Azimuthal Periodicity Research Articles

Radial Transport in the Earth’s Radiation Belts: Linear, Quasi-linear, and Higher-order Processes

Observational studies of the Earth’s radiation belts indicate that Alfvénic fluctuations in the frequency range of 2–25 mHz accelerate electrons to relativistic energies. For decades, statistical models of radiation belts have quantified the impact of Alfvénic waves in terms of quasi-linear diffusion. However, quasi-linear models are inadequate to quantify Alfvénic radial transport occurring on timescales comparable to the azimuthal drift period of 0.1–10 MeV electrons. With recent advances in observational methodologies offering coverage of the Earth’s radiation belts on fast timescales, a theoretical framework that distinguishes between fast and diffusive radial transport can be tested for the first time in situ. In this report, we present a drift-kinetic description of radial transport for planetary radiation belts. We characterize fast linear processes and determine the conditions under which higher-order effects become dynamically significant. In the linear regime, wave–particle interactions are categorized in terms of resonant and nonresonant responses. We demonstrate that the phenomenon of zebra stripes is nonresonant and can originate from injection events in the inner radiation belts. We derive a radial diffusion coefficient for a field model that satisfies Faraday’s law and that contains two terms: one scaling as L 10 independent of the azimuthal number m, and a second scaling as m 2 L 6. In the higher-order regime, azimuthally symmetric waves with properties consistent with in situ measurements can energize 10–100 keV electrons in less than a drift period. This process provides new evidence that acceleration by Alfvénic waves in radiation belts cannot be fully contained within diffusive models.

Rotating Instabilities in Shrouded Low Pressure Turbine at Design and Off-Design Conditions

Abstract The present study aims to analyze the rotating instabilities that may occur inside shroud cavities above rotors of low-pressure turbine configurations. Unsteady simulations have been carried out at design and off-design conditions on two configurations, one being a multi-stage configuration. Unsteady flow structures, uncorrelated from blade passing frequencies and depending on operating points, are identified in every rotor tip shroud cavity. Similarities regarding flow patterns and interactions with the main flow are observed: hot spots of gas having different azimuthal extend come from the tip shroud cavity and rotate at the interface with the main flow path. The influence on the main flow and the origin of these instabilities are discussed. The study at off-design operating points deepens the analysis and allows the identification of physical parameters which drive the instability.

Flash-back, blow-off, and symmetry breaking of premixed conical flames

Hydrodynamic and thermal-diffusive effects subject premixed flames to intrinsic instabilities that strongly influence their shape and propagation/stabilization characteristics. However, the interaction and coupling of intrinsic flame dynamics with background flow gradients and boundary conditions remain poorly understood. This paper presents a global nonlinear bifurcation analysis of burner-stabilized laminar premixed conical flames with varying reaction rates and reactant diffusivities, respectively parameterized by the Damköhler number Da and the Lewis number Le. Using a dimensionless formulation of the reacting, weakly-compressible Navier–Stokes equations, the dynamics of flash-back, blow-off, and cellular instability are explored in a fully-coupled framework. Our analysis identifies steady conical flame states over a finite range of Da and Le, limited by saddle–node bifurcations corresponding to spontaneous blow-off of the axisymmetric flame below a Le-dependent lower critical Da value and spontaneous boundary-layer flash-back beyond a higher critical Da value. Furthermore, the analysis reveals that the conical flame loses its axisymmetry via circle–pitchfork bifurcations as Le or Da decrease or increase beyond respective critical values. These bifurcations are shown to correspond to stationary three-dimensional global modes describing steady polyhedral or tilted flame structures, each associated with distinct azimuthal periodicities. Statement of SignificanceThis work presents a bifurcation analysis of laminar premixed conical flames within a fully-coupled flame/flow model. By considering variations in reaction rate and reactant diffusivity, the analysis identifies saddle–node bifurcations corresponding to spontaneous flash-back and blow-off of the axisymmetric flame. It also identifies axisymmetry breaking bifurcations associated with transitions to steady three-dimensional polyhedral and tilted flame states. These global instabilities indicate that hydrodynamic and thermal-diffusive effects strongly influence a flame’s steady structure in the azimuthal dimension even when no instabilities appear in the plane of symmetry. As such, future analyses of flame dynamics should rule out symmetry-breaking behaviors before adopting a two-dimensional computational framework.

Mean structure of the supercritical turbulent spiral in Taylor-Couette flow.

