\subsubsection*{Vortex drift by a background vorticity gradient}

For understanding the stability of the CPCs, it is essential to generalize
the beta-drift as a force that acts in the direction of rising background
vorticity. For intuition, one can think of a simplistic scenario in
which the conserved potential vorticity (PV) is a superposition of
constant vorticity by a solid disk, counter-clockwise rotating vortex,
and of an unspecified background vorticity ($\omega$ in Fig.\ \ref{fig: beta-drift-fig})
that monotonically rises northward. A fluid parcel that starts on
the southern edge of the vortex, where background vorticity is small,
will be carried by the vortex circulation to the eastern edge, where
background vorticity is higher (Fig.~\ref{fig: beta-drift-fig}a).
To conserve PV, negative relative vorticity would be induced. The
opposite will happen with a northern parcel that will induce positive
vorticity while reaching the west side. This dipole of induced vorticity,
usually termed ``$\beta$-gyres''\citep{fiorino1989some,sutyrin1994intense},
will then shear the velocity field and thus generate a northward velocity
profile (Fig.~\ref{fig: beta-drift-fig}b). The mean force acting
on a vortex due to this phenomenon is proportional to the gradient
of background vorticity both in magnitude and in direction. Counter-clockwise
rotating vortices are pulled toward the highest ascent of background
vorticity, while clockwise rotating vortices are pulled toward the
highest descent. In the planetary context (considering $\omega$ as
the planetary vorticity), this results in cyclones being pulled poleward
and anticyclones equatorward.
\begin{figure}
\begin{centering}
\includegraphics[width=1\columnwidth]{Gavriel_Fig2}
\par\end{centering}
\caption{\textbf{Beta-drift schematic. }Gray contours represent streamlines
of a vortex. Blue contours represent lines of equal background vorticity
($\omega$), where their increasing thickness represents the gradual
increase in the magnitude of the background vorticity. \textbf{a},
The Lagrangian motion of two fluid parcels leads to a dipole of induced
vorticity due to the conservation of PV during the motion led by the
vortex. \textbf{b}, A velocity profile induced in the vortex by the
shearing due to the vorticity dipole illustrated in \textbf{a}.\label{fig: beta-drift-fig}}
\end{figure}


\subsubsection*{Meridional stability of circumpolar cyclones}

\begin{figure}
\begin{centering}
\includegraphics[width=1\columnwidth]{Gavriel_Fig3}
\par\end{centering}
\caption{\textbf{An illustration of the balance holding a circumpolar cyclone
(CPC) around the polar cyclone (PC).} $r$ is the distance from the
pole (note that $r$ defines the positive meridional direction of
gradients). The green curve is the vertical component of the planetary
vorticity $f$. The blue curve is the profile around the PC of the
idealized axisymmetric tangential velocity in the vortex ($v_{{\rm PC}}$).
$\xi_{{\rm PC}}$ is the relative vorticity due to the presence of
the PC. $\Omega$ is the planetary rotation rate. The red arrows are
vorticity gradient forces on the CPC, induced by the PC and by planetary
sphericity.\label{fig: balance cartoon}}
\end{figure}
If the only vorticity gradient present in the background of a cyclone
is due to the planetary sphericity, it will tend to move poleward
until this gradient vanishes precisely at the pole. Assuming a PC
is already present at the pole (Fig.~\ref{fig: balance cartoon}),
the PC will induce a vorticity gradient of itself around the pole.
The velocity profiles observed for the PCs of Jupiter and Saturn resemble
a solid-disk rotation in an inner region, and an exponential decay
outside of it (Extended Data Figs.~1 and~2). Where velocity decays
exponentially, relative vorticity is negative due to the functional
relation between velocity and vorticity (Methods, Extended Data Fig.~3).
As the vorticity of the PC must vanish far away, an annulus of positive
relative vorticity gradient should exist around a PC due to its presence.
However, $f$ has its maximum magnitude at the poles and is $0$ at
the equator. Therefore, its gradient ($\beta\equiv\frac{\partial f}{\partial r}$)\textbf{
}is always negative away from the pole. The magnitude of $\beta$
is highest near the equator and vanishes at the poles\citep{vallis2017atmospheric}.
These trends mean that theoretically, there can be a latitude where
a poleward migrating CPC will be in equilibrium since the gradients
of vorticity due to the PC and due to the planetary sphericity are
equal and opposite\citep{li2020}. We propose this criterion of whether
a PC can generate a vorticity gradient that opposes $\beta$ as a
separating threshold between two polar states: one state in which
a circumpolar ring of vortices can be stably held, and one in which
the planetary gradient of vorticity is always greater, where any incoming
CPC will be merged into the PC.

