Download Modelling magnetic fields in spiral galaxies

Moss: Magnetic fields in spiral galaxies
Modelling magnetic
fields in spiral galaxies
Leon Mestel: David Moss gives an
overview of large-scale magnetic
field strength and structure in
galaxies, focusing on theoretical
and modelling studies, with an
emphasis on the disc plane.
T
he first hints that our galaxy, the Milky
Way, might host a large-scale magnetic
field came in the late 1940s, from studies of the polarization of starlight. However,
optical polarization is a relatively blunt tool
and it was only in the late 1950s that radio
synchroton observations began to indicate the
presence of large-scale fields (e.g. Gardner and
Davies 1966). By the early 1980s observations
of linearly polarized radio emissions (PI) were
giving unambiguous evidence for the presence
of large-scale magnetic fields of microgauss
strength in the Milky Way and nearby spiral galaxies (e.g. Beck et al. 1996). Note that
small-scale (“random”, “turbulent”) fields are
also universally present, detectable by their
unpolarized synchrotron emission, and they
are at least as strong (usually stronger) than the
ordered component. Note also that polarization
measurements do not unambiguously determine
field direction – there is a 180° uncertainty. To
resolve this ambiguity, multiwavelength Faraday rotation measures (RM) are necessary, a
much more demanding requirement.
At present, there is detailed knowledge of field
strengths and structure in many nearby spiral
galaxies and Local Group irregulars – see figures 1, 2 and 3 for some examples. Total field
strengths vary from a few microgauss in radio
faint galaxies such as M31 and M33 (figure
2), to 50–100 µG in starburst galaxies such as
NGC 4038/9 and in nuclear starbursts such as
seen in NGC 1097.
It appears that ordered magnetic fields are universally found in well-observed spiral galaxies, and generally trace (maybe rather patchily)
trailing spiral patterns, with pitch angles typically in the range 10–40°. Usually the degree of
polarization within optical spiral arms is only
a few percent, and regular magnetic fields are
found predominantly in between optical arms,
with small-scale fields dominating within the
arms, e.g. figure 3. The pitch angles usually are
A&G • October 2012 • Vol. 53 1: Polarization vectors and total intensity contours for M31 at 4.8 GHz. (Courtesy R Beck, MPIfR, Bonn)
2: Polarization vectors and polarized intensity contours overlaid on an optical image of NGC 6946.
(Courtesy R Beck, MPIfR, Bonn)
not identical with those of the optical arms. In
barred galaxies, field vectors tend to be aligned
with the strong streaming velocities.
Disc fields
Disc fields are often predominantly axi­
symmetric, with small admixtures of higher
modes, but in a number of cases the azimuthal
symmetry cannot be established definitively.
However, earlier belief that a significant proportion of fields were dominated by m = 1 azimuthal
modes has now weakened considerably. It is
difficult to determine clearly the symmetry of
galactic fields with respect to the disc plane (suitable galaxies seen edge-on are rare), but the data
are generally consistent with even (quadru­polar)
symmetry in the very few cases where determination is feasible. There is evidence for galactic
winds influencing field morphology in galactic
halos, by the “X-shaped” field patterns seen far
from the disc plane in a few suitably aligned
“edge-on” galaxies. In galaxies with massive bars and non-circular motions, field lines
approximately follow the gas flow, e.g. figure 3.
Signatures of large-scale magnetic fields are
currently seen in galaxies with redshifts z ≲ 2
5.1
Moss: Magnetic fields in spiral galaxies
– i.e. by ~3 Gyr after time zero: technological
advantages may soon push this “first field” time
back very considerably in favourable cases.
In general, studies of galactic magnetism have
one notable advantage compared with those of
solar and stellar magnetism: galaxies are comparatively transparent and, in favourable cases,
details of rotation and magnetic fields inside
galaxies can be determined. However, the largescale structure of the Milky Way field is quite
uncertain – largely because of our location in
the disc plane. There are strong claims for the
presence of one large-scale field reversal, and
weaker claims for others (e.g. van Eck et al.
2011). However, it remains possible that these
are all local features.
This review is primarily concerned with theoretical and modelling issues, and so only a very
broad-brush observational background has
been given. More details can be found in many
recent reviews, such as Beck et al. (1996) and
Beck (2011, 2012). It is short, and so it is written
substantially from the viewpoint of the author
and his collaborators. The emphasis is on relatively simple models that attempt to describe
the basic processes that give rise to the observed
fields. Attention is concentrated on modelling
the global structure of magnetic fields near the
disc plane, and fields in halos are not systematically addressed. Significant work has also been
done by groups in Germany and Poland, among
others. A brief account is given of this work and
of more complex and holistic modelling of the
interstellar medium is given in the penultimate
section of this paper.
