DKE: a fast numerical solver for the 3-D relativistic bounce-averaged electron Drift Kinetic Equation

Transcription

1 DKE: a fast numerical solver for the 3-D relativistic bounce-averaged electron Drift Kinetic Equation Joan Decker and Y Peysson Association EURATOM-CEA sur la Fusion CEA-Cadarache, F-38 Saint Paul-lez-Durance, France Plasma Science and Fusion Center Massachussetts Institute of Technology, Cambridge, MA 39, USA February 7, 5 jodecker@alum.mit.edu yves.peysson@cea.fr

2 Abstract A new original code for solving the 3-D relativistic and bounce-averaged electron drift kinetic equation is presented. It designed for the current drive problem in tokamak with an arbitrary magnetic equilibrium. This tool allows self-consistent calculations of the bootstrap current in presence of other external current sources. RF current drive for arbitrary type of waves may be used. Several moments of the electron distribution function are determined, like the exact and effective fractions of trapped electrons, the plasma current, absorbed RF power, runaway and magnetic ripple loss rates and non-thermal bremsstrahlung. Advanced numerical techniques have been used to make it the first fully implicit reverse time 3-D solver, particularly well designed for implementation in a chain of code for realistic current drive calculations in high β p plasmas. All the details of the physics background and the numerical scheme are presented, as well a some examples to illustrate main code capabilities. Several important numerical points are addressed concerning code stability and potential numerical and physical limitations.

3 Contents Introduction 4 Tokamak geometry and particle dynamic 7. Coordinate system Momentum Space Configuration Space Particle motion Arbitrary configuration Circular configuration Kinetic description of electrons 3. Boltzman equation; Gyro- and Wave-averaging Guiding-Center Drifts and Drift-Kinetic Equation Drift Velocity from the Conservation of Canonical Momentum Drift Velocity from the Expression of Single Particle Drift Case of Circular concentric flux-surfaces Steady-State Drift-Kinetic Equation Small drift approximation Small Drift Ordering Low collision limit and bounce averaging Fokker-Planck Equation Drift-Kinetic Equation Flux conservative representation General formulation Dynamics in Momentum Space Dynamics in Configuration Space Bounce-averaged flux calculation Up to first order term: the Drift Kinetic equation Moments of the distribution function Flux-surface Averaging Density Current Density Power Density Associated with a Flux Stream Function for Momentum Space fluxes

4 CONTENTS CONTENTS Ohmic electric field Fraction of trapped electrons Runaway loss rate Magnetic ripple losses Non-thermal bremsstrahlung Detailed description of physical processes 8 4. Collisions Linearized collision operator Electron-electron collision operators Electron-ion collision operators Bounce Averaged Fokker-Planck Equation Bounce Averaged Drift Kinetic Equation Ohmic electric field Conservative Form for the Ohmic Electric Field Operator Bounce Averaged Fokker-Planck Equation Bounce Averaged Drift Kinetic Equation Radio frequency waves Conservative formulation of the RF wave operator RF Diffusion coefficient for a Plane Wave Integration in k-space Incident Energy Flow Density Narrow Beam Approximation Normalized Diffusion Coefficient Bounce Averaged Fokker-Planck Equation Bounce Averaged Drift-Kinetic Equation Modeling of RF Waves Numerical calculations 5. Bounce integrals Grid definitions Momentum space Configuration space Time grid definition Discretization procedure Zero order term: Fokker-Planck equation First order term: Drift kinetic equation Zero order term: the Fokker-Planck equation Momentum dynamics Spatial dynamics Grid interpolation Discrete description of physical processes Collisions Ohmic electric field Up to first order term: the Drift Kinetic equation

5 CONTENTS CONTENTS 5.5. Grid interpolation Momentum dynamics Discrete description of physical processes Initial solution Zero order term: the Fokker-Planck equation Up to first order term: the Drift Kinetic equation Boundary conditions Zero order term: the Fokker-Planck equation Up to first order term: the Drift Kinetic equation Moments of the Distribution Function Flux discretization for moment calculations Numerical integrals for moment calculations Algorithm 3 6. Matrix representation Zero order term: the Fokker-Planck equation Up to first order term: the Drift Kinetic equation Inversion procedure Incomplete matrix factorization Zero order term: the Fokker-Planck equation Up to first order term: the Drift Kinetic equation Normalization and definitions Temperature and density Time Momentum, velocity, and kinetic energy Maxwellian electron momentum distribution Poloidal flux coordinate Drift kinetic coefficient Momentum convection and diffusion Radial convection and diffusion Fluxes Current density Power density Electron runaway rate Electron magnetic ripple loss rate Units Examples Ohmic conductivity Runaway losses Lower Hybrid Current drive Electron Cyclotron Current drive Introduction Grid size effects

6 CONTENTS CONTENTS Electron trapping effects Momentum-space dynamics Coupling to propagation models Conclusion Fast electron radial transport Fast electron magnetic ripple losses Maxwellian bootstrap current Conclusion 37 9 Acknowledgements 3 A Curvilinear Coordinate Systems 3 A. General Case u,u,u A.. Vector Algebra A.. Tensor Algebra A. Configuration space A.. System R,Z,φ A.. System r,θ,φ A..3 System ψ,s,φ A..4 System ψ,θ,φ A.3 Momentum Space A.3. System p,p,ϕ A.3. System p,ξ,ϕ B Calculation of Bounce Coefficients for Circular Concentric FS 336 B. Calculation of λξ B.. Series Expansion B.. Calculation of the Integrals J m B..3 Truncated Expression B. Calculation of s B.3 Calculation of {Ψ} B.4 Calculation of b C Effective trapped fraction for Circular Concentric FS 343 D Cold Plasma Model for RF Waves 35 D. Cold Plasma Model D.. Wave Equation and Dispersion Tensor D.. Dispersion Relation D..3 Polarization components D..4 Power flow D..5 Conclusion D. Lower Hybrid Current Drive D.. Electrostatic Dispersion Relation D.. Cold Plasma Limit

7 CONTENTS CONTENTS D..3 Lower Hybrid Waves D..4 Polarization D..5 Determination of Θ b,lh k D..6 Determination of Φ LH bp D..7 Determination of Φ LH bt D..8 LH Diffusion Coefficient D.3 Electron Cyclotron Current Drive D.3. Polarization D.3. Determination of Θ b,ec k D.3.3 Determination of Φ EC b in the low density limit D.3.4 EC Diffusion Coefficient E Alternative discrete cross-derivatives coefficients 368 F MatLab File List 379 5

8 List of Figures. Coordinates systems p,p,ϕ and p,ξ,ϕfor momentum dynamics Coordinates system R,Z,φ Coordinates system r,θ,φ Coordinates system ψ,s,φ Guiding center velocity definition Domain in configuration space where magnetic ripple well takes place for Tore Supra tokamak Directions of incident electron and emitted photon with respect to the local magnetic field direction Grid definition for the momentum dynamics Chang and Cooper weightingfunction Lower Hybrid boundary problem Trapped domain and related flux connections Momentum flux connections for grid point in the trapped region 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 Trapping and detrapping process induced by radial transport Trapped domain for the first order distribution g Qualitative shape of matrix B for the Fokker-Planck equation Qualitative shape of matrix Ĝ for the drift kinetic equation Typical arrangement of non-zero matrix coefficients in the first columns and rows in matrix N corresponding to the Fokker-Planck equation 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 slowingdown process at constant ξ. By definition values of all coefficients on the main diagonal are one Matrix factorization principle. Dashed areas correspond to non-zero coefficients

9 LIST OF FIGURES LIST OF FIGURES 6.6 Reduction of the non-zero elements for the L and Û matrices, by increasing δ lu. Values of δ lu are indicated on the top of each subfigure. For δ lu =, the inversion becomes instable 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 Normalized Ohmic resistivity as function of the inverse aspect ratio ɛ Contour plot of the electron distribution function at ɛ = Contour plot of the stream lines at ɛ = Electron distribution function averaged over the perpendicular momentum direction at ɛ =.363. Parallel and perpendicular temperatures of the electron distribution function are also shown Normalized Ohmic runaway rate as function of the inverse aspect ratio ɛ 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 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 Variation of the current drive efficiency with the upper mometum limit of the integration domain Variation of the current drive efficiency with the main ion charge in the plasma Variation of the current drive efficiency with the amplitude of the quasilinear diffusion coefficient for the Lower Hybrid current drive problem Contour plot of the electron distribution function for D LH = Contour plot of the electron distribution function for D LH = Contour plot of the electron stream function for D LH = Contour plot of the electron distribution function at ɛ = Contour plot of the Lower Hybrid quasilinear diffusion coefficient at ɛ = Electron distribution function averaged over the perpendicular momentum direction at ɛ =.363. The perpendicular and parallel temperatures are also shown ECCD in DIII-D ρ =., D EC =.5, N =.3, Y =.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

10 LIST OF FIGURES LIST OF FIGURES 7.8 ECCD in DIII-D ρ =., D EC =.5, N =.3, Y =.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 = p max, n ξ = ECCD in DIII-D D EC =.5, N =.3, Y =.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 ɛ = r/r p ; temperatures, densities and Z eff are kept constant across the plasma ECCD in DIII-D ρ =., D EC =.5, N =.3, Y =.98. D electron distribution function a, parallel distribution function b and perpendicular temperature c; blue thin lines represent f init, red thick lines represent f, and green dashed contours represent D EC ECCD in DIII-D θ b =, P EC = MW, N =.3, f EC = MHz. Current and power densities deposition profiles. 3D calculation with n p = n ξ =,n ψ = 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 Pitch-angle grid for 3-D JET current drive simulation Momentum grid for 3-D JET current drive simulation Momentum grid step for 3-D JET current drive simulation. Circles correspond to the flux grid, while stars to the distribution function half-grid 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 D contour plot of the poloidal magnetic flux surfaces as calculated for JET tokamak by the code HELENA Momentum dependence of the relativistic Maxwellian distribution function at ρ.36, and relation between velocity v and momentum p. The deviation from the main diagonal indicates that above p = 4, relativistic effects become important D contour plot in momentum space of the Lower Hybrid quasilinear diffusion cofficient at ρ.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 On the left side, D contour plot of the radial diffusion rate at ρ.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 at ξ =

11 LIST OF FIGURES LIST OF FIGURES 7.3 Relative particle conservation of the drift kinetic code for the 3 D JET Lower Hybrid current drive simulation Flux surface averaged power density profiles for collision, RF and Ohmic electric field absorption for the 3 D JET Lower Hybrid current drive simulation Flux surface averaged current density profiles for the 3 D JET Lower Hybrid current drive simulation D contour plot of the electron distribution function at ρ.36 for JET Lower Hybrid current drive Electron distribution function averaged over the perpendicular momentum direction at ρ.36 for JET Lower Hybrid current drive. The perpendicular and parallel temperatures are also shown D contour plot of the electron distribution function at ρ.78 for JET Lower Hybrid current drive Electron distribution function averaged over the perpendicular momentum direction at ρ.78 for JET Lower Hybrid current drive. The perpendicular and parallel temperatures are also shown Ion and electron temperature and density profiles, and effective charge profile used for calculating the Tore Supra magnetic equilibrium with HE- LENA. 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 D contour plot of the poloidal magnetic flux surfaces as calculated for Tore Supra tokamak by the code HELENA Flux surface averaged current density profiles for the 3 D Tore Supra Lower Hybrid current drive simulation 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 D contour plot of the electron distribution function at ρ.44 for Tore Supra Lower Hybrid current drive Electron distribution function averaged over the perpendicular momentum direction at ρ.44 for Tore Supra Lower Hybrid current drive. The perpendicular and parallel temperatures are also shown Bootstrap current profile given in the Lorentz model limit by the drift kinetic code and different analytical formulaes Effective trapped fraction as given by the by the drift kinetic code in the Lorentz limit and by coefficient L 3 from analytical expression see the text for more details 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 Pitch-angle dependence of f and g at ρ.4354, as given by the drift kinetic code and analytical expressions, for the Lorentz model limit

12 LIST OF FIGURES LIST OF FIGURES 7.48 First order distribution F averaged over the perpendicular momentum direction p as fonction of p at ρ.4354, as given by the drift kinetic code and analytical expressions, for the Lorentz model limit Contour plot of f at ρ.4354, as given by the drift kinetic code for the Lorentz model limit Contour plot of g at ρ.4354, as given by the drift kinetic code for the Lorentz model limit 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 Effective trapped fraction as given by the by the drift kinetic code and the HELENA magnetic equilbrium code for the tokamak Tore Supra Exact trapped fraction as given by the by the drift kinetic code for the tokamak Tore Supra Pitch-angle dependence of f and g at ρ.4354, as given by the drift kinetic code for the tokamak Tore Supra First order distribution F averaged over the perpendicular momentum direction p as fonction of p at ρ.4354, as given by the drift kinetic code for the tokamak Tore Supra Contour plot of f at ρ.4354, as given by the drift kinetic code for the tokamak Tore Supra Contour plot of g at ρ.4354, as given by the drift kinetic code for the tokamak Tore Supra B. Bounce averaging coefficient λ C. Bootstrap current coefficient κ as a function of the highest terms M and N kept in the series C. Bootstrap current coefficient κ as a function of the highest terms M and N kept in the series, for M = N

