Form of operator $B \cdot \nabla$ in $(\psi , \theta , \zeta )$ coordinates

The usefulness of the contravariant form [Eq. (250] of the magnetic field lies in that it allows a simple form of $\mathbf{B} \cdot \nabla$ operator in a coordinate system. (The operator $\mathbf{B}_0 \cdot \nabla$ is usually called magnetic differential operator.) In $(\psi , \theta , \zeta )$ coordinate system, by using the contravariant form Eq. (250), the operator is written as

$$\mathbf{B} \cdot \nabla f = - \Psi' (\nabla \zeta \times \nabla \psi) \cdot \nabla f (\psi, \... ...\Psi' q (\nabla \psi \times \nabla \theta) \cdot \nabla f (\psi, \theta, \zeta)$$

$$= - \Psi' \mathcal{J}^{- 1} \left( \frac{\partial}{\partial \theta} + q \frac{\partial}{\partial \zeta} \right) f.$$ (255)

Next, consider the solution of the magnetic differential equation, which is given by

$$\mathbf{B} \cdot \nabla f = h.$$ (256)

where $h = h (\psi, \theta, \zeta)$ is some known function. Using Eq. (255), the magnetic differential equation is written as

$$\left( \frac{\partial}{\partial \theta} + q (\psi) \frac{\partial... ...rtial \zeta} \right) f = - \frac{1}{\Psi'} \mathcal{J}h (\psi, \theta, \zeta) .$$ (257)

Note that the coefficients before the two partial derivatives of the above equation are all independent of $\theta$ and $\zeta$. This indicates that different Fourier harmonics in $\theta$ and $\zeta$ are decoupled. As a result of this fact, if $f$ and the right-hand side of the above equation are Fourier expanded respectively as

$$f (\psi, \theta, \zeta) = \sum_{m, n} f_{m n} (\psi) e^{i (m \theta - n \zeta)},$$ (258)

(note that, following the convention adopted in tokamak literature[6], the Fourier harmonics are chosen to be $e^{i (m \theta - n \zeta)}$, instead of $e^{i (m \theta + n \zeta)}$), and

$$- \frac{1}{\Psi'} \mathcal{J}h (\psi, \theta, \zeta) = \sum_{m, n} \gamma_{m n} (\psi) e^{i (m \theta - n \zeta)},$$ (259)

then Eq. (257) can be readily solved to give

$$f_{m n} = \frac{\gamma_{m n}}{i [m - n q]} .$$ (260)

The usefulness of the straight line magnetic coordinates $(\psi , \theta , \zeta )$ lies in that, as mentioned in the above, it makes the coefficients before the two partial derivatives both independent of $\theta$ and $\zeta$, thus, allowing a simple solution to the magnetic differential equation.