Scattering of Surface Waves by the Edge of a Small Undulation on a Porous Bed in an Ocean with Ice-cover

Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investigated within the framework of linearized water wave theory. The effect of surface tension at the surface below the ice-cover is neglected. There exists only one wave number propagating at just below the ice-cover. A perturbation analysis is employed to solve the boundary value problem governed by Laplace＇s equation by a method based on Green＇s integral theorem with the introduction of appropriate Green＇s function and thereby evaluating the reflection and transmission coefficients approximately up to first order. A patch of sinusoidal ripples is considered as an example and the related coefficients are determined.

Water Wave Scattering by a Nearly Circular Cylinder Submerged Beneath an Ice-cover

Assuming linear theory, the two-dimensional problem of water wave scattering by a horizontal nearly circular cylinder submerged in infinitely deep water with an ice cover modeled as a thin-elastic plate floating on water, is investigated here. The cross-section of the nearly circular cylinder is taken as r=a（1＋δC（θ））, where a is the radius of the corresponding circular cross-section of the cylinder, δ is a measure of small departure of the cross-section of the cylinder from its circularity and C（θ） is the shape function. Using a simplified perturbation technique the problem is reduced to two independent boundary value problems up to first order in δ. The first one corresponds to water wave scattering by a circular cylinder submerged in water with an ice-cover, while the second problem describes wave radiation by a submerged circular cylinder and involves first order correction to the reflection and transmission coefficients. The corrections are obtained in terms of integrals involving the shape function. Assuming a general Fourier expansion of the shape function, these corrections are evaluated approximately. It is well known that normally incident wave trains experience no reflection by a circular cylinder submerged in infinitely deep water with an ice cover. It is shown here that the reflection coefficient also vanishes up to first order for some particular choice of the shape function representing a nearly circular cylinder. For these cases, full transmission occurs, only change is in its phase which is depicted graphically against the wave number in a number of figures and appropriate conclusions are drawn.

Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green＇s integral theorem in terms of difference of potential functicns across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.

Construction of Wave-free Potential in the Linearized Theory of Water Waves

Various water wave problems involving an infinitely long horizontal cylinder floating on the surface water were investigated in the literature of linearized theory of water waves employing a general multipole expansion for the wave potential. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace's equation (for non-oblique waves in two dimensions) or two-dimensional Helmholz equation (for oblique waves) satisfying the free surface condition and decaying rapidly away from the point of singularity. The method of constructing these wave-free potentials is presented here in a systematic manner for a number of situations such as deep water with a free surface, neglecting or taking into account the effect of surface tension, or with an ice-cover modelled as a thin elastic plate floating on water.

Oblique Wave-free Potentials for Water Waves in Constant Finite Depth

In this paper, a method to construct oblique wave-free potentials in the linearised theory of water waves for water with uniform finite depth is presented in a systematic manner. The water has either a free surface or an ice-cover modelled as a thin elastic plate. For the case of free surface, the effect of surface tension may be neglected or taken into account. Here, the wave-free potentials are singular solutions of the modified Helmholtz equation, having singularity at a point in the fluid region and they satisfy the conditions at the upper surface and the bottom of water region and decay rapidly away from the point of singularity. These are useful in obtaining solutions to oblique water wave problems involving bodies with circular cross-sections such as long horizontal cylinders submerged or half-immersed in water of uniform fmite depth with a free surface or an ice-cover modelled as a floating elastic plate. Finally, the forms of the upper surface related to the wave-free potentials constructed here are depicted graphically in a number of figures to visualize the wave motion. The results for non-oblique wave-free potentials and the upper surface wave-free potentials are obtained. The wave-free potentials constructed here will be useful in the mathematical study of water wave problems involving infinitely long horizontal cylinders, either half-immersed or completely immersed in water.

Gravity Wave Generated by Initial Axisymmetric Disturbance at the Surface of an Ice‑covered Ocean with Porous Bed

This paper is concerned with the generation of gravity waves due to prescribed initial axisymmetric disturbances created at the surface of an ice sheet covering the ocean with a porous bottom.The ice cover is modeled as a thin elastic plate,and the bottom porosity is described by a real parameter.Using linear theory,the problem is formulated as an initial value problem for the velocity potential describing the motion.In the mathematical analysis,the Laplace and Hankel transform techniques have been used to obtain the depression of the ice-covered surface in the form of a multiple infnite integral.This integral is evaluated asymptotically by the method of stationary phase twice for a long time and a large distance from the origin.Simple numerical computations are performed to illustrate the efect of the ice-covered surface and bottom porosity on the surface elevation,phase velocity,and group velocity of the surface gravity waves.Mainly the far-feld behavior of the progressive waves is observed in two diferent cases,namely initial depression and an impulse concentrated at the origin.From graphical representations,it is clearly visible that the presence of ice cover and a porous bottom decreases the wave amplitude.Due to the porous bottom,the amplitude of phase velocity decreases,whereas the amplitude of group velocity increases.

