Second (2004) Workshop on thin current sheets. Ion- or electron-dominated thin current sheets

Transcription

1 Second (4) Workshop on thin current sheets Ion- or electron-dominated thin current sheets P. H. Yoon (Univ. Maryland) Outline of the Talk Harris versus non-harris Equilibrium Modified-Two-Stream versus Lower-Hybrid-Drift (MTSI vs LHDI) Instabilities Implication for Current Sheet Dynamics /Reconnection Onset

2 D vs 3D Reconnection D Reconnection Harris equilibrium with initial perturbation GEM reconnection problem Harris equilibrium with noise Low-amplitude saturation of tearing mode

3 3D Reconnection z z Jye Jye Jyi LHDI x Jyi y x y z z x y x y Patchy Reconnection D Reconnection 3

4 D reconnection in Harris current sheet without initial perturbation only leads to very small magnetic island formation, or it takes a long time before any sizable islands form (see Daughton s presentation). In 3D, the excitation of LHDI near the current sheet edge leads to thin electron current layer formation and increased temperature anisotropy, thus promoting the rapid onset of tearing instability (see Daughton s presentation. Also Shinohara s 3D simulation presented at 4 Fall AGU). The result may be a patchy reconnection with neutral sheets scattered in equitorial plane. LHDI excitation may be followed by longer time-scale, long wavelength kink (flapping) perturbation Kelvin-Helmholtz instability seen in yz plane D simulation, but contemporary 3D simulations are not capable of faithfully resolving both LHDI and the flapping motion. Over longer time scale, patchy reconnection may eventually evolve into quasi D structure. GEM Reconnection Challenge type of approach is to bypass the complicated 3D process and put the system directly in quasi-d stage. 4

5 Problem with Harris model: Although 3D reconnection mediated by LHDI leads to much faster reconnection than D, the process may still not be fast enough to explain explosive release of energy. LHDI operates near the outer current sheet edge, thus cannot explain EM fluctuations seen in the neutral sheet during MRX experiment. Since LHDI is located outside, it cannot be the source of anomalous resistivity either, e.g., MRX. Since LHDI is seen to lead to narrow electron current layer in the late stage of simulation, it is highly desirable to model such a current layer embedded within thicker ion plasma sheet as a quasi-equilibrium problem. 5

6 . Harris versus non-harris Equilibrium Particle distributions Harris model: F j = n π 3/ α 3 j ( δe v /α j +( δ) e (v V j) /α j sech z L Particle distributions non-harris model: F i = n [Ψ(z)] τ/(+τ) π 3/ α 3 i ( δe v /α i +( δ) e (v V i) /α i sech µ z L F e = n π 3/ α 3 e[ψ(z)] /(+τ) ( δe v /α e +( δ) e (v V e) /α e sech ν z L µ = ( + τ)u +U Ψ(z) = δ +( δ) sechν (z/l) δ +( δ) sech µ (z/l), ν = ( + τ) ( + U)τ, τ = T e T i, U = V i V e Note that Harris model is recovered if U =/τ or V i /T i = V e /T e. 6

7 To briefly summarize, our non-harris model is constructed on the basis of the usual canonical mometum and total Hamiltonian, and is an isotropic model. Thus, it is essentially the same as Harris model, except that in the classical Harris solution V i /V e and T i /T e are not independent parameters, but they are related to each other. The deviation in our model is to treat these two as independent. As a result, our current sheet model is slightly charged. Among the non-dimensional parameters, U = V i / V e and τ = T e /T i are the most important. U =/τ: Harris U =/τ +, where > : Ion-dominated non-harris U =/τ, where < /τ: Electron-dominated non- Harris 7

8 Harris Fi Harris Fe Z Vy - - Z Vy - non-harris Fi non-harris Fe Z 4 Vy - - Z Vy - Ion-dominated current sheet non-harris Fi non-harris Fe Z Vy - - Z Vy - Electron-dominated current sheet 8

9 Harris non-harris B x (z) =B tanh(z/l) φ(z) =, eφ(z) T i = τ +τ ln Ψ(z) n(z) n = δ +( δ) sech z L n(z) n = [Ψ(z)] τ/(+τ) δ +( δ) sech µ z L v y i V i = v y e V e = ( δ) sech (z/l) δ +( δ) sech (z/l) v y i V e = U ( δ) sechµ (z/l) δ +( δ) sech µ (z/l) v y e V e = ( δ) sechν (z/l) δ +( δ) sech ν (z/l) 9

