Wave Triads Research Articles

Thermal versus mechanical topography: an experimental investigation in a rotating baroclinic annulus

We present a series of experimental investigations in which a differentially-heated annulus was used to investigate the effects of topography on rotating, stratified flows. In particular, we investigate blocking effects via azimuthally varying differential-heating and compare them to previous experiments utilising partial mechanical barriers. The thermal topography used consisted of a flat patch of heating elements covering a small azimuthal extent of the base, forming an equivalent of a partial barrier, to study the difference between blocked and unblocked flow. These azimuthally-varying heating experiments produced results with many similarities to our previous experiments with a mechanical barrier, despite the lack of a physical obstacle or formation of bottom-trapped waves. In particular, a unique flow structure was found when the drifting flow and the topography interacted in the form of an “interference” regime at low Taylor number, but forming an erratic “irregular” regime at higher Taylor number. This suggests that blocking may be induced by either or both of a thermal or mechanical inhomogeneity. Evidence of coherent/persistent resonant wave triads was noted in both kinds of experiment, though the component wavenumbers of the wave-triads and their impact on the flow was found to depend on the topography in question.

Wave Breaking Type as a Typical Sign of Nonlinear Wave Transformation Stage in Coastal Zone

Experimental data demonstrate dependence of wave breaking type on wave amplitude-frequency-phase structure that is determined by nonlinear wave transformation. The spilling and plunging breaking waves are differed in the symmetry of the wave profile and the ratio of the amplitudes of the 1st and 2nd harmonics in breaking wave. The wave profile symmetry, in turn, is determined by the phase shift between these harmonics. In spilling breaking waves, it is close to zero, which corresponds to symmetrical waves. In plunging breaking waves, the phase shift is negative, which corresponds to the forward shifted 2nd harmonic. The ratio of the amplitudes of the 1st and 2nd harmonics in spilling breaking waves is less than that in plunging breaking waves. The periodic energy exchange between harmonics during near resonant triad wave interactions is the reason for the change in the types of wave breaking for waves propagating above a gentle inclined bottom. Due to the different structure, plunging breaking waves contribute to the erosion of the cross-shore underwater profile, while spilling breaking waves contribute to the accumulation of sand on it.

Nonlinear Rossby Wave–Wave and Wave–Mean Flow Theory for Long-term Solar Cycle Modulations

Abstract The Schwabe cycle of solar activity exhibits modulations and frequency fluctuations on slow timescales of centuries and millennia. Plausible physical explanations for the cause of these long-term variations of the solar cycle are still elusive, with possible theories including stochasticity of the alpha effect and fluctuations of the differential rotation. It has been suggested recently in the literature that there exists a possible relation between the spatiotemporal structure of the solar cycle and the nonlinear dynamics of magnetohydrodynamic (MHD) Rossby waves at the solar tachocline, including both wave–wave and wave–mean flow interactions. Here we extend the nonlinear theory of MHD Rossby waves presented in a previous article to take into account long-term modulation effects due to a recently discovered mechanism that allows significant energy transfers throughout different wave triads: the precession resonance mechanism. We have found a large number of Rossby–Hauwirtz wave triads whose frequency mismatches are compatible with the solar cycle frequency. Consequently, by analyzing the reduced dynamics of two triads coupled with a single mode (five-wave system), we have demonstrated that in the amplitude regime in which precession resonance occurs, the energy transfer throughout the system yields significant long-term modulations on the main ∼11 yr period associated with intratriad energy exchanges. We further show that such modulations display an inverse relationship between the characteristic wave amplitude and the period of intratriad energy exchanges, which is consistent with the Waldmeier law for the solar cycle. In the presence of a constant forcing and dissipation, the five-wave system in the precession resonance regime exhibits irregular amplitude fluctuations, with some periods resembling the grand minimum states.

