Frequency-wavenumber Space Research Articles

Dynamic control of Lamb modes in phononic crystals

This study explores the possibilities of temporal control of one-dimensional phononic crystal consisting of a homogeneous piezoelectric plate on which metallic electrodes are structured. Previous works [1] have shown that Lamb-type guided modes in these periodically structured piezoelectric ceramic plates exhibit a tunable frequency dispersion that can be controlled by external electric circuits connected to the electrodes. In this study we develop a device enabling changes in the electric boundary conditions (EBC) from one condition to another by making use of switches driven by a microcontroller. This device agility is exploited to perform dynamic control of the crystal properties. To our knowledge, this is the first study devoted to the dynamic modulation of Lamb-type guided waves. Here, we do not simply change the EBCs as a function of time, but we create a "wave" of modulation of properties. Broadband translation of Lamb-wave dispersion curves in the frequency-wavenumber space (f, k) are observed, depending on the modulation speed. Experimentally, the translation of the S0 mode is detected. The relationship between the position of static bands and the position of bands after temporal modulation obeys a very simple vector rule in the (f, k) space. [1] Phys. Rev. B 99, 094302 (2019).

Spectral proper orthogonal decomposition analysis of a spatially developing underexpanded round jet at Re = 45 000

This study conducts a comprehensive analysis of the coherent structures within a spatially developing underexpanded axisymmetric jet at a Reynolds number of 45 000, utilizing high-fidelity implicit large eddy simulations (iLES) in conjunction with spectral proper orthogonal decomposition (SPOD). In the frequency–wavenumber space, the global SPOD analysis identifies three distinct coherent structures, corresponding to three different mechanisms, namely, Kelvin–Helmholtz (KH), Orr, and lift-up. Their salient characteristics are discussed in detail. Local SPOD analysis further explores the streamwise evolution of these coherent structures, revealing that the influence of KH mechanism is confined to the near field, while the lift-up mechanism persists and dominates the energy content beyond the potential core, with the streaks of azimuthal wavenumbers one and two being the most energetic. The reconstruction of turbulent kinetic energy (TKE) and Reynolds shear stress from SPOD modes is assessed, and the first few azimuthal modes with low wavenumbers and frequencies are found crucial for capturing the dominant features of the flow. It is found that only the m = 0 and m = 1 modes contribute to the TKE at the centerline. The Reynolds shear stress reconstruction quality is comparable to TKE, but with a negligible contribution from the m = 0 mode. The azimuthal mode m = 1 captures the slope of the actual Reynolds shear stress profile in the vicinity of centerline, while m = 2 and higher modes capture the peak location of the actual profile.

A theoretical study on the bulk viscosity in a compressible flow of an electrically conducting gas in the presence of magnetoacoustic waves

Only a few studies in the current literature are dedicated to the analysis of the effect of a bulk viscosity (i.e., second viscosity coefficient), and little is known about how to determine this quantity for high frequency compressible flows involving magnetic effect. The dissipation phenomenon associated with rapid expansions in the flow and the consequent attenuation of magnetoacoustic waves by the bulk viscosity has been neglected by several works on gas dynamics at high frequency. In this paper, we present a theoretical study on a compressible flow of an electrically conducting barotropic gas in the presence of a bulk viscosity. The governing equations represent a coupling between hydrodynamics and Maxwell's equations in the context of magnetohydrodynamics (MHD), and the relevant physical parameters of the flow are presented after an appropriate dimensional analysis of these equations. First, an analysis of small perturbations around an equilibrium state of the electrically conducting gas results in a system of linearized equations in the frequency-wavenumber space. The dispersion relation for magnetic waves is determined in terms of magnetic and the bulk viscosity effects. Second, based on the dispersion relation for magnetoacoustic waves, we propose a new expression to estimate the bulk viscosity in terms of the gas barotropic coefficient, the wavenumber, the wave frequency, the orientation of the applied field, and the Euler magnetic parameter, which measures the relative importance between magnetic and thermodynamic pressures. The behavior of the bulk viscosity is examined as a function of the wavenumber for different magnetic Euler numbers and field orientation, revealing how the rate of energy dissipation associated with the bulk viscosity can be controlled by varying the intensity of an applied magnetic field and its orientation. We show that understanding magnetoacoustic waves not only provides a tool for estimating the bulk viscosity in plasma flows in terms of these wave parameters, but also offers a potential pathway to manipulating both the magnitude and orientation of a magnetic field in order to reduce the rate of energy dissipation in most of plasma flows. This rate of dissipation produced by the rapid expansion of a recombining gas can be orders of magnitude larger than the one produced by the standard shear viscosity. These insights into bulk viscosity in compressible MHD flow at high frequency have potential applications in reacting plasmas where magnetoacoustic waves are damped by bulk viscosity, in production and dissipation of turbulence and in phenomenon of sonoluminescence as occurs in liquid containing gas bubble oscillating at fairly high frequency.

