A model of oscillatory rough turbulent boundary layer flow

Approximate analytical solutions of the boundary layer equation are obtained in closed form for oscillatory rough turbulent flow. The solutions are based on a time-varying eddy viscosity, and the aim of the study is to assess the effects of these time variations on the properties of the wave boundary layer. The flow and the eddy viscosity are made interdependent by a closure assumption which relates the peak value of viscosity in the wave cycle to the peak value of shear velocity. Instantaneous vertical profiles of horizontal velocity and shear stress, and time series of the bed shear stress, are presented for a typical case study. In addition, the wave drag coefficient, the boundary layer thickness and the phase lead of peak bed shear stress over peak free-stream velocity, are determined as functions of both the relative roughness and the parameter governing the magnitude of the time variations in viscosity. Reasonable agreement is demonstrated with previous experimental and theoretical results.     

一种三次多项式湍黏性底边界层的流场分布

本文通过假定底边界层湍黏性的三次多项式参数化形式,基于简化的Navier–Stokes方程,并利用超几何方程的性质,推导出了湍流粗糙底边界层的速度解析解。同时,得到了底边界层内其他的动力参数,如底剪应力、Ekman传输、Ekman抽吸及近底部速度分布场,从理论上讨论了均匀混合底边界层特征量分布特征。通过数值结果分析,进一步得出底边界层的总速度、亏损速度及其剪应力受平均流的角频率和地球自转影响比较大;而底边界层的动力结构对于底边界层顶部粗糙度不敏感。该涡黏性模式从理论上丰富了底边界层涡黏性的形式,为底边界层的动力系统研究提供了借鉴和理论参考。     

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.     

Bottom friction and its effects on periodic long wave propagation

A new set of Boussinesq-type equations describing the free surface evolution and the corresponding depth-integrated horizontal velocity is derived with the bottom boundary layer effects included. Inside the boundary layer the eddy viscosity gradient model is employed to characterize Reynolds stresses and the eddy viscosity is further approximated as a linear function of the distance measured from the seafloor. Boundary-layer velocities are coupled with the irrotational velocity in the core region through boundary conditions. The leading order boundary layer effects on wave propagation appear in the depth-integrated continuity equation to account for the velocity deficit inside the boundary layer. This formulation is different from the conventional approach in which a bottom stress term is inserted in the momentum equation. An iterative scheme is developed to solve the new model equations for the free surface elevation, depth-integrated velocity, the bottom stress, the boundary layer thickness and the magnitude of the turbulent eddy viscosity. A numerical example for the evolution of periodic waves propagating in one-dimensional channel is discussed to illustrate the numerical procedure and physics involved. The differences between the conventional approach and the present formulation are discussed in terms of the bottom frictional stress and the free surface profiles.     

On extracting current profiles from vertically integrated numerical models

A transformation method is presented by which current profiles (of tidal or wind-induced origin) can be extracted at any horizontal position and moment in time from a vertically integrated, two-dimensional, hydrodynamic numerical model. An arbitrary vertical variation of eddy viscosity can be included in the method, which can incorporate a no-slip bottom boundary condition. The technique assumes that the sea is homogeneous.The method is used to improve the representation of bottom stress within the two-dimensional model, whereby the bottom stress is no longer related simply to the depth-mean current as in the “conventional” two-dimensional, vertically integrated model.Idealized calculations for a range of eddy viscosity profiles, show that elevations, current profiles, and time series of current extracted from this “enhanced” two-dimensional numerical model are in good agreement with currents obtained from a full three-dimensional model.     

