Double Tearing Modes Research Articles

Nonlinear evolution of weakly coupled double tearing modes with equilibrium sheared toroidal flow

Abstract The nonlinear evolution of weakly coupled DTMs with and without sheared toroidal flow is systematically investigated with the three-dimensional, toroidal-geometry, and nonlinear MHD code CLT. Due to the explosive growth of the weakly coupled DTMs, the pressure crashes are much more severe than those caused by the strongly coupled DTMs. The weakly coupled DTMs can easily be decoupled by sheared toroidal flow, and then the mutually driven term between the two tearing modes becomes periodically positive and negative, leading to periodic growth. The interaction slows the rotation between the two tearing modes when the phase difference is close to 180 degrees. When the islands are sufficiently large, the interaction can overcome the decoupling effect of the shear flow, and the two tearing modes then lock with each other. Due to the different coupling effects of the two tearing modes, the threshold island width or perturbed magnetic field for mode-locking increases with increasing separation. The coupling effect between the two tearing modes is also related to the resistivity. If the resistivity is high, the coupling effect between the two tearing modes can be huge. A stronger shear flow is needed to decouple them, and even a small island can lead to mode-locking in high-resistivity plasma.

Influence of electron cyclotron current drive on the pressure crashes caused by the 2/1 double tearing modes

Abstract A module with self-consistent evolution of driven current is developed and coupled with the resistive-MHD equations in the three-dimensional, toroidal, and nonlinear simulation code (CLT). The driven current equation is solved with a second-order accuracy symmetric scheme, which exhibits good conservation properties. With the new module, we find that the driven current can self-consistently concentrate inside the magnetic island when the parallel diffusion of the driven current is sufficiently large. The efficiency of the driven current on tearing mode suppression will then be much higher than those with stationary distributions. With the new module, the influence of electron cyclotron current drive (ECCD) on the nonlinear evolution of the 2/1 double tearing modes (DTMs) is investigated. When co-ECCD deposits on the outer resonant surface, the local magnetic shear is reduced, and the growth rates of the DTMs decrease; if ctr-ECCD deposits on the outer resonant surface, the local magnetic shear increases, and the DTMs become more unstable. However, things will be different if ECCD deposits on the inner resonant surface since the local magnetic shear is negative. The co-ECCD deposited on the inner resonant surface increases the negative shear and then promotes the growth of the DTMs; while the ctr-ECCD suppresses the DTMs. It is also found that the off-axis and central pressure crashes associated with the 2/1 DTMs can be converted to each other by properly depositing the driven current. To convert a central crash to an off-axis crash, the co-ECCD should be deposited on the outer resonant surface, or the ctr-ECCD deposited on the inner resonant surface. While, the co-ECCD should be deposited on the inner rational surface, or the ctr-ECCD deposited on the outer rational surface to convert an off-axis crash to a central crash. The co- or ctr-ECCD should be larger than a threshold for such transitions, and the threshold value is mainly determined by the location of the inner resonant surface.

Effect of Shear Flow on the Double Tearing Mode Induced by Resonant Magnetic Perturbation

Effect of Shear Flow on the Double Tearing Mode Induced by Resonant Magnetic Perturbation

Effect of toroidal mode coupling on explosive dynamics of m/n = 3/1 double tearing mode

The CLT code was used to quantitatively study the impact of toroidal mode coupling on the explosive dynamics of the m/n = 3/1 double tearing mode. The focus of this study was on explosive reconnection processes, in which the energy bursts and the main mode no longer dominates when the separation between two rational surfaces is relatively large in the medium range. The development of higher m and n modes is facilitated by a relatively large separation between two rational surfaces, a small q min (the minimum value of the safety factor), or low resistivity. The relationships between the higher m and n mode development, explosive reconnection rate, and position exchange of 3/1 islands are summarized for the first time. Separation plays a more important role than q min in enhancing the development of higher m and n modes. At a relatively large separation, the good development of higher m and n modes greatly reduces the reconnection rate and suppresses the development of the main mode, resulting in the main mode not being able to develop sufficiently large to generate the position changes of 3/1 islands.

