Quasi-monochromatic Wave Packet Research Articles

Transient internal wave excitation of resonant modes in a density staircase

The density of the ocean generally increases continuously with depth as a consequence of variations in salinity and temperature. In some regions, however, the density profile of the ocean adopts a (double diffusive) staircase structure in which successive layers of uniform density fluid are separated by rapid density jumps. Previous work has theoretically examined the transmission and reflection of periodic internal (gravity) waves incident upon a density staircase. This predicted the existence of transmission spikes (global modes) for certain combinations of frequency and horizontal wave number in which the incident waves transmit perfectly across a density staircase. It was hypothesized that the transmission spikes occur when the incident waves resonate with natural modes of disturbances in the staircase. Here we derive theory to investigate the interactions between incident internal waves and modes. We demonstrate a close correspondence between the frequency for incident waves at a transmission spike and the real part of the frequency of modes at the same horizontal wave number. However, the frequency of the corresponding modes has a negative imaginary part corresponding to exponential decay of the modes in time. We perform numerical simulations to examine the impact of this near-resonant coupling when a vertically localized, quasimonochromatic internal wave packet interacts with a density staircase. In a range of simulations with fixed incident wave frequency and varying horizontal wave number, the measured transmission coefficient does not exhibit transmission spikes, but decreases monotonically with increasing horizontal wave number about the critical wave number separating strong and weak transmission. We show this occurs because the incident wave excites modes that then slowly transmit energy above and below the staircase at a rate consistent with the predicted decay rate of the modes. This rate is slower for staircases with more steps with the decay time increasing as the cube of the number of steps. Published by the American Physical Society 2024

Modeling Wave Packet Dynamics and Exploring Applications: A Comprehensive Guide to the Nonlinear Schrödinger Equation

The nonlinear Schrödinger (NLS) equation stands as a cornerstone model for exploring the intricate behavior of weakly nonlinear, quasi-monochromatic wave packets in dispersive media. Its reach extends across diverse physical domains, from surface gravity waves to the captivating realm of Bose–Einstein condensates. This article delves into the dual facets of the NLS equation: its capacity for modeling wave packet dynamics and its remarkable breadth of applications. We illuminate the derivation of the NLS equation through both heuristic and multiple-scale approaches, underscoring how distinct interpretations of physical variables and governing equations give rise to varied wave packet dynamics and tailored values for dispersive and nonlinear coefficients. To showcase its versatility, we present an overview of the NLS equation’s compelling applications in four research frontiers: nonlinear optics, surface gravity waves, superconductivity, and Bose–Einstein condensates. This exploration reveals the NLS equation as a powerful tool for unifying and understanding a vast spectrum of physical phenomena.

Regimes of Resonant Interactions of Electrons with Auroral Kilometric Radiation

We analyze the resonant interaction of energetic electrons with auroral kilometric radiation (AKR). Possible regimes of such an interaction are studied on the basis of a numerical solution of the particle motion equations in a given field of a quasi-monochromatic AKR wave packet. It is shown that for realistic wave amplitudes 0.2–0.4 V/m, the interaction can be highly nonlinear. As a result of nonlinear interaction, both a substantial (approximately twofold) acceleration of particles and a decrease in their energy (by about 20–50%) are possible. With realistic plasma and wave packet parameters, electrons with initial energies of the order of 10 keV are accelerated, and for electrons with initial energies of several tens of keV, the interaction leads to a decrease in energy.

The dependence of the time delay on the shape of the wave packet tunneling through the potential barrier

The problem of quasi-monochromatic wave packet tunneling through a potential rectangular barrier is studied. It is shown that the approach to calculating the time delay of a wave packet called ``phase time’’ is not universally correct though well known in the literature. In this paper we discuss the mathematical reasons for this discrepancy, and substantiate another, more correct analytical method for calculating the time delay in the approximation of wide barriers. This time depends on the shape of the envelope complex wave function falling on the barrier. Analytical results for the Gaussian wave packet are presented.

