Resonant Wave Interactions Research Articles

The turbulent cascade of inertia-gravity waves in rotating shallow water

In this work we study features of inertia-gravity wave turbulence in the rotating shallow water equations. On examining the dynamics of waves with varying rotation rates, we find that the turbulent cascade of waves is strongest at low rotation rates, forming a $k^{-2}$ energy spectrum, and a rich distribution of shocks in physical space. At high rotation rates, the forward cascade of waves weakens along with a steeper energy spectra and vanishing of shocks in physical space. The wave cascade is seen to be scale-local, resulting in a noticeable time interval for energy to get transferred from domain scale to dissipative scale. Furthermore, we find that the vortical flow has a non-negligible effect on the wave cascade, especially at high rotation rates. The vortical flow assists in the forward cascade of waves and shock formation at high rotation rates, while the waves by themselves in the absence of the vortical flow lack a forward cascade and shock formation at such high rotation rates. On investigating the physical space structures in the vortical flow and their connections to the wave cascade, we find that strain-dominant regions, that are located around the boundaries of coherent vortices, are the physical space regions that contribute majorly to the forward cascade of waves. Our results in general highlight intriguing features of dispersive inertia-gravity wave turbulence that are qualitatively similar to those seen in three-dimensional homogeneous isotropic turbulence and are beyond the predictions of asymptotic resonant wave interaction theory.

Acoustofluidic tweezers via ring resonance.

Ring resonator (RR) devices are closed-loop waveguides where waves circulate only at the resonant frequencies. They have been used in sensor technology and optical tweezers, but controlling micron-scale particles with optical RR tweezers is challenging due to insufficient force, short working distances, and photodamage. To overcome these obstacles, an acoustofluidic RR-based tweezing method is developed to manipulate micro-sized particles that can enhance particle trapping through the resonance interaction of acoustic waves with high Q factor (>3000), more than 20 times greater than traditional acoustic transducers. Particles can be precisely manipulated within the RR by adjusting the signal phase, with trapping amplified by enlarging the connected waveguide. Rapid particle mixing is achieved when particles are placed between the waveguide and RR. The signal path is strengthened by strategically positioning the RR in a two-dimensional plane. Acoustofluidic RR tweezers have immense potential for advancing applications in biosensing, mechanobiology, lab-on-a-chip, and cell-cell communication research.

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

Nonlinear spatial evolution of degenerate quartets of water waves

In this manuscript we investigate the Benjamin–Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. We show that the maximal energy transfer consistently occurs for smaller side-band separation than the fastest linear growth rate. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.

Numerical Integration and Angular Spectrum Modeling of Resonant and Non-Resonant Wave Interaction with a Solid Plate

The angular spectrum (AS) model is widely used to calculate the pressure transmitted through, and reflected from, a fluid-immersed solid plate, using a baffled piston as source. However, narrow peaks (infinite peaks for the lossless case) in the piston aperture function and in the plane wave transmission and reflection coefficients, in addition to a highly oscillatory integrand, cause challenges in the numerical integration. To overcome the challenges related to the aperture function peak, the AS model is reformulated using integration by parts. The peaks in the transmission and reflection coefficients are identified as symmetric and antisymmetric Scholte modes, and a numerical integration method is provided enabling integration over these peaks. The AS-model is also used to obtain the full-wave diffraction correction for a pulse-train transmitted through, or reflected from, the plate, which is highly relevant in pulse-transmit or pulse-receive measurements. The diffraction effects caused by the plate are calculated by suppressing resonance effects in the plate and obtaining the corresponding plane wave pressure transmission and reflection coefficients. A hybrid Gauss-Filon adaptive numerical integration method is used to calculate the pressure. These calculations are conducted in APAS, a MATLAB program developed in the current work and provided as open-source supplementary material.

Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known “Benjamin-Feir resonances”. We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.

Influence of the Resonant Interaction of Surface Magnetostatic Waves with Exchange Modes on the EMF Generation in YIG/Pt Structures

Influence of the Resonant Interaction of Surface Magnetostatic Waves with Exchange Modes on the EMF Generation in YIG/Pt Structures

The 3-wave resonant interaction model: spectra and instabilities of plane waves

The three wave resonant interaction model (3WRI) is a non-dispersive system with quadratic coupling between the components that finds application in many areas, including nonlinear optics, fluids and plasma physics. Using its integrability, and in particular its Lax Pair representation, we carry out the linear stability analysis of the plane wave solutions interacting under resonant conditions when they are perturbed via localised perturbations. A topological classification of the so-called stability spectra is provided with respect to the physical parameters appearing both in the system itself and in its plane wave solution. Alongside the stability spectra, we compute the corresponding gain function, from which we deduce that this system is linearly unstable for any generic choice of the physical parameters. In addition to stability spectra of the same kind observed in the system of two coupled nonlinear Schrödinger equations, whose non-vanishing gain functions detect the occurrence of the modulational instability, the stability spectra of the 3WRI system possess new topological components, whose associated gain functions are different from those characterising the modulational instability. By drawing on a recent link between modulational instability and the occurrence of rogue waves, we speculate that linear instability of baseband-type can be a necessary condition for the onset of rogue wave types in the 3WRI system, thus providing a tool to predict the subsequent nonlinear evolution of the perturbation.

