Effects of Plasma Boundary Shape on Explosive Bursts Triggered by Tearing\\ Mode in Toroidal Tokamak Plasmas with Reversed Magnetic Shear

Haoyu Wang(王浩宇), Zheng-Xiong Wang(王正汹), Tong Liu(刘桐)$^{\hyperlink{s}{*}}$, and Xiao-Long Zhu(朱霄龙)

doi:10.1088/0256-307X/40/7/075201

⇦Chinese Physics Letters, 2023, Vol. 40, No. 7, Article code 075201⇨ Effects of Plasma Boundary Shape on Explosive Bursts Triggered by Tearing Mode in Toroidal Tokamak Plasmas with Reversed Magnetic Shear Haoyu Wang (王浩宇), Zheng-Xiong Wang (王正汹), Tong Liu (刘桐)*, and Xiao-Long Zhu (朱霄龙) Affiliations Key Laboratory of Materials Modification by Beams (Ministry of Education), School of Physics, Dalian University of Technology, Dalian 116024, China Received 10 April 2023; accepted manuscript online 31 May 2023; published online 27 June 2023 ^*Corresponding author. Email: liutong@dlut.edu.cn Citation Text: Wang H Y, Wang Z X, Liu T et al. 2023 Chin. Phys. Lett. 40 075201 Abstract Numerical research is conducted to investigate the effects of plasma boundary shape on the tearing mode triggering explosive bursts in toroidal tokamak plasmas. In this work, $m/n=2/1$ mode is responsible for the triggering of the explosive burst. Plasma boundary shape can be adjusted via the adjustment of the parameters triangularity ${\delta}$ and elongation ${\kappa}$. The investigations are conducted both under low $\beta$ (close to zero) and under finite $\beta$ regimes. In the low $\beta$ regime, triangularity and elongation both have stabilizing effect on the explosive burst, and the stabilizing effect of elongation is stronger. Under a large elongation (${\kappa =2.0}$), the elongation effect can evidently enhance the stabilizing effect in a positive triangularity regime, but barely affects the stabilizing effect in a negative triangularity regime. In the finite $\beta$ regime, the explosive burst is delayed in comparison with that in the low $\beta$ regime. Similar to the low $\beta$ cases, the effects of triangularity and elongation both are stabilizing. Under a large elongation (${\kappa =2.0}$), the elongation effect can evidently enhance the stabilizing effect on the explosive burst in a positive triangularity regime, but impair the stabilizing effect in a negative triangularity regime. The explosive burst disappears in the large triangularity case (${\delta =0.5}$), indicating that the explosive burst can be effectively prevented in experiments via carefully adjusting plasma boundary shape. Moreover, strong magnetic stochasticity appears in the negative triangularity case during the nonlinear phase.

