Sawtooth Oscillations Research Articles

Note on a paper by Omta et al. on sawtooth oscillations

We discuss a recently proposed model for the sawtooth oscillations of atmospheric carbon observed in ice age cores. The model at its simplest arises from a single reaction involving just two species, oceanic biomass and nutrient, and bears similarity to models which have been proposed to explain oscillations in glycolysis. We show that the sawtooth behaviour, and associated spiking of one of the constituents, is associated with the existence of a conservative nonlinear oscillator with an asymmetric potential at high values of the energy, and we give asymptotic descriptions of the solutions. We extend the analysis to a more complicated model which includes competition between planktonic species.

Dawn–dusk asymmetry in the Kelvin–Helmholtz instability at Mercury

The NASA MErcury Surface, Space ENvironment, GEochemistry and Ranging (MESSENGER) spacecraft entered orbital phase around Mercury on 18 March 2011. A surprising consistent feature in the data returned is large-scale vortices that form exclusively on the dusk side of the magnetosphere. Here we present global kinetic hybrid simulations that explain these observations. It is shown that vortices are excited by a Kelvin-Helmholtz instability near the subsolar point, which grows convectively along the dusk-side magnetopause. Virtual time series along a track approximating a flyby of the MESSENGER show correspondence with the satellite data; the data contain sawtooth oscillations in plasma density, flow and magnetic field, and exhibit the observed dawn-dusk asymmetry. It is shown that asymmetry between dawn and dusk at Mercury is controlled by the finite gyroradius of ions and by convection electric fields. Mercury's magnetosphere offers a natural laboratory for studying plasma regimes not present in other planetary magnetospheres or the laboratory.

On the formation and stability of long-lived impurity-ion snakes in Alcator C-Mod

Long-lived (1, 1) ‘snake’ modes were discovered nearly three decades ago, but basic questions regarding their formation, stability, and superb particle confinement—shown by surviving tens to hundreds of sawtooth cycles—have remained unanswered. High-resolution spectroscopic imaging diagnostics permit studies of heavy-impurity-ion snakes with unprecedented temporal and spatial resolution, making it possible to positively identify the SXR signals with specific ion charge states and to infer, for the first time, the perturbed impurity density, Zeff, and resistivity at the centre of these long-lived helical modes. The results show a new scenario for the formation of heavy-impurity-ion snakes, which can begin as a broad 1/1 kink asymmetry of the central impurity-ion density, that grows and undergoes a seamless transition to a large crescent-shaped helical island-like structure inside q < 1, with a regularly sawtoothing core. This type of formation departs strongly from the nonlinear island model based on a modified Rutherford equation proposed originally to describe the pellet-induced snakes and expanded further to account for the impurity effects (e.g. and ). These new high-resolution observations show details of their evolution and the accompanying sawtooth oscillations that suggest important differences between the density and temperature dynamics, ruling out a purely pressure-driven process. Instead, many features arise naturally from nonlinear interactions in a 3D MHD model that separately evolves the plasma density and temperature.

CPAP for Obstructive Sleep Apnea in the Post-Operative Setting: An Oximetry Evaluation Study

Objective: To study the prevalence and associations of nocturnal hypoxia following surgery in PAP-treated (CPAP or bilevel pressure support) OSA patients. Patients and Methods: PAP-treated OSA patients presenting for elective surgery (n=38) between August 2007 and October 2008 underwent overnight oximetry monitoring on their prescribed PAP therapy on postoperative night one. The lowest oxygen (O2) saturation (SpO2), time spent with SpO2 <89% and the presence of sleep disordered breathing (saw-tooth oscillations) were determined from the oximetry tracings. Need for supplemental O2, nursing/ respiratory interventions, and complications were recorded. Records were reviewed for the type and duration of surgery and anesthesia, net fluid balance, narcotic / sedative use, demographics and co-morbidities. Results: Seven of 38 subjects (18.4%) spent at least 30 minutes with SpO2 <90% during the night (hypoxic group=HG) on their prescribed PAP therapy. The severity of baseline OSA (total and supine apnea-hypopnea index), pre-CPAP SpO2 (asleep on diagnostic PSG and awake evening after surgery), net fluid balance, and total IV narcotic dose were associated with the HG. Nursing interventions and addition of O2 were more common in the HG, though other adverse events were not. Review of the oximetry tracings suggested 71% of the HG, and 42% of the non-hypoxic group, had findings of uncontrolled sleep apnea. Conclusions: Despite the use of prescribed PAP therapy the night following surgery, some OSA patients become hypoxic. Postoperative oximetry monitoring may be indicated in PAP-treated OSA patients undergoing surgery

