motivate both the geometrical framework and the subsequent mathematical

assumptions that are used to derive the reduced equation set (23.5).

The schematic picture for magnetically constrained convective motions

Julien, Knobloch & Tobias

348

that arises from theory (Chandrasekhar 1961; Julien et al. 2000) and numer-

ical simulations (Weiss et al. 1990, 1996) is one consisting of small aspect-

ratio convection cells where A := L/H 1 (Fig. 23.1). These cells align

themselves with the imposed magnetic ¬eld r as a consequence of the mag-

netic analogue of the Taylor-Proudman constraint (Taylor 1923, Julien et al.

2000). The tilted nature of the small aspect-ratio cells now implies the ex-

istence of two scales of motion in the vertical direction:

• a small scale z , controlled solely by the imposed ¬eld and to ¬rst-order

˜

una¬ected by the presence of boundaries in the vertical. This scale is of

the same order as the horizontal length scale L, and corresponds to the

vertical cross-section of the cells (Fig. 23.1);

• a large scale Z of the order of the layer depth H on which convective

motions are driven through buoyancy forcing (Fig. 23.1).

Mathematically, this picture suggests a dependence on four spatial variables

x ≡ (x, y, z , Z) and a multiple scales expansion in the vertical with the

˜

corresponding derivatives in equations (23.1)“(23.3) interpreted as ‚z = ‚z +

˜

A‚Z . However, we ¬nd that because of the magnetic analogue of the Taylor-

Proudman constraint, the solution cannot depend, at leading order, on the

small scale along r. Therefore a more concise mathematical description is

obtained by rotating the coordinate system for the small scales so that it is

aligned with r, while maintaining the alignment of Z with gravity (Fig. 23.1).

Hence x = R‘ (x, y, z )T , ∇ = R’1 ∇, u = R‘ u, and b = R‘ b; R‘ is the

T

˜ ‘

unitary rotation matrix in the (x, z) plane.

Note that the spatial dependence is now represented by the three vari-

ables (x , y ) and Z (because no ¬elds may exhibit z dependence); hence

(x , y , Z) comprises a non-orthogonal coordinate system. Note also that the

characteristic scales are su¬ciently small compared to the domain size to

justify the use of periodic boundaries in (x , y ).

We now summarize the derivation leading to the reduced equations (23.5)

from the Boussinesq equations (23.1)“(23.3). An equivalent derivation based

on an expansion in the Chandrasekhar number Q (for the vertical ¬eld case

alone) is given in Julien et al. (1999). Using Ma 1 as a small parameter,

we write R = Ma’1 R , and take Re, Pe and Rm ≈ O(1). The aspect ratio is

set, without loss of generality, as A ≡ Ma, with the ¬‚uid variables u ≈ O(1),

b ≈ O(Ma), π ≈ O(Ma’2 ). Finally

T = T (Z) + Ma θ(x , y , Z, t) , (23.4)

where the overbar denotes the spatial average over small scales and time:

Nonlinear magnetoconvection in the presence of a strong oblique ¬eld 349

„

’1

lim „ dx dy dt. The reduced equations are deduced from (23.1)“

„ ’∞ 0

(23.3) using asymptotic expansions in Ma for the rescaled ¬‚uid variables,

e.g.,

u = u 0 + Ma u 1 + Ma2 u 2 + · · · .

At leading order in the momentum, heat and induction equations one

readily deduces

• a mean hydrostatic balance ‚Z Π0 = R T0 , where Π0 (Z) ≡ π 0 and T0 (Z) ≡

T 0 are slowly varying mean quantities.

• an invariance of the motion with respect to the small scales along r ≡ z

leading to non-divergent motions in planes perpendicular to r. We are

therefore free to adopt the streamfunction representation u 0⊥ = z —∇ Ψ0

and b 0⊥ = z — ∇ A0 .

• the absence of any O(1) mean ¬‚ows or mean ¬elds. These quantities are

generated at higher orders with u ≈ O(Ma) and b ≈ O(Ma2 ).

