Nonlinear Resonant Interaction Research Articles

Triad resonance between gravity and vorticity waves in vertical shear

Weakly nonlinear theory is used to explore the effect of vertical shear on surface gravity waves in three dimensions. An idealized piecewise-linear shear profile motivated by wind-driven profiles and ambient currents in the ocean is used. It is shown that shear may mediate weakly nonlinear resonant triad interactions between gravity and vorticity waves. The triad results in energy exchange between gravity waves of comparable wavelengths propagating in different directions. For realistic ocean shears, shear-mediated energy exchange may occur on timescales of minutes for shorter wavelengths, but slows as the wavelength increases. Hence this triad mechanism may contribute to the larger angular spreading (relative to wind direction) for shorter wind-waves observed in the oceans.

Universality of sea wave growth and its physical roots

Assuming resonant nonlinear wave interactions to be the dominant physical mechanism of growing wind-driven seas we propose a concise relationship between instantaneous wave steepness and time or fetch of wave development expressed in dimensionless wave periods or lengths. This asymptotic physical law derived from the first principles of the theory of weak turbulence does not contain wind speed explicitly. The validity of this law is illustrated by results of numerical simulations, in situ measurements of growing wind seas and wind-wave tank observations. The impact of this new view of sea-wave physics is discussed in the context of conventional approaches to wave modelling and forecasting.

Resonant ensembles of stationary quasi-harmonic waves in one-dimensional crystals

Using a simple mathematical model, built on geometric representations of central and noncentral interactions between material particles in a one-dimensional anharmonic chain, nonlinear resonant interactions between quasi-harmonic waves are investigated in the so-called harmonic approximation. The investigation is carried out with standard asymptotic nonlinear dynamics methods. In the first-order approximation, resonant wave triads are established that are formed at a characteristic quadratic nonlinearity of the system provided that the phase-matching conditions are satisfied. It is demonstrated that the resonant triads can be of only three different types and each resonant triad can consist of only one longitudinal and two transverse oscillation modes. In the general case, a nontrivial interaction between different resonant triplets of three different types and spectral scales is implemented in a chain. Cascade processes of the energy exchange between oscillation modes are characterized by both the complicated dynamics typical of Hamiltonian nonintegrable dynamic systems and the presence of Lyapunov-stable multiwave steady-state motions. In ideal crystalline structures, such steady-state coherent wave ensembles can significantly affect the specific heat and other phenomenological parameters of a system, especially at low temperatures. Therefore, the theoretical and experimental study of these ensembles is of great importance.

Nonlinear wave interactions of kinetic sound waves

Abstract. We reconsider the nonlinear resonant interaction between three electrostatic waves in a magnetized plasma. The general coupling coefficients derived from kinetic theory are reduced here to the low-frequency limit. The main contribution to the coupling coefficient we find in this way agrees with the coefficient recently presented in Annales Geophysicae. But we also deduce another contribution which sometimes can be important, and which qualitatively agrees with that of an even more recent paper. We have thus demonstrated how results derived from fluid theory can be improved and generalized by means of kinetic theory. Possible extensions of our results are outlined.

NONLINEAR DYNAMICS OF MAGNETOHYDRODYNAMIC ROSSBY WAVES AND THE CYCLIC NATURE OF SOLAR MAGNETIC ACTIVITY

The solar dynamo is known to be associated with several periodicities, with the nearly 11/22 yr cycle being the most pronounced one. Even though these quasiperiodic variations of solar activity have been attributed to the underlying dynamo action in the Sun's interior, a fundamental theoretical description of these cycles is still elusive. Here, we present a new possible direction in understanding the Sun's cycles based on resonant nonlinear interactions among magnetohydrodynamic (MHD) Rossby waves. The WKB theory for dispersive waves is applied to magnetohydrodynamic shallow-water equations describing the dynamics of the solar tachocline, and the reduced dynamics of a resonant triad composed of MHD Rossby waves embedded in constant toroidal magnetic field is analyzed. In the conservative case, the wave amplitudes evolve periodically in time, with periods on the order of the dominant solar activity timescale (~11 yr). In addition, the presence of linear forcings representative of either convection or instabilities of meridionally varying background states appears to be crucial in balancing dissipation and thus sustaining the periodic oscillations of wave amplitudes associated with resonant triad interactions. Examination of the linear theory of MHD Rossby waves embedded in a latitudinally varying mean flow demonstrates that MHD Rossby waves propagate toward the equator in a waveguide from –35° to 35° in latitude, showing a remarkable resemblance to the structure of the butterfly diagram of the solar activity. Therefore, we argue that resonant nonlinear magnetohydrodynamic Rossby wave interactions might significantly contribute to the observed cycles of magnetic solar activity.

