Uneven Bottom Topography Research Articles

Time-dependent flexural gravity wave scattering due to uneven bottom in the paradigm of blocking dynamics

The present study investigates the scattering of flexural gravity waves due to uneven bottom topography in the context of wave blocking. Emphasis is given to analyzing the effects of multiple propagating wave modes on the solution procedures. The mathematical model is developed for two scenarios: a bottom step and a submerged rectangular breakwater. For the bottom step case, the complete solution in terms of the velocity potential is obtained using the eigenfunction expansion method. Subsequently, the solution associated with the wave transformation by the bottom step is extended to the case of a submerged rectangular breakwater using symmetry characteristics of the velocity potential. The energy balance relation is derived in both cases using the conservation of energy flux in the presence of multiple propagating wave modes. Wave blocking occurs for four different frequencies in both the cases of the bottom step and the submerged breakwater due to variations in water depth. This makes the problem more complex as, depending on the frequency, multiple propagating wave modes can exist in either the reflected region, the transmitted region, or both. The transmitted wave amplitude associated with the lower wavenumber within the blocking frequencies exceeds unity, and this excess energy is balanced by the corresponding energy transfer rate. Additionally, removable discontinuities are observed at the blocking frequencies in the scattering coefficients, where group velocity ceases. In the context of floating ice sheets, the deflection is analyzed in the time domain for frequencies within and outside the blocking limits.

Dissipation of incident wave energy by two floating horizontal porous plates over a trench type bottom

ABSTRACT This work investigates the scattering of oblique incident waves by two floating horizontal porous plates over a trench-type sea bed to study the role of different pairs of barriers in dissipating the incident wave energy. This problem is modelled based on Darcy's law for flow past a porous structure. Havelock's expansions of water wave potentials, suitable matching conditions, and the algebraic least-square method handle the boundary value problem. The role of the horizontal porous plates is studied by analysing the scattering coefficients, energy loss, hydrodynamic wave force and free surface elevation through graphs which show a periodic oscillatory pattern as a function of gap length. The study reveals that if the porous effect parameter of plates or the length of plates is increased, more energy is dissipated and the plates experience less wave force, and the free surface elevation on the lee side is decreased. The present results also signify that as compared to rigid plates, significant changes are found in wave scattering coefficients, hydrodynamic force and free surface elevation due to the consideration of one porous plate along with one rigid plate and also two porous plates, where a pair of porous plates are more effective as compared to one porous plate. The present study reveals that the scattering of water waves by two floating horizontal porous plates over uneven bottom topography plays a vital role in constructing floating structures having more staying power in harsh wave environments.

Wash waves generated by ship moving across a depth change

Unsteady wash waves generated by a ship with constant speed moving across an uneven bottom topography is investigated by numerical simulations based on a Mixed Euler–Lagrange (MEL) method. The transition is accomplished by the ship traveling from the deep water into shallow water via a step bottom. A small tsunami would be created after this transition. However, the unsteady wave-making resistance induced by this new phenomenon has not been well documented by literature. Therefore, the main purpose of the present study is to quantify the effects of an uneven bottom on the unsteady wash waves and wave-making resistance acting on the ship. An upwind differential scheme is commonly used in the Euler method to deal with the convection terms on free-surface condition to prevent waves in upstream. Evidently, it cannot be applied to the present problem when the ship-generated waves could propagate upstream. A MEL method is therefore employed to investigate the upstream wave generated by the ship moving over the uneven bottom. The central differential scheme provides more accurate results, but it is not unconditionally stable. Simulation results show that the hydrodynamic interaction between the ship and the uneven bottom could initiate an upstream tsunami, as well as unsteady wave-making resistance on ships. The unsteady wave-making resistance oscillates periodically, and the amplitude and period of the oscillations are highly dependent on speed and water depth.