Experimental study of turbulent oscillatory boundary layers in an oscillating water tunnel

A high-quality experimental study including a large number of tests which correspond to full-scale coastal boundary layer flows is conducted using an oscillating water tunnel for flow generations and a Particle Image Velocimetry system for velocity measurements. Tests are performed for sinusoidal, Stokes and forward-leaning waves over three fixed bottom roughness configurations, i.e. smooth, “sandpaper” and ceramic-marble bottoms. The experimental results suggest that the logarithmic profile can accurately represent the boundary layer flows in the very near-bottom region, so the log-profile fitting analysis can give highly accurate determinations of the theoretical bottom location and the bottom roughness. The first-harmonic velocities of both sinusoidal and nonlinear waves, as well as the second-harmonic velocities of nonlinear waves, exhibit similar patterns of vertical variation. Two dimensionless characteristic boundary layer thicknesses, the elevation of 1% velocity deficit and the elevation of maximum amplitude, are found to have power-law dependencies on the relative roughness for rough bottom tests. A weak boundary layer streaming embedded in nonlinear waves and a small but meaningful third-harmonic velocity embedded in sinusoidal waves are observed. They can be only explained by the effect of a time-varying turbulent eddy viscosity. The measured period-averaged vertical velocities suggest the presence of Prandtl's secondary flows of the second kind in the test channel. Among the three methods to infer bottom shear stress from velocity measurements, the Reynolds stress method underestimates shear stress due to missed turbulent eddies, and the momentum integral method also significantly underestimates bottom shear stress for rough bottom tests due to secondary flows, so only the log-profile fitting method is considered to yield the correct estimate. The obtained bottom shear stresses are analyzed to give the maximum and the first three harmonics, and the results are used to validate some existing theoretical models.     

An analytic solution to the wave bottom boundary layer governing equation under arbitrary wave forcing

Existing models of the wave bottom boundary layer have focused on the vertical and temporal dynamics associated with monochromatic forcing. While these models have made significant advances, they do not address the more complicated dynamics of random wave forcing, commonly found in natural environments such as the surf zone. In the closed form solution presented here, the eddy viscosity is assumed to vary temporally with the bed shear velocity and linearly with depth, however, the solution technique is valid for any eddy viscosity which is separable in time and space. A transformation of the cross-shore velocity to a distorted spatial domain leads to time-independent boundary conditions, allowing for the derivation of an analytic expression for the temporal and vertical structure of the cross-shore velocity under an arbitrary wave field. The model is compared with two independent laboratory observations. Model calculations of the bed shear velocity are in good agreement with laboratory measurements made by Jonsson and Carlsen (1976, J. Hydraul. Res., 14, 45–60). A variety of monochromatic, skewed, and asymmetric wave forcing conditions, characteristic of those found in the surf zone, are used to evaluate the relative effects on the bed shear. Because the temporal variation of the eddy viscosity is assumed proportional to the bottom shear, a weakly nonlinear interaction is created, and a fraction of the input monochromatic wave energy is transferred to the odd harmonics. For a monochromatic input wave, the ratio of the third harmonic of velocity at the bed to the first is <10%. However, for a skewed and asymmetric input wave, this ratio can be as large as 30% and is shown to increase with increasing root-mean-square input wave acceleration. The work done by the fluid on the bed is shown to be a maximum under purely skewed waves and is directed onshore. Under purely asymmetric waves, the work done is significantly smaller and directed offshore.     

Rational approach to the bottom shear stress in tidal planetary boundary layer flow

The friction velocity associated with the maximum bottom shear stress in neutrally stable tidal planetary boundary layer flow is presented. The directions of the bottom shear stresses for the anticlockwise and clockwise rotating components are also presented. The results are obtained by using similarity theory and are given for flow conditions in the rough, smooth and transitional smooth-to-rough turbulent regime. An approximation for the maximum bottom shear stress by disregarding the rotation of the velocity in the boundary layer as the seabed is approached is also presented.     

