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    {\small\rm\zihao{6}\songti\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm \zihao{6}\songti suppl.\hfill Y. Q. Lou: Circumnuclear Starburst Rings in Spiral Galaxies  %%（作者：题目名称）
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\zihao{5}\songti

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    \zihao{6}\songti
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    Vol.44~ Suppl.
    \hfill\hspace{8mm} %%% if ``October/January'' then \hspace{8/10 mm} July
    \small ACTA ASTRONOMICA SINICA
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      Feb.,~2003 \\[-1.5 true mm] %%January
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\vs\vs\vs \bc {\Large\bf
Circumnuclear Starburst Rings in Spiral Galaxies   %%（文题）
%\\[1ex]
%$\!\!^*$
}

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\vs\vs{\normalsize Y.Q. Lou}  %%（本文作者）

\vs
 (\nao)\\
 ({\footnotesize {\it Physics Department,
           The Tsinghua Center for Astrophysics,
           Tsinghua University, Beijing}~ 100084}) \\
 ({\footnotesize {\it Department of Astronomy and Astrophysics,
The University of Chicago,
5640 South Ellis Avenue, Chicago, Illinois }60637, USA )\\
Email: lou@oddjob.uchicago.edu, louyq@tsinghua.edu.cn }
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
We advance a physical scenario for the origin of circumnuclear
starburst rings on kiloparsec (kpc) scales in Seyfert galaxies that
involve gravitationally excited spiral fast magnetohydrodynamic (MHD)
density waves at the modified inner Lindblad resonance (mILR) in a
central magnetized gas disk. Within corotation, an effective damping
of these sustained fast MHD density waves will continuously reduce the
disk angular momentum and thus lead to a ringlike accumulation of gas
materials and magnetic flux inside the mILR which, during a critical
stage, would be vulnerable to massive star formation at smaller scales.
Depending on the effectiveness of MHD density wave damping, several
qualitatively different outcomes for central activities in Seyfert
galaxies may develop.
%(1) a very active starburst ring with a less active
%core for a strong damping;
%(2) a moderate starburst ring with a moderate core
%for an intermediate damping; and
%(3) a prominent active galactic nucleus with a much weaker starburst
%ring for a very weak circumnuclear wave damping on kpc scales.

\end{quotation}

\para{\large\bf{1}~~ A Brief Overview}

\vs\vs
We mainly focus on circumnuclear starburst activities on kiloparsec
(kpc) scales in Seyfert galaxies.
%although starburst activities are most violent and
%dramatic in colliding galaxies.
We shall use multiwavelength observations
of the starburst ring in the nearby barred spiral galaxy NGC1097 as an
example to illustrate the basic idea. We emphasize that magnetohydrodynamic
(MHD) density waves should play an important role and therefore
radio-continuum observations in the circumnuclear region with high
resolution and sensitivity are extremely valuable in understanding
the formation and dynamics of circumnuclear starburst rings. Here,
we offer several important clues that lead to our scenario on
circumnuclear starburst rings in spiral galaxies.

The density wave theory initiated by Lin \& Shu (1964)
%, nearly four decades ago,
has stimulated fruitful research on various aspects of
large-scale structures and dynamics of spiral galaxies.
%(Binney \& Tremaine 1987; Lin 1987; Bertin \& Lin 1996).
There is now a
growing appreciation for the density wave theory in understanding the
disk stability, the angular momentum transport and the mass transfer
in a differentially rotating disk. In essence, if one treats a stellar
disk as a fluid, a spiral density wave involves epicyclic oscillations,
self-gravity, and stellar velocity dispersion mimicked as an effective
sound speed in a differentially rotating disk. The dispersion relation
can be readily derived in the WKBJ or tight-winding approximation. A
density wave carries a negative or positive angular momentum inside or
outside the corotation radius, respectively.

In a series of papers by Goldreich \& Tremaine (1979, 1982),
%(1978a,b; 1979; 1982),
the galactic density wave theory was adapted to the Keplerian dust disk
around the Saturn. Specifically, the gravitational potential perturbation
of the Mimas satellite excites outgoing long trailing density
waves at the inner Lindblad resonance (ILR). Without dissipations, such
waves would be reflected at the so-called $Q-$barrier in the form of
incoming short trailing density waves. By inevitable viscous damping
in the dust disk of Saturn, these density waves, carrying negative
angular momentum within corotation, effectively remove angular momentum
from the disk. During the lifetime of the solar system, dust materials
in the disk have been gradually cleared up to form the widest Cassini
division. Likewise, other satellites of Saturn give rise to
corresponding gaps at appropriate resonance locations to form the
Saturn ring system we observe today.

