A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches

The Boussinesq equation appears as the zeroth-order term in the shallow water flow expansion of the non-linear equation describing the flow of fluid in an unconfined aquifer. One-dimensional models based on the Boussinesq equation have been used to analyse tide-induced water table fluctuations in coastal aquifers. Previous analytical solutions for a sloping beach are based on the perturbation parameter, _N=αcotβ (in which β is the beach slope, α is the amplitude parameter and is the shallow water parameter) and are limited to tan⁻¹(α)βπ/2. In this paper, a new higher-order solution to the non-linear boundary value problem is derived. The results demonstrate the significant influence of the higher-order components and beach slope on the water table fluctuations. The relative difference between the linear solution and the present solution increases as and α increase, and reaches 7% of the linear solution.     

Experimental investigation on sea ripple evolution over sloping beaches

This paper reports on a wave flume experimental campaign carried out to investigate the appearance, the growth and the migration of small scale bedforms on a sloping sandy bed due to both regular and random waves. A Vectrino Profiler along with a structured light approach were used for velocity and morphodynamic measurements at two positions, one located above the horizontal bed, and the other one above the sloping beach. The velocity was computed by phase averaging the velocity measurements. Several velocity profiles were analyzed, identifying an offshore-directed steady current that extends from few centimeters above the bottom for all the analyzed water column. Ripple geometry was measured by a structured light approach and compared with that predicted by several models to shed light on the effects induced by the sloping beach on the shape and asymmetry. Along the sloping beach, the ripples appeared strongly asymmetric with the onshore half wavelengths smaller than the offshore ones. Finally, ripple geometry and migration triggered by regular waves were compared with those generated by random waves with comparable flow orbital amplitude showing a good agreement.     

Modelling coastal ground‐ and surface‐water interactions using an integrated approach

Beach water table fluctuations have an impact on the transport of beach sediments and the exchange of solute and mass between coastal aquifer and nearby water bodies. Details are given of the refinement of a dynamically integrated ground‐ and surface‐water model, and its application to study ground‐ and surface‐water interactions in coastal regions. The depth‐integrated shallow‐water equations are used to represent the surface‐water flow, and the extended Darcy's equation is used to represent the groundwater flow, with a hydrostatic pressure distribution being assumed to apply for both these two types of flows. At the intertidal region, the model has two layers, with the surface‐water layer being located on the top of the groundwater layer. The governing equations for these two types of flows are discretized in a similar manner and they are combined to give one set of linear algebraic equations that can be solved efficiently. The model is used to predict water level distributions across sloping beaches, where the water table in the aquifer may or may not decouple from the free water surface. Five cases are used to test the model for simulating beach water table fluctuations induced by tides, with the model predictions being compared with existing analytical solutions and laboratory and field data published in the literature. The numerical model results show that the integrated model is capable of simulating the combined ground‐ and surface‐water flows in coastal areas. Detailed analysis is undertaken to investigate the capability of the model. Copyright © 2009 John Wiley & Sons, Ltd.     

Wave-induced setup of the mean surface over a sloping beach

A new theoretical approach for the wave-induced setup over a sloping beach is presented that takes into consideration the explicit variations of the surface waves due to bottom slope and viscosity. In this way, the wave forcing of the mean Lagrangian volume fluxes is calculated without assuming that the local depth is constant. The analysis is valid in the region outside the surf zone and is based on the shallow-water assumption. A novel approach for separating the viscous damping of the waves from the frictional damping of the mean flow is introduced, where the mean Eulerian velocity is applied in the bottom stress for the mean fluxes. In the case where the onshore Lagrangian mean transport is zero, a new formula is derived for the Eulerian mean free surface slope, in which the effects of bottom slope, viscous wave damping and frictional bottom drag on the mean flow are clearly identified. The analysis suggests that viscous damping of the waves and frictional dissipation of the Eulerian near-bed return flow could lead to setup outside the surf zone.     

