Resonant Wave-wave Interactions Research Articles

Turbulence spreading by resonant wave-wave interactions: A fractional kinetics approach.

This paper is concerned with the processes of spatial propagation and penetration of turbulence from the regions where it is locally excited into initially laminar regions. The phenomenon has come to be known as "turbulence spreading" and witnessed a renewed attention in the literature recently. Here, we propose a comprehensive theory of turbulence spreading based on fractional kinetics. We argue that the use of fractional-derivative equationspermits a general approach focusing on fundamentals of the spreading process regardless of a specific turbulence model and/or specific instability type. The starting point is the Hamiltonian of resonant wave-wave interactions, from which a family of scaling laws for the asymptotic spreading is derived. Both three- and four-wave interactions are considered. The results span from a subdiffusive spreading in the parameter range of weak chaos to avalanche propagation in regimes with population inversion. Attention is paid to how nonergodicity introduces weak mixing, memory and intermittency into spreading dynamics, and how the properties of non-Markovianity and nonlocality emerge from the presence of islands of regular dynamics in phase space. Also we resolve an existing question concerning turbulence spillover into gap regions, where the instability growth is locally suppressed, and show that the spillover occurs through exponential (Anderson-like) localization in case of four-wave interactions and through an algebraic (weak) localization in case of triad interactions. In the latter case an inverse-cubic behavior of the spillover function is found. Wherever relevant, we contrast our findings against the available observational and numerical evidence, and we also commit ourselves to establish connections with the models of turbulence spreading proposed previously.

Numerical Investigation on Temporal Evolution Behavior for Triad Resonant Interaction Induced by Steady Free-Surface Flow over Rippled Bottoms

Investigating the wave hydrodynamics of free-surface flow over rippled bottoms is a continuing concern due to the existence of submarine multiple sandbars and ambient flow in coastal and estuarial areas. More attention to free-surface wave stimulation has been received from the perspective of resonant wave-wave interaction, which is an intensive way for wave energy transfer and a potential way for wave component generation. However, the basic behavior of the triad resonant interaction of this problem is still limited and unclear. In this study, the triad resonant interaction induced by steady free-surface flow over rippled bottoms is numerically investigated by means of the High-Order Spectral (HOS) method. By considering the interactions among free-surface waves, ambient current, and rippled bottoms, the numerical model is applied for this situation based on Zakharov equation with ambient flow term. The temporal evolution of the triad resonant wave amplitude has been numerically investigated and compared well with multiple-scale expansion perturbation theory. Specifically, the temporal evolution behaviors of all six triad resonant wave components are confirmed by both numerical simulation and nonlinear perturbation analysis.

Monte Carlo Simulation of Random Waves at Finite Depth Based on the Joint Distribution of Wave Amplitude and its Associated Period in Nonlinear Random Waves of Finite Bandwidth

This study aims to develop a new methodology to generate random waves from a joint distribution of wave amplitude and its associated period. In doing so, the wave spectrum was derived from the joint distribution of wave amplitude and associated period and compared with the Wallops spectrum, which has been most frequently referred to in the literature. Numerical simulation shows that the nonlinearity of random waves makes a significant difference in the wave spectrum. These differences are summarized by a somewhat excessive peak period of the semi-empirical wave spectrum and underestimating wave energy over the high-frequency band. These behavioral characteristics of wave spectrum derived from the joint distribution of wave amplitude and its associated period are conceived to be due to the intrinsic property of the nonlinear random waves. The rationale of this reasoning can be found in the fact that non-negligible amount of wave energy shifts to the relatively short and long period band due to the resonant wave-wave interaction as nonlinearity is getting more pronounced as widely known in the coastal engineering community after studies of Hasselmann(1967) and Phillips(1980) on the developments of wind waves. Among the differences mentioned above, an excessively high peak period can be fully predicted by recalling that the wave period is getting shortened due to the phase difference between the free mode waves among the higher-order harmonic components appearing in random waves due to resonant wave-wave interaction and the resulting destructive interaction.