Origin of galactic fields
Two basic ideas have been proposed to explain
the origin of the large-scale magnetic fields
seen in spiral galaxies. The conceptually simpler idea is that fields of scales greater than that
of proto­galaxies are created in the early stages
of the universe. These fields are then stretched
and distorted by the galactic differential rotation, to give the basic fields seen today – which
may nevertheless be influenced by large-scale
non-circular motions, interstellar turbulence
and other flows. This idea encounters several
fundamental difficulties. Most fundamental perhaps is the “winding problem”. Given
that a generous upper limit for the primordial
field strength is O(10 –12) G – other estimates
are much smaller – to amplify this field to the
observed microgauss strengths, even after
allowing for compression during the collapse of
the protogalaxy, would require so much winding by the differential rotation that the resulting
pitch angles p (tan p = Br /Bf), would have p ≲ 1°,
whereas typical observed values are around 20°.
Moreover, such a field is rapidly expelled to near
the perimeter of the galaxy (the “flux expulsion
effect”) and would then be inconsistent with
RM measures. Further, if there is sufficient field
5.2
Moss: Magnetic fields in spiral galaxies
3: Polarization vectors
and total intensity
contours overlaid on
an optical image of
NGC 1097 at 4.8 GHz.
(Courtesy R Beck,
MPIfR, Bonn)
al. 1994, Moss et al. 2012), when this field is
already of microgauss strength. Alternatively,
turbulence could tangle and amplify a weak
relic field, in this way also providing a strong
small-scale seed field. Thus there may not be
such a fundamental distinction between these
scenarios. A further possibility is that fields
could be generated by the Biermann “battery”
mechanism, or even a dynamo in the first generation of stars, and subsequently ejected into
the interstellar medium (ISM); a more detailed
discussion is given in Sokoloff and Moss (2012).
4: Typical
steady-state
distribution of
the largescale toroidal
magnetic field
of M31. (From
Moss et al.
1998)
Early modelling
dissipation (reconnection) to restrict the field
winding sufficiently to yield the desired pitch
angles, the fields will be much too weak.
There is also a parity problem: a component
of a primordial field that is parallel to the disc
plane will have even parity with respect to the
plane, but this component will not survive the
winding process. A component parallel to the
rotation axis will have odd parity, which will
be subsequently preserved. In contrast, galaxy fields appear to have even symmetry with
respect to the plane. This leads to detailed consideration of in situ generation mechanisms. It is
now widely accepted that galactic discs are suitable sites for large-scale dynamos (see “Dynamo
theory” below) to operate. In particular, galactic dynamo theory generally predicts fields of
even parity with respect to the disc plane, and
that field vectors near the disc are offset from the
gas flow vectors – both features are in agreement
with observations. The most readily accessible
formulation of dynamo theory is mean-field
dynamo (MFD) theory – described below, but
note that less restrictive approaches are also
being developed and are summarized later.
Dynamo theory
In its simplest form, a dynamo is a mechanism
by which an infinitesimally small “seed” magnetic field can be amplified to finite magnitude,
and maintained indefinitely. In modern form,
astrophysical dynamo theory can be traced
to the seminal paper of Parker (1955), who
showed that mirror antisymmetric cyclonic
turbulence together with differential rotation
can drive dynamo action. This paper was specifically directed to generation of the field in
the solar convective envelope; it was then recognized that the mechanism could also operate
in galactic discs (Parker 1969). Solution of the
full dynamo problem would involve solution of
the MHD equation
∂B* ––– = ∇ × (u* × B* – ηm∇ × B*)(1)
∂t
where u* is the total fluid velocity (rotation,
large-scale streaming and small-scale turbulence), B* is the total magnetic field and hm is the
microscopic diffusivity. This should be coupled
with solution of the hydrodynamic equation,
and possibly the thermodynamic equations.
Given the wide range of spatial and temporal
scales involved, there is no prospect of a solution of this “full” problem without substantial
approximation and simplification. The most
dramatic, and most accessible and fruitful, simplication is mean-field dynamo theory, in which
equation 1 becomes
∂B ––– = ∇ × (αB + u × B – η∇ × B)(2)
∂t
Here equation 1 has been averaged over
some scale, and B, u are the resulting mean
fields, representing averages over these scales.