Stochastic solitons in a two-layer fluid system

In this paper, stochastic one- and two-soliton solutions for the stochastic Benjamin–Ono equation modeling the nonlinear random waves in a two-layer fluid system with and without the uneven bottom topography are derived via the Gaussian white noise functional approach and the Hirota method. The effects of Gaussian white noise on the propagation and the interaction of the stochastic solitons are discussed. In the stochastic-one-soliton case, impact on the velocity is enlarged with the increase of the intensity of Gaussian white noise while the soliton amplitude and shape keep unchanged during the propagation. We also investigate the results of the stochastic-two-soliton interaction and the effects of Gaussian white noise on the interaction of stochastic two solitons. With the increase of Gaussian white noise, the velocity of two solitons causes the corresponding change, while the soliton amplitudes and shapes keep unchanged before and after the interaction. • Successfuly obtaining the stochastic one-soliton, two-soliton and N-soliton solutions of the stochastic Benjamin–Ono equation. • Investigating the effect of Gaussian white noise on propagation and interactions of the stochastic solitons. • The impact on the velocity can be greater as the intensity of Gaussian white noise increases in the stochastic one-soliton case. • The amplitude keep unchange along the propagation in the stochastic one-soliton case. • There is a change over the velocities of the stochastic two solitons when there exists white noise.

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman’s approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether’s theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.

GROUP CLASSIFICATION OF THE TWO-DIMENSIONAL GREEN–NAGHDI EQUATIONS WITH A TIME-DEPENDENT BOTTOM TOPOGRAPHY

The two-dimensional Green–Naghdi equations are studied for the case of uneven bottom topography. The bottom topography function can depend on time. Group classification of these equations with respect to the function describing the bottom topography is performed using an algebraic approach.

The one-dimensional Green–Naghdi equations with a time dependent bottom topography and their conservation laws

This paper deals with the one-dimensional Green–Naghdi equations describing the behavior of fluid flow over an uneven bottom topography depending on time. Using Matsuno’s approach, the corresponding equations are derived in Eulerian coordinates. Further study is performed in Lagrangian coordinates. This study allowed us to find the general form of the Lagrangian corresponding to the analyzed equations. Then, Noether’s theorem is used to derive conservation laws. As some of the tools in the application of Noether’s theorem are admitted generators, a complete group classification of the Green–Naghdi equations with respect to the bottom depending on time is performed. Using Noether’s theorem, the found Lagrangians, and the group classification, conservation laws of the one-dimensional Green–Naghdi equations with uneven bottom topography depending on time are obtained.

Polygon-70: Data Interpretation in the Light of New Concepts about Internal Waves

We analyze our concepts about ocean dynamics and, in particular, internal waves. Our concepts have changed over the 50 years after the Polygon-70 experiment. We now understand that mesoscale eddies exist in the ocean and currents in these eddies frequently exceed the mean currents. We understand that internal tides are generated due to the interaction of tidal currents with uneven bottom topography and spread as perturbations of velocity along characteristic surfaces. The waves fill the entire ocean but are not confined to the fluctuations of the density pycnocline. An interpretation of the data on internal waves published in the 1970s has been revised in the light of our new concepts and knowledge.

Symmetries, Conservation Laws, Invariant Solutions and Difference Schemes of the One-dimensional Green-Naghdi Equations

The paper is devoted to the Lie group properties of the one-dimensional Green-Naghdi equations describing the behavior of fluid flow over uneven bottom topography. The bottom topography is incorporated into the Green-Naghdi equations in two ways: in the classical Green-Naghdi form and in the approximated form of the same order. The study is performed in Lagrangian coordinates which allows one to find Lagrangians for the analyzed equations. Complete group classification of both cases of the Green-Naghdi equations with respect to the bottom topography is presented. Applying Noether's theorem, the obtained Lagrangians and the group classification, conservation laws of the one-dimensional Green-Naghdi equations with uneven bottom topography are obtained. Difference schemes which preserve the symmetries of the original equations and the conservation laws are constructed. Analysis of the developed schemes is given. The schemes are tested numerically on the example of an exact traveling-wave solution.

