A&A 448, L33-L36 (2006)
DOI: 10.1051/0004-6361:200600011
A. Shukurov1,3 - D. Sokoloff2,3 - K. Subramanian3 - A. Brandenburg4
1 - School of Mathematics and Statistics, University of Newcastle, Newcastle
upon Tyne, NE1 7RU, UK
2 -
Department of Physics, Moscow
University, 119992 Moscow, Russia
3 -
Inter-University Centre for Astronomy and
Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
4 -
Nordita, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
Received 23 December 2005 / Accepted 19 January 2006
Abstract
Aims. Nonlinear behaviour of galactic dynamos is studied, allowing for magnetic helicity removal by the galactic fountain flow.
Methods. A suitable advection speed is estimated, and a one-dimensional mean-field dynamo model with dynamic -effect is explored.
Results. It is shown that the galactic fountain flow is efficient in removing magnetic helicity from galactic discs. This alleviates the constraint on the galactic mean-field dynamo resulting from magnetic helicity conservation and thereby allows the mean magnetic field to saturate at a strength comparable to equipartition with the turbulent kinetic energy.
Key words: magnetic fields - turbulence - ISM: magnetic fields - galaxies: ISM
The suppression of the -effect can be
a consequence of the conservation of magnetic helicity in a
medium of high electric conductivity
(see Brandenburg & Subramanian 2005a for a review).
In a closed system, magnetic helicity can only evolve on the Ohmic
time scale which is proportional to
;
in galaxies, this time scale by far exceeds the Hubble time.
Since the large-scale magnetic field necessarily has non-zero helicity
in each hemisphere through the mutual linkage of poloidal and toroidal fields, the
dynamo also has to produce small-scale helical magnetic fields with
the opposite sign of magnetic helicity.
Unless the small-scale magnetic field can be transported out of the system,
it quenches the
-effect together with the mean-field dynamo.
Blackman & Field (2000) first suggested that the losses of the
small-scale magnetic helicity through the boundaries of the dynamo region can
be essential for mean-field dynamo action. Such a helicity flux can result
from the anisotropy of the turbulence combined with large-scale velocity shear
(Vishniac & Cho 2001; Subramanian & Brandenburg 2004) or the
non-uniformity of the -effect (Kleeorin et al. 2000). The
effect of the Vishniac-Cho flux has already been confirmed in simulations,
allowing the production of significant fields,
(Brandenburg 2005).
Here we suggest another simple mechanism where the advection of small-scale
magnetic fields (together with the associated magnetic helicity) away from the
dynamo region allows healthy mean-field dynamo action. This effect naturally
arises in spiral galaxies where magnetic field is generated in the multi-phase
interstellar medium. The mean magnetic field is apparently produced by the
motions of the warm gas (Sect. 4.3 in Beck et al. 1996), which is in a
state of (statistical) hydrostatic equilibrium with a scale height of
(e.g. Korpi et al. 1999). However, some of the gas
is heated by supernova explosions producing a hot phase whose isothermal scale
height is 3 kpc.
The hot gas leaves the galactic disc, dragging along the small-scale part of
the interstellar magnetic field. Thus, the disc-halo connection in spiral
galaxies represents a mechanism of transport of small-scale magnetic fields
and small-scale magnetic helicity from the dynamo active disc to the galactic
halo. As we show here, this helps to alleviate the catastrophic
-quenching under realistic parameters of the interstellar medium.
The fountain flow drives gas mass flux from the disc,
through both surfaces, where f is the area
covering factor of the hot gas,
is the radius of the galactic
disc with vigorous supernova activity, and
is the hot gas
density. A lower estimate of the area covering factor is given by the volume
filling factor of the hot gas at the disc midplane, f=0.2-0.3 (e.g. Korpi
et al. 1999), because the scale height of the galactic disc is
comparable to the size of the hot cavities. This yields
(cf. Norman & Ikeuchi 1989 who obtain
).
The dynamo model discussed below refers to quantities averaged over scales
exceeding the size of the hot cavities, and so it treats the multi-phase
interstellar medium in an averaged manner. Then it is appropriate to introduce
an effective fountain speed as the one that drives the same mass flux as above
by advecting the diffuse interstellar gas at its mean density,
(i.e. number density of
)
The hot gas that leaves the galactic disc carries magnetic fields of
scales smaller than the size of the hot cavities
(0.1-1 kpc); these are mostly turbulent magnetic fields
(of scales
). The time
scale of the removal of the small-scale magnetic fields is of order
(with
the scale height of
the warm gas layer which hosts the mean-field dynamo), which is comparable to
the turbulent diffusion time of the mean field,
,
with
the turbulent
magnetic diffusivity. Hence, the leakage of the small-scale magnetic helicity
produced by the galactic fountain can significantly affect the mean-field
dynamo.
