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In this paper, we have studied the influence of resonance magnetic perturbation (RMP) on the double tearing modes induced by asymmetric magnetic perturbation in a 2D geometry based on the MHD model...

In tokamak operation, the control of dangerous MHD instabilities, possibly in real-time scenarios, must rely on prompt robust diagnostics of the state and stability of the system. The set of magnetic signals measured on the outside of the plasma boundary, based on the Zakharov–Shafranov, Shkarowsky, Wootton (ZSSW) current moments, has been always used for reliable monitoring of key characteristics of the instantaneous equilibrium condition, such as the quantities $\Delta _{\mathrm{ Sh}}$ , the Shafranov centroid shift, $\beta _{p} $ related to the thermal energy content, and $l_{i}$ related to the current profile peakedness. In addition, the fast pickup coils monitor the external magnetic field fluctuations due to internal MHD activity, however, without the possibility of radial localization of the source. Here, we explore the potential usefulness of more complete use of ZSSW moments in association with the information from fast B perturbation signals to detect tearing stability conditions. For clarity, we set up an analysis of the measurable response to tearing perturbations based on an exact equilibrium model, which is an extension of the Solov’ev case with the addition of an equilibrium, nonuniform plasma rotation $\Omega (\psi)$ . The relation of the selected (externally measurable) ZSSW moments to the calculated stability index is mapped for different rotation values. The footprint of the stability condition $\Delta ' on some current moments on the outer surface can then identify stability boundaries for different rotation conditions. This first discussion on an idealized exact model is proposed for testing the concept for application to realistic equilibria since it relies on a few externally monitorable quantities and very basic assumptions on the tearing modes physics.

The fast pressure crash associated with the nonlinear evolution of DTMs is investigated using a three-dimensional toroidal and nonlinear MHD code CLT. Our simulation results indicate that the fast pressure crash is not directly related to the explosive growth of DTM because the explosive growth of the DTM takes place in the earlier nonlinear phase and the fast pressure crash happens in the much later nonlinear stage. It is also found that the explosive growth lasts much longer than the fast pressure crash. At almost the end of the explosive growth phase, the magnetic fields around the magnetic axis become stochastic, and a strong radial flow is generated. The radial flow can effectively transfer the hot plasma around the magnetic axis into the outer cold region and consequently leads to the fast crash of the pressure in the core. As a result, the dependence of the crash time on the resistivity is very weak. It is also found that the geometry of Tokamaks has very weak influence on the fast pressure crash.

In the advanced Tokamak magnetic field configuration, the instability of the double tear mode excited by the diamagnetic shear configuration plays an important role. At the same time, due to the fusion reaction and auxiliary heating means, a large number of fast particles will be generated. The existence of fast particles will affect the growth of double tear mode. In the context of resistive magnetic fluids, the role of fast particle specific pressure in the instability of the double tear mode was analyzed, and the CLT-K program was used to simulate the instability of the fast particle specific pressure to the double-tearing mode to explore the physical mechanism of its impact.

Observations of solar flare ribbons show significant fine structure in the form of breaking wave-like perturbations and spirals. The origin of this structure is not well understood, but one possibility is that it is related to the tearing instability in the flare current sheet. Here we study this connection by constructing an analytical three-dimensional magnetic field representative of an erupting flux rope with a flare current sheet below it. We introduce small-scale flux ropes representative of those formed during a tearing instability in the current layer, and use the squashing factor on the solar surface to identify the shape of the presumed flare ribbons and fine structure. Our analysis suggests there is a direct link between flare-ribbon fine structure and flare current sheet tearing, with the majority of the ribbon fine structure related to oblique tearing modes. Depending upon the size, location and twist of the small-scale flux ropes, breaking wave-like and spiral features within the hooks and straight sections of the flare ribbon can be formed that are qualitatively similar to observations. We also show that the handedness of the spirals/waves must be the same as the handedness of the hooks of the main ribbon. We conclude that tearing in the flare current layer is a likely explanation for spirals and wave-like features in flare ribbons., Comment: ApJ, in press, 16 pages, 14 figures