MHD stability of spherical tokamak equilibria with non-monotonic q-profiles

We use the 3D magnetohydrodynamic (MHD) code M3D-C1 [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)] to examine the MHD stability and subsequent evolution of NSTX shot 129169. This discharge had a period with a non-monotonic safety factor profile, q (reversed shear), which was terminated by a MHD event that abruptly lowered the central safety factor, q0, and greatly reduced the peakedness of the pressure profile. We show that the equilibrium just before the MHD event occurred was linearly unstable to many pressure-driven infernal modes. Modes with toroidal mode number n≥3 all had rational surfaces very close to the minimum value of q. However, a non-resonant pressure-driven (1, 1) mode was also present, and this dominated the nonlinear evolution. The final state in the simulation, after the MHD activity subsided, had a reduced and flattened pressure profile and a nearly monotonic q-profile, in qualitative agreement with experimental results. The initial state was also unstable to the resistive interchange criteria in the reversed-shear region, but the final state was stable everywhere. The “double tearing mode” (DTM) does not appear to play a role in the MHD activity of this discharge. In Appendix A, we show that in a torus, the DTM is strongly stabilized by pressure, but it is destabilized in cylindrical geometry (which has been the most extensively analyzed in the literature).

Explosive dynamics of double tearing mode in Tokamak

Using the CLT code, the resistivity dependence of the reconnection rate during the explosive phase at various separations of two rational surfaces of m/n = 3/1 double tearing mode is investigated quantitatively. Our study focuses on the explosive reconnection process where the exchange of island positions takes place and no secondary island forms. The negative dependence of explosive reconnection rate on resistivity in low resistivity and the systematic study of the effect of the separation on the resistivity dependence in high resistivity have been studied for the first time. The negative dependence is qualitatively different from the results in some relative studies where it usually exhibits a positive dependence on the resistivity or is independent of the resistivity. The negative dependence in two regions with a low resistivity, with a high resistivity and a large separation is caused by different reasons: one is the thickness of the current sheet, and the other is the separation.

Numerical investigation of the 2/1 double tearing mode in EAST with the CLT code

The pressure crashes observed in shot No. 71326 in the Experimental Advanced Superconducting Tokamak are numerically investigated with the three-dimensional, toroidal, and full resistive-magnetohydrodynamics code (CLT). Based on the experimental observations, the pressure crash is caused by the nonlinear evolution of the m/n = 2/1 double tearing mode (DTM), where n and m are the toroidal and poloidal mode numbers, respectively. However, we find that the m/n = 2/1 DTM is stable based on the safety factor (q) profile from the equilibrium code EFIT, which indicates that the original q profile is somewhat inconsistent with the actual profile due to q measurement uncertainty. Since there is no motional Stark effect diagnostic for this shot, the local information of the magnetic field is missing, which leads to the largest contribution to the discrepancy. If other information is perfectly known and the q profile is the only uncertainty, then we could provide some information for the EFIT reconstruction by comparing our simulation results with electron cyclotron emission signals to constrain the uncertainty of the q profile to a much smaller region. The influence of plasma rotation and the two-fluids effect is also discussed.

Non-linear dynamics of the double tearing mode

3D non-linear magnetohydrodynamics simulations of a double tearing mode with the JOREK code are presented in the context of trying to better understand the benign termination of runaway electron beams observed in some experiments. It is shown that the non-linear behavior qualitatively depends on the resistivity η via its effect on how fast secondary, non-linearly destabilized, tearing modes grow relative to the primary mode. Within a certain range of η, a violent and global relaxation is observed, consistent with the “Kadomtsev-predicted” reconnection region extending from almost the very center up to the edge of the plasma.

