Showing papers on "Waves and shallow water published in 2018"

Shallow water bathymetry mapping using Support Vector Machine (SVM) technique and multispectral imagery

Abstract: Satellite imagery along with image processing techniques prove to be efficient tools for bathymetry retrieval as they provide time and cost-effective alternatives to traditional methods of water de...

Equatorial Magnetohydrodynamic Shallow Water Waves in the Solar Tachocline

Abstract: The influence of a toroidal magnetic field on the dynamics of shallow water waves in the solar tachocline is studied. A sub-adiabatic temperature gradient in the upper overshoot layer of the tachocline causes significant reduction of surface gravity speed, which leads to trapping of the waves near the equator and to an increase of the Rossby wave period up to the timescale of solar cycles. Dispersion relations of all equatorial magnetohydrodynamic (MHD) shallow water waves are obtained in the upper tachocline conditions and solved analytically and numerically. It is found that the toroidal magnetic field splits equatorial Rossby and Rossby-gravity waves into fast and slow modes. For a reasonable value of reduced gravity, global equatorial fast magneto-Rossby waves (with the spatial scale of equatorial extent) have a periodicity of 11 years, matching the timescale of activity cycles. The solutions are confined around the equator between latitudes ±20°–40°, coinciding with sunspot activity belts. Equatorial slow magneto-Rossby waves have a periodicity of 90–100 yr, resembling the observed long-term modulation of cycle strength, i.e., the Gleissberg cycle. Equatorial magneto-Kelvin and slow magneto-Rossby-gravity waves have the periodicity of 1–2 years and may correspond to observed annual and quasi-biennial oscillations. Equatorial fast magneto-Rossby-gravity and magneto-inertia-gravity waves have periods of hundreds of days and might be responsible for observed Rieger-type periodicity. Consequently, the equatorial MHD shallow water waves in the upper overshoot tachocline may capture all timescales of observed variations in solar activity, but detailed analytical and numerical studies are necessary to make a firm conclusion toward the connection of the waves to the solar dynamo.

Flexural-gravity wave dynamics in two-layer fluid: blocking and dead water analogue

Abstract: Flexural-gravity wave characteristics are analysed, in the presence of a compressive force and a two-layer fluid, under the assumption of linearized water wave theory and small amplitude structural response. The occurrence of blocking for flexural-gravity waves is demonstrated in both the surface and internal modes. Within the threshold of the blocking and the buckling limit, the dispersion relation possesses four positive roots (for fixed wavenumber). It is shown that, under certain conditions, the phase and group velocities coalesce. Moreover, a wavenumber range for certain critical values of compression and depth is provided within which the internal wave energy moves faster than that of the surface wave. It is also demonstrated that, for shallow water, the wave frequencies in the surface and internal modes will never coalesce. It is established that the phase speed in the surface and internal modes attains a minimum and maximum, respectively, when the interface is located approximately in the middle of the water depth. An analogue of the dead water phenomenon, the occurrence of a high amplitude internal wave with a low amplitude at the surface, is established, irrespective of water depth, when the densities of the two fluids are close to each other. When the interface becomes close to the seabed, the dead water effect ceases to exist. The theory developed in the frequency domain is extended to the time domain and examples of negative energy waves and blocking are presented.

An experimental decomposition of nonlinear forces on a surface-piercing column: Stokes-type expansions of the force harmonics

Abstract: Wave loading on marine structures is the major external force to be considered in the design of such structures. The accurate prediction of the nonlinear high-order components of the wave loading has been an unresolved challenging problem. In this paper, the nonlinear harmonic components of hydrodynamic forces on a bottom-mounted vertical cylinder are investigated experimentally. A large number of experiments were conducted in the Danish Hydraulic Institute shallow water wave basin on the cylinder, both on a flat bed and a sloping bed, as part of a European collaborative research project. High-quality data sets for focused wave groups have been collected for a wide range of wave conditions. The high-order harmonic force components are separated by applying the ‘phase-inversion’ method to the measured force time histories for a crest focused wave group and the same wave group inverted. This separation method is found to work well even for locally violent nearly-breaking waves formed from bidirectional wave pairs. It is also found that the th-harmonic force scales with the th power of the envelope of both the linear undisturbed free-surface elevation and the linear force component in both time variation and amplitude. This allows estimation of the higher-order harmonic shapes and time histories from knowledge of the linear component alone. The experiments also show that the harmonic structure of the wave loading on the cylinder is virtually unaltered by the introduction of a sloping bed, depending only on the local wave properties at the cylinder. Furthermore, our new experimental results reveal that for certain wave cases the linear loading is actually less than 40 % of the total wave loading and the high-order harmonics contribute more than 60 % of the loading. The significance of this striking new result is that it reveals the importance of high-order nonlinear wave loading on offshore structures and means that such loading should be considered in their design.