DOI:10.1088/0256-307X/40/7/075201 © 2023 Chinese Physics Society Article Text The tearing mode (TM) is one of dangerous magnetohydrodynamic (MHD) instabilities in present and future magnetic fusion devices, and as such, it has become a hot topic in the field of magnetic confinement fusion research.[1,2] TM can substantially enhance the radial heat transport by triggering fast field line reconnection, resulting in a great degradation of the confinement. The reversed magnetic shear (RMS) configuration is considered to be a promising candidate for achieving long-pulse steady-state performance operation in tokamak plasmas.[3,4] Many tokamak experiments have shown that the RMS configuration contributes to formation of an internal transport barrier, which allows for a high plasma pressure to be maintained in the core region.[5,6] On the other hand, the RMS configuration has a unique nonmonotonic safety factor $q$-profile, which results in the development of pairs of MHD instabilities at rational surfaces with the same helicities. These instabilities are referred to as double tearing mode (DTM).[7-13] Ishii et al. conducted numerical research on the nonlinear evolutions of DTM instabilities for $q$-profiles with different separations ($\Delta r_{\rm s}$) between the two rational surfaces.[14,15] In their research, they found that in the case of large $\Delta r_{\rm s}$, two single tearing modes with weak coupling effects saturate separately, while in the case of small $\Delta r_{\rm s}$, strongly coupled DTMs with fast linear growth rates eventually reach saturation at low levels of perturbation energy.[14,15] It is noteworthy that in the case of intermediate $\Delta r_{\rm s}$, the perturbation energy of DTM experiences an explosive growth in the nonlinear stage, which may be related to the major disruption observed in JT-60U experiments.[14,15] Further research was conducted on the trigger mechanism of this explosive burst based on Ishii's results.[16] It was discovered that the coupling of DTM magnetic islands on both rational surfaces can induce asymmetrical flux driven reconnection. This reconnection of the magnetic field can result in the triangular deformation of the magnetic island structure. If the degree of island deformation becomes sufficiently strong, it can lead to the onset of a type of explosive magnetic reconnection event. Wang et al. discovered that even in the case of large $\Delta r_{\rm s}$, the explosive burst can still be triggered by neo-classical DTM with a high fraction of bootstrap current.[17] Zhang et al. investigated the threshold and time scales of pressure crashes associated with DTMs.[18,19] Fast reconnection of magnetic field lines caused by DTM in off-axis sawtooth phenomena, as well as explosive bursts, have been reported in many tokamak experimental discharges with RMS configurations, including TFTR,[20,21] JT-60U,[22] Tore Supra,[23] RTP,[24] and ASDEX Upgrade[25] tokamaks. The DTM instability is inevitable for future advanced tokamaks with RMS configurations, including ITER and CFETR. Therefore, it is crucial to investigate DTM and to develop methods for their control. Extensive researches have been conducted to investigate different techniques for avoiding the explosive burst and for controlling DTM. These techniques include externally applied toroidal shear flow,[26] electron cyclotron current drive,[27-29] resonant magnetic perturbation,[30-33] and three-dimensional MHD spectroscopy.[34,35] However, most of the results reported have neglected the significant toroidal effects and boundary shape effects on MHD instability. Experimental investigations in literature have revealed that plasma triangularity ($\delta$) strongly influences electron heat transport and the nature of turbulent fluctuations in the TCV tokamak.[36,37] Previous numerical results found that plasma boundary shape can influence linear properties of the DTM.[38] Thus, it is necessary to investigate the effects of plasma boundary shape on the nonlinear dynamics of the explosive bursts triggered by DTM. In this work, nonlinear simulations are carried out to investigate the effects of plasma boundary shape on the TM triggering explosive bursts in toroidal tokamak plasmas. It is found that triangularity and elongation both have stabilizing effect on the explosive bursts. In the finite $\beta$ regime, the explosive burst can be effectively avoided via adjusting plasma boundary shape. In this Letter, we use the global three-dimensional nonlinear kinetic-MHD hybrid initial value code M3D-K for numerical simulation.[39,40] The M3D-K code used consists of two parts: the finite element method is used to solve the resistive MHD equations, and the energetic component is treated by drift-kinetic equations. Since this work does not involve the energetic-particle part, we only employ the following resistive MHD equations: \begin{align} &\frac{{\partial \boldsymbol{B}}}{\partial t}=-{\nabla \times \boldsymbol{E }}, \tag {1}\\ &{\nabla \times \boldsymbol{B}=}\mu_{0}{\boldsymbol{J }}, \tag {2}\\ &{\boldsymbol{E}+\boldsymbol{v}\times \boldsymbol{B}=}\eta {\boldsymbol{J }}, \tag {3}\\ &{\nabla \cdot \boldsymbol{B}=0}, \tag {4}\\ &\frac{\partial \rho }{\partial t}+\nabla\cdot(\rho{\boldsymbol{v}})=0, \tag {5}\\ &\rho \frac{d{\boldsymbol{v}}}{dt}={\boldsymbol{J\times B}}-\nabla p+\mu \nabla ^{2}{\boldsymbol{v }}, \tag {6}\\ &\frac{dp}{dt}=-\gamma p\nabla \cdot {\boldsymbol{v}}+\rho \nabla \cdot\Big({K}\cdot \nabla \frac{p}{\rho}\Big). \tag {7} \end{align} More details about the model can be found in Refs. [39,40]. The key parameters for the simulation are as follows: aspect ratio $\epsilon =R_{0}/a=4.80$, toroidal magnetic field $B_{0}=1.25$ T, central number density $n_{0}=1.0\times {10}^{19}$/m$^{-3}$, Alfvén speed $v_{\scriptscriptstyle{\rm A}}=B_{0}/(\mu_{0}\rho_{0})^{1/2}$, Alfvén time $\tau_{\scriptscriptstyle{\rm A}}=R_{0} / {\epsilon v_{\scriptscriptstyle{\rm A}}}$, Alfvén frequency $\omega_{\scriptscriptstyle{\rm A}}={\epsilon v_{\scriptscriptstyle{\rm A}}}/R_{0}$, the ratio of the volume-averaged plasma pressure to the magnetic pressure $\beta ={2\mu_{0}\langle P \rangle }/B_{0}^{2}$, the plasma resistivity $\eta =5\times {10}^{-6}$ and the plasma viscosity $\nu =1\times {10}^{-6}$. Numerical Results. In order to investigate the effect of plasma boundary shape, two parameters (elongation $\kappa$ and triangularity $\delta$) are adjusted for easily controlling the plasma boundary shape. Figure 1 illustrates the four different types of plasma boundary shapes obtained by adjusting the two parameters. Here, ${\kappa =1.0}$ and ${\delta =0}$ represent the circular cross-section case. To begin with, the nonlinear evolution of the DTMs is investigated under circular cross-section as the baseline case. During the RMS discharges in advanced tokamaks, burst-like disruptions are frequently observed in connection with MHD activities around the $q=2$ rational surfaces. To simulate these nonlinear events, we used the $q$-profile shown in Fig. 2(a) and focused on the $n = 1$ cases. For this study, we adopted a fixed $q$-profile with $q_{0}=2.88$ and $\Delta r_{\rm s}=r_{\rm s1}-r_{\rm s2}=0.343$. Under this profile, Fig. 2(b) demonstrates the nonlinear evolution of the kinetic energy for the DTM with a circular cross-section (triangularity ${\delta =0}$ and elongation ${\kappa =1.0}$) in intermediate $\Delta r_{\rm s}$ cases when $\beta$ is close to zero. It is clear that an explosive burst of the DTM's perturbed kinetic energy occurred after the Rutherford phase around ${t=1300}\tau_{\scriptscriptstyle{\rm A}}$.

