Expression of axisymmetric equilibrium magnetic field

A general axisymmetric magnetic field, which is not necessarily an equilibrium magnetic field, is given by Eq. (8), i.e.,

$$\mathbf{B}= \nabla \Psi \times \nabla \phi + g \nabla \phi,$$ (54)

For the above axisymmetric magnetic field to be consistent with the force balance equation (43), additional requirements for $\Psi$ and $g$ are needed, i.e., $\Psi$ is restricted by the GS equation and $g$ must be a function of only $\Psi$. Thus, an axisymmetric equilibrium magnetic field is determined by specifying two functions, $\Psi = \Psi (R, Z)$ and $g = g (\Psi)$. The function $\Psi$ is determined by solving the GS equation with specified boundary condition. Note that the GS equation contains two free functions, $P (\Psi)$ and $g (\Psi)$, both of which must be specified before the GS equation can be solved. (The way to specify the function $g (\Psi)$ to obtain desired global properties, such as total toroidal current and safety factor, will be discussed later.) This feature makes the process of solving GS equation different from solving usual partial differential equations where only boundary conditions are needed to be specified. For most cases, the terms on the right-hand side of the GS equation are nonlinear about $\Psi$ and thus the GS equation is a two-dimensional (about the cylindrical coordinates $R$ and $Z$) nonlinear partial differential equation for $\Psi$.

Equation (54) indicates that the poloidal and toroidal equilibrium magnetic field can be written as

$$\mathbf{B}_p = \nabla \Psi \times \nabla \phi,$$ (55)

and

$$\mathbf{B}_{\phi} = g \nabla \phi,$$ (56)

respectively.