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MAGNETIC FIELD RECONNECTION FROM FIRST PRINCIPLES TO LATEST RESULTS

by Forrest Mozer

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Reconnection is the process that occurs when magnetized plasmas flow into each other. It produces

a. Change of topologyb. Particle acceleration

Reconnection occurs at the magnetopause, on the sun, on all scales in astrophysics (accretion disks, etc.) and in laboratory plasmas.

RECONNECTION

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QUESTIONS ABOUT MOVING FIELD LINES AND RECONNECTION

1. Why should one think about magnetic field lines that move?

2. What are the necessary conditions for field lines to move with ExB/B2?

3. Do magnetic field lines move with ExB/B2 in a vacuum, or is plasma needed to satisfy the frozen-in condition before field lines can move with ExB/B2?

4. If all field lines move with ExB/B2 everywhere, can there be reconnection?

5. Does the magnetic field line at point A move with the ExB/B2 velocity if the frozen-in condition is violated somewhere else along that field line?

6. What are the necessary conditions for being in the reconnection region?

7. The electron diffusion region is the place where reconnection occurs. Has any experiment seen the electron diffusion region?

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Consider

Two magnetic field lines at time t1

They move with ExB/B2 velocity

Ions and electrons move with ExB/B2

At later times B and plasma move to t2…

At t5, magnetic field lines reconnect

Plasma, B ejected vertically at later times

PLASMA AND FIELD LINE MOTION

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Consider

Two magnetic field lines at time t1

They move with ExB/B2 velocity

Ions and electrons move with ExB/B2

At later times B and plasma move to t2…

At t5, magnetic field lines reconnect

Plasma, B ejected vertically at later times

PLASMA AND FIELD LINE MOTION

WHAT IS WRONG WITH THIS CARTOON?

No perpendicular currents if ions and electrons move together

jperp ≠ 0 and jperp·Eperp > 0 on large scale

No reconnection if B lines move with ExB/B2 everywhere

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GEOMETRY AT TIME t6 IF FIELD LINES MOVE WITH ExB/B2

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THE GENERALIZED OHMS LAW

In two fluid theory, the equations of motion for a unit volume of plasma are:

Ions nimi(Ui/t+Ui·Ui) = niZe(E+UixB)/c-·Pi+Pie (1) Electrons neme(Ue/t+Ue·Ue) = -nee(E+UexB)/c-·Pe+Pei (2)

Pi, Pe= ion and electron pressure tensors Pie= momentum transferred between ions and electrons

Subtract (2) from (1) assuming • neglect of quadratic terms • electrical neutrality• ignore me/mi terms

Gives THE GENERALIZED OHM’S LAW

Equivalently, because j(c/ne) = Ui Ue

E+UexB = c·Pe/en + (mec2/ne2)j/t + j

E+UixB = cjxB/en c·Pe/en + (mec2/ne2)j/t + j

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FIELD LINE VELOCITY FROM FIRST PRINCIPLES

The task is to show the conditions under which field line motion with velocity ExB/B2 causes the magnetic field MAGNITUDE and DIRECTION to evolve in time in a manner consistent with Maxwell’s equations.

MAGNITUDE AT t+δt CONSISTENT WITH MAXWELL’S EQUATIONS Consider an infinitesmal surface in the x-y plane having B = BZ perpendicular to that surface.

Because ·B = 0, the number of field lines is conserved, so

δBZ/δt + ·(Bv) = 0 (equation of continuity) (1)

Because v = ExB/B2, the components of Bv are (Bv)X = EY and (Bv)Y = -EX. So ·(Bv) = δEY/δx – δEx/δy which is the z-component of xE. Thus, the conservation equation is just Faraday’s law. So,

without approximation and in the presence or absence of plasma, the magnitude of the magnetic field is always that expected from Maxwell’s equations if magnetic field lines move with the ExB/B2 velocity.

It is noted that any velocity v' satisfying ·(Bv') = 0 may be added to ExB/B2 without modifying equation 1. Thus, there are an infinite number of magnetic field line velocities that preserve the magnitude of the field.

