Turbulent flow structure in experimental laboratory wind-generated gravity waves

This paper is the third part of a report on systematic measurements and analyses of wind-generated water waves in a laboratory environment. The results of the measurements of the turbulent flow on the water side are presented here, the details of which include the turbulence structure, the correlation functions, and the length and velocity scales. It shows that the mean turbulent velocity profiles are logarithmic, and the flows are hydraulically rough. The friction velocity in the water boundary layer is an order of magnitude smaller than that in the wind boundary layer. The level of turbulence is enhanced immediately beneath the water surface due to micro-breaking, which reflects that the Reynolds shear stress is of the order u *w 2. The vertical velocities of the turbulence are related to the relevant velocity scale at the still-water level. The autocorrelation function in the vertical direction shows features of typical anisotropic turbulence comprising a large range of wavelengths. The ratio between the microscale and macroscale can be expressed as λ/Λ=a Re Λ n, with the exponent n slightly different from -1/2, which is the value when turbulence production and dissipation are in balance. On the basis of the wavelength and turbulent velocity, the free-surface flows in the present experiments fall into the wavy free-surface flow regime. The integral turbulent scale on the water side alone underestimates the degree of disturbance at the free surface. © 2012 Elsevier B.V.

Study of the turbulence in the air-side and water-side boundary layers in experimental laboratory wind induced surface waves

This study detailed the structure of turbulence in the air-side and water-side boundary layers in wind-induced surface waves. Inside the air boundary layer, the kurtosis is always greater than 3 (the value for normal distribution) for both horizontal and vertical velocity fluctuations. The skewness for the horizontal velocity is negative, but the skewness for the vertical velocity is always positive. On the water side, the kurtosis is always greater than 3, and the skewness is slightly negative for the horizontal velocity and slightly positive for the vertical velocity. The statistics of the angle between the instantaneous vertical fluctuation and the instantaneous horizontal velocity in the air is similar to those obtained over solid walls. Measurements in water show a large variance, and the peak is biased towards negative angles. In the quadrant analysis, the contribution of quadrants Q2 and Q4 is dominant on both the air side and the water side. The non-dimensional relative contributions and the concentration match fairly well near the interface. Sweeps in the air side (belonging to quadrant Q4) act directly on the interface and exert pressure fluctuations, which, in addition to the tangential stress and form drag, lead to the growth of the waves. The water drops detached from the crest and accelerated by the wind can play a major role in transferring momentum and in enhancing the turbulence level in the water side.On the air side, the Reynolds stress tensor's principal axes are not collinear with the strain rate tensor, and show an angle α σ≈=-20°to-25°. On the water side, the angle is α σ≈=-40°to-45°. The ratio between the maximum and the minimum principal stresses is σ a/σ b=3to4 on the air side, and σ a/σ b=1.5to3 on the water side. In this respect, the air-side flow behaves like a classical boundary layer on a solid wall, while the water-side flow resembles a wake. The frequency of bursting on the water side increases significantly along the flow, which can be attributed to micro-breaking effects - expected to be more frequent at larger fetches. © 2012 Elsevier B.V.

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.

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.

Influence of mean water depth and a subsurface sandbar on the onset and strength of wave breaking

Wave breaking in the open ocean and coastal zones remains an intriguing yet incompletely understood process, with a strong observed association with wave groups. Recent numerical study of the evolution of fully nonlinear, two-dimensional deep water wave groups identified a robust threshold of a diagnostic growth-rate parameter that separated nonlinear wave groups that evolved to breaking from those that evolved with recurrence. This paper investigates whether these deep water wave-breaking results apply more generally, particularly in finite-water-depth conditions. For unforced nonlinear wave groups in intermediate water depths over a flat bottom, it was found that the upper bound of the diagnostic growth-rate threshold parameter established for deep water wave groups is also applicable in intermediate water depths, given by k(0) h greater than or equal to 2, where k(0) is the mean carrier wavenumber and h is the mean depth. For breaking onset over an idealized circular arc sandbar located on an otherwise flat, intermediate-depth (k(0) h greater than or equal to 2) environment, the deep water breaking diagnostic growth rate was found to be applicable provided that the height of the sandbar is less than one-quarter of the ambient mean water depth. Thus, for this range of intermediate-depth conditions, these two classes of bottom topography modify only marginally the diagnostic growth rate found for deep water waves. However, when intermediate-depth wave groups ( k(0) h greater than or equal to 2) shoal over a sandbar whose height exceeds one-half of the ambient water depth, the waves can steepen significantly without breaking. In such cases, the breaking threshold level and the maximum of the diagnostic growth rate increase systematically with the height of the sandbar. Also, the dimensions and position of the sandbar influenced the evolution and breaking threshold of wave groups. For sufficiently high sandbars, the effects of bottom topography can induce additional nonlinearity into the wave field geometry and associated dynamics that modifies the otherwise robust deep water breaking-threshold results.

