Sawtooth Collapse Research Articles

Impulsive reconnection dynamics in collisionless laboratory and space plasmas

There are some phenomena involving magnetic reconnection in collisionless laboratory and space plasmas that are characterized not only by fast growth but also by an impulsiveness, that is, a sudden increase in the time derivative of the growth rate. Three examples are the sawtooth collapse in tokamaks, the sudden enhancement of the cross‐tail current density just prior to substorm onset in the Earth's magnetotail, and the impulsive phase of a solar flare. Not only is the peak growth rate of these phenomena large (near‐Alfvénic), they are triggered suddenly, with the growth rate increasing from small to large values very rapidly. Traditional steady state reconnection models, such as those due to Sweet‐Parker or Petschek, do not include time dependency, and hence cannot account for the impulsiveness. It is demonstrated that in collisionless plasmas, the effects of electron pressure gradients and Hall currents (referred to collectively as Hall MHD effects) can provide a plausible explanation not only of the fast growth but also of the impulsiveness. The geometrical structure of thin current sheets realized during the dynamical evolution of such collisionless systems changes spontaneously from a Y point in the early nonlinear regime (characteristic of the models of Sweet‐Parker and Syrovatsky) to an X point (characteristic of the Petschek model). However, the underlying physics of reconnection is qualitatively different from that in resistive MHD models. In Hall MHD, the electron and ion dynamics are decoupled from each other in the reconnection layer. Such decoupling is forbidden in resistive MHD. In Hall MHD, while the spatial scale of the parallel electric field (that is, the component of the electric field parallel to the background magnetic field) is the ion skin depth, the spatial scale of the parallel current density is narrower, determined by the Lundquist number (or the electron skin depth), if resistivity (or electron inertia) is the dominant mechanism that breaks field lines. Analytical results as well as high‐resolution numerical simulations are presented, and the predictions of the model are compared with data from laboratory experiments as well as space.

Observation of magnetic field perturbations during sawtooth activity in tokamak plasmas

Sawtooth activity is a prominent example of a global plasma instability which is observed in virtually all tokamak devices. Despite numerous experimental and theoretical investigations, the phenomenon is still barely understood. As far as experimental effort is concerned, much attention has been paid to soft x-ray emission from the plasma and to its analysis in terms of two-dimensional contour plots, because it is thought to reflect the shape and temporal behaviour of magnetic flux surfaces during a sawtooth cycle. Recently, more direct methods of detecting sawtooth-related changes in the magnetic field structure have become available and have added new facets to the general picture. In this paper, some observations made on the Jülich tokamak TEXTOR by means of a Faraday rotation diagnostic technique will be reported. First, in correlation with the sawtooth collapse a localized periodic perturbation of the magnetic field with principal mode numbers m = 1 and n = 0 has been detected which, in the presence of an m = n = 1 island, may give rise to magnetic field line stochastization and thereby contribute significantly to a rapid expulsion of electronic energy from the plasma core region. Second, the so-called precursor oscillations prior to a sawtooth crash have been investigated and estimates have been obtained for the growth rate and width of a magnetic island forming immediately before the collapse.

Density fluctuations during the sawtooth crash in the Tore Supra tokamak

Using a laser scattering experiment, transient density fluctuations related to the sawtooth crash were measured, for about 100 sawteeth, during similar and stationary ohmic discharges of the Tore Supra tokamak. Simultaneously, the evolutions of electron temperature, density and soft x-ray emission were recorded to analyse the plasma behaviour during the crash. In particular, tomographic reconstructions of soft x-ray signals were used to draw emissivity maps and to investigate in detail the sawtooth collapse. Two bursts of density fluctuations, spatially localized within the q = 1 surface, have been identified. A first burst appears together with the fast displacement of the hot core at the beginning of the crash (phase I), and a second stronger burst develops when the central temperature collapses (phase II). The levels and propagation directions of these fluctuations have been systematically compared with the direction of the initial hot core motion. From this analysis, it is deduced that fluctuations propagate from the hot core compression zone along the q = 1 separatrix with a velocity that is about five to 10 times larger than the hot core motion. Levels are strongly correlated to the hot core motion, indicating spatially localized fluctuations. Using these results and a numerical model, it is shown that fluctuations detected during phase I are principally located in the vicinity of the hot core compression zone while those correlated to phase II are in two more central regions, where the temperature collapses.

