Impact of diverted geometry on turbulence and transport barrier formation in 3D global simulations of tokamak edge plasma

Transcription

1 1 Impact of diverted geometry on turbulence and transport barrier formation in 3D global simulations of tokamak edge plasma D. Galassi, P. Tamain, H. Bufferand, C. Baudoin, G. Ciraolo, N. Fedorczak, Ph. Ghendrih, N. Nace and E. Serre

2 2 Edge plasma turbulence: an ITER-relevant topic Turbulent transport of energy and particles Impact on confinement time, and possibly on the transition from Low to High confinement mode tokamak performances Determines the Scrape-Off Layer (SOL) width and heat flux poloidal asymmetries heat flux at target plates q < 10 MW m 2

3 3 Modelling edge plasma turbulence: reducing to 2D 2D Fluid turbulence codes Examples: TOKAM2D, HESEL Focused on a region with bad curvature, unstable to interchange turbulence: p B > 0 Based on flute assumption TOKAM2D density time slice (courtesy of N. Nace) k k k 0 Averaged parallel dynamics Advantages: Relatively low computational time Good agreement with experimental results (e.g. Y. Sarazin et al. PoP 1998) Able to reproduce transport barriers (e.g. J.J. Rasmussen PPCF2016) acting on key parameters

4 4 Including parallel dynamics 3D codes in circular (limiter) geometry Examples: GBS (Switzerland), BOUT++ (UK), GEMR(Germany), TOKAM3X (France) (P. Tamain et al. PET conference 2013) Put in evidence poloidal asymmetries in turbulent transport: k k Qualitatively recover the features of parallel dynamics in the SOL Maintain a good description of blob dynamics and statistical properties of turbulence 2π 2πqR > 0

5 5 5 TOKAM3X can handle diverted magnetic configurations We need to extend turbulent simulations towards a realistic geometry since: H-mode experimentally observed preferentially in divertor configuration Transverse transport in the divertor region is believed to affect the heat flux profile at the target Temperature and density fluctuations in the divertor could affect recycling and sputtering Objective: understanding the effect of complex geometry on turbulent transport

6 6 6 TOKAM3X: 3D flux-driven, fluid turbulence code Mass conservation e t N +. Γ J b +. N u E + u B = S N + D N 2 N Parallel momentum conservation t Γ +. Γ 2 N b +. Γ(u i E+u B ) = 2 N + D Γ 2 Γ Vorticity conservation t W +. W Γ N b +. Wu i E =. Nu B e Nu B +. J b + D W 2 W Ohm s law Vorticity definition Drifts η NJ = N N Φ W =. u E = 1 B 2 B Φ B 2 Φ + N N i/e B B u B = ±2Ti/e B 3 Interchange (RBM) Drift Waves Kelvin Helmholtz Here isothermal version, no neutrals Boundary conditions (P. Tamain JCP 2016) Bohm conditions in direction ψ = 0 Periodic in φ Periodic in θ for closed field lines k 1/ρ L δ π/2

7 7 7 Simulated magnetic geometries Flux expansion X-point DIVERTOR COMPASS like LIMITER Same parameters, Different safety factor q = 1 2π B φ dθ R Bθ

8 8 8 Turbulent structure shape varies strongly on a flux surface Ballooned character of turbulence is immediately visible Turbulence appears also in the divertor (J. R. Harrison et al. JNM 2015) In this region, turbulent structures become radially elongated at the X-point

9 9 9 Flux tube shape in divertor geometry If the flute assumption is valid, a turbulent structure must be homogeneous over a flux tube s B d S = 0 Flux tube cross section conserved since B = 0 S 1/B R Flux expansion f x (θ, ψ) = ψ θ, ψ mp ψ(θ, ψ) Magnetic shear (N. R. Walkden et al. PPCF 2013) s cyl = r dq q dr Focusing on a flux surface k θ 1 δθ δr S f x/r

10 10 Turbulent structures are (almost) field-aligned 10 Bottom (close to X-point) We focus on the deformation on a closed flux surface We calculate locally: k θ 2π λ And we compare to the flux tube shape Bottom (close to X-point) 2π/(2k θ ) To a first order, Flute assumption justified!