We study the effects of the simplest contribution to the flux,
The dynamics of
is controlled by
Eq. (2).
We argue that the main contribution to
comes from the integral scale of the turbulence
.
For Kolmogorov turbulence, we have the following spectral scalings:
and
,
so that
.
Moreover, numerical results of Brandenburg & Subramanian (2005b) indicate that
even
is dominated by the larger scales.
This justifies the estimate
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(8) |
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Figure 1:
Evolution of the field strength at z=0 obtained by solving
Eqs. (4)-(7)
with vertical advection (solid line, CU=0.3)
and without it (dashed line, CU=0), for
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We illustrate
in the inset of Fig. 1 the effect of varying the strength of
the advection velocity on the dynamo.
The steady-state strength of the mean field grows with
U0 as the advection strength increases to about CU=0.3.
For stronger advection,
,
the mean field
initially grows slower but still attains a steady state
strength slightly exceeding
.
Stronger advection,
CU > 1, affects the dynamo adversely since the mean field is removed too
rapidly from the dynamo active region.
A good compromise between rapid growth and large saturation field strength
is reached for
,
which is close to the values expected for spiral galaxies.
Modest advection does not noticeably affect the spatial distribution of magnetic field.
The profiles of
and
shown in
Fig. 2 for the steady state do not differ much
from
solutions
of the kinematic dynamo equations (cf. Ruzmaikin et al. 1988).
The corresponding profiles of
and
shown
in the lower panel, indicate that the suppression of
in stronger
near the disc midplane (at small z), where magnetic field is stronger.
The steady-state strength of
,
can be estimated
from magnetic helicity conservation.
Averaging Eq. (4), with
,
over
z on 0<z<h (with the mean denoted with angular brackets) yields
![]() |
Figure 2:
Plots of
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These conclusions follow from Fig. 1, where
moderate advection drastically changes the mean field levels achievable at
and prevents catastrophic quenching of the dynamo.
Excessive advection, however, hinders the dynamo as it
removes the mean field from the dynamo active region.
The steady-state strength of the mean magnetic field obtained in our model is
of order
,
which is a factor of
several weaker than what is observed; we made no attempt here to refine the model.
What is important, the mechanism suggested here resolves the problem of
catastrophic quenching of the dynamo.
The applicability of the vacuum boundary conditions (7) to a system with gas outflow from the disc can be questionable. Analysis of dynamo models with outflow (Bardou et al. 2001) suggests that reasonable changes to the boundary conditions do not affect the dynamo too strongly, but this aspect of our model should be further explored.
We have neglected the intrinsic difference of the behaviours of the mean and turbulent magnetic fields near the disc surface. Since the horizontal size of the hot cavities is larger than the scale of the turbulent magnetic field but smaller than the scale of the mean magnetic field, the Lorentz force can resist the advection of the mean field more efficiently than that of the turbulent field. Furthermore, large-scale magnetic field loops drawn out by the fountain flow can be detached from the parent magnetic lines by reconnection, so that the flow will carry mostly small-scale fields. Therefore, a more plausible (albeit less conservative) model would include advection of the small-scale (but not the large-scale) magnetic field. In such a model, the effect discussed here can be even better pronounced.
Brandenburg et al. (1995) argue that the fountain flow can
transport the large-scale magnetic field into the halo by topological
pumping if the hot gas forms a percolating cluster in the disc and if
the turbulent magnetic Reynolds number CU in the fountain flow
exceeds 20; our estimate given below is
,
and we
expect that the small-scale magnetic fields are removed from the disc
more efficiently than the mean field.
The idea that advection of
small-scale magnetic fields can help the galactic dynamo may be
more robust than our particular model of dynamo quenching
that involves the magnetic -effect.
For example, if the dynamo coefficients are quenched due to
the
suppression of Lagrangian chaos by the small-scale magnetic fields (Kim 1999),
their advection out of the galaxy
will still allow the dynamo to operate efficiently.
Acknowledgements
We are grateful to David Moss for helpful comments. K.S. acknowledges the hospitality of NORDITA. This work was supported by the Royal Society, RFBR grant 04-02-16094 and INTAS grant 2021.