A method for determining the critical tearing energy in rubber-like materials is proposed. In this method, the energy required for crack propagation in a rubber-like material is determined by the change of recovered elastic energy which is obtained by deducting the dissipated energy due to different inelastic processes from the total strain energy applied to the system. Hence, the classical method proposed by Rivlin and Thomas using the pure shear tear test is modified using the actual stored elastic energy. The total dissipated energy is evaluated using cyclic pure shear and simple shear dynamic experiments at the critical stretch level. To accurately estimate the total dissipated energy, the unloading rate is determined from the time the crack takes to grow an increment. A carbon-black-filled natural rubber is examined in this study. In cyclic pure shear experiment, the specimens were cyclically loaded under quasi-static loading rate of $$0.01~{\rm {s}}^{-1}$$ 0.01 s - 1 and for different unloading rates, i.e. $$0.01$$ 0.01 , $$0.1$$ 0.1 and $$1.0~{\rm {s}}^{-1}$$ 1.0 s - 1 . The simple shear dynamic experiment is used to obtain the total dissipated energy at higher frequencies, i.e. $$0.5$$ 0.5 -$$18~{\rm {Hz}}$$ 18 Hz which corresponds to unloading rates $$0.46$$ 0.46 -$$16.41~{\rm {s}}^{-1}$$ 16.41 s - 1 , using the similarities between simple and pure shear deformation. The relationship between dissipated energy and unloading stretch rate is found to follow a power-law such that cyclic pure shear and simple shear dynamic experiments yield similar result. At lower unloading rates (i.e. $${\dot{\lambda }}_{\rm {U}} < 1.0~{\rm {s}}^{-1}$$ λ ˙ U < 1.0 s - 1 ), Mullins effect dominates and the viscous dissipation is minor, whereas at higher unloading rates, viscous dissipation becomes significant. At the crack propagation unloading rate $$125.2~{\rm {s}}^{-1}$$ 125.2 s - 1 , the viscous dissipation is significant such that the amount of dissipated energy increases approximately by $$125.4\%$$ 125.4 % from the lowest unloading rate. The critical tearing energy is obtained to be $$7.04~{\rm {kJ}}/{\rm {m}}^{2}$$ 7.04 kJ / m 2 using classical method and $$5.12~{\rm {kJ}}/{\rm {m}}^{2}$$ 5.12 kJ / m 2 using the proposed method. Hence, the classical method overestimates the critical tearing energy by approximately $$37.5\%$$ 37.5 % .

Tearing mode creates magnetic islands and easily causes a disruption in the tokamak. A new method to control the tearing mode has been proposed. The external resonant magnetic perturbation (RMP) coils are used to create modulated magnetic perturbation in certain magnetic island phase region. In the region of - 0.5 π to - π , RMP has an accelerating and stabilizing effects on tearing mode. To verify the mechanism, two variable frequency current-pulse power supplies have been developed, which have the maximum frequency of 8 kHz and the current of 3 kA. In this paper, the phase-detection-based and frequency feedback control are introduced. A fast island phase identification theory is used in the feedback control system to provide the magnetic island phase information in real time. It uses a poloidal array of Mirnov probes to identify the O-point of the magnetic island. There is a current amplitude feedback control to adjust the strength of RMP field. A bypass is used in the power supply to establish the output current quickly in initial period and protect the inverter. Finally, the feedback control system and the new power supplies have been used on the J-TEXT tokamak and two control methods are compared.

Accurate real-time measurements of magnetic island and electron cyclotron current drive (ECCD) locations are essential for efficient suppression of the neoclassical tearing mode (NTM). To determine these locations, many control systems rely on motional Stark effect constrained equilibria reconstruction and real-time Thomson scattering with TORBEAM current drive evaluation and therefore require time-intensive cross-calibration of at least two different diagnostics. Here we present a simpler, proof-of-concept analysis that uses only a single diagnostic (a radial array electron cyclotron emission radiometer) for the simultaneous determination of both the radial position of a magnetic island and the deposition location of ECCD. Measurements are compared with the modified Rutherford equation to demonstrate the effect of ECCD alignment on NTM suppression.

The stabilization of tearing modes with rf driven current benefits from the cooperative feedback loop between rf power deposition and electron temperature within the island. This effect, termed rf current condensation, can greatly enhance and localize the current driven within magnetic islands. It has previously been shown that the condensation effect opens the possibility of passive stabilization with broad rf profiles, as would be typical of LHCD for steady state operation. Here, we show that this self-healing effect can be dramatically amplified by operation in a hot ion mode, due to the additional electron heat source provided by the hotter ions.

We report the experimental observation of seed magnetic island formation by nonlinear three-wave coupling of magnetic island triplets. In this experiment, disruptive 2,1 islands are seeded by the coupling of 4,3 and 3,2 tearing modes to a central 1,1 sawtooth precursor. Three-wave interactions between these modes are conclusively identified by bispectral analysis, indicating fixed phase relationships in agreement with theory. This new observation of this seeding mechanism has important implications for future reactors that must operate in stable plasma equilibria, free of disruptive 2,1 islands.