Fig. 1. The plasma boundary for (a) circular cross-section (${\kappa =1.0}$, ${\delta =0}$), (b) large elongation cross-section (${\kappa =2.0}$, ${\delta =0}$), (c) large negative triangularity cross-section (${\kappa =1.0}$, ${\delta =-0.5}$), and (d) large triangularity cross-section (${\kappa =1.0}$, ${\delta =0.5}$).

Fig. 2. (a) Initial non-monotonic safety-factor $q$-profile. (b) Nonlinear evolution of kinetic energy for the DTM in intermediate $\Delta r_{\rm s}$ cases when the $\beta$ is low.

In order to verify the nonlinear dynamics of the explosive burst, the magnetic topology and perturbed stream function $U$ at different moments are displayed in Fig. 3, where $U$ is the velocity stream function and is related to the incompressible part of the plasma velocity by the following equation: \begin{align} {\nu =}R^{2}\epsilon \nabla _{\bot }U\times \nabla \varphi +\nabla \chi +\nu_{\varphi }\nabla \varphi . \tag {8} \end{align} The Poincare plot is constructed by selecting a hyperplane or surface within the phase space, known as the Poincare section. It can reveal the period behavior of magnetic field in tokamak device in this work. Figures 3(a)–3(d) display the Poincare plots of perturbed field lines at different time slices for triangularity ${\delta =0}$ and elongation ${\kappa =1.0}$. Figures 3(e)–3(h) exhibit the DTM structure of the velocity stream function $U$ at four time slices for triangularity ${\delta =0}$ and elongation ${\kappa =1.0}$. With the development of the DTM, the $m/n=2/1$ magnetic islands are observed to gradually expand with time. The $m/n=3/1$ islands also appear due to the toroidal coupling effect. Right after the explosive burst, the islands on the both $q=2$ rational surfaces exchange their positions, which is consistent with the previous work.[8] Similar MHD activity has been observed by Fredrickson et al. in early TFTR experiments.[21] As the outer islands grow large, they constantly push inward, leading to large deformation of island structure.[16] At the same time, the perturbed velocity stream function extends to core region.

