T. H. C. Herbers

Generation and propagation of infragravity waves

The generation and propagation of infragravity waves (frequencies nominally 0.004–0.04 Hz) are investigated with data from a 24-element, coherent array of pressure sensors deployed for 9 months in 13-m depth, 2 km from shore. The high correlation between observed ratios of upcoast to downcoast energy fluxes in the infragravity (FupIG/FdownIG) and swell (Fupswell/Fdownswell) frequency bands indicates that the directional properties of infragravity waves are strongly dependent on incident swell propagation directions. However, FupIG/FdownIG is usually much closer to 1 (i.e., comparable upcoast and downcoast fluxes) than is Fupswell/Fdownswell, suggesting that upcoast propagating swell drives both upcoast and downcoast propagating infragravity waves. These observations agree well with predictions of a spectral WKB model based on the long-standing hypothesis that infragravity waves, forced by nonlinear interactions of nonbreaking, shoreward propagating swell, are released as free waves in the surf zone and subsequently reflect from the beach. The radiated free infragravity waves are predicted to be directionally broad and predominantly refractively trapped on a gently sloping shelf. The observed ratios FseaIG/FshoreIG of the seaward and shoreward infragravity energy fluxes are indeed scattered about the theoretical value 1 for trapped waves when the swell energy is moderate, but the ratios deviate significantly from 1 with both low- and high-energy swell. Directionally narrow, shoreward propagating infragravity waves, observed with low-energy swell, likely have a remote (possibly trans-oceanic) energy source. High values (up to 5) of FseaIG/FshoreIG, observed with high-energy swell, suggest that high-mode edge waves generated near the shore can be suppressed by nonlinear dissipation processes (e.g., bottom friction) on the shelf.

Infragravity-Frequency (0.005–0.05 Hz) Motions on the Shelf. Part II: Free Waves

Abstract In Part I, the energy levels of ocean surface waves at infragravity frequencies (nominally 0.005–0.05 Hz) locally forced by swell in 13-m water depth were shown to be predicted accurately by second-order nonlinear wave theory. However, forced infragravity waves were consistently much less energetic than free infragravity waves. Here, in Part II, observations in depths between 8 and 204 m, on Atlantic and Pacific shelves, are used to investigate the sources and variability of free infragravity wave energy. Both free and forced infragravity energy levels generally increase with increasing swell energy and decreasing water depth, but their dependencies are markedly different. Although free waves usually dominate the infragravity frequency band, forced waves contribute a significant fraction of the total infragravity energy with high energy swell and/or in very shallow water. The observed h−1 variation of free infragravity energy with increasing water depth h is stronger than the h−1/2 dependence pre...

Momentum balances on the North Carolina inner shelf

Four months of moored current, pressure, temperature, conductivity, wave, and wind observations on the North Carolina shelf indicate three dynamically distinct regions: the surf zone, the inner shelf between the surf zone and the 13-m isobath, and the midshelf. In the surf zone the along-shelf momentum balance is between the cross-shelf gradient of the wave radiation stress and the bottom stress. The linear drag coefficient in the surf zone is about 10 times larger than seaward of the surf zone. On the inner shelf the along-shelf momentum balance is also frictional; the along-shelf wind stress and pressure gradient are balanced by bottom stress. In the cross-shelf momentum balance the pressure gradient is the superposition of roughly equal contributions from the Coriolis force (geostrophy) and wave setdown from shoaling, unbroken surface gravity waves. At midshelf the along-shelf momentum balance is less frictional and hence flow accelerations are important. The cross-shelf momentum balance is predominantly geostrophic because the greater depth and smaller bottom slope at midshelf reduce the importance of wave setdown. The cross-shelf density gradient is in thermal wind balance with the vertical shear in the along-shelf flow in depths as shallow as 10 m. The dominant along-shelf momentum balances provide a simple estimate of the depth-averaged, along-shelf current in terms of the measured forcing (i.e., wind stress, wave radiation stress divergence, and along-shelf pressure gradient) that reproduces accurately the observed cross-shelf variation of the depth-averaged, along-shelf current between the surf zone and midshelf.