Two-dimensional approximation for tide-induced watertable fluctuations in a sloping sandy beach

The prediction of watertable fluctuations in a coastal aquifer is important for coastal management. However, most previous approaches have based on the one-dimensional Boussinesq equation, neglecting variations in the coastline and beach slope. In this paper, a closed-form analytical solution for a two-dimensional unconfined coastal aquifer bounded by a rhythmic coastline is derived. In the new model, the effect of beach slope is also included, a feature that has not been considered in previous two-dimensional approximations. Three small parameters, the shallow water parameter (ε), the amplitude parameter (α) and coastline parameter (β) are used in the perturbation approximation. The numerical results demonstrate the significant influence of both the coastline shape and beach slopes on tide-driven coastal groundwater fluctuations.     

Numerical Study of Tide-Induced Airflow and Salt–Fresh Water Dynamics in Unconfined Aquifers

This study employed a coupled water-air two-phase flow and salt water transport model to analyze the behaviors of generated airflow in unsaturated zones and the fluctuations of salinity at the salt–fresh water interface in a two-layered unconfined aquifer with a sloping beach surface subjected to tidal oscillations. The simulation results show that as the new dynamic steady state including effects of tidal fluctuations is reached through multiple tidal cycles, the dispersion zone in the lower salt water wedge is broadened because fresh water/salt water therein flows continuously landward or seaward during tidal cycles. The upper salt–fresh water interface exhibits more vulnerable to the tidal fluctuations, and the variation of salinity therein is periodic, which is irrelevant to the hydraulic head but is influenced by the direction and velocity of surrounding water-flow. With the tidal level fluctuating, airflow is mainly concentrated in the lower permeable layer due to the restraint of the upper semi-permeable layer, and the time-lag between the pore-air pressure and the tidal level increases with distance from the coastline. The effect of airflow in unsaturated zones can be transmitted downward, causing both the magnitude of salinity and its amplitude in the upper salt–fresh water interface to be smaller for the case with airflow than without airflow due to the resistance of airflow to water-flow. Sensitivity analysis reveal that distributions of airflow in unsaturated zones are affected by the permeability of the upper/lower layer and the van Genuchten parameter of the lower layer, not by the van Genuchten parameter of the upper layer, whereas the salinity fluctuations in the salt–fresh water interface are affected only by soil parameters of the lower layer.     

Tidal effects on groundwater dynamics in unconfined aquifers

The variation of seawater level resulting from tidal fluctuations is usually neglected in regional groundwater flow studies. Although the tidal oscillation is damped near the shoreline, there is a quasi‐steady‐state rise in the mean water‐table position, which may have an influence on regional groundwater flow. In this paper the effects of tidal fluctuations on groundwater hydraulics are investigated using a variably saturated numerical model that includes the effects of a realistic mild beach slope, seepage face and the unsaturated zone. In particular the impact of these factors on the velocity field in the aquifer is assessed. Simulations show that the tidal fluctuation has substantial consequences for the local velocity field in the vicinity of the exit face, which affects the nearshore migration of contaminant in coastal aquifers. An overheight in the water table as a result of the tidal fluctuation is observed and this has a significant effect on groundwater discharge to the sea when the landward boundary condition is a constant water level. The effect of beach slope is very significant and simplifying the problem by considering a vertical beach face causes serious errors in predicting the water‐table position and the groundwater flux. For media with a high effective capillary fringe, the moisture retained above the water table is important in determining the effects of the tidal fluctuations. Copyright © 2001 John Wiley & Sons, Ltd.     

A simple life-cycle method for predicting extreme shoreline erosion

The dynamical responses of a shoreline over long-term (years or decades) is a non-linear and time-dependent random process. It is affected by both longshore and cross-shore sediment transports. The former tends to cause cumulative changes in the mean shoreline position while the latter usually only leads to beach profile fluctuations relative to the moving mean beach profile. Due to the time-dependency of the process the life-cycle approach is ideally suited to formulate the probability distribution of extreme shoreline erosion. A model based on such approach and using standard Monte Carlo simulation techniques has been reported by Dong and Chen (1999). In this paper a simplified procedure is developed by introducing the assumption that the longshore and cross-shore processes are statistically independent. This then allows the probability distribution of the extreme erosion to be calculated in terms of the marginal probability distributions of the maximum recessions due to purely longshore and purely cross-shore transport. This method was applied to two idealised shoreline configurations and its usefulness for engineering applications is evaluated by comparison with the full Monte Carlo method.     

