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Current Sheet and Vortex Singularities: Drivers of Impulsive Reconnection A. Bhattacharjee, N. Bessho, K. Germaschewski, J. C. S. Ng, and P. Zhu Space Science Center Institute for the Study of Earth, Oceans, and Space University of New Hampshire Isaac Newton Institute, Cambridge University, August 9, 2004 High-Performance Computing Tools • Magnetic Reconnection Code (MRC), based on extended two- fluid (or Hall MHD) equations, in a parallel AMR framework (Flash, developed at the University of Chicago). • EPIC, a fully electromagnetic 3D Particle-In-Cell code, with explicit time-stepping. MRC is our principal workhorse and two-fluid equations capture important collisionless/kinetic effects. Why do we need a PIC code? • Generalized Ohm’s law 1 2 dJ di E v B J de J B pe S dt n • Thin current sheets, embedded in the reconnection layer, are subject to instabilities that require kinetic theory for a complete description. Adaptive Mesh Refinement Effectiveness of AMR High effective resolution Example: 2D MHD/Hall MHD Efficiency of AMR Level # grids # grid points 0 1 70225 1 83 146080 2 103 268666 3 153 545316 4 197 1042132 5 404 1926465 6 600 1967234 Grid points in adaptive simulation: 6976118 Grid points in non-adaptive simulation:268730449 Ratio 0.02 Impulsive Reconnection: The Trigger Problem Dynamics exhibits an impulsiveness, that is, a sudden change in the time-derivative of the reconnection rate. The magnetic configuration evolves slowly for a long period of time, only to undergo a sudden dynamical change over a much shorter period of time. Dynamics is characterized by the formation of near-singular current sheets in finite time. Examples Sawtooth oscillations in tokamaks and RFPs Magnetospheric substorms Impulsive solar and stellar flares Sawtooth crash in tokamaks (Yamada et al., 1994) Sawtooth events in MST (Almagri et al., 2003) Magnetospheric Substorms Current Disruption in the Near-Earth Magnetotail (Ohtani et al., 1992) Impulsive solar/stellar flares 2D Hall MHD: m=1 sawtooth instability Two-Field Reduced Model for large guide field and low plasma beta (Schep/Pegoraro/Kuvshinov 1994) (Grasso/Pegoraro/Porcelli/Califano 1999) Equilibrium: Resolving the current sheet zoom zoom current density Current sheet collapse, s = 0 1/current sheet width t magnetic flux function Island width t Island equation d cJ k 2 4 Ý 2 2 2 dt L x 4de x Ý 2 s 2 Ý x s or x L exp x 1 c d 2 1 c d 2 L L J e J e 2 de / 3 1/ 3 If x and L attain constant values and are of order of de (or s ), the island equation becomes (Ottaviani & Porcelli 1993 for s = 0 ): 2 ˆ d 1 ˆ c ˆ4 dtˆ 2 4 1/ 3 2 / 3 ˆ with t L t , L kde or kde s , / L . ˆ Island equation c.f. simulation Solid: simulation, dashed: island equation, cJ = 0.025, s 0.2, de 0.1, k 0.5, L 0.0024 Scaling of the reconnection rate: Is it Universal? Consider scaling of the inflow V in~ fVAd velocity: It has been argued that f ~ 0.1 [Shay et al., 2004], in a universal asymptotic regime, independent of system size and dissipation mechanism. Using the island equation in the asymptotic regime: c J L k3 / 2 f ~ 10 de 2 de / 3 1/ 3 Note that L is of the order of de (or s ). f depends on parameters de, s and k. It also depends weakly on time through (t) ~ 1 in the nonlinear phase. Numerically f is seen to be of the order of 0.1 for certain popular cloices of simulation parameters, but this is not universal. Magnetospheric Substorms Observations (Ohtani et al. 1992) Hall MHD Simulation Cluster observations: Mouikis et al., 2004 Bx B0tanh(z /a) vA /c 0.2 a L z /2 mi /me 25 J y y Ti 5Te di / a 0.1 E yb 0.01B0 (vA /c)(1 cos x / Lx ) Ey/B0 (c/vA) 0.02 0.01 0 0 10 20 30 