Kinetic Modeling of Magnetic Reconnection in Space and Astrophysical

Systems

J. F. Drake

University of Maryland

Large Scale Computation in Astrophysics

Newton Institute 2004

Recent Collaborators

• Marc Swisdak (NRL)

• Mike Shay (U. Maryland)

• Michael Hesse (GSFC)

• Cindy Cattell (U. Minn.)

Collisionless reconnection is ubiquitous

• Inductive electric fields typically exceed the Dreicer runaway field– classical collisions and resistivity not important

• Earth’s magnetosphere– magnetopause

– magnetotail

• Solar corona– solar flares

• Laboratory plasma– sawteeth

• astrophysical systems?

Resistive MHD Description

• Formation of macroscopic Sweet-Parker layer

•Slow reconnection•sensitive to resistivity•macroscopic nozzle

V ~ ( /L) CA ~ (A/r)1/2 CA << CA

• Petschek-like open outflow configuration does not appear in resistive MHD models with constant resistivity (Biskamp ‘86)• Why Sweet-Parker?

Singular magnetic island equilibria

• Equilibria that form as a consequence of reconnection are singular– Sweet-Parker current layers reflect this underlying singularity

• Allow reconnection to produce a finite magnetic island ( ) • Shut off reconnection ( = 0) and evolve to relaxed state

– Formation of singular current sheet

• Consequence of flux conservation and requirement that magnetic energy is reduced (Waelbroeck, 1989)€

≠0

Overview• MHD Reconnection rates 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– observational evidence in magnetosphere

• Non-MHD physics at small spatial scales produces fast reconnection– coupling to dispersive waves critical

• Electron-positron reconnection– No dispersive waves– Fast reconnection from turbulent outflow jet??

• Mechanism for strong particle heating during reconnection?

Kinetic Reconnection

• Coupling to dispersive waves in dissipation region at small scales produces fast magnetic reconnection– rate of reconnection independent of the mechanism which breaks

the frozen-in condition

– fast reconnection even for very large systems• no macroscopic nozzle

• no dependence on inertial scales

Generalized Ohm’s Law

• Electron equation of motion

•MHD valid at large scales•Below c/pi or s electron and ion motion decouple

•electrons frozen-in•whistler and kinetic Alfven waves control dynamics

•Electron frozen-in condition broken below c/pe

•Non-gyrotropic pressure tensor dominates

c/pic/pes scales

€

4π

ωpe2

dr J

dt=

r E +

1

c

r v i ×

r B −

1

nec

r J ×

r B +

1

ne∇ •

r r p e −η

r J

Electroninertia

whistlerwaves

kinetic Alfvenwaves

Kinetic Reconnection: no guide field

• Ion motion decouples from that of the electrons at a distance from the x-line– coupling to whistler and kinetic Alfven waves

• Electron velocity from x-line limited by peak speed of whistler– exceeds Alfven speed

c/pi

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

Bx = B0 tanh(x/w)

w = 0.5 c/pi

• Birn, et al., JGR, 2001, and companion papers

Rates of Magnetic Reconnection

• Rate of reconnection is the slope of the versus t curve

• All models which include the Hall term in Ohm’s law yield essentially identical rates of reconnection– Reconnection insensitive to mechanism that breaks frozen-in condition

• MHD reconnection is too slow by orders of magnitude

Birn, et al., 2001

Reconnection Drive

• Reconnection outflow in the MHD model is driven by the expansion of the Alfven wave

– Alfvenic outflow follows simply from this picture

• Coupling to other waves in kinetic and two-fluid models– Whistler and kinetic Alfven waves drive outflow from x-line

• Dispersive waves

Why is wave dispersion important?

• Quadratic dispersion character

~ k2

Vp ~ k– smaller scales have higher velocities

– weaker dissipation leads to higher outflow speeds

– flux from x-line ~vw

» insensitive to dissipation

:

Wave dispersion and the structure of nozzle• Controlled by the variation of the wave phase speed with

distance from the x-line

– increasing phase speed

•Closing of nozzle•MHD case since Bn and CA increase with distance from the x-line

- decreasing phase speed

•Opening of the nozzle•Whistler or kinetic Alfven waves v ~ B/w

Whistler Driven Reconnection: weak guide field

• At spatial scales below c/pi whistler waves rather than Alfven waves drive reconnection. How?

