A continuum model for the linear wave propagation in ice-covered oceans: An approximate solution

In order to model the wave-in-ice climate on a geophysical scale, a continuum viscoelastic model has been developed [Wang, R., Shen, H.H., 2010. Gravity waves propagating into an ice-covered ocean: a viscoelastic model. J. Geophys. Res. 115, C06024. doi:10.1029/2009JC005591]. In this model, two modes were identified to be dominant, each for the low or high elasticity range, respectively. In the intermediate elasticity range, both modes have comparable attenuation rates and wave numbers, and they could be co-dominant. Inspired by the Eigenfunction Expansion Matching Method, this paper presents an approximate approach to solve the wave propagation characteristics in the whole physical range of elasticity. A monochromatic wave propagating from semi-infinite open water into semi-infinite ice-covered water is considered. Only two transmitted wave modes are included in the eigenfunction expansions for the ice-covered water. These two modes correspond to the dominant mode under relatively low or high elasticity, respectively. Evanescent wave modes in open water are ignored. By minimizing the matching errors, the least square solution is obtained. For the pure elastic ice cover, the results are compared with the exact solution from the thin elastic plate model. The comparison is in good agreement when the wave period is larger than 10 s. Mild discrepancies exist when the period is smaller than 10 s. The present model provides the simplest way to describe wave propagation characteristics in different types of ice covers.     

海冰对海浪影响研究综述

受全球气候变化的影响,极区海浪尤其是北极海浪在过去几十年发生了显著的变化,使得海冰边缘区海冰与海浪的相互作用愈发显著。本文从物理海洋学的角度出发,较系统地总结了海冰对海浪作用研究的国内外现状,从理论和实测的角度分别探讨了海冰对海浪能量的耗散及其引起的波动频散关系的变化,同时分析了当前海冰覆盖海域海浪的数值模拟与现场观测研究,指出了未来开展有冰海域海浪数值模拟与预报所面临的主要问题,并对该方向今后的研究做出展望。总体来看,尽管海冰对海浪作用的机理复杂且与海冰类型高度相关,但是海冰对海浪能量的衰减与传播距离基本呈指数关系,并且海冰会一定程度上影响海浪的传播速度。未来依然需要更多不同海冰类型下海浪的观测数据以开展进一步的机理分析、模型检验和参数校准,进而实现高精度的业务化预报。     

Unevenly compressed flexural-gravitity waves

Free flexural-gravity waves unevenly compressed in an ice-covered basin have been studied using a linear formulation. The conditions were specified allowing determination of the bounds of the angular area of wave disturbances and of the oscillation frequency intervals due to compression. The paper considers the distribution of the wave characteristics over frequency depending on compressive stresses and the direction of wave propagation.Translated by Vladimir A. Puchkin.     

Experimental Investigation on Breakup of Ice Floe on Waves

Experimental investigation of for floe breakup on waves by use of non-refrigerated breakablematerials has been carried out.The incident wave heights for breakup of ice floe with different lengths,andthe influence of rigidity on reflection and transmission coeffcients are discussed.The experimental resultsshow that the ratio of the ice floe length,L_i,to the wavelength,L,is one of the significant factor affectingthe minimum wave height to cause fracture of ice floe,and another key factor is the ratio of the ice floethickness,h_i and L_i.     

Calculation of the nonlinear energy transfer through the wave spectrum at the sea surface covered with broken ice

The nonlinear energy transfer through the wave spectrum is studied on the basis of the previously obtained explicit equation for matrix elements of a four-wave kinetic integral. The equation describes the evolution of a system of gravity waves at the surface of a sea of finite depth with a uniform distribution of broken ice over the sea surface. Particular attention is paid to the analytical part of the algorithm of the calculation of the kinetic integral. This part differs from the standard algorithm by a set of prominent features of the dispersion relation for wave oscillations in the ice-covered water. The kinetic integral for the system under consideration is calculated, and the results are compared with the results obtained for the ice-free water.     

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.     

