The solar tachocline:

theoretical issues

Jean-Paul ZahnObservatoire de Paris

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

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

Governing equations (thin layer approximation)

hydrostatic equilibrium

geostrophic balance

transport of heat

conservation of angular momentum

meridional motions - anelastic approximation

variables separate:

radiative spreading

Radiative spreading

(Elliott 1997)

at solar age

boundary conditions (top of radiation zone)

initial conditions

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)

t1/4 t1/2

Brun & Zahn

Prandtl /K ~10-4

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

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)

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)

ReReii=0

ReReoo=70,000

laminar

turbulent

Example: nonlinear shear instability

But what causes there the turbulence?

To prevent spread of tachocline:

a process that tends to smooth out differential rotation in latitude

Anisotropic turbulent transport

Magnetic torquing

Can tachocline spread be prevented by fossil field ?

(Gough & McIntyre 1998)

advection of angular momentumis balanced by Lorentz torquein boundary layer of thickness

outward diffusion of fieldis 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 1997and MacGregor & Charbonneau 1999;included in Sule, Arlt & Rüdiger 2004 )

Magnetic confinement ?

stationary solution

B = 13,000 G

= = 4.375 1011 cm2/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)

Magnetic confinement ?

Answer strongly depends on initial conditions

Example with initial field threading into convection zone

(Brun & Z)

/

Back to the turbulent tachocline

In most tachocline models convection and convective overshoot have been ignored

Is this justified?

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

A new picture of the tachocline emerges

convection adiabatic

tachocline subadiabatic

the tachocline is located in the overshoot region

overshoot

quiet radiation zone

there, main cause of turbulence: convective overshoot

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

Effect of an oscillatory poloidal field

(fast tachocline)

2D simulations

() and Bpol(, t) imposed at top turbulent diffusivities

(Forgács-Dajka & Petrovay 2001, 2002)

a field of sufficient strength confines () to the overshoot region

Bpol= 2600 G for = = 1010 cm2/s

substantial time and latitude

dependence of tachocline thickness

penetration depth of periodic field:(2/cyc)1/2 = 0.01 r0 for = 109 cm2/s

Subsequent work adds migrating field,meridional circulation and (r) profile(Forgács-Dajka 2004)

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

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