Pieter Bart Smit

Propagating waves on the ocean surface can be represented as a stochastic process whose statistics are characterized by a spectrum. This paper reviews methods for measuring the wave spectrum and related quantities. Observations begin by sensing fluid dynamical properties of the sea surface over space and/or time. Visual observations, collected routinely since the mid‐18th century, comprise the longest‐running wave record. Nearshore measurement methods continue to advance, including traditional pressure and acoustic sensing as well as newer technologies like distributed acoustic sensing and LiDAR. Detailed small‐scale wave physics can now be explored with measurement techniques using light, including stereo‐imaging and polarimetry. Reductions in the size, cost, and power consumption of microelectronics have propagated through ocean wave instrumentation, most notably in wave buoys. Global networks of freely drifting miniature wave buoys offer novel observational capabilities. Remote sensing techniques based on radar and LiDAR continue to evolve and are widely deployed from land, ships, aircraft, autonomous vehicles, and satellites. Spaceborne altimeters form one of the most important records of wave height, and new spaceborne sensors now observe directional spectra globally with sampling akin to traditional altimetry. Aircraft and autonomous systems provide strategic sampling capabilities for detailed process studies and access to extreme storm environments. The quality and quantity of ocean wave measurements have never been greater. This review aims to help make sense of it all.

Nonlinear deep-water gravity waves can be described by Stokes-type perturbation expansions, but these are not unique. Multiple formulations exist that differ in how the perturbation amplitude is defined. We introduce a parameterized family of solutions and examine how different choices affect key wave characteristics—wave height, action, energy, momentum, and variance—in both deterministic and stochastic frameworks. For the parameterized form that aligns with the limit of the Zakharov equation for a single free-wave component, fourth-order corrections vanish from action and momentum, yielding a form that generalizes naturally to irregular and wave fields interacting with the environment. We discuss why different formulations arise in the literature and argue that while in general application dependent, this generalization property makes this the preferred form when considering the interaction of waves with the environment.

Peak periods estimated from finite-resolution frequency spectra are necessarily discrete. For wind-generated surface gravity waves, conflicting considerations of robust (quasi)-stationary statistics, and high spectral resolution, combined with the inverse relation between frequency and period, this typically implies that swell periods (above 10 s) are resolved at best at s intervals. Here, we consider a method to improve peak period estimates for finite-resolution spectra. Specifically, we propose to define the peak period based on continuous spectra derived from a spline-based interpolation of the discretely sampled monotone cumulative distribution function. The method may directly be applied to existing discrete spectra—the original time-domain data (which may not be available) are not required. We compare reconstructed spectra and derived peak periods to parametric shapes and field data. Peak estimates are markedly improved, allowing for better tracking of, e.g., swells. The proposed method also marginally improves spectral levels and shape for a given discretely sampled estimate.

Historically, the sparseness of in situ open‐ocean wave and weather observations has severely limited the forecast skill of weather over the ocean with major social and economic consequences for coastal communities and maritime industries. Ocean surface waves, specifically, are important for the interaction between atmosphere and ocean, and thus key in modeling weather and climate processes. Here, we investigate the improvements achievable from a large distributed sensor network combined with advances in assimilation strategies. Wave spectra from a global network of over 600 Sofar Spotter buoys are assimilated into an operational global wave forecast via optimal interpolation to update model spectra to best fit observations. We demonstrate end‐to‐end improvements in forecast skill of significant wave height of 38%, and up to 45% for other bulk parameters. This shows distributed observations of the air‐sea interface, with advances in assimilation strategies, can reduce uncertainty in forecasts to dramatically improve earth system modeling.

