A&A 385, 32-38 (2002)
DOI: 10.1051/0004-6361:20020079
S. Poedts1,
- A. D. Rogava2,3,4
1 - Centre for Plasma Astrophysics, K.U. Leuven,
Celestijnenlaan 200B, 3001 Leuven, Belgium
2 - Abastumani Astrophysical Observatory, Kazbegi ave. 2a,
Tbilisi-380060, Georgia
3 - Tbilisi State University, Chavchavadze ave. 2,
Tbilisi-380028, Georgia
4 - Abdus Salam International Centre for Theoretical Physics,
34014 Trieste, Italy
Received 22 June 2000 / Accepted 15 January 2002
Abstract
In this paper we argue that the peculiar magnetic spiral
structure of the giant, face-on spiral galaxy IC 342 may be
evidence for velocity shear induced magnetohydrodynamic (MHD)
density wave transformations.
Key words: magnetohydrodynamics (MHD) -
waves - galaxies: individual: IC 342 - galaxies: kinematics
and dynamics -
galaxies: magnetic fields
The universal appearance of shear flows in astrophysics implies
that the velocity shear induced "nonmodal" processes (Trefethen
et al. 1993), associated with the non-self-adjoint character of
the governing linear dynamics, may substantially influence the
physics of the corresponding astrophysical objects which may
explain their different and remarkable observational appearance.
One notable nonmodal process, that was recently disclosed,
(Chagelishvili et al. 1996), is associated with shear flows
sustaining more than one mode of wave motion. In such
systems the shear-induced drift of the wave vector
evokes a temporal variation of the eigenfrequencies (Chagelishvili
et al. 1994) and leads to a shear-induced coupling of the
corresponding wave modes maintained by the "ambient" shear flow.
Under favorable physical conditions the shear-induced coupling
leads to mutual transformations of different waves into each
other.
The advantage of the nonmodal approach is that in many interesting astrophysical situations it makes it possible to reduce the complicated mathematical layout of the problem to the solution of a relatively simple and solvable set of ordinary differential equations (ODEs). In the case of wave transformation processes it is also possible to further reduce this set of ODEs to a pair of second order, ordinary differential equations describing the dynamics of two coupled linear oscillators. The term "coupled oscillations'' refers to the case where two (or more) oscillators, on equal footing, are coupled (e.g. by a set of springs) so that the motion of each of them is affected by the other(s). This remarkable common mathematical basis with the well-known oscillatory system of Classical Mechanics helps to understand qualitatively the wave interaction processes in fluid mechanics and plasma physics and in all those astrophysical instances, where these processes presumably appear (Rogava et al. 2000).
This kind of mechanism (hereafter referred as shear-induced transformations or SITs) may be operational in a wide variety of astrophysical objects. In a pulsar magnetosphere, for example, it may ensure the conversion of nonescaping Langmuir oscillations, maintained by the highly relativistic magnetospheric electron-positron plasma, into the escaping radio emission of pulsars (Mahajan et al. 1997). In the solar atmosphere it may account for the generation of solar wind MHD waves, for the heating of the solar corona and the acceleration of the solar wind (Poedts et al. 1998, 1999; Rogava et al. 2000). Other potential astrophysical applications (see the discussion of Mahajan & Rogava 1999) include planetary atmospheres and interiors, planetary rings, stellar atmospheres, various kinds of accretion flows (accretion disks, disk winds, accretion columns), jets in AGN's, etc. In other words, in all astrophysical objects where some kind of shear flow is present and where the flow can sustain two or more modes of wave motion, velocity shear-induced wave transformations are very likely to occur.
Another prominent astrophysical example of a magnetized
astrophysical shear flow is a galactic gaseous disk. Fan and Lou
recently demonstrated that galactic discs can nourish both slow
(0.2 km s-1) and fast (
20 km s-1) MHD
density waves (Fan & Lou 1996, 1997). Moreover, their numerical
results contained strong evidence for the existence of coupling
between these two modes of MHD density waves. Recently it was
explicitly confirmed (Rogava et al. 1999) that the coupling
actually exists, that it is exclusively due to the presence of the
velocity shear and that it can lead to mutual transformations of
the waves into each other.
It is a fascinating challenge to disclose the first observational evidence of velocity-shear-induced transformations over several Megaparsecs, in some of the nearby spiral galaxies! The aim of this paper is to argue that the giant, face-on spiral galaxy IC 342 (from the Maffei1 subgroup of the Local Group of galaxies) may be the first astrophysical object where one can find the experimental evidence for the velocity shear induced wave transformations.
