Statistics of Lorenz force in kinematic stage of magnetic dynamo at large Prandtle number S.S.Vergeles Landau Institute for Theoretical Physics in collaboration with M.Chertkov, I.V.Kolokolov and V.V.Lebedev

High conductivity B >> E Typical velocities v are less than speed of sound: fluid is incompressible Plasma in turbulent state Magnetohydrodynamic equations: Prandtle number Diffusion Interstellar plasma in Galactic disk Pr

Kinematic stage Magnetic field is weak Schekochihin et.al, “Simulation of small-scale turbulent dynamo”, Astrophys. J. 612, 276 (2004) Saturated state Strong magnetic field

Kinematic stage at subviscous scales with some initial conditions Solution in Fourier space: In Batchelor regime velocity Statistics of does not depend on magnetic field Its correlation time is finite and is order of inverse Lyapunov exponent A.P.Kazancev, Sov.Phys. JETP 26, 1031 (1968) M.Chertkov, G.Falkovich, I.Kolokolov and M.Vergassola, “Small-Scale turbulent dynamo”, Phys. Rev. Lett. 83, (1999) Statistics of matrix W at times >> λ -1

Statistics of fluid element deformation. Entropy function S. -- orthogonal matrices Incompressibility of flow Lyapunov exponents : Main Lyapunov exponent Lagrangian trajectory Joint probability distribution function at times, when Entropy function S is convex and achieves minimum at

Statistics of passive scalar blob dimensions Initial condition: Blob with dimension l Matrix of “inertia moment” I(0)=1 Matrix I sets correlation volume at given velocity realization E.Balkovsky and A.Fouxon, PRE, 60, 4164 (1999) Vergeles S.S., JETP, 102, 685 (2006) Eigenvalues of I If ρ 2 grows, ρ 2 >-ln Pe, then If ρ 2 decreases, ρ 2 < -ln Pe, then

Single magnetic ring Initial condition:external flow Volume, which ring occupies Total magnetic energy, which is formed at optimal fluctuation  Strength of magnetic field does not increase  Flux of magnetic field is soncerved

Ensemble of rings Initial statistics of magnetic field distribution: Gaussian statistics with pair correlation function in Fourier-space Initial concentration of rings is much larger than inverse initial volume Really, this is not necessary condition Growth of magnetic energy energy with time   =0 does not grows Estimate for magnetic force: ~ ~ ~ M.Chertkov et al., Phys. Rev. Lett. 83, (1999) If ρ 2 grows, ρ 2 >-ln Pe If ρ 2 decreases, ρ 2 < -ln Pe

Statistics of Lorenz force One point moments Averaged only over statistics of initial field distribution Transition occurs at Expected tail of PDF middle size grows: dimensions of magnetic ring middle size decreases: dimensions of magnetic ring   To get moments of Lorenz force, one should implement averaging over statistics of initial field distribution and statistics of velocity M.Chertkov, G.Falkovich and I.Kolokolov, Intermittient Dissipation of a Passive Scalar in Turbulence, Phys.Rev.Lett., 80, 2121 (1998)

Possible mechanisms of magnetic dynamo halt 1.Arising correlation in magnetic rings distribution 2. Correlation between magnetic field and velocity field Value |Y| should decrease during transition from kinematic regime to saturated state After transition to saturated state short range order should appear

Conclusions Qualitative picture of magnetic dynamo at kinematic stage in smooth velocity field is proposed This picture survives at nonsmooth velocity field at high Prandtle number Possible mechanisms of magnetic energy growth halt are discussed M.Chertkov & V.Lebedev, PRE, 90, (2003) Verma (2004)