Linear Wave Field Research Articles

Interpretation of the spectral wave forecast model results using the phase-resolving model

The paper presents an interpretation of the results of spectral wave-forecast model using the phase-resolving model. Spectral models provide the information on the evolution of the potential energy distribution in terms of angle and frequency though the information about the geometry and statistical wave characteristics in such models are not available. This information has to be extracted through the additional, often unsubstantiated, hypotheses. The proposed computational procedure transforms spectral information into a two-dimensional wave field which consists of a set of linear modes with randomly distributed phases is proposed. The wave field is not realistic since it does not have non-linear properties, for example, various asymmetry properties such as increased kurtosis. Afterwards the linear wave field reproduced on the basis of the wave spectrum is set as the initial condition for the exact phase-resolving model. The exact models formally suitable for such calculations are cumbersome and inefficient and that practically restricts their broad and regular application. This restriction can be overcome by using a new type of 3D wave simulation based on 2D equations. The 2D model reproduces the statistical characteristics of the wave field similar to the results of the 3D exact model and runs several times faster. The examples of using the developed method of interpretation of the spectral wave forecast in the Baltic Sea are demonstrated.

Second-order near trapping of water waves by a square array of vertical columns in bi-chromatic seas

Near trapping of water waves by an array of closely spaced columns is a hydrodynamic phenomenon with practical applications. A numerical investigation is conducted for the second-order interaction of bi-chromatic waves with a square array of vertical columns. The cross-sectional shape of the columns can be either circular or square with rounded corners. Numerical results indicate that a four-column cluster can amplify free-surface elevation through a near-trapping mechanism. Three lower-order near-trapped modes, namely p=0, 2 and 3 modes, are primarily concerned, among which the amplification corresponding to the p=2 mode is most pronounced. In addition, the four-column cluster can create both the linear and second-order near trappings. The latter could occur when the frequency sum of a pair of incident waves is consistent with the near-trapping frequency. The linear and second-order near-trapped modes own common characteristics. The second-order near-trapping response is predominated by the ‘potential wave’ component, while the ‘quadratic wave’ component plays a secondary role. Meanwhile, the effect of frequency difference on the mode shape in the second order is not evident. It is also noted that a decrease in the corner ratio can enhance the hydrodynamic interference effect among the columns and, in turn, evidently promote the near-trapping response for the p=2 mode. An approximate approach is developed for the second-order near-trapping response, in which the ‘quadratic wave’ component is evaluated exactly. In contrast, the complicated ‘potential wave’ component corresponding to a pair of incident waves with distinct frequencies is approximated by that in mono-chromatic seas with the mean frequency. This approach is based on the hypothesis of weakly scattering of linear wave fields. Numerical results indicate that such approximation offers a reasonable alternative for the near-trapped modes in the low-frequency region.

On the Functionality of Radar and Laser Ocean Wave Sensors

Ocean wave design criteria are required for the design of offshore platforms and floating systems, which are derived using in situ measurements. However, there is uncertainty regarding the performance of the instruments used for the in situ measurements. The main instruments used by the offshore industry are the Datawell Directional Waverider buoy and Rosemount WaveRadar, with Laser instruments also having been used for specific studies. Recent reports indicate measurements from these three instruments differ in the order of 10% but given the quite disparate nature of the measurements made by these instruments, it is far from clear what the source of this difference is. This paper investigates the wave measurement principles of Radar and Laser instruments using linear wave field simulations to better understand how the instruments perform. The Radar and Laser simulations include modeling electromagnetic signal beam reflections from water surfaces of an area equal to their footprint sizes, considering their beam characteristics and antenna pattern. The study confirms that the Radar underestimates spectral levels at frequencies above 0.5 Hz due to its significantly larger footprint at the water sea surface compared to the Laser (5.25 m vs. 0.15 m). The Laser performs well for almost the entire frequency range for all the cases considered.

The Attenuation Effect of Jet Filament on the Eastward Mesoscale Eddy Lifetime in the Southern Ocean

Abstract The interaction between the Antarctic Circumpolar Current (ACC) jets and mesoscale eddies plays a prominent role in the dynamics of the ACC. However, few studies have investigated the influence of the jets on the statistics of the mesoscale turbulence field. This study combines a mesoscale eddy detection algorithm with the jet-detection technique to extract their interaction in the Southern Ocean from 1993 to 2016. It is found that stronger jet filaments can effectively shorten eddy lifetime, thus introducing a negative correlation between eddy lifetime and matched jet filament velocities. It can also be seen along the pathway of the ACC whether the eddy is located upstream or downstream of topography. A series of numerical experiments suggest that the stronger zonal jet plays the role of a waveguide to facilitate the stronger and faster eastward linear Rossby wave field induced by the mesoscale eddy. A stronger eastward Rossby wave phase speed, enabled by the jet-induced Doppler shift, results in a more rapid loss of eddy energy and shortens the eddy lifetime.

