Horizontal Orbital Velocities Research Articles

Dispersive Wave Focusing on a Shear Current: Part 1—Linear Approximations

We consider the evolution and kinematics during dispersive focussing, for a group of waves propagating atop currents varying with depth. Our analysis assumes long-crested linear waves propagating at arbitrary angles relative to the current. Although low steepness is assumed, the linear model is often a reasonable approximation for understanding rogue waves. A number of analytical approximate relations are derived assuming different sub-surface current profiles, including linearly varying current, exponentially varying current, and currents of arbitrary depth profile which are weakly sheared following the approximation of Stewart and Joy (Deep Sea Res. Abs. 21, 1974). The orbital velocities are likewise studied. While shear currents have modest influence on the motion of the envelope of the wave group, they significantly change wave kinematics. Horizontal orbital velocities are either amplified or suppressed depending on whether the shear is opposing or following, respectively. To illustrate these phenomena we consider a real-world example using velocity profiles and wave spectra measured in the Columbia River estuary. Near the surface at the point where focusing occurs, horizontal orbital velocities are, respectively, increased and decreased by factors of 1.4 and 0.7 for focusing groups propagating on following and opposing shear (respectively, upstream and downstream in the earth-fixed reference system). The implications for the forces a focusing wave group can exert on vessels and installations are profound, emphasising the importance of considering current profiles in maritime operations.

Plant trait-mediated drag forces on seedlings of four tidal marsh pioneer species

Salt marshes play an important role in coastal protection by reducing the impact of waves and shoreline erosion risks. While mature vegetation is responsible for the persistence and stability of marsh ecosystems, seedling survival of pioneer species is especially crucial for marsh propagation. Marsh seedlings, however, may be threatened by climate change induced increased coastal storm surge intensity and accompanying (extreme) wave conditions, imposing stronger drag forces on marsh seedlings. We test the hypothesis that drag forces experienced by seedlings increase with horizontal orbital velocity (Uw) in a species-specific manner, and that the drag forces experienced are individual-plant trait-mediated. To test our hypotheses, seedlings of four contrasting pioneer marsh species (Bolboschoenus maritimus, Schoenoplectus tabernaemontani, Spartina anglica, and Puccinellia maritima) were exposed to storm wave conditions in a flume, where Uw and experienced drag forces were measured. Linear mixed effect models demonstrated that seedling’s susceptibility to storm wave conditions is at least partly mediated by individual plant traits. Drag forces experienced by seedlings tended to increase with Uw, and with stem length and diameter. The interplay of both traits was complex, with increasing stem length being the most important trait accounting for increases in drag forces experienced at low to moderate Uw, while the stem diameter became more important with increasing Uw. Furthermore, experienced drag forces appeared to be affected by species-specific traits such as rigidity and leaf growth, being highest for Bolboschoenus maritimus and lowest for Puccinellia maritima. Our results provide important mechanistic insights into the drivers of tidal marsh seedling vulnerability to storm wave conditions due to experienced drag, both based on the traits of individual plants and species-specific ones. This type of knowledge is of key importance when modelling saltmarsh establishment and resilience under climate change.

Anchor Forces on Coir-Based Artificial Seagrass Mats: Dependence on Wave Dynamics and Their Potential Use in Seagrass Restoration

Seagrasses represent an essential part of the coastal environment and are hence the target of many coastal restoration projects. Artificial seagrass (ASG) mats may facilitate seagrass growth, making them a captivating option for restoration projects. However, little is known about the forces occurring on mats deployed in marine environments and especially on how these forces are transmitted to the anchoring points. Here, we present a study of prototype biodegradable coconut-mesh mats as base layer for ASG meadows and investigate the forces that occur at the anchors. We test the performance of three mesh types under wave forcing using two different anchor configurations without ASG and subsequently test ASG mats of one mesh type under wave forcing and a 4-anchor configuration to assess the effect of the ASG on anchor loading as a function of incident orbital velocities. We found that the mat composition plays a more important role than the number of anchors in anchor load reduction. The anchor forces were 2–4 times higher at front anchors compared to rear anchors, relative to wave propagation direction, and were also considerably higher in that direction compared to the opposite direction. With ASG, the forces increased compared to the highest measured forces without ASG. The forces on the anchors were almost fully dominated by the drag on the ASG based on material properties, ASG reconfiguration and flow conditions. We derive a relation between horizontal orbital velocities and expected forcing on the anchor based on ASG properties and the corresponding area of each anchor and discuss relevant criteria for the design of ASG mats. This should help to assess the loading on anchors deployed for restoration under specific site conditions and chosen materials.