Synchrotron radio emissions reveal magnetic fields and relativistic
cosmic-ray electrons in spiral galaxies. For nearby spiral galaxies,
which can be mapped in details, total radio-continuum emissions
show large-scale spiral structures that largely follow optical spiral
arms. Likewise, polarized radio-continuum emissions also show
large-scale spiral structures in associations with optical spiral
arms, yet with two distinct possibilities. In the case of M51 (e.g,
Neininger 1992; Berkhuijsen et al. 1997),
polarized radio emissions are
strongest along the dust lanes at the inner edges of optical arms. In
contrast, for NGC 6946 (e.g., Beck \& Hoernes 1996; Frick et al. 2000),
polarized spiral arms are interlaced with the optical spiral arms. With
these facts in mind, we proposed (Fan \& Lou 1996; Lou \& Fan 1998) the
scenario of fast and slow MHD density waves (FMDWs and SMDWs) that can
exist in the magnetized disk of interstellar medium (ISM). Recently, we
have further shown that SMDWs can also persist extensively in a disk
with a flat rotation curve in the form of stationary logarithmic spirals
with displaced optical and magnetic arms (Lou \& Fan 2002; Shen \& Lou
2003).

As a thin circumnuclear disk involves magnetic field and differential
rotation, it is natural to study the roles of circumnuclear MHD
density waves, their excitation at the ILR, and their effective
damping to form circumnuclear starburst rings on kpc scales and
central AGNs (Lou et al. 2001).

\para{\large\bf{2}~~ Theoretical Formulation and Main Results}

\vs\vs
With a high stellar velocity dispersion ( $^>_\sim$ 100\,km\,s$^{-1}$)
in the central bulge and a low gas temperature, the circumnuclear
region of a spiral galaxy contains a thin
%\footnote{Stability against bar-type instabilities is meant.
%Starbursts involve gravitational collapses on subscales along an
%accumulated circular zone (e.g., Elmegreen 1994).}
gas disk within a few kpcs. The gas rotation curve
in an extended radial range is determined by the total mass
distribution in the galaxy (Meaburn et al. 1981; Blackman 1981;
Gerin et al. 1988; Ondrechen et al. 1989) and the {\it background}
magnetic field in the circumnuclear disk plane of a strength $^>_\sim$
several tens of $\mu$G (Beck et al. 1999) is presumed to be axisymmetric
and azimuthal
%\footnote{It is an
%idealization to posit a purely azimuthal background magnetic field,
%based on which we perform analysis for spiral MHD density waves.
%At much larger radii into the barred region, the observed
%large-scale regular magnetic field strongly deviates from
%an azimuthal orientation. Even
%In the circumnuclear domain
%under consideration, observed magnetic field deviations
%are significant. The results of a perturbation analysis
%is only indicative. The effectiveness of this approach may
%be questionable in the fully nonlinear realm.}
to avoid the magnetic-field winding dilemma.
%(Roberts \& Yuan 1970). Lou \& Fan 1998)
%The observed circumnuclear pattern involves spiral FMDWs
%in such a background and is thus nonaxisymmetric and unsteady.
This circumnuclear magnetized gas disk is embedded with molecular
clouds (Gerin et al. 1988) with a mean-free path $l_c$ and a velocity
dispersion $v_c$. Spiral FMDWs involve collectively the magnetized
gas disk embedded with clouds. For NGC 1097, the large-scale
{\it barred spiral structure outside} the central gas disk rotates at
a pattern speed $\Omega_p$ and gives rise to an {\it external} periodic
gravitational potential $\phi^E$ as felt by the magnetized central disk.
%; this effect would be more pronounced when the large-scale
%galactic spiral structure contains an extended massive
%conspicuous bar (Ondrechen et al. 1989; Barth et al. 1995;
%Beck et al. 1999).
In cylindrical coordinates $(r,\theta,z)$, our model is developed
using the standard two-dimensional equations for coplanar MHD density
waves in the central magnetized gas sheet (Lou \& Fan 1998) at $z=0$
with $\phi^E$ varying slowly in $r$ as the inhomogeneous driving term.
%(Goldreich \& Tremaine 1978a,b; Yuan \& Cheng 1991).

