Introduction Particle Simulation Discussion Full orbit simulations of impurity transport in spherical tokamaks with strongly-sheared electric fields C. G. Wrench1, E. Verwichte1 , K. G. McClements2 1 Centre for Fusion, Space and Astrophysics University of Warwick Coventry, CV4 7AL, UK c.g.wrench@warwick.ac.uk 2 EURATOM/CCFE Fusion Association Culham Science Centre Abingdon, Oxfordshire, OX14 3DB, UK TTG Conference, Córdoba 7th September 2010 O2.09 TTG 2010 1/17 C. G. Wrench Introduction Particle Simulation Discussion Introduction • Advanced tokamak operating scenarios use strongly sheared electric fields to suppress turbulence (Chen 2001) • In spherical tokamaks (STs), such as MAST, radial electric fields associated with edge transport barriers can vary on length scales of the order of the ion Larmor radius (Meyer, 2009; Shaing, 1998) • Standard neoclassical theory not valid • We use full-orbit test-particle approach to study collisional impurity transport in vicinity of transport barriers in STs O2.09 TTG 2010 2/17 Radial electric field as measured in the edge of MAST (Meyer, 2009) C. G. Wrench Introduction Particle Simulation Discussion Radial E-field: effect on collisionless orbits • Impact of a sheared radial 1 electric field on collisionless orbits is to: fraction • Reduce orbit width of trapped ions • See: Shaing (1998) and Tau (1993). • Orbit squeezing shown by Shaing et al. (1998) to affect neoclassical heat conductivity • Collisional particle transport also affected O2.09 TTG 2010 3/17 0.6 Vertical Position [m] • Increase trapped particle 0.8 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0.2 0.4 0.6 0.8 1 1.2 Major Radius [m] Demonstration of orbit squeezing for an Ar12+ ion. C. G. Wrench Introduction Particle Simulation Discussion Test Particle Simulation • CUEBIT: Full orbit test particle simulation code • Solves the Lorentz-Langevin equation: mz dv mz = Ze (E + v × B) − (v − uf ) + mz a(t) | {z } |τ dt {z } Lorentz Langevin (1) using an implicit finite difference method • Solves test particle (Z 2 nz ni ) trajectories for a prescribed MAST-like equilibrium • All fields and bulk ion profiles prescribed and assumed static • Electrostatic potential prescribed and of the form: Φ = Φ0 arctan Ψ − Ψ1 ∆Ψ (2) • We assume that the dominant terms in the bulk ion force balance at the transport barrier are ∇p and ZeEr O2.09 TTG 2010 4/17 C. G. Wrench Introduction Particle Simulation Discussion Simulation Results: α-particles 4 (a) 1 x 10 (b) 2 1.5 1 (d) 1 0.5 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0 4 2.5 −0.5 −1 0.2 0.4 0.6 0.8 1 1.2 Major radius [m] Particle density [m−3] Height [m] 0.5 Particle density [m−3] 2.5 x 10 (c) 2 −0.5 1.5 −1 1 0.2 0.4 0.6 0.8 1 1.2 Major radius [m] 0.5 0 0 0.1 0.2 0.3 0.4 Minor radius [m] 0.5 0.6 Simulation of 104 α-particles without, (a) and (b), and with, (c) and (d), a sheared radial electric field. O2.09 TTG 2010 5/17 C. G. Wrench Introduction Particle Simulation Discussion Simulation Results: Ne10+ particles 4 (a) 1 x 10 (b) 4 1 (d) 2 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0 4 6 −0.5 −1 0.2 0.4 0.6 0.8 1 1.2 Major radius [m] Particle density [m−3] Height [m] 0.5 Particle density [m−3] 6 x 10 (c) −0.5 4 −1 2 0 0 0.2 0.4 0.6 0.8 1 1.2 Major radius [m] 0.1 0.2 0.3 0.4 Minor radius [m] 0.5 0.6 Simulation of 104 Ne10+ -particles without, (a) and (b), and with, (c) and (d), a sheared radial electric field. O2.09 TTG 2010 6/17 C. G. Wrench Introduction Particle Simulation Discussion Simulation Results: W20+ particles 4 (a) 1 x 10 (b) 4 3 1 (d) 2 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0 4 5 −0.5 −1 0.2 0.4 0.6 0.8 1 1.2 Major radius [m] Particle density [m−3] Height [m] 0.5 Particle density [m−3] 5 x 10 (c) 4 −0.5 3 −1 2 0.2 0.4 0.6 0.8 1 1.2 Major radius [m] 1 0 0 0.1 0.2 0.3 0.4 Minor radius [m] 0.5 0.6 Simulation of 104 W20+ -particles without, (a) and (b), and with, (c) and (d), a sheared radial electric field. O2.09 TTG 2010 7/17 C. G. Wrench Introduction Particle Simulation Discussion Simulation Results 10+ 2+ Vertical position [m] 20+ Ne He W 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 −1 0.5 1 1.5 Major radius [m] 0.5 1 1.5 Major radius [m] 0.5 1 1.5 Major radius [m] Poloidal distribution of 104 impurity ions after 100 ms. Red line indicates location of peak radial electric field. O2.09 TTG 2010 8/17 C. G. Wrench Introduction Particle Simulation Discussion Scaling with particle parameters Consider collisional equation of motion (Langevin equation): mz dv = Ze (E + v × B) − (v − uf ) + mz a(t) mz (3) dt τ To make analytics tractable, we assume Cartesian geometry and E = Ex = const. Solve Ze dvx + νvx = Ex + Ωvy (4) dt mz dvy + νvy = −Ωvx (5) dt