Resonant Four-wave Interaction Research Articles

Kinetic equations in a third-generation spectral wave model

Abstract

The Phillips spectrum and a model of wind-wave dissipation

We consider an extension of the kinetic equation developed by Newell & Zakharov (A.C. Newell and V.E. Zakharov. The role of the generalized Phillips' spectrum in wave turbulence. Phys.Lett.A, 372:4230-4233, 2008). The new equation takes into account not only the resonant four-wave interactions but also the dissipation associated with the wave breaking. A dissipation function that depends on the spectral energy flux is introduced into the equation. This function is determined up to a functional parameter, which optimal choice should be made based on comparison with the experiment. A kinetic equation with this dissipation function describes the transition from the Kolmogorov-Zakharov spectrum to the Phillips spectrum usually observed experimentally. The version of the dissipation function expressed in terms of the energy spectrum can be used for wave modeling and prediction of sea waves.

Alfvén wave collisions, the fundamental building block of plasma turbulence. I. Asymptotic solution

The nonlinear interaction between counterpropagating Alfvén waves is the physical mechanism underlying the cascade of energy to small scales in astrophysical plasma turbulence. Beginning with the equations for incompressible MHD, an asymptotic analytical solution for the nonlinear evolution of these Alfvén wave collisions is derived in the weakly nonlinear limit. The resulting qualitative picture of nonlinear energy transfer due to this mechanism involves two steps: first, the primary counterpropagating Alfvén waves interact to generate an inherently nonlinear, purely magnetic secondary fluctuation with no parallel variation; second, the two primary waves each interact with this secondary fluctuation to transfer energy secularly to two tertiary Alfvén waves. These tertiary modes are linear Alfvén waves with the same parallel wavenumber as the primary waves, indicating the lack of a parallel cascade. The amplitude of these tertiary modes increases linearly with time due to the coherent nature of the resonant four-wave interaction responsible for the nonlinear energy transfer. The implications of this analytical solution for turbulence in astrophysical plasmas are discussed. The solution presented here provides valuable intuition about the nonlinear interactions underlying magnetized plasma turbulence, in support of an experimental program to verify in the laboratory the nature of this fundamental building block of astrophysical plasma turbulence.

A Generalized Multiple Discrete Interaction Approximation for resonant four-wave interactions in wind wave models

For several decades, the Discrete Interaction Approximation (DIA) for nonlinear resonant four-wave interactions has been the engine of third-generation wind-wave models. The present study presents a Generalized Multiple DIA (GMD) which expands upon the DIA by (i) expanding the definition of the representative quadruplet, (ii) formulating the DIA for arbitrary water depths, (iii) providing complimentary deep and shallow water scaling terms and (iv) allowing for multiple representative quadruplets. The GMD is rigorously derived to be an extension of the DIA, and is backward compatible with it. The free parameters of the GMD are optimized holistically, by optimizing full model behavior in the WAVEWATCH III® wave model as reported in a companion paper. Here, a cascade of GMD configurations with increasing complexity, accuracy and cost is presented. First, the performance of these configurations is discussed using idealized test cases used to optimize the GMD. It is shown that in deep water, GMD configurations can be found which remove most of the errors of the DIA. The GMD is also capable of representing four-wave interactions in extremely shallow water, although some remaining spurious behavior makes applications of this part of the GMD less suitable for operational wave models. Finally, several GMD configurations are applied to an idealized hurricane case, showing that results from idealized test cases indeed are representative for real-world applications, and confirming that such GMD configurations are economically feasible in operational wind wave models. Finally, the DIA results in surprisingly large model errors in hurricane conditions.

A conservative nonlinear filter for the high-frequency range of wind wave spectra

Filtering of the high-frequency part of a wind wave spectrum may be useful in a numerical wind wave model for various reasons. First, it can be used to augment (or be part of) a parameterization of the resonant nonlinear interactions, that are essential to third-generation wind wave models. Second, when combined with a dynamic time stepping scheme for source term integration, it may result in smoother (and hence faster) wave model integration. In this study, such a filter is proposed, based on the traditional Discrete Interaction Approximation (DIA) for the resonant four-wave nonlinear interactions. This filter retains all conservative properties of the interactions. For small time steps and/or smooth spectra, it is formulated as a traditional source term. For larger time steps and/or non-smooth spectra it is formulated as a filter. This formulation guarantees stability of the filter itself and will enhance overall computational stability in a full wave model. The stability properties of this filter are illustrated using traditional wave growth computations. Examples are given where the filter improves model economy, and where it is shown to remove spurious high-frequency noise from a wave model.