A New Probabilistic Wave Breaking Model for Dominant Wind‐Sea Waves Based on the Gaussian Field Theory

AbstractThis study presents a novel method for obtaining the probability of wave breaking (Pb) of deep water, dominant wind‐sea waves (i.e., waves made of the energy within ±30% of the peak wave frequency) derived from Gaussian wave field theory. For a given input wave spectrum, we demonstrate how it is possible to derive a joint probability density function between wave phase speed (c) and horizontal orbital velocity at the wave crest (u) from which a model for Pb can be obtained. A nonlinear kinematic wave breaking criterion consistent with the Gaussian framework is further proposed. Our model would allow, therefore, the application of the classical wave breaking criterion (i.e., wave breaking occurs if u/c > 1) in spectral wave models which, to the authors' knowledge, has not been done to date. Our results show that the proposed theoretical model has errors in the same order of magnitude as six other historical models when assessed using three field data sets. With optimization of the proposed model's single free parameter, it can become the best performing model for specific data sets. Although our results are promising, additional, more complete wave breaking data sets collected in the field are needed to comprehensively assess the present model, especially in regards to the dependence on phenomena such as direct wind forcing, long wave modulation, and wave directionality.

Implementation of Open Boundaries within a Two-Way Coupled SPH Model to Simulate Nonlinear Wave–Structure Interactions

A two-way coupling between the Smoothed Particle Hydrodynamics (SPH) solver DualSPHysics and the Fully Nonlinear Potential Flow solver OceanWave3D is presented. At the coupling interfaces within the SPH numerical domain, an open boundary formulation is applied. An inlet and outlet zone are filled with buffer particles. At the inlet, horizontal orbital velocities and surface elevations calculated using OceanWave3D are imposed on the buffer particles. At the outlet, horizontal orbital velocities are imposed, but the surface elevation is extrapolated from the fluid domain. Velocity corrections are applied to avoid unwanted reflections in the SPH fluid domain. The SPH surface elevation is coupled back to OceanWave3D, where the originally calculated free surface is overwritten. The coupling methodology is validated using a 2D test case of a floating box. Additionally, a 3D proof of concept is shown where overtopping waves are acting on a heaving cylinder. The two-way coupled model (exchange of information in two directions between the coupled models) has proven to be capable of simulating wave propagation and wave–structure interaction problems with an acceptable accuracy with error values remaining below the smoothing length h S P H .

APPLICATION OF OPEN BOUNDARIES WITHIN A TWO-WAY COUPLED SPH MODEL TO SIMULATE NON-LINEAR WAVE-STRUCTURE INTERACTIONS

A two-way coupling between the fully non-linear potential flow (FNPF) solver OceanWave3D and the Smoothed Particle Hydrodynamics (SPH) solver DualSPHysics is presented. At the coupling interfaces within the SPH domain, an open boundary formulation is applied. An inlet and outlet zone are filled with bu er particles. At the inlet, horizontal orbital velocities and surface elevations calculated with OceanWave3D are imposed on the bu er particles. At the outlet, horizontal orbital velocities are imposed, but the surface elevation is extrapolated from the fluid domain. Velocity corrections are applied to avoid unwanted reflections in the fluid domain. The SPH surface elevation can be coupled back to OceanWave3D, where the original solution is overwritten. The coupling methodology is validated using a 2-D test case of a floating box. Additionally, a 3-D proof of concept is shown where overtopping waves are acting on a heaving cylinder. The 2-way coupled model proofs to be capable of simulating wave propagation and wave-structure interaction problems with an acceptable accuracy with RMSE values remaining below the smoothing length h.

Computational Fluid Dynamics for Modeling Gravity Currents in the Presence of Oscillatory Ambient Flow

Gravity currents generated by lock release are studied in the case of initially quiescent ambient fluid and oscillating ambient fluid (regular surface waves). In particular, the dynamics of the density currents are investigated by means of CFD numerical simulations. The aim is to evaluate the influence of the ambient fluid velocity field on the observed mixing and turbulent processes. Results of two different turbulence closure models, namely the standard k − ε turbulence model and the LES model, are analyzed. Model predictions are validated through comparison with laboratory measurements. Results show that the k − ε model is able to catch the main current propagation parameters (e.g., front velocity at the different phases of the evolution of the current, gravity current depth, etc.), but that a LES model provides more realistic insights into the turbulent processes (e.g., formation of interfacial Kelvin–Helmholtz billows, vortex stretching and eventual break up into 3D turbulence). The ambient fluid velocity field strongly influences the dynamics of the gravity currents. In particular, the presence of an oscillatory motion induces a relative increase of mixing at the front (up to 25%) in proximity of the bottom layer, and further upstream, an increase of the mixing process (up to 60%) is observed due to the mass transport generated by waves. The observed mixing phenomena observed are also affected by the ratio between the gravity current velocity v f and the horizontal orbital velocity induced by waves u w , which has a stronger impact in the wave dominated regime ( v f / u w < 1).

