the solar tachocline: theoretical issues jean-paul zahn observatoire de paris
TRANSCRIPT
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