Fig. 3. (a)–(d) The Poincare plots of magnetic field lines at different time slices for triangularity ${\delta =0}$ and elongation ${\kappa =1.0}$. (e)–(h) The structure of velocity stream function $U$ of DTM at four time slices for triangularity ${\delta =0}$ and elongation ${\kappa =1.0}$.

Fig. 4. [(a), (b)] Nonlinear evolution of kinetic energy for DTM with the low $\beta$ under different values of triangularity $\delta$. (c) Nonlinear evolution of kinetic energy for the DTM with the low $\beta$ under different values of elongation $\kappa$.

Based on the results shown in Fig. 2(b), the effect of plasma boundary shape on the explosive burst is investigated. Plasma boundary shape can be adjusted by changing the parameters of triangularity $\delta$ and elongation $\kappa$ based on the circular cross-section case. The numerical simulation results presented in Fig. 4 were obtained by changing one of the triangularity $\delta$ and elongation $\kappa$ meanwhile fixing the other. When triangularity $\delta$ is increased while elongation $\kappa$ is fixed at 1.0, the occurrence of the burst in Figs. 4(a) and 4(b) is slightly delayed (negative $\delta$ represents the deformation of negative triangularity). By comparing Figs. 4(a) and 4(b), it can be observed that the kinetic energy for the DTM follows the same trend while changing triangularity $\delta$. The same trend indicates that the positive and negative triangularity values of plasma boundary shape have almost the same influence on the nonlinear dynamics of the explosive burst under low $\beta$ regime. For changing the elongation $\kappa$ and fixing triangularity ${\delta =0}$, the explosive burst is largely delayed as shown in Fig. 4(c), meaning that the elongation has an evident stabilizing effect on the explosive burst. Further research was conducted based on the above simulation results. Figure 5 depicts the kinetic energy for the DTM with elongation ${\kappa =2.0}$ under different values of triangularity $\delta$. When fixing ${\kappa =2.0}$ and $\delta$ is varied, the explosive bursts exhibit similar trends between positive- and negative-triangularity cases, but the delay of the explosive burst is much obvious in the positive triangularity cases. This means that the elongation effect can evidently enhance the stabilizing effect on the explosive burst in a positive triangularity regime, but hardly affects the stabilizing effect in a negative triangularity regime. Figure 6 provides a more comprehensive illustration of the burst time of the DTM with the low $\beta$ under different values of triangularity $\delta$ and elongation $\kappa$.

Fig. 5. [(a), (b)] Nonlinear evolution of kinetic energy for the DTM with the elongation ${\kappa =2.0}$ under different values of triangularity $\delta$.

Fig. 6. Burst time of the DTM with the low $\beta$ under different values of triangularity $\delta$ and elongation $\kappa$.

Fig. 7. Nonlinear evolution of kinetic energy for the DTM with the triangularity ${\delta =0}$ and elongation ${\kappa =1.0}$, including $\beta =0$ and $\beta =0.5\%.$