Wave-induced kinematics inside submerged porous structures

A simple model for predicting the vertical distribution of wave-induced horizontal orbital velocities inside submerged porous structures is presented. A velocity profile is formulated based on the exponential profile of the nonlinear wave theory. This velocity distribution is then fitted to the numerically-predicted depth-integrated velocity given by a Boussinesq model assuming uniform porosity. For the sake of comparison, experimental measurements of horizontal velocities were taken in a laboratory flume. The proposed model was also compared with the results of a version of the commercial program FLOW-3D dealing with linear waves in a simple manner. A good agreement of the model results with the experiments was found as well as an improved behaviour with respect to this program. Moreover, increased velocities were identified at the downstream slope of the rubble mound caused by recirculating flow.

Investigating sediment re-mobilisation due to wave action by means of ADCP echo intensity data: Field data from the Tromper Wiek, western Baltic Sea

The Tromper Wiek is a semi-enclosed bay in the western Baltic Sea where gravel extraction by means of anchor hopper dredging has been carried out for many years. After an initial refilling of the extraction craters with material <2 mm that had been spilled during the extraction process, the craters are relatively stable for several years. Acoustic Doppler current profiler echo intensity raw data in combination with wave, current, and wind data are used to correlate sediment re-mobilisation events with hydrodynamic and meteorological conditions in the bay. Near-bottom orbital movements generated by surface waves entering the bay from the open sea proved to be the key mechanisms of sediment re-mobilisation. The strong correlation between wind speed from certain directions and local wave height, and between wave height and maximum horizontal orbital velocity at the bottom, enabled the frequency, duration, and intensity of sediment re-mobilisation events to be estimated from local wind data.

Sedimentary signatures and processes during marine bolide impacts: a review

Studies of submarine impact craters resulting from impacts of comets or asteroids demonstrate that the presence of water and the physical properties of target rocks have a major influence on sedimentary processes associated with meteorite impacts. This results in difference in sedimentary signature of bolide impacts in marine environments compared to subaerial impact craters. In subaerial impacts, the targets are commonly hard rocks, frequently of igneous and/or metamorphic origin, whereas in submarine impacts, the targets are mostly unconsolidated or poorly lithified sediments, or sedimentary rocks, with high volumes of pore water. Such differences result in variability in crater morphology and in sedimentary processes inside and outside the impact area. Impacts in shallow-water marine (neritic) environments produced craters with low or absent rims and wide and shallow brims, as characterize by both the Montagnais (on the Scotian shelf), the Mjølnir (in the Barents Sea), 45 and 40 km in diameter, respectively, and the Chesapeake Bay (90 km in diameter). Lack of elevated rims is thought to be the result of current reworking and resurge of the water back into the excavated cavity, as the water in the crater is vaporized. During this process, resurge gullies can be cut across the rim, while mass- and debris-flows, turbidites, and other gravity deposits are produced as results of tsunami and crater-wall and central high collapse, during and after the crater excavation stage. Such deposits are found both within and outside the crater structure. The only difference between gravity deposits triggered by an impact or other rare events, such as earthquakes, is the admixture of various melt particles and possible enrichments in iridium in the former. Impacts near the shelf edge may cause partial collapse of the continental margin as shown by the Montagnais and Chicxulub impacts. Some of the gravity and debris flows generated by margin collapse may be channelized, with final deposits up to several hundred meters thick, extending for hundreds of kilometers from the impact site. Other impact features such as shatter cones, tektites, spherules and Ni-spinels, shocked quartz, isotropication, and partial melts, are common to both submarine and subaerial impacts. Theoretical calculations of the destructive forces of mega-tsunami waves triggered by meteorite impact in the ocean greatly exceed those based on geologic evidence. A dearth of gigantic tsunami evidence for the Chicxulub impact outside of the Gulf of Mexico, where from theoretical modelling the maximum near bottom orbital velocity of water flow crossing the deep North Atlantic basin should have been >1 m/s, could be the result of the mitigating effect of the bathymetry of surrounding area, causing wave diffraction and interference. Calculated maximum horizontal orbital velocity near the seafloor at the shelf for the Montagnais impact is 22 m/s for a 200-m-high wave, 5.5 m/s for a 50-m-high wave at 500 km, decreasing to 0.5 m/s at 1000 km, strong enough to scour the deep ocean bottom and produce distinct erosion surfaces and disconformities in marine sedimentary record. However, lack of cores across impact horizon prevents confirmation of occurrence of such bottom water flows.

