Algorithm Development for the Full Two-Fluid Plasma System

1. Algorithm Development for the Full Two-Fluid Plasma System University of Washington Department of Aeronautics & Astronautics John Loverich Ammar Hakim Uri Shumlak

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

3. 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 Motivation Finite volume methods and discontinuous Galerkin methods have been used extensively in fluid mechanics. We would like to apply the same methods to Maxwell’s equations for the purpose of simpler algorithm design.

4. 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.

5. Full Two-Fluid Plasma Model: Maxwell’s Equations Ampere’s Law Faraday’s Law Poisson’s Equation Magnetic Flux The fluids are coupled to each other through the electromagnetic fields.

6. Maxwell’s 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.

7. Perfectly Hyperbolic Maxwell’s Equations Another approach to dealing with the divergence conditions is to use the perfectly hyperbolic Maxwell’s 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.

8. 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 …

9. Discontinuous Galerkin Method Constant term Linear variation Solution does not need to be continuous at cell edges

10. 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,

11. 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.

12. 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

13. Collisionless Reconnection Image borrowed from Journal of Geophysical Research, Vol. 106, No. A3, Pg 3721-3735, March 1, 2001

14. Collisionless Reconnection The following simulation is based off a widely explored collisionless magnetic reconnection problem called the GEM challenge.

15. 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 T=25/Wci

16. 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 T=25/Wci

17. 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

18. 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.