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At least 253 records · Page 14

The nonlinear instability in flap-lag of rotor blades in forward flight

The nonlinear flap-lag coupled oscillation of torsionally rigid rotor blades in forward flight is examined using a set of consistently derived equations by the asymptotic expansion procedure of multiple time scales. The regions of stability and limit cycle oscillation are presented. The roles of parametric excitation, nonlinear oscillation, and forced excitation played in the response of the blade are determined.

Tong, P.↗

Heat Transfer and Interfacial Flow Physics of Microgravity Flow Boiling in Single-Side-Heated Rectangular Channel with Subcooled Inlet Conditions – Experiments Onboard the International Space Station

This study is the culmination of a long-term collaborative effort between researchers from the Purdue University Boiling and Two-Phase Flow Laboratory (PU-BTPFL) and the NASA Glenn Research Center to investigate gravitational effects on flow boiling and flow condensation. The science and design concepts for this large-scale effort were initiated in 2011 and included several studies detailing various aspects of two-phase fluid physics in both Earth gravity and microgravity, culminating in construction of the large-scale experimental facility named “Flow Boiling and Condensation Experiment (FBCE)”. The experiment was launched to the International Space Station (ISS) in August 2021. Following the successful installation of FBCE, equipped with the Flow Boiling Module (FBM), onboard the ISS and completion of several safety checks, flow boiling experiments were performed for five months from February 2022 until July 2022. This resulted in a large flow boiling database covering broad ranges of operating parameters and heating configurations spanning several research objectives. This study investigates microgravity flow boiling of n-perfluorohexane with subcooled inlet in a single-side-heated rectangular channel of dimensions 114.6-mm heated length, 2.5-mm heated width, and 5.0-mm height. Key operating parameters investigated are mass velocity (199.90 – 3200.13 kg/m 2 s), inlet subcooling (0.10 – 45.76°C), and inlet pressure (113.30 – 164.29 kPa). Images and image sequences acquired via high-speed-video are presented to elucidate the interfacial flow physics. To analyze and explain the effects of various parameters in microgravity, heat transfer results are presented as flow boiling curves, streamwise profiles of wall temperature and heat transfer coefficient, and parametric trends of local and averaged heat transfer coefficient. Mass velocity and inlet subcooling significantly influenced most of the aforementioned aspects of flow boiling, whereas effects of inlet pressure were comparatively insignificant. Although the data and observed flow physics might be different, the parametric effects and trends in microgravity are similar to vertical upflow in Earth gravity. Some cases, especially low mass velocities, high heat fluxes, and large degrees of inlet subcooling, experienced temporally anomalous flow behaviors caused by two-phase flow instabilities manifesting as flow reversals and resulted in deviations in overall trends. Severe thermodynamic non-equilibrium is observed throughout the channel. Overall, FBCE’s ISS experiments were successful for subcooled inlet with single-sided heating of rectangular channel, and the collected data well established the various effects on flow boiling physics in highly controlled long-term microgravity conditions.

flow boiling↗

Parametric interactions between Alfven waves and sonic waves

Parametric interactions between a sonic pump wave and a weak Alfven wave are studied. It is shown that if the Alfven-wave velocity Va is small relative to the sonic-wave velocity Vs there is a time-growing instability leading to the increase of the Alfven wave at the expense of the sonic wave. This phenomenon can be of importance in solar and stellar physics. For Va large relative to Vs, the interaction is of the stop-band type.

Elachi, C.↗

Aeroelastic Stability of A Soft-Inplane Gimballed Tiltrotor Model In Hover

Soft-inplane rotor systems can significantly reduce the inplane rotor loads generated during the maneuvers of large tiltrotors, thereby reducing the strength requirements and the associated structural weight of the hub. Soft-inplane rotor systems. however, are subject to instabilities associated with ground resonance, and for tiltrotors this instability has increased complexity as compared to a conventional helicopter. Researchers at Langley Research Center and Bell Helicopter-Textron, Inc. have completed ail initial study of a soft-inplane gimballed tiltrotor model subject to ground resonance conditions in hover. Parametric variations of the rotor collective pitch and blade root damping, and their associated effects oil the model stability were examined. Also considered in the study was the effectiveness of ail active swash-plate and a generalized predictive control (GPC) algorithm for stability augmentation of the ground resonance conditions. Results of this study show that the ground resonance behavior of a gimballed soft-inplane tiltrotor can be significantly different from that of a classical soft-inplane helicopter rotor. The GPC-based active swash-plate was successfully implemented, and served to significantly augment damping of the critical modes to an acceptable value.

