# Stability of Rotating, Charged Fluids: Generalization of the Høiland
Conditions
in Newtonian Non-conductive Case

Kris Schroven<EMAIL_ADDRESS>Vladimír Karas  Jiří Horák  Astronomical
Institute, Czech Academy of Sciences, Boční II 1401, 14100 Prague, Czech
Republic  Audrey Trova<EMAIL_ADDRESS>Eva Hackmann
University of Bremen, Center of Applied Space Technology and Microgravity
(ZARM), 28359 Bremen, Germany

(August 28, 2024)

###### Abstract

We study the conditions for stability of electrically charged, non-conductive
perfect fluid tori with respect to linear perturbations. To this end we employ
Lagrangian perturbation formalism and we assume a system where the fluid
orbits a central body. Gravitational field of the latter is described in the
Newtonian framework. We first formulate the criteria valid for a general, non-
axisymmetric situation, and then we concentrate on the axisymmetric model in
more detail. In the latter case we generalize the Høiland criterion of
stability to non-vanishing electric charge and classify special examples.
Toroidal structures with constant angular momentum distribution are found to
be linearly stable. Subsequently, like in the uncharged case, rotating charged
fluids are found to be unstable with respect to non-axisymmetric
perturbations.

##  I Introduction

Equilibrium configurations of a rotating fluid and the conditions for their
stability play an important role in astrophysics [1, 2]. Previously, we
discussed a special case of electrically charged, non-conductive,
energetically bound figures of fluid equilibria around black holes [3, 4, 5,
6, 7, 8, 9]. A possibility of (even small) non-vanishing electrical charge
greatly enlarges the parameter space of the system. The astrophysical
motivation to explore the equilibrium structures of electrically charged,
orbiting fluid arises from the fact that combined action of irradiation and
rapid rotation are expected to lead to electric currents and charge
separation. Unfortunately, no single approach is adequate to tackle in full
generality the complex medium that emerges in these circumstances (however,
see e.g. reference [10] for discussion). We adopt the fluid description in
terms of magnetohydrodynamical scenario, nonetheless, even in this framework
the introduction of electromagnetic terms requires additional simplifying
assumptions about their microphysical origin [11, 12]. A globally neutral
astrophysical system can develop separated regions with non-zero net charge of
the fluid which occur in vast range of systems from planetary neighborhood to
accreting black holes. Numerical integration of the equations of motion is
probably the only way to solve these structures under astrophysically
realistic circumstances, but it often fails to give full understanding. On the
other hand, analytical techniques are restricted to highly symmetrical
configurations. Here we adopt the latter approach and we confine ourselves to
axially and symmetrical, stationary perfect fluids.

A follow-up question arises: under which conditions the equilibrium solutions
are stable. In this paper we will adopt the traditional approach to the
stability analysis in the linear regime, which was applied previously to the
neutral fluid and ideal magnetohydrodynamic equations (see e.g. [1, 13, 14,
15, 16]). This resulted in the Høiland equation. Here, we use the method of a
linear Lagrangian perturbation theory to set up general stability equations.
We can show, that like in the uncharged case, rotating charged fluids are
unstable with respect to non-axisymmetric perturbations. For axisymmetric
perturbations we can formulate a corresponding generalized version of the
Høiland criterion relevant to our specific setup of a charged, non-conductive
fluid in an electromagnetic background field.

The Lagrangian perturbation approach will be introduced in section II. In
section III we will introduce the fluid equations for a non-conductive,
charged fluid immersed in an electromagnetic background field. The Lagrangian
approach is then used in section IV to set up the general linear stability
equations for the charged fluid and the instability of rotating, charged
fluids with regards to non-axisymmetric perturbations is discussed briefly.
The generalized Høiland conditions for a charged, non-conductive fluid are
derived in section V. These generalized conditions are afterwards discussed
for selected special cases of model parameters, like angular momentum
distributions and background fields, which were discussed in previous works
concerning the existence of bound, charged fluid structures. A conclusion is
given in section VII.

##  II Lagrangian perturbation approach

While Eulerian perturbation theory describes the change of fluid variables at
a fixed point in space, the Lagrangian approach relates fluid variables in
equilibrium to the perturbed ones by a Lagrangian displacement vector field.
This results in a description of the change of the fluid variables with
respect to a frame dragged along with the displacement [13]. The Lagrangian
formalism was introduced by Lebovitz [17] and later on widely used to discuss
the stability of (magnetohydrodynamic) fluids in a Newtonian and general
relativistic setup (e.g. [18, 13, 19, 14, 16, 20]).

We denote the Eulerian physical coordinate by $x$, and the Lagrangian initial
coordinate by $r$. In Lagrangian perturbation theory, the dynamical variable
is the Lagrangian displacement field $\xi(r,t)$, arising from deviations of a
fluid line from its original course due to small perturbations of the fluid

$\displaystyle x(t):=r+\xi(r,t).$ (1)

It is worth mentioning that $\xi(r,t)=0$ leads to $r$ be the same as the usual
comoving coordinate at initial time $t=0$.

The Lagrangian perturbation approach describes the perturbation of a fluid
element with respect to a „Lagrangian frame“– a frame that is embedded in the
fluid and dragged along the perturbation direction. The Lagrangian
perturbation $\Delta Q$ of an arbitrary tensor quantity $Q$ is then related to
the Eulerian perturbation $\delta Q$ by [13]:

${\Delta Q=\delta Q+\mathcal{L}_{\xi}Q\,},$ (2)

where the Lie derivative $\mathcal{L}_{\xi}$ is given by

$\mathcal{L}_{\xi}f=\xi^{j}\nabla_{j}f,$ (3)

for scalar fields $f$ and by

$\displaystyle\mathcal{L}_{\xi}u^{i}=\xi^{j}\nabla_{j}u^{i}-u^{j}\nabla_{j}\xi^{i},\,$
(4)
$\displaystyle\mathcal{L}_{\xi}u_{i}=\xi^{j}\nabla_{j}u_{i}+u_{j}\nabla_{i}\xi^{j}.$
(5)

for contra- and co-variant vector fields $u^{i}$, $u_{i}$. Here, the Einstein
summation notation is used.

In order to express the conservation of mass and equation of state in terms of
Lagrangian displacement following [13] we consider

$\displaystyle\Delta u^{i}=\partial_{t}\xi^{i},$ (6)
$\displaystyle\Delta\rho=-\rho\nabla_{i}\xi^{i},\,$ (7)

where $u^{i}$ is the fluid velocity and $\rho$ is the density. In addition, by
utilizing the adiabatic assumption, as well as a polytropic equation of state,
one finds that

$\displaystyle\Delta p$ $\displaystyle=\frac{\gamma p}{\rho}\Delta\rho$ and
$\displaystyle\Delta S=0\;.$ (8)

Here $p$ is the pressure and $\gamma=(\partial\ln p/\partial\ln\rho)_{S}$ is
the adiabatic index, while $S$ is the entropy of the fluid.

###  II.1 Canonical energy and canonical displacements

An important fundamental property of the Lagrangian perturbation approach is
gauge freedom related to the displacement vector field $\xi$. In fact, there
is a class of trivial displacements $\eta$, for which the corresponding
Eulerian change in pressure, density, velocity and entropy of fluid vanishes.
As a result, two canonical displacements $\xi^{i}$ and $\hat{\xi}^{i}$
correspond to the same physical perturbation if they differ only by a trivial
displacement:

$\displaystyle\xi^{i}=\hat{\xi}^{i}+\eta^{i}\;.$ (9)

However, there is a subtlety here. Even though $\xi$ and $\hat{\xi}$ describe
identical physical perturbations, the canonical energy $E_{c}$ is not
preserved

$\displaystyle E_{c}(\xi^{i})\neq E_{c}(\hat{\xi}^{i}+\eta^{i}).$ (10)

The canonical energy is defined in terms of the symplectic structure
$W(\xi,\partial_{t}\xi)$ of a Lagrangian displacement and its time derivative
[13]

$\displaystyle
E_{c}:=\frac{1}{2}W(\xi,\partial_{t}\xi)=\frac{1}{2}H(\xi,\xi)\,,$ (11)

The canonical energy plays a key role in the formulation of the stability
criterion for a fluid configuration in the following sense [21]:

The configuration is

  1. 1.

dynamically stable, if $\quad\,E_{c}(\xi)>0$,

  2. 2.

dynamically unstable, if $E_{c}(\xi)=0$,

  3. 3.

secularly unstable, if $\quad\;\;E_{c}(\xi)<0$

for all initial conditions (in the last case, in particular, for non-
axisymmetric conditions).

This classification has a straightforward interpretation: if the canonical
energy is positive for all possible configurations, the perturbation can only
die away over time, while $E_{c}$ is decreasing. In the case that the
canonical energy becomes negative for any possible configuration, the
perturbations does not die away, as $E_{c}$ can only decrease and the system
is dynamically unstable.

We restrict the Lagrangian displacements to a subclass of so-called canonical
displacements, which are orthogonal to all trivial displacements with respect
to the symplectic product,

$\displaystyle W(\xi,\eta)=0,\quad\forall\eta,$ (12)

where $\eta$ is the trivial displacement [13]. Therefore, for this subclass of
canonical displacements the canonical energy is invariant.

