Crossing Sea States Research Articles

On the spectral wave climate of the French Atlantic Ocean

The accurate description of wave climate at different spatio-temporal scales requires the application of advanced statistical models together with a good knowledge of geophysical processes. Furthermore, accurate modelling of multimodal sea states is of primary importance for most of offshore and coastal activities, such as wave energy device optimization, maritime design practice and for safety at sea. The present manuscript is conceived within this scientific framework proposing to assess wave climate of a complex area of French Atlantic Ocean bordering seas. The main “Spectral Climatology Types” of the area are identified as resulting of defined combinations of wave systems detected through directional spectra and partitioned wave systems analysis. The presence of swell systems is evaluated quantitatively on the whole area, characterized by different regimes of multimodal sea-states at strong regional variability and put into relation with main meteorological forcing active on the area at different spatial scales. Celtic Sea exhibits a marked regional characterization with a prevalence of multimodal conditions given by different combination of wave systems with increasing contribution of swell systems moving from North-West to South. A significant presence of crossing sea states is observed in South Celtic Sea especially in proximity of bathymetric slope. The South Bay of Biscay is influenced by fully developed swell systems enhanced by refraction effects caused by both the coastline and bathymetry gradients. English Channel and North Sea show complex sea-states conditions induced by local topography features together with wind channelling and tide effects able to trigger geophysical processes at a sub-scale responsible for the development of multimodal seas. Crossing and bimodal seas are also found to be influenced by bathymetry gradients acting directly on the directional spectrum shape as well as by tide due to tidal current-induced refraction effect on wave propagation. No generalized significant trends are detected within the investigated spatial domain for the wave spectrum integral quantities; locations sited at northern Celtic Sea show a downward Significant Wave Height, while only locations confined at eastern English Chanel only shows a Peak Period upward trend.

Characterization of extreme wave fields during Mediterranean tropical-like cyclones

The Mediterranean Sea is a primary source of food, ecosystem services and economic activities and one of the most active cyclogenetic regions in the world, where the influence of orographic and morphological features of the relatively small basin plays an important role. Together with the explosive cyclogenesis, tropical-like cyclones (also called Mediterranean Hurricanes or Medicanes) are among the strongest types of storms that can be found in the Mediterranean basin, occurring predominantly in the Ionian, Balearic and Tyrrhenian sub-basins. Similarly to tropical cyclones (Hurricanes or Typhoons), these cyclonic structures are characterized by strong rotating and translating wind fields, which often lead to a combination of remotely generated swell waves and locally generated wind waves, often referred to as crossing sea states. Despite the well-known potential of Medicanes to cause significant damage near islands and coastal zones, which is predicted to intensify as a result of climate change, to date the characterization of maximum individual waves generated during these events is still lacking. In this study, we carry out the first analysis of the large-scale geographical distribution of wave maxima within the wave fields generated during three recent Medicane events using the WAVEWATCH III® spectral wave model forced by ERA5 reanalysis winds, also investigating the influence of crossing sea states on the maximum wave amplitudes with novel statistical formulations developed for such conditions. Our results show that, as in the case of tropical cyclones, several regions of the cyclone field are characterized by crossing sea states, whose role in the formation of the maximum individual waves occurring near the eye of the storm was found to be confined. Furthermore, extreme wave predictions accounting for the local crossing conditions yield differences up to 5% compared to standard statistical distributions.

Modeling “crossing sea state” wave patterns in layered and stratified fluids

Crossing sea states, with two or more wave trains propagating at an oblique angle to each other, constitute a plausible generation mechanism of rogue waves in the oceans. This underlying principle of enhanced modulation instability is extended to layered and stratified fluids. Long wave-short wave resonance in a two-layer flow and three-wave interaction in a fluid with constant buoyancy frequency are studied. Maxima in the growth rates imply preferred configurations of crossing sea states. A cascading mechanism is employed to demonstrate the emergence of breathers beyond linear modulation instability.

Space-time statistics of extreme ocean waves in crossing sea states

The study of extreme ocean waves has gained considerable interest in recent years, due to their importance for offshore design and navigation safety, and several theoretical approaches have been developed for their statistical description. However, in the case of crossing seas, where two or more wave systems of different characteristics are present, a full understanding of the main physical mechanisms responsible for the occurrence of very high individual waves is still lacking. As a consequence, the prediction of extremes in such conditions currently relies on integrated parameters of the total sea state, such as the spectral wave steepness. In this study, to gain further insight into the role of the crossing wind sea and swell wave systems in producing extreme individual waves, we investigate realistic sea states during typhoon Kong-rey (2018) using an ensemble of numerical simulations obtained from a phase-resolving wave model based on the High-Order Spectral (HOS) method. The reliability of the numerical fields is assessed for the first time with stereo wave measurements of the sea surface elevation field collected from an offshore platform in the area of interest. We show that, in specific conditions, space-time extreme crest heights in crossing seas can be larger than in unimodal seas due to second-order bound wave interactions between the wind sea and the swell. To improve existing prediction capabilities, we propose a novel formulation for the wave steepness in crossing seas, which includes nonlinear effects up to the second order and accounts for the spectral parameters of the interacting wave systems.

