Classical Tearing Mode Research Articles

Two-dimensional effects in the problem of tearing modes control by electron cyclotron current drive

The design of means to counteract robustly the classical and neoclassical tearing modes in a tokamak by localized injection of an external control current requires an ever growing understanding of the physical process, beyond the Rutherford-type zero-dimensional models. Here a set of extended magnetohydrodynamic nonlinear equations for four continuum fields is used to investigate the two-dimensional effects in the response of the reconnecting modes to specific inputs of the localized external current. New information is gained on the space- and time-dependent effects of the external action on the two-dimensional structure of magnetic islands, which is very important to formulate applicable control strategies.

Symmetry breaking and self-consistent rotation of magnetic islands in neoclassical viscous regimes

Classical or neoclassical tearing modes (NTMs) perturb the ideal axisymmetry of tokamaks. As a consequence of symmetry breaking a neoclassical toroidal viscosity (NTV) appears, that depends on the island amplitude. This work shows that in the low collisionality regimes NTV has a key role in determining self-consistently the magnetic island velocity and at the same time modifies significantly the ion polarization current effects on NTM instability. This finding can provide a better understanding of the mechanism of onset of NTMs, observed experimentally, and improve the concepts for their control or avoidance.

Locking of neoclassical tearing modes by error fields and its stabilization by RF current

The locking of neoclassical tearing modes (NTMs) by error fields is studied numerically. In the regime with low mode frequency and large plasma viscosity, the required field amplitude for mode locking is found to be proportional to the plasma viscosity and the mode frequency but inversely proportional to the square of the magnetic island width and the Alfven velocity, being similar to that of the classical tearing mode. This indicates that NTMs will be locked to low amplitude error fields in a fusion reactor. The stabilization of NTMs by RF current in the presence of a static helical field is therefore further investigated. The applied helical field allows one to control the location of the island's o-point to be in the RF wave deposition region, to enable the NTM stabilization by RF current after mode locking. When the island is large enough to be locked by a small amplitude helical field in the desired phase, the island is reduced to a smaller width by RF current compared with the case without the helical field. This suggests a possible way to enhance the stabilization of NTMs by RF current.

Resistive stability of a plasma with runaway electrons

In tokamak disruptions the Ohmic current is often replaced by a current of runaway electrons, which is likely to be more peaked in the center of the discharge than the predisruption current. This raises the question of the resistive stability of the postdisruption plasma, where the equilibrium current is entirely carried by the runaway electrons while the cold (∼10eV) background plasma is relatively resistive. It is found that the linear properties of the classical tearing mode are essentially determined by the cold bulk plasma, and the growth rate is approximately the same as in a plasma without runaways but with the same current profile. The nonlinear saturation amplitude is different, however. In a symmetric plasma slab, the saturated island size is larger when the current is carried by runaway electrons than in the Ohmic case, and undergoes a nonlinear bifurcation when the stability index Δ′ exceeds a critical value.

Thermal Transport Effects on Drift-Tearing Mode

The parameter dependence of the linear growth rate of collisional drift-tearing mode is investigated numerically and analytically using reduced two-fluid equations. Especially, the effect of parallel thermal transport on the growth rate is clarified. It is newly found that the thermal transport parallel to the ambient magnetic field line destabilizes the collisional drift-tearing mode and the diamagnetic drift frequency acts as an instability source in some parameter regime. With large parallel thermal transport coefficient, the collisional drift-tearing mode could be unstable even if the classical tearing mode is stable.

Nonlinear growth of marginally unstable tearing modes

In this work, the nonlinear growth of the classical tearing mode is revisited, using reduced magnetohydrodynamic numerical simulations in cylindrical geometry, emphasizing the limiting scenario of a marginally unstable mode, i.e., 0∼Δ′a<4, where a is the minor radius and Δ′ is the stability parameter [H. Furth, J. Killeen, and M. N. Rosenbluth, Phys. Fluids 6, 459 (1963)]. Conventionally, the mode becomes unstable when Δ′>0 and, eventually, a steady state island width (saturated island width) is achieved. It is found that, contrary to theoretical predictions, no steady state solution is obtained for typically low temperature plasmas with perpendicular viscosity and magnetic Reynolds number below a certain threshold. Plasma inertial forces are identified as the physical mechanism behind such nonlinear evolution.

