Gyrokinetic simulations including the centrifugal force in a strongly rotating tokamak plasma F.J. Casson, A.G. Peeters, Y. Camenen, W.A. Hornsby, A.P. Snodin, D. Strintzi, G.Szepesi CCFE Turbsim, July 2010
Outline GKW: The numerical tool When is strong rotation important? Gyro-kinetics in the rotating frame Coriolis force (previous work) Centrifugal force (this talk) New drift, enhanced trapping, modified equilibrium Consequences for drift waves [This talk is mostly contained in a paper submitted to PoP (2010)] Consequences for heat and particle transport Momentum transport (earlier work, if time)
GKW: The numerical tool Gyro-kinetic flux tube code Spectral (perpendicular), local δf Non-linear, kinetic electrons Fully electromagnetic perturbations General geometry (with CHEASE) Collisions, multiple species MPI scales to 8192+ processors Formulated in rotating frame http://gkw.googlecode.com/ A.G. Peeters, Y. Camenen, F.J. Casson, W.A. Hornsby, A.P. Snodin, D. Strintzi, G. Szepesi The non-linear gyro-kinetic flux tube code GKW Comp Phys Comm, 180, 2650, (2009)
Strong rotation For strong rotation, keep the centrifugal force (Going round a corner throws you sideways) Rotation is strong when the toroidal velocity of the plasma is of the order of the thermal velocity Mach Number: Caution:
Gyro-kinetics in a rotating frame Elegant formulation using the co-moving system [1] Guiding centre velocity Parallel velocity Inertial term A rigid body toroidal rotation is assumed Toroidal background rotation Angular frequency [1] A.J. Brizard PoP (1995)
Comments Toroidal background rotation Angular frequency Choose the rotation of the frame to be the rotation of the plasma on the flux surface we are modelling In a local model the co-moving system yields compact equation similar in form to the non rotating system. One does not have to deal with a large flow across the grid (the large ExB velocity of strong toroidal rotation is transformed away) Not suited for a global description since a gradient in the rotation would lead to a time dependent metric
Consequences of strong rotation (1) Centrifugal drift Peeters PRL 2007 Coriolis drift Coriolis drift influences momentum transport (Peeters PRL 2007) Not discussed here as it appears at weak rotation when Enhanced trapping Unlike the Coriolis drift the centrifugal drift has a component along the field and therefore accelerates the particles outward Energy equation: A.G. Peeters et al. Physics of Plasmas 16, 042310, (2009)
Consequences of strong rotation (2) Modified equilibrium equation Normal trapping. Any distribution (like the Maxwell) which is isotropic is an equilibrium distribution Mass dependent centrifugal force different for electrons and ions Must retain a background electro-static potential which is a function of the poloidal angle in order to satisfy quasi-neutrality Assume Maxwellian in velocity A.G. Peeters et al. Physics of Plasmas 16, 042310, (2009)
Consequences of strong rotation (3) Solve the equilibrium equation Find density is not a flux surface quantity Centrifugal energy (species dependant) Centrifugal potential found by solving for quasineutrality (Numerically for a plasma with more than 2 species) Density gradient also varies Choice for density definition location (Here we use LFS)
Choose where to define the density GKW gives two choices for R0: Low field side (used for all results later) Magnetic axis Physics is independent of this choice, but it is only possible to hold R / Ln constant at one location when scanning over rotation
Consequences of strong rotation (4) Density and rotation not independent parameters Some care required in interpreting results GA-STD case
Inertial Terms (full) (2 species plasma)
Source term - comments In order that the density gradient at R0 has the intuitive meaning of being in the radial direction, the radial derivative of R0 (at constant theta) must be kept in the source. The other parts of the derivative of the density exponential cancel with parts of the term