13 List of Tables 7. Ohmic conductivity as function of the e-e collision model Ohmic conductivity as function of the effective charge using the linearized e-e collision model Runaway rate as function of the effective charge using the Maxwellian e-e collision model Lower Hybrid current drive efficiencies η LH in a pure hydrogen plasma from various D relativistic Fokker-Planck codes Lower Hybrid current drive efficiencies A m/w in a pure hydrogen plasma from various D relativistic Fokker-Planck codes

14 Chapter Introduction The determination of the electron distribution function has a crucial importance in the tokamak plasma physics, since the toroidal current density profile that is mainly driven by electrons is intimately linked to the magnetic equilibrium and confinement performances []. Therefore, accurate and realistic calculations must be carried out, with the additional requirement of an optimized numerical approach, in order to reduce as much as possible both memory storage and computer time consumptions. The latter point is especially important, since kinetic calculations must be incorporated in a chain of codes for selfconsistent determination of all plasma properties []. In this document, an extensive presentation of the fast solver for the linearized electron drift kinetic is presented. This is a completely new tool based on previous numerical developments [3], that is designed for realistic calculations of the electron distribution function in the plasma region where the weak collision banana regime holds. It incorporates the major physical ingredients that must be taken into account for describing the corresponding physics in a fusion reactor, namely relativistic corrections, trapped particle effects, arbitrary magnetic equilibrium for high β p regimes. For this purpose, both zero and first order kinetic equations with respect to the small drift approximation are solved, which allows to determine self-consistently boostrap current with any type of external current source RF, Ohmic,... at any point of the momentum space, and not only at the trapped-passing boundary as done in a previous attempt [4]. Basically, the code gives access to the neoclassical physics dominated by collisions between charged particles, for non-maxwellian electron distribution functions. Therefore, it is particularly well suited for accurate current drive estimates in advanced tokamak regimes, including ITER, where locally, bootstrap current may strongly interplay with external current sources ITB, edge pedestal in H-mode... Besides these physical properties, the code offer also the possibility to incorporate any type of fast electron radial transport collision, turbulence or wave induced, which may be a key ingredient for the local control of plasma properties [5]. Written in a fully conservative form, the code naturally conserve the electron density, but also momentum for the current drive problem, keeping first order term of the Legendre polynomial expansion of the Beliaev-Budker collision operator [6]. As usual, several useful moments of the electron distribution function are calculated, namely the current density, the absorbed power, the fraction of trapped electrons, the magnetic ripple losses [7] and the bremsstrahlung

15 . Introduction emission [8]. Advanced numerical techniques have been used, so that memory storage requirement can be strongly reduced, while keeping fast convergence rate. For this purpose the electron Drift Kinetic equation is solved by the standard finite difference technique, which has proven so far to be the fastest numerical approach among all possible alternative techniques. Furthermore, this method is particularly well suited when large discontinuities of the diffusion or convection rates have to be considered, a case that occurs frequently when kinetic and ray-tracing calculations are coupled. Since in most cases, the steady-state solution is seeked with respect to the largest time scale collision or fast electron radial transport, the appropriate technique is the well known upwind time differencing, corresponding to the fully implicit time scheme, whose characteristic is to be almost unconditionally stable with respect to the time step value t. Nevertheless, the code offers also the possibility to investigate time dependent problems, with the usual Crank-Nicholson time differencing, which enables accurate time evolution. The bounce-averaged Drift Kinetic equation is basically a 3 D problem, D in momentum space slowing down, pitch-angle et D in configuration space radial dimension. Up to now, in order to reduce memory storage requirements, the numerical time scheme was based on the operator splitting technique, where both momentum and spatial dynamics evolved separately. If this approach turns out to be very fruitful, it has the drawback to slow down considerably the convergence towards the steady state solution, since only small time steps may be used for numericaly stable convergence. Therefore, the advantage to use fully implicit time scheme for each sub-space in hindered by this strong limitation, especially when radial transport of fast electrons must be taken into account. In order to avoid this problem, a fully implicit time scheme is considered, where both momentum and spatial dynamics are simultaneaously considered, so that no limitation occurs on the time step, which may be several order of magnitude larger than the collision reference time. However, this method requires a new technique for matrix inversion, in order to keep memory storage at an acceptable level. Indeed, with usual mesh sizes, the standard LU matrix factorization techniques does not hold anymore since matrices requirement may reach several giga-bytes. An alternative approach is therefore absolutely necessary. This critical point has been addressed by using advanced inversion techniques, based on incomplete LU factorization with drop tolerance. Since most of the off-diagonal coefficients of the matrices L and U are very small, one may take advantage to remove them so that the sparsity of the matrices can be greatly enhanced. Memory storage requirements can be therefore reduced drastically by one or two orders of magnitude with this pruning method, depending upon the initial matrix preconditioning, while only coefficients that are relevant of the physics problem here addressed are kept. Furthermore, computer time consumption can be also reduced, since the number of non-zero coefficients is considerably reduced. This method is similar to the strongly implicit method used for factorizing nine diagonal matrices [9], except that in this case, no restriction takes place regarding the number of diagonals. However, to take full benefit of this approach, the non-zero elements The energy transport time scale is usually on order on magnitude larger than the largest characteristic time for current drive calculations, except in tokamaks of small size, where time ordering here considered in the model must be likely revisited 3

16 . Introduction of the matrix which is inverted must lie predominantly along diagonals. Therefore, it may be applied for solving the zero and first order bounce-averaged Fokker-Planck equations, whose structures are well suited for this purpose, though coefficients arrangement can be complex, owing to the radial dependence of the internal trapped-passing boundary in momentum space, especialy when transport in configuration space has to be considered. This approach has been very successfuly implemented for the electron Drift Kinetic problem in tokamaks, using the MatLab language, which provides a built-in function for incomplete LU factorization with drop tolerance, and several very efficient iterative inversion tools, like the Conjugate Gradient Squared method for solving the system of linear equations. It is important to recall that this method is also available in FORTRAN programming language, under the package name SPARSEKIT that has also parallel processing capabilities []. Moreover, the very compact MatLab programming syntax allows to design the code structure in an original way, using multidimensional objects that describe simultaneously momentum and configuration space dynamics, but also wave-particle interaction. This makes the code particularly robust and easy to maintain. In the document, the physics and numerics issues of the code are detailed, and an extensive discussion of the underlying assumptions is presented. A specific attention is paid to derive matrix coefficients in a fully consistent manner, a crucial issue especially for an accurate and robust estimate of the current drive efficiency for the various methods used in tokamaks. Some examples are shown to illustrate code performances, though still numerous possibilities remain to be investigated but are beyond the purpose of this document. Aside from present day code capabilities, it is important to notice that the new numerical approach, here used, gives access to new physics domains that have never been studied accurately like wave-induce radial transport []. Furthermore, since the algorithm used is fast and stable, possible extensions to 4 D problems may be foreseen like in the plateau collision regime current drive at the very plasma edge, as well as studies of the difficult problem of electrons that are locally trapped at different spatial positions, like in stellarator. In addition the code may be extended quite in a straightforward manner to the multi-species problem, taking into account for example of the non-linear damping of the α-particles produced by fusion reactions on the electron population. However, the ion physics requires to perform orbit-averaging instead of bounce-averaging, because of the large banana width of some particles, a challenging issue for kinetic solver based on a finite difference technique. Such a requirement is crucial for describing torque induced by waves. Nevertheless, beside this difficulty, the code is already fully designed to take benefit from parallel processing, if the dynamics of various species must be studied. In particular, non-uniform momentum and pitch-angle grids are already implemented, so that refined calculations can be performed for the ions at low velocity, while accurate ones up to relativistic energies may be considered for the electrons. Useful informations are available on the website of Pr. Yousef Saad at the following internet address saad/ 4

17 Chapter Tokamak geometry and particle dynamic. Coordinate system General and specific properties of curvilinear coordinate systems are detailed in Appendix A. In this work, vectors are written in bold characters, like v, except unit vectors, which are covered with a hat, like v... Momentum Space Because we consider gyro-averaged kinetic equations, it is important to use coordinates with rotational symmetry in order to reduce the dimensionality of the problem. Two different momentum space coordinates system are considered here: First, the cylindrical coordinate system p,p,ϕ, where p is the component of the momentum along the magnetic field, and p is the component perpendicular and ϕ is the gyro-angle. This system is defined in A. and shown in Fig... The cylindrical coordinate system is the natural system for wave-particle interaction, or also the effect of the electric field. Second, the spherical coordinate system p, ξ, ϕ, where pis the magnitude of the momentum, and ξ is the cosine of the pitch-angle. This system is defined in A.47 and shown in Fig.. as well. The spherical coordinate system is the natural system for collisions. It is the primary system, used in the Drift Kinetic code, for an accurate description of collisions... Configuration Space The particular toroidal geometry of tokamaks requires to use specific coordinates, in order to make use of symmetry properties such as axisymmetry, and takes into account the fluxsurface magnetic configuration. Three different configuration space coordinates systems are considered here: 5

18 . Tokamak geometry and particle dynamic.. Coordinate system y ϕ ξ O p cos ξ p p p b z ϕ cmcm x Figure.: Coordinates systems p,p,ϕ and p,ξ,ϕfor momentum dynamics First, the toroidal coordinate system R,Z,φ, where R is the distance from the axis of the torus, and Z the distance along this axis. This φ coordinates system and the corresponding local orthogonal basis vectors R, Ẑ, are defined in A.6 and shown in Fig... This coordinate system conserves the largest generality in the magnetic geometry. Second, the poloidal polar coordinate system r, θ, φ assumes the existence of a toroidal axis at constant position R p,z p which is typically the plasma magnetic axis, corresponding to the position of an extremum of the poloidal magnetic flux ψ which can be arbitrarily chosen as ψ =. This coordinates system and the corresponding local orthogonal basis vectors r, θ, φ are defined in A.94 and shown in Fig..3. Third, the flux coordinate system ψ,s,φ is the natural system when we describe particles which are confined to a given flux surface ψ. This coordinates system and the corresponding local orthogonal basis vectors ψ,ŝ, φ are defined in A.36 and shown in Fig..4. The vector ψ is perpendicular to the flux surface, while ŝ is parallel to the surface, and included in the poloidal plane. The distance s is the length along the poloidal magnetic field lines. We can choose its origin as being at the position of minimum B-field amplitude within a flux-surface. B ψ,s = min s {B ψ,s} = B ψ. Note that from now on, and all along this paper, the subscript refers to quantities evaluated at the position of minimum B-field on a given flux-surface. The direction 6

19 . Tokamak geometry and particle dynamic.. Coordinate system cmcm Figure.: Coordinates system R,Z,φ. cmcm Figure.3: Coordinates system r,θ,φ. 7

20 . Tokamak geometry and particle dynamic.. Coordinate system cmcm Figure.4: Coordinates system ψ,s,φ. of evolution of s is counter-clockwise and the limits s min ψ and s max ψ are set at the position of maximum magnetic field B ψ,s s min = B ψ,s s max = max {B ψ,s} = B max ψ. The system ψ,θ,φ is an alternative to the previous system, which is used to implement numerically the calculation of the bounce coefficients. One advantage is that the θ grid is now independent of ψ, which simplifies the numerical calculations. On the other hand, the contravariant vectors ψ and θ are not orthogonal, and therefore are not respectively colinear with the covariant vectors X/ ψ and X/ θ. The properties of this curvilinear system are detailed in Appendix A. We also define, for geometrical purposes, a flux-function ρψ which coincides with the normalized radius on the horizontal Low Field Side LFS mid-plane. Indeed, in an axisymmetric system, using the functions R ψ,θ and Z ψ,θ, we define ρψ as ρψ = R ψ, R p R max R p.3 with ρ by construction, and where R max = R ψ max, is the value of R on the separatrix as it crosses the mid-plane. Here a p = R max R p is defined arbitrarily as the plasma minor radius since this definition merges with the exact one for circular concentric flux-surfaces. The D outputs from the axisymmetric equilibrium code HELENA are given on the ψ,θ grid []. The system ψ,θ,φ will be used from now on. 8

21 . Tokamak geometry and particle dynamic.. Particle motion φ b v cos B P B cmcm v s s Figure.5: Guiding center velocity definition. Particle motion.. Arbitrary configuration Transit or Bounce Time Normalized Expression The transit, or bounce time, is defined as the time for a passing particle to complete a full orbit in the poloidal plane, and for a trapped particle to complete half a bounce period. Note that this is possible only in the approximation of zero banana width. Otherwise, the bounce motion would be no longer symmetric in the forth and back motions, and both would need to be accounted for. We define then τ b ψ = smax s min ds v s = smax s min ds B v.4 B P where v s is the guiding center velocity along the poloidal field lines, and v is its velocity parallel to the magnetic field. B is the magnitude of the magnetic field, while B P is the magnitude of its poloidal component as shown in Fig..5. The limits s min and s max are defined in. for passing electrons, and are the positions, along the field lines, of turning points for trapped electrons. The differential arc length ds along the poloidal field line is generally expressed in curvilinear coordinates u,u,u 3 as A.3 ds = g ij du i du j.5 where the g ij are the metric coefficients, defined in A.. In the ψ,θ,φ coordinates, the variations dψ and dφ are essentially zero along the poloidal field line. As a consequence,.5 becomes ds = g dθ.6 9