Validity of the resonant four-wave interaction system in a model for surface water waves on an infinite deep sea

Models for the resonant four-wave interaction play a big role in the description of gravity-driven surface water waves. The question however, if such models, really describe the evolution of surface water waves is a difficult task, according to the fact that the original water wave problem contains quadratic terms and a resonance at the wave vector k = 0 . As a first step in answering the validity question positively we prove here that solutions of a model for surface water waves on an infinite deep sea really can be described by the associated systems for the resonant four wave interaction. We believe that the method developed in this paper applies to the problem of surface water waves on an infinite deep sea, too.

The WRT method for the computation of non-linear four-wave interactions in discrete spectral wave models

An overview is given of the WRT method for the computation of weakly resonant non-linear four-wave interactions in a gravity wave spectrum and its application in discrete spectral wave models. The WRT method is based on Webb's [Webb, D.J., 1978. Nonlinear transfer between sea waves. Deep-Sea Res., 25, 279–298.] transformation of the Boltzmann integral and the numerical method introduced by Tracy and Resio [Tracy, B.A., Resio, D.T., 1982. Theory and calculation of the nonlinear energy transfer between sea waves in deep water. WIS technical report 11. US Army Engineer Waterways Experiment Station, Vicksburg, Mississippi, USA, 47 pp.]. It is shown that Webb's method produces an attractive set of integrable equations. Moreover, the Jacobian term arising from the integration over the frequency delta-function in the Boltzmann integral has a singularity well outside the energy containing part of the wave spectrum. A description is given of methods for computing the integration space for a given discrete spectral grid, both for deep and finite depth water. Thereafter, the application of Webb's method to discrete spectral wave models is described, followed by a summary of techniques reducing the computational workload while retaining sufficient accuracy. Finally, some methods are presented for the optimal inclusion of the WRT method in operational discrete spectral wave prediction models.

Validation of the Theoretical Spectrum of Baroclinic Inertia–Gravity Waves

Abstract Assuming nonlinear dynamics of baroclinic inertia–gravity (BIG) waves to be dominated by resonant four-wave interactions, recent studies yielded simple analytical expressions for the energy spectra. The only parameter remaining unknown is a nondimensional constant of proportionality α, referred to as a “Kolmogorov constant” in wave turbulence theory. The goals of the present study are 1) to validate the theoretical spectrum of the kinetic wave energy by comparing it with observed frequency spectra of horizontal velocities and 2) to obtain the first empirical estimate of α for this model.

Analysis of a class of simplified models for nonlinear gravity wave interactions

Kolmogorov-type cascade processes have long been postulated to explain power-law falloff of the spectrum of gravity wave autocorrelations. In an effort to understand these processes more deeply, simplified models of the nonlinear Hasselmann resonant four-wave interaction have been introduced. These models respect the fundamental structural aspects of that interaction, but are otherwise chosen for computational simplicity. In this paper the concept of quasilocality within the context of such models is extended to its logical limit in one spatial dimension. This results in differential rather than integral equations for the wave fields. One scaling solution to the new equations has a spectral exponent independent of the details of the model interaction, and is stable to small perturbations. In addition, under weak restrictions on the first and second derivatives of the model interactions, a further pair of two-parameter families of scaling solutions exist. The spectral exponents in this case depend on the details of the model, but unstable as well as stable perturbations to these solutions are supported.

Resonant four-wave parametric interactions: Adiabatic formulation.

The density-matrix approach, in the quasistatic or adiabatic limit, is applied to the case of resonant four-wave interactions in four-level systems resulting in explicit analytical expressions for the nonlinear optical parametric and nonparametric polarizations, including ac Stark effects. The results are applicable to any closed-linkage diagram of this type but are applied only to the specific case of resonant frequency tripling. The results revealed the presence of up to six Stark-shifted nonparametric resonances and the presence of saturation and interference effects on the parametric susceptibility. In particular, the third-order susceptibility is shown to saturate to a very small value under high-field conditions which restricts the range of useful pump intensities. Under near-resonant to resonant conditions, these imply that because of either nonparametric loss present at the pump or harmonic frequencies or power-dependent dispersive and saturation effects, conversion efficiencies higher than 25% may be difficult to realize in practice.