##  III A charged, non-conductive fluid in the presence of an electromagnetic
field

A charged, non-conductive fluid in the vicinity of an electromagnetic
background field is described by the following set of equations; the Euler
equation of a charged, non-conductive fluid and the continuity equations for
mass and charge, respectively

$\displaystyle\frac{Du_{a}}{Dt}+\frac{\nabla_{a}p}{\rho}+\nabla_{a}\Phi_{G}-\frac{q}{\rho}\left(F_{ab}u^{b}-\nabla_{a}\Phi_{E}\right)=0\;,$
(13) $\displaystyle\partial_{t}{\rho}+\nabla_{b}(\rho u^{b})=0\;,$ (14)
$\displaystyle\partial_{t}{q}+\nabla_{b}(qu^{b})=0\;.$ (15)

Here, the external gravitational and electrostatic field are described by
their potentials $\Phi_{\mathrm{G}}$ and $\Phi_{\mathrm{E}}$, respectively.
Magnetic field is represented by the antisymmetric Maxwell tensor $F_{ab}$
($F_{ab}=\nabla_{a}B_{b}-\nabla_{b}B_{a}$) and $\rho$ and $q$ are mass and
charge densities. Each term in Eq. (13) represents different force, that
arises due to internal pressure gradients, gravity of the central compact
body, and the electromagnetic forces acting on the charged fluid medium,
respectively [22, 4]. Finally, we introduced the total time derivative

$\displaystyle\frac{D}{Dt}:=\partial_{t}+u^{j}\nabla_{j}$ (16)

in Eq. (13). In the present discussion we neglect the fluid self-gravity and
the electric current self-interaction. For simplicity reasons we introduce the
specific charge

$\displaystyle\frac{q}{\rho}:=\hat{q}\;.$ (17)

From equations (14) and (15) we find that

$\displaystyle\frac{D}{Dt}\hat{q}=0\,,$ (18)

which basically describes that the specific charge is preserved along the
fluid course (i.e. the specific charge is a material invariant).

Within the setup of our charged, non-conductive fluid we will use the
Lagrangian perturbation approach, described above. Next to the expressions for
the Lagrangian perturbation of the fluid quantities pressure, mass density,
entropy and fluid velocity, already introduced in [13] (see Eqs. (6)–(8)), we
receive for the Lagrangian perturbation of the specific charge

$\displaystyle\Delta\hat{q}$ $\displaystyle=0\,.$ (19)

Additionally, we derive the orthogonality condition between the canonical and
trivial displacements in our charged setup, by closely following the procedure
used in [13]:

$\varepsilon^{ijk}\nabla_{i}\hat{q}\nabla_{j}\left(\Delta_{\hat{\xi}}u_{k}-\hat{q}F_{kb}{\hat{\xi}}^{b}\right)=0\;,$
(20)

where $\Delta_{\hat{\xi}}$ is the Lagrangian perturbation related to the
displacement $\hat{\xi}$. Our stability discussion will be restricted to the
subclass of canonical displacements $\hat{\xi}^{i}$ that satisfy Eq. (20).

##  IV Stability discussion with the help of the canonical Energy

It is our aim in this section to first derive the canonical energy of the
perturbed charged fluid system, and afterwards use it for further stability
discussions, by following the approach used in [13]. We are only interested in
perturbing the fluid and not the external gravitational or electromagnetic
field. Therefore, we assume that the Eulerian perturbations of these external
fields vanish, $\delta
F_{ab}=\delta\nabla_{a}\Phi_{G}=\delta\nabla_{a}\Phi_{E}=0$. Furthermore, we
assume, that all discussed systems have a vanishing charge and mass density at
infinity. With these assumptions, the Lagrangian perturbation of the Euler
equation (13) is given by

$\displaystyle\begin{split}\Delta_{\mathrm{E}}=&\partial^{2}_{t}\xi_{a}+\left(2g_{ab}u^{c}\nabla_{c}-\hat{q}F_{ab}\right)\partial_{t}\xi^{b}+\left(u^{b}\nabla_{b}\right)^{2}\xi_{a}\\\
&+\frac{1}{\rho}\left(\nabla_{b}\xi^{b}\nabla_{a}p-\nabla_{a}\xi^{b}\nabla_{b}p-\nabla_{a}(\gamma
p\nabla_{b}\xi^{b})\right)\\\
&+\xi^{b}\nabla_{b}\nabla_{a}\Phi_{G}+\hat{q}\xi^{b}\nabla_{b}\nabla_{a}\Phi_{E}\\\
&-\hat{q}\left(F_{ab}u^{\gamma}\nabla_{\gamma}\xi^{b}+\xi^{b}\nabla_{b}F_{a\gamma}u^{\gamma}\right)=0.\
\end{split}$ (21)

Without electric and magnetic fields, this result agrees with the previous
works. However even with them, the equation is of the form

$\displaystyle
A_{ab}\partial^{2}_{t}\xi^{b}+B_{ab}\partial_{t}\xi^{b}+C_{ab}\xi^{b}=0\,.$
(22)

If $A_{ab}$, $B_{ab}$ and $C_{ab}$ are hermitian, anti-hermitian and
hermitian, respectively (see Appendix A for proof), Eq. (21) can be derived
from a variational principle with the action potential

$\begin{split}I:=&\int\mathscr{L}\,dV\\\
=&\frac{1}{2}\int\dot{\xi}^{a}A_{ab}\,\dot{\xi}^{b}+\dot{\xi}^{a}\,B_{ab}\,\xi^{b}-\xi^{a}\,C_{ab}\,\xi^{b}\,dV\,.\end{split}$
(23)

We can now calculate the canonical energy $E_{c}$ of the perturbed system,
using the Lagrangian $\mathscr{L}$. From (11) we obtain

$\displaystyle E_{c}=\frac{1}{2}H(\xi_{a},\xi^{a})$
$\displaystyle=\int\dot{\xi}^{a}\left(\frac{\partial\mathscr{L}}{\partial\dot{\xi}^{a}}-\mathscr{L}\right)\;dV$
(24)

By using Eq. (21), the expression for the canonical energy in our set-up is
explicitly given by

$\displaystyle\begin{split}E_{c}=\frac{1}{2}\int&\left\\{\rho\,\lvert\dot{\xi}\rvert^{2}-\rho\lvert
u^{\gamma}\nabla_{\gamma}\xi\rvert^{2}+\gamma
p\lvert\nabla_{a}\xi^{a}\rvert^{2}+{\xi^{a}}^{\ast}\nabla_{a}p\nabla_{b}\xi^{b}+\xi^{b}\nabla_{b}p\nabla_{a}{\xi^{a}}^{\ast}+{\xi^{a}}^{\ast}\xi^{b}\rho\left[\nabla_{b}\nabla_{a}p+\nabla_{b}\nabla_{a}\Phi_{G}\right]\right.\\\
&\left.+{\xi^{a}}^{\ast}\xi^{b}\rho\left[\hat{q}\nabla_{b}\nabla_{a}\Phi_{E}-\frac{\hat{q}}{2}\left(\nabla_{a}F_{b\gamma}+\nabla_{b}F_{a\gamma}\right)u^{\gamma}\right]-\rho\frac{\hat{q}}{2}\left({\xi^{a}}^{\ast}F_{ab}u^{\gamma}\nabla_{\gamma}\xi^{b}+\xi^{a}F_{ab}u^{\gamma}\nabla_{\gamma}{\xi^{b}}^{\ast}\right)\right\\}dV\;,\end{split}$
(25)

where we used $\nabla_{a}(\rho u^{a})=0$ and assumed that the mass density,
charge density and pressure of the fluid vanishes at the surface of the
considered system.

Equation (25) could be used to check the stability of the normal modes of the
system, where the normal mode under discussion is represented by a specific
$\xi^{a}$.

###  IV.1 Instability with regards to non-axisymmetric perturbations

The classical rotating, isentropic or nonisentropic, uncharged fluid is
unstable with regards to non-axisymmetric perturbations [21]. Perturbation
modes of negative energy transport angular momentum and energy to infinity,
while the rotating fluid is slowing down. The energy transfer is inspired by
variation of various multipoles caused by the perturbations. This instability,
however, is irrelevant for slowly rotating fluids [21].

The discussion by Friedman and Schutz [21] can be extended to our case of a
charged rotating fluid, leading to the same instability. To check this, we use
the following ansatz for the non-axisymmetric perturbation of the fluid in
cylindrical coordinates:

$\displaystyle\xi^{a}=$ $\displaystyle\zeta^{a}(r,z)\mathrm{e}^{im\phi},$
$\displaystyle\dot{\xi}^{a}=$
$\displaystyle\dot{\zeta}^{a}(r,z)\mathrm{e}^{im\phi}\;.$ (26)

The “amplitudes” $\zeta^{a}(r,z)$ and $\dot{\zeta}^{a}(r,z)$ can also depend
on $m$ but should be bounded everywhere, independent of $m$. They are chosen
such that $\zeta^{\phi}$ vanishes and condition (20) is fulfilled

$\displaystyle\zeta_{a}$ $\displaystyle=f\nabla_{a}\hat{q}\,,$
$\displaystyle\dot{\zeta}_{a}$
$\displaystyle=\frac{2}{im}\partial_{a}\left(r\Omega\zeta_{r}-F_{b\phi}\zeta^{b}\right)\
,$ (27)

where $f=f(r,z)$ has to be a smooth function, which is constant along the
fluid trajectories $(\partial_{t}+\mathcal{L}_{u})f=0$. Accordingly, the
divergence of $\xi^{a}$ becomes

$\displaystyle\nabla_{a}\xi^{a}=\nabla_{a}\zeta^{a}(r,z)\mathrm{e}^{im\phi}\;.$
(28)

Plugging the ansatz into Eq. (25) leads to

$\displaystyle\begin{split}E_{c}=K&-\int\rho\hat{q}\Omega\;\zeta^{a}\zeta^{b}\left(\nabla_{a}\nabla_{b}A_{\phi}-\nabla_{a}\nabla_{b}\Phi_{E}\right)\,dV\\\
&-m^{2}\int{\rho\Omega^{2}\lvert\zeta\rvert^{2}\,dV}\,.\end{split}$ (29)

Here, $K$ is a term that includes all terms from perturbing the uncharged
fluid with the given ansatz, which are independent from $m$. According to the
named assumptions, both $K$ and the first integral in Eq. (29), which contains
the electromagnetic effects and is independent of $m$ as well, are bounded
from above. Therefore $E_{c}$ will eventually become negative for sufficiently
high frequencies in the non-axisymmetric perturbation.

