Magnetic Structures in Electron-scale Reconnection Domain

Ilan Roth

Space Sciences

UC Berkeley, CA

Thanks: Forrest Mozer

Phil Pritchett

Dynamical Processes in Space Plasmas

Eyn Bokkek, Israel, 10-17 April 2010

Fundamental plasma processes with global

implications may occur in a narrow layer

Magnetic Reconnection

Magnetic shears - electron dominated region

What can we learn about

electron scale structures without (full) simulations?

Symmetric Configuration à la texbook cartoon

Classical Symmetric Crossing à la observations- Mozer, 2002

Hall

Hall

reconnect

reconnect

Collective plasma scales determine the different (nested) layers:

Outer: - Hall effect – ions decouple from B

Intermediate: e- inertia (pressure)

Inner: break(s) the e-

Innermost: frozen-in condition

Main purpose: assessing the non ideal effects of Ohms Law

Environment: electron (current) velocity >> mass velocity

ee cd /

ii cd /

eie dmm )/(eieie dmmdd 2/12 )/(/

4)

8(

2 )B(BBJv

Bp

cp

dt

d

Two Fluid: coupling (B,v)

“Ion” fluid

Electron fluid

cne

p

necdt

d

e

m eee BvEj

BJv

Sheared field, Inhomogeneous Plasma

General coupling between

Shear Alfven

Compressional Alfven

Slow Magneto-Acoustic

modified on short scales by

(mainly) electron effects

Two (extreme) approaches

• Lowest approximation of the electron dynamics + follow ion dynamics

• Lowest approximation of the ion dynamics + follow electron dynamics

A. Faraday and Ohm’s law couple

magnetic and velocity fields

MHD:

Magnetic field is frozen in the fluid drift

}{ BVB t

jΒVE c/

EB ct

Magnetic field – fictitious diagram of lines in R3

satisfying specific rules. MHD – approximate description of magnetic field motion in a plasma fluid.

Knot - closed loop of a non-self-intersecting curve, transformed via continuous deformation of R3 upon itself, following laws of knot topology - pushed smoothly in the surrounding viscous fluid, without intersecting itself (stretching or bending).

MHD field evolves as a topological transformation of a knot. MHD dynamics forms equivalent knot configurations with a set of knot invariants.

All KNOT deformations can be reduced to a sequence of Reidemeister “moves”: (I) twist (II) poke , and (III) slide.

Type 1 Type 2

Type 3

Knot topology described through knot diagrams

Reidemeister moves

Reidemeister moves preserve several invariants of the knot or link represented by their diagram - topological information.

MHD invariants: (cross) helicity, generalized vorticity, Ertel,…

Every knot can be uniquely decomposed as a knot sum of prime knots, which cannot themselves be further decomposed - Schubert (1949)

Prime knots

Characterization based on crossing number – Tait 1877

Flux-rope is a KNOT

MHD Turbulence forms a LINK- Collection of knots

Reconnection is NOT a KNOT: it forms a KNOT SUM

HELIOSPHERE

MHD (KNOT) can be broken via several physical processes

Various physical regions

Reconnection: topological transition

Diffusion: violation of frozen–in condition

Dissipation: conversion of em energy

(no consensus on definitions)

Parallel electric field is observed in tandem with density gradients

Localized electric field over

scale ≤ de=c/ωe – electron inertia effect?

Mozer +, 2005

Electron diffusion region: 0 BvE e

filamentary currents on scale ≤ de=c/ωe – dissipation region due to electron inertia effect?

ELECTRON PHYSICS COVERS LARGE SPATIAL SCALES.

