Near-bed Wave Orbital Velocity Research Articles

Mobilisation thresholds for coral rubble and consequences for windows of reef recovery

Abstract. The proportional cover of rubble on reefs is predicted to increase as disturbances increase in intensity and frequency. Unstable rubble can kill coral recruits and impair binding processes that transform rubble into a stable substrate for coral recruitment. A clearer understanding of the mechanisms of inhibited coral recovery on rubble requires characterisation of the hydrodynamic conditions that trigger rubble mobilisation. Here, we investigated rubble mobilisation under regular wave conditions in a wave flume and irregular wave conditions in situ on a coral reef in the Maldives. We examined how changes in near-bed wave orbital velocity influenced the likelihood of rubble motion (e.g. rocking) and transport (by walking, sliding or flipping). Rubble mobilisation was considered as a function of rubble length, branchiness (branched vs. unbranched) and underlying substrate (rubble vs. sand). The effect of near-bed wave orbital velocity on rubble mobilisation was comparable between flume and reef observations. As near-bed wave orbital velocity increased, rubble was more likely to rock, be transported and travel greater distances. Averaged across length, branchiness and substrate, loose rubble had a 50 % chance of transport when near-bed wave orbital velocities reached 0.30 m s−1 in both the wave flume and on the reef. However, small and/or unbranched rubble pieces were generally mobilised more and at lower velocities than larger, branched rubble. Rubble also travelled further distances per day (∼2 cm) on substrates composed of sand than rubble. Importantly, if rubble was interlocked, it was very unlikely to move (< 7 % chance) even at the highest velocity tested (0.4 m s−1). Furthermore, the probability of rubble transport declined over 3 d deployments in the field, suggesting rubble had snagged or settled into more hydrodynamically stable positions within the first days of deployment. We expect that snagged or settled rubble is transported more commonly in locations with higher-energy events and more variable wave environments. At our field site in the Maldives, we expect recovery windows for binding (when rubble is stable) to predominantly occur during the calmer north-eastern monsoon when wave energy impacting the atoll is less and wave heights are smaller. Our results show that rubble beds comprised of small rubble pieces and/or pieces with fewer branches are more likely to have shorter windows of recovery (stability) between mobilisation events, and thus be good candidates for rubble stabilisation interventions to enhance coral recruitment and binding.

Empirical formulas for near-bed wave orbital velocity parameters involved in maximum wave load in random wave trains

This paper presents empirical methods utilizing recently developed formulas (Kawamata and Kobayashi., 2022; Kawamata et al., 2018) to predict the maximum wave forces on near-bed structures and wave-induced movement ratios of isolated rocks in random wave trains. The methods assume that the maximum wave load occurs when the velocity semi-amplitude defined as half the difference between successive negative and positive peaks of the near-bed wave orbital velocity is maximized. The empirical formulas for the velocity-waveform parameters at the maximum velocity semi-amplitude in a random wave train were derived from the near-bed velocities and surface elevations measured in laboratory wave flumes. The laboratory formula for the maximum velocity semi-amplitude was reviewed and improved with near-bed wave orbital velocities and surface elevations estimated from pressure measurements under a wider range of wave conditions in the field. The newly developed formulas for the velocity-waveform parameters at velocity semi-amplitude maxima, combined with the previously developed formulas, showed reasonable agreement with the maximum wave forces measured on the artificial reef models under random waves in a laboratory wave flume as well as with the movement ratio of quarry rocks (median mass = 0.40 t) observed in a field test.

A new expression for free stream velocity with skewness and asymmetry

A new expression for skewed and asymmetric free stream velocities (near-bed wave orbital velocities at the edge of the wave boundary layer) is established in this paper. A comparison of the new expression with others is examined, and the new expression is found to be more satisfactorily fitted to wave orbital velocities with comparatively strong skewness and weak asymmetry than those presented by Abreu et al. (2010) and Malarkey and Davies (2012). Methods for determining the parameters in the new expression based on the Ursell number Ur are presented, and consequently, two empirical formulae are recommended for estimating bedload and suspended load sand transport rates related to wave skewness, respectively. Reasonability of the new expression for irregular waves is discussed by comparing the predicted time averaged quantities to those calculated by measured data, and good agreement is found.

