Doppler-shifted Cyclotron Resonance Research Articles

Application of simulations and theory of sub-cyclotron frequency modes to DIII-D

Theoretical and numerical study of Alfvén eigenmodes (AEs) in the sub-cyclotron frequency range explains the observed frequency scaling with beam parameters in experiments across different devices. Global Alfvén (GAEs) and compressional Alfvén eigenmodes (CAEs) are frequently excited during neutral beam injection (NBI) in the National Spherical Torus Experiment (NSTX/NSTX-U), as well as other beam-heated devices such as MAST and DIII-D. These modes are driven unstable through the Doppler shifted cyclotron resonance with the NBI ions and can be excited in ITER due to super-Alfvénic velocities and anisotropy of the fast ion distribution. Numerical models and theory for sub-cyclotron frequency modes, previously developed and used to study the excitation of GAEs/CAEs in the NSTX(-U), have been successfully applied to explain the DIII-D observations. Simulations for DIII-D demonstrate that the modes with ω/ω ci ∼ 0.6, previously mis-identified as CAEs, have shear polarization and should be identified as the GAEs. Simulation results match the observed frequencies and estimated toroidal mode numbers in DIII-D experiments. Good agreement is found between scaling of the experimentally observed GAE frequencies with NBI parameters for NSTX, NSTX-U and DIII-D and scaling predicted by the theory and simulations. The DIII-D simulation results also demonstrate that GAEs can be excited for larger aspect-ratio devices and smaller (sub-Alfvénic) beam injection velocities, therefore they can be unstable in ITER.

Identification of slow-wave ion cyclotron emission on JT-60U

Ion cyclotron emissions (ICEs) are generally considered to be emissions of fast waves driven by velocity-space instabilities due to fast ions. However, on JT-60U, measured properties of ICEs observed at frequencies lower than the bulk ion cyclotron frequency (L-ICEs) do not match the fast-wave dispersion relation, indicating the possibility of L-ICEs being slow-wave emissions. In this paper, we show that L-ICE observation on JT-60U can be explained in terms of a slow-wave emission driven by fast ions, i.e. a slow-wave ICE. To investigate whether the slow wave can be driven by fast ions, its linear growth rate has been calculated in a typical discharge using a wave dispersion code. This wave dispersion code is capable of performing calculations with an arbitrary spatially-localized velocity distribution. For the growth rate calculation, we have directly used fast ion velocity distributions evaluated by an orbit following Monte-Carlo simulation. It is found that negative-ion-source neutral beam (N-NB) injected fast ions can destabilize the slow wave. Its frequency and wavenumber at high linear growth rates are close to experimental observations of L-ICE. In addition, observed L-ICE frequencies agree reasonably well with frequencies based on the slow-wave dispersion relation at measured toroidal wavenumbers in several discharges. Moreover, the observed frequencies also agree with the Doppler-shifted cyclotron resonance frequencies for the N-NB injected ion. Therefore, L-ICE in JT-60U is identified as a slow-wave ICE driven by the N-NB injected fast ions and distinguished from other ICEs through a measurement of its frequency and wavenumber.

Hybrid simulations of sub-cyclotron compressional and global Alfvén eigenmode stability in spherical tokamaks

A comprehensive numerical study has been conducted in order to investigate the stability of beam-driven, sub-cyclotron-frequency compressional Alfvén eigenmodes (CAEs) and global Alfvén eigenmodes (GAEs) in low-aspect-ratio plasmas for a wide range of beam parameters. The presence of CAEs and GAEs has previously been linked to anomalous electron temperature profile flattening at high beam powers in NSTX experiments, prompting a further examination of the conditions necessary for their excitation. Linear simulations have been performed with the hybrid MHD–kinetic initial value code HYM in order to capture the general Doppler-shifted cyclotron resonance that drives the modes. Three distinct types of modes were found in the simulations—co-CAEs, cntr-GAEs, and co-GAEs—with differing spectral and stability properties. The simulations revealed that unstable GAEs are more ubiquitous than unstable CAEs, which is consistent with experimental observations, as they are excited at lower beam energies and generally have larger growth rates. Local analytic theory is used to explain key features of the simulation results, including the preferential excitation of different modes based on beam injection geometry and the growth rate dependence on the beam injection velocity, critical velocity, and degree of velocity space anisotropy. The background damping rate is inferred from the simulations and analytically estimated for relevant sources absent from the simulation model, indicating that co-CAEs are closer to marginal stability than modes driven by the cyclotron resonances.