The large-scale laminar/turbulent spiral patterns that appear in the linearly unstable regime of counter-rotating Taylor-Couette flow are investigated from a statistical perspective by means of direct numerical simulation. Unlike the vast majority of previous numerical studies, we analyse the flow in periodic parallelogram-annular domains, following a coordinate change that aligns one of the parallelogram sides with the spiral pattern. The domain size, shape and spatial resolution have been varied and the results compared with those in a sufficiently large computational orthogonal domain with natural axial and azimuthal periodicity. We find that a minimal parallelogram of the right tilt significantly reduces the computational cost without notably compromising the statistical properties of the supercritical turbulent spiral. Its mean structure, obtained from extremely long time integrations in a co-rotating reference frame using the method of slices, bears remarkable similarity with the turbulent stripes observed in plane Couette flow, the centrifugal instability playing only a secondary role. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical transactions paper (Part 2)'.

Density waves in protoplanetary discs excited by eccentric planets: linear theory

ABSTRACT Spiral density waves observed in protoplanetary discs have often been used to infer the presence of embedded planets. This inference relies both on simulations as well as the linear theory of planet–disc interaction developed for planets on circular orbits to predict the morphology of the density wake. In this work, we develop and implement a linear framework for calculating the structure of the density wave in a gaseous disc driven by an eccentric planet. Our approach takes into account both the essential azimuthal and temporal periodicities of the problem, allowing us to treat any periodic perturbing potential (i.e. not only that of an eccentric planet). We test our framework by calculating the morphology of the density waves excited by an eccentric, low-mass planet embedded in a globally isothermal disc and compare our results to the recent direct numerical simulations (and heuristic wavelet analysis) of the same problem by Zhu and Zhang. We find excellent agreement with the numerical simulations, capturing all the complex eccentric features including spiral bifurcations, wave crossings, and planet-wave detachments, with improved accuracy and detail compared with the wavelet method. This illustrates the power of our linear framework in reproducing the morphology of complicated time-dependent density wakes, presenting it as a valuable tool for future studies of eccentric planet–disc interactions.

Stewartson layer instability and triadic resonances in rotating sphere with oscillating inner core

This work is devoted to an experimental study of the stability of a steady flow in a rotating spherical cavity with an oscillating core. The case of circular core oscillations in a plane perpendicular to the rotation axis is considered. The steady flow structure strongly depends on the dimensionless oscillation frequency. In the frequency range where there are no inertial waves, the flow resembles a classical Taylor column with an almost uniform distribution of the angular velocity. With an increase in the oscillation amplitude above a certain threshold level, a two-dimensional azimuthal wave is excited at the column boundary. The wave velocity is determined by the intensity of the liquid differential rotation and does not depend on the Ekman number. In the case of oscillation with a frequency less than twice the rotation rate, the inertial waves propagate in fluid bulk and introduce a significant correction to the steady velocity profile. At high amplitudes, the triadic subharmonic resonance of inertial waves is excited. The triadic interactions simultaneously manifest themselves in two regions of the spherical shell, generating secondary waves with the same frequencies, but different wavenumbers: subharmonic waves with shorter (respectively, longer) wavelengths are observed in outer (respectively, inner) regions of the shell. In some cases, the third (intermediate) region with subharmonic waves of intermediate wavelength can be observed. The development of the triadic resonance leads to a strongly nonlinear response in the form of a system of steady vortices with well-defined azimuthal periodicity.

Mode splitting of spin waves in magnetic nanotubes with discrete symmetries

We investigate how geometry influences spin dynamics in polygonal magnetic nanotubes. We find that lowering the rotational symmetry of nanotubes by decreasing the number of planar facets, splits an increasing number of spin-wave modes, which are doubly degenerate in cylindrical tubes. This symmetry-governed splitting is distinct form the topological one recently observed in cylindrical nanotubes. Doublet modes with half-integer or integer multiple azimuthal periods of the number of facets, split to singlet pairs with lateral standing-wave profiles of opposing mirror-plane symmetries. Furthermore, the polygonal geometry facilitates the hybridization of modes with different azimuthal periods but the same symmetry, manifested in avoided level crossings. These phenomena, unimaginable in cylindrical geometry, provide new tools to control spin dynamics on nanoscale. The presented concept can be generalized to nano-objects of versatile geometries and order parameters, offering new routes to engineer dynamic response in nanoscale devices.

