Transformation of internal waves over an uneven bottom: Analytical results

The transformation of internal waves over the oceanic shelf of variable depth is studied analytically within a linear theory of a two-layer flow. It is shown that, at a specific character of depth variation, the internal wave propagates without reflection from the slope even if it is sufficiently steep. The properties of such progressive waves are studied—their form and the current structure in the upper and lower layers. The transformation of the wave propagating from the open ocean, where the depth is assumed to be constant, is considered. It is shown that the wave is transformed at the shelf edge and does not change its form in the course of time during its further propagation over the shelf. The height and form of the internal wave are calculated at the interface of the transition of the two-layer flow into the one-layer flow. Applications of the developed analytical theory to the estimation of internal wave transformation over a real shelf are discussed.     

f—平面上的宽陆架诱导阻尼波

由于陆架区资源的开发和利用以及污染、地震海啸和风暴潮预报等等的实际需要,近廿年来,随着浅海动力学中的陆架动力现象越来越受到人们的关注,陆架诱导波的研究也得到了相应的进展.特别是,我国东临宽广的浅水陆架区域,它蕴藏着丰富的石油和水产等资源,又不断遭到风暴潮的袭击,显然,宽陆架诱导阻尼波的探讨是具有特殊重要意义的.     

Continental shelf waves off the Fukushima coast

Past observations and theories have indicated the importance of the constitution of the lowest-mode of shelf waves to the velocity field. However, significant contributions of the higher mode waves to current velocity fluctuations in the vicinity of the coast are suggested in observational results obtained along the Fukushima coast in Pt. I of this study (Kubota et al., 1981). To understand the importance of the higher modes, the generation of shelf waves is investigated theoretically by two methods. First, the generation of long shelf waves by monochromatic forcing is examined, and it is concluded that near the coast the second mode's contribution to the longshore velocity is the largest for the Fukushima coast. Second, the response of shelf waves to broad-band forcing is investigated by taking the dispersive characteristics of shelf waves into consideration. It is concluded that shelf waves with zero group velocity are selectively excited if the forcing has a broad-band spectrum. According to observational results obtained along the Fukushima coast, the wind spectrum has a broad peak at about 100 hours (Kubota et al., 1981). Since the third mode of shelf waves has zero group velocity around the period of 100 hours, the third mode can be selectively generated off the Fukushima coast. From this it is suggested that the Fukushima coast is in the forced region and that observed current fluctuations are motions associated with the second- and third-mode shelf waves.     

Effects of longshore shelf variations on barotropic continental shelf waves,slope currents and ocean modes

Effects of continental shelf bends, converging depth contours and changing depth profiles are discussed. Some analysis is carried out for previously unstudied cases. Separate oceanic interior and shelf flow problems are formulated for a sufficiently narrow shelf. The ocean interior ‘sees’ only an integrated shelf effect, typically increasing shelf-edge amplitudes, retarding longshore Kelvin-wave propagation and increasing natural mode periods by 0 (10%). On the local shelf, the flow matches to the ocean interior and is nondivergent. Effects on shelf waves and slope currents depend subtly on the nature of the longshore variations. Curvature and contour convergence do not per se imply scaterring or generation of shelf waves. Indeed, any depth h(ξ) where ▽² ξ(x,y) = 0 (a condition approximating longshelf uniformity in the topography's convexity) supports essentially the same shelf waves as do straight depth contours (DAVIS, 1983), and slope currents follow depth contours. Scattering results rather from breaks in analyticity of the depth profile. Hence calculations for small isolated features (necessarily highly convex or concave) may overestimate scattering, and superposition for realistic topography may lead to much self-cancellation among scattered waves. Otherwise, examples show a strong preference for scattering into adjacent mode numbers and into any shelf wave mode near to its maximum frequency. A shelf sector, where the maximum shelf wave frequency maxω is less than the frequency ω of an incident shelf wave, causes substantial scattering unless maxω and ω are very close. Adjustment of slope currents to changed conditions takes place through (and over the decay distance of) scattered shelf waves.     