The physical theory of the large-scale structure of spiral
galaxies is based on the well-known density wave scenario. The
stellar disk is mainly visible in red and near infrared bands,
contains relatively old stars and sustains moderate strength,
large-scale density wave structures. Thermal and magnetic energy
densities are of comparable magnitude: for typical nearby galaxies
(such as M51, NGC 6946 and IC 342), the Alfvén ()
and sound
(
)
speeds are comparable and fall in the range
10-20 km s-1. This pledges the existence of both fast
(
20 km s-1) and slow (
0.2 km s-1) MHD
density waves in such a disk system. An important feature of MHD
density waves, which helps to distinguish them from one another,
concerns the phase characteristics of mass density and azimuthal
magnetic field perturbations (Fan & Lou 1996, 1997). In
particular, for fast MHD density waves the enhancement of the
surface gas mass density and the parallel component of the
magnetic field are more or less in phase, while for slow MHD
density waves there is a significant (
)
phase
difference between the mass density and the azimuthal magnetic
field perturbations.
Fan and Lou applied this criterion in the analysis of large-scale structures within nearby individual spiral galaxies. The "Whirlpool galaxy" M51, for example, has been identified as a clear case of a galactic gaseous disk with fast MHD density waves. Recently, these authors performed an analogous study for the spiral galaxy NGC 2997 which revealed its close resemblance with the M51 case (Fan & Lou 1999). Moreover, since according to the MHD-density-wave scenario the spiral pattern of NGC 2997 is identified with fast MHD density waves, Fan and Lou were able to predict the existence of a (still to be discovered) moderate H1 gas arm associated with the "magnetic arm" in the radio continuum.
Recently, it was discovered that there is another possible class
of late-type, gas-rich spiral galaxies with coherent large-scale
magnetic spiral arms located between the optical spiral arms
(NGC 6946 is an example of this class). Both its optical and
magnetic spiral arms occupy a disk angular radius of 4 arcmin, where the galactic gaseous disk rotates almost rigidly
(i.e., the rotation speed
increases almost linearly).
Further out the rotation curve slowly bends and approaches a
constant value
km s-1, which remains
roughly constant in the radial range from 4-11 arcmin. This fact
allowed Fan and Lou to conclude (Fan & Lou 1999) that the gaseous
disk in NGC 6946 maintains slow MHD density waves.
However, there is at least one example of a spiral galaxy where
optical and radio continuum observations indicate the presence of
both kinds of MHD density waves! This is the spiral galaxy
IC 342, seen behind the Milky Way in the Camelopardalis
constellation and physically belonging to the Maffei1 subgroup of
the Local Group of galaxies. IC 342 is one of the largest galaxies
in the northern sky: probably the third largest in apparent size,
preceded only by the Andromeda and Triangulum galaxies. It is of
the Hubble type Scd with a well developed spiral structure and is
almost face on. Its distance is unclear (given by de Vaucouleurs
(1979) to be 3.1 Mpc and by Newton (1980a,b) to be 4.5 Mpc).
The rotation velocity
increases linearly with radius
up to about 5 kpc, beyond which it becomes flat and remains
flat as far out as can be established. A remarkable
feature of this galaxy is that it has relatively weak density
waves between 11 kpc and 15 kpc. So it seems that density
waves might not have much influence on gas motions within the
outer ring of its gaseous disk and the gas can be assumed to be
axisymmetrically distributed. The optical spiral structure of
IC 342 is predominantly confined to the inner, rigidly rotating
region (
). The size of the radio-continuum emission
region is considerably larger than the optical one (see e.g.,
Fig. 9 in Grave & Beck 1988). The results of Krause et al.
(1989) imply that ridges of polarized emission in the inner disk
are located in the interarm regions of the optical spiral
arms. In the outer disk, on the contrary, magnetic arms extend far
outside the optical spiral pattern, but H1 is distributed in a
symmetric way implying the existence of a physical link between
these patterns. Moreover, H1 images show a much larger
(
)
spiral pattern that extends into the disk portion
with a largely flat rotation curve. On the basis of the above
observational evidence Fan & Lou (1999) suggested that IC 342
seems to feature both kinds of MHD density waves! In particular,
it exhibits slow MHD density waves in the inner (rigidly rotating)
region of the galactic gaseous disk and fast MHD density waves in
the outer (differentially rotating) region. However, it must be
noted that the positioning of polarization ridges may be related
to processes of compression and shear induced by optical arm
gas flows and/or with the lack of homogeneity of turbulent
diffusion in the disk (Rohde & Elstner 1998; Elstner et al.
