NONLINEAR COMPUTATION OF LABORATORY DYNAMOS DALTON D. SCHNACK Center for Energy and Space Science Science Applications International Corp. San Diego, CA 92121
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
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??
THE RFP MAGNETIC FIELD • Toroidal Z-pinch • Positive toroidal field in center • Negative toroidal field at edge • Near Taylor “relaxed state”
THE NEED FOR A LAB “DYNAMO” • Axisymmetric (Taylor) model: • After reversal, positive flux is affected by resistive diffusion • Depends on BC: • Bz(a) = const. < 0 ==> positive flux decays, total flux becomes negative • Flux = const. ==> reversal is lost • Require 3-D dynamical flux generation mechanism ==> “RFP Dynamo”
Relaxed State Nonlinear Relaxation RFP Dynamo Diffusion Instabilities 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
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 THEORY GAP “Zero-D” Relaxation Theory Full 3-D Nonlinear Computations
COMPUTATIONAL ISSUES • Resistive MHD • S > 104 • 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
COMPUTATIONS OF THE RFP DYNAMO • Toroidal effects not dominant => doubly periodic cylindrical geometry • Use resistive MHD model • External drive (voltage, current, and flux sources)
THE FIRST DYNAMO SIMULATION Sykes and Wesson, Proc. 8th Euopean Conf. On Controlled Fusion and Plasma Physics, Prague, p. 80 (1977) • 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
RESISTIVE MHD • Some simulations use the force-free model:
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
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?
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
CYCLIC TAYLOR RELAXATION • Diffusion drives plasma away from preferred state • Nonlinear modes drive it back
RELAXED PROFILES Decaying discharge
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
I Periodic BC I Line-tied BC z z r “Toroidal” direction “Toroidal” direction r Cylindrical RFP Spheromak SAME DYNAMO IN SPHEROMAK Sovinec • Spheromak is RFP with line-tied BCs • RFP: poloidal flux => toroidal flux • Spheromak: toroidal flux => poloidal flux
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)
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??
IT’S ALL ABOUT OHM’S 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
THE NIMROD CODE http://www.nimrodteam.org • 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
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 > 104 • Complete MPI implementation • Applications • Wide variety of fusion problems • Merging flux tubes (reconnection) • Collimation of magnetic field structures (astrophysical jets)
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