So far, the numerical results are all obtained under the condition of $\beta$ values close to zero. It is known that the plasma $\beta$ has a stabilizing effect on linear DTM.[11,37] In the following cases, we conducted numerical simulations on the nonlinear evolution of DTM with a finite $\beta =0.5\%$. Figure 7 illustrates the effect of changing $\beta$ values on the nonlinear evolution of kinetic energy for DTM in a circular cross-section (with $\delta =0$ and $\kappa =1.0$). In this part of the calculation, we continue to use the previous $q$-profile. It is evident that changing $\beta$ has an impact on both the linear and nonlinear stages of DTM instability. The overall effect of plasma $\beta$ on the explosive burst is stabilizing effect, which can be seen from Fig. 7 that the explosive burst is evidently delayed when increasing $\beta$. Here, the influence of triangularity $\delta$ and elongation $\kappa$ on nonlinear DTM instability at a finite $\beta =0.5\%$ is also investigated. The nonlinear evolutions of kinetic energy for DTM under different plasma boundary shapes are displayed in Fig. 8. It is found that increasing triangularity $\delta$ slightly delays the occurrence of the explosive burst both in positive and in negative triangularity regimes, which is similar to the results in low $\beta$ cases shown in Fig. 4. Figure 8(c) illustrates the influence of elongation $\kappa$ on the kinetic energy evolution of nonlinear DTM when triangularity ${\delta =0}$. It is found that increasing elongation $\kappa$ can evidently delay the occurrence of the explosive burst, and the bursting intensity is reduced on some level, indicating that elongation has a more effective stabilizing effect on the explosive burst under finite $\beta$ than that under low $\beta$.

Fig. 8. [(a), (b)] Nonlinear evolution of kinetic energy for DTM with the $\beta =0.5\%$ under different values of triangularity $\delta$. (c) Nonlinear evolution of kinetic energy for DTM with the $\beta =0.5\%$ under different values of elongation $\kappa$.

Similar to the above low $\beta$ investigations, the further research of simultaneously adjusting elongation and triangularity is also conducted here. Figure 9 depicts the kinetic energy for DTM with elongation ${\kappa =2.0}$ under different values of triangularity $\delta$. For the negative triangularity cases shown in Fig. 9(a), different from the low $\beta$ regime, the explosive burst has been brought forward instead of delay with increasing triangularity $\delta$. For the positive triangularity cases shown in Fig. 9(b), the explosive burst is evidently delayed. More inspiringly, the explosive burst disappears in the ${\delta =0.5}$ case. The nonlinear evolution of DTM instability directly enters into the saturation stage. This means that the elongation effect can evidently enhance the stabilizing effect on the explosive burst in a positive triangularity regime, but somehow impair the stabilizing effect in a negative triangularity. Thus, it is expected that the explosive burst could be effectively prevented in experiments via appropriately adjusting plasma boundary shape. Figure 10 provides a more comprehensive illustration of the burst time of DTM with the $\beta =0.5\%$ under different values of triangularity $\delta$ and elongation $\kappa$.

Fig. 9. [(a), (b)] Nonlinear evolution of kinetic energy for DTM with the $\beta =0.5\%$, elongation ${\kappa =2.0}$ under different values of triangularity $\delta$.

Fig. 10. The burst time of DTM with the $\beta =0.5\%$ under different values of triangularity $\delta$ and elongation $\kappa$.

