Example Tokamak Discharge

Figure: 14.2 Experimental normalized pressure ( $\hat{p}= \mu_0\,p/B_0^{\,2}$) and safety-factor profiles in KSTAR discharge #18594 at time $t=6450$ ms. The points in the right-hand panel show the locations of the $n=1$ rational surfaces.

Table: 14.1 Properties of $n=1$ rational surfaces in KSTAR discharge #18594 at time $t=6450$ ms. Here, $s(r)$ is defined in Equation (14.129). Only those surfaces with ${\mit \Psi }_N(r_k)<0.995$ are listed.
$k$ $m_k$ $r_k/r_{100}$ ${\mit\Psi}_N(r_k)$ $s(r_k)$ $g(r_k)$
1 2 $0.692$ $0.669$ $1.51$ $0.998$
2 3 $0.852$ $0.862$ $2.53$ $0.999$
3 4 $0.938$ $0.947$ $3.06$ $0.999$
4 5 $0.993$ $0.993$ $1.27$ $0.999$

KSTAR discharge #18594 [29] is a typical H-mode [34] discharge in a mid-sized tokamak. Figure 14.1 shows the equilibrium magnetic flux-surfaces in this discharge at time $t=6450$ ms, at which time $B_0=1.79$ T, $R_0 = 1.80$ m, $r_{100}=0.596$ m, and $q_{95}=4.04$. Figures 14.2 and 14.3 show the corresponding pressure, safety-factor, and $g$ profiles [29]. Of course, the equilibrium flux-surfaces shown in Figure 14.1, combined with the $p$ and $g$ profiles specified in Figures 14.2 and 14.3, constitute a solution of the Grad-Shafranov equation, (14.33). Note that the safety-factor becomes infinite at the edge of the plasma, due to the presence of a magnetic X-point on the bounding magnetic flux-surface. (See Figure 14.1.) In principle, there are an infinite number of $n=1$ rational surfaces lying within the plasma. However, if we truncate the plasma at ${\mit\Psi}_N=0.995$ (i.e., at the magnetic flux-surface that contains 99.5% of the poloidal magnetic flux contained by the last closed flux-surface), which is the standard approach, then there are only four such surfaces. The properties of these surfaces are listed in Table 14.1.

Table 14.2 specifies the elements of the $n=1$ E-matrix in KSTAR discharge #18594 at time $t=6450$ ms, calculate according to the procedure set out in Section 14.12. As expected, it can be seen that the matrix is Hermitian. Moreover, all of the diagonal elements of the matrix are negative, which indicates that the $n=1$ tearing mode are all classically stable. (This is not surprising because the classical drive is absent from our calculation.) Finally, the off-diagonal elements of the matrix are all substantial, indicating that there is significant coupling between different poloidal harmonics.

Table: 14.2 Elements of $n=1$ E-matrix in KSTAR discharge #18594 at time $t=6450$ ms.
$k$ $k'$ ${\rm Re}(E_{kk'})$ ${\rm Im}(E_{kk'})$ $k$ $k'$ ${\rm Re}(E_{kk'})$ ${\rm Im}(E_{kk'})$
1 1 $-4.81$ $+0.00$ 3 1 $+0.589$ $+1.22$
1 2 $-0.133$ $+1.41$ 3 2 $+2.04$ $-3.19$
1 3 $+0.589$ $-1.22$ 3 3 $-15.3$ $+0.00$
1 4 $+1.02$ $-0.646$ 3 4 $+8.13$ $+2.84$
2 1 $-0.133$ $-1.41$ 4 1 $+1.02$ $+0.646$
2 2 $-7.92$ $+0.00$ 4 2 $-0.726$ $+0.399$
2 3 $+2.04$ $+3.19$ 4 3 $+8.13$ $-2.84$
2 4 $-0.726$ $-0.399$ 4 4 $-17.4$ $+0.00$

In order to determine the effective tearing stability index for an $n=1$ toroidal tearing mode that reconnects magnetic flux at a particular rational surface in the plasma, let us assume that the responses of the other rational surfaces exhibit strong shielding. This assumption can be justified a posteriori. Our assumption implies that very little magnetic reconnection is driven at the other rational surfaces. In this case, it is reasonable to calculate the responses of these surfaces using linear theory. (The linear approximation is valid as long as the widths of the magnetic island chains driven at the other rational surfaces are less than the corresponding linear layer widths. See Section 5.16.)