Modified formula for predicting the maximum wave force on seabed objects

A modified wave-force formula, which predicts the maximum horizontal wave force on seabed objects, was developed using the velocity semi-amplitude, defined as half the difference between successive negative (offshore) and positive (onshore) peaks in the near-bed wave orbital velocity, and the corresponding peak-to-peak period. The formula was tested by analyzing wave-force data from three differently shaped structure models under random wave conditions and comparing its prediction accuracies with those of the Morison equation and the conventional wave-force formula based on the positive peak velocity. The maximum forces for individual zero-down-crossing velocity waveforms were predicted by applying each of the respective formulas with force coefficient(s) empirically pre-determined as a function of the Keulegan—Carpenter number. For all wave conditions and model structures, the modified formula was more accurate than the Morison equation, whereas the conventional formula was less accurate. Although the absolute relative prediction error increased as the velocity skewness deviated from 0.5, the increase in the error was least pronounced for the modified formula. The results also demonstrated that the highest maximum wave forces in random wave trains can be predicted with reasonable accuracy by the proposed formula using the highest value of the velocity semi-amplitude and the corresponding period.

Coral composition and bottom-wave metrics improve understanding of the patchiness of cyclone damage on reefs

Coral reefs are likely to be exposed to more intense cyclones under climate change. Cyclone impacts are spatially highly variable given complex hydrodynamics, and coral-specific sensitivity to wave impacts. Predicting reef vulnerability to cyclones is critical to management but requires high resolution environmental data that are difficult to obtain over broad spatial scales. Using 30m-resolution wave modelling, we tested cyclonic and non-cyclonic wave metrics as predictors of coral damage on 22 reefs after severe cyclone Ita impacted the northern Great Barrier Reef, Australia in 2014. Analyses of coral cover change accounting for the type of coral along a gradient of vulnerability to wave damage (e.g., massive, branching, Acroporids) excluded cyclone-generated surface wave metrics (derived from wave height) as important predictors. Increased bottom stress wave environment (near-bed wave orbital velocity) due to Ita (Ita-Ub) explained spatial patterns of 17% to 46% total coral cover loss only when the initial abundance of Acroporids was accounted for, and only when exceeding 35% cover. Greater coral losses occurred closer to the cyclone path irrespective of coral type. Massive and encrusting corals, however, had losses exacerbated in higher non-cyclonic bottom-wave energy environments (nc-Ub). The effect of community composition on structural vulnerability to wave damage was more important predicting damage that the magnitude of the cyclone-generated waves, especially when reefs are surveyed well beyond where damaging waves are expected to occur. Exposure to Ita-Ub was greater in typically high nc-Ub environments with relatively low cover of the most fragile morphologies explaining why these were the least affected overall. We reveal that the common surface-wave metrics of cyclone intensity may not always be able to predict spatial impacts and conclude that reef vulnerability assessments need to account for chronic wave patterns and differences in community composition in order to provide predictive tools for future conservation and restoration.

Improved Calculation of Nonlinear Near-Bed Wave Orbital Velocity in Shallow Water: Validation against Laboratory and Field Data