Magnetic reconnection is a key fundamental process in collisionless plasmas that explosively converts magnetic energy to plasma kinetic and thermal energies through a change of magnetic field topology in an electron-scale central region called the electron diffusion region. Past simulations and observations demonstrated that this process causes efficient energy conversion through the formation of multiple macro-scale or micro-scale magnetic islands/flux ropes. However, how these different spatiotemporal scale phenomena are coupled is still poorly understood. In this study, to investigate the turbulent evolution of magnetic reconnection, we perform a new large-scale fully kinetic simulation of a thin current sheet considering a power-law spectrum of initial fluctuations in the magnetic field as frequently observed in the Earth’s magnetotail. The simulation demonstrates that during a macro-scale evolution of turbulent reconnection, the merging of macro-scale islands results in reduction of the rate of reconnection as well as the aspect ratio of the electron diffusion region. This allows the repeated, quick formation of new electron-scale islands within the electron diffusion region, leading to an efficient energy cascade between macro- and micro-scales. The simulation also demonstrates that a strong electron acceleration/heating occurs during the micro-scale island evolution within the EDR. These new findings indicate the importance of non-steady features of the EDR to comprehensively understand the energy conversion and cascade processes in collisionless reconnection.

Dynamic mitigation for the tearing mode instability in the current sheet in collisionless plasmas is demonstrated by applying a wobbling electron current beam. The initial small amplitude modulations imposed on the current sheet induce the electric current filamentation and the reconnection of the magnetic field lines. When the wobbling or oscillatory motion is added from the electron beam having a form of a thin layer moving along the current sheet, the perturbation phase is mixed and consequently the instability growth is saturated remarkably, like in the case of the feed-forward control.

In this work, a surface current plasma model is coupled to a volume integral formulation for studying the interaction between tokamak tearing modes (TMs) and the machine metallic structures surrounding the plasma. By evaluating the passive response on a set of pickup probes, TMs' amplitude is estimated.

Tearing modes are a major concern for large tokamak devices. Therefore their detection and characterization is of importance for timely countermeasures to avoid significant impact on the discharge or at least prevent possible machine damage. In case of phase-locked tearing modes, the poloidal variation of induced magnetic field fluctuation depends on the amplitudes of and the phase difference between the individual modes. This affects mode detection and identification when not considered appropriately. The phase between phase-locked (2, 1) and (3, 1) tearing modes in ASDEX Upgrade has been determined from local electron temperature and magnetic measurements independently. It is shown that the modes can be in phase at any poloidal position starting from the low field side plasma midplane over the plasma top to the high field side midplane. This observation invalidates the widespread assumption that phase-locked tearing modes are in phase near the low field side midplane. Dependence of the in-phase position on both, the plasma pressure and the plasma rotation velocity, is observed.

The effect of externally applied resonant magnetic perturbations (RMPs) on the local equilibrium plasma current density profile is studied numerically based on two-fluid equations in simplified cylindrical geometry. It is found that a moderate RMP below its penetration threshold, via non-linear mode coupling, induces a parallel electric field around its rational surface that can significantly change the local flux-surface-averaged current density gradient. At a given RMP amplitude, the modification of the current density profile increases with increasing electron temperature, and it significantly depends on the bi-normal electron fluid velocity at the resonant surface. The effect of this modification on the magnetic island growth is demonstrated by the example of small m/n = 2/1 islands (m/n being the poloidal/toroidal mode numbers), driven by an unfavorable plasma current density profile and bootstrap current perturbation. The 2/1 mode growth is stabilized by moderate static 4/2 or 6/3 RMPs if the local electron fluid velocity is in the ion drift direction or sufficiently large in the electron drift direction. These results reveal that a weakly three-dimensional equilibrium, containing a moderate 4/2 RMP and the associated shielding current, can be more stable against the 2/1 mode, which often causes tokamak plasma major disruptions.

Numerical simulation of tearing mode suppression by applying non-resonant magnetic perturbations (non-RMPs) has been carried out in the J-TEXT tokamak based on the NIMROD (Non-Ideal Magnetohydrodynamics with Rotation, Open Discussion) code. In the present work, we investigate the growth of n = 1 and n = 2 (n is the toroidal mode number) magnetohydrodynamics (MHD) instabilities, mainly as tearing modes, in response to different modes and amplitudes of non-RMPs generated by the external helical coils. It is found that the tearing mode can be stabilized by non-RMPs, especially for the non-RMPs with mode m/n = -1/1 (m is the poloidal mode number) and coil current I MP = 4kA.

