H. Sugama

J.-K. Park

R." Kanno

S. "Satake

National Institute for Fusion Science (NIFS-Repository)

         Gyrokinetic theory for helical plasma with an equilibrium-scale radial electric field (Er0) predicts enhancement of the zonal flow (ZF) response, where Er0 drives the poloidal rotation of helical-ripple-trapped particles with reduced radial displacements of drift orbits. The poloidal ExB rotation due to Er0 is introduced in the gyrokinetic equation in terms of the poloidal Mach number (Mp). If the ion mass is changed with the ion temperature and the radial electric field fixed, the Er0 dependence of the ZF response is translated into an isotope effect through Mp. Indeed, a neoclassical transport analysis for the the Large Helical Device (LHD) confirms that magnitude of the equilibrium radial electric field estimated for the deuterium discharge is almost identical to that in the hydrogen plasma, if other parameters are the same. The isotope effect on ZF response is, thus, expected to appear through Er0 generated by the neoclassical transport, and would play a favorable role in reducing the turbulent transport because the turbulent transport regulation by the stronger zonal flows is anticipated. The gyrokinetic simulation of the ZF enhancement by Er0 in helical systems was carried out by means of the gkv code, and shows effective amplification of the ZF response for Mp=0.2-0.3 which is relevant to LHD experiments. The isotope effect on ZF response, thus, appears through Er0 generated by the neoclassical transport, and is expected to play a favorable role in reducing the turbulent transport.      In the present study, long-time behaviors of the ZF response are investigated where an extended version of gkv code is applied. The ITG turbulence simulation in case with the equilibrium-scale radial electric field in helical systems is also investigated. While the uniform and constant Er0 leads to the Doppler shift of the ITG mode frequency, the linear instability growth rate is unaffected. Nevertheless, the nonlinear gkv simulation presents reduction of a turbulent fluctuation level and transport flux in case with Er01).      Zonal flow amplitudes presented in the previous study have shown good agreement with the theoretical estimate in the long wavelength limit. According to the theoretical analysis, after many poloidal rotations, radial drift motion of trapped particles causes finer velocity-space structures of the perturbed distribution function with the higher kr. In contrast, the previous simulations were carried out only one poloidal rotation which is much shorter than the time scale for development of the fine velocity-space structures. Therefore, we have investigated a long time behavior of zonal flow response. In the long time simulation run for the single helicity case [Fig. 1 (a)], for all wavenumbers considered here, one finds oscillatory behaviors of the ZF response due to the ExB rotation of the helical ripple trapped particles. In the single-helicity case with no transition particles, Er0 effectively drives the poloidal rotation of helical ripple trapped particles. The measured oscillation period is close to that of the poloidal rotation. In the inward-shifted model case, the ZF amplitudes quite slowly decreases in time [see Fig. 1 (b)]. One finds that the relative amplitude among different wavenumber modes varies in time, where the smaller kr modes decay faster. This is clearly found by plotting relative potential amplitudes (normalized by that of the lowest wavenumber component) which continue to grow in time.  Fig.2 shows a resultt of the nonlinear gkv simulation for the ITG turbulence in helical systems with Er0, where ratio of the ZF amplitude, Z, and the turbulent fluctuations, T, is plotted as a measure for effectiveness of the zonal flow generation by turbulence. In the case with Mp = 0.3, the ZF amplitude normalized by the turbulent fluctuations (Z/T) increases after the saturation of the instability growth at t=40 Ln/vti, even though almost the same values of T are observed before the saturation. The obtained results suggest that the ZF response enhancement due to the equilibrium-scale radial electric field plays a positive role in the turbulence suppression and the transport reduction.   1)  Watanabe, T.