Stabilization of Alfvén Eigenmodes in DIII-D via Controlled Energetic Ion Density Ramp and Validation of Theory and Simulations.

Fast-ion driven Alfvén waves with frequency close to the ion cyclotron frequency (f=0.58f_{ci}) excited by energetic ions from a neutral beam are stabilized via a controlled energetic ion density ramp for the first time in a fusion research plasma. The scaling of wave amplitude with injection rate is consistent with theory for single mode collisional saturation near marginal stability. The wave is identified as a shear-polarized global Alfvén eigenmode excited by Doppler-shifted cyclotron resonance with fast ions with sub-Alfvénic energetic ions, a first in fusion research plasmas.

Analytic stability boundaries for compressional and global Alfvén eigenmodes driven by fast ions. I. Interaction via ordinary and anomalous cyclotron resonances

Conditions for net fast ion drive are derived for beam-driven, sub-cyclotron compressional (CAE) and global (GAE) Alfvén eigenmodes, such as those routinely observed in spherical tokamaks such as NSTX(-U) and MAST. Both co- and counter-propagating CAEs and GAEs are investigated, driven by the ordinary and anomalous Doppler-shifted cyclotron resonance with fast ions. Whereas prior results were restricted to vanishingly narrow distributions in velocity space, broad parameter regimes are identified in this work which enable an analytic treatment for realistic fast ion distributions generated by neutral beam injection. The simple, approximate conditions derived in these regimes for beam distributions of realistic width compare well to the numerical evaluation of the full analytic expressions for fast ion drive. Moreover, previous results in the very narrow beam case are corrected and generalized to retain all terms in ω/ωci and |k∥/k⊥|, which are often assumed to be small parameters but can significantly modify the conditions of drive and damping when they are non-negligible. Favorable agreement is demonstrated between the approximate stability criterion, simulation results, and a large database of NSTX observations of cntr-GAEs.

Review of Controlled Excitation of Non-linear Wave-Particle Interactions in the Magnetosphere

Controlled experiments involving injection of 0.5 Hz – 8 kHz electromagnetic waves into the Earth’s magnetosphere have played an important role in discovering and elucidating wave-particle interactions in near-Earth space. Due to the significant engineering challenges of efficiently radiating in the ELF/VLF: 300 Hz – 30 kHz band, few experiments have been able to provide sustained transmissions of sufficient power to excite observable effects for scientific studies. Two noteworthy facilities that were successful in generating a large database of pioneering and repeatable observations were the Siple Station Transmitter in Antarctica and the High Frequency Active Auroral Research Program (HAARP) facility in Alaska. Both facilities were able to excite Doppler shifted cyclotron resonance interactions leading to linear and nonlinear wave amplification, triggering of free running emissions, and pitch angle scattering of energetic electrons. Amplified and triggered waves were primarily observed on the ground in the geomagnetic conjugate region after traversal of the magnetosphere along geomagnetic field aligned propagation paths or in the vicinity of the transmitter following two traversals of the magnetosphere. In several cases, spacecraft observations of the amplified and triggered signals were also made. The observations show the amplifying wave particle interaction to be dynamically sensitive to specific frequency and also specific frequency-time format of the transmitted wave. Transmission of multiple coherent waves closely spaced in frequency showed that the wave particle interaction requires a minimum level of coherency to enter the nonlinear regime. Theory and numerical simulations point to cyclotron resonance with counter streaming particles in the 10-100 keV range as the dominant process. A key feature of the nonlinear interaction is the phase-trapping of resonant particles by the wave that is believed to drive non-linear wave amplification and the triggering of free-running emissions. Observations and modeling of controlled wave injections have important implications for naturally occurring whistler mode emissions of hiss and chorus and the broader phenomena of radiation belt dynamics. A review of observational, theoretical, and numerical results is presented and suggestions for future studies are made.

Global Alfvén eigenmode scaling and suppression: experiment and theory

The spherical tokamak NSTX has been upgraded to include a second neutral beam line, with three independent beam sources, and to be capable of higher toroidal fields and longer duration plasmas (Ono et al 2015 Nucl. Fusion 55 073007). In this paper we describe some of the initial observations of the affect that the higher field and the modified fast-ion distributions have had on the nature of the global Alfvén eigenmodes (GAE). We also report that the GAE excited through a Doppler-shifted ion cyclotron resonance (DCR) were suppressed in a large number of shots with the injection of a small amount of high pitch (V||/V) fast ions, consistent with the predictions of an analytic theory (Gorelenkov et al 2003 Nucl. Fusion 43 228). We show that the experimental scaling of the GAE frequency and toroidal mode numbers with toroidal field is qualitatively consistent with the predictions of the analytic theory, providing validation for the DCR model. The observed suppression of GAE has also been reproduced in simulations with the hybrid ideal stability code HYM (Belova et al 2017 Phys. Plasmas 24 042505).