Statistical properties of surface gravity waves and freak wave occurrence in crossing sea states

Statistical properties of surface waves and freak wave shapes in crossing sea states are systematically studied, and particularly a new index is proposed to measure the occurrence probability of extreme waves. The proposed new index is proven to be a more accurate indicator of freak wave occurrences in crossing sea states than the Benjamin-Feir index that has been widely applied for seas with single-peak spectra. Our work can shed light on the generation mechanism and negative impact of freak waves, and advance the development of warning systems for marine disasters.

Non-degenerate multi-rogue waves and easy ways of their excitation

In multi-component systems, several rogue waves can be simultaneously excited using simple initial conditions in the form of a plane wave with a small amplitude single-peak perturbation. This is in drastic contrast with the case of multi-rogue waves of a single nonlinear Schrödinger equation (or other evolution equations) that require highly specific initial conditions to be used. This possibility arises due to the higher variety of rogue waves in multi-components systems each with individual eigenvalue of the inverse scattering technique. In theory, we expand the limited class of Peregrine-type solutions to a much larger family of non-degenerate rogue waves. The results of our work may explain the increased chances of appearance of rogue waves in crossing sea states (wind generated ocean gravity waves that form nonparallel wave systems along the water surface) as well as provide new possibilities of rogue wave observation in a wide range of multi-component physical systems such as multi-component Bose–Einstein condensates, multi-component plasmas and in birefringent optical fibres.

Estimating ocean wave directional spreading using wave following buoys: a comparison of experimental buoy and gauge data

Directional spreading of ocean waves plays an important role in various aspects of ocean engineering, such as wave-induced loads, nonlinear wave evolution, and wave breaking. Wave following buoys, which are widely deployed across the oceans, offer the potential to measure directional wave properties. To assess the accuracy of directional spreading estimates made using buoy measurements, we compared estimates based on experimentally obtained buoy and wave gauge measurements, first presented in McAllister and van den Bremer (J Phys Oceanogr 50:399–414, 2020) Buoy and gauge measurements were recorded at the same locations and in identical sea states, allowing for a like-for-like comparison. We examine experiments with both following (unimodal in direction) and crossing (bimodal in direction) sea states. In addition to this, we use synthetic wave data to investigate the effects of wave generation and nonlinearity on spreading estimates. Our results show that while directional estimates produced using buoy measurements are reasonably accurate in following sea states, they struggle to identify distinct directional peaks in crossing sea states. We find that spreading estimates made using buoy measurements tend to underestimate the degree of directional spreading by approximately 9-14% in following sea states, which is most apparent in narrowly spread conditions.

ANALYTICAL SOLUTIONS OF COUPLED NONLINEAR SCHRODINGER EQUATIONS FOR TWO NON LINEARLY INTERACTING STOKES WAVE TRAINS OVER INFINITE DEPTH

An attempt to find the exact analytical solutions of the two coupled nonlinear Schrodinger equations of 3rd order occurring from the oblique interaction of two capillary gravity wave trains in the case of crossing sea states in deep water is the main premise of the present paper. The solutions obtained here are due to the nonlinear interaction of two Stokes wave trains in one spatial dimension. Graphs have been plotted to investigate the influence of capillarity on the amplitudes of such wave trains. From 3D figures it has been observed that the capillarity has diminishing influence on the amplitudes of the either wave packet.

Wind effect on the evolution of two obliquely interacting random wave trains in deep water

A pair of coupled nonlinear evolution equations for the spectral functions are derived in a situation of crossing sea states characterized by two narrowband Gaussian random surface wave systems in the presence of uniform wind flow. These two equations are employed to perform stability analysis of two initially homogeneous wave spectra subject to infinitesimal perturbations. It is found that the results of the stability analysis remain qualitatively similar with the corresponding deterministic situation. The notable difference is that the growth rate of instability reduces slightly due to the effect of randomness. But, it is much higher than the growth rate of a single wave system. As the spectral bandwidth increases the growth rate of instability decreases.

