Poloidal angle used in tearing mode theory

Next, we discuss a new poloidal angle often used in the tearing mode theory, which is defined by

$$\chi = \theta - \frac{n}{m} \zeta .$$ (261)

Then it is ready to verify that the Jacobian of $(\psi, \chi, \zeta)$ coordinates is equal to that of $(\psi , \theta , \zeta )$ coordinates [proof: $(\mathcal{J}')^{- 1} = \nabla \psi \times \nabla \chi \cdot \nabla \zeta = \na... ... \zeta = \nabla \psi \times \nabla \theta \cdot \nabla \zeta =\mathcal{J}^{- 1}$]. The component of $\mathbf {B}$ along $\nabla \chi$ direction (a covariant component) is written

$$B^{(\chi)} \equiv \mathbf{B} \cdot \nabla \chi$$

$$= - \Psi' (\nabla \zeta \times \nabla \psi + q \nabla \psi \times \nabla \theta) \cdot \nabla (\theta - n \zeta / m)$$

$$= - \Psi' (\nabla \zeta \times \nabla \psi \cdot \nabla \theta) + \Psi' \frac{n}{m} (q \nabla \psi \times \nabla \theta \cdot \nabla \zeta)$$

$$= - \Psi' \mathcal{J}^{- 1} + \Psi' \frac{n}{m} q\mathcal{J}^{- 1}$$

$$= \Psi' \mathcal{J}^{- 1} \left( q \frac{n}{m} - 1 \right) .$$ (262)

At the resonant surface $q = m / n$, Eq. (262) implies $B^{(\chi)} = 0$. On the other hand, the covariant component of $\mathbf {B}$ in $\theta$ direction is written

$$B^{(\theta)} \equiv \mathbf{B} \cdot \nabla \theta$$

$$= - \Psi' (\nabla \zeta \times \nabla \psi + q \nabla \psi \times \nabla \theta) \cdot \nabla \theta$$

$$= - \Psi' \nabla \zeta \times \nabla \psi \cdot \nabla \theta$$

$$= - \Psi' \mathcal{J}^{- 1} .$$ (263)

Using (263), equation (262) is written

$$B^{(\chi)} = B^{(\theta)} \left( 1 - \frac{n}{m} q \right) .$$ (264)

In the cylindrical limit, $\mathcal{J}$ is a constant independent of the poloidal angle. Then Eq. (263) indicates that $B^{(\theta)}$ is also a constant on a magnetic surface.