The regime of multi-stage trapping in free-electron lasers operating in the super-radiant and SASE regimes

The use of electron–wave interaction systems consisting of several tapered sections is considered as a method of efficiency enhancement of free-electron lasers (FELs) operating in different regimes of emission of a short wave pulse from a short electron bunch. These regimes are the principally multi-frequency self-amplified spontaneous emission regime traditionally used in short-wavelength FELs and the regime of electron–wave group synchronism, in which super-radiation of a quasi-monochromatic wave packet propagating together with the electron bunch occurs. In both the cases, the use of multi-stage trapping of electrons in the bunch by the radiated wave provides a significant (at least by an order of magnitude) increase in efficiency as compared to the saturated-stage efficiency in regular systems.

Transversely modulated wave packet

The wave packet consisting of two harmonic plane waves with the same frequencies, but with different wave vectors is considered. The dispersion relation of a packet is structurally similar to the dispersion relation of a relativistic particle with a nonzero rest mass. The possibilities of controlling the group velocity of a quasi-monochromatic wave packet by varying the angle between the wave vectors of its constituent waves and of creating a one-dimensional spatial structure in the region of wave packet propagation are discussed. The interaction of two transversely modulated wave packets is considered.

Turbulence versus Fire-hose Instabilities: 3D Hybrid Expanding Box Simulations

Abstract The relationship between a decaying plasma turbulence and proton fire hose instabilities in a slowly expanding plasma is investigated using three-dimensional hybrid expanding box simulations. We impose an initial ambient magnetic field along the radial direction, and we start with an isotropic spectrum of large-scale, linearly polarized, random-phase Alfvénic fluctuations with zero cross-helicity. A turbulent cascade rapidly develops and leads to a weak proton heating that is not sufficient to overcome the expansion-driven perpendicular cooling. The plasma system eventually drives the parallel and oblique fire hose instabilities that generate quasi-monochromatic wave packets that reduce the proton temperature anisotropy. The fire hose wave activity has a low amplitude with wave vectors quasi-parallel/oblique with respect to the ambient magnetic field outside of the region dominated by the turbulent cascade and is discernible in one-dimensional power spectra taken only in the direction quasi-parallel/oblique with respect to the ambient magnetic field; at quasi-perpendicular angles the wave activity is hidden by the turbulent background. These waves are partly reabsorbed by protons and partly couple to and participate in the turbulent cascade. Their presence reduces kurtosis, a measure of intermittency, and the Shannon entropy, but increases the Jensen–Shannon complexity of magnetic fluctuations; these changes are weak and anisotropic with respect to the ambient magnetic field and it is not clear if they can be used to indirectly discern the presence of instability-driven waves.

Time-dependent reflection at the localization transition

A short quasi-monochromatic wave packet incident on a semi-infinite disordered medium gives rise to a reflected wave. The intensity of the latter decays as a power law $1/t^{\alpha}$ in the long-time limit. Using the one-dimensional Aubry-Andr\'{e} model, we show that in the vicinity of the critical point of Anderson localization transition, the decay slows down and the power-law exponent $\alpha$ becomes smaller than both $\alpha = 2$ found in the Anderson localization regime and $\alpha = 3/2$ expected for a one-dimensional random walk of classical particles.

The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio

We examine the wave-induced flow of small-amplitude, quasi-monochromatic, three-dimensional, Boussinesq internal gravity wavepackets in a uniformly stratified ambient. It has been known since Bretherton (J. Fluid Mech., vol. 36 (4), 1969, pp. 785–803) that one-, two- and three-dimensional wavepackets induce qualitatively different flows. Whereas the wave-induced mean flow for compact three-dimensional wavepackets consists of a purely horizontal localized circulation that translates with and around the wavepacket, known as the Bretherton flow, such a flow is prohibited for a two-dimensional wavepacket of infinite spanwise extent, which instead induces a non-local internal wave response that is long compared with the streamwise extent of the wavepacket. One-dimensional (horizontally periodic) wavepackets induce a horizontal, non-divergent unidirectional flow. Through perturbation theory for quasi-monochromatic wavepackets of arbitrary aspect ratio, we predict for which aspect ratios which type of induced mean flow dominates. We compose a regime diagram that delineates whether the induced flow is comparable to that of one-, two- or compact three-dimensional wavepackets. The predictions agree well with the results of fully nonlinear three-dimensional numerical simulations.

