Deep Water Regime Research Articles

Lattice Boltzmann simulation of breaking waves

Water wave breaking at the ocean’s surface is a turbulent phenomenon and encompasses a wide range of physical processes such as generation of bubbles/drops with different spatio-temporal scales. In the present study, breaking of periodic waves in deep water regime has been simulated using phase-field lattice Boltzmann method (LBM). Compared to direct Navier–Stokes equation based numerical models, the present LB model is Poisson-free and explicit in time integration. Both the flow field and interface are solved using LBM with two sets of distribution functions. The normalized pressure-velocity formulation is used for the flow field, whereas the conservative Allen–Cahn equation is used to capture the interface. An impulsively started wave approach with Stokes third order wave profile has been used to initiate the breaking process. Through systematic variation of Weber number and initial steepness, the effect of surface tension on the kinematic, geometric, and energy dissipation characteristics has been studied for three breaker types, viz., spilling, weak plunging, and plunging breakers. The results are in good agreement with the past experimental observations and direct numerical simulation. In particular, the obtained breaking strength for different Weber numbers agrees well with the semi-empirical relation used in a wind-wave forecasting models.

A guide for selecting periodic water wave theories - Le Méhauté (1976)’s graph revisited

This note provides guidelines for selecting appropriate analytical periodic water wave solutions for applications, based on two physical parameters, namely, frequency dispersion parameter (water depth divided by wavelength) and the nonlinearity parameter (wave height divided by wavelength). The guidelines are summarized in a graphic format in the two-parameters space. The new graph can be viewed as the quantification of the well-known Le Méhauté (1976)’s graph with quantitative demarcations between applicable analytical periodic water wave solutions. In the deep water and intermediate water regimes the fifth-order Stokes wave theories (Zhao and Liu, 2022) are employed for the construction of the graph, while in the shallower water regime, Fenton (1999)’s higher order cnoidal wave theories are used. The dividing lines between the applicable ranges of Stokes wave and cnoidal wave theories are determined by the values of Ursell number, a ratio between the nonlinearity and frequency dispersion.

Non-homogeneous approximation for the kurtosis evolution of shoaling rogue waves

Bathymetric changes have been experimentally shown to affect the occurrence of rogue waves. We recently derived a non-homogeneous correction to the spectral analysis, allowing us to describe the evolution of the rogue wave probability over a shoal. Here, we extend this work to the evolution of the excess kurtosis of the surface elevation, that plays a central role in estimating rare event probabilities. Furthermore, we provide an upper bound to the excess kurtosis. In intermediate and deep water regimes, a shoal does not affect wave steepness nor bandwidth significantly, so that the vertical asymmetry between crests and troughs, the excess kurtosis and the exceedance probability of wave height stay rather constant. In contrast, in shallower water, a sharp increase in wave steepness increases the vertical asymmetry, resulting in a growth of both the tail of the exceedance probability and the excess kurtosis.

Changing sea-surface and deep-water conditions in the southern Cape Verde Basin during the mid-Pleistocene to Holocene

