Simulation of Plasma Blobs in Realistic Tokamak Geometry R. Jorge 1 , F. Halpern 2 , P. Ricci 2 , N. Loureiro 1 and C. Silva 1 1 IPFN, IST Lisbon and 2 CRPP, EPFL Lausanne Contact Information: Theory and Modeling Group, IPFN Instituto Superior T´ ecnico Av. Rovisco Pais, 1 1049-001 Lisboa , Portugal Phone: (+351) 218 417696 Email: rogerio.jorge@ist.utl.pt Abstract Understanding Scrape-off Layer SOL (Scrape-off Layer ) turbulence is crucial for the success of the entire fusion by magnetic confinement program. SOL dynam- ics determines the overall confinement and performance of future tokamaks (such as ITER), governs the heat load on the vessel wall, regulates the impurity dynamics, the plasma refueling and the removal of fusion ashes. The GBS (Global Bragin- skii Solver ) code [1] numerically integrates the “drift-reduced” Braginskii equations equations, allowing us to model the evolution of the SOL plasma dynamics in 3D, without separation between perturbations and equilibrium. In this work, the GBS code has been ported to the SOL configuration of the ISTTOK tokamak, where it already has been experimentally observed the ballooning character of turbulence in the SOL with a clear distinction between LFS and HFS [2] Thanks to the use of GBS simulations, we have theoretically investigated the turbulent regimes, the saturation mechanisms and the formation of the SOL profile in ISTTOK and assessed our find- ings with the numerical results. It has allowed us to identify which instabilities are the more influential in this region and what is the impact of the different operational SOL parameters. Introduction Nuclear fusion is a promising source of energy to respond to the increasing world energy demand. Although nowadays there are several different ma- chines with different geometries and configurations to achieve ignition we shall focus in the tokamak device, with special attention to ISTTOK con- figuration. This is a large aspect ratio tokamak with R = 46 cm, a =8.5 cm and a circular cross-section which uses a poloidal (graphite) limiter [2]. Here, we apply the GBS (Global Braginskii Solver) code to simulate the SOL (Scrape-off Layer) environment of this device and compare with the existing results. The GBS code is based on the drift-reduced Braginskii equations and models the SOL plasma turbulence with proper boundary conditions derived from the sheath dynamics [1]. It has been validated against several experiments and configurations. Figure 1: Typical snapshot of GBS simulations at a poloidal plane halfway between the limiter. Plot shows contour of density at a specific time slice. GBS Model Equations The SOL simulations evolve the drift-reduced Braginskii equations [1] in a limit that can capture the resistive-ballooning dynamics, while remain- ing relatively simple: large aspect ratio and circular flux surfaces. As turbulence in the SOL is characterized by time variations on a time scale much slower than the ion gyromotion and spatial variations on a scale of the order ρ s = c s /ω i , the ion Larmor radius at the sound speed c s (with ω i = eB/(m i c)) which is smaller than L ⊥ (the characteristic perpendicular length), we adopt the drift-reduced ordering ∂ ∂t ≃ V E ×B ·∇≃ ρ 2 s L 2 ⊥ ω i ≪ ω i , (1) which leads to use for perpendicular velocities V ⊥i = V E ×B + V ∗i + V pol and V ⊥e = V E ×B + V ∗e , where V E ×B =(−∇φ × B)/B 2 is the E × B drift velocity, V ∗i,e is the ion/electron diamagnetic and V pol is the ion polarization velocity. The used drift-reduced equations used are ∂n ∂t = − R Bρ s0 [φ,n] −∇ ‖  nV ‖e  + 2n B  C (T e − φ)+ T e n C (n)  , (2) ∂ Ω ∂t = − R Bρ s0 [φ, Ω] − V ‖i ∇ ‖ Ω+ B 2 j ‖ ∇ ‖ log(n) + B 2 ∇ ‖ j ‖ + B 3n C (G i )+2B  C + C (n) n  (T e + τT i ) , (3) ∂U ‖e ∂t = − R Bρ s0 m e m i  [φ,V ‖e ]+ V ‖e ∇ ‖ V ‖e  −  1.71∇ ‖ + ∇ ‖ log(n)  T e + ∇ ‖ φ − 2 3 ∇ ‖ G e + νj ‖ , (4) ∂V ‖i ∂t = − R Bρ s0 [φ,V ‖i ] − V ‖i ∇ ‖ V ‖i − 2 3 ∇ ‖ G i −  ∇ ‖ + ∇ ‖ log(n)  (T e + τT i ) , (5) ∂T e ∂t = − R Bρ s0 [φ,T e ] − V ‖e ∇ ‖ T e − 2 3 T e ∇ ‖ V ‖e + 4 3 T e B 7 2 C (T e ) + 4 3 T e B  T e n C (n) − C (φ)  + 2 3 T e 0.71  ∇ ‖ + ∇ ‖ log(n)  j ‖ , (6) ∂T i ∂t = − R Bρ s0 [φ,T i ] − V ‖i ∇ ‖ T i + 2 3 T i j ‖ ∇ ‖ log(n) − 2 3 T i ∇ ‖ V ‖e + 4 3 T i B  T e C (n) n + C  T e + τ 5 2 T i − φ  , (7) where Ω= ω + τ ∇ 2 ⊥ T i , U ‖e = m e m i V ‖e + β e 2 ψ and at each equation for the time derivative of the field A we add a source S A = A f e − x−x s σ 2 s where x s is the radial position of the source, A f its strength and σ s the width and also add diffusion operator D A . It is introduced the curvature operator C (f )= B/2(∇× b/B ), the Poisson Bracket [φ,f ]= b · (∇φ ×∇f ) and the adi- mensionalized resistivity ν = e 2 nR/(m i σ ‖ c s0 (σ ‖ =1.96n 0 e 2 τ e /m e ). We normalize n to the equilibrium density n 0 , T e to the SOL background tem- perature T e0 , φ to T e0 /e, ψ to c s0 m i β/2e, V ‖i to c s0 =  T e0 /m i and the time t to R/c s0 . Lengths in the perpendicular direction are normalized to ρ s0 = c s0 /ω i . Results The simulations performed contain (N x = 64, N y = 512, N z = 32) points in the (x,y,z ) direction. For a typical simulation, 32 cores are used through the course of one week (approximately 5000 CPU hours). The ballooning character of the turbulence can be asserted through a look at a typical den- sity snapshot in a poloidal plane (fig. 1). Figure 2: Typical snapshot of GBS simulations at a toroidal plane. Plot shows contour of ion parallel velocity averaged over the simulation period. There is a preferred poloidal angle where the gradient lengths in the radial direction increase (defined as θ = π ) showing a typical ballooning charac- ter at the LFS (low field side). At the HFS (high field side) in θ =0 the radial extension of the modes is smaller. Through simulations performed with no interchange drive (ballooning terms turned off) this behavior is not observed. In order to quantify this variation in the simulations, we look at the variation of L p e = p e /∇p e (where p e = nT e ). As we can see, the distinction between the LFS and HFS is clear. The error bars are intro- duced to represent the 95% confidence interval of the fit to the expression p e = p 0 e −x/L pe Figure 3: Poloidal variation of L p fitting GBS results to the expression p e = p 0 e −x/L pe . These results have been confirmed through linear simulations that use the linearized drift-reduced Braginskii equations and the gradient removal hy- pothesis [3]. These have provided a framework to identify the fastest grow- ing instabilities as a function of the parameters characterizing the tokamak SOL region and we have found that at the LFS there is a predominance of ballooning modes, while at the HFS the drift wave instability dominates. A scan on ν,β and q was performed within the linear solver and there was no significant change in predominant modes nor the variation and value of L p . Figure 4: Density contour on a poloidal plane of GBS simulations averaged over the simu- lation period without drift wave coupling and without interchange drive (ballooning terms) respectively. Experimentally, the different modes discussed here have been observed at ISTTOK [2]. Measurements of the electron density N e , floating potential V f = φ − 3T e and electron temperature T e have been performed at ISTTOK SOL with different plasma currents I p . Through the dimensionless param- eter η e = L n /L T e we have compared the results obtained from the non- linear simulations and the experiment showing a remarkable agreement of η e ≃ 0.85. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 1 r-a (cm) Te (eV) Te Data Fit General model: f(x) = LT0*exp(-x/LT) Coefficients (with 95% confidence bounds): LT = 0.9827 (0.6015, 1.364) LT0 = 18.62 (15.85, 21.4) R-square: 0.9858 Figure 5: Fit to ISTTOK experimental results at the SOL providing L T =0.98 cm which in combination with the obtained value of L n =0.834 cm yields η =0.85. References [1]P. Ricci, F. D. Halpern, S. Jolliet, J. Loizu, A. Mosetto, A. Fasoli, I. Furno, and C. Theiler. Simulation of plasma turbulence in scrape-off layer conditions: the gbs code, simulation results and code validation. Plasma Physics and Controlled Fusion, 52(124047), 2012. [2] C. Silva, H. Figueiredo, P. Duarte, and H. Fernandes. Characterization of the poloidal asymmetries in the isttok edge plasma. Plasma Physics and Controlled Fusion, 53, 2011. [3]P. Ricci and B. N. Rogers. Plasma turbulence in the scrape-off layer of tokamak devices. Physics of Plasmas, 20, November 2013. View publication stats View publication stats