Generation of internal waves by a barotropic tide in the coastal zone

Within the framework of the linear theory of long waves, we study internal waves generated by a barotropic tide in a two-layer ocean of variable depth taking into account the influence of the Coriolis force. Barotropic waves run over an extended unevenness of the bottom at an arbitrary angle. This unevenness is regarded as a model of the continental slope and shelf. We establish the dependences of the amplitudes of generated internal waves on the angle of incidence of the barotropic tide, topography of the bottom, and stratification. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

Winter dynamics on the northern Portuguese shelf. Part 2: bottom boundary layers and sediment dispersal

The northern Portuguese shelf, between 41°N and 42°N, is characterised by the presence of a mid-shelf mud deposit, the Douro Mud Patch (DMP). Observations conducted between July 1996 and June 1999 under the framework of project OMEX II–II, are used to examine the impact of the dynamic processes in the lower water column, particularly on the DMP. The typical wave conditions observed during the winter maintain a highly energetic environment, capable of promoting the erosion of fine sediments at mid-shelf depths. However, the bottom nepheloid layers generated beneath these waves only extend a few meters above the bottom and are contained within rocky outcrops with similar heights that fringe the outer shelf. Each year there are about ten storms, mostly associated with southerly winds that create downwelling conditions over the shelf. The waves associated with these storms produce shear velocities over 3 cm/s at mid-shelf and bottom nepheloid layers which extend a few tens of meters above the bottom and spread offshore, over the outer shelf and upper slope. A rough estimate suggests that these events account for an offshore export of about 20×10⁶ kg of fine sediments each year (equivalent to 1–2% of sediments trapped at DMP).     

Internal Wave Generation by Tidal Flow over a Continental Shelf Slope

The generation process of internal waves by strong tidal flow over a continental shelf slope is reproduced using a multi-level numerical model. On the basis of the numerical results, the crucial role of the tidal advection effect in the generation process of internal waves is demonstrated. The close relation between the resulting internal waveform and the strength of the tidal advection effect is also examined. The barotropic forcing on the internal wave actually works within a relatively small horizontal scale over the top of the continental shelf slope. When the maximum internal Froude number at the shelf break (Fr_m) is less than about 0.6, the amplitude of the resulting internal wave is almost proportional to F_rm. When F_rm is more than about 0.6, however, the amplitude of the resulting internal wave becomes larger than predicted by linear theory. In particular, when F_rm is more than unity, the time period during which the shoreward propagating internal wave stays in the barotropic forcing region becomes much longer. Consequently, the internal wave is significantly amplified with the horizontal scale approaching that of the barotropic forcing, which concentrates in a relatively small region over the top of the continental shelf slope. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical analysis of the transformation of tsunami waves in the south crimea shelf area

The paper analyses the transformation of tsunami-type solitary waves, propagating from the abyssal part of the Black Sea towards its shelf zone. The study is performed by solving numerically unidimensional non-linear equations for non-dispersive long waves, using the finite-difference slope and shelf, with the full wave reflection prescribed at a 10 m depth contour. The non-linearity of the process is shown to throughly impact the reflection of waves by the shore and the shape of the reflected wave. Tsunami wave heights have been seen to increase by several times in the Black sea shelf area. Translated by Vladimir A. Puchkin.     

Response of the suspended sediment transport system to continental shelf dynamics

Surface currents influenced by a wind-driven upwelling event in San Pedro Bay moved total suspended matter (TSM) confined to the inner shelf on 19 April 1978 seaward, so that by 27 April surface TSM had increased over the outer shelf. Near-bottom concentrations of TSM also increased across the shelf during this time. This is explained by sediment resuspended by large surface waves being advected from the inner shelf seaward by the mean flow after this flow had turned from southeasterly to southerly when upwelling ceased on 26 April. These complex shelf dynamics contribute to the off-shelf transport of mud to the slope and deep basins.     

Generation mechanism of higher mode nondispersive shelf waves by wind forcing

The higher mode predominance in the current velocity fields associated with wind-induced shelf waves in the nondispersive regime is studied with a special attention to the effect of the geographical boundary, e.g. wide strait or wide bank areas. The effect of such large topographic change is represented by wind forcing with a finite dimension near the geographical boundary. The time development processes of the wind-induced shelf waves is examined in the context of an initial-value problem, where a spatially finite wind stress is applied att=0. Various modes of shelf waves excited at the boundary start propagating simultaneously and develop monotonically within the forcing region. After the passage of such wave, the energy of wind is used to maintain the attained equilibrium condition, i.e. the steady shelf circulation. The current evolution of the lower mode is restricted to the earlier stage because of the large propagation speed. In contrast, the higher mode waves can travel slowly within the forcing region so that the kinetic energy is supplied from wind stress for a long time before the equilibrium condition is established. Consequently, the observation at the fixed point near the geographical boundary would show that the higher mode waves gradually dominate as time goes on, i.e. for the long-term forcing.     