Using acoustic telemetry to expand sonar escapement indices of Chinook salmon to in-river abundance estimates

Acoustic telemetry was combined with a project that uses sonar and drift gillnetting methods to estimate Chinook salmon Oncorhynchus tshawytscha escapement in the Nushagak River, Alaska. The sonar project uses dual-frequency identification sonars (DIDSONs) to count passing fish and drift gillnetting to apportion sonar estimates to species. These estimates are indices because the river’s width (∼300 m) and uneven bottom topography allow for only a third of the river to be sampled. This range is enough to fully enumerate sockeye salmon O. nerka, the dominate species, but not Chinook salmon, which are known to migrate beyond the sampling range. Acoustic telemetry was used to determine what proportion of Chinook salmon traveled within the sampling range of the sonar project. We inserted acoustic tags into Chinook salmon ∼13 km downriver and deployed an array of acoustic receivers at the sonar site to track tagged fish. From 2011 to 2014, 799 Chinook salmon were tagged. The tagged fish used the entire river width while migrating through the acoustic array exhibiting a wide variety of behaviors that included moving straight through the array, making multiple up and down trips, holding, and crossing over from one side of the river to the other. On average, 57% of tagged fish traveled through regions sampled by the sonar with annual percentages of 65% (2011), 54% (2012), 64% (2013), and of 47% (2014). These proportions were used to expand the sonar-derived indices to in-river abundance estimates.

Diffraction of water waves by an array of vertical barriers and heterogeneous bottom

The interaction of head waves with an infinite row of identical, equally spaced, rectangular breakwaters is investigated in the presence of uneven bottom topography. Using linear water wave theory and matched eigenfunction expansion method, the boundary value problem is transformed into a system of linear algebraic equations which are numerically solved to know the velocity potentials completely. Utilizing this method, reflected and transmitted wave energy are computed for different physical parameters along with the wave field in the vicinity of breakwaters. It is observed that the wave field becomes more complicated when the incoming wavelength becomes smaller than the channel width. A critical ratio of the gap width to the channel width, corresponding to the inflection point of the transmitted energy variation, is identified for which 1/3 of the total energy is transmitted. Similarly, depending on the incident wavelength, there is a critical breakwater width for which a minimum energy is transmitted. Further, the accuracy of the computed results is verified by using the derived energy relation.

Water wave scattering by an array of rectangular breakwaters on a step bottom topography

Water wave scattering by an array of identical rectangular breakwaters is investigated in the presence of a heterogenous bottom boundary in the cases of both infinite and semi-infinite fluid domains. Boundary value problem is formulated by employing small amplitude water wave theory and is reduced to a system of linear algebraic equations using matching eigenfunction expansion method. These equations are solved numerically to perceive the complete solution of the problems under present interest. The reflection and transmission coefficients are computed in different modes, and the energy relation is used to check the accuracy of computed results. Effects of different parameters on the total reflected and transmitted wave energy are computed and analyzed. Further, the surface motion is computed to see the wave phenomena in the vicinity of the breakwaters. The unknown coefficients appearing in velocity potentials are estimated, the frequency of resonance mode is resolved, and resonance surface modes are plotted in the case of the semi infinite fluid domain. This study is useful for the breakwater designer to install breakwaters near an uneven bottom topography.

Hamiltonian structure for two-dimensional extended Green–Naghdi equations

The two-dimensional Green–Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original Green–Naghdi (GN) model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

Coupled acoustic mode equations in the ocean waveguide with rough sea surface and elastic bottom

Sound wave scattering caused by rough surfaces has been the subject of investigation for many decades. The problem has no exact analytic solution and approximate approaches are needed. In the ocean, scattering of sound by perturbed ocean boundaries couples acoustic modes and increases attenuation. In this paper, approximate coupled mode equations are analyzed using a small perturbation approach. The analytic solution connecting mode shapes with coupling mode coefficients has been derived. It is shown that the coupled mode matrix can be approximated by a linear function of one parameter, which is the perturbation of the sea surface. The equations are applied for one-direction sound propagation with a rough sea surface. For this one parameter model, the coupled mode differential equation has an easy solution in the form of a matrix exponent and can be useful for many applications. The proposed method can be easily extended to the case where modes are coupling by uneven bottom topography including elastic properties for compressional and shear waves.

