1. Magneto-rotational instability in the solar core and Ap star envelopes Rainer Arlt Astrophysikalisches Institut Potsdam and Günther Rüdiger, Rainer Hollerbach

2. Solar rotational evolution Wind model by Stępień (1988) Wind model by Stępień (1988)

3. The solar tachocline and the core Thompson et al. 2003 from various sources Thompson et al. 2003 from various sources

4. Stellar radiative envelopes Star of spectral type A and B Star of spectral type A and B Small convective core Small convective core Extensive radiative zone Extensive radiative zone 10% of these stars have magnetic fields 10% of these stars have magnetic fields These 10% are slow rotators These 10% are slow rotators

5. Differential-rotation decay Rotation of solar core is slow and uniform Rotation of solar core is slow and uniform Rotation period has increased by factor of 10 during life Rotation period has increased by factor of 10 during life Viscosity too small to reduce rotation homogeneously throughout the Sun Viscosity too small to reduce rotation homogeneously throughout the Sun Magnetic Ap stars rotate much slower than „normal“ A stars Magnetic Ap stars rotate much slower than „normal“ A stars Did MRI reduce the internal rotation of Sun and Ap stars? Did MRI reduce the internal rotation of Sun and Ap stars?

6. Magneto-rotational instability Angular velocity decreasing with axis distance Angular velocity decreasing with axis distance Magnetic field or arbitrary geometry Magnetic field or arbitrary geometry  Instability with growth rate of the order of 

7. Lower limit for MRI k of most un- stable mode depends on B Diffusive decay rate increases with k  MRI sup- pressed below certain B in Gauss

8. Upper limit for MRI Wavelength of most un- stable MRI mode exceeds object size in kG

9. Numerical simulations Spherical spectral code (Hollerbach 2000) Spherical spectral code (Hollerbach 2000)

10. Initial conditions Vertical cut through radiative zone Vertical cut through radiative zone Left: Magnetic field Left: Magnetic field Right: Angular velocity Right: Angular velocity

13. Differential-rotation decay – close-up Rm = 10 4 Pm = 1 Ra = 0

14. Differential-rotation decay – close-up

15. Resolution at high Reynolds number t = 1 rotation t = 4 rotations t = 8 rotations

16. Resolution at high Reynolds number t = 1 rotation t = 8 rotations t = 4 rotations

17. Differential-rotation decay Steepness of rotation profile versus time Steepness of rotation profile versus time Initially Initially Rayleigh-stable Rayleigh-stable

18. Differential-rotation decay time versus Reynolds number

19. Effect of negative buoyancy Rm = 2·10 4 Pm = 1 Ra = -10 8

20. Effect of negative buoyancy

21. Differential-rotation decay Extrapolation to stellar parameters Extrapolation to stellar parameters Decay time of 10-100 million years Decay time of 10-100 million years Short compared with the age of the Sun (5 billion years)  MRI may have provided the enormous angular- momentum transport for slow-down Short compared with the age of the Sun (5 billion years)  MRI may have provided the enormous angular- momentum transport for slow-down Considerable fraction of Ap star ages (life-time < 10 9 yr)  MRI may still be operating in them. Considerable fraction of Ap star ages (life-time < 10 9 yr)  MRI may still be operating in them.

22. The End