= ∇ — (U — B) + ∇ — E ’ ∇ — ·∇ — B , where E = u — b . (10.1)

‚t

In order to get signi¬cant mean ¬elds on a relevant (i.e. non-di¬usive)

timescale we need the “±-e¬ect” ± , de¬ned by the ansatz E = ± · B, to

be of order |u|, i.e. independent of ·, the magnetic di¬usivity. While ±

is straightforward to calculate when the small-scale Rm is small (see, e.g.

Mo¬att 1978), it is much harder to see how to proceed when the small-

scale Rm is large. In the Parker (1955) picture ¬eld lines are twisted and

rotated by a helical “cyclonic event”. For events shorter than the turnover

time we can say that E is proportional to ’H, where H is the helicity.

But if such an event persists longer than a turnover time the constant of

proportionality may change sign due to multiple rotations. Thus even the

sign of the e¬ect is not certain, and there are other problems associated

with the possible nonlocal dependence of E on B. Nonetheless, one can

imagine an experiment in which a uniform magnetic ¬eld B0 permeates a

region of homogeneous MHD turbulence. There is no large scale dynamo

but E can be calculated as a function of B0 . It is crucial to understand

how ± depends on B0 . We expect it to reduce with increasing ¬eld (“±-

quenching”), but when do Lorentz forces become important and initiate

Dynamo processes: the interaction of turbulence and magnetic ¬elds 149

this quenching? There is considerable controversy over this question. To

1

¬x ideas, de¬ne BE , the equipartition ¬eld strength, as (µ0 ρ|u2 |) 2 . Then

we can all agree that because of the symmetry under sign change of B, we

expect some functional dependence for large Rm of the form

±(B) = F(Rm |B|2 /BE ).

2

a

(10.2)

The controversy resides in the value of the exponent a. If a 1 then the

large-scale ¬elds can reach equipartition values with relative ease, while if

a is not small the mean ¬eld mechanism shuts down when |B| is still well

below BE , making the timescales for the production of large-scale ¬elds

inordinately long.

Before looking at recent simulations which cast light on the value of a, we

¬rst deal with the formula for ± in MHD turbulence originally put forward

by Pouquet, Frisch & L´orat (1976) and revisited by Blackman & Field

e

(2000). We begin with ¬‚uctuating magnetic and velocity ¬elds b, u. Then a

uniform ¬eld B is added, and this has the e¬ect of changing the ¬‚uctuating

¬elds to b + b , u + u , where b , u obey the equations

B · ∇b

‚u 1

= ’∇p +

‚t µ0 ρ

+ small(?) di¬usion terms . (10.3)

‚b

B · ∇u

=

‚t

To ¬nd the mean e.m.f. proportional to B, we can assume isotropy, so that

±ij = ±δ ij . Thus we have (the dots denoting time derivatives)

E = ±B = u — b + u — b

„c (10.4)

™ ™

≈ u — b + u — b dt where „c is a “correlation time” .

0

If „c is short relative to other timescales then we can use (10.3) to obtain

„c

u · ∇ — u ’ (µ0 ρ)’1 b · ∇ — b .

±≈’ (10.5)

3

There are many assumptions made in this derivation, not least the one that

equates correlation times for velocity and magnetic ¬elds. Nonetheless the

expression (10.5) does have the satisfying characteristic that if the “turbu-

√

lence” takes the form of Alfv´n waves, for which u = ±b/ µ0 ρ, then E

e

must vanish. Unfortunately the formula has been interpreted by many au-

thors as giving a model of the e¬ects of large imposed ¬elds on ±, with u, b

considered as the actual ¬elds. In fact the formula can be justi¬ed only for

small |B|, since equations (10.3) can then be linearized; and where the ¬eld

Proctor

150

b has nothing to do with B but is preexisting. Nonetheless it is useful as a

guide to the initial growth rate of a large-scale ¬eld in the presence of MHD

turbulence. It should be emphasised that the induction equation remains

linear irrespective of the e¬ects of the Lorentz force, and so the last term in

(10.5) can only arise from magnetic ¬elds that do not owe their existence to

the imposed ¬eld B. This is not the situation considered by Mo¬att (1978)

and others.

Whether or not the above result remains true for large imposed ¬elds,

there remains the crucial question posed above: what is the form of the

function F de¬ned in (10.2), and what in particular is the crucial exponent

a? In general terms we expect that F(X) decreases with X, and ∼ X ’β

as X ’ ∞, with β ≥ 1. The existence of large-scale ¬elds of signi¬cant

amplitude suggests that a is small, while numerical calculations of idealized

problems suggest that a ∼ 1, which must lead to signi¬cant problems with

the large-scale ¬elds. In consequence these calculations have been criticized

as inapplicable to real MHD turbulence. Nonetheless there are several theo-

retical reasons for supposing a signi¬cant, and the critics have not yet found

a de¬nitive solution to the di¬culty.

The theoretical backing for a to be signi¬cant is provided by Gruzinov &

Diamond (1994, 1995). They consider a situation in which magnetic and

velocity ¬elds are statistically stationary. This implies that the time deriva-

tive of the mean magnetic helicity a · b vanishes, where a is the magnetic

potential de¬ned by b = ∇ — a, ∇ · a = 0. The equations for a and b are

‚a

= (u — B) + (u — b) ’ ∇χ ’ ·∇ — b

‚t (10.6)

‚b

= ∇ — (u — B) + ∇ — (u — b) ’ ∇ — (·∇ — b)

‚t

‚

where χ is the electrostatic potential. Setting (a · b) = 0, we obtain after

‚t

some manipulation

B · (u — b) = B · E = ’· b · ∇ — b,

and so we have the exact result (not depending on any assumptions con-

cerning small Rm or short correlation times)

± = ’|B|’2 · b · ∇ — b. (10.7)

It should be noted here that the ¬eld b is now the total small scale ¬eld;

there is no approximation involving small |B|. Gruzinov & Diamond use

(10.7) in combination with (10.5) to give a relation between ± and |B| of

Dynamo processes: the interaction of turbulence and magnetic ¬elds 151

the form

± = ±0 (1 + Rm |B|2 /BE )’1 ,

2

where ±0 is the kinematic ±-e¬ect that holds when Lorentz forces are neg-

ligible. which suggests that a = 1. Although this result is very appealing,

it must be recalled that the de¬nitions of b in (10.5) and (10.7) are not

obviously compatible. Further calculations establish that the part of E pro-

portional to gradients of B (the ˜turbulent di¬usivity™) only depends on

2

|B |/BE .

2

The physical picture that backs up the theory has been elaborated by

Cattaneo & Hughes (1996), and recently given support by Brandenburg

(2001). The basic idea is simple. The dynamical e¬ects of the magnetic

¬eld on the ¬‚ow must be felt when the Lorentz forces become signi¬cant.

In ¬‚ows of astrophysical interest, Rm 1 even on the small scales, and in

this case |b| |B|. In fact when the ¬elds are su¬ciently weak the growth

of small-scale ¬eld is limited by di¬usion in regions of ¬‚ow convergence, and

so we expect |b| ∼ Rm |B| if the ¬eld is in sheets. When we have ¬‚ux-tube-

1/2