Linear and Weakly Nonlinear Energetics of Global Nonhydrostatic Normal Modes

Abstract Here the theory of global nonhydrostatic normal modes has been further developed with the analysis of both linear and weakly nonlinear energetics of inertia–acoustic (IA) and inertia–gravity (IG) modes. These energetics are analyzed in the context of a shallow global nonhydrostatic model governing finite-amplitude perturbations around a resting, hydrostatic, and isothermal background state. For the linear case, the energy as a function of the zonal wavenumber of the IA and IG modes is analyzed, and the nonhydrostatic effect of vertical acceleration on the IG waves is highlighted. For the nonlinear energetics analysis, the reduced equations of a single resonant wave triad interaction are obtained by using a pseudoenergy orthogonality relation. Integration of the triad equations for a resonance involving a short harmonic of an IG wave, a planetary-scale IA mode, and a short IA wave mode shows that an IG mode can allow two IA modes to exchange energy in specific resonant triads. These wave interactions can yield significant modulations in the dynamical fields associated with the physical-space solution with periods varying from a daily time scale to almost a month long.

Infragravity Wave Generation by Wind Gusts

AbstractThrough the analysis of propagation times of infragravity wave packets along ray paths, reanalysis data, and our field measurements in the East Mediterranean, we find evidence of deep water infragravity wave generation by offshore storms. We confirmed the results also using deep water pressure cell measurements in the Pacific. The known nearshore generation mechanism showed large discrepancies with the observed infragravity energy near Aogashima, Japan, during winter. A new model of deep water infragravity wave generation is developed, based on nonlinear interactions of wind wave triads with submesoscale wind oscillations. The observed underprediction of infragravity waves is resolved using this new gustiness‐based model. The new source term is found to be of importance during strong storms in the open ocean and underlines the importance of accounting for submesoscale wind oscillations in wind wave models.

Parametric interaction of gravity-capillary wave triads under radiation pressure of ultrasound

Nonlinear parametric coupling of gravity-capillary waves (GCW) by ultrasound wave impinging on a liquid surface is studied experimentally. Modulation of the plane wave radiation pressure by dynamically curved surface of the liquid is considered as a mechanism of the coupling. The standing GCW modes are excited near antinodes by ultrasound beam of carrier frequency 1.030 MHz and coupled parametrically by a plane ultrasound wave with a frequency 1.035 MHz. The beam was modulated at resonance frequencies of GCW overtones f2=5.265 Hz and f4=7.873 Hz while intensity of the plane wave was modulated at frequency fp=2.657 Hz that corresponds to the nonlinear parametric resonance fp=2f2-f4. The amplitudes of the parametric interaction of GCW triads are measured and compared with critical condition for GCWs explosive instability.

Effects of following and opposing vertical current shear on nonlinear wave interactions

A Navier-Stokes solver in OpenFOAM® is combined with the Volume of Fluid (VOF) surface capturing method to investigate the wave interaction with depth-varying currents in intermediate and shallow waters. A special attention is paid to the separate effect of vertical current shear on near resonant triad wave interactions. It was found that in the presence of following vertical current shear, the wave exhibits a sharper crest and flatter trough, and the opposite is true in the presence of opposing vertical current shear. Our model results indicate that the wave steepness at which the current shear starts to affect the crest elevation is greater in deeper water than in shallower water. We found that adding vertical current shear to the uniform current further enhances the relative harmonic wave energy and the extent of triad interaction in the following current while weakens them in the opposing current. As a result, following and opposing current shear may cause wave to break at a lower and higher sea state respectively. Due to the increased wave nonlinearity in the presence of a following current shear, a linear superposition of the individual wave and current velocities is no longer adequate to represent the total horizontal velocity close to the free surface.

Time-Reversal Analogy by Nonlinear Acoustic–Gravity Wave Triad Resonance

Time reversal of free-surface water (gravity) waves due to a sudden change in the effective gravity has been extensively studied in recent years. Here, we show that an analogy to time-reversal can be obtained using nonlinear acoustic-gravity wave theory. More specifically, we present a mathematical model for the evolution of a time-reversed gravity wave packet from a nonlinear resonant triad perspective. We show that the sudden appearance of an acoustic mode in analogy to a sudden vertical oscillation of the liquid film, can resonate effectively with the original gravity wave packet causing energy pumping into an oppositely propagating (time-reversed) surface gravity wave of an almost identical shape.

On Bragg resonances and wave triad interactions in two-layered shear flows

The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways: (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the higher-order spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both on the theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of the study of these kinds of problems, to incorporate piecewise constant background shear.

