Neeraj Jain 1 , Surja Sharma 2 University of Maryland, College Park, Maryland

1. Electron-magnetohydrodynamic (EMHD) Simulations of Collisionless Reconnection with Multiple X-points Neeraj Jain1, Surja Sharma2 University of Maryland, College Park, Maryland

2. Hesse et al., JGR 106, 3721(2001) Reconnection: Interplay of Scales • 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

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, 245002 (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. z Geometry and Profiles z y • 2-D geometry (/y = 0; x-z plane) x 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 Eigen functions for =1.0, kx=0.4 (for maximum growth rate) Growth rate vanishes beyond kx =1/  Purely growing mode

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. Lz • Simulation domain : 0 –Lz –Lx 0 Lx • Simulation parameters: • Initial Perturbations (many wavelengths along x and localized along z):

13. Perturbed energy evolution 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 =1

14. Spatial Structures and Eigen-modes 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. 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

15. Out of plane current

16. Outflow Velocity

17. Out of plane magnetic field

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

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 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 (.5-1.5 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 = .08-.4 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. v v e i d i d e (Hesse et al., JGR 2001) (Xiao et. Al., GRL 2007) Emerging Picture of Reconnection Region

30. wygant