11 D shape of turbulent structures Density and electric potential turbulent structures have a finite length in the parallel direction L πqr Typical extension in parallel direction ~ ½ poloidal turn k 2π 2πqR 2π 10 4 ρ L k 2π 10 ρ L

12 12 12 Transverse E B fluxes are proportional to flux expansion Macroscopically, the poloidal variation of the structure shape has an impact on fluxes u E ψ 1 B θφ Rk θ Φ f x u E ψ mp k θ f x k θ mp R mp R Transverse fluxes not proportional to radial pressure gradient Not easily described by a diffusive approach: Γ ψ D(ψ, θ) N ψ

13 13 13 Getting rid of geometric effects: a remapping operation We are interested in transverse transport across the flux surfaces 1/f x Remap fluxes in magnetic space Normalizing fluxes to the flux expansion: usual picture of a ballooned transport Still, a peculiar flux pattern is visible in the divertor region

14 14 14 Turbulent and mean-field components of the flux ψ N. u E B t,φ ψ = N t,φ u E B t, φ + ψ N. u E B t, φ Mean-field flux Already included in 2D transport codes as SOLEDGE2D, SOLPS, EDGE2D,UEDGE.. Turbulent flux Included in other 3D turbulent codes, but rarely with realistic geometry (D. Galassi et al. NF 2017)

15 15 15 Significant turbulent transport in the outer divertor leg Outer divertor leg is unstable p B > 0 Turbulent transport towards the far SOL ~20% midplane Potentially interesting for the spreading of λ q Broad density profile at the outer target

16 16 16 Lower fluctuation level in the vicinity of the X-point Fluctuations are damped in the vicinity of the X-point (turbulent transport inhibited) Both in open and closed field lines, (width ~20 Larmor radii) In this region: Elevated magnetic shear s Elevated k θ Both are stabilizing conditions for the interchange modes (N. R. Walkden et al. NME 2016)

17 17 17 Spontaneous transport barrier in divertor geometry COMPASS-like DIVERTOR Enhanced pressure gradient in the closed flux surfaces, near the separatrix LIMITER

18 18 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities 18 A narrow and stable transport barrier Turbulent transport efficiency R b = Γ E ψ θ,φ ψ ψ Γ E + ΓDiff θ,φ (E. Floriani et al. PPCF2013) R b = 0 strong barrier R b = 1 no barrier Transport barrier with R b 0.3, varying in the range Stable character Small width ( 3 ρ L )

19 19 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities 19 Damping of fluctuation amplitude Barrier damping effect r/a 0.99 r/a 0.95 r/a 0.9 Electric potential fluctuation amplitude reduced up to 50% The level of fluctuations is reduced over the whole flux surface

20 20 20 Transport barrier acts mainly on low-amplitude fluctuations Positive skewness S = 3 N N t,φ 2 N N t,φ t,φ 3/2 t,φ Intermittent nature of turbulence Low-amplitude events affected by the barrier Effects of the barrier visible in ~15 ρ L outside the separatrix

21 21 21 Two possible candidates for the barrier formation s cyl s cyl Linearly stabilizing for interchange instabilities magnetic shear s cyl = r q dq dr Extremely strong in divertor in the vicinity of the X-point BUT turbulence has a non-local character!

