Gyrokinetic Large Eddy Simulations

Transcription

1 Gyrokinetic Large Eddy Simulations A. Bañón Navarro 1, P. Morel 1, M. Albrecht-Marc 1, D. Carati 1, F. Merz 2, T. Görler 2, and F. Jenko 2 1 Laboratoire de Physique Statistique et des Plasmas Université Libre de Bruxelles 2 Max-Planck-Institut für PlasmaPhysik Garching bei München Wolfgang Pauli Institute, Vienna: Gyrokinetics for ITER II 1/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

2 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

3 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

4 Local GK simulations: Perpendicular directions (respect to B) are spectral Direct Numerical Simulation (DNS): resolves all active scales k y model simula+on small scales large scales k x Large Eddy Simulation (LES): resolves large scales + models smallest ones objective: decrease numerical effort GK DNS can go up to N x N y N z N v N µ = up to CPUh 1, and even more (Trinity). 1 T. Görler, et al, Phys. Rev. Lett., 100 (2008). 4/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

5 gyrokinetic equation Distribution function f (k x, k y, z, v, v, t) evolution: t f = L[f ] }{{} linear drive local in k + N[f, f ] }{{} nonlinear E B advection couples k to k k D[f ] }{{} dissipations local in k Applying a Fourier cutoff filter in perp. plane (k x, k y ) to remove the smallest scales from the distribution leads to a closure problem: t f = L[f ] + N[f, f ] D[f ] T Sub-grid term: T = N[f, f ] N[f, f ] Describes the effect of the under-resolved scales on the largest scales 5/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

6 Sub-grid term: T = N[f, f ] N[f, f ] contains under-resolved information f Comparisons: free energy diagnostics E f 2. t f }{{} f filtered GK equation: = L[f ] }{{} f + N[f, f ] }{{} D[f ] }{{} f 2 f T }{{} f 2 filtered free energy equation: t E }{{} f = G }{{} f f 2 f D f }{{} f 2 T T }{{} f f 2 6/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

7 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

8 Sub-grid term: T = N[f, f ] N[f, f ] contains under-resolved information f Can we simply ignore the small scales? Free energy spectral density E kx : 300 Reference DNS with 128N x 64N y t E = G D vs Reduced DNS with 48N x 24N y : t E f = G f D f T T }{{} neglect free energy accumulated at smallest scales free energy E kx in A.U. Te0 units reference Nx = 128,Ny = 64 w/o model Nx = 48,Ny = k x ρ i No, even if the filter does not remove many scales 8/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

9 Sub-grid term: T = N[f, f ] N[f, f ] contains under-resolved information f What is the role of the sub-grid term? consider DNS 128N x 64N y + test filter consider resolved free energy: t E f = G f D f T T with T T the sub-grid contribution free energy E in Te0 A.U. units G f T T D f G f T T D... f time in R 0/v Ti units Sub-grid term dissipate free energy 9/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

10 kto Agreement with nonlinear free energy transfers studies k from Free energy is subject to a (strongly) local, forward cascade 2 A. Bañón Navarro, et al. Phys. Rev. Lett., 106, (2011). 10/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

11 Model approximates sub-grids, depending only on resolved scales: Model dissipates free energy T M[f ] perpendicular hyper-diffusions 3 k 4 : M[c, f ] = c k 4 f Free energy contribution: Satisfies: T T M M < 0 3 S. A. Smith and G. W. Hammett, Phys. Plasmas, 4 (1997). 11/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

12 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

13 Model: M[c, f ] = c k 4 f Unknown free parameter: c Free energy spectra vs c : Cyclone Base Case (ITG) c too small not enough dissipation c too strong overestimates injection c = good agreement plateau for c [0.25, 0.625] free energy E ky in Te0 A.U. units %!! $'! $!! #'! k y spectrum #!! DNS, Nx = 128, Ny = LES, c =0.0 '! LES, c =0.025 LES, c =0.375 LES, c =1.0!!!"#!"$!"% k yρ i!"&!"'!"( 13/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

14 Model: M[c, f ] = c k 4 f Unknown free parameter: c Free energy spectra vs c : k x spectrum Cyclone Base Case (ITG) c too small not enough dissipation c too strong overestimates injection c = good agreement free energy E kx in Te0 A.U. units )(! )!! #(! #!! '(! '!! (! DNS, Nx = 128, Ny = LES, c =0.0 LES, c =0.025 LES, c =0.375 LES, c =1.0 * plateau for c [0.25, 0.625] Numerical cost DNS/30! *!!"#!"$!"% k xρ i!"& ' '"# 14/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

15 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

16 Global quantities (Q) are truncated: heat flux,... k y δ y Q δ xy Q Q δ x Q k x In a LES, only the resolved part of Q is directly accessible Q The unresolved part of Q has to be estimated δq Q can be approximated by decaying power laws in the LES spectra: Q kx δ xy Q << δ x Q, δ y Q A x kx αx Q ky A y ky αy 16/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

