Physics of tearing mode rotation and slow-down in the RFX-mod tokamak

The effects of ion diamagnetic drift on the high-order harmonic modes in rotating tokamak plasmas are numerically investigated by using a cylindrical reduced magnetohydrodynamic model. It is found that the ion diamagnetic drift has a stabilizing effect on the high-order tearing modes in the small ion diamagnetic drift regime and can excite the high-order Kelvin–Helmholtz (KH) instability in the large ion diamagnetic drift regime. The effects of ion diamagnetic drift flow on the tearing modes are different from those of the poloidal shear flow. Moreover, the combined effect of the two flows on tearing modes and KH modes depends on their relative direction. The eigenmode structures of the tearing modes and KH modes under the influence of ion diamagnetic drift flow are also presented. Finally, the numerical results are verified reasonably by the scalings on resistivity.

Geodesic mode instability driven by the electron current in tokamak plasmas

In the present paper, we numerically investigate the influence of electron and ion diamagnetic drifts on the nonlinear evolution of kink modes in low-resistivity plasma through the updated CLT code based on bi-fluid model. In the linear stage, electron or ion diamagnetic drift leads to a finite and constant mode rotation frequency. The mode frequency changes differently for cases dominated by electron or ion diamagnetic drift. For cases with dominating electron diamagnetic drift, the mode frequency significantly increases in the nonlinear stage; for cases with dominating ion diamagnetic drift, the mode frequency reduces to zero and even changes its sign. The sudden change of the mode frequency is due to the multiple-X-line reconnection process in low-resistivity plasma. During multiple-X-line reconnection, secondary islands merge into the 1/1 main island due to electron or ion diamagnetic drift, and the X-point of the 1/1 main island suddenly moves to another region. Then the local values of the electron or ion diamagnetic term change dramatically, leading to the sudden change of the mode frequency. In high-resistivity plasma, multiple-X-line reconnection does not occur, and the mode frequency will not change its sign. Meanwhile, if the mode rotates too fast, secondary islands will quickly merge into the 1/1 main island and will not develop to a large amplitude.

Tearing modes (TM) are a significant challenge for the operation of large tokamak devices as they cause confinement degradation, can lead to a disruption and are part of the disruption process itself. Detecting and characterising tearing modes is basis for both, improved understanding of their evolution as well as adequate countermeasures. The most harmful tearing mode has the toroidal mode number n = 1, which, due to toroidal coupling, usually consists of different helicities, referred to as poloidal harmonics. Determining the poloidal harmonics requires modelling the magnetic measurements with different challenges for rotating and locked modes. Rotating modes produce frequency-dependent shielding currents in conducting structures that modify the perturbation field. Locked modes require pick-up coils measuring mainly the radial perturbation field component B r , of which there are far fewer than Mirnov coils in ASDEX Upgrade. The agreement between B r coils and Mirnov coils in a model description can be validated in the low frequency range, where all coils observe the modes and shielding currents are important. We employ a three-dimensional finite element method model to calculate the expected magnetic measurements of the perturbation field produced by an n = 1 tearing mode with a single helicity and frequency. The TMs are represented by helical perturbation currents at the corresponding resonant surfaces while the rest of the plasma is treated as vacuum. We show that, besides the vacuum vessel and the passive stabilisation loop, it is crucial to include other in-vessel components of ASDEX Upgrade. The poloidal composition of an n = 1 TM is determined by the linear superposition of modelled TMs with single helicities that best matches the measurements by a poloidal array of Mirnov coils. The resulting amplitudes and phases of the poloidal harmonics give simulated measurements that are found to be in agreement with their measured values for all coil types.

The linearized resistive boundary layer equations, including electron diamagnetic drift terms in general toroidal geometry, are solved numerically to obtain the dispersion relations of the resistive ballooning modes as well as the resistive interchange and tearing modes in both highly collisional and semicollisional regimes. Numerical solutions with high accuracy are obtained using two different numerical methods. Various physical effects, such as electron diamagnetic drift, parallel and perpendicular compression, and average curvature, are investigated in a wide range of equilibrium parameters. The present work is believed to be the first numerical study of the resistive ballooning modes with finite electron diamagnetic drift effects. These numerical solutions cover a wider range of parameter spaces than thus far available from analytical and other numerical methods. The new solutions contain previously known results as special cases. From the present study it is found that for the resistive ballooning and tearing modes, which have similar forms of dispersion relation, ion sound compression gives a substantial stabilizing effect, while ion-polarization compression gives a destabilizing effect and the stabilizing effect of favorable average curvature is significantly reduced as the diamagnetic drift effect increases. For the resistive interchange modes, these numerical results are shown to be in good agreement with previous results.

