------------------------------------------------------------------------
From: Benjamin Chandran  chandran@virgil.physics.uiowa.edu 
To: gcnews@aoc.nrao.edu


%astro-ph/0009184
\documentclass[11pt]{aastex}
\usepackage{emulateapj5}
\begin{document}
\bibliographystyle{plain}

\submitted{Submitted to ApJ}
\title{Equilibrium and stability of strong vertical magnetic
fields at the Galactic center}


\author{ B. D. G. Chandran}
 
\affil{Dept. Physics \& Astronomy, University of Iowa, Iowa City, IA}


\begin{abstract}

Observations of narrow radio-emitting filaments near the Galactic
center have been interpreted in previous studies as evidence of a
pervasive vertical (i.e. $\perp$ to the Galactic plane) milliGauss
magnetic field in the central $\sim 150$ pc of the Galaxy.  Such a
magnetic field would have important implications for the central
environments of spiral galaxies and would, for somewhat complex
reasons, be a key piece of evidence in support of a pre-Galactic
origin of the Galactic magnetic field.  This paper addresses the
question of whether  such an intense vertical magnetic field, whose
pressure significantly exceeds the thermal pressure of the surrounding
medium, could be confined.  A simple cylindrically symmetric model for
the equilibrium in this central region is proposed in which horizontal
(i.e. $\parallel$ to the Galactic plane) magnetic fields embedded in
an annular band of molecular material of radius $\sim 150$ pc are
wrapped around vertical magnetic fields threading hot plasma. Since
orthogonal magnetic fields can not easily interpenetrate, the
horizontal magnetic field in this equilibrium can transfer the weight
of the molecular material to the hot plasma and vertical magnetic
field, potentially providing confinement. The stability of this
equilibrium is studied indirectly by treating the equilibrium as an
infinite isothermal slab, with high-density plasma representing the
partially ionized molecular clouds suspended above low-density plasma,
and with the orthogonal magnetic fields in the high- and low-density
plasmas perpendicular to the effective gravitational acceleration $\bf
g$ (true gravitational acceleration minus the centripetal acceleration
of Galactic rotation).  Perturbations of the interface are found to be
unstable on wavelengths shorter than a critical wavelength that
depends upon $g$ and the sound and Alfv\'en speeds in the high-density
plasma. The results of the slab-equilibrium calculation are
extrapolated on a qualitative basis to argue that the proposed
cylindrically symmetric equilibrium is unstable if the magnetic
pressure of the vertical magnetic field $B_{\rm vert}$ greatly exceeds
the thermal pressure in the molecular material, or if $B_{\rm vert}
\gg 50\mu$G for Galactic-center parameters.  If this qualitative
argument is correct, then milliGauss vertical fields can not be stably
confined using the weight of the molecular material in the kind of
equilibrium that has been proposed. If an equilibrium can not be found
that can stably confine pervasive vertical milliGauss magnetic fields
at the Galactic center, then a powerful argument will be added in
favor of other interpretations of the Galactic-center radio filaments.

\end{abstract}

\section{Introduction}

Observations of the central $\sim 150$ pc of the Galaxy indicate that
the magnetic fields, plasma, and molecular gas within this central
region have properties in striking contrast to the interstellar medium
(ISM) of the Galactic disk.  For example, the Galactic center hosts a
phenomenon that is unique within the Galaxy: narrow, 
``vertical'' (i.e. $\perp$ to the Galactic plane) filaments of radio
emission (Morris 1996). The
polarization of the filaments of the Radio Arc
indicates that the ambient magnetic field is parallel to the
filaments (Tsuboi et~al.\ 1986, Reich 1994, Tsuboi et~al.\ 1995), and this
alignment is presumed to be typical of the other filaments as well. The
filaments are typically tens of parsecs long and only a fraction
of a parsec wide, and in virtually all cases appear to be in contact
with a molecular cloud at some point along their length. The absence
of bending of these filaments by the randomly moving clouds has been
interpreted as evidence of a milliGauss lower limit to the field
strength in the filaments (Yusef-Zadeh \& Morris 1987---see
section \ref{sec:review}). The apparent
impossibility of forming or confining narrow filaments with such
tremendous magnetic pressures (well in excess of the thermal pressures,
as will be discussed below) suggests that if a mG vertical field is present, it
pervades the central 150 pc of the Galaxy, and the filaments are
simply those flux tubes that possess large populations of relativistic
electrons (Morris \& Serabyn 1996). The presence of such a strong
field would suggest the presence of similar fields in the central
regions of other spiral galaxies, and would also be a key
piece of evidence in favor of a proto-Galactic or
primordial (e.g., Kulsrud et~al.\ 1997, Howard \& Kulsrud 1997)
origin of the Galactic magnetic field (Chandran et~al.\ 2000).