Energetic-particle-modified global Alfvén eigenmodes

Fully self-consistent hybrid MHD/particle simulations reveal strong energetic particle modifications to sub-cyclotron global Alfvén eigenmodes (GAEs) in low-aspect ratio, NSTX-like conditions. Key parameters defining the fast ion distribution function—the normalized injection velocity v0/vA and central pitch—are varied in order to study their influence on the characteristics of the excited modes. It is found that the frequency of the most unstable mode changes significantly and continuously with beam parameters, in accordance with the Doppler-shifted cyclotron resonances which drive the modes, and depending most substantially on v0/vA. This unexpected result is present for both counter-propagating GAEs, which are routinely excited in NSTX, and high frequency co-GAEs, which have not been previously studied. Large changes in frequency without clear corresponding changes in the mode structure are signatures of an energetic particle mode, referred to here as an energetic-particle-modified GAE. Additional simulations conducted for a fixed MHD equilibrium demonstrate that the GAE frequency shift cannot be explained by the equilibrium changes due to energetic particle effects.

A quasilinear operator retaining magnetic drift effects in tokamak geometry

The interaction of radio frequency waves with charged particles in a magnetized plasma is usually described by the quasilinear operator that was originally formulated by Kennel & Engelmann (Phys. Fluids, vol. 9, 1966, pp. 2377–2388). In their formulation the plasma is assumed to be homogenous and embedded in a uniform magnetic field. In tokamak plasmas the Kennel–Engelmann operator does not capture the magnetic drifts of the particles that are inherent to the non-uniform magnetic field. To overcome this deficiency a combined drift and gyrokinetic derivation is employed to derive the quasilinear operator for radio frequency heating and current drive in a tokamak with magnetic drifts retained. The derivation requires retaining the magnetic moment to higher order in both the unperturbed and perturbed kinetic equations. The formal prescription for determining the perturbed distribution function then follows a novel procedure in which two non-resonant terms must be evaluated explicitly. The systematic analysis leads to a diffusion equation that is compact and completely expressed in terms of the drift kinetic variables. The equation is not transit averaged, and satisfies the entropy principle, while retaining the full poloidal angle variation without resorting to Fourier decomposition. As the diffusion equation is in physical variables, it can be implemented in any computational code. In the Kennel–Engelmann formalism, the wave–particle resonant delta function is either for the Landau resonance or the Doppler shifted cyclotron resonance. In the combined gyro and drift kinetic approach, a term related to the magnetic drift modifies the resonance condition.

ELF/VLF Radio Signals Observed at Low-latitude Station Srinagar, India (<I>L</I> = 1.28)

This communication reports dynamic spectra of different ELF/VLF emissions such as hiss, chorus, hisstriggered chorus and whistler-triggered emissions. We also discuss their various observed features. It is argued that most of the emissions are generated during Doppler-shifted cyclotron resonance interaction between the whistler mode wave and energetic electrons. Resonance energy of the participating electron and interaction length are evaluated to explain the generation mechanism of some of these emissions observed at the Indian station Srinagar.

Non-linear modulation of short wavelength compressional Alfvén eigenmodes

Most Alfvénic activity in the frequency range between toroidal Alfvén eigenmodes and roughly one half of the ion cyclotron frequency on National Spherical Torus eXperiment [Ono et al., Nucl. Fusion 40, 557 (2000)], that is, approximately 0.3 MHz up to ≈1.2 MHz, are modes propagating counter to the neutral beam ions. These have been modeled as Compressional and Global Alfvén Eigenmodes (CAE and GAE) and are excited through a Doppler-shifted cyclotron resonance with the beam ions. There is also a class of co-propagating modes at higher frequency than the counter-propagating CAE and GAE. These modes have been identified as CAE, and are seen mostly in the company of a low frequency, n = 1 kink-like mode. In this paper, we present measurements of the spectrum of these high frequency CAE (hfCAE) and their mode structure. We compare those measurements to a simple model of CAE and present a predator-prey type model of the curious non-linear coupling of the hfCAE and the low frequency kink-like mode.