22 . Tokamak geometry and particle dynamic.. Particle motion The velocity and momentum are related through the relativistic factor γ p introduced in Sec , and therefore, we have v v p p = ξ.7 in the weak relativistic regime of tokamak plasmas, where the pitch-angle cosine ξ is defined in A.47 We get τ b ψ = π θmax v ξ θ min dθ ξ g π ξ where ξ is the pitch angle cosine at the position θ of minimum B-field B B P.8 θ θ B = B ψ.9 and the limits θ min and θ max will be calculated in the next subsection. The bounce time can be normalized as such: with and τ b ψ,ξ = πr p q ψ λψ,ξ. v ξ λψ,ξ = θmax q ψ θ min q ψ π dθ g ξ π R p ξ B B P. dθ g B. π R p B P The bounce time is normalized to the transit time of particles with parallel momentum only, such that λψ, ± =. The covariant metric element g is given by A.-A., which is in the ψ,θ,φ system becomes A.9 g = J ψ φ = r ψ.3 r Consequently, the normalized bounce time takes the form λψ,ξ = θmax dθ r B ξ q ψ θ min π ψ r R p B P ξ.4 with q ψ = π dθ r B π ψ.5 r R p B P

23 . Tokamak geometry and particle dynamic.. Particle motion Particle Motion in the Magnetic Field The particle motion along the magnetic field lines exhibits one constant of the motion, the energy or the total momentum p, and an adiabatic invariant, the magnetic moment µ. They are given by the equations p = p + p.6 µ = p m e B.7 such that, as a function of the moment component p,p at the location θ of minimum B-field, we have p + p = p + p.8 p B ψ,θ = p B ψ.9 Using the transformation A.5-A.5 from p,p to p,ξ, the system.8-.9 becomes p = p. ξ B ψ,θ = ξ B ψ. We get an expression for ξ as a function of ξ : ξ ψ,θ,ξ = σ Ψ ψ,θ ξ. where σ = sign ξ = sign v, and Ψ ψ,θ is the ratio of the total magnetic field B to its minimum value B Ψ ψ,θ B ψ,θ.3 B ψ The trapping condition is given by ξ < ξ T ψ, where ξ T ψ is the pitch angle, defined at the minimum B ψ on a given flux-surface, such that the parallel velocity of the particle vanishes at the maximum B max ψ. An expression for ξ T ψ can then be obtained from.: setting ξ ξ T,B = B max ψ =, we get ξ T ψ = B ψ B max ψ.4 The turning points are θ min ψ,ξ = θ max ψ,ξ = π θ T min π θ T max for passing particles for trapped particles for passing particles for trapped particles.5.6

24 . Tokamak geometry and particle dynamic.. Particle motion We can determine the turning angles θ T min ψ,ξ and θ T max ψ,ξ as the position where ξ ψ,θ,ξ =. At this position, we have B = B b ψ,ξ, where B b ψ,ξ is then given by. B b ψ,ξ = B ψ ξ.7 so that where θ is given by.9. θ T min ψ,ξ = θ B = B b θ < θ [π].8 θ T max ψ,ξ = θ B = B b θ > θ [π].9 Calculation of λψ,ξ From the Output Data of Equilibrium Codes The numerical calculation of λψ,ξ can be carried from the output of any magnetic equilibrium code. In the kinetic code here considered, we use HELENA for magnetic flux surface calculations [], since it is used in the the CRONOS tokamak simulation package []. Data are assumed to be the parametrization of the flux-surfaces R ψ, θ and Z ψ, θ, and the three components of the magnetic field B R ψ,θ, B Z ψ,θ and B φ ψ,θ. From these components we derive directly the toroidal and poloidal components of the field, as well as the total field: and also B T ψ,θ = B φ ψ,θ B P ψ,θ = BR ψ,θ + B Z ψ,θ B ψ,θ = BT ψ,θ + B P ψ,θ.3 R p = R,θ.3 Z p = Z,θ.3 We also have an expression for r r ψ,θ = R ψ,θ R p + Z ψ,θ Z p.33 and, using relation r = r R R + r ẐẐ R ψ,θ Rp = r R + Z ψ,θ Zp r Ẑ.34 that can be easily deduced from vector relation in Fig.., we get an expression for the scalar product ψ R R ψ,θ R p + ψ ψ Ẑ Z ψ,θ Z p r =.35 r ψ

25 . Tokamak geometry and particle dynamic.. Particle motion In a toroidal axisymmetric geometry, the magnetic field can be expressed generally as so that B = I ψ φ + ψ φ.36 We also have I ψ B T = I ψ φ = R B P = ψ φ = ψ R and therefore so that B T = I ψ φ = B φ φ.39 B P = ψ φ = B P ŝ.4 φ B P = φ ψ φ = φ ψ.4 ψ = φ B P φ = R φ B P.4 and we have the projections ψ R = R R φ B P = RB Z.43 ψ Ẑ = RẐ φ B P = RB R.44 Finally, the expressions for the normalized bounce time λ and q that are used in numerical calculations are B [R R p + Z Z p ] with λψ,ξ = θmax dθ q ψ θ min π q ψ = π dθ π R p B R Z Z p B Z R R p B [R R p + Z Z p ] R p B R Z Z p B Z R R p ξ ξ where R,Z,B R,B Z and B are functions of ψ,θ, and ξ is a function of ψ,θ,ξ given by.. Safety Factor q ψ way as The averaged safety factor q is defined in Ref. [3] in a general q ψ = I ψ δv R 4π.47 δψ where V is the volume enclosed by a flux-surface and denotes the flux-surface average. 3

26 . Tokamak geometry and particle dynamic.. Particle motion It can be expressed as where the Jacobian J is given by A.95 π dθ π dφ J q ψ = I ψ π π R.48 J = ψ θ φ Rr = ψ ψ r = r B P ψ r.49 where.4 is used We obtain π dθ r q ψ = I ψ π ψ r B P R.5 and, using.36, we finally have q ψ = π dθ r B π ψ T.5 r R B P The expression of q ψ and its relation to q ψ in the simplified case of circular concentric flux-surfaces will be addressed in sub-section... Using.33 and.35, we find the expression q ψ = π dθ π that is convenient for the numerical evaluation. Toroidal Extent of Banana Orbits [ R R p + Z Z p ] B T R B R Z Z p B Z R R p.5 We are interested in calculating the toroidal extent of banana orbits, that is, the toroidal angle corresponding to the path done by a trapped particle between two turning points. It is given by φ = φ max φ min = φmax φ min dφ.53 and can be expressed as a function of the length element dl along the path, using A.98 lφmax dφ lφmax φ = dl φ dl φ = dl φ.54 lφ min R lφ min 4

27 . Tokamak geometry and particle dynamic.. Particle motion The poloidal and toroidal elements are related through the local angle of the magnetic field, dl φ dl θ = B T.55 B P so that φ = Using A.97, we get Defining the integral lφmax lφ min φ = dl φ q T ψ,ξ = θmax θ min dφ lθmax dl φ = B T dl θ.56 lθ min R B P θmax θ min r B dθ ψ T.57 r R B P dθ π ψ r we find that the toroidal extent of banana orbits is Note that at the trapped/passing limit, we have r R B T B P.58 φ π = q T ψ,ξ.59 φ lim ξ ξ T π = q T ψ,ξ T = q ψ.6 Therefore, we retrieve the interpretation of the safety factors, which is the number of toroidal rotations φ/π for one poloidal rotation. Bounce Average In order to reduce the dimension of kinetic equations, it is important to define an average over the poloidal motion, which anihilates the term that accounts for the time evolution of the variations of the distribution function along the field lines. The natural average is {A} = τ b [ ] σ T smax s min ds v s A.6 where the sum over σ applies to trapped particles only. It can be rewritten in terms of the normalized bounce time λ using expression. [ ] {A} = θmax dθ g B ξ λ q π R p B P ξ A.6 or {A} = λ q [ σ ] σ T T θ min θmax θ min 5 dθ r B ξ π ψ r R p B P ξ A.63

28 . Tokamak geometry and particle dynamic.. Particle motion using relation.3. Another expression uses the output data from equilibrium codes. Following the work in the previous section, we find or explicitely {A} = λ q [ {A} = ] [ σ T θmax θ min ] σ T θmax θ min dθ π θmax θ min dθ π B [R R p + Z Z p ] R p B R Z Z p B Z R R p B [R R p + Z Z p ] ξ R p B R Z Z p B Z R R p ξ dθ B [R R p + Z Z p ] π R p B R Z Z p B Z R R p ξ ξ A.64 ξ ξ A.65 The bounce averaging of momentum-space operators in the kinetic equations leads to a set of coefficients that all have a similar structure, denoted λ k,l,m and λ k,l,m, which are define as { ξ ψ,θ,ξ k } Ψ l R ψ m ψ,θ = λ k,l,m ψ,ξ.66 R ψ,θ λψ,ξ and where ξ { ξ ψ,θ,ξ k } σ σ Ψ l R ψ m ψ,θ = λ k,l,m ψ,ξ R ψ,θ λψ,ξ ξ.67 R ψ R ψ,θ.68 Note that by definition, λ,, = λ. In addition, λ k,l,m = λ k,l,m for passing particles for trapped particles.. Circular configuration Parametrization of the Flux-Surfaces.69 In this case, we have ψ = ψ r and therefore it is easier to work in the r,θ,φ coordinate to account for the symmetry in the problem. The normalized radius is ρψ = r a p.7 We have now so that A.3 ψ = r.7 g = r.7 6

29 . Tokamak geometry and particle dynamic.. Particle motion Magnetic Field The toroidal field is and the poloidal field is where is now only a function of r or ρ. The total field is then and can be written as with B T r,θ = B P r,θ = I r Rr,θ ψ r R r,θ ψ r = dψ r dr = dψ ρ a p dρ.75 B r,θ = I r + ψ r B r,θ = B r B r = R r,θ R R r,θ I r + ψ r R.78 Consequently, we ratio of magnetic fields Ψ as defined in.3 becomes Ψ r,θ = R R r,θ.79 and is a function of r only. Safety factor B B P = I r + ψ r ψ r The safety factor given by expression.5 becomes = + I r ψ r.8 q r = π dθ r π R B T B P = r R p B T B P π dθ R p π R.8 The averaged value of R p /R is evaluated in B.7. It gives π dθ R p π R =.8 r/r p 7

30 . Tokamak geometry and particle dynamic.. Particle motion so that r B q ψ = T.83 r/r p R p B P Note that in the factor r/r p is usually neglected, which is valid only in the large aspect ratio approximation, i.e. when the inverse aspect ratio ɛ defined as is much less than unity. Particle Motion Using relation A.95, ɛ = r R p.84 R r,θ = R p + r cos θ.85 and recalling that the minimum B-field B corresponds to the poloidal angle value in that case θ =.86 we find Therefore, the expression.79 becomes and using relation.77 R min r = R p r = R p ɛ.87 R max r = R p + r = R p + ɛ = R r.88 Ψ ρ,θ = + ɛ + ɛcos θ.89 B max r = B r + ɛ.9 ɛ expression.4 is ξt r = ɛ.9 + ɛ The pitch-angle cosine ξ is then given by combining relations. and.89 ξ r,θ,ξ = σ + ɛ ξ + ɛcos θ.9 and the the turning angles are obtained from expression.7, or in the present notation B r,θ T = B b r,ξ = B r ξ.93 Using relation.89, one obtains B r + ɛ = B r + ɛcos θ T ξ.94 8

31 . Tokamak geometry and particle dynamic.. Particle motion and then so that and finally by symmetry ξ = + ɛcos θ T + ɛ = ɛ cos θ T + ɛ [ ] θ T = arccos ξ ξt θ T min = θ T.97 θ T max = θ T Bounce Time Using.7, the normalized bounce time reduces to with, using definition.5 λr,ξ = ɛ q θmax θ min dθ ξ B.98 π ξ B P π dθ B q r = ɛ.99 π B P Because B/B P only a function of r, as seen in.8, and can be taken out of the integrals, we get finally λr,ξ = θmax θ min dθ ξ π ξ. This integral can be performed analytically in a series expansion whose coefficients are calculated in B.. Note that in the case where B T B P and in the large aspect ratio approximation ɛ, we have q r q r, which explains the notations, and the introduction of pseudo safety factor like q. Other new definitions of pseudo safety factors will be introduced throughout the next sections, based on similar arguments. 9

32 Chapter 3 Kinetic description of electrons 3. Boltzman equation; Gyro- and Wave-averaging In the kinetic description, electrons are described by a distribution function f r, p, t, which gives the density in phase space of particles with a momentum p at a position r and at time t. The particle conservation equation in phase space is the Boltzmann equation f t + v rf + q e [E r,t + v Br,t] p f = f t 3. C where f t C f 3. C is the collision operator. The fields E r,t and Br,t are assumed to consist of timeindependent macroscopic fields E r and Br and fields associated with plane waves. E r,t = E r + Br,t = Br + Ẽ k e ik r ωt dk 3.3 B k e ik r ωt dk 3.4 Because we are interested in solving the kinetic equation on the bounce and collisional time scales, we need to average over the faster time scales, which are the gyromotion and the wave oscillation. Performing a time-averaging π/ω dt of the equation 3. removes the fast wave time scale from the equation, to give f t + v [ ] rf + q e E r + v Br p f = f π/ω t dt q e [Ẽk + v B ] k p f C k 3.5 3