Oblique Wave Scattering by An Undulating Porous Bottom in A Two-Layer Ice-Covered Fluid

The present study analyzes the reflection and transmission phenomenon of water-waves in a two-layer ice-covered system. The upper layer is covered by an ice-sheet, whereas the bottom of the lower layer is undulated and permeable. By using regular perturbation analysis and Fourier transform technique, the problem is solved and the first order reflection and transmission coefficients are determined. It is found that these coefficients depend on the shape as well as the permeability of the undulating bottom. Therefore, from the practical viewpoint, an undulating bottom topography is considered to determine all the aforesaid coefficients. The role of various system parameters, such as porosity, angle of incidence and ice parameters, are discussed to analyze the transformation of incident water wave energy from one layer to another layer. The outcomes are demonstrated in graphical forms.     

西北航道夏季气旋登陆期间海冰对波浪的影响

西北航道是指从北大西洋经加拿大北极群岛进入北冰洋,再进入太平洋的航道,是连接大西洋和太平洋的捷径。为了探讨西北航道通航期极端天气条件下强风及海冰对波浪场的影响机制,建立并验证了考虑海冰影响下的西北航道风浪演化模型,并以2012年8月北极气旋登陆期间为例探讨西北航道通航期波浪特性及波能流密度的时空演化及其对风和海冰的响应。研究结果表明,北极夏季海冰大多分布于西北航道以北海域,而风向大部分集中在SSW(南偏西22.5°)至SW(南偏西45°),西北航道海冰的存在并不会引起有效风区的明显减少,也不会引起无冰海域波能流的明显减小(不超过5%)。但是,当风向变为北向风时,无冰海域波能流减小幅度最多高达62%。最后,综合海冰和波浪要素的时空分布,提出了极端天气条件下西北航道通航期的最佳适航路线,为西北航道的夏季安全通航提供了科学依据。     

Development and validation of harmonic wave group single-run procedure for RAO with comparison to regular wave and transient wave group procedures using URANS

A harmonic wave group single run seakeeping procedure is developed, validated and compared with regular wave and transient wave group procedures, using an unsteady Reynolds averaged Navier–Stokes solver, although all procedures can be also implemented using experiments or potential flow. Incoming waves are generated by linear superposition of potential solution for a number of component waves. The regular wave procedure requires multiple runs, whereas single run procedures obtain the response amplitude operators (RAO) for a range of frequencies at a fixed speed, assuming linear ship response. The transient wave group procedure provides continuous RAO curves, while the harmonic wave group procedure obtains discrete transfer functions without focusing. Results are presented for heave and pitch response amplitudes and phases for the DTMB model 5512 in head waves. Verification and validation studies are performed for the transient wave group procedure. Validation is achieved at the average interval of 9.54 (%D). Comparisons of the procedures show that the harmonic wave group procedure is the most efficient, saving 75.8% on the computational cost compared with the regular wave procedure. Error values from all procedures are similar at 4 (%D). Harmonic wave group results are validated for a wide range of the Froude numbers, with satisfactory results.     

Interaction of uniform current with a circular cylinder submerged below an ice sheet

The problem of a uniform current passing through a circular cylinder submerged below an ice sheet is considered. The fluid flow is described by the linearized velocity potential theory, while the ice sheet is modelled through a thin elastic plate floating on the water surface. The Green function due to a source is first derived, which satisfies all the boundary conditions apart from that on the body surface. Through differentiating the Green function with respect to the source position, the multipoles are obtained. This allows the disturbed velocity potential to be constructed in the form of an infinite series with unknown coefficients which are obtained from the boundary condition. The result shows that there is a critical Froude number which depends on the physical properties of the ice sheet. Below this number there will be no flexural waves propagating to infinity and above this number there will be two waves, one on each side of the body. When the depth based Froude number is larger than 1, there will always be a wave at far upstream of the body. This is similar to those noticed in the related problem and is different from that in the free surface problem without ice sheet. Various results are provided, including the properties of the dispersion equation, resistance and lift, ice sheet deflection, and their physical features are discussed.     