We investigate the asymptotic scaling of the growth rate and of the characteristic layer width of reduced-MHD tearing modes occurring in thin current sheets when reconnection depends on two non-ideal parameters. For this purpose, we use a new multi-precision finite difference eigensolver. The viscous-resistive regime, the warm-resistive regime that includes both resistivity and electron temperature effects, the warm-inertial regime in which a finite electron inertia replaces resistivity in allowing reconnection, and the inertial-resistive regime that includes both electron inertia and resistivity are investigated. Previous analytical results of the first three regimes are recovered. For all regimes, the scalings of the width of the reconnecting layer are provided in the different limits of the wavelength spectrum, and general estimates for the fastest growing modes are obtained and generalized to different magnetic equilibria. Implications for the disruption of evolving current sheets are discussed.

We investigate the effects of runaway electron current on the dispersion relation of resistive magnetohydrodynamic modes in tokamaks. We present a new theoretical model to derive the dispersion relation, which is based on the asymptotic analysis of the resistive layer structure of the modes. It is found that in addition to the conventional resistive layer, a new runaway current layer can emerge whose properties depend on the ratio of the Alfv\'en velocity to the runaway electron convection speed. Due to the contribution from this layer, both the tearing mode and kink mode will have a real frequency in addition to a growth rate. The derived dispersion relation has been compared with numerical results using both a simplified eigenvalue calculation and a M3D-C1 linear simulation, and good agreement is found in both cases., Comment: 10 pages, 6 figures

Despite high demonstrated efficiency, lower-hybrid current drive (LHCD) has not been considered localized enough for neoclassical tearing mode (NTM) stabilization in tokamaks. This assessment must be reconsidered in view of the RF current condensation effect. We show that an island with a central hot spot induces significant localization of LHCD. Furthermore, in steady state tokamaks where a significant amount of current is provided by LHCD, passive stabilization of NTMs may occur automatically, particularly as islands become large, without requiring precise aiming of the wave power., 26 pages, 6 figures, pre-print

Young stars are surrounded by a circumstellar disk of gas and dust, within which planet formation can occur. Gravitational forces in multiple star systems can disrupt the disk. Theoretical models predict that if the disk is misaligned with the orbital plane of the stars, the disk should warp and break into precessing rings, a phenomenon known as disk tearing. We present observations of the triple star system GWOrionis, finding evidence for disk tearing. Our images show an eccentric ring that is misaligned with the orbital planes and the outer disk. The ring casts shadows on a strongly warped intermediate region of the disk. If planets can form within the warped disk, disk tearing could provide a mechanism for forming wide-separation planets on oblique orbits., 63 pages, 4+13 Figures, 6 Tables, published at Science

Binary systems exert a gravitational torque on misaligned discs orbiting them, causing differential precession which may produce disc warping and tearing. While this is well understood for gas-only discs, misaligned cirumbinary discs of gas and dust have not been thoroughly investigated. We perform SPH simulations of misaligned gas and dust discs around binaries to investigate the different evolution of these two components. We choose two different disc aspect ratios: A thin case for which the gas disc always breaks, and a thick one where a smooth warp develops throughout the disc. For each case, we run simulations of five different dust species with different degrees of coupling with the gas component, varying in Stokes number from 0.002 (strongly coupled dust) to 1000 (effectively decoupled dust). We report two new phenomena: First, large dust grains in thick discs pile up at the warp location, forming narrow dust rings, due to a difference in precession between the gas and dust components. These pile ups do not form at gas pressure maxima, and hence are different from conventional dust traps. This effect is most evident for St ~ 10 - 100. Second, thin discs tear and break only in the gas, while dust particles with St > 10 form a dense dust trap due to the steep pressure gradient caused by the break in the gas. We find that dust with St < 0.02 closely follow the gas particles, for both thin and thick discs, with radial drift becoming noticeable only for the largest grains in this range., Accepted for publication in MNRAS, 11 pages, 9 figures

Male and female genitalia usually evolve rather quickly and yield remarkable diversity in comparison with other characters. Since natural selection influences shapes indirectly through corresponding properties, links between genital features and their functional mechanisms are critically important, in order to understand evolutionary mechanisms of morphological diversity. In this chapter, we numerically modeled hyper-elongated male and female genitalia especially focusing on their physical properties previously observed in cassidine beetles. These male beetles bear an elongated flagellum and female genitalia have a helically coiled spermathecal duct including some reversal turns called knots. The flagellum was demonstrated to have stiffness gradient, which results in a specific mechanical behavior of the entire copulatory system, because the flagellum stiffness properties might influence its motion within the female spermathecal duct. These properties might be partially due to rubber-like protein resilin that is assumed to be an almost perfect elastomer in insects. Resilin is an ideal material for elastic joints that are subjected to repeated cyclical stress. During the lifetime of an insect, resilin shows neither tearing nor fatigue, when stressed within its natural limits. The resilin, almost unique among biological materials returns back to its original position, when the stress is relieved. In this chapter, the simple sphere-on-the-flat geometry was used in a microindentation experiment as one with the most precise determined boundary conditions. Then the mechanical response of resilin was simulated using a generalized Maxwell model with two characteristic time constants and alternatively using a 1D model with just one characteristic time constant. We also applied newly-developed method of dimension reduction allowing us utilization of functional model along with the generalized Maxwell model.