As one can see from Eq. (29), the electromagnetic interactions of the fluid do
not contribute any term that would depend on the perturbation frequency $m$.
Following the discussion in Friedman and Schutz [21], we assume that a set of
non-axisymmetric perturbations exist that do not die away in time, therefore
leading to instability.

##  V Axisymmetric perturbations: The generalization of the Solberg-Høiland
conditions

We saw that rotating, charged, non-conductive fluid structures will slow down
over time under non-axisymmetric perturbations. We will now turn our focus on
the special case of axisymmetric perturbations, that preserve the angular
momentum of the system. In the non-charged fluid case, this stability
discussion leads to the Solberg-Høiland conditions or otherwise the Rayleigh
criterion in the special case of fluids with an overall constant entropy $S$.

Under certain conditions for the perturbations in the fluid, the first order
variation $\delta^{1}E_{\mathrm{tot}}$ of the total energy of a charged fluid
vanishes (the solution to the Euler equation (13) has an extremal total energy
only with respect to this class of perturbations), and the canonical energy
coincides with the second order variation $\delta^{2}E_{\mathrm{tot}}$ of the
total energy of the fluid. Since the variation $\bm{\delta}E_{\mathrm{tot}}$
of the total energy is a physical property of our configuration, $E_{c}$ will
loose its gauge dependence under this conditions, and a discussion of
canonical and trivial displacements is not necessary anymore. Perturbations of
this class conserve the fluid mass and charge, leave the system adiabatic and
further conserve the redefined fluid’s angular momentum
($\Delta\hat{u}_{\phi}=\Delta(u_{\phi}+\hat{q}A_{\phi})=0$; see appendix B). A
conservation of mass and charge is automatically satisfied under any
perturbations, if the continuity equation for mass and charge (see Eqs.
(14)–(15)) are valid.

In this chapter we will focus the discussion on adiabatic, axisymmetric
perturbations that conserve the redefined angular momentum $\hat{u}_{\phi}$ of
the fluid. In this case $E_{c}$ coincides with the second variation of the
total energy of the system and we can follow the approach used by Tassoul [1].

###  V.1 Case of axial symmetry

Axissymmetry (independence of the angular coordinate $\phi$ corresponding to
that symmetry) leads to a conservation constant. For an uncharged fluid, the
conserved quantity is the angular momentum of the fluid $L=u_{\phi}$. In our
case of a charged, non-conductive fluid, affected by an electromagnetic
background field, the conserved quantity is a redefined angular momentum
$\hat{L}=\hat{u}_{\phi}$. It can be directly derived (in e.g. cylindrical or
spherical coordinates) from the $\phi$-component of the charged Euler equation
(13), while using Eq. (18):

$\displaystyle\frac{D}{Dt}u_{\phi}-\hat{q}F_{\phi b}u^{b}=0\,,$ (30)
$\displaystyle\Rightarrow\;$
$\displaystyle\frac{D}{Dt}\left(u_{\phi}+\hat{q}A_{\phi}\right)=\frac{D}{Dt}\hat{u}_{\phi}=0\,,$
(31)

where we assumed, that the electromagnetic background field is stationary.

Furthermore, the magnetic vector potential can be restricted to its
$\phi$-component in our axisymmetric case ( $A_{a}=A^{\varphi}\eta_{a}$, while
$\eta^{a}=\hat{e}_{\varphi}^{a}$ corresponds to the Killing vector for axial
symmetry).

In order to use the conservation of $\hat{u}_{\phi}$ in the following
stability analysis, we will rewrite the charged Euler equation (13) in terms
of

$\hat{u}_{\mu}=u_{\mu}+\hat{q}A_{\mu}\,.$ (32)

It becomes:

$\displaystyle\begin{split}\left(\hat{u}^{b}-\hat{q}A^{b}\right)\nabla_{b}\left(\hat{u}_{\mu}-\hat{q}A_{\mu}\right)-\hat{q}F_{\mu
b}\left(\hat{u}^{b}-\hat{q}A^{b}\right)&\\\
+\underbrace{\frac{\nabla_{\mu}p}{\rho}+\nabla_{\mu}\Phi_{G}+\hat{q}\nabla_{\mu}\Phi_{E}}_{T_{2}}&=0\,,\end{split}$
(33)
$\displaystyle\begin{split}\hat{u}^{b}\nabla_{b}\hat{u}_{\mu}-\hat{q}\left[A^{b}\nabla_{b}\hat{u}_{\mu}+\hat{u}^{b}\nabla_{b}A_{\mu}\right]+\hat{q}^{2}A^{b}\nabla_{b}A_{\mu}&\\\
-\hat{q}\left[\nabla_{\mu}A_{b}-\nabla_{b}A_{\mu}\right](\hat{u}^{b}-\hat{q}A^{b})+T_{2}&=0\,,\end{split}$
(34)
$\displaystyle\hat{u}^{b}\nabla_{b}\hat{u}_{\mu}-\hat{q}\left[A^{b}\nabla_{b}\hat{u}_{\mu}+\hat{u}^{b}\nabla_{\mu}A_{b}\right]+\hat{q}^{2}A^{b}\nabla_{\mu}A_{b}+T_{2}$
$\displaystyle=0\,.$ (35)

In the case of circular fluid motion and a magnetic vector field restricted to
its $\phi$-component, the second term in the equation above reduced to:

$\displaystyle\begin{split}&\left[A^{b}\nabla_{b}\hat{u}_{\mu}+\hat{u}^{b}\nabla_{\mu}A_{b}\right]\\\
&=g_{\phi\phi}\hat{u}^{\phi}{A^{\phi}}_{,\mu}+{g_{\phi\phi}}_{,\mu}\hat{u}^{\phi}A^{\phi}-2\cdot\left(\frac{1}{2}{g_{\phi\phi}}_{,\mu}\hat{u}^{\phi}A^{\phi}\right),\end{split}$
(36) $\displaystyle=g_{\phi\phi}\hat{u}^{\phi}{A^{\phi}}_{,\mu}.$ (37)

Finally, we end up with the following new version of the Euler equation (13)
in terms of $\hat{u}^{b}$:

$\displaystyle\begin{split}\hat{u}^{b}\nabla_{b}\hat{u}_{\mu}-\hat{q}\left(\hat{u}_{\phi}{A^{\phi}}_{,\mu}-\nabla_{\mu}\Phi_{E}\right)&\\\
+\frac{1}{2}\hat{q}^{2}(\left|A_{\phi}\right|^{2})_{,\mu}+\frac{\nabla_{\mu}p}{\rho}+\nabla_{\mu}\Phi_{G}&=0\,.\end{split}$
(38)

###  V.2 Rayleigh criterion for the charged, non-conductive fluid

We are only interested in the perturbation of the fluid, which is why we do
not consider any perturbation of the external gravitational or electromagnetic
field. Therefore, $\delta
F_{ab}=\delta\nabla_{a}\Phi_{G}=\delta\nabla_{a}\Phi_{E}=0$. A linear
perturbation of our new expression of the Euler equation (38) leads in the
case of an initially unperturbed fluid in circular motion to:

$\displaystyle E_{c}=$
$\displaystyle\delta^{2}E=\Delta_{\mathrm{E}}=\int\xi^{a}M_{ab}\xi^{b}\,dV\hskip
305.86711pt$
$\displaystyle\begin{split}=&\int\xi^{a}\left[\frac{\nabla_{b}\hat{L}^{2}}{2g_{\varphi\varphi}^{2}}\partial_{a}g_{\varphi\varphi}+\frac{\nabla_{a}p}{\rho}\left(\frac{\nabla_{b}\rho}{\rho}-\frac{\nabla_{b}p}{\gamma
p}\right)-\nabla_{b}\hat{q}\nabla_{a}\Phi_{E}-\frac{1}{2}\nabla_{b}\hat{q}^{2}(\left|\textbf{A}\right|^{2})_{,a}+\nabla_{b}(\hat{q}\hat{L}){A^{\phi}}_{,a}\right]\xi^{b}\rho
dV\\\ &+\underset{\approx 0}{\underbrace{\int\frac{(\delta p)^{2}}{\gamma
p}dV}}+\underset{=0}{\underbrace{\int_{\partial V}\xi^{a}\delta
p\,d\hat{n}_{a}}}\,.\end{split}$ (39)

where we assumed axisymmetric perturbations, which preserve the redefined
angular momentum: $\Delta\hat{L}=0$. Furthermore, we neglect the integral over
$(\delta p)^{2}$. This approximation is valid, if the characteristic time of
the disturbance exceeds the travel time of a sound wave in the perturbed area.
This approximation basically neglects any possible p–modes, that could arise
due to the perturbation [1].