One of the most pressing questions regarding the plausibility of
the milliGauss-field hypothesis
is whether such a strong magnetic field could be stably
confined. The volume-dominant phase of the central region is hot plasma at a
temperature $T\sim 10^8$~K which fills an elliptical region
$150 \times 270$ pc (FWHM) with a major axis that is tilted $20^\circ$
with respect to the Galactic plane (Yamauchi et~al.\ 1990).  The
density of this plasma was estimated to be $0.03- 0.06 \mbox{
cm}^{-3}$ by Yamauchi et~al.\ (1990) and $\sim 0.3-0.4$ by Koyama
et~al.\ (1996).  The mass-dominant phase of the central region is
molecular gas with $n > 10^4 \mbox{ cm}^{-3}$, $T \sim 70$~K, a filling factor 
in excess of 0.1, and a vertical
thickness of $\sim 30 $ pc (Bally et~al.\ 1988, Morris \& Serabyn
1996).  If the magnetic field strength 
is 1 mG, then the magnetic pressure $B^2/8\pi$ is
$\sim 4\times 10^{-8} \mbox{ dyne/cm}^2$, which far exceeds the
thermal pressure of both the hot plasma, $p_{\rm hot} \sim 4 \times
10^{-10}- 4\times 10^{-9}\mbox{ dyne/cm}^2$, and the molecular
material, $p_{\rm cloud} \sim 10^{-10} \mbox{ dyne/cm}^2$.  The
thermal pressure of the ambient medium is thus unable to confine
milliGauss fields, whether they are ubiquitous or 
in the form of isolated narrow flux tubes.
In addition, since the hot plasma is itself unconfined, it 
would be unable to confine a ubiquitous vertical field even
if its pressure were greater than the magnetic pressure.


The only force capable of confining a pervasive vertical mG field in
this central region is the weight of the molecular
material. Interestingly, the polarization of dust emission 
indicates that the molecular material is
threaded by horizontal (i.e., $\parallel$ to the Galactic plane)
magnetic fields (Morris \& Serabyn 1996).  Because this horizontal
field ${\bf B}_{\rm cloud}$ is perpendicular to the vertical magnetic
field in the hot plasma ${\bf B}_{\rm vert}$, it is difficult for the
two types of material to interpenetrate, and thus the
weight of the molecular material has the potential to prevent
a strong vertical field from expanding. In this paper, the
cylindrically symmetric equilibrium of figure~\ref{fig:equil} is
proposed as a starting point for investigating the ability of the
molecular material and horizontal magnetic field to confine the
vertical field.  It is assumed in this model 
that the molecular material is confined
to a homogeneous annular band.

\begin{figure*}[h]
\vspace{10cm}
\special{psfile=f1.eps  voffset=50 hoffset=110
                          vscale=70    hscale=70     angle=0}
\caption{A simplified model for an equilibrium of strong
vertical fields at the Galactic center. \label{fig:equil}  }
\end{figure*}


To make a compact analytic treatment possible, the stability of the
cylindrically symmetric equilibrium is explored under a number of
additional simplifying approximations. Most significantly, the
equilibrium is treated as an infinite slab, symbolically depicted in
figure~\ref{fig:slab} and described
in section \ref{sec:eq}, with the effective acceleration of gravity $\bf
g$ taken to be the true gravitational acceleration minus the
centripetal acceleration associated with Galactic rotation, and with
the magnetic field everywhere perpendicular to ${\bf g}$.  It is
assumed that the partial ionization within the molecular clouds is
sufficiently high and that the frequencies of any instabilities are
sufficiently small compared to the neutral-ion collision frequency
that ambipolar diffusion can be ignored. The molecular clouds are then
treated as a fully ionized dense MHD plasma.  The hot plasma
is taken to be less dense than the partially ionized clouds, but is
for simplicity taken to have the same temperature as the clouds, with
the weight of the clouds supported primarily by magnetic
pressure. Although the isothermal model is not faithful to the large
difference in temperature between the two phases, the temperature
difference is not expected to play an important
role in the global gravitational stability of the equilibrium. A very
similar system in which magnetized plasma is suspended above a vacuum
has been investigated by Gratton et~al.\ (1988)
and Gonzalez \& Gratton (1990). The present paper builds upon
results from these previous papers.
\begin{figure*}[h]
\vspace{10cm}
\special{psfile=f2.eps  voffset=50 hoffset=110
                          vscale=50    hscale=50     angle=0}
\caption{The isothermal-slab equilibrium.
\label{fig:slab} }
\end{figure*}

In section \ref{sec:stability} the slab equilibrium is found to be
unstable to a global gravitational mode in which the interface between
the high- and low-density plasmas is perturbed. The stability analysis
involves solving for the displacement in a boundary layer at the
interface between the high-density and low-density plasmas.  Because
the magnetic field in the high-density plasma is $\perp$ to the
magnetic field in the low-density plasma, there is always field-line
bending regardless of the direction of a perturbation's wave-vector in
the plane parallel to the interface.  For this reason, at sufficiently
small wavelengths the mode is stabilized by magnetic tension.
However, if $\beta_{\rm cloud} = 8\pi p_{\rm cloud}/B_{\rm cloud}^2$
is small, then there are unstable modes at wavelengths that are small
compared to the scale height $L_+$ of the dense plasma in the slab
equilibrium of figure~\ref{fig:slab}. This is because as $\beta_{\rm
cloud}$ is reduced, the dense plasma becomes increasingly compressible
along the magnetic field, and compressions allow gravity to play a
greater destabilizing role.

In section \ref{sec:implications}, the stability analysis of the slab
equilibrium is extrapolated to the cylindrically symmetric equilibrium
of figure~\ref{fig:equil} under the assumptions that the vertical and
radial scale heights of the molecular material in
figure~\ref{fig:equil} are comparable ($\sim L_+$), and that
any instability of the slab equilibrium at wavelengths $\ll L_+$ is
also present in the cylindrically symmetric equilibrium, since on 
scales $\ll L_+$ the curvature and vertical scale height of the cylindrical
equilibrium are unimportant. Since the slab equilibrium is unstable
at wavelengths $\ll L_+$ when $\beta_{\rm cloud} \ll 1$, stability requires that
the magnetic pressure of the vertical field $B_{\rm vert}^2/8\pi$ not
be much greater than the thermal pressure of the molecular material
$p_{\rm cloud}$---otherwise, pressure balance would require that the
magnetic pressure within the cloud $B_{\rm cloud}^2/8\pi$ be $\gg
p_{\rm cloud}$, i.e. $\beta_{\rm cloud} \ll 1$.  This means that
$B_{\rm vert}$ can not be much greater than $50 \mu$G.  If 
the extrapolation from the slab to the cylindrically symmetric
equilibrium is correct, and if another equilibrium can not be found in
which pervasive mG vertical magnetic fields can be stably confined,
then there will be a powerful argument for questioning the presence of
mG vertical magnetic fields in the central 150 pc of the Galaxy. 