Extrapolation and inversion of near-field propeller noise measurements

This paper describes an experimental method for making high resolution measurements of the tonal noise from a UAV propeller in the acoustic near-field and then methods of using this data to predict blade loading, via inversion, and the radiated ‘steady loading’ and ‘thickness’ tones, via extrapolation. The experimental method uses a traversing probe microphone to measure the acoustic pressure along an axial line outboard of the propeller tip. The acoustic pressures were then ensemble averaged over many propeller revolutions. Because of the azimuthal periodicity of the ensemble-average pressure field produced by the propeller, the measured data was used to infer the pressure on a cylinder enclosing the propeller. These measurements are compared with a noise prediction made using data from a computational fluid dynamics simulation of the same propeller and are found to be in reasonable agreement – with differences attributed to the small differences in the predicted and actual blade loading distributions. Because of the high amplitude and evanescent nature of the near-field pressure produced by a propeller, measurements were made in a non-anechoic environment and were relatively unaffected by background noise or reverberation effects. The near-field acoustic data was used as input to an inverse problem to estimate the aerodynamic loading distribution on the propeller blades. The method was applied first to the numerical data for validation and then to the experimental data as a demonstration. The reconstructed loading distributions were then used as input to a method for calculating the radiated acoustic field and the predictions show good agreement with a direct projection using the measured near-field experimental data.

Conditions of appearance and dynamics of the modified two-stream instability in E × B discharges

The large differential drift motion between electrons and ions that is created by the E × B current can produce different instabilities, such as the electron cyclotron drift instability, perpendicular to the magnetic field, and the Modified Two-Stream Instability (MTSI), with a component along the magnetic field. In this paper, we derive and validate a stability condition for the apparition of the MTSI modes in 2D particle-in-cell simulations of E × B discharges in the radial-azimuthal plane of a Hall thruster. We verify that, by choosing properly the domain dimensions, it is possible to capture correctly the MTSI growth and its corresponding number of azimuthal periods. In particular, we show that an azimuthal length that is smaller than a certain threshold prevents the MTSI from growing. Moreover, we show that the MTSI growth does not depend on the plasma density, but is affected by the axial electric field (perpendicular to the simulation domain). Additionally, we show that during its linear growth in the early times of the simulations, the MTSI produces an enhanced heating of the electrons in the magnetic field direction as well as an increased cross field mobility. For longer times, in the nonlinear regime, the system evolves toward a more chaotic state with the presence of structures that mostly exhibit large azimuthal wavelengths.

The effect of streaks on the instability of jets

Abstract

SAR Image Formation Method with Azimuth Periodically Missing Data Based on RELAX Algorithm

Synthetic aperture radar (SAR) is a widely used remote sensing observation technique. However, SAR raw echo data may be lost during the process of data acquisition by radar platform. In this paper, the imaging problem of SAR echo signal with periodically missing data along the azimuth is analyzed and a novel imaging method is proposed. Firstly, the problem of artificial artifact targets caused by periodically missing data is explained in detail, and the corresponding mathematical model is established. Then, the recovery method based on the RELAX algorithm with periodic notches data is proposed. In addition, when the size of two-dimensional (2D) echo data are large, block restoration along the azimuth is proposed to reduce the amount of calculation. Finally, the advantages of the algorithm proposed in this paper is demonstrated by the points target simulated SAR echo data processing and the real raw SAR echo data processing. When the azimuth periodically missing data rate is 50%, the SAR echo data can be recovered and the well-focused image can be obtained. Comparing the image entropy value and structural similarity index (SSIM) of the focused image, it proves the superiority of the proposed algorithm in solving the imaging problem of SAR azimuth periodically missing data.

Optical vortex with multi-fractional orders

Recently, optical vortices (OVs) have attracted substantial attention because they can provide an additional degree of freedom, i.e., orbital angular momentum (OAM). It is well known that the fractional OV (FOV) is interpreted as a weighted superposition of a series of integer OVs containing different OAM states. However, methods for controlling the sampling interval of the OAM state decomposition and determining the selected sampling OAM state are lacking. To address this issue, in this Letter, we propose a FOV by inserting multiple fractional phase jumps into whole phase jumps (2π), termed as a multi-fractional OV (MFOV). The MFOV is a generalized FOV possessing three adjustable parameters, including the number of azimuthal phase periods (APPs), N; the number of whole phase jumps in an APP, K; and the fractional phase jump, α. The results show that the intensity and OAM of the MFOV are shaped into different polygons based on the APP number. Through OAM state decomposition and OAM entropy techniques, we find that the MFOV is constructed by sparse sampling of the OAM states, with the sampling interval equal to N. Moreover, the probability of each sampling state is determined by the parameter α, and the state order of the maximal probability is controlled by the parameter K, as K * N. This work presents a clear physical interpretation of the FOV, which deepens our understanding of the FOV and facilitates potential applications, especially for multiplexing technology in optical communication based on OAM.