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FIELD LINE VELOCITY FROM FIRST PRINCIPLES

DIRECTION AT t+δt CONSISTENT WITH MAXWELL’S EQUATIONS

Consider two surfaces, S1 and S2, that are perpendicular to the magnetic field at times t and t + δt. At time, t, a magnetic field line intersects the two surfaces at points a and b. Thus, the vector (b – a) is parallel to B(t). At the later time, t + δt, the points a and b have moved at velocities ExB/B2(a) and ExB/B2(b) to points a’ and b’. What are the constraints on these motions that cause (b’ - a’) to be parallel to B(a, t+δt), i.e., that give (b’ - a’) x B(a, t+δt) = 0?

(b’ − a’)/ε = B + B·(ExB/B2)δtAlso

B(a, t + δt) = B + (δB/δt) )δt + ((ExB/B2)·)BδtAfter taking the cross product and simplifying, one gets

B x (xE||) = 0

IF Ell = 0, ExB/B2 MOTION CAUSES THE FIELD TO EVOLVE IN A MANNER CONSISTENT WITH MAXWELL’S EQUATIONS

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CONCLUSIONS

1. A necessary condition is that Ell ≠ 0 in the magnetic field reconnection region.

2. From E+UexB = cpe/en + (mec2/ne2)j/t + j, the left side of this equation is non-zero because Ell ≠ 0, so

· Electrons do not move with the ExB velocity. i.e., this is the

“electron diffusion region.” “Electrons are demagnetized.” · A term on the right side of this Generalized Ohm’s Law must be non-zero to support the parallel electric field. WHICH TERM?

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SPATIAL SCALES OF RECONNECTION

DIFFERENT PHYSICS OCCURS ON DIFFERENT SPATIAL SCALES

• Ion scales c/ωpI ~ 100 km at the sub-solar magnetopause cjxB/en on right side of the Generalized Ohm’s Law becomes important to decouple

ion motion and to allow perpendicular currents. Because this term is perpendicular to B, Ell = 0 so magnetic field lines and electrons move with ExB/B2

• Electron scales c/ωpe ~ 2 km at the sub-solar magnetopause The remaining terms on the right side of the Generalized Ohm’s Law can become

important, so Ell can be non-zero and reconnection can occur.

• Debye scales λDebye ~ 0.1 km

Many large (~150 mV/m) fields seen on this scale. They are mostly perpendicular to B.

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COMPUTER SIMULATION OF RECONNECTION

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ION SCALES, ~ 100 -1000 KM

HALL MHD PHYSICS IS DUE TO ADDITION OF jXB term. IT ALLOWS FOR PERPENDICULAR CURRENTS AND POSITIVE jperp·Eperp ON LARGE SCALE, BUT IT DOES NOT ALLOW FOR MAGNETIC FIELD LINES TO RECONNECT.

THIS PHYSICS IS UNDERSTOOD FROM:– Computer simulations (the first prediction, eg., Shay, M.A., J.F. Drake, B.N. Rogers, and R.E. Denton J. Geophys. Res., 106, 3759, (2001))– Wind measurements (Oieroset et al, Nature (London), 412, 414, (2001))– Geotail measurements (Nagai, T. et al, J. Geophys. Res., 106, 25929, (2001))– Polar measurements (F.S. Mozer, S.D. Bale, T.D. Phan, Phys. Rev. Lett., 89, 015002, (2002))– Cluster measurements (Cluster separations allow exploring this scale with four spacecraft, as exemplified by recent publications by Vaivads, et al, Phys. Rev. Lett., 93(10), 105001 (2004), Runov et al, (2003), and Wygant, et al, in publication, (2004))– Recently observed in the MRX lab reconnection experiment

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POLAR OBSERVATION OF THE ION SCALE

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COMPARISON OF COMPUTER SIMULATION AND MAGNETOPAUSE DATA

Simulation

Polar Data

Density (#/cc)

Bz (nT): GSM Coords

Bx (nT)

By (nT)

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ELECTRON SCALES ~ 1-10 KM

OBSERVED ONLY BY ELECTRIC FIELD EXPERIMENTS ON THE POLAR AND CLUSTER SATELLITES

Scudder, J.D., F.S. Mozer, N.C. Maynard, and C.T. Russell, J. Geophys. Res., 107, 1294 (2002)

Mozer, F.S., S.D. Bale, T.D. Phan, J.A. Osborne, Phys. Rev. Lett., 91, 245002, (2003)

Appear in satellite data as ~100 msec large perpendicular and parallel electric fields. No observations exist of magnetic fields and plasmas on this time scale and no multiple spacecraft data exists.

