Influence of turbulence on the dynamo threshold B. Dubrulle, GIT/SPEC N. Leprovost, R. Dolganov, P. Blaineau J-P. Laval and F. Daviaud

Basic equations Maxwell equations Navier-Stokes equations Field strectching Field diffusion Competition characterized by magnetic Reynolds number Dynamo if Rm > Rmc (Instability)

« Classical dynamo » paradigm Pas dynamo Dynamo Em t t Rmc Rm No dynamo Dynamo Indicator:

Dynamos in the Universe Def: Magnetic field generation through movement of a conductor In the universe…. stars, galaxies planets Control Parameters:

Problem Turbulent flow: Multiplicative noise Classical linear instability Mean Flow Fluctuation

« Classical turbulence » paradigm Mean Field argument: Mean Field equation: Mean Field dispersion: Mean Field instability: Turbulence creates dynamo « most of the time » « Helical turbulence is good for dynamo »

Numerical test ? Schekochihin et al, 2004 Ponty et al, 2005, 2006 Laval et al, 2006 Re Rm Pm=1 Whithout mean velocityWith mean-velocity

*Karlsruhe Experimental test ?

Dynamos with low “unstationarity”: success Riga Karlsruhe R Stieglitz, U. Müller, Phys.of Fluids,2001A. Gailitis et al., Phys. Rev. Lett., 2001

Experiments with unstationarity… “TM60”, no dynamoField “TM28”, dynamo No dynamo Dynamo VKS Experiment Sodium Measure Optimisation Kinematic code

…Failure!

Turbulence increases threshold With respect to time-averaged! Explanation: numerics Simulation with time averaged velocity Simulation with real velocity

Explanation: theory Kraichnan model Mean Field Theorie Perturbative computation (Petrelis, Fauve) ( =0) (with mean velocity fiel) ( =0) Dynamo only for Who is right ??????

Importance of the order parameter MFT, Petrelis, Fauve: transition over KM : transition over =0… No dynamo (MFT, Petrelis) non zero…Dynamo (KM) Vote: What is good order parameter?

Problem Turbulent flow: Multiplicative noise Classical linear instability Mean Flow Fluctuation

Troubles Model equation: Problem: how to define threshold? Instability threshold depends on moment order!!! Etc, etc... Solution: work with PDF and Lyapunov exponent

Stochastic approach Basic Equations Approximation 1 Approximation 2 Noise delta-correlated in timeMean Flow

Fokker-Planck Equation with Equation for P(B,x,t)

Mean-Field Equation beta effect (turbulent diffusion) Alpha effect Helicity if isotropy Mean Field Theory Equation Stability governed by alpha et beta….!!!???

Isotropic case Mean Field Magnetic energy

Stationary solutions Always solution! Other solution: Z: normalization D: space dimension a et  : coefficients Lyapunov exponent!

Bifurcations Non-zero Solution (normalisation)Most probable value  0aD No dynamo Intermittent Dynamo Turbulent Dynamo

New theoretical turbulent paradigm Rm Rm1Rm2 No dynamo Intermittent Dynamo Turbulent Dynamo Pas dynamo Dynamo Em t t Rmc Turbulent Laminar Rm

The Lyapunov exponent… Orientation (<0) (zero if =0) >0 and proportional to noise ( KM effect) Unstable Direction Rmc Expected result Stable Direction Noise intensity Rmc Leprovost, Dubrulle, EPJB 2005

Illustration: Bullard Homopolar Dynamo Noise intensity Intermittent Dynamo No Dynamo Leprovost, PhD thesis

Discussion Noise influences threshold through mu AND vector orientation Influence of alpha and beta through vector orientation Threshold different from Mean Field Theory prediction Dangerous to optimize dynamo experiments from mean field! Turbulent threshold can be very different from « laminar » ones

Simulations Time-average of velocity field computed through Navier-Stokes Type of simulation MHD-DNS Kinematic Noisy Computed through NS 0 Markovian noise (F,tc, ki)

Numerical code Spectral method Integration scheme: Adams-Brashford Resolution: 64*64*64 to 256*256*256 Forcing with Taylor-Green vortex Constant velocity forcing Cf Ponty et al, 2004, 2005

Time-averaged vs real dynamo Laval, Blaineau, Leprovost, Dubrulle, FD: PRL dynamo windows

Results for noisy delta-correlated Forcing at ki=1 Forcing at ki=16 Linear in (  -1) (Fauve-Petrelis)

Results for noisy tc=0.1 Forcing at ki=1 Forcing at ki=16

Results for noisy tc=1 s Forcing at ki=1

Summary of noisy ki=1 ki=16 DNS Tc=1 Compa DNS Stochastic noise k=1 Tauc=1 s Summary of noisy simulations

Interpretation   Kinetic energy of of the Velocity Fluid Rm* Rm Universal curve

In VKS  =30  =0.97 Definition of a universal « control parameter »

Comparison stochastic/DNS Compa DNS Stochastic noise k=1 Tauc=1 s Summary of noisy simulations Tauc ki=1 ki=16

Comparison DNS and mean flow Laval, Blaineau, Leprovost, Dubrulle, Daviaud (2005) Dynamo CM No dynamo Intermittent Dynamo

Conclusions In Taylor-Green, turbulence is not favourable to dynamo Large scale turbulence (unstationarity) increases dynamo threshold -> desorientation effect Small scale turbulence decreases dynamo threshold-> « friction » Turbulence looks like a large scale noise Bad influence through desorientation effect Possible transition to dynamo via intermittent scenario In natural objects: importance of Coriolis force (kills large-scale) Possibility of stochastic simulations to replace DNS