Image Formation of Azimuth Periodically Gapped SAR Raw Data with Complex Deconvolution

The phenomenon of periodical gapping in Synthetic Aperture Radar (SAR), which is induced in various ways, creates challenges in focusing raw SAR data. To handle this problem, a novel method is proposed in this paper. Complex deconvolution is utilized to restore the azimuth spectrum of complete data from the gapped raw data in the proposed method. In other words, a new approach is provided by the proposed method to cope with periodically gapped raw SAR data via complex deconvolution. The proposed method provides a robust implementation of deconvolution for processing azimuth gapped raw data. The proposed method mainly consists of phase compensation and recovering the azimuth spectrum of raw data with complex deconvolution. The gapped data become sparser in the range of the Doppler domain after phase compensation. Then, it is feasible to recover the azimuth spectrum of the complete data from gapped raw data via complex deconvolution in the Doppler domain. Afterwards, the traditional SAR imaging algorithm is capable of focusing the reconstructed raw data in this paper. The effectiveness of the proposed method was validated via point target simulation and surface target simulation. Moreover, real SAR data were utilized to further demonstrate the validity of the proposed method.

Intrusive polynomial‐chaos approach for stochastic problems with axial symmetry

We present and validate a computational approach that enables the quantification of time-dependent uncertainty in axially symmetric electromagnetic (EM) problems, in the context of a unique simulation. In essence, the finite-difference time-domain (FDTD) method, adapted to model bodies of revolution (BOR), is combined with truncated polynomial-chaos (PC) expansions of the involved field components, so that the stochastic nature of the latter is modelled reliably, when media with random electric properties need to be considered. The developed approach features two distinct advantages: First, by exploiting the azimuthal periodicity of the investigated problems' geometry, the high computational burden of three-dimensional (3D) simulations is reduced. Second, the large number of simulations that are normally required by Monte-Carlo (MC) methodologies for the extraction of statistical features is avoided, thanks to the integrated PC approximations. A number of numerical tests are conducted, in order to verify the validity and performance of the suggested stochastic algorithm.

Shear Wave Splitting Discloses Two Episodes of Collision‐Related Convergence in Western North America

AbstractSeismic anisotropy imposes first‐order constraints on the strain history of crust and upper mantle rocks. In this study, we analyze the mantle seismic anisotropy of the Western Canada Sedimentary Basin using a new shear wave spitting data set consisting of 1,333 teleseismic arrivals from 82 seismic stations. The resulting 332 high‐quality measurements yield a regional mean apparent splitting time (i.e., the magnitude of anisotropy) of 1.1 ± 0.3 s and an average fast orientation (i.e., the direction of anisotropy) of 54.6° ± 17.2°, which favor a two‐layer anisotropic model based on the 90° back azimuthal periodicity in both parameters. The northeast trending fast orientations dominate the lower layer at lithospheric depths and are approximately parallel to the present‐day absolute plate motions (APMs; i.e., <35°) due to the active asthenospheric flow. On the other hand, deviations from the APMs along the Canadian Rocky Mountain foothills could reflect disrupted mantle flow surrounding a southwestward migrating cratonic lithosphere. Also revealed are two elongated upper‐layer anisotropic anomalies in the lithosphere that are spatially correlated with Moho depths. Their characteristics suggest frozen‐in anisotropy imprinted along two convergent boundaries: (1) the Paleoproterozoic Snowbird Tectonic Zone that separates northeast (north) from northwest (south) fast directions and (2) the foothills of the Rocky Mountains that exhibit northeast trending orientations consistent with those of the APMs, maximum crustal stress, and electromagnetic anisotropy. Compressions associated with the Cordilleran orogenesis could be responsible for the spatial changes in the shear wave anisotropy from the foothills to the cratonic interior.