In section \ref{sec:review} the
arguments used to infer milliGauss fields at the Galactic center are
reviewed, and in section \ref{sec:conclusion} the main conclusions of
the paper are summarized.

\section{Slab equilibrium}
\label{sec:eq} 


In the slab equilibrium, 
the gravitational acceleration ${\bf g}$ is taken to be in the $-{\bf \hat{y}}$ 
direction [following the notation of Gratton et~al.\ (1988)], and
$g$ is assumed constant. 
It is also assumed that the equilibrium magnetic
field within the high-density dense medium has a single direction that is
$\perp$ to $\bf \hat{y}$, and that
the equilibrium magnetic field within the low-density plasma has a single 
direction
$\perp$ to $\bf \hat{y}$. These two directions, however, are not
the same. The transition between the two phases
is taken to occur in a narrow layer centered at $y=0$.

The equilibrium pressure $p$, field strength $B$, and density $\rho$ then 
satisfy
the equation
\begin{equation}
\frac{d}{dy}\left(p + \frac{B^2}{8\pi}\right) = -\rho g.
\label{eq:eq} 
\end{equation} 
The plasma is assumed to have an isothermal equation of state. The
isothermal sound speed $C_S = \sqrt{p/\rho}$ is then constant for all values of 
$y$.
The dense medium (at $y>0$) is taken to have an exponential profile,
\begin{eqnarray}
\rho & = & \overline{ \rho}_+ e^{-2 y/L_+} \label{eq:rho+}, \\
p & = & \overline{ p}_+ e^{-2 y/L_+} \label{eq:p+}, \mbox{ \hspace{0.3cm} and}\\
B & = & \overline{ B}_+ e^{-y/L_+} \label{eq:B+}, \\
\end{eqnarray} 
where $\overline{ \rho}_+$, $\overline{ p}_+$, and $\overline{ B}_+$ are
constants.
The value of the Alfv\'en speed $C_A = B/\sqrt{4\pi \rho}$ 
within the dense plasma, denoted  $C_{A+}$, is then constant. If
\begin{equation}
M_{A+} = \frac{C_A+}{C_S},
\end{equation} 
then equation~(\ref{eq:eq}) implies that
\begin{equation}
L_+ = \frac{C_{S}^2(2 + M_{A+}^2)}{g}.
\label{eq:L+} 
\end{equation} 

A similar solution is taken to hold within the low-density plasma 
($y<0$), with 
\begin{eqnarray}
\rho & = & \overline{ \rho}_- e^{-2 y/L_-} \label{eq:rho-}, \\
p & = & \overline{ p}_- e^{-2 y/L_-} \label{eq:p-}, \mbox{ \hspace{0.3cm} and}\\
B & = & \overline{ B}_- e^{-y/L_-} \label{eq:B-}, \\
\end{eqnarray} 
where $\overline{\rho}_-$, $\overline{p}_-$, and $\overline{B}_-$ are constants,
\begin{equation}
L_- = \frac{C_{S}^2(2 + M_{A-}^2)}{g},
\label{eq:L-} 
\end{equation} 
$C_{A-}$ is the constant value of $C_A$ 
within the low-density plasma, and $M_{A-} = C_{A-}/C_{S}$.

It is assumed that the transition between the solution of
equations~(\ref{eq:rho+}) through (\ref{eq:B+}) and the solution of
equations~(\ref{eq:rho-}) through (\ref{eq:B-}) occurs over a
distance of $\sim\epsilon L_+$, where
\begin{equation}
\epsilon \ll 1.
\label{eq:eps} 
\end{equation} 
The magnetic field changes direction within the transition layer,
but remains in the $xz$-plane. 
Pressure equilibrium across the interface requires that
as $\epsilon \rightarrow 0$
\begin{equation}
\overline{p}_+ + \frac{\overline{ B}_+^2}{8\pi}
= \overline{p}_- + \frac{\overline{ B}_-^2}{8\pi}
\label{eq:pressure_balance} 
\end{equation} 


\section{Global gravitational instability of the slab equilibrium}
\label{sec:stability} 