The Polar electric field experiment has catalogued several hundred such events, so they are frequently observed.

NECESSARY CONDITIONS FOR THE ELECTRON DIFFUSION REGION 1. Ell ≠ 0 2. jperp·Eperp >> 0 3. Scale size ~ c/ωpe

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POLAR OBSERVATION OF THE ELECTRON DIFFUSION REGION

• Reconnection magnetic field changes in steps

• Current filamentary

• At largest filament, see 60 mV/m electric field

lasting for ~75 msec (width ~c/ωpe).

• Ell ~ 8 mV/m

• jperp·Eperp/n ~ 1 MeV per particle per second

• Major density change at this time.

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POLAR OBSERVATION OF THE ELECTRON DIFFUSION REGION

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POLAR OBSERVATION OF THE ELECTRON DIFFUSION REGION

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ELECTRON DIFFUSION REGION EVENTS NEAR THE SUB-SOLAR MAGNETOPAUSE, 2001-2003

-40

-30

-20

-10

0

10

20

30

40

8 9 10 11 12 13 14 15 16

MAGNETIC LOCAL TIME

MA

GN

ET

IC L

AT

ITU

DE

, d

egre

es

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FOUR-SATELLITE OBSERVATIONS OF ELECTRON DIFFUSION REGIONS

4 SC CORRELATIONS?DATE TIME E field Density B E SC1 E SC2 E SC3 E SC4 Re MLT LAT Comments

12/21/2003 7:14:04 20 50 10 40 12.55 17.1 25.4 Several events.12/21/2003 7:45:38 yes yes yes 40 90 30 100 12.95 17.2 23.2 1/3/2004 13:09:00 yes yes yes 20 20 30 25 12.14 16.5 27.4 Durations ~ 0.3 sec

1/23/2004 11:43:20 yes for 3 yes for 3 yes 20 15 25 40 10.38 15.3 38 B steps, EDR in 3 SC1/23/2004 11:53:33 yes yes for 3 yes for 3 25 40 50 25 10.54 15.3 36.9 Turbulent E field2/4/2004 1:00:00 ? no ? 70 100 10 4.8 1.3 -26 4.8 Re, near midnight

2/13/2004 19:32:10 yes yes no 15 20 25 15 8.38 12.7 52.53/6/2004 8:39:22 yes yes yes 15 20 10 10 11.57 12.3 28.8

3/24/2004 23:21:25 20 0 70 0 5.3 22.3 -41 5.3 Re, near midnight.4/4/2004 0:36:20 yes yes yes 15 120 15 20 13.86 10.1 15.4

4/25/2004 7:03:20 yes some yes <10 70 <10 30 11.34 9.1 28.2 Two good EDR events4/30/2004 1:49:23 yes yes yes 20 20 20 30 11.87 8.7 25 1/2 sec widths12/21/2004 0:18:20 yes yes some 10 30 10 25 13.2 20.8

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EXAMPLES OF ELECTRON DIFFUSION REGION CANDIDATES IN FOUR SATELLITE DATA

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ELECTRIC FIELDS IN GSE FROM THE FOUR CLUSTER SPACECRAFT

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THREE SECONDS OF EY, DENSITY, AND BY FROM FOUR SPACECRAFT

NOTES:

SINGLE POINT PEAKS OF EY

ΔEY OF 40, 90, 30, 70 mV/m

ΔE CORRELATES WITH Δn AND BY

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FOUR SPACECRAFT TIMING OF ELECTRIC FIELD PULSES AT 0745:38

ANALYSIS ASSUMES PLANAR, STATIC WAVEFRONT THAT PASSES OVER THE FOUR SPACECRAFT

• nX, nY, nZ = (0.9260, -0.3526, 0.1352)

