Sawteeth In Tokamaks Research Articles

Core inductive electric field during sawtooth crashes on DIII-D

Sawtooth crashes on tokamak plasmas exhibit relaxation much faster than resistive time scales via a mechanism not fully understood. Using core magnetic measurements from the Radial Interferometer-Polarimeter (RIP) diagnostic on the DIII-D tokamak, Grad–Shafranov equilibria constrained by internal magnetic measurements that have high time resolution ( <1 µs) can be computed, allowing analysis of how equilibrium parameters such as safety factor q, current density J, and parallel electric field E∥ , particularly on-axis, evolve. At the sawtooth crash, on-axis safety factor q 0 is observed to rise by 5% but remain below 1 throughout the cycle, and on-axis current density J 0 is observed to drop by 5%. On-axis parallel electric field E∥(0) is found to be balanced by ηJ0 (resistivity times on-axis current density) except during the 200 µs crash period, where E∥(0) reaches 22 V m−1, exceeding ηJ0 by a factor of more than 2000. These first measurements in tokamak plasmas verify that generalized Ohm’s law is not balanced during the crash by resistive effects alone; this is a finding expected due to the relaxation being much faster than resistive timescales. Measurement of the electric field during the tokamak sawtooth serves to illuminate the physical mechanisms at work.

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.

Development of a reduced model for energetic particle transport by sawteeth in tokamaks

The sawtooth instability is known for inducing transport and loss of energetic particles (EPs), and for generating seed magnetic islands that can trigger tearing modes. Both effects degrade the overall plasma performance. Several theories and numerical models have been previously developed to quantify the expected EP transport caused by sawteeth, with various degrees of sophistication to differentiate the response of EPs at different energies and on different orbits (e.g. passing vs. trapped), although the analysis is frequently limited to a single time slice during a tokamak discharge. This work describes the development and initial benchmark of a framework that enables a reduced model for EP transport by sawteeth retaining the full EP phase-space information. The model, implemented in the ORBIT hamiltonian particle-following code, can be used either as a standalone post-processor taking input data from codes such as TRANSP, or as a pre-processor to compute transport coefficients that can be fed back to TRANSP for time-dependent simulations including the effects of sawteeth on EPs. The advantage of the latter approach is that the evolution of the EP distribution can be simulated quantitatively for sawtoothing discharges, thus enabling a more accurate modeling of sources, sinks and overall transport properties of EP and thermal plasma species for comprehensive physics studies that require detailed information of the fast-ion distribution function and its evolution over time.

Sensitivity of magnetic field-line pitch angle measurements to sawtooth events in tokamaks.

The sensitivity of the pitch angle profiles measured by the motional Stark effect (MSE) diagnostic to the evolution of the safety factor, q, profiles during the tokamak sawtooth events has been investigated for Korea Superconducting Tokamak Advanced Research (KSTAR). An analytic relation between the tokamak pitch angle, γ, and q estimates that Δγ ∼ 0.1° is required for detecting Δq ∼ 0.05 near the magnetic axis (not at the magnetic axis, though). The pitch angle becomes less sensitive to the same Δq for the middle and outer regions of the plasma (Δγ ∼ 0.5°). At the magnetic axis, it is not straightforward to directly relate the γ sensitivity to Δq since the gradient of γ(R), where R is the major radius of the tokamak, is involved. Many of the MSE data obtained from the 2015 KSTAR campaign, when calibrated carefully, can meet these requirements with the time integration down to 10 ms. The analysis with the measured data shows that the pitch angle profiles and their gradients near the magnetic axis can resolve the change of the q profiles including the central safety factor, q0, during the sawtooth events.

Non-linear dynamics of compound sawteeth in tokamaks

Compound sawteeth is studied with the XTOR-2F code. Non-linear full 3D magnetohydrodynamic simulations show that the plasma hot core is radially displaced and rotates during the partial crash, but is not fully expelled out of the q = 1 surface. Partial crashes occur when the radius of the q = 1 surface exceeds a critical value, at fixed poloidal beta. This critical value depends on the plasma elongation. The partial crash time is larger than the collapse time of an ordinary sawtooth, likely due to a weaker diamagnetic stabilization. This suggests that partial crashes result from a competition between destabilizing effects such as the q = 1 radius and diamagnetic stabilization.