Considering the meridional balance on a CPC, we define 

\begin{equation}
F_{\theta}\equiv\frac{\partial\xi_{{\rm PC}}}{\partial r}+\beta,\label{eq: Meridonal balance}
\end{equation}
which is proportional to the net meridional force on a CPC ($F_{{\rm PC}}-F_{\beta}$
in Fig.~\ref{fig: balance cartoon}). It is required that $F_{\theta}=0$
for a CPC to be in a meridional balance. Moreover, if $F_{\theta}$
is negative, the force is in the poleward direction; positive $F_{\theta}$
pushes cyclones equatorward.
\begin{figure*}[t]
\begin{centering}
\includegraphics[width=0.7\textwidth]{Gavriel_Fig4}
\par\end{centering}
\caption{\textbf{Latitudes of equilibrium in the Gas Giants. a}, Curves of
$F_{\theta}$ as a function of latitude. The red curves are calculated
for the north (solid) and south (dashed) poles of Jupiter (J-NP and
J-SP, respectively); the blue curves represent the north (solid, S-NP)
and south (dashed, S-SP) poles of Saturn. Only Jupiter has points
of equilibrium in which $F_{\theta}$ is zero. A stable balance for
Jupiter's poles is achieved at the equilibrium points farther from
the pole. \textbf{b}, The latitude of equilibrium as a function of
the ratio between the planetary radius ($a$) and the radius of the
PC ($R_{{\rm PC}})$, and of the ratio between the planetary rotation
rate ($\Omega$) and the rotation rate of the PC ($\omega_{{\rm PC}}$).
Only the stable solutions for $F_{\theta}=0$ are considered. The
white area on the left side of the contour is where equilibrium can
not be achieved. The values representing the curves in \textbf{a}
are shown as points. Both poles of Saturn are in the region with no
solution. \label{fig: results}}
\end{figure*}
In Fig.~\ref{fig: results}a, $F_{\theta}$ is plotted as a function
of latitude. The four curves are drawn according to relative vorticity
gradients calculated from idealized profiles of the PCs' tangential
velocity (Methods). These profiles are determined according to the
maximum velocities and the radii of maximum velocity evaluated for
the respective PCs on the north and south poles of Jupiter and Saturn\citep{grassi2018first,baines2009saturn}
(Methods). Temporal and local variations of the velocity fields are
not taken into account. The $\beta$ profiles are calculated according
to the respective planetary radii and rotation rates of Jupiter and
Saturn (Methods).

It is evident (Fig.~\ref{fig: results}a) that both poles of Saturn
cannot sustain a meridional equilibrium, and therefore do not have
CPCs. In contrast, each pole of Jupiter exhibits two equilibrium points.
However, the equilibrium point closer to the pole (in each of the
red curves) is unstable. This is because a perturbation in the latitude
of the CPC poleward from that point will further pull it to merge
with the PC due to the negative vorticity gradient poleward of that
point. A perturbation away from the pole will bring the vortex to
the farther point of equilibrium. That point is in stable equilibrium,
situated at latitude $\sim84^{\circ}$ for both poles. The circumpolar
ring observed at J-NP lies approximately along $83^{\circ}$N, while
at J-SP, it is roughly at $84^{\circ}$S\citep{tabataba2020long}.
This agreement between the calculated latitudes of equilibrium (Fig.~\ref{fig: results})
and observations and the lack of such equilibrium on Saturn, support
the suggested mechanism as the stabilizing balance that holds the
CPCs of Jupiter stable.

It can be seen (Fig.~\ref{fig: results}b) that the stable equilibrium
is achieved farther from the poles when the PC rotates faster (smaller
$\Omega/\omega_{{\rm PC}}$). This is because the vorticity gradient
of the PC is proportional to its rotation rate, so faster rotating
PCs can ``overcome'' the planetary vorticity gradient for greater
distances from their centers. Larger polar vortices (smaller $a/R_{{\rm PC}}$)
also result in farther latitudes of equilibrium. In this case, this
is due to their vorticity gradient profiles being stretched farther.
This, however, has a limitation. A PC too big, relative to the planet,
will be in a state where its region of positive vorticity gradient
is too far from the pole, where $\beta$ dominates. In these cases,
equilibrium cannot be reached, as is the case of the Saturnian poles.

