1)}_{\phi}-n\cos(2\theta_0)/(2\kappa_S)\;\delta x_S\\
e_6&=&\sin\theta_0\;\beta^{(1)}_{\theta}-\cos\theta_0\;\alpha^{(1)}_{\phi}
\end{array}
$$
the Hessian $d^2H_\xi|_\NN(x_e)$ block diagonalizes into $1\times 1$
blocks and two $3\times 3$ blocks, and $L_\NN$ block diagonalizes into
$2\times 2$ blocks and one $6\times 6$ block.
\end{prop}

\begin{proof}
  The proof is similar to that for a single ring (see Section
  \ref{singlering} and Proposition \ref{diagonering}).
\end{proof}

The mode $\ell=1$ gives a $3\times 3$ block from which,
unfortunately, we can not derive a useful formula for stability analogous
to that for a single polar vortex. However, we can derive formulae
for the stability of the other modes, and thereby obtain the
following sufficient condition for instability, illustrated by
Figures \ref{ring2poles} and \ref{fig:r2p-instability}.

\begin{theo}
\label{ring+2poles/stab}
A $\CC_{nv}(R,2p)$ relative equilibrium with $n\geq 4$ and $\mu\neq 0$
is linearly unstable if
$$
\kappa_N (1+\cos\theta_0)^2+\kappa_S (1-\cos\theta_0)^2 <
c_n-(n-1)(1+\cos^2\theta_0),
$$
where
$$
c_n=\cases{n^2/4 & if $n$ is even, \cr (n^2-1)/4 & if $n$ is odd,}
$$
and is stable with respect to the $\ell \geq 2$ modes if
this inequality is reversed.
\end{theo}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
    \begin{center}
      \includegraphics[width=3in,height=3in,angle=0]{newring2.eps}
     \caption{The relative equilibria $\CC_{nv}(R,2p)$ in the Northern
     hemisphere are unstable `below'
     this ruled surface in $(\theta_{0},\kappa_S,\kappa_N)$-space,
     shown in the figure for $n=4$. Above the surface the relative
     equilibrium is stable with respect to all the $\ell \geq 2$ modes.}
     \label{ring2poles}
     \end{center}
 \end{figure}
%%%%%%%%%%%

%%%%%%%%%%%
\begin{proof}
  The proof is similar to that for Theorem \ref{ring+pole/stab}.
  Following the notations of the proof of Theorem
  \ref{ring+pole/stab}, we have
  $\lambda^{(\ell)}_\phi=n\ell(n-\ell)/2$ and
  $$
  \lambda^{(\ell)}_\tet=\frac{n}{2\sin^2\tet_0}\left[
    -(\ell-1)(n-\ell-1)+(n-1)\cos^2\tet_0+\kappa_N(1+\cos\tet_0)^2+\kappa_S
    (1-\cos\theta_0)^2 \right].
  $$
  The relative equilibrium is linearly unstable if there exists
  $\ell\geq 2$, such that $\lambda^{(\ell)}_\tet<0$.  Since the
  highest $\lambda^{(\ell)}_\tet$ is for $\ell=[n/2]$, the relative
  equilibrium is linearly unstable if
  $$
  -([n/2]-1)(n-[n/2]-1)+(n-1)\cos^2\tet_0+\kappa_N(1+\cos\tet_0)^2+\kappa_S
  (1-\cos\theta_0)^2 < 0,
  $$
  and is stable with respect to the $\ell \geq 2$ modes if
 this inequality is reversed. This gives the desired criterion.
\end{proof}

\paragraph{Stability of the $\ell \geq 2$ modes}
From the theorem we can deduce the following results about the
(in)stability of the $\CC_{nv}(R,2p)$ relative equilibria with
respect to the $\ell \geq 2$ modes. These modes only occur for $n
\geq 4$.  We continue to assume the ring lies in the Northern
hemisphere.

\begin{itemize}

\item In the limiting case as the ring converges to the North pole
($\tet_0=0$), for all values of $\kappa_S$ the relative equilibria
are linearly unstable if $\kappa_N < \frac14(c_n-2n+2)$. This
agrees with the instability of a ring and single pole when
`$\kappa < \kappa_0$' in Proposition \ref{ring+pole/stab}.