Circumferential modal coupling analysis of orthogonally stiffened cylindrical shell coupled with internal rectangular plate using semi-analytical wavenumber–frequency method

Orthogonally stiffened cylindrical shell coupled with internal rectangular plate is widely applied in engineering industries. The significant effects of the internal plate on vibration signatures of the shell have been demonstrated. Little attention has been paid to vibration analysis in the wavenumber–frequency domain. This paper aims to conduct the circumferential modal coupling analysis of the coupled model using the semi-analytical wavenumber–frequency method. The coupled model is disassembled into the stiffened cylindrical shell and rectangular plate. The interface potential between the shell and rectangular plate is derived based on a unified variational modeling procedure, in which the transformation of the coordinate system must be conducted. The displacement fields of the shell are expanded with the Fourier series along the circumferential direction. The vibratory field in wavenumber–frequency space is analytically obtained. The numerical method is utilized to illustrate the convergence and accuracy of the proposed method. The results demonstrate that the internal plate affects modal characteristics and then results in the alterations of the number and the shifts of resonance peaks. The phenomenon can be demonstrated by the circumferential modal contribution of the vibration response. Besides, it should be noted that the dominant circumferential modes for the input energy and vibrational energy are inconsistent owing to the strong shell–plate coupling effect.

Filtering property of periodic in-filled trench barrier for underground moving loads

The filtering properties of a 2-dimensional (2D) periodic in-filled trench barrier system subjected to anti-plane and in-plane moving loads are investigated separately by combining the spatial Fourier transform and periodic structure theory. By the aid of spatial Fourier transform, the governing equation of the periodic system is reduced in dimension. Based on the periodic structure theory, the closed-form solution of the dispersion equation for anti-plane wave is obtained; while for in-plane wave, the State-Space-Transfer-Matrix-Method (SSTMM) is introduced to solve the dispersion relation. Furthermore, the attenuation zones (AZs) varying with the transformed wave number, i.e., the AZs-kx graph, are obtained. Comparing the load-speed-line cluster of the moving excitation source with the obtained AZs-kx graph, the vibration isolation regions of the 2D periodic in-filled trench barrier in frequency-wave number space under moving loads can be found. To validate the present theoretical results, the performances of a 2D finite periodic in-filled trench barrier embedded in soil matrix and subjected to anti-plane and in-plane moving loads are numerically simulated, respectively. It is found that the AZs-kx graph plays a crucial role in revealing the isolation mechanism of the periodic in-filled trench barrier under moving loads. Theoretically speaking, it can be used to derive the vibration isolation regions for arbitrary excitation time histories of the moving-load inputs. The methodology proposed in the present paper can be used to design the continuous-type periodic wave barrier for isolating ambient vibration induced by moving loads.

On the use of a linear array EMAT for remote thickness gauging using the reflected modes on a steel pipe

Guided wave nondestructive testing refers to a set of fast and reliable techniques used to scan structures such as plates or pipes over medium to long range distances. The structural filtering method uses high-order modes and their cutoff frequency-thickness products to estimate the thickness along an inspection line. A corroded region, which can be approximated by a local loss of thickness, will act as a filter on the propagating modes. Because of their cutoff frequency-thickness product, some modes will not be able to propagate and will be either reflected or converted. Shear horizontal guided waves are the main candidate for structural filtering due to the simplicity of interpretation of their dispersion curves and their low sensitivity to non-viscous surface fluid loading. Electromagnetic acoustic transducers (EMAT) provide fast and efficient SH wave transduction but the most common designs, namely, periodic and permanent magnet EMAT is a mono element probe allowing modes to be excited only around a fixed wavelength. Recent work has shown that a linear array EMAT can be combined with a phase velocity excitation to allow almost complete control of the emitted waves in the frequency wavenumber space. In this paper, the linear array EMAT developed is used to estimate the minimum thickness of a defect machined on a 10.2 mm thick steel pipe with an external diameter of 323.8 mm. The measurements and finite element simulations carried out enabled us to estimate the thickness in several positions, with a maximum error of 0.4 mm and locate the defect with an error of 10 mm over a propagation distance of 600 mm.