Without $\phi^E$ driving and away from the resonances, the
dispersion relation for free FMDWs in the WKBJ or
tight-winding regime is
%(Shu 1970; Fan \& Lou 1996) is
$$
(\omega-m\Omega)^2\approx\kappa^2+k^2
(C_A^2+C_S^2+v_c^2-2\pi G\mu_{0}/|k|)\ ,\eqno(1)
$$
where $v_c^2$ mimics an ``effective pressure'' due to
random cloud velocities $v_c$, $\omega$ is the angular frequency
in an inertial frame of reference, $m\geq 0$ is an integer for
the number of spiral arms, $\Omega$ is the disk angular speed,
$\kappa\equiv [(2\Omega/r)d(r^2\Omega)/dr]^{1/2}$ is the
epicyclic frequency, $G$ is the gravitational constant, $\mu_{0}$
is the background surface mass density, $k$ is the radial
wavenumber, $C_S$ is the sound speed, and $C_A$ is the Alfv\'en
speed (Lou \& Fan 1998). The relevant FMDW amplitude equation
(Fan \& Lou 1999) is
$$
{d\over dr} \bigg [ {r\mu_0 k\over (\omega-m\Omega)^2}
\bigg(C_S^2+v_c^2+C_A^2-{\pi G \mu_0 \over |k|}\bigg)
|\tilde v_r|^2\bigg]=0\ ,\eqno (2)
$$
where $\tilde v_r(r)$ is the magnitude of the radial velocity $v_r$.
Equation (1) contains the familiar short- and long-branches of FMDWs
(Lou \& Fan 1998). For FMDWs, one has $B_{\theta}\mu\cong\mu_{0}b_{\theta}$,
where $\mu$ is the surface mass density perturbation, and $b_{\theta}$ and
$B_{\theta}$ are the perturbation and background azimuthal magnetic fields,
respectively.
%the enhancements of $\mu$ and $b_{\theta}$ are in phase.
%(Fan \& Lou 1996; Lou \& Fan 1998).

The MHD stability against axisymmetric ring fragmentation requires
%the MHD version of Toomre's $Q$ parameter
$Q_M\equiv \kappa (C_A^2+C_S^2+v_c^2)^{1/2}/(\pi G\mu_{0})>1$ in a
magnetized self-gravitating gas disk (Lou \& Fan 1998, 2002;
Shen \& Lou 2003). The radial group velocity of FMDWs becomes
$$
F_G=-{\partial\omega\over\partial k}
\cong -{\hbox{sgn}(k)[(C_A^2+C_S^2+v_c^2)|k|-\pi G\mu_{0}]
\over (\omega-m\Omega)}\ ,\eqno(3)
$$
with $F_G>0$ and $F_G<0$ for outgoing and incoming waves.
Here, $k>0$ and $k<0$ represent leading and trailing spirals.
Inside corotation of FMDWs, wave energy travels away from the
disk center for long-trailing and short-leading FMDWs; wave
energy travels toward the disk center for long-leading and
short-trailing FMDWs. Outside corotation of FMDWs, the
directions of energy flow reverse for all these FMDW types.

The angular momentum flux carried by a FMDW now becomes
%(Fan \& Lou 1999)
$$
{\cal F}_{J}=-{\pi mkr\mu_0\over (\omega-m\Omega)^2}
\bigg(C_S^2+v_c^2+C_A^2-{\pi G\mu_0\over |k|}\bigg)|
\tilde v_r|^2\ ,\eqno (4)
$$
with the surface densities of angular momentum ${\cal J}^F
\equiv m\mu_{0}|\tilde v_r|^2/[2(\omega-m\Omega)]$, of energy
${\cal E}^F=\omega {\cal J}^F/m$ and of wave action
${\cal N}^F={\cal J}^F/m$. Inside and outside corotation,
${\cal J}^F$ is negative and positive, respectively. By
eqns (2) and (4), the angular momentum flux of free FMDWs is conserved.
The corotation at $\omega-m\Omega(r_c)=0$ is forbidden to wave
access by the $Q_M$-barrier when $Q_M$$^>_\sim 1$, and in the WKBJ
regime, the slightly modified Lindblad resonances occur at
$\Gamma\equiv\kappa^2-(\omega-m\Omega)^2+m^2C_A^2f(r)/r^2=0$
with $f(r)\sim {\cal O}(1)$ being a dimensionless factor.
Outside the $Q_M$-barrier, long-FMDWs exist only within
the two modified Lindblad resonances, while short-FMDWs propagate
within and outside the two modified Lindblad resonances as in the
hydrodynamic case.
%(Goldreich \& Tremaine 1978a).

