On the spatial instability of internal waves in continuously stratified flows

A stationary parallel current in a continuously stratified incompressible fluid of finite depth is considered. The instability of internal waves is studied with the assumption that the current has only a vertical velocity shear.Translated by Vladimir A. Puchkin.     

Temporal Variability of High Vertical Wavenumber Shear over the Izu-Ogasawara Ridge

Recent numerical studies (Hibiya et al., 1996, 1998, 2002) showed that the energy cascade across the internal wave spectrum down to small dissipation scales was under strong control of parametric subharmonic instabilities (PSI) which transfer energy from low vertical mode double-inertial frequency internal waves to high vertical mode near-inertial internal waves. To see whether or not the numerically-predicted energy cascade process is actually dominant in the real deep ocean, we examine the temporal variability of vertical profiles of horizontal velocity observed by deploying a number of expendable current profilers (XCPs) at one location near the Izu-Ogasawara Ridge. By calculating EOFs, we find the observed velocity profiles are dominated by low mode semidiurnal (∼double-inertial frequency) internal tides and high mode near-inertial internal waves. Furthermore, we find that the WKB-stretched vertical scales of the near-inertial current shear are about 250 sm and 100 sm. The observed features are reasonably explained if the energy cascade down to small dissipation scales is dominated by PSI.     

Modeling wave�Ccurrent bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.     

The effect of tidal currents on internal waves

Internal waves were observed by measuring temperature variations of several subsurface layers at the innermost part of Suruga Bay from December 1968 to October 1971. Spectral energy densities of temperature fluctuations were computed from the records of the measurements. In the shorter period range from one minute to one hour, peaks of energy density were found occasionally in the range shorter than the minimum of VÄisÄlÄ periods computed from the vertical distribution of water density. It has been generally understood, however, that the periods of internal waves in a stable stratum should be within the range between the inertial and VÄisÄlÄ periods.The measurements of tidal currents in the surface and lower layers, which were undertaken simultaneously with the temperature measurements, revealed that the short-period oscillations were associated with the increase of current velocity and of vertical shear of current at the pycnocline.It is considered that observed periods shorter than the minimum of VÄisÄlÄ period are not real but apparent periods due to the Doppler effect, because the waves are generated in the velocity shear of tidal current and the source is moving towards the station with the tidal current.     

An analysis of horizontal dispersion due to the drift current with an Ekman layer

An analytical method for describing horizontal matter dispersion in shear currents is presented using a tensor expression from the point of view that matter dispersion due to the shear effect should be one of the principal mixing dilution processes. Although the behavior of horizontal dispersion is considerably more complicated than common longitudinal dispersion, the present study elucidates the vertical structure of dispersion and the dispersing process from the initial to the stationary stage, besides the usual depth-averaged dispersion coefficient at the stationary stage. As one of the typical applications of horizontal dispersion, dispersion due to the pure drift current with an Ekman layer is examined theoretically using the present method. This examination reveals that the displacement of the centroid and the major axis of dispersion are twisted in the vertical direction more than the direction of the current vector forming the Ekman spiral; that the variance increases in proportion to the third power of the elapsed time; and that the dispersion coefficient at the stationary stage remains constant, independent of the depth normalized by an Ekman layer thickness. Such dependence of the dispersion coefficient in the steady current is shown to be different from that in the oscillatory current, which is inversely proportional to the depth normalized by a Stokes layer thickness. This is considered to be induced by the difference of the vertical profiles of the first order moment in both currents, that is, the shear region of the first order moment is restricted around the floor by the alternation of the current shear in the oscillatory current while it is diffused in the whole depth in the steady current.     

On the vertical distribution of

The aim of this paper is to present an analytical expression for the vertical distribution of the correlation between the horizontal ( ) and vertical ( ) wave velocity components. This quantity, , which appears explicitly in the time-averaged momentum balance equations, has been shown to play an important role in the vertical distribution of wave-induced currents.The proposed formulation for is based on an identity that relates the effective (wave) shear stress to the effective (wave) normal stresses ( ² and ²) and to the vorticity of the oscillatory flow gw. This general expression has been applied to simplified situations and has been shown to degenerate into other existing formulations with comparable simplifying assumptions, viz. irrotational waves in shallow water over an arbitrary bottom topography and breaking waves over a horizontal bottom.The model has also been applied to the case of waves interacting with a depth-varying current over a horizontal bottom, in which preliminary results have been obtained for a simplified situation invoking linear (small-amplitude) wave theory.     