The next order in Ma produces the reduced PDEs in the non-orthogonal

coordinate system, valid in the strong ¬eld limit:

D0t ’ Re’1 ∇⊥ ∇⊥ Ψ0 = R sin ‘ ‚y θ0 + rz ‚Z ∇⊥ A0 + J⊥ (A0 , ∇⊥ A0 )

2 2 2 2

D0t ’ Re’1 ∇⊥ w0 = R cos ‘ θ0 + rz ‚Z b03 + J⊥ (A0 , b03 )

2

D0t ’ Rm’1 ∇⊥ A0 = rz ‚Z Ψ0

2

D0t ’ Rm’1 ∇⊥ b03 = rz ‚Z w0 + J⊥ (A0 , w0 )

2

(23.5)

D0t ’ Pe’1 ∇⊥ θ0 = ’ cos ‘ w0 + sin ‘ ‚y Ψ0 ‚Z T 0

2

‚Z (cos ‘ w0 + sin ‘ ‚y Ψ0 )θ0 = Pe’1 ‚ZZ T 0 ,

where D0t = ‚t + J⊥ [Ψ0 , •] with Jacobian J⊥ (f, g) := ‚x f ‚y g ’ ‚y f ‚x g.

Equations (23.5a-f) represent coupled equations for the magnetically aligned

vorticity ζ = ∇⊥ Ψ0 , velocity w0 , current j = ∇⊥ A0 , and magnetic ¬eld b03 .

2 2

All these ¬elds are sustained through buoyancy forcing whose distribution is

governed by (23.5e). For ‘ = 0, buoyancy forcing cannot sustain Ψ0 and A0

which necessarily decay without an additional external source. For this case

alone the only surviving nonlinearity occurs in (23.5f) which describes the

distortion of the mean temperature T 0 via vertical convective ¬‚ux w0 θ0 =

(cos ‘w0 + sin ‘‚y Ψ0 )θ0 . The reduced equations are dynamically coupled

on the large scale Z by vertical stretching due to the presence of r.

Julien, Knobloch & Tobias

350

23.2.1 Computational and Theoretical Advantages

The reduced equations (23.5) have several appealing features that are not

present in the full Boussinesq equations (23.1)“(23.3). These include

• a relaxation of spatial resolution requirements. This occurs as a conse-

quence of a reduction of vertical order of the equations with respect to

the large-scale variable Z. It follows that the precise details of the vertical

boundary conditions are not distinguished, and that any (mechanical or

magnetic) boundary layers are passive and need not be resolved. This

fact is already known from linear theory (Chandrasekhar 1961) and was

established in the weakly nonlinear regime by Clune & Knobloch (1993).

Julien et al. (2000) show that even in the strongly nonlinear regime the

exact form of the mechanical boundary layer can be deduced from the

interior (bulk) solution. Note that thermal boundary layers, if present,

are retained through (23.5f).

• a relaxation of the timestepping/CFL criterion. It can be shown (in an

analysis similar to Embid & Majda 1998) that the reduced equation set

¬lters out the small scale fast Alfv´n waves. These waves, which must

e

be resolved in the Boussinesq equations, do not interact with the slow

convective dynamics.

• existence of exact analytic solutions due to the simpli¬ed nonlinearities.

23.3 Exact Single-Mode Solutions

For single-mode solutions of the form F (Z) exp(iωt + ik⊥ · x ⊥ ) plus its

complex conjugate, all nonlinearities in (23.5a-e) vanish identically with the

exception of the convective ¬‚ux term in (23.5f). The resulting equations can

be reformulated into a single nonlinear complex eigenvalue problem for the

vertical structure, namely

1

iω + Re’1 k⊥ (iω + Rm’1 k⊥ )W

D2 W ’ 2 2

(23.6)

2

rz

ˆ

R KPe (iω + Rm’1 k⊥ )(’iω + Pe’1 k⊥ ) cos2 ‘kx + ky

2 2

2 2

+ W =0.

ω 2 + Pe’2 k⊥ + 2k⊥ |W |2

4 2 2

ˆ2

rz k⊥