Reflection of Ocean Surface Gravity Waves from a Natural Beach

Abstract The energy of seaward and shoreward propagating ocean surface gravity waves on a natural beach was estimated with data from an army of 24 bottom-mounted pressure sensors in 13-m water depth, 2 km from the North Carolina coast. Consistent with a parameterization of surface wave reflection from a plane sloping beach by Miche, the ratio of seaward to shoreward propagating energy in the swell-sea frequency band (0.044–0.20 Hz) decreased with increasing wave frequency and increasing wave height, and increased with increasing beach-face slope. Although most incident swell-sea energy dissipated in the surf zone, reflection was sometimes significant (up to 18% of the incident swell-sea energy) when the beach face was steep (at high tide) and the wave field was dominated by low-energy, low-frequency swell. Frequency-directional spectra show that reflection of swell and sea was approximately specular. The ratio of seaward to shoreward propagating energy in the infragravity frequency band (0.010–0.044 Hz) v...

Alongshore momentum balances in the nearshore

The one-dimensional, time-averaged (over many wave periods) along- shore momentum balance between forcing by wind and breaking waves and the bottom stress is examined with field observations spanning a wide range of con- ditions on a barred beach. Near-bottom horizontal currents were measured for 2 months at 15 locations along a cross-shore transect extending 750 m from the shoreline to 8-m water depth. The hourly averaged bottom stress was estimated from observed currents using a quadratic drag law. The wave radiation stress was estimated in 8-m depth from an array of pressure sensors, and the wind stress was estimated from an anemometer at the seaward end of a nearby pier. The combined wind and wave forcing integrated over the entire cross-shore transect is balanced by the integrated bottom stress. The wind stress contributes about one third of the forcing over the transect. Analysis of the momentum balances in different cross-shore regions shows that in the surf zone, wave forcing is much larger than wind forcing and that the bottom drag coefficient is larger in the surf zone than farther seaward, consistent with earlier studies.

Spectral evolution of shoaling and breaking waves on a barred beach

Field observations and numerical model predictions are used to investigate the effects of nonlinear interactions, reflection, and dissipation on the evolution of surface gravity waves propagating across a barred beach. Nonlinear interactions resulted in a doubling of the number of wave crests when moderately energetic (about 0.8-m significant wave height), narrowband swell propagated without breaking across an 80-m-wide, nearly flat (2-m depth) section of beach between a small offshore sand bar and a steep (slope = 0.1) beach face, where the waves finally broke. These nonlinear energy transfers are accurately predicted by a model based on the nondissipative, unidirectional (i.e., reflection is neglected) Boussinesq equations. For a lower-energy (wave height about 0.4 m) bimodal wave field, high-frequency seas dissipated in the surf zone, but lower-frequency swell partially reflected from the steep beach face, resulting in significant cross-shore modulation of swell energy. The combined effects of reflection from the beach face and dissipation across the sand bar and near the shoreline are described well by a bore propagation model based on the nondispersive nonlinear shallow water equations. Boussinesq model predictions on the flat section (where dissipation is weak) are improved by decomposing the wave field into seaward and shoreward propagating components. In more energetic (wave heights greater than 1 m) conditions, reflection is negligible, and the region of significant dissipation can extend well seaward of the sand bar. Differences between observed decreases in spectral levels and Boussinesq model predictions of nonlinear energy transfers are used to infer the spectrum of breaking wave induced dissipation between adjacent measurement locations. The inferred dissipation rates typically increase with increasing frequency and are comparable in magnitude to the nonlinear energy transfer rates.