\item At the opposite extreme, when the ring is at the equator
($\tet_0=\pi/2$) they are linearly unstable if $\kappa_N+\kappa_S
< c_n-n+1$.  The right hand side of this inequality is
non-negative for all positive integers $n$, and so the
`equatorial' $\CC_{nv}(R,2p)$ relative equilibria are unstable if
the total polar vorticity has opposite sign to that of the ring.
If $\kappa_N+\kappa_S > 0$ then the critical ratio of the total
polar vorticity to the total ring vorticity needed to stabilize
the $\ell \geq 2$ modes grows linearly with $n$.

\item For all $n\geq 4$ the relative equilibria are unstable for
all latitudes in the Northern hemisphere if $\kappa_N <
\frac14(c_n-2n+2)$ \emph{and} $\kappa_N+\kappa_S < c_n-n+1$.  In
particular, for $n \geq 7$ the relative equilibria are unstable
for all $\theta_0$ if $\kappa_N < 0$ and $\kappa_S < 0$.

\end{itemize}


\begin{figure}[ht]
\psfrag{aa}{$\kappa_S$} %
\psfrag{bb}{$\kappa_N$} %
\psfrag{uu}{\small \parbox{18mm}{Unstable at all latitudes}} %
\psfrag{eq}{\parbox{20mm}{\small Unstable near equator}}%
\psfrag{np}{\small \parbox{20mm}{Unstable near North pole}}
    \begin{center}
      \includegraphics[width=2in,height=2in]{nostability.eps}%
      \qquad%
      \includegraphics[width=2in,height=2in]{nostability2.eps}%
     \caption{(a) The shaded regions depict the values of the polar
     vorticities for which \emph{all} the relative equilibria in the Northern hemisphere are
     unstable: the dark region is for $n=4$, the next for $n=8$ and the lightest for $n=10$.
     (b) Demonstrates that above each shaded region of (a) the corresponding relative
     equilibria near the North pole are unstable, while to the
     right it is the relative equilibria near the equator which are unstable.
     The instabilities are all due to the $\ell\geq2$ modes.}
     \label{fig:r2p-instability}
     \end{center}
\end{figure}

To determine whether there are in fact \emph{stable} relative
equilibria it is necessary to evaluate the eigenvalues arising
from the $\ell=1$ mode.  This is work in progress, and preliminary
numerical investigation suggests:
\begin{itemize}
    \item For all $n$ and for all sufficiently large and positive polar vorticities
    there are ranges of $\theta_0$ with elliptic relative
    equilibria;
    \item For all $n$ and for $\kappa_N$ sufficiently positive and
    $\kappa_S<0$ there are Lyapounov stable relative equilibria in
    the Northern hemisphere.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Stability of two aligned rings: $\CC_{nv}(2R)$}
\label{2R}

In this section we consider relative equilibria $x_e$ of symmetry
type $\CC_{nv}(2R)$, that is configurations formed of two
`aligned' rings of $n$ vortices each. We can assume without loss
of generality that the vorticities of the vortices in the first
and second ring are  $1$ and  $\kappa$, respectively, and we
denote their co-latitudes by $\tet_0$ and $\tet_1$. We can also
assume that the ring of vorticity $1$ and co-latitude $\tet_0$
lies in the Northern hemisphere, $\tet_0\in (0,\pi/2]$.  The first
question to answer is \emph{for which values of the parameters
$(\tet_0,\tet_1,\kappa)$ is the configuration $\CC_{nv}(2R)$ a
relative equilibrium?}  It was shown in \cite{LMR01} (p.~126) that
for given $\kappa>0$ and each $\mu$ with $|\mu| < n|1+\kappa|$
there is at least one solution for $(\theta_0,\theta_1)$ with
$n\cos\theta_0+n\kappa\cos\theta_1=\mu$ and with
$\theta_0<\theta_1$ and at least one with $\theta_1<\theta_0$. We
now make this more precise.