In Eq. (39), the canonical energy is now expressed in form of a perturbation
matrix $M_{ab}$. As long as $M_{ab}$ is positive definite, $E_{c}$ will be
positive for all $\xi^{a}$, and the system can be considered stable. The two
new stability conditions are therefore given by the conditions for a matrix
being positive definite. According to Tassoul [1], these conditions are:

$\displaystyle\operatorname{Tr}(M_{ab})=M^{a}_{a}\geq 0,$
$\displaystyle\det\left(M_{ab}\right)$ $\displaystyle\geq 0\,.$ (40)

The pressure and density of a perfect fluid are connected to the entropy $S$
as
$\alpha\nabla_{a}S=-\left(\frac{\nabla_{a}\rho}{\rho}-\frac{\nabla_{a}p}{\gamma
p}\right)$ (see e.g. [1]). By using the found expression for $M_{ab}$ in (39),
the two stability conditions take the form:

$\displaystyle\begin{split}&\operatorname{Tr}\left(M_{ab}\right)\\\
&\quad=g^{ab}\left[\left(\hat{q}{A^{\phi}}_{,a}-{\left(\frac{1}{g_{\varphi\varphi}}\right)}_{,a}\hat{L}\right)\nabla_{b}\hat{L}-\alpha\frac{\nabla_{a}p}{\rho}\nabla_{b}S\right.\\\
&\left.\quad+\left(E_{a}-\hat{q}(\left|\textbf{A}\right|^{2})_{,a}+\hat{L}{A^{\phi}}_{,a}\right)\nabla_{b}\hat{q}\right]\geq
0\,,\end{split}$ (41) $\displaystyle\begin{split}&\det\left(M_{ab}\right)\\\
&\quad=\varepsilon^{ab}k^{1}_{a}k^{2}_{b}\;\varepsilon^{\alpha\beta}\ell^{1}_{\alpha}\ell_{\beta}^{2}+\varepsilon^{ab}k^{1}_{a}k^{3}_{b}\;\varepsilon^{\alpha\beta}\ell^{1}_{\alpha}\ell_{\beta}^{3}\\\
&\quad+\varepsilon^{ab}k^{2}_{a}k^{3}_{b}\;\varepsilon^{\alpha\beta}\ell^{2}_{\alpha}\ell_{\beta}^{3}\geq
0\,,\end{split}$ (42)

with

$\displaystyle k^{1}_{a}=$
$\displaystyle\hat{q}{A^{\phi}}_{,a}-{\left(\frac{1}{g_{\varphi\varphi}}\right)}_{,a}\hat{L}\,,$
$\displaystyle\ell^{1}_{b}=$ $\displaystyle\nabla_{b}\hat{L}\,,$ (43)
$\displaystyle k^{2}_{a}=$ $\displaystyle\nabla_{a}p\,,$
$\displaystyle\ell^{2}_{b}=$ $\displaystyle-\alpha\nabla_{b}S\,,$ (44)
$\displaystyle k^{3}_{a}=$ $\displaystyle
E_{a}-\hat{q}(\left|\textbf{A}\right|^{2})_{,a}+\hat{L}{A^{\phi}}_{,a}\,,$
$\displaystyle\ell^{3}_{b}=$ $\displaystyle\nabla_{b}\hat{q}\,.$ (45)

In the case of a vanishing electromagnetic field, the stability conditions
(41) and (42) reduce to the Solberg-Høiland conditions. If aditionally the
entropy is constant, the stability conditions further reduce to the Rayleigh
criterion:

$\bm{\nabla}(\hat{L}^{2})\geq 0\,.$ (46)

###  V.3 Specific cases

In this section we will apply the new found stability conditions for a non-
conductive, charged fluid in an electromagnetic background field to several
special cases with a given electromagnetic field and angular momentum
distribution. We assume a barotropic equation of state for all cases, which
leads to a constant entropy distribution throughout the fluid structure. All
discussed special cases that follow were examined in previous works concerning
the existence of bound charged fluid structures, either in a classical or
general relativistic setup. This discussion below is meant to evaluate, if the
found bound structures are stable concerning axisymmetric perturbations [3, 4,
11, 12, 5, 6, 7]. An exemplary depiction of such bound fluid structures is
presented in Fig. 1.

Figure 1: Exemplary depiction of a gravitationally bound, rotating fluid
structure (blue) immersed in a background magnetic field (grey arrows aligned
with the vertical direction). The central body (black circle) can be
electrically charged, which adds an electric field to the system (not depicted
here). The fluid circulates in the azimuthal direction around the central body
with a given velocity distribution.

#### Case I: Constant angular momentum, homogeneous magnetic field.

In this case, the cylindrical coordinate system is used. In this coordinate
system, the vector potential for a homogeneous magnetic field is given by:

$\displaystyle A_{\phi}$ $\displaystyle=\frac{1}{2}B_{z}r^{2},$ and
$\displaystyle A^{\phi}$ $\displaystyle=\frac{1}{2}B_{z}\,.$ (47)

We consider a magnetic field only, therefore setting $\Phi_{E}=0$. The two
conditions for stable structures in case of a constant angular momentum
distribution and under axisymmetric perturbations reduce in this set up to:

$\displaystyle\operatorname{Tr}(M_{ab})$
$\displaystyle=(\hat{q}B_{z})^{2}+\frac{L}{r^{3}}B_{z}\partial_{r}(\hat{q}r^{2})\;,$
(48) $\displaystyle\det(M_{ab})$ $\displaystyle=0\;.$ (49)

According to this result, a charged fluid structure can be considered stable
to axisymmetric perturbations, if the specific charge distribution – or more
precisely $\hat{q}r^{2}$ – increases outwards in case of a repulsive Lorentz
force or decreases outwards in case of an attractive Lorentz force.

This is the direct generalization of the classical uncharged Polish doughnut
set up with a constant angular momentum distribution. Since uncharged ideal
fluid structures are considered stable, if the specific angular momentum in
increasing outwards, a small fluid charge can stabilize or destabilize the
structure.

#### Case II: Constant angular momentum, charged central object.

In this case, the spherical coordinate system is used. In this coordinate
system, the electric potential arising from the charge of the central object
is given by:

$E_{a}=\left(\frac{Q}{r^{2}},0,0\right)\;.$ (50)

We consider no magnetic field, therefore setting $A_{a}=0$. The two conditions
for stable structures for a constant angular momentum distribution and under
axisymmetric perturbations reduce in this set up to:

$\displaystyle\operatorname{Tr}(M_{ab})$
$\displaystyle=\partial_{r}{\hat{q}}E_{r}\;,$ (51) $\displaystyle\det(M_{ab})$
$\displaystyle=0\;.$ (52)

According to this result, a charged fluid structure can be considered stable
to axisymmetric perturbations, if the charge increases (same charge as the BH)
or decreases (opposite charge of the BH) outwards. This means, that in case of
a repulsive electric force on the fluid, the repulsive force should fall off
slower than the attractive gravitational force and vice versa for the case of
an attractive electric force on the fluid.

#### Case III: Constant angular momentum, charged central object immersed in a
homogeneous magnetic field.

Considered is a homogeneous magnetic field and a central object with charge.
The two conditions for stable structures for a constant angular momentum
distribution and under axisymmetric perturbations reduce to:

$\displaystyle\operatorname{Tr}(M_{ab})=$
$\displaystyle\partial_{a}\hat{q}E^{a}+(\hat{q}B_{z})^{2}+LA^{\phi}g^{ab}\frac{\partial_{a}g_{\phi\phi}}{g_{\phi\phi}^{2}}\partial_{b}(g_{\phi\phi}\hat{q})\;,$
(53)
$\displaystyle\begin{split}\det(M_{ab})=&\hat{q}A^{\phi}\frac{\hat{L}}{g_{\phi\phi}^{2}}\;\left(g^{ab}\partial_{a}g_{\phi\phi}\partial_{b}g_{\phi\phi}E^{c}\partial_{c}\hat{q}\right.\\\
&\left.-\partial_{a}g_{\phi\phi}E^{a}g^{cd}\partial_{c}g_{\phi\phi}\partial_{d}\hat{q}\right)\;.\end{split}$
(54)

In cylindrical coordinates, this leads to:

$\displaystyle\operatorname{Tr}(M_{ab})$
$\displaystyle=\partial_{a}\hat{q}E^{a}+(\hat{q}B_{z})^{2}+\frac{L}{r^{3}}B_{z}\partial_{r}(\hat{q}r^{2})\;,$
(55) $\displaystyle\det(M_{ab})$
$\displaystyle=\hat{q}B_{z}\;E^{z}\partial_{z}\hat{q}\,\frac{2L}{r^{2}}+(\hat{q}B_{z})^{2}\;E^{z}\partial_{z}\hat{q}\;.$
(56)

Generally one can deduce form this, that the fluid structure is more likely to
be stable, if only repulsive electromagnetic forces occur ($\hat{q}E^{a}>0$
and $\hat{q}B_{z}L>0$). At the same time the charge density of the fluid
structure should increase outwards.

