Distribution of Flexural Strains in a Sea-Ice Cover Caused by the Uniform Motion of an Area of Constant Pressure

The space distribution of flexural strains induced in a floating ice cover by its oscillations caused by the motion of an area of constant pressure is investigated. The dependences of flexural strains in front the load, in the loaded area, and in the wave track on the thickness of ice, velocity of propagation of pressure, and the type of its distribution are analyzed. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 6, pp. 3–16, November–December, 2005.     

The influence of the bottom contour on the deformed state of the ice cover due to the motion of the submarine

The authors have previously determined that the effectiveness and failure pattern of the ice cover caused by flexural-gravity waves generated by a submerged body motion near the bottom ice can greatly depend on the depth of the water area. In its turn, the presence of a ledge on the ice surface may affect a wave propagation pattern. This paper presents an experimental study of the bottom contour influence on the deflection and length of flexural-gravity waves. The authors describe a numerical model for the analysis of the deformed state of ice caused by hydrodynamic loads due to a submarine motion, taking into account the bottom contour. The experiments are carried out in the ice tank. The results of calculations and experiments are compared.     

Large-time response of ice cover to a load moving along a frozen channel

Unsteady response of an ice cover in a channel with vertical walls is studied for large times. The ice deflection is caused by a load moving along the frozen channel at a constant speed. The ice cover is modelled as a thin elastic plate clamped to the walls of the channel. The time-dependent problem is solved by using the Fourier transform along the channel and the method of separating variables. In the system moving along the channel together with the load, the large-time deflection of the ice cover consists of steady deflection and standing waves in front and behind the load. The number of waves, their frequencies and wavenumbers depend on the speed of the load and the values of the critical speeds for the channel. The number of the waves and their amplitudes are calculated for a given load and its speed. The maximum stress in the ice as a function of the load speed is estimated.     

Influence of ice compression on components of the velocity of motion of liquid under the ice cover in a traveling periodic flexural gravity wave of finite amplitude

By the method of multiple scales, we obtain (to within the third order of smallness) the asymptotic expansions for the components of the velocity of motion of liquid under a floating ice cover in the process of propagation of periodic surface flexural gravity waves of finite amplitude under the conditions of ice compression. We study the dependences of the distributions of the velocity components along the wave profile on the compressive forces and the parameters of the initial harmonic. It is shown that the amplitude values of the velocity components decrease and the phase shift of oscillations increases as the compressiveforces increase.     

Bed load transport on the shoreface by currents and waves

Tide-driven bed load transport is an important portion of the net annual sediment transport rate in many shoreface and shelf environments. However, bed load transport under waves cannot be measured in the field and bed load transport by currents without waves is barely measurable, even in spring tidal conditions. There is, consequently, a strong lack of field data and validated models. The present field site was on the shoreface and inner shelf at 2 to 8.5 km offshore the central Dutch coast (far outside the surfzone), where tidal currents flow parallel to the coast. Bed load transports were carefully measured with a calibrated sampler in spring tidal conditions without waves at a water depth of 13–18 m with fine and medium sands. The near-bed flow was measured over nearly a year and used for integration to annual transport rates. An empirical bed load model was derived, which predicts bed load transports that are a factor of > 5 smaller than predicted by existing models. However, they agree with laboratory data of sand and gravel transport in currents near incipient motion. The damped transport rates may have been caused by cohesion of sediment or turbulence damping due to mud or biological activity. The annual bed load transport rate was calculated using a probability density function (pdf) derived from the near-bed current and orbital velocity data which represented the current and wave climate well when compared to 30 years of data from a nearby wave station. The effect of wave stirring was included in the transport calculations. The net bed load transport rate is a few m²/year. This is much less than predicted in an earlier model study, which is partly due to different bed load models but also due to the difference in velocity pdf. The annual transport rate is very sensitive to the probability of the largest current velocities.     

