Analysis of nonlinear ship-induced 3d wave fields using nonlinear fourier transforms

In the past decade, observations in the German estuaries such as the rivers Elbe and Weser show increasingly serious damage to bank protection structures (groins and revetments). This damage is caused mainly by waves induced by the passing of big container ships in the shallow and narrow maritime waterways. These ship-induced 3D wave fields consist of long-periodic primary and short-periodic secondary wave components. Due to missing design approaches for the load of long-period waves on rubble-mound revetments, the current risk assessment for protective structures in maritime waterways is based on short-period, wind-induced waves. Therefore, the structures do not ensure sufficient stability against the long-period ship-induced wave loads within the estuaries. Within the research project “Parameterization of nonlinear ship-induced 3D wave fields for the hydraulic design of protective structures in maritime waterways (PaNSiWa)”, we apply nonlinear Fourier transforms (NFTs) on experimentally generated ship waves in maritime waterways. The objective of the project is to provide better understanding of the underlying nonlinear structure of the long-period primary waves and to separate the nonlinear spectral basic components within the ship-wave data from their nonlinear wave-wave interactions. In this paper, we present first analyses of the decomposition of ship-wave measurements from experimental tests and the identification of hidden solitons within the long-period primary ship wave

When Do JONSWAP Spectra Lead to Soliton Gases in Deep Water Conditions?

When a large number of solitons dominates the dynamics of a system, scientists describe this collective behaviour of solitons as a soliton gas. Soliton gases are currently the subject of intense practical and theoretical investigations. The existence of soliton gases has been confirmed in experiments, but is not clear what kind of sea states might lead to soliton gases. Therefore, in order to determine the wave parameters for sea states that lead to soliton gases, large numbers of surface wave elevations are generated by the well-known JOSNWAP model in this paper. Here, we only discuss soliton gases in deep water governed by the nonlinear Schrödinger (NLS) equation. The nonlinear Fourier transform (NFT) with vanishing boundary conditions is applied to the simulated ocean surface waves. The resulting nonlinear Fourier spectrum is used to calculate the energy of radiation waves and solitons. We investigate which JONSWAP parameters result in sea states that can be characterized as soliton gases, and find that a large Phillip’s parameter α, a large peak enhancement parameter γ and a short peak period TP are important factors for soliton gas conditions. The results allow researchers to estimate how likely soliton gases are in deep waters. Furthermore, we find that the appearance of rogue waves is slightly increased in highly nonlinear sea states with soliton gas-like conditions.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Team Michel VerhaegenTeam Sander Wahl

Analysis of nonlinear ship-induced 3d wave fields using nonlinear fourier transforms

In the past decade, observations in the German estuaries such as the rivers Elbe and Weser show increasingly serious damage to bank protection structures (groins and revetments). This damage is caused mainly by waves induced by the passing of big container ships in the shallow and narrow maritime waterways. These ship-induced 3D wave fields consist of long-periodic primary and short-periodic secondary wave components. Due to missing design approaches for the load of long-period waves on rubble-mound revetments, the current risk assessment for protective structures in maritime waterways is based on short-period, wind-induced waves. Therefore, the structures do not ensure sufficient stability against the long-period ship-induced wave loads within the estuaries. Within the research project “Parameterization of nonlinear ship-induced 3D wave fields for the hydraulic design of protective structures in maritime waterways (PaNSiWa)”, we apply nonlinear Fourier transforms (NFTs) on experimentally generated ship waves in maritime waterways. The objective of the project is to provide better understanding of the underlying nonlinear structure of the long-period primary waves and to separate the nonlinear spectral basic components within the ship-wave data from their nonlinear wave-wave interactions. In this paper, we present first analyses of the decomposition of ship-wave measurements from experimental tests and the identification of hidden solitons within the long-period primary ship wave.Team Sander Wahl

Water-depth identification from free-surface data using the KdV-based nonlinear Fourier transform

We propose a novel method to determine the average water depth from shallow, weakly nonlinear water waves that are approximated by the Korteweg-de Vries equation. Our identification method only requires free-surface measurements from two wave gauges aligned in the direction of wave propagation. The method we propose is based on comparing solitonic components in wave packets, which are computed using the nonlinear Fourier transform (NFT) (typical time-series data often contains at least some solitonic components, even when these components are not directly visible). When the correct water depth is used for the normalisation of the wave, the solitonic components found by the NFT remain constant as the wave packet propagates, whereas any other water depth will result in solitonic components that do not remain constant. The basic idea is thus to iteratively determine the water depth that leads to a best fit between the solitonic components of time series measurements at two different gauge positions. We present a proof-of-concept on experimental bore data generated in a wave flume, where the identified water depth is within 5% of the measured value.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Team Sander Wahl

Nonlinear Fourier classification of 663 rogue waves measured in the Philippine Sea