Sawtooth modulation of the poloidal field in TEXTOR under ohmic heating conditions

By means of far-infrared polarimetry the variation in the internal magnetic field structure during sawtooth activity has been measured in the TEXTOR tokamak with high spatial resolution and improved signal-to-noise ratios. Utilizing the reproducibility of ohmic plasmas as well as the uniformity of sawtooth oscillations under appropriate quasi-stationary conditions, the given number of vertical probing beams has been multiplied by running discharges at different major radii, and the noise levels of the Faraday rotation signals have been reduced by forming a coherent average of many sawtooth periods. In agreement with previous investigations it is found that the safety factor on axis has a mean value q0=0.73+or-0.1 and undergoes a change delta q0 approximately=0.06 during the sawtooth ramp phase. A new observation is the periodic appearance of a poloidal held perturbation in correlation with the sawtooth collapse which is localized at the large major radius side of the q=1 surface. The persistence of the corresponding signal deformations after averaging some hundred sawtooth periods suggests that this perturbation is axisymmetric with principal mode numbers m=1 and n=0. In the presence of an m=n=1 island, it may give rise to magnetic field line stochastization and thus may play an important role in the fast sawtooth crash.

MHD activity in JET hot ion H mode discharges

The MHD effects observed in the hot ion H modes in the pre-divertor configuration of JET, including those generated in the preliminary tritium experiment, are described. Some observations were found to be similar to those in high beta regimes while others are new and appear to be pertinent to high performance discharges only. The high performance phase is largely sawtooth free and dominated by fishbone activity, which increases in amplitude throughout this phase. During termination of the high performance phase, the growth of a large variety of MHD activity with low mode numbers is observed. Also, edge instabilities possibly associated with much larger mode numbers are seen in the Dalpha emission. In some cases, the unusual structure of two central m=n=1 islands was found, and resistive MHD modelling indicates that this observation is consistent with a nearly flat, non-monotonic q profile. In some discharges a sawtooth collapse immediately followed by an edge localized mode (ELM) is observed at, or shortly following, the termination of the high performance

Calculation of the post-crash state and 1 1/2 D simulation of sawtooth cycles

A model for sawtooth collapse, based on Taylor relaxation of the plasma core in a Tokamak, is described. The relaxation is assumed to take place within a central zone determined by the instability of localized resistive interchange modes, and is accompanied by an ideal MHD adjustment of the external region so as to maintain force balance across the interface. Using an aspect ratio expansion, equations determining the post-crash equilibrium are derived and solved in terms of equilibrium quantities prior to collapse. The crash model contains no arbitrary parameters. This model has been installed in the '1 1/2 D' resistive evolution code LARS, and in the paper we carry out a quantitative comparison of the simulation results with experimental data, in particular with the TEXTOR tokamak where detailed measurements are readily available.

Dynamics of the sawtooth collapse in tokamak plasmas.

Fast reconnection such as driven by electron inertia qualitatively changes the Kadomtsev picture of the sawtooth collapse in tokamak discharges. The collapse occurs in two steps, a fast Kadomtsev-type reconnection followed by a rapid reformation of a ${q}_{0}<1$ configuration. The latter is driven by the strong flows generated during the Kadomtsev phase. The theory provides a natural explanation of the main experimental observations including the snake phenomena.

Heat Flow during Sawtooth Collapse in Tokamak Plasmas

Heat flow during sawtooth collapse was studied on the WT-3 tokamak by using temporal evolution of soft X-ray intensity profile in the poloidal cross section in a lower hybrid current driven plasma as well as an electron cyclotron heated plasma. Two phases in sawtooth collapses were observed. In the first phase, the hottest spot that is the peak of the soft X-ray distribution approaches the inversion surface and heat flows out through a narrow gate on the inversion surface. In the second phase, the hottest spot stays on the inversion surface, and heat flows out through the whole inversion surface. This suggests that magnetic reconnection as predicted by Kadomtsev's model occurs in the first phase, but in the second phase, a different mechanism dominates heat flow.