Radiation and diffraction of linear water waves by an infinitely long submerged rectangular structure parallel to a vertical wall

The radiation and the diffraction of linear water waves by an infinitely long floating rectangular structure submerged in water of finite depth with leeward boundary being a vertical wall are analyzed in this paper by using the method of separation of variables. Analytical expressions for the radiated and diffracted potentials are derived as infinite series with unknown coefficients determined by the eigenfunction expansion matching method. The expressions for wave forces and hydrodynamic coefficients are given. A comparison is made between the results obtained by the present analytical solution and those obtained by the boundary element method. By using the present analytical solution, the hydrodynamic influences of the submergence, the width, the thickness of the structure, and the distance between the structure and the wall on the wave forces and hydrodynamic coefficients are discussed in detail.     

Water wave scattering by bottom undulations in an ice-covered two-layer fluid

The scattering of plane surface waves by bottom undulations in an ice-covered ocean modelled as a two-layer fluid consisting of a layer of fresh water of lesser density above a deep layer of salt water, is investigated here by using a simplified perturbation analysis. In such a two-layer fluid there exist waves of two different modes, one with higher mode propagates along the interface and the other with lower mode propagates along the ice-cover. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes so that transfer of wave energy from one mode to another takes place. The first-order reflection and transmission coefficients of two different modes are obtained due to incident waves of again two different modes by employing Fourier transform technique in the mathematical analysis. For sinusoidal bottom topography these coefficients are depicted graphically against the wavenumber. These figures show how the transfer of energy from one mode to another takes place.     

The interaction of flexural-gravity waves with a submerged rigid disc

In this paper, the interaction of flexural-gravity waves with a submerged disc is studied. The problem is solved by transforming it into a two-dimensional hypersingular integral equation. Initially, wave scattering problem is studied followed by radiation and radiation-diffraction problems. The effects of the rigidity of the ice cover and submergence depth of the rigid disc are investigated for all the three cases. It is observed that the submergence of a rigid disc causes significant changes in the scattered as well as radiated wave profiles. However, the presence of the rigid disc creates high-frequency resonance. The present study appears to be useful for understanding the nature of the flow physics in presence of a rigid disc beneath the ice-covered surface.     

Mooring forces and motion responses of pontoon-type floating breakwaters

The experimental and theoretical investigations on the behaviour of pontoon-type floating breakwaters are presented. A two-dimensional finite element model is adopted to study the behaviour of pontoon-type floating breakwaters in beam waves. The stiffness coefficients of the slack mooring lines are idealized as the linear stiffness coefficients, which can be derived from the basic catenary equations of the cable. The theoretical model is supported by an experimental programme conducted in a wave flume. The motion responses and mooring forces are measured for three different mooring configurations, and the results are reported and discussed in detail in this paper. The wave attenuation characteristics are presented for the configurations studied.     

Laboratory Study on the Interaction Between Regular Obliquely Incident Waves and Vertical Walls

The characteristics of wave forces are studied based on physical model tests with regular waves. The ratio of obliquely incident wave forces to normally incident wave forces on unit length of a vertical wall is related with various factors. A linear reduction of the mean force of obliquely incident waves is confirmed with an increase in the relative caisson length. Also the characteristics of reflection coefficient of diagonal waves are discussed.     

