Threedimensional stability of the solar tachocline

1
Three-dimensional stability of the solar
tachocline

• Rainer ArltAniket SuleGünther Rüdiger

2
Solar interior

• Radiative core belowconvection zone
• 0...0.7 solar radii
• No differential rotation
• Except transition layerto convection zone
• tachocline

3
Differential rotation

• Combination of radial and latitudinalshear
• Hydrodynamicallystable?Re R 2 ? ? ?

4
Differential rotation
Watson (1981) 28
5
Magnetic-field belts

• Tachocline often considered place for strong
  magnetic fields
• Ask for stability of toroidal magnetic fields
• Tayler (1973) found instability for
  non-axisymmetric modes
• What are the maximum possible fields?

6
Magnetic-field belts

• Linearized incompressible equationsU
  (0, 0, Rm r sin? ? ) Pm ???B (0, 0, S B?) S
  Lundquist number

7
Magnetic-field belts
B?
B?

• Different radialthickness
• Different magn.Prandtl numbersPm ?/?
• Different rotationprofiles

Vertical cross-sections
8
Equator symmetry
9
Magnetic Prandtl number
10
And in the Sun?

• Density 0.25 g/cm3length scale 0.1
  R?diffusivity 1000 cm2/s
• At Rm 1012 ? B 10 Gausssmall!
• Sunspot emergence at low latitudes and tilt angle
  of bipolar groups requires105 Gauss, if in
  tachocline

11
With differential rotation
12
Thickness of belts
13
Maximum field for turbulent values

• Large-scale field near surface stable?
• Rm ? 1500
• Stability analysis covers that value
• Bmax ? 1000 G

14
Field belts at different latitudes
19
60
15
Summary

• Toroidal field of gt 10-100 G unstable
• Hydrodynamic instability not affected by weak
  magnetic fields
• Magnetic instability changes weakly with
  differential rotation
• Tachocline bad place for field storage