\subsubsection*{Zonal stability of circumpolar cyclones}

\begin{figure*}[!t]
\begin{centering}
\includegraphics[width=0.7\textwidth]{Gavriel_Fig5}
\par\end{centering}
\caption{\textbf{Zonal stability of CPCs. a}, Schematic for the equilibrium
between CPC2 and the adjacent vortices in the zonal direction, titled
CPC1 and CPC3. $x$ is the distance from CPC1 in the zonal direction.
The radius of maximum velocity for a CPC is $R_{{\rm _{CPC}}}$. The
distance between two neighboring cyclones is $L$. $\xi_{{\rm CPC}}$
is the vorticity profile produced by the presence of CPCs 1 and 3.
The magenta and green curves are the vorticity gradients in the zonal
direction induced by CPC1 and CPC3, respectively. The rejection force
induced by each curve is illustrated as an arrow at the center of
CPC2. \textbf{b}, The second derivative of $\xi_{{\rm CPC}}$ in the
zonal direction, as a function of the distance $L$. This second derivative
is evaluated at a distance $L$ from CPC1.\textbf{ c},\textbf{d},
Plots representing an unstable and a stable case (respectively) of
the vorticity gradient profiles as a function of $x$.\textbf{ b}-\textbf{d},
For generalization of the plots, the hatted variables are non-dimensional
(Methods). \label{fig: zonal eq}}
\end{figure*}
Next, we investigate how many cyclones can fit in a circumpolar ring.
All cyclones in the circumpolar ring are of nearly similar size and
strength, with a comparable space between adjacent pairs\citep{tabataba2020long,adriani2020two,grassi2018first}.
Thus, in the following analysis, a ring of equally spaced identical
cyclones is considered. Then, it is postulated that the cyclones are
sustained by mutual rejection between their two neighboring vortices
in the zonal direction, according to the force derived by their respective
vorticity gradients (Fig.~\ref{fig: zonal eq}a). CPC2 is assumed
to be exactly in the middle between CPCs 1 and 3, such that the zonal
forces applied on it are equal and opposite. The remaining question,
however, is whether this equilibrium is stable. For understanding
the stability criterion for a ring of CPCs, it is insightful to consider
the second derivative in the zonal direction of the vorticity (only
the vorticity induced by CPCs 1 and 3) at the center of CPC2, as a
function of $L$ (Fig.~\ref{fig: zonal eq}b). When $L$ is too short,
a small perturbation in the location of CPC2 to the left encounters
a negative vorticity gradient (Fig.~\ref{fig: zonal eq}c), which
pulls the vortex further to the left. The opposite happens with a
perturbation to the right. This constitutes an unstable equilibrium
that will exist as long as the second derivative of the vorticity
is positive. When the second derivative of the vorticity is negative,
a stable equilibrium is formed (Fig.~\ref{fig: zonal eq}d). The
limiting distance, $L_{{\rm lim}}=4.54R_{{\rm CPC}}$ (Methods), in
which the second derivative vanishes, is the minimal distance between
CPCs that can maintain a stable equilibrium. This means that the vortex
centers have to be more than $L_{{\rm lim}}$ apart in order for the
CPCs to be in a sustainable configuration.

The available space for CPCs in the circumferential ring is approximately
$2\pi a(90^{\circ}-\left|\theta_{{\rm eq}}\right|)\pi/180^{\circ}$,
where $\theta_{{\rm eq}}$ is the latitude of equilibrium for the
respective pole (Fig.~\ref{fig: results}b). Therefore, the maximal
number of vortices to fit in the ring can be estimated according to
\begin{equation}
N_{{\rm max}}=\frac{2\pi a(90^{\circ}-\left|\theta_{{\rm eq}}\right|)\pi/180^{\circ}}{4.54R_{{\rm _{CPC}}}}.\label{eq: Nmax equation}
\end{equation}
Here, values for $R_{{\rm _{CPC}}}$ at the north and south poles
of Jupiter are evaluated by Juno's JIRAM imager\citep{adriani2020two}.
Higher values inside the variability range of $R_{{\rm _{CPC}}}$
were taken for the smallest constraint (Methods). Inserting the numbers
for the north and south poles to equation~(\ref{eq: Nmax equation})
results in $N\sim11.05$ and $\sim7.26$, respectively. However, as
$N$ should be an integer describing the maximal number of stable
CPCs, these numbers are rounded down to give

\begin{equation}
\begin{aligned}N_{{\rm max,N}}\sim & 11, & N_{{\rm max,S}}\sim7\end{aligned}
,
\end{equation}
where the subscripts N and S represent the north and south poles.

