Axisymmetric standing waves occur across a wide range of free surface flows. When these waves reach a critical height (steepness), wave breaking and jet formation occur. For travelling surface gravity waves, wave breaking is generally considered to limit wave height and reversible wave motion. In the ocean, the behaviour of directionally spread waves lies between the limits of purely travelling (two dimensions) and axisymmetric (three dimensions). Hence, understanding wave breaking and jet formation on axisymmetric surface gravity waves is an important step in understanding extreme and breaking waves in the ocean. We examine an example of axisymmetric wave breaking and jet formation colloquially known as the ‘spike wave’, created in the FloWave circular wave tank at the University of Edinburgh, UK. We generate this spike wave with maximum crest amplitudes of 0.15–6.0 m (0.024–0.98 when made non-dimensional by characteristic radius), with wave breaking occurring for crest amplitudes greater than 1.0 m (0.16 non-dimensionalised). Unlike two-dimensional travelling waves, wave breaking does not limit maximum crest amplitude, and our measurements approximately follow the jet height scaling proposed by Ghabache et al. (J. Fluid Mech., vol. 761, 2014, pp. 206–219) for cavity collapse. The spike wave is predominantly created by linear dispersive focusing. A trough forms, then collapses producing a jet, which is sensitive to the trough's shape. The evolution of the jets that form in our experiments is predicted well by the hyperbolic jet model proposed by Longuet–Higgins (J. Fluid Mech., vol. 127, 1983, pp. 103–121), previously applied to jets forming on bubbles.

We study the evolution of unidirectional water waves from a randomly forced input condition with uncorrelated Fourier components. We examine the kurtosis of the linearised free surface as a convenient proxy for the probability of a rogue wave. We repeat the laboratory experiments of Onorato et al. (Phys. Rev. E, vol. 70, 2004, 067302), both experimentally and numerically, and extend the parameter space in our numerical simulations. We consider numerical simulations based on the modified nonlinear Schrödinger equation and the fully nonlinear water wave equations, which are in good agreement. For low steepness, existing analytical models based on the nonlinear Schrödinger equation (NLS) are found to be accurate. For cases which are steep or have very narrow bandwidths, these analytical models over-predict the rate at which excess kurtosis develops. In these steep cases, the kurtosis in both our experiments and numerical simulations peaks before returning to an equilibrium level. Such transient maxima are not predicted by NLS-based analytical models. Above a certain threshold of steepness, the steady-state value of kurtosis is primarily dependent on the spectral bandwidth. We also examine how the average shape of extreme events is modified by nonlinearity over the evolution distance, showing significant asymmetry during the initial evolution, which is greatly reduced once the spectrum has reached equilibrium. The locations of the maxima in asymmetry coincide approximately with the locations of the maxima in kurtosis.

We have investigated steep three-dimensional surface gravity wave groups formed by dispersive focusing using a fully nonlinear potential flow solver. We find that third-order resonant interactions result in rapid energy transfers to higher wavenumbers and reduced directional spreading during focusing, followed by spectral broadening during defocusing, forming steep wave groups with augmented kinematics and a prolonged lifespan. If the wave group is initially narrow-banded, quasi-degenerate interactions arise, characterised by energy transfers along the resonance angle, ${\pm }35.26^{\circ }$, of the Phillips ‘figure-of-eight’ loop. Spectral broadening due to the quasi-degenerate interactions facilitates non-degenerate interactions, characterised by oblique energy transfers at approximately ${\pm }55^{\circ }$ to the spectral peak. We consider the influence of steepness, finite depth, directional spreading and the high-wavenumber tail on spectral evolution. Steepness is found to augment both the quasi-degenerate and non-degenerate interactions similarly. However, a reduction in depth is found to weaken the quasi-degenerate interactions more severely than the non-degenerate interactions. We observe that increased directional spreading reduces spectral evolution, partially because wave groups with more spreading focus for a shorter duration due to linear dispersion. However, we also find that directional spreading reduces the peak rates of energy transfer. Inclusion of the high-wavenumber tail of the Joint North Sea Wave Project spectrum further reduces rates of energy transfer compared with a Gaussian wavenumber spectrum. Thus, directional spreading and the high-wavenumber tail may be integral to a form of spectral equilibrium that reduces rapid energy transfers during a steep wave event.