Swell and Slanting-Fetch Effects on Wind Wave Growth

Abstract Wind-sea generation was observed during two experiments off the coast of North Carolina. One event with offshore winds of 9–11 m s−1 directed 20° from shore normal was observed with eight directional stations recording simultaneously and spanning a fetch from 4 to 83 km. An opposing swell of 1-m height and 10-s period was also present. The wind-sea part of the wave spectrum conforms to established growth curves for significant wave height and peak period, except at inner-shelf stations where a large alongshore wind-sea component was observed. At these short fetches, the mean wave direction θm was observed to change abruptly across the wind-sea spectral peak, from alongshore at lower frequencies to downwind at higher frequencies. Waves from another event with offshore winds of 6–14 m s−1 directed 20°–30° from shore normal were observed with two instrument arrays. A significant amount of low-frequency wave energy was observed to propagate alongshore from the region where the wind was strongest. The...

Nonlinear shoaling of directionally spread waves on a beach

The shoaling of directionally spread surface gravity waves on a gently sloping beach with straight and parallel depth contours is examined with weakly dispersive Boussinesq theory. In this second-order theory, energy is transferred from the incident waves to components with both higher and lower frequencies in near-resonant nonlinear triad interactions. Directional spreading of the incident waves causes a weak detuning from resonance that is of the same order as the detuning owing to dispersion. Boussinesq theory predictions of the evolution of a single triad (i.e., two primary wave components shoaling from deep water forcing a secondary wave component) are compared to predictions of dispersive finite depth theory for a typical range of beach slopes, incident wave amplitudes, frequencies, and propagation directions. The dependencies of the predicted secondary wave growth on primary wave incidence angles are in good agreement. Whereas the sum frequency response is insensitive to the (deep water) spreading angle of the primary waves, the difference frequency (infragravity) response is significantly reduced for large spreading angles. A stochastic formulation of Boussinesq wave shoaling evolution equations is derived on the basis of the closure hypothesis that phase coupling between quartets of wave components is weak. In this approximation the second- and third-order statistics of random, directionally spread shoaling waves are described by a coupled set of evolution equations for the frequency alongshore wavenumber spectrum and bispectrum. It is shown that a smooth overlap with solutions of dispersive finite depth theory exists in the limit of small beach slope and weak nonlinearity.

Observations of infragravity waves

Infragravity-wave (periods of one-half to a few minutes) energy levels observed for about 1 year in 8-m water depth in the Pacific and in 8- and 13-m depths in the Atlantic are highly correlated with energy in the swell-frequency band (7- to 20-s periods), suggesting the infragravity waves were generated locally by the swell. The amplification of infragravity-wave energy between 13- and 8-m depth (separated by 1 km in the cross shore) is about 2, indicating that the observed infragravity motions are dominated by free waves, not by group-forced bound waves, which in theory are amplified by an order of magnitude in energy between the two locations. However, bound waves are more important for the relatively few cases with very energetic swell, when the observed amplification between 13- and 8-m depth of infragravity-wave energy was sometimes 3 times greater than expected for free waves. Bispectra are consistent with increased coupling between infragravity waves and groups of swell and sea for high-energy incident waves.

Directional spreading of waves in the nearshore

Observations of surface gravity waves shoaling between 8-m water depth and the shoreline on a barred beach indicate that breaking results in an increase in the directional spread of wave energy, in contrast to the directional narrowing with decreasing depth predicted by refraction theory (Snells law). During low-energy wave conditions, when breaking-induced wave energy losses over the instrumented transect are small, the observed mean propagation direction and spread about the mean both decrease with decreasing depth, consistent with the expected effects of refraction. Nonlinearity causes high-frequency components of the spectrum to become directionally aligned with the dominant incident waves. During high-energy wave conditions with significant wave breaking on the sand bar, the observed mean directions still decrease with decreasing depth. However, the observed directional spreads increase sharply (nominally a factor of 2 for values integrated over the swell-sea frequency range) between the outer edge of the surf zone and the crest of the sand bar, followed by a decrease toward the shoreline. Observations on a nonbarred beach also show directional broadening, with spreads increasing monotonically from the outer edge of the surf zone to a maximum value near the shoreline. Although the mechanism is not understood, these spatial patterns of directional broadening suggest that wave breaking causes significant scattering of incident wave energy into obliquely propagating components.