Three-dimensional wave breaking

Abstract

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 exchanges¹. 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, 'travelling-standing-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.

Fig. 1. Surface elevations of maximally steep 3D wave groups differ in space and time.

a–c,g–i, Top and bottom rows show the measured surface elevation of maximally steep non-breaking waves at the time of maximum amplitude with corresponding images above and below. a–c, Directionally spread wave groups (Δθ = 0°) with σ_θ = 0° (a), 20° (b) or 40° (c). g–i, Crossing wave groups (σ_θ = 0°) with Δθ = 90° (g), 135° (h) or 180° (i). d,e, Time series (d) and corresponding frequency spectra (e) measured at the intended point of the linear focus (x = 0 and y = 0) for maximally steep wave groups with fixed Δθ = 0° (top) and fixed σ_θ = 20° (bottom). The line colours go from dark to light as σ_θ and Δθ are increased. f, Visualization of the directional parameter space of our experiments (Δθ = 0°; σ_θ = 0°, 10° or 20°; Extended Data Table 1). Markers with black outlines correspond to the experiments shown in a–c and g–i. In a–c and g–i, the waves travel from left to right.

Fig. 2. The global steepness at which breaking onset occurs increases as a function of directional spreading.

a,b, Breaking-onset steepness for unimodal (Δθ = 0°) (a) and bimodal (b) directionally spread focused wave groups. In b, σ_θ = 0° (circles with a saltire), 10° (circles with a cross) or 20° (open circles). c, 3D surfaces of the breaking-onset steepness of all the wave groups as a function of spreading width σ_θ and crossing angle Δθ. The markers denote measured values of global steepness $S_{\mathrm{M}}^{\star }$. Annotations denote the regimes of the different breaking phenomena observed during the experiments (‘Wave-breaking mechanisms’). The grey shaded area represents the transition between regimes.

Fig. 3. Wave-breaking onset can be parameterized using single-parameter measures of directional spreading.

a–f, Breaking-onset steepness plotted as a function of single-parameter measures of directional spreading (Δθ = 0°; σ_θ = 0°, 10° or 20°; Extended Data Table 1). a,d, Contour maps of how the measures of directional spreading Ω₀ (a) and Ω₁ (d) vary as functions of σ_θ and Δθ, with markers to demonstrate where our experiments are located within this parameter space. b,e, Maximum values of measured local slope ∣∇η∣^⋆ at breaking onset plotted as a function of Ω₀ (b) and Ω₁ (e). The horizontal grey dashed, dotted-dashed and dotted lines correspond to the maximum slopes for a progressive wave ($\tan \left(30^{°}\right)$), a periodic axisymmetric standing wave ($\sqrt{2}/2$) and a 2D periodic standing wave ($\tan \left(45^{°}\right)$), respectively. c,f, Measured global steepness $S_{\mathrm{M}}^{\star }$ at breaking onset plotted as a function of Ω₀ (c) and Ω₁ (f). The horizontal grey dashed lines correspond to S = 0.34 obtained for 2D waves in ref. . Blue markers are approximate values of the breaking steepness $S_{\mathrm{M}}^{\star }$ for an axisymmetric standing wave taken from ref. . See Extended Data Table 2 for the coefficients of the parametric fits shown by the black dotted lines in b, c and f. The black dashed line in b is the parametric fit to the data but shifted vertically so that it is equal to $\tan \left(30^{°}\right)$ at Ω₀ = 0. Error bars correspond to the truncation error associated with the first-order central differencing used to the calculate slope of ∣∇η∣^⋆ from gauges spaced at 0.1 m intervals (Methods). The errors of the global steepness $S_{\mathrm{M}}^{\star }$ are negligibly small and not shown (Methods).

Fig. 4. Three wave-breaking regimes are identified for 3D waves.

Illustrations of the three different wave-breaking phenomena: type I overturning ‘travelling-wave breaking’, type II vertical-jet forming ‘standing-wave breaking’ and type III ‘travelling-standing-wave breaking’. In type III, a near-vertical-jet emanates from a fast-moving ridge that forms as the crossing wave crests constructively interfere. Corresponding images were captured during experiments.

Fig. 5. For 3D waves, breaking onset does not limit crest height.

a, Post-breaking-onset behaviour. The graph shows the measured wave amplitude as a function of input amplitude, both normalized by values at the breaking onset, for experiments carried out at 112.5, 125 and 150% of the input breaking-onset steepness for wave groups with directional width σ_θ = 20° and crossing angles Δθ = 0°, 45°, 90° or 135°. The black dashed line is $a_{\mathrm{M}}/a_{\mathrm{M}}^{\star }=a_0/a_0^{\star }$. b–e, Surface elevation (side on, viewed from the x–z plane) at the times of maximum surface elevation, which correspond to the purple markers in a for σ_θ = 20° and Δθ = 135°. Error bars correspond to ±1 standard deviation (Methods).

Extended Data Fig. 1. Experimental set-up.

Clockwise from the leftmost image, high-density wave gauge array close-up (left), wave tank (top right), a schematic diagram showing wave tank set-up (bottom right), and a detailed diagram of the high-density gauge array in its 6 locations (bottom middle). Photograph: (top right) © Dave Morris (CC BY).

Extended Data Fig. 2. 3D wave steepness and slope.

Diagram showing 3D definitions of zero-crossing wave height H for the mean and normal directions (denoted H_x and H_y, respectively), wavelength λ (k = 2π/λ), and local slope $\mid \nabla \eta \mid =\sqrt{{\eta }_x^2+{\eta }_y^2}$ (where η_x = ∂η/∂x and η_y = ∂η/∂y) for two example wave groups: a unidirectional wave group propagating in the x-direction (panel a) and an axisymmetric wave group (panel d), which are respectively shown by the solid and dashed lines in panels (b) and (c). Both wave groups have the same global steepness S.