Water waves generated by instantaneous disturbances at the bed of a sloping beach

The two-dimensional problem of the generation of water waves due to instantaneous disturbances prescribed at the bed of a beach sloping at an arbitrary angle is studied here. It is formulated in terms of an initial-boundary-value problem for the velocity potential describing the motion in the fluid region assuming the linear theory. Using the Laplace transform in time and the Mellin transform in distance, the problem is reduced to solving a difference equation whose method of solution is of considerable importance in the literature. The form of the free surface is obtained in terms of a multiple infinite integral that is evaluated by the method of steepest-descent. For some prescribed forms of the disturbance at the bed of the beach, the free surface is depicted in a number of figures for different beach angles. It is observed that as the beach angle decreases, the maximum wave height increases, which is plausible.     

Tidal boundary conditions in SEAWAT

SEAWAT, a U.S. Geological Survey groundwater flow and transport code, is increasingly used to model the effects of tidal motion on coastal aquifers. Different options are available to simulate tidal boundaries but no guidelines exist nor have comparisons been made to identify the most effective approach. We test seven methods to simulate a sloping beach and a tidal flat. The ocean is represented in one of the three ways: directly using a high hydraulic conductivity (high-K) zone and indirect simulation via specified head boundaries using either the General Head Boundary (GHB) or the new Periodic Boundary Condition (PBC) package. All beach models simulate similar water fluxes across the upland boundary and across the sediment-water interface although the ratio of intertidal to subtidal flow is different at low tide. Simulating a seepage face results in larger intertidal fluxes and influences near-shore heads and salinity. Major differences in flow occur in the tidal flat simulations. Because SEAWAT does not simulate unsaturated flow the water table only rises via flow through the saturated zone. This results in delayed propagation of the rising tidal signal inland. Inundation of the tidal flat is delayed as is flow into the aquifer across the flat. This is severe in the high-K and PBC models but mild in the GHB models. Results indicate that any of the tidal boundary options are fine if the ocean-aquifer interface is steep. However, as the slope of that interface decreases, the high-K and PBC approaches perform poorly and the GHB boundary is preferable.     

EXPERIMENTAL STUDY ON INCIPIENT MOTION OF SEDIMENT PARTICLES ON GENERALIZED SLOPING FLUVIAL BEDS

1 INTRODUCTIONWhen water flows over a fluvial bed, hydro-dynandc force induced by the flow is acting on thesediment particles lying on the bed. A further increase in flow velocity results in an increase in themagnitude of this fOrce; and sediment particles begin to move if a situation is eventu8lly reached whenthe hydro-dynandc force exceeds a certain critical value. This initial movement of sediment pallicles istermed inciPient motion. The erosion and sedimentation of nuvial beds can be…     

Structure of numerically simulated katabatic and anabatic flows along steep slopes

Direct numerical simulation (DNS) is applied to investigate properties of katabatic and anabatic flows along thermally perturbed (in terms of surface buoyancy flux) sloping surfaces in the absence of rotation. Numerical experiments are conducted for homogeneous surface forcings over infinite planar slopes. The simulated flows are the turbulent analogs of the Prandtl (1942) one-dimensional laminar slope flow. The simulated flows achieve quasi-steady periodic regimes at large times, with turbulent fluctuations being modified by persistent low-frequency oscillatory motions with frequency equal to the product of the ambient buoyancy frequency and the sine of the slope angle. These oscillatory wave-type motions result from interactions between turbulence and ambient stable stratification despite the temporal constancy of the surface buoyant forcing. The structure of the mean-flow fields and turbulence statistics in simulated slope flows is analyzed. An integral dynamic similarity constraint for steady slope/wall flows forced by surface buoyancy flux is derived and quantitatively verified against the DNS data.     

Theoretical analysis of wave-induced gravel movements

One of the primary objectives of coastal research is the erection and testing of mechanically sound, predictive models for the two- and three-dimensional form of beach and nearshore bathymetries in order to account for and to predict coastal response to changing wave, tidal and sea-level conditions. The relationship between flow parameters and the mass transport of non-cohesive sedimentary particles has long been of importance in fluvial, aeolian and marine environments, but despite the existence of many numerical beach models, any of which might be developed to account for longer-term evolution, it is apparent that fundamental problems remain, not only in the representation of hydrodynamic and sediment dynamic processes but also in the choice of appropriate forcing functions for long-term simulations in real-time or quasi-real-time. An alternative is developed here in which frequency domain representations of a simplified set of process and response algorithms are presented, so that hydrodynamic, sediment dynamic and geomorphological parameters are related by a simple series of spectral gain functions. In particular, we concentrate on the relationship between the nearbed orthogonal flow spectrum and the resulting sediment mass transport rate. Numerical solutions, using both non-linear and linearized forms of the momentum equation, are presented.     