to find: Ex ν 1 (6) vx = v⊥ e −νt sin Ωt + B Ω 1 + ν 2 /Ω2 1 Ex (7) vy = v⊥ e −νt cos Ωt − B 1 + ν 2 /Ω2 Note: x-direction is a proxy for the radial direction O2.09 TTG 2010 9/17 C. G. Wrench Introduction Particle Simulation Discussion Scaling with particle parameters: Drag force drift Drag force drift in the same direction as the electric field: vx = v⊥ e −νt sin Ωt + 1 Ex ν B Ω 1 + ν 2 /Ω2 (8) This implies that there is: • an inward-pinch in the presence of a negative radial electric field (Ex < 0) and • an outward-pinch in the presence of a positive electric field. In tokamaks we have that ν 2 Ω2 , therefore, drag force drift is 1/2 vdrag O2.09 TTG 2010 10/17 Ex m e 3 ni ln Λ 'Z √ i 3/2 6 2π 3/2 20 Ti B 2 (9) C. G. Wrench Introduction Particle Simulation Discussion Vertical position [m] Scaling with particle mass 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 0.5 1 Major radius [m] 1.5 −1 0.5 1 Major radius [m] 1.5 −1 0.5 1 Major radius [m] 1.5 0.5 1 Major radius [m] 1.5 4 4 x 10 Ar10+ Ne10+ Mo10+ W10+ 3.5 Particle density [m−3] 3 Above: Poloidal distributions for (left to right) Ne10+ , Ar10+ , Mo10+ and W10+ . Left: Density profiles in minor radius for impurities with varying masses but the same charge state (same species as above). 2.5 2 1.5 1 0.5 0 0 O2.09 TTG 2010 0.1 0.2 0.3 0.4 0.5 0.6 Averaged minor radial coordinate [m] 11/17 0.7 C. G. Wrench Introduction Particle Simulation Discussion Characterising transport: Diffusion coefficients Diffusivities are estimated from simulated particle flux and density gradient: Γz (10) Deff = − dnz /dx E =Emax Assuming particle flux is composed of usual diffusion and pinch terms, we may relate drag force drift to transport: Γz = −D dnz − vnz dx (11) D +1 (12) implies Deff = nz v Γz where we take υ to be drag force drift velocity at peak of electric field. Therefore: Deff 1 = (13) D 1 + αZ O2.09 TTG 2010 12/17 C. G. Wrench Introduction Particle Simulation Discussion Scaling of effective diffusivities with particle mass and charge 2.5 1 CUEBIT data Linear fit 2 0.8 0.7 0.6 D0/Deff − 1 Normalised effective diffusion coefficient 0.9 0.5 1.5 1 0.4 0.3 0.5 0.2 0.1 0 0 20 40 60 80 100 120 Particle mass number 140 160 180 200 0 1 2 3 4 5 6 7 Particle charge number 8 9 10 11 Normalised effective diffusion Normalised effective diffusion coefficient for various charge states of coefficient for various impurities neon. species, all with Z=10+ . Diffusion coefficients are normalised to measured coefficient in the absence of an electric field O2.09 TTG 2010 13/17 C. G. Wrench Introduction Particle Simulation Discussion Scaling with electric field parameters Variation of effective diffusion coefficient with change in electric field width and strength for C6+ ions. Scaling is of the form: b1 (E −E0 ) ∆r Deff = D0 ∆r0 0.121E ∆r m2 s−1 = 9.436 0.159ρL 0.08 Electric field width [m] 0.07 0.06 0.6 0.05 0.4 0.2 0.04 0.8 0.03 0.02 0.3 0.01 0 0 0.7 0.9 −1 O2.09 TTG 2010 0.5 −2 −3 −4 −5 −6 Peak electric field strength [kVm−1] 14/17 −7 Here ∆r is the electric barrier width in meters and electric field strength, E , is measured in kVm−1 C. G. Wrench Introduction Particle Simulation Discussion Comparison of Carbon Profiles with Experiment 18 6 x 10 Scaled CUEBIT C6+ density Particle density [m−3] 5 4 3 2 1 0 0 Carbon density profile in MAST as measured by the Charge Exchange Diagnostic. Image courtesy J. McCone. O2.09 TTG 2010 15/17 0.1 0.2 0.3 0.4 Minor radius [m] 0.5 0.6 Scaled C6+ density profile as computed by CUEBIT. C. G. Wrench Introduction Particle Simulation Discussion Concluding remarks • A strongly sheared electric field can strongly modify collisional transport • We have used full-orbit test-particle approach to study collisional impurity transport in spherical tokamaks with strongly-sheared electric fields • Collisional drag in presence of radial E < 0 produces local inward pinch, which increases with Z but is independent of impurity mass • Negative implication of this is the retention of highly ionised species, e.g. tungsten • Effective diffusion scales with the electric field width with a power proportional to the electric field strength O2.09 TTG 2010 16/17 C. G. Wrench Introduction Particle Simulation Discussion References • Chen, H., Hawkes, N. C., Ingesson, L. C., Peacock, N. J. and Haines, M. G., Nuclear Fusion 41 pp. 31, 2001. • Hinton, F. L. and Kim, Y.-B., Physics of Plasmas 2, pp. 159, 1995. • Meyer et al., Nuclear Fusion 49 104017, 2009. • Shaing, K. C. and Hazeltine, R. D., Physics of Fluids 4, pp. 2547, 1992. • Shaing, K. C., Aydemir, A. Y. and Hazeltine, R. D., Physics of Plasmas 5, pp. 3680, 1998. • Tau, Y. -Q., et al., Phys. Fluids B 5, pp. 344, 1993. O2.09 TTG 2010 17/17 C. G. Wrench