Gratton et~al.\ (1988) showed that if the equilibrium
of section \ref{sec:eq}  is perturbed by the Lagrangian
displacement vector 
\begin{equation}
{\bf \xi} = \overline{ \bf \xi}(y) \exp(ik_x x
+ ik_z z -iwt),
\label{eq:defxi} 
\end{equation} 
 then the $y$-component of $\overline{ \bf \xi}(y)$,
denoted $\zeta$, satisfies the equation
\[
\frac{d}{dy}\left[ H\left(\frac{1}{M} - 1\right) \frac{d\zeta}{dy}\right]
+ k_T^2\zeta\left[ H\left(1 -\frac{g^2 k_T^2}{M w^4}\right) \right.
\]
\begin{equation} 
\left.- g\frac{d}{dy}\left(\rho + \frac{H}{Mw^2}\right)\right] = 0,
\label{eq:zeta1} 
\end{equation} 
where 
\begin{eqnarray} 
{\bf k}_T & = &k_x{\bf \hat{x}} + k_z {\bf \hat{z}},
\label{eq:kt} \\
H & = & -\rho(w^2 - k_\parallel^2 C_A^2) \label{eq:H}, \\
k_\parallel & = & \frac{{\bf k}_T \cdot {\bf B}}{B} \label{eq:kpar} ,
\mbox{ \hspace{0.3cm} and}\\
M & = & 1 - \frac{k_T^2(C_A^2 + C_S^2)}{w^2} + 
\frac{k_\parallel^2 k_T^2 C_A^2 C_S^2}{w^4} \label{eq:M} .
\end{eqnarray} 
Equation~(\ref{eq:zeta1}) is valid for all $y$.\footnote{Contrary to
what is suggested by Gonzalez \& Gratton (1990),
equation~(\ref{eq:zeta1}) is invalid for an isothermal equilibrium if
the fluctuations obey an adiabatic equation of state, $p\rho^{-\gamma}
= \mbox{ constant}$, with $\gamma \neq 1$. If the equilibrium and
perturbations are constrained to obey different equations of state,
then additional terms appear in equation~(\ref{eq:zeta1}).}  (It is
assumed that the narrow transition layer is sufficiently wide that the
viscous and resistive terms can be neglected.)


Upon introducing  the dimensionless variables
\begin{eqnarray}
\tilde{y}  & = & y/L_+  \label{eq:yt}, \\
\alpha_+ & = & \frac{1}{k_T L_+} \label{eq:alpha+},  \\
\cos \psi & = & \frac{k_\parallel}{k_T}, \\
h & = & - \frac{H}{\rho w^2} \label{eq:h}, \\
\Omega^2 & = & \frac{w^2}{k_T g} \label{eq:Omega}, \mbox{ \hspace{0.3cm} and} \\
\tilde{\rho} & = & \rho/ \overline{ \rho}_+ ,\\
\end{eqnarray} 
one can rewrite equation~(\ref{eq:zeta1})  as 
\[
\frac{d}{d\tilde{y}}\left[\tilde{\rho} h \left( \frac{1}{M} - 1 \right)
\frac{d\zeta}{d \tilde{y}}\right]
\]
\begin{equation}  + \alpha_+^{-2} \zeta
\left\{ \tilde{\rho} h \left(1 - \frac{1}{\Omega^4 M}\right)
+ \frac{\alpha_+ }{\Omega^2} \frac{d}{d \tilde{y} }\left[
\tilde{\rho} \left(1 - \frac{h}{M}\right)\right]\right\} = 0.
\label{eq:zeta2} 
\end{equation} 
Solutions to equation~(\ref{eq:zeta2}) with 
\begin{eqnarray} 
\alpha_+ & \sim & {\cal O}(1), \mbox{\hspace{0.3cm} and} \\
\Omega & \sim & {\cal O}(1) 
\end{eqnarray} 
will now be found for the equilibrium
described in section \ref{sec:eq}, where $\sim {\cal O}(1)$ mean
``of order 1,'' whereas any positive power of $\epsilon$
is taken to be small.
To find such solutions, one expands $\zeta$ 
in powers of~$\epsilon$,
\begin{equation}
\zeta = \zeta^{(0)} + \epsilon \zeta^{(1)} + \epsilon^2 \zeta^{(2)}  + \dots,
\label{eq:zetap} 
\end{equation} 
solves for $\zeta$ for $\tilde{y} \gg
\epsilon$ (within the dense plasma), for $\tilde{y} \ll
-\epsilon$ (within the low-density plasma), and for $-1 \ll \tilde{y} \ll 1$
(within the boundary layer separating the two phases), and then
matches the solutions in the three regions at each order to obtain the
eigenvalues of $w$. These are given by
an expansion in powers of $\epsilon$,
\begin{equation}
w = w^{(0)} + \epsilon w^{(1)} + \epsilon^2 w^{(2)} + \dots,
\end{equation}
and thus $h$, $M$, and $\Omega$  are also expansions in
$\epsilon$. To determine $w^{(0)}$, it is
sufficient to find $\zeta^{(0)}$ in the three
regions and $\zeta^{(1)}$ in the boundary layer, as will now be shown.




\subsection{Solution for $\tilde{y} \gg \epsilon$}

For $\tilde{y} \gg \epsilon$, the values of $h$, $M$, $\alpha^+$, and
$\Omega$ in equation~(\ref{eq:zeta2})
are constant in $y$ and thus $\zeta$
is an exponential,
\begin{equation}
\zeta = \zeta_+ e^{\tilde{y} + i\tilde{k}_+ \tilde{y} },
\label{eq:zp} 
\end{equation} 
where 
\begin{equation}
\tilde{k}_+ ^2 = \frac{M_+ h_+ - 2\alpha_+(M_+ - h_+) \Omega^{-2} 
- h_+ \Omega^{-4}}{\alpha_+^2h_+(1 - M_+)} - 1,
\label{eq:ky+} 
\end{equation}
$h_+$ and $M_+$ are the constant values of $h$ and $M$ for
$\tilde{y} \gg \epsilon$ (Gratton et~al.\ 1988), and $\zeta_+$,
$h_+$, $M_+$, $\tilde{k} _+$ are expansions in powers of $\epsilon$.
For the surface modes considered in this paper,
$\tilde{k}_+^2$ is negative, and
$\tilde{k} _+$ is assigned 
the positive imaginary root so that the total kinetic
energy of the mode is finite. [There are in addition 
internal modes with
positive $\tilde{k} _+^2$ (Gratton et~al.\ 1988).]