33 3. Kinetic description of electrons 3.. Boltzman equation; Gyro- and Wave-averaging where f = π/ω dt f 3.6 is the wave-period averaged distribution function. The time derivative in the first term of 3.5 implicitely refers to times longer than the wave period ω. Under the assumption of a strong magnetic field, such that the gyrofrequency Ω e Ω e = q eb γm e 3.7 is much larger than both the collisional frequency and the bounce frequency, we can expand the distribution function f = f + f + f with a small parameter The zero order equation becomes δ ω b Ω e ν c Ω e 3.9 We have, in the p,p,ϕ space defined in Appendix A, q e v B r p f = 3. v = with the momentum being given by relation A.4 and the gradient by expression A.39 In this system, by definition, p γ p,p me 3. p = p + p 3. p f = f f f + + ϕ 3.3 p p p ϕ B r = B r 3.4 so that the gyromotion operator becomes q e v Br p = Ω e p p f The equation 3. becomes consequently = Ω e ϕ 3.5 f ϕ = 3.6 3

34 3. Kinetic description of electrons 3.. Boltzman equation; Gyro- and Wave-averaging and therefore f is independent of ϕ. The first order equation is f t + v rf + q eer p f + q e v Br p f = f π/ω t dt q e [Ẽk + v B ] k p f C k 3.7 The last term in the equation 3.7 has been calculated by Lerche for a uniform plasma, in the form of a quasilinear operator Q f. We can rewrite C f f = t 3.8 C Q π/ω f = dt k q e [Ẽk + v B ] k p f 3.9 Performing the gyro-averaging π dϕ on the kinetic equation 3.7, we find, using 3.6, that and π π dϕ v r f = dϕ f t = f t π 3. dϕ v r f = v gc r f 3. where v gc is the electron velocity along the guiding center. Concerning the electric field, we decompose the gradient in momentum space using 3.3 π π [ ] f f dϕ q e E r p f = q e E r dϕ + f + p p p ϕ ϕ 3. and, using we obtain π π = q e f p E r f +q e E r p + q e p E r π π dϕ dϕ f ϕ ϕ 3.3 dϕ = 3.4 dϕ f ϕ ϕ = f π π dϕ ϕ π ϕ = f dϕ = 3.5 dϕ q e E r p f = q e E r f p 3.6 3

35 3. Kinetic description of electrons3.. Guiding-Center Drifts and Drift-Kinetic Equation The gyromotion term is averaged to zero so that we get finally π dϕ q e v Br p f = Ω e π f t + v gc r f + q e E r p f = C dϕ f ϕ = 3.7 f + Q f 3.8 This equation is called electron drift-kinetic equation. Renaming the guiding-center distribution function f = f r,p,p,t, E r = E r and Br = B r, we get where we define an electric field operator f t + v gc f = C f + Qf + E f 3.9 E f = q e E r p f 3.3 Implicitely, the time scale here considered is so that t π/ω,π/ω e. 3. Guiding-Center Drifts and Drift-Kinetic Equation As shown in previous section, for axisymmetric plasmas, the electron drift kinetic equation may be expressed in the general form f t + v gc f = C f + Qf + E f 3.3 where f = f p, ξ, ψ, θ, t is the guiding-center distribution function. In tokamaks, it can be shown that the guiding center velocity v gc may be decomposed into a fast parallel motion along the field lines, and a vertical drift velocity v D across the magnetic flux surfaces v gc = v b + vd 3.3 From the general expression.36 of the magnetic field B, one obtains in the ψ,s,φ coordinates system, B = I ψ φ + ψ φ 3.33 B = I ψ R φ ψ R ŝ 3.34 As shown in Appendix A, the gradient in ψ,s,φ coordinates is = ψ ψ + s s + φ φ = ψ ψ + ŝ s + φ R φ

36 3. Kinetic description of electrons3.. Guiding-Center Drifts and Drift-Kinetic Equation and recalling that the constants of the motion are the total energy or momentum p as defined in.6 and the magnetic moment µ as given by relation.7, following conservations laws µ s = 3.36 [ ] p s + µbm e = 3.37 are satisfied. 3.. Drift Velocity from the Conservation of Canonical Momentum The toroidal canonical momentum is also a constant of the motion because of axisymmetry. It is expressed as P φ = R [γm e v φ + q e A φ ] 3.38 where A φ is the toroidal component of the vector potential. From the relation B = A 3.39 and the expression A.7 of a rotational in ψ,s,φ coordinates, we get [ B = R s RA φ ] R φ A s ψ [ + R φ A ψ ψ ] R ψ RA φ ŝ [ + ψ ψ A s ψ ] Aψ φ 3.4 s ψ with In axisymmetric plasma, this reduces to A ψ = A ψ 3.4 A s = A ŝ 3.4 A φ = A φ 3.43 B = R s RA φ ψ ψ + R [ ψ ψ RA φŝ ψ A s ψ s ] Aψ φ 3.44 ψ so that B s = ψ R ψ RA φ

37 3. Kinetic description of electrons3.. Guiding-Center Drifts and Drift-Kinetic Equation In addition, we know from expression 3.34 that so that be obtain B s = ψ R 3.46 RA φ ψ = 3.47 Because the toroidal canonical momentum is a constant of the motion, we have which can be decomposed into v gc P φ = 3.48 v gc Rγm e v φ + v gc q e RA φ = 3.49 Using relation A.69, we get [ v gc q e RA φ = v gc ψ ψ + ŝ s + φ ] q e RA φ 3.5 R φ which in axisymmetric systems gives v gc q e A φ = q e v gc [ ψ RA ] φ ψ + ŝ RA φ s 3.5 Since B ψ =, we have from relation 3.44 RA φ / s = and therefore, using expression 3.47, v gc q e A φ = q e v gc ψ 3.5 The only velocity accross the flux-surfaces is the drift velocity we are looking for, so that we get, using relation 3.3 and the equation 3.49 becomes v gc q e A φ = q e v D ψ 3.53 q e v D ψ = v gc Rγm e v φ 3.54 Assuming a priori that v vd, a condition that holds in tokamaks, this equation reduces to v D ψ = q e v B B Rγm ev φ = v Ω e B Rv φ 3.55 where we used that γ/ s = because of the conservation of energy. The toroidal velocity is related to the parallel velocity by v φ = B φ B v = I ψ RB v

38 3. Kinetic description of electrons3.. Guiding-Center Drifts and Drift-Kinetic Equation so that Rv φ = I ψ B v 3.57 Since I ψ is a flux function, it can be taken out of the gradient, so that v D ψ = v Ω e I ψb 3.. Drift Velocity from the Expression of Single Particle Drift The guiding-center drift velocity due to the magnetic field gradient and curvature is v B 3.58 v D = v Ω + v B B e B 3.59 and its component perpendicular to the flux-surface can be written as v D ψ = Ω e v + v Inserting the expression 3.34 of the magnetic field, we find v D ψ = Ω e v + v Using 3.35, the equation 3.6 becomes v D ψ = Ω e v + v ψ B B 3.6 B ψ [ B I ψ ŝ + ψ R φ ] B 3.6 ψ B R [ I ψ B Under the assumption of axisymmetry, we are left with v D ψ = v Ω + v ψ I ψ e B R s + ψ R With the definition.7 of the magnetic moment µ, we rewrite v D ψ = ψ I ψ Ω e B v R + µb B m e s We have, using the conservation of magnetic momentum 3.36, v + µb B m e s = B v s + B µb s m e Using the conservation of energy 3.37, we get v + µb B m e s = v B s B s = v [ B v = v B s 36 s v v B B s v B s ] ] B φ

39 3. Kinetic description of electrons3.. Guiding-Center Drifts and Drift-Kinetic Equation and finally, the equation 3.64 becomes In addition, and, using axisymmetry, v D ψ = v I ψ ψ Ω e R s B so that we can rewrite 3.69 as expression which is the same as v B = I ψ φ + ψ φ = I ψ R φ ψ R s B = ψ R s v D ψ = v Ω e I ψb v B Case of Circular concentric flux-surfaces In this case, ψ = ψ r and therefore and which gives In addition, and, because ψ = ψ r r 3.73 v Dr = v D r = v D ψ ψ 3.74 v Dr = v I ψ Ω e ψ B B = ψ R v B s = ψ R s we find s = σ ψ r θ and B = ψ Rr θ so that finally v Dr = v I ψ v = v I ψ Ω e Rr θ B r RB θ When the toroidal field dominates, B I ψ /R and v Dr v v r θ Ω ŝ θ = ψ r = σ ψ v Ω

40 3. Kinetic description of electrons 3.3. Small drift approximation 3..4 Steady-State Drift-Kinetic Equation Using expressions 3.3, 3.3 and 3.58 or 3.7, we obtain in steady-state which can be rewritten as v b f v Ω e I ψb f v s s + v I ψ ψ Ω e R s v B v B f ψ f ψ = C f + Qf + E f 3.8 = C f + Qf + E f 3.83 with v s = v b ŝ Small drift approximation We recall the electron drift kinetic equation may be expressed as f v s s + v ψ I ψ Ω R s v B f ψ = C f + Qf + E f 3.85 Each of these terms corresponds to a time evolution, and is therefore associated with a time-scale: Motion along magnetic field lines. The time scale here is the bounce time τ b for trapped electrons, or the transit time τ t for circulating ones, which can be deduced directly from expression.. For circulating electrons τ t = πr p q v ξ πqr p v Te 3.86 taking λ in that case, and q q that is valid circular plasma cross-sections, and v ξ = v v Te for thermal electrons. Since v ɛv ɛv Te for trapped electrons, as the consequence of the magnetic moment conservation, it turns out that τ b may be deduced directly from τ t τ b = πr p q v ξ πqr p 3.87 ɛvte Consequently, τ b τ t, since ɛ, a result which is the consequence of the progressive slowing down of the parallel velocity as far as the electron approaches the turning point. Vertical drift across magnetic field lines. The time scale here is the drift time τ d, which corresponds to the time for an electron to drift across the plasma. It is then given by ψa τ d = dψ Ω [ e R v ] 3.88 v I ψ ψ s B 38

41 3. Kinetic description of electrons 3.3. Small drift approximation using expression 3.69 for the radial component v D ψ of the drift velocity v D as defined in Using relation I ψ as defined in.37, and the fact that dψ/ ψ = dr for circular concentric plasma cross-sections, τ d Ω e v Te ap dr B T [ s B ] 3.89 and since in that magnetic configuration / s π/r p, ones obtains finally τ d π Ω ap e vte R p B B T dr 3.9 or τ d π Ω e vte R p a p 3.9 Consequently, the small drift parameter δ d is defined as the ratio δ d τ t τ d q ρ L a p ρ L a p 3.9 where the thermal Larmor radius ρ L = v Te /Ω e has been introduced. Collisions. The Fokker-Planck theory considers the cumulative effect of many smallangle collisions in calculating the rate of change of a particle distribution. From this theory, the characteristic time scale τ c corresponds for Coulomb collisons to deflect an electron s path by a significant angle, on the order of π/. This is the electron thermal collision time τ c whose expression is τ c = ν c = 4πε m e v3 Te q 4 en e ln Λ 3.93 where ln Λ is the well known Coulomb logarithm, a slowly varying function of the plasma temperature and density. The collision time scale τ c holds for circulating electrons. For trapped ones, it is more physical to consider an alternate collision time scale determined not by the time for deflection of the path by π/, but by the time needed for the electron to be deflected so that it is no longer on a trapped orbit. In the limit ɛ, we can approximate the change of the pitch angle necessary to make trapped particles become untrapped ɛ ξ ξ T + ɛ ɛ 3.94 Because the small-angle collisions produce a random-walk change in the pitch-angle ξ, the effective collision time for detrapping τc eff. or τ dt is approximately deduced from relation νc eff. ν c ξ ν c 3.95 ɛ and τ dt = τ eff. c ɛτ c

42 3. Kinetic description of electrons 3.4. Low collision limit and bounce averaging Constant electric field acceleration. From the relation 3.3, it is straightforward to estimate that the time scale associated to constant electric field acceleration is where the well known Dreicer field E D τ e m ev Te q e E τ c E D E 3.97 E D = ν c m e v Te q e 3.98 is introduced. Here only circulating particle are concerned. Quasilinear diffusion. In a similar approach, τ ql m ev Te D ql Dql τ c 3.99 D ql where D ql = ν c m e v Te Small Drift Ordering In the small drift expansion δ d, where the small parameter is defined by relation 3.9 f = f + f + 3. the first order equation is v s f s = C f + Qf + E f 3. which is usually referred to as the Fokker-Planck equation. The second order equation is f v s s + v I ψ ψ Ω e R s v which is referred to as the electron Drift Kinetic equation. B f ψ = C f + Qf + E f Low collision limit and bounce averaging 3.4. Fokker-Planck Equation The Fokker-Planck Equation is v s f s = C f + Qf + E f 3.3 In the low collision regime, which is characterized by the condition ν τ b τ dt 3.4 4