Figure 2: The figures of equilibrium and the corresponding distribution of
pressure surfaces in charged slender tori immersed in the dipole magnetic
field corresponding to two Families lying in the equatorial plane (I and II);
their tentative existence was originally discussed by Slaný et. al [4]. Top
left panel: These tori correspond to ‘Family I’ solutions with the specific
angular momentum distribution
$\ell=\ell_{\mathrm{c}}[(r/r_{\mathrm{c}})\sin\theta]^{K_{1}}$. The horizontal
axis shows the parameter $K_{1}$; the parameter
$(\ell_{\mathrm{c}}/\ell_{\mathrm{K,c}})^{2}$ shown on the vertical axis
measures how much the rotation of the torus deviates from the Keplerian value
at the torus center due to Lorentz force; the value
$(\ell_{\mathrm{c}}/\ell_{\mathrm{K,c}})^{2}$ corresponds to uncharged tori,
values greater/smaller than one correspond respectively to
positively/negatively charged tori. The dashed areas denote regions in the
parameter space, where the criteria for stability are not satisfied. The graph
demonstrates that stable configurations do exist, however they are
considerably restricted by the stability criterion and limited to a small
region of the parameter space. Bottom left panel: Example of the equi-pressure
surfaces of negatively charged tori of finite size, whose parameters
correspond to the cross shown in the top panel. Even though the center of the
torus is stable, our criteria limit significantly possible sizes of the tori.
Out of the three configurations shown by the thick black solid line, only the
innermost one corresponds to an entirely stable torus. The corresponding
panels in the right column have been constructed in the way analogous to the
left column but for the case of Family II (off-equatorial tori). In the latter
case, we do not find any closed equi-pressure surface that would enclose a
region of stability.

#### Case IV: Constant angular velocity, homogeneous magnetic field.

In this case, the cylindrical coordinate system is used. In this coordinate
system, the vector potential for a homogeneous magnetic field is given by:

$\displaystyle A_{\phi}$ $\displaystyle=\frac{1}{2}B_{z}r^{2},$ and
$\displaystyle A^{\phi}$ $\displaystyle=\frac{1}{2}B_{z}\,.$ (57)

We consider a magnetic field only, therefore setting $\Phi_{E}=0$. The two
conditions for stable structures in case of a constant angular velocity
distribution and under axisymmetric perturbations reduce in this set up to:

$\displaystyle\operatorname{Tr}(M_{ab})$
$\displaystyle=\left(2\Omega+\hat{q}B_{z}\right)^{2}+\partial_{r}{\hat{q}}B_{z}\,r\Omega\;,$
(58) $\displaystyle\det(M_{ab})$ $\displaystyle=0\;.$ (59)

According to this result, a charged fluid structure can be considered stable
to axisymmetric perturbations, if the charge increases (angular momentum of
the fluid aligned to the magnetic field) or decreases (angular momentum of the
fluid anti-aligned to the direction of the magnetic field) outwards.

#### Case V: Constant angular velocity, charged central object.

In this case, the spherical coordinate system is used. In this coordinate
system, the electric potential arising from the charge of the central object
is given by:

$E_{a}=\left(\frac{Q}{r^{2}},0,0\right)\;.$ (60)

We consider no magnetic field, therefore setting $A_{a}=0$. The two conditions
for stable structures for a constant angular velocity distribution and under
axisymmetric perturbations reduce in this set up to:

$\displaystyle\operatorname{Tr}(M_{ab})$
$\displaystyle=4\Omega^{2}+\partial_{r}{\hat{q}}E_{r}\,r\Omega\;,$ (61)
$\displaystyle\det(M_{ab})$
$\displaystyle=2r\Omega^{2}\cos\theta\,E_{r}\left[r\cos\theta\,\partial_{r}\hat{q}-\sin\theta\,\partial_{\theta}\hat{q}\right]\;.$
(62)

According to this result, a charged fluid structure can be considered stable
to axisymmetric perturbations, if the charge increases (same charge as the BH)
or decreases (opposite charge of the BH) outwards. Structures along the
equatorial plane seem stable, as long as the charge does not significantly
change along the angle $\theta$, or, if it does, decreases (same charge as the
BH) or increases (opposite charge of the BH) towards the equatorial plane.

#### Case VI: Constant angular velocity, charged central object, homogeneous
magnetic field.

This time a homogeneous magnetic background field is assumed as well as a
charged central object.

Assuming the following charge distribution is motivated by the former work
[7]. In this paper bound charged fluid structures were found with a charge
distribution of

$\displaystyle\hat{q}=$ $\displaystyle f(S),$ $\displaystyle S=$
$\displaystyle-\Phi_{E}+\Omega A_{\phi}\;$ (63)

in a general relativistic setting.

The two conditions for stable structures in case of a constant angular
velocity distribution and under axisymmetric perturbations reduce in this set
up to:

$\displaystyle\begin{split}\operatorname{Tr}(M_{ab})=&\frac{g^{ab}\partial_{a}g_{\phi\phi}\partial_{b}g_{\phi\phi}}{g_{\phi\phi}}\left(\Omega+\hat{q}A^{\phi}\right)^{2}\\\
&+f^{\prime}(S)g^{ab}\left(E_{a}+\Omega\partial_{a}A_{\phi}\right)\left(E_{b}+\Omega\partial_{b}A_{\phi}\right),\end{split}$
(64) $\displaystyle\det(M_{ab})=$ $\displaystyle
f^{\prime}(S)\frac{\left(\Omega+\hat{q}A^{\phi}\right)^{2}}{g_{\phi\phi}}\left(\varepsilon^{ab}\partial_{a}g_{\phi\phi}(E_{b}+\Omega\partial_{b}A_{\phi})\right)^{2}.$
(65)

Both conditions are fulfilled, if $f^{\prime}(S)\geq 0$.

This setup was discussed in the general relativistic case by Kovář et al.
[11]. The equatorial tori fulfill the stability conditions in contrast to the
chosen setup for the polar clouds. This could be a hint, that the found polar
cloud solutions are actually unstable with respect to axisymmetric
perturbations. A general relativistic discussion is needed for a sound answer.

#### Case VII: Magnetic dipole field

Employing the spherical coordinates $\\{r,\theta,\phi\\}$, the background
magnetic field can be derived from the vector potential

$\displaystyle A_{\phi}$ $\displaystyle=\frac{\mathcal{M}}{r}\sin^{2}\theta,$
and $\displaystyle A^{\phi}$ $\displaystyle=\frac{\mathcal{M}}{r^{3}}\,.$ (66)

In addition, if the electric field vanish, $\Phi_{E}=0$, the two conditions
for stable equilibrium structures under the axisymmetric perturbations reduce
to

$\displaystyle\mathrm{Tr}\left(M_{ab}\right)=$
$\displaystyle\frac{1}{r^{2}\sin^{2}\theta}\Bigg{[}\left(\frac{2\ell}{r}+\hat{q}\partial_{r}A_{\phi}\right)\left(\partial_{r}\ell+\hat{q}\partial_{r}A_{\phi}\right)$
$\displaystyle+\frac{1}{r^{2}}\left(2\ell\cot\theta+\hat{q}\partial_{\theta}A_{\phi}\right)\left(\partial_{\theta}\ell+\hat{q}\partial_{\theta}A_{\phi}\right)+$
$\displaystyle+\ell\left(\partial_{r}A_{\phi}\partial_{r}\hat{q}+\frac{1}{r^{2}}\partial_{\theta}A_{\phi}\partial_{\theta}\hat{q}\right)\Bigg{]}$
(67)

and

$\displaystyle\mathrm{det}\left(M_{ab}\right)=$
$\displaystyle\frac{2\ell^{2}}{r^{4}\sin^{4}\theta}\Bigg{[}\frac{1}{r}\partial_{\theta}A_{\phi}-\left(\partial_{r}A_{\phi}\right)\cot\theta\Bigg{]}$
$\displaystyle\times\Bigg{[}\left(\partial_{r}\ell+\hat{q}\partial_{r}A_{\phi}\right)\partial_{\theta}\hat{q}+$
$\displaystyle\quad\quad\left(\partial_{\theta}\ell+\hat{q}\partial_{\theta}A_{\phi}\right)\partial_{r}\hat{q}\Bigg{]}.$
(68)

In the case of a rigid-body rotation.
$\Omega=\ell/(r\sin\vartheta)^{2}=\mathrm{const}$, a fluid equilibrium exists
when the specific charge can be expressed as function of the vector potential
only, that is when $\hat{q}=f(A_{\phi})$. In that case, the stability
conditions (67) and (68) coincide with Eqs. (64) and (65) given above.
Consequently, such configuration are therefore stable as long as
$f^{\prime}(A_{\phi})>0$.

##  VI Discussion

More general equilibrium configurations, with specific angular momentum
constant on cylinders $r\sin\vartheta=\mathrm{const}$ have been studied by
Slaný et al [4]. In particular, they consider angular momentum distribution
given by a power-law functions of the form

$\ell=K_{2}\left(r\sin\theta\right)^{K_{1}+1}$ (69)

with $K_{1}$ and $K_{2}$ being constant parameters ($K_{1}=\pm 1$ correspond
to constant angular velocity and constant angular momentum, respectively; when
$K_{2}=\sqrt{GM}$ and $K_{1}=-1/2$, Eq. (69) describes Keplerian flows). It is
also assumed that the distribution of specific charge can be expressed as

$\hat{q}=Cr^{\lambda}\sin(\theta)^{\delta}$ (70)

with $C$, $\lambda$ and $\delta$ being yet another constants. The functions
(69) and (70) describe equilibrium toroidal structures only when the
parameters $K_{1}$, $\lambda$ and $\delta$ satisfy the condition of
integrability. In order to reduce a large number of free parameters, Slaný et
al [4] consider four particular relations among the constants $K_{1}$,
$\lambda$ and $\delta$ for which they are able to find bound fluid
configurations. They referred to as the family I, II, III, and IV,
respectively. In general, all four families allow equatorial tori, with
pressure maxima located in the equatorial plane. These tori are similar to
thick-disks solutions of purely hydrodynamic equations, modified by presence
of Lorentz force. In addition, the families II – IV permit also solutions
where the Lorentz force pushes the pressure maxima out of the equatorial plane
allowing off-equatorial “levitating” tori. As noted by Slaný et al [4], such
configurations exist only for negative charge (i.e., when $\mathcal{M}q<0$).