Evidence of the complete magnetic reconnection during a sawtooth collapse in a tokamak.

Statistical analysis of a large set of current profile data obtained for sawtoothing discharges in the DIII-D tokamak indicates that the safety factor (inverse rotational transform) in the plasma center, ${\mathit{q}}_{0}$, is close to 1.0. The increase of ${\mathit{q}}_{0}$ associated with the sawtooth crash is found to be of the order of 0.05. The results of time resolved measurements of the current profile and the soft x-ray emissions support a model of the sawtooth instability based on total reconnection of the m=1 magnetic island.

Asymmetry of radial heat pulse propagation at TEXTOR

Electron heat pulse propagation following sawtooth collapse has been simultaneously observed along two opposite radial directions. For a given flux surface, the arrival of the heat pulse at the high field side (HFS) is delayed by up to 1 ms as compared with the arrival at the low field side (LFS). The observed delay agrees with an estimate of the time required for parallel electron heat transport along the magnetic field lines connecting the corresponding LFS and HFS isobars. This suggests that the primary radial heat transport occurs on the LFS, whereas the HFS is fed subsequently from the LFS by temperature equilibration along the connecting magnetic field lines

Sawtooth heat pulse propagation in tokamaks

Standard analysis treats the sawtooth-induced heat pulse as a diffusive phenomenon by linearizing the diffusion equation around the unperturbed state of the plasma. This implies that the heat pulse is governed by the incremental thermal diffusivity. This approach has been criticized by Frederickson et al. (1990 Phys. Rev. Lett. 65, 2869), who claim that turbulence associated with the sawtooth collapse affects the propagation of the heat pulse. However, this criticism is based on an incorrect application of the standard diffusive models. Furthermore, it is shown that their modelling with a time-dependent diffusivity (the 'ballistic' heat pulse) does not describe the measurements. The authors demonstrate that the measurements impose a temporary change of the effective thermal diffusivity during the heat pulse which is similar to the implicit time behaviour imposed by the linearized treatment.

Interpretation of heat and density pulse measurements in JET in terms of coupled transport

The perturbations of electron density and temperature profiles in a tokamak following a sawtooth collapse are considered. An analytic model for the interpretation of such perturbations is presented. It is shown that the perturbation can be decomposed into two contributions, which are eigenmodes of the linearized coupled diffusion equations for particles and energy. The approximations made in the analytic treatment are checked using computer simulations. Measurements of heat and density pulses in JET are used to illustrate the power of the new approach. It is shown that with the coupled equations an improved description of the heat and density pulses is obtained. The analysis yields the four diffusion coefficients in the linearized transport matrix. The non-zero off-diagonal elements explain certain salient features of the measurements, notably a marked decrease of the local density which occurs during the maximum of the temperature pulse.

The change in the safety factor profile at a sawtooth collapse

Perturbations of the electron density and the poloidal magnetic field induced by sawtooth instabilities are studied using interferometric and polarimetric data. The change in the safety factor profile at a sawtooth collapse is deduced. It is found that sawteeth do not lead to complete reconnection of the poloidal flux since ( Delta q0)collapse<1-q0. A comparison of the rate of change of q0 between sawteeth with field diffusion calculations assuming neo-classical resistivity also suggests that complete reconnection does not take place.

Evidence of coupling of thermal and particle transport from heat and density pulse measurements in JET

The evolution of electron temperature and density perturbations in a Tokamak following a sawtooth collapse is considered. An integrated numerical analysis of such perturbations is presented, which uses the complete time evolution after the collapse. The method takes into account coupling between heat and particle transport, the effects of the inward propagating density pulse due to a change of recycling at the edge when the heat pulse reaches the limiter, as well as the observed damping of the heat pulse, mainly due to electron-ion energy exchange. Heat and density pulses in JET have been simulated with the method, yielding the four elements of the linearized transport matrix. Certain features of the measurements, notably an initial decrease in the local density, can be explained by the presence of nonzero off-diagonal terms in the matrix.

