Three-dimensional instability of shear flows with inflection-free velocity profiles in stratified media with a high Prandtl number

In stably stratified media with a Prandtl number Pr ≫ 1, vertical scales of the density (ℓ) and horizontal velocity variation (L) are quite different, ℓ/L = O(Pr^−1/2) ≪ 1, and this influences the flow stability. In particular, shear flows without inflection points on the velocity profile are unstable even in an ideal incompressible fluid. The maximum instability growth rate for sufficiently small ℓ/L is of the same order as in homogeneous mixing layers, with mainly three-dimensional rather than two-dimensional oscillations increasing in a wide range of parameters. This paper focuses on the three-dimensional instability of such flows. It is shown that the spectrum of unstable oscillations is essentially anisotropic in the case of a relatively weak stratification when the bulk Richardson number J ≤ O[(ℓ/L)^3/2]. The results of the asymptotic analysis are illustrated by calculations for a model flow in a two-layer medium (ℓ = 0) as well as for flows with values of ℓ/L corresponding to a temperature or salinity stratification of the water.     

Instability caused by the coalescence of two modes of a one-layer coastal current with a surface front

Stability of a one-layer density-driven coastal current with a surface front whose potential vorticity is uniformly zero is examined, and it is shown that such a current can be unstable if the stability condition derived by Paldor (1983) is not satisfied,i.e., the current width,L, is larger than 2/3U/f, wheref is the Coriolis parameter andU is the current velocity at the front. This instability is caused by the coalescence of a frontal trapped and coastal trapped waves. The energetics of this instability and the effect of potential vorticity gradient are also discussed.     

A numerical study of three-dimensional wave interaction with a square cylinder

In this study, a three-dimensional numerical model is used to study the wave interaction with a vertical rectangular pile. The model employs the large eddy simulation (LES) method to model the effect of small-scale turbulence. The velocity and vorticity fields around the pile are presented and discussed. The drag and inertial coefficients are calculated based on the numerical computation. The calculated coefficients are found to be in a reasonable range compared with the experimental data. Additional analyses are performed to assess the relative importance of drag and initial effects, which could be quantified by the force-related Keulegan and Carpenter (KC) number: KC_f=UT/(4πL). Here U is the maximum fluid particle velocity, T the wave period and L the length of structure aligned with the wave propagation direction. For small KC_f, the effective drag coefficient is proportional to 1/KC_f, provided the wavelength is much longer than the structural length. When wavelength is comparable to the structure dimension, the effective drag coefficient would be reduced significantly due the cancellation of forces, which has been demonstrated by numerical results.     

海洋白浪寿命的定义及测量结果

通过对国内外白浪研究和应用的分析,首次提出了有效白浪寿命的定义,给出了计算白浪寿命的公式及测量方法和结果,并报告了以此方法在渤海实测的结果,得到了白浪寿命ＴＬ与海面风速Ｕ１０的关系为ＴＬ＝０．２６Ｕ１０以及白浪寿命概率分布近于瑞利分布等。     

On the relation between the growth rate of water waves and wind speed

Various wind velocitiesu _*,U _/2,U andU ₁₀ are correlated to the measured growth rate of water waves , whereu _* is the friction velocity of the wind, andU _/2,U andU ₁₀ are the wind speeds respectively at the heights /2, and 10m above sea surface (: wave length). It is shown that within a range of the dimensionless wind speed, 0.1<u _* /C<0.6, there are no appreciable differences in the correlations, whereC is the phase velocity of water waves. The present relation between andU shows qualitatively similar properties as the one obtained by Al'Zanaidi and Hui (1984); the growth rate for waves with rough surface is larger than that with smooth surface. However, our present relations give, for the both waves with different surface roughness, larger values by factors 1.71.8 than those given by Al'Zanaidi and Hui's relation.     

