Collisionless Magnetic Reconnection

J. F. Drake

University of Marylandpresented in honor of

Professor Eric Priest September 8, 2003

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)

Singular magnetic island equilibria

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

– Formation of singular current sheet

• Equilibria which form as a consequence of reconnection are singular – Waelbroeck’s ribbon is generic– Sweet-Parker current layers reflect this underlying singularity

• Generic consequence of flux conservation and requirement that magnetic energy is reduced

€

≠0

Limitations of the MHD Model

• 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– Results seem to scale to large systems

• 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

•not Alfven waves•Electron frozen-in condition broken below c/pe

c/pic/pes scales

Jpne1

BJnec1

Bvc1

EdtJd4

ei2pe

rrrrrrr

−∇+×−×+=π

Electroninertia

whistlerwaves

kinetic Alfvenwaves

Kinetic Reconnection

• 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 phase 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• MHD reconnection is too slow by orders of magnitude

Birn, et al., 2001

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

:

Whistler signature

• Magnetic field from particle simulation (Pritchett, UCLA)

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

Observational Support for Whistler Wave Role in Reconnection

• Recent encounter of Wind spacecraft with reconnection site in the Earth’s magnetotail (Oeierset, et al., 2001)

Magnetic Field Data from Wind

• Out-of-plane

magnetic fields seen as expected from standing whistler

Conditions for 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

Fast versus slow reconnection

• Structure of the dissipation region with strong guide field– 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•No large scale nozzle in kinetic reconnection

•Rate of reconnection insensitive to system size vi ~ 0.1 CA

3-D Magnetic Reconnection

• Turbulence, anomalous resistivity and energetic particle production– self-generated gradients in pressure and current near x-line and slow shocks

may drive turbulence– particle energization from turbulent particle acceleration?

• In a system with anti-parallel magnetic fields secondary instabilities play only a minor role– current layer near x-line is completely stable

• Strong secondary instabilities in systems with a guide field – 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 strong anomalous resistivity and electron heating

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 simulation with 670 million 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 Holes

• Localized region of positive potential in three space dimensions– ion and electron dynamics essential– dynamic structures (spontaneously form, grow and die)– Parallel electric fields in collisionless plasma are not uniformly

distributed along B• Dissipation occurs in 3-D localized structures

B

Electron Energization

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

Observational evidence

• Electron holes and double layers have long been observed in the auroral region of the ionosphere– Temerin, et al. 1982, Mozer, et al. 1997– Auroral dynamics are not linked to magnetic

reconnection

• Recent observations suggest that such structures form in essentially all of the boundary layers present in the Earth’s magnetosphere– magnetotail, bow shock, magnetopause

• Electric field measurements from the Polar spacecraft indicate that electron-holes are always present at the magnetopause (Cattell, et al. 2002)

• d

Intense currents

Kivelson et al., 1995

Satellite observations of electron

holes

• Magnetopause observations from the Polar

spacecraft (Cattell, et al.,

2002)

Theory/observational comparison of E ||

Polar observations(data exhibits both hole

polarities)

Simulation data

z

Scaling of electron holes

• Scale size Lh

• Velocity Vh

€

⇒ Lh = 2πVh

ω pe

mi

me

⎛ ⎝ ⎜

⎞ ⎠ ⎟

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E-hole scaling: comparison with Polar data

• Polar data from 5 magnetopause crossings– Direct measurements of Vh and Lh

– Compare with theory prediction• Reasonable agreement

• High velocity holes have larger scale lengths

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

– Can have fast reconnection with a guide field

• 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

• Generic mechanism for dissipating wave and magnetic field energy in nearly collisionless plasma systems?

– Can MHD turbulence produce fast reconnection?

Collaborators

• M.A. Shay• M. Swisdak• B. D. Jemella

• B. N. Rogers

• A. Zeiler

• C. Cattell

University of Maryland

Dartmouth College

IPP-Garching

University of Minnesota

′0004.0=

-Elongated current layer increases in length with tearing mode stability parameter .′

Structure of Current layer in resistive MHD

0.92

1.32

4.92

8.11

20.93

3.13

IwLLy =Area Conservation

Flux Conservation

IR LLL +=

Length of Ribbon

Scaling of Current Ribbon

w

y

LR LI

L

t = 0

t > 0

• Release of magnetic energy implies ribbon formation• Waelbroeck’s equilibrium theory more important than was recognized earlier

€

LR = L 1− U f /U i( )

€

Biy = B f w

Whistler Driven Reconnection

• 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

Transverse electric field

• Transverse electric field takes the form of a wake– Remains in phase with the hole– A nonlinear, current-driven lower hybrid wave

• Controls transverse structure• k|| << k

• more nonlocal than E|| in the direction of B

x

z

Ex

Scaling of electron holes

• Scale size

• Velocity

• The electron drift speed Vd is unknown – Bursts of holes from polar last only around 0.1s

– Insufficient time to measure the electron distribution functions

• Eliminate Vd

€

Lh = 2πVd / ωpe

€

Vh = (me / mi)1/ 3Vd

€

⇒ Lh = 2πVh

ω pe

mi

me

⎛ ⎝ ⎜

⎞ ⎠ ⎟

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