Effect of sheared equilibrium plasma rotation on the classical tearing mode in a cylindrical geometry

The effect of sheared equilibrium plasma rotation on the stability of tearing modes in an Ohmic (low plasma β) regime is investigated. It is found, by means of numerical MHD simulations in a cylindrical geometry, that plasma rotation in the equivalent toroidal direction can result either in the increase or in the decrease of the instability growth rate. Perpendicular plasma viscosity and plasma rotation shear at the modes’ rational surface play a key role on assessing the effect of shear flow. While destabilizing for low viscosity plasmas (ratio of the resistive to viscous diffusion time scales τR∕τV≪1), for viscous plasmas (τR∕τV>1) shear flow reduces the growth rate. Above a given threshold in the rotation shear (that depends on the ratio τR∕τV) a tearing mode, unstable in the absence of rotation, can be stabilized. The effect of sheared toroidal flow on mode stability, in both viscosity regimes that are considered in the paper, is qualitatively independent of the aspect ratio.

Neoclassical bootstrap current in solar plasma loops

From the similarities in magnetic configurations and plasma behaviors between tokamaks and solar current-carrying plasma loops, we apply the theory of neoclassical bootstrap current in tokamaks to the solar plasma loops. We present a simplified expression of the bootstrap current in the solar plasma loops and find that there may be a considerable component of the neoclassical bootstrap current in some compact current-carrying solar flare loops; e.g. the fraction of the bootstrap current is up to 44.6% of a flare loop of the event that occurred on Aug. 25, 1999. We suggest that the neoclassical effect changes the current distribution and affects the instability of solar plasma loops. Based on the data analysis of SXT/Yohkoh, HXT/Yohkoh, GOES, and NoRP, we find that the timescale of the neoclassical tearing modes is consistent with the rising time of the impulsive phase during the event, while the timescale of the classical tearing modes is much longer than that of the event, which may provide important evidence of the bootstrap current and help us understand the mechanisms of the eruptive events, such as solar flares, prominence, and CMEs.

Increasing the Tokamak Pressure Limit: Tearing Mode Experiments in DIII-D

Since its reconfiguration in 1986, DIII-D has performed a number of experiments involving resistive magnetohydrodynamic (MHD) stability. These were and are directed to understand the conditions in which confinement and beta reducing tearing mode islands form, how to avoid them, and if unavoidable, how to stabilize them. Coils for correction of toroidal nonaxisymmetry have been developed to avoid error field locked mode islands. Basic classical tearing mode stability physics has been confirmed with a state-of-the-art ensemble of profile diagnostics, MHD equilibrium reconstruction, and stability code analysis. Neoclassical tearing mode thresholds and seeding are now much better understood with future large higher field devices expected to be “metastable.” DIII-D is the leader in sophisticated real-time alignment of stabilizing electron cyclotron current drive on otherwise unstable rational surfaces. In all, DIII-D experiments are showing how higher stable beta with good confinement can be maintained without tearing mode islands limiting the plasma pressure.

Effect of sheared flows on classical and neoclassical tearing modes

The influence of toroidal sheared equilibrium flows on the nonlinear evolution of classical and neoclassical tearing modes is studied through numerical solutions of a set of reduced generalized MHD equations that include viscous force effects based on neoclassical closures. In general, differential flow is found to have a strong stabilizing influence leading to lower saturated island widths for the classical tearing mode and reduced growth rates for the neoclassical mode. Velocity shear, on the other hand, is seen to make a destabilizing contribution.

The spherical tearing mode

The spherical tearing mode is the three‐dimensional analog of the classical tearing mode in two dimensions. In three dimensions, loops of field lines connecting magnetic nulls play the role of separatrices. A numerical study of the tearing instability of a three‐dimensional analytic equilibrium containing a closed spherical tearing surface composed of null‐null lines is presented. A new fast tearing instability is identified. Implications of this model for global models of reconnection at the dayside magnetopause are discussed.

Study of nonlinear mode coupling during neoclassical tearing modes using bispectrum analysis

It is sometimes observed that several neoclassical tearing modes (NTMs) are present at the same time. This can happen both at the mode onset of a new NTM or during the nonlinear saturated phase of a given mode. In order to analyse whether nonlinear mode coupling plays any role before or after the onset of NTMs, we use a data analysis tool known as the bispectral technique. This technique has been used for identifying the nonlinear three wave interactions in turbulent plasmas. We have used this technique for the study of NTMs in the tokamak à configuration variable (TCV) and in the Joint European Torus (JET). The analysis during the saturated phase of a 3/2 NTM in a JET discharge suggests a strong coupling between the 3/2, 4/3 and 7/5 modes. This coupling could be responsible for the observed stabilization effect of the 4/3 mode on the 3/2 mode (Sauter O et al 2002 Plasma Phys. Control. Fusion 44 1999). It also demonstrates that this analysis tool can be used to identify high m/n modes, beyond the capabilities of the standard method using poloidal or toroidal probe arrays. As this technique allows a more accurate evaluation of the existence of nonlinear mode coupling, it has been used to study and confirm that no significant nonlinear coupling occurred to trigger two specific NTM onset cases in TCV and JET, where a classical tearing mode and a crash after a long sawtooth period are responsible for the triggering of the necessary seed islands, respectively.

