type structures, the ampli¬cation factor is larger but the dynamical e¬ects

smaller. In either case, when the small-scale ¬eld reaches the equiparti-

tion value we expect a signi¬cant change in the dynamo process. Thus the

physical picture predicts a ≈ 1. Although the small-scale ¬eld is highly

intermittent the crucial mechanism of dynamo generation occurs precisely

where the small-scale ¬elds are being produced “ and so such intermittency

is unlikely to a¬ect the value of a signi¬cantly.

These ideas have their origin in simpler studies in two dimensions (e.g.

Vainshtein & Cattaneo 1992) investigating the e¬ects of the Lorentz force on

the di¬usion rate of an imposed large scale ¬eld. Here there is no dynamo,

but similar considerations suggest that the stretching properties of the ¬‚ow

are a¬ected, leading to a decrease in the turbulent di¬usivity, In that work

it is argued that the conservation of the mean square magnetic potential in

the absence of di¬usion, together with the requirement that the turbulent

di¬usion have a value independent of ·, requires the small-scale magnetic

¬eld to exist on di¬usive length scales. There is a clear analogy in three

dimensions with the conservation of magnetic helicity. This leads via (10.7)

to the requirement of magnetic ¬elds on di¬usive scales in order that ± not

depend on the di¬usivity. It is notable that there is no similar conservation

law for mean square potential in three dimensions, and that the turbulent

di¬usivity in this case is a¬ected much less by the imposed ¬eld (one can

see that unknotted ¬eld lines can slip through highly conducting material

in 3D without a¬ecting the ¬‚ow much). One would expect the helicity

Proctor

152

constraint to have an e¬ect on this process; the situation remains unclear.

The reduction in the ±-e¬ect occurs on this view because the Lorentz forces

prevent the smallest scales of the magnetic ¬eld from reaching di¬usion

levels. In addition, when the magnetic Prandtl number is of order unity, as

may be appropriate for the Sun, the MHD turbulence spectrum may contain

a signi¬cant proportion of Alfv´n waves, for which u and b are parallel and

e

which thus give no contribution to E. When the magnetic Prandtl number

is very large, as may be the case in galaxies, then of course there are no

Alfv´n waves and the equilibration mechanism is di¬erent, perhaps leading

e

to smaller values of a, as shown in recent work by Schekochihin, Cowley,

Maron & Malyshkin (2002).

The idea that a is signi¬cant is given support from three very di¬erent

numerical studies. The ¬rst (Brandenburg 2001) considers ¬‚ow in a periodic

domain, forced by a helical body force on a small scale. There is eventual

growth of signi¬cant large-scale ¬elds, which are force-free and can grow to

large size free of dynamical constraints. While increasing Rm leads to more

rapid initial growth, the time taken for ¬nal equilibration also increases. The

±-e¬ect is calculated by solving a short-time initial value problem, and by

superposing a uniform mean ¬eld and calculating E directly. Both methods

(see Figure 10.3) yield a signi¬cant dependence on Rm in the ±-quenching

formula, with a ≥ 1. Brandenburg also ¬nds that the turbulent di¬usivity

is quenched, but that the dependence on Rm is rather weaker, as suggested

above. The remaining studies were carried out by Cattaneo & Hughes (1996)

and Cattaneo, Hughes & Thelen (2002). In the ¬rst, a kinematic ¬‚ow is

forced that has the form of the so-called CP-¬‚ow of Galloway & Proctor

(1992). A fully three-dimensional calculation is undertaken, starting from

this velocity ¬eld with an imposed uniform ¬eld in the z-direction. Only

that part of the ±-e¬ect which derives from fully three-dimensional, that

is nonlinearly driven, ¬‚ows is evaluated by direct calculation of E and the

results show that a ∼ 1 for the quenching properties. The magnitude of the

turbulent ¬‚uctuations, however, scarcely changes with the imposed ¬eld.