40 t/A 2 /(B0a) 1 0 0 10 20 30 40 t/A Ux Uz t=5.8 Highly Compressible Ballooning Mode in Magnetotail (Voigt model) x = -1 to -16 RE z = -3 to 3 RE ky= 25*2 e = 126 growth rate:0.2 t=29 ky and Dependence Euler equation for incompressible flow: v v v p t v 0 The incompressibility condition produces a Poisson’s equation for pressure 2 p (v v) Integro-differential equation for pressure 1 (v v) p(x,t) dx + b.c. 4 x x Solutions to Euler are essentiallynonlocal. t=.49 High-symmetry flow (Pelz 1997) t=0 t=.33 Vorticity in the high-symmetry flow Vorticity 2D and 1D cuts Growth of vorticity Distortion of vortices vorticity pressure Sufficient condition for finite-time singularity Consider flow along x -axis ( y z 0). By Kida symmetry vy vz 0, 0 Euler equation v x px Ý Define x vx , = y vy , = z vz ( 0) Differentiating Euler, obtain exact equation 2 p xx Ý If p xx 0 following a Lagrangian element, then will be singular in finite time. Sufficient condition (Eulerian) Origin is a • velocity null point • point of intersection of vorticity null lines Kida symmetries ensure p x (0) pxx (0) p xxx (0) 0 From the exact equation (along y z 0), by Taylor expansion pxxxx (0) 2 2 p xx (x) Ý x O(x 4 ), 2 where x vx . If there exists a range 0 x X(t) in which p xxxx is positive and increasing rapidly enough, there is a finite-time singularity. Resistive Tearing Modes in 2D Geometry Equilibrium Magnetic field assumed to be either infinite or periodic along z B xBP tanhy / a BT z ˆ ˆ x Neutral line at y=0 y Time Scales A a/VA a41/2 /BP R 4a 2 /(c 2 ) Lundquist Number S R / A Tearing modes S 3 / 5 (slower) or S 1 / 3 (faster) (Furth, Killeen and Rosenbluth, 1963) Collisionless Tearing Modes at the Magnetopause (Quest and Coroniti, 1981) • Electron inertia, rather than resistivity, provided the mechanism for breaking field lines, but the magnetic geometry is still 2D. • Growth rates calculated for “anti-parallel”T( 0 B ) and “component” ( T 0 ) tearing. It was shown that B growth rates for “anti-parallel” merging were significantly higher. • Model provided theoretical support for global models in which global merging lines were envisioned to be the locus of points where the reconnecting magnetic fields were locally “anti-parallel” (e.g. Crooker, 1979) Magnetic nulls in 3D play the role of X-points in 2D Spine Fan (Lau and Finn, 1990) Towards a fully 3D model of reconnection • Greene (1988) and Lau and Finn (1990): in 3D, a topological configuration of great interest is one that has magnetic nulls with loops composed of field lines connecting the nulls. • The null-null lines are called separators, and the “spines” and “fans” associated with them are the global 3D separatrices where reconnection occurs. • For the magnetosphere, the geometrical content of these ideas were already represented in the vacuum superposition models of Dungey (1963), Cowley (1973), Stern (1973). However, the vacuum model carries no current, and hence has no spontaneous tearing instability. The real magnetopause carries current, and is amenable to the tearing instability. Equilbrium B-field Perturbed B-lines Equilibrium Perturbed Field lines penetrating the spherical tearing surface Equilbrium Current Density, J Features of the spherical tearing mode • The mode growth rate is faster than classical 2D tearing modes, scales as S-1/4 (determined numerically from compressible resistive MHD equations). • Perturbed configuration has three classes of field lines: closed, external, and open (penetrates into the surface from the outside). • Tearing eigenfunction has global support along the separatrix surface, not necessarily localized at the nulls. • The separatrix is global, and connects the cusp