Non-breaking Waves Research Articles

WHEN DO WAVES BECOME TURBULENT?

Babanin and Haus (2009) recently presented evidence of high levels of turbulence induced by steep non-breaking waves. They proposed a Reynolds-like threshold wave parameter (a2ω/ν=3000, a wave amplitude, ω wave angular frequency, ν water viscosity) for the spontaneous occurrence of turbulence beneath surface waves. This contradicts the common assumed basis of existing irrotational wave theories and their classical experimental validation. Many laboratory wave experiments were carried out in the early 1960's (e.g. Wiegel, 1964). In those experiments no evidence of turbulence was reported and steep waves behaved as predicted by the high order irrotational wave theories within the accuracy of the theories and experimental techniques at the time. The spontaneous generation of turbulence under waves can have serious consequences for wave modelling, where the irrotational flow assumption has secured its place in engineering design. This contribution describes unique flow visualisation experiments for large scale steep non-breaking waves using conventional dye techniques in the wave boundary layer extending above the wave trough level. The measurements showed no evidence of turbulent mixing for waves up to a2ω/ν=7000. There is presently no evidence that water waves become spontaneously turbulent (up a2ω/ν=7000) except within the bed boundary layers, under wind forcing or at breaking. Excellent agreement was found with higher order irrotational theories. Orbital velocities, Stokes drift and Stokes coefficients were measured and compared with theoretical values, suggesting that conventional theories underestimate unforced monochromatic wave non-linearity, although the corrections remain small.

A note on the direct injection of turbulence by breaking waves

We investigate the turbulence induced by wave-breaking at the ocean surface. Two recent models use a mechanism of direct depth injection of turbulent kinetic energy (TKE) by breaking waves. Those models aim to reproduce the near-surface mean and turbulent properties, in particular the TKE dissipation rates. Of critical importance are the injection depth of each breaking wave and the size distribution of those breaking waves. The models by Sullivan et al. (2007) and by Kudryavtsev et al. (2008) have very different parameterizations, and those differences are reviewed here and compared to available observations. Using realistic parameterizations in these models leads to TKE injections too shallow to compare to observations, in particular for developed seas. The near-surface turbulence is thus still not well understood to the zeroth order. For instance, whether developed seas produce deeper or shallower mixing than young seas is neither well understood nor well modelled. Additional dedicated measurements as well as investigations of breaking non-breaking wave interactions are needed.

Nonbreaking waves cause near‐surface turbulence

Surface ocean turbulence is predominantly generated by two forces: near‐surface winds and breaking waves. Where waves crash, strong downwelling can induce mixing of the upper few dozen meters, while winds establish larger turbulent vortices. A third known mechanism exists for generating near‐surface turbulence, the propagation of nonbreaking surface waves. However, it is typically overlooked as being too minute to matter. Recent research has suggested that nonbreaking waves may be a more important mechanism than previously thought. Using a laboratory wave tank and a numerical turbulence model, Savelyev et al. found that propagating nonbreaking waves not only can induce near‐surface eddies but also can modulate and feed into existing turbulent dynamics.

Wave Overtopping at Smooth Impermeable Steep Slopes with Low Crest Freeboards

AbstractExtensive knowledge is available on the overtopping behavior of traditional smooth impermeable sea defense structures, such as mildly sloping dikes and vertical walls, both typically featuring a high crest freeboard to reduce wave overtopping. A particular design application emerges in the development of wave energy converters of the overtopping type, where maximization of wave overtopping is required, i.e., smooth impermeable steep sloping structures with low crest freeboards subjected to nonbreaking waves. To date, relatively limited knowledge is available on the overtopping behavior of those structures. In this study, the average overtopping rate obtained from new experiments has been analyzed and compared with existing prediction methods. This study contributes to a better knowledge on the overtopping behavior of the steep low-crested structures, which is positioned in between that of mildly sloping dikes and vertical walls and that of structures with zero crest freeboards and relatively large...

