1. NONLINEAR COMPUTATION OF LABORATORY DYNAMOS DALTON D. SCHNACK Center for Energy and Space Science Science Applications International Corp. San Diego, CA 92121

2. OUTLINE Brief summary of laboratory dynamo Role of large scale numerical simulations Examples of simulations of laboratory dynamo –Field reversal –Discharge sustainment –Confinement scaling The need for two-fluid modeling –Issues for numerical implementation The NIMROD code –Features and status –Summary

3. OVERVIEW OF LAB DYNAMO Laboratory dynamo is a nonlinear driven system Plasma is resistive Poloidal flux (toroidal current) sustained by applied voltage Toroidal flux is sustained by a dynamo Mediating dynamics –Nonlinear behavior of long wavelength MHD instabilities –Driven by resistive evolution of mean fields –Nonlinear evolution converts poloidal flux => toroidal flux –Like differential rotation Finite resistivity makes flux conversion irreversible No conversion of toroidal flux => poloidal flux –Poloidal flux supplied by external circuit Is it really a dynamo??

4. THE RFP MAGNETIC FIELD Toroidal Z-pinch Positive toroidal field in center Negative toroidal field at edge Near Taylor relaxed state

5. THE NEED FOR A LAB DYNAMO Axisymmetric (Taylor) model: –After reversal, positive flux is affected by resistive diffusion –Depends on BC: B z (a) = const. positive flux decays, total flux becomes negative Flux = const. ==> reversal is lost Require 3-D dynamical flux generation mechanism ==> RFP Dynamo

6. LAB DYNAMO AND RELAXATION More in common with relaxation theory (Taylor) than turbulent dynamo theories Fully nonlinear, but only a handful of coherent modes Low dimensionality makes realistic computations possible Relaxed State Diffusion Instabilities Nonlinear Relaxation RFP Dynamo

7. NEED FOR LARGE SCALE COMPUTATIONS Early theories invoked small scale turbulent processes –Unsuccessful in explaining experiments Full 3-D nonlinear simulations with realistic BCs required –An example where large scale computations led the way for theory and experiment Zero-D Relaxation Theory Full 3-D Nonlinear Computations

8. COMPUTATIONAL ISSUES Resistive MHD –S > 10 4 –Effieiency of spatial representation Long time scale evolution –Efficient implicit time advance Importance of boundary conditions –Applied voltages –Non-ideal boundaries –Applied magnetic fields Effective diagnostics Comparison with experimental data

9. COMPUTATIONS OF THE RFP DYNAMO Toroidal effects not dominant => doubly periodic cylindrical geometry Use resistive MHD model External drive (voltage, current, and flux sources)

10. THE FIRST DYNAMO SIMULATION Resistive MHD, S ~ 200, R/a = 1/2 Cylindrical geometry, 14 X 13 X 13 grid! Shows cyclic oscillations and sustainment of reversed field The RFP dynamo is very robust Subsequent calculations are refinements of this work Sykes and Wesson, Proc. 8th Euopean Conf. On Controlled Fusion and Plasma Physics, Prague, p. 80 (1977)

11. RESISTIVE MHD Some simulations use the force-free model:

12. ALGORITHMS Staggered finite differences in radius –Preserve annihilation properties of div and curl operators Dealiased pseudospectral in periodic coordinates (,z) Boundary conditions: –Conducting wall with applied voltages –Resistive wall –Concentric resistive walls Time advance –Leap-frog for waves –Predictor-corrector for advection –Semi-implicit for large time steps

13. TIME ADVANCEMENT ISSUES Fundamental differences in algorithm requirements? –Laboratory dynamo A few low order modes Relatively long times scales (compared with Alfvén time) Dictates implicit methods –Astrophysical dynamo Dominated by advection Dictates explicit methods Is a common code appropriate?

