I have studied the nonlinear dynamics of the radiative zone of a rotating

solar-type star when a latitudinal shear is imposed by an overlying convec-

tion zone. This study is valid provided that the star is far from break-up (i.e.

that the centrifugal force is small compared to the gravitational potential).

Dynamics of the solar tachocline 137

Fig. 9.2. Numerical solution of the system (9.1) for a solar-type star rotating 10

10’2 ). The left panel shows the angular velocity,

times faster than the sun (»

which increases with depth through angular-momentum conservation. Note how

the latitudinal variation of the angular velocity is small compared to its radial

variation. The interior rotation rate is 5.26 times that imposed at the surface at

the equator. The right panel shows the temperature ¬‚uctuations. Note that even

when the stellar oblateness is of order of 10’3 , the temperature ¬‚uctuations remain

of order of 10’6 through e¬cient heat di¬usion.

I found that few approximations can be safely used in this study: the

nonlinear advection terms and the e¬ects of the centrifugal force must be

carefully included in momentum equation. However, in the limit where the

star is far from the break-up point, the perturbations to the hydrostatic

background are found to be small indeed, which justi¬es the linearization of

the equation of state.

Two asymptotic limits were found, which depend on the value of the pa-

rameter » = σNh /„¦2 . In the case of a slowly rotating star (with »

2

1) the

c

hydrostatic background acquires a small ellipticity and the angular velocity

pro¬le is viscously dominated. The meridional ¬‚ow velocities are of order of

the local Eddington-Sweet velocity (e.g. Spiegel & Zahn, 1992) and take the

shape of alterning dipolar cells reminiscent of the Holton layer structure. In

the case of a fast rotating star (yet far from breakup), the temperature ¬‚uc-

tuations are determined by an advection-di¬usion balance which limits their

amplitude to roughly » Th ; this value is independent of the rotation rate.

The angular velocity pro¬le and the ¬‚uctuations in the gravitational poten-

tial can be determined independently through the momentum equation and

the Poisson equation. It is found that the angular velocity increases sharply

with depth, as expected from equation (9.2), and varies little with latitude.

Similarly, the perturbation to the gravitational potential vary little with lat-

Garaud

138

itude, which suggests the possibility of approximating this limit analytically

as a one-dimensional problem. This is under current investigation.

Finally, the e¬ects of the boundary conditions on the problem (and in

particular the presence of a lower rigid boundary) remain to be carefully

analysed.

To summarize the ¬rst part of this paper, I have shown that it is possible

to study semi-analytically (in some cases) and numerically the problem of

shear propagation into the solar radiative zone in a self-consistent way, when

taking into account isotropic viscosity only†. The main result is the follow-

ing: as Spiegel & Zahn predicted, in this isotropic case the shear imposed by

the convection zone penetrates all the way into the solar core. The failure to

reproduce observations therefore suggests that other dynamical phenomena

must be present in the solar radiative zone.

9.3 The other half of the problem: nonlinear interaction between

a large-scale ¬eld and ¬‚ows in a rotating sphere

Having studied the di¬culty of hydrodynamical models to explain the struc-

ture of the solar tachocline, Gough & McIntyre (1998) suggested an alterna-

tive theory, namely that the observations could be reproduced through the

existence of a large-scale fossil ¬eld in the solar radiative zone. As McGregor

& Charbonneau (1999) showed, such a ¬eld can indeed impose a uniform ro-

tation throughout most of the radiative zone and con¬ne the shear to a thin

tachocline provided none of the ¬eld lines are anchored into the convection

zone‡: the ¬eld must be entirely con¬ned to the radiative zone. Studies in

the non-magnetic case following the lines described in the ¬rst part of this

paper seem to suggest that shear-driven baroclinic imbalance leads to down-

welling ¬‚ows near the poles and the equator, with a localized upwelling in

mid-latitudes (in regions of little shear). This phenomenon is illustrated in

Fig. 9.3. Gough & McIntyre combined these two results and suggested that

baroclinically driven ¬‚ows could indeed lead to the con¬nement of the ¬eld

through nonlinear advection, and proposed a new model of the tachocline

based on this idea. However, only a fully nonlinear numerical study can

verify whether this dynamical balance could indeed lead to the observed

rotation pro¬le.