Combining Split‐Beam and Dual‐Frequency Identification Sonars to Estimate Abundance of Anadromous Fishes in the Roanoke River, North Carolina

AbstractRiverine hydroacoustic techniques are an effective method for evaluating abundance of upstream migrating anadromous fishes. To use these methods in the Roanoke River, North Carolina, at a wide site with uneven bottom topography, we used a combination of split‐beam sonar and dual‐frequency identification sonar (DIDSON) deployments. We aimed a split‐beam sonar horizontally to monitor midchannel and near‐bottom zones continuously over the 3‐month spring monitoring periods in 2010 and 2011. The DIDSON was rotated between seven cross‐channel locations (using a vertical aim) and nearshore regions (using horizontal aims). Vertical deployment addressed blind spots in split‐beam coverage along the bottom and provided reliable information about the cross‐channel and vertical distributions of upstream migrants. Using a Bayesian framework, we modeled sonar counts within four cross‐channel strata and apportioned counts by species using species proportions from boat electrofishing and gill netting. Modeled estimates (95% credible intervals [CIs]) of total upstream migrants in 2010 and 2011 were 2.5 million (95% CI, 2.4–2.6 million) and 3.6 million (95% CI, 3.4–3.9 million), respectively. Results indicated that upstream migrants are extremely shore‐ and bottom‐oriented, suggesting nearshore DIDSON monitoring improved the accuracy and precision of our estimates. This monitoring protocol and model may be widely applicable to river systems regardless of their cross‐sectional width or profile.Received July 16, 2014; accepted November 21, 2014

Wave propagation in underwater acoustic waveguides with rough boundaries

In underwater acoustic waveguides a pressure field can be decomposed over three kinds of modes: the propagating modes, the radiating modes, and the evanescent modes. In this paper, we analyze the effects produced by a randomly perturbed free surface and an uneven bottom topography on the coupling mechanism between these three kinds of modes. Using an asymptotic analysis based on a separation of scales technique we derive the asymptotic form of the distribution of the forward mode amplitudes. We show that the surface and bottom fluctuations affect the propagating-mode amplitudes mainly in the same way. We observe an effective amplitude attenuation which is mainly due to the coupling between the propagating modes themselves. However, for the highest propagating modes this mechanism is stronger and due to an efficient coupling with the radiating modes.

Numerical study on the effects of uneven bottom topography on freak waves

A numerical model is built by using an improved VOF method coupled with an incompressible Navier–Stokes solver. Exploiting the model, the freak wave formation due to the dispersive focusing mechanism is investigated numerically without uneven bottoms and in presence of uneven bottoms. During the freak wave transformation over an uneven bottom in finite water, combined effects of shoaling, refraction and reflection can modify the external characteristics of freak waves, and also can complicate the energy transfers. Furthermore, wavelet analysis method is adopted to analyze the behavior of the instantaneous energy structure of freak waves. It is found that when the bottoms vary in height, the external characteristic parameters and high frequency energy show a similar trend, but the value may be quite different due to the difference in local characteristic of the bottom.

NWF: Propagation of Tsunami and its Interaction with Continental Shelf and Vertical Wall

In this article, tsunamis represented as solitary waves was simulated using the fully nonlinear free surface waves based on Finite Element method developed by Sriram et al. (2006). The split up of solitary wave while it propagates over the uneven bottom topography is successfully established. Wave transmission and reflection over a vertical step introduced in the bottom topography is in good agreement with the experimental results from Seabra-Santos et al. (1987). The wave transformation over a continental shelf with different smooth slopes reveals that the solitary wave reflection increases while the continental slope varies from flat to steep. The interaction of the solitary wave with a vertical wall for different wave steepness has been analysed. The reflected shape of the profile is in good agreement with the observation made by Fenton and Rienecker (1982) and an increase in wave celerity is observed.

Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves.

The stochastic Benjamin-Ono equation is introduced, which models the propagation of nonlinear random waves in a two-layer fluid system with and without uneven bottom topography. In the case of the flat bottom, the effect of the external random flow field on the evolution of both soliton and periodic wave is investigated. In particular, the mean value and the correlation function of these nonlinear wave fields are calculated exactly under the assumption that the flow obeys the Gaussian stochastic process with a white noise. It is found that in the limit of large time, the mean value of an algebraic soliton approaches a Gaussian wave packet whereas that of a periodic wave is represented by Jacobi's theta function. In the case of the uneven bottom, a perturbation analysis is performed to evaluate the mean value of an algebraic soliton under the influence of random change of bottom topography. The large time asymptotic of the soliton is shown to exhibit a Gaussian wave packet with a small amount of the phase shift caused by the interaction between the soliton and the random bottom topography.