the solar tachocline: theoretical issues jean-paul zahn observatoire de paris
TRANSCRIPT
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The solar tachocline:
theoretical issues
Jean-Paul ZahnObservatoire de Paris
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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
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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
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Governing equations (thin layer approximation)
hydrostatic equilibrium
geostrophic balance
transport of heat
conservation of angular momentum
meridional motions - anelastic approximation
variables separate:
radiative spreading
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Radiative spreading
(Elliott 1997)
at solar age
boundary conditions (top of radiation zone)
initial conditions
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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
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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
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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)
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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?
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To prevent spread of tachocline:
a process that tends to smooth out differential rotation in latitude
Anisotropic turbulent transport
Magnetic torquing
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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 )
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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)
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Magnetic confinement ?
Answer strongly depends on initial conditions
Example with initial field threading into convection zone
(Brun & Z)
/
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Back to the turbulent tachocline
In most tachocline models convection and convective overshoot have been ignored
Is this justified?
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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
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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
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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
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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)
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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
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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