Reconstruction of the Atlantic Meridional Overturning Circulation (AMOC) based on long-term records spanning multiple glacial-interglacial transitions is essential for understanding the interplay between ocean dynamics and global climate change. However, despite valuable previous studies, many uncertainties remain, especially regarding the distribution of glacial water masses in the Atlantic and the AMOC intensity. Such long-term reconstructions are highly constrained by sedimentary geochemical records. Here, we present isotope, geochemical, and micropaleontological records from the core ANS-33047 retrieved from the southern part of the Cape Verde Basin at a water depth of 4027 m. The data reflect the glacial–interglacial changes in surface and near-bottom conditions in the area over the last 513 kyr. Within the upper ocean layer, enhanced bioproductivity associated with northward ITCZ migration has been determined during MIS 13/12, early MIS 11, 9, 7, and MIS 1. A gradual deepening of the mixed layer is inferred from MIS 5 to the present. Stronger deep-water upwelling and shoaling of the mixed layer over the study site preceded these intervals. In the deep-water environment, we propose three different regimes within the interglacials: a “classical model” and “local productivity effect” associated with a stronger advection of paleo-NADW into the study area (MIS 11, 5, 1, warm early phases of MIS 9 and 7, MIS 13/12 boundary), as well as “amplified paleo-AABW influence” (late MIS 13 and 7). The bottom-water chemistry was affected by organic matter degradation coinciding with intervals of northward ITCZ migrations and enhanced surface bioproductivity. The most prominent signal of southern-sourced deep water was reconstructed during the late MIS 12, MIS 8, and at the MIS 10/9 boundary. During MIS 3 and sub-stage 12b, the study area was bathed by a mixture of paleo-NADW and paleo-AABW. A comparison of the carbon isotope records from the northeastern and southeastern equatorial Atlantic demonstrates a larger difference in the deep-water properties within glacials in contrast to interglacials. • ITCZ shifts northwards and bioproductivity is enhanced during interglacial intervals. • Increased surface bioproductivity affects δ 13 C composition of paleo-NADW. • Three deep-water regimes occurrs within interglacials. • Stronger meridional variability of deep-water circulation in cold climatic phases.

Effect of surface-active contaminants on radial thermocapillary flows.

We study the thermocapillary creeping flow induced by a thermal gradient at the liquid-air interface in the presence of insoluble surfactants (impurities). Convective sweeping of the surfactants causes density inhomogeneities that confers in-plane elastic features to the interface. This mechanism is discussed for radially symmetric temperature fields, in both the deep and shallow water regimes. When mass transport is controlled by convection, it is found that surfactants are depleted from a region whose size is inversely proportional to the interfacial elasticity. Both the concentration and the velocity fields follow power laws at the border of the depleted region. Finally, it is shown that this singular behavior is smeared out when molecular diffusion is accounted for.

Statistics of single and multiple floaters in experiments of surface wave turbulence

We present laboratory experiments of surface wave turbulence excited by paddles in the deep water regime. The free surface is seeded with buoyant particles that are advected and dispersed by the flow. Positions and velocities of the floaters are measured using particle tracking velocimetry. We study the statistics of velocity and acceleration of the particles, mean vertical displacements, single-particle horizontal dispersion, and the phenomenon of preferential concentration. Using a simple model together with the experimental data, we show that the time evolution of the particles has three characteristic processes that dominate the dynamics at different times: drag by surface waves at early times, trapping by horizontal eddies at intermediate times, and advection by a large-scale mean circulation at late times.

Waves on a vortex: rays, rings and resonances

We study the scattering of surface water waves with irrotational draining vortices. At small depth, this system is a mathematical analogue of a rotating black hole and can be used to mimic some of its peculiar phenomena. Using ray-tracing methods, we exhibit the existence of unstable orbits around vortices at arbitrary depth. These orbits are the analogue of the light rings of a black hole. We show that these orbits come in pairs, one co-rotating and one counter-rotating, at an orbital radius that varies with the frequency. We derived an explicit formula for this radius in the deep-water regime. Our method is validated by comparison with recent experimental data from a wavetank experiment. We finally argue that these rings will generate a discrete set of damped resonances that we characterize and that could possibly be observed in future experiments.

A deep water shelly fauna from the uppermost Ordovician in northwestern Hunan, South China and its paleoecological implications