The Lagrangian mass transport by non-dissipating surface gravity wavepackets consists of the Stokes drift and the wave-induced return flow. We examine how directional spreading and density stratification affect this mass transport for an isolated non-dissipating wavepacket in deep water using a perturbation expansion. For an unstratified ocean, we show that the net displacement by the return flow is finite, negative, the same at all vertical levels and inversely proportional to the depth for spanwise-infinite packets representing unidirectional (two-dimensional) seas, but zero for spanwise-localised packets representing directionally spread seas (three-dimensional). We resolve this difference by demonstrating that a transition between two-dimensional-like (finite) and three-dimensional-like (zero) displacement occurs on a time scale inversely proportional to the degree of directional spreading. For a stratified ocean, we show that in two dimensions the net displacement profile by the return flow oscillates slowly with depth, with a wavelength dependent on the ratio of buoyancy frequency to the surface wave group velocity, and infinite displacements are predicted when the surface wavepacket resonantly excites internal waves. In three dimensions, the net displacement remains zero in the presence of stratification, but finite-time displacements may be appreciably altered.

Owing to the interplay between the forward Stokes drift and the backward wave-induced Eulerian return flow, Lagrangian particles underneath surface gravity wave groups can follow different trajectories depending on their initial depth below the surface. The motion of particles near the free surface is dominated by the waves and their Stokes drift, whereas particles at large depths follow horseshoe-shaped trajectories dominated by the Eulerian return flow. For unidirectional wave groups, a small net displacement in the direction of travel of the group results near the surface, and is accompanied by a net particle displacement in the opposite direction at depth. For deep-water waves, we study these trajectories experimentally by means of particle tracking velocimetry in a two-dimensional flume. In doing so, we provide visual illustration of Lagrangian trajectories under groups, including the contributions of both the Stokes drift and the Eulerian return flow to both the horizontal and the vertical Lagrangian displacements. We compare our experimental results to leading-order solutions of the irrotational water wave equations, finding good agreement.

We examine the flows induced by horizontally modulated, vertically confined (or guided), internal wavepackets in a stratified, Boussinesq fluid. The wavepacket induces both an Eulerian flow and a Stokes drift, which together determine the Lagrangian transport of passive tracers. We derive equations describing the wave-induced flows in arbitrary stable stratification and consider four special cases: a two-layer fluid, symmetric and asymmetric piecewise constant (‘top-hat’) stratification and, more representative of the ocean, exponential stratification. In a two-layer fluid, the Stokes drift is positive everywhere with the peak value at the interface, whereas the Eulerian flow is negative and uniform with depth for long groups. Combined, the net depth-integrated Lagrangian transport is zero. If one layer is shallower than the other, the wave-averaged interface displaces into that layer making the Eulerian flow in that layer more negative and the Eulerian flow in the opposite layer more positive so that the depth-integrated Eulerian transports are offset by the same amount in each layer. By contrast, in continuous stratification the depth-integrated transport due to the Stokes drift and Eulerian flow are each zero, but the Eulerian flow is singular if the horizontal phase speed of the induced flow equals the group velocity of the wavepacket, giving rise to a single resonance in uniform stratification (McIntyre, J. Fluid Mech., vol. 60, 1973, pp. 801–811). In top-hat stratification, this single resonance disappears, being replaced by multiple resonances occurring when the horizontal group velocity of the wavepacket matches the horizontal phase speed of higher-order modes. Furthermore, if the stratification is not vertically symmetric, then the Eulerian induced flow varies as the inverse squared horizontal wavenumber for shallow waves, the same as for the asymmetric two-layer case. This ‘infrared catastrophe’ also occurs in the case of exponential stratification suggesting significant backward near-surface transport by the Eulerian induced flow for modulated oceanic internal modes. Numerical simulations are performed confirming these theoretical predictions.