Characterisation and extraction of a Rayleigh-wave mode in vertically heterogeneous media using quaternion SVD

We propose a method that identifies a mode of Rayleigh waves and separates it from body waves and from other modes, using quaternions to represent multi-component data. Being well known the abilities of quaternions to handle rotations in space, we use previous results derived from Le Bihan and Mars (2004) [1] to prove that a Rayleigh-wave mode recorded by an array of vector-sensors can be approximated by a sum of trace-by-trace rotating time signals. Our method decomposes the signal into narrow-frequency bands, which undergo both a velocity correction and a polarisation correction. The aim of these corrections is to reduce the mode of interest to a quasi-monochromatic wave packet with infinite apparent velocity and quasi-circular polarisation. Once written in quaternion notation, we refer to this wave packet as “quaternion brick”. Based on theoretical considerations, we prove that this quaternion brick maps into the first quaternion eigenimage of the quaternion singular value decomposition. We apply this method to synthetic datasets derived from two vertically heterogeneous models to extract the fundamental mode and we prove that it is correctly separated from either a higher mode of propagation or body waves with negligible residual. Results are presented in both time–offset and frequency–phase slowness domains.

Two-dimensional Morlet wavelet transform and its application to wave recognition methodology of automatically extracting two-dimensional wave packets from lidar observations in Antarctica

Waves in the atmosphere and ocean are inherently intermittent, with amplitudes, frequencies, or wavelengths varying in time and space. Most waves exhibit wave packet-like properties, propagate at oblique angles, and are often observed in two-dimensional (2-D) datasets. These features make the wavelet transforms, especially the 2-D wavelet approach, more appealing than the traditional windowed Fourier analysis, because the former allows adaptive time-frequency window width (i.e., automatically narrowing window size at high frequencies and widening at low frequencies), while the latter uses a fixed envelope function. This study establishes the mathematical formalism of modified 1-D and 2-D Morlet wavelet transforms, ensuring that the power of the wavelet transform in the frequency/wavenumber domain is equivalent to the mean power of its counterpart in the time/space domain. Consequently, the modified wavelet transforms eliminate the bias against high-frequency/small-scale waves in the conventional wavelet methods and many existing codes.Based on the modified 2-D Morlet wavelet transform, we put forward a wave recognition methodology that automatically identifies and extracts 2-D quasi-monochromatic wave packets and then derives their wave properties including wave periods, wavelengths, phase speeds, and time/space spans. A step-by-step demonstration of this methodology is given on analyzing the lidar data taken during 28–30 June 2014 at McMurdo, Antarctica. The newly developed wave recognition methodology is then applied to two more lidar observations in May and July 2014, to analyze the recently discovered persistent gravity waves in Antarctica. The decomposed inertia-gravity wave characteristics are consistent with the conclusion in Chen et al. (2016a) that the 3–10h waves are persistent and dominant, and exhibit lifetimes of multiple days. They have vertical wavelengths of 20–30km, vertical phase speeds of 0.5–2m/s, and horizontal wavelengths up to several thousands kilometers in the mesosphere and lower thermosphere (MLT). The variations in the extracted wave properties from different months in winter indicate a month-to-month variability in the gravity wave activities in the Antarctic MLT region.

The mean flow and long waves induced by two-dimensional internal gravity wavepackets

Through theory supported by numerical simulations, we examine the induced local and long range response flows resulting from the momentum flux divergence associated with with a two-dimensional Boussinesq internal gravity wavepacket in a uniformly stratified ambient. Our theoretical approach performs a perturbation analysis that takes advantage of the separation of scales between waves and the amplitude envelope of a quasi-monochromatic wavepacket. We first illustrate our approach by applying it to the well-studied case of deep water surface gravity waves, showing that the induced flow, UDF, resulting from the divergence of the horizontal momentum flux is equal to the Stokes drift. For a localized surface wavepacket, UDF is itself a divergent flow and so there is the well-known non-local response manifest in the form of a deep return flow beneath the wavepacket. For horizontally periodic and vertically localized internal wavepackets, the divergent-flux induced flow, uDF, is found from consideration of the vertical gradient of the vertical flux of horizontal momentum associated with the waves. Because uDF is itself a non-divergent flow field, this accounts entirely for the wave-induced flow; there is no response flow. Our focus is upon internal wavepackets that are localized in the horizontal and vertical. We derive a formula for the divergent-flux induced flow that, as in this case of surface wavepackets, is itself a divergent flow. We show that the response is a horizontally long internal wave that translates vertically with the wavepacket at its group velocity. Scaling relationships are used to estimate the wavenumber, horizontal extent, and amplitude of this induced long wave. At higher order in perturbation theory we derive an explicit integral formula for the induced long wave. Thus, we provide validation of Bretherton's analysis of flows induced by two-dimensional internal wavepackets [F. P. Bretherton, “On the mean motion induced by gravity waves,” J. Fluid Mech. 36, 785–803 (1969)] and we provide further analyses that give intuitive insights into the physics governing the properties of the induced long wave. In particular, consistent with momentum conservation, we show that the horizontally-integrated horizontal flow associated with the long wave is given exactly by the horizontal integral of uDF. However, qualitatively different from horizontally periodic internal waves, the vertical profile of the induced horizontal flow across the horizontally localized wavepacket is positive at the leading edge and negative at the trailing edge. These results are validated by the results of numerical simulations of a Gaussian wavepacket initialized in an otherwise stationary ambient.

