List of Figures

2.1 Coordinates systems ( p_∥,p_⊥,φ) and (p,ξ,φ)for momentum dynamics
2.2 Coordinates system (R,Z,ϕ).
2.3 Coordinates system (r,θ,ϕ).
2.4 Coordinates system (ψ,s,ϕ).
2.5 Guiding center velocity definition
3.1 Domain in configuration space where magnetic ripple well takes place for Tore Supra tokamak
3.2 Directions of incident electron and emitted photon with respect to the local magnetic field direction
5.1 Grid definition for the momentum dynamics
5.2 Chang and Cooper weightingfunction
5.3 Lower Hybrid boundary problem
5.4 Trapped domain and related flux connections
5.5 Momentum flux connections for grid point 1 in the trapped region
5.6 Momentum flux connections for grid point 4 in the counter-passing region
5.7 Heuristic magnetic ripple modeling. The trapped/supertrapped boundary varies as well as the collision detrapping threshold are both functions of the radial location
5.8 Trapping and detrapping process induced by radial transport
5.9 Trapped domain for the first order distribution g
6.1 Qualitative shape of matrix ^ Bbb B for the Fokker-Planck equation
6.2 Qualitative shape of matrix ^ G for the drift kinetic equation
6.3 Typical arrangement of non-zero matrix coefficients in the first 2000 columns and rows in matrix ^ ℕ corresponding to the Fokker-Planck equation
6.4 Values of the non-zero matrix coefficients after diagonal preconditionning for matrix ^ ℕ^′ corresponding to the Fokker-Planck equation. Dot points correspond to pitch-angle process at constant p, while full line for slowing-down process at constant ξ₀. By definition values of all coefficients on the main diagonal are one
6.5 Matrix factorization principle. Dashed areas correspond to non-zero coefficients.
6.6 Reduction of the non-zero elements for the ^ L and ^ U matrices, by increasing δ_lu. Values of δ_lu are indicated on the top of each subfigure. For δ_lu = 10^-2, the inversion becomes instable.
6.7 Memory storage requirement reduction by increasing the δ_lu parameter, for the Lower Hybrid current drive problem. The rate of convergence towards the steady state solution is given, using the biconjugate gradients stabilized method to solve the system of linear equations. Here only a local analysis is considered at a given radial position
7.1 Normalized Ohmic resistivity as function of the inverse aspect ratio ϵ
7.2 Contour plot of the electron distribution function at ϵ = 0.31623
7.3 Contour plot of the stream lines at ϵ = 0.31623
7.4 Parallel projection, parallel and perpendicular temperatures of the electron distribution function at ϵ = 0.31623
7.5 Normalized Ohmic runaway rate as function of the inverse aspect ratio ϵ
7.6 Variations of the Lower Hybrid current and power densities, ratio between the RF and collision absorbed power density, and the current drive efficiency with the grid size. Here uniform pitch-angle and momentum grids are considered. Detailed aspect of the simulation are given in the text
7.7 Variations of the memory storage requirement and the time elapsed for kinetic calculations with the grid size. Here uniform pitch-angle and momentum grids are considered. Detailed aspect of the simulation are given in the text
7.8 Variation of the current drive efficiency with the upper mometum limit of the integration domain
7.9 Variation of the current drive efficiency with the main ion charge in the plasma
7.10 Variation of the current drive efficiency with the amplitude of the quasilinear diffusion coefficient for the Lower Hybrid current drive problem
7.11 Contour plot of the electron distribution function for D_LH = 10
7.12 Contour plot of the electron distribution function for D_LH = 2
7.13 Contour plot of the electron stream function for D_LH = 2
7.14 Contour plot of the electron distribution function at ϵ = 0.31623
7.15 Contour plot of the Lower Hybrid quasilinear diffusion coefficient at ϵ = 0.31623
7.16 Electron distribution function averaged over the perpendicular momentum direction at ϵ = 0.31623. The perpendicular and parallel temperatures are also shown
7.17 ECCD in DIII-D (ρ = 0.1,D_EC = 0.15,N_∥ = 0.3,Y = 0.98). Output density (a), normalized current density (b), normalized absorbed power density (c), normalized current drive efficiency (d), ratio of power absorbed to power lost on collisions (e), as a function of grid size (n_p = n_ξ).
