Center for Magnetic Reconnection Studies: Present Status, Future Plans

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Center for Magnetic Reconnection Studies Present
Status, Future Plans

• Amitava Bhattacharjee
• The University of Iowa
• PSACI PAC, Princeton, June 5 2003

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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)

• University of Iowa A. Bhattacharjee (PI and
  Director), B. Chandran, N. Bessho, L.-J. Chen, K.
  Germaschewski, Z. W. Ma, J. Maron, C. S. Ng, P.
  Zhu
• 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

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Presentation Plan

• CMRS Present Status, Future Plans A.
  Bhattacharjee (25 minutes)
• The Magnetic Reconnection Code Within the FLASH
  Framework R. Rosner (10 minutes)
• CMRS/TOPS Collaboration D. Keyes (5 minutes)

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Outline of this talk

• Magnetic Reconnection Code (MRC) its attributes
• What did we propose to do with the MRC?
• What have we done so far (2001-03)? (Items in
  blue will be covered in talk, items in green are
  in the presentation folder)
• Wave driven bursty reconnection in a
  compressible plasma (Fitzpatrick,
  Bhattacharjee, Ma, Linde, and Rosner)
• m1 sawtooth instability in 2D and 3D
  (Germaschewski, Ma, Ng, Bhattacharjee, Linde,
  Malyshkin, and Rosner)
• 3D hydrodynamics Finite-time vortex
  singularity? (Germaschewski, Bhattacharjee, and
  Grauer)
• Ballooning instability of the Earths
  magnetotail mechanism for substorm onset (Zhu,
  Ma, and Bhattacharjee)

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Outline of this talk (continued)

• Growth of magnetic helicity in a turbulent
  astrophysical dynamo (Maron, Blackman, and
  Chandran)
• In addition, in order to assess the importance
  of kinetic effects, we have developed a fully 3D
  electromagnetic PIC code (Bessho) and applied it
  to study
• Impulsive reconnection dynamics in the
  magnetotail (Bessho, Bhattacharjee, and
  Chandran)
• Shock acceleration during solar flares (Bessho
  and Chandran)

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Outline of this talk (continued)

• Innovative computational methodologies
• Fully implicit solvers for Hall MHD (Dobrian,
  Germaschewski, Keyes, and Smith)
• Development of Gradient Particle MHD (GPM) a
  new simulation methodology for nonlinear
  MHD/Hall MHD (Maron and Howes)
• Response to 2002 PSACI PAC recommendations
• CMRS outreach
• Roadmap for the next 3-year phase (beyond 2004)

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What we have also done, but is not covered in
this talk or the presentation folder

• Error-field induced reconnection in tokamaks
  detailed comparison with theoretical predictions
  (Fitzpatrick, presented in 2002)
• The spherical tearing mode (Hu, Bhattacharjee,
  and Greene, presented in 2002)
• Scaling of quasi-steady Hall MHD reconnection
  (Wang, Bhattacharjee, and Ma, presented in 2002.
  Also, recent work by Fitzpatrick on
  incompressible Hall MHD)
• Fast magnetic reconnection via jets and current
  microsheets (using FLASH-MRC) (Watson and Craig)
• Anisotropic MHD turbulence in MHD and EMHD (Ng,
  Bhattacharjee, Germaschewski, and Galtier)

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Magnetic Reconnection Code (MRC)

• A fully 3D code which integrates the equations
  of Hall MHD
• Geometry slab (2002), cylindrical and toroidal
  (2003)
• Massively parallel with Adaptive Mesh
  Refinement (AMR)
• 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 must perform and scale well, be highly
  portable, and be easy to maintain, modify and
  evolve into the future
• Framework defined by FLASH---developed by
  active collaboration with computer scientists

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Adaptive Mesh Refinement
zoom
Implicit solution of an elliptic test
problem, using fast-adaptive-composite algorithm
with PetSc's Krylov-Schwarz solvers for the level
solves
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Adaptive Mesh Refinement
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Adaptive Mesh Refinement Load Balancing
Sample coverage with adaptive grids, distribution
to processors (colors)
Underlying Peano-Hilbert space-filling curve used
for load balancing
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What did we propose to do with the MRC?

