A Comparison of 2D PIC Simulations of Collisionless Reconnection

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A Comparison of 2D PIC Simulations of
Collisionless Reconnection

• Mike Harrison
• Thomas Neukirch
• Michael Hesse
• March 19, 2008

School of Mathematics and Statistics The
University of St. Andrews
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Harris Sheet vs Force-Free Harris Sheet
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Investigate the transition from an anti-parallel
initial field Configuration through to a
Force-Free field configuration.
Investigate the Morphology of the Off-Diagonal
components of the Pressure Tensor.
Motivation
Compare the Differences in the Reconnection
process between Initial Configurations with
Constant guide-field and cases with spatially
varying shear fields
Compare the Reconnection Rates For different
initial magnetic field Configurations.
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PIC Simulation
Integrate the equations of motion
Calculate the charge and current densities
Calculate the force on each particle
Integrate the field equations on the grid
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Pushing the Particles

• Interpolate field quantities to nearest integer
  grid point.

dx
Bx(ix,iz) Ez(ix,iz)
dz
By(ix,iz)
Ey(ix,iz)
Bz(ix,iz) Ex(ix,iz)
Bz(ix-1,iz) Ex(ix-1,iz)

• Interpolate field quantities to particle
  positions.

xe(ixe), ze(iz1)
xe(ixe),ze(ize1)
1-fze
particle position
fze
xe(ixe), ze(ixe)
xe(ix1), ze(ize)
fxe
1-fxe

• Use implicit solver to solve the equation of
  motion and obtain new particle positions and
  velocities.

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Calculating the Densities

• Calculate the number densities and drift speeds
  (current densities) on the grid.

dns(ixe,ize1,k) vxs(ixe,ize1,k)

dns(ixe1,ize1,k) vxs(ixe1,ize1,k)

particle position
dns(ixe,ize,k) vxs(ixe,ize,k)

dns(ixe1,ize,k) vxs(ixe1,ize,k)

• Summing up density contributions for each species
  on each integer grid point weighted according to
  the proportional areas found from the new
  particle positions.

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Calculating the New Electric Field and Magnetic
Fields on the Grid

• Solve Maxwells equations on the grid for the new
  electric and magnetic field components.
• Solve the difference equations
• Write in terms of an implicit factor (we usually
  use )
• After this the cycle begins again.

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Normalisations and Simulation Parameters

• Times are normalised to inverse ion cyclotron
  frequency.
• Lengths are normalised to the ion inertial
  length.
• The magnetic field is normalised to its max value
  
• The ratio of the electron plasma frequency and
  electron cyclotron frequency is set to a
  numerical value of 5.
• A time step is used.
• Density is normalised to its max value
• Velocities are normalised against the Alfven
  speed

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Normalisations and Simulation Parameters

• The electric field is normalised against
  .
• The mass ratio is equal to 1.
• The grid spacing should satisfy the condition
• There are four particle species, two of ions and
  electrons. The first set of ion and electron
  species establishes the pressure and currents.
• In each simulation run ions and
  electrons are used for the foreground species and
  for the background ions and
  electrons.
• Periodic boundary conditions are employed at
  and and
  and

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Vlasov Equilibrium Theory
- The single particle distribution
functions for each species
-Probability of finding particles
within the 6 dimensional volume element
centred at the point
Distribution functions depend only on constants
of motion
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Anisotropic Maxwellian DistributionFunction
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The Component of the Current Density with
the Flux Function Overplotted
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Extended Distribution Function

• Have to find periodic solutions of the
  differential equations.
• The case gives anti-parallel field
  configuration
• The case gives linear force-free
  equilibrium (Bobrova et al.)
• Can use the parameter to vary the shear
  field while keeping
  constant
• Can investigate the transition from a pressure
  balanced equilibrium to a force-free equilibrium.

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Case

• Magnetic field solution is a reverse field
  configuration at
• Force balance maintained by plasma pressure.
• Can add varying magnitudes of constant guide
  field.

We have investigated the reconnection process as
you make the transition from zero guide field
through to . Also
compared this to the linear force-free case.
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The Component of the Current Density with
the Flux Function Overplotted with Guide Field0.1
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The Structure of the Diffusion Region

• The electron equation of motion
• The y component of the electric field can be
  written as
• Is found that the gradients of the off-diagonal
  terms of the pressure tensor are the dominant
  contributions to at the X-Point.

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Example for Linear Force-Free Case
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Simulation Snapshots at Time of Max Reconnection
Force-Free
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The Off-Diagonal Components of the
PressureTensor -
Force-Free
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The Off-Diagonal Components of the
PressureTensor -
Force-Free
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New Run with Initial Perturbation for Zero Guide
Field Case
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Reconnected Flux and Reconnection Rates
We expect strong guide field cases to have a
slower reconnection rate than those with weak
guide field.
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Future Work

• Carry out comparison reconnection simulations for
  equilibria with spatially varying shear field
  .
• This can be done by using the extended
  distribution function for different values of
  and solving for periodic solutions.
• Carry out a similar investigation of the
  reconnection rates.
• Carry out a similar investigation of the
  off-diagonal components of the pressure tensor as
  you make the transition from no shear field to a
  force-free field.
• Carry out force free simulations with low

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Example Equilibria
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Example Equilibria
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Examples Equilibria
Linear Force-Free Field
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