Study of a low β classical tearing mode in DIII-D

The tearing mode stability of low β plasmas is studied experimentally in the DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] tokamak. The linear and nonlinear characteristics of the plasma are measured and compared with theoretical predictions. In contrast to the neoclassical tearing mode, which occurs at high β and is dominated by the bootstrap current effect, in the low β regime the neoclassical bootstrap current effect is minimal. Thus the stability properties are more amenable to comparison with presently available theoretical codes. In the linear phase, the onset of the instability has been found to be predictable from equilibrium reconstruction to agree with the availability of the tearing mode free energy. In the nonlinear phase, the temperature and its fluctuation amplitude and phase have been fitted to the prediction of achieving a perturbed three-dimensional equilibrium with an assumed perturbation eigenfunction.

A new general method for solving the resistive inner layer problem

Classical resistive tearing mode stability in toroidal plasma configurations can be determined by matching the solutions between the ideal and resistive layers. A technique developed for the outer ideal region [S. A. Galkin et al., Phys. Plasmas 7, 4070 (2000)] is generalized and successfully applied to the inner resistive layer problem with irregular singularities. The new method is shown to agree with previous published results and can be used over an extended parameter range. A generalization of the traditional resistive layer equations for nonuniform inertia is also modeled using a variable density distribution and solved using the new technique. The influence of the total mass and density profiles on the matching conditions is studied and discussed.

Predictive capability of MHD stability limits in high performance DIII-D discharges

Results from an array oftheoretical and computational tools developed to treat theinstabilities of most interest for high performance tokamak dischargesare described. The theory and experimental diagnostic capabilitieshave now been developed to the point where detailed predictions can beproductively tested so that competing effects can be isolated andeither eliminated or confirmed. The linear MHDstability predictions using high quality discharge equilibriumreconstructions are tested against the observations for theprincipal limiting phenomena in DIII-D: L mode negative central shear(NCS) disruptions, H mode NCS edge instabilities, and tearing andresistive wall modes (RWMs) in long pulse discharges. In the case ofpredominantly ideal plasma MHD instabilities, agreement between thecode predictions and experimentally observed stability limits andthresholds can now be obtained to within several per cent, and thepredicted fluctuations and growth rates to within the estimatedexperimental errors. Edge instabilities can be explained by a newmodel for edge localized modes as predominantly ideal instabilitieswith low to intermediate toroidal mode number. Accurate idealcalculations are critical to demonstrating RWM stabilization by plasmarotation, and the ideal eigenfunctions provide a good representationof the RWM structure when the plasma rotation slows. Idealeigenfunctions can then be used to predict stabilization usingactive feedback. For non-ideal modes, the agreement in some cases ispromising. Δ' calculations, for example, indicate that some dischargesare linearly unstable to classical tearing modes, consistent with theobserved growth of islands in those discharges. Nevertheless, thereis still a great deal of improvement required before the non-idealpredictive capability can routinely approach levels similar to thosefor the ideal comparisons.

Generic structure of externally driven tearing modes instabilities

A major limit to steady state and advanced high βp operation of tokamaks of reactor class is due to the onset of tearing modes that develop magnetic and may cause loss of energy confinement or a major disruption. Here the structure of a classical problem about the effects of external control helical fields is analysed and it is shown to offer a general paradigm of response of low order classical and neoclassical tearing modes to a wide class of external perturbations. New results of principle on the structural stability of the response model are obtained, leading to a clear interpretation of the role of “seed islands” in the onset of neo-classical tearing modes and the role of finite ion larmor radius corrections to Ohm’s law.