张力腿平台涡激运动模型试验与数值分析

为揭示张力腿平台涡激运动响应规律,采用数值模拟与模型试验相结合的方法研究了全水深系泊张力腿平台涡激运动响应。根据张力腿平台主尺度参数,按照几何相似制作了水池试验模型,在满足运动相似和动力相似的条件下开展了均匀流、剖面流模型试验,并将模型试验结果与数值模拟结果进行了对比,验证了数值模拟与模型试验结果的一致性。分析结果表明在均匀流作用下张力腿平台涡激运动的锁定区间在5.5<U_r<8.5,来流角对涡激运动影响较大。剖面流作用下平台涡激运动规律与均匀流作用基本一致,但剖面流造流引起的能量分散,使平台在XY平面的运动轨迹变得不规律,系泊系统提供的回复力对涡激运动幅值有抑制作用,来流角和流速对张力腿或立管模态影响明显。论文得到结论对于张力腿平台的工程设计有一定借鉴意义。     

A simple analytical model for surface water waves on a depth-varying current

A perturbation analysis is presented in which a series of small amplitude regular waves co-exist with an arbitrarily sheared current, U(z). Assuming that the current velocity is weak, i.e. U(z)/c=O(ε), the solution is extended to O(ε²), where c is the phase velocity and ε=ak the wave steepness. This provides a first approximation to the non-linear wave-current interaction, and allows simple explicit solutions for both the modified dispersion relation and the water-particle kinematics to be derived. These solutions differ from the existing irrotational models commonly used in design and, in particular, highlight the importance of the near-surface vorticity distribution. These results are shown to be in good agreement with laboratory data provided by Swan et. al. [J. Fluid Mech (2001, in press)]. Perhaps more surprisingly, good agreement is also achieved in a number of strongly non-linear wave-current combinations, where the results of the present analytical solution are compared with a fully non-linear numerical wave-current model.     

A global wave parameter database for geophysical applications. Part 1: Wave-current–turbulence interaction parameters for the open ocean based on traditional parameterizations

Ocean surface mixing and drift are influenced by the mixed layer depth, buoyancy fluxes and currents below the mixed layer. Drift and mixing are also functions of the surface Stokes drift U_ss, volume Stokes transport T_S, a wave breaking height scale H_swg, and the flux of energy from waves to ocean turbulence Φ_oc. Here we describe a global database of these parameters, estimated from a well-validated numerical wave model, that uses traditional forms of the wave generation and dissipation parameterizations, and covers the years 2003–2007. Compared to previous studies, the present work has the advantage of being consistent with the known physical processes that regulate the wave field and the air–sea fluxes, and also consistent with a very large number of in situ and satellite observations of wave parameters. Consequently, some of our estimates differ significantly from previous estimates. In particular, we find that the mean global integral of Φ_oc is 68 TW, and the yearly mean value of T_S is typically 10–30% of the Ekman transport, except in well-defined regions where it can reach 60%. We also have refined our previous estimates of U_ss by using a better treatment of the high frequency part of the wave spectrum. In the open ocean, U_ss  0.013U₁₀, where U₁₀ is the wind speed at 10 m height.     

Barotropic instability of a viscous boundary jet

The barotropic instability of a boundary jet on a beta plane is considered with emphasis on the effect of internal viscosity. An eigenvalue problem for the disturbance equations and its inviscid version are solved by the aid of numerical methods, and instability characteristics are determined as functions of the Reynolds numberR for various values of the beta-parameter. Typical disturbance structures (eigenfunctions) are also computed. Numerical examples show that the minimum critical Reynolds numberR _cr for instability is smaller than 100. At a Reynolds number of the order of hundreds, there appears a second mode of instability in addition to the first unstable mode originating atR _cr ; a kind of ‘resonance’ between the first and second eigenvalues occurs at the particular value ofR. The neutral stability curves are accordingly multi-looped. Although each of the two unstable modes asymptotically approaches its inviscid counterpart asR→∞, the asymptotic approach to the inviscid limit is rather slow and the effect of varyingR is conspicuous even atR∼O (10⁴). It is thus demonstrated that the Reynolds number is an essential stability parameter for real boundary jets. The main part of the material contained in this paper was presented at 1981-Autumn Assembly of the Oceanographical Society of Japan.     