Relaxation oscillations properties in Goodwin's business cycle model

We consider Goodwin's time delay model of the business cycle described by neutral delay differential equation with fixed investment time lag. We investigate the properties of relaxation (sawtooth) oscillations, detected experimentally by Strotz et al. (1953). We show that the shape of relaxation fluctuations (amplitude, average value of income over the period of oscillation, time of income rise and fall) depends on the initial function. We also present an analytical sawtooth solution of the delay equation obtained by the method of steps. Finally, we show the existence of irregular dynamics of income.

Sawtooth-driven neoclassical tearing modes in HL-2A plasmas

Neoclassical tearing modes (NTMs) of m = 2, n = 1 induced by sawtooth crashes in tokamak plasmas are studied based on the results of HL-2A tokamak experiments. A model of sawtooth-driven Δ′ for NTMs due to the toroidal coupling effect is proposed and applied to the observed growth of 2/1 islands with the sawtooth oscillation observed in the HL-2A experiment. The neoclassical tearing island growth is calculated based on the model and the result is found to be in good agreement with experimental data.

Observation of fast ion behaviour with a neutron emission profile monitor in MAST

Preliminary measurements of neutron emissivity at the Mega Amp Spherical Tokamak (MAST) along collimated lines-of-sight show a clear correlation between the neutron emissivity temporal and spatial evolution and the evolution of different MHD instabilities. In particular, the variations in neutron emissivity during sawtooth oscillations are compared with changes in the classical fast ion slowing-down time, while fast ion losses are observed in bursts during fishbones or as a continuous process during long-lived modes.

Elevation angle alignment of quasi optical receiver mirrors of collective Thomson scattering diagnostic by sawtooth measurements

Localized measurements of the fast ion velocity distribution function and the plasma composition measurements are of significant interest for the fusion community. Collective Thomson scattering (CTS) diagnostics allow such measurements with spatial and temporal resolution. Localized measurements require a good alignment of the optical path in the transmission line. Monitoring the alignment during the experiment greatly benefits the confidence in the CTS measurements. An in situ technique for the assessment of the elevation angle alignment of the receiver is developed. Using the CTS diagnostic on TEXTOR without a source of probing radiation in discharges with sawtooth oscillations, an elevation angle misalignment of 0.9° was found with an accuracy of 0.25°.

Rapid Collapse of a Plasma Sawtooth Oscillation in the HT-7 Tokamak

The rapid collapse of a sawtooth oscillation which is characterized by the absence of any discernible precursor oscillation in the HT-7 tokamak has been observed in detail on a fast time scale. The contour plot of the raw data of soft-x-ray signals shows that sawtooth crash simultaneously on both the high field side and the low field side with symmetry. Only spatial structure of sawtooth exists in the whole sawtooth period. A larger heat outflow and profile of emission intensities flatten during sawtooth crash by means of soft x-ray tomography. The plausible interpretation is the stochastization of magnetic field lines lead to a transition from semicollisional to collisionless reconnection regimes during non-linear pre-crash phase in which the growth rate is rapidly increased and no any discernible precursor oscillation.