According to the analysis of Chapter 5, the linear response of the $k$th rational surface is characterized by

$\displaystyle {\mit\Delta\Psi}_k = S_k^{\,1/3}\,\skew{5}\hat{\mit\Delta}(Q_k,Q_...
...k},Q_{e\,k}, Q_{i\,k}, \tau_k,D_k, P_{\varphi\,k}, P_{\perp\,k})\,{\mit\Psi}_k,$ (14.123)


$\displaystyle S_k$ $\displaystyle = \frac{\tau_{R\,k}}{\tau_{H\,k}},$ (14.124)
$\displaystyle \tau_{R\,k}$ $\displaystyle = \mu_0\,r_k^{\,2}\,\sigma_{ee}(r_k)\,{\cal Q}_{ee}(r_k),$ (14.125)
$\displaystyle \sigma_{ee}(r)$ $\displaystyle = \frac{n_e(r)\,e^2\,\tau_{ee}(r)}{m_e},$ (14.126)
$\displaystyle \tau_{H\,k}$ $\displaystyle = \frac{R_0}{B_0\,g(r_k)}\,\frac{\sqrt{\mu_0\,\rho(r_k)}}{n\,s(r_k)},$ (14.127)
$\displaystyle \rho(r)$ $\displaystyle = m_i\,n_i + m_I\,n_I,$ (14.128)
$\displaystyle s(r)$ $\displaystyle = \frac{d\ln q}{d\ln r},$ (14.129)
$\displaystyle Q_k$ $\displaystyle = S_k^{\,1/3}\,\omega\,\tau_{H\,k},$ (14.130)
$\displaystyle Q_{E\,k}$ $\displaystyle = -S_k^{\,1/3}\,n\,{\mit\Omega}_E(r_k)\,\tau_{H\,k},$ (14.131)
$\displaystyle Q_{e\,k}$ $\displaystyle = -S_k^{\,1/3}\,n\,{\mit\Omega}_{\ast\,e}(r_k)\,\tau_{H\,k},$ (14.132)
$\displaystyle Q_{i\,k}$ $\displaystyle = -S_k^{\,1/3}\,n\,{\mit\Omega}_{\ast\,i}(r_k)\,\tau_{H\,k},$ (14.133)
$\displaystyle \tau_k$ $\displaystyle = -\frac{ {\mit\Omega}_{\ast\,e}(r_k)}{{\mit\Omega}_{\ast\,i}(r_k)},$ (14.134)
$\displaystyle D_k$ $\displaystyle = S_k^{\,1/3}\left(\frac{\tau_k}{1+\tau_k}\right)^{1/2}\,\frac{d_{\beta}(r_k)}{r_k},$ (14.135)
$\displaystyle d_\beta(r)$ $\displaystyle = \frac{\sqrt{(5/3)\,m_i\,[T_e+(n_i/n_e)\,T_i+(n_I/n_e)\,T_I]}}{e\,B_0\,g},$ (14.136)
$\displaystyle P_{\varphi\,k}$ $\displaystyle = \frac{\tau_{R\,k}}{\tau_{\varphi\,k}},$ (14.137)
$\displaystyle P_{\perp\,k}$ $\displaystyle = \frac{\tau_{R\,k}}{\tau_{\perp\,k}},$ (14.138)
$\displaystyle \tau_{\varphi\,k}$ $\displaystyle = \frac{r_k^{\,2}}{{\mit\Xi}_{\perp\,i}(r_k)},$ (14.139)
$\displaystyle \tau_{\perp\,k}$ $\displaystyle = \frac{r_k^{\,2}}{D_{\perp}(r_k)}.$ (14.140)

Here, $S_k$ is the Lundquist number, $\tau_{R\,k}$ the resistive diffusion time [see Equation (5.49)], and $\tau_{H\,k}$ the hydrodynamic time [see Equation (5.43)], at the $k$th rational surface. Moreover, $n_e(r)$, $n_i(r)$, $n_I(r)$ are the equilibrium number density profiles of the electrons, majority ions, and impurity ions, respectively, whereas $T_e(r)$, $T_i(r)$, and $T_I(r)$ are the corresponding temperature profiles, and $m_e$, $m_i$, and $m_I$ the corresponding species masses. Furthermore, $\omega$ is the real frequency of the tearing mode in the laboratory frame, whereas the E-cross-B, electron diamagnetic, and majority ion diamagnetic frequency profiles, ${\mit\Omega}_E(r)$, ${\mit\Omega}_e(r)$, and ${\mit\Omega}_i(r)$, respectively, are defined in Equations (A.77) and (A.78). The electron-electron collision time, $\tau_{ee}(r)$, is specified in Equation (A.23), and $e$ is the magnitude of the electron charge. The dimensionless function ${\cal Q}_{ee}(r)$, defined in Equation (A.81), specifies the reduction in the plasma electrical conductivity due to the presence of impurity ions and trapped particles. (See Section 2.20.) In addition, $\tau_{\varphi\,k}$ and $\tau_{\perp\,k}$ are the toroidal momentum confinement time [see Equation (5.50)] and the particle confinement time, respectively, at the $k$th rational surface, whereas ${\mit\Xi}_{\perp\,i}(r)$ and $D_\perp(r)$ are the ion perpendicular momentum diffusivity and perpendicular particle diffusivity profiles.

