Angular momentum transport and mixing in rotating stars

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Angular momentum transport and mixing in rotating
stars

• Jean-Paul Zahn
• Observatoire de Paris

Second Corot-Brazil Workshop Ubatuba, 2-6
November 2005
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Why bother about rotation in stars?

• Rotation is the main cause of mixing in stellar
  radiation zones
• It plays a major role in the generation and decay
  of magnetic field
• Rotation intervenes in the mass loss

hence its impact on stellar and galactic evolution
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In convection zones
- Very efficient mixing, due to turbulence

• Angular momentum transport
• due to the turbulent stresses ? differential
  rotation

Red Giant
Sun
Brun Toomre 2002
Brun et al. 2005
Massive parallel simulations - no simple
prescription (yet)
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Mixing processes in radiation zones
Main cause (differential) rotation

• - Rotational mixing of type I

Matter and angular momentum are transported by
the same processes meridional circulation and
turbulence
- Rotational mixing of type II
Mixing is caused by circulation and
turbulence, but another process (magnetic field,
waves) intervenes in the transport of angular
momentum
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Mixing processes in radiation zones rotational
mixing of type I

• Meridional circulation

Classical picture circulation is due to thermal
imbalance caused by perturbing force
(centrifugal, etc.) Eddington
(1925), Vogt (1925), Sweet (1950), etc
Eddington-Sweet time
Revised picture after a transient phase of about
tES, circulation is driven by the loss (or
gain) of angular momentum Busse (1981), JPZ
(1992), Maeder Z (1998)
No AM loss no need to transport AM ? weak
circulation
AM loss by wind need to transport AM to surface
? strong circulation
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Turbulence caused by differential rotation
By vertical shear W(r) (baroclinic instability)
- if maximum of vorticity linear instability
- if no maximum of vorticity finite amplitude
instability
reduced by radiative diffusion
- stabilizing effect of stratification
Richardson criterion
turbulence if
from which one deduces the turbulent diffusivity
(if ? cst)
Townsend 1959 Dudis 1974 JPZ 1974 Lignières et
al. 1999
K thermal diffusion n viscosity N
buoyancy frequency
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Turbulence caused by differential rotation
By horizontal shear W(?) (barotropic instability)

• Assumptions
• instability acts to suppress its cause, i.e.
  W(?)

• turbulent transport is anisotropic (due to
  stratification) Dh ?? Dv

Maeder 2003 Mathis, Palacios Z 2004
Main weakness no firm prescription for Dh
? 2 important properties - erodes
stabilising effect of stratification
Talon Z 1997
- changes advection of chemicals into vertical
diffusion
Chaboyer Z 1992
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Rotational mixing of type I - the observational
test

• The same processes (circulation and turbulence)
  are responsible for the mixing
  of chemical elements
• and for the transport of angular momentum
• Zahn (1992), Maeder Zahn (1998)

• Quite successful with early-type stars
• Talon et al. 1997 Maeder Meynet 2000
  Talon Charbonnel 1999

? For late-type stars, predicts - fast
rotating core ? helioseismology
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Rotation profiles in the Sun
predicted by standard rotational mixing
observed through acoustic sounding
tachocline
Talon (1997), Matias Zahn (1998)
GONG
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Rotational mixing of type I - the observational
test

• The same processes (circulation and turbulence)
  are responsible for the
  mixing of chemical elements
• and for the transport of angular momentum
• Zahn (1992), Maeder Zahn (1998)

• Quite successful with early-type stars
• Talon et al. 1997 Maeder Meynet 2000
  Talon Charbonnel 1999

For late-type stars, predicts - fast
rotating core ? helioseismology

• strong destruction of Be in Sun
• (may be explained by tachocline mixing)

• - mixing correlated with loss of angular
  momentum ? Li in tidally locked binaries
• ? little dispersion in the Spite plateau

? Another, more powerful process is responsible
for the transport of angular momentum
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Rotational mixing of type II

• Circulation and turbulence
  are responsible
  for the mixing of chemical elements
• Another process operates for the transport of
  angular momentum
• has indirect impact on mixing, by shaping the
  rotation profile