Freak or rogue waves are so called because of their unexpectedly large size relative to the population of smaller waves in which they occur. The 25.6 m high Draupner wave, observed in a sea state with a significant wave height of 12 m, was one of the first confirmed field measurements of a freak wave. The physical mechanisms that give rise to freak waves such as the Draupner wave are still contentious. Through physical experiments carried out in a circular wave tank, we attempt to recreate the freak wave measured at the Draupner platform and gain an understanding of the directional conditions capable of supporting such a large and steep wave. Herein, we recreate the full scaled crest amplitude and profile of the Draupner wave, including bound set-up. We find that the onset and type of wave breaking play a significant role and differ significantly for crossing and non-crossing waves. Crucially, breaking becomes less crest-amplitude limiting for sufficiently large crossing angles and involves the formation of near-vertical jets. In our experiments, we were only able to reproduce the scaled crest and total wave height of the wave measured at the Draupner platform for conditions where two wave systems cross at a large angle.

For sufficiently directionally spread surface gravity wave groups, the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (J. Fluid Mech. vol. 13, 1962, pp. 481–504), can turn into a set-up. Using a multiple-scale expansion for two crossing wave groups, we examine the structure and magnitude of this wave-averaged set-up, which is part of a crossing wave pattern that behaves as a modulated partial standing wave: in space, it consists of a rapidly varying standing-wave pattern slowly modulated by the product of the envelopes of the two groups; in time, it grows and decays on the slow time scale associated with the translation of the groups. Whether this crossing wave pattern actually enhances the surface elevation at the point of focus depends on the phases of the linear wave groups, unlike the set-down, which is always negative and inherits the spatial structure of the underlying envelope(s). We present detailed laboratory measurements of the wave-averaged free surface, examining both single wave groups, varying the degree of spreading from small to very large, and the interaction between two wave groups, varying both the degree of spreading and the crossing angle between the groups. In both cases, we find good agreement between the experiments, our simple expressions for the set-down and set-up, and existing second-order theory based on the component-by-component interaction of individual waves with different frequencies and directions. We predict and observe a set-up for wave groups with a Gaussian angular amplitude distribution with standard deviations of above $30{-}40^{\circ }$ ($21{-}28^{\circ }$ for energy spectra), which is relatively large for realistic sea states, and for crossing sea states with angles of separation of $50{-}70^{\circ }$ and above, which are known to occur in the ocean.

We examine the wave-induced flow of small-amplitude, quasi-monochromatic, three-dimensional, Boussinesq internal gravity wavepackets in a uniformly stratified ambient. It has been known since Bretherton (J. Fluid Mech., vol. 36 (4), 1969, pp. 785–803) that one-, two- and three-dimensional wavepackets induce qualitatively different flows. Whereas the wave-induced mean flow for compact three-dimensional wavepackets consists of a purely horizontal localized circulation that translates with and around the wavepacket, known as the Bretherton flow, such a flow is prohibited for a two-dimensional wavepacket of infinite spanwise extent, which instead induces a non-local internal wave response that is long compared with the streamwise extent of the wavepacket. One-dimensional (horizontally periodic) wavepackets induce a horizontal, non-divergent unidirectional flow. Through perturbation theory for quasi-monochromatic wavepackets of arbitrary aspect ratio, we predict for which aspect ratios which type of induced mean flow dominates. We compose a regime diagram that delineates whether the induced flow is comparable to that of one-, two- or compact three-dimensional wavepackets. The predictions agree well with the results of fully nonlinear three-dimensional numerical simulations.

A simple geometry that exhibits near motion trapping is tested experimentally, along with perturbed versions of the structure. The motion of the freely floating structure and the surrounding wave field is tracked and the near-motion-trapped mode is found, characterised by a slowly decaying heave motion with very small linear radiation of energy. It is found that the latter property is a better discriminator of the perturbed geometries as viscous damping masks fine differences in radiation damping as far as the motion of the structure is concerned. The magnitude of this viscous damping is reasonably well predicted by a simple Stokes oscillatory boundary layer analysis.

