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The solar tachocline theoretical
issues Jean-Paul Zahn Observatoire de Paris
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Internal rotation of Sun
Importance for stellar physics
? If motions in this layer (circulation,turbulen
ce) ? transport of chemical elements
(He Li, Be, B)
? Role in solar dynamo generation/storage
of toroidal field
tachocline
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Why is the tachocline so thin?
Assumed settings (early 90's) convection
penetration establish a quasi-adiabatic
stratification (2D sim. Hurlburt et al. 1986,
1994)
convection penetration
? adiabatic
subadiabatic
tachocline
the tachocline (or part of it) is located below,
in the stably stratified radiation zone
? it should spread through radiative
diffusion (EAS JPZ 1992)
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Governing equations (thin layer approximation)
hydrostatic equilibrium
geostrophic balance
meridional motions - anelastic approximation
transport of heat
conservation of angular momentum
variables separate
radiative spreading
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Radiative spreading
boundary conditions (top of radiation zone)
initial conditions
at solar age ?
(Elliott 1997)
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Radiative spreading - effect of (isotropic)
viscosity
conservation of angular momentum
? ? t1/4
? ? t1/2
in numerical simulations, radiative spread can
be masked by viscous spread (in Sun Prandtl
?/K 10-6)
Prandtl ?/K 10-4
Brun Zahn
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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)
conservation of angular momentum
Stationary solution ? tachocline thickness
ventilation time
(Elliott 1997)
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Cause of turbulence?
? a local instability due to the ?(?) profile ?
non-linear shear instability (Speigel Zahn
1992)
linear shear instability (due to max in
vorticity) (Charbonneau et al. 1999,
Garaud 2001)
linear shear instability 3D (shallow-water)
(Dikpati Gilman 2001)
linear MHD instability (with toroidal field)
(Gilman Fox 1997 Dikpati Gilman 1999
Gilman Dikpati 2000, 2002)
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 ?
Example nonlinear shear instability
Laboratory evidence Couette-Taylor
experiment, in regime where AM increases
outwards
laminar
turbulent
? shear turbulence decreases shear it is a
diffusive process (Wendt 1933 Taylor 1936
Richard 2001)
Rei0 Reo70,000
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)
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 ?
Can tachocline circulation prevent field from
diffusing into CZ? If not, field would imprint
differential rotation in RZ (Ferraros law)
(Gough McIntyre 1998)
Gough McIntyres model (slow tachocline)
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
NB. circulation plays crucial role (neglected by
Rüdiger Kitchanitov 1997 and MacGregor
Charbonneau 1999 included in Sule, Arlt
Rüdiger 2004 )
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Magnetic confinement ?
differential rotation imposed at top dipole field
rooted in deep interior non-penetrative boundaries
2D axisymmetric (Garaud 2002)
stationary solution ? B 13,000 G n h
4.375 1011 cm2/s
? 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. Garauds talk)
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Magnetic confinement ?
Answer strongly depends on initial conditions
Example with initial field threading into
convection zone
?/K 10-2 ?/h 10-2
(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)
plumes overshoot a fraction of pressure
scale-height

• even at high Péclet number, overshooting plumes
  are unable to establish a quasi-adiabatic
  stratification
• (see also Rempel 2004)

overshoot
? tachocline is located in the overshoot region
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A new picture of the tachocline emerges
convection
? adiabatic
overshoot
tachocline
subadiabatic
quiet radiation zone
? the tachocline is located in the overshoot
region
? 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 n,
h (Forgács-Dajka Petrovay 2001, 2002)
penetration depth of periodic field (2?/?cyc)1/2
0.01 r0 for h 109 cm2/s
? a field of sufficient strength
confines ?(?) to the overshoot region Bpol
2600 G for h n
1010 cm2/s
? substantial time and latitude
dependence of tachocline thickness
Subsequent work adds migrating field, meridional
circulation and h(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
? no need anymore to look for another
instability
AM transport is achieved through
turbulence (Miesch)
or/and
AM transport occurs through magnetic
stresses (Forgács-Dajka Petrovay)
Fast or slow tachocline? Observations will decide
!
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What we need to understand and to improve
Gough McIntyre model
validation through realistic simulations
all others
improve representation of turbulent transport
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
Miesch's model
why does convection act differently on AM
in bulk of CZ and in overshoot region ?
apply ?(?) on top, rather than enforce it in
situ
Forgács-Dajka Petrovay model
further refine, confront with observations