A fully nonlinear non-dispersive energy balance for surfzone waves is derived based on the nonlinear shallow water equations to study the nearshore dynamics of infragravity (IG) waves. Based on simulations of waves on a relatively moderate and mild beach slope with a non-hydrostatic wave-flow model (SWASH), the new theory shows that spatial gradients in IG energy flux are nearly completely balanced by the combined effect of bottom stresses and predominantly nonlinear triad interactions. The new balance confirms many features of existing weakly nonlinear theories, and yields an improved description in the inner surfzone where waves become highly nonlinear. A gain of IG energy flux throughout the shoaling and outer surfzones is driven by triad interactions between IG waves and pairs of sea-swell (SS) waves. The IG energy flux decreased in the inner surfzone, primarily through an energy cascade to the swell-band and superharmonic frequencies where wave energy is ultimately dissipated. Dissipation by bottom friction was weak on both slopes. The IG wave breaking, characterized by triads between three IG or two IG waves and one SS wave, was significant only deep inside the surfzone of the mild slope. Even though IG waves broke on the mild slope, nonlinear interactions between IG waves and pairs of SS waves were responsible for at least half of the net IG flux loss.

Historically, the sparseness of in situ open-ocean wave and weather observations has severely limited the forecast skill of weather over the ocean with major social and economic consequences for coastal communities and maritime industries. Ocean surface waves, specifically, are important for the interaction between atmosphere and ocean, and thus key in modeling weather and climate processes. Here, we investigate the improvements achievable from a large distributed sensor network combined with advances in assimilation strategies. Wave spectra from a global network of over 600 Sofar Spotter buoys are assimilated into an operational global wave forecast via optimal interpolation to update model spectra to best fit observations. We demonstrate end-to-end improvements in forecast skill of significant wave height of 38\%, and up to 45\% for other bulk parameters. This shows distributed observations of the air-sea interface, with advances in assimilation strategies, can reduce uncertainty in forecasts to dramatically improve earth system modeling.

We consider the recent conclusion of Kouskoulas and Toledo [Phys. Fluids 32, 076603 (2020)] on the existence of resonant triads for surface waves propagating in a uniform current and show that their conclusion is based on a misinterpretation of the geometry of the wave components. We provide a proof of nonexistence for the more general case of waves traveling at arbitrary directions in the horizontal plane.

Wave forecasting in ocean and coastal waters commonly relies on spectral models based on the spectral action balance equation. These models assume that different wave components are statistically independent and as a consequence cannot resolve wave interference due to statistical correlation between crossing waves, as may be found in, for instance, a focal zone. This study proposes a statistical model for the evolution of wave fields over non-uniform currents and bathymetry that retains the information on the correlation between different wave components. To this end, the quasi-coherent model (Smit & Janssen, J. Phys. Oceanogr., vol. 43, 2013, pp. 1741–1758) is extended to allow for wave–current interactions. The outcome is a generalized action balance model that predicts the evolution of the wave statistics over variable media, while preserving the effect of wave interferences. Two classical examples of wave–current interaction are considered to demonstrate the statistical contribution of wave interferences: (1) swell field propagation over a jet-like current and (2) the interaction of swell waves with a vortex ring. In both examples cross-correlation terms lead to development of prominent interference structures, which significantly change the wave statistics. Comparison with results of the SWAN model demonstrates that retention of cross-correlation terms is essential for accurate prediction of wave statistics in shear-current-induced focal zones.

Observations collected as part of the Cascadia Amphibious Array (Oregon‐Washington coast) show that ocean surface infragravity (IG) waves (periods from 30 to 250 s) in the Pacific Ocean originate from the coastal zone. With the deepest observations at 1909 m, we identify tidal modulation and a dramatic reduction in energy as IG waves cross the shelf break. Observations are consistent with radiation from an isotropic, nearshore source, and provide quantitative verification linking oceanic IG fluctuations with nearshore motions. Our analysis demonstrates that trapping of IG waves on the shelf break strongly reduces energy radiation into ocean basins. Further, for narrow continental shelves (U.S. West Coast), energy levels past the shelf break depend only weakly on shelf geometry. Consequently, oceanic energy levels can be related to coastal sources by scaling relations predicted by geometric optics, simplifying the effort to estimate the impact of radiated IG‐waves on remote sites (e.g., to estimate seismic noise levels or impact on ice shelves).