22 22 Two possible candidates for the barrier formation E B shear u E θ = d dr u E θ Large-scale equilibrium drive Driving mechanisms Turbulence drive: Reynolds stress t < u θ E > θ = d dr < u ψ E θ ue > θ Moreover, a residual Reynolds stress can be driven by magnetic shear (see N. Fedorczak et al. NF 2012) sinergy E B and magnetic shear 22 Reynolds stress (P.H. Diamond and Y.-B. Kim., Phys. Fluids B 1991)

23 23 23 Ongoing work suggests magnetic shear is the main player In a circular geometry, an artificial magnetic shear is imposed with a divertor-like average q profile Transport barriers are triggered at the position where the magnetic shear is maximum. The E B shear increases instead at the separatrix. Analogies Stability Width ~ characteristic q decay length Differences Lower amplitude Two peaks instead of one

24 24 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities 24 Conclusions TOKAM3X simulations in diverted configuration show non-trivial effects of geometry on turbulent transport. Turbulent structures are almost field aligned, they have a non-zero parallel wave-number due to ballooning. Transverse fluxes are proportional to flux expansion. In order to understand turbulent transport, we have to remap to magnetic space. A significant turbulent transport is observed in the outer divertor leg, while turbulence is damped in the vicinity of the X-point. Unlike in limiter configuration, a spontaneous, stable transport barrier buildup near the separatrix, reducing turbulent transport by 70% at its centre.

25 Backslides

26 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities Simulation parameters Geometry Physics ρ = ρ L a = D N,Γ,W = 2, η = 1, T i = T e = 1 ρ L 2 ω ci Particle source: gaussian-shaped in the radial direction, centered at the inner boundary of the domain (edge) and poloidally constant Numerics N θ = 512 (SOL and core edge) N r = 64 (SOL + core edge) N φ = 32

27 27 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities 27 Parallel flow trends are qualitatively recovered Parallel Mach number is studied as global effect of cross-field transport The main features of the parallel flows in the SOL are recovered (D. Galassi et al. NF 2017) Stagnation point between LFS midplane and LFS X-point Mach number close to 0.4 at the TOP of the machine Both small-scale turbulent flow and largescale flows contribute to the build-up of this parallel flow scheme HFS X-pt HFS mp TOP LFS mp LFS X-pt (N. Asakura et al. JNM 2007)

28 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities 28 Similar turbulence properties only far from X-point and separatrix Density fluctuations PDF at LFS midplane LIMITER r < a Skewness 0 (P. Tamain CPP 2014) r > a Skewness > 0 DIVERTOR

29 29 Not just an effect of metrics 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities 29 Does flux expansion have a direct effect on turbulent transport? Inner Shift No Shift Outer Shift Confinement improved by a low flux expansion at the LFS midplane Different causes have been detected Higher global curvature g in the inner shift case Stronger magnetic shear in the outer shift case can stabilize turbulence

30 30

31 31 Linear analysis of the system (P. Tamain PhD thesis)

32 32 u E ψ θ N θ Φ B

33 33 Residual Reynolds stress in divertor configuration Π < v r 2 > θ < d v E θ dr τ > θ < v r 2 > θ < s θf θ > θ (N. Fedorczak Re stress Negative viscosity Re stress induced by magnetic shear NF 2012) Re stress depends on blobs shape Top-bottom asymmetry introduced by the X-point causes a radial change of the average Reynolds stress Source of poloidal flows

34 12 th June 2017 IAEA Headquarters 8 th Technical Meeting on Theory of Plasma Instabilities 10 The vorticity equation contains information about the transport of poloidal momentum t W +. W Γ N b +. Wu i E =. Nu B e Nu B +. J b + D W 2 W Average on flux surface <. > θ,φ, Closed field lines t < W >+<. Wu E i > = <. Nu B W ~ 2 (Φ + ln (N)) 2 x e Nu B > +< D W 2 W > 0 < W > y = < 2 (Φ + ln (N)) 2 x + 2 (Φ + ln (N)) 2 y r x θ y > y = d dx < (vy ExB + v y ) > y t d dx < (v y ExB+v y ) > y + d2 dx 2 < (v y ExBv x ExB + v y v x d 3 ExB > y = D W dx 3 < (v y ExB+v y ) > y Reynolds stress Diamagnetic Reynolds stress