\subsection{Solution for $\tilde{y} \ll -\epsilon$}

For $\tilde{y} \ll - \epsilon$, 
\begin{equation}
\zeta = \zeta_- e^{\tilde{y}(L_+/L_-) + i\tilde{k}_- \tilde{y} },
\label{eq:zm} 
\end{equation}
where
\begin{equation}
\tilde{k} _- ^2 = \frac{M_- h_- - 2\alpha_+(L_+/L_-)(M_- - h_-)\Omega^{-2}
- h_-\Omega^{-4}}
{\alpha_+^2 h_- (1-M_-)} - \frac{L_+^2}{L_-^2},
\label{eq:km} 
\end{equation} 
$h_-$ and $M_-$ are the constant values of $h$ and $M$ for
$\tilde{y} \ll -\epsilon$, and $\zeta_-$, $h_-$, $M_-$, and $\tilde{k} _-$
are expansions in powers of $\epsilon$.
In the case of interest, $\tilde{k}_-^2$ is negative, and
$\tilde{k} _-$ is assigned 
the negative imaginary root so that the total kinetic
energy of the mode is finite.



\subsection{Solution in the boundary layer at $\tilde{y} = 0$}

Let 
\begin{equation}
\tilde{y} \equiv \epsilon u.
\label{eq:u} 
\end{equation} 
In terms of $u$, equation~(\ref{eq:zeta2}) becomes
\[
\epsilon^{-2}\frac{d}{du}\left[\tilde{\rho} h \left( \frac{1}{M} - 1 \right)
\frac{d\zeta}{d u}\right] + 
\epsilon^{-1}\alpha_+^{-1}\Omega^{-2}\zeta \frac{d}{d u }\left[
\tilde{\rho} \left(1 - \frac{h}{M}\right)\right]
\]
\begin{equation} 
 + \alpha_+^{-2} \zeta \tilde{\rho} h \left(1 - \frac{1}{\Omega^4 M}\right)
 = 0
\label{eq:zeta3} 
\end{equation} 
Because the transition between the high- and low-density phases is taken
to occur over an interval in $\tilde{y}$ of width $\epsilon$,
one has 
\begin{equation}
\frac{d \tilde{\rho} } {du} \sim \frac{d h}{du} \sim \frac{dM}{du} \sim {\cal 
O}(1).
\end{equation} 
(Although $h_-$ and $M_-$ are large, they are considered $\ll \epsilon^{-1}$.)

At lowest order the solution to equation~(\ref{eq:zeta3}) is given
by
\begin{equation}
\tilde{\rho} h^{(0)} \left(\frac{1}{M^{(0)}} - 1\right) \frac{d\zeta _{\rm BL} 
^{(0)}}{du} = c_1,
\label{eq:zbl0} 
\end{equation} 
where $c_1$ is a constant. For the boundary-layer solution 
to match onto the outer solutions, $d\zeta _{\rm BL}^{(0)}/du$ must
vanish as $u\rightarrow \pm \infty$, so that 
\begin{equation}
c_1 = 0,
\label{eq:c1} 
\end{equation}
which implies that
\begin{equation}
\zeta _{\rm BL}^{(0)} = \rm constant.
\label{eq:zbl0.5} 
\end{equation} 

At next order  the solution to equation~(\ref{eq:zeta3}) can be
integrated to yield
\begin{equation}
\tilde{\rho} h^{(0)} \left(\frac{1}{M^{(0)}} - 1\right) \frac{d\zeta _{\rm BL} 
^{(1)}}{du}
+ \frac{\zeta _{\rm BL}^{(0)} \tilde{\rho} }{\alpha_+ (\Omega^{(0)})^2}
\left( 1 - \frac{h^{(0)}}{M^{(0)}}\right) = c_2,
\label{eq:zbl1} 
\end{equation} 
where $c_2$ is a constant.

\subsection{Asymptotic matching of the three solutions to obtain the eigenvalues 
of $w$}

The dense-plasma and boundary-layer solutions must match at each value
of $u$ within the interval
$1 \ll u \ll \epsilon^{-1}$. It is sufficient to match 
the two solutions at $u \sim \epsilon^{-1/2}$. 
The dense-plasma solution can be expanded within the matching
region as
\begin{equation}
\zeta_+ = \zeta_+^{(0)} + \epsilon u \zeta_+^{(0)} (1 + i \tilde{k} _+) + 
\epsilon \zeta_+^{(1)}
+ \dots
\end{equation}
The boundary-layer solution is 
\begin{equation}
\zeta_{\rm BL} = \zeta _{\rm BL} ^{(0)} + \epsilon \zeta _{\rm BL} ^{(1)} + 
\dots
\end{equation} 
Matching at order $\epsilon^0$ requires that
\begin{equation}
\zeta_+^{(0)} = \zeta_{\rm BL} ^{(0)} \equiv \zeta_0.
\label{eq:zeta0} 
\end{equation} 
>From  equation~(\ref{eq:zbl1}) it can be seen that $|\zeta _{\rm BL} ^{(1)}|$
increases linearly with $|u|$ when $|u|\gg 1$. Thus, 
matching at order $\epsilon^{1/2}$ requires
that 
\begin{equation}
\frac{d\zeta _{\rm BL} ^{(1)}}{du} = \zeta_0(1 + i \tilde{k} _+)
\mbox{ \hspace{0.3cm} for $ u \sim \epsilon^{-1/2}$} .
\label{eq:match1} 
\end{equation}