2000).
Recently, inspired by the Fan & Lou theory of galactic MHD density waves (Fan & Lou 1997, hereafter referred as FL97), we reconsidered their mathematical model for the study of local perturbations in galactic gaseous disks (Rogava et al. 1999).
In FL97 the authors consider a thin, differentially rotating
gaseous disc, embedded in a large-scale azimuthal magnetic field
and sustained by its self-gravity. The equilibrium state of the
system was specified by the constant surface mass density
,
the constant speed of sound
,
the constant
vectors of the angular rotation velocity
,
the large-scale background azimuthal field
,
and by the linearized (locally
plane-parallel) mean velocity field
,
with
and
the first and the second Oort
constants, respectively. In the present paper we adopt and use the
same equilibrium model.
The excitation and time evolution of compressible MHD
perturbations in this system are governed by the set of equations
[
]:
![]() ![]() |
(1) |
![]() ![]() |
(2a) |
![]() ![]() |
(2b) |
![]() |
= | ![]() |
(3) |
![]() |
(4) |
![]() |
Figure 1:
Temporal evolution of density (![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
In this case, Kelvin's transformation of variables x'=x,
,
z'=z, t'=t is sufficient to
recover the class of nonexponentially evolving perturbations that
is overlooked in the framework of the standard normal-mode
approach. The switch to these variables transforms the initial
spatial inhomogeneity of the system (1-5) to a time inhomogeneity,
because
and
.
Hence, one can look for solutions in
the form of Spatial Fourier Harmonics (SFH) (Chagelishvili
et al. 1994), bearing the form
,
where
kx' and ky' are components of the wavenumber
vector. Employing this ansatz one can effectively reduce the
system to a set of first order ordinary differential equations
with time-dependent coefficients (Fan & Lou 1997) for the
perturbation amplitudes (
,
,
,
,
,
)
of the
relevant Fourier components (the superscript ' is hereafter
dropped). Moreover, Rogava et al. (1999) proved that this system
is further reducible to the following pair of second order
ordinary differential equations:
![]() |
= | ![]() |
(5a) |
![]() |
= | ![]() |
(5b) |
The physical coupling in the system (5) is exclusively due
to the differential character of the motion. This is obvious since
only the presence of the shear (,
leading to
and
)
makes the coefficients
of the system (5) time-dependent (Rogava et al. 1999).
However, the presence of the sheared motion (differential rotation
in the galactic gaseous disk) is a necessary but not a sufficient
condition for the shear-induced coupling to "inflame" efficient
mutual transformations of MHD density waves into each other. For
transformations to be efficient it is also necessary that
the Alfvén speed
and the sound speed
have
approximately the same magnitude, i.e.
.
In
parallel shear flows, where both self-gravity and rotational
effects are absent (
), the condition
is a necessary condition for wave transformations to be
efficient (Chagelishvili et al. 1996). It is not yet clear whether
this criterion remains the same or whether it is somehow affected
by the presence of the self-gravitation (
)
and
Coriolis parameter (
)
in galactic gaseous discs.
But it is known that these physical factors (self-gravity and
nonzero Coriolis parameter) lead to the appearance of the
transient Jeans instability and "epicyclic shaking", respectively
(Fan & Lou 1997).
![]() |
Figure 2:
Evolution of the total energy of the above perturbation (![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
A systematic study of this issue is currently in preparation
(Poedts & Rogava 2001) and preliminary results show that
condition
(
)
remains qualitatively
valid for the galactic case as well (see below). Different
combinations of the system parameters must lead to rather
different interferences of the wave transformation and the
transient amplification processes, so that the outcome of the
whole set of the shear-induced processes in galactic gaseous disks
can be diverse. It is obvious that this kind of detailed study for
the plausible ranges of involved parameter values is important for
developing the quantitative theory.
To the best of our knowledge, it is impossible to find in the literature any definite information about the values of all those physical parameters for the galaxy IC 342 which are important in the perspective of the efficiency of shear-induced wave transformations. That is why we can hardly make a choice of parameters, closely related to the situation in the different regions of this galaxy. Hence, it is still too early to try to develop a concrete model for these processes and to point out possible diagnostics.
Additional uncertainty comes from our current inability to guess the identity and location of "primary" MHD modes for the further onset of the SITs. Observations suggest, however, that IC342 involves a slow wave in the inner (nearly rigid rotation) region and a fast wave in the outer, differentially rotating region. So if the working hypotheses is that the shear-induced wave transformations are responsible for the presence of both these types of waves, it is reasonable to state that the transformations take place in the outer part, where the flow of the galactic gas is sheared.