Observation and measurement of the bottom boundary layer flow in the prebreaking zone of shoaling waves

In this paper, the characteristics of the bottom boundary layer flow induced by nonlinear, asymmetric shoaling waves, propagating over a smooth bed of 1/15 uniform slope, is experimentally investigated. Flow visualization technique with thin-layered fluorescent dye was first used to observe the variation of the flow structure, and a laser Doppler velocimeter was then employed to measure the horizontal velocity, U.The bottom boundary layer flow is found to be laminar except within a small region near the breaking point. The vertical distribution of the phase-averaged velocity U at each phase is non-uniform, which is directly affected by the mean velocity, . The magnitude of increases from zero at the bottom to a local positive maximum at about z/δ2.02.5 (where z is the height above the sloping bottom and δ is the Stokes layer thickness), then decreases gradually to zero at z/δ6.07.0 approximately, and finally becomes negative as z/δ increases further. Moreover, as waves propagate towards shallower water, the rate of increase in the maximum onshore oscillating velocity component is greater than that of the offshore counterpart except near the breaking point. The free stream velocities in the profiles of the maximum onshore and offshore oscillating velocity components, and are found to appear at z/δ≥6.0. This implies that, if the Stokes layer thickness is used as a length scale, the non-dimensionalized boundary layer thickness remains constant in the pre-breaking zone. Although is greater than and the asymmetry of the maximum free stream velocities (i.e. ) increases with decrease of water depth, a universal similar profile can be established by plotting z/δ versus ( ) or ( ). The final non-dimensional profile is symmetric and unique for the distributions of the maximum onshore and offshore oscillating velocity components within the bottom boundary layer, which are induced by nonlinear, asymmetric shoaling waves crossing the pre-breaking zone.     

Comparison of Ocean Surface Wind Stress Computed with Different Parameterization Functions of the Drag Coefficient

Depending on the choice of reference wind speed, the quantitative and qualitative properties of the drag coefficient may vary. On the ocean surface, surface waves are the physical roughness at the air-sea interface, and they play an important role in controlling the air-sea exchange processes. The degree of dynamic influence of surface waves scales with wavelength. Drag coefficient computed with the reference wind speed at an elevation proportional to the wavelength (for example, U _λ/2) is fundamentally different from the drag coefficient computed with the wind speed at fixed 10 m elevation (U ₁₀). A comparison has been carried out to quantify the difference in wind stress computation using several different parameterization functions of the drag coefficient. The result indicates that the wind stress computed from U ₁₀ input using a drag coefficient referenced to U _λ/2 is more accurate than that computed with drag coefficient functions referenced to U ₁₀.     

Modification of the damping function in the k–ε model to analyse oscillatory boundary layers

A simple relationship has been developed between the wall coordinate y⁺ and Kolmogorov's length scale using direct numerical simulation (DNS) data for a steady boundary layer. This relationship is then utilized to modify two popular versions of low Reynolds number k–ε model. The modified models are used to analyse a transitional oscillatory boundary layer. A detailed comparison has been made by virtue of velocity profile, turbulent kinetic energy, Reynolds stress and wall shear stress with the available DNS data. It is observed that the low Reynolds number models used in the present study can predict the boundary layer properties in an excellent manner.     

Validating a turbulence closure against estuarine microstructure measurements

A simple k–ε turbulence closure is introduced which has no stability functions but instead a Richardson number-dependent turbulent Prandtl number. Its free parameters are determined in a comparison with microstructure observations from a stratified and sheared tidal estuary and laboratory measurements. The closure is able to simulate observed turbulent dissipation rates (ε) and turbulent length scales (l_th) in regions of strong mean shear and small gradient Richardson number (R_g) to within factors of 2–3. It fails in regions of small shear and large R_g, presumably because of the dominance of internal wave-driven mixing. Additional simulations with a k–ε closure with stability functions taken from Canuto et al. [Canuto, V.M., Howard, A., Cheng, Y., Dubovikov, M.S., 2001. Ocean turbulence I: one-point closure model. Momentum and heat vertical diffusivities. J. Phys. Oceanogr. 31, 1413–1426] and with the closure of Baumert and Peters [Baumert, H., Peters, H., 2004. Turbulence closure, steady state, and collapse into waves. J. Phys. Oceanogr. 34, 505–512] show poor performance. Establishing a valid 1:1 comparison of simulated and observed ε and l_th requires nudging the model velocity and density toward observed values because free model integrations quickly diverge from the observations. Steady state gradient Richardson numbers are constrained to a range of 0.18–0.25, while flux Richardson numbers are constrained to the range of 0.1–0.22. The closure output is rather insensitive to such parameter variations. The simulations are sensitive, however, to the treatment of the observed velocity and density used to nudge the model. Good closure performance requires averaging the measured tidal flow over about an hour, a time scale for which conventional numerical models of estuarine circulations should be able to match observed shears. In the closure simulations the TKE balance stays close to a production–dissipation balance. The time rate of change and vertical diffusion of TKE are small, of the same order of magnitude, and vary in magnitude relative to each other systematically across the water column.     