On resistive magnetohydrodynamic studies of sawtooth oscillations in tokamaks

A fundamental requirement for the validity and accuracy of any large-scale computation is sufficiently well-resolved length and time scales relevant to the problem under study. Ironically, despite the enormous computational resources available today, poorly resolved length scales in sophisticated nonlinear calculations are not uncommon. Using the internal kink mode that is responsible for tokamak sawtooth oscillations as an example, consequences of not resolving in sufficient detail the linear and nonlinear layer widths of the resistive n = 1 mode and its nonlinear spectrum are examined. Poor radial and spectral resolution are shown to cause nonphysical, large-scale stochasticity that can be erroneously associated with a fast temperature collapse and sawtooth crash. With the assistance of a nonlinear mode coupling model, a sufficiently well-resolved toroidal spectrum is shown to require at least an order of magnitude more toroidal modes than is commonly used at dissipation levels relevant to today's tokamaks. A subgrid-scale model is introduced that helps with the spectral resolution problem by reducing the required number of degrees of freedom from that of a well-resolved direct numerical simulation.

Development of magnetohydrodynamic modes during sawteeth in tokamak plasmas

A dynamical analysis applied to a reduced resistive magnetohydrodynamics model is shown to explain the chronology of the nonlinear destabilization of modes observed in tokamak sawteeth. A special emphasis is put on the nonlinear self-consistent perturbation of the axisymmetric m = n = 0 mode that manifests through the q-profile evolution. For the very low fusion-relevant resistivity values, the q-profile is shown to remain almost unchanged on the early nonlinear timescale within the central tokamak region, which supports a partial reconnection scenario. Within the resistive region, indications for a local flattening or even a local reversed-shear of the q-profile are given. The impact of this ingredient in the occurrence of the sawtooth crash is discussed.

The q-profile effect on high-order harmonic q = 1 tearing mode generation during sawtooth crashes

The effect of q-profiles on the excitation of high-order harmonic q=1 tearing modes during sawtooth crashes is investigated by a collisionless fluid model with the electron inertia term in Ohm’s law. It is found that for a flat q-profile in the core region, the high-order harmonics, such as m/n=2/2 and/or m/n=3/3 modes, comparable to or stronger than the m/n=1/1 component, can be excited during tokamak sawteeth. The stronger the magnetic shear on the q=1 surface is, the more unstable the higher-m modes are. For smoothly monotonously increased q-profiles, a lower q value on the plasma edge tends to easily excite higher-m harmonics at the same level as the m = 1 mode simultaneously. The spatial characteristics of the eigenmodes in the cases with the typical q-profiles are also discussed. In addition, the basic feature of the magnetic island structures in the nonlinear evolution is numerically obtained, which is consistent qualitatively with the experimentally reconstructed phenomenon.

Transport and reconnection in tokamak sawteeth.

The core of a tokamak discharge often undergoes periodic relaxation oscillations, sawteeth, as the steepening current and temperature profiles are flattened by fast reconnection events. Careful analysis of the electron temperature evolution over this cycle gives an estimate of the energy dissipated in the electrons during reconnection and a measure of the transport characteristic (energy flux versus temperature gradient) over the range of parameters occurring over the remainder of the cycle. The energy dissipated is consistent with estimates of the loss of poloidal magnetic energy. The transport characteristics exhibit a wide range of behaviors.

Stabilization of sawteeth in tokamaks with toroidal flows

Sheared toroidal flows with a magnitude of the order of the sound speed are shown to stabilize sawteeth in tokamaks. In the absence of flows, tokamak equilibria in which the central safety factor q is less than unity are unstable to resistive tearing modes (resistive internal kink modes) with toroidal mode number n=1. As the ratio β of the plasma pressure to the magnetic field pressure increases, the growth rate of the n=1 mode rises because of the increasing pressure gradient. However, the addition of a toroidal flow to the equilibrium has a stabilizing effect. As the magnitude of the toroidal flow approaches the sound speed, the n=1 resistive tearing mode can be completely stabilized by the flow, eliminating sawteeth.

Onset of high-n ballooning modes during tokamak sawtooth crashes

A new phenomenon has been found during the nonlinear stage of the tokamak sawtooth crash in relatively high β plasmas. The m/n=1/1 magnetic island evolution gives rise to convection of the pressure inside the q=1 radius and builds up steep pressure gradient across the island separatrix, and thereby trigger ballooning instabilities below the threshold at the equilibrium. Effects of the ballooning modes on the magnetic reconnection process during the sawtooth crash are discussed.

Transport in the Sawtooth Collapse

The rapid temperature collapse in tokamak sawtooth oscillations having incomplete magnetic reconnection is generally thought to occur through ergodization of the magnetic field. An experiment in JET using injected nickel indicates that this explanation is improbable. {copyright} {ital 1997} {ital The American Physical Society}