Estimating space–time wave statistics using a sequential sampling method and Gaussian process regression

The traditional Monte Carlo sampling of waves requires generating tens of thousands of random waves to achieve stable statistics in the tail of the distribution, which is computationally costly for numerical simulations and time-consuming and often impractical for experiments. To improve the sampling efficiency, we present a sequential sampling method, which predicts the wave statistics based on the nonlinear response of deterministic wave groups. This data-driven method starts with parameterising the linear random wave field with a series of Gaussian wave groups. The nonlinear system response from the water wave system is then approximated by observations through the nonlinear evolution of wave groups with carefully designed initial conditions. A sequential sampling method is introduced during this sampling procedure, which determines the next best sampling point based on the previous observations and the distance metric. We examine the performance of the proposed method with Monte Carlo simulation of random wave fields as well as a grid search sampling strategy. The results show the proposed method can achieve computational cost savings over several orders of magnitude for numerical simulations. The results also suggest such a sequential sampling method can be used to determine the test matrix of the wave experiments with better coverage over the parameter space.

Wave-by-wave prediction in weakly nonlinear and narrowly spread seas using fixed-point surface-elevation time histories

The development of accurate, real-time predictive models for weakly nonlinear multidirectional wave fields would enable a significant reduction in the cost of wave energy, when combined with control strategies which use the prediction to increase power absorbed. An existing phase-resolved model for predicting weakly nonlinear long-crested wave fields is extended to sea-states with small spreading. When numerically advancing long-crested waves for prediction in spread seas, optimum prediction with the model is obtained when the waves are propagated at an offset angle (shown to be equal to the standard deviation of the component headings in the underlying sea) to the true mean wave direction. The model further exploits concurrent surface-elevation records from multiple adjacent fixed-point locations to obtain a prediction which is shown to be significantly more accurate than using individual input records. Furthermore, the prediction is unaffected by changes in mean wave direction up to a range of 30°, for a prediction using measurements from three points. The model is tested on both synthetic linear and second-order nonlinear wave fields where the directional and bulk wave statistics are assumed to be known. In the nonlinear case, bound harmonics are first estimated using narrow-band approximations and removed. The resulting linear records are then padded at both ends with half a NewWave-type extension to eliminate the step discontinuity in the records before advancing them in both space and time. The bound harmonics are re-calculated and re-inserted into the prediction signal as a final step, if required. The overall computational cost is very small.

A spatio-temporal noise robust filtering method for separation of 2D incident and reflected irregular waves

A spatio-temporal noise robust filtering method is proposed to separate 2D linear wave fields into incident and reflected irregular waves using water surface elevation gauges. The method is based on phased array methodologies, using delay-and-sum beamforming techniques with optimal noise attenuation. The method is developed for an arbitrary number of gauges and can be applied in real-time to surface elevation signals corrupted with measurement noise. The impact of the number of gauges and the distance between them on the performance achievable with the method is thoroughly analysed and a methodology to choose their optimum values is proposed. The robustness of the proposed method with respect to measurement noise is compared with the real-time method proposed by Frigaard and Brorsen and is shown to be adequate for noise levels higher than 1%.Separation errors smaller than 3% and 7% are obtained for simulated signals without noise and with 10% added noise, respectively.

A WKB derivation for internal waves generated by a horizontally moving body in a thermocline

An approximate solution for the steady linear internal wave field generated by a localized, horizontally moving source in a thermocline is derived using ray and WKB (Wentzel–Kramers–Brillouin) theory. The waves are assumed to be steady in a reference frame moving with the source velocity. This solution is shown to agree in the limit of propagating waves with an existing solution that is expressed in terms of an eigenfunction expansion. The WKB method is shown to reproduce all the factors in the eigenfunction solution, not just the eigenfunctions themselves. It also reveals the physical significance of the terms in the eigenfunction solution. Furthermore, the WKB solution suggests an alternate way to compute the wave field numerically that is computationally more efficient.