Increased nutrient supply to the Southern Ocean during the Holocene and its implications for the pre-industrial atmospheric CO2 rise

Abstract: A rise in the atmospheric CO_2 concentration of ~20 parts per million over the course of the Holocene has long been recognized as exceptional among interglacials and is in need of explanation. Previous hypotheses involved natural or anthropogenic changes in terrestrial biomass, carbonate compensation in response to deglacial outgassing of oceanic CO_2, and enhanced shallow water carbonate deposition. Here, we compile new and previously published fossil-bound nitrogen isotope records from the Southern Ocean that indicate a rise in surface nitrate concentration through the Holocene. When coupled with increasing or constant export production, these data suggest an acceleration of nitrate supply to the Southern Ocean surface from underlying deep water. This change would have weakened the ocean’s biological pump that stores CO_2 in the ocean interior, possibly explaining the Holocene atmospheric CO_2 rise. Over the Holocene, the circum-North Atlantic region cooled, and the formation of North Atlantic Deep Water appears to have slowed. Thus, the ‘seesaw’ in deep ocean ventilation between the North Atlantic and the Southern Ocean that has been invoked for millennial-scale events, deglaciations and the last interglacial period may have also operated, albeit in a more gradual form, over the Holocene.

Shallow water equations for equatorial tsunami waves

Abstract: We present derivations of shallow water model equations of Korteweg–de Vries and Boussinesq type for equatorial tsunami waves in the f -plane approximation and discuss their applicability. This article is part of the theme issue ‘Nonlinear water waves’.

On the limiting Stokes wave of extreme height in arbitrary water depth

Abstract: Both Schwartz (J. Fluid Mech., vol. 62 (3), 1974, pp. 553–578) and Cokelet (Phil. Trans. R. Soc. Lond., vol. 286 (1335), 1977, pp. 183–230) failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Pade approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.

An improved moist-convective rotating shallow-water model and its application to instabilities of hurricane-like vortices

Abstract: We show how the two-layer moist-convective rotating shallow water model (mcRSW), which proved to be a simple and robust tool for studying effects of moist convection on large-scale atmospheric motions, can be improved by including, in addition to the water vapour, precipitable water, and the effects of vaporisation, entrainment, and precipitation. Thus improved mcRSW becomes cloud-resolving. It is applied, as an illustration, to model the development of instabilities of tropical cyclone-like vortices.

Well-Balanced Second-Order Finite Element Approximation of the Shallow Water Equations with Friction

Abstract: This paper investigates the approximation of the shallow water equations with topography and friction, using continuous finite elements. A new, second-order, parameter-free, well-balanced and posit...

On the Wave Bottom Shear Stress in Shallow Depths: The Role of Wave Period and Bed Roughness

Abstract: Lagoons and coastal semi-enclosed basins morphologically evolve depending on local waves, currents, and tidal conditions. In very shallow water depths, typical of tidal flats and mudflats, the bed shear stress due to the wind waves is a key factor governing sediment resuspension. A current line of research focuses on the distribution of wave shear stress with depth, this being a very important aspect related to the dynamic equilibrium of transitional areas. In this work a relevant contribution to this study is provided, by means of the comparison between experimental growth curves which predict the finite depth wave characteristics and the numerical results obtained by means a spectral model. In particular, the dominant role of the bottom friction dissipation is underlined, especially in the presence of irregular and heterogeneous sea beds. The effects of this energy loss on the wave field is investigated, highlighting that both the variability of the wave period and the relative bottom roughness can change the bed shear stress trend substantially.