Viscous Effects on Nonlinear Double Tearing Mode and Plasmoid Formation in Adjacent Harris Sheets

In this paper, we study the effects of viscosity on the evolution of the double tearing mode (DTM) in a pair of adjacent Harris sheets based on the resistive MHD model in the NIMROD code. Similar to the tearing mode in the conventional single Harris sheet, a transition is observed in the generation of both normal and monster plasmoids at Prandtl number Pr=1. In the Pr<1 regime of the DTM, normal plasmoids (small plasmoids) are generated along with monster plasmoid, whereas in the single tearing mode (STM) cases, such a generation is not observed. When Pr is above the critical value, the generation of monster plasmoid is halted. Correspondingly, in the Pr<1 regime, a quadrupolar flow advects along the poloidal direction, but in the Pr>1 regime this flow advection is inhibited.

Nonlinear interaction between double tearing mode and Kelvin–Helmholtz instability with different shear flows

The nonlinear interaction between the double tearing mode (DTM) and Kelvin–Helmholtz (KH) instabilities with different shear flow profiles has been numerically investigated via the use of a compressible magnetohydrodynamics (MHD) model. We focus on KH instabilities in weak and reversed magnetic shear plasmas with strong stabilizing effect of field line bending. Results show that KH instabilities coupled with DTMs occur in these plasmas and the KH mode dominates the instability dynamics, suggesting the crucial role of weak magnetic shear in the formation of high-mode harmonics. For symmetric flows, an asymmetric forced magnetic reconnection configuration is maintained during the growth phase, leading to interlocking of the modes. Additionally, this investigation of the DTM-KH instability interaction contributes to our understanding of the nonlinear reconnection mechanism in the regime of weak and reversed magnetic shear plasmas, which is relevant for astrophysical and fusion studies.

Effects of plasma radiation on the nonlinear evolution of neo-classical tearing modes in tokamak plasmas with reversed magnetic shear

Effects of plasma radiation on the nonlinear evolution of neo-classical double tearing modes (NDTMs) in tokamak plasmas with reversed magnetic shear are numerically investigated based on a set of reduced magnetohydrodynamics (MHD) equations. Cases with different separations Δ rs = |r s2–r s1| between the two same rational surfaces are considered. In the small Δ rs cases, the plasma radiation destabilizes the NDTMs and makes the kinetic energy still grow gradually in the late nonlinear phase. Moreover, the NDTM harmonics with high mode numbers reach a high level in the presence of plasma radiation, forming a broad spectrum of MHD perturbations that induces a radially broadened region of MHD turbulence. As a result, the profiles of safety factors also enter a nonlinear oscillation phase. In the intermediate Δ rs case, the plasma radiation can advance the explosive burst of kinetic energy that results from the fast driven reconnection between the two rational surfaces, because it can further promote the destabilizing effects of bootstrap current perturbation on the magnetic island near the outer rational surfaces. In the large Δ rs case, through destabilizing the outer islands significantly, the plasma radiation can even induce the explosive burst in the reversed magnetic shear configuration where the burst cannot be induced in the absence of plasma radiation.

Oscillation dynamics of m/n= 3/1 double tearing mode

This study systematically investigates the roles of the plasma viscosity and resistivity in the oscillation dynamics during the decay phase of the m/n = 3/1 double tearing mode using the Ci-Liu-Ti (CLT) code. The primary objectives of this research are to examine the driving and suppressing mechanisms of the oscillation. The oscillation and steady-state are the result of the competition between the external injection and the reconnection annihilation of magnetic flux during the decay phase. In a regime with a higher viscosity (or a lower resistivity), the steady-state arises from the significant damping (weak generation) of plasma flows, resulting in the formation of saturated islands. In a regime with a lower viscosity (or a higher resistivity), the suppression of the oscillation amplitude can be attributed to a strong residual flow that quickly takes the injected magnetic flux away toward to the reconnection region, which caused no enough accumulated magnetic flux to drive oscillations and the system evolves toward a steady-state configuration. The steady-state condition results in the generation of a narrow radial vortex region which promotes formation of internal transport barriers. The upper threshold of the resistivity within the low-resistivity regime to achieve a steady-state decreases as the viscosity increases.

