Wave-induced transport in the coastal zone can play an important role in the distribution of nutrients, plankton, and pollutants including plastic litter. The effect of the water depth, unsteadiness of the wave field, and the rotation of the Earth on the wave-induced drift is investigated in this paper. We derive a model for the time-dependent drift velocity in finite-depth water for a time-varying wave forcing, where we include the effect of rotation and a constant eddy viscosity. The solution is given in the form of a finite-depth Ekman-Stokes kernel. Convolution of this kernel with a time series of the Stokes drift gives the wave-induced Eulerian-mean velocity. The resulting unsteady flow contains damped inertial oscillations, where the amplitude of the oscillations is influenced by the rate of change of the wave forcing. Both a decrease in water depth (lower (Formula presented.)) and a decrease in the relative importance of rotation compared to viscosity (lower (inverse) wave Ekman number (Formula presented.)) lead to faster decay of these oscillations. Applying the model to buoy data shows a large impact on the predicted transport in finite depth. ... Read more

The study of nearshore wave-induced currents, which play a critical role in marine transport, has motivated numerous laboratory experiments, and yet, the understanding of cross-shore wave-induced currents under controlled laboratory conditions remains incomplete. For the first time, 3D Particle Tracking Velocimetry is applied in a laboratory flume to measure Lagrangian wave-induced currents in front of a slope under five different regular wave conditions. The wave-induced velocity profiles evolve over time, reaching a quasi-equilibrium after approximately one hour. In most cases, the observed profiles do not align with the theoretical Stokes or conduction solutions. The surface drift is consistently smaller than theoretically predicted, and in some cases even negative, indicating the presence of a strong Eulerian-mean return current in the upper portion of the water column. The observed patterns cannot be explained solely by the relative water depth kh and wave steepness ka, leading to the hypothesis that convection processes contribute to these discrepancies. Further investigation of visually observed coherent convective structures, such as vortex trains, will be undertaken. ... Read more

Understanding the transport processes in the ocean is essential for predicting the distribution of nutrients, plankton, and pollutants such as plastic debris, as well as for search-and-rescue operations. Surface gravity waves and the Earth’s rotation both play important roles in this transport. A particle beneath a wave undergoes an unclosed elliptical motion, resulting in a net drift in the direction of wave propagation known as the Stokes drift. The sum of the Stokes drift and the Eulerian-mean velocity yields the Lagrangian-mean velocity, which is the velocity at which particles are transported and is referred to as the drift velocity. An Eulerian-mean velocity can arise under various circumstances, including as a result of the Earth’s rotation and in the finite domain of a wave flume, where an Eulerian return current develops to satisfy volume conservation. This thesis aims to improve our understanding of wave-induced transport in the world’s seas and oceans by examining the influence of the Earth’s rotation on this transport.

In Chapter 2, the first laboratory experiments of wave-induced drift under rotation are conducted by placing a purpose-built wave flume on a rotating table. The surface drift induced by deep-water waves is measured for a range of values of
the (inverse) wave Ekman number (Ek⁻¹), which is a non-dimensional number that quantifies the relative importance of the effects of rotation and viscosity. At larger rotation rates, inertial oscillations are observed in the particle trajectories. Averaging over the full trajectories, the mean surface drift is directed to the right of the direction of wave propagation (for counterclockwise rotation), with reduced magnitude compared to the non-rotating case. An increase in the (inverse) wave Ekman number (larger relative effect of rotation) leads to a decrease in the magnitude of the drift. This trend is in agreement with theoretical predictions, but the magnitude of the measured drift is larger than predicted by two steady models. At lower (inverse) wave Ekman numbers (smaller relative effect of rotation), an unsteady model provides improved agreement by accounting for the partial sampling of inertial oscillations.

In Chapter 3, a theoretical model is developed for the time-dependent wave-induced drift on a rotating Earth in response to a time-varying wave forcing in finite-depth water. The solution is given in the form of the so-called finite-depth kernel, which yields the wave-induced Eulerian-mean velocity through convolution with a time series of the Stokes drift. The wave-induced drift velocity under rotation consists of damped inertial oscillations. The rate of change of the wave forcing controls the amplitude of the oscillations, with a faster time variation leading to more prominent inertial oscillations. The damping of the oscillations is governed by a combination of the water depth (kh) and the relative importance of rotation and viscosity (Ek⁻¹), where a decrease in either (shallower water or smaller relative effect of rotation) leads to faster decay. Applying the finite-depth kernel to buoy data shows that unsteadiness is crucial for obtaining accurate predictions of transport at early times (up to a few days), whereas on longer timescales the steady approximation may suffice. Including the effect of finite depth has a large impact on both the predicted trajectories and, even more so, on the vertically-integrated transport.

In Chapter 4, the wave-induced drift in intermediate water depth is investigated over the entire water column in a second set of experiments using a newly built rotating facility known as the Delta Transport Processes Laboratory (DTP-Lab). The depth-uniform return current that develops in a wave flume to compensate the Stokes drift is found to be crucial for the amplitude of the inertial oscillations. The Lagrangian-mean velocity oscillates between a profile well described by the sum of the Stokes drift and the depth-uniform return current and the corresponding profile with opposite sign throughout the central part of the water column. Averaging over this oscillation yields a mean drift that is drastically reduced compared to the drift without rotation throughout the fluid interior. To account for the return current, the model developed in Chapter 3 is modified by incorporating the depth-uniform return current into the initial condition. The resulting modified model accurately predicts the drift velocity at early time, representing a significant improvement over existing models, but fails to capture the subsequent damping of the oscillations at later times. For smaller relative water depths (lower kh), the observed decay is substantially faster than predicted, whereas for the largest relative water depth (close to the deep-water regime) the measurements do not exhibit the gradual decay predicted by the model.

