1. Electron-magnetohydrodynamic (EMHD) Simulations of Collisionless Reconnection with Multiple X-points
Neeraj Jain1, Surja Sharma2
University of Maryland, College Park, Maryland
We are attempting to understand the dynamics of the electron current layer which is formed at the center of the reconnection region using electron MHD model. Present efforts are concentrated on 2-d dynamics of current layers. Today I shall present some of the preliminary results we have obtained. The results want for better understanding and I would appreciate inputs from you to understand them better.

2. Reconnection: Interplay of Scales
Hesse et al., JGR 106, 3721(2001)
Decoupling of electrons from ions at scale lengths (ion diffusion region)
Demagnetization of electrons at scale lengths (electron diffusion region)
Separation of scales (two scale structure )
Reconnection starts at electron scales and couples to ion scales
So let me start with a picture of reconnection in the magnetotail. The dipolar field lines of earth magnetic filed are stretched by the solar wind towards the night side to form a long tail. The tail extends beyond lunar orbit. The structure on the night side contains north lobe, south lobe plasma sheet and neutral sheets. The magnetic field in two lobes are in opposite directions and therefore need of current layer in the the filed reversal region. The opposite field lines reconnect at neutral sheet and form X-line. The region around the X-line where field lines reconnect is called diffusion region. Recent observation by cluster mission have shown that the width of the diffusion region is around one skin depth. These observations show the existance of hall currents alsoobservationBut near the earth at around 20 Re

3. Observations of electron scale structures
Cluster observation in tail at ~ 18 Re `
Bifurcated current sheet ~ 3-5 de each
[Wygant et al., JGR 110, A09206 (2005)]
Cluster observation of electron diffusion region near sub-solar magnetopause
Average width of individual electric field spike ~ 0.3 de
Total spatial structure ~ few de [Mozer et al., GRL 32, L24102 (2005)]
Polar observations of electron diffusion region at sub-solar magnetopause
Filamentary magnetopause currents with widths ~ several de
[Mozer et al., PRL 91, (2003)]
Cluster observations at magnetopause
Thin current layer ~ 20 Km ~ 5 de [Andre et al., GRL 31, L03803 (2004)]

4. Electron Dynamics in Reconnection
Reconnection starts at scales
Very initial phase is
electron dynamics dominated
Electron dynamics can change the structure
of the electron current layer before ion
dynamics is important
MMS aims to resolve structures at short electron scales.
Crucial questions:
a. Expected Structures?
b. Minimum scales to be resolved associated with these structures?
Studies at electron scales required.
We present simulations of electron dynamics dominated very initial phase of
Collision-less reconnection using Electron-MHD model.

5. Electron-Magnetohydrodynamics (EMHD)
Description of magnetohydrodynamic motion of electron fluid in the
presence of self consistent and external electric and magnetic fields.
Applicability:
Fast time scales phenomena (ci <<  <<  pe2 /  ce)
the upper limit ensures that displacement current can be ignored!
Short length scales (ce << k-1 <<  ci , di)
Unmagnetized stationary ions.

6. EMHD: Governing Equations
Curl of electron momentum equation:
Ampere’s law (displacement current ignored)
Normalization:
For kde << 1, mag. field is Frozen
in electron flow; electron inertia is
Important to break frozen-in condition
Electron flow is incompressible;
no density perturbations

7. Geometry and Profiles 2-D geometry (/y = 0; x-z plane)
n0=constant (reasonable approximation for thin embedded current sheet)
[Sergeev et al., JGR 98, 17345(1993)]
BC: Periodic in x while perturbations vanish at z-boundaries

8. Linear Eigen-mode Studies
Two coupled linear equations:
Local Dispersion relation:
(perturbation scale length is much smaller than equilibrium scale length)
Now since
The expression in the square root is positive for the chosen equilibrium
profiles and therefore local modes are stable.

9. Nonlocal Analysis-I Growth rate vanishes beyond kx =1/ 
Purely growing mode
Eigen functions for =1.0, kx=0.4
(for maximum growth rate)

10. Nonlocal Analysis-II Threshold can be obtained analytically.:
Eigen values  are either purely real or purely imaginary
 real and imaginary parts of  goes to zero simultaneously.
Putting =0 reduces the equations to,
which gives confined solution only when kx=1.

