Angelino, P.

Bottino, A.

Hatzky, R.

Jolliet, S.

Sauter, O.

Tran, T.

Villard, L.

English

MPG.PuRe

Plasma elongation and magnetic shear effects in nonlinear simulations of ITG-zonal flow turbulence

Plasma elongation and magnetic shear effects in nonlinearsimulations of ITG-zonal flow turbulenceP. Angelino1, A. Bottino2, R. Hatzky3, S. Jolliet1, O. Sauter1, T.M. Tran1, L. Villard11 Centre de Recherches en Physique des Plasmas, EPFL, Lausanne, Switzerland2 Max Plank Institut für Plasmaphysik, Garching, Germany3 IPP and Computer Center of the Max-Planck-Gesellschaft, Garching, GermanyZonal flow dampingRecent global gyrokinetic simulations [1] have shown the importance of the zonal (n = 0,m = 0) component of the perturbation on ITG turbulence: while the steady (zonal flow)component is efficient in reducing turbulence, an oscillatory component at the GeodesicAcoustic Mode (GAM) frequency appears to be correlated with less efficient turbulencesuppression. In this paper we examine geometrical effects on steady zonal flows and GAMoscillations. The initial zonal flow component (n = 0,m = 0) of an electrostatic per-turbation is found to linearly damp and a residual flow level is found [2]. Therefore, weexpect the E×B velocity [vE, normalized to the initial value vE (0)], generated by a pureaxisymmetric density perturbation, to behave as:vEvE (0) = (1−AR) e−γGt cos (ωGt) +AR ; AR = 11 + 1.6q2s (r/R)−1/2(1)where AR is the residual, at the radial position r [2]. The other parameters are R themajor radius and qs the safety factor value. The frequency and the decaying rate of thevelocity perturbation are respectively:ωG = vtiR ; γG = ωG exp(−q2s) (2)A more accurate calculation [3], in the fluid limit, gives:ωG =(1 +12q2s)1/2√2γPρmR2 ; γG = ωG exp(−q2s −12) (3)where γ is the adiabatic index or ratio of specific heats, P is the total plasma pressureand ρm is the mass density. We make the substitution√γP/ρm = cs, as in [4]. Here cs isthe sound speed. A kinetic expression for the frequency and the growth rate is derived in[5]:ωG =(74+ τ)1/2 vti√2R ; γG = ωG exp[−q2s2(74+ τ)](4)33rd EPS Conference on Plasma Phys. Rome, 19 - 23 June 2006 ECA Vol.30I, P-1.195  (2006)Here τ is the ratio of the electron temperature over the ion temperature and vti ≡√2kBTimithe ion thermal velocity. One of the main differences between the fluid and the kineticresults is that the fluid expression for the frequency includes effects of the parallel dynam-ics, which is neglected in the kinetic calculation. Geometrical effects, such as the plasmaelongation and triangularity, are ruled out in both cases by the circular magnetic surfaceassumption. We investigate the role of the parallel dynamics and of the plasma elongationwith the electrostatic global gyrokinetic code ORB5 (details on the model can be foundin [1][6]). A zonal flow damping test has been performed with the ORB5 in linear mode,which means that nonlinear terms in particle trajectory equations have been suppressed.In order to reproduce the results of Hinton and Rosenbluth theory, we solve only forthe n = 0,m = 0 component of the electrostatic potential, the other modes are Fourierfiltered. The initial condition has been prepared in order to obtain an axisymmetric iondensity perturbation. The perturbation of the ion density has been set to δni = n0 sin (πρ),where δn0 is chosen so that 〈vE〉ρ (t = 0) = 0.07 vti.Role of the parallel dynamicsA β ∼ 0, circular concentric magnetic surfaces, large aspect ratio (R/a = 10) equilib-rium has been supplied by the ideal MHD equilibrium solver code CHEASE. The physicalparameters are: major radius R = 5 m, minor radius a = 0.5 m, magnetic field on axisB0 = 1.44 T. Density, ion and electron temperature profiles are flat with Ti = Te. Thesize of the system is fixed by Lx ≡ 2a/ρs = 80, where ρs is the ion Larmor radius. Twocases (a) and (b), with different safety factor profiles, have been used for the test. The qsvalues at the radial positions (ρ = r/a) chosen for the test are:ρ = 0.5 ρ = 0.7(a) 1.15 1.33(b) 1.70 1.96(5)The results of the ORB5 simulations are plotted in Fig. 1. In these figures we plot theE × B velocity normalized at the initial value vE/vE (0) as a function of time. Time isnormalized to the ion cyclotron frequency Ωci. As a reference we also plotted the residualevaluated from Eq. (1) and the exponential decay predicted by Eq. (??). In all cases theresults are in good agreement with the residual predicted by