To better understand the difference of elongation effects in the positive and negative triangularity regimes, the magnetic topology and perturbed velocity stream function under two different parameters (${\kappa =2.0}$, ${\delta =-0.5}$ and ${\kappa =2.0}$, ${\delta =0.5}$) are displayed in Figs. 11 and Fig. 12, respectively. By comparing magnetic topology between negative and positive triangularity cases, it is observed that the negative triangularity cross section is more prone to instability, with the magnetic island undergoing an exchange of inner and outer positions at last. Moreover, the ${m/n=3}/1$ and ${m/n=4/1}$ islands also occur during the nonlinear phase for the negative triangularity case. The strong coupling between ${m=2 3 4}$ harmonics contributes to the generation of the magnetic stochasticity. The magnetic islands, as well as the perturbed velocity stream function, under negative triangularity regime are far more distorted than those under positive triangularity regime. For the positive triangularity cross-section case, the magnetic islands do not exchange positions, leading to the absence of the explosive burst. The DTM islands reach saturation at the end with dominant ${m/n=2}/1$ harmonic and no magnetic stochasticity appeared. The difference between the positive and negative triangularity regimes may be because of the change of the coupling strength due to the Grad–Shafranov shift. Under the finite plasma $\beta$ regime, the magnetic axis will move further outward along the major radius in comparison with that under low $\beta$ regime due to the Grad–Shafranov shift, which can evidently strengthen the coupling effect among different harmonics. The positive triangularity, along with a large elongation, weakens the influence of this shift to some extent. Nonetheless, the negative triangularity will aggravate the influence of this shift and cause strong coupling effect among different harmonics, leading to severe magnetic stochasticity. Conclusion. Effects of plasma boundary shape on the TM triggering explosive bursts are investigated in RMS toroidal tokamak plasmas using nonlinear kinetic-MHD hybrid initial value code M3D-K in this work. The $m=2$ harmonic is the dominant unstable harmonic responsible for the triggering of the explosive burst according to the adopted $q$-profile, and the $n = 1$ cases are investigated in this work. Plasma boundary shape can be adjusted by changing the parameters of triangularity $\delta$ and elongation $\kappa$ based on the circular cross-section case. The investigations are conducted both under low $\beta$ (close to zero) and under finite $\beta$ regime. Under low $\beta$ regime, with increasing triangularity without elongation, the explosive burst is slightly delayed for both the negative and positive triangularity cases. For changing the elongation without triangularity, the explosive burst is largely delayed, meaning that the elongation has an effective stabilizing influence on the explosive burst. Under a large elongation (${\kappa =2.0}$), the explosive bursts are also delayed by increasing triangularity for both the positive- and negative-triangularity cases. While the delay is quite large in the positive triangularity cases. This means that the elongation effect can evidently enhance the stabilizing effect on the explosive burst in a positive triangularity regime, but barely affects the stabilizing effect in a negative triangularity regime. Under finite $\beta$ regime, the linear and nonlinear stages of DTM instability are influenced in comparison with the low $\beta$ regime. The overall effect of plasma $\beta$ on the explosive burst is stabilizing effect. Similar to the low $\beta$ cases, increasing triangularity $\delta$ slightly delays the explosive burst in both positive and negative triangularity regimes. Increasing elongation $\kappa$ can evidently delays the explosive burst, and the bursting intensity is reduced, indicating that elongation has a more effective stabilizing effect on the explosive burst under finite $\beta$ than that under low $\beta$. Under a large elongation (${\kappa =2.0}$), for the negative triangularity cases, the explosive bursts are brought forward with increasing triangularity $\delta$. For the positive triangularity cases, the explosive bursts are largely delayed. Moreover, the explosive burst disappears in the large triangularity case (${\delta =0.5}$). This means that the elongation effect can evidently enhance the stabilizing effect on the explosive burst in a positive triangularity regime, but impair the stabilizing effect in a negative triangularity. Therefore, the explosive burst is expected to be prevented in experiments by carefully adjusting plasma boundary shape. Moreover, strong magnetic stochasticity can be found in the negative triangularity case during the nonlinear phase.

Fig. 11. (a)–(d) The Poincare plots of magnetic field lines at different time slices for triangularity ${\delta =-0.5}$ and elongation ${\kappa =2.0}$. (e)–(h) The DTM structure of velocity stream function $U$ at four time slices for triangularity ${\delta =-0.5}$ and elongation ${\kappa =2.0}$.

Fig. 12. (a)–(d) The Poincare plots of magnetic field lines at different time slices for triangularity ${\delta =0.5}$ and elongation ${\kappa =2.0}$. (e)–(h) The DTM structure of velocity stream function $U$ at four time slices for triangularity ${\delta =0.5}$ and elongation ${\kappa =2.0}$.