Naively these fields might be expected to correspond approximately to the observed regular fields. The key quantity is the quantity a,
which parameterizes the generative effects of
cyclonic turbulence; h is the turbulent resistivity. The parameters h and a thus represent
subgrid modelling; both may be tensor quantities. This approach was pioneered in Potsdam
in the 1960s and 1970s and a comprehensive
treatment is given in Krause and Rädler (1980).
The simplest forms of galactic dynamos are
driven by the joint effects of cyclonic turbuA&G • October 2012 • Vol. 53
lence (in this approximation, the alpha-effect)
and differential rotation. These are conveniently
summarized by dynamo numbers Ra = a0L/h0
and Rw = (rdW/dr)0L2 /h0 , where r is cylindrical
radius, L is a suitable length scale and subscript zero denotes a representative value. In
many cases these can be combined into a single
dynamo number |D| = Ra Rw , and in most physically relevant examples dynamo action occurs
when D exceeds some threshold value. When
applied to galactic discs, even simple models give
results that are broadly consistent with observations (e.g. Ruzmaikin et al. 1988, Beck et al.
1996). Some form of nonlinear dynamical feedback into equation 2 is required to limit fields
at finite magnitude, but after considerable dispute it is becoming accepted that “catastrophic
quenching” – in which the growth of large-scale
fields is limited at very low strengths – does not
inevitably occur. Outflows from the disc – galactic winds and fountains, for example – probably play an important role. Refinements to the
theory include the inclusion of several explicit
forms of nonlinear dynamical feedback, such as
buoyancy, cosmic rays and galactic winds. The
latter, besides taking part in the basic dynamo
action, may influence field structure in the halo
regions above and below the galactic disc.
Until relatively recently, computations in three
spatial dimensions has been quite challenging,
and much work has used axisymmetric models
that are effectively 2D (indeed some seminal
early studies reduced the problem to one spatial
dimension). An alternative approach to model
efficiently nonaxisymmetric fields in thin discs
has become known as the “no-z” model. This
traces its origin to Subramanian and Mestel
(1993), and by replacing spatial derivatives perpendicular to the disc plane by powers of 1/h,
where h is the disc semi-thickness, the problem
is again reduced to two spatial dimensions. This
approach is used in the hybrid models discussed
below, among others.
In the remainder of this review, the general
philosophy adopted first is that plausible and
useful results in modelling spiral galaxies can
be obtained by taking the simplest form of MFD
theory, while bearing in mind that additional
A&G • October 2012 • Vol. 53 effects may need to be included. Another mechanism appeals to the effects of buoyant motions
in the disc driven, for example, by “bubbles”
from sites of multiple supernovae explosions
(e.g. Ferrière 1998), or by inflation of bubbles
by cosmic rays. In a first approximation, these
models can also be studied by a quantity analogous to the alpha effect (e.g. Moss et al. 1999b).
MFD modelling has the advantage that substantial exploration of parameter space can be
made with limited computational resources. Of
course, the cost is in the uncertainty of parameterization of small-scale processes.
An alternative approach is known as direct
numerical simulation (DNS), which attempts
to model some smaller-scale MHD, dynamical and thermodynamical processes. Very
substantial computing resources are required,
and parameterization of transport processes at
small scales is still required. These studies are
described briefly below.
Seed fields
Dynamos need a “seed field” to be present
initially, which is subsequently amplified and
organized; thus discussion of the origin of
galactic fields is incomplete without consideration of possible seed fields. Typical growth
times (e-folding times) for galactic MFDs are
typically 5 × 108 –109 yr, so unless the primordial field strength is near its rather optimistic
upper limit of O(10 –12) G, there is insufficient
time for growth to contemporary observed
field strengths. In fact, the detection of strong
organized fields out to redshifts in excess of
unity provides an even stronger constraint on
primordial seed field strength. A more promising approach is to recognize that turbulence
will rapidly (timescale O(106) yr in a galactic
disc) drive small-scale dynamo action, producing disordered fields at the scale of the turbulence and in approximate equipartition with the
kinetic energy of the turbulent motions – i.e.
at least O(10 –6) G. Any such small-scale field
can then be organized into contemporary largescale fields by large-scale dynamo action. Signs
of such organization typically appear after a
few galactic rotations – say 1~2 Gyr (Beck et
Early quantitative results required substantial
analytical and/or numerical approximation
(e.g. Ruzmaikin et al. 1988). A comprehensive
historical introduction and summary of early
work is given in Krause and Rädler (1980). One
important result that persists in more sophisticated models is that in a thin-disc geometry
fields with even parity with respect to the galactic plane are the first to be excited as dynamo
numbers are increased.