Global well-posedness of a system of nonlinearly coupled KdV equations of Majda and Biello

This paper addresses the problem of global well-posedness of a coupled system of Korteweg-de Vries equations, derived by Majda and Biello in the context of nonlinear resonant interaction of Rossby waves, in a periodic setting in homogeneous Sobolev spaces $\dot H^s$, for $s\geq 0$. Our approach is based on a successive time-averaging method developed by Babin, Ilyin and Titi [1].

Multicomponent long-wave-short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons.

We consider a general multicomponent (2+1)-dimensional long-wave-short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multisoliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one- and two-soliton solutions and studied their dynamics briefly.

Electron scattering and nonlinear trapping by oblique whistler waves: The critical wave intensity for nonlinear effects

In this paper, we consider high-energy electron scattering and nonlinear trapping by oblique whistler waves via the Landau resonance. We use recent spacecraft observations in the radiation belts to construct the whistler wave model. The main purpose of the paper is to provide an estimate of the critical wave amplitude for which the nonlinear wave-particle resonant interaction becomes more important than particle scattering. To this aim, we derive an analytical expression describing the particle scattering by large amplitude whistler waves and compare the corresponding effect with the nonlinear particle acceleration due to trapping. The latter is much more rare but the corresponding change of energy is substantially larger than energy jumps due to scattering. We show that for reasonable wave amplitudes ∼10–100 mV/m of strong whistlers, the nonlinear effects are more important than the linear and nonlinear scattering for electrons with energies ∼10–50 keV. We test the dependencies of the critical wave amplitude on system parameters (background plasma density, wave frequency, etc.). We discuss the role of obtained results for the theoretical description of the nonlinear wave amplification in radiation belts.

Nonlinear spatio-temporal instability regime for electrically forced viscous jets

This paper considers the problem of nonlinear instability in electrically driven viscous axisymmetric jets with respect to spatial and temporal growing disturbances in the presence of a uniform or non-uniform applied electric field. The mathematical modeling for the jets, which uses the original electrohydrodynamics equations (Melcher and Taylor, 1969) [8], is based on the nonlinear mechanics that govern the liquid jet due to tangential electric field effects. At the linear stage, we found that a particular jet of fluid could exhibit the Rayleigh and Conducting flow Instabilities for the spatial and temporal evolution of the disturbance. For the nonlinear regime of the problem, we studied the resonant instability and nonlinear wave interactions of certain modes that satisfy the dyad resonant condition. The nonlinear wave interactions in the jet provided a significant change in the fluid flow properties that extend notably the available understanding of the problem at the linear stage. It was found that the nonlinear resonant instability provides an amplifying effect on the magnitude of the disturbances which evolves the jet to reduce significantly its radius at a shorter axial location. For the case of higher viscosity fluid, the electric field in the jet was found to be increasing spatially and temporally when nonlinear wave interactions were taken into account during the resonant instability. The resulting nonlinear solutions for the jet thickness, jet׳s electric field, jet׳s surface charge and jet velocity are presented and discussed.

Nonlinear resonance interaction between three capillary-gravity waves on the plane charged fluid surface

Three-wave interaction between capillary-gravity waves on a uniformly charged free fluid surface is analyzed using second-order analytic calculations. The time evolution of the wave amplitudes in the state of nonlinear resonance is studied. It is shown that the number of three-wave resonances is infinite and their exact locations for waves of finite amplitude depend on the initial conditions.

