A unified criterion for initiation of sediment motion and inception of sheet flow under water waves

A unified criterion is developed for initiation of non-cohesive sediment motion and inception of sheet flow under water waves over a horizontal bed of sediment based on presently available experimental data. The unified threshold criterion is of the single form, U-o = 2 pi C[1 + 5(T-R/T)(2)](-1/4), where U-o is the onset velocity of sediment motion or sheet flow, T is wave period, and C and T-R are the coefficients. It is found that for a given sediment, U-o initially increases sharply with wave period, then gradually approaches the maximum onset velocity U-o = 2 pi C and becomes independent of T when T is larger. The unified criterion can also be extended to define sediment initial motion and sheet flow under irregular waves provided the significant wave orbital velocity and period of irregular waves are introduced in this unified criterion.

Remote sensing of ocean waves by along-track interferometric synthetic aperture radar

A new wave retrieval method for the Along-Track Interferometric Synthetic Aperture Radar (AT-InSAR) phase image is presented. The new algorithm, named parametric retrieval algorithm (PRA), uses the full nonlinear mapping relations. It differs from previous retrieval algorithms in that it does not require a priori information about the sea state or the wind vector from scatterometer data. Instead, it combines the observed AT-InSAR phase spectrum and assumed wind vector to estimate the wind sea spectrum. The method has been validated using several C-band and X-band HH-polarized AT-InSAR observations collocated with spectral buoy measurements. In this paper, X-band and C-band HH-polarized AT-InSAR phase images of ocean waves are first used to study AT-InSAR wave imaging fidelity. The resulting phase spectra are quantitatively compared with forward-mapped in situ directional wave spectra collocated with the AT-InSAR observations. Subsequently, we combine the parametric retrieval algorithm (PRA) with X-band and C-band HH-polarized AT-InSAR phase images to retrieve ocean wave spectra. The results show that the ocean wavelengths, wave directions, and significant wave heights estimated from the retrieved ocean wave spectra are in agreement with the buoy measurements.

Validation of RADARSAT-2 fully polarimetric SAR measurements of ocean surface waves

C band RADARSAT-2 fully polarimetric (fine quad-polarization mode, HH+VV+HV+VH) synthetic aperture radar (SAR) images are used to validate ocean surface waves measurements using the polarimetric SAR wave retrieval algorithm, without estimating the complex hydrodynamic modulation transfer function, even under large radar incidence angles. The linearly polarized radar backscatter cross sections (RBCS) are first calculated with the copolarization (HH, VV) and cross-polarization (HV, VH) RBCS and the polarization orientation angle. Subsequently, in the azimuth direction, the vertically and linearly polarized RBCS are used to measure the wave slopes. In the range direction, we combine horizontally and vertically polarized RBCS to estimate wave slopes. Taken together, wave slope spectra can be derived using estimated wave slopes in azimuth and range directions. Wave parameters extracted from the resultant wave slope spectra are validated with colocated National Data Buoy Center (NDBC) buoy measurements (wave periods, wavelengths, wave directions, and significant wave heights) and are shown to be in good agreement.

Third-order Stokes wave solutions for interfacial internal waves in a three-layer density-stratified fluid

Interfacial internal waves in a three-layer density-stratified fluid are investigated using a singular method, and third-order asymptotic solutions of the velocity potentials and third-order Stokes wave solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory. as expected, the third-order solutions describe the third-order nonlinear modification and the third-order nonlinear interactions between the interfacial waves. The wave velocity depends on not only the wave number and the depth of each layer but also on the wave amplitude.