Long-term statistics of potentially hazardous sea states in the North Sea 1958–2014

Significant wave height and mean wave period are two of the most commonly used parameters to describe wave climate, wave climate variability, and their potential long-term changes. While these parameters are generally useful to characterize the distribution of waves within a given sea state, they provide less information about potentially high-risk situations. Over the recent years, a number of criteria were suggested that are considered to better characterize high-risk situations and which could bear a potential for the development of safety warning indices. Based on a multi-decadal high-resolution wind-wave hindcast, a climatology of such parameters is developed for the North Sea covering the years 1958–2014. More specifically, average conditions, inter-annual variability and long-term changes for unusually steep, rapidly developing and crossing sea states are considered. Generally, there are pronounced spatial variations in the frequency of such sea states, while over time, there is some seasonal and inter-annual variability but no substantial long-term trend could be identified.

Modulational Instability and Rogue Waves in Crossing Sea States

AbstractWave statistical properties and occurrence of extreme and rogue waves in crossing sea states are investigated. Compared to previous studies a more extensive set of crossing sea states are investigated, both with respect to spectral shape of the individual wave systems and with respect to the crossing angle and separation in peak frequency of the two wave systems. It is shown that, because of the effects described by Piterbarg, for a linear sea state the expected maximum crest elevation over a given surface area depends on the crossing angle so that the expected maximum crest elevation is largest when two wave systems propagate with a crossing angle close to 90°. It is further shown by nonlinear phase-resolving numerical simulations that nonlinear effects have an opposite effect, such that maximum sea surface kurtosis is expected for relatively large and small crossing angles, with a minimum around 90°, and that the expected maximum crest height is almost independent of the crossing angle. The numerical results are accompanied by analysis of the modulational instability of two crossing Stokes waves, which is studied using the Zakharov equation so that, different from previous studies, results are valid for arbitrary-bandwidth perturbations. It is shown that there is a positive correlation between the value of kurtosis in the numerical simulations and the maximum unstable growth rate of two crossing Stokes waves, even for realistic broadband crossing sea states.

Extreme Waves in Crossing Sea States

The evolution of crossing sea states and the emergence of rogue waves in such systems are studied via numerical simulations performed using a higher order spectral (HOS) method to solve the free surface Euler equations with a flat bottom. Two classes of crossing sea states are analyzed: one using directional spectra from the Draupner wave crossing at different angles, another considering a Draupner-like spectra crossed with a narrowband JONSWAP state to model spectral growth between wind sea and swell. These two classes of crossing sea states are constructed using the spectral output of a WAVEWATCH III hindcast on the Draupner rogue wave event. We measure ensemble statistical moments as functions of time, finding that although the crossing angle influences the statistical evolution to some degree, there are no significant third-order effects present. Additionally, we pay particular attention to the mean sea level measured beneath extreme crest heights, the elevation of which (set up or set down) is shown to be related to the spectral content in the low wavenumber region of the corresponding spectrum.

The set-down and set-up of directionally spread and crossing surface gravity wave groups

For sufficiently directionally spread surface gravity wave groups, the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (J. Fluid Mech. vol. 13, 1962, pp. 481–504), can turn into a set-up. Using a multiple-scale expansion for two crossing wave groups, we examine the structure and magnitude of this wave-averaged set-up, which is part of a crossing wave pattern that behaves as a modulated partial standing wave: in space, it consists of a rapidly varying standing-wave pattern slowly modulated by the product of the envelopes of the two groups; in time, it grows and decays on the slow time scale associated with the translation of the groups. Whether this crossing wave pattern actually enhances the surface elevation at the point of focus depends on the phases of the linear wave groups, unlike the set-down, which is always negative and inherits the spatial structure of the underlying envelope(s). We present detailed laboratory measurements of the wave-averaged free surface, examining both single wave groups, varying the degree of spreading from small to very large, and the interaction between two wave groups, varying both the degree of spreading and the crossing angle between the groups. In both cases, we find good agreement between the experiments, our simple expressions for the set-down and set-up, and existing second-order theory based on the component-by-component interaction of individual waves with different frequencies and directions. We predict and observe a set-up for wave groups with a Gaussian angular amplitude distribution with standard deviations of above $30{-}40^{\circ }$ ($21{-}28^{\circ }$ for energy spectra), which is relatively large for realistic sea states, and for crossing sea states with angles of separation of $50{-}70^{\circ }$ and above, which are known to occur in the ocean.

SPECTRAL PARTITIONING AND SWELLS IN THE BLACK SEA

The swell climate of the Black Sea has been determined using a long-term 31-year wave hindcast with the third-generation spectral wave model SWAN in combination with spectral partitioning. This technique enables decomposing wave spectra into individual wave systems representing wind seas or swells and computing integral wave parameters of each partition. Results are presented of the partition technique and of spatial and seasonal characteristics of wind sea and swell systems. In addition, the average amount of swell energy and the occurrence probability of dangerous crossing sea states are determined.