It is the first time to document the trilobite Mucronaspis ( Songxites ) wuningensis and the brachiopod Paromalomena - Aegiromenella Assemblage from the Xinkailing Bed (Hirnantian, uppermost Ordovician) in Taoyuan and Cili counties, northwestern Hunan Province. Synecological analysis shows that this is a deep water shelly fauna of South China during the Hirnantian. It lived on the upper Jiangnan Slope in northwestern Hunan, adjacent to the southeast margin of the Upper Yangtze Region, and belongs to the Hirnantia fauna with cool and deep water, and low diversity affinity, similar to its contemporary shelly faunas from the southern Shaanxi Slope adjacent to the north margin of the Upper Yangtze Region, the northern Chongqing depression (new name), and the Jiangxi-Anhui-Jiangsu gentle slope of the Lower Yangtze Region. Paleoecologically, it could be attributed to the BA 4–5 benthic regime (about 60–150 m deep, and deepest to 200 m), and was significantly different from the trilobite M . ( S .) mucronata and the typical Hirnantia fauna while the latters may have habited in BA 2–3 benthic regimes (about 5–60 m deep). A new paleogeographic distribution pattern of the Hirnantian shallow and deep water shelly faunas of South China is herein proposed for the first time. Assuming that the depositional rate was constant in northwestern Hunan during the end Ordovician, the duration of the Xinkailing Bed might be only 12 thousand years according to the thickness of the Wufeng Formation and the absolute age values of those relevant graptolitic biozones. Such a short time interval may indicate that the global environmental change during the crisis was much shorter than previously thought, and its influence on the deep water regime was significantly shorter than that on the shallow water regime.

Serre-type Equations in Deep Water

This manuscript is devoted to the modelling of water waves in the deep water regime with some emphasis on the underlying variational structures. The present article should be considered as a review of some existing models and modelling approaches even if new results are presented as well. Namely, we derive the deep water analogue of the celebrated SERRE–GREEN–NAGHDI equations which have become the standard model in shallow water environments. The relation to existing models is discussed. Moreover, the multi-symplectic structure of these equations is reported as well. The results of this work can be used to develop various types of robust structure-preserving variational integrators in deep water. The methodology of constructing approximate models presented in this study can be naturally extrapolated to other physical flow regimes as well.

Two-dimensional numerical study on the effect of water depth on resonance behaviour of the fluid trapped between two side-by-side bodies

The effect of water depth on gap resonance of two side-by-side rectangular boxes is examined in a 2D numerical wave flume at various gap widths and body drafts. The numerical model used is based on the volume of fluid (VOF) method for the solution of Navier–Stokes equations combined with free surfaces. The water depth covered ranges from shallow to deep water regimes. It is found that the resonant frequencies and wave heights depend apparently on water depth. For a given body draft and gap width, the resonant wave frequency shifts to higher values whereas the resonant wave height inside the gap decreases as water depth increases. The amount of reduction of wave height is found to be dependent on the depth to draft and gap width to breadth ratios. It is observed that the effect of water depth on resonance frequency is very small when the ratio of water depth to body draft is equal to or greater than six, while the effect of water depth on resonant water level diminishes when this ratio increases beyond 10. Furthermore, it is observed that the variation trends of the resonant frequency and the reduction of the resonant wave height with water depth are similar at different drafts and gap widths. The present results suggest that the water depth, body draft and the ratio of water depth to body draft are the key factors that influence the resonant responses. It is also found that the potential flow model not only over-predicts the resonant wave heights but also fails to predict the variation trends of the resonant wave height with water depth.

Effect of RANSbased Turbulence Models on Nonlinear Wave Generation in a TwoPhase Numerical Wave Tank

Ocean waves are the most important exciting source acting on marine structures such as ships, offshore platforms, wave energy converters and wavebreakers. In order to efficiently design the aforementioned structures, accurate modelling of these waves is of importance. In this paper a two dimensional Numerical Wave Tank (NWT) has been established based on the Reynoldsaveraged NavierStokes (RANS) equations for viscous, incompressible fluid and Volume of Fluid (VOF) method and a commercial software code ANSYS FLUENT (Release 15.0) has been used to numerically investigate ocean wave generation. Impact of different turbulence models such as standard k-ɛ, realizable k-ɛ, Shear Stress Transport (SST) and Reynolds Stress Models (RSM) on the generated ocean surface waves were investigated. With all uncertainties associated with various numerical setting aspects, experimental wave measurements over a wide range of wave conditions covering intermediate and deep water regimes have been conducted in a physical wave basin to validate the numerical results. The excessive generation of eddy viscosity resulted from using eddy viscosity turbulence models especially at the free surface interface, leads to a significant unphysical damping on the generated waves. Good numerical agreement with both experimental measurements and analytical wave theory was successfully achieved either with the RSM or implementing artificial turbulence damping at the airwater interface with the SST model.

