Vertical structure of hydrophysical characteristics and internal waves near the shelf boundary

Short and high frequency internal waves, propagating through horizontal gradients of temperature formed by long and low-frequency waves, induce vertical transferof heat with effective coefficients 10⁻⁴ – 10⁻³ m²/s and lead to a fine structure formation. Horizontal turbulence with seals of order 100 m produce the same effect in the presence of internal waves. The same is true for other substances, such as salt, oxygen, organic matter. It is suggested that this augmenting of vertical transports due to internal waves is an essential factor for high biological productivity in shelf zones.     

Coupling between internal and surface waves

In a fluid system in which two immiscible layers are separated by a sharp free interface, there can be strong coupling between large amplitude nonlinear waves on the interface and waves in the overlying free surface. We study the regime where long waves propagate in the interfacial mode, which are coupled to a modulational regime for the free-surface mode. This is a system of Boussinesq equations for the internal mode, coupled to the linear Schrödinger equations for wave propagation on the free surface, and respectively a version of the Korteweg-de Vries equation for the internal mode in case of unidirectional motions. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler’s equations, as described in (Benjamin and Bridges, J Fluid Mech 333:301–325, 1997, Craig et al., Comm Pure Appl Math 58:1587–1641, 2005a, Zakharov, J Appl Mech Tech Phys 9:190–194, 1968), using the perturbation theory for the modulational regime that is given in (Craig et al. to appear). We focus in particular on the situation in which the internal wave gives rise to localized bound states for the Schrödinger equation, which are interpreted as surface wave patterns that give a characteristic signature of the presence of an internal wave soliton. We also comment on the discrepancies between the free interface-free surface cases and the approximation of the upper boundary condition by a rigid lid.     

Analytical solutions of long nonlinear internal waves: Part I

The Gardner equation is an extension of the Korteweg–de Vries (KdV) equation. It exhibits basically the same properties as the classical KdV, but extends its range of validity to a wider interval of the parameters of the internal wave motion for a given environment. In this paper, we derive exact solitary wave solutions for the generalized Gardner equation that includes nonlinear terms of any order. Unlike previous studies, the exact solutions are derived without assuming their mathematical form. Illustrative examples for internal solitary waves are also provided. The traveling wave solutions can be used to specify initial data for the incident waves in internal waves numerical models and for the verification and validation of the associated computed solutions.     

Englacial kames near Jeziorany (Warmia — western Masurian Lakeland,Poland): Morphology,internal structure and origin

Groups of hills on a regional Vistulian (Weichselian) deglaciation plateau (Warmia, NE Poland) are covered by till of up to a dozen metres thick. It is the same till of the main stadial of the Vistulian glaciation that outcrops in the upland. The internal parts of the hills consist of fine-grained sands. These glacial landforms are referred to as englacial kames; they result from infilling of englacial caverns with sand. The orientation of the kame fields and the form pattern within these fields coincide with the system of primary crevasses in the ice, as it was reconstructed on the basis of the orientation of postglacial crevasse forms. The origin of the englacial caverns at crossing points of crevasse surfaces in the glacier is discussed. Structural analysis (relationship with a circular lineament, local differentiation of the complete profile of the Pleistocene deposits, possibility of occurrence of faults — festoon glacitectonics) suggests that the origin of these landforms is related to movements in the substratum.     

A model of regional fluid flow: Sand concentration factors and effective lateral and vertical permeabilities

Permeabilities of sand and shale differ by five to seven orders of magnitude. In the depositional pattern characteristic of the Gulf Coast, lenticular sand bodies occur randomly dispersed within a shale matrix, even though they can be grouped into stratigraphic sequences, contributing to net permeability of the sand-shale system and thus facilitating upward flow of pore water. In addition, a substantial lateral flow develops as fluid flow is directed toward sand bodies, thus concentrating fluid from a wide source area into sand bodies of limited lateral and vertical extent. Calculations reported here determine the effect of randomly distributed sands on both vertical and lateral permeabilities of composite sandshale systems. Effective permeabilities depend only on mean sand percent and local variance about the mean, because the effects of sand body shapes and their orientation and dip are not incorporated in the analysis. Results indicate that at a mean sand percentage of 20–50% a large lateral migration of fluid flow develops as sand bodies attract flow from a region two to seven times the radius of the sand body itself. For a mean sand percentage in excess of 50–60%, large vertical migration results and thus hydrocarbon trapping should not be expected for sand percentages in excess of 50–60%. Results of these calculations are in accord with observations that most larger oil accumulations on the Gulf Coast are found in regions having sandiness between about 20 and 50%.     

