Screening of resonant magnetic perturbation penetration by flows in tokamak plasmas based on two-fluid model

Consider a low density ohmically heated tokamak discharge with electron density $n_{e}\approx 2\times 10^{19}\rm\ m^{-3}$, toroidal magnetic field $B_{0}=2$ T and inverse aspect ratio $\varepsilon=a/R_{0}=0.5\rm m/2\rm m=0.25$. This will lead to the Alfvén speed $v_{\rm a}\approx 6.9\times 10^{6}\rm\ m/s$ and Alfvén time $\tau_{\rm a}\approx 7.24\times 10^{-8}\rm\ s$. The corresponding Alfvén frequency is $\omega_{\rm a}\approx 1.38\times 10^{7}\rm\ Hz$. Otherwise stated, other plasma parameters are set as follow, $\tau=1$, $\beta_{e}=0.01$, $\eta=10^{-6}$, $\nu=10^{-7}$, $\mu=10^{-6}$, $\chi_{\parallel}=10$ and $\chi_{\perp}=10^{-6}$. The radial mesh number is set as $\rm N_{r}=2048$. In this work, the nonlinear simulations only include single helicity perturbations with higher harmonics (m/n=3/2, 0 $\leq$ m $\leq$ 18), in addition to the changes in the equilibrium quantities (m/n=0/0 component). To simulate the mode penetration process, a linearly stable equilibrium safety factor $q$ profile and the normalized plasma pressure $p$ profile are given in figure 1, with the $q=3/2$ resonant surface located at $r=0.402$.

To begin with, the role of seed island width on the onset of NTM is verified to ensure the neoclassical current effect implemented properly. The seed island width is considered by the initial magnetic perturbation in a form of $\widetilde{\psi}_{t=0}=\psi_{00}(1-r)^{2}$. The nonlinear evolution of magnetic island width for different initial magnetic perturbations are presented in the left panel of figure 2. The solid traces are for bootstrap current fraction $f_{b}=0.3$ (NTM) and dotted one for $f_{b}=0.0$ (TM). For the classical TM, even a very large seed island width is given, the island width still fades with time, illustrating that the TM in this q profile is linearly stable. Taking the bootstrap current into consideration, it is seen that there is a threshold for the mode to grow, manifesting that the nonlinearly unstable NTM is triggered for a larger seed island width. In experiments, the RMP coils are commonly used to seed a magnetic island. Then the onset of NTM by RMP is tested. The RMP is turned on from the very beginning with the amplitude of $\delta B_{r}/B_{0}=3.75\times 10^{-5}$. In figure 2 (right), in the presence of RMP, the nonlinear evolution of island width for $f_{b}=0.3$ is shown. After applying the RMP, the originally stable TM can be forced driven unstable, as the island width grows even without any seed island. Then the RMP is turned off at different time when island width grows to a moderate magnitude, to testify if it is the driven reconnection or the onset of NTM. It turns out that, if the RMP is turned off before the island width is sufficiently large, the magnetic island still could not grow spontaneously. It confirms again that the existence of the critical island width for triggering the NTM, which is consistent with the theory.

Based on the above results, effects of different RMP amplitudes are then scanned for $f_{b}=0.0$ and $f_{b}=0.3$ with zero rotation (electric drift $\omega_{\rm E}$ and diamagnetic drift $\omega_{\rm dia}$ are both zero). For $f_{b}=0.0$ (figure 3 left), the saturated island width is positively related to the RMP amplitude, a typical driven reconnection. However, for $f_{b}=0.3$ (figure 3 right), a mode penetration like phenomenon can be observed. If the RMP amplitude is relatively small, the magnetic island is going to saturate at a very small magnitude. Once the RMP amplitude is sufficiently large, the final island width would be very large and keep almost the same even further increasing the RMP amplitude. This phenomenon is consistent with the “enduring mystery in experiments” reported by Hu et al, i.e. non-zero penetration threshold at zero plasma natural frequency[33, 34]. On the other hand, this phenomenon is different from the so-called mode penetration, and we will demonstrate it next.