Nonlinear evolution equations in crossing seas in the presence of uniform wind flow

The effect of uniform wind flow in a situation of crossing sea states is studied here on the stability analysis for a pair of obliquely interacting uniform wave trains. It is observed that the growth rate of instability in crossing seas in the presence of wind flow is much higher than that in the absence of wind flow. It is also found that the growth rate of instability increases as the wind flow velocity increases up to some critical velocity beyond which the wave becomes linearly unstable and consequently the present analysis does not remain valid. The growth rate of instability obtained here is also greater than that for a single wave packet propagating in the presence of uniform wind flow.

Wind-forced modulations in crossing sea states over infinite depth water

The present work is motivated by the work of Leblanc [“Amplification of nonlinear surface waves by wind,” Phys. Fluids 19, 101705 (2007)] which showed that Stokes waves grow super exponentially under fair wind as a result of modulational instability. Here, we have studied the effect of wind in a situation of crossing sea states characterized by two obliquely propagating wave systems in deep water. It is found that the wind-forced uniform wave solution in crossing seas grows explosively with a super-exponential growth rate even under a steady horizontal wind flow. This is an important piece of information in the context of the formation of freak waves.

Nonlinear Evolution Equations for Broader Bandwidth Wave Packets in Crossing Sea States

Two coupled nonlinear equations are derived describing the evolution of two broader bandwidth surface gravity wave packets propagating in two different directions in deep water. The equations, being derived for broader bandwidth wave packets, are applicable to more realistic ocean wave spectra in crossing sea states. The two coupled evolution equations derived here have been used to investigate the instability of two uniform wave trains propagating in two different directions. We have shown in figures the behaviour of the growth rate of instability of these uniform wave trains for unidirectional as well as for bidirectional perturbations. The figures drawn here confirm the fact that modulational instability in crossing sea states with broader bandwidth wave packets can lead to the formation of freak waves.

Modulational instability in crossing sea states over finite depth water

Nonlinear evolution equations are derived in a situation of crossing sea states characterized by water waves having two different spectral peaks. The nonlinear evolution equations derived here are valid for any water depth except for shallow water depth case. These evolution equations are then employed to study the instability properties of two Stokes wave trains considering both unidirectional and bidirectional perturbations. Figures have been plotted showing the growth rate of instability for various depths of water and for different values of the angle of interaction of the two wave systems. All the figures serve as an evidence to the fact that freak waves can be formed as a result of modulational instability in crossing sea states over finite depth water. It is observed that the growth rate of instability in crossing sea states situation over finite depth water is much higher than that for infinite depth case and it increases with the decrease of the depth of water.

Freak waves under typhoon conditions

Benjamin‐Feir Index (BFI) and directional spread are measures of nonlinear four‐wave interactions and resultant indices of possible conditions for freak waves. Temporal‐spatial distributions of BFI and directional spread are examined with numerical simulations of a spectral wave model using typhoon conditions. The spatial distributions of wave characteristics such as significant wave height, wave period, BFI and directional spread are different from each other around the eye of the typhoon. BFI is significantly large in the fourth quadrant of the typhoon, while waves are steep and have narrow frequencies and directional spectra. Freak waves resulting from nonlinear four‐wave‐wave interactions have a greater potential of occurring in the fourth quadrant of the typhoon than in the other quadrants. Furthermore, crossing sea states from two‐wind‐wave systems can be observed behind the eye of the typhoon. The crossing, two‐wind systems are also dangerous sea states, although as observed, they are closer to linear wave conditions. Finally, the characteristics of possible freak wave conditions during typhoons are verified with field data.

Did the Draupner wave occur in a crossing sea?

The ‘New Year Wave’ was recorded at the Draupner platform in the North Sea and is a rare high-quality measurement of a ‘freak’ or ‘rogue’ wave. The wave has been the subject of much interest and numerous studies. Despite this, the event has still not been satisfactorily explained. One piece of information that was not directly measured at the platform, but which is vital to understanding the nonlinear dynamics is the wave's directional spreading. This paper investigates the directionality of the Draupner wave and concludes it might have resulted from two wave-groups crossing, whose mean wave directions were separated by about 90 ° or more. This result has been deduced from a set-up of the low-frequency second-order difference waves under the giant wave, which can be explained only if two wave systems are propagating at such an angle. To check whether second-order theory is satisfactory for such a highly nonlinear event, we have run numerical simulations using a fully nonlinear potential flow solver, which confirm the conclusion deduced from the second-order theory. This is backed up by a hindcast from European Centre for Medium-Range Weather Forecasts that shows swell waves propagating at approximately 80 ° to the wind sea. Other evidence that supports our conclusion are the measured forces on the structure, the magnitude of the second-order sum waves and some other instances of freak waves occurring in crossing sea states.