Multiscale issues in modeling magnetic reconnection J. F. Drake University of Maryland IPAM Meeting on Multiscale Problems in Fusion Plasmas January 10, 2005

Magnetic energy dissipation in the universe The conversion of magnetic energy to heat and high speed flows underlies many important phenomena in nature –solar and stellar flares –magnetospheric substorms –disruptions in laboratory fusion experiments More generally understanding how magnetic energy is dissipated is essential to model the generation and dissipation of magnetic field energy in astrophysical systems –accretion disks –stellar dynamos –supernova shocks Known systems are characterized by a slow buildup of magnetic energy and fast release –trigger? –mechanism for fast release? –Mechanism for the production of energetic particles?

Magnetic Free Energy A reversed magnetic field is a source of free energy x x x x x x x x x x x x x J B Can imagine B simply self-annihilating What happens in a plasma? How does magnetic reconnection work?

Frozen-in Condition In an ideal plasma (  =0), the fluid moves so that the magnetic flux through any fluid element is preserved.

Energy Release from Squashed Bubble Magnetic field lines want to become round magnetic tension

Energy Release (cont.) Evaluate initial and final magnetic energies –use conservation law for ideal motion magnetic flux conserved area for nearly incompressible motion L R w W f ~ (w 2 /L 2 ) W i << W i Most of the magnetic energy is released

Flow Generation Released magnetic energy is converted into plasma flow Alfven time  A is much shorter than observed energy release time

Magnetic Reconnection Strong observational support for this general picture

Resistivity and the multiscale problem The frozen-in condition implies that in an ideal plasma (  =0) no topological change in the magnetic field is possible –tubes of magnetic flux are preserved –Breaking of magnetic field lines requires resistivity or some other dissipation process As in fluid systems, dissipation can only be important at small spatial scales Breaking of field lines occurs at very small spatial scales where the magnetic field reverses  dissipation region Release of energy in a macroscopic system depends on the complex dynamics of a boundary layer –Typically kinetic and turbulent –Reconnection is inherently a multiscale problem whose description is a computational challenge

Expulsion of the core temperature during sawteeth in tokamaks Reconnection is broadly important in fusion experiments The “sawtooth crash” is an important example –Periodic expulsion of the plasma from the core of tokamaks Yamada, et al, 1994

Characteristic Times Resistive Time Alfven Time Release Time Laboratory Tokamaks sec ~ 1  sec 50  sec Solar Flares ~ 10 4 years ~ 0.1 sec ~ 20 min Magnetosphere  100 sec ~ 30 min resistive time

Resistive Magnetohydrodynamic (MHD) Theory Formation of macroscopic Sweet-Parker layer Slow reconnection sensitive to resistivity macroscopic nozzle V ~ (  /L) C A ~ (  A /  r ) 1/2 C A << C A

Failure of the MHD model Resistive MHD reconnection rates are too slow to explain observations –solar flares –sawtooth crash –magnetospheric substorms Some form of anomalous resistivity is often invoked to explain discrepancies –strong electron-ion streaming near x-line drives turbulence and associated enhanced electron-ion drag Non-MHD physics at small spatial scales produces fast reconnection –coupling to dispersive waves critical Mechanism for strong particle heating during reconnection?

Role of dispersive waves Coupling to dispersive waves at small scale is key to understanding magnetic reconnection –rate of reconnection insensitive to the mechanism that breaks the frozen-in condition –fast reconnection even for large systems no macroscopic nozzle

Generalized Ohm’s Law Electron equation of motion MHD valid at large scales Below c/  pi electron and ion motion decouple electrons frozen-in Whistler and kinetic Alfven waves are dispersive Electron frozen-in condition broken below c/  pe c/  pi c/  pe ss scales Electron inertia whistler waves kinetic Alfven waves

Kinetic Reconnection Ion motion decouples from that of the electrons at a distance from the x-line –ion outflow width electron current layer and outflow width Whistler and kinetic Alfven waves control the dynamics in the inner region c/  pi c/  p e