Effects of resistivity and viscosity on dynamic evolution and radial position change of m/n = 3/1 double tearing mode

The effects of the plasma resistivity and viscosity on the dynamic evolution of the m/n = 3/1 double tearing mode (DTM) are studied and analyzed quantitatively using the CLT (Ci-Liu-Ti, which means magnetohydrodynamics in Chinese) code. In this work, we mainly focus on the change in the radial positions and the oscillatory dynamics of the magnetic islands grown on the two rational surfaces. We conduct a systematic investigation on the effect of viscosity on the DTM dynamics, which has rarely been studied before. From the results of the study, it is observed that the time required for entering the explosive phase decreases with decreasing viscosity. In the nonlinear phase, the kinetic energy exhibits an oscillatory behavior due to the magnetic flux injection and magnetic reconnection, and the oscillation amplitude is suppressed for a large viscosity due to dissipation. The effects of the plasma resistivity and viscosity on the change in the radial positions of magnetic islands are systematically explained. The change in the radial positions of magnetic islands occurs in an abrupt growth phase before the kinetic energy reaches its maximum value. Multiple position changes take place with a relatively higher reconnection rate and magnetic flux injection at low viscosity damping. A large range of radial vortices formed as a result of the change in the positions may have a positive effect on the transport.

Fast magnetic reconnection and driven plasma rotation in reversed central magnetic shear configuration

Numerical calculations have been carried out to study the nonlinear growth of the double tearing mode (DTM) in the reversed central magnetic shear configuration for medium-size tokamak plasma parameters, based on two-fluid equations and large aspect ratio approximation. Three different regimes of the DTM growth are found. (a) annular crash regime, existing for a small distance but a not too large plasma rotation frequency difference between two resonant surfaces. In this regime the plasma pressure between two resonant surfaces is flattened due to the fast magnetic reconnection in tens of microseconds, in agreement with experimental observations. In addition, a large plasma rotation shear is generated around the edge of the pressure flattening region right after the fast magnetic reconnection. (b) Core crash regime, existing for a medium distance and a relatively low rotation frequency difference between two resonant surfaces, in which the plasma pressure is flattened over a large region up to the magnetic axis during the fast magnetic reconnection in tens of microseconds, in agreement with experimental observation too. (c) No crash regime, existing for a sufficiently large distance and/or frequency difference between two resonant surfaces. In this regime the mode grows slowly in the nonlinear phase and saturates at a finite amplitude, causing a local flattening of the plasma pressure at the resonant surface but without fast crashes.

Neutron-yield reduction in sawtooth crashes from n = 2 resonant magnetic perturbations in EAST

A significant reduction in neutron yield in the plasma core has been observed in EAST neutral beam injection into plasmas with a low-edge safe factor of q95=3.5 and a n = 2 resonant magnetic perturbation (RMP) during a sawtooth crash. Harmonics of the m/n = 1/1 mode (m and n being the poloidal and toroidal mode numbers) develop simultaneously. A sawtooth crash is triggered when the current of the RMP exceeds a critical value; saturated fishbone phase occurs with small RMP currents. Soft x-ray (SXR) tomographic patterns obtained during sawtooth crashes have annular hot-ring shapes outside the q = 1 surface that are similar to those of double tearing modes triggered by off-axis sawtooth crashes in TFTR reversed magnetic shear plasma. M3D nonlinear simulations confirmed the initial penetration of the n = 2 RMP in the plasma core and further demonstrated a complete sawtooth crash with the appearance of a right annular hot ring near the q = 1 surface. Reconstruction of the trajectory of SXR signals before the crash shows chaotic behavior. Moreover, a high-frequency mode was seen to appear just before the sawtooth crash, accompanied by an edge geodesic acoustic mode. This edge mode is free in the saturated fishbone phase. The role of high-frequency mode in the sawtooth crash remains unclear.

