Gas-kineitc MHD Numerical Scheme and Its Applications to Solar Magneto-convection

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; Cell-center boundary

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 BGKMHD code

19. Non-magneto convection-2 • statistical properties: • Fluxes • Averages • rms of ρp T • rms of vx vy vz

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

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

22. Magneto-convection-2 • More cases: B0=6.70Beq B0=2.83Beq B0=2.12Beq B0=1.41Beq B0=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 (ρvy vz); viscous; magnetic BzBy • 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 Magnetic field+strength Velocity+temperature fluctuations CASE B0

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

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

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. Thanks!!