vertical velocity, while k⊥ ≡ |(kx , ky , 0)| is the wavenumber in the plane

perpendicular to r. The constant K is given by

’1

ω 2 + Pe’2 k⊥

1 4

K= dZ (23.7)

ω 2 + Pe’2 k⊥ + 2k⊥ |W |2

4 2

0

Nonlinear magnetoconvection in the presence of a strong oblique ¬eld 351

Fig. 23.2. Panel (a) (left) shows the (time-averaged) Nusselt number K for steady

(dashed line) and oscillatory (solid line) convection for ‘ = 0, 10—¦, 20—¦ , 30—¦ as a

function of the scaled Rayleigh number Ra when ζ = 0.1 and σ = 1.1. Panel (b)

(right) shows the development of an isothermal core in the mean temperature pro¬le

T (Z) with increasing Ra when ζ = 0.1, ‘ = 0.

and for steady patterns is to be identi¬ed with the Nusselt number; for

oscillatory patterns K represents the time-averaged Nusselt number.

Equation (23.6) is to be solved subject to impenetrable boundary condi-

tions W = 0. Once this is done the mean temperature pro¬le T (Z) can be

found from the relation

2

2k⊥

|W |2 = ’KPe .

DT 1 + 2 (23.8)

’2 k 4

ω + Pe ⊥

23.4 Results

The solutions of the problem (23.6-23.8) depend on the parameter set (R ,

Re, Pe, Rm), wavenumber k⊥ and ‘. For contact with previous work (Julien

et al. 1999, 2000), we set U = κ/L (the thermal di¬usive velocity scale). It

follows that Pe = 1, Re’1 = σ (Prandtl number), Rm’1 = ζ (magnetic

Prandtl number), and A ≡ Ma = (σζQ)’1/4 . We also de¬ne the scaled

Rayleigh number Ra = R /ζ. We solve this problem on a discretized one-

dimensional mesh using an iterative Newton-Raphson-Kantorovich scheme

with O(10’10 ) accuracy in the L2 norm of W (Z) and the corresponding

eigenvalues K and ω.

We present ¬rst the results for a vertical magnetic ¬eld (‘ = 0), and then

discuss the oblique ¬eld (‘ = 0) in the so-called perpendicular case, i.e.,

Julien, Knobloch & Tobias

352

when the roll axes are perpendicular (ky = 0) to the plane containing g

and ˆ. All results are obtained with the wavenumber k⊥ = 1, σ = 1.1 and

r

ζ = 0.1.

In Fig. 23.2a we show the (time-averaged) Nusselt number K and fre-

quency for both steady and overstable convection when ‘ = 0 as a function

of the scaled Rayleigh number Ra . Observe that solutions can be obtained

for highly supercritical Rayleigh numbers and that K increases monoton-

ically with increasing Ra . We also ¬nd that the frequency ω saturates.

For both steady and oscillatory convection the temperature gradients are

con¬ned to thinner and thinner boundary layers at the top and bottom as

Ra increases; this process occurs more rapidly in the steady case. At the

same time the bulk of the layer becomes more and more isothermal (see

Fig. 23.2b). Midplane symmetry implies that these boundary layers are

identical and that the isothermal interior has temperature T = 1/2.

Calculations show that these results are not changed qualitatively when

the magnetic ¬eld is tilted, provided that the tilt angle ‘ is not too large.

However, with increasing tilt both steady and oscillatory convection become

less e¬cient at transporting heat, and the Rayleigh number dependence of

the Nusselt number becomes weaker (Fig. 23.2a). The increase in tilt angle

leads to a larger Lorentz force, which in turn leads to a suppression of the

heat transport. The resulting dependence on the tilt angle is much stronger

in the oscillatory regime since ohmic di¬usion now has only a ¬nite time to

reduce the Lorentz force due to ¬eld distortion before the ¬‚ow reverses. In

contrast in the steady case the Lorentz force exerts a much weaker e¬ect and

the reduction of the Nusselt number is largely due to a geometrical e¬ect:

the strong oblique magnetic ¬eld inclines the convection cells relative to the

vertical allowing them more time to lose their upward buoyancy to adjacent

descending ¬‚uid.

