Algorithm Development for the Full Two-Fluid Plasma System

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Algorithm Development for the Full Two-Fluid
Plasma System
University of Washington Department of
Aeronautics Astronautics

• John Loverich
• Ammar Hakim
• Uri Shumlak

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Overview

• Motivation
• Full Two-Fluid Model
• Preserving Divergence
• Potential formulation
• Auxiliary variables
• Discontinuous Galerkin Method
• Collisionless Reconnection

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Motivation

• MHD is invalid in many plasma regimes
• Microinstabilities and anomalous transport
• Lower Hybrid Drift instability
• Modified Two-Stream instability
• Electron Kelvin Helmholtz instability
• Weibel instability
• Two-fluid stability - FRC, z-pinch
• Collisionless reconnection

Finite volume methods and discontinuous Galerkin
methods have been used extensively in fluid
mechanics. We would like to apply the same
methods to Maxwells equations for the purpose of
simpler algorithm design.
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Full Two-Fluid Plasma Model 5 Moment Fluid
Equations
Species Continuity
Species Momentum
Species Energy
There are two fluids, electron fluid and ion
fluid, each with complete inviscid fluid
Equations Lorentz force source terms. Higher
moments of the Vlasov (collisionless Boltzmann)
equation can be taken to improve the plasma model.
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Full Two-Fluid Plasma Model Maxwells Equations
Amperes Law
Faradays Law
Poissons Equation
Magnetic Flux
The fluids are coupled to each other through the
electromagnetic fields.
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Maxwells Equations Mixed Potential Formulation
The finite volume method absolutely required
divergence cleaning in order to get proper
solution to problems with in plane magnetic
fields.
The potential equations can be used to ensure the
divergence equations are satisfied.
The potential equations are re-written as 16
first order equations so that Riemann solvers can
be applied.
The Lorentz gauge condition must still be
satisfied. Errors in this constraint remain
small.
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Perfectly Hyperbolic Maxwells Equations
Another approach to dealing with the divergence
conditions is to use the perfectly hyperbolic
Maxwells equations
Auxiliary variables are used to propagate errors
in the solution out of the domain at some
pre-determined speed.
We have not yet noticed any unphysical effects
produced by the auxiliary variables.
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Why use the discontinuous Galerkin method?
Source terms included naturally Temporal
accuracy long time integration Spatial accuracy
High order methods are good at balancing
sources and fluxes near equilibrium (this is very
important in two-fluid equations) Divergence
Finite volume methods require divergence
cleaning to gain solutions to problems with in
plane magnetic fields Explicit easy to
parallelize Efficiency Higher order methods can
be computationally more efficient
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Discontinuous Galerkin Method
Constant term
Linear variation
Solution does not need to be continuous at cell
edges
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Discontinuous Galerkin Method
Start with a general balance law,
The Q are represented as a linear combination of
basis functions
Multiply the balance law by the same set of basis
functions and integrate over a volume element,
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Discontinuous Galerkin Method
Move the derivative off the flux F and onto the
basis functions using integration by parts,
In regular geometries with orthogonal basis
function the equation becomes.
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Discontinuous Galerkin Method
We still have a few things to evaluate.
Surface Fluxes Approximate Riemann
Flux Integrals Gaussian Quadrature Time
Derivatives Runge-Kutta methods
Extension to general geometries is very easy!
Calculate Jacobians at each quadrature
point Calculate basis function gradients in
global coordinates Calculate a local mass matrix
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Collisionless Reconnection
Image borrowed from Journal of Geophysical
Research, Vol. 106, No. A3, Pg 3721-3735, March
1, 2001
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Collisionless Reconnection
The following simulation is based off a widely
explored collisionless magnetic reconnection
problem called the GEM challenge.
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Collisionless Reconnection
After 25/Wci the 2nd order solution differs
substantially from the 3rd order due to the
formation of a large magnetic island in the 2nd
order solution.
Total electron current at T25/Wci
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Collisionless Reconnection
At a resolution of 512X256 the 2nd and 3rd order
methods are essentially the same. The 3rd order
method achieves a correct solution at lower grid
resolution.
Total electron current at T25/Wci
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Collisionless Reconnection
Comparison of reconnected magnetic flux for the
full two-fluid solution using 3rd order
discontinuous Galerkin method against solutions
published by M. Shay, Journal of Geophysical
Research, Vol. 106 No A3, Pg 3759-3772
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Conclusion

• We are interested in the two-fluid plasma model
  because MHD is inadequate.
• A discontinuous Galerkin method for the two-fluid
  plasma system has been described.
• Techniques that preserve divergence have been
  successfully applied to the two-fluid system.
• The algorithm produces results in agreement with
  other techniques.