2.1 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 .

2.1.1 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 , 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.212) and shown in Fig. 2.1. The cylindrical coordinate system is the natural system for wave-particle interaction, or also the effect of the electric field.

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  Figure 2.1: Coordinates systems and for momentum dynamics

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• Second, the spherical coordinate system , where pis the magnitude of the momentum, and ξ is the cosine of the pitch-angle. This system is defined in (A.247) and shown in Fig. 2.1 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.

2.1.2 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 flux-surface magnetic configuration. Three different configuration space coordinates systems are considered here:

• First, the toroidal coordinate system , where R is the distance from the axis of the torus, and Z the distance along this axis . This coordinates system and the corresponding local orthogonal basis vectors are defined in (A.61) and shown in Fig. 2.2. This coordinate system conserves the largest generality in the magnetic geometry.

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  Figure 2.2: Coordinates system (R,Z,ϕ).

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• Second, the poloidal (polar) coordinate system assumes the existence of a toroidal axis at constant position 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 = 0). This coordinates system and the corresponding local orthogonal basis vectors are defined in (A.94) and shown in Fig. ??.

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  Figure 2.3: Coordinates system (r,θ,ϕ).

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• Third, the flux coordinate system 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 are defined in (A.136) and shown in Fig. 2.4

  . The vector is perpendicular to the flux surface, while 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.

  (2.1)

  Note that from now on, and all along this paper, the subscript 0 refers to quantities evaluated at the position of minimum B-field on a given flux-surface. The direction of evolution of s is counter-clockwise and the limits s_min and s_max are set at the position of maximum magnetic field

  (2.2)

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  Figure 2.4: Coordinates system (ψ,s,ϕ).

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• The system 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 ρ which coincides with the normalized radius on the horizontal Low Field Side (LFS) mid-plane. Indeed, in an axisymmetric system, using the functions R and Z, we define ρ as

  (2.3)

  with 0 ≤ ρ ≤ 1 by construction, and where R_max = R 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 2 - D outputs from the axisymmetric equilibrium code HELENA are given on the grid [?]. The system will be used from now on.