Post on 12-Jan-2016
description
Gas-kineitc MHD Numerical Gas-kineitc MHD Numerical Scheme and Its Applications to Scheme and Its Applications to
Solar Magneto-convectionSolar Magneto-convection
Tian ChunlinBeijing 2010.Dec.3
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
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
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.
Finite Volume Mthod
• divide the whole computational domain into small volumes;
• apply conservations on these volumes;
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WWn
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Cell-center
gas-kinetic BGK solver
• Botlzmann equation vs. BGK equation
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gas-kineitc BGK scheme-2
• use distribution function to get fluxes
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Boltzmann <=> Navier-Stokes
Merits of BGK Scheme
• positivity; entropy condition; ... • smartly introduce dissipation;• robust and accurate scheme for supersonic flows.
Extenstion to MHD• implementation of additional terms by arbitrory sch
eme will introduce disspation and dispersion.
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Non-magnetic part by BGK-NS solver;Gravity term by consistent calculations;Magnetic part by gas-kinetic theory based flux splitting method.
Gas-kinetic based flux splitting Scheme
According to the direction of micro particles, the flux is split into two parts.
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Flux-splitting
• slope limiter• reconstruction
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gas-kinetic theory based fluxsplitting method for MHD, usingMaxwellian.
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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.
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
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
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
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=5Code: Gas kinetic BGK MHD code
Magneto-convection-1
• Initial magnetic field: uniform vertical lines• Boundary conditions: vertical lines• Parametric: different initial magnetic
strength. B0=3.53Beq B0=0.70Beq
Horizontal mean flows-phenomenon
• Unexpected under two circumstances– Small box; – After imposing strong magnetic field;
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
Effects of resolution
3:1, 138x134x204, Sandwich model
5:1, 64x64x64
Horizontal flow
Circular bubbles