Expression of $k_{\parallel }$ and $k_{\theta }$

Next, we derive the expression for the parallel and poloidal wave-number of a perturbation of the form

$$\delta A = \delta A_0 (\psi) \exp [i (m \theta - n \zeta - \omega t)],$$ (317)

where $(\psi, \theta, \zeta)$ are the flux coordinators, with $\theta$ and $\zeta$ being the generalized poloidal and toroidal angles. The parallel (to the local equilibrium magnetic field) wave vector, $k_{\parallel }$, is defined by

$$k_{\parallel} = \frac{\Delta \ensuremath{\operatorname{ph}}}{\Delta l},$$ (318)

where $\Delta \ensuremath{\operatorname{ph}}$ is the phase angle change of the perturbation when moving a distance of $\Delta l$ along the local equilibrium magnetic field. According to Eq. (317), the phase change can be written as

$$\Delta \ensuremath{\operatorname{ph}} = m \Delta \theta - n \Delta \zeta,$$ (319)

where $\Delta \zeta$ and $\Delta \theta$ are the change in the toroidal and poloidal angles when we move a distance of $\Delta l$ along the magnetic field. Use Eq. (319) in Eq. (318), giving

$$k_{\parallel} = \frac{m \Delta \theta - n \Delta \zeta}{\Delta l} .$$ (320)

Noting that the safety factor is given by

$$q = \frac{\Delta \zeta}{\Delta \theta}$$ (321)

(which is exact since we are using flux coordinator, in which magnetic field lines are straight on $(\theta, \zeta)$ plane), Eq. (320) is written as

$$k_{\parallel} = \frac{m / q - n}{\Delta l / \Delta \zeta} .$$ (322)

In the approximation of large aspect ratio, $\Delta l$ can be approximated by $\Delta l \approx R_0 \Delta \zeta$, where $R_0$ is the major radius of the magnetic axis. Using this, the above equation is written as

$$k_{\parallel} \approx \frac{m - n q}{q R_0} .$$ (323)

(Remarks: I should use the exact expression for $\Delta l$ to derive an exact expression for $k_{\parallel }$, I will do this later.) Equation (323) indicates that $k_{\parallel }$ is zero on the resonant surface.

Similarly, the component of the wave vector along the $\theta$ direction is written as

$$k_{\theta} = \frac{\Delta \ensuremath{\operatorname{ph}}}{\Delta l_p}$$

$$= \frac{m \Delta \theta - n \cdot 0}{\Delta l_p},$$ (324)

where $\Delta l_p$ is the poloidal arc length when the poloidal angle changes by $\Delta \theta$. If the equal-arc poloidal angle is used, then $\Delta l_p = L_p / (2 \pi) \Delta \theta$, where $L_p$ is the poloidal circumference of the magnetic surface. Using this, Eq. (324) is written as

$$k_{\theta} = \frac{m}{L_p / 2 \pi} .$$ (325)

If a circular flux surface is assumed, then the above equation is written as

$$k_{\theta} = \frac{m}{r},$$

where $r$ is the radius of the flux surface.