Closed-form solutions describing the behaviour of two-dimensional planar turbulent rising plumes and fountains from horizontal planar area and line sources in unconfined quiescent environments of uniform density are proposed. Extending the analysis on axisymmetric releases by van den Bremer & Hunt (J. Fluid Mech., vol. 644, 2010, pp. 165–192) to planar releases, the local flux balance parameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\varGamma =\varGamma (z)$ is instrumental in describing the bulk behaviour of steady Boussinesq and non-Boussinesq planar plumes and the initial rise behaviour of Boussinesq planar fountains as a function of height $z$. Expressions for the asymptotic virtual source correction are developed and the results elucidated by ‘scale diagrams’ (cf. Morton & Middleton, J. Fluid Mech., vol. 58, 1973, pp. 165–176) showing certain characteristic heights for different source conditions. These diagrams capture all the different manifestations of plume behaviour, encompassing fountains, jets, source-momentum-dominated or ‘forced’ plumes, pure plumes and source-buoyancy-dominated or ‘lazy’ plumes, and their associated key features. Other flow features identified include a gravity-driven deceleration regime and a mixing-driven regime for forced fountains. Deceleration in lazy fountains is purely gravity-driven. The results can be shown to be valid for both Boussinesq and non-Boussinesq plumes (but not for non-Boussinesq fountains) thus resulting in universal solutions valid for both cases provided the entrainment velocity is unaffected by non-Boussinesq effects. This paper presents and explores these universal solutions. An accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 245–258) examines the implications for non-Boussinesq plumes. The existing solutions of Lee & Emmons (J. Fluid Mech., vol. 11, 1961, pp. 353–368) generalized herein are valid for a constant entrainment coefficient $\alpha $. New results for an entrainment coefficient that varies linearly with $\varGamma (z)$ and thus captures experimental values far more realistically are presented for forced plumes.

In an accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 210–244) closed-form solutions, describing the behaviour of two-dimensional planar turbulent rising plumes from horizontal planar area and line sources in unconfined quiescent environments of uniform density, that are universally applicable to Boussinesq and non-Boussinesq plumes, are proposed. This universality relies on an entrainment velocity unmodified by non-Boussinesq effects, an assumption that is derived in the literature based on similarity arguments and is, in fact, in contradiction with the axisymmetric case, in which entrainment is modified by non-Boussinesq effects. Exploring these solutions, we show that a non-Boussinesq plume model predicts exactly the same behaviour with height for a pure plume as would a Boussinesq model, whereas the effects on forced and lazy plumes are opposing. Non-intuitively, the non-Boussinesq model predicts larger fluxes of volume and mass for lazy plumes, but smaller fluxes for forced plumes at any given height compared to the Boussinesq model. This raises significant questions regarding the validity of the unmodified entrainment model for planar non-Boussinesq plumes based on similarity arguments and calls for detailed experiments to resolve this debate.

Closed-form solutions describing the behaviour of buoyant axisymmetric turbulent rising plumes and fountains, emitted vertically from area sources in unconfined quiescent environments of uniform density, are proposed in a form that is universally applicable to Boussinesq and non-Boussinesq plumes. This paper, thereby, generalizes the results obtained separately for steady Boussinesq and non-Boussinesq plumes, including asymptotic virtual source corrections. The flux balance parameter Γ = Γ(z), a local Richardson number, is instrumental in describing the behaviour of steady plumes and the initial rise behaviour of fountains with height z. Non-dimensional graphs (cf. the ‘scale diagrams’ of Morton & Middleton, J. Fluid Mech., vol. 58, 1973, pp. 165–176) are plotted, showing certain characteristic heights for different source conditions, characterized by one single source flux balance parameter, giving a unique representation of the behaviour of Boussinesq fountains and both Boussinesq and non-Boussinesq plumes. Finally, a length scale has been identified that characterizes the height over which non-Boussinesq effects are important for lazy plumes rising from area sources.