The matching region for the boundary-layer and low-density-plasma solutions is
taken to be $u \sim - \epsilon^{-1/2}$, in which the low-density-plasma solution
is
\begin{equation}
\zeta_- = \zeta_-^{(0)} + \epsilon u \zeta_-^{(0)} \left(\frac{L_+}{L_-} + i 
\tilde{k} _-\right)
+ \epsilon \zeta_-^{(1)} + \dots
\end{equation}
Matching at order $\epsilon^0$ requires that
\begin{equation}
\zeta_-^{(0)} = \zeta _{\rm BL} ^{(0)} = \zeta_0.
\end{equation}
At order $\epsilon^{1/2}$, matching requires that
\begin{equation}
\frac{d \zeta _{\rm BL} ^{(1)}}{du} = \zeta_0\left(\frac{L_+}{L_-} + i
\tilde{k} _-\right) \mbox{ \hspace{0.3cm} for $u \sim -
\epsilon^{-1/2}$}.
\label{eq:match2} 
\end{equation} 

Combining equations~(\ref{eq:zbl1}),~(\ref{eq:match1}), and~(\ref{eq:match2}) 
gives 
\[
\overline{ \rho}_+ h_+^{(0)}\left(\frac{1}{M_+^{(0)}} - 1\right)
(1 + i \tilde{k} _+) - \overline{ \rho}_-h_-^{(0)}\left(
\frac{1}{M_-^{(0)}} -1\right) \left(\frac{L_+}{L_-} + i \tilde{k}_-\right)
\]
\begin{equation} 
= -\frac{\overline{ \rho}_+}{\alpha^+ (\Omega^{(0)})^2}
\left(1 - \frac{h_+^{(0)}}{M_+^{(0)}}\right) + 
\frac{\overline{ \rho}_-}{\alpha^+ (\Omega^{(0)})^2}
\left(1 - \frac{h_-^{(0)}}{M_-^{(0)}}\right).
\label{eq:match3} 
\end{equation} 
This is the dispersion relation for $w^{(0)}$,
which is in general different from the result of
Gonzalez \& Gratton (1990) for an isothermal-plasma/vacuum interface,
but which reduces to their result in a special case of 
interest, as will now be discussed.

If one assumes that ${\bf k} $ is $\perp$ to $\bf B$ in 
the low density plasma, that 
$\overline{ \rho}_-/\overline{ \rho}_+ \ll 1$, and
that $M_{A-} \gg 1$, then 
$L_+/L_- \ll 1$,  and,  to lowest order in $\overline{ \rho}_-/\overline{ 
\rho}_+$,
$\tilde{k}_- = -ik_T L_+$ and [from
equation~(\ref{eq:match3})]
\begin{equation}
 h_+^{(0)}\left(\frac{1}{M_+^{(0)}} - 1\right)
(1 + i \tilde{k} _+)
= -\frac{1}{\alpha^+ (\Omega^{(0)})^2}
\left(1 - \frac{h_+^{(0)}}{M_+^{(0)}}\right).
\label{eq:2mass} 
\end{equation} 
Equation~(\ref{eq:2mass}) 
is the same as equation~(13) of Gonzalez \& Gratton~(1990)
when the magnetic field in the vacuum in their model, which occupies the
half-space $y<0$, is orthogonal
to ${\bf k}_T$. If one in addition assumes that $\cos \psi = 1$
in the dense plasma, which implies that the magnetic fields
in the high-density and low-density plasmas are orthogonal,
then equation~(\ref{eq:2mass}) has
the following roots,
\begin{equation}
w^2 = \frac{k_T^2 C_{A+}^2 (C_{A+}^2 + 2C_S^2) \pm
\sqrt{k_T^4 C_{A+}^8 + 4 g^2 k_T^2 (C_{A+}^2 + C_S^2)^2}}{2(C_{A+}^2+C_S^2)},
\label{eq:omega} 
\end{equation} 
which is the dispersion relation found by
Gonzalez \& Gratton (1990) for a plasma-vacuum interface,
when $\cos \psi = 1 $ in the plasma at $y>0$ and $\cos \psi = 0$
in the vacuum. The equivalence of the dispersion relations
in this special case arises because in both the plasma/vacuum
and plasma/plasma systems, field line bending at $y<0$ plays
no role, and the mode is governed by 
gravity and field-line bending within the plasma at $y>0$.
As shown by Gonzalez \& Gratton (1990),
the $-$ sign in equation~(\ref{eq:omega}) 
corresponds to an instability when 
\begin{equation}
k_T < \frac{g}{C_{A+}^2} \sqrt{1 + M_A^2}.
\label{eq:kcrit} 
\end{equation}
In the low-$\beta$ limit ($\beta \equiv 2/M_A^2$), there is an
instability provided
\begin{equation}
k_T \lesssim \frac{g}{C_{A+} C_S}.
\label{eq:kcrit2} 
\end{equation} 
Gonzalez \& Gratton (1990) derived formulas for the
most unstable wave number $k_{\rm m}$ and maximum growth rate $\gamma_{\rm m}$. 
In the low-$\beta$ limit these reduce to 
\begin{eqnarray} 
k_{\rm m} &  = & \displaystyle \frac{g}{C_{A+}^{3/2} C_S^{1/2}} \mbox{ 
\hspace{0.3cm} and}\\
\gamma_{\rm m} &  = & \frac{g}{C_{A+}^2}.
\end{eqnarray} 