Dynamics of Trapped Solitary Waves for the Forced KdV Equation

Abstract: The forced Korteweg-de Vries equation is considered to investigate the impact of bottom configurations on the free surface waves in a two-dimensional channel flow. In the study of shallow water waves, the bottom topography plays a critical role, which can determine the characteristics of wave motions significantly. The interplay between solitary waves and the bottom topography can exhibit more interesting dynamics of the free surface waves when the bottom configuration is more complex. In the presence of two bumps, there are multiple trapped solitary wave solutions, which remain stable between two bumps up to a finite time when they evolve in time. In this work, various stationary trapped wave solutions of the forced KdV equation are explored as the bump sizes and the distance between two bumps are varied. Moreover, the semi-implicit finite difference method is employed to study their time evolutions in the presence of two-bump configurations. Our numerical results show that the interplay between trapped solitary waves and two bumps is the key determinant which influences the time evolution of those wave solutions. The trapped solitary waves tend to remain between two bumps for a longer time period as the distance between two bumps increases. Interestingly, there exists a nontrivial relationship between the bump size and the time until trapped solitary waves remain stable between two bumps.

Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation

Abstract: Abstract In this article, the generalized shallow water wave (GSWW) equation is studied from the perspective of one dimensional optimal systems and their conservation laws (Cls). Some reduction and a new exact solution are obtained from known solutions to one dimensional optimal systems. Some of the solutions obtained involve expressions with Bessel function and Airy function [1,2,3]. The GSWW is a nonlinear self-adjoint (NSA) with the suitable differential substitution, Cls are constructed using the new conservation theorem.

Wave transformation across a macrotidal shore platform under low to moderate energy conditions

Abstract: We investigate how waves are transformed across a shore platform as this is a central question in rock coast geomorphology. We present results from deployment of three pressure transducers over four days, across a sloping, wide (~200 m) cliff-backed shore platform in a macrotidal setting, in South Wales, United Kingdom. Cross shore variations in wave heights were evident under the predominantly low to moderate (significant wave height < 1.4 m) energy conditions measured. At the outer transducer 50 m from the seaward edge of the platform (163 m from the cliff) high tide water depths were 8+ m meaning that waves crossed the shore platform without breaking. At the mid platform position water depth was 5 m. Water depth at the inner transducer (6 m from the cliff platform junction) at high tide was 1.4 m. This shallow water depth forced wave breaking, thereby limiting wave heights on the inner platform. Maximum wave height at the middle and inner transducers were 2.41 and 2.39 m respectively and significant wave height 1.35 m and 1.34 m respectively. Inner platform high tide wave heights were generally larger where energy was up to 335% greater than near the seaward edge where waves were smaller. Infragravity energy was less than 13% of the total energy spectra with energy in the swell, wind and capillary frequencies accounting for 87% of the total energy. Wave transformation is thus spatially variable and is strongly modulated by platform elevation and the tidal range. While shore platforms in microtidal environments have been shown to be highly dissipative, in this macro-tidal setting up to 90% of the offshore wave energy reached the landward cliff at high tide, so that the shore platform cliff is much more reflective.

Effect of seawater temperature, pH, and nutrients on the distribution and character of low abundance shallow water benthic foraminifera in the Galápagos

Abstract: In order to help predict the effects of anthropogenic stressors on shallow water carbonate environments, it is important to focus research on regions containing natural oceanographic gradients, particularly with respect to interactions between oceanography and ecologically sensitive carbonate producers. The Galapagos Archipelago, an island chain in the eastern equatorial Pacific, spans a natural nutrient, pH, and temperature gradient due to the interaction of several major ocean currents. Further, the region is heavily impacted by the El Nino-Southern Oscillation (ENSO) and the Galapagos exhibited widespread coral bleaching and degradation following the strong ENSO events of 1982-1983 and 1997-1998. These findings are coupled with reports of unusually low abundances of time-averaged benthic foraminiferal assemblages throughout the region. Foraminifera, shelled single-celled protists, are sensitive to environmental change and rapidly respond to alterations to their surrounding environment, making them ideal indicator species for the study of reef water quality and health. Here, statistical models and analyses were used to compare modern shallow water benthic foraminiferal assemblages from 19 samples spanning the Galapagos Archipelago to predominant oceanographic parameters at each collection site. Fisher α diversity indices, Ternary diagrams, Canonical Correspondence Analysis, regression tree analysis and FORAM-Index (FI; a single metric index for evaluating water quality associated with reef development) implied a combined impact from ENSO and upwelling from Equatorial Undercurrent (EUC) waters to primarily impact foraminiferal abundances and drive assemblage patterns throughout the archipelago. For instance, repeated ENSO temperature anomalies might be responsible for low foraminiferal density, while chronically high nutrients and low aragonite saturation and low pH-induced by EUC upwelling and La Nina anomalies-likely inhibited post-ENSO recovery, and caused foraminiferal assemblages to exhibit a heterotrophic dominance in the southern archipelago. What resulted are low FI values in the southern collection sites, indicating environments not conducive to endosymbiont development and/or recovery.

