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LagrangianStatistics

of 3D MHDConvection

J. Pratt,W.-C. Muller

BoussinesqSimulation

Lagrangiansimulation

Lagrangian Statisticsof 3D MHD Convection

J. Pratt, W.-C. Muller

March 1, 2011


of 3D MHDConvection




Our approach to the Dynamo Problem

dynamo action: amplification of magnetic fields byturbulent flows, generation of large scale structures

collaboration with the group of Schussler et al. whosimulate solar convection (MURaM)

detailed treatment of turbulence: simulation of Boussinesqmagnetoconvection

flows not dominated by boundary conditions:pseudo-Rayleigh-Benard, fully periodic, no kz = 0 modes

Lagrangian particle simulation


of 3D MHDConvection




Boussinesq MHD Convection Equations

In Fourier space the non-dimensionalized Boussinesq MHDconvection equations, solved by pseudo-spectral calculation:(d

dt+ νk2

)ωk = ik × [v × ω + (∇× b)× b]

k+ ikθk × g(

d

dt+ ηk2

)bk = ik × [v × b]

k(d

dt+ κk2

)θk = − [v × ikθ]

k+ (vz)k

vk =ik

k2× ωk , ∇ · v = 0 , ∇ · b = 0

Convective motion defines the characteristic length and timescales: L = T∗/∇T0, tb = 1√

αg|∇T0|


of 3D MHDConvection




Physically realistic parameters

Re is limited by grid size.

Resolving the different numerical scales remains achallenge for the field of magnetoconvection simulation.

in convection zone MHD conv. sim.

Re ∼(Llarge

Lsmall

)4/31013 2 · 103 - 9 · 103

Rem = Re ν/η 1011 3 · 103 - 1.8 · 104

Pe = Re ν/κ 1013- 1014 2 · 103 - 9 · 103

Pr = ν/κ 10−5-10−7 1

PrM = ν/η 10−1-10−7 0.5 -2

Ra = αg∆TL30/νκ 1023 2.5 · 105 - 5.0 · 105


of 3D MHDConvection




Steady-state MHD convectionsustained by dynamo at resolution 5123

Lagrangian particle simulation during steady-state plasma convection


of 3D MHDConvection




Lagrangian statistics

Sawmill and Yeung (1994) hydrodynamic turbulence,Schumacher (2008) hydrodynamic convection,Busse-Muller (2007) MHD turbulence

Lagrangian studies follow single particles (or pairs ofparticles) and examine how they diffuse (or separate).

tetrads: anchor particle + three particles separated fromthe anchor in each of the three directions.

anchor particles distributed on deformed cubic grid,665500 particles total


of 3D MHDConvection




Lagrangian statistics

t/tb

95 100 105 110 115

24

68

12

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Ev

EB

ET

Particles are launched and followed during steady-stateplasma convection.

The highly variable nature of the convection drive causes afluctuation in global energy.

Extensive averaging of internal data blocks is necessary toreduce statistical noise.


of 3D MHDConvection




Order-n method for averaging overinternal data blocks

Dubbeldam et al. A new perspective on the order-nalgorithm for computing correlation functions. MolecularSimulation, Vol. 35, No. 12. (2009), pp. 1084-1097.

Several hundred ‘windows’ are necessary to get reasonablestatistical convergence.

Averaging over internal data blocks is only possible forsingle-particle Lagrangian statistics.


of 3D MHDConvection




Lagrangian Velocity Autocorrelations

We look at the VACF for clarification of diffusion/dispersionbehaviors, particularly to describe ballistic and diffusive regimes.The VACF 〈v(0)v(t)〉 has a differential relation to the diffusion:

d

dt〈dr(0)dr(t)〉 = 2

∫ t

0〈v(0)v(τ)〉dτ (1)

the visualization of relaxation of fluctuations over longtimes and distances.

for Brownian motion 〈v(t)v(0)〉 ∼ 〈v(0)2〉e−t/τc .

a single exponential is a good fit for hydrodynamicturbulence, for example: Yeung and Pope (1989), Satoand Yamamoto (1987)


of 3D MHDConvection




Lagrangian Velocity Autocorrelation

t/τη

ln <

v(t)

⋅v(0

)>/<

v2 >

vx

vy

vz

v

0 20 40 60 80

−3.

0−

2.5

−2.

0−

1.5

−1.

0−

0.5

0.0

1024 internal data blocks averaged

no change in sign in the VACF

for MHD convection, one exponential → poor fit

nonlinear least-squares-fit:〈v(t)v(0)〉 = a1 exp(−t/τ1) + a2 exp(−t/τ2)


of 3D MHDConvection




Lagrangian Diffusion

t/τη

(x[i]

−x[

0])^

2/N

(le

ngth

2 /η2 )

diff 1 diff 2tb~2

~1

xyz

1e−01 1e+00 1e+01 1e+02 1e+03

1e−

031e

+01

1e+

05

256 internal data blocks displayed

clear ballistic phase (slope 2), diffusive phase


of 3D MHDConvection




Lagrangian Acceleration Autocorrelation

t/τη

<a(

t)⋅a

(0)>

/<a2 >

ax

ay

az

a

0 2 4 6 8

0.0

0.4

0.8

classic recognizable shape1 2

1Figure 2 of R Kubo Rep. Prog. Phys. 29 255 1966.2Figure 8 of Yeung and Pope 1989, Fig 8 of Sawford 1990


of 3D MHDConvection




Lagrangian PDFs reflect intermittent behavior

vi

ln P

(vi)

sim. with rare events

averaged over 52 runs

−2 −1 0 1 2

−4

−3

−2

−1

0 vxvy

vz

vi

ln P

(vi)

sim. with rare events

averaged over 110 runs

−4 −2 0 2 4

−4

−3

−2

−1

0 vxvy

vz

Asymmetrical PDFs obtained when the averaging includes onlya small number of intermittent events associated with formationof large-scale magnetic structuresshape in extreme wings is typical; see isotropic turbulenceMordant et al. Phys. Rev. Lett. 2002 and hydrodynamicconvection Schumacher 2009


of 3D MHDConvection




Results and Summary

in diffusion: clear ballistic regime with length dependenton the system parameters (kinetic, magnetic, andtemperature dissipation)

two correlation times τ1 ∼ τη and τ2 ∼ tbacceleration autocorrelation functions that on average looksimilar to hydrodynamic turbulence.

Asymmetrical PDFs, obtained from averaging over only afew intermittant events, indicate formation of large-scalemagnetic structures in the flow

lagrangian statistics of 3d mhd convection - ens-lyon.fr filelagrangian statistics of 3d mhd...

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