March 1, 2011
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
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
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 10 13 2 10 3-9 10 3 Re m = Re ν/η 10 11 3 10 3-1.8 10 4 Pe = Re ν/κ 10 13-10 14 2 10 3-9 10 3 Pr = ν/κ 10 5-10 7 1 Pr M = ν/η 10 1-10 7 0.5-2 Ra = αg T L 3 0 /νκ 1023 2.5 10 5-5.0 10 5
Steady-state MHD convection sustained by dynamo at resolution 5123 W.-C. Mu ller particle during steady-state plasma convection
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, 665500 particles total
statistics 2 4 6 8 12 Ev EB ET o oo o o o o o o o 95 100 105 110 115 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
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. 1084-1097. Several hundred windows are necessary to get reasonable statistical convergence. Averaging over internal data blocks is only possible for single-particle statistics.
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)
Velocity Autocorrelation ln <v(t) v(0)>/<v 2 > 3.0 2.5 2.0 1.5 1.0 0.5 0.0 vx vy vz v 0 20 40 60 80 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 )
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
Acceleration Autocorrelation <a(t) a(0)>/<a 2 > 0.0 0.4 0.8 ax ay az a 0 2 4 6 8 classic recognizable shape 1 2 t/τη 1 Figure 2 of R Kubo Rep. Prog. Phys. 29 255 1966. 2 Figure 8 of Yeung and Pope 1989, Fig 8 of Sawford 1990
PDFs reflect intermittent behavior ln P(vi) 4 3 2 1 0 vx vy vz sim. with rare events averaged over 52 runs ln P(vi) 4 3 2 1 0 vx vy vz sim. with rare events averaged over 110 runs 2 1 0 1 2 4 2 0 2 4 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. 2002 and hydrodynamic convection Schumacher 2009 v i
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