In order to illustrate our results, we apply the stability criteria on both
types of the solutions. We first investigate stability properties of
equatorial tori corresponding to Family-I solutions. The relation between the
parameters $\lambda$, $\delta$ and $K_{1}$ are (see Ref. [4])

$\lambda=-\textstyle{\frac{3}{2}}\left(K_{1}-1\right),\quad\delta=0.$ (71)

The equilibrium configurations are fully described by three independent
parameters: the exponent in the angular momentum distribution $K_{1}$, the
charge parameter of the torus $C$ and the maximal pressure $p_{\mathrm{c}}$
(i.e. the value of the pressure at the torus center). The Lorentz force,
proportional to $C$, determines a departure of the fluid angular momentum at
the center of the torus $\ell_{\mathrm{c}}$ from the local Keplerian value
$\ell_{\mathrm{K,c}}$ and thus also the value of the parameter $K_{2}$. Here,
we take the ratio $(\ell_{\mathrm{c}}/\ell_{\mathrm{K,c}})^{2}$ as a free
parameter, instead of $C$, because of its immediate physical meaning.

Starting with slender tori ($p_{\mathrm{c}}\rightarrow 0$), in left-top panel
of Fig. 2, we plot possible configurations in the parameter space of $K_{1}$
vs $(\ell_{\mathrm{c}}/\ell_{\mathrm{K,c}})^{2}$. The horizontal line labeled
as $q=0$ corresponds to uncharged hydrodynamic slender tori, for which
$\ell_{\mathrm{c}}=\ell_{\mathrm{K,c}}$. The blue-shaded and red-shaded areas
below and above this line correspond to allowed combinations of these two
parameters for negatively- ($\mathcal{M}q<0$) and positively-
($\mathcal{M}q>0$) charged tori, respectively. These regions are bounded by
dashed black lines, where the topology of the central point changes from
maximum to saddle. Clearly only the former one gives reasonable solutions.

Using expressions (67) and (68), we determine regions in the parameter space,
where either $\mathrm{Tr}(M_{ab})$ or $\mathrm{det}(M_{ab})$ is negative.
These areas correspond to hatched regions in the plot. According to the
discussion in previous sections, we know that even slender tori corresponding
to these parameters cannot be stable. Naturally, the same applies also to
bigger tori with finite thickness, as their equi-pressure surfaces surround
the ones of the slender tori. In other words, a stability of slender tori is a
necessary condition for stability of tori with finite thickness, whose maximal
pressure points are at the same locations. Therefore it follows that the only
stable region in the parameter space left for possible stable configurations
corresponds to negatively-charged tori with $K_{1}$ in the range $[-1,-0.5]$,
where at least slender tori with negligible cross-section are stable.

We next examine tori of finite thickness with parameters corresponding to this
region. The left bottom panel of Fig. 2 shows an example of the equi-pressure
surfaces in the poloidal plane for the negatively charged torus with angular
momentum at the center $\ell_{\mathrm{c}}^{2}=0.922\ell_{\mathrm{K,c}}^{2}$
and $K_{1}=-0.6$. The region of negative determinant $\mathrm{det}(M_{ab})$ is
shown in the plot as a hatched area, the trace is positive everywhere. We
observe that although the center of the torus lies in the stable area, our
stability conditions put a serious constrain on the possible radial extend of
the torus.

Next, we investigate stability of the off-equatorial (“levitating”) tori. We
first concentrate on the Family-II solutions fully determined by three
independent parameters: the radial angular momentum distribution $K_{1}$,
height of the torus maximal pressure point above the equatorial plane
$\cos\theta_{\mathrm{c}}$ and the central pressure $p_{\mathrm{c}}$. The
family II corresponds to the charge distributions of the form

$\hat{q}=Cr^{3/2}(\sin\theta)^{-3K_{1}}$ (72)

(see Ref. [4]) and the force balance at the torus center determines values of
the two remaining constants $K_{2}$ and $C$. Again we start by discussing
stability of slender tori in the limit $p_{\mathrm{c}}\rightarrow 0$. The
center of the torus corresponds to the pressure maximum when $K_{1}<-1/2$,
independently of the height of the torus above the equatorial plane. In
addition, both $K_{1}$ and $\cos\theta_{\mathrm{c}}$ determine the stability
of the torus center. The regions where the trace and determinant of the matrix
$M_{ab}$ are negative are shown in the top right panel of Fig. 2 as
differently hatched areas. Clearly, no combination of $K_{1}$ and
$\cos\theta_{\mathrm{c}}$ gives a stable configuration. Consequently, neither
tori of finite thickness can be stable. In the bottom right panel of Fig. 2
this finding is illustrated by an example of the structure of equi-pressure
surfaces for $K_{1}=-2.25$ and $\theta_{\mathrm{c}}=1.1$. The regions of
negative trace and determinant of the matrix $M_{ab}$ are shown in the plot by
differently hatched areas, showing that even the maximal pressure points of
the toroidal structures are located well inside the unstable areas.

These finding brinks up a natural question: Do stable charged off-equatorial
tori exist at all? To answer this question, we decide to go beyond the
original analysis of Slaný et al. and let both $K_{1}$ and $\lambda$ to be
independent parameters. We examine stability of slender off-equatorial tori at
arbitrary height ($\sin^{2}\theta_{\mathrm{c}}\leq 1$). The result is shown in
Fig. 3. The blue-shaded area corresponds to parameters giving off-equatorial
toroidal configurations. The families II, III and IV introduced by Slaný et
al. [4] correspond to the three color lines indicated in the plot. We also
hatched the region where the trace or determinant of the stability matrix
$M_{ab}$ is negative in full range of height,
$0\leq\theta_{\mathrm{c}}\leq\pi$. Clearly, even with this more general
approach, no combination of $K_{1}$, $\lambda$ and $\theta_{\mathrm{c}}$ may
give a stable slender torus and our question thus has a negative answer. The
same necessarily applies to tori of finite thickness, as for their stability,
a stability of slender tori with the same $K_{1}$, $\lambda$ and
$\theta_{\mathrm{c}}$ is necessary condition.

Finally, let us remark that the construction of off-equatorial tori can be
still further generalized by allowing for different powers of $r$ and
$\sin\theta_{\mathrm{c}}$ in the prescribed angular momentum distribution
(69). It is likely that for some particular values of these exponents a stable
toroidal configuration will exist. Deeper explorations of this possibility is
however behind the scope of this paper, where we mainly intent to illustrate
power of our stability criteria. We wish to return to this question in our
future work.

Figure 3: Three families (II, III and IV) also allow for off-equatorial tori
in the regions of $K_{1}$–$\lambda$ parameter plane shown in the graph.
Interestingly, none of these equilibrium configurations satisfy the stability
criterion introduced in the present paper, as indicated by hatches covering
the entire plane (see the main text for details). Each colored line
corresponds to the range of $K_{1}$ values permitted by the equilibrium
condition for three distinct families.

##  VII Conclusions

We investigated the linear stability of charged, non-conductive perfect fluid
structures immersed in an electromagnetic background field concerning non-
axisymmetric and axisymmetric perturbations in the Newtonian regime. In order
to do so, we used the Lagrangian perturbation formalism introduced by Lebovitz
[17].

We found that the charge distribution within a non-conductive fluid cannot
stabilize the general instability of ideal fluids with respect to non-
axisymmetric perturbations, in agreement with the original argument by
Friedman and Schutz [21].

In the case of axisymmetric perturbations, we found a generalization of the
Høiland conditions for stable fluid structures for our case of charged, non-
conductive fluids. The generalized conditions were used to make specific
statements concerning the stability of seven special cases of electromagnetic
background fields and angular momentum distributions of a charged fluid
structure. The seven cases were chosen either by their simplicity and direct
generalization of a classical non-charged fluid structure (Polish doughnut
with constant angular momentum) or because these setups had been discussed
concerning the existence of bound charged fluid structures in previous works,
and the question of stability of these structures needed to be answered.

Charged fluid structures with a constant angular momentum distribution are
found to be stable with respect to axisymmetric perturbations if the charge
distribution within the structure increases outwards and gives rise to a
repulsive Lorentz force everywhere. We could show that, while the found
charged fluid structures discussed in [11] and some charged fluid structures
of family I in [4] are stable in the case of equatorial tori, the solutions
for polar clouds appear to be unstable.

Stability could also not be confirmed for the case of off-equatorial tori that
had been discussed in Slaný et al. [4] with the magnetic dipole background
field. On the other hand, modifying the assumed profile of the radial
distribution of the charge density or the magnetic field structure will likely
lead to stable configurations.

###### Acknowledgements.

We wish to acknowledge the continued support from the Czech Science Foundation
EXPRO program (VK and JH, ref. 21-06825X) and the CTA-CZ research
infrastructure of the Czech Ministry of Education, Youth and Sports (ref.
LM2023047). AT and EH acknowledge the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) funded under the project number 420243324.
Moreover, EH acknowledges the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2123
QuantumFrontiers - 390837967. We further thank Shokoufe Faraji, Jiří Kovář and
Petr Slaný for inspiring discussions.