Corresponding to figure 3 (right), the saturated magnetic island width is shown as a function of RMP amplitude in figure 4 (left). As the RMP amplitude increases, the saturated island width increases slowly first. Once the RMP amplitude exceeds a threshold, there is a jump of the saturated island width. This process can be divided into two parts, i.e. the driven reconnection phase for smaller RMPs and the onset of NTM for large RMPs. Next, including the effect of equilibrium ${\bf E\times B}$ flow, scan over the RMP amplitude is performed again. The equilibrium ${\bf E\times B}$ flow is considered by a poloidal momentum source $\Omega_{s}(r)=1/r*d\phi_{0}/dr$. Here, $\delta$ is set to be zero, so only the ${\bf E\times B}$ flow is considered. For a 6.4 kHz plasma rotation, the saturated island width versus the RMP amplitude is presented in figure 4 (right). This is a typical mode penetration case, as the RMP amplitude increases above a threshold, the island width boosts to a large magnitude. There is one main distinction between the left and right panel of figure 4, even though they look similar. That is, below the threshold, the island width barely changes for the case with plasma rotation but slowly increases for the case without rotation. The increase regime for the case without rotation is due to the forced reconnection driven by the externally applied RMP. However, the regime below the threshold for the case with rotation is because of the screening effect of the plasma flow. The corresponding eigenmode structures of perturbed poloidal magnetic flux and current density for the case without/with plasma rotation are plotted in the left/right column of figure 5, respectively. It can be clearly observed that the RMP is screened inside the resonant surface for the case with rotation, from the fact that the magnetic perturbation is nearly zero for $r<r_{s}$. In addition, a strong shielding current is formed at the resonant surface, which is consistent with the experimental results on TEXTOR tokamak[42]. Theses characteristics are the same as the small locked island (SLI) observed in J-TEXT tokamak[43, 44], indicating that the SLI is a suppressed state before the mode penetration. In Fitzpatrick’s linear theory[22], the transition from the linear suppressed island to the non-linear island state is irreversible, i.e. in the linear response there is only downward bifurcation[45]. However, the SLI is a phenomenon that a non-linear magnetic island return to the linear state where the island is suppressed. Thus, perhaps the branch of this kind of upward bifurcation is missed in the previous theory. For the case without rotation, other other hand, the m/n = 3/2 component of magnetic perturbation is induced and kept by the 3/2 RMP at the boundary.

All in all, the above simulation results may provide another possible explanation for the “enduring mystery”; considering the effect of bootstrap current, the onset of NTM can lead to a mode penetration like phenomenon, a driven connection regime plus a NTM regime. As a result, it seems there is a finite threshold for mode penetration if not carefully distinguished.

In this subsection, effects of diamagnetic drift flow on the RMP penetration are numerically studied in comparison with the ${\bf E\times B}$ electric drift flow. The equilibrium ${\bf E\times B}$ flow velocity can be directly obtained by $v_{\rm E0}=\partial\phi_{0}/\partial r$, and the equilibrium electron diamagnetic flow velocity can be calculated by $v_{\rm dia0}=-\delta\partial p_{0}/\partial r$, where the subscript 0 is for equilibrium quantities. Therefore, the natural plasma rotation could be obtained by the sum of these two effects. By changing the value of $\delta$, different amplitude of diamagnetic drift flow can be implemented.

As shown in figure 6, the screening effects of the two types of flow on the RMP penetration are compared. The solid lines are for cases with a equilibrium ${\bf E\times B}$ flow frequency $\omega_{\rm E0}$=6.4 kHz and diamagnetic flow frequency $\omega_{\rm dia0}$=0, while dotted lines for $\omega_{\rm dia0}$=6.4 kHz and $\omega_{\rm E0}$=0. Thus, for both cases the natural frequency is $\omega_{0}=\omega_{\rm E0}+\omega_{\rm dia0}$=6.4 kHz. It makes no difference on the penetration threshold, no matter what kind of the flow it is, which means the penetration threshold only depends on the rotation difference between the resonant surface and the RMP. Corresponding to the two penetrated case in figure 6, the temporal evolution of the flow frequencies $\omega_{\rm E}$, $\omega_{\rm dia}$, $\omega_{\rm E}$+$\omega_{\rm dia}$ and the frequency of the phase of the island $\omega_{\rm ph}$ is illustrated in figure 7. It shows a good agreement for the flow frequency and the phase frequency after mode penetration, indicating that the magnetic island and the flow are coupled in accordance with the frozen-in theorem. For the case ($\omega_{\rm E0}$=6.4 kHz, $\omega_{\rm dia0}$=0), the $\omega_{\rm E}$ decreases with time and drops to zero at the moment penetration occurs. For the case ($\omega_{\rm E0}$=0, $\omega_{\rm dia0}$=6.4 kHz), since the $\omega_{\rm dia}$ is a rigid flow effect which is mainly proportional to the pressure gradient, it doesn’t change at the beginning but starts to decrease as the magnetic island grows, where the pressure gradient is flattened. In consequence, the $\omega_{\rm E}$ will rotate at the opposite direction to cancel the diamagnetic flow.

