7.4 Electron Cyclotron Current drive

7.4.1 Introduction

Fast and accurate kinetic calculations of electron cyclotron current drive (ECCD) are of great importance given that ECCD provides the most controllable and adjustable known source of CD in tokamaks, and is used in a wide range of advanced tokamak experiments, including to achieve fully non-inductive steady state operation and stabilize neoclassical tearing modes . The calculation of ECCD requires to use all the features and power of the the DKE code, including the fast implicit treatment of trapped particles and momentum space fluxes at the trapped/passing boundary. Indeed, wave-induced trapping and momentum exchange with trapped particles are a dominant aspect of off-axis ECCD as they explain the Ohkawa effect.

We illustrate ECCD calculations using an idealized DIII-D scenario with R_p = 1.67 m, a_p = 0.67 m, B_t = 2 T, Z_eff = 2, T_e0 = 4 keV and n_e0 = 3 × 10¹⁹ m^-3. The temperature and density profiles are either parabolic or constant, as specified for each following simulation. The poloidal magnetic field is assumed to be negligible. For simplicity, the plasma is assumed to be circular without Shafranov shift. The primary ECCD parameters are X-mode polarization, N_∥ = 0.3, f_EC = 110 MHz P_0EC = 1.0 MW and Gaussian spectrum width ΔN_∥ = 0.02. For these parameters, an ideal EC beam propagating from the low field side on the outboard midplane (θ_b = 0) would be absorbed at second harmonic near the plasma center, and the peak in the power deposition profile would be located at a normalized radius ρ = 0.1. At this location, the frequency ratio y₂ = 2ω_ce∕ω, which defines the position of the resonance curve in momentum space (along with N_∥), is y₂ = 0.98. The normalized diffusion coefficient is D_0EC^new = 0.15. In the following sets of calculation, the relaxation is let to evolve completely freely: no normalization between time steps and no forced Maxwellian at p = 0.

7.4.2 Grid size effects

In the first set of calculations, we consider the primary ECCD case (ρ = 0.1, θ_b = 0, N_∥ = 0.3, ΔN_∥ = 0.02, y₂ = 0.98, D_0EC^new = 0.15) and vary the size of the momentum grid , which is taken to be uniform (Fig 7.17). The maximum momentum grid point is set at p_max = 10. The computed electron density increases slightly then stabilizes during the relaxation and reaches an output value n_eout with (∕n_e ≤ 6%). This difference is quite small given that a EC diffusion coefficient of D_0EC^new = 0.15 is rather large. The output density does not rapidly converge to 1 when the grid size in increased (graph a) which indicates that the primary reason for the density increase is not the precision of the discretization scheme. The the flux-surface averaged driven current densiry (graph b) converges towards a normalized asymptotic value j_∞^EC = 2.4 × 10^-2 (en_ev_Te). However, there are oscillations in the evolution of j^EC with which account for most of the disparity with j_∞. These oscillations can be attributed to the sharp Gaussian-like evolution of the diffusion coefficient in momentum space, with a characteristic variation size that is smaller than a grid step. The gaussian shape of the spectrum being somewhat smoother than the LH square spectrum, the variations in j^EC are less important than for the previous LH case. The calculation of the the flux-surface averaged density of power absorbed (graph c) is however quite robust and does not depend much on the grid size. It rapidly converges to a value p_∞^EC = 1.8 × 10^-2 (n_em_eν_ev_Te²). The evolution of the normalized figure of merit η^EC ≡ j^EC∕p^EC mostly follows the variations of j^EC (graph d). It converges to a value of η_∞^EC = 1.3 (e∕m_eν_ev_Te). For a 100 × 100 grid, the mean error on j^EC and η^EC is about 10%. It reduces to less than 5% for a 200 × 200 grid. The ratio of the power absorbed from ECW to the power lost on collisions expectedly tends to 1 as the grid size increases (graph e), until the grid size reaches about 150 × 150, where it undergoes an unexplained jump to about 1.1.

The requirements of ECCD calculations on the maximum momentum p_max are far more easily satisfied than for LHCD. To illustrate this, we consider the same case (ρ = 0.1, θ_b = 0, N_∥ = 0.3, ΔN_∥ = 0.02, y₂ = 0.98, D_0EC^new = 0.15) and fix n_ξ = 100. The maximum momentum p_max and the momentum grid size n_p are varied proportionally so that the momentum step size remains unchanged and the discretization effects are therefore removed. The results are shown on Fig. 7.18. None of the relevant integral quantities (n_eout, j^EC, p^EC, p^EC∕p^col., η^EC) varies significantly for p_max∕p_Te ≥ 7. For comparison, the maximum value of p_⊥ on the resonance curve is p_⊥,max^res = 2.7p_Te and the maximum value of p is p_max^res = 6.4p_Te.