A new instrument system to investigate sediment dynamics on continental shelves

A new instrumented tripod, the GEOPROBE system, has been constructed and used to collect time-series data on physical and geological parameters that are important in bottom sediment dynamics on continental shelves. Simultaneous in situ digital recording of pressure, temperature, light scattering, and light transmission, in combination with current velocity profiles measured with a near-bottom vertical array of electromagnetic current meters, is used to correlate bottom shear generated by a variety of oceanic processes (waves, tides, mean flow, etc.) with incipient movement and resuspension of bottom sediment. A bottom camera system that is activated when current speeds exceed preset threshold values provides a unique method to identify initial sediment motion and bed form development.

Data from a twenty day deployment of the GEOPROBE system in Norton Sound, Alaska, during the period September 24 – October 14, 1976 show that threshold conditions for sediment movement are commonly exceeded, even in calm weather periods, due to the additive effects of tidal currents, mean circulation, and surface waves.     

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.     

Current-induced scour around a vertical pile in cohesive soil

Stability of many ocean structures is affected by seabed scour induced by under-currents. The depth of scour is an important parameter for determining the minimum depth of foundations as it reduces the lateral capacity of the foundations. A review of the literature reveals that there is not much information available in the field of scour in cohesive soils. Hence, a detailed laboratory testing programme on model piles of diameters 50 mm to 110 mm embedded in soft silty clay soil was carried out in a wave flume of 30 m long, 2.0 m wide and 1.7 m deep, which has the capability of simulating steady currents. Scour around the pile due to steady streaming is monitored by using special instrumentation. A procedure has been suggested to predict the ultimate scour depths based on the observed variation in scour depth over a limited time period. The study indicates that the ultimate scour depth is controlled by diameter of obstruction, current velocity, model Reynolds number, flow Froude number, shear stress, and soil characteristics. Based on these results, a few functional relationships are suggested between scour depth and other parameters like Reynolds number, Froude number, and strength of the soil bed.     

Numerical Simulation of Spatial Lag Between Wave Breaking Point and Location of Maximum Wave-Induced Current

A quasi three-dimensional numerical model of wave-driven coastal currents with the effects of surface rollers is developed for the study of the spatial lag between the location of the maximum wave-induced current and the wave breaking point.The governing equations are derived from Navier-Stokes equations and solved by the hybrid method combining the fractional step finite different method in the horizontal plane with a Galerkin finite element method in the vertical direction.The surface rollers effects are considered through incorporating the creation and evolution of the roller area into the free surface shear stress.An energy equation facilitates the computation process which transfers the wave breaking energy dissipation to the surface roller energy.The wave driver model is a phase-averaged wave model based on the wave action balance equation.Two sets of laboratory experiments producing breaking waves that generated longshore currents on a planar beach are used to evaluate the model's performance.The present wave-driven coastal current model with the roller effect in the surface shear stress term can produce satisfactory results by increasing the wave-induced nearshore current velocity inside the surf zone and shifting the location of the maximum longshore current velocity landward.     

Development of the summer breeze circulation in the west part of the Black Sea

On the basis of the results of regional reanalysis of the atmospheric circulation presented with a resolution of 9 km, the process of formation of breezes is studied for the case of weak synoptic background activity in the rectilinear part of the west coast of the Black Sea for the period 01–04.07.2007. It is shown that the gravitational currents, breeze fronts, and intense internal waves are formed under these conditions in the troposphere during the daytime. The Hovm?ller diagrams of the wind velocity and the maps of the vertical sections of potential temperature, vertical velocity of the air masses, and other parameters are presented. On the basis of these diagrams and maps, we obtain quantitative estimates of the wind velocity in the breeze and of the velocity of propagation of the breeze front and compare these estimates with the available literature data.     

Non-linear effects in the packets of internal waves in the ocean

The mechanisms for the generation of mean density and current velocity fields in a medium with a vertical shear, conditioned by the non-linearity of packets of internal waves are analysed. Modulation instability of internal waves is studied allowing for the earth's rotation. A non-linear Schrödinger's equation for the evolution of the envelope has been derived. Also corrections to the mean density and current velocity in the approximation, quadratic by the wave's amplitude, non-oscillating on the wave's time scale have been obtained. The longitudinal modulation conditions have been analysed.Translated by V. Puchkin. UDK 551.466.8.     