We investigate tearing modes (TM) driven by current density gradient in collisionless tokamak plasmas by using the electromagnetic gyrokinetic simulation code ORB5. We elucidate the TM width by simulations for flat profiles, as the absence of background diamagnetic flows implies a small rotation speed, while finite gradients are included to investigate the TM rotation. For flat profiles, the initial saturation width of nonlinearly driven magnetic islands is related to the TM linear growth rate; however, large islands in the initial saturation phase are prone to current density redistribution that reduces the island width in the following evolution. Island-induced E×B and diamagnetic sheared flows develop at the separatrix, able to destabilize the Kelvin–Helmholtz instability (KHI). The KHI turbulence enhances a strong quadrupole vortex flow that reinforces the island decay, resulting in a strong reduction of the island width in an eventual steady state. This process is enhanced by trapped electrons. For finite gradients profile, the TM usually rotates in the electron diamagnetic direction but can change direction when the ion temperature gradient dominates the other gradients. The reduced growth of the TM by diamagnetic effects results in a moderate island size, which remains almost unchanged after the initial saturation. At steady state, strong zonal flows are nonlinearly excited and dominate the island rotation, as expected from previous theoretical and numerical studies. When β is increased, the TM mode is suppressed and a mode with the same helicity but with twisting parity, coupled with the neighboring poloidal harmonics, is destabilized, similar to the kinetic ballooning mode.

Measurements are made of the propagation frequency ω of tearing mode magnetic islands in the local Er=0 frame in co-injected neutral beam heated H-mode discharges in the DIII-D tokamak [J. L. Luxon et al., Plasma Physics and Controlled Fusion Research (International Atomic Energy Agency, Vienna, 1987), Vol. 1, p. 159]. Propagation is found to be in the ion diamagnetic drift direction with a frequency more than that of the diamagnetic drift frequency ωi* for modes near the axis and less than ωi* off-axis. This is consistent with the predicted stabilizing polarization current threshold for neoclassical tearing modes for all modes observed only if the condition 0<ω<ωi* is modified by |ωe*| or ηi effects, for example.

A two-dimensional three-component (2D/3C) electron magnetohydrodynamic (EMHD) model is implemented to investigate the linear behavior of collisionless tearing modes in slab geometry. Owing to nonuniformity of thermal pressure and plasma density, the electron diamagnetic drift and Biermann battery effects are involved. The linear structures, growth rate, and real frequency are analyzed with a thin current sheet in the electron inertia scale. The ratio of the electron current to the total current in equilibrium can notably promote the growth of the tearing mode in EMHD. More numerical results then show that the effect of the pressure gradient on the tearing mode is dependent on the plasma beta, stabilizing the mode in a low enough beta limit but destabilizing it with the higher beta. The frequency of the mode caused by the pressure gradient is found to be increasing with it. The Biermann battery effect slightly stabilizes the tearing mode in low beta plasma but is indicated to be significant in much higher beta conditions.

Using Time History of Events and Macroscale Interactions during Substorms observations (radial distance r from 9 to 35 Earth radii, RE), we investigate ion and electron contributions to the cross‐tail current density in the magnetotail current sheet. We analyze plasma pressure measurements (including the contribution from high‐energy particles) and estimate the magnitudes of ion and electron diamagnetic drifts. In the downtail, r > 15RE, region, ion (electron) diamagnetic drifts are shown to provide more than 50% (less than 25%) of the cross‐tail current density at the neutral plane, Bx=0. Conversely, in the near‐Earth region, r≤15RE, the ion (electron) diamagnetic drift contribution to the cross‐tail current density is 20% (50%). The directly measured duskward (dawnward) component of the ion (electron) velocity, vyi (−vye), where y is the GSM direction, is very small (quite large) in the downtail region but large (small) in the near‐Earth region. This systematic discrepancy between the expected values of vyi, −vye (based on estimates of diamagnetic drifts) and the direct measurements of the velocity, vyi, −vye, is consistent with a contribution to the total velocity by an E × B drift caused by an electric field oriented parallel to the x axis, Ex. To decrease the ion (increase the electron) total drift to agree with the measured flows in the downtail region and increase (decrease) this total drift to match the measurements in the near‐Earth region, this Ex would need to be directed earthward at r > 15RE and tailward at r≤15RE. Such an Ex distribution is consistent with the equatorial projection of the Harang discontinuity.

