induced by kinematic dynamo studies and the MHD of steady ¬‚ows. In a

turbulent ¬‚ow the ¬eld lines are stretched at almost all locations, as nearby

trajectories of the ¬‚ow particles separate exponentially. Thus there is a

strong mechanism for enhancing magnetic energy locally. In the fully devel-

oped state, although there will be some intermittency due to cancellation

caused by folding of the trajectories, the ¬‚uid will be permeated with ¬eld

to a much greater extent than would appear at the surface. This is because,

at a boundary, areas of surface particles are not conserved. We therefore

expect to ¬nd tangled ¬elds with a fractal dimension between 1 and 2, even

in the limit · ’ 0. These ¬elds will exert a signi¬cant dynamical in¬‚uence

on the ¬‚ow, and so can be expected to be at equipartition levels. Such a

“magnetic fondue” can be glimpsed in the magnetic carpet, but the basic

arguments apply to all scales where the magnetic Reynolds numbers are

large, and where the turnover time (which, rather than the di¬usion time, is

the appropriate time for growth of the ¬elds) are not too large. The mech-

anisms at larger scales are likely to be a¬ected by Coriolis forces; however

the magnetic carpet, and numerical simulation, (Cattaneo 1999, Cattaneo

Dynamo processes: the interaction of turbulence and magnetic ¬elds 145

& Hughes 2001) show that large-scale rotation is not necessary for dynamo

action. The problem, therefore, is to understand how the large-scale ob-

served ¬elds can be generated (the “±” part of an ±-ω dynamo), in these

magnetically dominated ¬‚ows. The answers are still highly controversial.

This paper reviews recent theoretical ideas and associated numerical work

in an attempt to throw light on the di¬culties.

10.2 Field structure in kinematic dynamos at large Rm

The ±-e¬ect, or mean ¬eld dynamo, has long been a mainstay of theories of

the solar cycle, and it is still widely used today. The text by Mo¬att (1978)

gives an excellent overview of early applications, while more recent references

can be found in Weiss (1994). There are two basic assumptions; that there

is scale separation between ˜mean™ and ˜¬‚uctuating™ ¬elds; and that the

averaged e.m.f. induced by the small-scale ¬elds is a local function of the

mean magnetic ¬eld and its derivatives. The ¬rst assumption would seem

reasonable, but the second is harder to justify in the interesting case where

the magnetic Reynolds number is very large, even on the smallest scales.

(When the small-scale Rm 1 then a rigorous theory can be constructed;

see e.g. Mo¬att™s book). There are several important consequences of large

Rm (Galloway & Proctor 1992, Cattaneo et al 1995):

’1

• Field structures are highly intermittent, with length scales ∼ Rm 2 .

• These structures do not depend much on the value of Rm , but on the

topology of the ¬‚ow pattern. Only the thickness of the structures depends

on Rm .

• These small-scale ¬elds can be self sustaining; that is, there is a small-scale

dynamo.

The smallest scales of the ¬eld appear very rapidly in kinematic com-

putations at high Rm “ in fact after a few turnover times L/U . However

the growth rate of dynamo disturbances does depend on Rm , but typically

appears to reach a limiting value independent of Rm as Rm ’ ∞, though

precise computation becomes very di¬cult owing to to the small length

scales involved. This limiting growth rate is typically of order L/U ; these

are known as fast dynamos (see, e.g. Childress & Gilbert 1995). In spite of

the di¬culty in resolving the smallest structures, we ¬nd that scaling laws

for the eigenfunctions are established accurately at much lower Rm and can

be accurately calculated. Such laws give a power-law distribution for in-

tegrated quantities such as R1 ≡ | B |2 / |B|2 , which ∼ Rm , where γ is

γ

a constant of order unity depending on the ¬‚ow, as shown in Figure 10.1.

Proctor

146

Fig. 10.1. Behaviour of the quantity R1 de¬ned in the text as a function of Rm

for three di¬erent dynamo ¬‚ows (from Cattaneo et al. c 1995 by the American

Physical Society). The greater the slope, the greater the energy in the small-scale

¬elds.