Effects of time-filtering the Navier–Stokes equations

The underlying premise of temporal large eddy simulation (TLES) is that the attenuation of high-frequency content also attenuates the high-wavenumber content. Yet, to date, the effect in wavenumber space of removing high-frequency oscillations by time-domain filtering is not well understood. In this work, we numerically investigate the relationship between the frequency and wavenumber with particular attention to the role of the temporal residual-stress in TLES. Moreover, since under-resolved simulations that use high-order, non-dissipative numerical methods require some measure of artificial dissipation for stabilization, we also discuss the regularization term with practical relevance to under-resolved applications of TLES. Specifically, we analyze the effects of Eulerian time-domain filtering with a causal exponential filter on homogeneous isotropic turbulence. The data are generated by direct numerical simulation of the Navier–Stokes equations, which are driven to maintain an average Reynolds number (Reλ) of 200. A priori, Fourier transformations of the velocity fields were performed in order to compute the unfiltered and filtered energy and dissipation spectra in both wavenumber space and wavenumber–frequency space. Furthermore, the amount of unresolved dissipation of an insufficiently resolved simulation was approximated in an attempt to estimate the required additional artificial dissipation. The results indicate that the numerically motivated stabilization term can be reduced due to temporal filtering. Moreover, it has been shown that a sharp cutoff in the frequency domain does not translate into a sharp cutoff in the wavenumber space. Thus, a hybrid model that combines temporal filtering for the residual-stress and spatial filtering for stabilization might be advantageous.

Flow dynamics and noise generation mechanisms in supersonic underexpanded rectangular and planar jets

In this study, large eddy simulations of two rectangular jets with different aspect ratios and one planar jet with spanwise periodic boundaries are carried out in an attempt to investigate and clarify some key issues regarding the flow dynamics and noise generation mechanisms for the screech tone. The substantial effects of the nozzle configuration on shock structures and noise are first revealed in detail. It is found that when the lateral confinement is weakened, the spacing of the shock cells increases and the jet oscillates more intensively in the minor-axis plane, increasing the noise level and altering the screeching frequency. By analyzing the pressure fluctuations of the shear layer in the wavenumber-frequency space, different kinds of waves in these supersonic jets are examined. Importantly, it is revealed that the guided jet wave should play a dominant role in closing the resonance loops rather than the acoustic wave. Moreover, the energy-containing structures with pertinent frequencies are extracted by employing the reduced-order variational mode decomposition, and some underlying flow dynamics are presented. Especially, a novel mechanism has been identified: the low-frequency stretching motions of shock cells have a significant modulation effect on the screeching amplitude in the planar jet. Furthermore, with the aid of the multi-process acoustic theory, the characteristics of physical noise sources are diagnosed, particularly the source mechanisms related to the screech tone. The general structures and distribution of the kinematic noise source Sβ and entropy noise source Se are presented. Sβ mainly exhibits a vortex-like structure near the nozzle, while Se exhibits a lamellar bilayer structure. The spatiotemporal correlations between the physical sources and far-field noise show that the dominant mechanisms for the screech tone rely on the nozzle configuration and the screech tone tends to be produced by multiple sources in all cases.

A Model of Energy and Spectral Shape for the Internal Gravity Wave Field in the Deep Sea: The Parametric IDEMIX Model

Abstract The spectral description of the energy of oceanic internal gravity waves is generally represented by the Garrett–Munk (GM) model, a function with a power-law decrease of spectral energy in wavenumber–frequency space. Besides the slopes of these power laws, the spectrum is expressed as a function of energy and a bandwidth parameter that fixes the range of vertical modes excited in the respective state. Whereas concepts have been developed and agreed upon of what processes feed the wave spectrum and what dissipates energy, there is no explanation of what shapes the spectral distribution, i.e., how the power laws come about and what sets the bandwidth. The present study develops a parametric spectral model of energy and bandwidth from the basic underlying energy balance in terms of forcing, propagation, refraction, spectral transfer, and dissipation. The model is an extension of the IDEMIX (Internal Wave Dissipation, Energy and Mixing) models where bandwidth was taken as a constant parameter. The current version of the model is restricted to single-column mode and the slopes of the spectral power laws are fixed. A coupled system of predictive equations for energy and bandwidth (for up- and downward propagating waves) results. The equations imply that bandwidth relates to energy by a power law with an exponent given by the dynamical parameters. It agrees favorably with energy, bandwidth, and slope data from previously published fits of the GM model to Argo float observations. Numerical solutions of the coupled energy–bandwidth model in stand-alone modus are presented.