Laboratory study of instability-driven mixing of fluid mud under surface wave motion

Motivated by the role of interfacial instabilities in sediment resuspension in coastal areas, this paper provides quantitative measurements of fluid mud density profile during motion of a surface wave over a muddy bed in a wave flume. Following a fluidization process, a quasi-standing interfacial wave grew owing to a resonant wave interaction with the surface wave. In the process, the quasi-standing wave reached a maximum amplitude and then approached a steady state. The long-time behavior of the resonantly generated interfacial wave and the changes in vertical density profile during wave motion were recorded. Increasing the surface wave frequency led to a higher initial growth rate of the interfacial wave within the experimental range, but the faster growth rate did not result in a larger final amplitude. The results show that excitation of the interfacial wave results in increasing water turbidity such that the water column becomes turbid in a matter of a few minutes. In general, the change in the fluid density profile is highly correlated with the quasi-standing interfacial wave amplitude during the resonant interaction. The amount of entrained mud particles into the clear water by the end of each experiment was determined. The ultimate amplitude of the quasi-standing interfacial wave was found to be a major factor in sediment resuspension.

Mechanism of vorticity amplification by elastic waves in a viscoelastic channel flow

Inertia-less viscoelastic channel flow displays a supercritical nonnormal mode elastic instability due to finite-size perturbations despite its linear stability. The nonnormal mode instability is determined mainly by a direct transition from laminar to chaotic flow, in contrast to normal mode bifurcation leading to a single fastest-growing mode. At higher velocities, transitions to elastic turbulence and further drag reduction flow regimes occur accompanied by elastic waves in three flow regimes. Here, we demonstrate experimentally that the elastic waves play a key role in amplifying wall-normal vorticity fluctuations by pumping energy, withdrawn from the mean flow, into wall-normal fluctuating vortices. Indeed, the flow resistance and rotational part of the wall-normal vorticity fluctuations depend linearly on the elastic wave energy in three chaotic flow regimes. The higher (lower) the elastic wave intensity, the larger (smaller) the flow resistance and rotational vorticity fluctuations. This mechanism was suggested earlier to explain elastically driven Kelvin-Helmholtz-like instability in viscoelastic channel flow. The suggested physical mechanism of vorticity amplification by the elastic waves above the elastic instability onset recalls the Landau damping in magnetized relativistic plasma. The latter occurs due to the resonant interaction of electromagnetic waves with fast electrons in the relativistic plasma when the electron velocity approaches light speed. Moreover, the suggested mechanism could be generally relevant to flows exhibiting both transverse waves and vortices, such as Alfven waves interacting with vortices in turbulent magnetized plasma, and Tollmien-Schlichting waves amplifying vorticity in both Newtonian and elasto-inertial fluids in shear flows.

Investigation of influence of Kerr and detuning parameters over instability gain in the three wave resonant interaction of stimulated Raman backscattering

Investigation of influence of Kerr and detuning parameters over instability gain in the three wave resonant interaction of stimulated Raman backscattering

Kinetic Interaction between an Electron Flow with a Wide Velocity Spread and a Short-Adjusted Slipping Wave Pulse at the Cherenkov Resonance

In this study, kinetic interaction at the Cherenkov resonance between an electromagnetic wave pulse and a flow of electrons possessing a wide velocity spread at the scale of the characteristic range of the resonant electron wave interaction is considered. Due to the absence of a distribution function slope in the range of velocities corresponding to the electron wave’s resonance, an electron’s flow is a nearly stable media from the point of view of its interaction with a long enough wave pulse. In this paper, we explain our findings on the process of electron interaction with potential relief where the wave pulse is so short that the characteristic scale of the wave amplitude’s inhomogeneity and the profile of the potential relief is comparable to the wavelength. We show that if an appropriate slippage between the phase and group velocities of the wave is provided, then the reflection process of particles from “fast” and “slow” close-to-resonance velocity fractions becomes non-symmetrical. This can provide a mechanism of amplification of short intensive wave pulses with electron flows with very large velocity spreads.