Nonlinear internal resonance interaction between surface waves and internal waves in an inhomogeneous laminated liquid

Eight nonlinear resonances may arise in a two-layer liquid with a free surface, as follows from asymptotic calculations of the second order of smallness. Two of them involve waves generated by either surface, and six mixed resonances involve waves generated by different surfaces. Of these six resonances, two are degenerate and the remaining four are secondary combination resonances. However, not all of them are observed in practice. In the mixed combination resonances, interaction arises only between surface waves provided that the internal wave (which does not change its parameters but is necessary for the resonance to occur) has a catalytic influence.

Thermal electron acceleration by localized bursts of electric field in the radiation belts

AbstractIn this paper we investigate the resonant interaction of thermal ∼10–100 eV electrons with a burst of electrostatic field that results in electron acceleration to kilovolt energies. This single burst contains a large parallel electric field of one sign and a much smaller, longer‐lasting parallel field of the opposite sign. The Van Allen Probe spacecraft often observes clusters of spatially localized bursts in the Earth's outer radiation belts. These structures propagate mostly away from the geomagnetic equator and share properties of soliton‐like nonlinear electron acoustic waves: a velocity of propagation is about the thermal velocity of cold electrons (∼3000–10,000 km/s), and a spatial scale of electric field localization along the field lines is about the Debye radius of hot electrons (∼5–30 km). We model the nonlinear resonant interaction of these electric field structures and cold background electrons.

Sub-harmonic resonant wave interactions in the presence of a linear interfacial instability

This work considers the role of nonlinear sub-harmonic resonant wave interactions in the development of interfacial waves, which may be under the influence of a linear interfacial instability, in an inviscid two-fluid stratified flow through a horizontal channel. We begin by examining the case of resonant interactions between one linearly unstable mode and its sub-harmonic that may be linearly stable or unstable. Using the method of multiple scales, we derive the nonlinear interaction equations governing the time evolution of interacting wave amplitudes. These nonlinear equations account for the combined effects of both the nonlinear resonant interaction and the linear instability. We show that through this nonlinear coupling, the linearly stable sub-harmonic mode can achieve faster than exponential growth. It is found that such a mechanism is capable of generating large-amplitude long waves that are stable by linear stability analysis. The analytical predictions are cross validated by comparisons with direct nonlinear numerical simulations based on a more general perturbation based spectral method. Good agreement between the analytical and numerical solutions is observed. The more complicated case where a single mode is simultaneously involved in multiple (sub-harmonic and triad) resonances is also investigated numerically. The results demonstrate that chains of resonances can permit the energy generated by the linear instability, among high wavenumber components, to be passed across the spectrum to the longest wave components creating an efficient mechanism for the generation of large-amplitude long waves from unstable short waves.

Nonlinear degenerate resonance interaction of waves on a charged liquid surface

The physics of nonlinear degenerate resonance energy exchange between waves on the flat free charged surface of a conducting liquid is analytically (asymptotically) studied up to the second order of smallness. A set of differential equations for the evolution of the amplitudes of nonlinearly resonantly interacting waves is derived. It turns out that nonlinear computations (taking into account the dependence of the wave frequency on the finite amplitude) yield an infinite number of degenerate resonances, although computations based on frequencies found in the linear theory give a finite number of resonances. In nonlinear computations, the positions of the degenerate resonances depend on the surface charge density (or on the external electric field normal to the free surface of the liquid) in contrast to the results of linear computations (based on frequencies found in the linear theory). It is found that as the wavenumber of an exact degenerate resonance is approached (that is, in the vicinity of this number), the direction of energy transfer changes sign: now the energy is transferred from a shorter wave to a longer one and not the reverse.

Overview of Stimulated Brillouin Scattering Effect and Various Types of Method to Eliminate this Effect

Stimulated Brillouin scattering (SBS) is a resonant nonlinear optical interaction with the material that results in transmitted light being scattered back towards the input. Although high power lasers are available to overcome the intrinsic loss of standard single mode optical transmission fibers (0.2 to 0.3 dB/km) but SBS places an upper limit on the optical power that can be transmitted through the link. Usually, SBS normally has a lower threshold power (≤1.4 mW) than other nonlinear effects. In this paper we see the SBS effect in optical fiber transmission system and different types of methods to eliminate this effect. Here we propose two new approaches to eliminate this effect.