Instead of giving a detailed analysis, here we wish to present
only the illustrative example of SMW-FMW transformations, which is
shown in Figs. 1 and 2. Figure 1 displays the evolution of the
density (solid line) and the azimuthal magnetic field (dashed
line) perturbations. It is clear that the perturbation is
initially a pure SMW with substantial phase difference between
these two oscillations and that it is transformed into an almost
pure FMW, with a higher frequency and with oscillations of
and
which are, now, approximately in phase.
The evolution of the total energy of the same perturbation (solid
line) is shown in Fig. 2. The
graph is plotted together
with the time-dependent dispersion curves of the SMW
(
,
dashed line) and the FMW (
,
dashed-dotted line) effective normal frequencies. The energy
evolution is adiabatic (Chagelishvili et al. 1994;
Chagelishvili et al. 1996), following the SMW/FMW
dispersion curve before/after the transformation.
![]() |
Figure 3:
Evolution of the total energy (![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
For these illustrative plots we deliberately have chosen values of
the Coriolis and the self-gravity parameters (
and
)
so that the influence of the "epicyclic shaking" and
the transient Jeans instability, respectively, would be negligible
in comparison with the shear induced transformations. The
emergence of these phenomena and their interference with the SITs
needs a detailed separate study and is intended to be published
elsewhere (Poedts & Rogava 2001).
The high efficiency of the SITs, visible in Figs. 1 and 2, is
guaranteed by the condition
(i.e.
), which, as
it has been discovered by Chagelishvili et al. (1996), implies
the most favorable regime for efficient transformations of SMWs
into FMWs. When the ratio of Alfvén and sound speeds is either
larger or smaller than one, the SITs become visibly less
pronounced. In other words, the wave mode coupling becomes less
efficient for
.
In order to illustrate the importance
of this condition, we made numerical runs for different values of
this parameter
,
keeping all other parameters fixed
at the same values as in Figs. 1 and 2. In Fig. 3 the energy
plots (analogous to Fig. 2) are shown for
,
0.75,
1.25, and 1.5, respectively. One can see from the evolution of
the total energy that, when the Alfvén speed is either larger or
smaller than the speed of sound, the efficiency of the
transformation drastically diminishes. The further we are from the
optimum condition
,
the more the adiabatic energy graph
tends to follow the dispersion curve for the initial SMW mode,
i.e., a smaller fraction of the initial SMW is transformed into
the FMW.
It is a remarkable fact that according to extensive observational
data for typical nearby spiral galaxies (including IC 342) the
Alfvén speed
and the sound speed
are indeed of
comparable magnitude and fall in the range
10-20 km s-1.
This allows us to argue that shear-induced MHD density wave
couplings in the magnetized, self-gravitating, differentially
rotating gaseous disc of IC 342 should lead to mutual
transformations of these waves. The physical picture, as it seems
plausible to us, may be the following: slow MHD density waves are
excited within the inner (rigidly rotating) region and keep their
identity throughout the whole inner disk, since velocity shear is
absent and there is no way to transform them into the fast MHD
density waves. However, as soon as rotation becomes differential
and the flow of the galactic gas becomes sheared, conditions for
the shear-induced transformation of "primary" slow MHD density
waves into the "secondary" fast MHD waves become favorable,
providing the condition
is fulfilled. The
efficiency of the transformation is rather sensitive to the value
of the ratio
,
it is the strongest (almost
complete) when
and becomes less pronounced for either
and
.
The visibility of the transformation
phenomenon also depends on whether the physical mechanism
launching MHD density waves in the rigidly rotating inner disk is
producing pure slow MHD density waves or, rather, some mixture of
both slow and fast waves. Generally, it is possible that fast and
slow MHD density waves coexist in the almost rigidly rotating part
of a spiral galaxy. In this case, transformations of both wave
modes will still occur in the differentially rotating region of
the galactic gaseous disk, but the resulting wave bundle will
remain to be a mixture of both magnetoacoustic waves so that the
presence of transformations will be hardly visible.
