Non-Inductive Startup and Flux Compression in the Pegasus Toroidal Experiment John B. O Bryan University of Wisconsin Madison NIMROD Team Meeting July 31, 2009
Outline 1 Introduction and Motivation 2 Modeling Basic Modeling Considerations Perpendicular Thermal Conduction 3 Results Zero-Beta Flux Compression Scaling Study Axisymmetric Finite-Beta Flux Compression Results Preliminary Relaxation Results 4 Future Work Outline Non-Inductive Startup in Pegasus 2/22
Non-Solenoidal Startup Is Being Investigated on Pegasus Spherical tokamaks have limited V-s for ohmic drive DC helicity injection through washer-gun plasma sources Small gun configuration: can be mounted through a diagnostic port High-Z impurities are trapped within the gun aperture Up to 4kA injected current per gun with ion temperatures up to 50eV Plasma guns can be mounted in two configurations In the divertor region At the device outboard midplane, which is the configuration of interest Figure: Pegasus Cut Away Introduction and Motivation Non-Inductive Startup in Pegasus 3/22
Up to Three Plasma Guns May Be Used Simultaneously Injection Radius, R = 0.70 m Span of Plasma Gun Array, 2b = 0.30 cm Vertical offset between gun array and anode of approximately 40cm. Current channels relax into a tokamak-like plasma with higher toroidal current. Flux compression applied on the relaxed plasma results in current amplification. Figure: Pegasus Cross Section Introduction and Motivation Non-Inductive Startup in Pegasus 4/22
Initially Modeled a Single-Turn Current Loop Analogous to discharges where the current loop terminates on an anode directly behind the plasma gun. Experimentally observed current amplification: 1kA injected current 2kA plasma current Rate of current loop compression towards the inboard side of the device is dependent upon the injected current. Current multiplication also somewhat dependent upon the injected current. Used as a mockup of the relaxed plasma state for the flux compression studies Introduction and Motivation Non-Inductive Startup in Pegasus 5/22
Basic Modeling Considerations Figure: Poloidal Flux Vacuum magnetic fields are generated by modeling external coils with simple current loops. Flux compression is achieved by uniformly incrementing B z on the domain boundary. The vertical magnetic field increases linearly with time. Current injection is modeled by adding an ad-hoc force density in a poloidally (& toroidally) localized spot that acts as a modification to Ohm s Law. E + v B = η (J J inj ) Modeling Non-Inductive Startup in Pegasus 6/22
A Demagnetization Correction Is Included to Prevent Unphysically Large Perpendicular Thermal Conduction in Regions of Low-Temperature (ω cs τ s < 1) x s = ω csτ s κ,s = nstsτs m s γ 0 δ 0 κ,s = nstsτs m s γ 1 x 2 s + γ 0 x 4 s + δ 1x 2 s + δ 0 lim κ,s = κ,s Ts 5/2 x 1 lim κ,s = nstsτs γ 1 x 1 m s xs 2 Ts 1/2 B 2 NIMROD normally uses the high-magnetization (x s 1) formulation for perpendicular thermal conductivity. In regions of low temperature (x s < 1), e.g. edge regions, κ > κ. This can lead to a severe loss of confinement. These simulations have a cold background plasma, so the full formulation [1] for κ will realistically limit perpendicular thermal transport and enhance confinement. The correction does not signficantly alter the total energy of the system unless suppressing significant heat loss across the domain boundary. [1] S.I. Braginskii, Transport Processes in a Plasma, Reviews of Plasma Physics. 1965. Modeling Non-Inductive Startup in Pegasus 7/22
Without the Demagnetization Correction, κ Deviates from the High Magnetization Limit at ω cs τ s 5 Modeling Non-Inductive Startup in Pegasus 8/22
