Numerical results of MHD continua

The eigenfrequency of Eq. (213), $\overline{\omega}^2$, as a function of the radial coordinate gives the continua for the equilibrium. It can be proved analytically that the eigenfrequency of Eq. (213), $\overline{\omega}^2$, is a real number (I do not prove this). Making use of this fact, we know that a crude method of finding the eigenvalue of Eq. (213) is to find the zero points of the real part of the determinant of $\mathbf{E}$. Since, in this case, both the independent variables and the value of the function are real, the zero points can be found by using a simple one-dimension root finder. This method was adopted in the older version of GTAW (bisection method is used to find roots). In the latest version of GTAW, as mentioned above, the generalized eigenvalue problem in Eq. (213) is solved numerically by using the zggev subroutine in Lapack library. (The eigenvalue problem is solved without the assumption that $\overline{\omega}^2$ is real number. The eigenvalue $\overline{\omega}^2$ obtained from the routine is very close to a real number, which is consistent with the analytical conclusion that the eigenvalue $\overline{\omega}^2$ must be a real number.)

Figure 16 plots the eigenfrequency of Eq. (213) as a function of the radial coordinate $\overline {\Psi }$. The result is calculated in the slow sound approximation, thus giving only the Alfven branch of the continua. Also plotted in Fig. 16 are the Alfven continua in the cylindrical limit. As shown in Fig. 16, the Alfven continua in toroidal geometry do not intersect each other, thus forming gaps at the locations where the cylindrical Alfven continua intersect each other.

The first gap, which is formed due to the coupling of sound wave and Alfven wave, starts from zero frequency. This gap is called BAE gap since beta-induced Alfven eigenmode (BAE) can exist in this gap. The second gap is called TAE gap, which is formed mainly due to the coupling of $m$ and $m + 1$ poloidal harmonics. The third gap is called EAE gap, which is formed mainly due to the coupling of $m$ and $m + 2$ poloidal harmonics. The fourth gap is called NAE gap, which is formed due to the coupling of $m$ and $m + 3$ poloidal harmonics. A gap can be further divided into sub-gaps according to the two dominant poloidal harmonics that are involved in forming the gap. For example, a sub-gap of the TAE gap is the one that is formed mainly due to the coupling of $m = 1$ and $m = 2$ harmonics. For the ease of discussion, we call this sub-gap ``$(1, 2)$ sub-gap'', where the two numbers stand for the poloidal mode numbers. The frequency range of a sub-gap is defined by the frequency difference of the two extreme points on the continua. The radial range of the sub-gap can be defined as the radial region whose center is the location of one of the extreme points on the continua, width is the half width between the neighbor left and right extreme points.

Figure 16: $n = 1$ Alfven continua in toroidal geometry (red dots)(calculated in slow sound approximation) and in cylindrical geometry limit for $m = 0, 1, 2, 3, 4$, and $5$ (calculated by using Eq. (220)). The equilibrium used for this calculation is for EAST shot #38300 at 3.9s (G-eqdsk filename g038300.03900, which was provided by Dr. Guoqiang Li). (main ions are deuterium, impurity ions are assumed to be absent).

Figure 17 compares the continua of the full ideal MHD model with those of slow sound and zero $\beta$ approximations. The results indicate that the slow sound approximation eliminates the sound continua while keeps the Alfven continua almost unchanged. The zero $\beta$ approximation eliminates the BAE gap.

Figure 17: (a) full continua (b) slow sound approximation of the continua (c) zero $\beta$ approximation of the continua. Other parameters: toroidal mode number $n = 1$, the range of poloidal harmonics number is truncated within $[- 10 : 10]$. The equilibrium used for this calculation is for EAST shot 38300 at 3.9s.

(Numerical results indicate that the eigenvalue $\overline{\omega}^2$ is always grater than or equal to zero. Can this point be proved analytically?)

In order to verify the numerical convergence about the number of the poloidal harmonics included in the expansion, we compares the results obtained when the poloidal harmonic numbers are truncated in the range $[- 10, 10]$ and those obtained when the truncation region is $[- 15, 15]$. The results are plotted in Fig. 18, which shows that the two results agree with each other very well for the low order continua in the core region of the plasma. For continua in the edge region or higher order continua, there are some discrepancies between the two results. These discrepancies are due to that higher order poloidal harmonics are needed in evaluating the continua for those cases.

Figure 18: Comparison of the results obtained when the poloidal harmonic numbers are truncated in the range $[- 10, 10]$ (solid circles) and those obtained when the truncation region is $[- 15, 15]$ (cross marks). The equilibrium used for this calculation is for EAST shot #38300 at 3.9s.

The $n = 4$ Alfven continua are plotted in Fig. 19, which shows that there are more TAE gaps than those of the $n = 1$ case. The number of gaps is roughly given by $n (q_{\ensuremath{\operatorname{edge}}} - q_{\ensuremath{\operatorname{axis}}})$ for a monotonic $q$ profile[10].

Figure 19: $n = 4$ Alfven continua (in slow sound approximation). The poloidal harmonic numbers are truncated in the range $[- 15, 20]$. The equilibrium used for this calculation is for EAST shot 38300 at 3.9s. The corresponding Alfven continua in the cylindrical limit are also plotted.

Remarks: If, instead of the definition (), we define the $\langle \ldots \rangle_{m' m}$ operator as

$$\langle W \rangle_{m' m} = \frac{1}{2 \pi} \int_0^{2 \pi} W e^{i (m + m') \theta} d \theta,$$ (260)

then, can we still obtain correct results for the continuous spectrum? When I began to work on the calculation of the continuous spectrum, I noticed that the definition (), instead of (260), is adopted in Cheng's paper[3]. By intuition, I thought the definition (260) should work as well as (). Since the definition (260) is simpler than () (there is no additional minus in (260)), I prefer using the definition (260). This choice does not cause me trouble when I use symmetrical truncation of the poloidal harmonics. Trouble appears when I try using asymmetrical truncation of the poloidal harmonics. For example, when asymmetrical truncation of the poloidal harmonics is used (e.g. poloidal harmonics in the range $[- 10, 15]$), it is easy to verify analytically that the determinant of the resulting matrix for the case $\gamma = 0$ is zero for any values of $\omega^2$. It is obvious that this will not give correct results for the continuous spectrum. It took me two days to find out the definition in () must be used in order to deal with asymmetrical truncation of poloidal harmonics. In summary, the advantage of using the definition () over (260), is that the former can deal with the asymmetrical truncation of the poloidal harmonics, while the latter is limited to the case of symmetrical truncation.