Triggerless 2/1 neoclassical tearing modes (NTMs), i.e. 2/1 NTMs that originate from unstable safety factor profiles (with positive classical stability index at zero island width, i.e.Delta(t)(0) > 0) and saturate neoclassically under the effect of perturbed bootstrap current, have been observed reproducibly in TCV discharges with a strong near-axis electron cyclotron current drive (ECCD). An unexpected density dependence of the onset of these NTMs is newly observed based on the statistics of many TCV discharges, suggesting a certain range of density within which the NTMs can occur. The range is found to broaden with increasing near-axis ECCD power. Based on a different set of experiments and corresponding interpretative simulations with the modified Rutherford equation (MRE), a simple analytical model for Delta(t)(0) has been developed. This model proves to explain well the observed density dependence of mode onset, resulting from the density dependence of the stability of ohmic plasmas and of the ECCD efficiency. The simulations have also quantified various effects and reproduced self-consistently the entire island width evolution of the triggerless NTM, from the onset as a tearing mode at zero island width to the neoclassical saturation as an NTM. The standard terms of the MRE model used in the paper are relevant for NTMs that are seeded by other mechanisms.

Understanding the effect of Alfvén eigenmodes (AEs) and neoclassical tearing modes (NTMs) on fast ions is highly important for fusion reactors due to potentially strong resonant interactions between the fast ions and the modes. Here, we use the four-view fast-ion D-alpha (FIDA) diagnostic installed in the DIII-D tokamak to reconstruct the fast-ion velocity distribution at two radial positions during two sequential discharges with strong and weak mode activity, respectively. The velocity-space coverage of the diagnostics, however, only allows reliable reconstructions of fast ions with positive pitches. Therefore, we suggest new tomographic inversion methods relying on prior information outside the well-diagnosed region. We find that within the population of fast ions with positive pitches, ions, at all energies, are transported away from the measurement volumes. Comparisons between the reconstructions and kick model simulations, where the mode activity is considered, reveal that low-frequency modes such as the NTMs and low-frequency AEs contribute significantly to the positive-pitch fast-ion transport in the central measurement volume, whereas TAEs and EAEs become important farther out and are responsible for decreased fast-ion confinement.

New studies identify the critical parameters and physics governing disruptive neoclassical tearing mode (NTM) onset. An m/n = 2/1 mode in DIII-D that begins to grow robustly after a seeding event (edge localized mode ELM or sawtooth precursor and crash) causes the mode rotation to drop close to the plasma’s E r = 0 rest frame; this condition opens the stabilizing ion-polarization current ‘gate’ and destabilizes an otherwise marginally stable NTM. Our new experimental and theoretical insights and novel toroidal theory-based modeling are benchmarked and scalable to ITER and other future experiments. The nominal ITER rotation at q = 2 is found to be stabilizing (‘gate closed’) except for MHD-induced transients that could ‘open the gate’. Extrapolating from the DIII-D ITER baseline scenario (IBS) discharges, MHD transients are much more likely to destabilize problematic robustly growing 2/1 NTMs in ITER; this makes predictions of seeding and control of both ELMs and sawteeth imperative for more than just minimizing divertor pulsed-heat loading.

The 2/1 tearing mode instability during the plateau phase of the plasma current time evolution has been observed since the start of the Globus-M spherical tokamak operation. Because of a small amount of shots in a database, the poloidal beta dependence of the instability growth rate had been investigated with an insufficient accuracy. In this work, a more detailed analysis has been produced that proves the fact that the observed modes can be treated as neoclassical. The saturated island width as a function of poloidal beta has been obtained; a time evolution of the magnetic island has been calculated from the Mirnov magnetic coil signals and compared to the modified Rutherford theory, and the critical magnetic island width has been found for the neoclassical tearing mode (NTM) of the Globus-M tokamak.

The linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode are presented. CLT is an explicit finite difference code, while M3D-C1 is an implicit finite element code. Although the implementations of CLT and M3D-C1 are totally different, we find that the simulation results of the resistive-kink mode and the m / n = 2 / 1 tearing mode from M3D-C1 and CLT are almost the same, including the linear and nonlinear growth rates, the mode structures, the nonlinear saturation levels, the Poincare plots, and the scaling laws . This confirms that the nonlinear results for the 1/1 resistive-kink mode and 2/1 tearing mode are accurate and reliable.