When it comes to a larger rotation frequency, the situation is somewhat different. Similar to figure 6, the nonlinear evolution of the island width for a larger natural frequency $\omega_{0}$=12.8 kHz is presented in figure 8. It shows again that the penetration threshold is almost the same. However, there are two differences observed. First, the saturated island width is evidently smaller for the case ($\omega_{\rm E0}$=0, $\omega_{\rm dia0}$=12.8 kHz) than that of the case ($\omega_{\rm E0}$=12.8 kHz, $\omega_{\rm dia0}$=0), implying that the diamagnetic flow can drive a stabilizing effect on the magnetic island. Second, an oscillation phenomenon of the magnetic island is discovered after mode penetration. To further analyze this oscillation phenomenon, the island width and flow frequency of the oscillated case in figure 8 are plotted, as shown in figure 9. The periodical oscillation of diamagnetic flow frequency is observed as well in figure 9 (right). It should be noted that the oscillation of frequency lags behind the island width a bit, suggesting that the change of island width results in the change of the diamagnetic flow frequency at first. Since the diamagnetic flow is proportional to the pressure gradient, it can be straightforwardly inferred that this phenomenon is related to the change of plasma pressure. To proceed a further step, the contour plot of the pressure $p$ together with the poloidal magnetic flux $\psi$ is shown in figure 10, corresponding to the four red time points in figure 9. As the magnetic island grows larger (t=73000 and t=83200), the pressure gradient $\nabla_{\parallel}p$ with respect to the magnetic field inside the island becomes larger. For a smaller island width (t=77000 and t=88800), on the contrary, the $\nabla_{\parallel}p$ is smaller. This modification of pressure gradient can in return affect the island width by the $\delta\nabla_{\parallel}p$ term in equation (2), leading to the above oscillation phenomenon.

In order to further verify our conjecture, the effect of resistivity $\eta$ is then investigated. For a larger $\eta$, it turns out that the oscillation phenomenon disappears and the island width recovers as illustrated in figure 11 (upper left). It can be easily understood through equation (2). Since $\eta$ is a diffusive term , the effect of $\delta\nabla_{\parallel}p$ term, stabilizing the island and causing the oscillation, can be diffused to some extent with the increasing $\eta$, in much the same way as viscosity $\nu$ in the vorticity equation stabilizing the oscillation of rotation. In other words, the oscillation phenomenon is a result of the competition of the two terms $\delta\nabla_{\parallel}p$ and $\eta(j-j_{b})$. For the same reason, a smaller bootstrap current fraction $f_{b}$ can remove the oscillation by making the $\eta(j-j_{b})$ term larger, shown in figure 11 (upper right). As the ratio of parallel to perpendicular transport coefficient $\chi_{\parallel}/\chi_{\perp}$ is crucial to the process of pressure evolution, the effect of different $\chi_{\parallel}$ and $\chi_{\perp}$ values are studied. In figure 11 (lower left and right), the temporal evolution of the island width is plotted for different $\chi_{\parallel}$ and $\chi_{\perp}$. The results are intuitive, i.e. a smaller $\chi_{\parallel}$ or larger $\chi_{\perp}$ can eliminate the oscillation. This is because a smaller $\chi_{\parallel}/\chi_{\perp}$ can lower the energy transport level along the magnetic field lines, which would reduce the variation of $\delta\nabla_{\parallel}p$ term when the size of magnetic island changes.