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;

dttFtFx

WWn

n

t

t jjnn

1

2/12/11 1

boundary

Cell-center

gas-kinetic BGK solver

• Botlzmann equation vs. BGK equation

},,{ wvu

Qft

fcollision

c

cfg

ft

f

c

gfft

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0c Maxwellian ))()((2

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222

VvUu

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eg

gas-kineitc BGK scheme-2

• use distribution function to get fluxes

dguupUU

d[...])([...] uugpUU

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.

BBvB

gvBBvFvΣv

gBBΣvv

v

2

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)(

)(

)2

1()(

0)(

t

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Bpt

vt

d

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.

))((2

1

))((2

1

))((2

1

22

22

22

)(

)()(

rr

ll

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rr

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Flux-splitting

• slope limiter• reconstruction

2/12/12/1

12/1

)1(

)()(

if

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FFF

UFUFF

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.

BGK-MHD solver testing

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

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

Current status of Numerical Simulation of turbulent convection

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

Non-magneto convection-2

• statistical properties:– Fluxes– Averages– rms of ρp T– rms of vx vy vz

Magneto-convection-1

• Initial magnetic field: uniform vertical lines• Boundary conditions: vertical lines• Parametric: different initial magnetic

strength. B0=3.53Beq B0=0.70Beq

Magneto-convection-1

• B0=3.53Be

• B0=2.83Be

Magneto-convection-2

• More cases:

B0=6.70Beq B0=2.83Beq B0=2.12Beq B0=1.41Beq B0=0.35Beq

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

Horizontal mean flows-analysis-2

• Effect of aspect ratio

3.6PSHs 1/5

6.5PSHs 1/3

3.6 PSHs 1/1.5

Horizontal mean flows-analysis-3

Velocity+temperature fluctuations Magnetic field+strength

B0

CASE

Horizontal mean flows-anisotropy

Non temporally averaged!!!!!

Effects of resolution

3:1, 138x134x204, Sandwich model

5:1, 64x64x64

Horizontal flow

Circular bubbles

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.