The role of bottom friction in models of nonbreaking tsunami wave runup on the shore

The role of bottom friction in the runup of nonbreaking long waves on the shore is analyzed. The case of the normal incidence of monochromatic waves is considered. The relief of the model region consists of an even horizontal bottom area conjugated with a flat slope. The energy dissipation is estimated as the work of bottom friction forces over the wave field obtained using the known analytical solution based on the Carrier-Greenspan transforms. Estimates of energy losses for waves whose periods are typical for tsunami waves have been obtained. The energy dissipation is shown to be not concentrated in the shore line area as a rule. The question about the practicability of using partially reflecting boundary conditions on the coast to take into account the bottom friction in large-scale models of tsunami propagation is considered.

주기영향을 고려한 직립식 구조물의 월파량 산정 : 비쇄파조건

Two-dimensional hydraulic experiments for wave overtopping under non-breaking wave condition are conducted. The wave overtopping formula for vertical structure is suggested and the results are compared with EurOtop (2007). The relative water depth coefficient () shows that almost the same coefficient is obtained for certain range (kh > 1.55) regardless of relative water depth, that is, although the relative water depth becomes larger, the relative water depth coefficient is almost same. When the wave steepness becomes larger the wave steepness coefficient decreases. The overtopping formula are expressed by relative freeboard(R) and non-dimensional wave overtopping rate(Q) and this formula has the form of exponential function. In this formula, the effects of wave period on wave overtopping are quantitatively investigated and suggested through the relative water depth coefficient() and wave steepness coefficient().

An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations

We present a high-order well-balanced unstructured finite volume (FV) scheme on triangular meshes for modeling weakly nonlinear and weakly dispersive water waves over slowly varying bathymetries, as described by the 2D depth-integrated extended Boussinesq equations of Nwogu, rewritten here in conservation law form. The FV scheme numerically solves the conservative form of the equations following the median dual node-centered approach, for both the advective and dispersive part of the equations. For the advective fluxes, the scheme utilizes an approximate Riemann solver along with a well-balanced topography source term upwinding. Higher order accuracy in space and time is achieved through a MUSCL-type reconstruction technique and through a strong stability preserving explicit Runge–Kutta time stepping. Special attention is given to the accurate numerical treatment of moving wet/dry fronts and boundary conditions. The model is applied to several examples of non-breaking wave propagation over variable topographies and the computed solutions are compared to experimental data. The presented results indicate that the presented FV model is robust and capable of simulating wave transformations from relatively deep to shallow water, providing accurate predictions of the wave's propagation, shoaling and runup.

Evaluating austral summer mixed‐layer response to surface wave–induced mixing in the Southern Ocean

Three ocean general circulation models and a climate model are employed to evaluate the influences of the nonbreaking surface wave–induced mixing on the austral summer (December‐January‐February) mixed layer in the Southern Ocean. These models use either the Mellor‐Yamada turbulence closure scheme or the K profile parameterization scheme. The numerical results show that the nonbreaking surface wave–induced mixing can reduce sea surface temperature and increase subsurface temperature of the upper ocean, which can deepen the upper ocean mixed layer. For these four models, austral summer mixed‐layer depths in the cases with the wave‐induced mixing deepen by 6.2–21.5 m than those without in the Southern Ocean (30°S–65°S), much closer to the observation. These results indicate that incorporation of the wave‐induced mixing can improve the simulation of the austral summer upper mixed layer in the Southern Ocean. Although the responses of four models are different, all simulation results show notable improvements, which may imply that the nonbreaking surface wave–induced mixing is an indispensable physical process for ocean and climate models.

Comparison between vertical shear mixing and surface wave‐induced mixing in the extratropical ocean

Most parameterizations of vertical mixing are associated with local shear instability, which do not explicitly include the effects of surface waves. Here, we compared the performance of vertical mixing induced by vertical shear of the mean current and that by nonbreaking surface waves in the upper ocean through three numerical experiments. The vertical mixing from vertical shear alone was too weak especially in the extratropical ocean, and failed to produce a reasonable mixed layer depth and seasonal thermocline, which resulted in a large cold bias and an unrealistic seasonal cycle in the subsurface. Surface waves can enhance the vertical mixing in the upper ocean, and induce vertical mixing to sustain a reasonable upper ocean temperature structure especially in the extratropical ocean. Both the temperature structure and seasonal cycle were significantly improved by including the nonbreaking surface wave–induced vertical mixing, no matter whether shear effect was included or not. These results indicate that the vertical mixing from surface waves is more important than that associated with velocity shear of the mean current for the upper ocean especially in the extratropics.