Nixon, Mark W.↗

Aeroelastic Stability of a Soft-Inplane Gimballed Tiltrotor Model in Hover

Soft-inplane rotor systems can significantly reduce the inplane rotor loads generated during the maneuvers of large tiltrotors, thereby reducing the strength requirements and the associated structural weight of the hub. Soft-inplane rotor systems, however, are subject to instabilities associated with ground resonance, and for tiltrotors this instability has increased complexity as compared to a conventional helicopter. Researchers at Langley Research Center and Bell Helicopter-Textron, Inc. have completed an initial study of a soft-inplane gimballed tiltrotor model subject to ground resonance conditions in hover. Parametric variations of the rotor collective pitch and blade root damping., and their associated effects on the model stability were examined. Also considered in the study was the effectiveness of an active swashplate and a generalized predictive control (GPC) algorithm for stability augmentation of the ground resonance conditions. Results of this study show that the ground resonance behavior of a gimballed soft-inplane tiltrotor can be significantly different from that of a classical soft-inplane helicopter rotor. The GPC-based active swashplate was successfully implemented, and served to significantly augment damping of the critical modes to an acceptable value.

Nixon, Mark W.↗

A multi-dimensional dataset for two-phase instability in low pressure natural circulation based on direct transient local measurement

Thermal hydraulic instability can challenge the control and the safety of nuclear reactors. This experimental research captures and investigates the instability in a low-pressure (1–10 atmospheres) natural circulation loop with an adiabatic chimney installed above the heated section. Direct transient local measurement of two-phase flow is performed at multiple axial locations along the flow channel with thermocouples and conductivity probes traversed by customized high-precision mechanisms. Based on the periodicity of oscillations, a post-processing method is proposed to align the data captured non-simultaneously to reconstruct the instability behavior in one representative period. The averaging operation over multiple oscillation cycles is performed to reduce the random errors. Cyclic phenomena are clearly demonstrated by the post-processed data, and rapid void fraction change induced by the flashing is observed to be significant in the driving mechanism for low pressure oscillations. Parametric effects of inlet subcooling, heat flux, inlet flow restriction and system pressure are studied within the experiment range, and qualitative results are explained and compared with existing literature. The ratio between the oscillation period and the travelling time of the fluid particle through the test section is compared with the theoretical value for the density wave oscillation. The new dataset contains directly measured transient multi-dimensional two-phase data along the flow channel for natural circulation under density wave oscillations. The generated dataset is valuable for the validation of system analysis codes and CFD codes in the simulation of natural circulation phenomena and dynamic two-phase flow.

42 ENGINEERING↗

Parametric studies for tiltrotor aeroelastic stability in high-speed flight

The influence of several system design parameters on tiltrotor aeroelastic stability is examined for the high-speed (axial) flight mode. Coupling of the rotor flapping modes with the wing elastic modes produces a whirl motion, typical of tiltrotors, that can become unstable at high speeds. The sensitivity of this instability with respect to rotor frequencies, wing stiffness, forward wing sweep, and rotor thrust level is examined. Some important new trends are identified regarding the role of blade lag dynamics and forward wing sweep in tiltrotor aeroelastic stability. The blade lag frequency may be tuned to improve tiltrotor stability, and forward wing sweep is destabilizing because of changes in rotor force components associated with the sweep.

