1. Resistive Modes in CDX-U J. Breslau, W. Park. S. Jardin, R. Kaita – PPPL D. Schnack, S. Kruger – SAIC APS-DPP Annual Meeting Albuquerque, NM October 30, 2003

2. Abstract CDX-U is an attractive device to model to benchmark resistive MHD codes. Its small size and low temperature (S < 10 5 ) make it possible to simulate MHD events in it using actual experimental parameters in a reasonable time on present- day computers. The dominant MHD activity during normal operation is the sawtooth oscillation, a resistive internal kink mode with toroidal mode number n=1 and dominant poloidal mode number m=1 [1]. We model both the linear growth of the instability and the nonlinear reconnection event at the q=1 rational surface (the sawtooth crash) as a test problem for both the M3D [2] and NIMROD [3] codes. Depending on the resistivity model employed and on the initial value of q 0, the crash can either lead to a disruption or to a quiescent state with q on axis above 1. [1] D. Stutman et al., Plasma Phys. Control. Fusion 41, 867 (1999). [2] W. Park et al., Phys. Plasmas 6, 1796 (1999). [3] C.R. Sovinec et al., Phys. Plasmas 10, 1727 (2003).

3. Characteristics of the Current Drive Experiment Upgrade (CDX-U) Low aspect ratio tokamak (R 0 /a = 1.4 – 1.5) Small (R 0 = 33.5 cm) Elongation  ~ 1.6 B T ~ 2300 gauss n e ~ 4  10 13 cm -3 T e ~ 100 eV I p ~ 70 kA Soft X-ray signals from typical discharges indicate two predominant types of low-n MHD activity: –sawteeth –“snakes”

4. Equilibrium at t=12.25ms (as q 0 drops to 0.95) is used to initialize 3D runs TSC follows 2D (axisymmetric) evolution of typical CDX-U discharge The TSC Sequence

5. Typical Case Questions to investigate: –Linear growth rate and eigenfunctions –Nonlinear evolution disruption? stagnation? repeated reconnections? minor radius Equilibrium taken from a TSC sequence (Jsolver file). R/a = 1.4 q(0)  0.955 q(a) ~ 10 toroidal current density

6. Typical Baseline Parameters Lundquist Number S ~2  10 4 on axis. Resistivity  Spitzer profile  T eq -3/2. Prandtl Number Pr10 - 100 on axis. Viscosity  Constant in space and time. Perpendicular thermal conduction   ~200 m 2 /sec (CDX-U energy confinement time) Parallel thermal conduction  || =   for main (isotropic) case. =10 8 m 2 /sec for anisotropic NIMROD case. Plasma  ~10 -2 (low-beta). Density EvolutionTurned on for nonlinear phase.

7. Comparing the Codes Uses linear finite elements in-plane. Uses finite differences between planes. Partially implicit treatment allows efficient time advance but requires small time steps. Linear operation: full nonlinear + filtering, active equilibrium maintenance. Nonlinear operation: all components of all quantities evolve nonlinearly. M3D and NIMROD are both parallel 3D nonlinear extended MHD codes in toroidal geometry maintained by multi-institutional collaborations, and comprise the two members of the Center for Extended MHD Modeling (CEMM) SciDAC. Uses high-order finite elements in- plane. Uses Fourier decomposition in toroidal direction. Fully implicit treatment requires costly matrix inversions but allows large time steps. Linear operation: evolve perturbations to particular modes only. Nonlinear operation: perturbations to fixed equilibrium are evolved, with nonlinear couplings between modes. M3DNIMROD

8. 40  24 structured grid 4th order basis functions on quadrilateral elements Conducting wall Fourier decomposition toroidally; 10 or more modes retained 70 radial zones, up to 210 in  in unstructured mesh Linear basis functions on triangular elements Conducting wall Finite differences toroidally; 16 or more planes M3D Poloidal Meshes for the CDX-U Case

9. M3D RESULTS

10. Linear n=1 Eigenfunctions, Pr = 10 Large m = 1 mode at two minor radii Higher m components are insignificant  A = 3.76  10 -3 BB temperature incompressible poloidal velocity stream function J  isosurface

11. The Nonlinear Phase n = 1 first decays, then exceeds linear prediction. Higher n modes appear to couple nonlinearly, grow to disruption. linear prediction 17,367 Total kinetic energy Kinetic energy by mode number

12. Current Flattening in the Nonlinear Phase Before After J  (surface plot) /0/0 q(r)q(r) minor radius

13. Magnetic Field Poincaré Section Series for Nonlinear Phase  A (t-t 0 ) = 2.17  A (t-t 0 ) = 2.06  A (t-t 0 ) = 0.00  A (t-t 0 ) = 1.81 Initial state1,1 mode dominantHigh-m islands growingStochastic The narrowing of the current channel leads to a disruption.* *This result applies to a case with a current density that peaks in time. A more realistic resistivity model that more closely resembles the CDX-U experiment is now under study. It is not expected to yield the same current profile behavior, so the present result should not be taken as a prediction of disruption in the experiment.

14. NIMROD RESULTS

15. Linear Eigenfunctions n = 1 Pr = 10 = const.  ~ T eq -3/2 (fixed) Isotropic heat flux: –   = 200 m 2 /sec Small m = 1 Higher m components at inboard edge Density evolution (no effect on linear behavior)  A = 3.7  10 -4 BB JJ n P

16. Mode Structure Low aspect ratio Low-n field lines make more turns on inboard side Mode localized along equilibrium field line will have more structure on inboard side Higher-n ? Equilibrium field line with pitch m = 3, n = 1

17. Effect of toroidal mode number n Modes occur at edge of discharge Move slightly outboard with increasing n Growth rate increases with n n = 1 n = 2 n = 3 n = 4n = 5  A = 3.1  10 -4  A = 6.1  10 -3  A = 4.9  10 -3  A = 3.4  10 -3  A = 3.8  10 -4

18. Nonlinear Phase Linear n = 1 is dominantly 1/1 (“sawtooth”) Nonlinear n = 1 changes from 1/1 to 2/1 Finite-sized 2/1 mode grows nonlinearly in Rutherford regime No indication of high-n instability n = 1 n = 0 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 log 10 ( Kinetic Energy) Case 2: q 0 =0.97 log 10 ( Kinetic Energy) Case 1: q 0 =0.95 n = 10, 9,… most unstable Lower n (e.g., n = 1) now nonlinearly driven:  n=1 ~  n=9 +  n=10 Different from linear picture

19. n = 1 mode Changes After Reconnection Before sawtooth saturation After sawtooth saturation t = 2.5 X 10 -4 sec. t = 4.20 X 10 -4 sec. 1/1 with harmonics2/1 with harmonics

20. Conclusions There is a qualitative agreement between the linear eigenfunctions found by M3D and the 1,1 mode seen during the NIMROD run labeled “Case 2”. The M3D nonlinear result shows a possible route to disruption for tokamaks in which narrow current channels can form. NIMROD’s “Case 1” eigenfunctions, with q 0 = 0.95, are anomalous. This seems likely to be a result of the quality of the initial equilibrium, which is now under investigation.

21. Future Work Both codes should run converged linear and nonlinear studies on the same case for comparison. –M3D will switch on current drive to maintain equilibrium q profile. –More physically accessible and well-behaved equilibria, e.g. q 0 =0.97, will be considered. Several effects have been seen that merit follow-up study: –Presence and stability of high-m modes on inboard side in spherical tori, and their sensitivity to q 0. –Nonlinear triggering of higher-m modes off of 1,1 sawtooth crashes. –Current channel narrowing as a cause of disruptions.