-H., Sugama, H., Nunami, M.: Proceedings of the 23rd IAEA Fusion Energy Conference. Fig2: Zonal flow potential amplitudes normalized by the turbulent fluctuations Fig1: Long-time zonal flow response for (a) single-helicity and (b) inward-shifted cases. Control of the toroidal rotation is an important is-sue in tokamaks in order to improve the stability of theconﬁned plasmas. Recent studies have shown that thenon-axisymmetry as small as δB/B0 ∼ 10−4 can inducesigniﬁcant damping in toroidal rotation by the neoclassi-cal toroidal viscosity (NTV). To give a precise predictionof the toroidal rotation damping by small perturbation,a new simulation to evaluate NTV is developed using theδf Monte Carlo method.1) It is the ﬁrst application ofthe δf simulation to NTV calculation. The advantageof this method is that we can calculate NTV withoutrelying on the conventional approximations used in ana-lytic theories such as zero-orbit-width limit, mono-energytreatment with pitch-angle scattering operator, etc.We utilize FORTEC-3D neoclassical transportcode2) to this purpose. It solves the drift-kinetic equa-tion for δf in 5-dimensional phase space, where δf isthe deviation of plasma distribution function from localMaxwellian. One can directly evaluate NTV ⟨eζ · ∇ ·P⟩from δf as followings;⟨eζ · ∇ ·P⟩ = B0∑m,n̸=0nδm,n⟨δPBsin(mθ − nζ)⟩, (1)where δP = M∫d3vδf(v2⊥/2 + v2∥) measures the pres-sure anisotropy, M is ion mass, and δm,n are Fouriercomponents of magnetic ﬁeld in Boozer coordinates,B(r, θ, ζ) = B0[1 +∑m,n δm,n(r) cos(mθ − nζ)].Since it is the ﬁrst application of the δf simu-lation for NTV calculation in weakly-perturbed toka-maks, we have carried out detailed benchmark tests inE×B→ 0 case with analytic formulae,5) which are de-rived from bounce-averaged drift kinetic equation. Theone is Shaing’s asymptotic limit formula for the 1/νand Superbanana-plateau regimes,3) and the other isPark’s combined analytic formula which is applicablewide range of collisionality regimes.4) Figure 1 shows thedependence of NTV on the normalized collisionality ν∗in a tokamak with (a): (m,n) = (7, 3) single-helicityand (b): (m = 13, 14, 15, n = 3) multi-helicity pertur-bation cases, respectively. The perturbation amplitudeeach case is dm,n ∼ 10−3, and the resonant ﬂux surfaceon which the rotational transform q equals to m/n is lo-cated at r = 0.49a (Fig. 1(a)) and r = 0.70a (Fig. 1(b)),respectively. In each case, we found good agreement be-tween FORTEC-3D simulation and Park’s combined an-alytic formula. On the other hand, the asymptotic limitformulae is found to fail the dependence of NTV on col-lision frequency as ν∗ becomes lower. The benchmarkresult proved that proper model that takes account ofν∗-dependence of NTV is important to quantitative eval-uation of NTV in low-collisionality plasmas.Neoclassical viscosity is also important in transportstudies on helical plasmas. FORTEC-3D is also applica-ble to neoclassical poloidal and toroidal viscosity calcula-tions in LHD. We have shown that the neoclassical radialﬂux evaluated from linear combination of toroidal andpoloidal viscosities exactly equals to that calculated fromthe radial drift velocity, Γ =⟨∫d3vvd · ∇rˆδf⟩, whichdemonstrates the accuracy of the viscosity calculation inhelical conﬁgurations1). It is planned to study the de-pendence of the neoclassical viscosity on radial electricﬁeld, magnetic axis position, collision frequency. etc.1e-061e-051e-041e-031e-021e-04 1e-03 1e-02 1e-01 1e+00 1e+01NTV [N/m2 ](a) F(0.381)F(0.493)F(0.656)F(0.806)P(0.381)P(0.493)P(0.656)P(0.806)1/�SB-P1e-071e-061e-051e-041e-031e-04 1e-03 1e-02 1e-01 1e+00 1e+01NTV [N/m2 ]� *(b) F(0.606)F(0.706)F(0.794)P(0.606)P(0.706)P(0.794)1/�SB-PFig. 1: NTV dependence on ν∗ in tokamak with(a):single-helicity and (b):multi-helicity magnetic pertur-bation. The numerals on the legend describe the radialposition r/a. The ”F” curves are results from FORTEC-3D, ”P” symbols are Park’s formula, and straight linesare asymptotic limit formulae, respectively.1) Satake, S. et al., Plasma Phys. Control. Fusion 53(2011) 054018.2) Satake, S. et al., Plasma and Fusion Res. 3 (2008)S1062.3) Shaing, K. C. et al.., Nucl. Fusion 50 (2010) 025022.4) Park, J.-K. et al.., Phys. Rev. Lett 102 (2009) 065002.5) Satake, S. et al.., ”Neoclassical Toroidal ViscosityCalculations in Tokamaks using a δf Monte CarloSimulation and Their Veriﬁcations”, accepted to pub-lish in Phys. Rev. Lett.355§16. Development of Neoclassical Viscosity Simulation MethodSatake, S., Park, J.-K. (PPPL), Sugama, H., Kanno, R.

Development of Neoclassical Viscosity Simulation Method

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Development of Neoclassical Viscosity Simulation Method

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