Nixon, Mark W.↗

PPPL Report on Reduced Modeling of Fusion Alpha Transport in ARC Burning Plasmas

We are reporting on the modeling of fusion alpha particle transport in the planned ARC fusion device being designed by the CFS (Commonwealth Fusion Systems: https://cfs.energy). The ARC tokamak is designed to operate in a burning-plasma regime characterized by a substantial population of fusion-born alpha particles. Alfvén eigenmode (AE) stability is assessed both analytically and numerically, incorporating alpha-particle drive, ion Landau, and radiative damping from thermal species and collisional damping from trapped electrons. Regions of unstable and near-threshold AE activity are mapped across ARC’s operational parameter space. Linear stability analysis with NOVA indicates multiple, often marginally unstable AEs, extending to toroidal mode numbers up to n= 30. The present report focuses on the ARC flat-top operating point prior to the sawtooth event. Alpha-particle transport on timescales exceeding the neoclassical slowing-down time is assessed using the NUBEAM module [1][2] of the TRANSP code [3], employing transport coefficients derived from the RBQ quasilinear modeling (cf. Appendix B). These global simulations identify favorable and unfavorable operating regimes with respect to alpha confinement, pressure redistribution, and overall alpha-heating efficiency. We also evaluate additional transport mechanisms—including neoclassical tearing mode (TM)–induced stochasticity, sawtooth-driven redistribution, and toroidal-field ripple using the kick model (cf. Appendix C) which makes use of the guiding-center code ORBIT, see Section 5. The kick model is integrated into TRANSP to enable self-consistent predictions of alpha-driven current formation and sustainment within the ARC scenario. Sensitivity scans are performed over the mode frequency, rational-surface alignment, island width, mode amplitude, and proximity of the limiter to the plasma. Our study provides an initial, physics-based guidance for machine design, operational planning, and equilibrium control, ensuring adequate alpha confinement and robust self-heating performance in ARC. Our simulations mostly targeted worst case scenarios, e.g. for TMs and sawteeth. Overall, we expect benign effects for the ARC scenario investigated in this work on fusion alpha confinement and losses in the presence of AEs, tearing modes and sawteeth. This report addresses three thrusts identified at the outset. The first thrust focuses on analytic estimates of the parametric dependencies of EP relaxation based on local AE stability simulations (Section 3). The second thrust involves global evaluations of AE stability using the NOVA, RBQ, and NUBEAM codes (Section 4). Finally, we investigate alpha-particle transport driven by low-frequency instabilities associated with sawteeth and tearing modes (Section 5).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Influence of third-degree geometric nonlinearities on the vibration and stability of pretwisted, preconed, rotating blades

The governing coupled flapwise bending, edgewise bending, and torsional equations are derived including third-degree geometric nonlinear elastic terms by making use of the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. These equations are specialized for blades of doubly symmetric cross section with linear variation of pretwist over the blade length. The nonlinear steady state equations and the linearized perturbation equations are solved by using the Galerkin method, and by utilizing the nonrotating normal modes for the shape functions. Parametric results obtained for various cases of rotating blades from the present theoretical formulation are compared to those produced from the finite element code MSC/NASTRAN, and also to those produced from an in-house experimental test rig. It is shown that the spurious instabilities, observed for thin, rotating blades when second degree geometric nonlinearities are used, can be eliminated by including the third-degree elastic nonlinear terms. Furthermore, inclusion of third degree terms improves the correlation between the theory and experiment.

Subrahmanyam, K. B.↗

Influence of third-degree geometric nonlinearities on the vibration and stability of pretwisted, preconed, rotating blades

The governing coupled flapwise bending, edgewise bending, and torsional equations are derived including third-degree goemetric nonlinear elastic terms by making use of the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. These equations are specialized for blades of doubly symmetric cross section with linear variation of Pretwist over the blade length. The nonlinear steady state equations and the linearized perturbation equations are solved by using the Galerkin method, and by utilizing the nonrotating normal modes for the shape functions. Parametric results obtained for various cases of rotating blades from the present theoretical formulation are compared to those produced from the finite element code MSC/NASTRAN, and also to those produced from an in-house experimental test rig. It is shown that the spurious instabilities, observed for thin, rotating blades when second degree geometric nonlinearities are used, can be eliminated by including the third-degree elastic nonlinear terms. Furthermore, inclusion of third degree terms improves the correlation between the theory and experiment.

Subrahmanyam, K. B.↗

Relativistic stellar stability - Preferred-frame effects

In a previous paper, the PPN (parametrized post-Newtonian) formalism was used to analyze relativistic influences on stellar stability in nearly all metric theories of gravity. That analysis omitted all preferred-frame terms. In this paper, possible preferred-frame effects on stellar stability are examined and no new instabilities are found. Although terms linear in the preferred-frame velocity w (time-odd terms, analogous to viscosity and energy generation) change the shapes of the normal modes, their symmetry properties prevent them from changing the characteristic frequencies. Thus, no new vibrational or secular instabilities can occur. Terms quadratic in w do not change either the shapes of the normal modes or the characteristic frequencies for radial pulsations (except for the effects due to the renormalization of the gravitation constant which does not affect stability). Thus, they have no influence on radial stability. Terms quadratic in w do change both the normal modes and the characteristic frequencies of nonradial pulsations; but in the limit of a neutral mode these changes vanish.

