Recent studies of water waves propagating over sloping seabeds have shown that sudden transitions from deeper to shallower depths can produce significant increases in the skewness and kurtosis of the free surface elevation and hence in the probability of rogue wave occurrence. Gramstad et al. (Phys. Fluids 25 (12): 122103, 2013) have shown that the key physics underlying these increases can be captured by a weakly dispersive and weakly nonlinear Boussinesq-type model. In the present paper, a numerical model based on an alternative Boussinesq-type formulation is used to repeat these earlier simulations. Although qualitative agreement is achieved, the present model is found to be unable to reproduce accurately the findings of the earlier study. Model parameter tests are then used to demonstrate that the present Boussinesq-type formulation is not well-suited to modelling the propagation of waves over sudden depth transitions. The present study nonetheless provides useful insight into the complexity encountered when modelling this type of problem and outlines a number of promising avenues for further research.

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Periodic water waves generate Stokes drift as manifest from the orbits of Lagrangian particles not fully closing. Stokes drift can contribute to the transport of floating marine litter, including plastic. Previously, marine litter objects have been considered to be perfect Lagrangian tracers, travelling with the Stokes drift of the waves. However, floating marine litter objects have large ranges of sizes and densities, which potentially result in different rates of transport by waves due to the non-Lagrangian behaviour of the objects. Through a combination of theory and experiments for idealised spherical objects in deep-water waves, we show that different objects are transported at different rates depending on their size and density, and that larger buoyant objects can have increased drift compared with Lagrangian tracers. We show that the mechanism for the increased drift observed in our experiments comprises the variable submergence and the corresponding dynamic buoyancy force components in a direction perpendicular to the local water surface. This leads to an amplification of the drift of these objects compared to the Stokes drift when averaged over the wave cycle. Using an expansion in wave steepness, we derive a closed-form approximation for this increased drift, which can be included in ocean-scale models of marine litter transport.

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This paper examines experimentally the dispersion and stability of weakly nonlinear waves on opposing linearly vertically sheared current profiles (with constant vorticity). Measurements are compared against predictions from the unidirectional (1D + 1) constant vorticity nonlinear Schrödinger equation (the vor-NLSE) derived by Thomas et al. (Phys. Fluids, vol. 24, no. 12, 2012, 127102). The shear rate is negative in opposing currents when the magnitude of the current in the laboratory reference frame is negative (i.e. opposing the direction of wave propagation) and reduces with depth, as is most commonly encountered in nature. Compared to a uniform current with the same surface velocity, negative shear has the effect of increasing wavelength and enhancing stability. In experiments with a regular low-steepness wave, the dispersion relationship between wavelength and frequency is examined on five opposing current profiles with shear rates from 0 to −0.87 s⁻¹. For all current profiles, the linear constant vorticity dispersion relation predicts the wavenumber to within the 95 % confidence bounds associated with estimates of shear rate and surface current velocity. The effect of shear on modulational instability was determined by the spectral evolution of a carrier wave seeded with spectral sidebands on opposing current profiles with shear rates between 0 and −0.48 s⁻¹. Numerical solutions of the vor-NLSE are consistently found to predict sideband growth to within two standard deviations across repeated experiments, performing considerably better than its uniform-current NLSE counterpart. Similarly, the amplification of experimental wave envelopes is predicted well by numerical solutions of the vor-NLSE, and significantly over-predicted by the uniform-current NLSE.

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Uncertainty affects estimates of the power potential of tidal currents, resulting in large ranges in values reported for sites such as the Pentland Firth, UK. Kreitmair et al. (2019, R. Soc. open sci. 6, 180941. (doi:10.1098/rsos.191127)) have examined the effect of uncertainty in bottom friction on tidal power estimates by considering idealized theoretical models. The present paper considers the role of bottom friction uncertainty in a realistic numerical model of the Pentland Firth spanned by different fence configurations. We find that uncertainty in removable power estimates resulting from bed roughness uncertainty depends on the case considered, with relative uncertainty between 2% (for a fully spanned channel with small values of mean roughness and input uncertainty) and 44% (for an asymmetrically confined channel with high values of bed roughness and input uncertainty). Relative uncertainty in power estimates is generally smaller than (input) relative uncertainty in bottom friction by a factor of between 0.2 and 0.7, except for low turbine deployments and very high mean values of friction. This paper makes a start at quantifying uncertainty in tidal stream power estimates, and motivates further work for proper characterization of the resource, accounting for uncertainty inherent in resource modelling.

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The net movement of Lagrangian particles under water waves comprises a Stokes drift in the direction of wave propagation and an Eulerian return flow in the opposing direction. Accurate prediction of the Eulerian return flow in the ocean is of importance in modeling the transport of plastic pollution, oil, wreckage, and sediment. Herein, we derive a multiple-scales solution for the Eulerian mean flow under wave packets that is valid for all water depths, relative to both the length of the wave and the length of the wave packet. To validate this solution, we carry out particle tracking velocimetry experiments in a long flume to extract the mean motion from Lagrangian seeding particles under wave packets, finding good agreement. The extraction technique is able to deal with small background motion and subharmonic error waves associated with wave generation by the paddle, the latter being relatively large in finite-depth flume experiments. In finite depth, the return flow is forced by both the divergence of the Stokes transport on the wave-packet scale and the formation of a non-negligible mean set-down underneath the packet, which acts like a bounding streamtube in the form of a convergent-divergent duct. The magnitude of the horizontal return flow is thus enhanced, with particular relevance to transport in the finite-depth coastal environment.

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Stationary wave groups exist in a range of nonlinear dispersive media, including optics, Bose-Einstein condensates, plasma, and hydrodynamics. We report experimental observations of nonlinear surface gravity X waves, i.e., X-shaped wave envelopes that propagate over long distances with constant form. These can be described by the 2D+1 nonlinear Schrödinger equation, which predicts a balance between dispersion and diffraction when the envelope (the arms of the X) travel at ±arctan(1/2)≈±35.26° to the carrier wave. Our findings may help improve understanding the lifetime of extremes in directional seas and motivate further studies in other nonlinear dispersive media.

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Uncertainty affects estimates of the power potential of tidal currents, resulting in large ranges in values reported for a given site, such as the Pentland Firth, UK. We examine the role of bottom friction, one of the most important sources of uncertainty. We do so by using perturbation methods to find the leading-order effect of bottom friction uncertainty in theoretical models by Garrett & Cummins (2005 Proc. R. Soc. A 461, 2563-2572. (doi:10.1098/rspa.2005.1494); 2013 J. Fluid Mech. 714, 634-643. (doi:10.1017/jfm.2012.515)) and Vennell (2010 J. Fluid Mech. 671, 587-604. (doi:10.1017/S0022112010006191)), which consider quasi-steady flow in a channel completely spanned by tidal turbines, a similar channel but retaining the inertial term, and a circular turbine farm in laterally unconfined flow. We find that bottom friction uncertainty acts to increase estimates of expected power in a fully spanned channel, but generally has the reverse effect in laterally unconfined farms. The optimal number of turbines, accounting for bottom friction uncertainty, is lower for a fully spanned channel and higher in laterally unconfined farms. We estimate the typical magnitude of bottom friction uncertainty, which suggests that the effect on estimates of expected power lies in the range 25 to +30%, but is probably small for deep channels such as the Pentland Firth (5-10%). In such a channel, the uncertainty in power estimates due to bottom friction uncertainty remains considerable, and we estimate a relative standard deviation of 30%, increasing to 50% for small channels.@en