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Center for Magnetic Reconnection Studies: Final Report

Amitava Bhattacharjee Institute for the Study of Earth, Oceans, and Space

University of New HampshirePSACI PAC, Princeton, June 3 2004

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• 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)

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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Adaptive Mesh Refinement

Effectiveness of AMRExample: 2D MHD/Hall MHD

Efficiency of AMR

High effective resolution

Level # grids # grid points0 1 702251 83 1460802 103 2686663 153 5453164 197 10421325 404 19264656 600 1967234

Grid points in adaptive simulation: 5966118Grid points in non-adaptive simulation: 268730449Ratio 0.02

lo g

Scaling on a model reconnection problem

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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E + v × B =1

SJ + de

2 dJ

dt+

din

J × B −∇ •t p e( )

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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 studiesWhile 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….

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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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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Interrelationship between fusion, space, and astrophysical plasmas

An example…..

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. ExamplesSawtooth oscillations in tokamaksMagnetospheric substormsImpulsive solar and stellar flares

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Sawtooth crash in tokamaks (Yamada et al. 1994)

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Magnetospheric Substorms

Current Disruption in the Near-Earth Magnetotail

Ohtani et al. 1992

Solar corona

Magnetic field

http://uk.cambridge.org/assets/astronomy/encyclopedias/Fig5_28.jpg

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Impulsive solar/stellar flares

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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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Resolving the current sheet

zoomzoom

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t

Island width

magnetic flux function

S = 109

Jz

Jz, cut

zoom

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

Resistive MHDt=200 t=400 t=600

Poloidal velocity streamlines, Vz (color coded)

Hall MHDt=200 t=260 t=320

Two-fluid (or Hall MHD)

Resistive MHD

Growth rate(Time-history)

MRC Cylinder

Observations (Ohtani et al. 1992)

Hall MHD Simulation

Cluster observations: Mouikis et al., 2004

Hall MHD Ballooning Instabilities

Intermediate Regime for Instability: Compressional Stability at High

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Large but finite ky ballooning modes from initial-value studies:Towards a nonlinear theory of ballooning and tests of “detonationmodels”

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ky and Dependence

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

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Error-field induced reconnection in tokamaks

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Comparison of theory and simulation for different values of resistivity and viscosity

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.

Analytical 3D spherical equilibrium with spherical separator containing two

magnetic nulls

Unstable equilibrium

Equilibriumplus perturbation

Field linespenetrating thespherical tearingsurface

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

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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GGCM picture including solar wind open field lines draping the Earth’s magnetosphere