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Magneto-rotational instability in the solar core and Ap star envelopes Rainer Arlt Astrophysikalisches Institut Potsdam and Günther Rüdiger, Rainer Hollerbach Solar rotational evolution Wind model by Stępień (1988) The solar tachocline and the core Thompson et al. 2003 from various sources Stellar radiative envelopes Star of spectral type A and B Small convective core Extensive radiative zone 10% of these stars have magnetic fields These 10% are slow rotators Differential-rotation decay Rotation of solar core is slow and uniform Rotation period has increased by factor of 10 during life Viscosity too small to reduce rotation homogeneously throughout the Sun Magnetic Ap stars rotate much slower than „normal“ A stars Did MRI reduce the internal rotation of Sun and Ap stars? Magneto-rotational instability Angular velocity decreasing with axis distance Magnetic field or arbitrary geometry Instability of with growth rate of the order Lower limit for MRI k of most unstable mode depends on B Diffusive decay rate increases with k MRI suppressed below certain B in Gauss Upper limit for MRI Wavelength of most unstable MRI mode exceeds object size in kG Numerical simulations Spherical spectral code (Hollerbach 2000) u 0, and B 0 Initial conditions Vertical cut through radiative zone Left: Magnetic field Right: Angular velocity Differential-rotation decay – close-up Rm = 104 Pm = 1 Ra = 0 Differential-rotation decay – close-up Resolution at high Reynolds number t = 4 rotations t = 8 rotations t = 1 rotation Resolution at high Reynolds number t = 4 rotations t = 8 rotations t = 1 rotation Differential-rotation decay Steepness of rotation profile versus time Initially 0 1 (2 s ) 2q Rayleigh-stable Differential-rotation decay time versus Reynolds number Effect of negative buoyancy Rm = 2·104 Pm = 1 Ra = -108 Effect of negative buoyancy Differential-rotation decay Extrapolation to stellar parameters Decay time of 10-100 million years Short compared with the age of the Sun (5 billion years) MRI may have provided the enormous angularmomentum transport for slow-down Considerable fraction of Ap star ages (life-time < 109 yr) MRI may still be operating in them. The End