Kadomtsev Model Research Articles

On discriminating tokamak sawtooth crash models via localized density and temperature measurements

The core electron temperature drops rapidly during the sawtooth crash in tokamak plasmas, which causes heat loss and may lead to fast particle losses or even a disruption. Several models have been proposed for the periodic crash, including the Kadomtsev model with magnetic reconnection and the quasi-interchange model with the growth of higher-mode-number pressure-driven instabilities. 3D MHD simulations were performed for these two models with a goal to develop intuition and to predict qualitatively how different types of sawtooth will appear in various diagnostics. The structures of electron density ne and electron temperature Te show a dominant (1, 1) mode for the Kadomtsev case and a dominant (4, 4) mode for the quasi-interchange case. The oscillations of ne and Te have a positive correlation near the inversion layer for both cases, while their frequencies and amplitudes are different depending on the dominant modes. Particularly, for the Kadomtsev case, we find a relation between the amount of flux reconnected during a sawtooth event and ne or Te oscillations. Therefore, we connect recently developed measurement capabilities for ne and Te to the internal sawtooth behavior. We propose that this method of analysis can help in identifying the type of sawtooth in future experiments augmented by simulations.

Power-transfer and fixed-point analysis of sawtooth simulations of a current-carrying stellarator

Power-transfer and fixed-point analysis of previous NIMROD simulations (Roberds et al., Phys. Plasmas, vol. 23, issue 9, 2016, 092513) improved the understanding of the effect of 3D (non-axisymmetric equilibrium) magnetic fields on sawtooth oscillations in the Compact Toroidal Hybrid (CTH) experiment. Computing the locations of order-1 fixed points, their Greene's residues, and local values for the rotational transform results in a description of CTH sawteeth consistent with Kadomtsev's model. A power-transfer analysis quantifies the distribution of energy among toroidal Fourier modes and their nonlinear interactions. The Lorentz power transfer drives sawtooth growth, and it is unambiguously interpreted as the flow of energy from toroidal mode $n'$ to mode $n$ , catalysed by $\boldsymbol {B}_{n-n'}$ . It has been reported previously that the CTH sawtooth frequency increases with the 3D field strength. This is attributed to an increased growth rate of the internal kink that drives sawtooth oscillations. Here, 3D fields remove energy from the kink, eliminating the possibility that these fields are an additional energy source that drives growth. Instead, 3D fields catalyse energy transfer from large-to-small scales, where magnetic reconnection is stronger. It is proposed that this energy transfer increases the reconnection rate at small scales, which is consistent with the increased growth rate observed at higher 3D field strengths.

Magnetic reconnection rate during sawtooth crashes in ASDEX Upgrade

The radial velocity of the plasma core during the sawtooth crashes has been measured for the first time with electron cyclotron emission imaging diagnostic. The measurements have been compared with nonlinear two-fluid simulation. The comparison reveals good qualitative and quantitative agreement, which indicates that two-fluid effects (inertia and pressure gradient of electrons) are sufficient for the correct prediction of the experimental results. Contrarily, the crash time of the Kadomtsev model, which is based on a single-fluid picture of magnetic reconnection, disagrees with the experimental results.

Effect of sawtooth crashes on fast ion distribution in NSTX-U

During the 2016 experimental campaign of NSTX-U (Menard et al 2012 Nucl. Fusion 52 083015), long L-mode and reproducible sawtoothing plasmas have been achieved that were previously not accessible before the upgrade. This provides a good opportunity to study the effect of sawtooth crashes on fast ion confinement and redistribution in spherical tokamaks. The solid-state neutral particle analyzer (SSNPA) and fast ion D-alpha (FIDA) diagnostics on NSTX-U each have a tangentially-viewing instrument and a radially- or vertically-viewing instrument, which are mainly sensitive to passing and trapped fast ions, respectively. It has been observed on both diagnostics that passing particles are strongly expelled from the plasma core to the edge during sawtooth crashes, while trapped fast ions are weakly affected. The tangentially-viewing SSNPA observes large signal spikes at the sawtooth crashes because fast ions move to the edge and charge exchange with edge neutrals. The radially-viewing SSNPA data suggest that there is a small drop of trapped particles in the core. The tangentially-viewing FIDA (t-FIDA) system observes a depletion as large as 25% in the region inside the inversion radius, while an increase at the outer region. There is almost no change in the signals of the vertically-viewing FIDA system. The neutron emission can drop as much as 15% at the sawtooth crashes, accompanied by an increase of edge light. Simulations with the Kadomstev and Porcelli sawtooth models have been performed and compared with the measurements. The full reconnection Kadomtsev model overestimates the neutron rate drop at each sawtooth crash. When tuning the sawtooth input parameters, the partial reconnection Porcelli sawtooth model can qualitatively reproduce the neutron rate drop and t-FIDA signal drop in the core, but it fails to predict the t-FIDA signal increase at the edge.