Wave-induced mean flows in rotating shallow water with uniform potential vorticity

Abstract: Theoretical and numerical computations of the wave-induced mean flow in rotating shallow water with uniform potential vorticity are presented, with an eye towards applications in small-scale oceanography where potential-vorticity anomalies are often weak compared to the waves. The asymptotic computations are based on small-amplitude expansions and time averaging over the fast wave scale to define the mean flow. Importantly, we do not assume that the mean flow is balanced, i.e. we compute the full mean-flow response at leading order. Particular attention is paid to the concept of modified diagnostic relations, which link the leading-order Lagrangian-mean velocity field to certain wave properties known from the linear solution. Both steady and unsteady wave fields are considered, with specific examples that include propagating wavepackets and monochromatic standing waves. Very good agreement between the theoretical predictions and direct numerical simulations of the nonlinear system is demonstrated. In particular, we extend previous studies by considering the impact of unsteady wave fields on the mean flow, and by considering the total kinetic energy of the mean flow as a function of the rotation rate. Notably, monochromatic standing waves provide an explicit counterexample to the often observed tendency of the mean flow to decrease monotonically with the background rotation rate.

Wave Breaking in a Shallow Water Model

Abstract: We show wave breaking---bounded solutions with unbounded derivatives---in the nonlinear nonlocal equations which combine the dispersion relation of water waves and the nonlinear shallow water equations and which extend the Whitham equation to permit bidirectional propagation, provided that the slope of the initial data is sufficiently negative.

A study on the role of hydrogeology on the distribution of uranium in alluvial aquifers of northwest India

Abstract: A study was undertaken to decipher the uranium distribution in relation to a number of hydrogeological factors in groundwater of southwest Punjab. Existing geological information for the region suggests that the shallow alluvial aquifer extends up to 50–70 m below ground level (bgl) and is in turn underlain by a deeper aquifer which extends to a depth of 250 m bgl. The presence of clayey units limits the vertical mixing of groundwater between the shallow and deep aquifers. Water level data (averaged over 5 years period) indicates that the south and southwestern regions of the study area have shallow water levels (3–5 m bgl) while the north and northeast regions have deep water levels (20–28 m bgl). This difference in water levels is found to be increasing with time. Higher concentrations of uranium occur in the central, southern, and southwestern parts of the study area where the water table occurs at shallow depth. Groundwater in the northern and northeastern parts of the study area shows U concentration within permissible levels for potable use (< 30 μg/L) while the highest concentration of U (341 μg/L) was found in the central part of the study area. Seasonal variation in dissolved U concentration is found to be statistically significant. The observed increases in U concentrations during the post-monsoon season are due to the addition of bicarbonate from the root zone as well as increased dissolved oxygen, nitrate, and sulphate concentration (oxic condition) in the groundwater while the decrease in U concentration is attributed to quick recharge by precipitation through sand dunes and contribution of surface water. Deeper groundwater does not show much seasonal variation in dissolved U concentration. Correlation between U and other hydrochemical parameters was evaluated. Cluster analysis of the data also indicates the oxidative mobilization of U from the sediments. Based on the lithological, hydrogeological, and dissolved U data, a schematic map is prepared depicting the various factors affecting the U distribution in alluvial aquifers, which can also be applied to other regions of similar hydrogeological setup.

Topological soliton solutions for three shallow water waves models

Abstract: In this article, we investigate three distinct physical structures for shallow water waves models by the improved ansatz method. The method was improved and can be used to obtain more gener...