Three-dimensional wave breaking

Although a ubiquitous natural phenomenon, the onset and subsequent process of surface wave breaking are not fully understood. Breaking affects how steep waves become and drives air-sea exchanges 1 . Most seminal and state-of-the-art research on breaking is underpinned by the assumption of two-dimensionality, although ocean waves are three dimensional. We present experimental results that assess how three-dimensionality affects breaking, without putting limits on the direction of travel of the waves. We show that the breaking-onset steepness of the most directionally spread case is double that of its unidirectional counterpart. We identify three breaking regimes. As directional spreading increases, horizontally overturning 'travelling-wave breaking' (I), which forms the basis of two-dimensional breaking, is replaced by vertically jetting 'standing-wave breaking' (II). In between, 'travellingstanding-wave breaking' (III) is characterized by the formation of vertical jets along a fast-moving crest. The mechanisms in each regime determine how breaking limits steepness and affects subsequent air-sea exchanges. Unlike in two dimensions, three-dimensional wave-breaking onset does not limit how steep waves may become, and we produce directionally spread waves 80% steeper than at breaking onset and four times steeper than equivalent two-dimensional waves at their breaking onset. Our observations challenge the validity of state-of-the-art methods used to calculate energy dissipation and to design offshore structures in highly directionally spread seas.

Wave breaking continues to be at the forefront of ocean wave research 2-9 . Although it is a widely observable and ubiquitous natural phenomenon, the onset and subsequent process of wave breaking are not fully understood. Alongside not being fully understood, interest in wave breaking is also driven by the central role it plays in key oceanographic and air-sea interaction processes that, in turn, impact the world's climate 1,10 . As waves become very large (steep), breaking occurs, initiating an irreversible turbulent process. The breaking process is the main mechanism for dissipating wave energy in the ocean and affects the transfer of mass, momentum, energy and heat between the air and the sea. Understanding how and when energy is dissipated is crucial to the accurate modelling of ocean waves 11 and is one of the most pressing unresolved issues in wave forecasting 12 . Uncertainty also remains in how breaking waves affect the production of sea spray and bubble-mediated gas exchange, both key factors in climate modelling 1 . Wave breaking is thought to limit the size that waves can grow to, making it an important factor in the formation of extreme or rogue waves 8 . Breaking waves also constitute the most severe loading conditions for offshore structures 13 . This lack of understanding of wave breaking is a result of challenges associated with modelling breaking waves both numerically and experimentally. To fully resolve wave breaking numerically requires computationally costly, high-fidelity models such as direct numerical simulations of the Navier-Stokes equations (for example, ref. 14). Experimentally, although not without its complexities, producing breaking waves is somewhat simpler and quicker (for example, ref. 15). However, quantitatively measuring even simple, visually observable properties, such as surface elevation (at high spatial resolution), in the laboratory can be highly challenging, not to mention properties that are invisible to the naked eye, such as fluid kinematics. Both approaches to studying wave breaking (numerical and experimental) have, as a result of these computational and experimental limitations, been carried out primarily for two-dimensional (2D) conditions (in which waves propagate in only a single direction, sometimes referred to as unidirectional or long-crested waves). Thus, although the oceans are clearly three dimensional (3D), an assumption of two-dimensionality underpins most seminal and state-of-the-art research on wave breaking (for example, refs. 15,16). This assumption constitutes one of the major shortcomings in our current understanding of wave breaking.

Understanding when waves will break, wave-breaking onset, is the first step in fully understanding ocean wave breaking. Following a kinematic description of wave breaking (in which breaking occurs when the horizontal fluid velocity u at the crest of a wave is equal to the crest speed C), Stokes 17 first proposed a limiting waveform above which waves may become no steeper and breaking will occur. This limit for 2D periodic (monochromatic) progressive waves propagating on deep water occurs at a steepness of kH/2 = 0.44, where k is the wavenumber and H is the wave height. Waves in the ocean are not monochromatic but comprise a spectrum of interacting wave components of different frequencies and directions. The shape and bandwidth of the spectra that underlie a given set of waves can cause the steepness at which wave