Study of character of modulation instability in cyclotron resonance interaction of an electromagnetic wave with a counterpropagating rectilinear electron beam

In this paper, the interaction of a monochromatic electromagnetic wave with a counterpropagating electron beam moving in an axial magnetic field is considered. The purpose of this study is to investigate the conditions for occurrence of modulation instability (MI) in such a system and to determine at which parameters of the incident wave the MI is absolute or convective. Methods. Theoretical analysis of the MI character is carried out by studying the asymptotic form of unstable perturbations using the saddle-point analysis. The analytical results are verified by numerical simulations. Results. Theoretically, the boundary of change in the character of MI on the plane of input signal parameters (amplitude and detuning of the frequency from the cyclotron resonance) is determined. Numerical simulations confirm that as the signal frequency increases, the regime of self-modulation, which corresponds to the absolute MI, is replaced by the stationary single-frequency transmission corresponding to the convective MI. The numerical results coincide with the analytical ones for the system, which is matched at the end. The matching is implemented by smooth increasing of the guiding magnetic field in the region of electron beam injection. Conclusion. Determining the analytical conditions for the implementation of the absolute MI is of practical interest, since the emerging self-modulation can lead to the generation of trains of pulses with the spectrum in the form of frequency combs.

Four-wave resonant interaction of surface gravity waves in finite water depth

We investigated the four-wave resonant and quasi-resonant interactions in a special degenerated case, wherein bichromatic mother waves are generated to give birth to a daughter wave. Through theoretical and numerical analyses, we first discover the threshold value of water depth when the four-wave resonance diminishes. Our work can shed light on the limiting shallow water depth where the energy exchange due to four wave resonance creases.

Effects of finite non-Gaussianity on evolution of a random wind wave field.

We examine the long-term evolution of a random wind wave field generated by constant forcing, by comparing numerical simulations of the kinetic equationand direct numerical simulations (DNS) of the dynamical equations. While the integral characteristics of the spectra are in reasonably good agreement, the spectral shapes differ considerably at large times, the DNS spectral shape being in much better agreement with field observations. Varying the number of resonant and approximately resonant wave interactions in the DNS numerical scheme, we show that when the ratio of nonlinear and linear parts of the Hamiltonian tends to zero, the DNS spectral shape approaches the shape predicted by the kinetic equation. We attribute the discrepancies between the kinetic equationmodeling, on one side, and the DNS and observations, on the other, to the neglect of non-Gaussianity in the derivation of the kinetic equation.

Mutual enhancement of wind- and tide-induced near-inertial internal waves in Luzon Strait

Abstract Tide-induced near-inertial internal waves (NIWs) are generated by tide-topography interaction and are energized by internal tides through triadic resonant interaction of internal waves. They are located above topography and could be in close contact with wind-induced NIWs when the topography is a tall ridge, like in the Luzon Strait of the northern South China Sea (SCS). A natural question arises as to whether there is significant interaction between wind- and tide-induced NIWs. By using moored velocity observations, satellite-tracked surface drifter dataset, and idealized numerical simulations, we find that in the presence of tide-induced NIWs, the wind can inject slightly more near-inertial energy (NIE), while in the presence of wind-induced NIWs, significantly more tidal energy is transferred to NIWs. Thus, wind- and tide-induced NIWs can mutually enhance each other, producing more NIE than a linear superposition of that generated by wind and tide forcing alone. Increasing wind intensity and tidal excursion lead to saturation of NIE enhancement, while a taller ridge leads to stronger enhancement. The high mixed-layer NIE near Luzon Strait is mostly generated by the wind, while the mutual enhancement between wind- and tide-induced NIWs can further enhance this pattern. The interaction between wind- and tide-induced NIWs leads to an enhancement of 25% more NIE. If tide-induced NIWs are neglected, as is usually the case in the estimation of NIE, the total NIE will be underestimated by almost 50%. This might imply that tide-induced NIWs are important for the energetics of NIWs in Luzon Strait.

General higher—order breathers and rogue waves in the two‐component long‐wave–short‐wave resonance interaction model

AbstractGeneral higher order breather and rogue wave (RW) solutions to the two‐component long wave–short wave resonance interaction (2‐LSRI) model are derived via the bilinear Kadomtsev–Petviashvili hierarchy reduction method and are given in terms of determinants. Under particular parametric conditions, the breather solutions can reduce to homoclinic orbits, or a mixture of breathers and homoclinic orbits. There are three families of RW solutions, which correspond to a simple root, two simple roots, and a double root of an algebraic equation related to the dimension reduction procedure. The first family of RW solutions consists of bounded fundamental RWs, the second family is composed of bounded fundamental RWs coexisting with another fundamental RWs of different a bounded state ( being positive integers), while the third one has fundamental bounded RWs ( being nonnegative integers). The second family can be regarded as the superpositions of the first family, while the third family can be the degenerate case of the first family under particular parameter choices. These diverse RW patterns are illustrated graphically.