By a massive bar, a satellite, or a large-scale spiral structure,
an external gravitational potential $\phi^{E}(r,\theta, t)$ felt at
the central gas disk induces FMDWs at the modified Lindblad
resonances $r_M$. Around $r_M$, one writes $x\equiv (r-r_M)/r_M$ and
$\Gamma\equiv {\cal G}_Mx$ with $\hbox{sgn }{\cal G}_M=\pm 1$ for
{\it modified inner and outer Lindblad resonances} (mILR and mOLR).
In the WKBJ limit, the set of inhomogeneous FMDW equations
for small $x$ may be reduced to the familiar form in terms of the
disk self-gravity potential $\phi$ as
$$
{d^2\phi\over dx^2}-i\alpha_M{d\phi\over dx}-\beta_M x\phi
=i\alpha_M\Psi\ ,\eqno(5)
$$
where
$
\alpha_M\equiv -[2\pi G\mu_{0}r /(C_S^2+v_c^2+C_A^2)]_{r_M}
\hbox{sgn}(k),
$
$
\beta_M\equiv [r^2{\cal G}_M/(C_S^2+v_c^2+C_A^2)]_{r_M},
$
and
%the driving term
$
\Psi\equiv d\phi^{E}/dx+2m\Omega\, \phi^{E}/(m\Omega-\omega).
$
The solutions of eq (5) involving Airy functions with proper energy
flow directions are known; the physics of these solutions studied
previously (without magnetic field and clouds) in contexts of
large-scale galactic density waves
(Goldreich \& Tremaine 1979)
%Yuan \& Kuo 1997)
and of planetary rings (Goldreich \& Tremaine 1982) can now
be extended and applied to a circumnuclear magnetized gas disk of
a spiral galaxy.
%(our sign convention in this paragraph now)
%By (5), the magnitude of a FMDW is proportional to
%the strength of $\Psi$ involving $\phi^E$.

%In addition to a nonwave solution part (Goldreich \& Tremaine
%1978a) of
By solutions of eq (5), long-trailing FMDWs are dominantly
excited by $\phi^E$ at mILR and mOLR; these waves exist
between the two
%modified Lindblad
resonances and
propagate toward corotation after their generations at mILR
and mOLR. Reflections at the $Q_M$-barrier transform these
long-trailing FMDWs to short-trailing
FMDWs that travel away from corotation and continue across the mILR
and mOLR.
%if dissipations were not strong.
In the case of mILR, long-trailing FMDWs sustained by $\phi^{E}$ carries
a {\it negative} angular momentum toward corotation; a subsequent
reflection at the $Q_M$-barrier gives rise to short-trailing FMDWs again
carrying a {\it negative} angular momentum that may reach the disk center
until being disrupted by small-scale ($^<_\sim$ a few pc) nuclear processes.

\para{\large\bf{3}~~ Circumnuclear Spiral Arms and Starburst ``Ring" of NGC 1097}

\vs\vs NGC 1097 is a nearby barred spiral galaxy
%(Beck et al. 1999) in the southern sky
with a declination of $\sim -30^{\circ}$ and an inclination estimated
to range from $\sim 37^{\circ}$ to $57^{\circ}$. It is of Seyfert
%(both Seyfert 1 or 2 mentioned in the literature$^{1, 9}$)
type with an AGN known as a LINER
%low-ionization nuclear emission-line region;
(Barth et al. 1995) and might host a
supermassive black hole (SMBH) of
$\sim 2\times 10^{10}M_{\odot}$ (Rickard 1975). Estimates of
its distance range from $11.6-17$ Mpc.
It has a bright circumnulear starburst ``ring"
%(actually two-armed spirals in a high-resolution {\it HST}
%image of Barth et al. 1995)
with a diameter of $\sim 18''$ and a far infrared (FIR)
luminosity of $\sim 3\times 10^{10}L_{\odot}$ (Telesco 1988;
Gerin et al. 1988), forming stars at a
rate of $\sim 5\,M_{\odot}\,{\rm yr}^{-1}$ (Hummel et al. 1987).
The rotation curves were determined for $r\sim$ 4''---20'',
using H$_{\alpha}$\ 6563\AA\,and [N\,{\scriptsize II}]
6584\AA \ (Meaburn et al. 1981; Blackman 1981) as well as
CO (J=$1-0$) (Gerin et al. 1988), and for $r\sim $20''---6',
using H\,I 21\,cm despite a strong depletion in the bar
region (Ondrechen et al. 1989). For 10''$^<_\sim r^<_\sim 20''$, the
inferred rotation speed $V_{\theta}\sim 300\,{\rm km s}^{-1}$
%$V_{\theta}\sim 356\,{\rm km s}^{-1}$
remains roughly constant. The two dark dust lanes originating
at $\sim 4''-5''$ follow along the inner edges of the two
spiral arms and extend continuously out to the bar (Rickard 1975)
with large arm pitch angles of $\sim 50^{\circ}$ there. Shown in
Fig. 1 is our wavelet analysis (Lou et al. 2001) on an {\it HST}
image of central NGC 1097 (Barth et al. 1995) that reveals two
distinct circumnuclear spiral arms extended inward to $^<_\sim 1.5''$.