Tsunami and wind-driven wave forces in the Mediterranean Sea

The comparison between the impact forces generated from a tsunami wave and those from a wind-driven wave are presented here. Wind-driven wave forces are derived from established maritime theory going back to the nineteenth century. The understanding of tsunami impact forces is derived from more recent research correlated further by some recent model testing. The maritime theory on wave impact force for breaking and non-breaking waves is very well developed from breakwater/seawall engineering. It commenced with Stephenson and Hiro's nineteenth century rule of thumb equations, which was then followed in the twentieth century by Sainflou's theory (1928) for standing waves in deep water. Minikin's studies in the 1950s yielded results for breaking waves and noted high dynamic localised loads and these was followed in 1985 by Goda's equations applicable for both breaking and non-breaking waves. At Alderney, typical impact pressures of 40 t/m2 (400 kN/m2) for 20 m/s waves were recorded, although a Ciria document published in 1992 notes that the average wave pressure on sea walls varies from 150 kN/m2 down to 50 kN/m2 pressure. Users are cautioned about the extremely high wave forces associated with the Minikin method. In 1984, Blackmore and Hewson developed a method to estimate an average wave pressure from broken wave loads. The present analysis which was undertaken for the bathymetry and climatological data of the Mediterranean Sea then outlines the Malta scenario. It can act as a guide to structural engineers in selecting suitable values for the impact pressures that vertical walls may be subjected to for the cases of wind-driven or tsunami waves.

A novel adaptive σ coordinate model and its applications to wave and structure interaction

In this paper, a novel adaptive mesh σ coordinate model is proposed for the studies of Wave and Structure Interaction (WSI). The model is validated by using the case of a solitary wave movement in an open channel with constant water depth and the case of nonbreaking solitary wave propagating over a step. Numerical results agree well with the analytical solutions obtained based on the Boussinesq theory, the laboratory data and other numerical model results. The proposed model is then used to study a solitary wave interacting with a suspended fixed structure.

Improved Simulation of the South Asian Summer Monsoon in a Coupled GCM with a More Realistic Ocean Mixed Layer

Abstract Simulation and prediction of the South Asian summer monsoon in a climate model remain a challenge despite intense efforts by the atmosphere and ocean research community. Because the phenomenon arises from the interaction of the atmosphere with the upper ocean, a deficiency in the simulation of the latter can lead to a poor simulation of the atmospheric meridional circulation. This study demonstrates that a significant improvement can be obtained in the simulation of the summer monsoon by correcting a prevailing deficiency in the mixed layer simulation of the Indian Ocean. A particular physical process of the nonbreaking wave–ocean mixing parameterized as Bυ, which has not been considered in any climate model, is included in this study to enhance the vertical mixing in the upper ocean. Results show that the inclusion of this mixing process in a climate model leads to a better simulation of the ocean mixed layer, especially in the regions where the mixing was previously underestimated. The improved mixed layer simulation further results in stronger meridional differential heating, which drives stronger low-level monsoonal winds and results in stronger moisture transport and convergence, especially in the northern Indian Ocean. Moisture convergence into the Bay of Bengal is significantly enhanced and in general the spatial distribution of moisture is more consistent with observations. The directly driven monsoonal winds by the differential heating are further amplified by the resultant latent heating, which generates not only a wind amplitude comparable to the observations but also a correct vertical structure.

An improved calculation model for the wave-induced pore pressure distribution in a rubble-mound breakwater core

The spatial distribution of the wave-induced pore pressure height in the core of a conventional rubble-mound breakwater is studied in this work. Use is made of existing theoretical and experimental knowledge to establish a calculation model for the pressure distribution on the front core slope and the attenuation of pore pressures within the breakwater core. The new model formulae are derived empirically and calibrated by means of a non-linear regression analysis of pore pressure measurements in a large-scale conventional breakwater model, under non-overtopping and non-breaking wave conditions. They relate the spatial distribution of pore pressure height to the sea state in front of the breakwater considering a homogeneous breakwater core, but do not contain any material properties related to the porous flow resistance. The new calculation model predicts the pore pressure distribution with higher accuracy than the existing method generally employed and is applicable in a broad range of wave conditions. The calculation model is validated with additional experimental data from pore pressure measurements on a small-scale model, confirming the capability of predicting the pore pressure height under varying wave conditions. The comparison between both models moreover provides insight into the general applicability of the calculation model.