•Side view

•Whistler signature is out-of-plane magnetic field

Whistler signature

• Magnetic field from particle simulation (Pritchett, UCLA)

•Self generated out-of-plane field is whistler signature

Coupling to the kinetic Alfven wave: with a guide field

• Signature of kinetic Alfven wave is odd parity density perturbation

Kleva et al, 1995

Structure of plasma density

• Even parity with no guide field

• Odd parity with guide field– Kinetic Alfven

structure

Bz0=0

Bz0=1.0

Tanaka, 1996Pritchett, 2004

Fast versus slow reconnection

• Structure of the dissipation region– Out of plane current

No dispersive waves

With dispersive waves

Fast Reconnection in Large Systems•Large scale hybrid simulation

T= 160 -1

T= 220 -1

•Kinetic models yield Petschek-like open outflow configuration even in very large systems•Rate of reconnection insensitive to system size vi ~ 0.1 CA

•Many simulations in the literature are not large enough to enter the asymptotic regime confusion

Positron-Electron Reconnection

• No decoupling of the motion of the two species– No dispersive whistler waves– Displays Sweet-Parker structure– Reconnection rate in large systems?

Turbulent outflow jet in electron-positron reconnection

• Outflow jet goes unstable and becomes fully turbulent– Broadens outflow region to Petchek-like open outflow geometry

– Another mechanism for producing fast reconnection?• a laVishniac and Lazarian??

3-D Magnetic Reconnection

• Turbulence and anomalous resistivity– strong electron streaming near x-line leads to Buneman

instability and evolves into nonlinear state with strong localized parallel electric fields produced by “electron-holes” and lower hybrid waves

– resulting electron scattering produces anomalous resistivity that is sufficient to break the frozen-in condition

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

• fluctuations linked to current in layer

– Sawtooth crash in laboratory tokamaks• strong fluctuations peaked at the x-line

– Magnetic fluctuations in Magnetic Reconnection eXperiment (MRX)

• Particle simulations with up to 1.4 billion particles

• Bz=5.0 Bx, mi/me=100

• Development of strong current layer– Buneman instability evolves into electron holes

3-D Magnetic Reconnection: with guide field

y

x

Buneman Instability

• Electron-Ion two stream instability

• Electrostatic instability– (me/mi)1/3

pe

– k de ~ 1

– Vd ~ 1.8Vte

Initial Conditions:

Vd = 4.0 cA

Vte = 2.0 cA x

z

Ez

Formation of Electron holes

• Intense electron beam generates Buneman instability– nonlinear evolution into “electron holes”

• localized regions of intense positive potential and associated bipolar parallel electric field

x

z

Ez

B

Electron Drag

vx

vzElectron Distribution Functions

Scattered electrons

Accelerated electrons

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

€

∂p ez

∂t= −en0 Ez − e⟨˜ n ˜ E z⟩

€

Dz =c⟨˜ n ˜ E z⟩n0vAB0

• Drag Dz has complex spatial and temporal structure with positive and negative values

• Results not consistent with the quasilinear model

Electron drag due to scattering by parallel electric fields

y

x

Energetic electron production in nature

• The production of energetic electrons during magnetic reconnection has been widely inferred during solar flares and in the Earth’s magnetotail.– In solar flares up to 50% of the released magnetic energy appears in

the form of energetic electrons (Lin and Hudson, 1971)– Energetic electrons in the Earth’s magnetotail have been attributed to

magnetic reconnection (Terasawa and Nishida, 1976; Baker and Stone, 1976).

• The mechanism for the production of energetic electrons has remained a mystery– Plasma flows are typically limited to Alfven speed

• More efficient for ion rather than electron heating

Wind magnetotail observations

• Recent Wind spacecraft observations revealed that energetic electrons peak in the diffusion region (Oieroset, et al., 2002)– Energies measured up

to 300kev

– Power law distributions of energetic electrons

Requirements of a theoretical model

• Energetic electrons seem to be produced in the dissipation region near the x-line– Not coherent acceleration at a single x-line

• Electron energy exceeds the cross-tail potential drop

• Why power law distribution of energetic particles?– Fermi mechanism produces power law distributions at shocks– Does such a mechanism exist for electron heating during

reconnection?• Need repetitive interaction with magnetic neutral line(s)

– Turbulence seems unable to produce sufficient scattering– Trapping or scattering in magnetic islands can lead to repetitive acceleration– Power law energy distribution?