Determination of fractional energy loss of waves in nearshore waters using an improved high-order Boussinesq-type model

Fractional energy losses of waves due to wave breaking when passing over a submerged bar are studied systematically using a modified numerical code that is based on the high-order Boussinesq-type equations. The model is first tested by the additional experimental data, and the model's capability of simulating the wave transformation over both gentle slope and steep slope is demonstrated. Then, the model's breaking index is replaced and tested. The new breaking index, which is optimized from the several breaking indices, is not sensitive to the spatial grid length and includes the bottom slopes. Numerical tests show that the modified model with the new breaking index is more stable and efficient for the shallow-water wave breaking. Finally, the modified model is used to study the fractional energy losses for the regular waves propagating and breaking over a submerged bar. Our results have revealed that how the nonlinearity and the dispersion of the incident waves as well as the dimensionless bar height (normalized by water depth) dominate the fractional energy losses. It is also found that the bar slope (limited to gentle slopes that less than 1:10) and the dimensionless bar length (normalized by incident wave length) have negligible effects on the fractional energy losses.

Generation of internal and surface waves by seafloor movement in a two layer fluid system

Internal and surface waves generated by the deformations of the solid bed in a two layer fluid system of infinite lateral extent and uniform depth are investigated. An integral solution is developed for an arbitrary bed displacement on the basis of a linear approximation of the complete description of wave motion using a transform method (Laplace in time and Fourier in space) analogous to that used to study the generation of tsunamis by many researchers. The theoretical solutions are presented for three interesting specific deformations of the seafloor; the spatial variation of each seafloor displacement consists of a block section of the seafloor moving vertically either up or down while the time-displacement history of the block section is varied. The generation process and the profiles of the internal and surface waves for the case of the exponential bed movement are numerically illustrated, and the effects of the deformation parameters, densities and depths of the two layers on the solutions are discussed. As expected, the solutions derived from the present work include as special cases that obtained by Kervella et al. [Theor Comput Fluid Dyn 21:245-269, 2007] for tsunamis cased by an instantaneous seabed deformation and those presented by Hammack [J Fluid Mech 60:769-799, 1973] for the exponential and the half-sine bed displacements when the density of the upper fluid is taken as zero.

The effects of random surface waves on the steady Ekman current solutions

The response of near-surface current profiles to wind and random surface waves are studied based on the approach of Jenkins [1989. The use of a wave prediction model for driving a near surface current model. Dtsch. Hydrogr. Z. 42,134-149] and Tang et al. [2007. Observation and modeling of surface currents on the Grand Banks: a study of the wave effects on surface currents. J. Geophys. Res. 112, C10025, doi:10.1029/2006JC004028]. Analytic steady solutions are presented for wave-modified Ekman equations resulting from Stokes drift, wind input and wave dissipation for a depth-independent constant eddy viscosity coefficient and one that varies linearly with depth. The parameters involved in the solutions can be determined by the two-dimensional wavenumber spectrum of ocean waves, wind speed, the Coriolis parameter and the densities of air and water, and the solutions reduce to those of Lewis and Belcher [2004. Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift. Dyn. Atmos. Oceans. 37, 313-351] when only the effects of Stokes drift are included. As illustrative examples, for a fully developed wind-generated sea with different wind speeds, wave-modified current profiles are calculated and compared with the classical Ekman theory and Lewis and Belcher's [2004. Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift. Dyn. Atmos. Oceans 37, 313-351] modification by using the Donelan and Pierson [1987. Radar scattering and equilibrium ranges in wind-generated waves with application to scatterometry. J. Geophys. Res. 92, 4971-5029] wavenumber spectrum, the WAM wave model formulation for wind input energy to waves, and wave energy dissipation converted to currents. Illustrative examples for a fully developed sea and the comparisons between observations and the theoretical predictions demonstrate that the effects of the random surface waves on the classical Ekman current are important, as they change qualitatively the nature of the Ekman layer. But the effects of the wind input and wave dissipation on surface current are small, relative to the impact of the Stokes drift. (C) 2008 Elsevier Ltd. All rights reserved.

Self-organized Criticality Model for Ocean Internal Waves

In this paper, we present a simple spring-block model for ocean internal waves based on the self-organized criticality (SOC). The oscillations of the water blocks in the model display power-law behavior with an exponent of -2 in the frequency domain, which is similar to the current and sea water temperature spectra in the actual ocean and the universal Garrett and Munk deep ocean internal wave model [Geophysical Fluid Dynamics 2(1972) 225; J. Geophys. REs. 80 (1975) 291]. The influence of the ratio of the driving force to the spring coefficient to SOC behaviors in the model is also discussed.

