Annu. Rev. Astron. Astrophys. 1996. 34:
155-206
Copyright © 1996 by . All rights reserved |

**4.5. The Quenching Problem**

In recent years the feedback of the magnetic field on the turbulent diffusion and the -effect has become a topic of major concern. Piddington (1970) was the first to suggest that for large magnetic Reynolds numbers the magnetic fluctuations would be strong enough to suppress turbulent diffusion. This idea was rejected by Parker (1973), who argued that the development of strong small-scale fields is limited by reconnection, so that they do not hinder turbulent mixing of field and fluid. In fact, without turbulent diffusion the galactic differential rotation would wind up the field so tightly that it would not resemble the magnetic field structure of any observed galaxy (Section 8.3).

The results of the two-dimensional numerical MHD experiment of
Cattaneo & Vainshtein
(1991)
stimulated new interest in the problem of turbulent diffusion. They
found that
_{t} is
suppressed according to
_{t} =
*v l* / (1 + *R*_{m}
^{2} /
*B*_{eq}^{2}), where *R*_{m} =
*vL* /
is the magnetic Reynolds number based on the microscopic diffusivity.
Evidently,
_{t}
would be significantly reduced when
is comparable to *R*_{m}^{-1/2} *B*_{eq}.
In galaxies, *R*_{m} =
(10^{17}), so
_{t}
would essentially be zero.
Even if we used a Reynolds number based on ambipolar diffusion,
with *R*_{m}^{AD}
*O*(10^{3}),
_{t}
would still be too small. This type of quenching is much stronger than
the "traditional" quenching
(Moffatt 1972),
so something seems to be wrong (e.g.
Field 1996).

In three dimensions the turbulent motions would continue to entangle the magnetic field in the direction perpendicular to (Krause & Rüdiger 1975, Parker 1992). This has now also been demonstrated numerically (Nordlund et al 1994) as well as analytically (Gruzinov & Diamond 1994). In other words, turbulent diffusion is really not significantly suppressed at field strengths somewhat below the equipartition value. The decay of sunspots is a good example of this (Krause & Rüdiger 1975).

Vainshtein & Cattaneo
(1992),
Tao et al (1993)
suggested that the
-effect might also be
quenched dramatically,
=
_{kin} /
(1 + *R*_{m}
^{2} /
*B*_{eq}^{2}), where
_{kin} is the
kinematic value of Equation (3). The analysis of
Gruzinov & Diamond
(1994)
seems to support this result. On the other hand, the simulations of
Tao et al (1993),
as well as unpublished simulations by A Brandenburg, are reminiscent of
an earlier result by
Moffatt (1979),
that the -effect may
fluctuate strongly and never
converge to a finite value if *R*_{m} is large.

There is at present no conclusive resolution to this problem, but here
are some possibilities: (*a*) The conventional
-effect might still work
in reality, but the method used to estimate
from simulations is
inappropriate
(e.g. the boundary conditions preserve the magnetic flux, so the
-effect is forced to
have zero effect on the average field;
or the computational domain might be too small compared to the eddy size).
(*b*) The conventional
-effect is really
nonexistent, but instead some
other mechanism (e.g. an inverse cascade mechanism, incoherent
-effect, or
cross-helicity effect) generates large-scale fields in conjunction with
shear. (*c*) An important contribution to
comes from the Parker
instability: This mechanism would work especially for finite magnetic
fields.

A somewhat different problem was raised by Kulsrud & Anderson (1992), who suggested that the growth of large-scale fields is suppressed by ambipolar diffusion at small scales. However, before we can draw any final conclusions, nonlinear effects need to be included. These can be important for two reasons: The inverse cascade process is inherently nonlinear, and nonlinear ambipolar diffusion can lead to sharp magnetic structures (Brandenburg & Zweibel 1995), which would facilitate fast reconnection and rapidly remove magnetic energy at small scales.

The problem raised by
Vainshtein & Cattaneo
(1992) is related to the
assumption that most of the magnetic energy is at small scales,
i.e. < **B**^{2} >> > **B**^{2}.
This, however, is only a result of linear theory and is
not supported by observations (Section 3).
A recent simulation by
Brandenburg et al (1995a)
is relevant in this context. Here a large-scale field is generated with
<**B**^{2}> / <**B**^{2}>
0.5 >>
*R*_{m}^{-1/2}
0.1.
The dynamo works even in the presence of ambipolar diffusion, which
Kulsrud & Anderson
(1992)
thought to be effective in destroying large-scale dynamo action.
Here, the incoherent
-effect is much larger
than the coherent
effect, but the estimated value of the dynamo number is nevertheless
above the critical value, suggesting that conventional dynamo action
might also be at work.