Evolution of the central safety factor during stabilized sawtooth instabilities at KSTAR

A motional Stark effect (MSE) diagnostic has recently been installed in the KSTAR tokamak. A difficulty faced at KSTAR and common to other MSE diagnostics is calibration of the system for absolute measurements. In this report we present our novel calibration routine and discuss first results, evaluating the evolution of the the central safety factor during sawtooth instabilities. The calibration scheme ensures that the bandpass filters typically used in MSE systems are aligned correctly and identifies and removes systematic offsets present in the measurement. This is verified by comparing the reconstructed safety factor profile against various discharges where the locations of rational q surfaces have been obtained from MHD markers. The calibration is applied to analyse the evolution of q0 in a shot where the sawteeth are stabilized by neutral beam injection. Within the analysed sawtooth periods q0 drops below unity during the quiescent phase and relaxes close to or slightly above unity at the sawtooth crash. This finding is in line with the classical Kadomtsev model of full magnetic reconnection and earlier findings at JET.

The effect of sawtooth oscillations on the alpha particle distribution and energy balance in the ITER plasma

The mixing of toroidal plasma under the conditions of sawtooth oscillations is considered using the Kadomtsev model. A new mixing formula for the averaged distribution function of fast transit and trapped particles is proposed in the methodology of a kinetic equation averaged over drift trajectories. The proposed formula generalizes the known results for the case of non-circular magnetic surfaces, an arbitrary aspect ratio, and charged particle drift trajectories significantly deviating from the magnetic surfaces. The formula is applicable for a sufficiently wide class of instabilities. The 3D kinetic equation is numerically solved using the FPP- 3D computation code for parameters close to the ITER inductive scenario. The alpha particle distribution function and the power introduced by alpha particles in plasma when sawtooth oscillations occur are calculated. It is shown that such oscillations may change the energy input of a thermonuclear reaction in certain areas by several times.

Numerical simulation of ion-acoustic turbulence in the B. B. Kadomtsev model

The present report is devoted to the discussion of the properties of two equations relatively long ago proposed by B. B. Kadomtsev for the nonlinear theory of ion-acoustic plasma turbulence and some results of their solutions. We concern with these equations as objects of mathematical physics, focusing attention to both the approach to their study and arising numerical results.

Fast-ion transport and neutral beam current drive in ASDEX upgrade

The neutral beam current drive efficiency has been investigated in the ASDEX Upgrade tokamak by replacing on-axis neutral beams with tangential off-axis beams. A clear modification of the radial fast-ion profiles is observed with a fast-ion D-alpha diagnostic that measures centrally peaked profiles during on-axis injection and outwards shifted profiles during off-axis injection. Due to this change of the fast-ion population, a clear modification of the plasma current profile is predicted but not observed by a motional Stark effect diagnostic.The fast-ion transport caused by MHD activity has been studied in low collisionality discharges that exhibit strong modes. In particular due to sawtooth crashes, significant radial redistribution of co-rotating fast-ions is observed which can very well be described by the Kadomtsev model. In addition, first tomographic reconstructions of the central 2D fast-ion velocity space in the presence of sawtooth crashes allow the investigation of the pitch dependence of the mode-imposed redistribution: a stronger redistribution of mainly co-rotating fast ions is observed than of those with smaller pitch values.