##  Appendix A The (anti-)hermitian character of the operators in the
perturbed charged Euler equation

This section is dedicated to show that the operators $A_{ab}$, $B_{ab}$ and
$C_{ab}$, occurring in Eq. (21), are hermitian, anti-hermitian, and hermitian
respectively. Since this has already been shown by Friedman and Schutz [13] in
the uncharged fluid case, we will restrict the discussion here to those terms,
which arise from the electromagnetic interaction of the charged fluid with the
electromagnetic background filed.

The operators are given by:

$\displaystyle\eta^{a}A_{ab}\xi^{b}=$ $\displaystyle g_{ab}\eta^{a}\xi^{b}\;,$
(73) $\displaystyle\eta^{a}B_{ab}\xi^{b}=$
$\displaystyle\eta^{a}(2g_{ab}u^{c}\nabla_{c}-\hat{q}F_{ab})\xi^{b}\;,$ (74)
$\displaystyle\begin{split}\eta^{a}C_{ab}\xi^{b}=&\eta^{a}\left[\left(u^{b}\nabla_{b}\right)^{2}\xi_{a}+\frac{1}{\rho}\left(\nabla_{b}\xi^{b}\nabla_{a}p-\nabla_{a}\xi^{b}\nabla_{b}p\right.\right.\\\
&\left.-\nabla_{a}(\gamma
p\nabla_{b}\xi^{b})\right)+\xi^{b}\nabla_{b}\nabla_{a}\Phi_{G}+\hat{q}\xi^{b}\nabla_{b}\nabla_{a}\Phi_{E}\\\
&\left.-\hat{q}\left(F_{ab}u^{\gamma}\nabla_{\gamma}\xi^{b}+\xi^{b}\nabla_{b}F_{a\gamma}u^{\gamma}\right)\right]\;.\end{split}$
(75)

Since $A_{ab}$ is hermitian, and the electromagnetic contribution in $B_{ab}$
obviously anti-hermitian, ($F_{ab}$ is anti-hermitian), only the
electromagnetic terms in $C_{ab}$ need to be checked for their symmetry. These
terms are

$\hat{q}\eta^{a}\xi^{b}\nabla_{b}\nabla_{a}\Phi_{E}-\hat{q}\eta^{a}\left(F_{ab}u^{\gamma}\nabla_{\gamma}\xi^{b}+\xi^{b}\nabla_{b}F_{a\gamma}u^{\gamma}\right)\,.$
(76)

Since covariant derivatives commute, the electric term
($\hat{q}\nabla_{b}\nabla_{a}\Phi_{E}$) has to be hermitian. The magnetic
contribution
$\hat{q}\left(F_{ab}u^{\gamma}\nabla_{\gamma}+\nabla_{b}F_{a\gamma}u^{\gamma}\right)$
can be rewritten as follows

$\displaystyle\begin{split}q\eta^{a}&\left(F_{ab}u^{\gamma}\nabla_{\gamma}\xi^{b}+\xi^{b}\nabla_{b}F_{a\gamma}u^{\gamma}\right)\\\
&\quad=\frac{q}{2}\eta^{a}\xi^{b}\left[\nabla_{a}F_{bc}+\nabla_{b}F_{ac}\right]u^{c}\\\
&\quad-\frac{q}{2}\left[\eta^{a}F_{ab}u^{c}\nabla_{c}\xi^{b}+\xi^{a}F_{ab}u^{c}\nabla_{c}\eta^{b}\right]\\\
&\quad-\frac{1}{2}\nabla_{c}\left(qu^{c}\eta^{a}F_{ab}\xi^{b}\right)\,.\end{split}$
(77)

The last term under the volume integral over the system vanishes because we
demand the charge density at infinity to vanish, too. The two remaining terms
are symmetric in exchanging $\eta$ and $\xi$. It is therefore shown, that the
electromagnetic terms of $C_{ab}$ under an integral over the system volume are
also hermitian.

##  Appendix B Second variation of the total energy

The total energy of a charged, non-conductive fluid configuration is given by:

$\displaystyle E_{\mathrm{tot}}=T+U+W_{G}+W_{E}\;,$ (78)

where $T$ is the kinetic, $W_{G}=\int{\frac{1}{2}\Phi_{G}\rho\,dV}$ the mass
potential, $W_{E}=\int{\frac{1}{2}\hat{q}\Phi_{E}\rho\,dV}$ the electric
potential and $U$ the internal energy. The magnetic field does not contribute
to the total energy of the system, since the Lorentz-force $F_{ab}u^{b}$ is
always perpendicular the the fluid motion. The given total energy therefore
only has an additional electric potential term $W_{E}$ compared to the total
energy of an uncharged, perfect fluid.

Let us note that Friedman and Schutz [13] derived the variation of the total
energy until second order for an uncharged, adiabatic, perfect fluid under the
premise of mass conservation:

$\displaystyle\bm{\delta}E_{\mathrm{tot,nc}}=$
$\displaystyle\delta^{1}E_{\mathrm{tot,nc}}+\delta^{2}E_{\mathrm{tot,nc}}+\mathcal{O}(\xi^{3})$
(79) $\displaystyle=$
$\displaystyle\int{\left[\rho\xi^{i}\left(u^{j}\nabla_{j}u_{i}+\frac{1}{\rho}\nabla_{i}p+\nabla_{i}\Phi_{G}\right)+\rho
u^{\beta}\Delta u_{\beta}\right]dV}$ (80)
$\displaystyle+\frac{1}{2}\int{\rho\,\lvert\partial_{t}\xi\rvert^{2}-\rho\lvert
u^{\gamma}\nabla_{\gamma}\xi\rvert^{2}+\gamma
p\lvert\nabla_{a}\xi^{a}\rvert^{2}+2{\xi}^{a}\nabla_{a}p\nabla_{b}\xi^{b}+{\xi}^{a}\xi^{b}\rho\left[\nabla_{b}\nabla_{a}p+\nabla_{b}\nabla_{a}\Phi_{G}\right]dV}+\mathcal{O}(\xi^{3})\;.$
(81)

In the charged case, the variation of the electric potential until second
order needs to be added to the equation above. By taking into account, that
the overall charge of the system is conserved (see Eq. (15), and
$\Delta\hat{q}=0$) one gets analog to the variation of the gravitational
potential:

$\displaystyle\begin{split}\mathrm{\delta}W_{E}=&\int{\hat{q}\Delta\left[\frac{1}{2}\Phi_{E}\rho\right]\,dV}\\\
=&\int{\left[\hat{q}\rho\xi^{a}\nabla_{a}\Phi_{E}+\frac{1}{2}\hat{q}\rho\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}\Phi_{E}\right]\,dV}\\\
&+\mathcal{O}(\xi^{3}).\end{split}$ (82)

If the term $\rho u^{\beta}\Delta u_{\beta}$ is rewritten in the form $\rho
u^{\beta}\Delta\hat{u}_{\beta}$ in the charged case the variation of the total
energy up to second order will take the form:

$\displaystyle\begin{split}\bm{\delta}E_{\mathrm{tot}}=&\int{\left[\rho\xi^{i}\left(u^{j}\nabla_{j}u_{i}+\frac{1}{\rho}\nabla_{i}p+\nabla_{i}\Phi_{G}+\hat{q}\nabla_{i}\Phi_{E}-\hat{q}F_{ij}u^{j}\right)+\rho
u^{\beta}\Delta\hat{u}_{\beta}\right]dV}\\\
&+\frac{1}{2}\int\rho\,\lvert\partial_{t}\xi\rvert^{2}-\rho\lvert
u^{\gamma}\nabla_{\gamma}\xi\rvert^{2}+\gamma
p\lvert\nabla_{a}\xi^{a}\rvert^{2}+2{\xi}^{a}\nabla_{a}p\nabla_{b}\xi^{b}+{\xi}^{a}\xi^{b}\rho\left[\nabla_{b}\nabla_{a}p+\nabla_{b}\nabla_{a}\Phi_{G}+\hat{q}\nabla_{b}\nabla_{a}\Phi_{E}\right]\\\
&-\rho\hat{q}\xi^{a}\left(F_{ab}u^{\gamma}\nabla_{\gamma}\xi^{b}+\xi^{b}\nabla_{b}F_{a\gamma}u^{\gamma}\right)dV+\mathcal{O}(\xi^{3}).\end{split}$
(83)

The first term in the first integral must vanish according to the Euler
equation for charged fluids (see Eq. (13)). Like in the uncharged case, the
first order variation becomes zero, and the second order variation coincides
with the canonical energy, if $u^{\beta}\Delta\hat{u}_{\beta}$ vanishes. In
the case of a fluid in circular motion, this is the case, if the redefined
angular momentum $\hat{u}_{\phi}=u_{\phi}+\hat{q}A_{\phi}$ is conserved.

Equation (83) was derived by calculating the difference
$\rho\hat{q}u^{\beta}\Delta A_{\beta}$ between $\rho u^{\beta}\Delta
u_{\beta}$ and $\rho u^{\beta}\Delta\hat{u}_{\beta}$ to up to second order:

$\displaystyle\Delta A_{i}=\Delta(g_{ib}A^{b})=\Delta g_{ib}A^{b}+g_{ib}\Delta
A^{b}+\Delta g_{ib}\Delta A^{b}\,,$ (84)

where

$\Delta
g_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}+\nabla_{a}\xi^{\beta}\nabla_{b}\xi_{\beta}+\mathcal{O}(\xi^{3})$
(85)

and

$\displaystyle\Delta
A^{i}=\left(\delta_{j}^{i}-\nabla_{j}\xi^{i}+\nabla_{j}\xi^{\beta}\nabla_{\beta}\xi^{i}\right)\bar{A}^{j}-A^{i}+\mathcal{O}(\xi^{3})$
(86)

(see [13]). Here, $\bar{A}^{i}$ is the perturbed vector field
${A^{\prime}}^{i}$, framedragged along $\xi$. In our case, the magnetic vector
potential is not affected by the perturbations.