Mixed Rossby–Gravity Wave–Wave Interactions

AbstractMixed triad wave–wave interactions between Rossby and gravity waves are analytically derived using the kinetic equation for models of different complexity. Two examples are considered: initially vanishing linear gravity wave energy in the presence of a fully developed Rossby wave field and the reversed case of initially vanishing linear Rossby wave energy in the presence of a realistic gravity wave field. The kinetic equation in both cases is numerically evaluated, for which energy is conserved within numerical precision. The results are validated by a corresponding ensemble of numerical model simulations supporting the validity of the weak-interaction assumption necessary to derive the kinetic equation. Since they are generated by nonresonant interactions only, the energy transfers toward the respective linear wave mode with vanishing energy are small in both cases. The total generation of energy of the linear gravity wave mode in the first case scales to leading order as the square of the Rossby number in agreement with independent estimates from laboratory experiments, although a part of the linear gravity wave mode is slaved to the Rossby wave mode without wavelike temporal behavior.

Explosive Instability of Gravity-Capillary Waves under Ultrasound Radiation Pressure

The parametric interaction of ultrasonic and gravity-capillary waves (GCWs) on a liquid surface has been investigated. Harmonic modulation of the intensity of a plane ultrasonic wave incident on the liquid surface from the depth provides nonlinear parametric coupling of GCW wave triads. We determined the resonance conditions, providing spatiotemporal synchronism of interacting waves, and the threshold ultrasound intensities, at which an explosive-type instability develops in the system of standing and traveling GCWs.

Effect of Dynamical Phase on the Resonant Interaction Among Tsunami Edge Wave Modes

Different modes of tsunami edge waves can interact through nonlinear resonance. During this process, edge waves that have very small initial amplitude can grow to be as large or larger than the initially dominant edge wave modes. In this study, the effects of dynamical phase are established for a single triad of edge waves that participate in resonant interactions. In previous studies, Jacobi elliptic functions were used to describe the slow variation in amplitude associated with the interaction. This analytical approach assumes that one of the edge waves in the triad has zero initial amplitude and that the combined phase of the three waves φ = θ1 + θ2 − θ3 is constant at the value for maximum energy exchange (φ = 0). To obtain a more general solution, dynamical phase effects and non-zero initial amplitudes for all three waves are incorporated using numerical methods for the governing differential equations. Results were obtained using initial conditions calculated from a subduction zone, inter-plate thrust fault geometry and a stochastic earthquake slip model. The effect of dynamical phase is most apparent when the initial amplitudes and frequencies of the three waves are within an order of magnitude. In this case, non-zero initial phase results in a marked decrease in energy exchange and a slight decrease in the period of the interaction. When there are large differences in frequency and/or initial amplitude, dynamical phase has less of an effect and typically one wave of the triad has very little energy exchange with the other two waves. Results from this study help elucidate under what conditions edge waves might be implicated in late, large-amplitude arrivals.

A numerical investigation on nonlinear transformation of obliquely incident random waves on plane sloping bottoms

The statistical properties of obliquely incident irregular waves over a planar sloping bottom were investigated numerically by the well-known numerical wave model FUNWAVE 2.0. Irregular waves based on the averaged JONSWAP spectra with various wave heights and peak periods were simulated to propagate over planar bottoms. A wide range of incident angles from 0° though 75° were considered to study the influence of incident angles. It was found that incident angles have a significant influence on the wave nonlinearity. The wavelet-based bicoherence revealed that the degree of triad wave interactions of primary waves and the higher harmonics was apparently weakened by increasing wave incident angles. Importantly, using the simulated data, the Klopman wave height distribution was improved by incorporating the influence of obliquely incident angles. It was found that the improved Klopman distribution shows better performance for describing wave height distributions in shallow water depth. Moreover, two empirical formulae are recommended to reflect the relationship between the skewness and the asymmetry of waves and the Ursell number for obliquely incident waves on plane sloping bottoms.