The present results systematically reveal the effect of plasma boundary shape on the explosive burst and explore the optimal boundary shape for better suppressing the explosive burst. For the avoidance of the explosive burst, there are many other effective methods, such as electron cyclotron current drive[27,28] and resonant magnetic perturbation.[33] In the future, the control of the explosive burst through different methods will be further investigated together with the effect of the plasma boundary shape to provide better operating schemes for experiments. Acknowledgement. The authors acknowledge the Supercomputer Center of Dalian University of Technology for providing computing resources. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11925501, 12105034, and 12205034), and the Fundamental Research Funds for the Central Universities (Grant Nos. DUT22ZD215). References Chapter 3: MHD stability, operational limits and disruptionsChapter 6: Steady state operationQuasisteady High-Confinement Reversed Shear Plasma with Large Bootstrap Current Fraction under Full Noninductive Current Drive Condition in JT-60UJET Quasistationary Internal-Transport-Barrier Operation with Active Control of the Pressure ProfileProgress towards steady-state operation and real-time control of internal transport barriers in JETAdvanced tokamak research in DIII-DLinear analysis of the double-tearing modeFast Resistive Reconnection Regime in the Nonlinear Evolution of Double Tearing ModesCurrent point formation and magnetic reconnection process in nonlinearly destabilized double tearing modesEnergetic ion beta scaling of q ≳ 1 non-resonant modes in tokamak plasmas with a weak magnetic shear configurationMultiple MHD instabilities in high- β_N toroidal plasmas with reversed magnetic shearCore-crash sawtooth associated with m / n = 2/1 double tearing mode in TokamakNonlinear evolution and control of neo-classical tearing mode in reversed magnetic shear tokamak plasmasNonlinear evolution of double tearing modesStructure-Driven Nonlinear Instability of Double Tearing Modes and the Abrupt Growth after Long-Time-Scale EvolutionStructure-Driven Nonlinear Instability as the Origin of the Explosive Reconnection Dynamics in Resistive Double Tearing ModesNonlinear evolution of neo-classical tearing modes in reversed magnetic shear tokamak plasmasThe off-axis pressure crash associated with the nonlinear evolution of the m/n = 2/1 double tearing modeNumerical Studies of Fast Pressure Crash Associated with Double Tearing ModesOff-Axis Sawteeth and Double-Tearing Reconnection in Reversed Magnetic Shear Plasmas in TFTRNonlinear evolution of double tearing modes in tokamaksResistive instabilities in reversed shear discharges and wall stabilization on JT-60UNonlinear magnetohydrodynamic simulation of Tore Supra hollow current profile dischargesElectron Thermal Transport Barrier and Magnetohydrodynamic Activity Observed in Tokamak Plasmas with Negative Central ShearMHD phenomena in reversed shear discharges on ASDEX UpgradeInfluence of shear flows on dynamic evolutions of double tearing modesSuppression of explosive bursts triggered by neo-classical tearing mode in reversed magnetic shear tokamak plasmas via ECCDPrevention of electron cyclotron current drive triggering explosive bursts in reversed magnetic shear tokamak plasmas for disruption avoidanceControl of neoclassical tearing mode by synergetic effects of resonant magnetic perturbation and electron cyclotron current drive in reversed magnetic shear tokamak plasmasNonlinear evolution of multi-helicity neo-classical tearing modes in rotating tokamak plasmasAnalytical model of plasma response to external magnetic perturbation in absence of no-slip conditionQuasi-linear theory of forced magnetic reconnection for the transition from the linear to the Rutherford regimeA brief review: effects of resonant magnetic perturbation on classical and neoclassical tearing modes in tokamaksIdentification of multiple eigenmode growth rates in DIII-D and EAST tokamak plasmasIdentification of multiple eigenmode growth rates towards real time detection in DIII-D and KSTAR tokamak plasmasInfluence of Plasma Shape on Transport in the TCV TokamakThe effect of triangularity on fluctuations in a tokamak plasmaEffects of plasma boundary shape on the β_N threshold in the suppression of tearing mode in toroidal tokamak plasmas with reversed magnetic shearPlasma simulation studies using multilevel physics modelsGlobal hybrid simulations of energetic particle effects on the n=1 mode in tokamaks: Internal kink and fishbone instability

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