Stochastic electron motion driven by space plasma waves

Abstract. Stochastic motion of relativistic electrons under conditions of the nonlinear resonance interaction of particles with space plasma waves is studied. Particular attention is given to the problem of the stability and variability of the Earth's radiation belts. It is found that the interaction between whistler-mode waves and radiation-belt electrons is likely to involve the same mechanism that is responsible for the dynamical balance between the accelerating process and relativistic electron precipitation events. We have also considered the efficiency of the mechanism of stochastic surfing acceleration of cosmic electrons at the supernova remnant shock front, and the accelerating process driven by a Langmuir wave packet in producing cosmic ray electrons. The dynamics of cosmic electrons is formulated in terms of a dissipative map involving the effect of synchrotron emission. We present analytical and numerical methods for studying Hamiltonian chaos and dissipative strange attractors, and for determining the heating extent and energy spectra.

General multicomponent Yajima-Oikawa system: Painlevé analysis, soliton solutions, and energy-sharing collisions

We consider the multicomponent Yajima-Oikawa (YO) system and show that the two-component YO system can be derived in a physical setting of a three-coupled nonlinear Schrödinger (3-CNLS) type system by the asymptotic reduction method. The derivation is further generalized to the multicomponent case. This set of equations describes the dynamics of nonlinear resonant interaction between a one-dimensional long wave and multiple short waves. The Painlevé analysis of the general multicomponent YO system shows that the underlying set of evolution equations is integrable for arbitrary nonlinearity coefficients which will result in three different sets of equations corresponding to positive, negative, and mixed nonlinearity coefficients. We obtain the general bright N-soliton solution of the multicomponent YO system in the Gram determinant form by using Hirota's bilinearization method and explicitly analyze the one- and two-soliton solutions of the multicomponent YO system for the above mentioned three choices of nonlinearity coefficients. We also point out that the 3-CNLS system admits special asymptotic solitons of bright, dark, anti-dark, and gray types, when the long-wave-short-wave resonance takes place. The short-wave component solitons undergo two types of energy-sharing collisions. Specifically, in the two-component YO system, we demonstrate that two types of energy-sharing collisions-(i) energy switching with opposite nature for a particular soliton in two components and (ii) similar kind of energy switching for a given soliton in both components-result for two different choices of nonlinearity coefficients. The solitons appearing in the long-wave component always exhibit elastic collision whereas those of short-wave components exhibit standard elastic collisions only for a specific choice of parameters. We have also investigated the collision dynamics of asymptotic solitons in the original 3-CNLS system. For completeness, we explore the three-soliton interaction and demonstrate the pairwise nature of collisions and unravel the fascinating state restoration property.

Nonlinear resonance interaction of waves on a jet in a radial electric field

Nonlinear analytic asymptotic second-order calculations show that the motion of a charged jet relative to a material medium results in the appearance of rise-time periodic wave motions of the interface (Kelvin-Helmholtz instabilities), as well as an internal nonlinear resonance interaction of waves, whose parameters (the intensity and the characteristic time of interaction) depend on the physical parameters of the system: the electric charge density at the jet, its velocity relative the medium, the mass density, the values of the wavenumbers of the interacting waves, and the value of the surface tension coefficient of the boundary.

An extended equatorial plane: linear spectrum and resonant triads

The usual equatorial -plane supports a range of wave types, including trapped baroclinic Rossby waves, a Yanai wave and a Kelvin wave, which are known to be important in equatorial wave dynamics, both linear and nonlinear. Here, we extend this model to the whole sphere, using a Mercator projection, in which the Coriolis parameter is retained in full, but the metric coefficients are approximated by their values at the equator. We find that while the trapped baroclinic waves are preserved in this model, the barotropic waves are also trapped, with consequences for nonlinear resonant triad interactions.

Internal nonlinear resonance on a charged jet

Nonlinear analytical asymptotic calculations of the second order of smallness show that the motion of a charged jet in a material medium generates periodic wave motions of the jet-medium interface (Kelvin-Helmholtz instabilities), which grow in time. In addition, the motion of the jet gives rise to nonlinear internal resonance interaction of waves. The parameters of this interaction (intensity and characteristic time) depend on the physical parameters of the system: electric charge density of the jet, its velocity in the medium, mass density, wavenumbers of interacting waves, and the interfacial tension coefficient.