On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models

Nearshore morphodynamic models are computationally demanding, especially when the time scale of interest is weeks or longer. Hence, they often rely on a simple parameterization or non-linear wave theory to estimate the skewed-asymmetric shape of the near-bed, free-stream wave orbital motion, relevant to the prediction of onshore sand transport during mild wave conditions. Recently, Abreu et al. (2010) presented a simple analytical expression for this shape. Here, we present parameterizations to estimate the non-linearity parameter r and phase ϕ in this expression, such that the non-linear orbital motion can be estimated efficiently from values of the significant wave height Hs, wave period T, and water depth h, standard output of nearshore morphodynamic models. The parameterizations are based on a data set of 30.000+ field observations of the orbital skewness Su and asymmetry Au, collected under non-breaking and breaking wave conditions. Consistent with earlier observations, we find that the Ursell number, which includes Hs, T and h, describes the variability in Su and Au well and we use it to link Hs, T and h to r and ϕ. The comparison of our findings to another large field data set suggests that wave non-linearity depends weakly on wave directional spread and that our parameterizations may underestimate Su for narrow-banded swell and (unidirectional) laboratory conditions. Furthermore, the use of the parameterizations is not advised on bed slopes steeper than in our data set (i.e., >1:30).

Motion of water and sediment due to non-breaking waves in the swash zone

A long wave run-up theory is applied to the modelling of wave-induced flow velocities, sediment transport rates and bottom changes in a swash zone. First, the properties of the water tongue motion and the resulting lithodynamic response are analysed theoretically. Next, an attempt is made to run the model for the natural conditions encountered on the southern Baltic Sea coast. The Lagrangian swash velocities are used to determine the Eulerian phase-resolved bed shear stresses with a momentum integral method, after which the motion of sand is described by the use of a two-layer model, comprising bedload and nearbed suspended load. Seabed evolution is then found from the spatial variability of the net sediment transport rates. The results presented are limited to cases of the small-amplitude waves that seem to be responsible for accretion on beaches.

Suction Removal of Sediment from between Armor Blocks. III: Breaking Waves

When a sediment beach covered by stones or an armor layer is exposed to breaking waves, the turbulence generated by the breaking waves can cause mobilization and removal of the sediment underneath the stones. In two earlier studies by the Technical University of Denmark (DTU)-group, the suction removal of sediments from between armor blocks has dealt with the case of steady current and nonbreaking waves. The present study is an extension of these studies and the results will be presented in a similar way. The critical conditions for removal of sediment are determined. It is found that the onset of removal of sediment is governed by three parameters: (1) the sediment mobility (based on the sediment size, wave height, and wave period), (2) the ratio between the sediment size and the stone size, d/D, and (3) surf similarity parameter, ξ=tan(β)/(H0/L0)0.5. The variation of the critical mobility number for removal of sediment as function of d/D and ξ is determined for the range 0.001<d/D≤1.0 and 0.15<ξ<6.00. The experiments were made on 1∶2, 1∶14, and 1∶30 slopes. Spilling, plunging, and surging breakers were used in the experiments. Both one and two layers of armor stones and rectangular blocks were studied.

Turbulence beneath finite amplitude water waves

Babanin and Haus (J Phys Oceanogr 39:2675–2679, 2009) recently presented evidence of near-surface turbulence generated below steep non-breaking deep-water waves. They proposed a threshold wave parameter a 2ω/ν = 3,000 for the spontaneous occurrence of turbulence beneath surface waves. This is in contrast to conventional understanding that irrotational wave theories provide a good approximation of non-wind-forced wave behaviour as validated by classical experiments. Many laboratory wave experiments were carried out in the early 1960s (e.g. Wiegel 1964). In those experiments, no evidence of turbulence was reported, and steep waves behaved as predicted by the high-order irrotational wave theories within the accuracy of the theories and experimental techniques at the time. This contribution describes flow visualisation experiments for steep non-breaking waves using conventional dye techniques in the wave boundary layer extending above the wave trough level. The measurements showed no evidence of turbulent mixing up to a value of a 2ω/ν = 7,000 at which breaking commenced in these experiments. These present findings are in accord with the conventional understandings of wave behaviour.