These power laws demonstrate that the ¬eld distribution is fractal in na-

ture; and indeed if γ is not too small then at large Rm the smallest scales

of ¬eld are dominant, as is perhaps to be expected. One ¬nal aspect of

these kinematic fast dynamos deserves attention. When the dynamo ¬eld

exists on essentially the same scale as the velocity ¬eld, helicity of the ¬‚ow

is not necessary for e¬cient dynamo action. This shows that the fact that

the magnetic carpet ¬elds are on too short a timescale to notice the Sun™s

rotation does not rule out dynamo action as their cause. Such ¬elds will

not work as mean-¬eld dynamos (see below), because for them helicity is

essential. If buoyancy is the principal driving mechanism then helicity can-

not be introduced directly; it follows that for an e¬cient mean ¬eld dynamo

we require either rotation or inhomogeneity (giving gradients of large-scale

helicity).

10.3 Dynamical equilibration of small-scale dynamos

How large can a small-scale dynamo ¬eld get before the growth of the ¬eld

is halted by the dynamical e¬ects of the Lorentz force? In the solar context,

where the viscosity is small, we expect such e¬ects to occur when the mag-

netic energy density |B|2 /2µ0 is comparable with the kinetic energy density

Dynamo processes: the interaction of turbulence and magnetic ¬elds 147

Fig. 10.2. Finite time Liapunov exponents for a simple quasi-two dimensional dy-

namo (after Cattaneo et al. 1996). Lighter shades indicate greater stretching. (a)

Kinematic case, (b) dynamic case when Lorentz forces have reduced the stretching

properties of the ¬‚ow.

ρ|u|2 (equipartition). This expectation is con¬rmed by the results of sev-

eral calculations of model dynamos, and by the full MHD simulation of a

convective dynamo by Cattaneo (1999). At high values of Rm , as the ¬eld

amplitude grows, we must pass from a growth rate comparable with the

turnover time to one which is zero! How is this accomplished? One mech-

anism, which would hold for spatially constrained ¬‚ows, would be for the

kinetic energy to be reduced, thus reducing the magnetic Reynolds number

towards the critical value. This is most unlikely to happen when the kine-

matic Rm is far above critical, since that would demand a huge reduction in

the kinetic energy. Instead, these systems equilibrate in a much more subtle

way, which is almost invisible in the Eulerian statistics. An example is given

for a simpli¬ed model by Cattaneo, Hughes & Kim (1996), and examples

of ¬nite-time Liapunov exponents for the kinematic and dynamic cases are

shown in Figure 10.2. They reduce their e¬ciency as a dynamo by altering

their stretching properties, so that the Liapunov exponents go down, lead-

ing to less e¬cient energy growth, leaving cancellation e¬ects to mop up

such growth as remains. (It is possible that in some cases the cancellation

is enhanced, rather than the stretching reduced. But the detailed results

produced to date do not show this. Such enhancement is more likely to

be a consequence of two-dimensionalization of the ¬‚ow induced by a large-

scale ¬eld.) How long does it take for equilibration to be achieved? The

growth rates at high Rm are fastest for the smallest scales of motion, so one

could expect that each scale might become dynamically active after a time

Proctor

148

proportional to its turnover time. Magnetic energy reaches equipartition

successively at longer and longer scales. Finally we have “MHD turbulence”

with Lorentz forces important at all scales. The crucial question for the

coherent dynamo involved in the solar cycle is: can ¬elds which have a scale

much greater than that of the turbulent ¬‚ows grow at a substantial rate?

Thus we need to address the dynamical e¬ect of the Lorentz force on mean

¬eld growth.

10.4 Growth and equilibration of mean ¬elds

In this section we discuss the way in which large-scale (“mean”) ¬elds can

arise as a result of small-scale ¬‚uctuating motion. We ¬rst note that the

distinction between large and small scales is only clear when the small-scale

turbulence is homogeneous. Any systematic large-scale inhomogeneity will

ineluctably lead to components of the Fourier spectrum of the ¬eld on the

same scale. These are of a di¬erent nature, however, from the independently

generated ¬elds that form the cycle. The e¬ects of the small-scale on the

large-scale ¬elds may be seen by writing the magnetic ¬eld B = B + b and

the velocity ¬eld U = U+ u; then the induction equation for time derivative

of the mean ¬eld B becomes