Implementation of the Demagnetization Correction into NIMROD is Straightforward Fields added for (optional) toroidally-dependent perpendicular thermal conductivities. Modification to subroutines tirhs & terhs are largely trivial simple variable replacement. Exception: χ contribution [1] needs to be added before FFT. «n T γ 1 t + v T = p v + Q i + n h χ χ bb + χ I T [1] C.R. Sovinec et. al, Nonlinear Magnetohydrodynamics Simulation Using High-Order Finite Elements. J. Comput. Phys. 195, p. 355. 2004. κ,s is calculated normally, but moved before κ,s in the subroutine find kappa t. Magnetization is calculated using 3D fields, found using the usual NIMROD IFFT. κ,s coefficients are calculated from the κ,s and magnetization, using the previously given relations. Demagnetization correction is left optional Still update n=0 perpendicular thermal conduction Calculate κ,s in find kappa t for p model= aniso plltdep Modeling Non-Inductive Startup in Pegasus 9/22
Results from Zero-Beta Cases For a Lundquist number larger than some critical value (S 100), the current multiplication is roughly constant. The observed current amplification (M) is dependent upon the normalized injected current, I N (a is the current channel radius): I N = µ 0I inj 2πaB z M = I φ,max I inj The maximum observed current multiplication factor, M 2.09 occurs at I N 1. Larger injected currents compress slightly faster, but overall compression rate mostly dependent upon vertical field ramp. Results Non-Inductive Startup in Pegasus 10/22
The Variation of Maximum Current Multiplication Observed (With Respect to Lundquist Number) is Shown for Three Normalized Currents. Results Non-Inductive Startup in Pegasus 11/22
The Variation of Maximum Current Multiplication Observed (With Respect to I N ) is Shown in the Ideal Limit. Results Non-Inductive Startup in Pegasus 12/22
The temporal-evolution of the toroidal current for a zero-beta case (I N = 1.031) is shown. Lundquist number, S = 1.034 10 4 The current multiplication factor for this case is M = 2.086. The times by indicated by vertical bars correspond to the contours on the next slide. Results Non-Inductive Startup in Pegasus 13/22
Contours of the toroidal current and poloidal flux for a zero-beta case (I N = 1.031) are shown. t = 0 µs t = 7.39 µs t = 63.51 µs Results Non-Inductive Startup in Pegasus 14/22
Axisymmetric Finite-Beta Results Thermal Equilibrium Achieved Before Flux Compression t = 11.04 µs t = 5.75 µs Results Non-Inductive Startup in Pegasus 15/22
Axisymmetric Finite-Beta Results Thermal Equilibrium Achieved Before Flux Compression t = 40.82 µs t = 26.23 µs Results Non-Inductive Startup in Pegasus 16/22
The Toroidal Current Evolution is Similar to that of a Zero-Beta Case. Iφ [A] time [s] Results Non-Inductive Startup in Pegasus 17/22
Axisymmetric Finite-Beta Results Using Temperature Estimates t = 1.135 ms t = 1.007 ms Results Non-Inductive Startup in Pegasus 18/22
Axisymmetric Finite-Beta Results Using Temperature Estimates t = 3.877 ms t = 1.135 ms Results Non-Inductive Startup in Pegasus 19/22
Significant heating results from the flux compression. x10 4 2.5 3.0 3.5 I Total and n=0 Current vs. t 2 4 6 8 x10-3 t Internal Energy (tot, el, ion) vs. t E 0 20 40 60 80 100 120 2 4 6 8 x10-3 t Results Non-Inductive Startup in Pegasus 20/22
A curved boundary terminates the current channel for the relaxation simulations, instead of attempting to directly model the anode in NIMROD. x10-1 -6-4 -2 0 2 4 6 Z Finite Element Mesh -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 R Attempting to reproduce tokamak-like plasma Current Channels: I inj = 4 ka R inj = 0.70 m Z inj [ 0.35, 0.05] m D 6 cm Hopefully, relaxation study will provide insights for flux compression calculations Temperature and current profiles Lack of experimental measurements Numerical stability issues Results Non-Inductive Startup in Pegasus 21/22
Future Work Revisiting Zero-Beta Flux Compression Injection Radius Varying Current Channel Geometry Magnetic Prandtl Number Finite-Beta Flux Compression Sensitivity to peak temperature, resistivity & thermal conduction Effects on compression, both major and minor radial Need more insights into realistic temperature profiles Relaxation Study Reproduce tokamak-like plasma Insights into temperature and current profiles Future Work Non-Inductive Startup in Pegasus 22/22