Ni, W.-T.↗

Planetary waves can activate resonant drag instabilities in 3D dusty gaseous discs

Resonant drag instabilities (RDIs) in protoplanetary discs are driven by the aerodynamic back-reaction of dust on gas and occur when the relative dust-gas motion resonate with a wave mode intrinsic to the gas fluid. Axisymmetric models indicate that the RDI generates filamentary perturbations, leading to grain clumping and planetesimal formation. Motivated by these findings, we investigate the dust–gas interaction in a non-axisymmetric inviscid protoplanetary disc with an embedded low-mass planet ($M_{\mathrm{p}}\in [0.3, 3] \, \rm M_{\oplus }$, here $\, \rm M_{\oplus }$ is the Earth mass). We conduct global 3D high-resolution two-fluid simulations, with the dust being parametrized by the Stokes number $\mathrm{St}\in [0.01,0.5]$. We find that planetary waves (PWs; also known as Rossby waves), which propagate along the downstream separatrices of the horseshoe region, resonate with the streaming motion and trigger the RDI. The consequent development of a global-scale filamentary dust distribution does not sensitively depend on the Stokes number, nor does it depend on the fast dust settling that takes place in an inviscid disc. The rapid onset of this instability, which is comparable to the dynamical orbital time-scale, suppresses the formation of asymmetric structures in the dust in the vicinity of the planet (such as dust voids and filaments). Additionally, we find that the dust feedback enables buoyancy resonances in an otherwise non-buoyant (globally isothermal) disc. Therefore, our results provide the first numerical evidence of RDIs generation driven by planetary waves.

79 ASTRONOMY AND ASTROPHYSICS↗

Tunable Laser-Plasma Amplifier (Final Report)

Exploration at the laser intensity frontier has always offered new avenues for physics and reaching beyond this frontier is a grand challenge. Present-day petawatt-class lasers provide focused intensities on target of 10 22 W/cm 2 , corresponding to electric fields of 200 TV/m, while laser-plasma amplification opens a route for focused intensities well above 10 23 W/cm 2 . Intensities in this range provide the ability to test quantum electrodynamics in the unexplored low-energy, strong-field regime where signatures for new physics may arise. The behavior of matter under such extraordinary conditions is a rich and fascinating subject, not only in its own right in fundamental plasma physics, but for the many potential applications that promise to enrich the natural sciences in the future, including compact electron-beam, ion-beam particle accelerators, and ultra-bright X-ray sources. Although laser systems are now under construction internationally to access intensities of 10 23 W/cm 2 , the current technologies used appear to be fundamentally limited to these intensities. The realization of intensities beyond 10 23 W/cm 2 using parametric amplification in plasmas promises a breakthrough in high-energy density physics. Parametric amplification using Raman scattering in a plasma could provide the enabling technology for the generation of ultra-high-power laser pulses, but a more complete understanding of the nonlinear optics of plasmas is required. There is a significant gap in well-diagnosed laser-plasma instability studies of nonlinear plasma-wave phenomena, which are critical to understand for future laser-plasma devices. To achieve an efficient laser-plasma amplifier, plasma waves must be driven to large amplitude where significant energy can be rapidly transferred from the pump to the seed over the pulse duration of the seed. Simulations suggest that this nonlinear pump depletion regime can be achieved in the “pi-pulse” amplification regime. While simulations show this optimal regime with efficient amplification, it has remained elusive in experiments and there is a growing consensus within the community that thermal effects and pump beam limitations prevent laser-plasma amplifiers from progressing through the linear regime into the nonlinear pump depletion regime. Previous experiments have been significantly limited by the laser power available at the necessary wavelengths for the seed laser; therefore, the amplification is required to start in the linear regime where it is sensitive to many deleterious effects. The enabling technology (currently unique to plasma-wave amplification in the world) at the University of Rochester is the ability to provide a seed pulse with sufficient power (4 mJ/100 fs seed) to immediately drive nonlinear plasma waves into the pi-pulse regime and to tune its wavelength to optimize the efficiency of energy transfer. This in combination with the state-of-the-art OMEGA heater beams providing multiple kilojoules in a nanosecond to sufficiently heat the plasma make this system distinct from previous studies. These heater beams will provide, for the first time in Raman amplification studies, a homogeneous electron temperature high enough to prevent pump beam propagation issues that have plagued previous experiments. These systems will provide a platform for driving electron-plasma waves into the nonlinear regime where pump depletion and pulse shortening are predicted to lead to high amplification efficiencies (>30%). The Team has made significant progress through prior support from DOE Fusion Energy Sciences [DOE Office of Science Award Number DE-SC0016253 (2016-2022)]. This includes twenty-two peer-reviewed manuscripts, one patent, ten contributed talks presented at international conferences, and research that was highlighted as invited talks at fifteen international conferences. The broader impacts of this research are evident in the support of early career scientists, two Ph.D. theses, four current graduate students, a Masters Project, two undergraduate researchers, and an underrepresented minority student hired through the California Alliance for Minority Participation who now works as a Research Engineer in the group. This research met all of the funded research objectives and the highlights from primary Raman amplification thrust of this work are discussed below and form the foundation for the proposed research.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Test results from a simple, low-cost, pressure-fed liquid hydrogen/liquid oxygen rocket combustor

