3.7 Free boundary equilibrium problem

If the value of Ψ on the boundary of the computational box is unknown, how do you sovle the GS equation?

Suppose the boundary is a rectangle in (R,Z) plane. To numerically solve the GS equation within this boundary, we need the value of Ψ on the boundary. Therefore we need to adopt some initial guess, then solve the GS equation to get the value of Ψ within the computational box. Using the computed Ψ, we can calculate J_ϕ through Eq. (54). After this, all the current (current in the plasma and in the external coils) perpendicular to the poloidal plane is known, we can calculate the value of Ψ on the boundary of the box, Ψ_b, by using the Green function formulation:

(73)

Note that Ψ_b calculated this way usually differs from the initial guess of the value of Ψ on the boundary. Thus, we need to use the Ψ_b calculated this way as a new guess value of Ψ on the computational boundary and repeat the above procedures. The process is repeated until Ψ_b obtained in two successive iterations agrees with each other to a prescribed tolerance.

In solving the equilibrium problem, the current in external coils is given and known while the current distribution in the plasma is unknown. Therefore, solving the equilibrium problem actually corresponds to determining J_ϕ.