Applications of improved 2-D numerical wave tanks in wave breaking

Song and Banner (2002, henceforth referred to as SB02) used a numerical wave tank (developed by Drimer and Agnon, and further refined by Segre, henceforth referred to as DAS) to study the wave breaking in the deep water, and proposed a dimensionless breaking threshold that based on the behaviour of the wave energy modulation and focusing during the evolution of the wave group. In this paper, two modified DAS models are used to further test the SB02's results, the first one (referred to MDAS1) corrected many integral calculation errors appeared in the DAS code, and the second one (referred to MDAS2) replaced the linear boundary element approximation of DAS into the cubic element on the free surface. Researches show that the results of MDAS1 are the same with those of DAS for the simulations of deep water wave breaking, but, the different values of the wavemaker amplitude, the breaking time and the maximum local average energy growth rate delta(max) for the marginal breaking cases are founded by MDAS2 and MDAS1. However, MDAS2 still satisfies the SB02' s breaking threshold. Furthermore, MDAS1 is utilized to study the marginal breaking case in the intermediate water depth when wave passes over a submerged slope, where the slope is given by 1 : 500, 1 : 300, 1 : 150 or 1 : 100. It is found that the maximum local energy density U increases significantly if the slope becomes steeper, and the delta(max) decreases weakly and increases intensively for the marginal recurrence case and marginal breaking case respectively. SB02's breaking threshold is still valid for the wave passing over a submerged slope gentler than 1 : 100 in the intermediate water depth.

Numerical Simulation of Waves Generated by Seafloor Movements

Waves generated by vertical seafloor movements are simulated by use of a fully nonlinear two-dimensional numerical wave tank. In the source region, the seafloor lifts to a designated height by a generation function. The numerical tests show that file linear theory is only valid for estimating the wave behaviors induced by the seafloor movements with a small amplitude, and the fully nonlinear numerical model should be adopted in the simulation of the wave generation by the large amplitude seafloor movements. Without the background surface waves, many numerical tests on the stable maximum elevations eta(max)(0) are carried out by both the linear theory and the fully nonlinear model. The results of two models are compared and analyzed. For the fully nonlinear model, the influences of the amplitudes and the horizontal lengths on eta(max)(0) are stronger than that of the characteristic duration times. Furthermore, results reveal that there are significant differences between the linear theory and the fully nonlinear model. When the influences of the background surface waves are considered, the corresponding numerical analyses reveal that with the fully nonlinear model the eta(max)(0) near-linearly varies with the wave amplitudes of the surface waves, and the eta(max)(0) has significant dependences on the wave lengths and the wave phases of the surface waves. In addition, the differences between the linear theory and the fully nonlinear model are still obvious, aid these differences are significantly affected by The wave parameters of the background surface waves, such as the wave amplitude, the wave length and the wave phase.

Local Conentration of Waves due to Abrupt Alongshore Variation of Nearshore Bathymetry

Small-scale physical and numerical experiments were conducted to investigate the local concentration of waves (monochromatic and group) due to abrupt change of nearshore bathymetry in alongshore direction. Wave run-up motions along the shoreline were measured using an image analysis technique to compare localized concentration of wave energy, when waves propagate a over bathymetry composing rhythmic patterns of mild/steep slope bottom configurations. Measured alongshore variation of maximum wave run-up heights showed significant peak near the boundary, which has sudden alongshore change of depth, both under monochromatic and group wave trains. This phenomenon is found to be due to interaction of waves with neashore currents, which is further enhanced by excitation of long wave components by breaking of group waves. Furthermore, this paper discusses results of preliminary experiments carried out to test the effectiveness of several shore protection structure layouts in mitigating such wave concentrations. Numerical simulations were performed by using a model developed based on Nwogu (1993) Boussinesq-type equations; coupled with a transport equation to model energy dissipation due to wave breaking.