Resonant four-wave interaction in the field of pump waves of arbitrary intensity

A theoretical investigation is made of the influence of the asymmetry of the intensities of pump waves on the spectrum of weak radiation participating in a nondegenerate four-wave interaction in a resonant medium with a homogeneously broadened absorption line. It is shown that a difference between the intensities of pump waves results in additional modulation of the spectrum of reflected and transmitted waves. In particular, an increase in the asymmetry of the pump wave intensities increases the phase mismatch between the waves and reduces the central maximum in the weak radiation spectrum. When the difference between the intensities of the pump waves becomes sufficiently large, this central maximum of the reflected radiation disappears completely and the side maxima acquire a doublet structure. It is shown that the position of the maxima in the reflected radiation spectrum corresponds exactly to the minima of the phase matching between the waves. An analysis is made of the characteristics of a reflecting narrow-band optical filter based on a nondegenerate four-wave interaction in a resonant medium.

Self-reversal of the wavefront of high-power radiation in a light-induced four-wave interaction in a resonant medium

A wavefront reversal system is proposed for high-power optical radiation. This system is based on a resonant four-wave interaction without recourse to external pump waves. Stimulated scattering of the wave being reversed in a resonant medium placed inside a resonator creates two opposite pump waves and a parametric interaction between them generates reversed radiation. The efficiency of such self-reversal is comparable with the efficiency of conventional wavefront reversal as a result of a four-wave interaction in nonresonant media with a phase nonlinearity and it may reach 10−3.

Wave-front reversal in the case of resonant four-wave interaction in the field of time-separated reference pulses

Wave-front reversal in the case of resonant four-wave interaction in the field of time-separated reference pulses

Spectral characteristics of light wave reversal in a nondegenerate four-wave interaction in a resonant medium

An investigation is made of wavefront reversal of optical radiation in a nondegenerate four-wave interaction in a resonant medium. It is shown that for a pump intensity comparable with the saturation intensity of the absorption line of the resonant medium, the reversed wave spectrum has two side peaks positioned symmetrically relative to the central peak which corresponds to a degenerate interaction. These side peaks are attributable to regions of negative absorption (amplification) of weak radiation by the resonant medium situated in a high-power field of opposite pump waves. The dependences of the magnitudes and positions of these peaks on the pump intensity can be used to control the spectral and energy characteristics of the reflected radiation. The conditions for self-excitation of a reversing system based on resonant four-wave interaction are determined. In particular, the ranges of the pump intensity and the optical density of the resonant medium in which self-excitation of the reversing system may occur, are determined for a degenerate interaction.

Dependence of the efficiency of resonant four-wave interaction on the frequency difference of the pumping waves

Dependence of the efficiency of resonant four-wave interaction on the frequency difference of the pumping waves

Coupled nonlinear Schrödinger equations for Langmuir and elecromagnetic waves and extension of their modulational instability domain

A pair of coupled nonlinear Schrödinger equations for transverse and longitudinal waves has been derived. The coupling resulting from equalization of group velocities drives the Langmuir waves modulationally unstable for wavelengths shorter than (mi/me)½ λD and also extends the domain of modulational instability of electromagnetic waves when relativistic effects are taken into account. Instability is found to occur also for the perturbation wavenumber domain in which both the uncoupled Langmuir and electromagnetic waves are modulationally stable. This is shown to be caused by resonant four-wave interaction l + t → l′ + t′. The growth rate of the instability is, in general, of the order of but increases to the extent of a factor (c/vth)2 near the resonance Solitary wave solutions are given. Depending on the relative values of the self-modulation and coupling coefficients, the Langmuir or the transverse or both the waves may be localized in space.

Resonant Four-Wave Interaction in a Dispersive Medium

The resonant four-wave interaction is studied for a nonlinear two-dimensional Klein-Gordon system. A set of dynamical equations describing the slowly varying amplitudes is first derived for a spatially homogeneous case. Exact solutions to the set of equations are then obtained. They are classified into the following four types: (i) periodic solutions, (ii-1) solitary pulse solutions, (ii-2) a phase jump and three solitary pulse solutions, (iii) shock-like solutions. Some of these solutions, especially the type (iii) solutions, are quite different from those of the well-known nonlinear three-wave interaction.

Resonant Four-Wave Interaction of Electron-Plasma Oscillations

The time evolution of a uniformly turbulent ensemble of electron fluids is studied in the electrostatic approximation. A kinetic equation for the action density is obtained in the longwavelength limit, in which resonant four-wave processes cause the nonlinear transfer of energy in the oscillation spectrum. It is shown that the entire region of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ space (within the limits of the simple-fluid model) is accessible to resonant four-wave interactions. Such fourwave mode coupling serves as a mechanism for the transfer of wave energy into shorter wavelengths.