Nonlinear three-dimensional MHD simulations of tearing modes in tokamak plasmas

The comprehension of the dynamics of classical and neoclassical tearing modes is a key issue in high-performance tokamak plasmas. Avoiding these instabilities requires a good knowledge of all the physical mechanisms involved in their linear and/or nonlinear onset. Our tridimensional time evolution code XTOR, which solves the full magnetohydrodynamic (MHD) equations including thermal transport, is used to tackle this difficult problem. In this paper, to show the state of art in full-scale nonlinear MHD simulations of tokamak plasmas, we investigate the effect of plasma curvature on the tearing mode dynamics. For a realistic picture of this dynamics, heat diffusion is required in the linear regimes as well, as in the nonlinear regimes. We present a new dispersion relation including perpendicular and parallel transport, and show that it matches the linear and nonlinear regimes. This leads to a new tearing mode island evolution equation including curvature effects, valid for every island size in tokamak plasmas. This equation predicts a nonlinearly unstable regime for tearing instabilities, i.e. a regime which is linearly stable, but where the tearing mode can be destabilized nonlinearly by a finite-size seed island. These theoretical predictions are in good agreement with XTOR simulations. In particular, the nonlinear instability due to curvature effects is reproduced. Our results have an important impact on the onset mechanism of neoclassical tearing modes. They indeed predict that curvature effects lead to a resistive MHD threshold.

Reconnection: Space plasma simulations for multi-spacecraft satellite observations in the ISTP Era

In the ISTP era multi-spacecraft measurements have not only become a tool for phenomenological and correlative studies of the solar- terrestrial coupling. They also open new opportunities to test, verify or falsify basic physical concepts. For the use of multi-spacecraft measurements in this sense numerical modelling and simulations are becoming more and more a necessary tool for interpreting the observations correctly. We take the reconnection process as an example. Reconnection is supposed to release magnetic field energy from the sun, to couple the Earth's magnetosphere to the solar wind and to suddenly release previously stored magnetic energy in the course of magnetospheric substorms. Hence, reconnection is a key concept in the solar-terrestrial relationship. On the other hand it is difficult to theoretically grasp collisionless reconnection in space plasmas. This complicates its experimental verification.In this paper we present some selected essentially kinetic features of spontaneous reconnection through collisionless thin current sheets in two and three dimensions: These should be observable, e.g., at the onset of magnetospheric substorm. The kinetic approach is necessary due to the cross-scale coupling, inherent to collisionless reconnection. We demonstrate that microscopic magnetic islands of the Debye length scale are always present through thin current sheets. They are, however, reversible and occur at the level of thermal fluctuations. Macroscopic reconnection and island structure formation starts with irreversible energy transfer from fields to plasma. We demonstrate the different routes to macroscopic merging in two and three dimensions. In two dimensions Landau damping in the potential wells of magnetic islands causes the classical slow tearing mode instability. In three dimensions a sausage mode bulk current instability accelerates the transition to macroscopic reconnection. As a result three-dimensional magnetic islands are formed, which are limited in all three spatial directions. We, finally, demonstrate how multi-spacecraft observations of the ISTP era together with well posed computer simulations can help to understand col lisionless reconnection by means of in situ measurements in the Earth's magnetosphere.

Measuring Δ′ from electron temperature fluctuations in the Tokamak Fusion Test Reactor

A method is developed for determining directly from experimental data the classical tearing mode stability parameter Δ′. Specifically, an analytical fit function is derived for the electron temperature fluctuations (T̃e) in the vicinity of a magnetic island. Values of Δ′ determined from the fit function parameters for m/n=2/1, 3/2 and 4/3 modes (m and n are poloidal and toroidal mode numbers) are obtained using the high resolution T̃e profile data from major radius shift (“jog”) experiments on the Tokamak Fusion Test Reactor (TFTR) [D. Meade et al., Proceedings of the International Conference on Plasma Physics and Controlled Nuclear Fusion. Washington, District of Columbia, 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. I, pp. 9–24]. It is found that the n⩾2 modes have Δ′<0.

Stability of tearing modes in finite-beta plasmas

The linear stability of classical tearing modes in a finite-beta, cylindrical plasma is reconsidered. The different regimes of tearing mode instability are delineated by means of critical Lundquist numbers. The well-known criterion Δ′≳Δc due to Glasser, Greene, and Johnson [Phys. Fluids 18, 875 (1975)] defines a critical Lundquist number above which the plasma is stable. However, the physical importance of this Lundquist number is diminished by the fact that there is another critical Lundquist number, which is typically an order of magnitude smaller, above which the mode changes qualitatively in that the growth rate scales linearly with resistivity, and eventually becomes indistinguishable from resistive diffusion. The effect of parallel viscosity, which reduces the stabilizing coupling to the sound wave, is also considered. For large parallel viscosity, it is shown that Δc tends to zero. Thus, the stability criterion Δ′≳Δc for a finite-beta plasma is simply replaced by the criterion Δ′≳0. These analytical results are shown to be in good agreement with numerical results.