On the Effects of Non-Gaussian Wave-Slope Distribution on the Failure Probability of an Armour block of Rubble Mound Breakwater

This study examined the effect of non-Gaussian wave slope distribution on the failure probability of an armoring rock of rubble mound breakwater using numerical simulation. Numerical simulation was carried out using the Van der Meer equation and Level III reliability design method based on the Monte Carlo simulation. In doing so, modified Glukhovskiy wave height distribution and situ wave height data collected at Ulleungdo from January 1, 1979-December 31, 2019 were used. Tri-modal Gaissuian wave slope distribution was also used, which showed good agreements with situ wave data collected at Mangbang from April 26, 2017, to April 20, 2018. The probability coefficients of tri-modal Gaussian distribution were estimated using Matlab-based statistics and machine learning toolbox, MLE [Maximum Likelihood Estimates]. In the numerical simulation, the intrinsic limitations of Gaussian distribution were revealed, such as imposing a non-negligible probability mass even in the negative wave slope, under-shooting in longer and shorter waves, and over-shooting in mid-scale waves. In the case of failure probability of an armoring rock, Gaussian distribution was shown to give underestimated failure probability. The extent of underestimation was more considerable at Mangbang, where a tidal terrace of lower depth and gently varying slope was developed than at Ulleungdo. These differences were triggered by the presence of probability plateau formed in wave slope distribution whenever infra-gravity waves appear in random wave packet due to the resonant wave-wave interaction. Therefore, it can be easily conceivable that probability plateau has a significant effect on the armoring rock failure probability. Therefore, PIANC's recommendation that wave slope follows Gaussian distribution needs to be amended. Moreover, the reliability-based design of breakwater should be implemented based on tri-modal Gaussian wave slope distribution, which can accurately reproduce probability plateau.

Near-inertial dissipation due to stratified flow over abyssal topography

AbstractLinear theory for steady stratified flow over topography sets the range for topographic wavenumbers over which freely propagating internal waves are generated, and the radiation and breaking of these waves contribute to energy dissipation away from the ocean bottom. However, previous numerical work demonstrated that dissipation rates can be enhanced by flow over large scale topographies with wavenumbers outside of the lee wave radiative range. We conduct idealized 3D numerical simulations of steady stratified flow over 1D topography in a rotating domain and quantify vertical distribution of kinetic energy dissipation. We vary two parameters: the first determines whether the topographic obstacle is within the lee wave radiative range and the second, proportional to the topographic height, measures the degree of flow non-linearity. For certain combinations of topographic width and height, breaking occurs in pulses every inertial period, such that kinetic energy dissipation develops inertial periodicity. In these simulations, kinetic energy dissipation rates are also enhanced in the interior of the domain. In the radiative regime the inertial motions arise due to resonant wave-wave interactions. In the small wavenumber non-radiative regime, instabilities downstream of the obstacle can facilitate the generation and propagation of non-linearly forced inertial motions, especially as topographic height increase. In our simulations, dissipation rates for tall and wide non-radiative topography are comparable to those of radiative topography, even away from the bottom, which is relevant to the ocean where the topographic spectrum is such that wider abyssal hills also tend to be taller.

Exact discrete resonances in the Fermi-Pasta-Ulam–Tsingou system

In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the exchange of energy between normal modes. In the weakly nonlinear regime, those interactions may be responsible for the equipartition of energy in Fourier space. Here we consider analytically resonant wave-wave interactions for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) system. Using a number-theoretical approach based on cyclotomic polynomials, we show that the problem of finding exact resonances for a system of N particles is equivalent to a Diophantine equation whose solutions depend sensitively on the number of particles N, in particular on the set of divisors of N. We provide an algorithm to construct all possible resonances for N particles, based on two basic methods: pairing-off and cyclotomic, which we introduce and use to build up explicit solutions to the 4-wave, 5-wave and 6-wave resonant conditions. Our results shed some light in the understanding of the long-standing FPUT paradox, regarding the sensitivity of the resonant manifolds with respect to the number of particles N and the corresponding time scale of the interactions leading to an eventual thermalisation. In this light we demonstrate that 6-wave resonances always exist for any N, while 5-wave resonances exist if N is divisible by 3 and N ⩾ 9. It is known that in the discrete case 4-wave resonances do not produce energy mixing across the spectrum, so we investigate whether 5-wave resonances can produce energy mixing across a significant region of the Fourier spectrum by looking at the structure of the interconnected network of Fourier modes that can interact nonlinearly via resonances. We obtain that the answer depends on the set of odd divisors of N which are not divisible by 3: the size of this set determines the number of dynamically independent components, corresponding to independent constants of motion (energies). We show that 6-wave resonances connect all these independent components, providing in principle a restoring mechanism for full-scale thermalisation.

