A Fermi mechanism for electron acceleration during magnetic reconnection

J. F. Drake

University of Maryland and SSL

•M. Swisdak University of Maryland•H. Che University of Maryland•M.A. Shay University of Delaware

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– Energy releases from magnetars– 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– mechanism for fast release?– Why does reconnection occur as an explosion?

• Why does so much energy go into electrons?

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 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?

Energy Release from Squashed Bubble

• Magnetic field lines want to become round

magnetic tension

€

rF = −

r ∇(p +

B2

8π) +

14π

r B •

r ∇r B

Energy Release (cont.)

• Evaluate initial and final magnetic energies– use conservation law for ideal motion

• magnetic flux conserved

• area for nearly incompressible motion

RL

w

Wf ~ (w/L) Wi << Wi

•Most of the magnetic energy is released

Flow Generation

• Released magnetic energy is converted into plasma flow

π=ρ

8B

v21 2

2

2/12

A )4B

(vvπρ

≡≈

•Alfven time A is much shorter than observed energy release time

AA v/L=

Magnetic Reconnection

• Key features of this picture have been in space and laboratory observations

• Dissipation required to break field lines• Key issue is how newly reconnected field lines at very small

scales expand and release their tension

• d

Intense currents

Kivelson et al., 1995

Fast Flows at the

Magnetopause

Scurry et al. ‘94

Reconnection in Solar Flares

F. Shu, 1992

• X-class flare: ~ 100 sec.

• Alfven time:

• A L/cA ~ 10 sec.

=> Alfvenic Energy Release

RHESSI observations • Exploring timing of production of energetic electrons and

ions during flares

Jan 20, 2005 X7 flareKrucker/Hurford

Flares in high magnetic field neutron stars

Rhessi data: Hurley et al., 2005

• Magnetars: Isolated neutron stars with:– B ~ 1015 Gauss

– Strongest B-fields in universe.

• Giant Flare (SGR 1806-20)– Dec. 27, 2004, in our galaxy!

– Peak Luminosity: 1047 ergs/sec.

– Largest supernova: 4 x 1043 ergs/sec.

– Cause: Global crust failure and magnetic reconnection.

– Could be a source of short duration gamma ray bursts.

Resistive MHD Description

• Formation of macroscopic Sweet-Parker layer

•Slow reconnection not consistent with observations•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)

Hall Reconnection

• MHD model breaks down in the dissipation region at small spatial scales where electron and ion motion decouple

• Key is to understand how newly reconnected field lines expand at very small spatial scales where MHD no longer valid– The outflow from the x-line is driven by whistler and kinetic

Alfven waves dispersive waves– fast reconnection even for very large systems

• No ad hoc assumptions

• Key signatures of Hall reconnection have been measured by magnetospheric satellites and laboratory experiments

Hall versus MHD reconnection

– MHD model produces rates of energy release too slow to explain observations -- macroscopic nozzle a la Sweet-Parker

– Hall model produces fast reconnection as suggested by Petschek

Hall

MHD

Magnetic Reconnection Simulation

QuickTime™ and aBMP decompressor

are needed to see this picture.

Energetic electron production

• 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)• Why is the electron energy linked to the energy release?

– 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 spacecraft trajectory through the Earth’s magnetosphere

• d

Intense currents

Kivelson et al., 1995

Wind

Wind magnetotail observations

• 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

Electron acceleration by the reconnection electric field

• What is the structure of parallel electric fields during reconnection?

• Guide field reconnection produces deep density cavities that map the magnetic separatrix – Pritchett and Coroniti, 2004

• The parallel electric field is localized within these cavities– Cavities are microscopic in length

• Parallel electric fields are too spatially localized to be a significant source of large numbers of energetic electrons

E||

n

Failure of the single x-line model: sun

• Solar observations up to 50% of the energy can go into electrons– Parallel electric fields are

highly localized around the x-line

• Magnetic energy is not released at the x-line but downstream as the reconnected fields relax their stress

• X-line has negligible volume on the physical scale of the region where energy is released in the corona

• Can’t come close to explaining the large number of electrons gaining energy

Tsuneda 1997

Failure of the single x-line model: magnetosphere

• Energetic electrons should be accelerated by the electric field toward the dawn side of the magnetotail and energy would be limited to the potential drop across the tail (around 150 keV).

