Resonant Wave Interactions Research Articles

Resonant instability and nonlinear wave interactions in electrically forced jets

We investigate the problem of linear temporal instability of the modes that satisfy the dyad resonance conditions and the associated nonlinear wave interactions in jets driven by either a constant or a variable external electric field. A mathematical model, which is developed and used for the temporally growing modes with resonance and their nonlinear wave interactions in electrically driven jet flows, leads to equations for the unknown amplitudes of such waves. These equations are solved for both water and glycerol jet cases, and the expressions for the dependent variables of the corresponding modes are determined. The results of the generated data for these dependent variables versus time indicate, in particular, that the instability resulted from the nonlinear interactions of such modes is mostly quite strong but can also lead to significant reduction in the jet radius.

Wave motion in a gravity field on the free surface and stratification interface of a fluid with stratified inhomogeneity. Nonlinear analysis

Regularities of the nonlinear gravitational wave motion in a two-layer density-stratified fluid are investigated for a finite thickness of the upper, lighter, layer. The characteristics of the nonlinear internal resonant interaction of the gravity waves generated by the free surface of the upper layer and the medium interface are considered. It is shown that in second-order calculations both degenerate (two-wave) and secondary combined (three-wave) resonant interactions may be realized.

On the observations of unique low latitude whistler-triggered VLF/ELF emissions

A detailed analysis of the VLF/ELF wave data obtained during a whistler campaign under All India Coordinated Program of Ionosphere Thermosphere Studies (AICPITS) at our low latitude Indian ground station Jammu (geomag. lat. = 22°26′ N, L=1.17) has yielded two types of unusual and unique whistler-triggered VLF/ELF emissions. These include (1) whistler-triggered hook emissions and (2) whistler-triggered long enduring discrete chorus riser emissions in VLF/ELF frequency range during night time. Such types of whistler-triggered emissions have not been reported earlier from any of the ground observations at low latitudes. In the present study, the observed characteristics of these emissions are described and interpreted. Dispersion analysis of these emissions show that the whistlers as well as emissions have propagated along a higher geomagnetic field line path with L-values lying ∼L=4, suggesting that these triggered emissions are to be regarded as mid-latitude emissions. These waves could have propagated along the geomagnetic field lines either in a ducted mode or in a pro-longitudinal (PL) mode. The measured intensity of the triggered emissions is almost equal to that of the source waves and does not vary throughout the period of observation on that day. It is speculated that these emissions may have been generated through a process of resonant interaction of the whistler waves with energetic electrons. Parameters related to this interaction are computed for different values of L and wave amplitude. The proposed mechanism explains some aspects of the dynamic spectra.

The existence of a global attractor for the [formula omitted]-dimensional long wave–short wave resonance interaction equation

In this paper, we prove the existence of a global attractor and an asymptotic smoothing effect of the solution for the (2+1)-dimensional long wave–short wave resonance interaction equation.

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.

Fiske steps and Abrikosov vortices in Josephson tunnel junctions

We present a theoretical and experimental study of the Fiske resonances in the current-voltage characteristics of "small" Josephson junctions with randomly distributed misaligned Abrikosov vortices. We obtained that in the presence of Abrikosov vortices the resonant interaction of electromagnetic waves, excited inside a junction, with the ac Josephson current manifests itself by Fiske steps in a current-voltage characteristics even in the absence of external magnetic field. We found that the voltage positions of the Fiske steps are determined by a junction size, but the Fiske step magnitudes depend both on the density of trapped Abrikosov vortices and on their misalignment parameter. We measured the magnetic field dependence of both the amplitude of the first Fiske step and the Josephson critical current of low-dissipative small $Nb$ based Josephson tunnel junctions with artificially introduced Abrikosov vortices. A strong decay of the Josephson critical current and a weak non-monotonic decrease of the first Fiske step amplitude on the Abrikosov vortex density were observed. The experimentally observed dependencies are well described by the developed theory.

