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NONLINEAR COMPUTATION OF LABORATORY DYNAMOS DALTON D. SCHNACK Center for Energy and Space Science Science Applications International Corp. San Diego, CA 92121 Slide 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 Slide 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?? Slide 4 THE RFP MAGNETIC FIELD Toroidal Z-pinch Positive toroidal field in center Negative toroidal field at edge Near Taylor relaxed state Slide 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 Slide 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 Slide 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 Slide 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 Slide 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) Slide 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) Slide 11 RESISTIVE MHD Some simulations use the force-free model: Slide 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 Slide 13 TIME ADVANCEMENT ISSUES Fundamental differences in algorithm requirements? Laboratory dynamo A few low order modes Relatively long times scales (compared with Alfvn time) Dictates implicit methods Astrophysical dynamo Dominated by advection Dictates explicit methods Is a common code appropriate? Slide 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 Slide 15 DYNAMO/RELAXATION AT FINITE- Slide 16 SUSTAINMENT Slide 17 CYCLIC TAYLOR RELAXATION Diffusion drives plasma away from preferred state Nonlinear modes drive it back Slide 18 DYNAMO IS CHAOTIC Slide 19 RELAXED PROFILES Decaying discharge Slide 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 Slide 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 Slide 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) Slide 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?? Slide 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 Slide 25 DISPERSIVE WAVES Slide 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 Slide 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) Slide 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