10 δ =.4, τ =, U = /τ (Harris), U = 4 (non-harris) n(z).5 Harris n(z) ion-dominated non-harris.5 Φ <v> i <v> e.5 - z/l <v> i - z/l <v> e

11 δ =.4, τ =, U = /τ (Harris), U =.5 (non-harris) n(z).5 Harris n(z) el-dominated non-harris.5 Φ <v> i <v> e.5 - z/l <v> e.5 <v> i - z/l

12 There are other models for embedded electron current layer, examples being: Sitnov s forced current sheet model [JGR, 5, 39, ; GRL, 3, 3, 7,.9/3GL78, 3], Schindler and Birn s non-gaussian isotropic current sheet model [JGR, 7, A8, 93,.9/JA34, ], and numerically simulated current sheets [Pritchett and Coroniti, JGR,, 355, 995; Hesse etal., J. Geomag. Geoelectr., 48, 749, 996; Becker etal., JGR, 6, 38, ] These models are generally computed on the basis of numerical analysis. The model considered in this presentation is by far the simplest which lends itself to local and nonlocal stability analysis.

13 . MTSI and LHDI in Harris vs non-harris current sheet MTSI in Harris currentsheet Although the current she density/magnetic field gradient are mutually related, for historical reasons, instability theory which invokes only the cross-field current was developed first. This instability is known as the modified-two-stream instability (MTSI). In our formalism, MTSI assumes free energy source from unmagnetized drifting ions. The electrons are assumed to be stationary. Their distributions are given by F i e (v V i) /α i F e e v /α e The complete local stability analysis requires detailed computation of 3 3 linear dielectric matrix with EM effects included, and itinvolves Z functions and Bessel functions. Shown below is only the growth rate curve as a function of k y (currentdirection) and k x (magnetic field direction). The important point to note is that the maximum growth occurs for oblique angle of propagation. However, the maximum growth rate is relatively low (. of the lower-hybrid frequency). 3

14 growth rate kx. Schematics of MTSI mechanism is shown below. The free energy source is the ion cross-field drift, and the instability propagates at an angle to the current flow..5 ky.5 z Ion drift Vi x MTSI y 4

15 LHDI in Harris currentsheet MTSI is an incomplete theory since it ignores the inhomogeneity. A more complete theory is known as the lower-hybrid-drift instability (LHDI), and accounts for free energy from both the unmagnetized drifting ions and density inhomogeneity (which only the magnetized electrons feel). Thus, LHDI theory starts from ion and electron distributions, F i e (v V i) /α i F e v y LΩ e e v /α e The growth rate now shows perpendicular propagation angle as the most unstable mode, and the growth rate is an order of magnitude higher than MTSI. Thus, ion-drift induced MTSI feature is complete buried underneath the electron density gradient induced LHDI feature. This shows that ignoring density gradientwas a bad idea, and thatmtsi is nota good first-order theory. 5

16 growth rate kx Schematics of LHDI is shown below. Since the gradient is maximum at the edge, LHDI gets excited also at the edge. ky z Electron drift LHDI Density n(z) Ve Vi Ion drift x LHDI y 6

17 Since LHDI is the dominant instability in Harris current sheet, and since LHDI is situated at the edge not the neutral sheet proper, the only role LHDI can play in the reconnection process is via indirectmeans (i.e., nonlinear modification of currentsheet, as evidenced by recentsimulations). However, once the current sheet characteristics turns from Harris to ion-dominated sheet at the center, then MTSI-like off-angle peak growth rate feature begins to emerge. Since MTSI-like feature is directly related to ion drift, it is expected that this mode will also centralize near the neutral sheet. Electron-dominated current sheet can only lead to more and more LHDI-like feature, since it was shown that LHDI feature is directly related to density gradient, which in turn directly translates to electron drift. MTSI-like mode also happens to possess EM polarization. Thus, the excitation of MTSI-like oblique instability in the ion-dominated currentsheetmay be highly relevantto laboratory reconnection experiment (MRX) where large EM fluctuations are observed at the central region. 7

18 Transition from LHDI to MTSI in (ion-dominated) non-harris currentsheet. growth rate.5. U = /τ kx kx ky ky growth rate U = + /τ kx kx ky ky growth rate.5. U = 4 + /τ kx kx ky ky 8

19 3. Implication for reconnection onset Ion-dominated current sheet may be unstable to the excitation of MTSI at the neutral sheet. Current sheet edge is dominated by LHDI excitation. Instabilities operative at the neutral sheet may provide anomalous resistivity, and thus may lead to the onset of reconnection. z LHDI Density n(z) Ve MTSI x LHDI Vi y 9