Conservation laws and invariants of motion for nonlinear internal waves: part II

In this paper, we derive three conservation laws and three invariants of motion for the generalized Gardner equation. These conserved quantities for internal waves are the momentum, energy, and Hamiltonian. The approach used for the derivation of these conservation laws and their associated invariants of motion is direct and does not involve the use of variational principles. It can be easily applied for finding similar invariants of motion for other general types of KdV, Gardner, and Boussinesq equations. The stability and conservation properties of discrete schemes for the simulations of internal waves propagation can be assessed and monitored using the analytical expressions of the constants of motion that are derived in this work.     

Sand waves and sediment transport around the end of a tidal sand bank

Two detailed surveys have been made of the north-western end of the Haisborough Sand off the Norfolk coast using echosounder, 3-5 kHz reflection profiler and side-scan sonar. Asymmetrical sand waves indicate north-westerly sand movement on the southern side and south-easterly sand movement on the steeper northern side of the bank. Secondary, superimposed megaripples, which are probably better indicators of sediment movement, give evidence of a cross-bank component. Between the north-westerly and south-easterly facing sand waves on the tip of the bank there is a zone of symmetrical sand waves. These are usually taken to indicate zero net transport, but in this case the oblique orientation of megaripples in their troughs indicates transport parallel to the sand wave crests. This suggests the route by which sand travels around the end of the bank to form a roughly closed circulation. Sediment textural parameters support the notion that sand is winnowed from the foot of the bank on both sides and is transported to the middle with an overall net transport from the south to the north. Analysis of charts dating back to 1886 shows that the bank is stable within the error limits of position fixing, though that could allow more than 0-25 km shift to the north east in 100 years to pass undetected. A box model is drawn up for the estimated sediment fluxes around the end of the bank, and implications for residence times and circulation rates are drawn from it.     

近岸波浪传播的非静压数值模型

基于均匀网格,建立了沿水深积分的非静压波浪传播数值模型,模型的求解分静压步和动压步两部分。静压步的控制方程为全非线性浅水方程,采用有限体积格式求解,通过采用线性重构技术和全隐式离散底摩阻项,保证了格式的和谐性、守恒性和水深非负性,有效处理了海岸动边界问题。动压步通过应用有限差分方法求解泊松方程考虑动压力,使得模型具备模拟色散性波浪传播的能力。引入波浪破碎指标,波浪破碎后模型退化为静压模型,破碎波自动捕捉为间断。通过算例对所建立模型进行了验证。     

A non-hydrostatic model for the numerical study of landslide-generated waves

This paper proposes and demonstrates a two-layer depth-averaged model with non-hydrostatic pressure correction to simulate landslide-generated waves. Landslide (lower layer) and water (upper layer) motions are governed by the general shallow water equations derived from mass and momentum conservation laws. The landslide motion and wave generation/propagation are separately formulated, but they form a coupled system. Our model combines some features of the landslide analysis model DAN3D and the tsunami analysis model COMCOT and adds a non-hydrostatic pressure correction. We use the new model to simulate a 2007 rock avalanche-generated wave event at Chehalis Lake, British Columbia, Canada. The model results match both the observed distribution of the rock avalanche deposit in the lake and the wave run-up trimline along the shoreline. Sensitivity analyses demonstrate the importance of accounting for the non-hydrostatic dynamic pressure at the landslide-water interface, as well as the influence of the internal strength of the landslide on the size of the generated waves. Finally, we compare the numerical results of landslide-generated waves simulated with frictional and Voellmy rheologies. Similar maximum wave run-ups can be obtained using the two different rheologies, but the frictional model better reproduces the known limit of the rock avalanche deposit and is thus considered to yield the best overall results in this particular case.     

