1. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Two-Fluid Applications of NIMROD D. C. Barnes University of Colorado at Boulder November 17, 2004 HP1-10

2. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Outline Two-fluid MHD Time-implicit method – a stability theorem NIMROD-2F implementation and dispersion tests FRC application Conclusions

3. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Abstract A time-implicit, two-fluid model has been implemented and tested within NIMROD. Numerical issues are discussed. For this, we introduce the property of spectral fidelity which asserts that all real frequency physical modes will have a real nimerical frequency. Next, it is shown that this property follows from implicit time differencing with a particular, natural time centering. We recast this as a predictor- corrector problem to reduce this system to that amenable to solution using the NIMROD numerics. We have observed that only a single corrector step is required, so that the resulting time step is equivalent to a modified leap-frog NIMROD step. Results for linear waves on a uniform equilibrium show robust stability and excellent dispersion. Initial results of applying this algorithm to the Hall stability of long FRC’s will also be presented.

4. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Single/Two-Fluid “MHD” (1F/2F)

5. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Can Rewrite Hall Term

6. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ 1F  2F Plasma Features Waves (uniform, unbounded plasma) –Whistler –Kinetic Alfven –Low frequency electrostatic Drift waves  * stabilization of MHD modes Electron-Ion decoupling

7. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Physics of HMHD Full (warm 2 fluid) dispersion relation has 3 waves (cubic) This was given by Stringer (1963) and is also discussed by Swanson

8. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Modest Parameters Give Extreme Stiffness E.G. = 10 -8, k || /k ┴ =10 -4, d i /L x = 0.1  12 orders of magnitude in  10 -6 10 -4.01 1 100 10 4 10 6 10 8 0.001 0.01 0.1 1 10 k || d i  A ii ee

9. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Modest Parameters Give Extreme Stiffness E.G. = 10 -2, k || /k ┴ =10 -4, d i /L x = 0.1  12 orders of magnitude in  10 -6 10 -4 0.01 1 100 10 4 10 6 10 8 10 -5 10 -4 0.001 0.01 0.1 1 10 k || d i  A A IIC SA KAW MA W ii ee

10. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Extension of 1F Method to 2F Challenging 1F of form 2F is not! Challenge is to obtain method for low dissipation –Real frequency physical modes should give real frequency numerical modes

11. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ A Useful Stability Theorem To simulate low dissipation cases, need “spectral fidelity” –If physical system has only real frequency modes, numerical system will have only real frequency modes (or controlled damping would also work) One case of sufficient conditions

12. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Some Possible Schemes Both have vanishing damping for low frequencies. 1L has no damping for high frequencies as well, while 2L gives strongest damping when = ¾ (|Z| = 1/3).

13. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Using NIMROD Machine for This Form Use Predictor/Corrector method Advance all equations (slave) except momentum (master) with trial u*~u n+1 Error in momentum gives correction Linear operator from linearizing change in slave variables in change in u P/C required because not possible to incorporate exact linear change of slave variables –  ≠  2 (use compact stencil for 2 nd order operator) –Slave equations contain operator inversion

14. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Some Numerical Details Care in integration by parts because of different BC Correct BC is to require only u n = 0

15. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Waves in a Box Tests

16. History for KAW Calculation

17. Profile for KAW Calculation

18. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Numerical Results show good Dispersion

19. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Convergence Acceleration of P/C Can consider present scheme as preconditioner Apply GMRES to iteration Some success with partial GMRES (1 lag level) For larger problems, can use direct solve per block to get Schwarz preconditioner

20. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Extension of Linear Solve Instead of master/slave scheme, solve all equations simultaneously (8 unknowns/ node instead of 3) No P/C required NIMROD machinery supports this (almost) Almost same as scheme(s) of Chacón and Glasser

21. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ NIMROD Initialized with FRC Equilibrium

22. University of Colorado at Boulder Center for Integrated Plasma Studies http://cips.colorado.edu/ Conclusion NIMROD 2F well under way –Waves in box test (nearly) passed –Early version into CVS (update soon) –All thermoelectric terms in (nonlinear?) FRC application beginning –Need better separatrix conditions Algorithm improvements continue in parallel with applications –Finite m e –GMRES, Block direct + Schwarz –Alternative: fully couple all equations