Momentum-space dynamics of Dirac quasiparticles in correlated random potentials: Interplay between dynamical and Berry phases

We consider Dirac quasi-particles, as realized with cold atoms loaded in a honeycomb lattice or in a $\pi$-flux square lattice, in the presence of a weak correlated disorder such that the disorder fluctuations do not couple the two Dirac points of the lattices. We numerically and theoretically investigate the time evolution of the momentum distribution of such quasi-particles when they are initially prepared in a quasi-monochromatic wave packet with a given mean momentum. The parallel transport of the pseudo-spin degree of freedom along scattering paths in momentum space generates a geometrical phase which alters the interference associated with reciprocal scattering paths. In the massless case, a well-known dip in the momentum distribution develops at backscattering (respective to the Dirac point considered) around the transport mean free time. This dip later vanishes in the honeycomb case because of trigonal warping. In the massive case, the dynamical phase of the scattering paths becomes crucial. Its interplay with the geometrical phase induces an additional transient broken reflection symmetry in the momentum distribution. The direction of this asymmetry is a property of the Dirac point considered, independent of the energy of the wave packet. These Berry phase effects could be observed in current cold atom lattice experiments.

Anelastic internal wave reflection and transmission in uniform retrograde shear

We perform fully nonlinear simulations in two dimensions of a horizontally periodic, vertically localized, anelastic internal wavepacket in order to examine the effects of weak and strong nonlinearity upon wavepackets approaching a reflection level in uniform retrograde shear. Transmission, reflection, and momentum deposition are measured in terms of the horizontal momentum associated with the wave-induced mean flow. These are determined in part as they depend upon the initial wavenumber vector, \documentclass[12pt]{minimal}\begin{document}$\vec{k}=(k,m)$\end{document}k⃗=(k,m), which determines the modulational stability (if |m/k| ≳ 0.7) or instability (if |m/k| ≲ 0.7) of moderately large amplitude quasi-monochromatic internal wavepackets. Whether modulationally stable or unstable, the evolution of the wavepacket is determined by the height of the reflection level predicted by linear theory, zr, relative to the height, zΔ, at which weak nonlinearity becomes significant, and the height, \documentclass[12pt]{minimal}\begin{document}$z_b>z_\Delta$\end{document}zb>zΔ, at which linear theory predicts anelastic waves first overturn in the absence of shear. If \documentclass[12pt]{minimal}\begin{document}$z_r<z_\Delta$\end{document}zr<zΔ, the amplitude remains sufficiently small and the waves reflect as predicted by linear theory. If zr is moderately larger than \documentclass[12pt]{minimal}\begin{document}$z_\Delta$\end{document}zΔ, a fraction of the momentum associated with the wavepackets transmits past the reflection level. This is because the positive shear associated with the wave-induced mean flow can partially shield the wavepacket from the influence of the negative background shear enhancing its transmission. The effect is enhanced for weakly nonlinear modulational unstable wavepackets that narrow and grow in amplitude faster than the anelastic growth rate. However, as nonlinear effects become more pronounced, a significant fraction of the momentum associated with the wavepacket is irreversibly deposited to the background below the reflection level. This is particularly the case for modulationally unstable wavepackets, whose enhanced amplitude growth leads to overturning below the predicted breaking level. Because the growth in the amplitude envelope of modulationally stable wavepackets is retarded by weakly nonlinear effects, reflection is enhanced and transmission retarded relative to their modulationally unstable counterparts. Applications to mountain wave propagation through the stratosphere in the winter hemisphere are discussed.