While this last analysis can be explained intuitively, to get a more
accurate constraint, a 2D analysis is performed that does not treat
CPC2 as a singular point but instead considers the different influence,
weighted by the meridional velocity of CPC2, around the spread of
the vortex. The force acting on CPC2 by the presence of CPCs 1 and
3 is proportional\citep{rossby1948displacements} to the integral
\begin{equation}
F\sim\iintop v_{{\rm 2}}(\xi_{{\rm 1}}+\xi_{3})dS\label{eq: Rossbey integral}
\end{equation}
around CPC2. Here, $v_{{\rm 2}}$ is the meridional velocity of CPC2,
and $\xi_{{\rm 1(3)}}$is the vorticity of CPC1(3). The results from
this analysis are qualitatively similar to those shown in Fig.~\ref{fig: zonal eq};
however, the limiting distance for stability ($L_{{\rm lim}}$) is
found to be $5.87R_{{\rm CPC}}$ instead (Methods). This constrains
$N_{{\rm max}}$ further as CPC2 is now influenced more by regions
closer to CPCs 1 and 3. Using this value instead of $4.54R_{{\rm _{CPC}}}$
in equation~(\ref{eq: Nmax equation}) results in $N\sim8.54$ and
$\sim5.62$ for the north and south poles, such that
\begin{equation}
\begin{aligned}N_{{\rm max,N}}\sim8 & , & N_{{\rm max,S}}\sim5\end{aligned}
.
\end{equation}
This constraint correctly predicts the actual number of vortices in
the north and south poles of Jupiter. It is interesting to note that
the estimation of $N_{{\rm max,S}}$ is in between $5$ and $6$,
and, consistently, two years observations of the south pole found
a constant gap in the south polar ring\citep{tabataba2020long}, implying
on space that is slightly larger than that required for the five CPCs.
This gap was temporarily occupied with a sixth vortex around the time
of Juno's 18th perijove\citep{adriani2020two}, but as the formation
was not stable, this additional vortex disappeared by the time of
the 19th perijove, indicating that the 5.62 value has a dynamical
meaning. These predictions further support the described mechanism
for the stability of the CPCs on Jupiter.

\subsubsection*{Polar cyclones on the Gas Giants \label{sec:Discussion}}

The analysis presented here is based on the assumption that large-scale
movement of vortices is mainly due to advection of background vorticity
with the tangential velocity of the vortex, and that this movement
is proportional, both in magnitude and in direction, to the background
vorticity gradient. This logic implies that if the background vorticity
gradient is zero at the center of a vortex, that vortex will not move.
The gradient of planetary vorticity that acts on a circumpolar cyclone
in the meridional direction can be opposed under certain conditions
by the gradient of vorticity induced by the polar cyclone (equation~(\ref{eq: Meridonal balance})).
Such an equilibrium is shown (Fig.~\ref{fig: results}b) to be favored
for small and strong PCs, relative to their host planets. In the zonal
direction, it is shown (Fig.~\ref{fig: zonal eq}) that stability
can be sustained for up to a certain amount of vortices in a circumpolar
ring. More vortices can fit in the ring for poles in which the meridional
stability is achieved at latitudes farther from the pole, and for
poles with smaller CPCs. These analyses only treat the assumed highest-order
forces that control the stability of circumpolar vortices, and suggest
that the governing dynamics are controlled by 2D (barotropic) PV conservation.
There are other forces, such as ones deriving from a 3D structure
of a cyclone, that may be responsible for the small changes in the
locations, speeds, and sizes observed in the circumpolar vortices
of Jupiter\citep{tabataba2020long}.

While this study explains the existence of circumpolar cyclones on
Jupiter in contrast to the absence of such on Saturn, it does so by
considering the specifics of their observed corresponding PCs. However,
it does not explain the variation between the PCs of the planets.
Theoretically, Saturn could sustain circumpolar vortices if its PCs
were smaller or were to spin faster. Since the stability criterion
for a circumpolar ring in the North Pole of Saturn is nearly reached,
it may be that minor future variations in the polar conditions of
Saturn (e.g., due to its seasonality) would manifest in a circumpolar
ring or that such a ring existed in the past. It is also possible
that cyclic variations in the solar forcing may alternate the number
of circumpolar vortices between the north and south poles of Jupiter.
Nonetheless, the match of the meridional and zonal force balances
to the observations provides strong evidence that the physical balances
outlined in this study are responsible for setting the location, stability,
and number of circumpolar cyclones on the Gas Giants.

\section*{Acknowledgments}

We thank Keren Duer and Eli Galanti for insightful conversations.
This research has been supported by the Minerva Foundation with funding
from the Federal German Ministry for Education and Research and the
Helen Kimmel Center for Planetary Science at the Weizmann Institute
of Science.

\section*{Author contributions}

N.G. has designed the study, performed the calculations and written
the paper with support of Y.K.

\section*{Additional information}

\paragraph*{Correspondence and requests for materials}

should be addressed to N.G. (nimrod.gavriel@weizmann.ac.il)

\bibliographystyle{naturemag}
