1. The solar tachocline: theoretical issues Jean-Paul Zahn Observatoire de Paris

2. Internal rotation of Sun tachocline Importance for stellar physics  If motions in this layer (circulation,turbulence)  transport of chemical elements (He; Li, Be, B)  Role in solar dynamo: generation/storage of toroidal field

3. Why is the tachocline so thin?  it should spread through radiative diffusion (EAS & JPZ 1992) Assumed settings (early 90's): convection + penetration establish a quasi-adiabatic stratification (2D sim. Hurlburt et al. 1986, 1994) convection + penetration  adiabatic tachocline subadiabatic the tachocline (or part of it) is located below, in the stably stratified radiation zone

4. Governing equations (thin layer approximation) hydrostatic equilibrium geostrophic balance transport of heat conservation of angular momentum meridional motions - anelastic approximation variables separate: radiative spreading

5. Radiative spreading (Elliott 1997) at solar age  boundary conditions (top of radiation zone) initial conditions

6. Radiative spreading - effect of (isotropic) viscosity conservation of angular momentum in numerical simulations, radiative spread can be masked by viscous spread (in Sun Prandtl = /K ~10 -6 )   t 1/4   t 1/2 Brun & Zahn Prandtl /K ~10 -4

7. Why is the tachocline so thin?  spread can be prevented by anisotropic momentum diffusion due to anisotropic turbulence (Spiegel & Zahn 1992) (Elliott 1997) Stationary solution  tachocline thickness conservation of angular momentum ventilation time

8. Cause of turbulence? non-linear shear instability (Speigel & Zahn 1992) linear shear instability (due to max in vorticity) (Charbonneau et al. 1999, Garaud 2001) linear MHD instability (with toroidal field) (Gilman & Fox 1997; Dikpati & Gilman 1999; Gilman & Dikpati 2000, 2002)  a local instability due to the  (  ) profile ? linear shear instability 3D (shallow-water) (Dikpati & Gilman 2001) same, followed up in non-linear regime (Cally 2003; Cally et al. 2003; Dikpati et al. 2004)

9. Consistency check: does such turbulence prevent radiative spreading i.e. does it act to reduce differential rotation ? Geophysical evidence: in stratified turbulent media, angular momentum is transported mainly by internal gravity waves  turbulence acts to increase shear: not a diffusive process (Gough & McIntyre 1998; McIntyre 2002) Laboratory evidence: Couette-Taylor experiment, in regime where AM increases outwards  shear turbulence decreases shear: it is a diffusive process (Wendt 1933; Taylor 1936; Richard 2001) Re i Re i =0 Re o Re o =70,000 laminar turbulent Example: nonlinear shear instability But what causes there the turbulence?

10. To prevent spread of tachocline: a process that tends to smooth out differential rotation in latitude  Anisotropic turbulent transport  Magnetic torquing

11. Can tachocline spread be prevented by fossil field ? (Gough & McIntyre 1998 ) advection of angular momentum is balanced by Lorentz torque in boundary layer of thickness  outward diffusion of field is prevented by circulation at lower edge of tachocline; yields thickness  of tachocline Can tachocline circulation prevent field from diffusing into CZ? If not, field would imprint differential rotation in RZ (Ferraro’s law) Gough & McIntyre’s model (slow tachocline) NB. circulation plays crucial role (neglected by Rüdiger & Kitchanitov 1997 and MacGregor & Charbonneau 1999; included in Sule, Arlt & Rüdiger 2004 )

12. Magnetic confinement ? stationary solution  B = 13,000 G  =  = 4.375 10 11 cm 2 /s 2D axisymmetric (Garaud 2002) differential rotation imposed at top dipole field rooted in deep interior non-penetrative boundaries  signs of tachocline confinement, but high diffusivities required by numerics substantial diff. rotation in radiation zone circulation driven by Ekman-Hartmann pumping stratification and thermal diffusion added in subsequent work (cf. P. Garaud’s talk)

13. Magnetic confinement ? Answer strongly depends on initial conditions Example with initial field threading into convection zone (Brun & Z)   /  

14. Back to the turbulent tachocline In most tachocline models convection and convective overshoot have been ignored Is this justified?

15. Evidence for deep convective overshoot 3D simulations of penetrative convection (Brummell, Clune & Toomre 2002)  tachocline is located in the overshoot region even at high Péclet number, overshooting plumes are unable to establish a quasi-adiabatic stratification (see also Rempel 2004) plumes overshoot a fraction of pressure scale-height overshoot

16. A new picture of the tachocline emerges convection  adiabatic tachoclinesubadiabatic  the tachocline is located in the overshoot region overshoot quiet radiation zone  there, main cause of turbulence: convective overshoot

17. Modelisation of the turbulent tachocline 3D simulations  (r,  ) induced by body force randomly-forced turbulence (of comparable energy) (Miesch 2002) turbulence reduces horizontal shear  (  ) increases vertical shear  (r)  acts to stop spread of tachocline

18. Effect of an oscillatory poloidal field (fast tachocline) 2D simulations  (  ) and B pol ( , t) imposed at top turbulent diffusivities   (Forgács-Dajka & Petrovay 2001, 2002)  a field of sufficient strength confines  (  ) to the overshoot region B pol = 2600 G for  =  = 10 10 cm 2 /s  substantial time and latitude dependence of tachocline thickness penetration depth of periodic field: (2  /  cyc ) 1/2 = 0.01 r 0 for  = 10 9 cm 2 /s Subsequent work adds migrating field, meridional circulation and  (r) profile (Forgács-Dajka 2004)

19. The new picture of the tachocline the tachocline is the overshoot region the tachocline is turbulent turbulence is due to convective overshoot AM transport is achieved through turbulence (Miesch) AM transport occurs through magnetic stresses (Forgács-Dajka & Petrovay) or/and Fast or slow tachocline? Observations will decide !  no need anymore to look for another instability

20. What we need to understand and to improve why does convection act differently on AM in bulk of CZ and in overshoot region ? apply  (  ) on top, rather than enforce it in situ Miesch's model: Forgács-Dajka & Petrovay model: further refine, confront with observations all others: improve representation of turbulent transport Gough & McIntyre model: validation through realistic simulations Spiegel & Zahn model: establish whether such anisotropic turbulence does occur, and acts to reduce  (  ) Gilman, Dikpati & Cally MHD model: consistency check : is  (  ) is reduced in turbulent regime