Brillouin scattering of visible and hard X-ray photons from optically synthesized phonon wavepackets

We monitor how destructive interference of undesired phonon frequency components shapes a quasi-monochromatic hypersound wavepacket spectrum during its local real-time preparation by a nanometric transducer and follow the subsequent decay by nonlinear coupling. We prove each frequency component of an optical supercontinuum probe to be sensitive to one particular phonon wavevector in bulk material and cross-check this by ultrafast x-ray diffraction experiments with direct access to the lattice dynamics. Establishing reliable experimental techniques with direct access to the transient spectrum of the excitation is crucial for the interpretation in strongly nonlinear regimes, such as soliton formation.

Detecting optically synthesized quasi-monochromatic sub-terahertz phonon wavepackets by ultrafast x-ray diffraction

We excite an epitaxial SrRuO3 thin film transducer by a pulse train of ultrashort laser pulses, launching coherent sound waves into the underlying SrTiO3 substrate. Synchrotron-based x-ray diffraction (XRD) data exhibiting separated sidebands to the substrate peak evidence the excitation of a quasi-monochromatic phonon wavepacket with sub-THz central frequency. The frequency and bandwidth of this sound pulse can be controlled by the optical pulse train. We compare the experimental data to combined lattice dynamics and dynamical XRD simulations to verify the coherent phonon dynamics. In addition, we observe a lifetime of 130 ps of such sub-THz phonons in accordance with the theory.

Photon scattering in geometrical optics

A quasimonochromatic wave packet is scattered by a system of two level atoms. The transition frequency of atoms coincides with the frequency of the wave packet, their distances are proportional to that frequency. The time evolution can be calculated explicitly. Using Fourier–Weyl transform and after some renormalization, we establish the limit of geometrical optics. Applying the singular coupling limit, we arrive at the result: the main part of the wave packet is unperturbed, and the scattered part yields an intensity distribution in the six dimensional space of positions and wave numbers. This distribution contains all interferences. The Huygens–Fresnel theory yields an approximation of our result.

Optical excitation and detection of terahertz acoustic waves with semiconductor superlattices

Experimental results about light-to-sound transduction with semiconductor superlattices are presented. Picosecond ultrasonics with pump and probe incident on opposite sides of the substrate allows a full decoupling between generation and detection processes. Associating a metallic transducer, we show that superlattices generate quasi-monochromatic wave packets which can propagate over macroscopic distances in the underlying substrate but are also very selective and sensitive detectors. At the end, using two superlattices simultaneously as phonons generator and detector we evidence that these new transducers allow to perform acoustic experiments at the challenging frequency of 1THz.

Observation of Conical Waves in Focusing, Dispersive, and Dissipative Kerr Media

Excitation of unbalanced-Bessel beams by a gradual increase of nonlinearity in a water sample outlines the achievement of the first ever observed quasimonochromatic wave packet that propagates stably for hundreds of Rayleigh lengths in a focusing and dispersive Kerr medium, i.e., in the absence of spectral broadening and conical emission. A modulational instability analysis reveals the key role of nonlinear dissipation in quenching the growth of spatiotemporal unstable modes.

Propagation of General Wave Packets in Some Classicaland Quantum Systems

In quantum mechanics the center of a wave packet isprecisely defined as the center of probability. The center-of-probabilityvelocity describes the entire motion of the wave packet.In classical physics there is no precise counterpart to thecenter-of-probability velocity of quantum mechanics, in spite of the fact thatthere exist in the literature at least eight different velocities forthe electromagnetic wave. We propose a center-of-energy velocity todescribe the entire motion of general wave packets in classicalphysical systems. It is a measurable quantity, and is well definedfor both continuous and discrete systems. For electromagnetic wavepackets it is a generalization of the velocity of energy transport.General wave packets in several classical systems are studied and thecenter-of-energy velocity is calculated and expressed in terms of thedispersion relation and the Fourier coefficients. These systemsinclude string subject to an external force, monatomic chain anddiatomic chain in one dimension, and classical Heisenberg model inone dimension. In most cases the center-of-energy velocity reduces tothe group velocity for quasi-monochromatic wave packets. Thus it alsoappears to be the generalization of the group velocity. Wave packetsof the relativistic Dirac equation are discussed briefly.