5.5 Up to first order term: the Drift Kinetic equation

5.5.1 Grid interpolation

Spatial grid interpolation for gradient calculation The first order drift kinetic equation requires to calculate ⁽⁰⁾ defined as

(5.315)

where = RB_T . Here all quantities are determined on the poloidal position where the magnetic field B is minimum.

The spatial grid being non-uniform, radial derivative requires a specific treatment, for an accurate determination. Let ψ_-,ψ et ψ₊ the three neighbor radial positions where are calculated the distribution function f₀⁽⁰⁾ with Δψ_- = ψ_-- ψ, Δψ₊ = ψ₊ - ψ and Δψ = ψ₊ - ψ_-. A parabolic interpolation of the form y = aψ² + bψ + c is used for calculating the radial derivative dy∕dψ = 2aψ + b. The coefficients a,b and c being determined by values y_-,y and y₊ at grid points ψ_-,ψ and ψ₊, one can easily show that

(5.316)

where V is a Van der Monde matrix of order 3,

(5.317)

whose determinant is simply

(5.318)

Therefore

(5.319)

and applying this result for ⁽⁰⁾,

(5.322)

(5.323)

and

(5.324)

Momentum grid interpolation As indicated in Sec. 3.5.5, it is possible to keep the conservative form for the first-order drift kinetic equation. The main advantage is that the numerical differencing technique already used for the zero-order Fokker-Planck equation may be also employed for determining the numerical solution of this equation. The determination of the momentum _{l+1∕2,i+1∕2,j+1∕2}^(k) and pitch-angle _{l+1∕2,i+1∕2,j+1∕2}^(k) derivatives requires, as for f₀, interpolation techniques in order to evaluate ⁽⁰⁾ on flux grid at the radial position l + 1∕2. By analogy, one have to determine ⁽⁰⁾ on the following grid points

(5.325)

(5.326)

(5.327)

and

(5.328)

Using the weighting factors δ_p⁽⁰⁾ introduced for f₀, as shown in Sec. 5.4.3 which both are functions of ψ, one obtains for the grid point

Reordering coefficients, one obtains,

This double difference makes coefficient _{l+1∕2,i+1,j+1∕2} is second order correction, that is almost negligible when spatial gradients are weak. However, for strong gradients, this correction must be, in principle, considered.

A similar expression is obtained for the grid point , by replacing i + 1 → i in all above relations.

Finally, a similar approach may be used for the pitch-angle grid interpolation. For grid points ,

However, since by definition, δ_{ξ,l+1∕2,i+1∕2,j+1}⁽⁰⁾ = δ_{ξ,l-1∕2,i+1∕2,j+1}⁽⁰⁾, and δ_{ξ,l+1∕2,i+1∕2,j+1}⁽⁰⁾ = δ_{ξ,l+3∕2,i+1∕2,j+1}⁽⁰⁾, it turns out that

(5.339)

and a similar result is obtained for grid points , expressions are

(5.341)

5.5.2 Momentum dynamics

The starting point of the discrete representation of the first order drift kinetic equation is the conservative relation

(5.346)

(5.347)

(5.348)

(5.349)

(5.352)

(5.353)

Discrete expressions of the partial derivatives are,

(5.354)

(5.355)

(5.356)

(5.357)

(5.358)

(5.359)

and cross-derivatives

As for the zero-order Fokker-Planck equation, other derivatives in discrete form become

(5.361)

Since the distribution function is defined on the half grid, while fluxes on the full grid, it is necessary to interpolate, because in some derivatives, values of are taken on the full grid. As discussed in the previous section, interpolation procedure is more complex for than for f₀. Therefore, for terms proportional to _ξξ and _ξ

Gathering all terms in a matrix form

For the determination of matrix _ψ, it is useful to start from terms ₁ and ₂ that contain _{l+1∕2,i,j+1∕2}^(k+1) and _{l+1∕2,i+1,j+1∕2}^(k+1). coefficients. Because of the grid interpolation

(5.381)

(5.382)

(5.383)

5.5.3 Discrete description of physical processes

Collisions

Since first order drift kinetic terms may be expressed in a conservative form as for the zero order Fokker-Planck equation, the determination of the matrix elements is therefore straightforward. One obtains

(5.387)

and

(5.388)

where coefficients of the collision operator A,F and B_t are the same as defined for the zero order bounce averaged Fokker-Planck equation, in Sec. 5.4.4.

Concerning the first order Legendre correction for electron-electron collisions that ensures momentum conservation, one must calculate

on the distribution function grid, where

(5.389)

as shwon in Sec. 4.1.7.

Here, the collision integral has a similar form as for the zero order Fokker-Planck equation except that f₀ is just replaced by ₀ in all corresponding terms. Therefore, by definition, at the iteration number ,

l+1∕2,i+1∕2,j+1∕2	= -H
	×l+1∕2,i+1∕2,j+1∕2	(5.390)

where

(5.391)

Ohmic electric field

According to the bounce averaged expression,

(5.392)

and

(5.393)

where E_∥0,l+1∕2 is the parallel component of the Ohmic electric field along the magnetic field line direction normalized to the Dreicer field taken at the poloidal position where the magnetic field B is minimum, as explained in Sec.4.2.

Radio-frequency waves

From the expressions given in Sec. 4.3.8, the components of the tensor _p^RF are

(5.394)

and the components of the vector _p^RF are

(5.395)

where quasilinear diffusion coefficients _{b,n,l+1∕2,i+1∕2,j+1∕2}^RF(0)D and _{b,n,l+1∕2,i+1∕2,j+1∕2}^RF(0)F are

b,n,l+1∕2,i+1∕2,j+1∕2RF(0)D	=
×	Db,n,0,l+12RF,θb HH
	×T δ2	(5.396)

b,n,l+1∕2,i+1∕2,j+1∕2RF(0)F	=
	×Db,n,0,l+12RF,θb HH
	×T δ2	(5.397)

Here D_b,n,0,l+12^RF,θ_b, N _{∥res,l+1∕2,i+1∕2,j+1∕2}^θ_b, Θ _{k,θb,l+1∕2,i+1∕2,j+1∕2}^b, are similar to the case of the quasilinear diffusion coefficient for the zero-order Fokker-Planck equation, and are therefore given in Sec. 5.4.6. The definition of coefficients Ω_{0,l+1∕2,i+1∕2}, r_θb,l+1∕2, B_l+1∕2^θ_b, B_P,l+1∕2^θ_b, ξ _θb,l+1∕2² and Ψ _θb,l+1∕2 are also given in the same section.

All coefficients corresponding to different indexes may be obtained readily by performing the adequate index transformation, i + 1∕2 → and j + 1∕2 →