Inverse scattering transform for the Yajima-Oikawa equations with nonvanishing boundary conditions

Application of the Yajima-Oikawa (YO) equations describing the long wave--short wave resonance interaction to study of vector solitons of mixed bright-dark types in quasi-one-dimensional spinor Bose-Einstein condensates is considered. The inverse scattering transform for the YO equations with the mixed vanishing and nonvanishing boundary values is constructed. Examples of management of the high-frequency waves by changing initial and boundary values associated with the low-frequency waves are presented.

Resonant Wave Interactions in the Presence of a Diurnally Varying Heat Source

Abstract Resonant interactions among equatorial waves in the presence of a diurnally varying heat source are studied in the context of the diabatic version of the equatorial β-plane primitive equations for a motionless, hydrostatic, horizontally homogeneous and stably stratified background atmosphere. The heat source is assumed to be periodic in time and of small amplitude [i.e., O(ɛ)] and is prescribed to roughly represent the typical heating associated with deep convection in the tropical atmosphere. In this context, using the asymptotic method of multiple time scales, the free linear Rossby, Kelvin, mixed Rossby–gravity, and inertio-gravity waves, as well as their vertical structures, are obtained as leading-order solutions. These waves are shown to interact resonantly in a triad configuration at the O(ɛ) approximation, and the dynamics of these interactions have been studied in the presence of the forcing. It is shown that for the planetary-scale wave resonant triads composed of two first baroclinic equatorially trapped waves and one barotropic Rossby mode, the spectrum of the thermal forcing is such that only one of the triad components is resonant with the heat source. As a result, to illustrate the role of the diurnal forcing in these interactions in a simplified fashion, two kinds of triads have been analyzed. The first one refers to triads composed of a k = 0 first baroclinic geostrophic mode, which is resonant with the stationary component of the diurnal heat source, and two dispersive modes, namely, a mixed Rossby–gravity wave and a barotropic Rossby mode. The other class corresponds to triads composed of two first baroclinic inertio-gravity waves in which the highest-frequency wave resonates with a transient harmonic of the forcing. The integration of the asymptotic reduced equations for these selected resonant triads shows that the stationary component of the diurnal heat source acts as an “accelerator” for the energy exchanges between the two dispersive waves through the excitation of the catalyst geostrophic mode. On the other hand, since in the second class of triads the mode that resonates with the forcing is the most energetically active member because of the energy constraints imposed by the triad dynamics, the results show that the convective forcing in this case is responsible for a longer time scale modulation in the resonant interactions, generating a period doubling in the energy exchanges. The results suggest that the diurnal variation of tropical convection might play an important role in generating low-frequency fluctuations in the atmospheric circulation through resonant nonlinear interactions.

Interplay of Resonant and Quasi-Resonant Interaction of the Directional Ocean Waves

Abstract Recent experimental study of the evolution of random directional gravity waves in deep water provides new insight into the nature of the spectral evolution of the ocean waves and the relative significance of resonant and quasi-resonant wave interaction. When the directional angle containing half the total energy is broader than ∼20°, the spectrum evolves following the energy transfer that can be described by the four-wave resonant interaction alone. In contrast, in the case of a directionally confined spectrum, the effect of quasi-resonant wave–wave interaction becomes important, and the wave system becomes unstable. When the temporal change of the spectral shape due to quasi resonance becomes irreversible owing to energetic breaking dissipation, the spectrum rapidly downshifts. Under such extreme conditions, the likelihood of a freak wave is high.

Nonlinearly coupled KdV equations describing the interaction of equatorial and midlatitude Rossby waves

Amplitude equations governing the nonlinear resonant interaction of equatorial baroclinic and barotropic Rossby waves were derived by Majda and Biello and used as a model for long range interactions (teleconnections) between the tropical and midlatitude troposphere. An overview of that derivation is presented and geared to readers versed in nonlinear wave theory, but not in atmospheric sciences. In the course of the derivation, two other sets of asymptotic equations are presented: the long equatorial wave equations and the weakly nonlinear, long equatorial wave equations. A linear transformation recasts the amplitude equations as nonlinear and linearly coupled KdV equations governing the amplitude of two types of modes, each of which consists of a coupled tropical/midlatitude flow. In the limit of Rossby waves with equal dispersion, the transformed amplitude equations become two KdV equations coupled only through nonlinear fluxes. Four numerical integrations are presented which show (i) the interaction of two solitons, one from either mode, (ii) and (iii) the interaction of a soliton in the presence of different mean wind shears, and (iv) the interaction of two solitons mediated by the presence of a mean wind shear.

Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities

Bright and bright-dark type multisoliton solutions of the integrable N-coupled nonlinear Schrodinger (CNLS) equations with focusing, defocusing and mixed type nonlinearities are obtained by using Hirota’s bilinearization method. Particularly, for the bright soliton case, we present the Gram type determinant form of the n-soliton solution (n:arbitrary) for both focusing and mixed type nonlinearities and explicitly prove that the determinant form indeed satisfies the corresponding bilinear equations. Based on this, we also write down the multisoliton form for the mixed (bright-dark) type solitons. For the focusing and mixed type nonlinearities with vanishing boundary conditions the pure bright solitons exhibit different kinds of nontrivial shape changing/energy sharing collisions characterized by intensity redistribution, amplitude dependent phase-shift and change in relative separation distances. Due to nonvanishing boundary conditions the mixed N-CNLS system can admit coupled bright-dark solitons. Here we show that the bright solitons exhibit nontrivial energy sharing collision only if they are spread up in two or more components, while the dark solitons appearing in the remaining components undergo mere standard elastic collisions. Energy sharing collisions lead to exciting applications such as collision based optical computing and soliton amplification. Finally, we briefly discuss the energy sharing collision properties of the solitons of the (2+1) dimensional long wave-short wave resonance interaction (LSRI) system.

Equal energy phase space trajectories in resonant wave interactions

Adiabatic evolution of two and three resonantly interacting wave systems with nonlinear frequency/wave vector shifts is discussed. The corresponding Hamiltonian, depending on the coupling, detuning, and nonlinear frequency shift parameters may have a variable number of fixed points, i.e., the system can experience a topological change of phase space when these parameters vary in time or space. It is shown that the oscillation periods of two equal energy trajectories in these wave systems are equal and the difference between the action integrals of such trajectories is obtained analytically as a function of the system parameters. Based on these findings, a scheme of simultaneous adiabatic variation in the parameters is designed, such that any pair of initially equal energy trajectories continues to have the same energy at later times. These results are generalizations of a previous work [O. Polomarov and G. Shvets, Phys. Plasmas 13, 054502 (2006)] for a single, resonantly driven wave.

Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis

We investigate, via perturbation analyses, the mechanisms of nonlinear resonant interaction of surface-interfacial waves with a rippled bottom in a two-layer density-stratified fluid. As in a one-layer fluid, three classes of Bragg resonances are found to exist if nonlinear interactions up to the third order in the wave/ripple steepness are considered. As expected, the wave system associated with the resonances is more complicated than that in a one-layer fluid. Depending on the specifics of the resonance condition, the resonance-generated wave may be a surface or internal mode and may be transmitted or reflected. At the second order, class I Bragg resonance occurs involving two surface and/or internal waves and one bottom-ripple component. The interaction of an incident surface/internal wave with the bottom ripple generates a new surface or internal wave that may propagate in the same or the opposite direction as the incident wave. At the third order, class II and III Bragg resonances occur involving resonant interactions of four wave/ripple components: two surface and/or internal waves and two bottom-ripple components for class II resonance; three surface and/or internal waves and one bottom-ripple components for class III resonance. As in class I resonance, the resonance-generated wave in class II resonance has the same frequency as that of the incident wave. For class III resonance, the frequency of the resonant wave is equal to the sum or difference of the two incident wave frequencies. We enumerate and represent, using Feynman-like diagrams, the possible cases and combinations for Bragg resonance up to the third order (in two dimensions). Analytical regular perturbation results are obtained and discussed for all three classes of Bragg resonances. These are valid for limited bottom patch lengths and initial/finite growth of the resonant waves. For long bottom patches, a uniformly valid solution using multiple scales is derived for class I resonance. A number of applications underscoring the importance and implication of these nonlinear resonances on the evolution of ocean waves are presented and discussed. For example, it is shown that three internal/surface waves co-propagating over bottom topography are resonant under a broad range of Bragg conditions. The present study provides the theoretical basis and understanding for the companion paper (Alam, Liu & Yue 2008), where a direct numerical solution for the general nonlinear problem is pursued.