The nonlinear dynamics of the m/n = 2/1 double tearing mode (DTM) in ITER are systematically studied using the three-dimensional toroidal magnetohydrodynamic code, CLT. We carefully investigate the effects of the radial locations and magnetic shear strengths of the inner and outer rational surfaces r 1, r 2, s 1 and s 2, as well as the safety factor at the magnetic axis q 0 on DTM. It is found that the explosive burst takes place only with the moderate separation of the two rational surfaces or the stronger magnetic shear strength in which the strong interaction of magnetic islands in the two rational surfaces happens in the early nonlinear phase of the island development. The explosive burst can result from either the direct mutual driving associated with the fast growth island in the two rational surfaces or a strong nonlinear mode–mode coupling. For a large separation and a weak shear strength of the two rational surfaces, the magnetic islands saturate without strong interaction with each other, and (w in + w out)/2 is always below the separation Δr s. For a small separation, the kinetic evolution of DTM only exhibits an oscillation with a very low level and then decreases.

The 2D electron-magnetohydrodynamics (EMHD) dominant tearing mode in an electron-skin-depth-scale current sheet (ECS) is further studied. The resistive diffusion is proved to be insignificant at the scale. Electron inertia leads to the expansion of the “inner region” as well as a wider saturation island and the invalidity of the boundary layer approximation. The unstable tearing mode index Δ′ thus decreases dramatically from that in classical asymptotic theory. As for nonlinear evaluation, the inverse spectral cascade and the flattening of the m = 0 anti-parallel asymptotic magnetic field will result in an m = 1 final island after nonlinear coupling in a long ECS. A rapid normal saturation transition is observed and only expected for a larger wave number due to the growth rate dependence on the wave number being a single humped function. A linear analysis of the EMHD tearing mode is also presented for the force-free equilibrium. With a strong guide field, it shows that the tearing mode can be suppressed by the shear flow. Nonlinear simulation results with specific parameters then showed that the dynamic structures in the current sheet are consistent with the observation in the Earth’s turbulent magnetosheath.

An investigation of the characteristics of double-tearing modes (DTMs) and the influence of the kinetic effects of ions has been carried out with the gyrokinetic code GTC. A simplified linear fluid and a gyro-kinetic model has been used for the electron and ion dynamics, respectively. It was found that as the separation of the rational surfaces was increased, the growth rates of DTMs were enhanced and the DTM system tended to decouple into a system of two single-tearing modes. When the width between the rational surfaces is larger, the corresponding amplitude of the inner-tearing mode is smaller. We also demonstrate that the existence of thermal ions destabilizes the DTM. The toroidal effect also has a mild destabilizing effect ( ∼ 8 % ) on the growth of DTMs, which is consistent with the theoretical analysis.

The mechanism of rapid frequency chirping for neoclassical tearing modes (NTMs) is studied. Resonance between NTMs and trapped energetic ions can provide an additional torque to change the evolution of frequency. Whether the frequency rises or falls depends on the direction of island propagation. If the island propagates in the direction of ion diamagnetic drift, the frequency will be increased dramatically and rapidly. If the island propagates in the direction of electron diamagnetic drift, the frequency will be reduced to a lower value. The predicted chirping time is consistent with experimental results in DIII-D (Liu et al 2020 Nucl. Fusion 60 112009).

In JT-60U tokamak, low frequency modes (LFMs) have been reported recently when the m/n = 2/1 neoclassical tearing mode grows sufficiently. LFMs show the low mode frequencies (< 20 Hz) with the rotation direction to the counter-current direction (ctr-direction) toroidally even if the strong injection power to the co-current direction (co-direction) is applied by tangential neutral beam injections. As a candidate to drive the rotation to the ctr-direction, the neoclassical toroidal viscosity (NTV) torque is investigated in this study. Indeed, when a LFM is observed, the estimated "offset velocity" of the NTV torque is larger than the observed toroidal velocity around the q = 2 surface and results in the torque to the ctr-direction (ctr-torque). The torque balance between the NTV torque and the torque induced by the eddy current on the resistive wall is investigated. It is found that the rotation to the ctr-direction having the low mode frequency is obtained as the result of the ctr-torque to the offset velocity. Our investigation suggests the requirement to include the NTV torque in the modelling on the mode locking of neoclassical tearing modes in tokamaks.

The dynamo effect of tearing modes is derived using the resistive magnetohydrodynamics equations. The dynamo effect is divided into two components, parallel and perpendicular to the magnetic field. First, the force-free plasma is considered. It is found that the parallel dynamo effect drives opposite current densities on either sides of the rational surface ( k·B=0), completely flattening the λ=j·B/|B|2 profile near the rational surface. This may be a possible explanation for the Taylor relaxation mechanism. In contrast, farther from the rational surface, the parallel dynamo effect is much smaller, and the nonlinear dynamo form approximates the quasilinear form. Second, a pressure gradient is included. It is found that a finite λ gradient can exist after modification by the parallel dynamo effect. The perpendicular dynamo effect is found to eliminate the pressure gradient near the rational surface. Our results provide a further basis for the assumption that the current density is flat in the magnetic island for tearing mode theory.

