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The University of Chicago Center for Magnetic Reconnection Studies: Final Report Amitava Bhattacharjee Institute for the Study of Earth, Oceans, and Space University of New Hampshire PSACI PAC, Princeton, June 3 2004

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The University of Chicago University of New Hampshire: A. Bhattacharjee (PI and Director), N. Bessho, K. Germaschewski, J. Maron, C. S. Ng, P. Zhu University of Iowa: B. Chandran, L.-J. Chen, Z. W. Ma, J. Maron University of Chicago: R. Rosner (PI), T. Linde, L. Malyshkin, A. Siegel University of Texas at Austin: R. Fitzpatrick (PI), F. Waelbroeck, P. Watson TOPS collaborators : D. Keyes, F. Dobrian, B. Smith CMRS: Interdisciplinary group drawn from applied mathematics, astrophysics, computer science, fluid dynamics, plasma physics, and space science communities (supported also by DOE/ASCI, NASA, NSF)

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CMRS: What have we accomplished on the computing front? Principal Computational Deliverable: Magnetic Reconnection Code (MRC) We have developed the world’s first two-fluid (or Hall MHD) code in an Adaptive Mesh Refinement (AMR) framework. Its attributes are: A fully 3D code which integrates two-fluid/Hall MHD equations Massively parallel with Adaptive Mesh Refinement (AMR) Geometry: slab (2002) and cylindrical/toroidal (2004) (with AMR in the radial direction) Flexible boundary conditions: free as well as forced reconnection studies Options for equations of state Modular code, with the flexibility to change algorithms if necessary Code performs and scales well Framework defined by FLASH---developed by active collaboration with computer scientists Code can be used for diverse applications, not just magnetic reconnection

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The University of Chicago Adaptive Mesh Refinement

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Effectiveness of AMR Example: 2D MHD/Hall MHD Efficiency of AMR High effective resolution Level# grids# grid points 0170225 183146080 2103268666 3153545316 41971042132 54041926465 66001967234 Grid points in adaptive simulation: 5966118 Grid points in non-adaptive simulation: 268730449 Ratio0.02 log

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Scaling on a model reconnection problem

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CMRS: 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 U of C). ExPIC, a fully electromagnetic 3D Particle-in-Cell (PIC) code MRC is our principal workhorse and two-fluid equations capture important collisionless/kinetic effects. Why did 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. ExPIC is massively parallel and can do realistic electron-to-ion mass ratios.

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The University of Chicago What did we originally propose to do with the MRC? And what have we done so far?(√) Applications to astrophysical, fusion, and space plasmas Sawtooth oscillations in tokamaks( √) and magnetotail substorms ( √) Error-field studies in tokamaks ( √) Astrophysical applications: galactic dynamo ( √), transport of magnetic flux to the galactic center ( √) Direct simulations of laboratory magnetic reconnection experiments RFP dynamo studies While we have not completed all the tasks in our original proposal, we have carried out a number of tasks that were not in our proposal….

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What have we done so far also includes topics that are above and beyond what was proposed originally…. (marked in red) Sawtooth crashes in tokamaks (using two-field, four-field, and full two-fluid equations in slab and cylindrical geometry) Error-field induced islands in tokamaks Scaling of forced reconnection in the Taylor model (resistive and Hall MHD), with and without anomalous resistivity Discovery of compressional wave-driven bursty reconnection in the Taylor model Magnetotail substorms: role of reconnection and ballooning instabilities at onset 3D tearing instabilities involving nulls: discovery of the spherical tearing mode Mathematical and computational solution of the Parker problem in coronal physics Investigation of finite-time vortex singularity in Navier-Stokes equations (“Millenium prize problem”) Growth of magnetic helicity in a turbulent astrophysical dynamo Anisotropic MHD and electron MHD turbulence

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The University of Chicago SciDAC TOPS-CMRS collaboration CMRS team has provided TOPS with model 2D multicomponent MHD evolution code, and explicit solver TOPS has implemented fully nonlinearly implicit GMRES-MG-ILU parallel solver –in PETSc ’s FormFunctionLocal format using DMMG and automatic differentiation for Jacobian objects CMRS and TOPS reproduce the same dynamics on the same grids with the same time-stepping –up to a finite-time singularity due to collapse of current sheet (further dynamics falls below present uniform mesh resolution) TOPS code, being implicit, can choose timesteps an order of magnitude larger, with potential for higher ratio in more physically realistic parameter regimes –but is still slower in wall-clock time

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The University of Chicago Interrelationship between fusion, space, and astrophysical plasmas An example…..

