Simulations of Sawteeth in CTH. Nicholas Roberds August 15, 2015

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1 Simulations of Sawteeth in CTH Nicholas Roberds August 15, 2015

2 Outline Problem Description Simulations of a small tokamak Simulations of CTH 2

3 Sawtoothing Sawtoothing is a phenomenon that is seen in all tokamak devices Lots of previous numerical studies have focused on sawtoothing First MHD simulation of a sawtooth crash in the late 1970s. First simulation with successive sawtooth crashes in the late 1980s. 3

4 The Sawtooth Oscillation Cycle 1) q0 < 1 2) Resistive internal kink mode becomes unstable 3) Island grows exponentially well into the nonlinear phase Center of island will become the new magnetic axis 4) Flux inside q=1 is completely or partially reconnected Temperature, q, current density profiles are flattened 5) q0= 1 after crash with complete reconnection 6) Temperature, current density profiles peak again due to ohmic heating 7) Go to 1) 4

5 Sawteeth are observed in CTH The Compact Toroidal Hybrid (CTH) is a tokamak-stellarator hybrid Has a stellarator field with N=5 periodicity Has significant ohmically driven plasma current We would like to simulate CTH sawteeth 5

6 Sawtooth Simulations Initial conditions: Ideal MHD equilibrium from VMEC with q0 > 1 Start in the stable side of parameter space Electric field drives q0 < 1 Produces a self-consistent equilibrium For tokamak cases, can choose to evolve only n=0 until q0 is driven below 1 Increase toroidal resolution when q<1 Saves computational resources 6

7 ηχ-mhd Is Used in Simulations Temperature dependent kperp Value of kpll usually clamped to a large value Spitzer resistivity Must take considerations to prevent m=2 islands from blowing up after first crash q=2 near conducting wall Electron temperature offset of ~30 ev 7

8 Simulations with Periodic Sawteeth Careful choice of simulation parameters periodic or quasi-periodic sawtooth crashes Some important parameters Viscosity kperp kpll Iplasma 8

9 Small Tokamak, Baseline case 33 successive crashes 6 toroidal modes n=5 mode energy larger than n=3 mode energy (under-resolved) 30x20 finite elements of degree 3 S ~ 1.4E5 9

10 Small Tokamak, Baseline case 33 successive crashes Show movie 10

11 Small Tokamak, Low k_pll Relaxes to n=1 helical equilibrium with no sawtoothing k_pll ~ 1E7 at center with T dependence k_pll in previously shown cycling simulation uses k_pll=1e20 with k_pll_max=1e7 to remove T dependence in k_pll Both simulations have T dependant k_perp 11

12 Small Tokamak, Higher Resolution Periodic Cycles n=0 n=1 n=1 n=21 22 toroidal modes 30x30 finite elements of degree 3 S ~ 1.6E5 12

13 Small Tokamak, Higher Resolution Linear Phase VR component of the unstable n=1 mode Unstable mode is n=1 Has flow pattern of a rigid displacement Consistent with flow pattern of resistive internal kink mode 13

14 Small Tokamak, Higher Resolution Oscillations in Te Become Inverted Away from Core Electron temperature Inboard Side at phi=0 Time (ms) Outboard Side at phi=0 Time (ms) 14

15 Small Tokamak, Higher Plasma Current Activity seen after crashes Plasma current increased from 105 ka to 115 ka Activity can be seen after the complete reconnection of crash Can reduce kperp to eliminate this activity Possible explanation: Quasi-interchange mode unstable after crash Show movie 15

16 Small Tokamak, Higher Plasma Current Fastest growing mode just after crash is n=1 Flow pattern similar to quasi-interchange mode Quasi-interchange associated with a flat q profile, with q~1 Equilibrium at Step Flow Field of n=1 Mode 16

17 Small Tokamak, Higher Plasma Current Poincare plots just after crash look consistent with quasi-interchange Step Step

18 Small Tokamak, Higher Plasma Current Movies Poincare plots over Jphi Flow vectors over Jphi Electron temperature volumetric rendering 18

19 Simulation with CTH Equilibrium Field n=0 n=5 S ~ 1.2E5 43 toroidal modes 30x30 finite elements with degree 3 tmax = 5E-8 n=10 19

20 CTH, Linear Phase Unstable mode in CTH is represented with many Fourier numbers A tearing mode analogous to the small tokamak case grows The tearing mode is represented with fourier numbers n=1, 4,6,9,11,14,16,19,21,... CTH has a stellarator field period of 5 n is not a good quantum number when stellarator fields are added n=1 4,6 9,11 39,41 20

21 CTH, Linear Phase Comparison with tokamak linear phase n=1 4,6 9,11 39,41 n=1 21

22 Unstable mode in CTH Images show fourier components of VR of the unstable mode. Because CTH has a stellarator field period of N=5, many fourier numbers are needed to represent this tearing mode. n=1 n=4 n=6 n=9 n=11 n=14 Higher n values have finer structure 22

23 CTH, Nonlinear Phase Time step is under resolved Affects linear growth rates, influencing τsaw q0,min A simulation that cycles with tmax = 5E-8 may not cycle with tmax = 2E8 Using a very small maximum timestep might be important in simulations of any 3D device Tokamak-stellarator hybrids Perturbed Tokamaks RFPs 23

24 Discussion ηχ-mhd simulations demonstrating repeated sawtooth crashes have been done Convergence properties of CTH cases should be further explored Future work to vary stellarator field strength in simulations to reproduce experimental scalings of sawtooth period Future work may explore 2-fluid sawtoothing 24

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