Characterizing an Optically Induced Sub-micrometer Gigahertz Acoustic Wave in a Silicon Thin Plate.

Optically induced GHz-THz guided acoustic waves have been intensively studied because of the potential to realize noninvasive and noncontact material inspection. Although the generation of photoinduced guided acoustic waves utilizing nanostructures, such as ultrathin plates, nanowires, and materials interfaces, is being established, experimental characterization of these acoustic waves in consideration of the finite size effect has been difficult due to the lack of experimental methods with nm × ps resolution. Here we experimentally observe the sub-micrometer guided acoustic waves in a nanofabricated ultrathin silicon plate by ultrafast transmission electron microscopy with nm × ps precision. We successfully characterize the excited guided acoustic wave in frequency-wavenumber space by applying Fourier-transformation analysis on the bright-field movie. These results suggest the great potential of ultrafast transmission electron microscopy to characterize the acoustic modes realized in various nanostructures.

Development of a linear array electromagnetic acoustic transducer for shear horizontal guided wave inspection

Ultrasonic guided wave screenings have proven to be fast and reliable in detecting various types of defects in plate-like structures. Today, low frequency ultrasonic guided waves are routinely used to screen long sections of pipelines. For many years, ultrasonic guided waves at frequencies beyond the cutoff of the first high-order mode attracted interest in the research community due to the plurality of modes that can propagate. During operation at a frequency beyond the cutoff of the first high-order mode, the transduction mechanism becomes crucial for selectively exciting and detecting a single high-order mode or a group of high-order modes. Ultrasonic comb transducers allow to selectively excite and detect high-order ultrasonic guided waves at a desired wavelength. Linear array transducers are even more flexible, and allow virtually full control in the frequency wavenumber space. High-order shear horizontal (SH) modes have multiple potential applications including, for example, remote thickness gauging and crack monitoring. However, these modes are notoriously difficult to excite and detect using conventional piezoelectric transducers. Electromagnetic acoustic transducers (EMAT) make the excitation and detection of SH modes relatively simple. Periodic permanent magnet (PPM) EMAT can be used to selectively excite and detect high-order SH modes based on a desired wavelength. However, in some applications, it may be necessary to excite and detect high-order SH modes with more control in the frequency wavenumber space. In this paper, a novel EMAT linear array is proposed using a configuration similar to a PPM EMAT but with individual coils wrapped around the magnets. The performance of the EMAT linear array in transmission and reception is demonstrated in simulations and experimentally on a 9.53 mm steel plate. The results show a successful transmission and detection of SH0 to SH4 for a 300 to 800 kHz frequency range.

Two-dimensional effects on electrostatic instabilities in Hall thrusters. I. Insights from particle-in-cell simulations and two-point power spectral density reconstruction techniques

Using 2D particle-in-cell (PIC) simulations coupled to a fluid description of the gas dynamics, we study the electrostatic instabilities developing in the axial–azimuthal plane of a Hall thruster, during several periods of a low-frequency oscillation (the so-called breathing mode at 10 kHz). As done in experiments, the 2D PIC-MCC (Monte Carlo collision) code is coupled to an electrical circuit in order to partially damp the (otherwise large) discharge current fluctuations at the breathing mode frequency. The different electrostatic higher frequency modes that develop in the plasma are analyzed using a two-point power spectral density reconstruction method, which allows us to generate the dispersion diagrams (in the frequency-wavenumber space) along the axial and azimuthal directions and at different times during the low-frequency breathing mode oscillations. This technique allows us to distinguish between different well-identified instabilities: the electron cyclotron drift instability and its evolution toward an ion acoustic wave and the ion transit time instability. These instabilities are usually considered unidirectional (either axial or azimuthal); however, it is shown here that they exist in both directions. This two-dimensional character is instrumental in understanding where these instabilities grow and how they propagate in the thruster channel and plume. A theoretical discussion of this aspect is proposed in Paper II. The effects of (i) the azimuthal length of the simulation box and (ii) the electron temperature injection at the cathode are also discussed.