NOTE ON THE TWO-COMPONENT ANALOGUE OF TWO-DIMENSIONAL LONG WAVE – SHORT WAVE RESONANCE INTERACTION SYSTEM

AbstractAn integrable two-component analogue of the two-dimensional long wave – short wave resonance interaction (2c-2d-LSRI) system is studied. Wronskian solutions of 2c-2d-LSRI system are presented. A reduced case, which describes resonant interaction between an interfacial wave and two surface wave packets in a two-layer fluid, is also discussed.

On the theory of type III radio bursts in the nonhomogeneous interplanetary space (quasi-linear approximation)

Within the limits of geometrical optics frequency characteristics of perturbations of one-dimensionally non-uniform system “electron beam–solar wind plasma” are investigated in linear approximation on the basis of Maxwell equations closed by the derived constitutive equation. The beam is generated by the active region during solar flares and it appears as a source of type III radio emission in the interplanetary space. The appropriate dispersion equation is solved. Resonance interaction of wave with electron beam appears to happen only in two space points. Such transient (pointwise) mechanism of resonance throws light on one of the basic problems of physics of electron beams generated by solar flares: incomparably more long-term time of their existence compared to the time of existence resulting from the former theoretical estimates of velocity of beam energy loss on radiation within the limits of homogeneous medium. The degree and time of electron beam dissipation were determined in quasi-linear approximation.

Homoclinic orbits for some (2+1)-dimensional nonlinear Schrödinger-like equations

Chaos is closely associated with homoclinic orbits in deterministic nonlinear dynamics. In this paper, analytic expressions of homoclinic orbits for some (2+1)-dimensional nonlinear Schrodinger-like equations are constructed based on Hirota’s bilinear method, including long wave-short wave resonance interaction equation, generalization of the Zakharov equation, Mel’nikov equation, and g-Schrodinger equation are constructed based on Hirota’s bilinear method.

Possibility of the development of explosive instability in a two-stream gravitating medium

An expression for the energy of the density wave in a multicomponent gravitating medium is obtained that is similar to an appropriate expression in electrodynamics of continua. The possibility of birth of a wave triplet including both positive and negative wave energies in a nonequilibrium gravitating medium is demonstrated. It is shown that such a resonant interaction of waves leads to the development of explosive instability. An expression for the characteristic growth time of instability within a one-dimensional model is obtained by a method developed in plasma physics.

Resonance absorption, reflection, transmission of phonons and heat transfer through interface between two solids

The different mechanisms of resonant transport of phonons between two media in the presence of an impurity intermediate layer are described. Particular attention is focused on the resonance interaction of elastic waves with a two-dimensional defect on the contact boundary between two solids, on the multichannel interface phonon scattering, and on the experimentally observed nonmonotonic temperature dependence of the reduced heat flux. In the cases when there is a direct interaction between edge atoms of the matrix as non-nearest neighbors or when the impurities do not fill completely the 2D interface layer, an additional channel for the transmission of phonons through the interface opens. This additional transmission channel manifests itself as a transmission (or reflection or absorption) peak with an asymmetric line shape (the so-called Fano-like resonance for phonons due to interference between the two transmission channels). Some applications of the Fano-like interference phenomena in magnon heat conductivity are also discussed.

The global solution of the (2+1)-dimensional long wave–short wave resonance interaction equation

Abstract

Generation of zonal flow by resonant Rossby-Haurwitz wave interactions

Starting out with two interacting Rossby-Haurwitz waves, the generation of zonal flow is discussed. It is shown that zonal flow cannot be generated by first or second order interactions between two such waves, unless they are exchanging energy with a third wave within a resonant triad. The generation of zonal flow at second order through resonant triad interactions is subsequently established and studied.