matched to a ¬‚ux proportional to T 4 .

In the absence of convection there is a polytropic reference atmosphere

with T ∝ z and ρ ∝ z m . For a gas with γ = 5/3 the layer is superadiabat-

ically strati¬ed if m < 3/2: in the calculations described here m = 1 and

θ = d/z0 = 10, so that the density increases by a factor of (θ + 1) = 11

across the layer. When the equations are rendered dimensionless the system

is de¬ned by ¬ve dimensionless parameters. These are the Rayleigh number

R, which measures the superadiabatic gradient; the Chandrasekhar number

Q, which is proportional to the square of the imposed magnetic ¬eld, the

Prandtl number σ; the ratio ζ of the magnetic to the thermal di¬usivity; and

the aspect ratio ». Since both R and ζ are functions of depth it is convenient

ˆ

ˆ

to use the values R and ζ de¬ned at the middle of the layer (z = z0 + d/2).

All the compressible calculations presented here adopt the parameter values

ˆ

σ = 1 and ζ = 1.2. Thus the actual di¬usivity ratio, which is proportional

to the density, lies in the range 0.2 ¤ ζ ¤ 2.2; this mimics the e¬ects of

ionization near the solar surface, where ζ > 1 at depths between 2 000 and

20 000 km.

The governing equations are solved numerically for three-dimensional mo-

tion in a cuboidal box, using a mixed ¬nite-di¬erence pseudospectral code

that has been optimized for parallel processing, with up to 256 — 256 — 100

mesh points. The results described here were obtained using a 64-processor

partition of the Hitachi SR2201 computer at the University of Cambridge

High Performance Computing Facility.

The onset of instability, followed by an ordered array of weakly nonlinear

plumes and eventually by turbulent convection, can be studied either by

ˆ ˆ

increasing R for ¬xed Q or by decreasing Q for ¬xed R. Rucklidge et al.

(2000) set Q = 1 000, and ¬xed the temperature at the upper boundary. In

Modelling solar and stellar magnetoconvection 333

a small box, with » = 2, the initial pattern of steady plumes on a slightly

distorted hexagonal lattice, gave way to intermittent behaviour and eventu-

ˆ

ally to broad chaotic plumes as R was increased. The results are, however,

sensitive to the aspect ratio of the computational box. In a su¬ciently wide

box, with » = 8, a new e¬ect appears: there are distinct regions where the

¬eld is strong and convection is suppressed, separated from regions from

which magnetic ¬‚ux has been expelled by vigorous convecting plumes (Tao

et al. 1998).

22.3 Flux separation

The regime in which ¬‚ux separation occurs has been systematically explored

ˆ

by setting R = 100 000 and varying Q in a box with » = 8 and a ˜radiative™

thermal boundary condition at z = z0 (Weiss, Proctor & Brownjohn 2002).

For these parameter values, linear theory shows that the onset of convection

occurs as a stationary bifurcation at R ≈ 4 200. This is followed by a

ˆ

magnetically dominated regime, for Q ≥ 2200, with a pattern of steady

small-scale convection in narrow hexagonal cells. Figure 22.2 shows the

array of steady plumes for Q = 3 000. Owing to the strati¬cation, these

plumes expand as they rise, and they are surrounded by a network of cooler

sinking ¬‚uid. The rising plumes sweep magnetic ¬‚ux aside and concentrate it

into a network at the upper surface. The sinking ¬‚uid is focused into slender

falling plumes that impinge upon the lower boundary, where magnetic ¬‚ux

is concentrated at the centres of the rising plumes. Small-scale convection

remains stable down to Q = 1 600; the pattern illustrated in Figure 22.3 is,

however, time-dependent and alternate plumes wax and wane aperiodically

in vigour. Such spatially modulated oscillations were ¬rst identi¬ed in two-

dimensional calculations.

When Q = 1 400 this pattern is unstable. Rising plumes amalgamate,

expelling magnetic ¬‚ux to form a vigorously convecting and almost ¬eld-

free cluster. This is surrounded by a region where the magnetic ¬eld is

su¬ciently strong that only small-scale convection can occur. Thus magnetic

¬elds are segregated from the motion. This process of ¬‚ux separation is,

moreover, associated with hysteresis: Figure 22.4 shows a ¬‚ux-separated

solution for the same parameter values as Figure 22.3. Both types of solution

are stable in the intermediate regime with 2 000 ≥ Q ≥ 1 600. For moderate

¬eld strengths (1 400 ≥ Q ≥ 600) only ¬‚ux separated solutions are found.

The pattern for Q = 1 000 is shown in Figure 22.5. Finally, when the

imposed ¬eld is weak (Q ¤ 500), magnetic ¬‚ux is con¬ned to a narrow

network enclosing clusters of actively convecting and evolving plumes, as

Weiss

334

Fig. 22.2. Compressible magnetoconvection with R = 105 , σ = 1, ζ = 1.2 and

ˆ

ˆ

Q = 3 000 in a box with » = 4. The grey-scale image in the upper panel shows the

variation of |B|2 across the top and bottom of the layer; dark (light) regions denote

weak (strong) ¬elds. Temperature ¬‚uctuations are indicated on the sides of the

box and the arrows represent the tangential component of the velocity. The lower

panel shows the temperature gradient |‚T /‚z| at the upper and lower boundaries,

with dark (light) regions denoting weak (strong) gradients; note that |‚T /‚z| ∝ T 4

at the top. Rising and expanding plumes at the top boundary appear dark in the

upper panel and light in the lower panel. (After Weiss et al. 2002.)

shown in Figure 22.6. Magnetic ¬‚ux moves rapidly through this network,

like a ¬‚uid, giving rise to intense but ephemeral ¬elds at the corners. For

Q 200 convection becomes much more vigorous and magnetic ¬‚ux is

con¬ned to isolated ¬‚ux tubes, with intense magnetic ¬elds, that are almost

completely evacuated. The resulting computational di¬culties provide an

Modelling solar and stellar magnetoconvection 335

Fig. 22.3. Time-dependent small-scale convection for Q = 1 600. As Figure 22.2

but in a box with » = 8.

e¬ective lower bound to the values of Q that can be used in this series of

numerical experiments.

22.4 Small-scale dynamo action

The model calculations described above leave open the question of what

happens when Q 200, or when no net ¬eld is imposed. Is turbulent con-

vection capable of generating and maintaining a disordered magnetic ¬eld?

Although the relevant regimes cannot yet be reached in our compressible

runs, they are accessible if the Boussinesq approximation is adopted. In-

deed, Cattaneo (1999) has shown that turbulent convection can act as a

Weiss

336

Fig. 22.4. As Figure 22.3 but for a ¬‚ux-separated solution, again with Q = 1 600.

There is now a region with vigorously convecting plumes, separated by a front from

the magnetically dominated region.

small-scale dynamo if the magnetic Reynolds number, Rm , is su¬ciently

high. In this approximation the ¬‚uid is assumed to be incompressible and

density ¬‚uctuations only enter through the buoyancy term in the equation

of motion, so that both u and B are solenoidal. This simpli¬cation makes it

possible to model much more vigorous convection, with R = 5 — 105 , σ = 1

and ζ = 0.2, when Rm ≈ 1 200. The governing equations have been solved

numerically, subject to standard idealized boundary conditions, in very wide

boxes, with aspect ratios » = 10 and » = 20, requiring meshes with up to

10242 — 96 gridpoints.

Modelling solar and stellar magnetoconvection 337

Fig. 22.5. As Figure 22.4 but for Q = 1 000 when the convection is more vigorous

and the magnetic ¬eld is concentrated into a smaller fraction of the layer.