Fast wave-by-wave prediction of weakly nonlinear unidirectional wave fields

Accurate future prediction of wave fields can be used to optimize the performance of wave energy converters as well as various offshore operations. A highly efficient model using single, fixed-probe measurements for predicting unidirectional wave fields of up to second-order nonlinearity has been developed. The fundamental problem with the use of the Fast Fourier Transform (FFT) in forward prediction is the inherent assumption of periodicity of the finite-length record beyond its ends. The main aim here is to enhance the performance of the FFT algorithm during prediction of linear wave fields while retaining its computational efficiency. Given a finite-duration wave record up to the present time at a location upwave, it is observed that introducing a smooth transition of the record down to zero at both ends by adding half of a NewWave-type wave group improves the prediction at the point of interest dramatically compared to just using the raw record. These extensions can in turn serve as predictions for cases requiring future knowledge without the use of upwave measurements. These schemes are tested on their capability to predict synthetic wave fields at locations downwave and into the future. Comparisons are made by computing the numerical error within a theoretical predictable zone which is also estimated in this work. A second validation is done on weakly nonlinear sea states that include both sum and difference bound wave components calculated using exact second-order theory. A simulation method that initially linearizes the signal, then advances it in both space and time, and finally re-inserts estimated bound-harmonics based on a narrow-banded approximation proves to be very effective, yielding substantial improvements compared to assuming the signal is completely linear. The narrow-banded linearization/re-insertion is a very fast time-domain operation, hence the additional computational cost is very little. The nonlinear model is validated using synthetic wave records for waves in intermediate-depth water.

Energy Spectrum of Linear Internal Wave Field in the Vicinity of Continental Slope

The purpose of the research was to investigate two-dimensional modeling of efficiency of mixing, resulting from the reflection of a linear internal wave field (IWF) off a continental slope. Efficiency of deep ocean mixing was associated with the energy balance of the radiating IWF into an interior of the ocean in the vicinity of a sloping bottom topography. Since waves are generated not only at the fundamental frequency but also at all of its harmonics ωn = nω less than buoyancy frequency N and greater than Coriolis frequency f, our analysis includes, in general, an infinite number of discrete internal wave modes n satisfying the dispersion relationship for internal waves. However, since we are interested only in the radiating part of the field, the mode numbers are limited. Due to multiple singularities of order two caused by resonance in the vicinity of critical slope, the energy is visualized in Lδ -norm with δ > 2. Research results include the visualization of the effects of the continental slope and the Earth’s rotation on resulting energy in the vicinity of the slope.

Linear Wavefield Optimization Using a Modified Source

Linear Wavefield Optimization Using a Modified Source

The Ocean Mesoscale Regime of the Reduced-Gravity Quasigeostrophic Model

AbstractA statistical-equilibrium, geostrophic-turbulence regime of the stochastically forced, one-layer, reduced-gravity, quasigeostrophic model is identified in which the numerical solutions are representative of global mean, midlatitude, open-ocean mesoscale variability. Solutions are forced near the internal deformation wavenumber and damped linearly and by high-wavenumber enstrophy dissipation. The results partially rationalize a recent semiempirical stochastic field model of mesoscale variability motivated by a global eddy identification and tracking analysis of two decades of satellite altimeter sea surface height (SSH) observations. Comparisons of model results with observed SSH variance, autocorrelation, eddy, and spectral statistics place constraints on the model parameters. A nominal best fit is obtained for a dimensional SSH stochastic-forcing variance production rate of 1/4 cm2 day−1, an SSH damping rate of 1/62 week−1, and a stochastic forcing autocorrelation time scale near or greater than 1 week. This ocean mesoscale regime is nonlinear and appears to fall near the stochastic limit, at which wave-mean interaction is just strong enough to begin to reduce the local mesoscale variance production, but is still weak relative to the overall nonlinearity. Comparison of linearly inverted wavenumber–frequency spectra shows that a strong effect of nonlinearity, the removal of energy from the resonant linear wave field, is resolved by the gridded altimeter SSH data. These inversions further suggest a possible signature in the merged altimeter SSH dataset of signal propagation characteristics from the objective analysis procedure.

Preferential orientation of spheroidal particles in wavy flow

We report a theoretical study of the angular dynamics of small, non-inertial spheroidal particles in a linear wave field. We recover the observation recently reported by DiBenedetto et al. (J. Fluid Mech., vol. 837, 2018, pp. 320–340) that the orientation of these spheroids tends to a stable limit cycle consisting of a preferred value with a superimposed oscillation. We show that this behaviour is a consequence of finite wave amplitude and is the angular analogue of Stokes drift. We derive expressions for both the preferred orientation of the particles, which depends only on particle shape, and the amplitude of the oscillation about this preferred value, which additionally depends on the wave parameters and the depth of the particle in the water column.