南海北部2011年台风“纳沙”尾迹处近惯性内波特征的研究

In September 2011, Typhoon Nesat passed over a moored array of instruments recording current and temperature in the northern South China Sea（SCS）. A wake of baroclinic near-inertial waves（NIWs） commenced after Nesat passed the array. The associated near-inertial currents are surface-intensified and clockwise-polarized. The vertical range of NIWs reached 300 m, where the vertical range is defined as the maximum depth of the horizontal near-inertial velocity 5 cm/s. The current oscillations have a frequency of 0.709 9 cycles per day（cpd）, which is 0.025 f higher than the local inertial frequency. The NIWs have an e-folding time-scale of 10 d based on the evolution of the near-inertial kinetic energy. The depth-leading phase of near-inertial currents indicates downward group velocity and energy flux. The estimated vertical phase velocity and group velocity are 0.27 and 0.08 cm/s respectively, corresponding to a vertical wavelength of 329 m. A spectral analysis reveals that NIWs act as a crucial process to redistribute the energy injected by Typhoon Nesat. A normal mode and an empirical orthogonal function analysis indicate that the second mode has a dominant variance contribution of 81%, and the corresponding horizontal phase velocity and wavelength are 3.50 m/s and 420 km respectively. The remarkable large horizontal phase velocity is relevant to the rotation of the earth, and a quantitative analysis suggests that the phase velocity of the NIWs with a blue-shift of 0.025 f overwhelms that of internal gravity waves by a factor of 4.6.     

Numerical simulation of currents in a basin of variable depth with two straits

Within the framework of the linear theory of long waves, with regard for the turbulent viscosity, we study the development of tidal currents in a basin of variable depth with two straits. The problem is solved numerically. The velocity field on the strait-basin boundary is regarded as known. The numerical analysis is performed for different depths of the straits. We study the influence of the geometric characteristics of the basin on the amplitudes of the profile of free surface and wave velocity and establish the dependences of the wave characteristics on the period of current velocities in the strait and the parameters of the basin. In particular, it is shown that the increase in the period of current velocities in the strait leads to significant changes in the level and structure of currents. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 4, pp. 3–12, July–August, 2007.     

Transformation of the Indonesian Throughflow Water by Vertical Mixing and Its Relation to Tidally Generated Internal Waves

Numerical experiments with two-dimensional nonhydrostatic model have been performed to investigate tidally generated internal waves at the Dewakang sill at the southern Makassar Strait where two large-amplitude “bumps” of relatively shallow water exist. We investigate the effect of these features on vertical mixing, with emphasis on the transformation of the Indonesian throughflow (ITF) water properties. The result shows that large-amplitude internal waves are generated at both bumps by the predominant M₂ tidal flow, even though the condition of the critical Froude number and the critical slope are not satisfied. The internal waves induce such vigorous vertical mixing in the sill region that the vertical diffusivity attains a maximum value of 6 × 10⁻³ m²s⁻¹ and the salinity maximum and minimum core layers characterizing the ITF thermocline water are considerably weakened. Close examination reveals that bottom-intensified currents produced mainly by the joint effect of barotropic M₂ flow and internal tides generated in the concave region surrounding both bumps can excite unsteady lee waves (Nakamura et al., 2000) on the inside slopes of the bumps, which tend to be trapped at the generation region and grow into large-amplitude waves. Such generation of unsteady lee waves does not occur in case of one bump alone. Trapping and amplification of the waves in the sill region induce large vertical displacements (∼60 m) of water parcels during one tidal period, leading to strong vertical mixing there. Since the K₁ tidal currents are relatively weak, large-amplitude internal waves causing intense vertical mixing are not generated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of following and opposing vertical current shear on nonlinear wave interactions

A Navier-Stokes solver in OpenFOAM^® is combined with the Volume of Fluid (VOF) surface capturing method to investigate the wave interaction with depth-varying currents in intermediate and shallow waters. A special attention is paid to the separate effect of vertical current shear on near resonant triad wave interactions. It was found that in the presence of following vertical current shear, the wave exhibits a sharper crest and flatter trough, and the opposite is true in the presence of opposing vertical current shear. Our model results indicate that the wave steepness at which the current shear starts to affect the crest elevation is greater in deeper water than in shallower water. We found that adding vertical current shear to the uniform current further enhances the relative harmonic wave energy and the extent of triad interaction in the following current while weakens them in the opposing current. As a result, following and opposing current shear may cause wave to break at a lower and higher sea state respectively. Due to the increased wave nonlinearity in the presence of a following current shear, a linear superposition of the individual wave and current velocities is no longer adequate to represent the total horizontal velocity close to the free surface.     