Figure: 14.3 Experimental electron number density, electron temperature, majority ion number density, majority ion temperature, E-cross-B frequency, and $g$ profiles in KSTAR discharge #18594 at time $t=6450$ ms. The vertical dotted lines show the locations of the $n=1$ rational surfaces.

Equation (14.123) states that the linear response of the $k$th rational surface to a magnetic perturbation generated at another rational surface is governed by nine dimensionless parameters. These parameters are the Lundquist number, $S_k$, the normalized mode frequency, $Q_k$, the normalized E-cross-B frequency, $Q_{E\,k}$, the normalized electron diamagnetic frequency, $Q_{e\,k}$, the normalized ion diamagnetic frequency, $Q_{i\,k}$, the pressure gradient ratio parameter, $\tau_k$, the semi-collisional parameter, $D_k$, and the two magnetic Prandtl numbers, $P_{\varphi\,k}$ and $P_{\perp\,k}$. (Note that these parameters are called $Q$, $Q_E$, $Q_e$, $Q_i$, $\tau $, $D$, $P_{\varphi}$, and $P_\perp$, respectively, in Chapter 5.) The dimensionless layer response index, $\skew{5}\hat{\mit\Delta}_k$, can be calculated numerically as a function of these nine parameters by solving the Riccati differential equation, (5.121), subject to the boundary conditions (5.122) and (5.123).

Table: 14.3 $n=1$ resistive layer parameters in KSTAR discharge #18594 at time $t=6450$ ms.
$k$ $m_k$ $\tau_{H\,k}$ $S_k$ $\tau_k$ $P_{\perp\,k}$ $P_{\varphi\,k}$ $D_k$ $Q_{E\,k}$ $Q_{e\,k}$ $Q_{i\,k}$
1 2 $2.41\times 10^{-7}$ $1.52\times 10^{7}$ 1.51 8.05 40.24 2.41 $-3.85$ 1.11 -0.735
2 3 $1.34\times 10^{-7}$ $2.87\times 10^{7}$ 0.917 5.67 28.4 1.96 $-1.96$ 0.475 -0.518
3 4 $1.06\times 10^{-7}$ $3.98\times 10^{7}$ 1.00 4.91 24.5 1.86 $-0.78$ 1.78 -1.78
4 5 $1.30\times 10^{-8}$ $5.34\times 10^{7}$ 0.877 0.513 2.57 1.13 $+0.33$ 0.959 -1.09

Figure 14.3 shows the experimental number density, temperature, and E-cross-B frequency profiles in KSTAR discharge #18594 at time $t=6450$ ms [29]. The majority ions are deuterium, whereas the impurity ions are carbon-VI (i.e., $Z_I=6$). The majority ion and impurity ion number density profiles are calculated from the measured electron number density profile [see Equations (A.4) and (A.5)] on the assumption that the effective ion charge number, $Z_{\rm eff}$ [see Equation (A.3)], takes the value $2.0$ throughout the plasma. (This value is a best guess based on the measured stored energy.) The impurity ions are assumed to have the same temperature as the measured temperature of the majority ions. The E-cross-B frequency profile is deduced from the measured impurity ion toroidal angular velocity profile using the neoclassical theory outlined in Appendix A [15]. In particular, the impurity ion poloidal angular velocity profile is assumed to take its neoclassical value. (See Section A.7). Furthermore, the ion perpendicular momentum and perpendicular particle diffusivities are given the plausible values $1.0\,{\rm m}^2/{\rm s}$ and $0.2\,{\rm m}^2/{\rm s}$, respectively, throughout the plasma [34]. Finally, the values of the various $n=1$ resistive layer parameters, determined from the data shown in Figure 14.3, as well as the aforementioned assumptions, are given in Table 14.3.