43 3. Kinetic description of electrons 3.4. Low collision limit and bounce averaging where τ dt = ɛτ c is the collision detrapping time, as defined in the previous section, it is assumed that electrons circulating or trapped are able to complete their orbit in a time too short for collisions to deflect them from their orbit. As a consequence, the dominant term in the Fokker-Planck equation is simply so that f is constant along the field lines. Then, performing a bounce-averaging, we have { } [ ] f v s = τb s = τ b [ v s f s = 3.5 σ ] σ T T smax s min ds v s v f s s σ [f ] smax s min 3.6 For passing particles, the positions s min and s max coincide, so that [f ] smax s min = and the term vanishes. For trapped particles, the term also vanishes because of the sum over σ = ±, since, by definition, v = at the turning points s min and s max, and consequently f is independent of the sign of σ. Therefore, the bounce-averaged Fokker-Planck equation becomes with f constant along the field lines Drift-Kinetic Equation The drift kinetic equation is f v s s + v I ψ ψ Ω e R {C f } + {Qf } + {E f } = 3.7 s v In the low collisiona regime ν, the dominant term is which gives where B f v s s + v I ψ ψ Ω e R f = f ψ = C f + Qf + E f 3.8 s v B f ψ = 3.9 f = f + g 3. ds v v s Ω e I ψ B P s v B f ψ and g is a constant function along the field lines. Noting that 3. b ŝ = B P B 3. 4

44 3. Kinetic description of electrons 3.4. Low collision limit and bounce averaging we get f = v ds I ψ B Ω e s B = γm ei ψ f ds q e ψ s = v I ψ f Ω e ψ f ψ v B 3.3 where we used the fact that f / s =. Then, performing a bounce-averaging, we find again, using the same argument as in 3.6, that { } f v s = 3.4 s In addition, we have { v I ψ ψ Ω e R [ ] = τ b σ T s v smax s min = γm e I ψ f τ b q e ψ = γm e I ψ f τ b q e ψ B [ [ } f ψ σ v ds I ψ ψ v s Ω e R s ] smax σ ds s min s ] σ T T σ [ v ] smax B v B v B f ψ s min 3.5 Again, for passing particles, the positions s min and s max coincide, so that [ v /B ] s max s min = and the term vanishes. For trapped particles, the term also vanishes because v at the turning points s min and s max. Consequently, we find that the bounce-averaged drift kinetic equation becomes {C f } + {Qf } + {E f } = 3.6 where. f = f + g 3.7 and g is then given by {C g} + {Qg} + {E g} = f = v I ψ f Ω e ψ { } { } { } C f Q f E f using the fact that all operators are linear. 4

45 3. Kinetic description of electrons 3.5. Flux conservative representation 3.5 Flux conservative representation 3.5. General formulation The starting point of the flux conservative representation is the conservation of the total number of particles in the plasma, N = f X,P d 3 Xd 3 P 3. where X and P are respectively coordinates in configuration and momentum spaces. According to the systems which are used in the calculations, X = ψ,θ,φ and P = p,ξ,ϕ, { d 3 X = ψ ψ r Rr dψdθdφ 3. d 3 P =p dpdξdϕ as shown in Appendix A, one obtains Rr N = f ψ,θ,φ,p,ξ,ϕ ψ ψ p dψdθdφdpdξdϕ 3. r Using the transformation ξ B ψ,θ = ξ B ψ 3.3 that results from conservation of the magnetic moment and energy, Rr N = f ψ,θ,φ,p,ξ,ϕ ψ ψ p ξ B ψ,θ r ξ B ψ dψdθdφdpdξ dϕ 3.4 because ξdξ/b ψ,θ = ξ dξ /B ψ. Since at the zero order, f is constant along a magnetic field line, f = f is independent of the poloidal angle θ, where f is the bounce averaged distribution function. Hence N = f ψ,p,ξ p dψdpdξ θmax ξ B ψ,θ π π ξ B ψ dθ dφ dϕ Rr θ min ψ ψ r = 4π θmax θ min f ψ,p,ξ Rr ψ ψ r B ψ p dψdpdξ ξ B ψ,θ dθ 3.5 ξ taking into account, in addition, of the toroidal aximmetry, and cylindrical symmetry of the distribution along the magnetic field line direction. Here, θ min and θ max depends of the particle trajectory in the configuration space, wether they are passing or trapped. 43

46 3. Kinetic description of electrons 3.5. Flux conservative representation From plasma equilibrium, since where B P is the poloidal magnetic field, θmax θ min Rr ψ ψ r ξ θmax ξ B ψ,θdθ = θ min ψ R = B P 3.6 r ψ r ξ ξ B B P dθ = πλψ,ξ q ψ 3.7 Here, appears, as expected the normalized bounce time λψ,ξ and the factor q ψ introduced in Sec..., which in conjunction with B ψ characterizes the local shape of magnetic flux surface. Hence, N = 8π 3 f ψ,p,ξ q ψ B ψ λψ,ξ p dψdpdξ 3.8 From this expression, the Jacobian J of the coordinate system ψ,p,ξ may be simply defined as, q ψ J = J ψ J p J ξ = B ψ λψ,ξ p 3.9 where J ψ = q ψ /B ψ J p = p J ξ = λψ,ξ and the generic conservative form of the kinetic equation may be immediatly deduced 3.3 f t + ψ,p,ξ S = 3.3 where the phase space flux S at B = B min is decomposed into a diffusive and a convective part S = D f + F f 3.3 in the mean field theory. Here, D and F are respectively the diffusion tensor and convection vector in phase space. They can be expressed generally as D = D ψψ D ψp D ψξ D pψ D pp D pξ D ξψ D ξp D ξξ 3.33 and F = F p F ξ F ψ

47 3. Kinetic description of electrons 3.5. Flux conservative representation where each element is function of ψ,p,ξ. Here the gradient vector in the reduced ψ,p,ξ space is ψ = ψ / ψ = p = / p 3.35 ξ ξ = p / ξ so, following calculations given in Appendix A, ψ,p,ξ S f = J p = J p + J ψ = p p JS e p + J JS p J J ψ S ψ p S p λψ,ξ p ξ B ψ + λψ,ξ q ψ ψ JS e ξ + JS e ψ ξ J ψ ξ p ξ JS ξ ξ λψ,ξ S ξ q ψ B ψ λψ,ξ ψ S ψ 3.36 where ψ is taken on the magnetic flux surface where B is minimum, i.e., B = B. The first two terms correspond to the usual dynamics in momentum space at a given spatial position ψ, while the third one is associated to spatial transport at fixed p and ξ. It is interesting to note that spatial transport is not independent of the momentum dynamics through the parameter λψ,ξ. It corrects the spatial transport from the particle dynamics along the magnetic field line, since most particles tend to spend more time far from B = B. In the limit of strongly passing particles, λψ,ξ, and the spatial term becomes independent of ξ. It is interesting to cross-check the conservative nature of the transport equation is well ensured by performing the integral [ f t + ψ,p,ξ S ]Jdpdξ dψ = 3.37 or N t + ψ,p,ξ S Jdpdξ dψ = 3.38 Indeed, in that case, the variation of the total number of particles N/ t as a function of time depends only from boundary conditions. ψ,p,ξ S Jdpdξ dψ = I p I ξ + I ψ 3.39 where I ξ = I p = p p λψ,ξ p ξ p S p Jdpdξ dψ 3.4 ξ λψ,ξ S ξ 45 Jdpdξ dψ 3.4

48 3. Kinetic description of electrons 3.5. Flux conservative representation For I p, I ψ = B ψ λψ,ξ q ψ ψ = p p [ p S p = p max q ψ B ψ λψ,ξ ψ S ψ Jdpdξ dψ 3.4 p S p ] pmax S p p max Jdpdξ dψ q ψ B ψ λψ,ξ dξ dψ q ψ B ψ λψ,ξ dξ dψ 3.43 and assuming that lim p p S p =, one finds I p =. This condition is generally well fullfiled, except in strong runaway regimes, where above the Dreicer limit characterized by critical momentum p D, electrons gain energy up to very high energies, that are usually well beyond the domain of integration addressed in numerical calculations for the current drive problem. However, in this case, I p is given by S p at p max, where p max corresponds to the boundary of the momentum domain of integration. Its conservative nature is well ensured, since N/ t only depends of this parameter for p. The integration of ξ leads to I ξ = ξ λψ,ξ S ξ Jdpdξ dψ = λψ,ξ p ξ [ ξ λψ,ξ S ξ ] + q ψ B ψ pdpdψ = 3.44 which indicates that pitch-angle scattering never contributes to variations of N, and therefore the conservative nature of this term in the transport equation is also well demonstrated. Finally, I ψ = = B ψ q ψ λψ,ξ q ψ ψ [ q ψ B min ψ λψ,ξ ψ S ψ = q ψ a B ψ a ψ ψ a B ψ λψ,ξ ψ S ψ ] ψa p dpdξ Jdpdξ dψ λψ a,ξ S ψ ψ ap dpdξ 3.45 which only depends of edge values at ψ a since ψ = R B P = at ψ = when no particle is injected at the plasma center. Here, B P is the poloidal magnetic field where B is minimum. If S ψ ψ a =, I ψ =, and the total number of particles is conserved in the discharge. It is important to notice that the magnetic moment is intrinsically conserved in the equations, in particular for the radial transport part, through the pitch-angle dependence of the normalized bounce time λ. Therefore, spatial transport is valid not only for circulating particles satisfying p /p, but also for highly trapped electrons, i.e. when p /p. 46

49 3. Kinetic description of electrons 3.5. Flux conservative representation 3.5. Dynamics in Momentum Space Momentum space operator It is possible to recover the general bounce averaged transport equation in momentum space by another independent approach. Here, the momentum space dynamics of the kinetic equation is be expressed in conservative form as a flux divergence that may expressed according to the A.57 introduced in Appendix A p S p = Jp J p p i Sp i 3.46 where J p is the momentum space Jacobian associated with the momentum space coordinate system p, ξ, ϕ, described in A.47. Since the spherical system has the natural symmetry of collisions, the momentum space Jacobian is A.69 J p = p 3.47 so that, taking into account that the kinetic equation is gyroaveraged and therefore the coordinate ϕ disappears, the following expression for the divergence A.78 is obtained where by definition p S p = p p S p ξ p p ξ S ξ 3.48 S p = S p p 3.49 S ξ = S p ξ 3.5 In the mean-field theory, the momentum space fluxes may be expressed as the sum of diffusive and convective parts, S p = D p p + F p 3.5 with D p = F p = Dpp D pξ D ξp Fp F ξ D ξξ with The gradient vector p in the reduced coordinates system p,ξ in given by A.77 p p = 3.54 ξ p = p ξ ξ = p ξ

50 3. Kinetic description of electrons 3.5. Flux conservative representation so that and f ξ S p = D pp p + f D pξ p ξ + F pf 3.57 f ξ S ξ = D ξp p + f D ξξ p ξ + F ξf 3.58 Bounce-averaged operator The bounce averaged operator is { { p S p } = p } { } p S p ξ p p ξ S ξ 3.59 where the bounce averaging operation is defined in.6 [ ] {A} = θmax dθ r B ξ λ q θ min π ψ r R p B P ξ A 3.6 σ T and ξ is given along the trajectory by ξ ψ,θ,ξ = σ Ψ ψ,θ ξ with Ψ ψ,θ = B ψ,θ B ψ as shown in Sec... We find from 3.6 that in momentum space and we also get ξdξ = Ψξ dξ 3.63 ξ = Ψ ξ 3.64 Then, keeping in mind that ξ = σξ is independent of σ, we can transform as follows, { } ξ p ξ S ξ [ ] = θmax dθ r B ξ λ q σ θ T min π ψ ξ r R p B P ξ p ξ S ξ [ ] = θmax dθ r B σ λp σξ q σ θ T min π ψ ξ r R p B P Ψ S ξ [ ] = ξ λp ξ σ q θmax dθ r B σ σ θ T min π ψ Ψ S ξ r R p B P = { } σξ ξ λp ξ λσ S ξ 3.65 Ψξ 48

51 3. Kinetic description of electrons 3.5. Flux conservative representation using relation ξdξ = Ψξ dξ that is deduced from expression 3.6. Finally, we can rewrite the equation 3.59 in a conservative form as { p S p } = p p S p ξ p λp ξ λs ξ 3.66 where the following components are defined S p = {S p } 3.67 and S ξ { } σξ = σ S ξ Ψξ 3.68 Here, expression 3.66 is completely equivalent to the momentum transport equation deduced from particle conservation. However, this equivalence may be used only because the bounce-averaged operator is local, and does not depends of ψ Dynamics in Configuration Space Configuration space operator The operator that describes the spatial transport is given by relation f B ψ q ψ = t λψ,ξ q ψ ψ B ψ λψ,ξ ψ S ψ 3.69 r and since B P = ψ /R, it may be rewritten in the form f B ψ = R t ψ q ψ B P ψ λψ,ξ q ψ ψ B ψ λψ,ξ S ψ r 3.7 where R and B P are taken at the poloidal location where the magnetic field is minimum B = B. Much in the same way, S ψ = D ψψ ψf + F ψ f = D ψψ ψ f ψ + F ψ f = D ψψ R ψ B P ψ f ψ + F ψ f 3.7 between momentum and config- where the diffusion cross-terms D uration spaces have been neglected. pψ, D ψp,d ξψ and D ψξ 49

