Spatial grid interpolation for gradient calculation
The first order drift kinetic equation requires to calculate (0) defined as
![]() | (5.315) |
where = RBT . 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 δp(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, δξ,l+1∕2,i+1∕2,j+1(0) = δξ,l-1∕2,i+1∕2,j+1(0), and δξ,l+1∕2,i+1∕2,j+1(0) = δξ,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) |
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) |
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 Bt 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
,
![]() ![]() | = -![]() ![]() ![]() | ||
×![]() ![]() ![]() | (5.390) |
where
![]() | (5.391) |
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.
From the expressions given in Sec. 4.3.8, the components of the tensor pRF
are
![]() | (5.394) |
and the components of the vector pRF
are
![]() | (5.395) |
where quasilinear diffusion coefficients b,n,l+1∕2,i+1∕2,j+1∕2RF(0)D and
b,n,l+1∕2,i+1∕2,j+1∕2RF(0)F
are
![]() | = ![]() ![]() ![]() ![]() ![]() | ||
× | Db,n,0,l+12RF,θb
H![]() ![]() | ||
×![]() ![]() ![]() | (5.396) |
![]() | = ![]() ![]() ![]() ![]() ![]() | ||
×![]() ![]() ![]() | |||
×![]() ![]() ![]() | (5.397) |
Here Db,n,0,l+12RF,θb, N
∥res,l+1∕2,i+1∕2,j+1∕2θb, Θ
k,θb,l+1∕2,i+1∕2,j+1∕2b, 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 Ω0,l+1∕2,i+1∕2, rθb,l+1∕2, Bl+1∕2θb,
BP,l+1∕2θb, 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 →