A new parameterization for calculating the nonlinear near-bed wave orbital velocity in the shallow water was presented. The equations proposed by Isobe and Horikawa (1982) were modified in order to achieve more accurate predictions of the peak orbital velocities. Based on field data from Egmond Beach in the Netherlands, the correction coefficient and maximum skewness were determined as functions of the Ursell number. The obtained equations were validated against measurements from Egmond Beach, and with laboratory data from small-scale wave flume experiments at Delft University of Technology and from large-scale wave flume experiments at Delft Hydraulics. Inter-comparisons with other previously developed parameterizations were also carried out. The model simulations by the present study were in good agreement with the measurements and have been improved compared to the previous ones. For Egmond Beach, the root-mean-square errors for the peak onshore (uc) and offshore (ut) orbital velocities were approximately 21%. The relative biases were small, approximately 0.013 for uc and −0.068 for ut. The coefficient of determination was in the range between 0.64 and 0.68. For laboratory experiments, the root-mean-square errors in a range of 7.2%–24% for uc, and 7.9%–15% for ut.

Wind-wave and Tidally Driven Sediment Resuspension in a Macrotidal Basin

A coupled hydrodynamic-wave-sediment model is used to simulate the broad-scale tidal circulation, surface waves, and suspended sediment concentrations (SSC) in Minas Basin, a 70-km long tidal estuary in the Bay of Fundy, during winter and summer periods. The model hydrodynamics are validated using acoustic-Doppler current profile observations, the surface SSC predictions are compared to satellite observations, and model results indicate that strong seasonal signals in SSC can be explained in part by seasonal changes in fetch-limited surface waves generated by local winds over the basin. The spatial and temporal variability of SSC is evaluated in this study by focussing on different forcing conditions from waves and tidal currents, the two primary physical process that influence the response of sediments in suspension. Model predictions in the intertidal areas indicate that surface waves can increase the bed shear stress from tidal currents alone by up to 1–5 Nm− 2, causing excess bed shear stresses to be higher and result in higher SSC by 100–200 gm− 3 particularly during wind events that are stronger and more frequent in winter months. Resuspension of sediments on tidal flats is driven by the combination of shear stresses from near-bed wave orbital velocities and tidal currents, and transport of the suspended materials over deeper areas of the basin is driven by advection from the strong tidal currents.

Wave-current interactions in the continental shelf bottom boundary layer of the Australian North West Shelf during tropical cyclone conditions

The distant passage of an intense and large Category 3 Tropical Cyclone (TC) generated highly energetic surface waves in a large region across the Australian North West Shelf (NWS). Two sites on the continental shelf experienced maximum significant wave heights of up to 10 m with near-bed wave orbital velocities up to 0.7 m s−1 at 40 and 74 m depth. Concurrent current profiles were measured between 0.5 and 8.5 m above the seabed at these sites and therefore the observations presented in this study allowed a detailed analysis of the wave-current interactions in the continental shelf bottom boundary layer during highly energetic tropical cyclone conditions. These observations revealed a strong modification of the current profiles during the TC with an increase of the apparent bottom roughness by up to 2 orders of magnitude compared to the typical tide-dominated current conditions. This modification was predominantly due to wave-current interactions; however, the results also suggest that changes in bed forms during the TC conditions likely acted as a secondary mechanism contributing to the enhanced bottom roughness experienced by the mean currents. A comparison of the total mean bed shear stresses as a function of the waves and currents showed a somewhat larger enhancement due to the nonlinear wave-current interactions than predicted by common semi-empirical models. Overall, the enhancement of the mean bed shear stresses during the TC significantly reduced the mean current velocities in the bottom boundary layer, highlighting the importance of including wave-current interactions within ocean circulation models to accurately predict shelf currents during TC conditions.

Analytical approximate wave form for asymmetric waves

A simple analytical formulation that reproduces a skewed, nonlinear near-bed wave orbital velocity is presented. It contains four free parameters, where two are solely related to the velocity and acceleration skewnesses. The equation is compared with other models and is validated against field and laboratory experiments. The results reveal that it can simulate a wide range of nonlinear wave shapes, reproducing satisfactorily the measured nonlinear wave particle velocity. Also, the new expression overcomes some limitations of the other models. The new formulation is therefore capable of being used in many engineering applications that require the use of representative wave forms.