The fixed point set $\Fix(G_{x_e})$ is parametrized by
$x:=\cos\tet_0$ and $y:=\cos\tet_1$. Denote by $\tilde F$ the
restriction of a function $F$ to $\Fix(G_{x_e})$.  It is
straightforward to show that the Hamiltonian can be split in such
a way that
$$
H\ =\ H_{11}+\kappa H_{12}+\kappa^2 H_{22}
$$
where $H_{11},H_{12},H_{22}$ do not depend on $\kappa$, $\tilde
H_{11}$ does not depend on $y$ and $\tilde H_{22}$ does not depend
on $x$.  The following proposition shows that for almost every
pair $(\tet_0,\tet_1)$ there exists a unique $\kappa$ such that
the $\CC_{nv}(2R)$ configuration with parameters
$(\tet_0,\tet_1,\kappa)$ is a relative equilibrium.

%%%%%%%%%%%%%%%%%
\begin{prop}
\label{existlamda2R} Let $x_{e}$ be a $\CC_{nv}(2R)$ configuration
with parameters $(\tet_0,\tet_1,\kappa)$.
\begin{enumerate}
\item There exists a unique $\kappa\in\R^*$ such that $x_e$ is a
relative equilibrium if and only if both the following conditions
hold:
$$
\left(\frac{\partial \tilde H_{12}}{\partial y}-\frac{\partial
    \tilde H_{11}}{\partial x}\right) (\cos\theta_0,\cos\theta_1)\neq
0,\ \ \left(\frac{\partial \tilde H_{22}}{\partial y}-\frac{\partial
    \tilde H_{12}}{\partial x}\right) (\cos\theta_0,\cos\theta_1)\neq
0.
$$

\item The configuration $x_e$ is a relative equilibrium for {\it
all} $\kappa\in\R^*$ in the degenerate case when both the
following conditions hold:
$$
\left(\frac{\partial \tilde H_{12}}{\partial y}-\frac{\partial
    \tilde H_{11}}{\partial x}\right) (\cos\theta_0,\cos\theta_1) = 0,
\ \ \left(\frac{\partial \tilde H_{22}}{\partial
    y}-\frac{\partial \tilde H_{12}}{\partial x}\right)
(\cos\theta_0,\cos\theta_1) = 0.
$$

\item
In both cases the angular velocity $\xi$ of $x_e$ satisfies
$$
\xi\ =\ \frac{1}{n}\left(\frac{\partial \tilde H_{11}}{\partial
    x}(x_e)+\kappa\frac{\partial \tilde H_{12}}{\partial
    x}(x_e)\right).
$$
In particular, if in addition $\partial_x \tilde H_{11}(x_e)$ and
$\partial_x \tilde H_{12}(x_e)$ are non-zero, then there exists a
unique $\kappa\in\R^*$ such that $\xi = 0$ and so $x_e$ is an
equilibrium.

\end{enumerate}
\end{prop}
%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%
\begin{proof}
  Since $H+\xi\Phi$ is a $G_{x_e}$-invariant function
  (see Section \ref{intro}) the Principle of Symmetric Criticality
  \cite{P79} implies that $x_e$ is a relative equilibrium if and only if it is a critical
  point of $\tilde H+\xi\tilde\Phi$.
  It follows from $\tilde\Phi=n(x+\kappa y)$ that $d(\tilde
  H+\xi\tilde\Phi)(x_e)=0$ is equivalent to the pair of equations:
  $$
  \kappa\left(\frac{\partial \tilde H_{22}}{\partial
      y}(x_e)-\frac{\partial \tilde H_{12}}{\partial x}(x_e)\right) +
  \frac{\partial \tilde H_{12}}{\partial y}(x_e)-\frac{\partial \tilde
    H_{11}}{\partial x}(x_e)\ =\ 0
  $$
  $$
  \xi=\frac{1}{n}\left(\frac{\partial \tilde
      H_{11}}{\partial x}(x_e)+\kappa\frac{\partial \tilde
      H_{12}}{\partial x}(x_e)\right).
  $$
  The proposition follows easily from these.
\end{proof}
%%%%%%%%%%%%%%%%

For example, in the case $n=4$ the degenerate case occurs when the
two rings form the vertices of a cube. Hence for any values of the
vorticities of the two rings the \pg cube configuration\pd\ is a
relative equilibrium. However among this family of relative
equilibria only one is an equilibrium, namely the one for which
the two rings have the same vorticities, $\kappa=1$, which
corresponds to the $\oct_h(f)$ equilibrium \cite{LMR01}, a cube
formed of identical vortices.