Magnetic field ?
Internal gravity waves ?
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Role of a fosssil magnetic field
Does it prevent the spread of tachocline? Does it
enforce uniform rotation?
convection zone
tachocline void of magnetic field
magnetopause
Gough McIntyre 1998
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Role of a fossil magnetic field
Does it prevent the spread of tachocline? Does it
enforce uniform rotation?
Stationary solutions intermediate field case
(13000 G)
At high latitude poloidal field threads through
CZ enforces diff. rotation (Ferraros law)
Garaud 2002
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Role of a fossil magnetic field
Does it prevent the spread of tachocline? No
Does it enforce uniform rotation? No
Time-dependent solutions result strongly depends
on initial field
Brun Zahn 2005
Initial field connects with CZ
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Role of a fossil magnetic field
Does it prevent the spread of tachocline? No
Does it enforce uniform rotation? No
Initial field does not connect with CZ
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Role of a fossil magnetic field
Time-dependent solutions result strongly
depends on initial field
? No field
Initially (too) deeply buried poloidal field
?
Brun Zahn 2005
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Role of a fossil magnetic field
Time-dependent solutions result strongly
depends on initial field
? No field
Initially (too) deeply buried poloidal field
?
Brun Zahn 2005
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Role of a fossil magnetic field
Does it prevent the spread of tachocline? No
Does it enforce uniform rotation? No
Initial field is deeply buried in RZ
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Role of a fossil magnetic field
Probably not important in solar-type stars
But in A-type stars?
Initial random field relaxes in a mixed
poloidal/toroidal configuration
which then diffuses toward the surface
Polytrope n3 2 Msol Braithwaite Nordlund 2005
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Properties of internal gravity waves

• Propagate in stratified media
• restoring force ? buoyancy
• Excited by turbulence (e.g. in or close to
  convective zones)
• Conserve momentum (or angular momentum)
• if they are not damped
• ? transport AM to place where they are
  dissipated

buoyancy (Brunt-Väisälä) frequency oscillation
frequency of a displaced element in a stratified
region
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Excitation of internal waves
Analytical treatment Goldreich, Murray Kumar
1994 used by Talon Charbonnel 2003
2D simulations of penetrative convection Kiraga
et al. 2003
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Momentum transport by waves

• In stars, IGW are damped by thermal diffusion

flux at the base of the CZ
frequency in frame rotating with CZ
thermal diffusion
local frequency is Doppler shifted if
there is differential rotation
Waves transfer momentum from the region where
they are excited to the region where they
are dissipated
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Momentum transport by waves
- if prograde (mgt0) and retrograde (mlt0) waves
are equally excited and if there is no
differential rotation ? no net momentum
deposition

• if there is differential rotation, m and -m
  waves deposit their momentum
• at different locations
• ? waves increase the local differential rotation

high l waves are damped very close to the CZ
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Below the convection zone high-degree
waves
Talon Charbonnel 2005
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Below the convection zone high-degree
waves
Shear Layer Oscillation (SLO)
Talon Charbonnel 2005
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Momentum transport by waves

• if prograde (mgt0) and retrograde (mlt0) waves are
  equally excited and there is no differential
  rotation
• ? no net momentum deposition
• if there is differential rotation, m and -m
  waves deposit their momentum at different
  locations
• ? waves increase the local differential rotation

- high l waves are damped very close to the CZ -
low l, low frequency waves are damped in deep
interior
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Interior low-degree, low-frequency waves
Angular momentum extracted by solar wind
Talon Charbonnel 2005
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Interior low-degree, low-frequency waves
Angular momentum extracted by solar wind Effect
of SLO filtered out
Talon Charbonnel 2005
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Effect of IGW on 1.2 Msol star
with all other hydrodynamical transport
mechanisms included
Rotation profile
Li profile at 0.7 Gyr
Vi 50 km/s
Hyades ?
with IGW
type I
type I
0.2
0.5
0.7
Talon Charbonnel 2005
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Effect of IGW in the Sun
Rotational mixing type I (without IGW)
Rot. mixing type II (with IGW)
Rotation profile initial velocity 50 km/s
All transport mechanisms included, except
magnetic field
0.2, 0.5, 0.7, 1.0, 1.5, 3.0, 4.6 Gyr
Charbonnel Talon 2005
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standard model
Rotational mixing in radiation zones
rotational mixing type I
rotational mixing type II
microscopic diffusion
distribution of chemical elements
penetration, overshoot
meridional circulation
turbulent transport
magnetic field
convection
(in tachocline)
rotation
internal gravity waves
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Weakest points of present models
- Convective penetration into radiation zones
- Parametrisation of shear turbulence due to
differential rotation
- Power spectrum for IGW emitted at base of
convection zone
- Particle transport by IGW ?
- Role of magnetic field ?
CoRoT will put most valuable constraints
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