Internal amplitude, structure and identification of compressional and global Alfvén eigenmodes in NSTX

Fast-ions (e.g. fusion alphas and neutral beam ions) will excite a wide range of instabilities in ITER and a Fusion Nuclear Science Facility device. Among the possible instabilities are high frequency Alfvén eigenmodes (AEs) excited through Doppler-shifted cyclotron resonance with beam ions. High frequency AEs cause fast-ion transport, correlate with enhanced electron thermal transport and are postulated to contribute to ion heating. These high frequency modes have historically been identified as a mixture of compressional (CAE) and global (GAE) Alfvén eigenmodes, but distinguishing between the CAEs and GAEs has sometimes proven difficult. Identification is essential for understanding the extent of their effect, since the two types of modes have very different effects on resonant particle orbits. The effect on plasma performance of high frequency AEs is investigated in NSTX, facilitated by a recently upgraded array of 16 fixed-frequency quadrature reflectometers. Detailed measurements of high frequency AE amplitude and eigenmode structure were obtained in a high power (6 MW), beam-heated H-mode plasma that is very similar to those in which high frequency AE activity is shown to correlate with enhanced electron thermal transport. These measurements, which extend from the plasma edge to deep in the core, can be used in modelling the effects of the modes on electron thermal transport. The observed modes are identified by comparison of their frequency and measured toroidal mode numbers with local Alfvén dispersion relations. The modes identified as CAEs have higher frequencies (predominantly f> ∼600 kHz) and smaller toroidal mode numbers (|n| ⩽ 5) than the GAEs (predominantly f < ∼600 kHz, n = −6 to −8). Also, they are strongly core localized, in contrast with the GAEs, which also peak towards the plasma centre but have much broader radial extent.

On the theory of nonlinear dopplerons in metals

The possibility of the nonlinear wave propagation occurring in metals, such as cadmium, in the geometry where a constant magnetic field H is directed at an angle with respect to the hexagonal axis of the crystal has been investigated theoretically. In the linear regime in this geometry, there is a significant magnetic Landau damping, which prevents propagation of waves. It has been shown that the electron trapping by the magnetic field of a wave can substantially decrease the efficiency of this damping. As a result, the propagation of a doppleron, i.e., the mode related to the Doppler-shifted cyclotron resonance of electrons, becomes possible. It has been noted that a similar “bleaching” effect should also take place in some other metals.

Observation of global Alfvén eigenmode avalanche events on the National Spherical Torus Experiment

Instabilities excited by the fast-ion population on NSTX (Ono et al 2000 Nucl. Fusion 40 557) extend from low-frequency energetic particle modes (EPMs) at tens of kHz through toroidal Alfvén eigenmodes (TAEs) in the range 50–150 kHz to global and compressional Alfvén eigenmodes (GAE and CAE) in the frequency range 0.3–2.5 MHz, or roughly 0.1ωci to 0.7ωci. The GAE instabilities exhibit complex non-linear behaviour, including onset of strong growth above an amplitude threshold. This is conjectured to occur when resonance regions in phase space start to overlap, resulting in enhanced rapid growth and redistribution of energetic particles, a process referred to as an ‘avalanche’ (Berk et al 1995 Nucl. Fusion 35 1661). The GAE are suppressed following the avalanche, suggesting depletion of the fast-ion population resonantly driving the modes, and in some instances the GAE bursts appear to trigger lower frequency TAE avalanches or EPMs, suggesting some significant redistribution of fast ions in phase space has occurred. These are the first reported observations of avalanching behaviour for an instability driven through the Doppler-shifted cyclotron resonance. This paper also provides internal measurements of GAE structure showing that the mode amplitude peaks towards the plasma core.

Doppler-shifted cyclotron resonance of fast ions with circularly polarized shear Alfvén waves

The Doppler-shifted cyclotron resonance between fast ions and shear Alfvén waves (SAWs) has been experimentally investigated with a test-particle fast-ion (Li+) beam launched in the helium plasma of the Large Plasma Device [Gekelman et al., Rev. Sci. Instrum. 62, 2875 (1991)]. Left- or right-hand circularly polarized SAWs are launched by an antenna with four current channels. A collimated fast-ion energy analyzer characterizes the resonance by measuring the nonclassical spreading of the averaged beam signal. Left-hand circularly polarized SAWs resonate with the fast ions but right-hand circularly polarized SAWs do not. The measured fast-ion profiles are compared with simulations by a Monte Carlo Lorentz code that uses the measured wave field data.