Quantification of the impact of large and small-scale instabilities on the fast-ion confinement in ASDEX Upgrade

The confinement fast ions, generated by neutral beam injection (NBI), has been investigated at the ASDEX Upgrade tokamak. In plasmas that exhibit strong sawtooth crashes, a significant sawtooth-induced internal redistribution of mainly passing fast ions is observed, which is in very good agreement with the theoretical predictions based on the Kadomtsev model. Between the sawtooth crashes, the fishbone modes are excited which, however, do not cause measurable changes in the global fast-ion population. During experiments with on- and off-axis NBI and without strong magnetohydrodynamic (MHD) modes, the fast-ion measurements agree very well with the neo-classical predictions. This shows that the MHD-induced (large-scale), as well as a possible turbulence-induced (small-scale) fast-ion transport is negligible under these conditions. However, in discharges performed to study the off-axis NBI current drive efficiency with up to 10 MW of heating power, the fast-ion measurements agree best with the theoretical predictions that assume a weak level anomalous fast-ion transport. This is also in agreement with measurements of the internal inductance, a Motional Stark Effect diagnostic and a novel polarimetry diagnostic: the fast-ion driven current profile is clearly modified when changing the NBI injection geometry and the measurements agree best with the predictions that assume weak anomalous fast-ion diffusion.

Formation of plasmoids during sawtooth crashes

The nonlinear growth of the internal kink mode is studied numerically using reduced magnetohydrodynamic equations in cylinder geometry. For low Lundquist numbers, S < 107, the already well-known results have been reproduced: a m/n = 1/1 magnetic island (m: poloidal, n: toroidal mode number) grows while the original core shrinks until full reconnection is achieved. For higher S values, however, the dynamics is found to be qualitatively different from the well-known Kadomtsev's model (Kadomtsev 1975 Sov. J. Plasma Phys. 1 389). The growth of the 1/1 island causes the development of a very thin current sheet which becomes tearing unstable. The current sheet is thus broken up and secondary islands (plasmoids) form. These plasmoids strongly speed up the reconnection and eventually coalesce into one secondary island. The formation of a large secondary island stops the fast reconnection process, leading even to a partial reversal of this process. The final state of sawtooth reconnection is thus no longer an axis-symmetric equilibrium as in the case of complete reconnection for low S values, but a helical equilibrium with two coexisting magnetic islands.

Impurity behavior during sawtooth activity in tokamak plasmas

The transport of impurities by a sawtooth crash is simulated with the XTOR-2F code. Impurities are modeled as passive scalars, evolving in the compressible MHD flow inferred from the main MHD plasma. For a peaked impurity density profile, the non-linear kink flow of the sawtooth crash redistributes the profile efficiently and most of the particles in the peak inside the q = 1 surface are expelled. For an initially hollow impurity density profile, the crash leads to a significant penetration up to the magnetic axis. The results are compared with Kadomtsev's model. Despite essentially different mechanisms, the evolution of the particle content inside the q = 1 surface for Kadomtsev's model and for the non-linear case are virtually identical for the peaked profile, while the model slightly overestimates penetration for the hollow case.

Optimization of the number of soft x-ray arrays and detectors for the SST-1 tokamak by the tomographic method

An x-ray imaging system for tomographic reconstruction of two-dimensional emissivity is being developed for the SST-1 tokamak. In this article it is shown that for circular plasma, which will be generated in the first phase of operation of the machine, 7 arrays and 115 detectors are adequate to reconstruct the simulated data. Sawtooth phenomena depicted by both the Wesson model and the Kadomtsev model and the disruption phenomenon associated with m=2 mode are reconstructed well with the help of this imaging system.

Sawtooth studies with q-profile measurements on TFTR (abstract)a)

The q profile plays a key role for plasma equilibrium and theories of MHD instabilities. With the development of the motional Stark effect (MSE) diagnostic,b) accurate q(r,t) profiles have been measured and utilized for equilibrium and stability analysis. Data have been obtained for several conditions on TFTR including L mode, supershot, current ramps, noninductive current drive, and high βp. Sawtoothing discharges on TFTR have q(0) well below one, and it remains below one throughout the evolution of the sawtooth cycle. This is in contradiction with the widely believed Kadomtsev model. It has also been found that stabilization of sawteeth can be achieved with q(0)&amp;lt;1. The criteria for stabilization have been compared to several theoretical models such as resistive, ideal, and two-fluid MHD. The ω* stabilization from the two-fluid MHD model agrees well with the data from TFTR.