• Need to accelerate large numbers of electrons. How?– Solar observations

Electron acceleration

during reconnection

• Strongest bulk acceleration in low density cavities– Not at x-line!!– Pritchett 2004

• Length of density cavity increases with system size

• Maximum vparallel

increases with system size– Longer acceleration

region

vparallel

ne

Bz0=1.0

Structuring of the parallel electric field along separatrix: 2-D

• The parallel electric field remains non-zero in the low density cavities that parallel the magnetic separatrix– Drive strong parallel electron beams

• Strong electron beams break up Ep into localized structures– Electron holes and double layers– Most intense in density cavities

By=1.0

Electron-holes and double layers

• Structure of Ep along field line – Electron holes and double layers– Structures predominate in low

density cavity remote from the x-line

Electron heating

• Electron cooling in cavity accelerators– Well known from accelerator theory

• Cooling along direction of acceleration

• Overall large number of accelerated electrons• Strong acceleration within secondary island

– Multiple passes through acceleration region

Scaling of electron acceleration cavities

• Acceleration cavities are limited in length– Constraint from maximum in-plane electron currents and

associated magnetic fields

• Limit on the electron velocity in a single pass through the acceleration cavity

V|| = CAe(1 + )1/2

Production of energetic electrons

• Bz=1.0, three times in the simulation

• High energy tail from multiple interactions with x-line in secondary island

Location of energetic electrons

• Electrons with energy greater than 1.4mec2

• Most energetic electrons have multiple interactions with acceleration cavities

• Magnetic islands rather than turbulent scattering facilitates multiple interactions with acceleration regions

Electron acceleration in a secondary island

• Test particle acceleration in the secondary island is consistent with the large electron heating seen in the full simulation in this region– Island facilitates multiple

accelerations– Too much energy and the

particle escapes

Conclusions

• Fast reconnection requires either the coupling to dispersive waves at small scales or a mechanism for anomalous resistivity

• Coupling to dispersive waves– rate independent of the mechanism which breaks the frozen-in

condition– Open Petschek-like magnetic configuration– Supported by magnetospheric satellite observations

• Turbulence and anomalous resistivity– strong electron beams near the x-line drive Buneman instability– nonlinear evolution into “electron holes” and lower hybrid waves

• seen in the ionospheric and magnetospheric satellite measurements

Conclusions (cont.)

• Production of energetic electrons– Large scale density cavities that develop during reconnection with a

guide field become electron accelerators– All electrons crossing into these cavities undergo significant

acceleration• A multi-island reconnection picture will leader to acceleration of

essentially all electrons crossing magnetic separatrices– Requires many interacting islands and not a single large island

– Secondary islands facilitate multiple interactions of electrons with acceleration cavities and the production of very energetic electrons

• Explanation for observed powerlaw energy distributions remains an outstanding problem

Dispersive waves

• Geometry

• whistler

• kinetic Alfven

Api

y Ckc

k

= =

spi

y Ckc

k

= =

y0

0y kB

Bk ==

Parameter space for dispersive waves

• Parameters

1

1

y

none

whistlerwhistlerkinetic Alfven

kinetic Alfven

2y0y B/nT4π=

i

e2

y0

20

m

m

B

B)1( +=

•For sufficiently large guide field have slow reconnection

Rogers, et al, 2001

Dissipation mechanism• What balances Ep during guide field reconnection?

• In 2-D models non-gyrotropic pressure can balance Ep even with a strong guide field (Hesse, et al, 2002).

€

4π

ωpe2

dJz

dt= E z −

1

c(r v e ×

r B )z +

1

ne(∇ •

r r p e )z

Bz=0 Bz=1.0

y y

Satellite observations of electron

holes

• Magnetopause observations from the Polar

spacecraft (Cattell, et al.,

2002)