The Structure of the Parallel Electric Field and Particle Acceleration During Magnetic

Reconnection

• J. F. Drake• M.Swisdak• M. Shay

• M. Hesse

• C. Cattell

University of Maryland

GSFC

University of Minnesota

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

Overview• MHD model yields reconnection rates to low to explain

observations in nature– solar flares– sawtooth crash– magnetospheric substorms

• Non-MHD physics at small spatial scales produces fast reconnection– Focus on reconnection with a guide field and the coupling to dispersive

kinetic Alfven waves– Structure of the parallel electric field

• 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– Parallel electric field develops structure in the form of electron holes and

double layers– Observational evidence in from recent Cluster data

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

Coupling of reconnection to the kinetic Alfven wave

• 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

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

• Can calculate parameters for which whistler versus kinetic Alfven waves dominate dynamics or have no dispersive waves

No dispersive waves

With dispersive waves

Rogers et al, 2001

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

Dissipation with a guide field

• The breaking of the frozen-in condition and electron energization occurs in locations where the parallel electric field Ep is non-zero– Largest near x-line and

along low density side of separatrix

Ep

Transverse structure of Ep

• Transverse width scales with electron Larmor radius e

– Much smaller than had been believed

– Earlier theory was the ion scale s

– Why?

mi/me=400

mi/me=100

Structure of Ep

• The structure of Ep is controlled by the development of an electrostatic field that shorts out the parallel inductive electric field

• Earlier theories predicted the transverse scale s

– incorrect because presumed ions magnetized– For small transverse scales ions are unmagnetized

– Transverse scale is es

€

E p = −1

c

∂Az

∂t−

r b • ∇ϕ

€

(ρ e 02 ∇ 2 +1)

v b • ∇ϕ = −

T0

Te

˙ A zc

−T0

n0ed3v∫

v b • ∇ge (

r v )

€

1

T0

=1

Te

+1

Ti

€

e02 = ρ e

2 T0

Te

Dissipation mechanism• What balances Ep during guide field reconnection?

• Scaling with electron Larmor scale suggests the non-gyrotropic pressure can balance Ep (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=1.0

y

Electron acceleration

• Strongest bulk acceleration in low density cavities where Ep is non-zero– 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

Electron heating

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

• Cooling along direction of acceleration

• Strong heating along high density side of separatrix– Beams are injected into x-line from cavity accelerator– Scattered into outflow along high density separatrix

Electron distribution functions

• Cold energetic beam in cavity

• Hot streaming plasma ejected along high density separatrix

cavity

Outflow separatrix

• Bz=5.0 Bx, mi/me=100

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

Structuring of the Parallel Electric Field during 3-D Magnetic Reconnection with guide

field

y

x

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

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

• How far along separatrix does turbulence spread?

Electron drag due to scattering by parallel electric fields

y

x

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

• Strong electron beams break up Ep into localized structures– Electron holes and double layers– Most intense in density cavities– In 2-D not near x-line because beams in out-of-plane direction

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

Fine structure of holes and double layers

• Electron-holes correspond to depletions of the electron density– Modest enhancements of the local ion density– Hole velocities much slower than electron beam velocity

• Consistent with smaller scale 3-D simulations• Consistent with Buneman source

• Double layers correspond to bipolar charged layers

Ep

ne

Cluster observations

• Consistent with those on geotail and polar– Electron holes

– Double layers

• First direct measurement of consistency with electron beams

Cattell, et al, 2004

Cluster: electron velocity distributions

• Electron holes seen only with distinct, relatively cold beam feature– Only in top two

panels

– No electron holes seen in the four bottom panels

Modeling of Cluster data

• A small guide field (By~0.2Bx) is required to reproduce the beams and double layers in Cluster data -- holes?

Structure of parallel electric fields in Nature

• 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

• Observations and theory now suggest that such structures form in essentially all of the boundary layers present in the Earth’s magnetosphere where reconnection takes place– Magnetotail, magnetopause

• More generally, evidence is now strong that uniform, large-scale parallel electric fields do not exist in Nature– Dissipation always take place in localized non-linear structures– Consequences for understanding particle heating?

Conclusions

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

• The kinetic Alfven wave dominates in cases with a large guide field

• The transverse scale length of the region where the parallel electric field is non-zero is much smaller than thought previously– Scales with an effective electron Larmor radius

Conclusions (cont.)

• Non-gyrotropic pressure continues to balance the parallel electric field near the x-line (Hesse, et al, 2002)– Consequence of the small spatial scales

• Electron acceleration and heating is controlled by a spatially non-local region of depleted electron density (Tanaka, 1997; Pritchett and Coroniti, 2004)– Low density cavity is a region of extended non-zero

parallel electric field– An efficient plasma accelerator

– Electron heating results from the injection of energetic electrons from this cavity into the x-line

Conclusions (cont.)

• Strong electron beams in the low density cavity produce electron-holes and double layers even in a 2-D simulation

• Electron beams are seen in the magnetotail by Cluster when and only when electron holes and double layers are observed (Cattell, et al, 2004).– Consistent with theory

– Consistent with earlier auroral observations

– Large-scale parallel electric fields are unlikely to exist in Nature.