Investigation of neoclassical tearing mode and its suppression by electron cyclotron current drive (ECCD) has been carried out in HL-2M tokamak. The current driving capability of the electron cyclotron wave is evaluated. It is found that the deposition location can be effectively controlled by changing the poloidal angle. The validation of electron cyclotron wave heating and current driving has been demonstrated for the upper launcher port. We show that 3.0 MW and 2.5 MW modulated ECCD can completely stabilize (2,1) and (3,2) NTMs, respectively. The non-modulated ECCD, radial misalignment as well as current profile broadening have deleterious effect on the NTM stabilization. The time required for suppression of (3,2) mode is shorter than that required for the suppression of (2,1) mode. Moreover, the time needed for complete stabilization at different initial island width has been quantitatively presented and analyzed.

Simulations and theory are presented of an ITER locked mode thermal quench (TQ). In present experiments, locked mode disruptions have a long precursor phase, followed by a rapid termination and thermal quench, which can be identified with a resistive wall tearing mode (RWTM). In ITER, the RWTM will be slowed by the highly conductive vacuum vessel. The rapid termination might be absent, and the plasma could remain in the precursor phase. If the edge temperature is in the collisional regime, the TQ would proceed on a long timescale, limited by the RWTM to almost $100ms.$ This is an important self mitigating effect.

In the so-called “hybrid” shots characterized by shear-free core with safety factor slightly above 1, an interesting yet unexplained chirping activity has been observed in tokamaks with a high fast ion content. Namely, fishbone modes with m = n = 1 and downward frequency chirping are accompanied by frequency jumps of the tearing modes with m = n + 1 and n > 1 [here m(n) is the poloidal (toroidal) mode number]. In the present work, these related phenomena are attributed to a combination of the two impulsive toroidal torques: (1) recoil torque applied to the shear-free core during fishbone burst and (2) electromagnetic torque applied to the tearing layer. We have developed a quasi-linear model in which fast transients of the plasma toroidal rotation induced by these torques qualitatively reproduce the observed chirping dynamics.

The Electron Cyclotron Upper Launcher is an eight beamline ITER antenna whose main goal is to drive current locally, inside magnetic islands that may form on the q = 3/2 or 2 rational magnetic flux surfaces, in order to stabilize neoclassical tearing modes (NTMs). The primary vacuum boundary at the port plug extends into the port cell region through the ex-vessel mm-wave waveguide components, defining the so-called First Confinement System (FCS). Thermal expansion, seismic events and plasma disruption events result in displacements of the vacuum vessel, relative to the tokamak building, that are transferred to the FCS at its interfaces with the port plug. In absence of suitable inline waveguide bellows, the adaptation to such imposed displacements is provided by bending compliance of the straight waveguide sections. An analysis methodology for the ex-vessel components was developed which covers a preliminary set of applicable load combinations throughout the system life-cycle. This methodology describes the FCS numerical model developed to assess the system against independent events as well as the load combination strategy followed to combine the results. Some simplifications like the simulation of the Vertical Displacement Events (VDEs) by a static approach are considered in order to develop this methodology. The analysis of the results shows that the miter bends and the pieces of waveguides rigidly attached to the port cell ceiling are the most demanded components for most of the considered load combinations.

The formation of a magnetic island as a result of tearing instability can be interpreted as the bifurcation of an axisymmetric equilibrium configuration at which nested magnetic surfaces are preserved. The modification of the current density profile due to such bifurcation is studied using the Hamiltonian formalism. In the case of a long narrow island, the gradient profile changes to a profile with an extremum on the axis of the magnetic island.

In this work, we explore the influence of perturbative wavelengths, shorter than those usually considered, on the growth rate γ of the tearing modes. Thus, we adopt an extended form of Ohm’s law, which includes a finite relaxation time for the current density, due to inertial effects of charged species. In the long wavelength limit, we observe the standard γ of the tearing modes. However, in the short wavelength limit, we show that γ does not depend on the fluid resistivity any longer. Actually, we find out that γ now scales with the electron number density ne as γ ~ ne−3/2. Therefore, through a suitable combination of both limiting results, we show that the standard γ can be substantially amplificated, even by moderate shortenings of perturbative wavelengths. Further developments of our theory may contribute to the explanation of the fast magnetic reconnection of field lines, as observed in astrophysical plasmas.