7.4.3 Electron trapping effects

In the next set of calculation (Fig. 7.19), we keep the same ECCD parameters (θ_b = 0, N_∥ = 0.3, ΔN_∥ = 0.02, y₂ = 0.98, D_0EC^new = 0.15) but vary the radial location of ECCD. The density and temperature profiles are kept constant such that the only effect to consider is the increasing number of trapped particles as ϵ = r∕R_p increases. The output density increases slightly with ϵ (graph a) but remains acceptable. The density of driven current steadily decreases and reverses signs for ϵ ≥ 0.16 (graph b). This evolution indicates that the Ohkawa effect, which is due to ECW-induced trapping and drives a counter-current, eventually compensates and even dominates the Fisch-Boozer effect. In fact, the Ohkawa current density peaks at ϵ ≃ 0.25 and then goes back to 0 as ϵ further increases. The ohkawa effect is maximum where the EC resonance curve in momentum space is tangent to the trapped passing boundary, so that wave-induced trapping is maximum. For larger ϵ - or wider trapping region - most of the EC power is transferred directly to trapped electrons, which do not drive current. This explains why → 0 as ϵ ≥ 0.25 increases. The density of power absorbed slowly increases with ϵ (graph c). The increases is faster for ϵ ≥ 0.25, which can be expected since at these location, increasing coupling with trapped particles occur. Indeed, because of the fast bounce motion, this coupling is effectively distributed between trapped electrons travelling in both directions. This distribution divides the effective magnitude of the diffusion coefficient by two but also doubles the amount of resonant particles in the trapped region. As a result, quasilinear effects on ECCD with trapped particles are reduced, which explains the larger density of power absorbed. The evolution of the normalized figure of merit is similar as that of (graph d) and the ratio of he power absorbed from ECW to the power lost on collisions is quite stable, and very close to 1.

7.4.4 Momentum-space dynamics

In order to illustrate the momentum-space dynamics of ECCD, the 2D distribution function is plotted in the space (Fig. 7.20). The case considered here is again our primary example (ρ = 0.1, θ_b = 0, N_∥ = 0.3, ΔN_∥ = 0.02, y₂ = 0.98, D_0EC^new = 0.15). On graph a, the initial Maxwellian (thin blue contours) and the steady-state ECCD distribution (thick red contours) are displayed, as well as contour of the magnitude of the EC diffusion coefficient (green dashed contours). The distortion of the distribution from a Maxwellian in the vicinity of the resonant region is clearly visible. Collisions are trying to restore the Maxwellian and induce momentum-space fluxes that affect the entire momentum space. By integrating over p_⊥, we obtain the parallel distribution

(7.11)

shown on graph b. In the non-relativistic limit, the integration of F_∥ over p_∥ gives the driven current density. The Fisch-Boozer effect, resulting in an accumulation of particles with large, positive p_∥, appears clearly. This effect - understood as an asymmetric resistivity due to EC heating - is also illustrated on graph c, where the perpendicular temperature is plotted as a function of p_∥; the difference in temperature, due to asymmetric EC heating, induces the asymmetric resistivity and driven current. Note that EC induced diffusion in momentum space is mostly in the perpendicular direction, and therefore does not directly drive any current. It is always the collisional response to this diffusion which drives a current.

7.4.5 Coupling to propagation models

The code can be coupled to ray-tracing calculations. Iterations on the 3D ECCD calculations ensure that the power absorbed from the EC wave is consistent with the power travelling in the EC beam. Such calculations are very fast, thanks to the fully-implicit 3D algorithm. To illustrate this, we consider a simplified ECCD scenario with X-mode polarization, N_∥ = 0.3, f_EC = 110 MHz, P_0EC = 1.0 MW and Gaussian spectrum width ΔN_∥ = 0.02. The EC beam propagates from the low field side on the outboard midplane (θ_b = 0). The evolution of N_∥ due to toroidicity is neglected, and the cold plasma model is used to solve the dispersion relation. The DKE calculations are performed on a nψ × n_p × n_ξ = 26 × 100 × 100 grid, and consistency between travelling and absorbed power is reached after 4 iterations. The resultant current density and absorbed power density profiles are shown on Fig. 7.21. The current profile appears to be shifted slightly to the right compared to te power profile. This shift results from the fact that at ρ ≥ 0.1, resonant electrons are further in the tail of the distribution - meaning that they are less collisional - than at ρ ≤ 0.1, and therefore the local efficiency is higher for ρ ≥ 0.1. The location ρ = 0.1 corresponds to the peak in the absorbed power profile, which justifies taking this value for the previous simulations.

7.4.6 Conclusion

To sum up, we have shown that the DKE code accurately describes and calculates ECCD. It runs robustly in the free conservative mode. Grid parametric requirements are somewhat less restrictive than for LHCD, because the EC diffusion coefficient variations are somewhat smoother. It accounts correctly for trapped electron effects, such as the Ohkawa current. It can be used to simulate actual ECCD scenarios, including coupling and consistency with ray-tracing calculations.