Asymmetric Simulation – Pritchett, 2009

Violation of electron frozen-in condition

Elongated Electron Diffusion regions

?0/ vB mce

Magnetic field vs Electron Vorticity vs dissipation

B. Faraday and Ohm’s law couple

magnetic and velocity fields

eMHD:

Generalized vorticity field is frozen in

the electron fluid drift

jpnm

udt

d

e

mBu

cE e

11

}{EB t

uBGGuG

)/(};{ emct

vorticity

eMHD: Electron fluid:

}{ GuG t

MHD: “Ion” fluid

}{ BVB t

jBBB

cddt ee })1(){(]1[ 2222

coeffHallenc

une

4/

BuJ

Electron inertia Hall ee cd /

Homogeneous, incompressible electron fluid

BuBG )1()/( 22 edemc

Magnetic field slips with respect to the electron fluid

Generalized vorticity G is frozen in the electron drift u

Inhomogeneous electron fluid

Bnxn

Bxen

cu

Bc

uxenJ

oo

o

/)(]

)(4[

4)(

pe

c

t

)/1(})/{( GBG

ee cd /

ln))(/(])/(1[ 222 BBG ee dd

nxnx o /)()(

)1(

)()(

22edk

kBkk

Linear homogeneous

infinite plasma waves

Whistler branch

Generalized Vorticity – Inhomogeneous fluid

A. Incompressible Homogeneous Plasma; [n(x)=no]

zzyyo eBexBxB )()('; yoo BUUenJ

uenjlinearized o:

zyxe

xyxyze

bxGkbdx

dkd

bxBbdx

dkxBb

dx

dkd

)()](1[

)())(()](1[

22

222

''2

22

2

222

Electron inertia effect is manifested on the small spatial scale

Inclusion of ion dynamics in the limit

zyxe

xyxyze

bxBkbdx

dkd

bxBbdx

dkxBb

dx

dkd

)()](1[

)())(()](1[

22

222

''2

22

2

222

eMHD limit:

yzixxe

x"

xyi

ze

BbkdkVbdγτ

bBbBγτ

db)ετ/γ(d

y

][1[

][]11[

222

222222

1)/( 222 sc

Coupling of shear Alfven and compressional Alfven

im

Mirnov+, 2004

2/12/11 ~/;~~ iiiia mcdmLV

Eigenmodes: two components of the magnetic field

Unstable mode in a whistler regime

Califano, 1999

By=tanh(x/L)

de/L=1

bx

bz

'2222

''222

22

~)(]1[

)(])1(1[

Bn

ndikbxGkbd

bBbxBbd

ezyxe

xyxyze

ce

Increase in the effective electron skin depth

Compressibility - “Guiding” field: enhance the electron inertia effect

B. Compressible Homogeneous Plasma

nnnUnunejlinearized oo~;)~(:

C. Inhomogeneous, compressible plasma

zzyyo eBexBxB )()(

')](/[)();()()( yBx

oxUxUx

oenx

oJ

nxnx oo /)()(

o

yezyxeo

ozx

xyxyze

ceo

B

n

ndikbxGkbdx

dxxdnkbiiUbd

bBbBbdx

'2222

2

''222

22

~)(])([

)/)()()(/1(

])1()([

Density dips enhance the electron inertia effect

D. Inhomogeneous, compressible plasma – generalized configuration

zzyyo exBexBxB )()()(

')](/[)(;')](/[)(z

Bxo

xUyBx

oxU yz

nnnxnx oo /~~;/)()(

zyxeo

xyxyzozzee

ceo

bxGkbd

bBbBbBBdx

ddd

)(][

})]/1(1[{

222

''22'22

22

3D structure may enhance the electron inertia effect

E. Kinetic, incompressible, inhomogeneous plasma

xyexI

Rz

Ithe

zI

Rexex

I

IR

th

bdbZ

Zb

Zkv

k

dx

dd

bZ

Zkdbdb

Z

ZZ

kv

''22

2

22

222222

)])/(

1(1[

Attico +, 2002

SUMMARY

A. MHD satisfies the axioms of knot theory – both evolve preserving various invariants. Knot sum is equivalent to violation of frozen-in condition.

B. Density gradients/dips, compressibility, and thermal effects may have a significant effect on the electron vorticity, which determines the slipping of the magnetic field with respect to the electrons. These effects modify the structure of the magnetic field on the short-scale, forming current filaments, parallel electric fields, which violate the frozen-in condition and contribute to electron heating. These regions are ubiquitous and are observed outside of the x-points in the reconnection domain.