Synchrotron emissions (Ondrechen \& van der Hulst 1983)
%Hummel et al. 1987)
at 20 cm reveal that the intensity contour closely follows the
dust lanes along the bar and merges into the central region of
$r\sim 30''$ to $35''$. The total radio intensity ``ring" overlaps
with the circumnuclear optical/infrared ``ring" (Hummel et al. 1987;
Telesco 1988), implying intense magnetic fields and profuse
cosmic-ray electrons. Polarized radio emission at 22, 18, 6.2 and
3.5 cm (Beck et al. 1999) indicates that regular magnetic fields
are parallel to the two dust lanes along the bar and gradually
swirl into a trailing pattern. Relative to the circumnuclear
``ring", magnetic fields are inclined by $\sim 50^{\circ}$.
%(Beck et al. 1999).
Despite the limited resolution, it is indicative that a
trailing spiral magnetic-field pattern inside the
``ring" continues toward the nucleus. By this evidence
and our Fig. 1, it is clear that spiral FMDWs (Fan \&
Lou 1996; Lou \& Fan 1998, 2002) play a significant dynamic role in
the circumnuclear region of a barred spiral galaxy, and magnetic
fields provide an important radio diagnostics.
%The fact that in the ``ring" of NGC 1097, the radial and azimuthal
%components of magnetic field are comparable (Beck et al. 1999)
%means that the magnetic-field swirl pattern is not so tightly
%wound there. This perhaps relates to long-trailing FMDWs that are
%excited and sustained by $\phi^E$ at the mILR, and spiral FMDWs
%involve a {\it nonaxisymmetric} radial magnetic field $b_r$.
%Fig. 1a of Beck et al. (1999) with 15" resolution shows a central
%swirl with some axisymmetry, while their Fig. 2 with 6" resolution
%shows is already indicative of a central spiral pattern with
%nonaxisymmetry. The apparent central ``axisymmetric swirl" of their
%Fig. 1a most likely results from a larger beam size.
%otherwise the divergence-free condition of
%a coplanar magnetic field would be violated.
%In radio polarization studies of large-scale magnetic fields in
%nearby spiral galaxies, the earlier more regular ``axisymmetric"
%swirl patterns were later replaced by {\it nonaxisymmetric} spiral
%patterns. As an example, one may compare Fig. 7 of Harnett et al.
%(1989), Fig. 5 of Ehle \& Beck (1993), and Fig. 2 of Beck \&
%Hoernes (1996) in the case of NGC 6946.
For orientations along magnetic spiral arms,
%(not so well defined yet),
the tendency that magnetic field of central NGC 1097
seems to be axisymmetric instead of bisymmetric within
the circumnuclear ``ring" region
%(in the sense of direction as opposed to bisymmetric magnetic field)
may be understood from the intuitive picture of distorted unaligned
magnetic-field rings for FMDWs (Lou \& Fan 1998, 2002) analogous to
that of distorted stellar orbits for density waves (Kalnajs 1973).