Low-frequency internal waves and their influence on transmission loss variability

Analysis of the time series data collected from a stationary location in the continental shelf of the southeastern Arabian Sea during different month indicated prominent internal wave (IW) activity. Time evolution of temperature, resolved using Morlet wavelet technique, revealed that maximum energy was concentrated in the diurnal band at the density interface, whereas within the interior of thermocline, the dominant energy concentration shifted to semi-diurnal band. Both these harmonics have maximum amplitude (>15 m) during the pre-monsoon and monsoon season when the water column was highly stratified (>0.05 kg/m⁴), but they were not discernable in the temperature record when the stratification was weak (i.e., especially during winter). An acoustic propagation model based on ray theory, Bellhop (http:/oalib.hlsresearch.com/modes/acoustictoolbox/at.zip) was utilized to compute the transmission loss (TL) associated with the passage of low-frequency IWs. The TL was computed using the model considering (1) range-dependent and range-independent environmental scenario and (2) for different source and receiver depth configurations. Intermittent fading of acoustic signals was observed in the presence of IW. It was also observed that fading of signals very much depends on the source–receiver configuration.     

The initiation of sedimentary furrows by standing internal waves

Longitudinal helical vortices are generally considered to be the cause of longitudinal sedimentary bedforms. In tidal oscillatory flows, however, it is not clear how a regular system of vortices will become fixed for long enough to establish the bedforms. It is proposed that the presence of standing internal waves, with axes parallel to the tidal current, have provided the basic flow pattern on which longitudinal helical vortices develop in Southampton Water. The observed furrows then developed in a pattern determined by the standing internal waves.     

Sediment suspension under waves and currents: time scales and vertical structure

Field measurements of the vertical structure of near-bed suspended sediment concentrations were obtained from arrays of fast response optical backscatter suspended solids sensors to examine the time-dependent response of sediment resuspension to waves and currents and the constraints imposed by bedforms. Data were recorded from both a nonbarred, marine shoreface and a barred lacustrine shoreface, under both shoaling and breaking waves (significant heights of 0·25–1·50m; peak periods of 3 and 8 s) and in water depths of 0·5–5·0 m. Sediment concentrations are positively correlated with increasing elevation above the bed, but lagged in time. The time lag varies directly with separation distance between measurement locations and inversely with the horizontal component of the near-bed oscillatory velocity. Both the presence of wave groups and the settling velocities of the sediment particules in suspension influence the temporal changes in concentration at a given elevation. Sediment concentrations appear to respond more slowly to the incident wind-wave forcing with distance away from the bed as a result of two factors: (1) the sequential increase in concentration induced by a succession of large waves in a group; and (ii) the relative increase in finer sediments with smaller settling velocities. Bedforms interact with the near-bed horizontal currents to impose a distinct constraint upon the timing of suspension events relative to the phase of the fluid motion, and, therefore, the vertical structure of the suspended sediment concentration at a range of time scales. The near-bed concentrations appear to be strongly dependent upon the vertical convection of sediment associated with the ejection from the wave boundary layer of separation vortices generated in the lee of ripple crests. Concentration gradients in the presence of vortex ripples are large, as are the correlation between concentrations measured at different elevations within the fluid.     

A catastrophe mathematical model for uranium mineralization of hot-water origin

Presented in this paper are some primary data on the temperature, stratigraphical altitude and thickness of a certain uranium ore deposit. Taking these data as the parameters for the temperature, pressure and geochemical conditions of uranium mineralization under ideal geological conditions which constitute a finite variable series, in conjunction with the catastrophe theory developed by Rene Thom (1968), we have proposed a swallow-tail model fit to uranium mineralization. This model provides insight into the mechanism of uranium mineralization. Our study shows that the process of uranium mineralization is a discontinuous or catastrophic event. Temperature and pressure are considered important thermodynamic conditions which control the distribution pattern of uranium mineralization, and the principle of temperature-pressure correspondence for uranium mineralization has been also derived. Geological bodies of minerogenetic importance serve as favourable geochemical media for capturing ore-forming materials. Uranium mineralization of commercial importance depends on a delicate coupling between temperature,pressure, and geochemical condition. The loci of uranium mineralization in the model constitute the surfaceK, referred to as the surface of enrichment-mineralization, on which minerogenetic domain is expected. The model provides quantitatively clues to the process of uranium mineralization, i.e., a process from quantitative to qualitative changes.     