Turbulence and the nonlinear dynamics of sawteeth in tokamaks

Tokamaks typically show cyclic behaviour (i.e. sawtooth oscillations) and this suggests strongly that these systems may be describable in terms of two reduced degrees of freedom, or equivalently two 'effective' physical variables. This leads us to further explore an earlier model which is based on a thermal instability within a central core region which is assumed to be within the q=1 surface, as suggested by experimental and theoretical considerations. The proposed model should be regarded as a 'proof of principle' calculation which shows that sawtooth oscillations need not necessarily result from gross linear MHD instabilities (e.g. M=1) but may be a consequence of thermal instabilities associated with plasma turbulence. To identify the relevant dynamical variables and derive the equations of motion which they satisfy, we consider a full set of conservation equations along with certain constitutive relations applying to the turbulent transport of energy and particles. In particular, we formulate and apply a formal invariance principle relating to the structure of turbulent correlations. We assume the existence of the q=1 surface at r=r1 and that there are no particle sources within this core region. Adopting simple models for the spatial profiles of density, temperature and current and spatially averaging the equations within the core(r<or=r1), we develop two coupled equations for the time evolution of the averaged poloidal beta (equivalently the temperature) and the magnetic fluctuation level. Analytical and numerical studies show that the solutions of these equations exhibit limit-cycle behaviour. This is the result of the nonlinear interaction between the internal energy (i.e. pressure) of the plasma within the core and the magnetic turbulence. This interaction is the combined effect of the conservation laws, the turbulent constitutive relations governing the transport and crash physics, and the boundary conditions at r1.

Current profile flattening and hot core shift due to non-linear development of resistive kink mode

A new and interesting phenomenon that the toroidal current profile, which at first exhibits peaking due to ohmic heating, flattens around the magnetic axis is obtained through a full compressible resistive MHD simulation for a 3-D toroidal geometry. It is concluded that the current flattening is caused by non-linear excitation of the unstable m=1/n=1 resistive kink mode, specifically by the quasi-linear effect of the coupling of the plasma velocity and the magnetic field of the n=+or-1 mode. The time-scale of the current profile flattening is of the order of the MHD time-scale. When the current profile is largely flattened, the plasma dynamic pressure due to the strongly excited kink mode pushes the hot core plasma in the radial kink flow direction and the hot core starts deviating from the magnetic surface owing to the dynamic pressure and the plasma compressibility. Because of the reduction of the magnetic shear due to the current flattening in the q<1 region, magnetic field reconnection is driven by the strong kink flow and leads to the destruction of the magnetic field structure within the q=1 rational surface. Subsequently, the magnetic field configuration recovers to an axisymmetric profile. This can explain the fast crash phase of sawteeth in tokamaks. The effect of the thermal conduction on the evolution of such a process is also discussed

Role of compressibility on driven magnetic reconnection

Whether it is induced by an ideal (current driven) instability or by an external force, plasma flow causes a change in the magnetic field configuration and often gives rise to a current intensification locally, thereby a fast driven reconnection being driven there. Many dramatic phenomena in magnetically confined plasmas such as magnetospheric substorms, solar flares, magnetohydrodynamic (MHD) self-organization, and tokamak sawtooth crash, may be attributed to this fast driven reconnection. Using a fourth-order MHD simulation code it is confirmed that compressibility of the plasma plays a crucial role in leading to a fast (MHD time scale) driven reconnection. This indicates that the incompressible representation is not always applicable to the study of a global dynamical behavior of a magnetically confined plasma.

Stabilization of sawteeth in tokamaks by sheared poloidal flows

The resistive internal kink mode in a current profile with a central safety factor q(0)&amp;lt;1 is shown to be stabilized by sheared poloidal flows. The internal kink mode is found to be stabilized when the local gradient in the poloidal rotation frequency ω at the q=1 resonant surface r1 is larger than the local gradient in the shear Alfvén frequency ωA=k∥(r)VA, where VA is the Alfvén velocity and the parallel wave number k∥(r)=R−1[1−1/q(r)] with R the major radius of the torus and q(r) the safety factor profile in the minor radial coordinate r; i.e., ‖dω/dr‖r1≥‖dωA/dr‖r1. The internal kink is stabilized by sub-Alfvénic flows in which the peak velocity is only a very small fraction of the Alfvén velocity. When the shear in the rotation frequency at the resonant surface is too weak to linearly stabilize the internal kink mode, then the kink mode can still be nonlinearly stabilized, resulting in incomplete reconnection, if the shear in the rotation increases just outside the resonant surface at larger radii.