Moored fish cage dynamics in waves and currents

Recent work in the area of open ocean aquaculture system dynamics has focused separately upon either the response of fish cages in waves or the steady drag response due to ocean currents. In reality, however, forcing on these open ocean structures is a nonlinear, multidirectional combination of both wave and current profiles. At the University of New Hampshire-operated Open Ocean Aquaculture site, data were collected from a wave measurement buoy and a downward-looking Acoustic Doppler Current Profiler to characterize the surface elevation and water velocity profiles during an extreme northeast storm event. In addition to waves and currents, fish cage motion response in heave, surge, and pitch was inferred from accelerometer measurements during the same storm. The environmental data sets obtained during the peak of the storm were processed, analyzed, and used as input to a dynamic finite-element model. Simulations were performed using three load case scenarios: 1) in both waves and currents; 2) in waves only; and 3) in currents only. Model motion response results in both the time and frequency domain were compared with data obtained in situ . In addition to the motion response tests, the wave and current forcing influencing the mooring line tension response was also investigated. Analysis shows that in this case, the currents do not severely influence the oscillatory motion response, but do cause the cage to tilt, layback, and sink. The wave and current interaction effect did, however, influence the anchor line loads with a portion being attributed to nonlinear effects.     

浮冰在规则波和双色波下的运动特性研究

为研究浮冰在波浪下的运动特性,以单个聚氯乙烯（PVC）塑料圆板模拟浮冰,进行了一系列波浪试验研究。在规则波试验基础上,同时研究了双色波条件下的浮冰运动特性。结果显示：规则波条件下浮冰运动幅频响应算子首先随无量纲波长的增大而增大,当无量纲波长大于4.0后,运动幅频响应算子随无量纲波长的变化不明显,且趋于一定值;在双色波条件下,对应频率组成成分的浮冰运动幅频响应算子与规则波条件下随波长的变化规律一致。根据部分规则波试验结果提出预测浮冰慢漂速度的经验公式,并用已有的试验和余下的规则波与双色波试验结果进行验证。结果表明,经验公式对规则波和双色波条件下的浮冰慢漂速度的误差在20%以内,预测结果吻合较好。     

Velocities of motion of liquid particles under the floating ice cover in the case of propagation of periodic waves of finite amplitude

By using the velocity potential obtained by the method of multiscale asymptotic expansions to within the quantities of the third order of smallness, we study the dependences of the components of the velocity of motion of a homogeneous liquid under the floating ice cover on the thickness of the cover and its modulus of elasticity in the process of propagation of periodic waves of finite amplitude. It is shown that the presence of broken ice leads to a decrease in the moduli of components of the velocity of liquid particles and the phase delay of generated oscillations. The effect of the elasticity of ice becomes more pronounced as the wavelength of the initial harmonic decreases and manifests itself in the increase in the maximum values of the components of velocity and in the phase shift of oscillations in the direction of propagation of waves.     

Hydrodynamic Pressure of Liquid in the Process of Motion of a Region of Constant Disturbances over the Floating Ice Cover

We study the dependence of the hydrodynamic pressure induced in a liquid as a result of motion of a load of constant intensity over the floating ice cover on the flexural stiffness of ice and the character of distribution of the load over the moving region. The contribution of the deflection of ice in the vicinity of the load and the induced flexural and flexural-gravitational waves to the formation of the field of hydrodynamic pressure near the ice-water interface is estimated.__________Translated from Morskoi Gidrofizicheskii Zhurnal, No. 1, pp. 3–14, January–February, 2005.     

A note on the horizontal momentum exchange in combined waves and current

The horizontal exchange of momentum due to the organized motion in combined waves and current has been analyzed. The combination of the vertical orbital wave motion and the mean current gives a periodic variation in the horizontal velocity in addition to the wave orbital motion. This periodic variation, combined with the wave orbital motion, gives a significant contribution to the momentum exchange. Two examples are considered, the interaction of a pure wave motion and a current normal to the direction of wave propagation, and a wave driven longshore current with an undertow velocity profile. It is demonstrated that the new contribution changes the resulting momentum exchange considerably.     

Bed friction and dissipation in a combined current and wave motion

Two simple two-layer eddy viscosity models, which facilitate analytical solutions, are presented in order to describe the velocity field and associated shear stress in a combined current wave motion. The models, which have the same eddy viscosity in the current boundary layer, but different eddy viscosities in the wave boundary layer, cover together the whole rough turbulent regime. Straightforward definitions are made for the wave friction factor and the current friction factor for the combined motion, which are in accordance with the results for pure waves and pure currents. In this way one avoids the fictitious reference velocities and elliptic integrals which e.g. Grant and Madsen (1978, 1979) experienced. The two friction factors turn out to be functions of four dimensionless parameters. A detailed calculation procedure is presented. Comparison with laboratory experiments yields promising results. A new relation connecting dissipation and bed shear stress is also developed.     