Breaking of a tidal internal wave on a steep shelf as inferred from temperature measurements

The structure, evolution, and breaking of a tidal internal wave on a steep shelf are discussed on the basis of the data of temperature measurements. The bottom slope at the measurement site is close to the critical slope for a tidal wave. The tidal wave and other waves are inclined coastward. The tidal-wave amplitude increases monotonically with increasing horizon depth. The tidal wave is nonlinear in amplitude and turns over on the outer shelf. On the inner shelf, the internal wave is close in shape to rectangular and generates harmonics of its own. The harmonics make the tidal wave steeper and form solitary rises similar to bilateral bores. All these features ensure a more rapid sink for the internal-tide energy.     

Diurnal shelf waves off Hamada on San'in Coast

The tidal current data observed off Hamada on San'in coast have shown the diurnal tidal currents to be larger than the semidiurnal ones by a factor of 5–8, although the ratio (K₁+O₁)/(M₂+S₂) for the tidal heights at Hamada is 1.3. Furthermore, the diurnal currents are found to be more remarkable on the shelf slope than on the shelf. We consider such diurnal current features as being due to the vortical mode waves, and show that the broad shelf and steep shelf slope off San'in coast allow 1st-mode interior shelf waves (ISWs) at a diurnal-period. Using a simple shelf model, it is shown that ISWs occur in response to the seaward component of diurnal tidal oscillations on the shelf and their propagation originates from the western entrance of the Tsushima Straits.     

地形对陆架拦截波性质及机制的影响

陆架波的性质如频散关系、形成机制等受地形影响。研究地形对陆架波的性质影响具有重要意义。基于陆架拦截波理论,数值计算了分段线性地形下不同宽度陆架上陆架拦截波的频散关系、长波假设下波动的相速度、阻尼情况下的波动耗散率以及强迫波的外力影响因子。分析了陆架宽度及坡度对自由及强迫陆架拦截波性质的影响。陆架宽度影响陆架拦截波的频散关系。陆架变宽,使得长波频散曲线的斜率增大。陆架宽度的增加使第一模态陆架拦截波有明显的性质变化:相速度增大,波动受辐散影响的程度变大,摩擦衰减距离增大,且风应力旋度在波动的生成机制中起到的作用渐强。在宽的陆架上,研究陆架拦截波的生成及强迫波的振幅时,应充分考虑风应力旋度的作用。第二、三模态波动的相速度受陆架坡度的影响较大,但摩擦衰减距离基本都在200km左右,几乎不随陆架宽度改变,属于局地波。     

Applicability of numerical models to nonlinear dispersive waves

The applicability of three different wave-propagation models in nonlinear dispersive wave fields has been investigated. The numerical models tested here are based on three different wave theories: a fully nonlinear potential theory, a Stokes second-order theory, and a Boussinesq-type theory with an improved dispersion relation. Physical experiments and computations were conducted for wave evolutions during passage over a submerged shelf under various wave conditions. As expected, the fully nonlinear solutions agree better with the measurements than do the other solutions. Although the second-order solution has sufficient accuracy for smaller-amplitude wave cases, the truncation after the third harmonics causes significant discrepancies in wave form for larger waves. In addition, the second-order model markedly overestimates the first- and second-harmonic amplitudes in transmitted waves. The Boussinesq model provides excellent predictions of wave profile over the shelf even in larger wave cases. However, this model also overestimates the magnitudes of several higher harmonics in transmitted waves. These facts may indicate that energy transfer from bound components into free waves in these higher harmonics cannot be accurately evaluated by the Boussinesq-type equations.     

Amplitude variations of edge waves on a shelf slowly varying in the alongshore direction

Within the framework of the linear shallow-water theory, the dynamics of edge waves over a shelf characterized by a cylindrical bottom relief is investigated under the assumption that shelf parameters vary slowly in the alongshore direction. An asymptotic theory and an energy approach are used to calculate the amplitude of the edge wave. In the analytic form, the results are obtained for shelves of three different profiles with parameters varying along the shore: an infinite linear profile, a concave exponential profile, and a stepwise profile.     