内孤立波环境下圆柱和方柱受力特征——II.数值模拟

柱体受内波作用力的数值模拟大多针对圆柱展开,方柱在内波环境下受力特性研究较少。借助三维数值波浪水槽,采用大涡模拟（LES）技术研究了内孤立波的产生、传播及其对不同形状柱体的受力特性。对比分析了内孤立波波幅对圆柱和方柱受力的影响规律。研究表明,在分层流环境下,随着内孤立波波幅的增大,圆柱和方柱所受的水平作用力均增大;相同波幅情况下,无论在上层或下层水体,方柱表面的压力分布更不均匀,迎流面与背流面的压差更大,从而导致方柱相较于圆柱会受到更大的水平作用力。     

Sand transport model of barchan dune equilibrium

Erosion and deposition over a barchan dune near the Salton Sea, California, is modelled by book-keeping the quantity of sand in saltation following streamlines of transport. Field observations of near-surface wind velocity and direction plus supplemental measurements of the velocity distribution over a scale model of the dune are combined as input to Bagnold-type sand-transport formulae corrected for slope effects. A unidirectional wind is assumed. The resulting patterns of erosion and deposition compare closely with those observed in the field and those predicted by the assumption of equilibrium (downwind translation of the dune without change in size or geometry). Discrepancies between the simulated results and the observed or predicted erosional patterns appear to be largely due to natural fluctuation in the wind direction. Although the model includes a provision for a lag in response of the transport rate to downwind changes in applied shear stress, the best results are obtained when no delay is assumed. The shape of barchan dunes is a function of grain size, velocity, degree of saturation of the oncoming flow, and the variability in the direction of the oncoming wind. Smaller grain size or higher wind speed produce a steeper and more blunt stoss-side. Low saturation of the inter-dune sandflow produces open crescent-moon-shaped dunes, whereas high saturation produces a whaleback form with a small slip face. Dunes subject to winds of variable direction are blunter than those under unidirectional winds. The size of barchans could be proportional to natural atmospheric scales, to the age of the dune, or to the upwind roughness. The upwind roughness can be controlled by fixed elements or by the sand is saltation. In the latter case, dune scale may be proportional to wind velocity and inversely proportional to grain size. However, because the effective velocity for transport increases with grain size, dune scale may increase with grain size as observed by Wilson (1972).     

A condition for the existence of quasi-periodic nonlinear internal waves in the ocean shelf zone

Free internal gravity waves, which are typical of the shelf ocean zone are studied. A necessary condition for the existence of nonlinear wave disturbances quasi-periodic in time in a continuously stratified sea with variable depth H is found in the quasistatic and the “hard cap” approximation with respect to dissipative factors and latitudinal variation in the Coriolis parameter. The obtained assessment is equivalent to the condition obtained within the framework of linear theory for the case of the f-plane and a constant depth H.     

A rigorous finite volume model to simulate subaerial and submarine landslide-generated waves

This paper presents a new landslide-generated wave (LGW) model based on incompressible Euler equations with Savage-Hutter assumptions. A two-layer model is developed including a layer of granular-type flow beneath a layer of an inviscid fluid. Landslide is modeled as a two-phase Coulomb mixture. A well-balanced second-order finite volume formulation is applied to solve the model equations. Wet/dry transitions are treated properly using a modified non-linear method. The numerical model is validated using two sets of experimental data on subaerial and submarine LGWs. Impulsive wave characteristics and landslide deformations are estimated with a computational error less than 5 %. Then, the model is applied to investigate the effects of landslide deformations on water surface fluctuations in comparison with a simpler model considering a rigid landslide. The model results confirm the importance of both rheological behavior and two-phase nature of landslide in proper estimation of generated wave properties and formation patterns. Rigid slide modeling often overestimates the characteristics of induced waves. With a proper rheological model for landslide, the numerical prediction of LGWs gets more than 30 % closer to experimental measurements. Single-phase landslide results in relative errors up to about 30 % for maximum positive and about 70 % for maximum negative wave amplitudes. Two-phase constitutive structure of landslide has also strong effects on landslide deformations, velocities, elongations, and traveling distances. The complex behaviors of landslide and LGW of the experimental data are analyzed and described with the aid of the robust and accurate finite volume model. This can provide benchmark data for testing other numerical methods and models.     

三维非均匀结构体对面波的多次散射

在面波层析成像和传统面波勘探中,都假定地球或浅层探测对象是一个均匀分层的多层介质模型,而实际上,介质的横向不均匀性广泛存在于从地球直到超声尺度的各种范围内.这里采用模式耦合的方法,研究了均匀半空间和二层介质中(也可以是多层的),三维非均匀体对Rayleigh面波的多次散射,计算了频率和时间域中的散射波场,并分析了散射波场的特点,为散射面波的应用提供了初步的理论基础.