Strong rotation: Enhanced trapping (2 species plasma) Centrifugal potential traps electrons, but detraps ions. For both species: Velocity space plot for a trapped electron mode (TEM): Trapped region enlarges with rotation, so mode is enhanced
Modified threshold for TEM Growth rate spectrum ITG TEM GA-STD case, D plasma: ITG Dispersion relation shows TEM dominates at larger scales with rotation TEM
Nonlinear heat fluxes Zonal flow response (GAM) Rosenbluth-Hinton GA-STD case For ITG turbulence dominated case, heat transport increases with rotation Not included here: Stabilising effect of background ExB shear from sheared toroidal rotation
Behaviour near TEM threshold Threshold of dominant linear mode does not translate exactly to nonlinear case ITG mode seems more resilient in nonlinear phase Zonal flow reduction benefits ITG more than TEM? Coexistence and interaction of ITG / TEM modes
Particle and Impurity transport The Mach number for impurities is high even at low Mach number for the bulk ions: Diffusion Thermodiffusion Convection (inward) C.Angioni, A.G.Peeters, PRL 96, 095003 (2006) Impurities feel the centrifugal force more strongly Linear analysis of a single mode may be misleading: No interplay of scales No interplay of TEM / ITG modes All coexist in the nonlinear state Balance is important for particle flux [ Angioni PoP 2009 ] Choice of R/Ln needed to define diffusion coefficent
Locate the null flux (Back to the GA-STD case) Null flux state independent of choice of R0 Null flux state is a balance of scales, at each scale: Inward contribution from slow trapped electrons Outward contribution from fast trapped electrons Effect expected to be stronger for heavy impurities [ Angioni PoP 2009 ] More important to locate the null flux state due to strong density redistribution
Strong Rotation: Future work Quantify predictions of impurity transport with rotation and compare to experiment Well resolved non-linear simulations, collisions Ideal experiment: Rotation controlled independently from heating with dual NBI. Include E x B shear with strong rotation Regimes with less ITG dominance ETG - MAST relevance
Summary and Conclusions The flux tube model in the rotating frame naturally allows the inclusion of the centrifugal force Density gradient and rotation become coupled The centrifugal force leads to an enhanced trapping which promotes trapped electron modes In strong TEM regimes, increased electron heat transport is expected with strong rotation Particle pinch for ITG dominated cases examined Increased fraction of slow trapped electrons Stronger effect expected for impurity transport More nonlinear simulations needed These effects could be observed on MAST?
Momentum transport (local) symmetry breaking mechanism type / direction toroidal rotation gradient (Peeters PoP 05) toroidal rotation (Peeters PRL 07, Peeters PoP 09) ExB advection in Coriolis drift the background (+ kin. electrons) ExB sheared flow (Dominguez 93, Waltz PoP 07, Casson PoP 09) up-down asym. of the MHD equil. (Camenen PRL 09, Camenen PoP 09) ExB shearing asymmetry of perp. drifts and k diagonal part pinch residual stress residual stress outward generally inward inward/outward inward/outward -1 to -4 0.4 to 0.8 0 to 1 [small param. range explored] [linear sim. only] Pr = 0.8 to 1.2 higher for TEM magnitude lower with ExB shearing due to toroidal rotation ε, β, ΤΕΜ/ΙΤΓ, θ, R/Ln, mag. shear, s, s, mag. shear, main dep. / tested µαγ. σηεαρ, Ρ/Λν, e, q, Te/Ti, R/LT, B j ge, u' Ρ/ΛΤι, νεφφ n ef f, TEM/ITG q, mag. shear, R/Ln, TEM/ITG, sb, sj
Symmetry breaking by E x B shear Toroidal rotation Modification to diffusivity Reduction in effective diffusivity (opposite for negative magnetic shear) F.J.Casson, A.G.Peeters et Al., Phys. Plasmas, 16, 092303 (2009)
End
Reformulation of Brizard s equations yields Finite ρ expansion Parallel velocity ExB drift Coriolis Force Centrifugal force Curvature drift Coriolis drift Grad-B drift Centrifugal drift