17 free energy injection G ky in Te0 A.U. vti/r0 units regression interval DNS LES c =0.375 LES c =0.0 resolved scales unresolved kyρi Q Q + δ x Q + δ y Q Q + Kx DNS k x >K LES x A x k αx x + Ky DNS k y >K LES y A y k αy y G LES = 1.11 G DNS E LES = 1.10 E DNS G No Model = 1.38 G DNS E No Model = 2.1 E DNS Good agreement with model 17/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

18 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

19 Robustness of c = 0.375? Comparison between GyroLES and DNS - ω Ti = 8.0. free energy E kx in Te0 A.U. units $(! $!! )(! )!! #(! #!! '(! '!! (! DNS LES c =0.375 LES c =0.0 free energy E ky in Te0 A.U. units &'! &!! %'! %!! $'! $!! #'! #!! '! DNS LES c =0.375 LES c =0.0!!!"#!"$!"%!"& ' '"# kxρi!!!"#!"$!"%!"&!"'!"(!") kyρi strong turbulence: good agreement slightly overestimate of the free energy at small scales c should be increased a little 19/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

20 Robustness of c = 0.375? Comparison between GyroLES and DNS - ω Ti = 6.0. free energy E kx in Te0 A.U. units )!! #(! #!! '(! '!! (! DNS LES c =0.375 LES c =0.0 free energy E ky in Te0 A.U. units DNS LES c =0.375 LES c =0.0!!!"#!"$!"%!"& ' '"# kxρi kyρi weak turbulence: less satisfactory agreement c should be decreased develop alternative methods: dynamic calibration 20/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

21 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

22 model coefficient has to vary when varying external parameters Calibrate model parameters dynamically k y 1/ Δ Consider filtered simulation ( ): 1/Δ model Simula+on 1 t f = L[f ]+N[J 0 φ, f ] M[c,, f ] small scales large scales k x 22/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

23 model coefficient has to vary when varying external parameters Calibrate model parameters dynamically 1/Δ k y model Consider filtered simulation ( ): 1/Δ 1/ Δ ˆ Simula+on 1 Simula+on 2 t f = L[f ] + N[J 0 φ, f ] M[c,, f ] small scales large scales k x Consider filtered simulation ( ): t f = L[ f ] + N[ Ĵ 0 φ, f ] M[c,, f ] 23/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

24 model coefficient has to vary when varying external parameters Calibrate model parameters dynamically Consider filtered simulation ( ): small scales 1/Δ k y 1/Δ 1/ Δ ˆ large scales model Simula+on 1 + test filter Simula+on 2 k x t f = L[f ] + N[J 0 φ, f ] M[c,, f ] Apply a test-filter, ( = ): t f = L[ f ]+N[ Ĵ 0 φ, f ] T, M[c,, f ] Consider filtered simulation ( ): t f = L[ f ] + N[ Ĵ 0 φ, f ] M[c,, f ] 24/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

25 model coefficient has to vary when varying external parameters Calibrate model parameters dynamically Minimize difference between GyroLES with and GyroLES with and test filter: ( dλ c M[c,, f ] M[c, 2, f ] T, ) 0 optimized free parameter c can be done for many parameters M[c,, f ] = c a k n f M[c x, c y,, f ] = c x a k n x f + c y a k n y f Anisotropy 25/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

26 Outline Introduction to GyroLES 1 Introduction to GyroLES /31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

27 Comparison between GyroLES and DNS - ω Ti = 6.0. fre energy in A.U DNS M x,y M = 0 fre energy in A.U DNS M x, y M = k x i k y i weak turbulence: good agreement 27/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

28 Comparison between GyroLES and DNS - ω Ti = DNS M x,y M = DNS M x, y M = 0 fre energy in A.U free energy in A.U k x i k y i strong turbulence: good agreement slightly overestimate of the free energy at small scales Numerical cost DNS/20 28/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

29 Test dynamic procedure for different set of parameters: magnetic shear ŝ, safety factor q... (analyses in progress) Studies with ETG driven turbulence Studies with two kinetic species Implementation of more sophisticated models 29/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

30 Model is needed even if resolution is not decreased dramatically Analysis of DNS shows that models have to dissipate free energy A simple model M = c k 4 f has been successfully tested c has been calibrated by trial and error for a given set of parameters 4 Dynamic calibration of the amplitude of the model has been successfully tested for some parameters. 4 P. Morel, et al, submitted to Physics of Plasmas 30/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II

31 Thank you 31/31 A. Bañón Navarro, P. Morel, M. Albrecht-Marc et al. Gyrokinetics for ITER II