Rogue waves are sudden and extreme occurrences, with heights that exceed twice the significant wave height of their neighboring waves. The formation of rogue waves has been attributed to several possible mechanisms such as linear superposition of random waves, dispersive focusing, and modulational instability. Recently, nonlinear Fourier transforms (NFTs), which generalize the usual Fourier transform, have been leveraged to analyze oceanic rogue waves. Next to the usual linear Fourier modes, NFTs can additionally uncover nonlinear Fourier modes in time series that are usually hidden. However, so far only individual oceanic rogue waves have been analyzed using NFTs in the literature. Moreover, the completely different types of nonlinear Fourier modes have been observed in these studies. Exploiting twelve years of field measurement data from an ocean buoy, we apply the nonlinear Fourier transform (NFT) for the nonlinear Schrödinger equation (NLSE) (referred to NLSE-NFT) to a large dataset of measured rogue waves. While the NLSE-NFT has been used to analyze rogue waves before, this is the first time that it is systematically applied to a large real-world dataset of deep-water rogue waves. We categorize the measured rogue waves into four types based on the characteristics of the largest nonlinear mode: stable, small breather, large breather and (envelope) soliton. We find that all types can occur at a single site, and investigate which conditions are dominated by a single type at the measurement site. The one and two-dimensional Benjamin-Feir indices (BFIs) are employed to examine the four types of nonlinear spectra. Furthermore, we verify on a part of the data set that for the localized types, the largest nonlinear Fourier mode can be attributed directly to the rogue wave, and investigate the relation between the height of the rogue waves and that of the dominant nonlinear Fourier mode. While the dominant nonlinear Fourier mode in general only contributes a small fraction of the rogue wave, we find that soliton modes can contribute up to half of the rogue wave. Since the NLSE does not account for directional spreading, the classification is repeated for the first quartile with the lowest directional spreading for each type. Similar results are obtained.Team Michel Verhaege

Nonlinear Fourier Analysis of Free-Surface Buoy Data Using the Software Library FNFT

Nonlinear Fourier Analysis (NFA) is a powerful tool for the analysis of hydrodynamic processes. The unique capabilities of NFA include, but are not limited to, the detection of hidden solitons and the detection of modulation instability, which are essential for the understanding of nonlinear phenomena such as rogue waves. However, even though NFA has been applied to many interesting problems, it remains a non-standard tool. Recently, an open source software library called FNFT has been released to the public. (FNFT is short for “Fast Nonlinear Fourier Transforms”.) The library in particular contains code for the efficient numerical NFA of hydrodynamic processes that are approximately governed by the nonlinear Schroedinger equation with periodic boundary conditions. Waves in deep water are a prime example for such a process. In this paper, we use FNFT to perform an exemplary NFA of typhoon data collected by wave buoys at the coast of Taiwan. Our goals are a) to demonstrate the application of FNFT in a practical scenario, and b) to compare the results of a NFA to an analysis based on the conventional linear Fourier transform. The exposition is deliberately educational, hopefully enabling others to use FNFT for similar analyses of their own data.Accepted Author ManuscriptTeam Sander Wahl

Analysis of Bore Characteristics Using KdV-Based Nonlinear Fourier Transform

Bores propagating in shallow water transform into undular bores and, finally, into trains of solitons. The observed number and height of these undulations, and later discrete solitons, is strongly dependent on the propagation length of the bore. Empirical results show that the final height of the leading soliton in the far-field is twice the initial mean bore height. The complete disintegration of the initial bore into a train of solitons requires very long propagation lengths, but unfortunately these required distances are usually not available in experimental tests or nature. Therefore, the analysis of the bore decomposition for experimental data into solitons is difficult and requires further approaches. Previous studies have shown that by application of the nonlinear Fourier transform based on the Korteweg–de Vries equation (KdV-NFT) to bores and long-period waves propagating in constant depth, the number and height of all solitons can be reliably predicted already based on the initial bore-shaped free surface.Against this background, this study presents the systematic analysis of the leading-soliton amplitudes for non-breaking and breaking bores with different strengths in different water depths in order to validate the KdV-NFT results for non-breaking bores, and to show the limitations of wave breaking on the spectral results. The analytical results are compared with data from experimental tests, numerical simulations and other approaches from literature.Accepted Author ManuscriptTeam Sander Wahl

Comparative analysis of bore propagation over long distances using conventional linear and KdV-based nonlinear Fourier transform

In this paper, we study the propagation of bores over a long distance. We employ experimental data as input for numerical simulations using COULWAVE. The experimental flume is extended numerically to an effective relative length of x/h=3000, which allows all far-field solitons to emerge from the undular bore in the simulation data. We apply the periodic KdV-based nonlinear Fourier transform (KdV-NFT) to the time series taken at different numerical gauges and compare the results with those of the conventional Fourier transform. We find that the periodic KdV-NFT reliably predicts the number and the amplitudes of all far-field solitons from the near-field data long before the solitons start to emerge from the bore, even though the propagation is only approximated by the KdV. It is the first time that the predictions of the KdV-NFT are demonstrated over such long distances in a realistic set-up. In contrast, the conventional linear FT is unable to reveal the hidden solitons in the bore. We repeat our analyses using space instead of time series to investigate whether the space or time version of the KdV provides better predictions. Finally, we show how stepwise superposition of the determined solitons, including the nonlinear interactions between individual solitons, returns the analysed initial bore data.Team Sander Wahl