Magneto-rotational Magneto-rotational instability in the instability in the solar core and Ap solar core and Ap

star envelopesstar envelopes

Rainer ArltRainer ArltAstrophysikalisches Institut PotsdamAstrophysikalisches Institut Potsdam

and Günther Rüdiger, Rainer and Günther Rüdiger, Rainer HollerbachHollerbach

Solar rotational evolutionSolar rotational evolution

Wind model by StWind model by Stępień (1988)ępień (1988)

The solar tachocline and the The solar tachocline and the corecore Thompson et al. 2003 from various Thompson et al. 2003 from various

sourcessources

Stellar radiative envelopesStellar radiative envelopes

Star of spectral type A and Star of spectral type A and BB

Small convective coreSmall convective core Extensive radiative zoneExtensive radiative zone 10% of these stars10% of these stars

have magnetic fieldshave magnetic fields These 10% are slowThese 10% are slow

rotatorsrotators

Differential-rotation decayDifferential-rotation decay Rotation of solar core is Rotation of solar core is

slow and uniformslow and uniform Rotation period has increased by Rotation period has increased by

factor of 10 during lifefactor of 10 during life Viscosity too small to reduce rotation Viscosity too small to reduce rotation

homogeneously throughout the Sunhomogeneously throughout the Sun Magnetic Ap stars rotate much Magnetic Ap stars rotate much

slower than „normal“ A starsslower than „normal“ A stars Did Did MRIMRI reduce the internal rotation reduce the internal rotation

of Sun and Ap stars?of Sun and Ap stars?

Magneto-rotational Magneto-rotational instabilityinstability

Angular velocity decreasing withAngular velocity decreasing withaxis distanceaxis distance

Magnetic field or arbitrary Magnetic field or arbitrary geometrygeometry

Instability with growth rate of the Instability with growth rate of the order of order of

Lower limit for MRILower limit for MRIk of most un-stable mode depends on B

Diffusive decay rate increases with k

MRI sup-pressed below certain B

in Gauss

Upper limit for MRIUpper limit for MRI

Wavelength of most un-stable MRI mode exceeds object size

in kG

Numerical simulationsNumerical simulations Spherical spectral code (Hollerbach 2000)Spherical spectral code (Hollerbach 2000)

0 and ,0 Bu

Initial conditionsInitial conditions

Vertical cut Vertical cut through radiativethrough radiativezonezone

Left:Left:Magnetic fieldMagnetic field

Right:Right:Angular velocityAngular velocity

Diff

ere

nti

al-

rota

tion

Diff

ere

nti

al-

rota

tion

deca

y

deca

y –

clo

se-u

p–

clo

se-u

p

Rm = 104

Pm = 1Ra = 0

Diff

ere

nti

al-

rota

tion

Diff

ere

nti

al-

rota

tion

deca

y

deca

y –

clo

se-u

p–

clo

se-u

p

Resolution at high Reynolds Resolution at high Reynolds numbernumber

t = 1 rotation

t = 4 rotations

t = 8 rotations

Resolution at high Reynolds Resolution at high Reynolds numbernumber

t = 1 rotationt = 8 rotations

t = 4 rotations

Differential-rotation decayDifferential-rotation decay

Steepness of rotation profile versus Steepness of rotation profile versus timetime

InitiallyInitially

Rayleigh-stableRayleigh-stable

qs 2

0

)2(1

Differential-rotation decay Differential-rotation decay timetimeversus Reynolds numberversus Reynolds number

Effect of negative Effect of negative buoyancybuoyancy

Rm = 2·104

Pm = 1Ra = -108

Effect of negative Effect of negative buoyancybuoyancy

Differential-rotation decayDifferential-rotation decay Extrapolation to stellar parametersExtrapolation to stellar parameters Decay time of 10-100 million yearsDecay time of 10-100 million years Short compared with the age of the Short compared with the age of the

Sun (5 billion years) Sun (5 billion years) MRI may have MRI may have provided the enormous angular-provided the enormous angular-momentum transport for slow-downmomentum transport for slow-down

Considerable fraction of Ap star ages Considerable fraction of Ap star ages (life-time < 10(life-time < 1099 yr) yr) MRI may still MRI may still bebeoperating in them.operating in them.

The EndThe End