• 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, penetration of plasma into the
  magnetosphere of compact stellar objects
• Simulation of laboratory magnetic reconnection
  experiments (e.g., MRX at Princeton, VTF at MIT)

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Impulsive or Bursty Reconnection (Germaschewski,
Ma, Ng, Bhattacharjee, Linde, Malyshkin, and
Rosner 2003)
Dynamics exhibits an impulsiveness, that is, a
sudden increase in the time-derivative of the
growth rate.
In other words, the magnetic configuration
evolves slowly for a period of time, only to
undergo a sudden dynamical change over a much
shorter period of time. Dynamics is
characterized by the development of near-singular
current and vortex sheets in finite time.
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Wave driven bursty reconnection (Fitzpatrick,
Bhattacharjee, Ma, Linde, and Rosner)

• Discovered numerically using Flash-MRC for forced
  reconnection induced by a sinusoidal perturbation
  at the boundary (Taylors model).
• Standard asymptotic matching theory fails at
  early times.
• Developed novel Laplace transform treatment which
  does not rely on asymptotic matching and is thus
  able to capture early time dynamics.
• Analytical model reveals that at early times
  there is an additional contribution to the
  reconnection rate from compressional waves
  excited by the boundary perturbations.
• If boundary perturbation switched on rapidly,
  wave driven reconnection occurs in a sequence of
  bursts with amplitudes far greater than predicted
  by asymptotic matching theory.

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Wave driven reconnection comparison of theory
and simulation
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Scaling on a model reconnection problem
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2D Hall MHD m1 sawtooth instability
Model equations
(Ottaviani/Porcelli 1993) (Grasso/Pegoraro/Porcell
i/Califano 1999)
Equilibrium
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magnetic flux function
Growth of the island, ?s 0
Island width
t
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current density
Current sheet collapse, ?s 0
1/current sheet width
t
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Resolving the current sheet
de 0.1, ?s 0.2, ? 10-5
zoom
zoom
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magnetic flux function
Island width
t
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Resistive Hall MHD
magnetic flux function
de 0, ?s 0.1, ? 10-5
Island width
t
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Generalized island equation
,
(boundary layer width)
Additional dependence on
normalized by
,
in many cases of interest
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Generalized island equation
time normalized by
Nonlinear reconnection rate does depend on
,
in finite time
Including effects of ? and ?2, hyper-resistivity
as observed in simulations
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Hierarchy of collisionless reconnection models
Hall MHD
(with generalized Ohm's law)
Variables magnetic field B, velocity v, pressure
p
Four-Field Model
(Hazeltine et al. 1985, 1986, 1987 Aydemir 1991,
1992)
Variables magnetic potential ?, stream function
?, parallel speed v, pressure p
Two-Field Model
(Kuvshinov et al. 1994, 1998 Porcelli et al.
1993, 1998, 1999)
Variables magnetic potential ?, stream function ?
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Limitations of the two-field model
Two-field model can be derived from four-field
model in the large guide field (low beta) limit.
Low beta limit is a singular perturbation
which may fail as the current sheet width tends
to zero .
Effects from the neglected field variables
(parallel velocity v, pressure p) must be
included in the island equation which will change
the dependence of the critical time on parameters.
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Sawtooth instability in cylinder/torus

• MRC in cylindrical/toroidal geometry has been
  developed. It is a pseudopsectral parallel code
  but has no AMR yet.
• First application investigate the m1 sawtooth
  instabilty in a cylinder for parameters very
  similar to Aydemirs (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, diamagnetic shear
  flows that are strongly stabilizing.
• The runs reported were carried out on IBM Power4
  (Cheetah) at CCS, ORNL (special thanks to CCS
  staff for granting us a special class for our
  batch jobs)