Field Observation and Analysis of Wave-Current-Sediment Movement in Caofeidian Sea Area in the Bohai Bay, China

In order to study the mechanism of flow-sediment movement, it is essential to obtain measured data of water hydrodynamic and sediment concentration process with high spatial and temporal resolution in the bottom boundary layer (BBL). Field observations were carried out in the northwest Caofeidian sea area in the Bohai Bay. Near 2 m isobath (under the lowest tidal level), a tripod system was installed with AWAC (Acoustic Wave And Current), ADCP (Acoustic Doppler Current Profilers), OBS-3A (Optical Backscatter Point Sensor), ADV (Acoustic Doppler Velocimeters), etc. The accurate measurement of the bottom boundary layer during a single tidal period was carried out, together with a long-term sediment concentration measurement under different hydrological conditions. All the measured data were used to analyze the characteristics of wave-current-sediment movement and the BBL. Analysis was performed on flow structure, shear stress, roughness, eddy viscosity and other parameters of the BBL. Two major findings were made. Firstly, from the measured data, the three-layer distribution model of the velocity profiles and eddy viscosities in the wave-current BBL are proposed in the observed sea area; secondly, the sediment movement is related closely to wind-waves in the muddy coast area where sediment is clayey silt: 1) The observed suspended sediment concentration under light wind conditions is very low, with the peak value generally smaller than 0.1 kg/m3 and the average value being 0.03 kg/m3; 2) The sediment concentration increases continuously under the gales over 6-7 in Beaufort scale, under a sustained wind action. The measured peak sediment concentration at 0.4 m above the seabed is 0.15-0.32 kg/m3, and the average sediment concentration during wind-wave action is 0.08-0.18 kg/m3, which is about 3-6 times the value under light wind conditions. The critical wave height signaling remarkable changes of sediment concentration is 0.5 m. The results show that the suspended load sediment concentration is mainly influenced by wave-induced sediment suspension.     

An approximate solution of wave-modified Ekman current for gradually varying eddy viscosity

An approximate steady solution of the wave-modified Ekman current is presented for gradually varying eddy viscosity by using the WKB method with the variation of parameters technique. The parameters involved in the solution can be determined by the two-dimensional wavenumber spectrum of ocean waves, wind speed, the Coriolis parameter and the densities of air and water. The solution reduces to the exact solution when the eddy viscosity is taken as a constant. As illustrative examples, for a fully developed wind-generated sea with different wind speeds and a few proposed gradually varying eddy viscosities, the current profiles calculated from the approximate solutions are compared with those of the exact solutions or numerical ones by using the Donelan and Pierson wavenumber spectrum, the WAM wave model formulation for wind input energy to waves, and wave energy dissipation converted to currents. It is shown that the approximate solution presented has an elegant form and yet would be valid for any given gradually varying eddy viscosity. The applicability of the solution method to the real ocean is discussed following the comparisons with published observational data and with the results from a large eddy simulation of the Ekman layer.     

含变涡动系数的超浅海风暴潮模型

超浅海风暴潮模型提出后[2],对渤海风潮,作为超浅海问题,进行了数值研究[1]。其结果的分析和观测资料的比较都表明了该模型有一定的应用价值;故,对超浅海风暴潮模型作进一步的探讨是有一定意义的。尤其因为我国是一个多浅水域和多风暴潮的国家,这种研究就具有更重要的意义。     

Field observation and analysis of wave-current-sediment movement in Caofeidian Sea area in the Bohai Bay,China