– Observations indicate are more equally spread– Energies in the meV range are sometimes observed

Energetic electrons in a cross section of the

magnetotail

• IMP 7 & 8 data (Meng et al 1981)

• Electrons with energy 220kev-2.5MeV

– Exceeds potential drop across the tail

• Dawn-dusk asymmetry stronger during quiet times than active times

– Not consistent with traditional cross tail acceleration.

• During active times must have a diffusive process for energy gain in the tail

– Must be able to gain energy while moving in either direction across the tail

Erec

Failure of the single x-line model: magnetosphere

• Energetic electrons produced by parallel electric fields should be highly localized around the x-line and adjacent separatrices

– Electrons are broadly distributed in observational data

• Electron velocities are dominantly moving parallel to B

– Nearly isotropic at high energy in the data

A multi-island acceleration model

• A single open x-line does not produce the energetic electrons observed in the data

• The development of multiple magnetic islands is expected from theory and simulations of reconnection

• Observations of secondary magnetic islands with magnetospheric satellites and solar observations of localized downflows also call into question a single x-line model

Generation of multiple magnetic islands

• Narrow current layers spawn multiple magnetic islands in guide field reconnection

• In 3-D magnetic islands will be volume filling

Cluster magnetotail reconnection event

• Fields are noisy with identifiable discrete magnetic islands

Eastwood et al, 2007

TRACE observations of downflow blobs

• Data from the April 21, 2002, X flare

• Interpreted as patchy reconnection from overlying reconnection site

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

A Fermi electron acceleration mechanism inside contracting islands

• Energy is released from newly reconnected field lines through contraction of the magnetic island

• Reflection of electrons from inflowing ends of islands yields an efficient acceleration mechanism for electrons even when the parallel electric field is zero

• When an ambient guide field is present, electrons can gain energy while moving either into or out of the page crucial for explaining the tail observations.

CAx

Electron Dynamics in simulation fields

• Electrons follow field lines and drift outwards due to EXB drift– Eventually exit the magnetic island

• Gain energy during each reflection from contracting island– Increase in the parallel velocity

• Electrons become demagnetized as they approach the x-line– Weak in-plane field and sharp directional change– Scattering from parallel to perpendicular velocity

• Sudden increase in Larmor radius• Isotropic distribution consistent with observations? Probably

Energy Gain

• Calculate energy gain through multiple reflections from the contracting island

– Note that rate of increase of energy is independent of the mass• Should the energy gain of ions and electrons be comparable?

– The bulk ions don’t have time to bounce– Only super Alfvenic ions gain energy with multiple bounces

– Particle simulations of reconnection miss this mechanism because the electron velocities because of artificial mass ratios are only marginally above the Alfven speed

CAx

dεdt

=2εGCAx

δx

G =G(Bx,Bz)

δ x

PIC Simulations of island contraction

• Separating electron heating due to the Fermi mechanism from heating due to E|| during reconnection is challenging– Study the contraction of an isolated, flattened flux bundle (m i/me=1836)– E|| =0

• Strong increase in T|| inside the bundle during contraction T|| ~ 3T

• 60% of released energy goes into electrons

T||

Multi-island reconnection

• Large energy gains require interaction with multiple magnetic islands energy gain linked to geometrical change of island aspect ratio

• Consider a reconnection region with multiple islands in 3-D with a stochastic magnetic field– Electrons can wander from island to island

• Stochastic region assumed to be macroscopic

uup

CAx

x

y

Kinetic equation for energetic particles

• Ensemble average over multiple islands

• Steady state transport equation for electrons

– Similar to Parker’s equation for particle heating in a 1-D shock

– Contains no velocity scale powerlaw solutions

– Missing feedback on energetic particles on the island contraction

dεdt

=2ε3

AdcAx

dy

r∇•

ruf −

r∇ • κ (v)

r∇f =

1

3A

dcAx

dy

∂

∂vvf

A =<Gi

δyi

δxi

>

Linking energy gain to magnetic energy released

• Basic conservation laws– Magnetic flux BW = const.– Area WL = const.– Electron action VL = const.