A simple, low-cost rocket engine was designed, fabricated, and successfully hot fire tested over a wide range of interface conditions and operating parameters. The engine used low enthalpy hydrogen (45 to 70 R, 200 to 390 psia) and oxygen (139 to 163 R, 210 to 480 psia) propellants pressure-fed directly from facility cryogenic tanks. The engine demonstrated excellent performance, with 97% average combustion efficiency, and absence of combustion instabilities. Engine design chamber pressure was 300 psia, yielding about 16,500 pounds thrust at sea level with a 3:1 expansion ration test nozzle. The engine used a fixed-element injector based on TRW's unique coaxial pintle design, but was operated at 60%, 80%, and 100% thrust levels by throttling facility propellant valves. The engine was tested at propellant mixture ratios (O/F) from 5.8 to 8.4; design O/F was 6.6. To document combustion stability, in five tests RDX explosive pulse guns were detonated in radial and tangential directions across the combustion chamber during steady-state operation. The largest disturbance consisted of simultaneous detonation of a 20-grain radial gun and a 40-grain tangential gun. In no case was an instability, either feed system mode or chamber acoustic mode, excited. High-frequency piezoelectric pressure transducers documented stable recovery from disturbance overpressures within 40 milliseconds of peak pressure. A total of 67 firing tests, accumulating 149 seconds of firing time above 10% P(sub c), were performed. Since parametric testing required run durations of only 2 to 3 seconds, a heat sink combustion chamber was employed for most runs. To evaluate the feasibility of a low-cost ablative system for a flight engine design, one 20-second continuous firing was conducted with a silicone rubber chamber/throat/nozzle liner cast in one piece directly into the engine. The ablative engine operated at the equivalent of 309 seconds sea level specific impulse, when adjusted to a 98% efficient 6:1 expansion ration nozzle, and 430 seconds vacuum specific impulse, when adjusted to a 98% efficient 50:1 expansion ratio nozzle. This engine and test series represent an initial subscale demonstration of a new booster-class engine that eliminates the cost and complexity associated with regenerative cooling and typical engine cycles. This paper presents a description of the engine design and discussion and summary data plots of the performance measured during the parametric testing.