Figure 23.3a,b shows the corresponding results for the oscillatory mode

when ‘ = 65—¦ . The ¬gure reveals a remarkable behaviour: the Nusselt

number K initially increases rapidly with Ra as in the vertical magnetic

¬eld case, but then undergoes a hysteretic transition to a new state char-

acterized by a small Nusselt number, and one that decreases slowly with

increasing Ra . As this state is followed to larger Rayleigh numbers we see

that the mean temperature becomes almost piecewise linear (Fig. 23.3c),

with a limited isothermal core. The extent of this core quickly saturates, in

contrast to the case of a vertical ¬eld for which the isothermal core grows

continuously with Ra as the temperature gradients are compressed into

ever thinner thermal boundary layers (as in Fig. 23.2b). Evidently, in this

state increasing the heat input does not result in increased heat transport

Nonlinear magnetoconvection in the presence of a strong oblique ¬eld 353

Fig. 23.3. (a) (top left) The (time-averaged) Nusselt number K for oscillatory per-

pendicular rolls as a function of the scaled Rayleigh number Ra for ‘ = 65—¦ and

ζ = 0.1, σ = 1.1. (b) (bottom left) The corresponding frequency ω. Note the

hysteretic transition from the “vertical” convection mode to the “horizontal” con-

vection mode with increasing Ra . (c) (top right) Mean temperature pro¬les T (Z)

and (d) (bottom right) the convection amplitude as measured by |W (Z)| for in-

creasing values of Ra along the horizontal branch. Note the development of broad

boundary layers of approximate thickness 1/2K and a small isothermal core. These

properties are characteristic of the “horizontal” convection mode.

across the layer. Instead the added energy is all stored in the magnetic

¬eld perturbations (since the ¬eld strength is large this is achieved with

small deformation of the ¬eld); moreover, the perturbation magnetic ¬eld

suppresses the convective motion in the boundary layers near the top and

bottom (see Fig. 23.3d) thereby reducing the transport of heat across the

layer. In this regime (i.e., on the branch where the Nusselt number remains

low as Ra is increased) the system of perpendicular rolls therefore behaves

Julien, Knobloch & Tobias

354

much more like one with an imposed horizontal ¬eld (cf. Brownjohn et al.

1995). Similar results are obtained for parallel rolls (kx ≡ 0), although the

hysteretic transition is delayed to higher Ra .

Note that the Rayleigh number must exceed a critical value before the

“horizontal” convection mode sets in. This is because the ¬‚ow in the in-

terior must be strong enough to expel the magnetic ¬eld perturbation into

the boundary layers at the top and bottom; this expulsion occurs primar-

ily in the vertical direction because the vertical velocity in the interior is

much larger than the horizontal velocity. In steady convection the resulting

boundary layer thickness is determined by ohmic di¬usion and is there-

fore narrow (because ζ 1). In contrast in an oscillatory ¬‚ow the ¬‚ow

reversals prevent the formation of such narrow boundary layers and the

boundary layer thickness is determined by the perturbation Lorentz force

and not ohmic di¬usion. A number of conclusions follow immediately from

these considerations. First, the transition between the two regimes occurs

at lower Rayleigh numbers when ‘ is larger. Indeed, for small values of

‘ the transition to the lower “horizontal branch” does not occur (for this

value of ζ). This is so also if ζ is larger even though a “ horizontal branch”

may still be present. Moreover, since the ability of the magnetic ¬eld to

suppress oscillatory convection increases with decreasing ζ the value of the

Rayleigh number at which the transition from the “vertical ¬eld” regime to

the “horizontal ¬eld” regime takes place is an increasing function of ζ. This

argument also explains why the two convection regimes are only found in

oscillatory convection.

It is tempting to speculate about the possible role of the fully nonlinear

single-mode solutions discovered here for the structure of a sunspot. In the

sunspot umbra the magnetic ¬eld is nearly vertical, while in the penumbra

it is tilted. Since the critical Rayleigh number steadily decreases at small

Ma (large Q) as the tilt angle increases the supercriticality, or e¬ective

Rayleigh number, increases with radial distance from the spot centre. Since

the temperature di¬erence across the layer, here supposed to be constant,

determines the Rayleigh number convection will be less supercritical in the

spot centre than farther out (this is consistent with the observed contrast in