Nature-based Solutions (NbS) use natural processes to address social, economic, and environmental challenges, including climate change, and this study explores their application for coastal protection. Before the actual implementation of NbS structures, co-creation with stakeholders to identify and assess potential NbS solutions for the selected local site is essential. However, existing frameworks for selecting NbS measures in coastal protection remain limited in handling the complex interactions among multiple indicators and the inherent uncertainty in expert evaluations. To address this gap, we develop an integrated assessment framework specifically for NbS in coastal protection. We first identified 18 distinct coastal protection measures through literature reviews. The proposed framework combines the Fuzzy Delphi Method (FDM) and entropy-based intuitionistic fuzzy TOPSIS (IF-TOPSIS) to facilitate expert-driven multi-criteria decision analysis (MCDA), both of which use fuzzy theory to address uncertainty and ambiguity in expert judgments. The 18 consensus indicators from an initial pool of 63 evaluation indicators are obtained in FDM analysis. Subsequently, IF-TOPSIS is applied to rank the 18 measures against these evaluation indicators based on membership, non-membership, and hesitancy degrees. The proposed framework is demonstrated through a case study at the Golden Coast, Tainan, Taiwan to illustrate its practical applicability.

The separation of wind sea and swell is crucial for advancing wave dynamics research, improving wave forecasting, and optimizing the design of coastal and offshore structures. In this study, we highlight the limitations of the widely used wave age method for separating two-dimensional wind sea and swell. Specifically, under strong wind conditions, waves require extended durations to reach full development—an aspect not accounted for by the wave age method, which assumes fully developed seas and thus tends to overestimate wind sea. Furthermore, changes in wind direction and wave refraction in shallow waters can lead to misclassification. To overcome these issues, we propose a novel algorithm for directional spectral separation, grounded in wind wave growth theory and incorporating wave refraction effects. The proposed method improves separation accuracy and delivers more consistent results across a range of wave conditions.

The Korteweg–De Vries (KdV) equation is a partial differential equation used to describe the dynamics of water waves under the assumptions of shallow water, unidirectionality, weak nonlinearity and constant depth. It can be solved analytically with a suitable nonlinear Fourier transform (NFT). The NFT for the KdV equation is subsequently referred to as the KdV-NFT. The soliton part of the nonlinear Fourier spectrum provides valuable insights into the nonlinear evolution of waveforms by exposing the amplitudes and velocities of potentially hidden solitonic components. Under the KdV equation, the nonlinear spectrum evolves trivially according to simple analytic rules. This in particular reflects that solitons are conserved by the KdV equation. However, in reality, the nonlinear spectrum will change during evolution due to deviations from the KdV equation. For example, waves in the ocean are typically multi-directional. Furthermore, the water depth may range into the intermediate regime, e.g. depending on tides and peak periods. It is therefore uncertain how long the nonlinear spectrum of real-world data remains representative. In particular, it is unclear how stable the detected soliton components are during evolution. To assess the effectiveness of the KdV-NFT in representing water wave dynamics under non-ideal conditions, we generated numerical sea states with varying directional spreading in intermediate water (kh=1.036) using the High-Order Spectral Ocean (HOS-Ocean) model for nonlinear evolution. After applying the NFT to space series extracted from these evolving directional wave fields, we observe that the KdV-soliton spectra from the NFT are quite stable for cases with small directional spreading. We in particular observe that the largest soliton amplitude is (sometimes dramatically) more stable than the amplitude of the largest linear mode. For large directional spreading, the applicability is limited to short propagation times and distances, respectively.

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.

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 T_P 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.

Rogue waves are extreme waves in the ocean that appear from nowhere and disappear without a trace. They are usually modelled by the nonlinear Schrödinger equation (NLS), which describes nonlinear phenomena such as modulational instability and solitons on finite backgrounds. In this study, the periodic nonlinear Fourier transform (NFT) for the NLS equation is applied to simulate ocean surface waves in deep water. The temporal and spatial structures of surface waves are obtained by evolving JONSWAP time series using the NLS equation. Several parameters extracted from the NFT spectra of the initial time series are investigated as predictors for the maximum wave height during evolution. We investigate several parameters from the literature, and find that with suitably optimized coefficients, a NFT-based parameter based on the largest unstable mode has a good correlation with the overall maximum wave amplitude. This new spectral criterion can contribute to rogue wave forecasting under extreme sea states.