In order to study the mechanism of flow-sediment movement, it is essential to obtain measured data of water hydrodynamic and sediment concentration process with high spatial and temporal resolution in the bottom boundary layer （BBL）. Field observations were carried out in the northwest Caofeidian sea area in the Bohai Bay. Near 2 m isobath （under the lowest tidal level）, a tripod system was installed with AWAC （Acoustic Wave And Current）, ADCP （Acoustic Doppler Current Profilers）, OBS-3A （Optical Backscatter Point Sensor）, ADV （Acoustic Doppler Velocimeters）, etc. The accurate measurement of the bottom boundary layer during a single tidal period was carried out, together with a long-term sediment concentration measurement under different hydrological conditions. All the measured data were used to analyze the characteristics of wave-current-sediment movement and the BBL. Analysis was performed on flow structure, shear stress, roughness, eddy viscosity and other parameters of the BBL. Two major findings were made. Firstly, from the measured data, the three-layer distribution model of the velocity profiles and eddy viscosities in the wave-current BBL are proposed in the observed sea area; secondly, the sediment movement is related closely to wind-waves in the muddy coast area where sediment is clayey silt： 1） The observed suspended sediment concentration under light wind conditions is very low, with the peak value generally smaller than 0.1 kg/m^3 and the average value being 0.03 kg/m^3; 2） The sediment concentration increases continuously under the gales over 6-7 in Beaufort scale, under a sustained wind action. The measured peak sediment concentration at 0.4 m above the seabed is 0.15-0.32 kg/m^3, and the average sediment concentration during wind-wave action is 0.08-0.18 kg/m^3, which is about 3-6 times the value under light wind conditions. The critical wave height signaling remarkable changes of sediment concentration is 0.5 m. The results show that the suspended load sediment concentration is mainly influenced by wave-induced sediment suspension.     

A note on a theoretical model of oscillatory smooth turbulent boundary layers

An analytical theory which describes the motion in an oscillatory smooth turbulent boundary layer using a two-layer time invariant eddy viscosity model is presented. The eddy viscosity in the inner layer increases quadratically with the height above the wall. In the outer layer the eddy viscosity is taken as a constant.     

Contributions to dynamics of the Antarctic Circumpolar Current (A.C.C.) (I. Zonal transport)

Scaling of the equations of motion of the Antarctic Circumpolar Current indicates that the Rossby number and the Ekman number are 10⁻⁴ to 10⁻⁵ but the vertical Ekman number may reach unity in the bottom boundary layer. The equations of motion are integrated vertically from the surface to the bottom and averaged over a latitude circle. The resulting equation in the meridional direction is predominantly geostrophic, whereas the main terms of the equation in the zonal direction are the wind stress and the bottom stress. When the vertical eddy viscosity near the bottom is of the order of 10²cm²/sec, the total zonal transport through the Drake Passage computed from the balance of the wind stress and the bottom stress equals 260×10⁶m³/sec, the amount determined byReid andNowlin (1970) from observations. The northward transport reduces the eastward transport corresponding to the wind stress of the westerlies in the A. C. C. through the Coriolis' term in the vertically integrated equation of motion of the zonal direction. South of the Drake Passage, such reduction reaches about ten percent of the wind-driven transport mainly due to the peripheral water discharge. North of the Drake Passage, the northward transport may be generated by the effect of the South American coast which prevents free eastward movement of the A. C. C., causing a wake to the east. This transport may contribute to a part of the northward transport of the bottom water postulated byMunk (1966). The effect of the horizontal eddy viscosity in the zonal transport equation is negligible except near the Antarctic coast, if the eddy viscosity is less than 10⁹cm²/sec.     

数据同化反演风应力拖曳系数以及垂向涡动黏性系数的分布

所作的孪生实验表明:通过利用变分优化控制技术将气象学和海洋学(表层和次表层)的观测资料同化到海洋的埃克曼层模型中,可将未知的边界条件(风应力拖曳系数)和垂向涡动黏性系数的分布同时反演出来.     

Numerical study on vertical structures of undertow inside and outside the surf zone

Nearshore shoaling and breaking waves can drive a complex circulation system of wave-induced currents. In the cross-shore direction, the local vertical imbalance between the gradient of radiation stress and that of pressure due to the setup drives an offshore flow near the bottom, called ‘undertow’, which plays a significant role in the beach profile evolution and the structure stability in coastal regions. A 1DV undertow model was developed based on the relationship between the turbulent shear stress and t...     