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.

Computer tomography of M=1 mode during sawtooth oscillation with three soft X-ray detector arrays on the WT-3 tokamak

Soft X-rays (SXRs) emitted from plasma in the WT-3 tokamak were measured by three SXR detector arrays, viewing the plasma from three different angles. From these experimental data, the authors reconstructed the two dimensional (2-D) contour of constant SXR emissivity as a function of time, using the technique of computer tomography (CT). Computer simulation results show that data from three or more sets of detector arrays are required to reconstruct the 2-D emissivity structure with sufficient accuracy to distinguish Kadomtsev's model from the quasi-interchange (Wesson's) model of sawtooth crash. Further, it is shown that if the noise amplitude is more than 1.5%, reconstruction of the real 2-D SXR emissivity structure is difficult.

High resolution soft-x-ray tomography of sawtooth oscillations on the Tokamak de Varennes.

Up to five arrays of soft-x-ray detectors are used with the Fourier-Bessel harmonic reconstruction method to generate 2D images of sawtooth collapses during ohmic plasma discharges on the Tokamak de Varennes. For low-harmonic reconstructions (/ital M/less than or equal to2), a quasi-interchange behavior (Wesson model) is observed during the sawtooth crash, but a reconnectionlike behavior (Kadomtsev model) is recovered using higher harmonics (/ital M/greater than or equal to3) on the same data. This indicates that the Kadomtsev model is a better description for the sawtooth collapse, although the collapse time remains shorter than predicted.

Disruptive instabilities in the discharges of the TBR-1 small Tokamak

Minor and major disruptions as well as sawteeth oscillations (internal disruptions) were identified in the discharges of the small Tokamak TBR-1, and their main characteristics were investigated. The coupling of a growing m=2 resistive mode with an m=1 perturbation seems to be the basic process for the development of a major disruption, while the minor disruption could be associated with the growth of a stochastic region of the plasma between the q=2 and q=3 islands. Measured sawteeth periods were compared with those predicted by scaling laws and good agreement was found. The time necessary for the sawteeth crashes also agrees with the values expected from Kadomtsev's model. However, there are some sawteeth oscillations, corresponding to conditions of higher plasma Zeff, which showed longer crashes and could not be explained by this model.

Soft-x-ray tomography of sawtooth oscillations in the JIPP T-II tokamak.

Time evolutions of the soft-x-ray tomography of a tokamak plasma have been obtained during sawtooth oscillations by use of the Fourier-Bessel expansion technique. The colder plasma is carried to the center by a hydrodynamic flow, and becomes a new hot spot during the crash phase. The old hot core is pushed out and the new hot spot remains at the center after the crash. The observed sawtooth crash is consistent with neither the Kadomtsev model nor the Wesson model. The crash time is shorter as the current profile is broader.

Electric field profile in the presence of sawtooth activity in a tokamak

The two coupled diffusive equations for the perturbed electron temperature and poloidal magnetic field have been solved in the framework of the Kadomtsev model for the sawtooth activity. It is found that the time averaged toroidal electric field profile is radially nonuniform. A simple formula that describes the radial behavior of the toroidal electric field is proposed, and the effect on the power balance is presented.

Tomography of m=1 mode structure in tokamak plasma using least-square-fitting method and Fourier–Bessel expansions

A method is discussed for reconstruction of the emissivity profile from projections. The emissivity profile is expanded into Fourier–Bessel expansions, of which coefficients are determined by the use of least-square-fitting method with the help of the Akaike information criterion. Tomographic reconstruction of the m=1 mode structure is obtained in JIPP T-II tokamak during the precursor oscillation of the sawtooth crash. The m=1 mode structure fits Wesson model rather than the Kadomtsev model.