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**Center for Magnetic Self-Organization, August 4, 2004**

Magnetic Reconnection: Recent Developments and Connections with CMSO Research A. Bhattacharjee, K. Germaschewski, J. Dorelli, J. Greene, C. S. Ng, S. Hu, and P. Zhu Space Science Center Institute for the Study of Earth, Oceans, and Space University of New Hampshire Center for Magnetic Self-Organization, August 4, 2004

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Outline Some recent developments and key questions pertaining to reconnection in weakly collisional plasmas. Examples: Impulsive reconnection phenomena and its scaling (sawtooth crashes in tokamaks and RFPs, magnetotail substorms, and solar flares) Geometry and dynamics of 3D reconnection (applications to the dayside magnetosphere, solar corona, and laboratory experiments). These questions be studied using the experimental (MST, MRX, SSPX), theoretical, and computational (NIMROD, DEBS) resources of the CMSO group. Combined with in situ satellite observations of magnetospheric plasmas (WIND, POLAR, CLUSTER, and MMS), global imaging of solar plasmas (SOHO, TRACE, RHESSI, STEREO) and new simulation codes (Hall MHD/Two-fluid and Particle-In-Cell), there is now a unique opportunity to make progress on important, unresolved questions.

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**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 Thin current sheets, embedded in the reconnection layer, are subject to instabilities that require kinetic theory for a complete description.

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**Adaptive Mesh Refinement**

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**Effectiveness of AMR Example: 2D MHD/Hall MHD Efficiency of AMR**

High effective resolution Example: 2D MHD/Hall MHD Efficiency of AMR log Level # grids # grid points Grid points in adaptive simulation: Grid points in non-adaptive simulation: Ratio 0.02

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**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

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**Sawtooth crash in tokamaks (Yamada et al., 1994)**

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**Sawtooth events in MST (Almagri et al., 2003)**

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**Magnetospheric Substorms**

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**Current Disruption in the Near-Earth Magnetotail**

(Ohtani et al., 1992)

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**Impulsive solar/stellar flares**

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**2D Hall MHD: m=1 sawtooth instability**

Model equations: (Ottaviani/Porcelli 1993) (Grasso/Pegoraro/Porcelli/Califano 1999) Equilibrium:

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**Resolving the current sheet**

zoom zoom

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**Current sheet collapse, s = 0**

current density Current sheet collapse, s = 0 1/current sheet width t

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**magnetic flux function**

Island width t

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Island equation or If dx and dL attain constant values and are of order of de (or ), the island equation becomes (Ottaviani & Porcelli 1993 for rs = 0 ): with ,

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**Island equation c.f. simulation**

Solid: simulation, dashed: island equation, cJ = 0.025,

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**Scaling of the reconnection rate: Is it Universal?**

Consider scaling of the inflow 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: Note that dL is of the order of de (or ). f depends on parameters de, and k. It also depends weakly on time through l(t) ~ 1 in the nonlinear phase. Numerically f is seen to be of the order of 0.1, if one uses simulation parameters of the type used by Shay et al.

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**Role of CMSO experiments in resolving issues of scaling**

CMSO experiments can play an important role in elucidating scaling issues in quasi-steady (MRX, SSPX) as well as time-dependent reconnection (sawteeth in MST, combined with comparative tokamak studies). Strategy: vary ion/electron skin depths, plasma beta, guide field, resistivity, and boundary conditions. Theory and PIC simulations predict a range of scaling behavior. Identify and measure mechanisms (describable by two-fluid or more complete PIC models) of instability and turbulence of thin current sheets, and challenge simulations to produce them. (This appears to be well under way already.) Can fully kinetic features be incorporated in Hall MHD framework by appropriate closure relations on the anisotropic pressure tensor?

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**Sawtooth instability in cylinder**

Goal: To investigate the m=1 sawtooth instabilty in a cylinder for parameters very similar to Aydemir’s (1992) four-field simulation which showed near-explosive growth of islands, possibly accounting for the sawtooth trigger. This exercise, expected to be a routine benchmark, turns out to be a surprise: no explosive growth is seen. The Hall currents induce poloidally asymmetric shear flows that are strongly stabilizing. The runs reported were carried out on IBM Power4 (Cheetah) at CCS, ORNL.

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S = 109 Jz, cut zoom Jz

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**Poloidal velocity streamlines, Vz (color coded)**

Resistive MHD t=200 t=400 t=600 Hall MHD t=200 t=260 t=320

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MRC Cylinder Growth rate (Time-history) Two-fluid (or Hall MHD) Resistive MHD

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**Magnetospheric Substorms**

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**Observations (Ohtani et al. 1992)**

Hall MHD Simulation Cluster observations: Mouikis et al., 2004

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J y y

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**z = -3 to 3 RE growth rate:0.2 Ux Uz Highly Compressible t=5.8**

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

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ky and Dependence

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**Resistive Tearing Modes in 2D Geometry**

Equilibrium Magnetic field assumed to be either infinite or periodic along z x Neutral line at y=0 y Time Scales Lundquist Number Tearing modes (Furth, Killeen and Rosenbluth, 1963)

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**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” ( ) and “component” ( ) tearing. It was shown that 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)

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**Magnetic nulls in 3D play the role of X-points in 2D**

Spine Fan (Lau and Finn, 1990)

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**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.

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Equilbrium B-field

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Perturbed B-lines

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Equilibrium Perturbed Field lines penetrating the spherical tearing surface

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**Equilbrium Current Density, J**

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**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 SSPX, which has observed reconnection involving nulls.

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**Solar corona astron.berkeley.edu/~jrg/ ay202/img1731.gif**

Space/gifs/yokohflscl.gif

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**Solar corona: heating problem**

photosphere corona Temperature Density Time scale Magnetic fields (~100G) --- role in heating? ~ ~ Alfvén wave current sheets

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Parker's Model (1972) Straighten a curved magnetic loop Photosphere

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Reduced MHD equations low limit of MHD

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**Magnetostatic equilibrium**

(current density fixed on a field line) c. f. 2D Euler equation 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.

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**Footpoint Mapping Identity mapping: or e. g. uniform field**

For a given smooth footpoint mapping, does more than one smooth equilibrium exist?

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**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)

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**Implication An unstable but smooth equilibrium cannot relax to a**

second smooth equilibrium, hence must have current sheets.

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In 3-D torii (such as stellarators), current singularities are present where field lines close on themselves Solutions to the quasineutrality equation, General solution has two classes of singular currents at rational surface [i(ymn) = n/m] (Cary and Kotschenreuther, 1985; Hegna and Bhattacharjee, 1989). Resonant Pfirsch-Schluter current Current sheet

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**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.

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**Simulations of Parker's model**

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**Simulations of Parker's model**

bottom middle top + -

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**Simulations of Parker's model**

bottom middle top + -

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**Simulations of Parker's model**

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More general topology Parker's optical analogy (1990)

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Main current sheet

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