This thesis has advanced our understanding of wave-induced transport on a rotating Earth through a combination of theoretical modelling and laboratory experiments. The results provide the first laboratory observations of wave-driven inertial oscillations, confirming a phenomenon that had previously only been predicted theoretically and demonstrating that laboratory experiments in a rotating wave flume are a valuable tool for studying wave-induced transport. The findings show that the Earth’s rotation can substantially modify both the magnitude and direction of wave-induced transport, and that (non-wave-driven) background flows play a crucial role in determining the amplitude of the resulting inertial oscillations. The importance of these effects in observational and modelling contexts can be estimated using two non-dimensional parameters: the relative water depth kh and the (inverse) wave Ekman number Ek⁻¹. Together, these results highlight that rotation, unsteadiness, and finite depth can play an important role in determining transport in the ocean.
... Read more

Laboratory experiments were performed to investigate the attenuation of progressive deep-water waves by a mono-layer of loose- and close-packed floating spheres. We measured the decay distance of waves having different incident wave frequency and steepness. The attenuation of waves was strong if the surface concentration of particles was close-packed, with the decay distance being shorter for incident waves with higher frequency and steepness. The amplitude of the highest-frequency (2.0 Hz) and largest amplitude incident waves (with steepness 0.25) decayed by half over a distance of approximately 3 wavelengths. Theoretical models used previously in the study of surface wave damping by sea ice do not capture correctly the physics of wave attenuation by floating spheres. We developed a new theory that estimates the influence upon wave attenuation of turbulent dissipation resulting from oscillatory flow under a close-packing of spheres. This theory predicts that the wave amplitude decays as a power law, and gives a correct order-of-magnitude estimate of the observed decay distance. We explore the potential implications of these findings for the attenuation of progressive waves by (pancake) sea ice and for the indirect detection of marine plastic pollution from space. ... Read more

Marine pollution is a major global environmental problem. The transport and dispersion of marine pollution is driven by a wide range of hydrodynamic processes, including wave-induced currents (e.g., Stokes drift) that are generated by free-surface and internal gravity waves in density-stratified fluids. While the (Lagrangian-mean) Stokes drift is known to fundamentally change transport patterns, wave-induced Eulerian-mean currents, such as those generated in the presence of the Coriolis force due to the Earth's rotation, are generally less well understood. To address this, the Delta Transport Processes Laboratory (DTP-Lab), a multi-purpose lab with novel facilities and state-of-the-art equipment, is being constructed in the Hydraulic Engineering Laboratory at TU Delft. The DTP-Lab combines multiple components: a 4.40-m diameter turntable, which can support a (removable) 5-m long flume; a 12.7-m long stainless steel flume; a piston-type, wet-back, force-controlled surface wave generator; a pumping system to create any type of density stratification; and a 3D Particle Tracking Velocimetry (PTV) system. The design and construction of these components along with technical validation and performance tests are presented in this technical note. A scaling analysis demonstrates the suitability of the laboratory to investigate wave-induced current under rotation. The DTP-Lab will pioneer the combined experimental study of surface waves, density stratification and Coriolis forces. The DTP-Lab is presented here with the objective of giving practical information to future users and to describe its novelty and range of applications. ... Read more

Waves transport particles in the direction of wave propagation with the Stokes drift. When the Earth’s rotation is accounted for, waves induce an additional (Eulerian-mean) current that reduces drift and is known as the anti-Stokes drift. This effect is often ignored in oceanic particle-tracking simulations, despite being important. Although different theoretical models exist, they have not been validated by experiments. We conduct laboratory experiments studying the surface drift induced by deep-water waves in a purpose-built rotating wave flume. With rotation, the Lagrangian-mean drift deflects to the right (counterclockwise rotation) and reduces in magnitude. Compared with two existing steady theoretical models, measured drift speed follows a similar trend with wave Ekman number but is larger. The difference is largely explained by unsteadiness on inertial time scales. Our results emphasise the importance of considering unsteadiness when predicting and analysing the transport of floating material by waves. ... Read more

The presence of marine pollutants such as marine plastics has increased significantly over the last decades and poses a major environmental problem, in both the coastal and offshore area. Marine pollutants are transported, mixed and diffused in the ocean, which means the understanding and modelling of marine transport is key for mitigation purposes (Moulton et al., 2022). Additional to large scale and planetary currents that play a major role in marine transport, free surface waves, internal gravity waves in density stratified fluids and the Coriolis force due to the rotation of the Earth are also fundamental drivers of transport that need to be accounted for. The fundamental fluid mechanics processes associated with these are often not resolved in large-scale models, but are instead included in a parametrised form. However, some fundamental processes associated with wave-induced currents (e.g., Stokes drift) in rotating, density-stratified fluids with a free surface remain unclear and untested. In addition, parametrisation for different environments, forcings and time scales must be developed and tested before being implemented into models for them to reliably predict transport, accumulation and storage of marine pollutants. For this purpose, the Delta Transport Processes Laboratory (DTPLab) is being developed at TUDelft Hydraulic Engineering Laboratory. This laboratory pioneers the combined experimental study of surface waves, density stratification and Coriolis forces in a single laboratory. The DTPLab was designed with a multi- users and purposes vision, with interchangeable facilities and state-of-the-art measurement devices. This paper presents the DTPLab facilities (under construction) and equipment that make this laboratory unique in the world, and describes, as an example of what is feasible, a novel experiment that will be performed in this lab.</jats:p> ... Read more