We investigate exact nonlinear waves on surfaces locally approximating the rotating sphere for two-dimensional inviscid incompressible flow. Our first system corresponds to a $\beta$-plane approximation at the equator, and the second to a $\gamma$ approximation, with the latter describing flow near the poles. We find exact wave solutions in the Lagrangian reference frame that cannot be written down in closed form in the Eulerian reference frame. The wave particle trajectories, contours of potential vorticity and Lagrangian mean velocity take relatively simple forms. The waves possess a non-trivial Lagrangian mean flow that depends on the amplitude of the waves and on a particle label that characterizes values of constant potential vorticity. The mean flow arises due to potential vorticity conservation on fluid particles. Solutions over the entire space are generated by assuming that the flow far from the origin is zonal and there is a region of uniform potential vorticity between this zonal flow and the waves. In the $\gamma$ approximation, a class of waves is found that, based on analogous solutions on the plane, we call Ptolemaic vortex waves. The mean flow of some of these waves, which we can describe in highly nonlinear scenarios due to the exact nature of the solutions, resembles polar jet streams. Several illustrative solutions are used as initial conditions in the fully spherical rotating Navier–Stokes equations, where integration is performed via the numerical scheme presented in Salmon & Pizzo (Atmosphere, vol. 14, issue 4, 2023, 747). The potential vorticity contours found from these numerical experiments vary between stable permanent progressive form and fully turbulent flows generated by wave breaking.

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.

Two-dimensional turbulence transfers its energy towards the lowest mode in the domain, but domain geometry exerts a powerful control. On the sphere, with its three axes of rotational symmetry, angular momentum conservation prevents energy from entering the three lowest modes – those corresponding to the spherical harmonics $Y_1^0$ and $Y_1^{\pm 1}$ – because the amplitudes of these three modes are proportional to the three conserved components of the angular momentum vector. Non-spherical ellipsoids partly or completely break the rotational symmetry corresponding to angular momentum conservation. The flow on spheroids, which have only one axis of rotational symmetry, conserves only a single component of angular momentum. If the axis of symmetry is taken to be the $z$-axis, then only the $z$-component of angular momentum is conserved. Energy can flow into the other two lowest modes. The general triaxial ellipsoid breaks all rotational symmetries, thus angular momentum is not conserved, and energy can flow into any mode. We describe numerical experiments that confirm these predictions.

A light breeze rising over calm water initiates an intricate chain of events that culminates in a centimetres-deep turbulent shear layer capped by gravity–capillary ripples. At first, viscous stress accelerates a laminar wind-drift layer until small surface ripples appear. The surface ripples then catalyse the growth of a second instability in the wind-drift layer, which eventually sharpens into along-wind jets and downwelling plumes, before devolving into three-dimensional turbulence. In this paper, we compare laboratory experiments with simplified, wave-averaged numerical simulations of wind-drift layer evolution beneath monochromatic, constant-amplitude surface ripples seeded with random initial perturbations. Despite their simplicity, our simulations reproduce many aspects of the laboratory-based observations – including the growth, nonlinear development and turbulent breakdown the wave-catalysed instability – generally validating our wave-averaged model. But we also find that the simulated development of the wind-drift layer is disturbingly sensitive to the amplitude of the prescribed surface wave field, such that agreement is achieved through suspiciously careful tuning of the ripple amplitude. As a result of this sensitivity, we conclude that wave-averaged models should really describe the coupled evolution of the surface waves together with the flow beneath to be regarded as truly ‘predictive’.

The role of the Lagrangian mean flow, or drift, in modulating the geometry, kinematics and dynamics of rotational and irrotational deep-water surface gravity waves is examined. A general theory for permanent progressive waves on an arbitrary vertically sheared steady Lagrangian mean flow is derived in the Lagrangian reference frame and mapped to the Eulerian frame. A Lagrangian viewpoint offers tremendous flexibility due to the particle labelling freedom and allows us to reveal how key physical wave behaviour arises from a kinematic constraint on the vorticity of the fluid, inter alia the nonlinear correction to the phase speed of irrotational finite amplitude waves, the free surface geometry and velocity in the Eulerian frame, and the connection between the Lagrangian drift and the Benjamin–Feir instability. To complement and illustrate our theory, a small laboratory experiment demonstrates how a specially tailored sheared mean flow can almost completely attenuate the Benjamin–Feir instability, in qualitative agreement with the theory. The application of these results to problems in remote sensing and ocean wave modelling is discussed. We provide an answer to a long-standing question: remote sensing techniques based on observing current-induced shifts in the wave dispersion will measure the Lagrangian, not the Eulerian, mean current.

This paper presents laboratory measurements of surface transport due to non-breaking and breaking deep-water focusing surface wave packets and examines the dependence of the transport on the wave packet bandwidth, $\varDelta$. This extends the work of Deike et al. (J. Fluid Mech., vol. 829, 2017, pp. 364–391) and Lenain et al. (J. Fluid Mech., vol. 876, 2019, p. R1), where similar numerical and laboratory experiments were conducted, but the bandwidth was held constant. In this paper, it is shown that the transport is strongly affected by the bandwidth. A model for the horizontal length scale of the breaking region is proposed that incorporates the bandwidth, central frequency, the linear prediction of the slope at focusing and the breaking threshold slope of the wave packet. This is then evaluated with data from archived and new laboratory experiments, and agreement is found. Furthermore, the horizontal length scale of the breaking region implies modifications to the model of the energy dissipation rate from Drazen et al. (J. Fluid Mech., vol. 611, 2008, pp. 307–332). This modification accounts for differing trends in the dissipation rate caused by the bandwidth in the available laboratory data.