• Magnetic energy change with L

– Island contraction is how energy is released during reconnection

• Particle energy change with L

• Island contraction stops when

• Energetic electron energy rises until it is comparable to the released magnetic energy

L

w

WB =B2

4π

ΔL

L< 0

ε =−εL

L> 0

ε :

B2

4π⇒ βP : 1

Suppression of island contraction by energetic particle pressure

• Explore the impact of the initial on the contraction of an initially elongated island• With low initial island becomes round at late time• Increase in p|| during contraction acts to inhibit island contraction when the initial is

high contraction stops at firehose marginal stability

=0.3

=1.2

Kinetic equation with back-pressure

• Include the feedback of energetic particles on island contraction

– Energetic particles can stop island contraction through their large parallel pressure

• Steady state kinetic equation for electrons

• Can solve this equation numerically in reconnection geometry– Saturation of energetic particle production– Two key parameters:

• Initial plasma beta: 0=8πp0/B2

• Energy drive: A

r∇•

ruf −

r∇ • κ (v)

r∇f =

1

3A 1−

8πW

3B2

⎛

⎝⎜⎞

⎠⎟

1/2dcAx

dy

∂

∂vvf

v =cAx 1−8πW3B2

⎛

⎝⎜⎞

⎠⎟

1/2

Energetic electron spectra

• Powerlaw spectra at high energy

• The initial plasma beta, 0, controls the spectral index of energetic electrons

– For Wind magnetotail parameters where 0 ~ 0.16, v2f ~ E- 3.6

– For the solar corona where 0 is small, v2f ~ E-1.5

• Universal spectrum for low 0

• Results are insensitive to the drive A for strong drive

– Back pressure always reduces the net drive so that energy transfer to electrons is comparable to the released magnetic energy

Simulation geometry

The multi-island electron acceleration model explains many of the observations

• Magnetotail– Energy can exceed the cross-tail potential– Weak East-West asymmetry across the tail – Velocity distributions isotropic above a critical energy– Powerlaw energy distributions which match the Wind observations

• Soft spectra a consequence of the relatively large initial plasma pressure

• Solar corona– Large numbers of energetic electrons

• If island region is macroscopic

– Electron energy gain linked to the released magnetic energy– Powerlaw energy distributions consistent with the observations

• Harder limiting spectra of E-1.5 a result of the low initial plasma pressure

Critical issues in explaining the solar observations

• The electron numbers problem– The contracting island

region must be macroscopic

• energetic electrons gain a large fraction of the magnetic energy released

Island region

Can a similar Fermi process produce energetic ions?

• The Fermi mechanism if efficient only for ions with velocities above the Alfven speed

• Need a mechanism producing a seed distribution of energetic ions

• Observational evidence in the heliosphere of E-1.5 spectra of protons

10-11

10-9

10-7

10-5

10-3

10-1

101

103

1 10 100

tail retail 2:40:15 PM 1/22/06

*FWtail*tail+SWSW distribution1FWFW<FW>meanFWnetFWbkgsum core+tail quietFWPI upFW -26day to TS LECPFW -20day to TS LECPFW -26day to TS LECPFW -20day to TS LECPFW(Vr broadened)Tail with cutoff

Phase Space Density (s

3 /km

6 )

W Ion Speed/Solar Wind Speed

f(w) = fow -5

(in solar wind frame)

H+

quiet timetails

Solar wind protons

<R> = 4.86 AU

SWICS

ULEIS

1 AU

4.23 AU

94 AU

Core pickup protons

Common in the quiet solar wind (Gloeckler et al, 2006)Similarity to spectra from the Fermi mechanism is striking

Proton spectra of the form j = jo E -1.5 or equivalently f = fov -5 are often observed

Conclusions

• Acceleration of high energy electrons is controlled by a Fermi process within contracting magnetic islands

• Reconnection in systems with a guide field involves the interaction of many islands over a volume– Remains a hypothesis based on our 2-D understanding

• Averaging over these islands leads to a kinetic equation describing the production of energetic electrons that has similarities to that in particle acceleration in shocks