Two-Dimensional Solitons in Nonlocal Media: A Brief Review

This is a review addressing soliton-like states in systems with nonlocal nonlinearity. The work on this topic has long history in optics and related areas. Some results produced by the work (such as solitons supported by thermal nonlinearity in optical glasses, and orientational nonlinearity, which affects light propagation in liquid crystals) are well known, and have been properly reviewed in the literature, therefore the respective models are outlined in the present review in a brief form. Some other studies, such as those addressing models with fractional diffraction, which is represented by a linear nonlocal operator, have started more recently, therefore it will be relevant to review them in detail when more results will be accumulated; for this reason, the present article provides a short outline of the latter topic. The main part of the article is a summary of results obtained for two-dimensional solitons in specific nonlocal nonlinear models originating in studies of Bose–Einstein condensates (BECs), which are sufficiently mature but have not yet been reviewed previously (some results for three-dimensional solitons are briefly mentioned too). These are, in particular, anisotropic quasi-2D solitons supported by long-range dipole-dipole interactions in a condensate of magnetic atoms and giant vortex solitons (which are stable for high values of the winding number), as well as 2D vortex solitons of the latter type moving with self-acceleration. The vortex solitons are states of a hybrid type, which include matter-wave and electromagnetic-wave components. They are supported, in a binary BEC composed of two different atomic states, by the resonant interaction of the two-component matter waves with a microwave field that couples the two atomic states. The shape, stability, and dynamics of the solitons in such systems are strongly affected by their symmetry. Some other topics are included in the review in a brief form. This review uses the “Harvard style” of referring to the bibliography.

On the Stability of the Periodic Waves for the Benney System

We analyze the Benney model for interaction of short and long waves in resonant water wave interactions. Our particular interest is in the periodic traveling waves, which we construct and study in detail. The main results are that, for all natural values of the parameters, the periodic dnoidal waves are spectrally stable with respect to perturbations of the same period. For another natural set of parameters, we construct the snoidal waves, which exhibit instabilities, in the same setup. Our results are the first instability results in this context. On the other hand, the spectral stability established herein improves significantly on the work of Angulo, Corcho, and Hakkaev [Adv. Difference Equ., 16 (2011), pp. 523--550], which established stability of the dnoidal waves, on a subset of parameter space, by relying on the Grillakis--Shatah theory. Our approach, which turns out to give definite answer for the entire domain of parameters, relies on the instability index theory, as developed by Kapitula, Kevrekidis, and Sandstede [Phys. D, 3--4 (2004), pp. 263--282]; Kapitula, Kevrekidis, and Sandstede [Phys. D, 195 (2004), no. 3--4, 263--282], Phys. D, 201 (2005), pp. 199--201]; Lin and Zeng [Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs, 2021]; and Pelinovsky [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), pp. 783--812]. Interestingly, end even though the linearized operators are explicit, our spectral analysis requires subtle and detailed analysis of matrix Schrödinger operators in the periodic context, which support some interesting features.

On using nonlinear wave interactions in multi-directional seas for energy conversion on the ocean floor

This paper investigates possible utilization of energy in interacting surface waves on the seafloor. Theory suggests that resonant wave interactions in confused open seas can cause energetic downward traveling second-order pressure waves, which during stormy conditions can be large enough to excite seafloor microseisms (Longuet-Higgins, 1950). Our analysis of contemporaneous data from seafloor pressure and velocity sensors from an Ocean Observatories Initiative (OOI) installation in the Northern Pacific and collocated surface-wave hindcasts showed evidence of the theoretically predicted vertical resonances (Longuet-Higgins, 1950). We observed that area-averaged predicted dynamic pressures and particle velocities 200 m below the water surface and their measured counterparts 2.6 km below at the seafloor were well correlated at the predicted resonance frequencies. Our results further revealed occurrence of net vertical power transfer at resonance from surface to seafloor by means of the predicted double-frequency pressure waves. Motivated by these findings, we estimated year-long seafloor power availability at five selected deep-sea locations. Next, we evaluated upper bounds on potentially convertible area averaged yearly-mean power amounts by a small flexible sphere type device at the seafloor. At ∼400 mW m−2 to ∼15 Wm−2, these power levels could assist sensor operations on and from the deep ocean floor.