Dressler, G. A.↗

Heat Entrapment Effects Within Liquid Acquisition Devices

We introduce a model problem to address heat entrapment effects or the local accumulation of thermal energy within liquid acquisition devices. We show that the parametric space consists of six parameters, namely the Rayleigh and Prandtl numbers, the aspect ratio, and heat flux ratios for the bottom, side, and top boundaries of the enclosure. For the range of Ra considered 1 to 10(sup 9), beyond Ra on the order of 10(sup 5), convective instability is the dominant mode of convection in comparison to natural convection. The flow field transitions to asymmetric modes at Ra on the order of 10(sup 7). Direct numerical simulation of a large geometric length scale prototype for Ra on the order of 10(sup 9) shows that the flow field evolves from small wavelength instability which gives rise to nonlinear growth of thermals, propagation of the instability occurs via growth of secondary and tertiary modes, and a travelling wave mode occurs prior to asymmetry. The effect of a large aspect ratio is to increase the number of modes in the vertical direction. Due to the slow diffusion of heat in the prototype, asymptotic states are not readily attained, we show that dynamical similarity can be used for a model which allows the attainment of asymptotic states and that transition to a chaotic state occurs for Ra on the order of 10(sup 9) via a broadband power spectrum. These dynamical events show that for the baseline condition in which heat is absorbed from background laboratory environment, higher heat flux is absorbed at the top and bottom boundaries of the enclosure than a nominal value of 34.9 ergs per square centimeter -second.

Duval, W. M. B.↗

Flame flashback in a premixed dump combustor

A stainless steel, rectangular center-dump, premixed-prevaporized combustor was used at the NASA Lewis Research Center to study flashback in controlled conditions simulating gas turbine combustor operations under steady-state and transient conditions. The combustor allowed visual access to the combustion area, and was operated at inlet air temperatures from 600 to 850 K, premixer wall temperatures from 450 to 900 K, and average premixer velocity from 40 to 80 ft/s. A parametric study revealed a slight decrease in the flashback equivalence ratio as the inlet air temperatures increased. It also indicated that the premixer wall temperature and premixer velocity are not governing parameters of flashback. It is suggested that flashback occurs through the premixer wall boundary layer flow reversal caused by combustion instability.

Proctor, M. P.↗

Grey Galaxies’ as an endpoint of the Kerr-AdS superradiant instability

Kerr-AdS d+1 black holes for d ≥ 3 suffer from classical superradiant instabilities over a range of masses above extremality. We conjecture that these instabilities settle down into Grey Galaxies (GGs) — a new class of coarse-grained solutions to Einstein’s equations which we construct in d = 3. Grey Galaxies are made up of a black hole with critical angular velocity ω = 1 in the ‘centre’ of AdS, surrounded by a large flat disk of thermal bulk gas that revolves around the centre of AdS at the speed of light. The gas carries a finite fraction of the total energy, as its parametrically low energy density and large radius are inversely related. GGs exist at masses that extend all the way down to the unitarity bound. Their thermodynamics is that of a weakly interacting mix of Kerr-AdS black holes and the bulk gas. Their boundary stress tensor is the sum of a smooth ‘black hole’ contribution and a peaked gas contribution that is delta function localized around the equator of the boundary sphere in the large N limit. We also construct another class of solutions with the same charges; ‘Revolving Black Holes (RBHs)’. RBHs are macroscopically charged SO(d, 2) descendants of AdS-Kerr solutions, and consist of ω = 1 black holes revolving around the centre of AdS at a fixed radial location but in a quantum wave function in the angular directions. RBH solutions are marginally entropically subdominant to GG solutions and do not constitute the endpoint of the superradiant instability. Nonetheless, we argue that supersymmetric versions of these solutions have interesting implications for the spectrum of supersymmetric states in, e.g. N = 4 Yang-Mills theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Radiative damping of toroidal Alfvén eigenmode in low-shear plasmas

Instabilities of Alfvén eigenmodes (AEs) are of significant concern because they can enhance the cross-field transport of fusion-born alpha particles beyond the neoclassical level in magnetic fusion plasmas. The threshold value of alpha-particle pressure for exciting AEs depends critically on the damping rate of AEs. The damping mechanisms include kinetic damping due to interactions with thermal particles, continuum damping due to AE frequency crossing Alfvén continuum, and radiative damping due to emitting kinetic Alfvén waves (KAWs). The radiative damping is substantial and can even prevail in high-temperature burning plasmas [1]. We revisit the radiative damping analytic theory for TAE in plasmas with low positive magnetic shear, considering TAE with an eigenfrequency near the bottom of TAE-gap and with poloidal harmonics of the same sign (even TAE). In contrast to earlier papers, we provide the damping calculations in real space rather than Fourier space. This approach is straightforward technically and more enlightening from a physics standpoint for benchmarking numerical calculations of radiative damping. The parametric dependence of the resulting damping rate agrees with that of Refs. [2-5], but it has a smaller numerical factor in front of it.

Alpha-particle driven instability↗