Distribution of Siliceous-Walled Algae in Taylor Valley, Antarctica Lakes

The McMurdoDryValleysofAntarcticaare a unique environment characterized by extreme lows in temperature and precipitation, which supports a low diversity microbial and multicellular fauna and flora. Terrestrial biomass is largely limited to soil microbes and mosses, while perennially ice-covered lakes host aerobic and anaerobic microbial communities, algae, and a low diversity eukaryotic fauna. This study provides a scanning electron microscope survey of the distribution of siliceous-walled algae in the water columns and surface sediments of fourTaylorValleylakes. No patterns of distribution of algae, chrysophyte cysts and diatoms, are detected, suggesting that cores taken from perennially ice-covered lakes contain basin-wide records, rather than records specific to the lake depth or other lake-specific criteria. Since Taylor Valley lakes became perennially ice-covered, shifts in diatom assemblages in cores are more likely to record changes to sediment and microfossil transport, e.g. the dominance of eolian vs. stream input, rather than other ecological conditions. Basin-wide records are episodically overprinted by lake-specific events, as demonstrated by a marked increase of the stream diatom genus Hantzschia during a period of increased stream flow into East Lake Bonney.

Wave radiation by a horizontal circular cylinder submerged in deep water with ice-cover

Using the multipoles method,we formulate the problems of water wave radiation(both heave and surge)by a submerged circular cylinder in water with an ice-cover,the ice-cover being modelled as an elastic plate of very small thickness.This leads to an infinite system of linear equations which are solved numerically by standard techniques.The added-mass and damping coefficients for a heaving and surge problems are obtained and depicted graphically against the wave number for various values of flexural rigidity of the ice-cover to show the effect of the presence of ice-cover on these quantities.When the flexural rigidity and surface density of ice-cover are taken to be zero,the ice-cover tends to free-surface then the curves for added-mass and damping coefficients exactly coincide with the curves for the case of water with free surface.Also it is observed that the added-mass and damping coefficients for the two modes of motion are here also the same.

ICE JAMS IN A SMALL RIVER AND THE HEC-RAS MODELING

This paper describes a model of a 3.06km long river reach between two small reservoirs under both open flow and ice covering conditions for different operational settings of the stoplogs in the downstream reservoir. The HEC-RAS model developed by the Hydrological Engineering Center of US Army Corps of Engineers was used to compare different approaches in terms of flow velocity, water level and the Froude number. The impacts of heavily vegetated main channel and floodplain on ice accumulations were investigated. And it is shown that this vegetation plays a significant role in the formation of river ice jam during winter period and thus the vegetated channel has strong influence on ice flooding. In addition, the paper explores the impact bouh of the operation of the stoplogs during the winter period and the presence of the downstream dam on the accumulation of ice jam along this river reach.

FLEXURAL-GRAVITY WAVES DUE TO TRANSIENT DISTURBANCES IN AN INVISCID FLUID OF FINITE DEPTH

The dynamic response of an ice-covered fluid to transient disturbances was analytically investigated by means of integral transforms and the generalized method of stationary phase. The initially quiescent fluid of finite depth was assumed to be inviscid, incompressible, and homogenous. The thin ice-cover was modeled as a homogeneous elastic plate. The disturbances were idealized as the fundamental singularities. A linearized initial-boundary-value problem was formulated within the framework of potential flow. The perturbed flow was decomposed into the regular and the singular components. An image system was introduced for the singular part to meet the boundary condition at the fiat bottom. The solutions in integral form for the vertical deflexion at the ice-water interface were obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motion were explicitly derived for large time with a fixed distance-to-time ratio. The effects of the finite depth of fluid on the resultant wave pattems were discussed in detail. As the depth increases from zero, the critical wave number and the minimal group velocity first increase to their peak values and then decrease to constants.