Route to thermalization in the α-Fermi–Pasta–Ulam system

We study the original α-Fermi-Pasta-Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ(8), where ϵ is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

On the nonlinear shaping mechanism for gravity wave spectrum in the atmosphere

Abstract. The nonlinear mechanism of shaping of a high vertical wave number spectral tail in the field of a few discrete internal gravity waves in the atmosphere is studied in this paper. The effects of advection of fluid parcels by interacting gravity waves are taken strictly into account by calculating wave field in Lagrangian variables, and performing a variable transformation from Lagrangian to Eulerian frame. The vertical profiles and vertical wave number spectra of the Eulerian displacement field are obtained for both the case of resonant and non-resonant wave-wave interactions. The evolution of these spectra with growing parameter of nonlinearity of the internal wave field is studied and compared to that of a broad band spectrum of gravity waves with randomly independent amplitudes and phases. The calculated vertical wave number spectra of the vertical displacements or relative temperature fluctuations are found to be consistent with the observed spectra in the middle atmosphere.

Nonlinear wave-wave interactions and wedge waves

A tetrad mechanism for exciting long waves, for example edge waves, is described based on nonlinear resonant wave-wave interactions. In this mechanism, resonant interactions pass energy to an edge wave, from the three participating gravity waves. The estimated action flux into the edge wave can be orders of magnitude greater than the transfer fluxes derived from other competing mechanisms, such as triad interactions. Moreover, the numerical results show that the actual transfer rates into the edge wave from the three participating gravity waves are two-to three-orders of magnitude greater than bottom friction.

High frequency ion sound waves associated with Langmuir waves in type III radio burst source regions

Abstract. Short wavelength ion sound waves (2-4kHz) are detected in association with the Langmuir waves (~15-30kHz) in the source regions of several local type III radio bursts. They are most probably not due to any resonant wave-wave interactions such as the electrostatic decay instability because their wavelengths are much shorter than those of Langmuir waves. The Langmuir waves occur as coherent field structures with peak intensities exceeding the Langmuir collapse thresholds. Their scale sizes are of the order of the wavelength of an ion sound wave. These Langmuir wave field characteristics indicate that the observed short wavelength ion sound waves are most probably generated during the thermalization of the burnt-out cavitons left behind by the Langmuir collapse. Moreover, the peak intensities of the observed short wavelength ion sound waves are comparable to the expected intensities of those ion sound waves radiated by the burnt-out cavitons. However, the speeds of the electron beams derived from the frequency drift of type III radio bursts are too slow to satisfy the needed adiabatic ion approximation. Therefore, some non-linear process such as the induced scattering on thermal ions most probably pumps the beam excited Langmuir waves towards the lower wavenumbers, where the adiabatic ion approximation is justified.

Brief Review of the Turbulent Bremsstrahlung (Plasma-Maser) Effect

Because of nonlinear interaction between particles and waves, energy conversion between waves of large frequency difference can occur without particle population inversion or resonant wave-wave interaction. This turbulent bremsstahlung (plasma-maser) effect is important in determining the transport properties of weakly turbulent plasmas. Here recent advances in the theory of the plasma-maser effect are briefly reviewed.

Altimeter observations of baroclinic oceanic intertia–gravity wave turbulence

For a wide range of nonlinear wave processes - from capillary to planetary waves - theory predicts the existence of Kolmogorov-type spectral cascades of energy and other conserved quantities occuring via nonlinear resonant wave-wave interactions. So far, observations of wave turbulence (WT) have been limited to small-scale processes such as surface gravity and capillary-gravity waves.

On internal wave–shear flow resonance in shallow water

The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The study is focused on the most intense resonant interaction occurring when the phase velocity of internal waves matches the flow velocity at the surface. The perturbations of the shear flow are considered as ‘vorticity waves’, which enables us to treat the wave–flow resonance as the resonant wave–wave interaction between an internal gravity mode and the vorticity mode. Within the weakly nonlinear long-wave approximation a system of evolution equations governing the nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. At resonance the nonlinearity of the internal wave dynamics is due to the interaction with the vorticity mode, while the wave's own nonlinearity proves to be negligible. The equations derived are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the ‘fast’ solitary waves are limited from above; the crest of the limiting wave forms a sharp corner. The solitary waves of amplitude smaller than a certain threshold are shown to be stable; ‘subcritical’ localized pulses tend to such solutions. The localized pulses of amplitude exceeding this threshold form infinite slopes in finite time, which indicates wave breaking.