Numerical Experiments on the Breaking of Solitary Internal Wavesover a Slope–Shelf Topography

A theoretical study of the transformation of large amplitude internal solitary waves (ISW) of permanent form over a slope–shelf topography is considered using as basis the Reynolds equations. The vertical fluid stratification, amplitudes of the propagating ISWs, and the bottom parameters were taken close to those observed in the Andaman and Sulu Seas. The problem was solved numerically. It was found that, when an intense ISW of depression propagates from a deep part of a basin onto the shelf with water depth Hs, a breaking event will arise whenever the wave amplitude am is larger than 0.4(Hs − Hm), where Hm is the undisturbed depth of the isopycnal of maximum depression. The cumulative effect of nonlinearity in a propagating ISW leads to a steepening and overturning of a rear wave face over the inclined bottom. Immediately before breaking the horizontal orbital velocity at the site of instability exceeds the phase speed of the ISW. So, the strong breaking is caused by a kinematic instability of the propagating wave. At the latest stages of the evolution the overturned hydraulic jump transforms into a horizontal density intrusion (turbulent pulsating wall jet) propagating onto the shelf. The breaking criterion of the ISW over the slope was found. Over the range of examined parameters (0.52° < γ < 21.8°, where γ is the slope angle) the breaking event arises at the position with depth Hb, when the nondimensional wave amplitude a = am/(Hb − Hm) satisfies the condition a ≅ 0.8°/γ + 0.4. If the water depth Hs on a shelf is less than Hb, a solitary wave breaks down before it penetrates into a shallow water zone; otherwise (at Hs > Hb) it passes as a dispersive wave tail onto the shelf without breaking.

Waves and currents over a fixed rippled bed: 1. Bottom roughness experienced by waves in the presence and absence of currents

Several theoretical models have been advanced for turbulent boundary layers associated with wave and current flows over a rough bed. Although these models differ, particularly in the sophistication with which turbulence is modeled, they all share one common feature: the bottom roughness is characterized by a single roughness length scalekn. However, no experimental data have been presented which accurately characterize the bottom roughnesses for wave and current boundary layers such that this basic assumption can be considered verified. This paper and a companion paper [Mathisen and Madsen, this issue] present results of an experimental study to verify this basic assumption. In this paper, wave roughnesses (i.e., bottom roughnesses experienced by pure waves and waves in the presence of a current) are compared with the roughness experienced by a pure current. The roughnesses for pure currents and pure waves are shown to be the same if determination of the pure wave roughness from measurements of wave energy dissipation includes the phase difference between the near‐bottom horizontal orbital velocity and bottom shear stress. When these analysis procedures are extended to combined wave‐current flows, the roughness experienced by waves in the presence of a current is shown to be that obtained for pure waves.

A numerical model for shore‐normal sediment size variation on a macrotidal beach

AbstractThis paper describes a computer simulation model which is designed to predict the selective shore‐normal sorting of grain sizes in the nearshore environment. The model simulates wave shoaling, wave height attenuation due to frictional losses and breaking, using linear theory up to the break point and a breaker decay model in the surf zone. Peak horizontal orbital velocities at the bed are calculated from Stokes second‐order wave theory. The peak onshore and offshore velocities are used with the threshold expression of Komar and Miller (1975) to generate a spatial pattern of size variation of threshold grain diameter along a profile normal to the shore from deep water to the swash zone. The predicted grain size is used in an hydraulic interpretation of grain size distribution on the intertidal profile, based on the hydrodynamic variations over a tidal cycle on a macrotidal beach. The model is successful in predicting the broad pattern of increasing grain size in the onshore direction which has been observed in nature. Comparisons between measured and predicted grain size distributions indicate that the predictions of the model are better than those of previous models, but the model is more successful at predicting sediment size distributions than at predicting mean sizes on a beach profile.

NEAR-SURFACE ORBITAL VELOCITIES IN IRREGULAR WAVES

The paper deals with measurements of horizontal orbital velocities near the surface of mechanically generated waves in a wave flume. Due to the characteristics of most velocity probes, it is difficult or impossible to measure with a fixed probe in the area above the lowest trough. As the probe is not submerged continuously, failures or uncertainties in the measurements may occur. To overcome these limitations, a movable frame for the velocity probe was designed, which can be moved vertically up and down with the surface elevation by a disc rotor servo motor, controlled by a wave gauge. By that continuously velocities up to 3 cm below the surface could be measured. Theoretical velocities have been calculated for comparison with different simulation methods for irregular waves, based on linear wave theory.