By the late 1980s, increasing computer
resources meant that axisymmetric models,
with dynamo active discs embedded in largely
passive diffusive spherical halos, could be investigated (e.g. Stepinsky and Levy 1988, Elstner et
al. 1990, and many other papers). Except for one
or two rather artificial cases, the preference for
even parity modes persisted. The other important result was that pitch angles (the angles made
by the magnetic field vectors with the local tangent direction) are not very small – the winding
problem does not exist for these models.
These successes for dynamo theory were
encouraging, but it was becoming clear that such
generic modelling was not enough. The longterm aim of any astrophysical modelling must
be comparison with observed fields of specific
objects – both to verify the theory behind the
modelling and to give predictive value. As properties of dynamo models depend sensitively on
some physical parameters – especially large-scale
velocities, i.e. rotation curves W(r) and, in some
cases, non-circular velocities – this is only possible for certain well-observed galaxies. Thus the
next section will illustrate this aspect of the subject, although there have been surprisingly few
papers that systematically address these issues.
Modelling M31 and NGC 1365
M31 is a nearby galaxy with well-determined
magnetic field and relatively well-determined
rotation curve. The polarized emission observations have a peculiar feature in that there
appears to be a ring of field at 6–10 kpc galacto­
centric radius (figure 1). Moss et al. (1998) used
an axisymmetric embedded disc MFD code to
model this galaxy. They found an azimuthal
field distribution as shown in figure 4; the
details depended somewhat on relatively minor
5.3
Moss: Magnetic fields in spiral galaxies
5: Magnetic field vectors from a simulation
of the weakly barred galaxy IC4214,
superimposed on a greyscale representation
of the gas density. (From Moss et al. 1999)
features of the rotation curve, but the field
maximum in the ring at 6–10 kpc is a robust
feature. However, there is no “hole” in the field
distribution at 3–5 kpc, as suggested by the PI
observations. Subsequently, Han et al. (1998)
obtained RMs for lines of sight passing through
the “hole”, thus indicating the presence of a significant ordered field in these regions. Presumably this part of the field was not apparent in PI
measures because of local lack of cosmic-ray
electrons – as suggested in Moss et al. (1998).
This example provides some confidence in the
MFD modelling.
Attention then turned to barred spirals, where
in a few cases determinations of non-circular
velocities are available. An early attempt was
made for the weakly barred galaxy IC 4214
(Moss et al. 1999a), using a velocity field precomputed by a hydrodynamic code. Figure 5
gives magnetic field vectors from a typical simulation. Unfortunately there is no good observational determination of the magnetic field for
this galaxy (it is small and radio-weak), so the
desired comparison with observed fields could
not be made.
Moss et al. (2007) attempted to model the
well-observed and hydrodynamically modelled strongly barred spiral NGC 1365. As in
the previous study, field vectors were close to,
but offset slightly from, the velocity vectors.
Synthetic polarization maps were constructed
and compared with the observed PI distribution.
Agreement was promising but, unsurprisingly,
partial. Figure 6a shows observed B vectors
(from polarization measures, so ambiguous to
180°) and polarized intensity contours, while
figure 6b plots a synthetic map of polarized synchrotron intensity (contours) and polarization
planes from the mean-field modelling. Interestingly, in barred galaxies details of the alphaeffect appear relatively unimportant, and the
geometry of the field generated is largely determined by the strong streaming motions. Results
5.4
Moss: Magnetic fields in spiral galaxies
6: NGC 1365.
(a) Polarized intensity
contours and magnetic
vectors of polarized
radio emission at 6.2 cm
wavelength, smoothed to
25ʺ resolution.
(b) A synthetic map of
polarized synchrotron
intensity (contours) and
polarization planes at
6.2 cm, superimposed
on the optical image.
This synthetic map has
been smoothed to 25ʺ
resolution to match that of
the observed map shown
in (a). The dashed lines
are not discussed here.
(From Moss et al. 2007)
7: Field vectors in
disc plane for model
shown in figure 8b of
Moss et al. (2012).
(a): At time 2.3 Gyr.
(b): Statistically
steady configuration.
The unit of length is
10 kpc. Field vectors
are shown only at
a sample of grid
points.
desired effect is visible.
8: Statistically
steady magnetic
field in a model with
field injection only
in spiral regions,
that rotate with a
given pattern speed,
and in a central disc
region (as shown by
the closed contour).