The statistical distribution of nearbed wave orbital velocity in intermediate coastal water depth

The statistical distribution of wave orbital velocity in intermediate coastal water depth has been quantitatively determined from the comprehensive field velocity data collected near the seabed in this study. Two ocean ADV current meters, which were mounted at 0.5 m above the seabed on two separate stainless steel tripods sitting on the seabed, were used to measure instantaneous water particle velocities at a 2 Hz sampling rate for 17.07 min every hour in two coastal water depths of 11 m and 23 m in nine field deployments over a period of 2 years. The zero-crossing method is applied to analyse the field velocity data collected in each field deployment to obtain a large sample of wave orbital velocity amplitudes of individual waves. Based on the collected field velocity data, it is found that the histogram of instantaneous wave orbital velocities perfectly follows the Gaussian distribution as commonly assumed, while the histogram of wave orbital velocity amplitudes is less accurately described by the Rayleigh distribution than the modified Rayleigh and the Weibull distribution. It is also found that large orbital velocity amplitudes are generally overestimated by the Rayleigh distribution, but well predicted by the modified Rayleigh and the Weibull distribution. The expected value of maximum orbital velocity in a velocity record of finite size is also derived from the three distributions and found to agree well with the present field data.

Calculating wave-generated bottom orbital velocities from surface-wave parameters

Near-bed wave orbital velocities and shear stresses are important parameters in many sediment-transport and hydrodynamic models of the coastal ocean, estuaries, and lakes. Simple methods for estimating bottom orbital velocities from surface-wave statistics such as significant wave height and peak period often are inaccurate except in very shallow water. This paper briefly reviews approaches for estimating wave-generated bottom orbital velocities from near-bed velocity data, surface-wave spectra, and surface-wave parameters; MATLAB code for each approach is provided. Aspects of this problem have been discussed elsewhere. We add to this work by providing a method for using a general form of the parametric surface-wave spectrum to estimate bottom orbital velocity from significant wave height and peak period, investigating effects of spectral shape on bottom orbital velocity, comparing methods for calculating bottom orbital velocity against values determined from near-bed velocity measurements at two sites on the US east and west coasts, and considering the optimal representation of bottom orbital velocity for calculations of near-bed processes. Bottom orbital velocities calculated using near-bed velocity data, measured wave spectra, and parametric spectra for a site on the northern California shelf and one in the mid-Atlantic Bight compare quite well and are relatively insensitive to spectral shape except when bimodal waves are present with maximum energy at the higher-frequency peak. These conditions, which are most likely to occur at times when bottom orbital velocities are small, can be identified with our method as cases where the measured wave statistics are inconsistent with Donelan's modified form of the Joint North Sea Wave Project (JONSWAP) spectrum. We define the “effective” forcing for wave-driven, near-bed processes as the product of the magnitude of forcing times its probability of occurrence, and conclude that different bottom orbital velocity statistics may be appropriate for different problems.

Sediment transport on the Palos Verdes shelf over seasonal to decadal time scales

We combine direct observations, longer-term wave data, and model calculations to characterize resuspension and transport of fine-grained, effluent-affected sediment on the Palos Verdes shelf. Near-bed waves, currents, and suspended sediment concentrations were monitored during the winter of 1992–93 with a bottom tripod and current-meter mooring at a 63-m-deep site. Wave conditions that winter were moderate (∼2 year recurrence interval), and mean current was alongshelf to the northwest; currents were not significantly correlated with wave conditions. Seven wave events during the winter (December–March) produced near-bed wave orbital velocities at the study site in excess of 14 cm s −1, the observed threshold for significant resuspension. Three of these events occurred during the bottom tripod deployment and are characterized by the highest persistent suspended sediment concentrations in the tripod record. Suspended sediment flux was alongshelf to the northwest for 5 of the 6 wave events for which current data were available; one event occurred during low southeast currents. Measured suspended sediment concentration and grain size generally agree with values that were calculated using a shelf sediment transport model with no adjustment of parameters from values determined for two muddy sites on the northern California shelf. We extend our seasonal observations to a period of almost 2 decades by applying the observed thresholds for wave-driven resuspension to near-bed wave conditions calculated from NDBC Buoy 46025 surface wave data. An average of 10 resuspension events per year, with an average duration of 1.6 days, were identified at a water depth of 60 m; the number of events dropped to 3 per year at 90 m, beyond the shelf break. For the majority of these events, calculated net suspended sediment flux is toward the northwest (alongshelf) at an average rate of 140 kg m −1 h −1; about a third of the events have net southeastward flux at an average rate of 30 kg m −1 h −1. The calculated thickness of the resuspended surface layer of the bed was less than 1 cm for all events at 60 m.