11. Nonlocal Analysis-III
Growth rate maximized over kx
decreases rapidly for large values
of  (weak electron inertia limit).
electron inertia is important
for the instability.
For  > 5, growth rate < .001
(beyond the validity of EMHD)
simulation studies confined to  < 5
The value of kx at which growth
rate is maximum also decreases
with 

12. 2-D Simulation
We present simulations for half width of current sheet =1.
For larger values of , results are qualitatively similar.
–Lx Lx
–Lz Lz
Simulation domain :
Simulation parameters:
Initial Perturbations (many wavelengths along x and localized along z):

13. Perturbed energy evolution
=1
Initial transient phase during
which eigen structures form ,
Linear phase during
which growth of perturbations is dominated by
the maximally growing mode.
Slope of dashed red line (maximum linear
growth rate) matches with the slope of solid blue
line (simulation growth rate).
Nonlinear Phase
during which mean profiles are modified by
nonlinear processes leading to slower growth
and finally to the saturation
instability saturates

14. Spatial Structures and Eigen-modes
Color: Normal component (Bz ) of magnetic field
Lines: Magnetic Field Lines
Arrows: Electron flow vectors
From simulations,
Wavelength along x
From linear theory,
growth rate maximizes at
Form and scale length of spatial structures developed is similar to those of linear eigen modes.
Reconnection starts at multiple locations corresponding to the wavelength of maximally
growing mode.

15. Out of plane current

16. Outflow Velocity

17. Out of plane magnetic field

18. Physical Mechanism z Q y P x R At point P: At point Q: At point R:
Faraday
Law
Thinning of current sheet
At point P:
At point Q:
At point R:
spreading of current sheet

19. Current sheet and Outflow
Intensification and thinning at X-point
Bifurcation and later filamentation inside islands
Maximum length of the reconnection region is
10 skin depth which reduces to 6 skin depth
Outflow jets close to max. whistler speed
Jets split and flow along field lines
Collision of jets from neighbouring reconnection
sites

20. Single Mode Evolution only one reconnection site
Bifurcation; No filamentation; Longer reconnection region 26 skin depth periodic boundaries
Secondary islands for t > 25; [Daughton et al(2006) and Karimabadi et al (2007)]
Outflow jets do not split.

21. Multiple Reconnection Sites Limit the length of Reconnection Region
Length limted due to following:
a. oppositely directed outflow jets from neighbouring X-points
b. stability of these jets
c. collision of these jets
Scaling with :
length independent of initial current layer width
max. ~ 10 skin depth; min ~ 6 skin depth

22. Filamentation of Out of plane Current
Filamentation due to the secondary instabilities for multi-mode evolution
(not seen in single mode evolution)

23. Modification to Quodrupole Structure

24. 3 X-points

25. Summary One Reconnection Site:
1. Basic tendency is to form elongated structure along outflow direction
2. length of the reconnection region increases with simulation box size.
3. Bifurcation of current in the outflow region. No filamentation.
Multiple Reconnection Sites:
1. Oppositely directed outflow from neighbouring reconnection sites
limit the length of the reconnection region to 6-10 de, independent of
simulation box size and 
2.filamentary structures between the bifurcated arms of the current
sheet.
3. Modification to quodrupole structure of out of plane magnetic field
due to neighbouring reconnection sites.

26. Reconnection Rate

27. Tail Parameters N = .08-.4 cm-3 Nav=.2 cm-3 Fpe=4 kHz Fpi=.1 kHz
The current sheet disruption region is localized < 1 Re and the disruption
Of current is also partial typically involving the 20% of the cross tail current.
After a period of sluggish growth ( hr) the cross tail current density
Exhibits rapid impulsive growth during a short interval (< 1 min) just before
The onset of the expansion phase. Following the impulsive growth phase
Current disrupts on a very short time scales (approx 10 seconds).
N = cm-3
Nav=.2 cm-3
Fpe=4 kHz
Fpi=.1 kHz
De=12.5 km
Di=500 km
B0=20 nT
Fce=56 Hz
Fci=.03 Hz
ce=2.56 km
ci=128 km

28. Average profiles
Average profile spreads in z due to which effective shear width increases.
The instability saturates when effective shear width becomes so large that even the minimum value of wavenumber (longest wavelength) present in the system is not able to satisfy the linear instability criteria
Minimum value of wavenumber present in the system is
The maximum unstable value of shear width is given by
Which matches with the one observed in the simulation.

29. Emerging Picture of Reconnection Regiond e v i
(Xiao et. Al., GRL 2007)
(Hesse et al., JGR 2001)
So let me start with a picture of reconnection in the magnetotail. The dipolar field lines of earth magnetic filed are stretched by the solar wind towards the night side to form a long tail. The tail extends beyond lunar orbit. The structure on the night side contains north lobe, south lobe plasma sheet and neutral sheets. The magnetic field in two lobes are in opposite directions and therefore need of current layer in the the filed reversal region. The opposite field lines reconnect at neutral sheet and form X-line. The region around the X-line where field lines reconnect is called diffusion region. Recent observation by cluster mission have shown that the width of the diffusion region is around one skin depth. These observations show the existance of hall currents alsoobservationBut near the earth at around 20 Re

30. wygant