Benchmark of GTAW code

To benchmark GTAW code, we use it to calculate the continua and gap modes of the Solovev equilibrium and compare the results with those given by NOVA code. The Solovev equilibrium used in the benchmark case is given by

$$\Psi = \frac{B_0}{2 R_0^2 \kappa_0 q_0} \left[ R^2 Z^2 + \frac{\kappa_0^2}{4} (R^2 - R_0^2)^2 \right],$$ (258)

$$p_0 = p_0 (0) - \frac{B_0 (\kappa_0^2 + 1)}{{\textmu}_0 R_0^2 \kappa_0 q_0} \Psi, g = g_0,$$ (259)

with $B_0 = 1 T$, $R_0 = 1 m, g_0 = 1 m T$, $\kappa_0 = 1.5$, $q_0 = 3$, and $p_0 (0) = 1.1751 \times 10^4 \ensuremath{\operatorname{Pa}}$. The flux surface with the minor radius being $0.3 m$ (corresponding to $\Psi = 2.04 \times 10^{- 2} T m^2$) is chosen as the boundary flux surface. Main plasma is taken to be Deuterium and the number density is taken to be uniform with $n_D = 2 \times 10^{19} m^{- 3}$. Figure 3 compares the Alfven continua calculated by NOVA and GTAW, which shows good agreement between them.

Figure 3: Comparison of the $n = 1$ Alfven continua calculated by NOVA and our code. The continua are calculated in the slow sound approximation[7] and the equilibrium used is the Solovev equilibrium given in Eqs. (258) and (259).

A gap mode with frequency $f = 297 \ensuremath {\operatorname {kHz}}$ is found in the NAE gap by both NOVA and GTAW. The poloidal mode numbers of the two dominant harmonics are $m = 2$ and $m = 5$, which is consistent with the fact that a NAE is formed due to the coupling between $m$ and $m + 3$ harmonics. Before comparing the radial structure of the poloidal harmonics given by the two codes, a discussion about the assumption adopted in NOVA is desirable. As is pointed out by Dr. Gorelenkov, NOVA at present is restricted to up-down symmetric equilibrium and, for this case, it can be shown that the amplitude of all the radial displacement can be transformed to real numbers. For this reason, NOVA use directly real numbers for the radial displacement in its calculation. In GTAW code, the amplitude of the poloidal harmonics of the radial displacement are complex numbers. The Solovev equilibrium used here is up-down symmetric and the results given by GTAW indicate the poloidal harmonics of the radial displacement can be transformed (by multiplying a constant such as $(1 - i)$) to real numbers. After transforming the radial displacement to real numbers, the results can be compared with those of NOVA. Figure 4 compares the radial structure of the dominant poloidal harmonics $(m = 2, 3, 4, 5)$ given by the two codes, which indicates the results given by the two codes agree with each other well.

Figure 4: The dominant poloidal harmonics ( $m = 2, 3, 4, 5$) of a $n = 1$ NAE as a function of the radial coordinate. The solid lines are the results of GTAW while the dashed lines are those of NOVA. The corresponding poloidal mode numbers are indicated in the figure. The frequency of the mode $f = 297 \ensuremath {\operatorname {kHz}}$. The equilibrium is given by Eqs. (258) and (259).

Figure 5: Slow sound approximation of the $n = 1$ continua of the Solovev equilibrium given by Eqs. (258) and (259). Also plotted are the frequency of the NAE ( $f = 297 \ensuremath {\operatorname {kHz}}$) and the $m = 2$ and $m = 5$ continua in cylindrical limit.

Figure 6 plots the mode structure of the NAE on $\phi = 0$ plane, which shows that the mode has an anti-ballooning structure, i.e., the mode is stronger at the high-field side than at the low-field side.

Figure 6: Two dimension mode structure of the NAE in Fig. 4. The dashed line in the figure indicates the boundary magnetic surface and the small circle indicates the inner boundary used in the numerical calculation.

Figure 7: Real part (a), imaginary part (b), and absolute value of the amplitude (c) of the poloidal harmonics of a $n = 1$ TAE as a function of the radial coordinate. The frequency of the mode is $f = 93 \ensuremath {\operatorname {kHz}}$. The dominant poloidal harmonics are those with $m = 3$ and $m = 4$. The equilibrium is given by Eqs. (258) and (259).

Figure 8: Slow sound approximation of the continua of the Solovev equilibrium. Also plotted are the frequency of the TAE ( $f = 93 \ensuremath {\operatorname {kHz}}$) and the $m = 3$ and $m = 4$ continua in cylindrical limit. Toroidal mode number $n = 1$.

For the case that $p_0 (0) = 1.5 \times 10^4 \ensuremath{\operatorname{Pa}}$, a TAE with $f = 102 \ensuremath {\operatorname {kHz}}$ is found in the TAE gap. The radial dependence of the poloidal harmonics of the mode is plotted in Fig. 9. Figure 10 plots the frequency of the mode on the Alfven continua graphic.

Figure 9: Real part (a), imaginary part (b), and absolute value of the amplitude (c) of the poloidal harmonics of a $n = 1$ TAE as a function of the radial coordinate. The frequency of the mode is $f = 102 \ensuremath {\operatorname {kHz}}$. The dominant poloidal harmonics are those with $m = 3$ and $m = 4$. The equilibrium is given by Eqs. (258) and (259), (old)

Figure 10: Slow sound approximation of the continua of the Solovev equilibrium. Also plotted are the frequency of the TAE ( $f = 102 \ensuremath {\operatorname {kHz}}$) and the $m = 3$ and $m = 4$ continua in cylindrical limit. Toroidal mode number $n = 1$.(old)