The unit of length
is 10 kpc and the
correlation radius
is about 5 kpc (~0.5
in dimensionless
units). Field vectors
are shown only at
a sample of grid
points.
are sensitive to details of the hydrodynamical
model, which certainly has deficiencies and
cannot be regarded as definitive. For example,
the existing models do not include an adequate
representation of shocks and compression in
the gas arms, which from observations seem
to leave a significant imprint on the field. An
important outcome of this and earlier modelling
of (and observations of) strongly barred spiral
galaxies is that magnetic fields (tens of microgauss) in the innermost bar region are strong
enough to drive mass inflows of several solar
masses a year, sufficient to fuel the activity of
the nucleus. This in turn suggests that these systems require simultaneous hydrodynamical and
MHD modelling, rather than independently
determined velocities being inserted without
dynamical feedback.
A hybrid approach to mean-field
modelling
Standard mean-field modelling suffers from an
inherent restriction to modelling the regular
part of the field. Star-forming regions (SFRs)
are thought to be the site of intense turbulence
driven by supernovae explosions and winds
from massive hot stars – see below. These in turn
are thought to be especially favourable sites for
small-scale dynamo action (discussed above).
This view is strongly supported by the observed
tight correlation between unpolarized radio
emissions and far-infrared luminosity of starforming galaxies. Moss et al. (2012) attempted
to introduce a representation of such regions,
by continually injecting small-scale field of
approximate equipartition strength at random
locations in a standard thin-disc MFD model.
Differential rotation rapidly organizes this field
A&G • October 2012 • Vol. 53
(which is then maintained by the alpha effect),
and approximately equipartition strength fields
are found to be present after 1–2 Gyrs. Figure
7 shows field vectors at time 2.3 Gyr and in the
statistically steady configuration (“now”); both
large- and small-scale components are clearly
visible. In this example small-scale reversals are
present at the “present times” (ca. 13.2 Gyr); in
other models there are also global-scale reversals. (Some of these results were anticipated by
Poezd et al. 1993 in a 1D model.)
Arm–interarm fields
As mentioned at the start, a conspicuous feature
of some grand design galaxies is that the regular
field is found mostly in the regions between the
arms, and in the arms the turbulent field is much
stronger. Various explanations of this phenomenon have been proposed: in the MFD context
A&G • October 2012 • Vol. 53 Direct numerical simulations
these include higher diffusivity within the
arms, streaming motions along the arms, and
a delayed response of the dynamo to the effects
of stronger turbulence in the arms. Necessarily,
standard MFD theory can only make predictions about the regular component of field.
As an extension of the model of Moss et al.
(2012) discussed above, Moss et al. (in prep.)
investigated a model in which the small-scale
field is injected predominantly in spiral regions
(“arms”), that rotate with the pattern speed.
The motivation was the idea that as the arms
moved through the ISM, the injected smallscale fields are “left behind” and can be
organized by the differential rotation without
further disruption. Then small-scale fields will
dominate the arms, with more regular fields
in the interarm regions. Preliminary results
for a generic model are shown in figure 8 – the
By the late 1990s, available computing resources
had advanced to a point where it was possible to
simulate in a certain amount of detail the evolution of the ISM, while simultaneously modelling
the evolution of the magnetic field, the dynamics
of the multiphase gas, and the hydrodynamics.
This is known as DNS. This was then, and still
is today, a substantial undertaking and various
compromises need to be made. For example,
there is no possibility of modelling transport
processes explicitly at the molecular level – socalled subgrid modelling is required in some
form. There is inevitably a trade off between
the size of the region modelled and the effective spatial resolution, and detailed numerical
simulations are confined to Cartesian “shearing boxes” (i.e. with imposed velocity shear in
the place of differential rotation) representing
localized regions of the galactic disc. Possible
outcomes of such simulations include elucidation of the role of various mechanisms in driving
the turbulence in the ISM, and of the realization
of dynamo action. The “obvious” mechanism to
drive turbulence – supernovae explosions – is
found to be effective, but the magnetorotational
instability may also operate. Further, the instability of the cosmic-ray “gas” component of the
ISM in the disc can drive a thermal instability
(“buoyancy”), which in turn can drive dynamo
action, effectively through a form of alpha effect
as proposed by Parker (1992) (and exploited in a
mean-field model by Moss et al. 1999b).
For example, early modelling by Korpi et
al. (1999) used a box 0.5 × 0.5 × 2 kpc with a
grid resolution of (8 pc)3. The energy input
was from simulated supernova explosions and
some evidence for dynamo action was found.