The effect of the toroidal flow and flow shear on the m/n = 2/1 tearing mode (TM) has been investigated in J-TEXT tokamak, where m and n are the poloidal and toroidal mode numbers. It is found that the toroidal flow increment is linearly modulated by the bias current/return current. The frequency of the 2/1 TM is also found to be proportional to the toroidal flow during the bias phase, which is consistent with the theoretical expectation. The electron diamagnetic drift frequency is almost constant before and during the bias phase. Moreover, the statistical analysis shows that the saturated island width of the TM is more correlated to the toroidal flow amplitude than the flow shear, which reveals that the effect of the TM depends on the magnitude of the toroidal flow rather than the flow shear. This result may illustrate the process of the 2/1 TM controlled by the electrode biasing, and suggests that changing the magnitude of toroidal flow is a possible method for stabilizing the TM.

Drift kinetic effects of the neutral beam injection induced passing energetic particles (EPs) on the linear stability of the n = 1 tearing mode (TM) (with the dominant poloidal harmonic of m = 2) are numerically investigated utilizing the MARS-K code (Liu et al 2008 Phys. Plasmas 15 112503), in a tokamak plasma with finite equilibrium pressure and anisotropic thermal transport. In the low plasma pressure regime, it is found that co- (counter-) passing EPs stabilize (destabilize) the TM, agreeing with previous studies. However, as the plasma pressure increases beyond a critical value, it is found that co-passing EPs also destabilize the mode. An in-depth analysis reveals that the net effect of co-passing EPs is a result of competition between the stabilizing contribution from the non-adiabatic drift kinetic terms and the destabilizing contribution associated with adiabatic terms, with the latter becoming more dominant at higher equilibrium pressure. Non-perturbative magnetohydrodynamic-kinetic hybrid modeling also finds that co- and counter-passing EPs modify the TM eigenfunction differently, with the counter-passing EPs enhancing the sideband harmonics. Furthermore, effects of the plasma resistivity and toroidal rotation, as well as that of the equilibrium distribution of EPs in the particle pitch angle space, are also investigated, showing asymmetric results on the TM stability between the co- and counter-passing EPs. The first order finite orbit width correction is found to be stabilizing with co-passing EPs and destabilizing with counter-passing particles. Finally, drift resonances between passing EPs and the TM induce finite frequency to the mode and generate finite net torques inside the plasma, due to the neoclassical toroidal viscosity and the Reynolds stress associated with 3D perturbations.

Ideal MHD yields at best inconclusive predictions about the stability of the LHD heliotron for ⟨β⟩ ⩾ 3%. We investigate the impact of the drift stabilization of ballooning modes for the inward-shifted LHD configuration (vacuum magnetic axis R0 ∼ 3.5 m). The background equilibrium is considered anisotropic in which the neutral beam ions contribute about 1/4 fraction of the total diamagnetic beta, ⟨βdia⟩. A drift corrected ballooning mode equation obtained from the linearized gyrokinetic equation is expanded assuming that the hot particle drifts are much larger than the mode frequency. The fast particle pressure gradients contribute weakly to both the instability drive and the diamagnetic drift stabilization (which is dominated by the thermal ion diamagnetic drifts) for ⟨βdia⟩ ∊ [0, 4.8]%. In the single-fluid limit (diamagnetic drifts ignored), the thermal pressure gradients drive ballooning modes in a broad region encompassing the outer 60–90% of the plasma volume at ⟨βdia⟩ ≈ 4.8%. To stabilize these modes, we find that diamagnetic drift corrections must be invoked (mainly due to the thermal ions). The energetic ion diamagnetic drifts play a role only for low wave number values, kα ⩽ 8. It has been verified that the fast particle drift ordering imposed by the model is amply satisfied for on-axis hot particle to thermal density Nh0/Ni0 ≈ 1% even at high ⟨βdia⟩.