南海冬、夏季环流的三维数值模拟

本文利用一个斜压三维陆架海模式——HAMSOM模式对12月份和8月份的南海环流进行数值模拟,结果为:对上层流场,在12月份,在西沙群岛－中沙群岛海区间呈现一个气旋式环流,在越南中部东岸存在一支南向西边界流,在金兰湾的远海为一局地反气旋涡,在南海南部,主要表现为万安滩的气旋式大弯曲(气旋涡)及在北康暗沙北侧的反气旋涡;在8月份,在东沙群岛－中沙群岛－吕宋岛西侧海域间存在一大尺度的气旋涡,在南海西部主要表现为以西沙群岛南部的气旋涡与金兰湾－礼乐滩间的反气旋式大环流相对峙的局面,同时在万安滩东侧有－气旋涡.由于斜压效应、底形效应的作用,使冬、夏季的南海南部中层流场几乎与上层流场相反.     

Contributions to the dynamics of the Antarctic Circumpolar Current (A. C. C.) II. Integral relationship

Non-dimensional equations of motion are derived for the A.C.C. of the barotropic mode, including the bottom friction and the horizontal eddy viscosity. Integration of the vorticity equation along a streamline leads to the zeroth order stream function which is dependent only on depth divided by Coriolis parameter. Integration of the momentum equation along a streamline yields the relation between the momentum input by wind stress and its dissipation by the bottom friction and by the horizontal eddy viscosity. This relation determines the magnitude of the stream function. It explains differences in the total transport of the A.C.C. obtained byBryan andCox (1972), though it gives only one third of the total transport obtained byKamenkovich (1972) with his vertical eddy viscosity of 10²cm² s^?1. With 1 cm² s^?1 of this viscosity,Bryan andCox obtained the transport of about 650 or less than 32×10⁶m³s^?1 for constant or variable depth models, respectively. The higher transport is mainly due to broadening of the width of the A.C.C., whereas the lower value is due to its narrowing and meandering which in turn make the horizontal eddy viscosity more effective (by exercising friction on both sides of the A.C.C.) and the wind stress input smaller than the almost zonal streamlines for constant depth. In the Appendix dynamics of the bottom boundary layer is treated to give rational estimates of the bottom stress in terms of the geostrophic flow and is compared to the recent observations of the benthic boundary current in the Straits of Florida and off San Diego.     

Vertical structure of the M<Subscript>2</Subscript> tidal current in the Yellow Sea

The vertical structure of the M₂ tidal current in the Yellow Sea is analyzed from data acquired using an acoustic Doppler current profiler. The observed vertical profiles of the M₂ tidal current are decomposed into two rotating components of counter-clockwise and clockwise, and restructured using a simple one-point model with a constant vertical eddy viscosity. The analyzed results show that the internal fictional effect dominates the vertical structure of the tidal current in the bottom boundary layer. In the Yellow Sea, the effect of the bottom friction reduces the current speed by about 20–40% and induces the bottom phase advance by about 15–50 minutes. In the shallower coastal regions, the effects of bottom topography are more prominent on the vertical structure of tidal currents. The vertical profile of the tidal current in summer, when the water column is strongly stratified, is disturbed near the pycnocline layer. The stratification significantly influences the vertical shear and distinct seasonal variation of the tidal current.     

海—气相互作用与海流、风暴潮

从方法论上说,除潮汐以外,通常在处理海洋动力学问题时,大多撇开海洋对大气的影响,强调大气对海洋的主导作用,把大气运动当作诱发海水运动的唯一原动力,视海面风场为给定条件,而后用经验或半经验公式算出海面风应力场,作为施加于海水的强迫力。因此,一个成功的海浪、海流或风暴潮的预报,除了具备反映海水运动的主要物理性能的数学模型外,还必须以客观的、准确的海面风场的数值计算和预报为前提。由于问题的复杂性,迄今为止似乎还不能说在实用上已经提供了海面风的一种足够精确的估算或预报方法。海上气象观测资料,尤其是测风资料的稀少,给海面风应力的实际计算带来不少困难。