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Impulsive Reconnection: The Trigger Problem Dynamics exhibits an impulsiveness, that is, a sudden change in the time-derivative of the growth 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 and vortex sheets in finite time. Examples Sawtooth oscillations in tokamaks Magnetospheric substorms Impulsive solar and stellar flares

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The University of Chicago Sawtooth crash in tokamaks (Yamada et al. 1994)

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The University of Chicago Magnetospheric Substorms

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Current Disruption in the Near-Earth Magnetotail Ohtani et al. 1992

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Solar corona Magnetic field http://uk.cambridge.org/assets/astronomy/ encyclopedias/Fig5_28.jpg

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The University of Chicago Impulsive solar/stellar flares

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The University of Chicago Hierarchy of collisionless reconnection models 3D Hall cylindrical MHD Four-Field Model (Hazeltine et al. 1987, Aydemir 1992) Variables: magnetic field B, velocity v, pressure p Variables: magnetic potential , stream function , parallel speed v, pressure p Two-Field Model (Porcelli et al. 1999) Variables: magnetic potential , stream function (with generalized Ohm's law)

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The University of Chicago Resolving the current sheet zoom

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The University of Chicago t Island width magnetic flux function

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S = 10 9 JzJz J z, cut zoom

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Aydemir’s four-field simulations (1992): effect of electron pressure gradient in the generalized Ohm’s law causes near-explosive nonlinear growth of m=1 island. Sawtooth Oscillations in Tokamaks

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Resistive MHD t=200 t=400 t=600 Poloidal velocity streamlines, V z (color coded) Hall MHD t=200 t=260 t=320

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

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Observations (Ohtani et al. 1992) Hall MHD Simulation Cluster observations: Mouikis et al., 2004

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Hall MHD Ballooning Instabilities

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Intermediate Regime for Instability: Compressional Stability at High

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The University of Chicago Large but finite ky ballooning modes from initial-value studies: Towards a nonlinear theory of ballooning and tests of “detonation models”

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The University of Chicago k y and Dependence

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Possible Scenario of Substorm Onset: Near-Earth Ballooning Instability Induced by Current Sheet Thinning and/or Reconnection

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The University of Chicago Error-field induced reconnection in tokamaks

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The University of Chicago Comparison of theory and simulation for different values of resistivity and viscosity

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The Spherical Tearing Mode: A fully 3D model of reconnection (with J. M. Greene and S. Hu) Greene (1988) and Lau and Finn (1990): in 3D, topological configuration of great interest is one that has magnetic nulls with loops composed of field lines connecting the nulls. There are two types of nulls, type A and type B, and they come in pairs. The null-null lines are called separators, and these are analogous to closed field lines in toroidal plasmas. For the magnetosphere, these ideas were already represented in the vacuum superposition models of Dungey (1963), Cowley (1973), Stern (1973). But a vacuum carries no current, and hence no spontaneous tearing instability.

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Analytical 3D spherical equilibrium with spherical separator containing two magnetic nulls

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Unstable equilibrium Equilibrium plus perturbation Field lines penetrating the spherical tearing surface

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Breaking of the spherical tearing surface allows external field lines to penetrate into the surface

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Evidence of spherical tearing in a Global General Circulation Model (GGCM) simulation with northward IMF (with J. Dorelli and J. Raeder)

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The University of Chicago GGCM picture including solar wind open field lines draping the Earth’s magnetosphere

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