Destabilization of low-frequency modes (LFMs) driven by a thermal pressure gradient in EAST plasmas with q min ⩽ 2

Mode structures and excitation conditions for the low-frequency modes (LFMs) have been investigated in experimental advanced superconducting tokamak (EAST) plasmas with q min ⩽ 2. Two different stages/categories of the LFM instabilities are observed during the oscillation of annular/central collapse events: (I) the upward sweeping frequency of LFMs; (II) the upward frequency jumpsof LFMs. The annular/central events are triggered by the m/n = 2/1 double tearing modes with different q-profiles, while the LFMs are characterized by higher mode numbers m/n = 4/2, 6/3, …, where m and n are the poloidal and toroidal mode numbers, respectively. The maximum radial coverage of the LFMs is located in the annular region of 1.97 2.07 m with the normalized minor radius 0.2 0.4, while the higher-frequency (or upward sweeping frequency) branch is more localized to the radial position of 2 2.02 m (q min). The frequency characteristics of upward sweeps or upward jumps of the LFMs are mainly attributed to the change in the q-profile, e.g. the upward sweeping frequency in stage I is caused by q min decreasing. Accordingly, the linear wave properties of LFMs in EAST with weak/reversed magnetic shear are studied numerically and analytically based on a general fishbone-like dispersion relation. Without considering the contribution of energetic ions, it is shown that the LFM with Alfvénic polarization is an MHD-unstable kinetic ballooning mode with frequency of the order of the ion diamagnetic drift frequency. Several important factors for the excitation of LFM instability are analyzed: (1) the role of energetic ions is unimportant, and the LFMs can be excited under the two conditions of with/without energetic ions; (2) the higher τ = T e/T i with larger η i = L ni /L Ti are required, namely the normalized pressure gradient α ∝ (1 + τ)(1 + η i) should be large enough to overcome the stability effect of finite field line bending; (3) the weak/reversed shear q-profile with q min ⩽ 2 and suitable S ≡ (r/q)(q″)1/2 are required.

Stabilization of double tearing mode growth by resonant magnetic perturbations

It is well known that for non-monotonic profiles of the safety factor q with two q = m/n resonant surfaces inside the plasma (m/n being the poloidal/toroidal mode numbers), the low-m double tearing modes (DTMs) are usually unstable, especially for plasmas with a high bootstrap current fraction as required for the steady operation of advanced scenarios. The effect of applied resonant magnetic perturbations (RMPs) on the m/n = 2/1 DTM growth is investigated numerically in this paper using two-fluid equations. The DTM growth is found to be stabilized by moderate static m/n = 2/1, 4/2 or 6/3 RMPs below their penetration threshold if the distance between the two resonant surfaces and the local plasma rotation velocity at the outer resonant surface are sufficiently large. The outer magnetic island is stabilized due to the change of the local plasma current density gradient around the outer resonant surface caused by RMPs, while the inner island growth is stabilized by the bootstrap current perturbation in the negative magnetic shear region. The mode stabilization is more effective for a higher electron temperature, indicating a possible method to improve the DTM stability in a fusion reactor.

Effects of compressibility and guiding field on the Kelvin–Helmholtz instability in weakly reversed magnetic shear configurations

The effects of compressibility and guiding field on the Kelvin–Helmholtz (KH) instability in a double resonant surface system are numerically investigated by using a compressible magnetohydrodynamics model. The linear growth rate and the nonlinear saturation amplitude are analyzed for different parameters. A transition from double tearing modes to the KH instability occurs depending on the compressibility and guiding field with a weak magnetic shear. These results may deepen our understanding of the properties of the KH instability with double resonant surfaces in both space and laboratory plasmas.

Influence of equilibrium magnetic shear on the double tearing modes for different rational surface spacing

Influence of equilibrium magnetic shear on the double tearing modes for different rational surface spacing

Effect of kinetic ions on the toroidal double-tearing modes

We investigate the effects of kinetic ions on double-tearing modes (DTMs) using the gyrokinetic particle-in-cell simulation code GEM with a gyrokinetic ion/fluid electron hybrid model. It is found that the ion kinetic effects can decrease the growth rate of the DTMs. This effect is more significant for stronger coupling of DTMs with smaller distance between the rational surfaces. Kinetic ions can also enhance the coupling effect between the two rational surfaces. Energy transfer analyses between particles and wave fields show that the stabilizing effect of kinetic ions comes mainly from the perpendicular magnetic drift of ions in the coupling region and around the outer rational surface.