Later studies (e.g. Balsara et al. 2004, Piontek
and Ostriker 2007) modelled a smaller domain
of (0.2 kpc)3 at a resolution of about (1 pc)3.
5.5
Moss: Magnetic fields in spiral galaxies
Rather similarly, Hanasz et al. (2009a) studied local models driven by cosmic rays in a box
0.5 × 1 × 2 kpc at a linear spatial resolution of
about 10 pc. These and several other studies
were able to find evidence for local amplification of magnetic field, but the simulations were
necessarily restricted to regions of the disc that
are much too small to investigate issues of global
field amplification and structure. Gressel et al.
(2008) took a slightly different approach, finding evidence for large-scale field amplification,
and supporting the claim that catastrophic
quenching does not occur.
A different approach was attempted in a series
of papers that simulated in less detail the evolution of the global field (e.g. Hanasz et al. 2009b,
Kulpa-Dybel et al. 2011, among several others).
Here, a parameter­ization of cosmic-ray injection and modelling of cosmic-ray transport were
included, together with injection of weak dipolar
field during supernova explosions. (These fields
were assumed to have been generated by dynamo
action in the progenitor stars.) Global-scale
magnetic fields were generated and maintained
by these buoyancy-driven dynamos. Figure 9
shows edge-on and face-on synthetic polarization maps from Kulpa-Dybel et al. (2010) after
5 Gyr, when the dynamo is saturated.
Conclusions
Magnetic fields appear to be an almost ubiquitous feature of spiral galaxies, both at the
near-global scale, and also at the scale of the
turbulence. Their energy density is of the same
order as that of the turbulent motions in the
ISM and of cosmic rays. Thus it is essential for
a holistic understanding of the evolution of
these galaxies, with subsequent formation of
stars and planetary systems, that the generation
and structure of galactic fields be understood,
and that they are included as an intrinsic part
of studies of these phenomena.
A notable omission from this short review has
been the modelling of magnetic fields in galactic
halos. Fields may be transported from disc to
halo, and even to the intracluster medium, by
outflows such as galactic winds and fountains.
There is also the possibility of independent
dynamo action in the halo (e.g. Sokoloff and
Shukurov 1990).
It can be reasonably asserted that we have a
basic understanding of the mechanisms that
generate the magnetic fields observed in spiral
galaxies, although there is no consensus about
the prime driver of the cyclonic turbulence
essential for dynamo action: possibly it varies
depending on galaxy type and history. However, this understanding is primarily generic;
that is, there are few, if any, models that reproduce the fields of any specific galaxy in a comprehensive manner. Correspondingly, at best
only a very broad-brush predictive power exists.
This is particularly unsatisfactory when account
5.6
9: Edge-on (top) and
face-on (bottom)
synthetic polarization
maps at 6.2 cm, from the
barred galaxy model of
Kulpa-Dybel et al. (2010).
Polarized intensity
contours and polarization
angles are superimposed
on column density
contours (greyscale).
The time is 5 Gyr, when
the dynamo is saturated.
(With kind permission of
Dr Kulpa-Dybel)
is taken of the vast amounts of data that will be
available when the Square Kilometre Array is
fully operational. The fundamental issue is the
difficulty in inputting sufficiently detailed physics into global galaxy models and our ignorance
of some essential physics relevant to the local
models. The most promising approach may be
to develop local “in a box” models of regions
of the disc with different parameters – energy
input, differential rotation, disc thickness, etc –
and to use these to calculate reliable mean-field
tensors a and η to insert into global models.
(Indeed, Ferrière 1998 and Ferrière and Schmitt
2000 are pioneering papers in this area.) Alternatively, shell models of MHD turbulence could
be used to calculate systematically the meanfield coefficients locally as the computation proceeds. (Shell models of MHD turbulence follow,
in a computationally efficient manner, the evolution of representative Fourier modes of physical
quantities, while conserving magnetic helicity,
energy, etc – e.g. Frick and Sokoloff 1998.) Of
course, a good knowledge of the large-scale
motions (circular and non-circular) will also
be necessary. In such ways, if relevant properties can be determined for object galaxies, the
prospect of realistic predictive modelling may
be approached. Needless to say, any such programme will be distinctly ambitious. ●
David Moss, School of Mathematics, University of
Manchester, UK ([email protected]).
Acknowledgments: Rainer Beck, John Brooke and
Dmitri Sokoloff made valuable comments on an
early draft.
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