A linear Boltzmann equation to model wave scattering in the marginal ice zone

We present a linear Boltzmann equation to model wave scattering in the Marginal Ice Zone (the region of ocean which consists of broken ice floes). The equation is derived by two methods, the first based on Meylan et al. [Meylan, M.H., Squire, V.A., Fox, C., 1997. Towards realism in modeling ocean wave behavior in marginal ice zones. J. Geophys. Res. 102 (C10), 22981–22991] and second based on Masson and LeBlond [Masson, D., LeBlond, P., 1989. Spectral evolution of wind-generated surface gravity waves in a dispersed ice field. J. Fluid Mech. 202, 111–136]. This linear Boltzmann equation, we believe, is more suitable than the equation presented in Masson and LeBlond [Masson, D., LeBlond, P., 1989. Spectral evolution of wind-generated surface gravity waves in a dispersed ice field. J. Fluid Mech. 202, 111–136] because of its simpler form, because it is a differential rather than difference equation and because it does not depend on any assumptions about the ice floe geometry. However, the linear Boltzmann equation presented here is equivalent to the equation in Masson and LeBlond [Masson, D., LeBlond, P., 1989. Spectral evolution of wind-generated surface gravity waves in a dispersed ice field. J. Fluid Mech. 202, 111–136] since it is derived from their equation. Furthermore, the linear Boltzmann equation is also derived independently using the argument in Meylan et al. [Meylan, M.H., Squire, V.A., Fox, C., 1997. Towards realism in modeling ocean wave behavior in marginal ice zones. J. Geophys. Res. 102 (C10), 22981–22991]. We also present details of how the scattering kernel in the linear Boltzmann equation is found from the scattering by an individual ice floe and show how the linear Boltzmann equation can be solved straightforwardly in certain cases.     

Generating–absorbing sponge layers for phase-resolving wave models

The accurate generation and absorption of water waves in phase-resolving models are critical issues in representing nearshore processes. Here, we present a source function method for combined wave generation and absorption using modified sponge layers. This technique can be easily adapted to a wide variety of systems, and does not require the solution of Green's functions but rather the simpler knowledge of solutions for free waves. These solutions may be linear or nonlinear, regular or irregular, and generated waves can be made arbitrarily accurate through simple selection of sponge layer coefficients. Generating–absorbing sponge layer systems are shown to have a close correspondence to relaxation zones for wave generation if relaxation coefficients are chosen appropriately.     

Edge waves under ice cover at a straight coast with a sloping beach

Edge waves in an ice-covered sea at a straight coast with a sloping beach are analyzed within the linearized theory. Such waves propagate along the coast with an amplitude which exponentially decays offshore. The problem is examined without using the hydrostatic assumption. The seawater is considered to be a homogeneous, inviscid, nonrotating, and incompressible fluid. Ice with a uniform thickness is considered, with constant values of density, cylindrical rigidity, Poisson ratio, and compressive stress in the ice. The normal velocity at the bottom is zero; the linearized kinematic and dynamic boundary conditions are satisfied at the lower surface of the ice. Explicit solutions for the edge flexural-gravity waves and the corresponding dispersion equations are obtained and analyzed.     

Wave-ice interaction in the North-West Barents Sea

The results of field work on drift ice during wave propagation are analyzed and presented. The field work was performed in the Barents Sea, and the main focus of the paper is on wave processes in the MIZ. A model of wave damping in broken ice is formulated and applied to interpret the field work results. It is confirmed that waves of higher frequencies are subjected to stronger damping when they propagate below the ice. This reduces the frequency of most energetic wave with increasing distance from the ice edge. Difference of wave spectra measured in two relatively close locations within the MIZ is discussed. The complicated geometry and dynamics of the MIZ in the North-West Barents Sea allow waves from the Atlantic Ocean and south regions of the Barents Sea to penetrate into different locations of the MIZ.     

Second-Oorder Analytic Solutions of Nonlinear Interactions of Edge Waves on A Plane Sloping Bottom

Based on the full water-wave equation, a second-order analytic solution for nonlinear interaction of short edge waves on a constant plane sloping bottom is presented in this paper. For special case of slope angle b=p/2, this solution can be reduced to the same order solution of deep water gravity surface waves traveling along parallel coastline. Interactions between two edge waves including progressive, standing and partially reflected standing waves were also discussed. The unified analytic expressions with transfer functions for kinematic-dynamic elements of edge waves were also discussed. The random model of the unified wave motion processes for linear and nonlinear irregular edge waves is formulated, and the corresponding theoretical autocorrelation and spectral density functions of the first and second orders are derived. The boundary conditions for the determining determination of the parameters of short edge wave are suggested, that may be seen as one special simple edge wave excitation mechanism and an extension to the sea wave refraction theory. Finally some computation results are demonstrated.