Error Analysis of an Implicit Spectral Scheme Applied to the Schrödinger-Benjamin-Ono System

We develop error estimates of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a coupled nonlinear Schrödinger-Benjamin-Ono system that describes the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The accuracy of the numerical solver is checked using some exact travelling wave solutions of the system.

The confined Muskat problem: Differences with the deep water regime

We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip R × [−l,l]. This problem is known as the Muskat problem and is analogous to the two phase Hele-Shaw cell. The main goal of this paper is to compare the qualitative properties between the model when the fluids move without boundaries and the model when the fluids are confined. We find that, in a precise sense, the boundaries decrease the diffusion rate and the system becomes more singular.

Evaluating nitrogen isotopes as proxies for depositional environmental conditions in shales: Comparing Caney and Woodford Shales in the Arkoma Basin, Oklahoma

In this study, we assessed the use of stable nitrogen isotopes (δ15N) to evaluate differences in water column conditions during the deposition of the Woodford Shale and Caney Shale in the Arkoma Basin in Oklahoma by comparing bulk δ15N values. The Woodford Shale was mainly deposited under anoxic bottom water column conditions, whereas the Caney Shale shows evidence of deposition under a suboxic/dysoxic deep water regime. Therefore, these two units can be used to test the utility of δ15N as a proxy for the differentiation of these intervals based on depositional environment and post-depositional processes, and provide insights into differences that may impact their assessments as reservoir and source rocks. We measured δ15Nbulk, organic carbon isotopes (δ13Corg), total organic carbon (TOC) and total nitrogen (TN) values from a core from the Arkoma Basin that contains sections of both Caney Shale and Woodford Shale. Bulk δ15N values are higher in the Caney Shale (average δ15Nbulk=9.5±0.8‰) than in the Woodford Shale (average δ15Nbulk=3.1±0.7‰); TOC values in the Woodford Shale are slightly higher than those in the Caney Shale (6±2% and 4±2%, respectively). No significant differences were identified in the δ13Corg and TN values between the two units. Given the depositional history of these units, we suggest that the difference in δ15N between the Caney Shale and the Woodford Shale primarily reflects differences in depositional environment, and that changes in δ15Nbulk values due to diagenesis or catagensis are subordinate.

Stokes drift for inertial particles transported by water waves

We study the effect of surface gravity waves on the motion of inertial particles in an incompressible fluid. We perform analytical calculations based on perturbation expansions which allow us to predict the dynamics of inertial particles in the deep-water regime. We find that the presence of inertia leads to a non-negligible correction to the well-known horizontal Stokes drift velocity. Moreover, we find that the vertical sedimentation velocity is also affected by a drift induced by waves. The latter result may have some relevant consequences on the rate of sedimentation of particles of finite size. We underline that the vertical drift would also be observed in the (hypothetical) absence of the gravitational force. Kinematic numerical simulations are performed and the results are found to be in excellent agreement with the analytical predictions, even for values of the parameters beyond the perturbative limit.

Palaeoenvironmental aspects of Late OrdovicianSericoideashell concentrations in an impact crater, Tvären, Sweden

Frisk, Å.M. & Harper, D.A.T. 2010: Palaeoenvironmental aspects of Late Ordovician Sericoidea shell concentrations in an impact crater, Tvären, Sweden. Lethaia, Vol. 44, pp. 383–396. Numerous studies have reported the presence of the small, thin-shelled cosmopolitan rhynchonelliformean Sericoidea as being environmentally controlled and, together with its close relatives, characteristic of deep-water, distal, clastic Ordovician and Silurian settings. Assemblages of Sericoidea have been analysed from post-impact strata in a newly formed Late Ordovician impact crater. In the crater succession, colonization of benthic faunas can be monitored through the post-impact limestone, demonstrating a number of environmental preferences. Consequently, the crater, as a result of its restricted area, provides an experimental arena for faunal distributions to be correlated with specific environments. The continuous infilling of the crater following its formation reveals a transition from argillaceous mudstones to carbonates deposited in deeper-water environments to shallower regimes. Rhynchonelliformean brachiopods inhabited the crater depression very late after the impact and are entirely represented by the genus Sericoidea, occurring abundantly in the upper third of the existing crater infill. The deep-water regime that existed in the depression during the initial interval of crater formation had been substantially reduced. Clearly Sericoidea-bearing associations associated with shaly substrates did not merely favour and occupy deep-water environments as previously suggested. The unfavourable conditions triggered by the impact and the inhospitable aftermath allowed Sericoidea to exploit a less-crowded ecospace. This reorganization, following the catastrophe, from a deep-water related ecological niche to considerable shallower settings suggests that Sericoidea was a pioneer colonist displaying an opportunist r-strategy. The shell beds analysed are related to shallower water and this may, moreover, help unravel the dilemma of the general absence of Sericoidea in the deeper-water Foliomena fauna. □Dalby Limestone, impact crater, Late Ordovician, opportunists, Sericoidea, Tvären.