For orthogonal fields in the high-density and low-density plasmas, 
it is expected that the most unstable mode satisfies ${\bf k}_T \perp
{\bf B}_{-}$, where ${\bf B}_{-}$ is the magnetic field in the
low-density plasma, since on average for $|y|<L_+$ the magnetic field
is stronger in the low-density plasma, and since the condition ${\bf
k}_T\perp {\bf B}_{-}$ implies no field-line bending within the
low-density plasma (Gratton et~al.\ 1996).  Thus,
equation~(\ref{eq:kcrit}) and its low-$\beta$ limit,
equation~(\ref{eq:kcrit2}), are expected to be necessary conditions for
instability

\section{Implications of stability analysis for the galactic-center magnetic 
field}
\label{sec:implications} 

In section \ref{sec:stability} the instability criterion
[equation~(\ref{eq:kcrit})] for surface modes of the infinite-slab
equilibrium of figure~\ref{fig:slab} was derived. In order to
extrapolate qualitatively from the slab-equilibrium to the
two-dimensional equilibrium of figure~\ref{fig:equil}, it is assumed
that in the equilibrium of figure~\ref{fig:equil} the scale height of
the molecular material in the radial direction in the Galactic plane
is approximately the same as in the vertical direction.  It is also
assumed that instabilities of the infinite-slab equilibrium at
wavelengths  $\ll L_+$ are present in the two-dimensional
equilibrium of figure~\ref{fig:equil}, while instabilities of the
infinite-slab equilibrium at wavelengths $\gtrsim L_+$ are not relevant
to the 2D equilibrium since the vertical structure of the 2D
equilibrium would be important on such scales.  Given these
assumptions, equations~(\ref{eq:kcrit}) and (\ref{eq:kcrit2}) suggest
that the equilibrium of figure~\ref{fig:equil} is unstable to global
gravitational modes when the magnetic pressure of the vertical
magnetic field $B_{\rm vert}$ in the hot medium greatly exceeds the
thermal pressure of the molecular clouds $p_{\rm cloud}$.  The reason
is that when $B_{\rm vert}^2/8\pi \gg p_{\rm cloud}$, pressure balance
across the cloud/hot-plasma interface requires that $B_{\rm cloud}^2
/8\pi\gtrsim B_{\rm vert^2}/8\pi$, and that therefore $\beta_{\rm
cloud} \equiv 8\pi p_{\rm cloud}/ B_{\rm cloud}^2 \ll 1$. If
$\beta_{\rm cloud} \ll 1$, then equations~(\ref{eq:L+}) and
(\ref{eq:kcrit2}) imply that there are gravitational surface
instabilities at wavelengths significantly smaller than~$L_+$. For a
cloud density of $10^4 \mbox{ cm}^{-3}$ and temperature of
$70$~K, stability requires that the vertical magnetic field
not be $\gg 50\mu$G.

Given the assumptions underlying this argument, the condition that
$B_{\rm vert}$ can not be $\gg 50\mu$G is only suggestive. Moreover,
only one possible equilibrium has been considered, and other
equilibria might be stable. However, the arguments presented indicate
that it is not straightforward to use the gravitational weight of the
molecular material to confine milliGauss vertical magnetic fields.


\section{Dynamical estimates of the field strengths in radio filaments}
\label{sec:review} 

All of the radio filaments that have been studied in detail near the
Galactic center appear to be interacting with at least one randomly
moving molecular
cloud, and yet most are undeformed at the sites of interaction (Morris
\& Serabyn 1996). If the magnetic field were
dynamically insignificant, one would expect the turbulent
motions of the clouds to cause turbulent bulk motions in the
hot plasma that would deform the frozen-in magnetic field lines.
The lack of deformation gives a lower limit on the field strength in the
filaments that depends upon how many independently
moving clouds (or clumps within a cloud) interact with a single
filament, and also upon the details of how the clouds and filaments
interact.

If a vertical filament interacts with a single molecular cloud at a
single point along its length, then the lower limit on its field
strength is fairly small.  Since the vertical field $B_{\rm vert}$
threads the very tenuous hot plasma, there is very little mass to
anchor the ends of the vertical field line above and below the
Galactic disk. Any deformation in the field line in the galactic plane
by an interaction with a molecular cloud is carried away along the
field line at the Alfv\'en speed of the hot medium, $C_{A, {\rm
hot}}$.  In order for the visible deformations in the filaments to be
small, the Alfv\'en speed in the hot medium must be much larger than
the turbulent speed of the the clouds $V_{\rm cloud}$ with respect to
the ends of the filament.  The angle at which a field line is bent
will be $\sim V_{\rm cloud}/C_{A, {\rm hot}}$, as depicted in
figure~\ref{fig:deformation}.  The condition $C_{A, {\rm hot}} \gg
V_{\rm cloud}$ can be written
\begin{equation}
B_{\rm vert} \gg   n_{\rm hot}^{1/2} \left(\frac{V_{\rm cloud}}{2 \mbox{ 
km/s}}\right) \mu {\rm G},
\label{eq:b1} 
\end{equation} 
where $n_{\rm hot}$ is the number density of the hot plasma.
If $V_{\rm cloud} = 30$ km/s, and $n_{\rm hot} = 0.3 \mbox{ cm}^{-3}$,
then $B_{\rm vert}\gg 8 \mu$G. 
\begin{figure*}[h]
\vspace{10cm}
\special{psfile=f3.eps  voffset=50 hoffset=110
                          vscale=70    hscale=70     angle=0}
\caption{Because the field lines in the hot low-density plasma are not anchored
above or below the Galactic plane, they remain almost
straight if they interact with a cloud at only a single point along
their length provided the Alfv\'en speed in the hot plasma $C_{A, {\rm hot}}$
is $\gg V_{\rm cloud}$. \label{fig:deformation}   }
\end{figure*}