As a ¬rst step towards a complete numerical simulation of the tachocline

according to the Gough & McIntyre model, I have looked at the nonlinear

† Incidentally, it is clear that this type of analysis is not limited to the solar case, but can be

applied to other stars with a wide range of rotation rates, masses, and ages. It will be interesting

to compare the corresponding results to the asteroseismic observations of COROT.

‡ The shear would otherwise propagate along ¬eld lines according to Ferraro™s isorotation law.

Dynamics of the solar tachocline 139

Fig. 9.3. Numerical solution of the system (9.1) for a solar-type star rotating with

the observed solar angular velocity. The rigid bottom boundary was arti¬cially

placed at r = 0.97rc to mimic the presence of a con¬ned large-scale magnetic ¬eld.

The left panel shows the angular velocity, when the convection zone shear is imposed

at the top. The right panel shows the streamlines, with dotted lines representing

clockwise ¬‚ow and solid lines representing anti-clockwise ¬‚ow. Note the two-cell

structure with upwelling in mid-latitudes; note also the presence of an equatorial

boundary layer.

interaction of a dipolar magnetic ¬eld and shear-driven motions only, when

all thermal/compressibility e¬ects are neglected. In these simulations, the

¬‚uid is incompressible with constant density ρ. This allows me to determine,

through a simpli¬ed model, whether the idea of ¬eld con¬nement through

meridional motions of the type described by Gough & McIntyre is indeed

possible. In order to do this, I have created a numerical model in which

meridional ¬‚ows are created by the shear, not through baroclinic driving but

through Ekman pumping on the boundary. The interest of this approach

is that the geometry of the ¬‚ow in this simpli¬ed problem is qualitatively

similar to that shown in Fig. 9.3: it possesses a downwelling near the poles

and the equator, and upwells in mid-latitude.

The numerical procedure is the following. I solve the following system of

equations

Eµ 2

2(ˆ z — u)φ = Λ(j — B)φ +

e (∇ u)φ ,

ρ

[∇—(u — B)]φ + E· (∇2 B)φ = 0 ,

∇·u = 0 ,

∇·B = 0 ,

where B is the magnetic ¬eld, E· is the magnetic Ekman number and Λ

is the global Elsasser number de¬ned as Λ = B0 /ρrc „¦2 . This system is

2 2

c

solved in a spherical shell, where, as in the ¬rst part of this paper, the

outer boundary corresponds to the bottom edge of the convection zone and

the inner core is removed to avoid singularities. The outer boundary is

Garaud

140

now assumed to be impermeable in order to create arti¬cially an Ekman

layer at the interface with the convection zone which will drive the required

meridional ¬‚ows. A point dipole is placed at r = 0 such that the radial ¬eld

at the pole at r = rin is B0 . The regions outside the region of simulation

are supposed to be conductive so that the magnetic ¬eld satis¬es ∇2 B = 0

in the steady state case. As in the ¬rst set of simulations, the inner core is

rotating solidly with angular velocity „¦in where „¦in is determined through

the steady-state requirement that the angular-momentum ¬‚ux through the

inner boundary is null.

The results are now discussed for ¬xed di¬usive parameters, when only

the Elsasser number is varied. For low Elsasser number (Λ 1), the system

is dominated by the Coriolis forces and the magnetic ¬eld is mostly passive.

The shear imposed by the convection zone propagates deep into the radiative

zone along the rotation axis, thereby satisfying Proudman™s rotation law.

Two meridional circulation cells are created by Ekman pumping on the outer

boundary, with downwelling at the poles and the equator and upwelling in

mid-latitudes, as required; they burrow deep into the radiative zone. In the