Spontaneous m = 1 instability in the JET sawtooth collapse

The collapse phase of sawtooth oscillations in tokamaks is believed to be due to the rapid growth of an m = 1 instability. We show here that the onset of the rapid growth is spontaneous and cannot be described by the increase of the linear growth rate arising from development of the plasma equilibrium. This result calls for a reappraisal of the sawtooth problem.

Sawtooth reconnection

A model of magnetic field reconnection in tokamak sawteeth is given. The reconnection rate is determined by electron inertia rather than resistivity and this leads to a faster sawtooth collapse than Kadomtsev reconnection.

Measurement and analysis of neutron sawteeth on the Frascati Tokamak

The time evolution of neutron emission has been measured in sawtoothing discharges in the Frascati Tokamak (FT). The observed change in the neutron emission level during the sawtooth crash has been studied in terms of variations of the deuteron density and temperature. Changes of the peak values of these quantities as well as profile changes have been considered. It is shown that the occurrence of a net ion energy outflow through the mixing radius after a sawtooth collapse can explain the observed drop in the neutron emission. The model is also used to infer the ion thermal conductivity near the plasma centre which is found to be compatible with neoclassical predictions.

Heat pulse analysis in JET limiter and X-point plasmas

Measurements of the electron heat diffusivity χ of the JET plasma in a wide variety of discharge conditions are reported. χ is deduced from heat pulse propagation analysis, using the sawtooth collapse as the perturbation of the temperature profile. The measurements of the electron temperature are performed with a 12-channel ECE polychromator. In limiter discharges, scans of plasma current, density, input power and Zeff have been studied. It is shown that χ is independent of density and input power, while a clear dependence on Zeff and on the electron temperature Te is demonstrated. The scaling provides a good fit to the JET data. xhp denotes the normalized radius where χ is measured. This radius is coupled to qa. As a result, the xhp-dependence only parameterizes the combined effect of the plasma current and the profile shapes of temperature, density, safety factor, etc. Various transport models are discussed in the light of this scaling. Also, χ measurements in the Ohmic, L-mode and H-mode phases of X-point plasmas are reported. The χ values are on average 20% higher in the H-mode than in the Ohmic phase. In one discharge where χ was measured in all three phases, no significant changes could be demonstrated.

Heat-pulse propagation in tokamaks and the role of density perturbations

Studies of the propagation of heat pulses, launched by a sawtooth collapse or by modulated auxiliary heating, yield information on transport properties in the Tokamak. Corresponding values of the thermal conductivity are often found to be significantly larger than values obtained from power balance calculations. A full treatment of the heat pulse must, however, include possible coupling to an associated density pulse. Such effects are discussed in the context both of neoclassical transport theory and of a model of anomalous transport due to drift waves. It is found that the discrepancies between heat pulse and power balance measurements could arise from coupling between density and temperature perturbations due to the presence of off-diagonal terms in the transport matrix.

Numerical simulations of sawteeth in tokamaks

Numerical simulations of sawteeth in tokamaks have been carried out using reduced magnetohydrodynamics and a simple transport model. The electron temperature is evolved self-consistently, including Ohmic heating and a highly anisotropic thermal diffusivity. The sawtooth period and collapse time found in the simulations for Lundquist numbers S below 107 compare favourably with experimental results from small and medium size ohmically heated tokamaks. The sawtoothing is found to be sensitive to the values of the transport coefficients. In particular, the perpendicular viscosity must be comparable to, or larger than, the perpendicular thermal diffusivity for distinct relaxation oscillations to occur. To study the scaling with S, the viscosity and perpendicular thermal conductivity have been scaled as 1/S, and S has been varied. Modifications of the equilibrium, caused by the sawteeth, play an important role in the scaling of the collapse time and period with the Lundquist numnber. The self-consistently computed q-profiles are very flat in the central region where q ≈ 1. Outside the low-shear region, the shear rises sharply. This q-profile allows a resistive mode to be turned on quickly with a high growth rate. The deviation of the central q from unity over the sawtooth cycle decreases with increasing S, and the collapse time shows a weaker dependence than the S½ scaling of Sweet and Parker.