A Third-Generation Model for Wind Waves on Slowly Varying, Unsteady, and Inhomogeneous Depths and Currents

Abstract A full discrete spectral model for propagation generation and dissipation of wind waves for arbitrary depth, current and wind fields is presented (WAVEWATCH). This model incorporates all relevant wave-current interaction mechanisms including changes of absolute frequencies due to unsteadiness of depth and currents. The model furthermore explicitly accounts for growth and decay of wave energy and for nonlinear resonant wave-wave interactions. The numerical schemes for propagation are basically second-order accurate. Effect of refraction and frequency shifts (due to unsteadiness of depth and current) are calculated on a fixed grid, also using second-order schemes. This paper focuses on the governing equations and the numerical algorithms. Furthermore some results for academic and realistic cases are presented to illustrate some features and merits of the model.

Very large phase velocities of non two‐stream, meter scale irregularities in the high latitude E region ionosphere

We present experimental evidence on a new category of meter wavelength radio auroral irregularities characterized by a unique Doppler spectrum signature. The new echo type has been identified with the Scandinavian Twin Auroral Radar Experiment radars which are capable of observing simultaneously full Doppler spectra from the same scattering volume along two widely different azimuths. The new echoes were seen only by the Finland radar which observes at large angles to the electrojet flow and at times when the Norway radar observed very stable type 1 echoes with phase velocities near the ion acoustic speed. The new spectrum, named type 5, is composed of three main frequency bands, two being symmetric about zero Doppler shift near ±450 m/s and a third always at very large phase velocities near 1000 m/s, this resulting to a characteristic asymmetry in the spectrum. The Doppler polarity of the large phase velocity component is always at negative shifts in the eastward electrojet and at positive shifts in the westward electrojet. These echoes, which are not strong and last from a few minutes to tens of minutes, were observed during strong but also moderate geomagnetic conditions but, apparently, always in the presence of large and stable electric fields. The evidence suggested that the new echoes cannot be due to direct generation by the two‐stream instability and, most likely, a secondary generation mechanism is involved. Although there are some similarities with 50 MHz type 4 spectral observations, type 5 echoes have unique characteristics which we believe are sufficient to justify their classification into a different echo category. We have no satisfactory theoretical explanation for the new echoes. That the fast moving irregularities have frequencies about two times the frequency of the simultaneously observed ion acoustic waves led us to suggest that a nonlinear resonant wave‐wave interaction process might be in operation under some appropriate conditions in the plasma.

Resonant Wave-Wave Interactions of Electromagnetic and Electrostatic Waves in a Homogeneous Magnetized Plasma

General expressions of the matrix elements for nonlinear resonant wave-wave interactions of electromagnetic and electrostatic waves in a homogeneous magnetized plasma are derived from the Vlasov-Maxwell equations. The kinetic wave equations obtained for electromagnetic waves are represented by three-order tensors in the rotating and Cartesian coordinates. The waves are allowed to propagate at arbitrary angles to the uniform magnetic field, and no restrictions are imposed upon the mean Larmor radius, the frequencies, or the wave numbers. By electrostatic approximation of the dielectric tensor and the matrix elements the kinetic equations can be also applicable to the case in which the three waves are electrostatic altogether or partially electrostatic. The matrix elements in the limit of parallel or perpendicular propagation to the magnetic field are given.

Stochastic equations arising from test particle problems: III. Stability of randomly driven oscillators

As a model for resonant wave-wave interaction three nonlinearly coupled oscillators in a harmonic lattice are considered. Using a stochastic description the influence of the nonresonant oscillators on the energy transfer is studied in the weak coupling limit. Conditions for amplitude and energy instability and the stationary solutions are discussed.