Linear transition ripple migration and wave orbital velocity skewness: Observations

Field observations were made in 3–4 m water depth of linear transition ripple geometry and migration using a high‐resolution laser‐video bed profiling system and acoustic scanning sensors during both the growth and decay phases of an autumn storm event. Linear transition ripples are long‐crested, low‐steepness bedforms of the anorbital ripple type and were observed to occur here at relatively high wave energies just below the flatbed threshold, with wavelengths of 8.5±0.5 cm and heights of 0.3±0.1 cm. The maximum observed migration rate was 0.7 cm/min. Migration was offshore during storm growth and onshore during storm decay. The observed ripple migration velocities were highly correlated (r2 > 0.7) with nearbed wave orbital velocity skewness in both cross‐shore directions. During storm growth the incident wave spectrum was bimodal and the orbital velocity skewness was negative. During storm decay the wave spectrum was unimodal and the velocity skewness was positive. Bispectral analysis shows that the main contribution to negative velocity skewness during storm growth was due to a difference interaction between the two principal (sea and swell) components of the bimodal velocity spectrum. Positive skewness during storm decay was due to self‐self interaction of the narrowband residual swell. The negative velocity skewness observed during storm growth is consistent with prediction by a two‐frequency second‐order wave theory.

Increase of current bottom shear stress due to waves

The increase of current bottom shear stress in the presence of waves is studied. It is found that this increment is linearly proportional to the near-bed wave orbital velocity amplitude Aω and can be considered practically to be independent of the relative bed roughness k s A and the angle θ between the wave propagation and the current. A new model is presented to calculate the current bottom shear stress in the presence of waves and gives better agreements with a wide range of presently available experimental data than Bijker's model.

Measurements of the semi-diurnal drag coefficient over sand waves

Simultaneous measurements of tidal flow and pressure gradient over a 10 km distance have been used to estimate the frictional drag coefficient over sand waves in the southern North Sea. The measurements were made over a 27-day period in October–November 1988 at 52°10′N, 3°46′E, within a field of essentially two-dimensional sand waves approximately 3 m high and of wavelength 250 m. The M 2 drag coefficient for depth-averaged flow normal to sand wave crests is found to be 2.95 × 10 −3, in good agreement with values used in numerical models of the region. The uncertainty in this value is estimated to be of the order of 10%, primarily due to possible errors in the phase of the flow relative to the pressure gradient. The time series of daily-averaged semi-diurnal ( Z 2 ) drag coefficients over the 27-day period shows surprisingly little correlation with nearbed wave orbital velocities. The time of highest waves is associated with a decrease in drag coefficient rather than the increase predicted by most wave-current interaction theories and some previous observations. It is suggested that this behaviour is caused by stratification near the bed due to sediment resuspension under high waves. Predictions using the Glenn and Grant (1987, Journal of Geophysical Research, 92, 8244–8264) theory provide qualitative support for this hypothesis, though the magnitude of the predicted effect is smaller than observed. The relative insensitivity of the drag coefficient to wave conditions suggests that incorporating simple wave-current algorithms into numerical models may be misleading.