For $\tet_1=\pi-\tet_0$, the configuration has an extra symmetry
and its symmetry type is $D_{nh}(2R)$. Such a configuration is a
relative equilibrium if $\kappa=-1$, the two rings have opposite
vorticities. The existence and stability of such relative
equilibria were studied in \cite{LP02}.

With the help of the discussion of Section \ref{tworings}, we
performed a $G_{x_e}$-invariant isotypic decomposition and found that the
symmetry adapted basis for the symplectic slice at a $\CC_{nv}(2R)$
relative equilibrium with $n\geq3$ and $\mu\neq 0$ is
$$
\left( e_1,e_2,e_3,e_4,e_5,e_6,e_7,e_8,\left\lbrace B_\ell \mid
    2\leq \ell\leq (n-1)/2 \right\rbrace \right) $$
for $n$ odd, while
it is
$$
\left( e_1,e_2,e_3,e_4,e_5,e_6,,e_7,e_8,\left\lbrace B_\ell \mid
    2\leq \ell\leq n/2-1 \right\rbrace
  ,\alpha^{(n/2)}_{0,\tet},\alpha^{(n/2)}_{1,\tet},\alpha^{(n/2)}_{0,\phi},\alpha^{(n/2)}_{1,\phi}
\right) $$
for $n$ even, where
$$
\begin{array}{lll}
e_1&=&\alpha^{(0)}_{0,\phi}-\alpha^{(0)}_{1,\phi}\\
e_2&=&\kappa\sin\theta_1\;\alpha^{(0)}_{0,\tet}-\sin\theta_0\;\alpha^{(0)}_{1,\tet}\\
e_3&=&
\sin\theta_0\;\sin\theta_1\;(\kappa\cos\theta_1\;\alpha^{(1)}_{0,\tet}+\cos\theta_0\;\alpha^{(1)}_{1,\tet})\\
&&\quad
+\cos\theta_0\;\cos\theta_1\;(\kappa\sin\theta_1\;\beta^{(1)}_{0,\phi}+\sin\theta_0\;\beta^{(1)}_{1,\phi})\\
e_4&=&\kappa\cos\theta_1\;\alpha^{(1)}_{0,\tet}-\cos\theta_0\;\alpha^{(1)}_{1,\tet}\\
e_5&=&\kappa\sin\theta_1\;\beta^{(1)}_{0,\phi}-\sin\theta_0\;\beta^{(1)}_{1,\phi}\\
e_6&=&\sin\theta_0\;\sin\theta_1\;(\kappa\cos\theta_1\;\beta^{(1)}_{0,\tet}+\cos\theta_0\;\beta^{(1)}_{1,\tet})\\
&&\quad
-\cos\theta_0\;\cos\theta_1\;(\kappa\sin\theta_1\;\al^{(1)}_{0,\phi}+\sin\theta_0\;\al^{(1)}_{1,\phi})\\
e_7&=&\kappa\cos\theta_1\;\beta^{(1)}_{0,\tet}-\cos\theta_0\;\beta^{(1)}_{1,\tet}\\
e_8&=&\kappa\sin\theta_1\;\al^{(1)}_{0,\phi}-\sin\theta_0\;\al^{(1)}_{1,\phi}
\end{array}
$$
and
$$
\begin{array}{rcl}
B_\ell&=&\left\lbrace\alpha^{(\ell)}_{0,\theta}-\alpha^{(\ell)}_{1,\theta},\:\beta^{(\ell)}_{0,\phi}+\beta^{(\ell)}_{1,\phi},\:
  \alpha^{(\ell)}_{0,\phi}+\alpha^{(\ell)}_{1,\phi},\:\beta^{(\ell)}_{0,\theta}-\beta^{(\ell)}_{1,\theta},\right.\\
&&\qquad\left.
  \alpha^{(\ell)}_{0,\theta}+\alpha^{(\ell)}_{1,\theta},\:
\beta^{(\ell)}_{0,\phi}-\beta^{(\ell)}_{1,\phi},\:
  \alpha^{(\ell)}_{0,\phi}-\alpha^{(\ell)}_{1,\phi},\:\beta^{(\ell)}_{0,\theta}+\beta^{(\ell)}_{1,\theta}\right\rbrace.
\end{array}
$$
The adapted basis for $n=2$ is simply $(e_1,e_2,e_4,e_8)$.\\