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Search for finite-time singularities
Morf, Orszag, Frisch (1980) Pade-approximation,
complex time singularity, singularity yes Chorin
(1982) Vortex method, singularity yes Pumir,
Siggia (1990) Adaptive grid, singularity
no Bell, Marcus (1991) Projection method,
resolution (128)3 , singularity yes Brachet,
Meneguzzi, Vincent, Politano, Sulem (1992)
pseudospectral code, resolution (256)3 ,
Taylor-Green vortex with resolution (864) 3 ,
singularity no Kerr (1993) Chebyshev
polynomials, resolution 512 x 256 x192,
singularity yes Boratav, Pelz (1994, 1997) high
symmetry Kida flow, singularity yes
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High-symmetry flow (Pelz 1997)
t.49
t0
t.33
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pressure
vorticity
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Vorticity in the high-symmetry flow
Vorticity 2D and 1D cuts
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Growth of vorticity
Distortion of vortices
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Growth of Magnetic Helicity in a Turbulent Dynamo

• Weak magnetic field is introduced into a
  turbulent fluid subject to helical forcing.
• H(Hconcentrated at scales larger (smaller than or
  equal to) the stirring scale l.
• For thelicity is conserved, that is, H H
• For t1, resistivity slows growth of small-scale
  H keeps growing steadily.
• Astrophysical consequence growth of magnetic
  field on scales larger than l for tsufficiently fast to be a viable mechanism for
  the origin of the galactic magnetic field.

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Growth of magnetic helicity in a turbulent dynamo
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CMRS Outreach

• Mini-Workshop for Basic Studies in Fusion
  Science, DPP-APS, Long Beach, October 2001
  (organized by CMRS and LAPD group, UCLA)
• Magnetofluid Modeling Workshop, GA, San Diego
  August, 2002 (organized by F. Waelbroeck and
  other members of the CMRS, CEMM groups)
• Mini-Conference on Singularity formation in
  plasmas and fluids, DPP-APS, Orlando, November
  2002 (organized by CMRS)
• Extended Workshop on Magnetic Reconnection at
  Aspen Center for Physics, June 9-22 2003
  (organized by CMRS)
• IPELS, July 2003 (A. Bhattacharjee, Program
  Committee Chair)

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Response to 2002 PSACI PAC recommendations

• Closer linkage with the University of Chicagos
  Flash code and its adaptive mesh refinement
  techniques The MRC is developed at the
  University of Iowa in a local framework with its
  own AMR, and then integrated into the Flash
  framework (which uses a different AMR). This
  allows greater flexibility and independent
  testing and verification. (R. Rosner, next
  presentation)
• We have established a closer linkage, not only
  on framework matters but also in science (in wave
  driven reconnection and the sawtooth
  instability).
• a more aggressive use of supercomputers we
  have done so in 2002-03, both at NERSC and CCS.

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Response to 2002 PSACI PAC recommendations

• substantial interaction with the Keyes Center
  We have done so on fully implicit solvers as well
  as the PETSc framework/toolkit (D. Keyes, after
  R. Rosners presentation)
• a more aggressive use of supercomputers We
  have done so in 2002-03, now both at NERSC and
  CCS.
• We look forward to development of synergy with
  the Extended MHD project So do we, and we are
  waiting. We have nothing concrete to report yet.
• Develop a web page, modeled after FLASH Done.
  

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Roadmap for the next three-year phase (beyond
2004)

• News I have recently accepted the Paul
  Professorship in Space Science at the Institute
  for the Study of Earth, Oceans and Space (EOS) at
  the University of New Hampshire (UNH). Most of
  the current members of my group, K.
  Germaschewski, C. S. Ng and P. Zhu, as well as N.
  Bessho (as well as the CMRS cluster Zephyr) are
  moving to UNH. (In short, the Iowa branch of
  CMRS is relocating to New Hampshire.) Our
  research effort for the third year is and will
  remain on track. Our group will be strengthened
  further by (1) the addition of Associate
  Professor J. Raeder who is bringing his group on
  computational space science from UCLA to UNH, and
  (2) the generous research support of the Paul
  Endowment and EOS.

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Roadmap for the next three-year phase (beyond
2004)

• As the fusion SciDAC program redefines its
  compass, we continue to see ourselves play the
  following important roles
• (1) By means of careful, step-by-step
  theoretical investigations and state-of-the-art
  numerical experiments, determine the essential
  ingredients of integrated and predictive models
  for certain types of fusion phenomenology. (In
  some cases, like the sawtooth instability, it is
  not quite clear yet what the ingredients of an
  integrated model should be.)
• (2) Astrophysicists have considerable experience
  in the complex and realistic computational
  modeling of burning plasmas including transport
  effects and various equations of state. Even
  though the regimes are substantially different in
  fusion and astrophysical plasmas, similar
  theoretical and computational models are often
  used.

