PICMCC Simulation of a CuspConfined Plasma

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PIC/MCC Simulation of a Cusp-Confined Plasma

• By Christian Zuniga

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Overview

• Background on cusp-confined plasmas
• Previous simulation of the system
• Method of solution of PIC program
• Conclusions

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Introduction Magnetic Cusps

• First used as a possible confinement system for
  thermonuclear plasmas
• - MHD stability at high ß
• - High particle loss rate through the cusps
• Multi-polar cusps now used to confine lower
  temperature plasmas for ion-beam sources and
  basic plasma studies
• Particle leak-width at the cusp is still not well
  understood

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Cusp Leak-Width

• Both experiments and theories disagree on the
  size of the leak-width
• - 2
• -
• The leak-width has been attributed to numerous
  electrostatic, collision, and turbulence effects

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Previous Computer Simulation

• In the early 80s, Marcus, Knorr, and Joyce made a
  PIC program to simulate a plasma in a periodic
  picket fence.
• Program characteristics
• - 2 dimensional (drifts neglected)
• - Wire spacing 16 real length
• - 64-256
• - Collisionless low-ß plasma

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Simulation Results

• Relatively good agreement with experiments by
  Hershkowitz
• - Development of a quasi-steady state
• - Absence of any violent instabilities
• - Development of a deep potential channel
  between the wires
• - Occurrence of potential hills in front of the
  wires
• - Effective leak-width on the order of the
  hybrid radius (inconclusive)

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Motivation for PIC/MCC Simulation

• Electrostatic effects are believed to be
  important in the confinement of ions
• Electrostatic effects are thought to affect the
  leak width.
• Collisional effects also affect the leak width

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Computer Model

• Electrostatic approximation
• Deals with high frequencies and small space
  scales on the order of the Debye length
• Currents presumed small so that the self magnetic
  field is small
• Equations to solve for

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Computation Cycle
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Field Equations

• 5 pt difference approximation to Poissons
  equation solved in a 2 dim mesh using the FACR(1)
  algorithm by Hockney. The error term is
• higher order terms
• FACR algorithm is part of a class of algorithms
  (RES) that use direct methods to obtain the exact
  solution to a set of Ng difference equations in a
  number of ops proportional to (or better) and
  have a storage requirement proportional to Ng

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Poisson Solver

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Poisson Solver (Steps 1-2)

• Elimination of the odd lines
• Fourier Analysis on even lines only

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Poisson Solver (Step 3)

• Solution of harmonics by recursive cyclic
  reduction

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Poisson Solver (Steps 4-6)

• Fourier synthesis to find the potential on the
  even lines
• Charge modification on the odd lines
• Solution on the odd lines by recursive cyclic
  reduction

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Weighting

• Charge interpolation and field interpolation use
  the same weighting scheme to conserve momentum by
  avoiding the self-force
• For linear interpolation in 2 dimensions

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Lorentz Force Integrator

• Leap-frog Integrator (2nd order accurate)
• Separate electric and magnetic forces (Boris)

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Monte Carlo Collisions

• Plasma neutral collisions (Burger)
• Probability of a collision for each particle
  traveling with velocity v(t) in a time interval
  DT
• Determine in a random manner which particles
  will suffer a collision.
• Determine the velocities of the particle after a
  collision
• Use of null-collision method (Boeuf and Marode)
  to speed up calculations.

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Diagnostics

• Energy calculation
• Snapshots of the potential, particle positions
  and velocities
• Record of particles escaping the system in the
  cusp region

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Collision and Heating Times (Hockney)

• The collision time gives an estimate of the time
  for which the simulation can be run before
  collisional effects become important.
• The heating time is a measure of the failure of
  conservation of energy in the computer model.

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Particle Loading

• Electrons and ions loaded with a Maxwellian
  velocity distribution for each and are uniformly
  distributed in the upper area. Te/Ti 4-8
  Te1-4 eV
• Velocities calculated from

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Computer Quantities

• Charge to mass ratio preserved, but the charge
  contribution from each particles is obtained from

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Particle Handling

• Particles crossing a periodic boundary are placed
  an equal distance on the opposite boundary but
  with their z velocity reversed
• Particles crossing the upper boundary are
  reflected
• Particles crossing the lower boundary are
  counted. The electrons are replaced less
  frequently than the ions.

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Conclusions

• Presently the program is collisionless and is
  giving promising results
• Will also solve Poissons equation with the BC in
  y reversed.
• Collisions need to be added.
• Program requires further optimization