Trapped quasi-inertial waves in shear oceanic currents

We studied trapped long quasi-inertial waves in horizontally inhomogeneous flows with low Rossby numbers. A simple heuristic derivation of two equations for the wave amplitude is presented. These equations are true for strong and weak density stratifications. A spectral problem is formulated to find the frequencies of trapped waves based on the amplitude equations. Exact solutions of the hyperbolic problem for a free hyperbolic shear layer are found. It is shown that the location of the trapping area principally depends on the stratification. If the buoyancy frequency is greater than the inertial frequency, trapping occurs in the region of anticyclonic velocity shear; if the buoyancy frequency is smaller than the inertial frequency, trapping occurs in the region of cyclonic velocity shear. Thus, in the first case, the frequencies of the trapped waves are smaller than the inertial frequency, while, in the second case, they are greater. The intense wave activity observed in the regions of oceanic fronts and jet currents can be related to the existence of trapped waves.     

A New Class of Edge Baroclinic Waves and the Mechanism of Their Generation

Edge baroclinic waves are generated in a geostrophic flow with a vertical shear near a solid surface. The study investigates a new class of baroclinic waves in flows with horizontal and vertical shears and a linear distribution of potential vorticity. It is shown that taking account of the horizontal shear leads to the appearance of new features of wave dynamics. These include the nonmodal growth of energy in the initial stage of development, the time dependence of the vertical wave scale, and the possibility of generation of stationary or blocked waves. The horizontal shear makes the mechanism of generation of baroclinic waves by initial vortex perturbations more efficient. One important feature is associated with vortex paths, which are formed by the superposition of a baroclinic wave on the flow with horizontal shear.     

长江河口波-流共同作用下的全沙数值模拟

针对长江河口地形、水文、泥沙运动等复杂的特点,建立了波-流共同作用下的二维全沙及河床演变模型.在合理计算研究区域流场等的基础上,利用切应力概念确定悬沙扩散方程中的源函数;通过系列数值试验和实测资料的统计分析,在经典的泥沙临界起动速度中引入反映河床底质结构及固结程度的局地系数;选用由流速、盐度、含沙量浓度确定的泥沙颗粒絮凝沉降速度,从而提高长江口悬沙场数值模拟精度.在底沙输运计算中,提出一种较为合理确定有关参数的方法.通过洪、枯季大、中、小潮水文、泥沙资料和典型台风引起航槽冲淤变化的实测资料验证,表明该文提出的模型能较合理地反映长江河口流场、泥沙场及地形的演变.     

Experimental study of the instabilities of alongshore currents on plane beaches

A laboratory experiment on alongshore currents was conducted for two plane beaches, with gradients 1:40 and 1:100, to investigate the instability of alongshore currents. Complicated and strongly unstable alongshore current motions were observed. In order to clearly examine the spatial and temporal variations of the shear instability of the currents, digital images from a charge-coupled device (CCD) recorded the deformations of dye batches released in the surf zone. Some essential characteristics of the shear instability were obtained from analyses of images showing the temporal variation of the dye patches.A high-resolution spectral analysis technique (the maximum entropy method, or MEM) was used to analyze the dominant frequency of the observed oscillation, along with the trigonometric regression method for determining the variations of the oscillation strength in the cross-shore direction. The propagation speed of the dye patch was obtained by tracking the movement over time of fixed locations in the dye patch, such as its peak, in the longshore direction. This data was then fitted linearly.Alongshore and cross-shore velocity time series acquired from sensors showed clearly that large-amplitude, long-period (about 50 s or 100 s) oscillations were present for all sensors deployed in the cross-shore direction under regular and irregular wave conditions. The analysis found that the maximum shear wave amplitude was approximately one-sixth of the maximum for the mean alongshore current, and occurred approximately at the position of the maximum of the mean alongshore current for irregular waves. The spatial structure of the shear waves was studied by analyzing collected images of the dye patches. The phase velocity of the meandering movements was obtained by measuring the magnitude of the oscillations of the dye patches in the alongshore direction with respect to time. The results suggest that the propagation speed of the shear instability was approximately one-half to three-quarters of the maximum mean longshore current for both regular and irregular waves.Linear instability analysis theory was applied to the characteristics of alongshore current instability, which suggested that there are two instability modes related to the observed oscillations: the frontshear mode observed for the 1:100 slope, and the backshear mode observed for the 1:40 slope. Theoretical analyses agreed with the experimental results in both cases. The velocity profile of the mean longshore current was found to affect the instability mode significantly, leading to further investigations on the influence of the velocity profiles and to provide support for the above conclusions.