Barotropic tide in the northeast South China Sea

A moored array deployed across the shelf break in the northeast South China Sea during April-May 2001 collected sufficient current and pressure data to allow estimation of the barotropic tidal currents and energy fluxes at five sites ranging in depth from 350 to 71 m. The tidal currents in this area were mixed, with the diurnal O1 and K1 currents dominant over the upper slope and the semidiurnal M2 current dominant over the shelf. The semidiurnal S2 current also increased onshelf (northward), but was always weaker than O1 and K1. The tidal currents were elliptical at all sites, with clockwise turning with time. The O1 and K1 transports decreased monotonically northward by a factor of 2 onto the shelf, with energy fluxes directed roughly westward over the slope and eastward over the shelf. The M2 and S2 current ellipses turned clockwise and increased in amplitude northward onto the shelf. The M2 and S2 transport ellipses also exhibited clockwise veering but little change in amplitude, suggesting roughly nondivergent flow in the direction of major axis orientation. The M2 energy flux was generally aligned with the transport major axis with little phase lag between high water and maximum transport. These barotropic energy fluxes are compared with the locally generated diurnal internal tide and high-frequency internal solitary-type waves generated by the M2 flow through the Luzon Strait.     

南海北部陆架区内波的演变与耗散机制

海洋是多尺度强迫-耗散系统,机械能主要在大尺度输入,在小尺度耗散。在大、中尺度运动的能量向小尺度湍流传递过程中,内波扮演着重要角色。内波的生成和破碎可打破海洋动力平衡,而在陆架区,内波（主要是内孤立波）的浅化演变与耗散则是驱动湍流混合的关键过程。通过长期的理论、观测与数值模拟研究,目前已认识到内波浅化过程中主要发生如下演变：波形调制、极性转变、裂变、破碎与耗散。相较于直接发生破碎,浅化演变过程中的裂变及其引发的剪切不稳定和对流不稳定是内孤立波在陆架区的主要耗散机制,显著调制陆架区的跃层混合。从能量串级的角度讲,内孤立波浅化裂变为动力不稳定的高频内波是潮能串级的重要通道。本文简要回顾南海北部陆架区内波的研究历史,并着重总结内波在陆架区演变与耗散机制的研究进展。     

Large-scale ripple marks on the shelf margin of the Northern Okinawa Trough

Trains of large-scale ripple marks (megaripples and sand waves) were found on the Amakusa and East China Sea shelves bordering the northern Okinawa Trough. Side-scan sonar surveys were carried out in 1974 and 1976 to investigate sea-floor features lying along a proposed submarine cable line. Megaripples were found on the outer margin of the Amakusa shelf between depths of 140 and 200 m. The megaripples were especially well developed at a depth of 167 m. They were typically straight-transverse crested with asymmetrical profiles, and measured 7 to 15 m in wavelength and 0.4 to 1.4 m in waveheight. Formation of the megaripples on the Amakusa shelf is probably controlled by relatively complex oceanographic conditions. A secondary circulation associated with the Gotô-nada clock-wise Current may be responsible for formation of the ripple marks. Local vorticities generated in the coastal boundary layer as a result of curvature of the Gotô-nada Current are known to cause the complex flow pattern at the Gotô and Amakusa shelf margins. The main semidiurnal (M₂) tidal current may also interact with these fluid processes.On the East China Sea shelf, megaripples and sand waves were found between depths of 140 and 220 m. Sand waves (200 m in wavelength) were observed in seismic reflection profiles. Large-scale lunate megaripples were observed at a depth of 154 m by the side-scan sonar. They had wavelengths of 10 to 30 m and waveheights of 1 to as high as 3 m. It appears from the types and nature of distribution of the megaripples that they are responding to the present-day flow regime, and it is partly ascertained from our observations over an interval of two years that the megaripples appear to be short-term response elements compared wit hteh sand waves. We conclude that the megaripples on the East China Sea shelf are current-formed during peak typhoon flow in August to November. From their distribution, the long term path of the main flow of the Tsushima Current is inferred at the edge of the East China Sea shelf. An area of low sediment mud content (less than 20 per cent) coincides with this path giving further support to our interpretation.     

Study of intensive internal waves in the shelf zone of Morocco

Conditions for generating a train of intensive internal waves and a solitary internal wave, as well as the evolution and dissipation of shelf waves have been studied using temperature measurements at six buoy moorings and an array of distributed temperature meters towed in the Morocco shelf zone. Waves of each type have been numerically simulated. The paper's focus is the generation of a baroclinic tide in a shelf zone and the occurrence of a packet of intensive internal waves on the back slope of a baroclinic wave. Besides, it studies the mechanism responsible for the generation of a solitary internal wave of soliton type. Thein situ data and the model data are matched up and shown to be consistent, in terms of quality and quantity. Translated by Vladimir A. Puchkin.