Higher-order Boussinesq-type equations for interfacial waves in a two-fluid system

Interfacial waves propagating along the interface between a three-dimensional two-fluid system with a rigid upper boundary and an uneven bottom are considered. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. A set of higher-order Boussinesq-type equations in terms of the depth-averaged velocities accounting for stronger nonlinearity are derived. When the small parameter measuring frequency dispersion keeping up to lower-order and full nonlinearity are considered, the equations include the Choi and Camassa's results (1999). The enhanced equations in terms of the depth-averaged velocities are obtained by applying the enhancement technique introduced by Madsen et al. (1991) and Schaffer and Madsen (1995a). It is noted that the equations derived from the present study include, as special cases, those obtained by Madsen and Schaffer (1998). By comparison with the dispersion relation of the linear Stokes waves, we found that the dispersion relation is more improved than Choi and Camassa's (1999) results, and the applicable scope of water depth is deeper.

Strongly nonlinear long interfacial waves in uniform basic flows and their dispersion relationship

In this paper, long interfacial waves of finite amplitude in uniform basic flows are considered with the assumption that the aspect ratio between wavelength and water depth is small. A new model is derived using the velocities at arbitrary distances from the still water level as the velocity variables instead of the commonly used depth-averaged velocities. This significantly improves the dispersion properties and makes them applicable to a wider range of water depths. Since its derivation requires no assumption on wave amplitude, the model thus can be used to describe waves with arbitrary amplitude.

Second-order Stokes solutions for internal waves in three-layer density-stratified fluid

In this paper, internal waves in three-layer stratified fluid are investigated by using a perturbation method, and the second-order asymptotic solutions of the velocity potentials and the second-order Stokes solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory. As expected, the first-order solutions are consistent with ordinary linear theoretical results, and the second-order solutions describe the second-order modification on the linear theory and the interactions between the two interfacial waves. Both the first-order and second-order solutions derived depend on the depths and densities of the three-layer fluid. It is also noted that the solutions obtained from the present work include the theoretical results derived by Umeyama as special cases.

Second-order solutions for random interfacial waves under steady uniform currents

In the present research, the study of Song (2004) for random interfacial waves in two-layer fluid is extended to the case of fluids moving at different steady uniform speeds. The equations describing the random displacements of the density interface and the associated velocity potentials in two-layer fluid are solved to the second order, and the wave-wave interactions of the wave components and the interactions between the waves and currents are described. As expected, the extended solutions include those obtained by Song (2004) as one special case where the steady uniform currents of the two fluids are taken as zero, and the solutions reduce to those derived by Sharma and Dean (1979) for random surface waves if the density of the upper fluid and the current of the lower fluid are both taken as zero.

Spatial and temporal variations of sound speed at the PN section

Gridded sound speed data were calculated using Del Grosso's formulation from the temperature and salinity data at the PN section in the East China Sea covering 92 cruises between February 1978 and October 2000. The vertical gradients of sound speed are mainly related to the seasonal variations, and the strong horizontal gradients are mainly related to the Kuroshio and the upwelling. The standard deviations show that great variations of sound speed exist in the upper layer and in the slope zone. Empirical orthogonal function analysis shows that contributions of surface heating and the Kuroshio to sound speed variance are almost equivalent.

Statistical distribution of depth-integrated local horizontal momentum for second-order random ocean waves in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, statistical distributions of the depth- integrated local horizontal momentum components are derived by use of the characteristic function expansion method. The parameters involved in the distributions can be all determined by the water depth and the wave-number spectrum of ocean waves. As an illustrative example, a fully developed wind-generated sea is considered and the parameters are calculated for typical wind speeds and water depths by means of the Donelan and Pierson spectrum. The effects of nonlinearity and water depth on the distributions are also investigated.