A robust mechanism for strengthening of the Brewer–Dobson circulation in response to climate change: critical-layer control of subtropical wave breaking

Climate models consistently predict a strengthened Brewer–Dobson circulation in response to greenhouse gas (GHG)-induced climate change. Although the predicted circulation changes are clearly the result of changes in stratospheric wave drag, the mechanism behind the wave-drag changes remains unclear. Here, simulations from a chemistry–climate model are analyzed to show that the changes in resolved wave drag are largely explainable in terms of a simple and robust dynamical mechanism, namely changes in the location of critical layers within the subtropical lower stratosphere, which are known from observations to control the spatial distribution of Rossby wave breaking. In particular, the strengthening of the upper flanks of the subtropical jets that is robustly expected from GHG-induced tropospheric warming pushes the critical layers (and the associated regions of wave drag) upward, allowing more wave activity to penetrate into the subtropical lower stratosphere. Because the subtropics represent the critical region for wave driving of the Brewer–Dobson circulation, the circulation is thereby strengthened. Transient planetary-scale waves and synoptic-scale waves generated by baroclinic instability are both found to play a crucial role in this process. Changes in stationary planetary wave drag are not so important because they largely occur away from subtropical latitudes.

The role of baroclinic waves in the initiation of tropical cyclones across the southern Indian Ocean

Cases where tropical storms are initiated simultaneously along one latitude are investigated. It is argued that such structure arises as part of a baroclinic wave. A case from February 2008 is examined using European Centre for Medium-Range Forecasts (ECMWF) analyses; the birth of three tropical cyclones in the low-level cyclonic regions to the east of upper-level troughs suggests that the wave was instrumental for initiation. Archived satellite imagery and storm warnings reveal that baroclinic waves over the southern Indian Ocean accompany tropical cyclogenesis twice a season on average, mainly in late summer, when breaking Rossby waves on the subtropical westerly jet are closest to the Intertropical Convergence Zone (ITCZ). Copyright © 2012 Royal Meteorological Society

Sound waves and solitons in hot and dense nuclear matter

Assuming that nuclear matter can be treated as a perfect fluid, we study the propagation of perturbations in the baryon density. The equation of state is derived from a relativistic mean field model, which is a variant of the non-linear Walecka model. The expansion of the Euler and continuity equations of relativistic hydrodynamics around equilibrium configurations leads to differential equations for the density perturbation. We solve them numerically for linear and spherical perturbations and follow the propagation of the initial pulses. For linear perturbations we find single soliton solutions and solutions with one or more solitons followed by ""radiation"". Depending on the equation of state a strong damping may occur. We consider also the evolution of perturbations in a medium without dispersive effects. In this case we observe the formation and breaking of shock waves. We study all these equations also for matter at finite temperature. Our results may be relevant for the analysis of RHIC data. They suggest that the shock waves formed in the quark gluon plasma phase may survive and propagate in the hadronic phase. (C) 2009 Elseiver. B.V. All rights reserved.

Dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms

We consider formation of dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms. It is shown that for big enough initial inhomogeneity of density, interplay of nonlinear and dispersion effects leads to wave breaking phenomenon followed by generation of a train of dark solitons. Analytical theory is confirmed by numerical simulations.

Theory of small aspect ratio waves in deep water

In the limit of small values of the aspect ratio parameter (or wave steepness) which measures the amplitude of a surface wave in units of its wave-length, a model equation is derived from the Euler system in infinite depth (deep water) without potential flow assumption. The resulting equation is shown to sustain periodic waves which on the one side tend to the proper linear limit at small amplitudes, on the other side possess a threshold amplitude where wave crest peaking is achieved. An explicit expression of the crest angle at wave breaking is found in terms of the wave velocity. By numerical simulations, stable soliton-like solutions (experiencing elastic interactions) propagate in a given velocities range on the edge of which they tend to the peakon solution. (c) 2005 Elsevier B.V. All rights reserved.

Propagation of ship waves on a sloping bottom

The numerical model FUNWAVE was adapted in order to simulate the generation and propagation of ship waves to shore, including phenomena such as refraction, diffraction, currents and breaking of waves. Results are shown for Froude numbers equal to 0.8, 1.0 and 1.1, in order to verify the refraction of the wave pattern, identify breaking conditions and to investigate the wave generation scheme as a moving pressure at the free surface. © 2009 World Scientific Publishing Co. Pte. Ltd.

Waves generated by two or more ships in a channel

The numerical model FUNWAVE+Ship simulates the generation and propagation of ship waves to shore, including phenomena such as refraction, diffraction, currents and breaking of waves. The interaction of two wave trains, generated by ships moving either in the same direction at different speeds or in opposite directions, is studied. Focus is given to the wave orbital velocities and to the free surface pattern.