Magnetoconvection

1. Magnetoconvection 磯部洋明 2003/11/10 太陽雑誌会

2. Today I introduce: • “On the interaction between convetcion and magnetic field” Cattaneo, Emone, & Weiss 2003, ApJ, 588, 1183 - Systematic study of 3D Boussinesq convection • “Magnetic flux sepation in photospheric convection” Weiss, Proctor, & Brownjohn 2002, MNRAS, 337, 293 - Systematic study of 3D compressible convection

3. On the interaction between convetcion and magnetic field (Cattaneo et al. 2003) • Large scale 3D numerical experiment of non-linear Bossinesq magnetoconvection • Systematic survey of interaction of convection with imposed vertical magnetic field • As the field strength decrease, the solution falls in different regime: convection inhibited, oscillatory convection, vigorous magnetoconvection, and dynamo.

4. Boussinesq approximation • Density variation is considered only in the buoyancy term. • Up-down symmetry

5. Examples

6. Model • Cartesian box 10x10x1, no rotation • Initial condition: polytropic static solution • Boundary conditions • Periodic in horizontal direction • Rayleigh # R = 5x105 (Rc〜657.51 when B=0) • Kinetic Prandtl # σ=ν/κ = 1 • Magnetic Prandl # σm =ν/η = 5 => ζ= σ/ σm = 0.2 (magnetic Schmidt #) • Chandrasekhar # Q = Bσm /σ2

7. Linear analysis • Case of B=0 • convection sets in as a direct instability at R=Rc • Case of B≠0 and ζ≥1 • convection sets in as a direct instability for all value of B • Case of B≠0 and ζ<1 (as is the case here) • direct instability if B(or Q) is below critical value, and oscillatory instability if it is above it.

8. Oscillatory and stationary bifurcations No unstable mode Instability extends to a broad band of horizontal wave number => little indication of preffered horiozontal scale Oscillation Stationary convection * Rayleight # is fixed horizonta k

9. Critial wave number for the onset of convection Smaller horizontal scale in larger B case Large B

10. From dynamo action to magnetoconvection seed field = weak and small scale magnetic field with no net flux

11. Surface features: dynamo solution • meso-scale celler pattern • magnetic field (both positive and negative) concentrates on the corners of meso-scale pattern.

12. Surface features: vertical field imposed

13. Typical horizontal scales at the top boundary Temperature Diamonds: mesocale cells Squares: smaller cells Line: linear theory B

14. Temporal evolution of kinetic and magnetic energy

15. Probability Distibution Functions velocity B solid: dynamo dashed: case 4 (intermidiate), dotted: case6 (oscillatory)

16. Up-down symmetry Light; |B| at the top Dark: |B| at the bottom

17. 3D structure of strong magnetic field (dynamo solution)

18. Magnetic flux separation in photospheric convectionWeiss, Proctor, & Bronwjohn 2002, MNRAS, 337, 293 • systematic study of compressible magnetoconvection • Wide aspect ratio (8x8x1), strongly stratified (density ratio = 11) • A regime that does not apper in Boussinesq convection: flux separation

19. Flux separation B • First recognized by Tao et al (1998) • Magnetic field is expelled from broad and vigorous convecting plume. • Strong fields are associated with small-scale weak convection T

20. Model • Almost same as that of Cattaneo et al 2002 except for compressibility (density ratio = 11) • Slightly different top boundary (dT/dz=(θT)4: radiative boundary) • Magnetic field strengty is changed in the range of Q=200-3000. No dynamo regime. • Plasma beta > 50 for all cases. Magnetic pressure can be only locally important

21. Q=3000 (aspect ratio=4): small scale convection ←B at the top ←T flucturation at the side ←B at the bottom ←dT/dz at the top ←T flucturation at the side ←dT/dz at the bottom dT/dz∝T4 is a proxy for the intensity of surface radiation

22. Q=1400: flux separation

23. Q=2000: 2 solutions begin with small large perturbation begin with small scale perturbation

24. Q=1600: again, 2 solutions Flux separation is a kind of hysteresis.

25. Measure of hysteresis

26. Q=1000 and 500

27. Q=200 • Magnetic field concentrate at the cell boundaries. • No isolated flux tube is found • Run with smaller Q difficult because of numerical problem.

28. Discussion • Flux separation is a robust feature seen in broad range of parameters • Flux separation is hardly seen in Boussinesq calculations. Up-down asymmetry in compressible convection causes the higher velocity at the top, so the field can be swept into the upper network

29. Application to solar photosphere • Umbral dots • Light bridges

32. Magnetic sheet (no shear) in vigorously conevecting velosity field • 計算領域: 0<x<40, 0<y<40, -6<z<3.5 (unit: scale height at z=0) • 鉛直方向の境界条件: T=一定, Vz=0, stress free, perfect conducting • 水平方向は周期境界 • 光球付近(-0.5<z<0.5)に（放射）冷却 • 初期のRayleigh数 = 10000, Prandl数 = 0.1, 磁気Prandl数 =0.1 • 準定常状態に発達した対流中に磁気シートを挿入

33. （B ≈ 5×Beq） t=20 t=50 t=80 t=100 Isovolume of |B|, Vz at z=0 (gray), Temperature (side)

34. Vz(gray )and |B| (contour) at the phorosphere t=0 t=50 t=100 t=150 • Interchange mode is dominant. Convective plume is modified along the magneti field • No signature of undular mode in the nonlinear phase • Corresponding to the dark lane in the ephemeral regions?

35. B ≈ 0.2Beq • Magnetic field cannot change the structure of convection. • Strong B field (B>Beq) at the corner of the cell boundaries. • No mesoscale is seen, probably because of small aspect ratio

36. shear = 45°

37. shear = 90°