Wave spectroscopy in a driven granular material

Driven granular media constitute model systems in out-of-equilibrium statistical physics. By assimilating the motions of granular particles to those of atoms, by analogy, one can obtain macroscopic equivalent of phase transitions. Here, we study fluid-like and crystal-like two-dimensional states in a driven granular material. In our experimental device, a tunable magnetic field induces and controls remote interactions between the granular particles. We use high-speed video recordings to analyse the velocity fluctuations of individual particles in stationary regime. Using statistical averaging, we find that the particles self-organize into collective excitations characterized by dispersion relations in the frequency-wavenumber space. These findings thus reveal that mechanical waves analogous to condensed matter phonons propagate in driven granular media. When the magnetic coupling is weak, the waves are longitudinal, as expected for a fluid-like phase. When the coupling is stronger, both longitudinal and transverse waves propagate, which is typically seen in solid-like phases. We model the dispersion relations using the spatial distribution of particles and their interaction potential. Finally, we infer the elastic parameters of the granular assembly from equivalent sound velocities, thus realizing the spectroscopy of a granular material.

Efficient phase shift migration for ultrasonic full-matrix imaging of multilayer composite structures

Multilayer composite structures are essential for various fields but void defects inside will greatly reduce its mechanical properties. Ultrasonic full-matrix capture (FMC) is a useful imaging modality to detect defects in multilayer structures, but it also brings heavy computational burden. Though great progress has been made in the full-matrix imaging for homogeneous medium, most of these methods are invalid or inefficient for the cases of multilayer structures. To amend it, an efficient phase shift migration for ultrasonic full-matrix imaging of multilayer structures is proposed in this paper. The method regards FMC as data in three dimensions: receiver position, transmitter position and time, and transforms it into frequency-wavenumber (f-k) space by a three-dimensional fast Fourier transform. By downward extrapolating the wavefield on the surface, the wavefield at arbitrary depth in the measuring region is reconstructed, with a firstly introduced double-square-root vertical wavenumber. Then the imaging condition t = 0 is directly applied in the f-k space to produce the defect image. Computational complexity was reduced greatly, and the calculation time of the proposed method was only 1/6 of that of the well-established extended phase shift migration, in the case of the same resolution. Therefore, the proposed method has great potential in on-line ultrasound defect detection, especially for cases of large multilayer structures.

The energy flux of three-dimensional waves in the atmosphere: Exact expression for a basic model diagnosis with no equatorial gap

AbstractA model diagnosis for the energy flux of off-equatorial Rossby waves in the atmosphere has previously been done using quasi-geostrophic equations and is singular at the equator. The energy flux of equatorial waves has been separately investigated in previous studies using a space-time spectral analysis or a ray theory. A recent analytical study has derived an exact universal expression for the energy flux which can indicate the direction of the group velocity for linear shallow water waves at all latitudes. This analytical result is extended in the present study to a height-dependent framework for three-dimensional waves in the atmosphere. This is achieved by investigating the classical analytical solution of both equatorial and off-equatorial waves in a Boussinesq fluid. For the horizontal component of the energy flux, the same expression has been obtained between equatorial waves and off-equatorial waves in the height-dependent framework, which is linked to a scalar quantity inverted from the isentropic perturbation of Ertel’s potential vorticity. The expression of the vertical component of the energy flux requires computation of another scalar quantity that may be obtained from the meridional integral of geopotential anomaly in a wavenumber-frequency space. The exact version of the universal expression is explored and illustrated for three-dimensional waves induced by an idealized Madden-Julian Oscillation forcing in a basic model experiment. The zonal and vertical fluxes manifest the energy transfer of both equatorial Kelvin waves and off-equatorial Rossby waves with a smooth transition at around 10°S and around 10°N. The meridional flux of wave energy represents connection between off-equatorial divergence regions and equatorial convergence regions.

Effective wave motion in periodic discontinua near spectral singularities at finite frequencies and wavenumbers

We consider the effective wave motion, at spectral singularities such as corners of the Brillouin zone and Dirac points, in periodic continua intercepted by compliant interfaces that pertain to e.g. masonry and fractured materials. We assume the Bloch-wave form of the scalar wave equation (describing anti-plane shear waves) as a point of departure, and we seek an asymptotic expansion about a reference point in the wavenumber-frequency space using wavenumber separation as the perturbation parameter. Using the concept of broken Sobolev spaces to cater for the presence of kinematic discontinuities, we next define the “mean” wave motion via inner product between the Bloch wave and an eigenfunction (at specified wavenumber and frequency) for the unit cell of periodicity. With such projection-expansion approach, we obtain an effective field equation, for an arbitrary dispersion branch, near apexes of “wavenumber quadrants” featured by the first Brillouin zone. For completeness, we investigate asymptotic configurations featuring both (a) isolated, (b) repeated, and (c) nearby eigenvalues. In the case of repeated eigenvalues, we find that the “mean” wave motion is governed by a system of wave equations and Dirac equations, whose size is given by the eigenvalue multiplicity, and whose structure is determined by the participating eigenfunctions, the affiliated cell functions, and the direction of wavenumber perturbation. One of these structures is shown to describe the so-called Dirac points – apexes of locally conical dispersion surfaces – that are relevant to the generation of topologically protected waves. In situations featuring clusters of tightly spaced eigenvalues, the effective model is found to entail a Dirac-like system of equations that generates “blunted” conical dispersion surfaces. We illustrate the analysis by numerical simulations for two periodic configurations in R2 that showcase the asymptotic developments in terms of (i) wave dispersion, (ii) forced wave motion, and (iii) frequency- and wavenumber-dependent phonon behavior.