For NGC 1097, the mean magnetic field $B_{\theta}$
threads through the central thin H{\scriptsize I} gas disk
%(Ondrechen \& van der Hulst 1983; Ondrechen et al. 1989; Beck et al. 1999)
embedded with H$_2$ molecular clouds.
%(Gerin et al. 1988).
Spiral FMDWs interact with the magnetized H{\scriptsize I} gas
%\footnote{A partial ionization and sufficiently frequent collisions
%between neutral H{\scriptsize I} gas and charged particles are presumed here.
%such that magnetic field affects large-scale H{\scriptsize I} gas dynamics.}
and H$_2$ clouds together. Random motions of H$_2$ clouds provide an
``effective pressure" and a damping mechanism for FMDWs. The mILR is
identified at $r\sim 10''$, the outer rim
%\footnote{For the Cassini division in
%the Saturn ring system (Goldreich \& Tremaine 1978a), particles
%outside the Mimas' 2:1 ILR have been cleared by density waves
%over 5 billion years. In the disk region outside the
%circumnuclear starburst ring of a barred spiral galaxy, there
%may be some mechanisms in operation to replenish the gas disk
%to avoid an outright gap.}
of the circumnuclear ``ring". For $m=2$, $\Omega$ at mILR is
$2+\sqrt{2}$ times $\Omega_p$ in a flat rotation curve, the
corotation is located around 30''$^<_\sim r ^<_\sim$ 35'', and
the mOLR would be around 50''$^<_\sim r ^<_\sim $60''.
%(Meaburn et al. 1981; Blackman 1981; Ondrechen et al. 1989).
%The latter two estimates are only indicative, as both stellar
%and magnetized gas disks are involved at larger $r$.

\begin{figure}[h]
\center {\includegraphics [bb=0 0 561 516,scale=0.6]{spiralc.eps}}

   \parbox[tb]{13cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,1~~ A wavelet transformed and reconstructed WFPC2/HST (F555W
filter) image in false color for the central spiral arms around and
within the circumnuclear starburst ``ring" of NGC 1097 showing a
less than 2'' angular scale.}

    \end{figure}
Our emphasis is on wave processes around the mILR. Long-trailing
FMDWs are excited and sustained at the mILR by $\phi^{E}$ and
propagate toward corotation. They are reflected by the
$Q_M$-barrier around corotation and travel backward to the mILR
as short-trailing FMDWs. Inside corotation, both
long- and short-trailing FMDWs carry {\it negative} angular
momenta (Fan \& Lou 1999). By dissipations, these FMDWs
damp and deposit in the disk their negative angular momentum.
As the disk angular momentum is reduced continuously, disk
material outside the mILR gradually spirals inward, bringing
along magnetic flux meanwhile. By this process, disk materials
(H{\scriptsize I} and H$_{2}$ etc) and magnetic flux would accumulate
inside the mILR. Regions of high gas density and enhanced magnetic
flux naturally favor births of bright young massive stars as well
as star clusters (Elmegreen 1994); this gives rise to a
circumnuclear starburst ``ring" inside the mILR. Outside the
mILR, the strength of a long-trailing FMDW is stronger than
that of a returning short-trailing FMDW by wave damping.