A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control

Various forms of numerical shock instabilities are known to plague many contact and shear preserving approximate Riemann solvers, including the popular Harten-Lax-van Leer with Contact (HLLC) scheme, during high speed flow simulations governed by the Euler system of equations. In this paper we propose a simple and inexpensive novel strategy to prevent the HLLC scheme from developing such spurious solutions without compromising on its linear wave resolution ability. The cure is primarily based on a reinterpretation of the HLLC scheme as a combination of its well-known diffusive counterpart, the HLL scheme, and an antidiffusive term responsible for its accuracy on linear wavefields. In our study, a linear analysis of this alternate form indicates that shock instability in the HLLC scheme could be triggered due to the unwanted activation of the antidiffusive term appearing in its mass and interface-normal momentum flux component discretizations on interfaces that are not aligned with the shock front. This inadvertent activation results in weakening of the favourable dissipation provided by its inherent HLL scheme and causes unphysical mass flux variations along the shock front. To mitigate this, we propose a modified HLLC scheme that employs a simple differentiable pressure-ratio based multidimensional shock sensor to achieve smooth control of these critical antidiffusive terms near shocks. Using a linear perturbation analysis and a matrix based stability analysis, we establish that the resulting scheme, called HLLC-ADC (Anti-Diffusion Control), is shock stable over a wide range of freestream Mach numbers. Results from standard numerical test cases demonstrate that the HLLC-ADC scheme is indeed free from the most common manifestations of shock instability including the Carbuncle phenomenon without significant loss of accuracy on shear dominated viscous flows.

Nonlinear effects of micro-cracks on acoustic surface and wedge waves

Micro-cracks give rise to non-analytic behavior of the stress-strain relation. For the case of a homogeneous spatial distribution of aligned flat micro-cracks, the influence of this property of the stress-strain relation on harmonic generation is analyzed for Rayleigh waves and for acoustic wedge waves with the help of a simple micromechanical model adopted from the literature. For the efficiencies of harmonic generation of these guided waves, explicit expressions are derived in terms of the corresponding linear wave fields. The initial growth rates of the second harmonic, i.e., the acoustic nonlinearity parameter, has been evaluated numerically for steel as matrix material. The growth rate of the second harmonic of Rayleigh waves has also been determined for microcrack distributions with random orientation, using a model expression for the strain energy in terms of strain invariants known in a geophysical context.

Dynamic fragmentation of ice spheres: Two specific fracture patterns

Although ice materials are commonly found not only in our daily life on the earth but also in space, e.g. in the rings of Saturn, their fracture mechanisms, especially those in a dynamic range, have not been fully clarified. In our previous investigation, therefore, collision of a brittle ice sphere (diameter 25, 50 or 60 mm) that is free falling against a fixed plate of ice or polycarbonate has been experimentally observed using high-speed digital video cameras at a frame rate of up to 150,000 frames per second. It has been recognized that there exist basically only two specific fracture patterns generated by this dynamic impact: (i) “Top”-type fracture, normally at a relatively smaller impact speed, where only the bottom surface areas of a sphere are fragmented into small pieces by the impact at bottom and a relatively large top-shaped portion is left unbroken; and (ii) “orange segments”-type in which rather flat fracture planes extend approximately along the central axis of the sphere and split the sphere into three or four larger segments of comparable size. Preliminary comparison of the experimental findings with linear elastic wave fields obtained numerically by three-dimensional finite difference calculations suggests two distinct spatiotemporal scales: (1) The “top” fracture pattern is induced by the propagation of surface waves with relatively shorter wavelengths from the bottom along the free surface of the sphere, quickly producing fracture only near the bottom; (2) In the “orange segments”-type fracture, a longer contact time makes largely stressed regions enlarge more slowly and widely (quasi-statically-like) and larger fracture planes develop along the axis of the sphere. The above speculations seem to be supported by recent computations adopting the Discontinuous Galerkin (DG) method that treat more rigorously the mechanics of dynamic contact between the sphere and plate.

Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups

This paper provides a mass, momentum, and energy flux conservation analysis between the linear and nonlinear steady-state wave groups. Convergent high-order solutions for nonlinear wave groups with multiple steady-state near resonances in deep water have been obtained by means of the homotopy analysis method. The small divisors associated with nearly resonant components are transformed to singularities that are originally caused by exact resonances by a piecewise auxiliary linear operator. Both two primary components and other nearly resonant ones are considered in the initial guess to search for finite amplitude wave groups. It is found that as nonlinearity of wave groups increases, more wave components appear in the spectrum due to the nearly resonant interactions. The nonlinear wave fields change from the initial bi-chromatic waves that contain only two nontrivial primary components into the steady-state resonant waves that contain both two primary components and other nearly resonant ones. The conservation of mean rates of mass, momentum, and energy fluxes is established between the nonlinear wave groups and linear waves that are combined by two primary components with the same frequencies as in nonlinear wave groups. Comparison of the linear and nonlinear wave fields shows that the nearly resonant components influence the wave field distribution significantly: the nonlinear free surfaces have more peaked crests, steeper troughs, and more flatten wave nodes, and the related velocities at the crests and troughs increase more rapidly with the nonlinearity. All of these findings are helpful to enrich and deepen our understanding about nonlinear wave groups.

An Integrated Numerical Model for the Design of Coastal Protection Structures

In the present work, an integrated coastal engineering numerical model is presented. The model simulates the linear wave propagation, wave-induced circulation, and sediment transport and bed morphology evolution. It consists of three main modules: WAVE_L, WICIR, and SEDTR. The nearshore wave transformation module WAVE_L (WAVE_Linear) is based on the hyperbolic-type mild slope equation and is valid for a compound linear wave field near coastal structures where the waves are subjected to the combined effects of shoaling, refraction, diffraction, reflection (total and partial), and breaking. Radiation stress components (calculated from WAVE_L) drive the depth averaged circulation module WICIR (Wave Induced CIRculation) for the description of the nearshore wave-induced currents. Sediment transport and bed morphology evolution in the nearshore, surf, and swash zone are simulated by the SEDTR (SEDiment TRansport) module. The model is tested against experimental data to study the effect of representative coastal protection structures and is applied to a real case study of a coastal engineering project in North Greece, producing accurate and consistent results for a versatile range of layouts.

Information loss due to environmental variability and uncertainty in Bayesian localization of a narrowband source

Performance degradation of Bayesian localization of a low-frequency narrowband acoustic source due to variability and imperfect knowledge of the acoustic environment is investigated in a computational study. The environmental variability is modeled as arising from water column fluctuations associated with a diffuse random linear internal wave field in a shallow-water ocean waveguide. The ambient noise spatial cross-spectrum is represented by a Kuperman-Ingenito model. For the case of complex Gaussian internal wave spectral amplitudes, a closed-form expression is derived for the conditional pdf, given source location, of the signal spectral values received on an acoustic array. Examples computed for a vertical receiver array quantify localization performance degradation as an increase in the entropy of the Bayesian source-location posterior. The effects of model bias, model spectral uncertainty, and medium variability as determined by the internal wave power spectrum are quantified separately and jointly. Potential extensions to more general models of medium variability are discussed. [Work supported by ONR.]

Dynamics of Orographic Gravity Waves Observed in the Mesosphere over the Auckland Islands during the Deep Propagating Gravity Wave Experiment (DEEPWAVE)

Abstract On 14 July 2014 during the Deep Propagating Gravity Wave Experiment (DEEPWAVE), aircraft remote sensing instruments detected large-amplitude gravity wave oscillations within mesospheric airglow and sodium layers at altitudes z ~ 78–83 km downstream of the Auckland Islands, located ~1000 km south of Christchurch, New Zealand. A high-altitude reanalysis and a three-dimensional Fourier gravity wave model are used to investigate the dynamics of this event. At 0700 UTC when the first observations were made, surface flow across the islands’ terrain generated linear three-dimensional wave fields that propagated rapidly to z ~ 78 km, where intense breaking occurred in a narrow layer beneath a zero-wind region at z ~ 83 km. In the following hours, the altitude of weak winds descended under the influence of a large-amplitude migrating semidiurnal tide, leading to intense breaking of these wave fields in subsequent observations starting at 1000 UTC. The linear Fourier model constrained by upstream reanalysis reproduces the salient aspects of observed wave fields, including horizontal wavelengths, phase orientations, temperature and vertical displacement amplitudes, heights and locations of incipient wave breaking, and momentum fluxes. Wave breaking has huge effects on local circulations, with inferred layer-averaged westward flow accelerations of ~350 m s−1 h−1 and dynamical heating rates of ~8 K h−1, supporting recent speculation of important impacts of orographic gravity waves from subantarctic islands on the mean circulation and climate of the middle atmosphere during austral winter.