Multiple timescale calculations of sawteeth and other global macroscopic dynamics of tokamak plasmas

The M3D-C1 (Breslau et al 2009 Phys. Plasmas 16 092503) code is designed for performing three-dimensional nonlinear magnetohydrodynamics (MHD) calculations of a tokamak plasma that span the timescales associated with ideal and resistive stability as well as parallel and perpendicular transport. This requires a scalable fully implicit time advance where the time step is not limited by a Courant condition based on the MHD wave velocities or plasma flow but is chosen instead to accurately and efficiently resolve the physics. In order to accomplish this, we make use of several techniques to improve the effective condition number of the implicit matrix equation that is solved every time step. The split time advance known as the differential approximation (Caramana 1991 J. Comput. Phys. 96 484) reduces the size of the matrix and improves its diagonal structure. A particular choice of velocity variables and annihilation operators approximately splits the large matrix into three sub-matrices, each with a much improved condition number. A final block-Jacobi preconditioner further dramatically improves the condition number of the final matrix, allowing it to converge in a Krylov solver (GMRES) with a small number of iterations. As an example, we have performed transport timescale simulations of a tokamak plasma that periodically undergoes sawtooth oscillations (Von Goeler et al 1974 Phys. Rev. Lett. 33 1201). We specify the transport coefficients and apply a 'current controller' that adjusts the boundary loop-voltage to keep the total plasma current fixed. The short-time plasma response depends on the initial conditions, but the long-time behavior depends only on the transport coefficients and the boundary conditions applied.

Experimental study of toroidal Alfvén modes in the Globus-M spherical tokamak

In the experiments carried out on the Globus-M tokamak in regimes with injection of 26-keV neutral beams with a power of 0.75–0.85 MW, two branches of instabilities excited by fast ions were observed in the early stage of a discharge: a low-frequency energetic particle mode (EPM) in the frequency range of 5–30 kHz and a high-frequency mode in the range of 50–200 kHz, identified as a toroidal Alfven eigenmode (TAE). The TAE developed in the initial phase of the discharge at q(0) > 1 and terminated when sawtooth oscillations were excited at q(0) < 1. The spectrum and spatial localization of the mode agree with predictions of the linear theory. The modes observed in the Globus-M tokamak possess both properties common to other tokamaks and their own specific features.

Sawtooth Precursor Oscillations on DIII-D

The sawtooth oscillation, observed in tokamak plasmas with a central safety factor of less than unity, is a periodic disruptive instability characterized by a slow ramping of central plasma density and temperature, followed by a fast relaxation resulting in flattening of both profiles. Elongated neutral-beam-heated discharges on the DIII-D tokamak exhibit multiple precursor oscillations with mode number m/n = 1/1. The dominant m/n = 1/1 mode oscillates at the plasma rotation frequency. A downshifted mode also appears early in the sawtooth ramp. A normalization of electron cyclotron emission imaging data that removes the contribution of slow electron temperature profile evolution reveals that both modes are consistent with an underlying quasi-interchange plasma displacement.

Diamagnetic thresholds for sawtooth cycling in tokamak plasmas

The cycling dynamics of the internal kink mode, which drives sawtooth oscillations in tokamak plasmas, is studied using the three dimensional, non-linear magnetohydrodynamic (MHD) code XTOR-2F [H. Lütjens and J.-F. Luciani, J. Comput. Phys. 229, 8130 (2010)]. It is found that sawtooth cycling, which is characterized by quiescent ramps and fast crashes in the experiment, can be recovered in two-fluid MHD provided that a criterion of diamagnetic stabilization is fulfilled. The simulation results indicate that diamagnetic effects alone may be sufficient to drive sawteeth with complete magnetic reconnection in high temperature Ohmic plasmas.

New findings about sawtooth oscillations in LHCD plasmas on the HT-7 tokamak using tomography of soft x-ray signals

Two new physical phenomena have been observed during the sawtooth (both with mid-cursor and without precursor) crash in lower hybrid current drive (LHCD) plasmas on the HT-7 tokamak by using tomography of high-resolution soft-x-ray emission together with the singular value decomposition technique. Firstly, there is no displacement of the hot core during the mid-cursor-sawtooth (sawtooth oscillations with mid-cursor) crash; secondly, magnetic reconnection does not occur during the precursorless-sawtooth (sawtooth oscillations without precursor) crash. These observations demonstrate that the crashes of the two typical sawtooth oscillations (mid-cursor-sawtooth and precursorless-sawtooth) are inconsistent with both full reconnection theory and Wesson's theory.