Rosenbluth-Hinton.Table 6 and 7 give a summary of the frequencies ωG and decaying rates γG from the sim-ulations, compared to the values predicted by theory. We find an overall good agreementbetween numerical results and theory predicted values. The simulations highlight the ef-33rd EPS 2006; P.Angelino et al. : Plasma elongation and magnetic shear effects in nonlinear simulations of I... 2 of 40 0.2 0.4 0.6 0.8 100.51ρ = 0.7ORB5 dataAR form RH theory(1−AR)exp(−γG*t)+ARt⋅Ωci⋅10−4V E/VE(0)0 0.5 1 1.5 2−0.500.51ρ = 0.7ORB5 dataAR form RH theory(1−AR)exp(−γG*t)+ARt⋅Ωci⋅10−4V E/VE(0)Figure 1: E ×B velocity (vE) at ρ = 0.7 as a function of time, for low qs profile (a) and high qs profile(b). The solid line is the result of the ORB5 simulation in linear mode, with sin (piρ) perturbation. Thedotted line is the residual and the dashed line is the exponential decay from Rosenbluth-Hinton theory.fect of the parallel dynamics on ωG which is correctly included in the fluid model, [Eq. (3)],but not in the kinetic model, [Eq. (4)], where the qs dependence has been neglected.qs ωG Eq. (4) ωG Eq. (3) ωG (ORB5)1.15 0.0059 0.0059 0.00611.33 0.0059 0.0057 0.00581.70 0.0059 0.0055 0.00531.96 0.0059 0.0054 0.0053(6)qs1.151.331.701.96γG Eq. (4) γG Eq. (3) γG (ORB5)−0.0010 −0.0010 −0.0009−0.0005 −0.0006 −0.0006−0.0001 −0.0002 −0.0002−0.00002 −0.0001 −0.0001(7)Role of the plasma geometryIn order to investigate the role of plasma elongation, we run a simulation with the sameparameters as in the previous section [safety factor profile (a)], but with an elongationκ = 1.5. The resulting frequency, growth rate and residual are noted in Table 8:κ = 1.5qs = 1.15qs = 1.33ωG (ORB5) γG (ORB5) AR (ORB5)0.0060 −0.0008 0.1090.0058 −0.0007 0.098(8)Comparing these results with the ones in Table 6 and 7, we observe that the elongationalone has a minor effect on the frequency and damping rate. Now we introduce a more33rd EPS 2006; P.Angelino et al. : Plasma elongation and magnetic shear effects in nonlinear simulations of I... 3 of 4complex magnetic configuration which is reconstructed from a shot of the tokamak TCV.It is a D-shaped plasma with inverse aspect ratio ǫ ≡ a/R = 0.27, a = 0.24 m = 40 ρi,triangularity δ = 0.4 and magnetic field on axis B0 = 1.44 T. Density, ion temperatureand electron temperature profile are flat with Ti = Te. The size of the system is fixed byLx ≡ 2a/ρs = 80, where ρs is the ion Larmor radius. Two different values of the plasmaelongation are considered κ = 1 and κ = 1.5. The qs values at the radial positions chosenfor the test are qs (ρ = 0.5) = 2.20, qs (ρ = 0.7) = 3.10. The results are noted in Table 9and 10:κ = 1qs = 2.2qs = 3.1ωG (ORB5) γG (ORB5) AR (ORB5)0.0140 −0.0012 0.0490.0138 −0.0011 0.035(9)κ = 1.5qs = 2.2qs = 3.1ωG (ORB5) γG (ORB5) AR (ORB5)0.0128 −0.0013 0.0580.0124 −0.0012 0.042(10)For the case κ = 1, we write in Table 11 the values predicted by the kinetic and fluidtheories. A comparison between Table 11 and Table 9 shows that triangularity and smallaspect ratio effects introduce a substantial difference between the predicted γG values andthe values obtained by the simulations. Again, comparing Table 9 with Table 10 gives usan idea of the magnitude of the elongation effects.κ = 1qs = 2.2qs = 3.1ωG Eq. (4) γG Eq. (4)0.0149 −7 · 10−50.0146 −6 · 10−7ωG Eq. (3) γG Eq. (3)0.0167 −2 · 10−50.0167 −3 · 10−8AR Eq. (1)0.0460.028(11)The preliminary results presented suggest that geometrical effects can play a relevantrole on the physics of the zonal flow damping. The effects from the triangularity andaspect ratio together seem to be more important than from the elongation only. Furtherinvestigations are needed in order to separate the effects of triangularity and aspect ratio.References[1] P. Angelino et al. Plasma Phys. Control. Fusion, 48:557—571, 2006.[2] M. N. Rosenbluth and F. L. Hinton. Physical Review Letters, 80(4):724—727, 1998.[3] N. Winsor, J. L. Johnson, and J. M. Dawson. Phys. Fluids, 11:2448, 1968.[4] P. H. Diamond et al. Plasma Phys. Control. Fusion, 47:R35, 2005.[5] V. B. Lebedev et al. Physics of Plasmas, 3(8):3023—3031, 1996.[6] P. Angelino et al. Physics of Plasmas, 13(5):052304, 2006.33rd EPS 2006; P.Angelino et al. : Plasma elongation and magnetic shear effects in nonlinear simulations of I... 4 of 4

Plasma elongation and magnetic shear effects in nonlinear simulations of ITG-zonal flow turbulence

Abstract

Similar works

Full text

Available Versions

MPG.PuRe

MPG.PuRe

MPG.PuRe