For estimates, we take
$v_c\sim 35\,{\rm km\,s}^{-1}$ (Gerin et al.
1988)
%\footnote{There seems to be no escape for a high Mach number
%of molecular clouds in the gas disk. Resonance processes might
%continuously convert FMDW energy into random cloud motions.}
%the CO fitting would not be sensitive to this
%parameter; could range from 20 to 50\,km/s.
and $l_c\sim 100$ pc. The gross turbulent
dissipation coefficient $\nu_c$
%(including viscosity and resistivity)
is $\nu_c\sim v_cl_c/3\cong 3\times 10^{26}\,{\rm cm}^2\hbox{ s}^{-1}$.
For a $B\sim 40\mu$G (Beck et al. 1999), a
H{\scriptsize I} gas column density of $^<_\sim 10^{21}\,{\rm cm}^{-2}$
(Ondrechen et al. 1989), a H$_2$ gas column density
$\sigma_{0}\sim 6\times 10^{22}\,{\rm cm}^{-2}$ (Gerin et al. 1988)
%Kotaro 2000, private communication)
%(a surface mass density $\mu_{0}\sim 0.2\hbox{ g cm}^{-2}$)
and a disk thickness $h ^<_\sim 100$ pc, the Alfv\'en speed is
$C_A\sim 4\times 10^5\,{\rm cm s}^{-1}$. For a gas temperature
$T ^<_\sim 100\hbox{ K}^{\circ}$, the sound speed is $C_S$$^<_\sim
10^5\,{\rm cm s}^{-1}$.
%A rule of thumb for active star
%formation in a magnetized gas disk is
%$2\pi G^{1/2}\Sigma/B\gsim 1$ with $\Sigma$ being the surface
%mass density (Shu 2001, private communications). Given the
%above parameters, $2\pi G^{1/2}\Sigma/B\sim 8$.
The $Q_M$ estimate is uncertain, as the corotation region
30''$^<_\sim r ^<_\sim 35''$ for two-armed spiral FMDWs may involve both
stellar and magnetized gas disks;
%(Lou, Yuan \& Fan 2001);
we take $Q_M$$^>_\sim 1$. Right at the mILR ($r\sim 10''$), the short
wavelength is $\lambda_S=(C_A^2+C_S^2+v_c^2)/(G\mu_{0})\sim 300$ pc,
while the long wavelength $\lambda_L$ is infinite; at the reflection
point of the $Q_M$-barrier ($r\sim 30''$), they become equal
$\lambda_S=\lambda_L=2(C_A^2+C_S^2+v_c^2)/(G\mu_{0})\sim 600$ pc.
For a mean radial wave lengthscale of $\bar\lambda\sim 450$ pc, the
damping timescale is
$\tau_c\sim\bar\lambda^2/\nu_c\cong 6\times 10^{15}$ s. The wave timescale
$\tau_w$ for a FMDW to go from the mILR to $Q_M$-barrier and back to
the mILR is estimated to be $\tau_w\sim 3\times 10^{15}$ s. Thus, a
sizable fraction of negative angular momentum carried by long-trailing
FMDWs sustained at the mILR would have been deposited in the disk as a
remnant short-trailing FMDW returns to the mILR. As the disk angular
momentum is reduced, gas materials with frozen-in magnetic flux from
outside tend to accumulate inside the mILR to form a circumnuclear
starburst ``ring". An estimate of net mass inflow rate $\dot M$ involves
several uncertain aspects because the magnitude of excited FMDWs relates
to the strength of
%bar-potential
$\phi^E$ and the wave damping distribution depends
on the effective viscosity as well as nonlinear effects.
By taking a $|v_r|\sim 10\,{\rm km s}^{-1}$ and a
$|\omega-m\Omega|r\sim 30\,{\rm km s}^{-1}$ in
%formula
eq (4), an upper limit of $\dot M$ may be estimated by
$\dot M$$^<_\sim 10M_{\odot}\,{\rm yr}^{-1}$. For a stronger
$\phi^E$, nonlinear wave and damping effects may become
important and this upper limit may be raised.
The large pitch angle of $\sim 50^{\circ}$ of optical arms
(Rickard 1975) and magnetic field (Beck et al. 1999) outside
the ``ring" is interpreted in terms of long-trailing FMDWs
sustained by $\phi^{E}$ at the mILR. As a test, it may be possible
to detect a weaker short-trailing FMDW outside the ``ring" superposed
with the stronger long-trailing FMDW.
%If one were to regard the large
%magnetic-field pitch angle as a non-frozen-in steady state, which we
%think unlikely, then the upper limit of $\dot M$ would be
%$\sim 1M_{\odot}\,{\rm yr}^{-1}$ (Beck et al. 1999). Note the upper
%limit for $\dot M$ induced by turbulent viscosity with a flat rotation
%curve is also $\sim 1M_{\odot}\,{\rm yr}^{-1}$.
%Some simulations also show that viscous torques are negligible
%with respect to gravitational torques several kpc away from the
%nucleus, when a relatively strong bar or tidal interaction is
%present (Combes 1991). In terms of feeding active star formation,
%the relative importance of viscous and gravitational torques around
%circumnuclear rings remain unresolved.

\para{\large\bf{4}~~ Discussion}

\vs\vs
In our scenario, the effectiveness of MHD density wave damping can
give rise to qualitatively different outcomes for nuclear and
circumnuclear activities around the centers of magnetized spiral
galaxies. For a very weak damping, FMDWs excited at the mILR and
reflected by the $Q_M-$barrier would travel to the nucleus; and disk
processes on sub-kpc scales in the vicinity of the central SMBH would
destroy these FMDWs that carry negative angular momentum. In this case,
central accretions onto the SMBH should lead to an AGN with much less
circumnuclear activities on kpc scales.
For a very strong wave damping, FMDWs excited at the mILR are
largely damped without even reaching the $Q_M-$barrier and therefore
disk angular momentum is incessantly reduced {\it outside} the mILR.
In this situation, circumnuclear accretion of gas and magnetic flux
occurs somewhere inside the mILR. One thus expects a prominent
circumnuclear starburst ring without necessarily involving an AGN.
For an intermediate wave damping, a significant fraction of FMDWs
excited at the mILR can be damped outside the mILR with remnant
FMDWs coming towards the galactic neucleus. Thus, in addition to
a circumnuclear ``ring" of starbursts inside the mILR, remnant
short-trailing FMDWs travel across the mILR and be partially
enhanced by the $r^{-1}$ geometric converging effect.
%(Montenegro et al. 1999).
Damping of these remnant short-trailing FMDWs on sub-kpc scales
around the nucleus would then induce a net accretion of gas and
magnetic flux to fuel galactic nuclear activities.