Magnetosphere Sawtooth Oscillations Induced by Ionospheric Outflow

The sawtooth mode of convection of Earth's magnetosphere is a 2- to 4-hour planetary-scale oscillation powered by the solar wind-magnetosphere-ionosphere (SW-M-I) interaction. Using global simulations of geospace, we have shown that ionospheric O(+) outflows can generate sawtooth oscillations. As the outflowing ions fill the inner magnetosphere, their pressure distends the nightside magnetic field. When the outflow fluence exceeds a threshold, magnetic field tension cannot confine the accumulating fluid; an O(+)-rich plasmoid is ejected, and the field dipolarizes. Below the threshold, the magnetosphere undergoes quasi-steady convection. Repetition and the sawtooth period are controlled by the strength of the SW-M-I interaction, which regulates the outflow fluence.

Dynamics of fast ions during sawtooth oscillations in the TEXTOR tokamak measured by collective Thomson scattering

Experimental investigations of sawteeth interaction with fast ions measured by collective Thomson scattering on TEXTOR are presented. Time-resolved measurements of localized 1D fast-ion distribution functions allow us to study fast-ion dynamics during several sawtooth cycles. Sawtooth oscillations interact strongly with the fast-ion population in a wide range of plasma parameters. Part of the ion phase space density oscillates out of phase with the sawtooth oscillation during hydrogen neutral beam injection (NBI). These oscillations most likely originate from fast hydrogen ions with energies close to the full injection energy. At lower energies passing fast ions in the plasma centre are strongly redistributed at the time of sawtooth collapse but no redistribution of trapped fast ions is observed. The redistribution of fast ions from deuterium NBI in the plasma centre is found to vary throughout velocity space. The reduction is most pronounced for passing ions. We find no evidence of inverted sawteeth outside the sawtooth inversion surface in the fast-ion distribution function.

Effective fluid model of turbulent dynamics and transport in tokamak core plasmas

A relatively simple dynamic model of low-frequency plasma convection is proposed to simulate turbulent cross-field transport in the tokamak core. The model is based on a special set of nonlinear weakly dissipative adiabatically reduced MHD-like equations, which self-consistently describe both turbulent plasma fluctuations and the resulting nondiffusive transport. Transient regimes with a fast turn on, turn off, and spatial redistribution of electron-cyclotron heating, as well as regimes with sawtooth oscillations were simulated numerically at time intervals exceeding the plasma energy confinement time. The results obtained demonstrate fast nonlocal responses of the resulting transport processes to the fast changes in external conditions and are in reasonable agreement with experiments in various tokamaks.

Comment on “The importance of auroral features in the search for substorm onset process” by Syun-Ichi Akasofu, A. T. Y. Lui, and C.-I. Meng

Comment on “The importance of auroral features in the search for substorm onset process” by Syun-Ichi Akasofu, A. T. Y. Lui, and C.-I. Meng

Oscillation regimes of the internal kink mode in tokamak plasmas

The long-term dynamics of the internal kink mode are studied using the non-linear, two-fluid magnetohydrodynamics (MHD) code XTOR-2F (Lütjens et al 2010 J. Comput. Phys. 229 8130). The dynamics of the internal kink are studied in resistive MHD plus transport, and, in addition, also including some two-fluid effects. The simulation results yield a pattern of kink cycles or stationary states with m/n = 1/1 helicity. The threshold for sustained (non-decaying) kink cycles, which characterize sawtooth oscillations, is shown to depend on the plasma pressure, the current and energy diffusion times, and the ion and electron diamagnetic drifts. In particular, the diamagnetic flow stabilization extends the cyclic kink regime into a parameter space that is adequate to carry out studies of sawtooth oscillations in ohmic tokamak discharges.

Fast-ion redistribution due to sawtooth crash in the TEXTOR tokamak measured by collective Thomson scattering

Here we present collective Thomson scattering measurements of 1D fast-ion velocity distribution functions in neutral beam heated TEXTOR plasmas with sawtooth oscillations. Up to 50% of the fast ions in the centre are redistributed as a consequence of a sawtooth crash. We resolve various directions to the magnetic field. The fast-ion distribution is found to be anisotropic as expected. For a resolved angle of 39° to the magnetic field we find a drop in the fast-ion distribution of 20–40%. For a resolved angle of 83° to the magnetic field the drop is no larger than 20%.