• Particle distributions of energetic electrons take the form of powerlaws– The initial electron pressure dominantly controls the spectral indices of the

energy distributions• Low initial pressure as in the solar corona yields harder spectra than in the

magnetosphere

• Electrons gain a substantial fraction of the energy released during magnetic reconnection

The MHD Reconnection Rate Problem

• Reconnection rates too slow to explain observations– solar and stellar flares– sawtooth crash in fusion experiments– Storms in the Earth’s magnetosphere

• Ongoing scientific issue since the late 1950’s• The solution: non-MHD physics at the small spatial scales

drives fast reconnection– The one-fluid MHD model breaks down in the narrow boundary layers

that develop during magnetic reconnection– The motion of electrons and ions in the narrow boundary layers where

magnetic field lines break decouples Hall reconnection• New class of “dispersive” waves facilitates fast reconnection• Physics is confirmed in magnetospheric satellite observations and in

laboratory reconnection experiments.

10-3

10-1

101

103

105

107

109

1 10

1H1t|w0.89-1.11|M/Q0.90-1.10|ccM21H1d corrected for He++*1H1t*ccM21H1d corrected for He++ccM21H1d(saturation @W~1&spill)5.26^2*ccM21H1d(saturation @W~1&spill)H+peak corrected*H+peak correctedFWH+ACEtail*ULStailFWH+_ULEISF(

W) Phase Space Density (s

3/km

6)

W (Proton speed/Solar Wind speed)

1997.108-1999.1085.26 AU

F(W) = FoW

–5

(x27.7)

H+

wth

=.028

kappa=7

vH= 406. vHe= 416. VthHe= 17.1 AU= 5.26 SPE= 7.46 HLat= -7.09

SW_density(1/cm^3) Vth(km/s) Vkappa 0.1575 18.0 3.90000

2 4 7

SWICS Ulysses

SuprathermalPickup Ion Tail

Pickup Ion Core

SolarWind

Krimigis et al., AGU (Fall 2003)

CRS

CRS

10-5

10-4

10-3

10-2

10-1

100

101

102

1 2 3 4 5 6 7 8 9

1996.180-2000.180 corrected for Vsw

FW corr <=1<FW> cnts(W>2.39, <=1FW corr <=400sum core+tail baselinesum core+tail quiet412.3M21H1d|w2.00-2.39|M/Q1.00-1.00|*FWcoreFW corr <=1FW corr <=400

Phase Space Density (s

3 /km

6 )

W Ion Speed/Solar Wind Speed

f(w) = fow-5

SWICS Ulysses

H +

quiet timetail1996.5-2000.5

Pickup H +

core

j = joE-1.5

Decker et al., Science (2005)

Voyager 1 LECP2004:352-2005:144

ACR

Accelerated Pickup Ions

Quiet-time tails of the form j = jo E -1.5 or equivalently f = fov -5 are often observed

~45 AU ~85 AU

~5 AU

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

Positron-Electron Reconnection• No decoupling of the motion of the two species

– No dispersive whistler waves

• Displays Sweet-Parker structure but reconnection rate is high (Hesse, Bessho and Bhattacharjee).

• Scaling of reconnection rate to large systems?

Why is reconnection explosive?• Slow Sweet-Parker reconnection and fast Hall reconnection are

valid solutions for the same parameters

• Sweet-Parker solution does not exist below a critical resistivity For the solar corona the critical temperature is around 100 eV and the

reconnection rate will jump a factor of 105

Cassak et al2005

Ez

δ

Particle Scattering

• Increase of v|| within island

• Nearly constant vL within island

• Scattering from v|| to vL near the separatrix

• Isotropic particle distributions at high energy?

Powerlaw spectra

• Solve the kinetic equation in reconnection geometry – Fermi drive balances convective loss

• Powerlaw spectra -- as often seen in both solar and magnetospheric observations

• The energy integral diverges– Spectral index depends on the ratio of the aspect ratio of the island region (~0.1) to the

mean aspect ratio of individual islands. – In the strongly driven regime, < 3, the energy content of energetic electrons diverges

• Energy budget of electrons is important• Feedback of the energetic component on the reconnection process must be calculated

f (v) = −1v dv'

0

v

∫ fup(v')v'−1

=1 +3Δy

Δx A= 1 +

3Δy

Δx < Gi

δ yi

δ xi

>