McIntyre, M. E. & Palmer, T. N., 1985, Pure Appl. Geophys., 123, 964.

Mohebalhojeh, A. R. & Dritschel, D. G., 2001, J. Atmos. Sci., 58, 2411.

Norton, W. A., 1994, J. Atmos. Sci., 51, 654.

Plumb, R. A. & McEwan, A. D., 1978, J. Atmos. Sci., 35, 1827.

Polvani, L. M. & Plumb, R. A., 1992, J. Atmos. Sci., 49, 462.

Rhines, P. B., 1975, J. Fluid Mech., 69, 417.

Riese, M., et al., 2002, J. Geophys. Res., 107, no. 8179, p. CRI 7“1.

Schou, J., et al., 1998, ApJ, 505, 390.

Spiegel, E. A. & Zahn, J.-P., 1992, A&A, 265, 106. [SZ]

Stewart, R. W. & Thomson, R. E., 1977, Proc. Roy. Soc. Lond., A354, 1.

Thompson, M. J., et al., 1996, Science, 272, 1300.

Waugh, D. W. & Plumb, R. A., 1994, J. Atmos. Sci., 51, 530.

Waugh, D. W., et al., 1994, J. Geophys. Res., 99, 1071.

9

Dynamics of the solar tachocline

PASCALE GARAUD

Department of Applied Mathematics and Theoretical Physics

University of Cambridge, Cambridge CB3 9EW, UK

Douglas Gough & Michael McIntyre proposed, in 1998, the ¬rst global and

self-consistent model of the solar tachocline. Their model is however far

more complex than analytical methods can deal with. In order to validate

their work and show how well it can indeed represent the tachocline dynam-

ics, I report on progress in the construction of a fully nonlinear numerical

model of the tachocline based on their idea. Two separate and complementary

approaches of this study are presented: the study of shear propagation into a

rotating strati¬ed radiative zone, and the study of the nonlinear interaction

between shear and large-scale magnetic ¬elds in an incompressible, rotating

sphere. The combination of these two approaches provides good insight into

the dynamics of the tachocline.

9.1 Introduction

The tachocline was discovered in 1989 by Brown et al.; it is a thin shear

layer located at the interface of the uniformly rotating radiative zone and

di¬erentially rotating convective zone of the sun. Several issues about these

observations remain unclear. Why is the radiative zone rotating uniformly

despite the latitudinal shear imposed by the convection zone, and why is the

tachocline so thin? How can the tachocline operate the dynamical transi-

tion between the magnetically spun-down convection zone and the interior?

The ¬rst model of the tachocline was presented by Spiegel & Zahn (1992).

They studied the propagation of the convection zone shear into the radia-

tive zone under various hypotheses; in particular, they showed that in the

case where angular momentum in the tachocline was transported only by

isotropic viscosity the convection zone shear would propagate deep into the

radiative zone within a local Eddington-Sweet timescale (rather than a vis-

cous timescale) contrary to what is suggested by observations (Schou et al.,

131

Garaud

132

1998). Very roughly, the mechanism for shear propagation into a strati¬ed

region is the following: the existence of shear leads to a slight imbalance

in the hydrostatic equilibrium and thereby drives meridional ¬‚ows; these

can burrow into the radiative zone, transporting and redistributing angular

momentum deeper and deeper. Spiegel & Zahn then studied ways of con¬n-

ing the shear to a thin tachocline through angular-momentum transport by

anisotropic Reynolds stresses; however, in a ¬rst part of this paper I would

like to look a little more in detail at the isotropic case, as it can both be

used in further investigations of the Gough & McIntyre model, as well as in

more general studies of stellar rotation and rotational mixing.

9.2 One half of the problem: shear propagation into a rotating

strati¬ed ¬‚uid

In this ¬rst part, I will consider solar-type stars only and assume that their

radiative zone is a stable, isotropic ¬‚uid with uniform viscosity µv , and that

it has little in¬‚uence on the dynamics of overlying convection zone. As a

result, I will simply assume that the convection zone is imposing a given

shear to the underlying stably strati¬ed region. Also, I will assume that the

dynamical timescale of this system is short compared to the stellar evolution

timescale and the stellar spin-down timescale, so that I can limit my study

to the steady-state case. This assumption will be dropped in future works

on this subject. The equations describing this steady system are

1

ρh u · ∇u = ’∇˜ ’ ρh ∇¦ ’ ρ∇¦h + µv ∇2 u + µv ∇(∇·u) ,

˜˜

p

3

ρh Th u · ∇sh = ∇·(k∇T ) ,

˜

˜

p

˜ ρ

˜ T

= + ,

ph ρh Th

∇2 ¦ = 4πG˜ ,

˜ ρ

∇·(ρh u) = 0 , (9.1)

where ρ, p and T are respectively the total density, pressure and tempera-

ture, u = (ur , uθ , uφ ≡ r sinθ „¦) is the velocity ¬eld with respect to spherical

˜

polar coordinates (r, θ, φ), ¦ is the gravitational potential, and k is the ther-

mal conductivity. These equations are the ¬rst-order perturbation around

the non-rotating hydrostatic background equilibrium (denoted by su¬x h);

this is a good approximation, as we will see, provided the centrifugal force

is much smaller than the gravitational force. The background quantities

ρh , ph , ¦h and Th are extracted from the standard solar model calculated by

Dynamics of the solar tachocline 133

Christensen-Dalsgaard et al. (1991). The perturbed quantities are denoted

˜˜ ˜ ˜

by tildes, p, ¦, ρ and T . The full nonlinearity of the momentum advection

process is kept.

The boundary conditions used on the system are the following: the con-

vection zone shear (as it is observed in the sun) is imposed at the top bound-

ary and continuity of the stresses across the radiative-convective interface

imposes another two conditions (on the continuity of the radial derivatives

of the azimuthal and latitudinal velocities). A small impermeable core is

removed from the region of computation near the centre to avoid singular-

ities. This core is assumed to be rotating solidly, with a rotation rate „¦in

determined through the steady-state condition that the total ¬‚ux of angular

momentum through the boundary is null. The regions outside the domain of

simulation are assumed to be highly conductive so that they satisfy ∇2 T = 0,

˜

which provides the thermal boundary conditions to apply to the system.

Using the assumption of axisymmetry, I reduce the momentum equation

in (9.1) to:

Eµ 2

u · ∇ξ (ξ sinθ uφ ) = D (ξ sinθ uφ ) , (9.2)

ρh

˜

1‚ sinθ ‚ρh ‚ ¦ 1 ‚ ρ ˜ Eµ 2

’ =’ ’

ρh u2 + D (ξ sinθ ωφ ) ,

φ

ρh ‚z ρh ‚ξ ‚θ ‚θ ρh

where ξ = r/rc is the new radial coordinate normalized by the radius rc

of the star, z is the normalized cylindrical coordinate that runs along the

2

rotation axis, θ is the co-latitude, Eµ = µv /rc „¦c is the Ekman number,

= rc „¦2 (l¦h /lξ)’1 is the ratio of the centrifugal to gravitational forces, and

2

c

ω = ∇ — u is the vorticity. In this expression the following normalizations

˜

have been applied: [r] = rc , [u] = rc „¦c , [¦] = rc „¦2 , [T ] = 1 K, [ρ] =

2

c

’3

1 g cm , where rc is the radius of the radiative zone and „¦c is the typical

rotation rate of the star. The operator D2 is de¬ned as

‚2 sinθ ‚ 1‚