7.18 ECCD in DIII-D (ρ = 0.1,D_EC = 0.15,N_∥ = 0.3,Y = 0.98). Output density (a), normalized current density (b), normalized absorbed power density (c), normalized current drive efficiency (d), ratio of power absorbed to power lost on collisions (e), as a function of momentum grid limit p_max (n_p = 10p_max,n_ξ = 100).
7.19 ECCD in DIII-D (D_EC = 0.15,N_∥ = 0.3,Y = 0.98). Output density (a), normalized current density (b), normalized absorbed power density (c), normalized current drive efficiency (d), ratio of power absorbed to power lost on collisions (e), as a function of the inverse aspect ratio ϵ = r∕R_p; temperatures, densities and Z_eff are kept constant across the plasma.
7.20 ECCD in DIII-D (ρ = 0.1,D_EC = 0.15,N_∥ = 0.3,Y = 0.98). 2D electron distribution function (a), parallel distribution function (b) and perpendicular temperature (c); blue thin lines represent f_init, red thick lines represent f₀, and green dashed contours represent D_EC.
7.21 ECCD in DIII-D (θ_b = 0,P_EC = 1 MW,N_∥ = 0.3,f_EC = 110 MHz). Current and power densities deposition profiles. 3D calculation with n_p = n_ξ = 100,n_ψ = 26.
7.22 Radial grid for 3-D JET current drive simulation. Circles correspond to the normalized poloidal flux coordinate ψ, while crosses correspond to normalized radius ρ
7.23 Pitch-angle grid for 3-D JET current drive simulation.
7.24 Momentum grid for 3-D JET current drive simulation.
7.25 Momentum grid step for 3-D JET current drive simulation. Circles correspond to the flux grid, while stars to the distribution function half-grid
7.26 Ion and electron temperature and density profiles, and effective charge profile used for calculating the JET magnetic equilibrium with HELENA. Here hydrogen and tritium densities are zero (pure deuterium plasma) . The poloidal flux coordinate ψ as function of the normalized radius ρ in the equatorial mid-plane corresponds to the magnetic equilibrium code output
7.27 2-D contour plot of the poloidal magnetic flux surfaces as calculated for JET tokamak by the code HELENA
7.28 Momentum dependence of the relativistic Maxwellian distribution function at ρ ≃ 0.36, and relation between velocity v and momentum p. The deviation from the main diagonal indicates that above p = 4, relativistic effects become important
7.29 2-D contour plot in momentum space of the Lower Hybrid quasilinear diffusion cofficient at ρ ≃ 0.36. The relativistic curvature of the lower bound of the resonance domain avoid intersection with the region of trapped electrons. The two full straight lines correspond to trapped/passing boundaries at that radial position
7.30 On the left side, 2-D contour plot of the radial diffusion rate at ρ ≃ 0.36. The velocity threshold corresponds to a kinetic energy of 35 keV approximately in the MKSA units. On the right side, the velocity dependence of D_ψ⁽⁰⁾ at ξ₀ = 1
7.31 Relative particle conservation of the drift kinetic code for the 3-D JET Lower Hybrid current drive simulation
7.32 Flux surface averaged power density profiles for collision, RF and Ohmic electric field absorption for the 3-D JET Lower Hybrid current drive simulation.
