1 Turbulent Generation of Large Scale Magnetic Fields in Unmagnetized Plasma Vladimir P.Pavlenko Uppsala University, Uppsala, Sweden

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 2 Coworkers  Zhanna N.Andrushchenko  Martin Jucker

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 3 Outline  Some notes from Fluid Mechanics  Motivation  Modeling  Self-consistent description  Non-linear dynamics  Conclusions

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 4 Description of flows in a turbulent media

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 5 Motivations  Strong magnetic fields - Solar flares Magnetic field diffusivity Reconnection - Laser produced plasma Strong fields produced in unmagnetized plasma 70’s: Laser fusion experiments: Strong magnetic field observed in unmagnetized plasma Magnetic electron drift modes Magnetic electron drift modes

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 6 Modeling: Assumptions

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 7 Modeling: Equations

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 8 Model equations

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 9 Linear approximation  - purely growing for  - no linear instability for  - largest increment for

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 10 Comparison to electrostatic drift wave turbulence  Two-field vs. One-field model  Electron skin depth vs. Ion Larmor radius with electron temperature  Direct and inverse cascade vs. Inverse cascade

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 11 Large scale structures: Definitions  Zonal magneticfields  Magnetic streamers

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 12 Large scale magnetic fields generation

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 13 Large scale structures:

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 14 Self-consistent description  Define ”action-like invariant” or wave spectrum  Wave kinetic equation (WKE)  Doppler shifted frequency

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 15 Non-linear dynamics Large scale Drift-type wave Structures turbulence

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 16 Large scale fields generation - Recipe  Model equations  Wave spectrum  Quasi-linear analysis, linearized WKE  Response function  Dispersion relation

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 17 Large scale field generation – Hydrodynamic regime  Hydrodynamic regime - Monochromatic wave packet - Instability criterion - Explicit frequency

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 18 Large scale field generation – Kinetic regime  Kinetic regime - Resonance (purely growing) - Instability criterion Contrary to Langmuir turbulence Contrary to Langmuir turbulence

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 19 Large scale field generation – Modulation instability  Modulation instability - Restart from basic model equations - Pump wave and flows - Triad interactions sidebands - Explicit frequency - Well known Lighthill criterion

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 20 Large scale field generation - Summary  Explicit increments - Hydrodynamic regime - Modulational instability - Note  Instability criteria - Hydrodynamic, modulational - Kinetic regime

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 21 Non-linear dynamics Large scale Drift-type wave Structures turbulance

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 22 Shearing  Quasi-linear analysis  Wave kinetic equation  Diffusion in k-space, i.e. shearing  Large k Small scales Dissipation

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 23 Large scale pattern and predator- prey phenomena  Reduction of the basic equations dynamical model (zero-dimensional approach) with two principal components: minimal dynamical model (zero-dimensional approach) with two principal components: small-scale waves () + ”zonal” magnetic pattern () small-scale waves (prey) + ”zonal” magnetic pattern (predator)+ Lotka-Volterra system

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 24 Predator-prey dynamics of the system composed of zonal magnetic pattern and wave turbulence Results of the numerical analysis

2007, May 29 Workshop ’Reconnection and Turbulence’ Uppsala 25 Conclusions  Magnetic electron drift mode turbulence – model equations  Separation of scales: waves + ”fields”  Self-consistent description – wave kinetic equation  Waves ”Fields”: Generation  ”Fields” Waves: Shearing  Long term dynamics Predator-prey dynamics