52 3. Kinetic description of electrons 3.5. Flux conservative representation Case of circular concentric flux-surfaces In that case, ψ = r, and by definition D ψψ = D rr, F ψ = F r, since ψ is here just a label. Therefore, using relation ψ / ψ = / r, S ψ = D ψψ ψf + F ψ f = D ψψ ψ = D rr f r f ψ + F ψ f + F r f = S r 3.7 Furthermore, since one obtains, f t circ. r q r = r R p B B P 3.73 = R pb P λr,ξ r = = R ψ ψ λr,ξ R r ψ λr,ξ R r r rλr,ξ S r R p R rλr,ξ S r R rλr,ξ S r 3.74 Note that R may not be here simplified, since it is a function of r, which corresponds to the toroidal configuration. Dynamics in momentum and configuration spaces are also not decoupled, the normalized bounce time λr,ξ being on both sides of the radial derivative. Only for strongly circulating electrons, lim ξ f t circ. r = rr S r rr r 3.75 since lim ξ λr,ξ =, and the usual cylindrical conservative expression of the radial transport f circ. lim = rs r 3.76 ξ t r r is only found in the case ɛ, i.e. when R R p. r 5

53 3. Kinetic description of electrons 3.5. Flux conservative representation Bounce-averaged flux calculation Zero order term: the Fokker-Planck equation The bounce averaged Fokker-Planck equation is is given in the conservative form by relation 3.66, with the bounce-averaged fluxes 3.67 and 3.68 S p = {S p } 3.77 { } σξ = σ S ξ 3.78 Ψξ S ξ Because f is constant along a magnetic field line, we have f p,ξ = f p,ξ which is independent of θ and σ. Using the following identities we can rewrite σ { { { D pξ ξ p { σξ σ D pp f p f ξ Ψξ D ξp f p σξ ξ D ξξ Ψξ p σ } = {D pp } f p } = ξ p 3.79 { } σξ f D pξ 3.8 Ψξ σ ξ {F p f } = {F p }f 3.8 } { } σξ f = σ D ξp 3.8 Ψξ p } { } f ξ = ξ f σ ξ p Ψξ D ξξ 3.83 σ ξ { σξ Ψξ F ξ f } = σ where the bounce averaged flux is decomposed into S S p p = S ξ with S p = D pp S ξ = D ξp { σξ Ψξ F ξ } f 3.84 S p = D p,ξ + F p 3.85 f ξ p + p ξ f p + p D pξ D ξξ 3.86 f ξ + F p f 3.87 f ξ + F ξ f

54 3. Kinetic description of electrons 3.5. Flux conservative representation by defining the diffusion components and the convection components D pp = {D pp } 3.89 { } D σξ pξ = σ D pξ 3.9 Ψξ { } D σξ ξp = σ D ξp 3.9 Ψξ { } D ξ ξξ = Ψξ D ξξ 3.9 F p = {F p } 3.93 { } σξ = σ F ξ 3.94 Ψξ F ξ where the gradient vector in the reduced p,ξ momentum space is p p,ξ = ξ 3.95 with p = p 3.96 ξ = ξ p ξ Up to first order term: the Drift Kinetic equation In the first-order drift kinetic equation, the momentum space operator p S p f 3.98 where the fluxes are expressed as 3.5 may be decomposed as p S p f = p S p f + p S p g 3.99 According to 3.66, we can express the bounce-averaged operator as { } J p p S p f = p p S p p ξ λ S λ ξ ξ {J p p S p g} = p S p p ξ p λ ξ λs ξ

55 3. Kinetic description of electrons 3.5. Flux conservative representation where we need to evaluate the bounce-averaged fluxes 3.67 and 3.68 for f and g respectively { } S p = S p f 3. { S σξ ξ = σ S ξ f } 3.3 Ψξ and S p = {S p g} 3.4 { } σξ = σ S ξ g 3.5 Ψξ S ξ Because g is constant along a field line, we have g p,ξ = g p,ξ which is independent of θ and σ. Therefore, the fluxes for g have exactly the same expression as for f in the zero-order equation described in section. This is why the same notation in 3. is used, while the fluxes associated with f are noted S. Indeed, f has an explicit dependence upon θ, which can be isolated as follows: with fψ,θ,p,ξ = ξψ,θ,ξ Ψψ,θξ f ψ,p,ξ 3.6 f ψ,p,ξ = pξ I ψ q e B ψ f ψ,p,ξ ψ 3.7 We can note that f is antisymmetric in the trapped region, since f is symmetric and ξ is, of course, antisymmetric. As a result, only σ f can be taken out of the bounce 53

56 3. Kinetic description of electrons 3.5. Flux conservative representation averaging operator. Taking the bounce-average of each term, we find { D f } { pp = σ σ ξ } D f pp 3.8 p Ψξ p { ξ D f } { } ξ pξ = ξ p ξ p Ψ 3/ ξ D f pξ ξ { ξ + σ σ Ψ } p Ψ 3/ ξ 3 D pξ f 3.9 { } { F p f = σ σ ξ } F p f 3. Ψξ σ { σ { σξ D f ξp Ψξ p σξ ξ D f ξξ Ψξ p ξ where the following relation is used We can therefore rewrite S p } { ξ = } { } σξ σ F ξ f = Ψξ ξ = σ ξ ξ ξ = σ Ψξ ξ ξ ξ = σ Ψ ξ ξ } Ψ 3/ ξ D f ξp p 3. { } ξ = σξ 3 σ p Ψ ξ 3 D f ξξ ξ { } ξ + ξ Ψ D ξξ f 3. p { ξ } Ψ 3/ ξ F ξ Ψ ξ 4 Ψ ξ ξ f = D p f p,ξ + F p f f where the bounce averaged flux is decomposed into S S p p = S ξ with S p = S ξ = D pp D ξp f ξ p + p f ξ p + p D pξ D ξξ f + ξ f + ξ F p F ξ 3.5 f 3.6 f

57 3. Kinetic description of electrons 3.6. Moments of the distribution function by defining the diffusion components { D pp = σ σ ξ } D pp Ψξ and the convection components F p F ξ = { σ ξ = σ F p Ψξ { ξ Ψ 3/ ξ { D ξ } pξ = Ψ 3/ ξ D pξ { } D ξ ξp = Ψ 3/ ξ D ξp { } D σξ 3 ξξ = σ Ψ ξ 3 D ξξ } { ξ + Ψ pξ 3 Ψ 3/ } ξ F ξ + pξ 3 σ D pξ { σξ Ψ ξ Ψ } D ξξ } where we use the fact that σξ 3 may be taken out of the bounce averaged operator, since ξ 3 is an odd function of ξ. The gradient vector in the reduced p,ξ momentum space is p p,ξ = 3.4 with ξ p = p 3.5 ξ = ξ p 3.6 Moments of the distribution function 3.6. Flux-surface Averaging Surface densities ξ 3.6 We consider the flux-surface averaging of a surface quantity, such as a flux of a current, generally noted Γψ,θ. It is defined as the averaged flux of Γ through the infinitesimal poloidal surface ds ψ dsψ ds Γ ψ,θ Γ S ψ = dsψ ds 3.7 In the ψ,θ,φ system, the differential poloidal surface element is given by A. as introduced in Appendix A r ds = ψ ψ dψdθ φ 3.8 r 55

58 3. Kinetic description of electrons 3.6. Moments of the distribution function so that the infinitesimal poloidal surface element ds p ψ is r π r ds = ds pψ ds pψ ψ ψ dψdθ = dψ dθ r ψ ψ r and the flux-surface averaged flux in the toroidal direction is π dsp r ] Γ φ ψ = dθ dψ ψ ψ [ φ Γ ψ,θ r with ds p ψ dψ = = π π r dθ ψ ψ 3.3 r r dθ ψ 3.3 r R B P Defining the new pseudo saftey factor q as q ψ π dθ r π ψ r R B ψ B P 3.33 we get and ds p ψ dψ = πq ψ B ψ Γ φ ψ = π dθ r q ψ π ψ r R B ψ B P [ φ Γ ψ,θ ] Volume densities We consider the flux-surface averaging of a volume quantity, such as a power density, generally noted Φ ψ,θ. It is defined as the average value of Φ within the infinitesimal volume dv ψ dv ψ Φ V ψ = Φ ψ,θdv dv ψ dv 3.36 In the ψ,θ,φ system, the differential volume element is given by A. as introduced in Appendix A Rr dv = ψ ψ dψdθdφ 3.37 r so that the infinitesimal volume element dv ψ of a flux-surface is Rr π π Rr dv = dv ψ dv ψ ψ ψ dψdθdφ = dψ dθ dφ r ψ ψ r

59 3. Kinetic description of electrons 3.6. Moments of the distribution function and the flux-surface averaged quantity in the toroidal direction is with Φ V ψ = dv π π dθ dψ Rr dφ ψ ψ Φ ψ,θ 3.39 r dv ψ dψ = = π π π Rr dθ dφ ψ ψ r 3.4 π dθ dφ r ψ r B P 3.4 Under the assumption of axisymmetry, we get dv ψ dψ π = 4π dθ r π ψ r B P 3.4 = 4π R π dθ r B B ψ π ψ ψ r R p B P 3.43 Defining the new pseudo saftey factor q as q ψ π dθ r B π ψ ψ 3.44 r R p B P we get and finally dv ψ dψ = 4π R p q ψ B ψ 3.45 Φ V ψ = π dθ r B q ψ π ψ ψ Φ ψ,θ 3.46 r R p B P 3.6. Density Definition The electron density n e ψ,θ is given by the relation n e ψ,θ = π + dξ p dp f p,ξ,ψ,θ

60 3. Kinetic description of electrons 3.6. Moments of the distribution function Using the general expression 3.46 of the flux-surface averaging of a volumic quantity n e V ψ = q = π q = π q π dθ π ψ r p dp r B n e ψ,θ R p B P π dθ r B π ψ r R p B P π p dθ r B dp π ψ r R p B P + + dξ f ψ,θ,p,ξ [ ] σ=± T dξ f ψ,θ,p,ξ 3.48 where the trapping condition evaluated at the location θ is given by ξ < ξ T = B ψ,θ B ψ Using ξdξ = Ψξ dξ with the condition 3.7 on ξ ξ Ψ ψ,θ one get [ ] + σ=± T dξ = + [ ] σ=± T Ψ ψ,θ ξ ξ H ξ dξ 3.5 Ψ ψ,θ where H is the usual Heaviside function which is defined as H x = for x >, and H x = elsewhere. Note that the condition 3.5 is equivalent to so that, the integrals over θ and ξ may be permuted, θ min ψ,ξ θ θ max ψ,ξ 3.5 n e V ψ = π q p dp [ ] σ=± T θmax θ min + dξ dθ r B ξ π ψ r R p B P ξ f ψ,θ,p,ξ 3.53 where the bounce-averaging of the distribution appears naturally. Therefore, expression 3.37 can be rewriten in the simple form n e V ψ = π q q p dp + dξ λ {f ψ,θ,p,ξ }

61 3. Kinetic description of electrons 3.6. Moments of the distribution function Fokker-Planck Equation For the zero order distribution function, since f is constant along a field line, one obtains Drift Kinetic Equation n e V ψ = π q q f ψ,θ,p,ξ = f ψ,p,ξ 3.55 p dp + dξ λf ψ,p,ξ 3.56 When we consider the first order distribution function, we have f = f + g, where g is constant along a field line, and therefore its contribution n e V ψ has the same expression as for f. However, f has an explicit dependence upon θ, which is given by 3.6 where fψ,θ,p,ξ = ξψ,θ,ξ Ψψ,θξ f ψ,p,ξ 3.57 Therefore, the flux-surface averaged density contribution of f is + { } ñ e V ψ = π p q q ξ dp dξ λ f ψ,p,ξ 3.58 Ψψ,θξ = π q q p dp + { λ,, = σ σ dξ λ,, f ψ,p,ξ 3.59 ξ Ψψ,θξ } λ 3.6 according to the notation in Sec..., since f is antisymmetric in the trapped region. Since f and g have no definite symmetry properties, both can contribute to the density and n e V ψ = n e V ψ + n e V ψ + ñ e V ψ Current Density Definition The density of current carried by electrons is given by Jx = q e d 3 p vf x,p 3.6 so that the parallel current density is J x = q e d 3 p v f x,p 3.63 which becomes in ψ,θ,p,ξ phase space J ψ,θ = πq e p dp dξ pξ f ψ,θ,p,ξ 3.64 γm 59

62 3. Kinetic description of electrons 3.6. Moments of the distribution function Flux-Surface Averaging We are usually interested in the flux-surface averaged current density in the toroidal direction. It is generally given by 3.3 J φ ψ = q = q π π dθ r B [ φ b] π ψ J r R B ψ,θ P dθ r B π ψ J r R B ψ,θ B T P B 3.65 and finally, using.3 J φ ψ = q π dθ r B π ψ T J ψ,θ r R B P Ψ ψ,θ 3.66 Fokker-Planck Equation When we consider only the zero order distribution function, we have that f is constant along a field line, so that f ψ,θ,p,ξ = f ψ,p,ξ 3.67 where Consequently, we find where the condition ξ = σ J ψ,θ = πq e Ψ ψ,θ ξ 3.68 p dp = πq e p dp = πq e p dp H ξ ξ Ψ ψ,θ dξ pξ γm e f ψ,θ,p,ξ dξ pξ γm e f ψ,p,ξ dξ Ψ ψ,θ Ψ ψ,θ pξ γm e f ψ,p,ξ results from the equation 3.68 and means that only the particle who reach the position θ must be considered. Note that the integrand in the equation 3.69 is odd in ξ for trapped electrons, since f is symmetric in the trapped region. As a consequence, the contribution from trapped electrons vanishes, and 3.69 can be rewritten as J ψ,θ = πq e p dp dξ Ψ ψ,θ H ξ ξ T 6 pξ γm e f ψ,p,ξ 3.7

