Sawteeth in Tokamaks and their relation to other Two-Fluid Reconnection Phenomena

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1 Sawteeth in Tokamaks and their relation to other Two-Fluid Reconnection Phenomena S. C. Jardin 1, N. Ferraro 2, J. Chen 1, et al 1 Princeton Plasma Physics Laboratory 2 General Atomics Supported by the RPI SCOREC Center: Mark Shephard, Fan Zhang (RPI) Supported by U.S. DOE Contract No. DE-AC02-09CH

2 Modern MHD simulation codes are able to reproduce many qualitative features of sawtooth in tokamaks Sawtooth Cycle hottest in center current peaks becomes unstable reconnection event flattens current repeats Fast temperature crash m=1 precursor Periodic behavior

3 Why study sawteeth now? Greatly improved 2Fluid 3D Implicit MHD codes that can follow the entire sawtooth cycle with high accuracy New 2D ECE imaging diagnostics that give greatly improved time and space resolution of temperatures A quantitative description of sawteeth is important for ITER as they make the discharge non-stationary and can drive other modes. 3

4 M3D-C 1 is an implicit 3D 2F MHD code. Why implicit MHD? Tokamaks are normally MHD stable However, because of diffusion and external sources, the pressure and current profiles will be slowly evolving on the transport timescale (slow) These profiles will evolve towards an unstable state in which a MHD instability develops and causes a rapid reconnection event that involves MHD stability(sawtooth). After this reconnection event, the configuration can again return to a quiescent state where transport is again the dominant process. This phenomena can repeat as in the case of sawtooth oscillations in tokamaks. Since perpendicular transport, parallel transport, reconnection, and stability timescales are vastly different, an implicit treatment is needed. 4

5 Non-linear 3D M3D-C 1 calculations contain both transport and stability timescales in same calculation resistive or 2F MHD also 2 variable reduced MHD and 4 variable reduced MHD Fully implicit time advance (no Courant condition) can take very long time steps Δt ~ τ A total simulation time up to 10 5 or 10 6 τ A free-boundary plasma can be separatrix limited plasma is surrounded by low-temperature plasma vacuum simulation domain is interior to vacuum vessel need to specify: transport model (η, μ, χ) sources for energy and particles loop-voltage on vacuum vessel (with controller)

6 Numerical Methods in M3D-C 1 High-order triangular wedge finite elements: Unstructured triangles in poloidal plane Structured in toroidal angle Continuous derivatives in all directions C 1 Eigenvalues of 3D velocity matrix BEFORE and AFTER preconditioning Split-Implicit method breaks time advance into 3 parts: Implicit velocity advance using Schur compliment Implicit magnetic field advance Implicit temperatures and density advance Partial separation of wave types due to representation of vector fields and annihilation operators: V = R U ϕ + R ω ϕ + R χ 2 A = ψ ϕ + R ϕ f F ln Rz 0 ˆ Condition number reduced from to 30!! Block-Jacobi preconditioner uses direct solves of 2D matrices within each plane to precondition full 3D matrix. Scales well up to over 10,000 processors 6

7 Physical Model The following equations are solved for the plasma density, the magnetic field, the fluid velocity, and the e - and ion temperatures : n + 2 ( nv) = D n + Sn t B = E ib = 0 μ0j = B t V nm i( + V V ) + p = J B i Π μ i Π GV t 1 E+ V B= ηj+ [ J B pe] p n( Te + Ti) ne 3 Te 2 n + Te + nte = ηj e + QΔ 2 Vi t iv iq 3 Ti n + Ti + nti = μ : i QΔ 2 Vi t iv Π V iq q = nκ T nκ bbi T Πμ = μ V+ V e, i e, i ei, ( ) These are the two-fluid MHD equations, and include the time evolution of the density and separate temperature equations for each species. 7

8 Depending on the transport model: sawteeth or internal helical states Some transport models and geometry lead to periodic sawtooth oscillations. Shown here are 6 sawtooth events (only first depends on initial conditions). Bottom shows Poincare plots at 4 times indicated. Other transport models lead to stationary states: Caution. This result may not be converged! 8

9 We suspect there are 3 timescales that scale differently with resistivity The temperature crash is very short compared to the reconnection time. We are now performing scaling studies to verify this. Time scale value scaling Between sawteeth Reconnection time Temperature crash time 10,000 τ A η? 1,000 τ A η 1/2? 200 τ A η 0?

10 68 T e0 =.0183 DIII D shot : Run26 69 T e0 = T e0 = T e0 = T e0 =.0154 t = t = t = t = t = Temperature change from previous frame

11 Compound Sawteeth In some cases, system goes into complex periodic behavior with period τ = 16,000 τ A 1. Island growth 2. Thermal crash 3. New axis forms 4. Re-circularizes 5. New axis tries to form 6. Second Te crash 7. New axis rejected 8. Resymmetrizes

12 Future Plans (with K. Lackner, S. Guenter, and IPP Grad Student) Study importance of 2-fluid effects In momentum equation and Ohm s law Study the physical mechanism of the temperature crash Comparison with 2D ECE and other data Study seeding of NTMs by sawtooth Dependence on delta-prime and other equilibrium properties Dependence on geometry Study triggering of sawtooth by applying external (1,1) perturbation Interpret snakes and clarify how both sawteeth and snakes can be present in the same discharge Clarify under what conditions one would expect stationary interior (1,1) helical deformation instead of sawteeth 12

13 Basic reconnection studies Next 2 vgs show results published 5 years ago with this same code on 2- fluid Harris-sheet reconnection without and with a guide field. We still use this as a test problem when modifying the code. 13

14 GEM Harris reconnection 2-Fluid reconnection benchmarked against NIMROD and HiFi Convergence study shows h 5 14

15 Effect of Guide Field on Reconnection Scaling formula for max reconnection rate from many runs. Addition of guide field dramatically reduces the reconnection rate. 15