regions. Reconnection along the separatrix is spatially inhomogeneous. Provides a new framework for analysis of satellite data at the dayside magnetopause. Possibilities for SSX, which has observed reconnection involving nulls. Solar corona astron.berkeley.edu/~jrg/ ay202/img1731.gif www.geophys.washington.edu/ Space/gifs/yokohflscl.gif Solar corona: heating problem photosphere corona 3 6 Temperature ~ 5 10 K ~ 10 K 12 3 Density ~ 1023 m 3 ~ 10 m 4 ~ 20s Time scale ~ 10 s Magnetic fields (~100G) --- role in heating? Alfvén wave current sheets Parker's Model (1972) Straighten a curved magnetic loop Photosphere Reduced MHD equations J [ , ] [ A, J] 2 low t z A limit of [ , A] 2 A MHD t z B z B z A z --- magnetic field, ˆ ˆ ˆ v z --- fluid velocity, ˆ --- vorticity, 2 2 J A --- current d ensity , --- resistivity, --- viscosity, [ , A] y Ax x Ay Magnetostatic equilibrium J (current density [ A, J] 0 ,or B J 0 fixed on a field line) z with = 0. Field-lines are tied at z 0 , L . c. f. 2D Euler equation [ , ] 0 t A , J , z t Existence theorem: If is smooth initially, it is so for all Time. However, Parker problem is not an initial value problem, but a two-point boundary value problem. Footpoint Mapping x (z) X[x (0), z], x (L) X[x (0), L], dX A dY A with (X,Y, z) , ( X,Y,z) dz y dz x Identity mapping:x (L) x (0) e. g. uniform field B z or A const ˆ For a given smooth footpoint mapping, does more than one smooth equilibrium exist? A theorem on Parker's model For any given footpoint mapping connected with the identity mapping, there is at most one smooth equilibrium. A proof for RMHD, periodic boundary condition in x (Ng & Bhattacharjee,1998) Implication An unstable but smooth equilibrium cannot relax to a second smooth equilibrium, hence must have current sheets. In 3-D torii (such as stellarators), current singularities are present where field lines close on themselves • Solutions to the quasineutrality equation, J 0, J|| B J B JB 2 mn mne im in , B mn (m n ) mne im in p' mnmn amne im in • General solution has two classes of singular currents at rational surface [ymn) = n/m] (Cary and Kotschenreuther, 1985; Hegna and Bhattacharjee, 1989). mn amn ˆ mn p' (y y mn ) (m n ) mn – Resonant Pfirsch-Schluter current – Current sheet Reconnection without nulls or closed field lines Formation of a true singularity (current sheet) thwarted by the presence of dissipation and reconnection. Classical reconnection geometries involve: (i) closed field lines in toroidal devices (ii) magnetic nulls in 3D Parker’s model is interesting because it fall under neither class (i) or (ii). Questions: • Where do singularities form? What are the geometric properties of singularity sites and reconnection? Strategic issue for CMSO research. • Need more analysis and high-resolution simulations. Incompressible spectral element MHD code (under development by F. Cattaneo) may be very useful. Simulations of Parker's model Start with a uni form B field Apply constant footpoint t wisting (x ,0) 0, (x , L) 0 (x ) with [ 0 , 2 0 ] 0 Current l ayers appear after large distortion. Quasi-equilibrium at first, becomes unst able J grows faster in the middl e non equilibrium Simulations of Parker's model bottom middle top + Simulations of Parker's model bottom middle top + Simulations of Parker's model t J max J max0 q 2 d 2 qmax dmax 118.5 9.91 9.86 0.233 0.00028 7.946 0.708 131.6 16.5 10.9 0.930 0.00238 16.04 2.272 145.6 28.3 14.6 4.361 0.1471 32.94 5.997 150.5 35.3 16.4 21.98 13.486 83.39 79.01 with q J / z [ A, J ], d q 2 . More general topology Parker's optical analogy (1990) Main current sheet Qu i ckTi m e™ a nd a GIF de co mp re s so r a re ne ed ed to se e th is pi c tu re.