Lagrangian Statistics. of 3D MHD Convection. J. Pratt, W.-C. Müller. Boussinesq Simulation. Lagrangian. simulation. March 1, 2011

Transcription

1 March 1, 2011

2 Our approach to the Dynamo Problem dynamo action: amplification of magnetic fields by turbulent flows, generation of large scale structures collaboration with the group of Schüssler et al. who simulate solar convection (MURaM) detailed treatment of turbulence: of magnetoconvection flows not dominated by boundary conditions: pseudo-rayleigh-bénard, fully periodic, no k z = 0 modes particle

3 MHD Equations In Fourier space the non-dimensionalized MHD convection equations, solved by pseudo-spectral calculation: ( ) d dt + νk2 ω k = ik [v ω + ( b) b] k + ikθ k ĝ ( ) d dt + ηk2 b k = ik [v b] k ( ) d dt + κk2 θ k = [v ikθ] k + (v z ) k v k = ik k 2 ω k, v = 0, b = 0 Convective motion defines the characteristic length and time scales: L = T / T 0, t b = 1 αg T0

4 Physically realistic parameters Re is limited by grid size. Resolving the different numerical scales remains a challenge for the field of magnetoconvection. in convection zone MHD conv. sim. ( ) Llarge 4/3 Re L small Re m = Re ν/η Pe = Re ν/κ Pr = ν/κ Pr M = ν/η Ra = αg T L 3 0 /νκ

5 Steady-state MHD convection sustained by dynamo at resolution 5123 W.-C. Mu ller particle during steady-state plasma convection

6 statistics Sawmill and Yeung (1994) hydrodynamic turbulence, Schumacher (2008) hydrodynamic convection, Busse-Müller (2007) MHD turbulence studies follow single particles (or pairs of particles) and examine how they diffuse (or separate). tetrads: anchor particle + three particles separated from the anchor in each of the three directions. anchor particles distributed on deformed cubic grid, particles total

7 statistics Ev EB ET o oo o o o o o o o Particles are launched and followed during steady-state plasma convection. The highly variable nature of the convection drive causes a fluctuation in global energy. Extensive averaging of internal data blocks is necessary to reduce statistical noise. t/t b

8 Order-n method for averaging over internal data blocks Dubbeldam et al. A new perspective on the order-n algorithm for computing correlation functions. Molecular, Vol. 35, No. 12. (2009), pp Several hundred windows are necessary to get reasonable statistical convergence. Averaging over internal data blocks is only possible for single-particle statistics.

9 Velocity Autocorrelations We look at the VACF for clarification of diffusion/dispersion behaviors, 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 long times and distances. for Brownian motion v(t)v(0) v(0) 2 e t/τc. a single exponential is a good fit for hydrodynamic turbulence, for example: Yeung and Pope (1989), Sato and Yamamoto (1987)

10 Velocity Autocorrelation ln <v(t) v(0)>/<v 2 > vx vy vz v t/τ η 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) = a 1 exp( t/τ 1 ) + a 2 exp( t/τ 2 )

11 Diffusion (x[i] x[0])^2/n (length 2 /η 2 ) 1e 03 1e+01 1e+05 x y z ~2 ~1 diff 1 t b diff 2 1e 01 1e+00 1e+01 1e+02 1e+03 t/τ η 256 internal data blocks displayed clear ballistic phase (slope 2), diffusive phase

12 Acceleration Autocorrelation <a(t) a(0)>/<a 2 > ax ay az a classic recognizable shape 1 2 t/τη 1 Figure 2 of R Kubo Rep. Prog. Phys Figure 8 of Yeung and Pope 1989, Fig 8 of Sawford 1990

13 PDFs reflect intermittent behavior ln P(vi) vx vy vz sim. with rare events averaged over 52 runs ln P(vi) vx vy vz sim. with rare events averaged over 110 runs v i Asymmetrical PDFs obtained when the averaging includes only a small number of intermittent events associated with formation of large-scale magnetic structures shape in extreme wings is typical; see isotropic turbulence Mordant et al. Phys. Rev. Lett and hydrodynamic convection Schumacher 2009 v i

14 Results and Summary in diffusion: clear ballistic regime with length dependent on the system parameters (kinetic, magnetic, and temperature dissipation) two correlation times τ 1 τ η and τ 2 t b acceleration autocorrelation functions that on average look similar to hydrodynamic turbulence. Asymmetrical PDFs, obtained from averaging over only a few intermittant events, indicate formation of large-scale magnetic structures in the flow