\rmk Almost all $\CC_{nv}(2R)$ relative equilibria have a non-zero
momentum.  Indeed $\mu=0$ iff $x+\kappa y=0$, and from the
expression of $\kappa$ one can show that this last equation is the
equation defining an algebraic curve in variables $(x,y)\in
[0,1)\times (-1,1)\simeq\Fix \CC_{nv}$.

\bigskip

With respect to this basis $d^2H_\xi|_\NN(x_e)$ block diagonalises
into two $4 \times 4$ blocks for each $\ell = 1 \ldots
[(n-1)/2]$), together with two $2 \times 2$ blocks for $\ell =
n/2$ when $n$ is even. The linearisation $L_\NN$ block
diagonalises into half as many blocks of twice the size. In order
to calculate the stability of the relative equilibria, we ran a
\textsc{Maple} programme to compute numerically the eigenvalues of
each of the blocks of $d^2H_\xi|_\NN(x_e)$ and $L_\NN$. The
results are summarized for $n = 2 \ldots 6$ in Figure \ref{twins}.
Figure \ref{signeta} shows how the sign of $\kappa$ varies for
relative equilibria with different values of $\theta_0$ and
$\theta_1$.

%%%%%%%%%%%
\begin{figure}[tb]
    \begin{center}
      \includegraphics[width=4in,height=3in,angle=0]{sign6.eps}
      \caption{Sign of $\kappa$ and stability frontier for $\CC_{nv}(2R)$
        relative equilibria.  The degenerate case occurs when the
        curves $\kappa=0$ and $\kappa^{-1}=0$ intersect each other.
        The figure is for $n=6$, but is similar for other values of
        $n$ (concerning the sign of $\kappa$).}
\label{signeta}
\end{center}
 \end{figure}
%%%%%%%%%%%



\paragraph{Discussion of results}\

\begin{itemize}
\item The numerical results suggest strongly that the relative
equilibria $\CC_{nv}(2R)$ are never stable if the two rings lie in
the same hemisphere (Figure \ref{twins}) or have the same sign
vorticity (Figure \ref{signeta}). \item The stable configurations
are for $\theta_0$ close to $0$ and $\theta_1$ close to $\pi$, so
the ring of vorticity $1$ is close to the North pole and the other
ring is close to the South pole with a vorticity close to $-1$.
Thus the two rings `look like' two polar vortices. It is well
known that any configuration of two vortices is Lyapunov stable.
\item As $n$ increases the region of stability decreases in size.
Numerical experiments with $n \geq 7$ suggest that in these cases
the relative equilibria are never stable. \item For $n = 2$, $4$
and $6$ stability is first lost by a pair of imaginary eigenvalues
of the $\ell = n/2$ block of $L_\NN$ passing through $0$ and
becoming real. For $n = 3$ and $5$ close to the $D_{nh}(2R)$
relative equilibria a pair of imaginary eigenvalues of the $\ell =
(n-1)/2$ block passes through $0$ but remains on the imaginary
axis, so the stability changes from Lyapunov to elliptic. This
imaginary pair then collides with another pair, and all move off
the imaginary axis to form a complex quadruple and create
instability. It seems likely that this behaviour also occurs away
from the $D_{nh}(2R)$ relative equilibria, but in a region too
small to be seen in the figure.