14. RESULTS Dynamo mechanism identified –Cyclic relaxation reproduced Source of anomalous loop voltage identified –Dynamo enhancement in presence of resistive wall Sustainment mechanisms investigated –Feedback –Profile modification (current drive, helicity injection) –Time dependent BCs Transport –Dynamo (relaxation) produces stochastic fields –Balance between Ohmic heating and thermal loss along stochastic field lines –Obtained confinement scaling laws

15. DYNAMO/RELAXATION AT FINITE-

16. SUSTAINMENT

17. CYCLIC TAYLOR RELAXATION Diffusion drives plasma away from preferred state Nonlinear modes drive it back

18. DYNAMO IS CHAOTIC

19. RELAXED PROFILES Decaying discharge

20. TRANSPORT DUE TO DYNAMO 3-D nonlinear calculations including: - Poynting and Ohmic input - Anisotropic heat flux Parametric studies determine scaling of confinement with global parameters

21. SAME DYNAMO IN SPHEROMAK Spheromak is RFP with line-tied BCs –RFP: poloidal flux => toroidal flux –Spheromak: toroidal flux => poloidal flux r z I Periodic BC Cylindrical RFP r z I Line-tied BC Spheromak Toroidal direction Toroidal direction Sovinec

22. COMPUTATION OF MRI MRI in doubly periodic cylinder Lab dynamo: liquid Gallium S = 2.59 Pr = 1.5 X 10 -2 Rotating and outer boundaries m : 0 => 21 n : -21 => 21 Computed with DEBS code (old technology)

23. STATUS AND DIRECTIONS Resistive MHD computations of laboratory (RFP) dynamo are mature –Basic nature and consequences of RFP dynamo elucidated –Good agreement with experiment Recent measurements show resistive MHD is not sufficient for details –Hall dynamo –Diamagnetic dynamo Require electron (2-fluid) dynamics FLR effects?? Neutrals ?? (astrophysical dynamos) MRI??

24. ITS ALL ABOUT OHMS LAW Two-fluid relaxation theory (Mahajan & Yoshida, Steinhauer & Ishida, Hegna, Mirnov) –Flows enter on equal footing with magnetic fields –Flows are ubiquitous in laboratory plasmas –Importance to astrophysical plasmas?? Computational difficulty: –Dispersive waves can limit time step –Higher order operators –Require specialized linear algebra packages

25. DISPERSIVE WAVES

26. THE NIMROD CODE Resistive MHD –Development essentially complete –Thoroughly tested and benchmarked on a variety of problems –Mixed finite-element/pseudo-spectral representation –Semi-implicit formulation allows arbitrary time steps Two-fluid model –Explicit two-fluid model available –Hall term and electron pressure –Energetic minority ion species Semi-implicit two-fluid model being tested –Based on dispersive wave operators –Will allow large time steps Several FLR formulations under consideration –Full gyro-viscous stress tensor –Drift-MHD and the gyro-viscous cancellation http://www.nimrodteam.org

27. NIMROD FEATURES Flexible geometry and boundary conditions –Axially symmetric boundary with arbitrary poloidal (R, Z) cross-section –Slab with a variety of boundary conditions Highly accurate spatial representation –Arbitrary order finite elements in poloidal plane –Pseudo-spectral/FFT in toroidal dimension Accuracy for highly anisotropic problems –Highly accurate semi-implicit time advance –Routinely compute with CFL > 10 4 Complete MPI implementation Applications –Wide variety of fusion problems –Merging flux tubes (reconnection) –Collimation of magnetic field structures (astrophysical jets)

28. SUMMARY Laboratory dynamo can be accurately computed –Closely related to plasma relaxation –Resistive MHD model –Reveal underlying dynamics and consequences –Computations led theory and experiment –Possible because of low dimensionality Relevance to other fields? –Solar dynamo? –Formation and disruption of coronal structures –Other quasi-periodic phenomena? Extensions –Two-fluid modeling Necessary to explain detailed experimental measurements May account for ubiquitous plasma flows NIMROD and DEBS are available for collaborative applications http://www.nimrodteam.org