On the Effects of Finite Depth on Wind-Wave Spectra: 1. A Comparison with Deep-Water Equilibrium-Range Slope and Other Spectral Parameters

Spectral parameters calculated from wind-wave measurements in a finite-depth restricted-fetch estuary are compared with similar deep-water parameters. The equilibrium range of these finite-depth spectral data seems to be fitted more satisfactorily by the −3 slope predicted for constant depth by Kitaigorodskii et al. (1975) and measured for shoaling waves by Thornton (1977). Non-dimensional effective-fetch x˜e appears to be the parameter of choice for use in displaying other scaled spectral data (like wave energy ε and peak frequency ν) because it reconciles differences in ε and ν data for short (5–7 km) and long (20–42 km) fetches without having to alter the ε and ν data, but the results also suggest that using fetch as a scaling parameter may not be satisfactory. Finite-depth effects were clearly shown in the ε-x˜e data (the slope of the power-law relation was significantly larger than for deep-water relations) and in the ν-x˜e data [the slope was between the relations of Phillips (1977), Ross (1978) and Liu and Ross (1980), but well above these power-law lines]. There was remarkable agreement between this study's finite-depth ε-ν equilibrium data and the relations of Ross (1978) and Liu and Ross (1980) when kph was 0(1) or greater, with the largest departure when kph≤0.7 (where kp is the wavenumber associated with the spectral peak). In addition to the expected restriction to wave growth by bottom dissipation, refraction and shoaling, there is evidence in the data supporting the calculations of Hasselmann and Hasselmann (1980) that show that resonant wave-wave interaction cross-spectral transfer rates for finite-depth waves increase rapidly above the deep-water rates when kph<O(1), which may help explain the departure from the deep-water power-law relations discussed above.

A Parametric Wave Prediction Model

Measurements of fetch-limited wave spectra from various sources indicate an approximate invariance of the normalized spectral shape with fetch. It has been suggested from investigations of the spectral energy balance that this can be explained by the shape-stabilizing influence of nonlinear resonant wave-wave interactions, which are also responsible for the migration of the spectral peak to lower frequencies. Analyses of a series of further data sets obtained under non-uniform, non-stationary wind conditions show that the invariance of the spectral shape is not restricted to uniform-wind, fetch-limited situations, but applies generally for a growing wind sea. The observed shape invariance is exploited in a wave prediction model by projecting the full transport equation for the two-dimensional spectral continuum onto two variables characterizing the energy and frequency scales of the spectrum. Inspection of the resultant equations reveals further simplifications, enabling the system to be reduced to a single prediction equation for one scale variable, the peak frequency. This is feasible because of the rapid adjustment of the spectrum to a quasi-equilibrium level in which the atmospheric input is balanced by the nonlinear transfer of energy out of the central region of the spectrum to higher and lower frequencies. The balance occurs sufficiently rapidly to be treated as a local response process, thereby providing a relation between the energy level of the spectrum (characterized, for example, by Phillip' constant α), the peak frequency fm, and the local wind speed U (the latter two occurring only in the non-dimensional combination ν=Ufm/g). The directional distribution of the wave spectrum is also established locally and can be regarded as a given function of the non-dimensional frequency f/fm and ν. For the remaining independent scale parameter, the peak frequency, the dominant source term in the transport equation is determined by the nonlinear energy transfer, which can be computed rigorously. To lowest order, the one-parameter wave model is independent of the relative contributions of the atmospheric input and dissipation in the central region of the spectrum. However, because of lack of (consistent) direct measurements of the atmospheric input or dissipation, the quasi-equilibrium relation inferred between α and ν must be calibrated empirically, for example, by comparison with fetch-limited data. Within the scatter of the data, all data acts analyzed (with two exceptions, where the data were considered questionable) were reasonably consistent with a common α–ν relation. The residual scatter of the data is thought to be associated largely with small (sub-grid) scale inhomogeneities of the wind field and may represent a natural limitation of the accuracy achievable with deterministic wave models. A complete wave model would need to combine the proposed parametric model for growing wind seas with a swell propagation model.

Trapping of electromagnetic waves by nonlinear resonant interactions with ion sound waves

Resonant wave-wave interaction among one ion sound wave and two electro-magnetic waves in an isotropic plasma is studied. The emphasis is on the possibility of trapping the electromagnetic wave. Equations for the three-wave system are derived. One particularly interesting case is that for which the frequency of ion sound wave is much less than the frequency of electromagnetic waves. For this case it is shown that energy exchange takes place only between the two high frequency waves. The ion sound wave does not participate in the energy exchange process but acts as a kind of catalyst for the interaction. Simple solutions are obtained. It is found that the electromagnetic energy is trapped within a certain spatial region. The trapping width is found to depend, among other parameters, on the magnitude of ion sound wave perturbation. Possible application of the theory to topside ionospheric observations of field-aligned propagation is discussed.