Angular momentum transport and mixing in rotating stars

Angular momentum transport and mixing in rotating stars

Jean-Paul Jean-Paul Zahn

Observatoire de ParisObservatoire de Paris

Second Corot-Brazil Workshop Ubatuba, 2-6 November 2005Second Corot-Brazil Workshop Ubatuba, 2-6 November 2005

Why bother about rotation in stars?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

In convection zones

- Very efficient mixing, due to turbulence

- Angular momentum transport due to the turbulent stresses differential rotation

Red GiantSun

Massive parallel simulations - no simple prescription (yet)

Brun & Toomre 2002 Brun et al. 2005

Mixing processes in radiation zones

- 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

Main cause : (differential) rotation

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

Turbulence caused by differential rotation

By vertical shear r(baroclinic instability)

turbulence if

from which one deduces the turbulent diffusivity (if = cst)

K thermal diffusion; viscosity; N buoyancy frequency

- if maximum of vorticity: linear instability

- if no maximum of vorticity: finite amplitude instability

Townsend 1959Dudis 1974; JPZ 1974Lignières et al. 1999

reduced by radiative diffusion- stabilizing effect of stratification

Richardson criterion

Turbulence caused by differential rotation

By horizontal shear ( (barotropic instability)

Chaboyer & Z 1992

Assumptions:

- instability acts to suppress its cause, i.e. (

2 important properties: - erodes stabilising effect of stratification

- turbulent transport is anisotropic (due to stratification): Dh Dv

Main weakness: no firm prescription for Dh Maeder 2003 Mathis, Palacios & Z 2004

Talon & Z 1997

- changes advection of chemicals into vertical diffusion

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)

For late-type stars, predicts

- fast rotating core helioseismology

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

Rotation profiles in the Sun

GONG

predicted by standard rotational mixing

Talon (1997), Matias & Zahn (1998)

tachocline

observed throughacoustic sounding

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 momentumZahn (1992), Maeder & Zahn (1998)

· 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

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

Another, more powerful process is responsible for the transport of angular momentum

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 ?

Role of a fosssil magnetic field

Does it prevent the spread of tachocline?Does it enforce uniform rotation?

convection zone

tachoclinevoid of magnetic field

magnetopause

Gough & McIntyre 1998

Role of a fossil magnetic field

Does it prevent the spread of tachocline?Does it enforce uniform rotation?

Garaud 2002

Stationary solutions

intermediate field case (13000 G)

At high latitudepoloidal field threads through CZenforces diff. rotation(Ferraro’s law)


Does it prevent the spread of tachocline? No

Brun & Zahn 2005

Time-dependent solutions: result strongly depends on initial field

Initial field connects with CZ

Does it enforce uniform rotation? No



Initial field does not connect with CZ



Brun & Zahn 2005

QuickTime™ et undécompresseur Cinepak

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No field

Initially (too) deeply buriedpoloidal field

Time-dependentsolutions:result strongly depends on initial field

QuickTime™ et undécompresseur Cinepak

sont requis pour visionner cette image.



Initial field is deeply buried in RZ



Polytrope n=32 Msol

Braithwaite & Nordlund 2005

Probably not important in solar-type stars

But in A-type stars?

Initial random field relaxes in

a mixed poloidal/toroidalconfiguration

which then diffusestoward the surface

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

Excitation of internal waves

2D simulations ofpenetrative convectionKiraga et al. 2003

Analytical treatmentGoldreich, Murray & Kumar 1994used by Talon & Charbonnel 2003

Momentum transport by waves

• In stars, IGW are damped by thermal diffusion

local frequency is Doppler shifted if there is differential rotation

thermal diffusion

flux at the base of the CZ

frequency in frame rotating with CZ

Waves transfer momentum from the region where they are excited to the region where they are dissipated


- if prograde (m>0) and retrograde (m<0) waves are equally excited and if there is no differential rotation

no net momentum deposition

· high l waves are damped very close to the CZ

- if there is differential rotation, +m and -m waves deposit their momentum at different locations

waves increase the local differential rotation

Below the convection zone high-degree waves

Talon & Charbonnel 2005

Below the convection zone high-degree waves


Shear Layer Oscillation(SLO)


• 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

Interiorlow-degree, low-frequency waves


Angular momentumextracted by solar wind

Interiorlow-degree, low-frequency waves


Angular momentumextracted by solar wind

Effect of SLOfiltered out

Effect of IGW on 1.2 Msol star

Rotation profile Li profile at 0.7 Gyr


type I

type Iwith IGW

0.7

0.5

0.2

Vi = 50 km/s Hyades

with all other hydrodynamical transport mechanisms included

Effect of IGW in the Sun

Charbonnel & Talon 2005

Rotation profileinitial velocity 50 km/s

Rotational mixing type I(without IGW)

Rot. mixing type II(with IGW)

All transport mechanisms included,except magnetic field

0.2, 0.5, 0.7, 1.0, 1.5, 3.0, 4.6 Gyr

microscopic diffusion

distribution of chemical elements

meridional circulation turbulent transport

magnetic field

rotation

convection

internal gravity waves

penetration, overshoot

standard model

rotational mixing type I

rotational mixing type II

Rotational mixing in radiation zones

(in tachocline)

Weakest points of present models

- 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 ?

- Convective penetration into radiation zones

CoRoT will put most valuable constraints

Angular momentum transport and mixing in rotating stars

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