Numerical modeling of induction heating systems with load of azimuthal periodicity

The paper presents developed numerical modelling methodology of induction heating problems of disks with a simple and complex profile in 3D for calculating the coupled electromagnetic and thermal tasks in the parametric design language (ANSYS APDL). We demonstrate that the developed tools can be used for products of various sizes, as well as for other metal engineering components using the example of local electromagnetic processing of a metal disk and a diaphragm spring. The authors used development for parametric research and design of induction heaters. The described methodology has particular specificity, due to which it is possible to expand the scope for a number of engineering units with azimuthal periodicity property.

Scalable detection of photonic topological charge using radial phase grating

A method to detect the photonic topological charges (TCs) of optical vortex beams using a radial phase grating is proposed and demonstrated. The modulus of TCs can be obtained by the number of dark stripes of far-field diffraction patterns, and the sign of TCs is determined by the orientation of the patterns. The detection of TCs up to ±120 is demonstrated with this scheme. In addition, through investigating the evolution of patterns with various azimuthal periods of the grating and the distance between the centers of the grating and vortex beams, we show that this detection scheme has excellent alignment tolerance and does not have stringent requirements on the parameters of the grating.

Research on the Turbine Blade Vibration Base on the Immersed Boundary Method

This paper is concerned with the study of a kind of discrete forcing immersed boundary method (IBM) by which the loosely aero-elasticity coupled method is developed to analyze turbine blade vibration. In order to reduce the spurious oscillations at steep gradients in the compressible viscous flowing field, a five orders weighted essentially nonoscillatory scheme (WENO) is introduced into the flow solver based on large eddy simulation (LES). The three-dimensional (3D) full-annulus domain of the last two stages of an industrial steam axial turbine is adopted to validate the developed method. By the method, the process of grid generation becomes very simple and the unsteady data transferring between stator and rotor is realized without the process of being averaged or weighted. Based on the analysis of some important aerodynamic parameters, it is believed that hypothesis of azimuthal periodicity is not reasonable in this case and full-annulus passages model is more feasible and suitable to the research of turbine blade vibration. Meanwhile, the blade vibration data are also discussed. It is at about 65% of rotor blade height of the last stage that an inflection point is observed and the midspan region of the blade is the vulnerable part damaged potentially by the blade vibration.

Steady flow instability in an annulus with deflectors at rotational vibration

Experimental study and direct numerical simulation of the dynamics of an isothermal low-viscosity fluid are done in a coaxial gap of a cylindrical container making rotational vibrations relative to its axis. On the inner surface of the outer wall of the container, semicircular deflectors are regularly situated, playing the role of flow activators. As a result of vibrations, the activators oscillate tangentially. In the simulation, a 2D configuration is considered, excluding the end-wall effects. In the experiment, a container with a large aspect ratio is used. Steady streaming is generated in the viscous boundary layers on the activators. On each of the latter, beyond the viscous domain, a symmetric vortices pair is formed. The steady streaming in the annulus has an azimuthal periodicity. With an increase in the vibration intensity, a competition between the vortices occurs, as a result of which one of the vortices (let us call it even) approaches the activator and the other one (odd) rolls away and couples with the vortices from the neighbouring pairs. Streamlines of the odd vortices close on each other, forming a cog-wheel shaped flow that encircles the inner wall. Comparison of the experiment and the simulation reveals an agreement at moderate vibration intensity.

Investigation of the operating characteristics of a 12 stepped-cavity relativistic magnetron with axial extraction driven by an “F” transparent cathode using particle-in-cell simulations

We explore the performance of a 12 stepped-cavity relativistic magnetron with axial extraction (12 stepped-cavity RMDO) driven by an “F” transparent cathode (the “F” transparent cathode is a coaxial transparent cathode with two azimuthal periods of increased thickness and which looks like the letter “F,” so we call it “F” transparent cathode) through particle-in-cell (PIC) simulations. It is shown that using the “F” transparent cathode, an electronic efficiency of 70% with gigawatt output power is obtained while reducing the axial leakage current by about 50% compared to using the usual transparent cathode. Further PIC simulations demonstrate that frequency bifurcation occurs and mode switching can be achieved using several hundred kilowatts input RF power in the 12 stepped-cavity RMDO driven by an “F” transparent cathode. For example, it was found that using an applied driver power of 180 kW for 10 ns, the operating TE31 mode can be switched to the TE41 mode. It is also found that the secondary electron and backscattered electron emission and axial leakage current were two disturbing factors for the 12 stepped-cavity RMDO when it works at a stable operation mode but when the 12 stepped-cavity RMDO works near the critical magnetic field at the boundary between two modes, these two factors would lead to the operation modes changing.