Numerical simulation on the resonant magnetic perturbation penetration is carried out by the newly-updated initial value code MDC (MHD@Dalian Code). Based on a set of two-fluid four-field equations, the bootstrap current, parallel, and perpendicular transport effects are included appropriately. Taking into account the bootstrap current, a mode penetration-like phenomenon is found, which is essentially different from the classical tearing mode model. To reveal the influence of the plasma flow on the mode penetration process, E × B drift flow and diamagnetic drift flow are separately applied to compare their effects. Numerical results show that a sufficiently large diamagnetic drift flow can drive a strong stabilizing effect on the neoclassical tearing mode. Furthermore, an oscillation phenomenon of island width is discovered. By analyzing it in depth, it is found that this oscillation phenomenon is due to the negative feedback regulation of pressure on the magnetic island. This physical mechanism is verified again by key parameter scanning.

Stabilization of tearing modes and neoclassical tearing modes is of great importance for tokamak operation. Electron cyclotron waves (ECWs) have been extensively used to stabilize the tearing modes with the virtue of highly localized power deposition. Complete suppression of the m/n = 2/1 tearing mode (TM) by electron cyclotron resonance heating (ECRH) has been achieved successfully on the J-TEXT tokamak. The effects of ECW deposition location and power amplitude on the 2/1 TM suppression have been investigated. It is found that the suppression is more effective when the ECW power is deposited closer to the rational surface. As the ECW power increases to approximately 230 kW, the 2/1 TM can be completely suppressed. The island rotation frequency is increased when the island width is reduced. The experimental results show that the local heating inside the magnetic island and the resulting temperature perturbation increase at the O-point of the island play dominant roles in TM suppression. As the ECW power increases, the 2/1 island is suppressed to smaller island width, and the flow shear also plays a stabilizing effect on small magnetic islands. With the stabilizing contribution of heating and flow shear, the 2/1 TM can be completely suppressed.

This paper is the first in a series of three with the aim of addressing one of the controversial issues at the magnetopause, namely the location where reconnection first occurs during periods of a large interplanetary magnetic field By. In this first paper, the linear properties of the collisionless tearing mode are reexamined as a function of the guide field By using a formally exact approach for computing the nonlocal Vlasov stability of a current layer. Three distinct parameter regimes are identified depending on the degree to which electron orbits are modified by the guide field in the central region of the current layer. In the limit of both weak and strong guide field, the fastest‐growing tearing mode has a wave vector kx perpendicular to the direction of the current, in agreement with previous theoretical treatments. However, for intermediate values of the guide field where the electrons begin to transition to magnetized orbits, the fastest‐growing modes have a finite wave vector ky in the direction of the current. In this newly discovered regime, the so‐called drift tearing modes have finite real frequency and propagate in the direction of the electron diamagnetic drift with growth rates 10–50% larger than the conventional tearing instability. Maximum growth occurs for a propagation angle in the range θ = tan−1(ky/kx) ≈ 6–10°. These new predictions are confirmed using fully kinetic particle‐in‐cell simulations. The structure of the out‐of‐plane magnetic field perturbation predicted by nonlocal Vlasov theory is examined as a function of guide field. In the limit of a neutral sheet, the quadrupole structure has a characteristic scale near the electron meandering width and shows significant differences with the predictions of linear Hall MHD. The addition of a guide field strongly distorts the quadrupole structure and compresses the spatial extent. In the strong guide field limit, the width of the out‐of‐plane magnetic field perturbation is reduced to the electron gyroscale in the guide field. During the onset phase, these structures represent a distinct signature of the collisionless tearing mode that is significantly different than the typical ion‐scale quadrupole pattern from fast reconnection. Finally, we note that the tearing mode maintains a significant growth rate over a large range of guide field so that component merging cannot be ruled out based on linear theory.