粒子法による波の遡上計算

In the river and coastal engineering field, a lot of researchers have been dealt with wave run-up. Estimation of the height of wave run-up is very important for planning of disaster prevention, development of structures on the waterfront, and management from coastal zones to riverside. The authors simulate the wave run-up of a solitary wave under the conditions of breaking wave and nonbreaking wave using particle method. The accuracy validation of these results was done through comparison with the result from Synolakis's experiment. This study shows that the particle method is a useful method in estimating the relation between height of wave run-up and wave height.

Implementation of the vortex force formalism in the coupled ocean-atmosphere-wave-sediment transport (COAWST) modeling system for inner shelf and surf zone applications

The coupled ocean-atmosphere-wave-sediment transport modeling system (COAWST) enables simulations that integrate oceanic, atmospheric, wave and morphological processes in the coastal ocean. Within the modeling system, the three-dimensional ocean circulation module (ROMS) is coupled with the wave generation and propagation model (SWAN) to allow full integration of the effect of waves on circulation and vice versa. The existing wave-current coupling component utilizes a depth dependent radiation stress approach. In here we present a new approach that uses the vortex force formalism. The formulation adopted and the various parameterizations used in the model as well as their numerical implementation are presented in detail. The performance of the new system is examined through the presentation of four test cases. These include obliquely incident waves on a synthetic planar beach and a natural barred beach (DUCK’ 94); normal incident waves on a nearshore barred morphology with rip channels; and wave-induced mean flows outside the surf zone at the Martha’s Vineyard Coastal Observatory (MVCO).Model results from the planar beach case show good agreement with depth-averaged analytical solutions and with theoretical flow structures. Simulation results for the DUCK’ 94 experiment agree closely with measured profiles of cross-shore and longshore velocity data from Garcez Faria et al. (1998, 2000). Diagnostic simulations showed that the nonlinear processes of wave roller generation and wave-induced mixing are important for the accurate simulation of surf zone flows. It is further recommended that a more realistic approach for determining the contribution of wave rollers and breaking induced turbulent mixing can be formulated using non-dimensional parameters which are functions of local wave parameters and the beach slope. Dominant terms in the cross-shore momentum balance are found to be the quasi-static pressure gradient and breaking acceleration. In the alongshore direction, bottom stress, breaking acceleration, horizontal advection and horizontal vortex forces dominate the momentum balance. The simulation results for the bar/rip channel morphology case clearly show the ability of the modeling system to reproduce horizontal and vertical circulation patterns similar to those found in laboratory studies and to numerical simulations using the radiation stress representation. The vortex force term is found to be more important at locations where strong flow vorticity interacts with the wave-induced Stokes flow field. Outside the surf zone, the three-dimensional model simulations of wave-induced flows for non-breaking waves closely agree with flow observations from MVCO, with the vertical structure of the simulated flow varying as a function of the vertical viscosity as demonstrated by Lentz et al. (2008).

Simulation of Wave Overtopping of Maritime Structures in a Numerical Wave Flume

A numerical wave flume based on the particle finite element method (PFEM) is applied to simulate wave overtopping for impermeable maritime structures. An assessment of the performance and robustness of the numerical wave flume is carried out for two different cases comparing numerical results with experimental data. In the first case, a well‐defined benchmark test of a simple low‐crested structure overtopped by regular nonbreaking waves is presented, tested in the lab, and simulated in the numerical wave flume. In the second case, state‐of‐the‐art physical experiments of a trapezoidal structure placed on a sloping beach overtopped by regular breaking waves are simulated in the numerical wave flume. For both cases, main overtopping events are well detected by the numerical wave flume. However, nonlinear processes controlling the tests proposed, such as nonlinear wave generation, energy losses along the wave propagation track, wave reflection, and overtopping events, are reproduced with more accuracy in the first case. Results indicate that a numerical wave flume based on the PFEM can be applied as an efficient tool to supplement physical models, semiempirical formulations, and other numerical techniques to deal with overtopping of maritime structures.