Acknowledgements The author is grateful to P. K. Smolarkiewicz who carried

out simulations with his VLES code.

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22

Modelling solar and stellar magnetoconvection

NIGEL WEISS

Department of Applied Mathematics & Theoretical Physics,

University of Cambridge, Cambridge CB3 9EW, UK

Numerical experiments on three-dimensional convection in the presence of

an externally imposed magnetic ¬eld reveal a range of behaviour that can be

compared with that observed at the surface of the Sun (and therefore expected

to be present in other similar stars). In a strongly strati¬ed compressible

layer small-scale convection gives way to a regime with ¬‚ux separation as the

¬eld strength is reduced; with a weak mean ¬eld magnetic ¬‚ux is concentrated

into narrow lanes enclosing vigorously convecting plumes. Small-scale dy-

namos, generating disordered magnetic ¬elds, have been found in Boussinesq

calculations with very high magnetic Reynolds numbers; there is a gradual

transition from dynamo action to magnetoconvection as the strength of the

imposed ¬eld is increased.

22.1 Introduction

Thirty-seven years ago, when I was a postdoc at Culham, Roger Tayler told

me that he was sending a very bright young research student to spend the

summer there “ and so I ¬rst met Douglas. When I moved to Cambridge

a year later he was ¬nishing his Ph.D. and then he and Rosanne went o¬

to the States for a few years. We™ve been in close contact ever since they

returned to Cambridge and it has been a great pleasure having Douglas as

a colleague and a friend “ always stimulating and often argumentative, but

never causing any serious disagreement. So I am very glad to have a chance

of saying ˜Thank you™ here.

As we have already been reminded, Douglas™s third paper (Gough &

Tayler 1966) was on magnetoconvection. In those days we were all con-

cerned with linear theory and they used an energy principle to extend the

Schwarzschild criterion for convective stability in a strati¬ed atmosphere

and to establish a su¬cient criterion for stability in the presence of a mag-

329

Weiss

330

Fig. 22.1. Magnetoconvection at the solar photosphere. This G-band image, ob-

tained on the Swedish Vacuum Solar Telescope at La Palma by T. Berger & G.

Scharmer, shows a region about 120 000 km square. Granular convection is inter-

rupted by sunspots and pores, with strong magnetic ¬elds, and locally intense ¬elds

show up as small bright points in intergranular lanes. (Courtesy of T. Berger.)

netic ¬eld. Nowadays we are all involved with nonlinear problems. What is

exciting is that advances in observational techniques and high performance

computing have at last made it possible to compare ¬ne structures revealed

by high-resolution observations with corresponding features in numerical ex-

periments. So I shall present some recent computational results on nonlinear

magnetoconvection and relate them to small-scale ¬elds at the surface of the

Sun.

Let me begin with the observational motivation. Figure 22.1 shows a re-

markable image of the solar photosphere, obtained in the CH G-band at

4305 ˚. The granulation, corresponding to convective plumes with diame-

A

ters of about 1 000 km, is clearly visible, as are various magnetic features

embedded in it. The latter range from a large sunspot, where convection

Modelling solar and stellar magnetoconvection 331

is partially suppressed by a strong, predominantly vertical magnetic ¬eld,

through small, dark pores to bright points that denote the sites of locally

intense magnetic ¬elds. These concentrated ¬elds are embedded in the dark

lanes surrounding the bright granules, which form a network of colder sink-

ing gas, and especially at junctions in this network. On this scale, the Sun™s

rotation has no signi¬cant e¬ect.

Observations from the MDI instrument on SOHO have demonstrated

that magnetic ¬‚ux is constantly emerging through the solar surface to form

ephemeral active regions, largely uncorrelated with the ordered ¬‚ux associ-

ated with the 11-year activity cycle. This ¬‚ux is shredded by the granulation

to form small ¬‚ux elements which are transported by the supergranular ¬‚ow

until they end up in the network that encloses supergranules, where oppo-

sitely directed ¬elds eventually reconnect and disappear (Simon, Title &

Weiss 2001). This whole process raises the question of how and where all

this magnetic ¬‚ux is generated, to which I shall return.

In what follows, I shall ¬rst describe some general features of compress-

ible magnetoconvection. Then I shall focus on the issue of ¬‚ux separation,

which we have recently explored at Cambridge. After that, I shall discuss

the related problem of turbulent dynamo action and the maintenance of

disordered small-scale ¬elds, which has been studied in collaboration with

colleagues in Chicago. Finally, I shall attempt to summarize where we stand

and what we need to do next.

22.2 Compressible magnetoconvection

Nonlinear convection in a perfect gas, in the presence of an imposed vertical

magnetic ¬eld, is governed by the equation of motion:

Du

= ’∇p + ρg + j — B + Fvisc ,

ρ

Dt

together with the continuity equation

‚ρ

= ’div(ρu) ,

‚t

the equation of state p = RT ρ, the entropy equation

DS

= ’div(k∇T ) + Sdiss

ρT

Dt

and the induction equation

‚B

= curl (u — B) + ·∇2 B ,

‚t

Weiss

332

with div B = 0. Here u, B and j are the velocity, magnetic ¬eld and electric

current, while ρ, p, S and T are the density, pressure, entropy density and

temperature, respectively; g is the gravitational acceleration, R is the gas

constant, k is the thermal conductivity, · is the magnetic di¬usivity, Fvisc

is the viscous force and Sdiss includes both viscous and ohmic dissipation.

These equations are solved in a cuboidal region {0 ¤ x ¤ »d, 0 ¤ y ¤

»d, z0 ¤ z ¤ z0 + d}, subject to periodic lateral boundary conditions in

the horizontal x- and y-directions. At the upper and lower boundaries the

normal velocity and the tangential components of the viscous stress both

vanish, while the magnetic ¬eld is constrained to be vertical. The temper-

ature is ¬xed at the lower boundary (z = z0 + d); at the upper boundary

(z = z0 ), there are two possibilities: either the temperature is ¬xed or else