This last result was predicted previously by Cattaneo & Hughes (1996).

In the paper of Cattaneo et al. the CP ¬‚ow is again employed, but now

solutions are sought in a long periodic box in the z-direction (the original

¬‚ow being independent of z). The length of the box is chosen as 8 times

the period of the most rapidly growing mode; the latter then plays the role

of ¬‚uctuating ¬eld, while the mode with the same period as the box plays

the role of the large-scale ¬eld. Two di¬erent case are considered. In one

the initial condition has comparable energy in the small and large scales,

while in the other the large-scale energy is initially much greater. The ¬nal

Dynamo processes: the interaction of turbulence and magnetic ¬elds 153

Fig. 10.3. Graphs of various runs from the paper of Brandenburg (2001), showing

the reduction of ± with increasing B. The top ¬gure shows the result of solving a

short-time initial value problem, and the bottom ¬gure the value calculated from

imposing a uniform ¬eld. The results are very similar. The lower curves correspond

to greater values of Rm .

state appears to depend on these initial conditions. In the ¬rst case the

nonlinear interactions between di¬erent wavenumbers force rapid growth

of the large-scale ¬eld, although its natural growth rate is much less than

that of the small-scale ¬eld, but growth stops when the large-scale ¬eld has

Proctor

154

Fig. 10.4. Graphs of runs from the paper of Cattaneo et al. (2002). The di¬erent

lines refer to modes of di¬erent wavenumber. The initial energy of the largest scale

(K = 1) and the most unstable modes (K = 8) are comparable. Growth of the

K = 1 mode is accelerated above its kinematic rate between times t1 and t2 .

much lower energy than the short lengthscale mode. There seems to be a

further adjustment on a much longer timescale. In the second case the large

scale quickly equilibrates, leaving the other scales at lower values. Results

are shown in Figures 10.4, 10.5. It turns out that the evolution of the large

scale ¬eld can be discussed in terms of an ±-e¬ect. This is veri¬ed by looking

at similar short box calculations and evaluating the ±-e¬ect as in the earlier

Dynamo processes: the interaction of turbulence and magnetic ¬elds 155

Fig. 10.5. As for the previous ¬gure except that the energy in the largest scale is

initially much greater than that in the other scales. The magnetic ¬eld becomes

dynamically active at time td , and nonlinear saturation occurs at time ts .

paper described above. The two methods give very similar results, justifying

the interpretation. It is again found that the process of ±-quenching depends

strongly on Rm , as indeed does the initial value of ± for weak imposed ¬elds.

(see Figure 10.6). From the results (Figure 10.7) we can see that ± falls to

di¬usive values while the mean ¬elds are well below equipartition values.

Proctor

156

Fig. 10.6. The behaviour of the ± coe¬cient for ¬xed B as a function of Rm , in the

calculation of Cattaneo et al. with comparable initial large and small-scale ¬eld. It

can be seen that there is a strong reduction as Rm increases.

Fig. 10.7. The behaviour of the ± coe¬cient (y-axis) for Rm = 100 as a function of

2

B (x-axis), in the same calculation as for the previous ¬gure. It can be seen that

2

there is a strong reduction in ± when Rm B is of order unity.

The conclusion of these studies, that ±-quenching is very strong at large

values of Rm , of course makes it di¬cult to see how large-scale ¬elds could

arise on other than irrelevant di¬usive timescales. A possible chink in the

Dynamo processes: the interaction of turbulence and magnetic ¬elds 157

reasoning has been identi¬ed by Blackman & Field (2000), who argue that

the results depend strongly on the constraint that small-scale helicity (and

not just total helicity) is conserved. Such conservation is natural in model

experiments with periodic boundary conditions, but it may be that with

more realistic boundary conditions the separate conservation of small-scale

and large-scale magnetic helicity will be destroyed, allowing a decrease of

small-scale helicity, which may a¬ect the quenching process. Calculations