Water wave collapses over quasi-one-dimensional nonuniformly periodic bed profiles

Nonlinear water waves interacting with quasi-one-dimensional nonuniformly periodic bed profiles are studied numerically in the deep-water regime with the help of approximate equations for envelopes of the forward and backward waves. Spontaneous formation of localized two-dimensional wave structures is observed in the numerical experiments, which looks essentially as a wave collapse.

Rate of Iceland Sea acidification from time series measurements

Abstract. The Iceland Sea is one part of the Nordic Seas. Cold Arctic Water prevails there and the deep-water is an important source of North Atlantic Deep Water. We have evaluated time series observations of measured pCO2 and total CO2 concentration from discrete seawater samples during 1985–2008 for the surface and 1994–2008 for deep-water, and following changes in response to increasing atmospheric carbon dioxide. The surface pH in winter decreases at a rate of 0.0024 yr−1, which is 50% faster than average yearly rates at two subtropical time series stations, BATS and ESTOC. In the deep-water regime (>1500 m), the rate of pH decline is a quarter of that observed in surface waters. The surface seawater carbonate saturation states (Ω) are about 1.5 for aragonite and 2.5 for calcite, about half of levels found in subtropical surface waters. During 1985–2008, the degree of saturation (Ω) decreased at an average rate of 0.0072 yr−1 for aragonite and 0.012 yr−1 for calcite. The aragonite saturation horizon is currently at 1710 m and shoaling at 4 m yr−1. Based on this rate of shoaling and on the local hypsography, each year another 800 km2 of seafloor becomes exposed to waters that have become undersaturated with respect to aragonite.

Evolution equation for short surface waves on water of finite depth

We address the question of determining the evolution equation for surface waves propagating in water whose depth is much larger than the typical wavelength of the surface disturbance. We avoid making the usual approximation of supposing the evolution to be given in the form of a modulated wave-packet. We treat the problem by means of a conformal transformation allowing to explicitly find the Dirichlet-to-Neumann operator for the problem together with asymptotic expansions in parameters measuring the nonlinearity and depth. This allows us to obtain an equation in physical variables valid in the weakly nonlinear, deep-water regime. The equation is an integro-differential equation, which reduces to known cases for infinite depth. We discuss solutions in a perturbative setting and show that the evolution equation describes Stokes-like waves.

SPECTRAL FORMULATION OF THE TWO FLUID EULER EQUATIONS WITH A FREE INTERFACE AND LONG WAVE REDUCTIONS

A nonlocal spectral formulation of classic water waves is derived, and its connection to the classic water wave equations and the Dirichlet–Neumann operator is explored. The nonlocal spectral formulation is also extended to a two-fluid system with a free interface, from which long wave asymptotic reductions are obtained. Of particular interest is an asymptotically distinguished (2 + 1)-dimensional generalization of the intermediate long wave equation, which includes the Kadomtsev–Petviashvili equation and the Benjamin–Ono equation as limiting cases. Lump-type solutions to this (2 + 1)-dimensional ILW equation are obtained, and the speed versus amplitude relationship is shown to be linear in the shallow, intermediate, and deep water regimes.