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Roadmap for the next three-year phase (beyond
2004)

• (3) Synergistic ties with existing ASCI (such as
  Flash) and SciDAC Centers (such as TOPS and
  APDEC), or similar centers in space and
  geosciences, are value added to the fusion SciDAC
  effort because they make available major
  resources and expertise, especially in computer
  science and applied math. This is a two-way
  street because fusion/laboratory plasma physics
  has much to offer in the way of insights as well
  as challenges to neighboring fields of science
  and applied math.
• We would appreciate your guidance in
  answering this difficult question.

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The following is additional material for the
panel, not included in the talk. We will be glad
to take questions.
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Numerical schemes for MRC
Nessyahu,Tadmor 1990 Kurganov, Levy 2000
Central weighted ENO schemes

• Why central schemes?
• no (approximate) Riemann solver necessary
• straightforward to generalize to
  multidimensional systems
• high order
• properties like ENO, monotone, TVD with
  appropriate reconstruction

Divergence cleaning
Dedner et al, 2002

• Solutions
• constrained transport methods
• Hodge projection
• truncation-error method

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PIC Simulations of Reconnection and Shock
Acceleration
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Electromagnetic PIC Code
Linear weighting
Charge conserved method
(Villasenor and Buneman, 1992)
(i, j1)
(i1, j1)
(i1, j)
(i, j)
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Driven magnetic reconnection
2D Electromagnetic PIC code
Initial condition
Particles are injected
0
0
Boundary condition
Upper and lower boundary
Left and right boundary periodic
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d 0.1
i
t40
This 2D PIC simulation shows that the features of
impulsive and large reconnection, seen in Hall
MHD simulations with identical initial
conditions, are preserved, although the
microscopic current sheet structure shows some
differences.
t40
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Magnetosonic shock wave
3D Electromagnetic PIC code
z
B
Particles are injected
Particles are going out
Shock wave
0
q
x
upstream
downstream
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Observations of ballooning modes from the WIND
satellite at a substorm event and a linear
ballooning mode simulation (that does not rely on
the ballooning formalism)
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The start of large magnetic fluctuations and the
modulation of energetic ion fluxes observed in
Earth's magnetotail plasma sheet coincides with
the onset of an isolated substorm.Note the
impulsive relaxation of the radial pressure
gradient signified by the sudden reduction of the
differences between the duskward (90) and
dawnward (270) fluxes -- consistent with the
feature of ballooning instabilities.L.-J. Chen,
et al., 2003
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Power spectrograms of the magnetic field show
impulsive enhancement of the wave power in the
ballooning frequency range.The enhancement is
broadband, bursty, and simultaneous with the
impulsive pressure gradient reduction.L.-J.
Chen, et al., 2003
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Ballooning Instability of Near-Earth Magnetotail

• Both ideal and Hall MHD analyses find unstable
  parameter regimes in the near-earth magnetotail
  (using the analytical Voigt model and a numerical
  Hall MHD model of current sheets).
• 2D ideal MHD codes are developed to simulate the
  linear compressible ballooning mode with finite
  ky in the near-earth magnetotail.
• Cartesian and Flux coordinate systems
• Local and PETSC frameworks

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Features of Ballooning Mode in Near-Earth
Magnetotail Configuration

• Field-line geometry
• Dipole-type and X-type coexist
• Semi-open, sensitive to boundary conditions
• Low magnetic shear or shear-free,when
  conventional finite ky ballooning mode theory
  breaks down (analogues to infernal modes)
• Compression
• High-???plasma leads to large compression
• Significantly stabilizes the ballooning mode

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Ux
Uz
Highly Compressible Ballooning Mode in
Magnetotail (Voigt model) x -1 to -16 RE z
-3 to 3 RE ky 252? ?e 126 growth rate0.2
t5.8
t29
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POLAR observations of diffusion region rs
scaling in measured E// (Scudder and Ma)
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