Wavemaker theories for acoustic–gravity waves over a finite depth

Acoustic–gravity waves (hereafter AGWs) in ocean have received much attention recently, mainly with respect to early detection of tsunamis as they travel at near the speed of sound in water which makes them ideal candidates for early detection of tsunamis. While the generation mechanisms of AGWs have been studied from the perspective of vertical oscillations of seafloor and triad wave–wave interaction, in the current study, we are interested in their generation by wave–structure interaction with possible implication to the energy sector. Here, we develop two wavemaker theories to analyse different wave modes generated by impermeable (the classic Havelock’s theory) and porous (porous wavemaker theory) plates in weakly compressible fluids. Slight modification has been made to the porous theory so that, unlike the previous theory, the new solution depends on the geometry of the plate. The expressions for three different types of plates (piston, flap, and delta-function) are introduced. Analytical solutions are also derived for the potential amplitudes of the gravity, acoustic–gravity, evanescent waves, as well as the surface elevation, velocity distribution, and pressure for AGWs. Both theories reduce to previous results for incompressible flow when the compressibility is neglected. We also show numerical examples for AGWs generated in a wave flume as well as in deep ocean. Our current study sets the theoretical background towards remote sensing by AGWs, for optimised deep ocean wave-power harnessing, among others.

The Arithmetic Geometry of Resonant Rossby Wave Triads

Linear wave solutions to the Charney--Hasegawa--Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface $X$. The set of all resonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our work independently finds all resonant triads via a rational parametrization of $X$. We provide a fiberwise description of $X$ as a rational singular elliptic surface, yielding many new results about the set of wavevectors belonging to resonant triads. In particular, we show there is an infinite number of resonant triads (with relatively prime wavevectors) containing a wavevector $(a,b)$ with $a/b=r$, where $r$ is any given rationa...

Source term balance in a severe storm in the Southern North Sea

This paper presents the results of a wave hindcast of a severe storm in the Southern North Sea to verify recently developed deep and shallow water source terms. The work was carried out in the framework of the ONR funded NOPP project (Tolman et al. 2013) in which deep and shallow water source terms were developed for use in third-generation wave prediction models. These deep water source terms for whitecapping, wind input and nonlinear interactions were developed, implemented and tested primarily in the WAVEWATCH III model, whereas shallow water source terms for depth-limited wave breaking and triad interactions were developed, implemented and tested primarily in the SWAN wave model. So far, the new deep-water source terms for whitecapping were not fully tested in shallow environments. Similarly, the shallow water source terms were not yet tested in large inter-mediate depth areas like the North Sea. As a first step in assessing the performance of these newly developed source terms, the source term balance and the effect of different physical settings on the prediction of wave heights and wave periods in the relatively shallow North Sea was analysed. The December 2013 storm was hindcast with a SWAN model implementation for the North Sea. Spectral wave boundary conditions were obtained from an Atlantic Ocean WAVEWATCH III model implementation and the model was driven by hourly CFSR wind fields. In the southern part of the North Sea, current and water level effects were included. The hindcast was performed with five different settings for whitecapping, viz. three Komen type whitecapping formulations, the saturation-based whitecapping by Van der Westhuysen et al. (2007) and the recently developed ST6 whitecapping as described by Zieger et al. (2015). Results of the wave hindcast were compared with buoy measurements at location K13 collected by the Dutch Ministry of Transport and Public Works. An analysis was made of the source term balance at three locations, the deep water location North Cormorant, the inter-mediate depth location K13 and at location Wielingen, a shallow water location close to the Dutch coast. The results indicate that at deep water the source terms for wind input, whitecapping and nonlinear four-wave interactions are of the same magnitude. At the inter-mediate depth location K13, bottom friction plays a significant role, whereas at the shallow water location Wielingen also depth-limited wave breaking becomes important.

Parameter sensitivity and uncertainty analysis for a storm surge and wave model

Abstract. Development and simulation of synthetic hurricane tracks is a common methodology used to estimate hurricane hazards in the absence of empirical coastal surge and wave observations. Such methods typically rely on numerical models to translate stochastically generated hurricane wind and pressure forcing into coastal surge and wave estimates. The model output uncertainty associated with selection of appropriate model parameters must therefore be addressed. The computational overburden of probabilistic surge hazard estimates is exacerbated by the high dimensionality of numerical surge and wave models. We present a model parameter sensitivity analysis of the Delft3D model for the simulation of hazards posed by Hurricane Bob (1991) utilizing three theoretical wind distributions (NWS23, modified Rankine, and Holland). The sensitive model parameters (of 11 total considered) include wind drag, the depth-induced breaking γB, and the bottom roughness. Several parameters show no sensitivity (threshold depth, eddy viscosity, wave triad parameters, and depth-induced breaking αB) and can therefore be excluded to reduce the computational overburden of probabilistic surge hazard estimates. The sensitive model parameters also demonstrate a large number of interactions between parameters and a nonlinear model response. While model outputs showed sensitivity to several parameters, the ability of these parameters to act as tuning parameters for calibration is somewhat limited as proper model calibration is strongly reliant on accurate wind and pressure forcing data. A comparison of the model performance with forcings from the different wind models is also presented.