Influence of the Distortion of Vertical Wavenumber Spectra on Estimates of Turbulent Dissipation using the Finescale Parameterization: Eikonal Calculations

AbstractThe finescale parameterization, formulated on the basis of a weak nonlinear wave–wave interaction theory, is widely used to estimate the turbulent dissipation rate, ε. However, this parameterization has previously been found to overestimate ε in the Antarctic Circumpolar Current (ACC) region. One possible reason for this overestimation is that vertical wavenumber spectra of internal wave energy are distorted from the canonical Garrett-Munk spectrum and have a spectral “hump” at low vertical wavenumbers. Such distorted vertical wavenumber spectra were also observed in other mesoscale eddy-rich regions. In this study, using eikonal simulations, in which internal wave energy cascades are evaluated in the frequency-wavenumber space, we examine how the distortion of vertical wavenumber spectra impacts on the accuracy of the finescale parameterization. It is shown that the finescale parameterization overestimates ε for distorted spectra with a low-vertical-wavenumber hump because it incorrectly takes into account the breaking of these low-vertical-wavenumber internal waves. This issue is exacerbated by estimating internal wave energy spectral levels from the low-wavenumber band rather than from the high-wavenumber band, which is often contaminated by noise in observations. Thus, in order to accurately estimate the distribution of ε in eddy-rich regions like the ACC, high-vertical-wavenumber spectral information free from noise contamination is indispensable.

Direct numerical simulation of wave propagation in saturated random granular packings using coupled LBM-DEM

Poroelasticity theory predicts wave velocities in a saturated porous medium through a coupling between the bulk deformation of the solid skeleton and porous fluid flow. The challenge emerges below the characteristic wavelengths at which hydrodynamic interactions between grains and pore fluid become important. We investigate the pressure and volume fraction dependence of compressional- and shear-wave velocities in fluid-saturated, random, isotropic, frictional granular packings. The lattice Boltzmann method (LBM) and discrete element method (DEM) are two-way coupled to capture the particle-pore fluid interactions; an acoustic source is implemented to insert a traveling wave from the fluid reservoir to the saturated medium. We extract wave velocities from the acoustic branches in the wavenumber-frequency space, for a range of confining pressures and volume fractions. For random isotropic granular media the pressure-wave velocity data collapse on a single curve when scaled properly by the volume fraction.

Frequency diffusion of waves by unsteady flows

Abstract

Four-dimensional Frequency–Wavenumber Power Spectrum of a Strong Turbulence Obtained from Hybrid Kinetic Simulations

Abstract We carry out three-dimensional hybrid kinetic simulations of a strong decaying turbulence. The turbulence is initiated with a seed spectrum that includes Alfvén wave modes at low wavenumbers. From the temporal and spatial distribution of the simulation output in the quasi-steady phase, we calculate a four-dimensional frequency–wavenumber spectrum of the turbulence. Our analysis shows that kinetic Alfvén waves can be identified in the wavenumber–frequency space in the vicinity of the seed turbulence spectrum. They produce a power peak consistent with a linear dispersion relation. However, further away from the seed spectrum, where most of the wave–particle interaction takes place, the signature of the Alfvén modes disappears among other fluctuations that are not described by any dispersion relations. Furthermore, at higher wavenumbers at which the signature is still identifiable, its frequency broadening becomes comparable to the frequency itself. Therefore, the use of linear waves based on the conventional dispersion relation is not necessarily justified to describe the turbulence if their signature is still present and even more so when it disappears. We find that at larger rms amplitudes of the turbulence, the signature of the dispersion relation is confined to lower parallel wavenumbers. In the range where it is visible, the frequency broadening becomes greater at larger amplitudes. This suggests that stronger nonlinearity makes the fluctuations behave less like wave modes until the conventional wave-mode approach is no longer valid.