1. Gas-kineitc MHD Numerical Scheme and Its Applications to Solar Magneto-convection Tian Chunlin Beijing 2010.Dec.3

2. Outline Gas-kinetic MHD scheme –gas-kinetic shceme for hydrodynmics –exention to magntohydrodynamics Numerical simulations of turbulent magneto-convections in the Sun –stellar turbulent convections –magneto-convections

3. Gas-kinetic Scheme -- Introduction two ways to describe the gas –macro: density, pressure, temperature, etc. –micro: distribution of particles in phase space. governing equations –macro: Euler, Navier-Stokes, ideal MHD, resistive MHD. –micro: Boltzmann, BGK (non-magentic) Boltzmann Navier-Stokes –by defining non-equilibrium transport coefficients

4. Gas-kinetic Scheme Classification of numerical schemes –finite difference; finite volume; finite element,... –spectrum scheme –TVD, PPM, Reo, Godnov, Upwinding –grid, non-grid –gas-kinetic; particle smooth hydrodynamics; –...... gas-kineitc scheme is based on finte volume method: calculate the fluxes by gas-kinetic theory.

5. Finite Volume Mthod divide the whole computational domain into small volumes; apply conservations on these volumes; boundary Cell-center

6. gas-kinetic BGK solver Botlzmann equation vs. BGK equation Maxwellian

7. gas-kineitc BGK scheme-2 use distribution function to get fluxes Boltzmann Navier-Stokes

8. Merits of BGK Scheme positivity; entropy condition;... smartly introduce dissipation; robust and accurate scheme for supersonic flows.

9. Extenstion to MHD implementation of additional terms by arbitrory scheme will introduce disspation and dispersion. Non-magnetic part by BGK-NS solver; Gravity term by consistent calculations; Magnetic part by gas-kinetic theory based flux splitting method.

10. Gas-kinetic based flux splitting Scheme According to the direction of micro particles, the flux is split into two parts.

11. Flux-splitting slope limiter reconstruction gas-kinetic theory based flux splitting method for MHD, using Maxwellian.

12. BGK MHD solver non-magnetic part: BGK-NS under gravity solver magnetic part: gas-kinetic theory based flux splitting method, using solution of BGK equation Divergence free condition ensured by constrait tansport method. effects of gravity and Lorentz force included in the particle distribution function.

13. BGK-MHD solver testing BGK-MHD is a high order accuracy MHD solver for supersonic flows.

14. Applications of BGK MHD code to solar convections  Introduction –importance of convection –Existing simulations of solar convection  Numerical Results –Non magneto-convection –Interaction between turbulent convection and magnetic field. time evolution of magnetic structure horizontal mean flows effect of numerical resolution

15. Introduction-1  Why study it? –Efficient way for mixture and energy transport –common state of star matter sun: lower radiation envelope +upper convective envelope massive star: convective core giants: totally convective –Very important for understanding the stars:  Together with rotation to drive the dynamo  Generate p mode oscillations  Produce energetic waves  Move the footpoints of tubes  Why numerically? –Highly non-linear –It is a parabolic system –Complicated system: NS + Induction + radiation transfer  Why difficult (need huge computational resource)? –Multi length-scale: solar radius/molecular scale –Multi time-scale: thermal scale/ dynamical scale

16. Current status of Numerical Simulation of turbulent convection  Realistic simulation –Great success has been achieved Since Nordlund & Stein (1998) –including realistic EOS –including realistic radiation –realsitic parameters  Parametric study  ideal gas  simlified radiation  changing parameters

17. Current status of Numerical Simulation of turbulent convection

18. Non-magneto convection Configuration Initial hydrostatic state Open lower boundary Closed upper boundary Radiation treated by diffusion model Turbulence treated by SGS model Vertically 3.6PSH Aspect ratio: hrz/vtc=5 Code: Gas kinetic BGK MHD code

19. Non-magneto convection-2 statistical properties: –Fluxes –Averages –rms of ρp T –rms of v x v y v z

20. Magneto-convection-1 Initial magnetic field: uniform vertical lines Boundary conditions: vertical lines Parametric: different initial magnetic strength. B0=3.53Beq B0=0.70Beq

21. Magneto-convection-1 B0=3.53Be B0=2.83Be

22. Magneto-convection-2 More cases: B0=6.70Beq B0=2.83BeqB0=2.12BeqB0=1.41BeqB0=0.35Beq

23. Horizontal mean flows-phenomenon Unexpected under two circumstances –Small box; –After imposing strong magnetic field;

24. Horizontal mean flows-analysis Conservation law of y momentum At the lower boundary surface: –Advection (ρv y v z ); viscous; magnetic B z B y On the finite volume –Horizontal gradient of pressure

25. Horizontal mean flows-analysis- 2 Effect of aspect ratio 3.6PSHs 1/5 6.5PSHs 1/3 3.6 PSHs 1/1.5

26. Horizontal mean flows-analysis- 3 Velocity+temperature fluctuations Magnetic field+strength B0 CASE

27. Horizontal mean flows- anisotropy Non temporally averaged!!!!!

28. Effects of resolution  3:1, 138x134x204, Sandwich model 5:1, 64x64x64 Horizontal flow Circular bubbles

29. Summary  Numerical Scheme:  gas-kineitc scheme is based on finite volume Method  BGK MHD solver is robust and accurate  Magneto-convection  Realistic vs. parametric  Convections in a strong magnetic fields: time evolution of convective tube, horizontal mean flows.