developed by Paul Matthews and Louis Tao) and with Fausto Cattaneo and

Thierry Emonet at Chicago. We are grateful for support from PPARC.

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23

Nonlinear magnetoconvection in the presence of a

strong oblique ¬eld

KEITH JULIEN

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA

EDGAR KNOBLOCH & STEVEN M. TOBIAS

Department of Mathematics, University of Leeds, Leeds LS2 9JT, UK

Reduced partial di¬erential equations valid for convection in a strong imposed

magnetic ¬eld (vertical or oblique) are derived and discussed. These equa-

tions ¬lter out fast, small-scale Alfv´n waves, and are valid outside of passive

e

horizontal boundary layers. In the regime in which the convective velocities

are not strong enough to distort substantially the ¬eld, exact, fully nonlinear,

single-mode solutions exist. These are determined from the reduced PDEs

reformulated as a nonlinear eigenvalue problem whose solution also gives,

for each Rayleigh number, the time-averaged Nusselt number and oscilla-

tion frequency together with the mean vertical temperature pro¬le. In the

oblique case a hysteretic transition between two distinct convection regimes

is identi¬ed. Possible applications to sunspots are discussed.

23.1 Introduction

The study of convection in an imposed magnetic ¬eld is motivated primarily

by astrophysical applications, particularly by the observed magnetic ¬eld dy-

namics in the solar convection zone (Hughes & Proctor 1988). Applications

to sunspots (Thomas & Weiss 1992) have led several authors to investigate

the suppression of convection by strong “vertical” or “horizontal” magnetic

¬elds. However, the magnetic ¬eld in sunspots is neither vertical nor hori-

zontal, and this has led to recent nonlinear investigation of convection in an

oblique magnetic ¬eld (Matthews et al. 1992, Julien et al. 2000). Numerical

simulations of magnetoconvection are unable to reach the parameter values,

both in terms of ¬eld strengths and Reynolds number (Re), characteristic

of convection in sunspots. Indeed, the former compounds the prohibitive

temporal and spatial restrictions placed on high-Re simulations through

the presence of high frequency Alfv´n waves (if Ma

e 1, where the Mach

number Ma := U/VA and U , VA are the ¬‚ow and Alfv´n speeds) and the

e

345

Julien, Knobloch & Tobias

346

development of thin (magnetic) boundary layers (if Rm 1, where the

magnetic Reynolds number Rm := U L/·, L is the length scale and · is

the ohmic di¬usivity). Even with today™s state-of-the-art computers these

facts together with limitations on memory and computational speed place

substantial constraints on the accessible parameter range. We develop here,

from the primitive-variable Boussinesq equations, a reduced set of PDEs

valid in the strong ¬eld limit (Ma 1) where convective velocities are

not large enough to distort the imposed ¬eld. These equations have the

appealing property of ¬ltering fast Alfv´n waves and relaxing the need to

e

resolve (magnetic or mechanical) boundary layers which can be determined,

a posteriori, as a passive inner solution.

The reduced equations presented in (23.5) below admit exact, fully non-

linear, single-mode solutions for both two- and three-dimensional spatially

periodic convection. The degree of nonlinearity is characterized by the dis-

tortion of the mean temperature pro¬le, and the vertical structure can be

followed from onset to high Re (or equivalently large Rayleigh number Ra)

via a one-dimensional eigenvalue problem for the (time-averaged) Nusselt

number. The derivation of this problem can be performed analytically, al-

though the problem itself must be solved numerically. We ¬nd that in the

strong ¬eld limit all competing steady patterns are degenerate in the sense

that they transport the same amount of heat. This is so also for oscillatory

patterns. In the overstable case two distinct modes of convection are un-

covered. The ¬rst or “vertical ¬eld” mode is characterized by thin thermal

boundary layers and a Nusselt number that increases rapidly with the ap-

plied Rayleigh number; this mode is typical of steady convection as well. The

second or “horizontal ¬eld” mode is present in overstable convection only

and has broad thermal boundary layers and a Nusselt number that remains

small and approximately independent of the Rayleigh number. At large

Rayleigh numbers this regime is characterized by a piecewise linear temper-

ature pro¬le with a small isothermal core. The “horizontal ¬eld” mode is

favoured for substantial inclinations of the ¬eld and su¬ciently small ohmic

di¬usivity. The transition between the two regimes is typically hysteretic

and for ¬xed inclination and di¬usivity may occur with increasing Re.

The dimensionless Boussinesq equations describing magnetoconvection in

a plane horizontal layer are

Dt u = ’∇π + Ma’2 B · ∇B + RT ˆ + Re’1 ∇2 u (23.1)

z

Dt T = Pe’1 ∇2 T (23.2)

Dt B = B · ∇u + Rm’1 ∇2 B , (23.3)

Nonlinear magnetoconvection in the presence of a strong oblique ¬eld 347

z

z™

θ

x

^

r

g x™

H ˜L

L

θ

Z

˜L

Fig. 23.1. A schematic diagram illustrating the appearance of a small length scale

in the vertical due to magnetic alignment. The rotated coordinate system for the

small scales is also shown.

with Dt = ‚t + u · ∇ and ∇ · u = ∇ · B = 0. Here u = (u, v, w) is the velocity

¬eld in Cartesian coordinates (x, y, z) with z vertically upwards, T denotes

the temperature and π is the total (thermal and magnetic) pressure. The

dimensionless magnetic ¬eld is assumed to be the superposition B = r + b

of an imposed oblique ¬eld of unit strength and a three-dimensional ¬eld

b(x, y, z, t) due to the presence of convection. Here r = (sin ‘, 0, cos ‘),

where ‘ denotes the angle with respect to the vertical in the (x, z) plane

(Fig. 23.1). The equations have been nondimensionalized with respect to

characteristic horizontal length scale L, speed U , time scale L/U , magnetic

¬eld strength B0 , and temperature di¬erence ∆T . The resulting dimen-

sionless parameters are the Reynolds (Re), P´clet (Pe), magnetic Reynolds

e

(Rm), buoyancy (R), and the Alfv´n speed Mach (Ma) numbers.

e

23.2 Reduced PDE description for Ma 1

Given our primary focus on investigating magnetically constrained ¬‚ows we