GEM Reconnection Challenge National collaboration to explore reconnection with a variety of codes – MHD, two-fluid, hybrid, full-particle nonlinear tearing mode in a 1-D Harris current sheet B x = B 0 tanh(z/w) w = 0.5 c/  pi Birn, et al., 2001

Rates of Magnetic Reconnection Rate of reconnection is the slope of the  versus t curve All models that include the Hall term in Ohm’s law yield essentially identical rates of reconnection –Consequence of dispersive waves MHD reconnection is too slow by orders of magnitude

Why is wave dispersion important? Quadratic dispersion character  ~ k 2 V p ~ k –smaller scales have higher velocities –weaker dissipation leads to higher outflow speeds –flux from x-line ~vw »insensitive to dissipation

Fast reconnection in large systems Large scale hybrid simulation (Shay, et al., 1999) T= 160  -1 T= 220  -1 Rate of reconnection insensitive to system size v i ~ 0.1 C A No large scale nozzle in kinetic reconnection

3-D Magnetic Reconnection Turbulence and anomalous resistivity –2-D models produce strong electron streaming around the magnetic x-line Can such streams drive turbulence? Electron-ion streaming instability (Buneman) evolves into nonlinear state with strong wave turbulence Electron scattering produces enhanced electron-ion drag, (anomalous resistivity) that is sufficient to break magnetic field lines even without classical resistivity

Observational evidence for turbulence There is strong observational support that the dissipation region becomes strongly turbulent during reconnection –Earth’s magnetopause broad spectrum of E and B fluctuations –Sawtooth crash in laboratory tokamaks strong fluctuations peaked at the x-line –Magnetic fluctuations in Magnetic Reconnection eXperiment (MRX)

Particle simulations (PIC) with up to 1.4 billion particles Development of strong current layer Current layer becomes turbulent –Electron-ion streaming instability (Buneman) evolves into electron holes 3-D Magnetic Reconnection: with guide field y x

Turbulence and the formation of electron holes Intense electron beam generates Buneman instability –nonlinear evolution into “electron holes” localized regions of depleted electron density Seen in satellite observations in the magnetosphere x z EzEz B

Anomalous drag on electrons Parallel electric field scatter electrons producing effective drag Average over fluctuations along z direction to produce a mean field electron momentum equation –correlation between density and electric field fluctuations yields drag Normalized electron drag

Drag D z has complex spatial and temporal structure with positive and negative values Sufficient to break magnetic field lines during reconnection Electron drag due to scattering by parallel electric fields y x

The computational challenge Modeling reconnection in plasma systems (solar corona, fusion plasmas, the Earth’s magnetosphere) requires the description of the dynamics of the largest spatial scales – describes the buildup and storage of magnetic energy –MHD description adequate At the same time must include the dynamics of a microscale boundary layer –This dissipation region is both kinetic and turbulent Modeling the dissipation region –Including the coupling to dispersive waves to model fast reconnection requires a two-fluid or kinetic (PIC, gyrokinetic) description Modeling turbulence and anomalous resistivity –Kinetic (PIC) description down to Debye scales Modeling the production of energetic particles –Kinetic (PIC) description

Range of spatial scales Spatial ScalesMacro L c/  pi c/  pe L/(c/  pe ) Fusion plasma200cm5cm0.1cm2000 Solar Corona10 4 km10m0.2m 5  10 7 Earth’s magnetosphere 10 5 km50km1km10 5 Modeling kinetic turbulence requires even smaller spatial scales!! Even AMR codes will not be able to treat such disparate scales The development of innovative multiscale algorithms for handling such problems is an imperative

Conclusions Magnetic reconnection causes an explosive release of energy in plasma systems –similar to other types of explosions sonic flows –a difference is that the explosion is non-isotropic Fast reconnection depends critically on the coupling to dispersive waves at small scales –rate independent of the mechanism which breaks the frozen-in condition –rate independent of all kinetic scales ~ 0.1 C A –rate consistent with observations Modeling magnetic reconnection in a macroscale system requires the simultaneous treatment of a microscale boundary layer that is both collisionless and therefore inherently kinetic and turbulent –Describing the dynamics is a multiscale challenge

Outstanding Issues Onset Structure of slow shocks Electron heating Role of turbulence and anomalous resistivity