One can also speculate about the possible existence of galaxies
where either slow and/or fast MHD density waves might be excited
by a central bar (like NGC 6946 which appears to have a weak
central bar) or by a satellite galaxy (e.g., M51 and its companion
NGC 5195). This means that in different galaxies MHD wave
portraits of both rigidly rotating and differentially rotating
regions of the galaxy disk and associated MHD wave transformation
processes can be widely diverse. It is even possible that contrary
to the well-known WKB or tight-winding approximation theories,
which do not encompass the effect of mutual wave transformations,
there are spiral galaxies with fast MHD density waves in the
inner, rigidly rotating disk and slow MHD density waves in the
outer, differentially rotating disk. Everything depends on the
location and morphology of the sources of "primary" MHD waves, on
the specific rotation curves of galaxies and on the values of
in differentially rotating regions of galactic disks, where
transformations may or may not occur depending on the presence or
absence of the
condition.
Galactic interstellar medium is resistive, with turbulent
magnetic diffusion being about
cm2 s-1(Ruzmaikin et al. 1988; Elstner et al. 2000).
Therefore, magnetic diffusion must play an important role in the
dynamics of galactic gaseous disks. Certainly the coefficient is
not homogeneous throughout the whole disk but varies being higher
in spiral arms due to star formation activity and lower in
interarm regions. Irrespective of their identity both modes
of MHD density waves are subject to damping via magnetic
diffusion. But since the optical and radio continuum observations
seem to allow the presence of MHD waves in galactic disks, it
appears that despite the damping, there are other, probably more
robust physical processes, which guarantee the persistence of the
wave appearance. In our opinion the SITs can be one of these
processes. Unfortunately, currently, we are only at the very
beginning of the understanding of the real-space appearance of the
SITs. But from numerical simulations for simple, plane-parallel
MHD flows (Bodo et al. 2001) we recently learned that the SIT
persist even in the presence of sufficiently large magnetic
diffusion.
Certainly, a crucial test for any theoretical result or prediction is a comparison with observational data. Since the observational picture of magnetic field in galaxies is so complicated one can not even claim that MHD waves are the unique or even the most important process in spiral galaxies. Most probably the situation is very diverse in different galaxies. In the galaxy M51, for instance, which seems to exhibit the appearance of fast MHD density waves (Fan & Lou 1999), the structure of the magnetic field can also be well explained by turbulent or fast dynamo mechanisms (Elstner et al. 1990; Hanasz & Lesch 1998) and/or compressions and shears provoked by the spiral arms (Rohde & Elstner 1998; Elstner et al. 2000). This means that we, suggesting the SITs as a possible mechanism for the generation of the particular magnetic structure in IC 342, are fully aware that serious proofs in favor of this suggestion can be found only after detailed observational studies. That is why we consider this idea as a cautious hypotheses rather than a direct prediction.
Another efficient way to test the validity of our theoretical
model would be the knowledge about the pitch angle picture for the
IC 342. The observations of nearby galaxies (Beck & Hoernes 1996;
Ehle et al. 1996) disclose that the magnetic
structure is mainly spiral with quite large pitch angles, similar
to pitch angles of optical arms. The IC 342 observations give the
pitch angle value about
(Krause et al. 1989).
Unfortunately, our current model is limited to the study of the
temporal evolution of individual wave harmonics, and does not
allow to reproduce a picture of the azimuthal distribution for the
MHD waves in order to check whether the pitch angles can match the
observed value. The latter will become possible only after we will
have an adequate numerical model with an accurate real-space
portrait of the phenomenon.
It is well-known that about half of the total galactic magnetic field comes in the form of its random component (Ruzmaikin et al. 1988). This poses an interesting question: how does the presence of the random component influence the SITs taking place in galactic gaseous disks?! This question, being a part of the more general problem of the influence of random fields on the shear-induced physical processes, obviously requires a separate and comprehensive consideration. On the basic, theoretical level one needs to learn how to adjust the existing schemes of the nonmodal approach to the study of flows influenced by stochastic fields. Direct numerical investigations are also an option, since after having a numerical model for a gaseous disk with regular equilibrium magnetic field it would be easier to include random components into the model and to test their influence on shear-induced processes in a numerical way.
Therefore, it seems obvious that the hypothesis, argued in this paper, needs to be addressed further. The detailed observations of IC 342, as well as of other, similar kinds of spiral galaxies may give additional evidence in favor of (or contrary to) our interpretation. On the theoretical level the further development of this approach, with the possible inclusion of dissipative effects, random fields and with numerical studies of the "real-space'' appearance of MHD density waves, will help to "transform'' this hypothesis into a model with definite predictive power.
Acknowledgements
Our work was supported, in part, by the INTAS grant No. 97-0504. A.D.R. wishes to thank Abdus Salam International Centre for Theoretical Physics for supporting him, in part, through an Associate Membership Award.