The toroidal elliptic operator in Eq. (389) can be written
| (393) |
where haβ is defined by Eq. (151), i.e.,
| (394) |
Next, we derive the finite difference form of the toroidal elliptic operator. The finite difference form of the term (hψψΨψ)ψ is written
| (396) |
The finite difference form of (h𝜃𝜃Ψ𝜃)𝜃 is written
| (398) |
The finite difference form of (hψ𝜃Ψ𝜃)ψ is written as
| (399) |
Approximating the value of Ψ at the grid centers by the average of the value of Ψ at the neighbor grid points, Eq. (399) is written as
| (400) |
where
| (401) |
Similarly, the finite difference form of (hψ𝜃Ψψ)𝜃 is written as
| (403) |
| (404) |
and
| (405) |
where the Jacobian
| (406) |
The partial derivatives, R𝜃, Rψ, Z𝜃, and Zψ, appearing in Eqs. (403)-(406) are calculated by using the central difference scheme. The values of hψψ, h𝜃𝜃, hψ𝜃 and 𝒥 at the middle points are approximated by the linear average of their values at the neighbor grid points.