• BOUNDARY SPEED = 179 km/sec

• NORMAL DISTANCE BETWEEN TWO MEASUREMENT POINTS < 1.8 c/ωpe

spacecraft distance, km time, secSC1-SC2 276 1.163SC1-SC3 246 1.141SC1-SC4 264 0.419SC2-SC3 275 -0.022SC2-SC4 244 -0.744SC3-SC4 269 -0.722Average 262

skin depths 65

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SPACECRAFT LOCATIONS IN THE PLANE ON 12/21/03 AT 0745:38

-250

-200

-150

-100

-50

0

50

100

150

200

250

-250 -200 -150 -100 -50 0 50 100 150 200 250

X, km

Y, k

m

ELECTRON DIFFUSION REGIONS ARE STABLE IN SPACE OVER HUNDREDS OF KILOMETERS AND TIMES OVER MANY SECONDS

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PHYSICS OF PLASMAS VOLUME 11, NUMBER 10 OCTOBER 2004

Three-dimensional simulations of magnetic reconnection in slab geometryM. Onofri, L. Primavera, F. Malara, and P. Veltri

CURRENT ISOSURFACES

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SUMMARY - ANSWERS TO QUESTIONS

1. Why should one think about magnetic field lines that move? To visualize the evolution of the magnetic field geometry with time.2. What are the necessary conditions for field lines to move with ExB/B2? Ell = 03. Do magnetic field lines move with ExB/B2 in a vacuum, or is plasma needed

to satisfy the frozen-in condition before field lines can move with ExB/B2? Magnetic field lines move with ExB/B2 in a vacuum if Ell = 04. If all field lines move with ExB/B2 everywhere, can there be reconnection? No5. Does the magnetic field line at point A move with the ExB/B2 velocity if the

frozen-in condition is violated somewhere else along that field line? Yes6. What are the necessary conditions for being in the reconnection region? Ell ≠ 0, jperp·Eperp large, spatial scale ~ c/ωpe

7. The electron diffusion region is the place where reconnection occurs. Has any experiment seen the electron diffusion region?

Yes, the Electric Field Instruments on Polar and Cluster have seen hundreds of them.

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DEBYE SCALE ~ 0.1-1 KM

FIRST OBSERVATIONS RECENTLY REPORTED FROM ELECTRIC FIELD MEASUREMENTS ON POLAR (Mozer, F.S., S.D. Bale, and J.D. Scudder, 31, doi:10.1029/2004GL020062, (2004)

1-10 MILLISECOND DURATION, >100 mV/m AMPLITUDE, ELECTRIC FIELDS

NO MAGNETIC FIELD OR PLASMA DATA ON THIS TIME SCALE

VERIFIED IN SIMULATIONS (Ma, Z.W., J. Huang, J.D. Scudder, F.S. Mozer, Paper SM51D-02, Fall AGU meeting, San Francisco, (2004)

POSSIBLE PRECURSER THAT ESTABLISHES CONDITIONS FOR RECONNECTION (Scudder, J.D., Z.W. MA, F.S. Mozer, Paper SM53B-0426, Fall AGU meeting, San Francisco, (2004)

~ HUNDREDS OF POLAR OBSERVATIONS MADE ALONG THE FIELD LINE CONNECTED TO THE RECONNECTION REGION.

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POLAR OBSERVATION OF DEBYE SCALE STRUCTURES

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DEBYE SCALE EVENTS

0

2

4

6

8

10

0 6 12 18 24

MAGNETIC LOCAL TIME

GE

OC

EN

TR

IC A

LT

ITU

DE

, R

e

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INVARIANT LATITUDE VERSUS MAGNETIC LOCAL TIME FOR DEBYE EVENTS

0

10

20

30

40

50

60

70

80

90

0 6 12 18 24

MAGNETIC LOCAL TIME

INV

AR

IAN

T L

AT

ITU

DE

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DEBYE STRUCTURES HAVING |Epar/Eperp| > 0

0

1

2

3

4

5

6

7

8

9

10

0 6 12 18 24

MAGNETIC LOCAL TIME

GE

OC

EN

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IC A

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DE

, R

e