%\item The relative equilibria $D_{nh}(2R)$ with $n$ even lose
%their stability via a pitchfork bifurcation \cite{LP02}.  But the
%bifurcating branches do not lie in $\Fix(\CC_{nv}(2R))$:
%$D_{4h}(2R)$ and $D_{6h}(2R)$ break to respectively $D_{2h}(4R)$
%and $D_{3h}(4R)$, while $D_{2h}(2R)$ breaks to
%$\CC_{2v}(R_m,R_m^\prime)$ (two rings each of one vortex $+1$ and
% one $-1$).
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tb]
    \begin{center}
      \includegraphics[width=2in,height=4in,angle=0]{twinNeven.eps}
      \includegraphics[width=2in,height=4in,angle=0]{twinNodd.eps}

      \caption{Stability results for $\CC_{nv}(2R)$ relative equilibria.
        The curve plotted for each $n$ is the stability frontier: on
        one side the relative equilibria are Lyapunov stable (S),
        while on the other side they are linearly unstable (U), with a
        gap of elliptic stability (E) between the two in the odd case.
        For $n\geq 7$, it seems likely that the relative equilibria
        are all unstable.}
       \label{twins}
     \end{center}
 \end{figure}
%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Stability of two staggered rings: $\CC_{nv}(R,R')$}
\label{RR'}

In this section we consider relative equilibria formed of two
rings of $n$ vortices each of strengths $1$ and $\kappa$ and
co-latitude $\tet_0$ and $\tet_1$ respectively. They differ from
those of the previous section in that the rings here are  \pg
staggered\pd, that is they rotated relative to each other with an
offset of $\pi/n$. Their symmetry type is $\CC_{nv}(R,R')$. As in
the previous section we can assume without loss of generality that
the ring of vorticity $1$ and co-latitude $\tet_0$ lies in the
Northern hemisphere.

An analogue of Proposition \ref{existlamda2R} also holds for
$\CC_{nv}(R,R')$ configurations: for almost every pair
$(\tet_0,\tet_1)$ there exists a unique $\kappa$ such that the
corresponding $\CC_{nv}(R,R')$ configuration is a relative
equilibrium.  With the notation of the previous section, in the
non-degenerate case the angular velocity and $\kappa$ satisfy
$$
\begin{array}{rcl}
  \kappa & = & -(\partial_x \tilde H_{11}-\partial_y \tilde H_{12})
                /(\partial_y \tilde H_{22}-\partial_x \tilde H_{12})(x_e),\\
  \xi & = & n^{-1}(\partial_x \tilde H_{11} + \kappa\partial_x \tilde H_{12})(x_e).
\end{array}
$$
There exist also degenerate cases. When $\tet_1 = \tet_0$ the
configuration forms a single ring with $2n$ vortices with
$\kappa=1$: all the vortices have the same vorticity. These are
the relative equilibria of type $\CC_{2nv}(R)$ studied in Section
\ref{singlering}.  For $\tet_1 = \pi-\tet_0$, the configuration
has an extra symmetry and its symmetry type is $D_{nd}(R,R')$. In
this case $\kappa=-1$, the two rings have opposite vorticities.
The existence and stability of such relative equilibria were
studied in \cite{LP02}.

With the help of the discussion of Section \ref{tworings}, we found
that a symmetry adapted basis for the symplectic slice at a
$\CC_{nv}(R,R')$ relative equilibrium with $n\geq 3$ and $\mu\neq 0$
is given by:
$$
\left( e_1,e_2,e_3,e_4,e_5,e_6,e_7,e_8,\left\lbrace B_\ell \mid
    2\leq \ell\leq (n-1)/2 \right\rbrace \right)
$$
for $n$ odd, while for $n$ even one is given by:
$$
\begin{array}{r}
\left( e_1,e_2,e_3,e_4,e_5,e_6,e_7,e_8,\left\lbrace B_\ell \mid
    2\leq \ell\leq n/2-1 \right\rbrace,\right.\quad \quad \quad \\
  \left.\alpha^{(n/2)}_{0,\tet}-\alpha^{(n/2)}_{1,\tet},\alpha^{(n/2)}_{0,\phi}-\alpha^{(n/2)}_{1,\phi},
  \beta^{(n/2)}_{1,\theta},\beta^{(n/2)}_{1,\phi} \right),
\end{array}
$$
where the expressions of $e_1,\dots,e_8$ and $B_\ell$ remain as in
the previous section.  The corresponding symmetry adapted basis
for $n=2$ is simply $(e_1,e_2,e_3,e_6)$. As in the previous
section, we can show that almost all $\CC_{nv}(R,R')$ relative
equilibria have non-zero momenta.