63 3. Kinetic description of electrons 3.6. Moments of the distribution function Therefore, the flux-surface averaged current density becomes J φ ψ = q J φ ψ = πq e m e π dp p3 γ dθ r B π ψ T J ψ,θ r R B P Ψ ψ,θ π dθ q π The integrals over θ and ξ can be permuted J φ ψ = πq e q m e π ψ r r B T R B P Ψ ψ,θ 3.7 dξ Ψ ψ,θ H ξ ξ T ξ f ψ,p,ξ 3.73 dθ π dp p3 γ ψ r dξ H ξ ξ T ξ f ψ,p,ξ r B T 3.74 R B P We recognize the expression of the safety factor.5 so that J φ ψ = πq e q m e q dp p3 γ dξ H ξ ξ T ξ f ψ,p,ξ 3.75 Case of circular concentric flux-surfaces In that case, we showed in.83 that the safety factor is ɛ B q r = T 3.76 ɛ B P with ɛ = r/r p the inverse aspect ratio. In addition, q r becomes q r = = π π dθ π dθ π = ɛ B B P R p R r R p B B P r B R p B since R = R p + r cos θ, and B /B = R/R. We have then q r + ɛ q r = B T ɛ B B B P = ɛ B ɛ B P 3.78 In the case when B T B P, we retrieve the bounce-averaged coefficient s and in the large aspect ratio limit ɛ, q r lim ɛ q r = + ɛ B T B

64 3. Kinetic description of electrons 3.6. Moments of the distribution function Drift Kinetic Equation When we consider the first order distribution function, we have f = f + g, where g is constant along a field line, and therefore its contribution has the same expression as for f. However, f has an explicit dependence upon θ, which is given by 3.6 fψ,θ,p,ξ = ξψ,θ,ξ Ψψ,θξ f ψ,p,ξ 3.8 where Consequently, we find ξ = σ Ψ ψ,θ ξ 3.8 J ψ,θ = πq e p dp = πq e p dp = πq e p dp H ξ Ψ ψ,θ dξ pξ γm e f ψ,θ,p,ξ dξ pξ γm e ξ dξ ξ ξ Ψψ,θξ f ψ,p,ξ pξ γm e f ψ,p,ξ 3.8 where again the condition ξ Ψ ψ,θ 3.83 results from the equation 3.68 and means that only the particle who reach the poloidal position θ must be considered. Therefore, the flux-surface averaged current density contribution from f J φ ψ = q π dθ r B J π ψ T ψ,θ r R B P Ψ ψ,θ 3.84 becomes J φ ψ = πq e m dξ ξ ξ H dp p3 γ ξ π dθ q π ψ r Ψ ψ,θ Note that the condition 3.83 is equivalent to r B T R B P Ψ ψ,θ ξ f ψ,p,ξ 3.85 θ min ψ,ξ θ θ max ψ,ξ

65 3. Kinetic description of electrons 3.6. Moments of the distribution function so that, permuting the integrals over θ and ξ, we find We have then J m e θmax φ ψ = πq e q θ min dθ π dp p3 γ ψ r r B T R B P Ψ ψ,θ = R pi ψ R B B T dξ ξ f ψ,p,ξ r ξ 3.87 R B P Ψ ψ,θ ξ r B R R p B P R Ψ ψ,θ 3.88 Then, noting the the integrand in 3.87 is independent of σ, so that the sum over σ for trapped particles can be added, we obtain J ψ = πq e φ m e [ ] σ T dp p3 γ θmax θ min dθ π dξ ξ f ψ,p,ξ R p I ψ q R B [ ] r B ξ [ ] ψ R ξ Ψ ψ,θ 3.89 r R p B P ξ R ξ We recognize the expression of a bounce coefficients defined by the general relation.66 in Sec..., so that we get finally J ψ = πq e q R p B T φ m e q R B dp p3 γ dξ λ,, ξ f ψ,p,ξ 3.9 with λ,, = λ { ξ ξ } Ψ R R 3.9 Case of circular concentric flux-surfaces In that case, we showed in.99 that q is q r = ɛ B B P 3.9 with ɛ = r/r p the inverse aspect ratio. In addition, q r is q r = ɛ B ɛ B P and since we have then in the limit B P B. R = R p + ɛ 3.94 q R p B T = B T 3.95 q R B B 63

66 3. Kinetic description of electrons 3.6. Moments of the distribution function so that Also, in this case, Ψ ψ,θ = R R 3.96 { } ξ λ,, = λ,, = λ = s 3.97 using notations used in previous publications. The exact expression of s in terms of a series expansion is given in relation Power Density Associated with a Flux Definition The kinetic energy associated with a relativistic electron of momentum p is ξ E c = m e c γ 3.98 Then, the local energy density of electrons is εx = d 3 p m e c γ fx,p 3.99 The density of power absorbed through the process O, Pabs O, is Pabs O ε x = t = d 3 p m e c fx, p γ O t 3.3 O When the operator is described in conservative form, as the divergence of a flux f t = p S O p = p O p S O p + ξ p p ξ Sξ O 3.3 then the power density becomes Pabs O = πm ec p dp γ + [ dξ p p S O p ξ p p ξ Sξ O ] 3.3 The integration of the Sξ O term gives no contribution, since the particle energy is function of p only + and the equation 3.3 reduces to dξ [ ξ ξ Sξ O = ξ Sξ O + Pabs O = πm ec dξ γ p ] + = 3.33 p Sp O dp

67 3. Kinetic description of electrons 3.6. Moments of the distribution function Integrating by parts, we get + Pabs O = πm ec dξ Assuming that lim p p S O p [γ p Sp O ] dγ dp p Sp O dp =, and using 3.35 the equation 3.35 reduces to Flux-Surface Averaging dγ dp = + Pabs O ψ,θ = π dξ p γm e c 3.36 dp p3 γm e S O p 3.37 Starting from the general expression of the flux-surface averaging of a volume quantity 3.46, the flux-surface averaged power density Pabs O ψ is P O abs V ψ = q π V dθ r B π ψ Pabs O ψ,θ 3.38 r R p B P which becomes P O abs V ψ = π dp p3 π dθ r B γm e q π ψ r R p B P + dξ S O p 3.39 The sum over σ for trapped electrons can be added, using [ ] ξt dξ Sp O = dξsp O + dξsp O + ξt dξ Sp O σ=± ξ T ξ T T σ=± ξt = dξsp O + dξsp O + ξt dξ [ Sp O ξ + Sp O ξ ] ξ T ξ T = = ξt dξs O p + ξ T dξs O p + ξt ξ T dξs O p ξ dξs O p 3.3 where the trapping condition evaluated at the poloidal location θ is ξ < ξ T = B ψ,θ B max ψ 3.3 Using ξdξ = Ψξ dξ with the condition 3.7 on ξ ξ Ψ ψ,θ

68 3. Kinetic description of electrons 3.6. Moments of the distribution function we get that + dξ = + Ψ ψ,θξ dξ H ξ Note that the condition 3.3 is equivalent to ξ Ψ ψ,θ 3.33 so that, permuting the integrals over θ and ξ, we find P O abs V ψ = π θ min ψ,ξ θ θ max ψ,ξ 3.34 [ q ] σ=± dp p3 T γm e θmax θ min + dξ 3.35 dθ r B ξ π ψ r R p B P ξ SO p 3.36 We see that the bounce-averaging of the fluxes appears naturally, so that we can rewrite P O abs V ψ = π q q dp p3 γm e + dξ λ { Sp O } 3.37 Using the definition 3.67, we observe that the flux-surface averaged power density is calculated using the momentum flux component of the bounce-averaged kinetic equation: P O abs V ψ = π q q dp p3 γm e + dξ λs O p 3.38 Case of circular concentric flux-surfaces In that case, we showed in 3.9 that the coefficient q is with ɛ = r/r p. In addition, q r becomes q r = ɛ B B P 3.39 π dθ r B q r = π R p B P = ɛ B π dθ B B P π B = ɛ B π dθ R B P π R = ɛ B ɛ B P using the simple relation B/B = R /R and R = R p + ɛ. We have then q ψ q ψ = + ɛ

69 3. Kinetic description of electrons 3.6. Moments of the distribution function Fokker-Planck Equation The Fokker-Planck equation 3.7 solves for the zero-order distribution function f. The density of power transfered to f through the momentum-space mechanism O is then P O abs V ψ = π q q dp p3 γm e + dξ λs O p f 3.3 where S O p f is given by 3.87 S O p f = D O pp f p + ξ p D O pξ f ξ + F O p f 3.33 as- The momentum-space diffusion and convection elements D pp O, D O pξ sociated with a particular mechanism O are calculated in chapter 4. Drift Kinetic Equation and F O p The Fokker-Planck equation 6. solves for the first-order distribution function f = f +g 3.7. The densities of power transfered to f and g through the momentum-space mechanism O are then respectively where D pξ S O p f P O abs V ψ = π q q P O abs ψ = π V q q dp p3 γm e dp p3 γm e + + dξ λ S O p and S O p g are given by 3.87 and 3.6 S f p = S O p D pp f ξ p + p ξ p g = D pp O g p + D pξ f 3.34 dξ λs O p g 3.35 f + ξ F p The momentum-space diffusion and convection elements D O pp f 3.36 D O g pξ + F p O g 3.37 ξ, D O pξ, F p O, D pp, and F p associated with a particular mechanism O are calculated in chapter Stream Function for Momentum Space fluxes When transport in configuration space is ignored, and a steady-state regime is assumed to be reached, the Fokker-Planck equation reduces to the conservative equation 3.46 p S p = 3.38 Because S p is a divergence-free field vector, it can be expressed as the curl of a stream function S p = T p

70 3. Kinetic description of electrons 3.6. Moments of the distribution function The expression of a curl in momentum space p,ξ,ϕ is given by relation A.79 in Appendix A S p = p ξ ξ T ϕ + S ξ = p p pt ϕ p T p ξ ϕ S ϕ = ξ p p pt T p ξ p ξ p T ξ ξ ϕ Because S ϕ =, we can choose T ξ = T p =, which leads to and we can rewrite S p = ξ p ξ T ϕ S ξ = p p pt ϕ S p = T ϕ ϕ In order to give a physical meaning to T ϕ p,ξ,ψ, we define formally T ϕ ψ,p,ξ = K ψ,p,ξ Aψ,p,ξ where the function Ap,ξ is such that the flux of electrons between two contours A and A is equal to n e ψ A A. Lets consider a path γ between the contours A and A. The total flux of electrons through this path, which is in fact a surface, given the rotational symmetry in ϕ, is given by Γ = ds S p n S = ds T ϕ ϕ S = T ϕ dl ϕ C By rotational symmetry in ϕ, and using A.7, we get Γ = πp ξ T ϕ πp ξ T ϕ If we define we obtain K ψ,p,ξ n e ψ πp ξ Γ = n e ψ A A

71 3. Kinetic description of electrons 3.6. Moments of the distribution function and therefore the total flux between the contours A and A is equal to n e ψ A A. We call Aψ,p,ξ the stream function, and we get finally S p = n e ψ A πp ξ n e ψ S ξ = πp A ξ p Since there are no fluxes across the internal boundaries in the momentum space, this boundary coincide with a contour A, and therefore we can arbitrarily set this value to : Then A can be calculated by any of the integrals or A,ξ = Ap, ± = ξ ξ Aψ,p,ξ = πp dξ S p = πp dξ S p n e ψ n e ψ Aψ,p,ξ = π ξ n e ψ p p dp S ξ However, Aψ,p,ξ remains a function of ξ, which depends upon θ. Starting from the bounce-averaged fluxes, it is interesting to compute a function A ψ,p,ξ, such that A,ξ = Ap, ± = S p S ξ = = n e ψ πp A ξ n e ψ πp ξ A p 69

72 3. Kinetic description of electrons 3.6. Moments of the distribution function We first need to demonstrate the existence of such a function. Starting from S p, A ψ,p,ξ = πp ξ n e ψ where we used = πp n e ψ = ξ dξ = σ λ q [ ξ dξ λ q ] σ ξ dξ {S p } [ T [ ] = σ λ q σ T [ ] = σ λ q σ T [ ] = σ q λ q σ λ q ] σ π [ T ] σ π dξ H ξ T π π T θmax θ min dθ r B ξ π ψ r R p B P ξ S p dθ π H B r B ξ b B ψ A r R p B P ξ ξ dθ r B π ψ r R p B P Ψ dθ r B σ π ψ r R p B P Ψ dθ r B π ψ σa r R p B P θ min θ θ max B B b σ ξ A ξ ξ ξ dξ A ξ σa V Ψ ξ