This paper explores an idealized model of the ocean surface in which widely separated surface-wave packets and point vortices interact in two horizontal dimensions. We start with a Lagrangian which, in its general form, depends on the fields of wave action, wave phase, stream function and two additional fields that label and track the vertical component of vorticity. By assuming that the wave action and vorticity are confined to infinitesimally small, widely separated regions of the flow, we obtain model equations that are analogous to, but significantly more general than, the familiar system consisting solely of point vortices. We analyse stable and unstable harmonic solutions, solutions in which wave packets eventually coincide with point vortices (violating our assumptions), and solutions in which the wave vector eventually blows up. Additionally, we show that a wave packet induces a net drift on a passive vortex in the direction of wave propagation which is equivalent to Darwin drift. Generalizing our analysis to many wave packets and vortices, we examine the influence of wave packets on an otherwise unstable vortex street and show analytically, according to linear stability analysis, that the wave-packet-induced drift can stabilize the vortex street. The system is then numerically integrated for long times and an example is shown in which the configuration remains stable, which may be particularly relevant for the upper ocean.

An exact set of equations describing deep-water irrotational surface gravity waves, originally proposed by Balk (Phys. Fluids, vol. 8 (2), 1996, pp. 416–420), and studied in the case of standing waves by Longuet-Higgins (J. Fluid Mech., vol. 423, 2000, pp. 275–291) and Longuet-Higgins (Proc. R. Soc. Lond. A, vol. 457 (2006), 2001, pp. 495–510), are analytically examined and put in a form more suitable for practical applications. The utility of this approach is its simplicity. The Lagrangian is a low-order polynomial in the Fourier coefficients, leading to equations of motion that are correspondingly of low degree. The structure of these equations is examined, and the existence of solutions is considered. To gain intuition about the system of equations, a truncated model is first examined. Linear stability analysis is performed, and the evolution of the fully nonlinear system is discussed. The theory is then applied to fully resolved permanent progressive deep-water waves and a simple method for finding the eigenvalues and eigenvectors of the normal modes of this system is presented. Potential applications of these results are then discussed.

While it has long been recognized that Lagrangian drift at the ocean surface plays a critical role in the kinematics and dynamics of upper ocean processes, only recently has the contribution of wave breaking to this drift begun to be investigated through direct numerical simulations (Deike et al., J. Fluid Mech., vol. 829, 2017, pp. 364–391; Pizzo et al., J. Phys. Oceanogr., vol. 49(4), 2019, pp. 983–992). In this work, laboratory measurements of the surface Lagrangian transport due to focusing deep-water non-breaking and breaking waves are presented. It is found that wave breaking greatly enhances mass transport, compared to non-breaking focusing wave packets. These results are in agreement with the direct numerical simulations of Deike et al. (J. Fluid Mech., vol. 829, 2017, pp. 364–391), and the increased transport due to breaking agrees with their scaling argument. In particular, the transport at the surface scales with $S$, the linear prediction of the maximum slope at focusing, while the surface transport due to non-breaking waves scales with $S^{2}$, in agreement with the classical Stokes prediction.

Geometric, kinematic and dynamic properties of focusing deep-water surface gravity wave packets are examined in a simplified model with the intent of deriving a wave breaking threshold parameter. The model is based on the spatial modified nonlinear Schrödinger equation of Dysthe (Proc. R. Soc. Lond. A, vol. 369 (1736), 1979, pp. 105–114). The evolution of initially narrow-banded and weakly nonlinear chirped Gaussian wave packets are examined, by means of a trial function and a variational procedure, yielding analytic solutions describing the approximate evolution of the packet width, amplitude, asymmetry and phase during focusing. A model for the maximum free surface gradient, as a function of $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6E5}$, for $\unicode[STIX]{x1D716}$ the linear prediction of the maximum slope at focusing and $\unicode[STIX]{x1D6E5}$ the non-dimensional packet bandwidth, is proposed and numerically examined, indicating a quasi-self-similarity of these focusing events. The equations of motion for the fully nonlinear potential flow equations are then integrated to further investigate these predictions. It is found that a model of this form can characterize the bulk partitioning of $\unicode[STIX]{x1D716}-\unicode[STIX]{x1D6E5}$ phase space, between non-breaking and breaking waves, serving as a breaking criterion. Application of this result to better understanding air–sea interaction processes is discussed.

The Lagrangian transport due to non-breaking and breaking focusing wave packets is examined. We present direct numerical simulations of the two-phase air–water Navier–Stokes equations describing focusing wave packets, investigating the Lagrangian drift by tracking tracer particles in the water before, during and after the breaking event. The net horizontal transport for non-breaking focusing packets is well described by the classical Stokes drift, both at the surface and in the bulk of the fluid, where the e-folding scale of the evanescent vertical profile is given by the characteristic wavenumber. For focusing wave packets that lead to breaking, we observe an added drift that can be ten times larger than the classical Stokes drift for a non-breaking packet at the surface, while the initial depth of the broken fluid scales with the wave height at breaking. We find that the breaking induced Lagrangian transport scales with the breaking strength. A simple scaling argument is proposed to describe this added drift and is found to be consistent with the direct numerical simulations. Applications to upper ocean processes are discussed.

A simple criterion for water particles to surf an underlying surface gravity wave is presented. It is found that particles travelling near the phase speed of the wave, in a geometrically confined region on the forward face of the crest, increase in speed. The criterion is derived using the equation of John (Commun. Pure Appl. Maths, vol. 6, 1953, pp. 497–503) for the motion of a zero-stress free surface under the action of gravity. As an example, a breaking water wave is theoretically and numerically examined. Implications for upper-ocean processes, for both shallow- and deep-water waves, are discussed.