Steady-state migration and diffusion of hydrocarbons in complex stratigraphic sequences

A mean-field method is used to describe both average and fluctuating flow properties of water and oil in sedimentary basins. Comparison of oil and water flow shows the relative strengths of capillarity and buoyancy in controlling the mean and fluctuating components of flow. Effects of sandiness and water-wetness are studied as is the effect of decreasing porosity with increasing subsurface burial. Numerical estimates suggest that sandiness percentages between about 20 to 80% influence mean and fluctuating components of flow. Outside of this regime the end-member situations of shale or sand prevail.The extra effects of vertical buoyancy and capillarity generally make lateral diffusion less important for oil flow than for water flow.     

Numerical modelling of wind effects on breaking waves in the surf zone

Wind effects on periodic breaking waves in the surf zone have been investigated in this study using a two-phase flow model. The model solves the Reynolds-averaged Navier–Stokes equations with the k ? ?? turbulence model simultaneously for the flows both in the air and water. Both spilling and plunging breakers over a 1:35 sloping beach have been studied under the influence of wind, with a focus during wave breaking. Detailed information of the distribution of wave amplitudes and mean water level, wave-height-to-water-depth ratio, the water surface profiles, velocity, vorticity, and turbulence fields have been presented and discussed. The inclusion of wind alters the air flow structure above water waves, increases the generation of vorticity, and affects the wave shoaling, breaking, overturning, and splash-up processes. Wind increases the water particle velocities and causes water waves to break earlier and seaward, which agrees with the previous experiment.     

Turbulent kinetic energy budget in a gravel-bed channel flow

The present experimental investigation focuses on the characteristics of near bed turbulence in a fully rough, uniform open-channel flow over a gravel-type bed. Due to bed topography small scale heterogeneity, the flow is not uniform locally in the near bed region and a double averaging methodology is applied over a length scale much larger than the gravel size. The double-averaged Turbulent Kinetic Energy (TKE) budget derived in the context of the present flow over a gravel bed differs from the TKE budget written for flow over a vegetation canopy. The non-constant shape of the roughness function measured in our gravel bed leads to an additional bed-induced production term which is null for vertical roughness elements, such as simplified vegetation elements. The experimental estimation of the terms of the TKE budget reveals that the maximum turbulent activity takes place away from the reference plane, near the roughness crests. However, within the interface sublayer the work of the bed induced velocity fluctuations against the Reynolds stress is of the same magnitude as the main turbulence production term. Consequently, the characteristics of the TKE budget have similarities with uniform flows over canopies and strongly differ from uniform flows over smooth and transitionally rough flows over sedimentlike beds.     

Analysis of continuous and pulsed pumping of a phreatic aquifer

In a phreatic aquifer, fresh water is withdrawn by pumping from a recovery well. As is the case here, the interfacial surface (air/water) is typically assumed to be a sharp boundary between the regions occupied by each fluid. The pumping efficiency depends on the method by which the fluid is withdrawn. We consider the efficiency of both continuous and pulsed pumping. The maximum steady pumping rate, above which the undesired fluid will break through into the well, is defined as critical pumping rate. This critical rate can be determined analytically using an existing solution based on the hodograph method, while a Boundary Element Method is applied to examine a high flow rate, pulsed pumping strategy in an attempt to achieve more rapid withdrawal. A modified kinematic interface condition, which incorporates the effect of capillarity, is used to simulate the fluid response of pumping. It is found that capillarity influences significantly the relationship between the pumping frequency and the fluid response. A Hele-Shaw model is set up for experimental verification of the analytical and numerical solutions in steady and unsteady cases for pumping of a phreatic aquifer. When capillarity is included in the numerical model, close agreement is found in the computed and observed phreatic surfaces. The same model without capillarity predicts the magnitude of the free surface fluctuation induced by the pulsed pumping, although the phase of the fluctuation is incorrect.     