73 3. Kinetic description of electrons 3.6. Moments of the distribution function Now, starting from S ξ, we have A ψ,p,ξ = π ξ n e ψ = π ξ n e ψ = p dp σ λ q [ ] p p = σ λ q σ [ ] = σ λ q σ T [ ] = σ q λ q σ [ { } σξ p dp σ S ξ Ψξ p dp σ λ q ] σ θmax θ T min θmax T θ min T [ θmax dθ π dθ π θ min ψ r ψ r ] σ T θmax θ min dθ r B π ψ r R p r B σ R p B P Ψ r B σa R p B P dθ r B π ψ r R p p B P B P σ ξ Ψ A ξ p dp A p σ Ψ S ξ σa V and we find the same function A. The existence of a function A verifying is therefore demonstrated. We need now to demonstrate that A verifying leads to the bounce-averaged Fokker-Planck equation 3.66: { p S p } = p p = p p = λp pξ p S p ξ λp ξ λs ξ p n e ψ A πp ξ ξ λp ξ λ n e ψ πp ξ [ ] [ ] λn e ψ A λn e ψ A π λp ξ p π A p = 3.35 In conclusion, a stream function verifying A,ξ = Ap, ± = 3.35 has been found which leads to the bounce-averaged Fokker-Planck equation and which can be calculated from the bounce-averaged fluxes by either or relations. A ψ,p,ξ = πp n e ψ ξ dξ S p A ψ,p,ξ = π ξ n e ψ = πp n e ψ p ξ dξ S p 3.35 p dp S ξ

74 3. Kinetic description of electrons 3.6. Moments of the distribution function Ohmic electric field The electrical conductivity of the plasma σ e is defined as the ratio of the flux averaged current density J φ to the flux surface averaged parallel Ohmic electric field E φ, By definition, σ e = E φ ψ = π dθ q ψ π = π dθ q ψ π J φ E φ r ψ r R r ψ r R B [ φ b] E B ψ,θ P B ψ E B ψ,θ B T P B = π dθ r B q ψ π ψ T E ψ,θ r R B P Ψ ψ,θ Using R E ψ,θ = Ψ ψ,θ R E ψ where E ψ is the value at the minimum magnetic field B, one obtains or E φ ψ = q π = E ψ q dθ r B π ψ T E ψ r R B P Ψ ψ,θ π R R dθ r B π ψ T r R B P Ψ ψ,θ E φ ψ = E ψ R π p dθ r B ξ q R π ψ r R p B P ξ = E ψ q ψ [ ξ B T B T B ξ B T B B = E ψ q R π p R Ψ ψ,θ B T B R p R = E ψ q B T λσ q R B = E ψ q q R p dθ π R 3 R 3 π dθ π ψ r ] ψ r { σ ξ ξ [ ξ B T ξ B r B ξ R p B P ξ r B ξ R p B P ξ } Ψ 3 ψ,θ R 4 R 4 R R Ψ ψ,θ [ ξ ξ Ψ 3 ψ,θ R 3 R 3 ] R 4 R 4 R p R B T B λ, 3, ] 7

75 3. Kinetic description of electrons 3.6. Moments of the distribution function Case of circular concentric flux-surfaces In that case, and since R p /R = / + ɛ, using relation Ψ r,θ = R/R. Therefore, as λ,, = +ɛ ɛ q r q r = + ɛ E φ r = B T B λ,,e r E φ r = B T B = B T B λ,,e r ɛ ɛ E r 3.36 for circular concentric flux-surfaces. Moreover, in this limit, σ e = J φ E φ = J 3.36 E since with J = πq e m e J φ ψ,θ = B T + ɛ B ɛ J p dp dξ H ξ ξ T pξ γm f ψ,p,ξ In that case, the neo-classical conductivity can be either calculated from flux surface averaged quantity, or local values at B = B Fraction of trapped electrons The ratio between the number of trapped and passing electrons is an important quantity in the neoclassical transport theory, since the parallel viscosity responsible for reduction of the Ohmic conductivity and the bootstrap current level are both roughly proportional to this parameter. Therefore, under the influence of RF waves, its large variation will indicate unambiguously that significant macroscopic changes are to be expected on the current generation and the power absorption due to neoclassical effects. We could expect to encounter such circomstances especially when wave-particle interaction takes place in the near vicinity of the trapped-passing boundary. The starting point of the calculations is the determination of the flux averaged density n e. According to the definition of the electron momentum distribution function f, the local electron density n e ψ,θ is given by the relation n e ψ,θ = π + dξ 73 p dp f ψ,θ,p,ξ 3.365

76 3. Kinetic description of electrons 3.6. Moments of the distribution function Using the general expression 3.46 of the flux-surface averaging of a volumic quantity n e V ψ = q = π q π dθ π ψ r p dp = π q [ ] + r B n e ψ,θ R p B P π π p dθ dp π σ=± T dθ r B π ψ ψ r R p B P ψ r r B ψ R p B P + dξ f ψ,θ,p,ξ dξ f ψ,θ,p,ξ where the trapping condition evaluated at the location θ is given by ξ < ξ T = B ψ,θ B max ψ Using ξdξ = Ψξ dξ with the condition 3.7 on ξ ξ Ψ ψ,θ one get [ ] + σ=± T dξ = + [ ] σ=± T Ψ ψ,θ ξ ξ H ξ dξ Ψ ψ,θ Note that the condition is equivalent to so that, the integrals over θ and ξ may be permuted, θ min ψ,ξ θ θ max ψ,ξ 3.37 n e V ψ = π q p dp [ ] σ=± T θmax θ min + dξ dθ r B ξ π ψ r R p B P ξ f ψ,θ,p,ξ 3.37 where the bounce-averaging of the distribution appears naturally. Therefore, expression 3.37 can be rewriten in the simple form n e V ψ = π q q p dp + dξ λ {f ψ,θ,p,ξ }

77 3. Kinetic description of electrons 3.6. Moments of the distribution function and the exact trapped fraction F t is given by the ratio F t ψ = where λ is the normalized bounce time.. Since, {f} f + f +g, F t ψ = p dp +ξ T ξ T λ {f}dξ p dp + λ {f}dξ p dp [ +ξ T ξ T λ f + g ]dξ p dp + λ [ f + f + g ] dξ taking into account that f is an odd function of ξ in the trapped region. When f = f M = f M is a Maxwellian distribution on the magnetic flux surface ψ, F M t ψ = taking into account that g M g M, the zero order trapped fraction FM t which reduces to p dp +ξ T ξ T λf M dξ p dp [ λ f M + g M ]dξ = for trapped electrons. Neglecting the contribution of is given by Ft M ψ = p dp +ξ T ψ ξ T ψ λf M p,ψ dξ p dp + λf M p,ψ dξ F M t ψ = +ξt ψ ξ T ψ λdξ + λdξ = ξt ψ λdξ + λdξ In this limit, Ft M is only a function of the geometrical magnetic configuration, while is the general case, F t is a fully kinetic quantity. Case of circular concentric flux-surfaces In that case, the normalized bounce time is simply θmax dθ ξ λξ = π ξ [ J ξ,ξ T ] π ξ TJ ξ,ξ T θ min which may be expanded up to the second order with an excellent accuracy as shown in Appendix B.. Here, ɛ ξ T = ɛ with ɛ = r/r p the usual inverse aspect ratio. It is interesting to estimate the parametric dependence of Ft M for ɛ. For trapped particles, λξ [ ξ ξ K π ξ T ξt [ ]] ξ ξ T K ξ ξt E ξt

78 3. Kinetic description of electrons 3.6. Moments of the distribution function where K x and E x are complete elliptic integrals of the first and second kind. Hence, +ξt ψ λξ dξ = π Using the recurrence relation ɛ x[k x ɛk x E x]dx 3.38 n x n K x dx = n x n K xdx and since xe x dx = / according to formulaes 6.47 and 6.3 in Ref. [4], lim ɛ For circulating electrons, and +ξ T ψ From the relation λξ π +ξt ψ [ K ξ T ξ λξ dξ ɛ ɛ/ π [ ξ ξ K T ξ ]] ξ E T ξ λξ dξ = K x K x E x ɛ π ɛ x + ɛ x 4 dx K x x dx = E x x which is given by formula 5..9 of Ref. [4], λξ dξ = E ɛ ɛ π ɛ +ξ T ψ ɛ + π π ɛ ɛ + π ɛ ɛ ɛ K x E x x dx ɛ K x E x x 4 dx and using the indefinite integrals K x E x dx = E x x and E x x 4 dx = 9x 3 [ x E x + x K x ] 3.39 according to formulaes 5.3. and 5.. in Ref. [4], lim δ δ3 δ K x E x x 4 dx = δ3 K δ E δ + δ + δ E δ π 4 δ

79 3. Kinetic description of electrons 3.6. Moments of the distribution function so that up to the first order term, Consequently, λξ dξ + ɛ 3.39 lim ɛ FM t ψ ɛ.9 ɛ π and the ɛ dependence in the limit ɛ is well recovered, as expected from an intuitive explanation. It is worth noting that this result is well recovered by a simple Monte-Carlo technique, where the poloidal angle θ is taken to be a uniform random variable between and π, as well as ξ between and. Using the relation. which translates ξ to ξ at the minimum B value, and considering that the particle is trapped when ξ ξ T, the fraction of trapped particle found numerically is exactly Ft M ψ, while the distribution scales like λξ. It is important to precise that Ft M is not the effective trapped fraction Feff. t given by the well known relation F eff. t ψ = 3 h xdx xh found repeatedly in the litterature for the bootstrap current or the neoclassical conductivity, where h = B/B max and B max is the maximum value of the magnetic field B along the particle trajectory. This quantity results from the reduction of the conductivity due to trapped particles, or the onset of the bootstrap current. Its expression with notations used in the text is determined from the bootstrap current calculations with the Lorentz collision operator, as shown Sec It is important to notice that F eff. t is in principle not a fraction of trapped electrons, and in addition there is no demonstration that F eff. t is always satisfied for all magnetic configurations, as mentioned clearly in Ref. [5]. In fact the denomination effective trapped fraction F eff. t is quite confusing, since it applies only for Maxwellian regime, and is not established as a kinetic quantity like Ft M. This point is especially important when non-maxwellian distributions are considered for evaluating the bootstrap current. Consequently, F eff. t must not be used in such regimes, but only F t as an true physical sense for comparisons between different regimes Runaway loss rate When the Ohmic electric field exceeds the Dreicer level, a fraction of the electron population run away. The total number of electrons is therefore no more conserved, since the flux S p at p = p max, on the boundary of the integration domain. The runaway loss rate Γ R is given by the relation Γ R ψ,θ = S p ψ,p max,ξ dsp max

80 3. Kinetic description of electrons 3.6. Moments of the distribution function where element of surface dsp = pdξdϕ p according to the Appendix A. Therefore, since dϕ = π by symmetry, one obtains immediately Γ R ψ,θ = πp max + S p ψ,p max,ξ dξ Since all quantities are calculated at the spatial location where B is minimum, one have by definition in a straightforward manner Γ R ψ = πp max + S p ψ,p max,ξ λψ,ξ dξ where S p ψ,p max,ξ results from the solution of the bounce-averaged Fokker-Planck equation. The term λψ,ξ arises from the Jacobian J ξ. The flux-surface averaged runaway rate Γ R V is given by the relation and since one obtains Γ R V ψ = π q p max + dξ = Γ R V ψ = π q p max = π q p max = q q πp max π dθ r B π ψ ψ r R p B P Ψ ψ,θ ξ ξ H dξ [ Magnetic ripple losses ] σ T ξ θmax θ min + {S p ψ,p max,ξ} λψ,ξ q ψ dξ S p ψ,p max,ξ dξ dξ Ψ ψ,θ dθ r B ξ π ψ r R p B P ξ S p ψ,p max,ξ λψ,ξ S p ψ,p max,ξ dξ 3.4 Though magnetic ripple losses is a full 4 D problem, it can be considered in a simple manner by defining a super-trapped volume V p ST in momentum space, V p ST ψ = π p dp + H p p c H ξ ξ ST dξ 3.4 in which particle escape the plasma. A low energy, it is bounded by the collision detrapping when p p c, while the pitch-angle dependence results from the condition that only electrons whose banana tip enter the bad confinement region characterized by the well known criterion α are trapped in the magnetic well, in an irreversible manner. As shown in Fig. 3., even if this is a rather crude modeling, it captures most of the salient features of the physics. Therefore, all trapped electrons which in addition fullfils 78

81 3. Kinetic description of electrons 3.6. Moments of the distribution function cmcm Figure 3.: Domain in configuration space where magnetic ripple well takes place for Tore Supra tokamak the condition p /p ξ ST are super-trapped. Here, ξ ST is deduced from the intersection between the poloidal extend of the banana and the good confinement domain α on a given flux surface [7]. The pitch-angle threshold ξ ST depends therefore of the radial position ψ and close to the edge, lim ψ ψ a ξ ST = ξ T 3.4 which indicates that all trapped electrons are expected to escape the magnetic configuration. Furthermore, it is assumed that electrons, once in this magnetic well, do not contribute anymore to the overall momentum dynamics, which is obviously a very crude approximation. An heuristic description of this process may be obtained by introducing a Krook term restricted to the volume V p ST in the Fokker-Planck equation f t = ν d ST f ψ,p,ξ H p p c H ξ ξ ST 3.43 where ν d ST is the drifting time taken by super-trapped electrons for leaving the plasma. In order to reproduce the fact that super-trapped electrons are decoupled from the momentum dynamics, a simple method is to force ν d ST τ b. Without detailed knowledge of the local dynamics, ν dst is taken constant in V p ST, which is obviously a coarse approximation. However, in the limit ν d ST τ b, the shape of the distribution function becomes independent of ν dst, since by definition f ψ,p,ξ in the super-trapped domain. 79