Ponderomotive acceleration of electrons by a laser pulse in magnetized plasma

Electron acceleration by a circularly polarized Gaussian laser pulse in magnetized plasma is investigated in the limit of frozen refractive index. The electron acceleration depends on the ratio of laser frequency to electron cyclotron frequency, amplitude of the laser pulse and plasma density. Near Doppler shifted cyclotron resonance the electron acquires maximum energy. In this scheme, 0.10 MeV electrons can be effectively accelerated to 1–100 MeV using moderate intensity laser pulse.

Observation of fast-ion Doppler-shifted cyclotron resonance with shear Alfvén waves

The Doppler-shifted cyclotron resonance (ω−kzvz=Ωf) between fast ions and shear Alfvén waves is experimentally investigated (ω, wave frequency; kz, axial wavenumber; vz, fast-ion axial speed; Ωf, fast-ion cyclotron frequency). A test particle beam of fast ions is launched by a Li+ source in the helium plasma of the LArge Plasma Device (LAPD) [W. Gekelman, H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. Maggs, Rev. Sci. Instrum. 62, 2875 (1991)], with shear Alfvén waves (SAW) (amplitude δ B/B up to 1%) launched by a loop antenna. A collimated fast-ion energy analyzer measures the nonclassical spreading of the beam, which is proportional to the resonance with the wave. A resonance spectrum is observed by launching SAWs at 0.3–0.8ωci. Both the magnitude and frequency dependence of the beam-spreading are in agreement with the theoretical prediction using a Monte Carlo Lorentz code that launches fast ions with an initial spread in real/velocity space and random phases relative to the wave. Measured wave magnetic field data are used in the simulation.

Electron acceleration in the Van Allen radiation belts by fast magnetosonic waves

Local acceleration is required to explain electron flux increases in the outer Van Allen radiation belt during magnetic storms. Here we show that fast magnetosonic waves, detected by Cluster 3, can accelerate electrons between ∼10 keV and a few MeV inside the outer radiation belt. Acceleration occurs via electron Landau resonance, and not Doppler shifted cyclotron resonance, due to wave propagation almost perpendicular to the ambient magnetic field. Using quasi‐linear theory, pitch angle and energy diffusion rates are comparable to those for whistler mode chorus, suggesting that these waves are very important for local electron acceleration. Since pitch angle diffusion does not extend into the loss cone, these waves, on their own, are not important for loss to the atmosphere. We suggest that magnetosonic waves, which are generated by unstable proton ring distributions, are an important energy transfer process from the ring current to the Van Allen radiation belts.

Doppler-shifted cyclotron, doppleron–phonon, and pseudo-doppleron–phonon resonances in indium

Ultrasonic investigations have been carried out in geometry of the experiment corresponding to acoustic analogy of the Faraday effect with the use of a super pure crystal like indium. Lowering the temperature from 4.2 K to 1.28 K has revealed new peculiarities of in the magnetic field dependences of absorption and phase velocity of circularly polarized ultrasonic waves. The peculiarities have been interpreted as manifestation of multiple doppleron–phonon and Doppler-shifted cyclotron resonances. The discovery of these resonances has changed interpretation of some anomalies observed in the previous experiments.

Modelling substorm chorus events in terms of dispersive azimuthal drift

Abstract. The Substorm Chorus Event (SCE) is a radio phenomenon observed on the ground after the onset of the substorm expansion phase. It consists of a band of VLF chorus with rising upper and lower cutoff frequencies. These emissions are thought to result from Doppler-shifted cyclotron resonance between whistler mode waves and energetic electrons which drift into a ground station's field of view from an injection site around midnight. The increasing frequency of the emission envelope has been attributed to the combined effects of energy dispersion due to gradient and curvature drifts, and the modification of resonance conditions and variation of the half-gyrofrequency cutoff resulting from the radial component of the ExB drift. A model is presented which accounts for the observed features of the SCE in terms of the growth rate of whistler mode waves due to anisotropy in the electron distribution. This model provides an explanation for the increasing frequency of the SCE lower cutoff, as well as reproducing the general frequency-time signature of the event. In addition, the results place some restrictions on the injected particle source distribution which might lead to a SCE. Key words. Space plasma physics (Wave-particle interaction) – Magnetospheric physics (Plasma waves and instabilities; Storms and substorms)