Sea ice evolution and internal wave generation due to a tidal jet in a frozen sea

The purpose of this study is to investigate the influence of tidal currents on sea ice in Spitsbergen fjords which may cause rapid decrease of the ice thickness due to erosion and melting of the ice. The effect was studied in-situ near the narrow channel connecting the Van Mijen Fjord and Lake Vallunden. The strong jet-like tidal currents in the strait driven by semidiurnal tide continue into the lake preventing ice freezing along a narrow strip during high tide and relatively warm weather. Understanding the formation of open water regions or regions with thin ice is important for the safe transportation on ice. We estimate conditions and representative time over which strong tidal current influences ice thickness along a narrow strip in solid ice. Changes of tidal phase and decrease in air temperature influence freezing of the strip in one-two days. While the tidal flow leaves the strait it overflows a shallow bar and generates internal lee waves propagating downslope and mixing the water. Tidal forcing of internal waves was measured using pressure gauges and by scanning of the ice surface during flood and ebb phases. Internal waves were measured using three types of CTD instruments and an ADCP current meter. The generation of wave packets occurs every tidal cycle when the current flows into the lake, but no generation occurs during the ebb phase of the tide because the currents over the bar slope are low. Parameters of internal waves are estimated. Model simulations confirm generation of internal wave train by the tidal current descending downslope.     

Wave-Ice interaction

1 INTRODUcTIONlt is weIl known by whalers and sealers tha in iee covered seas wave is attenuatedas it lnoves inward from the ice edge. But Systematic obsewation on the attenuationphenomenon is rare. Some examples are given in the references [l-3]. In these referenceS,direct measurementS on wave attenuation dije to ice covers are repeded. Theoretical foundationis needed to explain these observations in order to generelize the resultS. These generalizedlesults can then be appIted to wave-…     

Numerical and simplified methods for the calculation of the total horizontal wave force on a perforated caisson with a top cover

The VOF method and the k–ε model, combined with the equation of state of air at constant temperature, have been used to calculate the total horizontal wave force caused by monochromatic waves acting on a perforated caisson with a top cover. From comparison of various parameters, such as the total horizontal force, the pressure difference on the front wall, the pressure on the back wall and the pressure on the top cover, between the numerical results and test data, it can be seen that the numerical results agree well with the test data. It is concluded that the method described in this paper can be utilized to calculate wave forces acting on perforated caissons with a top cover in the case of nonovertopping, nonbreaking waves. A simplified method to calculate the total horizontal force has been developed, based on test data, using a least-squares method. A comparison between the numerical results and the values calculated from the simplified equations shows good agreement. Therefore the simplified equations can be used in engineering applications to evaluate the total horizontal force on a perforated caisson with a top cover.     

An experimental method to identify a component of wave orbital motion in propeller effective inflow velocity and its effects on load fluctuations of a ship main engine in waves

Improvements of estimation accuracy on propeller torque fluctuations in waves will contribute assessments on safe operation of a ship main engine as in adverse sea condition. The propeller torque and thrust in waves can be estimated by propeller effective inflow velocity in waves, using the propeller open-water characteristics. Fluctuation components in the mathematical model of the propeller effective inflow velocity in waves can be composed of two components, respectively caused by ship surge motion and wave orbital motion at propeller position. In this study, an experimental method by the model test to directly identify the characteristics of the component by the wave orbital motion is newly proposed. Furthermore, the free-running model test in regular waves, using a simulator of the marine diesel engine which manages the shaft speed of the motor on a ship model as behaving the actual diesel engine, is carried out to obtain realistic torque fluctuations for comparisons of the estimated results applying the proposed identification method. Through comparisons of estimated fluctuations with the measured results, the proposed approach for the component of the inflow velocity due to wave orbital motion is successfully validated.     

A numerical study on local characteristics of predetermined irregular wave trains

In actual sea states, damage to offshore floating structures is usually caused by a few extreme waves or wave groups in an irregular wave train. Accurate simulation of the irregular wave trains can lay a solid foundation for understanding the local flow field and impact loads that would potentially cause such damage. This paper describes how the generation of a single extreme wave was investigated. Determination of the wave-maker motion for generating specified irregular wave trains is the key to this work. First, an experimental irregular wave train was decomposed into a certain number of small-amplitude waves. Fourier series expansion was performed to determine the amplitude and the initial phase angle of each wave component. Then a hydrodynamic transfer function was used to calculate the amplitude of the wave-maker motion associated with each wave component. Superposition was made on all the wave components to get the final wave-maker motion. During the numerical simulation, calculated horizontal velocity profiles of the extreme wave at different moments were analyzed and compared with experimental results, and a satisfactory agreement was obtained. In the simulation, VOF method was employed to capture the free surface, and a dissipation zone was used to deal with wave reflection.     