Taylor expansion up to second order results in this case in:

$\bar{A}^{i}=A^{i}(x^{a}+\xi^{a})=A^{i}+\xi^{a}\nabla_{a}A^{i}+\frac{1}{2}\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}A^{i}+\mathcal{O}(\xi^{3})\;.$
(87)

Calculating all terms up to second order results in:

$\Delta
A_{i}=\xi^{a}\nabla_{a}A_{i}+A_{a}\nabla_{i}\xi^{a}+\nabla_{i}\xi^{a}\xi^{b}\nabla_{b}A_{a}+\frac{1}{2}\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}A_{i}\,.$
(88)

This leads to

$\displaystyle\begin{split}&\int{\rho\hat{q}u^{\beta}\Delta A_{\beta}\,dV}\\\
&\quad=\int\hat{q}\rho
u^{\beta}\left(\xi^{a}\nabla_{a}A_{\beta}+A_{a}\nabla_{\beta}\xi^{a}+\nabla_{\beta}\xi^{a}\xi^{b}\nabla_{b}A_{a}\right.\\\
&\quad+\left.\frac{1}{2}\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}A_{\beta}\right)\,dV\end{split}$
(89)
$\displaystyle\begin{split}&\quad=\int{\rho\,\xi^{a}\,\hat{q}F_{a\beta}u^{\beta}\,dV}\\\
&\quad+\frac{1}{2}\int{\rho\hat{q}\xi^{a}\left(F_{ab}u^{\gamma}\nabla_{\gamma}\xi^{b}+\xi^{b}\nabla_{b}F_{a\gamma}u^{\gamma}\right)\,dV},\end{split}$
(90)

where we used $\nabla_{i}(\rho\hat{q}u^{i}=0)$ and assumed, that the charge
and mass density vanish at infinity. By adding the variation of the electric
potential to Eq. (79) and replacing $\rho u^{\beta}\Delta u_{\beta}$ by $\rho
u^{\beta}\Delta\hat{u}_{\beta}-\rho\hat{q}u^{\beta}\Delta A_{\beta}$ finally
leads to Eq. (83).

## References

  * Tassoul [2000] J.-L. Tassoul.  _Stellar Rotation_.  Cambridge University Press, 2000.
  * Mestel [2003] L. Mestel.  Stellar rotation: a historical survey.  In Michael J. Thompson and Jørgen Christensen-Dalsgaard, editors, _Stellar Astrophysical Fluid Dynamics_ , pages 75–98. Cambridge (UK): Cambridge University Press, 2003.
  * Kovář et al. [2011] J. Kovář, P. Slaný, Z. Stuchlík, V. Karas, C. Cremaschini, and J. C. Miller.  Role of electric charge in shaping equilibrium configurations of fluid tori encircling black holes.  _Phys. Rev. D_ , 84:084002, October 2011.  doi: 10.1103/PhysRevD.84.084002.  URL http://link.aps.org/doi/10.1103/PhysRevD.84.084002.
  * Slaný et al. [2013] P. Slaný, J. Kovář, Z. Stuchlík, and V. Karas.  Charged Tori in Spherical Gravitational and Dipolar Magnetic Fields.  _Astrophys. J. Suppl. S._ , 205(1):3, February 2013.  doi: 10.1088/0067-0049/205/1/3.  URL https://doi.org/10.1088/0067-0049/205/1/3.
  * Schroven et al. [2018] K. Schroven, A. Trova, E. Hackmann, and C. Lämmerzahl.  Charged Fluid Structures around a Rotating Compact Object with a Magnetic Dipole Field.  _Phys. Rev. D_ , 98:023017, July 2018.  doi: 10.1103/PhysRevD.98.023017.  URL https://link.aps.org/doi/10.1103/PhysRevD.98.023017.
  * Trova et al. [2018] A. Trova, K. Schroven, E. Hackmann, V. Karas, J. Kovář, and P. Slaný.  Equilibrium Configurations of a Charged Fluid around a Kerr Black Hole.  _Phys. Rev. D_ , 97:104019, May 2018.  doi: 10.1103/PhysRevD.97.104019.  URL https://link.aps.org/doi/10.1103/PhysRevD.97.104019.
  * Trova et al. [2020] A. Trova, E. Hackmann, V. Karas, K. Schroven, J. Kovář, and P. Slaný.  Influence of test charge and uniform magnetic field on charged fluid equilibrium structures.  _Phys. Rev. D_ , 101:083027, April 2020.  doi: 10.1103/PhysRevD.101.083027.  URL https://link.aps.org/doi/10.1103/PhysRevD.101.083027.
  * Kovář et al. [2008] J. Kovář, Z. Stuchlík, and V. Karas.  Off-equatorial orbits in strong gravitational fields near compact objects.  _Class. Quantum Grav._ , 25(9):095011, April 2008\.  doi: 10.1088/0264-9381/25/9/095011.  URL https://doi.org/10.1088/0264-9381/25/9/095011.
  * Kovář et al. [2020] J. Kovář, Y. Kojima, P. Slaný, Z. Stuchlík, and V. Karas.  Charged fluids encircling compact objects: force representations and conformal geometries.  _Class. Quantum Grav._ , 37(24):245007, December 2020.  doi: 10.1088/1361-6382/abbe70.
  * Bonitz, M. and Horing, N. and Ludwig, P. [2010] Bonitz, M. and Horing, N. and Ludwig, P.  _Introduction to Complex Plasmas_ , volume 59.  Springer-Verlag Berlin Heidelberg, 2010.  doi: 10.1007/978-3-642-10592-0.
  * Kovář et al. [2014] J. Kovář, P. Slaný, C. Cremaschini, Z. Stuchlík, V. Karas, and A. Trova.  Electrically charged matter in rigid rotation around magnetized black hole.  _Phys. Rev. D_ , 90:044029, August 2014.  doi: 10.1103/PhysRevD.90.044029.  URL https://link.aps.org/doi/10.1103/PhysRevD.90.044029.
  * Kovář et al. [2016] J. Kovář, P. Slaný, C. Cremaschini, Z. Stuchlík, V. Karas, and A. Trova.  Charged perfect fluid tori in strong central gravitational and dipolar magnetic fields.  _Phys. Rev. D_ , 93:124055, June 2016.  doi: 10.1103/PhysRevD.93.124055.  URL https://link.aps.org/doi/10.1103/PhysRevD.93.124055.
  * Friedman and Schutz [1978a] J. L. Friedman and B. F. Schutz.  Lagrangian perturbation theory of nonrelativistic fluids.  _Astrophys. J._ , 221:937–957, May 1978a.  doi: 10.1086/156098.
  * Friedman and Schutz [1975] J. L. Friedman and B. F. Schutz.  On the stability of relativistic systems.  _Astrophys. J._ , 200:204–220, August 1975.  doi: 10.1086/153778.
  * Balbus and Hawley [1991] S. A. Balbus and J. F. Hawley.  A Powerful Local Shear Instability in Weakly Magnetized Disks. I. Linear Analysis.  _Astrophys. J._ , 376:214, July 1991.  doi: 10.1086/170270.
  * Glampedakis and Andersson [2007] K. Glampedakis and N. Andersson.  Lagrangian perturbation theory for rotating magnetic stars.  _Mon. Not. R. Astron. Soc._ , 377(2):630–644, April 2007.  doi: 10.1111/j.1365-2966.2007.11625.x.  URL https://doi.org/10.1111/j.1365-2966.2007.11625.x.
  * Lebovitz [1961] N. R. Lebovitz.  The Virial Tensor and its Application to Self-Gravitating Fluids.  _Astrophys. J._ , 134:500, September 1961.  doi: 10.1086/147175.
  * Lynden-Bell and Ostriker [1967] D. Lynden-Bell and J. P. Ostriker.  On the Stability of Differentially Rotating Bodies.  _Mon. Not. R. Astron. Soc. 368_ , 136(3):293–310, July 1967.  doi: 10.1093/mnras/136.3.293.  URL https://doi.org/10.1093/mnras/136.3.293.
  * Chandrasekhar and Friedman [1972] S. Chandrasekhar and J. L. Friedman.  On the Stability of Axisymmetric Systems to Axisymmetric Perturbations in General Relativity, I. The Equations Governing Nonstationary, and Perturbed Systems.  _Astrophys. J._ , 175:379, July 1972.  doi: 10.1086/151566.
  * Friedman and Stergioulas [2011] J. Friedman and N. Stergioulas.  Stability of relativistic stars.  _B. Astron. Soc. India_ , 39:21–48, March 2011.  doi: 10.1142/9789814374774˙0007.
  * Friedman and Schutz [1978b] J. L. Friedman and B. F. Schutz.  Secular instability of rotating Newtonian stars.  _Astrophys. J._ , 222:281–296, May 1978b.  doi: 10.1086/156143.
  * Rezzolla and Zanotti [2013] L. Rezzolla and O. Zanotti.  _Relativistic Hydrodynamics_.  Oxford University Press, Oxford, England, September 2013.