Approximate central frequency and radial location of continua gap

In the cylindrical geometry, continua with different poloidal mode numbers will intersect each other, as shown in Fig. 1. Next, we calculate the radial location of the intersecting point of two continua with poloidal mode number $m$ and $m + 1$, respectively. In the intersecting point, we have

$$\omega_{A m}^2 = \omega_{A m + 1}^2,$$ (237)

i.e.

$$k_{\parallel m}^2 V_A^2 = k_{\parallel m + 1}^2 V_A^2$$ (238)

which gives

$$k_{\parallel m} = k_{\parallel m + 1},$$ (239)

or

$$k_{\parallel m} = - k_{\parallel m + 1}$$ (240)

Inspecting the expression for $k_{\parallel }$ in Eq. (323), we know that only the case in Eq. (240) is possible, which gives

$$\frac{m - n q}{q R_0} = - \frac{(m + 1) - n q}{q R_0},$$ (241)

which further reduces to

$$q = \frac{2 m + 1}{2 n} \equiv q_{\ensuremath{\operatorname{gap}}} .$$ (242)

The above equation determines the radial location where the $m$ continuum intersect the $m + 1$ continuum. Note that a mode with two poloidal modes has two corresponding resonant surfaces. For the case where the mode has $m$ and $m + 1$ poloidal harmonics, the resonant surfaces are respectively $q = m / n$ and $q = (m + 1) / n$. Note that the value of $q$ given in Eq. (242) is between the above two values.

In toroidal geometry, the different poloidal modes are coupled, and the continuum will ``reconnect'' to form a gap in the vicinity of the original intersecting point, as shown in Fig. . Therefore the original intersecting point, Eq. (242), gives the approximate location of the gap. Furthermore, using Eq. (242), we can determine the frequency of the intersecting point, which is given by

$$\omega^2 = k_{\parallel m}^2 V_A^2 = \left( \frac{n}{2 m + 1} \right)^2 \frac{V_A^2}{R_0^2},$$ (243)

which can be further written

$$\omega = \frac{1}{2 q_{\ensuremath{\operatorname{gap}}}} \frac{V_A}{R_0} .$$ (244)

According to the same reasoning given in the above, Eq. (244) is an approximation to the center frequency of the TAE gap. The frequency and the location given above are also an approximation to the frequency and location of the TAE modes that lie in the gap.

For the ellipticity-induced gap (EAE gap), which is formed due to the coupling of $m$ and $m + 2$ harmonics, the location is approximately determined by

$$k_{\parallel m} = - k_{\parallel m + 2},$$ (245)

which gives

$$q_{\ensuremath{\operatorname{gap}}} = \frac{m + 1}{n},$$ (246)

and the approximate center angular frequency is

$$\omega^2 = k_{\parallel}^2 V_A^2 = \left( \frac{n}{m + 1} \right)^2 \frac{V_A^2}{R_0^2},$$ (247)

which can be written as

$$\omega = \frac{2}{2 q_{\ensuremath{\operatorname{gap}}}} \frac{V_A}{R_0} .$$

Generally, for the gap formed due to the coupling of $m$ and $m + \Delta$ harmonics, we have

$$k_{\parallel m} = - k_{\parallel m + \Delta},$$

which gives

$$q_{\ensuremath{\operatorname{gap}}} = \frac{2 m + \Delta}{2 n},$$ (248)

and

$$\omega^2 = k_{\parallel}^2 V_A^2 = \left( \frac{\Delta n}{2 m + \Delta} \right)^2 \frac{V_A^2}{R_0^2} .$$ (249)

Equation (249) can also be written as

$$\omega = \frac{\Delta}{2 q_{\ensuremath{\operatorname{gap}}}} \frac{V_A}{R_0} .$$ (250)