For NGC 1097, in addition to the
starburst ``ring", circumnuclear tight-winding trailing
spiral arms of FMDWs are detected (Fig. 1) down to the
immediate environs of the nucleus ($r$$^<_\sim 1.5''$).
%Again, the wave damping reduces the disk angular momentum
%inside the ring and thus leads to nuclear accretion of
%materials and magnetic flux to fuel the central SMBH.
We identify this spiral structure inside the ``ring"
(Fig. 1) as remnant short-trailing FMDWs and suspect that
the damping of these FMDWs plays a key role in fueling the
central SMBH on sub-kpc scales. For future observational campaigns,
it is crucial that coordinated and dedicated multiwavelength
observations of circumnuclear regions of spiral galaxies would
provide more detailed information.
%to enrich the scenario advanced here.
While already indicative, it is anticipated that refined maps
of polarized radio-continuum emissions would eventually establish
specific correspondences of spiral structures of gas density and
magnetic field around and within the circumnuclear starburst ``ring",
analogous to those of galactic spiral structures on much
larger scales. In particular, we expect the total and polarized
radio-continuum spiral arms to closely follow the inner edges of
circumnuclear optical spiral arms of NGC 1097 shown in Fig. 1.

%The fact that in the circumnuclear ``ring" of NGC 1097, the radial
%and azimuthal components of magnetic field are comparable (Beck et
%al. 1999) means that the magnetic-field swirl pattern is not tightly
%wound there. This relates to long-trailing FMDWs that are excited
%and sustained by the periodic bar-potential $\phi^E$ at the mILR,
%and spiral FMDWs involve a nonaxisymmetric radial magnetic field
%$b_r$. Fig. 1a of Beck et al. (1999) with 15" resolution shows a
%central swirl with some axisymmetry, while their Fig. 2 with 6"
%resolution is already indicative of a central spiral pattern with
%nonaxisymmetry. The apparent central ``axisymmetric swirl" of their
%Fig. 1a most likely results from a larger beam size. Otherwise the
%divergence-free condition of a coplanar magnetic field would be violated.
%In radio polarization studies of large-scale magnetic fields in nearby
%spiral galaxies, the earlier more regular ``axisymmetric" swirl patterns
%were usually replaced later by nonaxisymmetric spiral patterns. As an
%example, one may compare figures of Klein et al. (1982), Fig. 7 of
%Harnett et al. (1989), Fig. 5 of Ehle \& Beck (1993), Fig. 2 of Beck
%\& Hoernes (1996) and figures of Frick et al. (2000) in the case of
%NGC 6946.

%In barred spiral galaxies, luminous circumnuclear rings of starbursts
%(Buta 1986; Telesco 1988) commonly appear. NGC 6951 is another example
%similar to NGC 1097 (Barth et al. 1995). Some galaxies with AGNs are
%also known to contain circumnuclear starburst rings (Simkin et al. 1980;
%Arsenault 1989; Barth et al. 1995). Recent wavelet analyses on several
%dozens of {\it HST} central images of spiral galaxies reveal bars, spirals,
%and barred-spirals etc. With this contribution, we hope to further
%stimulate more high-quality observations of total and polarized radio
%emissions from circumnuclear starburst or hot spot ``rings" to complement
%{\it HST} optical observations.

\vs\vn{\bf ACKNOWLEDGEMENTS}~~
This research was supported in part by grants from US NSF
(ATM-9320357 and AST-9731623) to the University of Chicago,
by the ASCI Center for Astrophysical Thermonuclear Flashes at the
University of Chicago under Department of Energy contract B341495,
by the Visiting Scientist Program at ASIAA
%Institute of Astronomy \& Astrophysics, Academia Sinica
(NSC-88-2816-M-001-0010-6),
%from the National Taiwan University (NSC89-2112-M002-037).
by the Special Funds for Major State Basic Science Research
Projects of China,
by the Collaborative Research Fund from the NSF of China
(NSFC) for Outstanding Young Overseas Chinese Scholars
(NSFC 10028306) at the National Astronomical Observatory,
Chinese Academy of Sciences,
and by the Yangtze Endowment from the Ministry of Education.
Affiliated institutions share this contribution.

\bc
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\ec

\vspace{-4mm}
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