D = 2+ 2 (9.3)

‚ξ ξ ‚θ sinθ ‚θ

The energy equation becomes, to ¬rst order in the thermodynamical per-

turbations

2

σNh ρh

ur = ∇2 T˜

Th 2 (9.4)

ξ

„¦c Eµ

where σ is the Prandtl number, Nh is the background buoyancy frequency.

Finally, the equation of state can be combined with the radial and latitu-

dinal components of the momentum equation to provide an expression for

Garaud

134

ρ:

˜

˜ ˜

1 ‚ρ˜ ρh 2 2 cosθ 2 ‚ ¦ 1 ‚T

uφ ’ ’

= rc „¦c . (9.5)

ρh ‚θ ph sinθ ‚θ Th ‚θ

Two standard approximations are often performed. The ¬rst one is the

Boussinesq approximation, commonly used in studies of the tachocline,

which is only justi¬ed when the thickness of the layer studied is much smaller

than the background density scale-height. The second approximation con-

sists in neglecting the e¬ects of the mean centrifugal force on the system by

supposing that its main contribution is a very small (negligible) oblateness

of the hydrostatic background.

At the time of the Mons conference I presented numerical and analytical

solutions of this system of equations and boundary conditions under both

approximations. It has since appeared that both approximations were highly

unjusti¬ed in this problem (as the bulk of the radiative zone spans many

scale-heights, and as the mean centrifugal force creates a global baroclinicity

of the system that must be taken into account) and lead to erroneous results.

I now present instead the solution to the complete problem, solving the

equations presented in (9.1). These equations are solved numerically, and

˜

the results suggest a scaling of the unknowns T , ur and uθ which depends

essentially on the parameter

» = σNh /„¦2 .

2

(9.6)

c

9.2.1 Slow rotating case (» 1)

˜

In the case of slow rotation, I ¬nd by studying the numerical results that T

and the poloidal components of the velocity ur,θ scale the following way:

˜

T = Th T ,

ur,θ = Eµ /(»ρh )ur,θ , (9.7)

where the quantities with bars are the scaled quantities, of order of unity. It

˜

is also found that ¦ is always of order of unity, which is expected. Note that

the scaling for the meridional motions is a local Eddington-Sweet scaling (see

Spiegel & Zahn, 1992). Applying this ansatz to the system of equations given

in (9.1), an expansion in powers of 1/» reveals that the angular-momentum

balance is dominated to zeroth order by viscous transport; thus

D2 (ξ sinθuφ ) = 0 , (9.8)

Dynamics of the solar tachocline 135

Fig. 9.1. Numerical solution of the system (9.1) for a solar-type star rotating 100

times slower than the sun (» 104 ). The quadrants show the radiative zone only

˜

and the imposed shear at the top of the radiative zone is solar-like (i.e. „¦cz =

(1 ’ 0.15 cos2 θ ’ 0.15 cos4 θ). The left panel shows the angular velocity, which

is viscously dominated. The interior rotation rate is 0.957 times that imposed at

the surface at the equator. The right panel shows the streamlines (dotted lines

represent a clockwise ¬‚ow, and solid lines represent an anti-clockwise ¬‚ow). The

contours are logarithmically spaced. The structure is reminiscent of Holton layers.

which determines the angular velocity pro¬le uniquely. Using this result in

the ¬rst order equations provides a relation between the temperature and

gravitational potential perturbations:

˜

ph ‚ ρh 2 ‚T d ln Th ‚ ¦

’

u = sinθ ,

ph φ

ρh ‚z ‚θ dξ ‚θ

˜

‚ 2˜ 4πGρh rc d ln ph cosθ 2 ‚ ¦ ‚T

∇ξ ¦ = ’ u’ + , (9.9)

sinθ φ ‚θ

‚θ gh dξ ‚θ

˜

which can be solved independently for T and ¦. Finally, the temperature

¬‚uctuations lead to meridional motions through

∇2 T .

ur (9.10)

ξ

Figure 9.1 shows the results of the numerical solutions for the angular ve-

locity pro¬le and the meridional motions corresponding to a slowly rotating

solar-type star (for which » 104 ).

Garaud

136

9.2.2 Fast rotating case (» 1)

In the case of fast rotation it is found that the correct scaling is

˜

T = » Th T ,

ur,θ = Eµ /ρh ur,θ . (9.11)

This time, I perform an asymptotic expansion in the small parameter ». In

this limit the temperature ¬‚uctuations are strongly damped by the rapid

heat di¬usion (as » 1 is equivalent to the small Prandtl number limit)

and the system reaches an equilibrium which is determined by the zeroth

order equations:

˜

ph ‚ ρh 2 d ln Th ‚ ¦

= ’ sinθ

u ,

ph φ

ρh ‚z dξ ‚θ

˜

ρ2 cosθ 2 ‚ ¦

‚2

∇ξ ¦ = 4πG h uφ ’ . (9.12)

‚θ ph sinθ ‚θ

˜

These equations can in principle be solved for u2 and ¦ and provide, to the

φ

next order in », the meridional ¬‚ow through the advection di¬usion balance:

u · ∇ξ (ξ sinθuφ ) = D2 (ξ sinθuφ ) , (9.13)

and, ¬nally, the temperature ¬‚uctuations through

u = ∇2 T . (9.14)

ξ

10’2 ) are

The results of the numerical simulations for small lambda (»

shown in Fig. 9.2.

9.2.3 Solar rotation rate

In the solar case, the parameter » varies between 0.1 and 1 in the region

between the two boundaries. Although the solution is closer to the fast

rotating case, the asymptotic analysis does not apply and the dynamics of

the system result from a complex interaction of the momentum balance, the

thermal energy advection-di¬usion balance and the Poisson equation.