Consistent nonlinear stochastic evolution equations for deep to shallow water wave shoaling

Nonlinear interactions between sea waves and the sea bottom are a major mechanism for energy transfer between the different wave frequencies in the near-shore region. Nevertheless, it is difficult to account for this phenomenon in stochastic wave forecasting models due to its mathematical complexity, which mostly consists of computing either the bispectral evolution or non-local shoaling coefficients. In this work, quasi-two-dimensional stochastic energy evolution equations are derived for dispersive water waves up to quadratic nonlinearity. The bispectral evolution equations are formulated using stochastic closure. They are solved analytically and substituted into the energy evolution equations to construct a stochastic model with non-local shoaling coefficients, which includes nonlinear dissipative effects and slow time evolution. The nonlinear shoaling mechanism is investigated and shown to present two different behaviour types. The first consists of a rapidly oscillating behaviour transferring energy back and forth between wave harmonics in deep water. Owing to the contribution of bottom components for closing the class III Bragg resonance conditions, this behaviour includes mean energy transfer when waves reach intermediate water depths. The second behaviour relates to one-dimensional shoaling effects in shallow water depths. In contrast to the behaviour in intermediate water depths, it is shown that the nonlinear shoaling coefficients refrain from their oscillatory nature while presenting an exponential energy transfer. This is explained through the one-dimensional satisfaction of the Bragg resonance conditions by wave triads due to the non-dispersive propagation in this region even without depth changes. The energy evolution model is localized using a matching approach to account for both these behaviour types. The model is evaluated with respect to deterministic ensembles, field measurements and laboratory experiments while performing well in modelling monochromatic superharmonic self-interactions and infra-gravity wave generation from bichromatic waves and a realistic wave spectrum evolution. This lays physical and mathematical grounds for the validity of unexplained simplifications in former works and the capability to construct a formulation that consistently accounts for nonlinear energy transfers from deep to shallow water.

Excitation of superharmonics by internal modes in non-uniformly stratified fluid

Theory and numerical simulations show that the nonlinear self-interaction of internal modes in non-uniform stratification results in energy being transferred to superharmonic disturbances forced at twice the horizontal wavenumber and frequency of the parent mode. These disturbances are not in themselves a single mode, but a superposition of modes such that the disturbance amplitude is largest where the change in the background buoyancy frequency with depth is largest. Through weakly nonlinear interactions with the parent mode, the disturbances evolve to develop vertical-scale structures that distort and modulate the parent mode. Because pure resonant wave triads do not exist in non-uniformly stratified fluid, parametric subharmonic instability (PSI) is not evident even though noise is superimposed upon the initial state. The results suggest a new mechanism for energy transfer to dissipative scales (from large to small vertical scale and with frequencies larger and smaller than that of the parent mode) through forcing superharmonic rather than subharmonic disturbances.

Equilibrium distribution of the wave energy in a carbyne chain

The steady-state energy distribution of thermal vibrations at a given ambient temperature has been investigated based on a simple mathematical model that takes into account central and noncentral interactions between carbon atoms in a one-dimensional carbyne chain. The investigation has been performed using standard asymptotic methods of nonlinear dynamics in terms of the classical mechanics. In the first-order nonlinear approximation, there have been revealed resonant wave triads that are formed at a typical nonlinearity of the system under phase matching conditions. Each resonant triad consists of one longitudinal and two transverse vibration modes. In the general case, the chain is characterized by a superposition of similar resonant triplets of different spectral scales. It has been found that the energy equipartition of nonlinear stationary waves in the carbyne chain at a given temperature completely obeys the standard Rayleigh–Jeans law due to the proportional amplitude dispersion. The possibility of spontaneous formation of three-frequency envelope solitons in carbyne has been demonstrated. Heat in the form of such solitons can propagate in a chain of carbon atoms without diffusion, like localized waves.