Multiscale, multiphysics modeling of turbulent transport and heating in collisionless, magnetized plasmas

Transcription

1 Multiscale, multiphysics modeling of turbulent transport and heating in collisionless, magnetized plasmas Michael Barnes Plasma Science & Fusion Center Massachusetts Institute of Technology Collaborators: W. Dorland (Maryland), F. I. Parra (MIT), G. W. Hammett (PPPL), I. G. Abel, A. A. Schekochihin (Oxford), S. C. Cowley (Culham), T. Gorler (IPP-Garching), F. Jenko (IPP-Garching),

2 Overview Collisionless plasma dynamics are hard Let s make life simpler: a mean field approach for turbulence and transport/heating in tokamak plasmas Numerical modeling approach: multiscale, hybrid kinetic/fluid code Trinity Issues and opportunities

3 Hot, magnetized plasmas

4 What do r P = df dt = C[f]

5 A tempting thought Think? Why think?! We have computers to do that for us. -- Jean Rostand

6 Range of scales Time scale (seconds) e 1 i L/v te L/v ti c Gyromotion Turbulence Mean profiles Collisions Space scale (meters) e i L Gyromotion Turbulence Mean Profiles Collisions

7 Expense (brute force) Time scale (seconds) Temporal grid: ~10 13 time steps Collisions Space scale (meters) Spatial grid: ~10 6 grid points x 3-D = grid points Velocity grid: ~10 grid points x 3-D = 10 3 grid points Total: ~10 34 total grid points

8 Scale separation macroscale Energy content microscale k, ω, p

9 Multiscale, multiphysics time Fast camera image of MAST plasma

10 Density Equilibrium macroscale spatial profile Radius

11 Microscale fluctuations GYRO simulation J. Candy and R. Waltz

12 Equilibrium macroscale speed distribution f = f(r, v,t) v v

13 Microscale fluctuations f v v GS2 simulation

14 Scale separation F macroscale f(r, v,t)=f + f Energy content microscale f k, ω, p

15 Macro drives micro f(r, v,t)=f + f F Energy content macroscale fixed in time microscale f k, ω, p

16 Micro feeds back into macro F macroscale f(r, v,t)=f + f Energy content average over microscale f k, ω, p

17 Coarse-grain average In space and in time time and moment approach in velocities

18 Multiscale model Turbulent fluctuations calculated in small regions of fine space-time grid embedded in coarse grid for mean quantities (implemented in TRINITY code)

19 Gyrokinetic description of + dr + dv = C[f s] Average over fast gyromotion and follow guiding center position Eliminates fast time scale and gyro-angle variable (6-D --> 5-D)

20 Multiscale gyrokinetics Decompose f into mean and fluctuating components: f = F + f Mean varies perpendicular to mean field on system size while fluctuations vary on scale of gyro-radius: r? ln F L 1 r? ln f 1 Fluctuations are anisotropic with respect to the mean field: r k ln f L 1

21 Mixing length estimates Density (n) Eddy size ~ ρ Δn n n L L Position Mixing length ~ (step size) (# steps) 1/2 Macro time scale ~ (step time) (# steps to mix over length L) ~ (L/ρ) 2 (step time)

22 Multiscale gyrokinetics Decompose f into mean and fluctuating components: f = F + f Mean varies perpendicular to mean field on system size while fluctuations vary on scale of gyro-radius: r? ln F L 1 r? ln f 1 Fluctuations are anisotropic with respect to the mean field: r k ln f L 1 => Turbulent fluctuations are low amplitude: f f => Mean profile evolution slow compared to turbulence: L 1 ln F t 2 3

23 Multiscale h + dr dt Gyrokinetic equation for h fi q + dr dt = hc[ f]i Fluid conservation equations for mean dynamics: V 0 = V 0 = S L V 0 Q = S p Fluxes are functions of fluctuating quantities

24 Expense (multiscale gyrokinetics) Time scale (seconds) Temporal grid: ~10 5 time steps Space scale (meters) mean-fluctuation separation Perpendicular spatial grid: ~10 5 grid points x 2-D = grid points Parallel spatial grid: ~10 grid points x 1-D = 10 grid points Velocity grid: ~10 grid points x 2-D v-space = 10 2 grid points Total: ~10 18 total grid points (10 6 savings)

25 TRINITY schematic Steady-state turbulent fluxes and heating Flux tube 1 GS2/GENE Flux tube 2 GS2/GENE Macro profiles Flux tube 3 GS2/GENE Transport solver Flux tube N GS2/GENE

26 TRINITY transport solver Transport equations are stiff, nonlinear PDEs: 3 = 1 (V 0 hq s r i)+... Heat Flux Stiff transport Q s = Q s [n(, t),t(, t);, t] Implicit treatment needed for stiffness Critical R/L T

27 Newton solve Challenge: requires computation of quantities like m+1 j m j + y m+1 y y m y =[{n k }, {p sk }, {L k }] T Local k j n ) n ) k Simplifying assumption: normalized fluxes depend primarily on gradient scale lengths Implicit treatment allows for time steps ~0.1 seconds (vs. turbulence sim time ~0.001 seconds)

28 Parallelization Calculating flux derivative approximations: at every radial grid point, simultaneously calculate aaaaaaa j (R/Ln aa ) m j and j (R/Ln a ) m j + using 2 different flux tubes Possible because flux tubes independent (do not communicate during turbulence calculation) Perfect parallelization use 2-point finite differences for n ) j j (R/Ln ) m j j (R/Ln ) m j +

29 Scaling Example calculation with 10 radial grid points: Evolve electron density, toroidal angular momentum, and ion/electron pressure profiles Simultaneously calculate fluxes for equilibrium profile and for perturbed profiles (one for each time-varying gradient scale length, i.e. 4) Total of 50 flux tube simulations running in parallel

30 Flux tube scaling GENE strong scaling GS2 strong scaling GENE weak scaling

31 Scaling Example calculation with 10 radial grid points: Evolve electron density, toroidal angular momentum, and ion/electron pressure profiles Simultaneously calculate fluxes for equilibrium profile and for perturbed profiles (one for each time-varying gradient scale length, i.e. 4) Total of 50 flux tube simulations running in parallel ~2-4k cores or more per flux tube => scaling to over 100k s processors with very high efficiency

32 Example: JET H-mode 10 radial grid points Costs ~120k CPU hrs (<10 clock hrs) Dens and temp profiles agree within ~15% across device Energy off by 5% Incremental energy off by 15%

33 Issues and opportunities

34 Issue: Scale separation? macroscale Energy content microscale k, ω, p

35 Issue: Scale separation? Energy content k, ω, p

36 Summary From T. S. Eliot s Little Gidding, We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time.

37 Opportunity Model energy flow out Back reaction of micro on macro Energy content and energy flow in k, ω, p

38 Issue: Computational expense Could reduce computational expense considerably by using a reduced model for the microscale physics fluid model proper orthogonal decomposition LES for kinetics These approaches share weakness that they are not universal This weakness can be ameliorated by the multiscale approach

39 Equilibrium macroscale spatial profile