As pointed out by Yusef-Zadeh \& Morris (1987) and Morris \& Serabyn (1996), 
if a filament interacts with clouds  at more than one point
along its length, as happens in the Radio Arc, 
then the lower limit on the field strength may be considerably greater than
in equation~(\ref{eq:b1}).
If two different clouds push in
different directions on a single filament, then to avoid
deformation the filament must either slip rapidly around the clouds
or dynamically oppose the force applied by the cloud.
In addition, if each cloud consists of independently moving clumps,
then a filament interacting with a single cloud may be simultaneously
forced in multiple directions at different points along its length
(Morris \& Serabyn 1996).
The minimum field strength implied by the observations then depends
upon the degree to which filaments can slip around clouds, the
amount of magnetic tension required for a filament to slice
through a cloud, and also the difference in the clouds' velocities
at different points along a single filament (i.e., on a single
line $\perp$ to the Galactic plane, the velocity dispersion 
may be less than the overall velocity dispersion).
If the condition for avoiding deformation is that 
the magnetic pressure must exceed the ram pressure of 
the clouds, then $B_{\rm vert} > 1 $ mG (Yusef-Zadeh \& Morris 1987).

\section{Conclusion}
\label{sec:conclusion} 

This paper addresses the question of whether a pervasive milliGauss
vertical magnetic field within the central 150 pc of the Galaxy can be
confined. A simple 2D model for the equilibrium in this central region
is proposed in which horizontal magnetic fields threading a ring of
molecular material enclose vertical magnetic fields embedded in hot plasma
(figure~\ref{fig:equil}). Since orthogonal
magnetic fields can not easily interpenetrate, the horizontal magnetic
field in this equilibrium can transfer the weight of the molecular
material to the hot plasma and vertical magnetic field,
potentially providing confinement. The stability of such an
equilibrium is studied indirectly by treating the equilibrium as an
infinite slab, with dense plasma (representing the partially
ionized molecular clouds) suspended above rarefied plasma, with
the orthogonal magnetic fields in the dense and rarefied plasmas in the $xz$
plane, and with the effective acceleration of gravity $g$ (true
gravitational acceleration minus the centripetal acceleration of
Galactic rotation) in the $-\hat{\bf y}$ direction.  Perturbations of
the interface between the dense and rarefied plasmas are found to be
unstable on wavelengths shorter than a critical wavelength that
depends upon $g$ and the sound and Alfv\'en speeds in the dense
plasma. The results of the slab-equilibrium calculation are
extrapolated on a qualitative basis to argue that the two-dimensional
equilibrium of figure~\ref{fig:equil} is unstable if the magnetic
pressure of the vertical magnetic field $B_{\rm vert}$ greatly exceeds
the thermal pressure in the molecular material, or if $B_{\rm vert}
\gg 50\mu$G for Galactic-center parameters. 
If this qualitative argument is correct, then
milliGauss vertical fields can not be stably confined using the weight
of the molecular material in the kind of equilibrium that has been
proposed. If an equilibrium can not be found that can stably confine
pervasive vertical milliGauss magnetic fields at the Galactic center,
then a powerful argument will be added in favor of other
interpretations (Morris 1996, Morris \& Serabyn 1996) of the Galactic-center
radio filaments.


\acknowledgements{I would like to thank Steve Cowley, Mark Morris,
and Eric Blackman for valuable discussions.}


\section{References}

Bally, J., Stark, A., Wilson, R., and Henkel, C. 1988, ApJ, 324, 223

Chandran, B. D. G., Cowley, S. C., \& Morris, M. 2000, ApJ, 528, 723

Gonzalez, A., and Gratton, J. 1990, Plasm. Phys. Contr. Nuc. Fus.,
32, 3

Gratton, F. T., Farrugia, C. J., \& Cowley, S. W. H. 1996, J. Geophys. Res., 
101, 4929

Gratton, J., Gratton, F., Gonzalez, A. 1988, Plasma Phys. Contr. Nuc. Fus.,
30, 435

Howard, A., \& Kulsrud, R. 1997, ApJ, 483, 648


Koyama, K., Maeda, Y., Sonobe, T., Takeshima, T., Tanaka, Y., \&
Yamauchi, S. 1996, Publ. Astron. Soc. Japan, 48, 249

Kulsrud, R. M., Cen, R.,  Ostriker, J. P.,  Ryu, D. 1997, ApJ, 480, 481

Morris, M. 1996, in {\em Unsolved Problems of the Milky Way},
eds. L. Blitz and P. Teuben (Netherlands: IAU,  p. 247)

Morris, M., and Serabyn, E. 1996, Ann. Rev. Astron. Astrophys., 34, 645

Reich, W. 1994, in {\em Nuclei of Normal
Galaxies: Lessons from the Galactic Center}, eds. Genzel, R.,
Harris, A. I. (Dordrecht: Kluwer, p. 55)

Sofue, Y., \& Fujimoto, M. 1987, Publ. Astron. Soc. Jpn., 39, 843

Tsuboi, M., Inoue, M. Handa, T., Tabara, H., Kato, T. et~al.\ 1986, 
Astron. J., 92, 818

Tsuboi, M., Kawabata, T., Kasuga, T., Handa, T., Kato, T. 1995,
Publ. Astron. Soc. Jpn, 47, 829


Yamauchi, S., Kawada, M., Koyama, K., Kunieda, H., \&  Tawara, Y. 1990,
ApJ, 365, 532

Yusef-Zadeh, F., and  Morris, M. 1987, Astron. J., 94, 1178

Zweibel, E., \& Heiles, C. 1997, Nature, 385, 131

\end{document}














































































































































































































------------- End Forwarded Message -------------