Barotropic instability of a boundary jet on a sloping bottom

Abstract

The stability of a shear flow on a sloping bottom in a homogeneous, rotating system was investigated by means of a laboratory experiment.

The basic flow was driven near a vertical wall of a circular container by a ring-shaped plate that contacted with a free surface of the working fluid and rotated relative to the fluid container. The velocity profile was asymmetric in the radial direction and had only one inflection point. The velocity profile was well expressed by a linear theory for the vertical shear layer.

The effect of the circular geometry was checked by comparing experimental results obtained in two fluid systems in which only the sign of the curvature was opposite and it was confirmed that circular geometry was not essential for the shear flow on the sloping bottom in this experiment.

It was found that the sloping bottom stabilizes the basic flow only when the drift direction of the topographic Rossby wave is opposite to that of the basic flow. The viscous dissipation in both the Ekman layer and the interior region was also important in determining the critical Rossby number.

The eddy fields caused by the instability can be classified into two types: One is the stationary eddy field in which a row of eddies moves along the basic flow without changing form. The other is the flow pattern in which eddies have finite life times and their configuration is not well organized. When the sloping bottom does not stabilize the basic flow, the former flow pattern is realized, otherwise the latter flow pattern appears.

The wave numbers of the eddies in the regular flow pattern were observed as a function of the Rossby number. The relation did not fit to linear preferred modes predicted by an eigenvalue problem.     

Bed-flow instability of the longshore current

An initially uniform longshore current on a plane erodible beach is considered and a linear stability analysis of the bed-flow system is performed in order to investigate the growth of alongshore periodic topographic features such as transverse or oblique bars. γ, numerical model based on the shallow water equations and a simple sediment transport formula is used. For a wide range of parameters instability is found, leading to the growth of large-scale topographic features (lengthscale of the order of the current width) downflow progressing. The growth rates and the dominant unstable mode depend mainly on R = c_d/β parameter, where c_d is the bottom friction coefficient and β is the beach slope. For a small R, say less than 0.1, instability is very weak, probably negligible. For R between 0.1 and 0.7 instability increases with R, leading typically to a quite simple transverse bars pattern. A further increase in R produces a far more complicated behaviour where complex patterns with downcurrent oriented oblique bars, bumps and holes can be dominant. In this region growth rates may either decrease or increase with R depending on the beach slope and the maximum Froude number of the basic flow, F. Usually, the most complex behaviour is found for gently sloping beaches. The physical mechanism of the instability is found to lie on the disturbances of potential vorticity caused by topographically induced differences in bottom friction. In this sense it is similar to the alternate bars growth in a river rather than the dunes or antidunes occurrence for 1D channel flow. The predictions of the model compare well with the available experimental data. The alongshore wavelength, γ, typically of the order of one to four times the width of the current, is close to four times for the most common values of R. The typical growth time is proportional to γ² and for a wavelength of 100 m can be of the order of one day, depending on the sediment transport rate. The migrational speed is inversely proportional to γ, in accordance to earlier field data reported by Sonu (1969) Collective movement of sediment in littoral environment.     

Sloped multi‐roller isolation devices for seismic protection of equipment and facilities

The rolling motion of mutually orthogonal rollers respectively sandwiched between two bearing plates in which one or both have V‐shaped sloping surfaces makes the sloped rolling‐type isolation device have an excellent in‐plane seismic isolation performance. In this study, the sloped rolling type isolation device in which a single roller moves between two V‐shaped sloping surfaces along each principle horizontal direction is refined by incorporating multi‐roller, built‐in damping, and pounding prevention mechanisms. The associated dynamic behavior is further clarified, and a simplified twin‐flag hysteretic model, which can be easily applied in most commercial computational tools is then proposed. Seismic simulation tests on the refined isolation devices (i.e. the sloped multi‐roller isolation devices) with different design parameters such as sloping angles of bearing plates and built‐in damping capabilities, together with a raised floor system by employing the sloped multi‐roller isolation devices, were conducted. Not only is the efficiency of the sloped multi‐roller isolation devices in seismically protecting the important objects, but also the practicability and accuracy of the proposed simplified numerical model in predicting the seismic responses of the sloped multi‐roller isolation devices is experimentally verified. Copyright © 2014 John Wiley & Sons, Ltd.