1. Angular momentum transport and mixing in rotating stars
Jean-Paul Zahn
Observatoire de Paris
Second Corot-Brazil Workshop Ubatuba, 2-6 November 2005

2. 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

3. In convection zones Angular momentum transport
- Very efficient mixing, due to turbulence
Angular momentum transport
due to the turbulent stresses  differential rotation
Sun
Red Giant
Brun & Toomre 2002
Brun et al. 2005
Massive parallel simulations - no simple prescription (yet)

4. 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

5. 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

6. 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
- stabilizing effect of stratification
reduced by radiative diffusion
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

7. 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

8. 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

9. Rotation profiles in the Sun
observed through
acoustic sounding
predicted by standard rotational mixing
tachocline
Talon (1997),
Matias & Zahn (1998)
GONG

10. 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

11. 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 ?

12. 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

13. 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
(Ferraro’s law)
Garaud 2002

14. 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
Initial field connects with CZ
Brun & Zahn 2005

15. 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

16. 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

17. 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

18. 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

19. 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 n=3
2 Msol
Braithwaite & Nordlund 2005

20. 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

21. Excitation of internal waves
Analytical treatment
Goldreich, Murray & Kumar 1994
used by Talon & Charbonnel 2003
2D simulations of
penetrative convection
Kiraga et al. 2003

22. 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

23. Momentum transport by waves
- if prograde (m>0) and retrograde (m<0) 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

24. Below the convection zone high-degree waves
Talon & Charbonnel 2005

25. Below the convection zone high-degree waves
Shear Layer Oscillation
(SLO)
Talon & Charbonnel 2005

26. Momentum transport by waves
if prograde (m>0) and retrograde (m<0) 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

27. low-degree, low-frequency waves
Interior
low-degree, low-frequency waves
Angular momentum
extracted by solar wind
Talon & Charbonnel 2005

28. low-degree, low-frequency waves
Interior
low-degree, low-frequency waves
Angular momentum
extracted by solar wind
Effect of SLO
filtered out
Talon & Charbonnel 2005

29. 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

30. Effect of IGW in the Sun Rotation profile initial velocity 50 km/s
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

31. Rotational mixing in radiation zones
standard model
rotational mixing type I
microscopic diffusion
rotational mixing type II
distribution of chemical elements
penetration, overshoot
meridional circulation
turbulent transport
magnetic field
convection
(in tachocline)
rotation
internal gravity waves

32. 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