Ice response to an underwater body moving in a frozen channel

Strains in the ice cover of a frozen channel, which are caused by a body moving under the ice at a constant speed along the channel, are studied. The channel is of rectangular cross section, the fluid in the channel is inviscid and incompressible. The ice cover is modeled by a thin viscoelastic plate clamped to the channel walls. The underwater body is modeled by a three-dimensional dipole. The intensity of the dipole is related to the speed and size of the underwater body. The problem is considered within the linear theory of hydroelasticity. For small deflections of the ice cover the velocity potential of the dipole in the channel is obtained by the method of images without account for ice deflection in the leading order. The problem of a dipole moving in the channel with rigid walls provides the hydrodynamic pressure on the upper boundary of the channel, which corresponds to the ice cover. This pressure distribution does not depend on the deflection of the ice cover in the leading approximation. The deflections of ice and the strains in the ice cover are independent of time in the coordinate system moving together with the dipole. The problem is solved numerically using the Fourier transform along the channel, the method of normal modes across the channel, and the truncation method for resulting infinite systems of linear equations. It was revealed that the strains in the ice strongly depend on the speed of the dipole with respect to the critical speeds of the hydroelastic waves propagating along the frozen channel. The width of the channel matters even it is much larger than the characteristic length of the ice cover.     

Transport Properties of Trapped Topographic Waves

In the Boussinesq approximation, we study weakly nonlinear topographic waves trapped by a flat slope of arbitrary orientation. We compute the mean currents induced by the waves due to the nonlinearity in the quadratic approximation with respect to the wave amplitude in the presence of dissipation of the wave energy into the turbulent motion. In the diffusion approximation, we determine the vertical distribution of the concentration of wave-suspended sediments. It is shown that the consumption of sediments across the isobaths is directed downward along the slope. At the same time, the consumption of sediments along the isobaths has the same direction as the projection of the horizontal wave vector.     

Interaction of Surface Waves in a Basin with Floating Broken Ice

The multiscale method is used to obtain asymptotic expansions up to the quantities of the third order for the elevations of the surface of the basin and the velocity potential of motion of liquid particles in the wave disturbances formed in the process of nonlinear interaction of periodic running waves of the first and second harmonics in a homogeneous ideal incompressible liquid of constant finite depth covered with broken ice. The dependences of the amplitude-phase structure of disturbances on the ice thickness, depth of the basin, and the parameters of interacting harmonics are investigated. We estimate the error of evaluation of the characteristics of the formed vertical displacement of the surface of the basin and nonlinear mass transfer introduced by neglecting the curvature of the wave profile in the expression for the velocity potential in deducing the kinematic and dynamic surface boundary conditions for nonlinear approximations.     

Downward transfer of momentum by wind-driven waves

A model for the downward transfer of wind momentum is derived for growing waves. It is shown that waves, which grow due to an uneven pressure distribution on the water surface or a wave-coherent surface shear stress have horizontal velocities out of phase with the surface elevation. Further, if the waves grow in the x-direction, while the motion is perhaps time-periodic at any fixed point, the Reynolds stresses associated with the organized motion are positive. This is in agreement with several field and laboratory measurements which were previously unexplained, and the new theory successfully links measured wave growth rates and measured sub-surface Reynolds stresses. Wave coherent air pressure (and/or surface shear stress) is shown to change the speed of wave propagation as well as inducing growth or decay. From air pressure variations that are in phase with the surface elevation, the influence on the waves is simply a phase speed increase. For pressure variations out of phase with surface elevation, both growth (or decay) and phase speed changes occur. The theory is initially developed for long waves, after which the velocity potential and dispersion relation for linear waves in arbitrary depth are given. The model enables a sounder model for the transfer to storm surges or currents of momentum from breaking waves in that it does not rely entirely on ad-hoc turbulent diffusion. Future models of atmosphere-ocean exchanges should also acknowledge that momentum is transferred partly by the organized wave motion, while other species, like heat and gasses, may rely totally on turbulent diffusion. The fact that growing wind waves do in fact not generally obey the dispersion relation for free waves may need to be considered in future wind wave development models.