Neoclassical tearing mode (NTM) can degrade plasma confinement or even cause disruptions in existing tokamaks. Stabilization of the NTMs by radio frequency (rf) current is investigated by the modified Rutherford equation (MRE) in this paper. In a range of parameters, the required rf current for mode stabilization is obtained, which is linearly proportional to the bootstrap current density for both modulated current drive (MCD) and non-modulated current drive (NMCD), linearly (quadratically) to the radial width of the rf current for MCD (NMCD), and quadratically to the radial deviation of the rf current from the rational surface for both MCD and NMCD.

Recently, it has been realized that magnetic plasma turbulence and magnetic field reconnection are inherently related phenomena. Turbulent fluctuations generate regions of a sheared magnetic field that become unstable to the tearing instability and reconnection, thus modifying turbulence at the corresponding scales. In this contribution, we give a brief discussion of some recent results on tearing-mediated magnetic turbulence. We illustrate the main ideas of this rapidly developing field of study by concentrating on two important examples -- magnetohydrodynamic Alfv\'en turbulence and small-scale kinetic-Alfv\'en turbulence. Due to various potential applications of these phenomena in space physics and astrophysics, we specifically try not to overload the text by heavy analytical derivations, but rather present a qualitative discussion accessible to a non-expert in the theories of turbulence and reconnection., Comment: A book chapter in AGU Book "Solar and Heliospheric Plasma Structures" (to appear)

Origami structures have been proposed as a means of creating three-dimensional structures from the micro- to the macroscale and as a means of fabricating mechanical metamaterials. The design of such structures requires a deep understanding of the kinematics of origami fold patterns. Here we study the configurations of non-Euclidean origami, folding structures with Gaussian curvature concentrated on the vertices, for arbitrary origami fold patterns. The kinematics of such structures depends crucially on the sign of the Gaussian curvature. As an application of our general results, we show that the configuration space of nonintersecting, oriented vertices with positive Gaussian curvature decomposes into disconnected subspaces; there is no pathway between them without tearing the origami. In contrast, the configuration space of negative Gaussian curvature vertices remains connected. This provides a new, and only partially explored, mechanism by which the mechanics and folding of an origami structure could be controlled.

In this chapter we explore the linear onset of one of the most important instabilities of resistive magnetohydrodynamics, the tearing instability. In particular, we focus on two important aspects of the onset of tearing: asymptotic (modal) stability and transient (non-modal) stability. We discuss the theory required to understand these two aspects of stability, both of which have undergone significant development in recent years.

International audience

Tearing modes are the resistive magnetohydrodynamic (MHD) modes, which occur in all conventional tokamaks. They are known to degrade plasma confinement by creating magnetic islands which often lead to plasma disruption if their growth is not controlled. They have been extensively studied both theoretically and experimentally, as controlling them is foremost priority for every tokamak, including ITER and future large size tokamaks. In the high diamagnetic-drift frequency regime, the drift mode couple to tearing modes resulting in drift-tearing modes. The growth rate of Drift-Tearing modes decreases as its real frequency increases. In ADITYA as well as in ADITYA Upgrade tokamak, the frequency spectra of Mirnov signal show multiple frequency bands identified as drift tearing modes. Interestingly, the higher frequencies have precisely been found to be the integral multiples of the fundamental frequency. Further analysis reveals that these frequencies do not belong to different modes but harmonics of a single mode. These harmonics are also observed in the density as well as impurity emission. It has been observed that the harmonics generation is strongly correlated with the presence of high energetic electrons, known as runaway electrons, in the plasma. When the magnetic island grows in size, the runaway electrons get trap in these islands. And as the linear growth rate of tearing modes increases in presence of REs, the island size increases further due to the presence of runaway electrons, leading to harmonics generation.

International audience; This study investigates the tearing of a thin notched sheet when two points on the sheet are pulled apart. The concepts that determine the crack trajectory are reviewed in the general anisotropic case, in which the energy of the fracture depends on the fracture direction. When observed as a flat sheet a purely geometric "tearing vector" is defined through the location of the crack tip and the pulling points. Both the Griffiths's criterion and the "maximum energy release rate criterion" (MERR) predict a fracture path that is parallel to the tearing vector in the isotropic case. However, for the anisotropic case, the application of the MERR leads to a crack path that deviates from the tearing vector, following a propagation direction that tends to minimize the fracture energy. In the case of strong anisotropy, it is more difficult to obtain an analytical prediction of the tearing trajectory. Thus, simple geometrical arguments are provided to give a derivation of a differential equation accounting for crack trajectory, according to the natural coordinates of the pulling, and in the case that the anisotropy is sufficiently weak. The solution derived from this analysis is in good agreement with previous experimental observations.