As for the aligned rings, we ran a \textsc{Maple} programme to
determine the stability of the relative equilibria. The results
are summarized in Figure \ref{twinds} for $n = 2 \ldots 6$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htp]
    \begin{center}
      \includegraphics[width=2in,height=4in,angle=0]{twind2.eps}
      \qquad
      \includegraphics[width=2in,height=4in,angle=0]{twind3.eps}
      \\
      \includegraphics[width=2in,height=4in,angle=0]{twind4.eps}
      \qquad
      \includegraphics[width=2in,height=4in,angle=0]{twind56.eps}
      \caption{Stability results for $\CC_{nv}(R,R')$ relative equilibria.
        The curves plotted are stability frontiers: on one side the
        relative equilibria are Lyapunov stable (S), while on the
        other side they are linearly unstable (U).  For $n\geq 7$, it
        seems likely that the relative equilibria are all unstable.}
       \label{twinds}
     \end{center}
 \end{figure}
%%%%%%%%%%%

\paragraph{Discussion of results}\

\begin{itemize}
\item Numerical experiments suggest that stable relative
equilibria only exist for $n \leq 6$.

\item For $n=5$ and $6$ the relative equilibria $\CC_{nv}(R,R')$
are stable only if the  two rings lie in the same hemisphere but
are sufficiently far apart.

\item For $n\leq4$ these stable regions extend to include relative
equilibria with the rings in different hemispheres. However,
contrary to the case $\CC_{nv}(2R)$, the stable regions are far
from the line $\theta_1=\pi-\theta_0$ corresponding to
$D_{nd}(R,R')$ relative equilibria.

\item For $n = 2$ and $3$ there is also a stable region with the
two rings in the same hemisphere and close to each other. This
includes the stable $\CC_{4v}(R)$ and $\CC_{6v}(R)$ relative
equilibria discussed in Section \ref{singlering}.

\item Note also that for $n\leq6$, there exist stable relative
equilibria (for some values of $\kappa$) in any neighbourhood of
$(\theta_0,\theta_1) = (0,0)$, that is with the two rings close to
the North pole.

\item A study of the sign of $\kappa$ shows that when $3 \leq n
\leq 6$ the relative equilibria with $\kappa<0$ are all unstable.
However for $n=2$ there exist relative equilibria with $\kappa$
positive in the stable region corresponding to the two rings both
being relatively close to the equator, but in opposite
hemispheres.

\item The tetrahedral equilibrium with all 4 vortices identical is
Lyapounov stable, and in Fig.~\ref{twinds} (with $n=2$) lies at
the point where the two stable regions meet the two unstable
regions on the $D_{2d}(R,R')$-locus.  The analogous point with
$n=3$ corresponds to the stable equilibrium consisting of 6
identical vortices lying at the vertices of an octahedron
$\oct_h(v)$.
\end{itemize}
\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Acknowledgements.}
This work was partially supported by the EC Research Training Network `MASIE'.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\small

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\bigskip

\hbox to 0.4\textwidth{\hrulefill}

\smallskip


 \parindent=0pt

  FLP: Institut Non Lin\'eaire de Nice, UMR 6618 CNRS, 1361 route des
  Lucioles, 06560 Valbonne, France.

  \texttt{laurent@inln.cnrs.fr}

  \medskip

  JM: Department of Mathematics, UMIST, Manchester M60 1QD, UK.

  \texttt{j.montaldi@umist.ac.uk}

  \medskip

  MR: Department of Mathematics and Statistics, University of Surrey,
  Guildford GU2 7XH, UK.

  \texttt{m.roberts@eim.surrey.ac.uk}

}

\end{document}

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\end{thebibliography}