The relationship between q profiles, transport, and sawteeth in tokamaks

The purpose of this paper is to investigate the possible relationships between transport properties such as thermal diffusivity and resistivity on the one hand, and the magnetic properties such as q profile and toroidal flux change on the other, in tokamaks that exist under macroscopically quasistationary conditions. It is experimentally well established that when the sources are held constant over times long compared with energy and particle confinement times, tokamak discharges can exist in a quasistationary state with or without periodic sawteeth superposed on the basic equilibrium. The principal aim of this study is to provide qualitative physical insight into the nature of such states. For simplicity, single-fluid equations are assumed and the analysis is restricted to a cylinder model. The complete set of conservation equations comprising continuity, pressure balance, Ohm’s law, and energy balance are used along with appropriate sources. The conductive energy loss is assumed to occur due to an anomalous thermal diffusivity. Following earlier time-dependent studies of tokamak transport phenomenology due to the authors [Haas and Thyagaraja, Plasma Phys. Controlled Fusion 28, 1093 (1986)], the friction force in the generalized Ohm’s law is assumed to be described by an effective resistivity tensor with its principal axes in the poloidal and toroidal directions. The toroidal resistivity is taken to be of order Spitzer while the poloidal component ηθ is assumed to be anomalously large compared with ηz. A range of phenomena involving Ohmic discharges, auxiliary heated discharges, and noninductively driven currents is investigated. Attention is drawn to the joint implications of the conservation laws and the general forms of the constitutive relations for the structure of the profiles and conditions for the existence of equilibria. In particular, it is shown that βp&amp;gt;1 is achievable in macroscopically steady conditions, only if a sufficiently strong particle source is present in addition to an energy source. It also follows from the analysis that low βp discharges require an anomalous thermal force type term in Ohm’s law, in addition to the anomalous poloidal resistivity. Recent experimental results on sawtoothing discharges are used to extend the theoretical considerations from strictly steady discharges to those involving sawteeth. A simple nonlinear dynamical model of sawtoothing is constructed and used to illustrate some of the features of sawtooth oscillations. The paper is intended to complement fuller numerical studies. It may help in understanding transport and large-scale relaxation processes like the sawtooth in tokamaks by providing a set of theoretical relations that can be subjected to a direct experimental test. It is a feature of the results that they are independent of the details of the turbulent dynamics, which are ultimately thought to be responsible both for the form and scaling of the constitutive properties considered.

Current sheet stability and accelerated reconnection of magnetic field lines

The stability of a narrow current sheet embedded in a sheared magnetic field to short wavelength microtearing modes is examined and a mechanism for the accelerated reconnection of magnetic field lines during the fast crash of sawteeth in tokamaks is presented. The stability of the sheet is shown to depend crucially on the sign of Il/ k′∥, where Il is the line current in the sheet and k∥ ≡dk∥(x)/dx is the rate of change of the parallel wave number in the direction of the inhomogeneity, as well as on three important characteristic lengths: Δr, the distance between the center of the current sheet and the resonant surface of the mode; the width σs of the sheet; and the resonant layer width wr of the mode. When Il/k′∥ is positive, the current sheet can be unstable only if Δr &amp;lt; 1/2 max(σs,wr); otherwise, the sheet is stable. Conversely, when Il/k∥ is negative the sheet can be unstable only if Δr &amp;gt; 1/2 max(σs,wr); otherwise, the sheet is stable. Expressions for the unstable spectrum and the growth rate of the unstable modes as a function of the relative magnitude of Δr, σs, and wr are presented. For narrow current sheets with σs&amp;lt;wr, the growth rate of the tearing mode actually increases as the plasma resistivity decreases. The stability analysis is applied to the current sheet generated around the X point of a q=1 mode during the fast crash of the central temperature during sawteeth in tokamaks. The sheet is unstable if σs is less than a critical width σsc/a ≡‖Is/I0‖ ×(4/qa), where Is/I0 is the ratio of the current Is in the sheet to the total current I0, a is the minor radius, and qa is the safety factor at the edge of the plasma. For typical TFTR tokamak [Plasma Physics and Controlled Nuclear Fusion Research, Kyoto, 1986 (IAEA, Vienna, 1986), Vol. 1, p. 421] parameters, σsc is a few ion gyroradii. The sheet generated by the q=1 mode in a hot tokamak like TFTR or JET is much narrower than σsc. As a consequence, the sheet is unstable to a broad spectrum of short wavelength perturbations with m, n&amp;gt;100, where m and n are the poloidal and toroidal mode numbers, respectively. These modes grow rapidly with a growth time of the order of 1 μsec, much less than the sawtooth crash time of 50–100 μsec. This short wavelength turbulence will likely broaden the current sheet to the marginally stable width σsc and thereby increase the rate of reconnection of magnetic field lines by the q=1 mode, resulting in a fast crash of the central temperature.

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.

Alfvén-ion-cyclotron instability and absence of sawteeth in tokamaks

Linear stability of the anisotropic energetic ion population, observed in recent experiments at the Joint European Torus [Phys. Rev. Lett. 60, 2148 (1988)], to Alfvén-ion cyclotron wave excitations is analyzed and the saturation mechanism is discussed. In the absence of external heating the anisotropic distribution lifetime is found comparable to the observed delay time in the monster sawtooth crash after the turn-off of the heating. It is suggested that the ponderomotive force of unstable waves is responsible for this delay. Saturated wave fields are found sufficiently large to stabilize the internal kink mode.