7.33 Flux surface averaged current density profiles for the 3-D JET Lower Hybrid current drive simulation.
7.34 2-D contour plot of the electron distribution function at ρ ≃ 0.36 for JET Lower Hybrid current drive
7.35 Electron distribution function averaged over the perpendicular momentum direction at ρ ≃ 0.36 for JET Lower Hybrid current drive. The perpendicular and parallel temperatures are also shown
7.36 2-D contour plot of the electron distribution function at ρ ≃ 0.78 for JET Lower Hybrid current drive
7.37 Electron distribution function averaged over the perpendicular momentum direction at ρ ≃ 0.78 for JET Lower Hybrid current drive. The perpendicular and parallel temperatures are also shown
7.38 Ion and electron temperature and density profiles, and effective charge profile used for calculating the Tore Supra magnetic equilibrium with HELENA. Here hydrogen and tritium densities are zero (pure deuterium plasma) . The poloidal flux coordinate ψ as function of the normalized radius ρ in the equatorial mid-plane corresponds to the magnetic equilibrium code output
7.39 2-D contour plot of the poloidal magnetic flux surfaces as calculated for Tore Supra tokamak by the code HELENA
7.40 Flux surface averaged current density profiles for the 3-D Tore Supra Lower Hybrid current drive simulation.
7.41 Magnetic ripple loss rate profile for Tore Supra tokamak in Lower Hybridcurrent drive regime, as calculated by two different methods (see the text for more details)
7.42 2-D contour plot of the electron distribution function at ρ ≃ 0.44 for Tore Supra Lower Hybrid current drive
7.43 Electron distribution function averaged over the perpendicular momentum direction at ρ ≃ 0.44 for Tore Supra Lower Hybrid current drive. The perpendicular and parallel temperatures are also shown
7.44 Bootstrap current profile given in the Lorentz model limit by the drift kinetic code and different analytical formulaes
7.45 Effective trapped fraction as given by the by the drift kinetic code in the Lorentz limit and by coefficient L₃₁ from analytical expression (see the text for more details)
7.46 Exact trapped fraction as given by the by the drift kinetic code in the Lorentz limit and by analytical expression (see the text for more details)
7.47 Pitch-angle dependence of ^ f ⁽⁰⁾ and g⁽⁰⁾ at ρ ≃ 0.4354 , as given by the drift kinetic code and analytical expressions, for the Lorentz model limit
7.48 First order distribution F_∥₀⁽¹⁾ averaged over the perpendicular momentum direction p_⊥ as fonction of p_∥ at ρ ≃ 0.4354 , as given by the drift kinetic code and analytical expressions, for the Lorentz model limit
7.49 Contour plot of ^ f ⁽⁰⁾ at ρ ≃ 0.4354 , as given by the drift kinetic code for the Lorentz model limit
7.50 Contour plot of g⁽⁰⁾ at ρ ≃ 0.4354 , as given by the drift kinetic code for the Lorentz model limit
7.51 Bootstrap current profile given by the drift kinetic code for the Tore Supra magnetic configuration and different corresponding analytical formulas (see the text for more details)
7.52 Effective trapped fraction as given by the by the drift kinetic code and the HELENA magnetic equilbrium code for the tokamak Tore Supra
7.53 Exact trapped fraction as given by the by the drift kinetic code for the tokamak Tore Supra
7.54 Pitch-angle dependence of ^ f ⁽⁰⁾ and g⁽⁰⁾ at ρ ≃ 0.4354 , as given by the drift kinetic code for the tokamak Tore Supra
7.55 First order distribution F_∥₀⁽¹⁾ averaged over the perpendicular momentum direction p_⊥ as fonction of p_∥ at ρ ≃ 0.4354 , as given by the drift kinetic code for the tokamak Tore Supra
7.56 Contour plot of ^ f ⁽⁰⁾ at ρ ≃ 0.4354 , as given by the drift kinetic code for the tokamak Tore Supra
7.57 Contour plot of g⁽⁰⁾ at ρ ≃ 0.4354 , as given by the drift kinetic code for the tokamak Tore Supra
C.1 Bootstrap current coefficient κ as a function of the highest terms M and N kept in the series.
C.2 Bootstrap current coefficient κ as a function of the highest terms M and N kept in the series, for M = N