A simple model of sediment initiation under waves

A simple model is developed to study the initial motion of sediment on a horizontal bed under non-breaking waves. The model is derived to be A=C(T−T₀) based on a wide range of experimental data collected in different flow regimes, where A is the nearbed semi-excursion of wave motion, T is the wave period, and C and T₀ are the coefficients dependent on sediment properties only. For a given sediment, the onset velocity of sediment motion derived from the model is shown to initially increase sharply with wave period T and then approach a constant. The flow Reynolds number Re corresponding to an initiated sediment is also calculated from the simple model and found to be a function of sediment properties and wave period. For the completeness of this study, the initial motion of light sediment under very short waves is also investigated. The present model agrees well with the available laboratory and field data.     

Wave friction factor rediscovered

The wave friction factor is commonly expressed as a function of the horizontal water particle semi-excursion (A _wb) at the top of the boundary layer. A _wb, in turn, is normally derived from linear wave theory by \fracU_\textwbT_\textw2p \frac{{{U_{\text{wb}}}{T_{\text{w}}}}}{{2\pi }} , where U _wb is the maximum water particle velocity measured at the top of the boundary layer and T _w is the wave period. However, it is shown here that A _wb determined in this way deviates drastically from its real value under both linear and non-linear waves. Three equations for smooth, transitional and rough boundary conditions, respectively, are proposed to solve this problem, all three being a function of U _wb, T _w, and δ, the thickness of the boundary layer. Because these variables can be determined theoretically for any bottom slope and water depth using the deepwater wave conditions, there is no need to physically measure them. Although differing substantially from many modern attempts to define the wave friction factor, the results coincide with equations proposed in the 1960s for either smooth or rough boundary conditions. The findings also confirm that the long-held notion of circular water particle motion down to the bottom in deepwater conditions is erroneous, the motion in fact being circular at the surface and elliptical at depth in both deep and shallow water conditions, with only horizontal motion at the top of the boundary layer. The new equations are incorporated in an updated version (WAVECALC II) of the Excel program published earlier in this journal by Le Roux et al. Geo-Mar Lett 30(5): 549–560, (2010).     

关于南海北部上层水团的分类及三维分布的研究

Using the fuzzy cluster analysis and the temperature-salinity(T-S) similarity number analysis of cruise conductivity-temperature-depth(CTD) data in the upper layer(0–300 m) of the northern South China Sea(NSCS), we classify the upper layer water of the NSCS into six water masses: diluted water(D), surface water(SS),the SCS subsurface water mass(U_S), the Pacific Ocean subsurface water mass(U_P), surface-subsurface mixed water(SU) and subsurface-intermediate mixed water(UI). A new stacked stereogram is used to illustrate the water mass distribution, and to examine the source and the distribution of U_P, combining with the sea surface height data and geostrophic current field. The results show that water mass U_P exists in all four seasons with the maximum range in spring and the minimum range in summer. In spring and winter, the U_P intrudes into the Luzon Strait and the southwest of Taiwan Island via the northern Luzon Strait in the form of nonlinear Rossby eddies, and forms a high temperature and high salinity zone east of the Dongsha Islands. In summer, the U_P is sporadically distributed in the study area. In autumn, the U_P is located in the upper 200 m layer east of Hainan Island.     

A method for evaluating statistical errors associated with logarithmic velocity profiles

A logarithmic velocity profile is often fitted to velocity data in order to calculate the friction velocity (u _*) and typify the surface texture by a roughness length (z _o ). A method is given for estimating the errors in these parameters as calculated by this method. An example is given in which the size of the error is compared with the fluctuations that typically occur in the time seriesu _*(t) andz _o (t).     

Instability of the Kuroshio in Luzon Strait: Effects of ridge topography and stratification

The spectral analyses of moored current velocities in the central Luzon Strait reveal northward (i.e., downstream of the Kuroshio) propagation of a frontal wave with a five-day period, with wave amplitude increasing northward. Estimated from both curve fitting and frequency domain Empirical Orthogonal Function methods, the characteristics of five-day variations have wave speeds ranging from 32 to 40 cm s⁻¹, wavelengths ranging from 130 to 150 km, and e-folding time scales for growth ranging from 0.8 to 3 days. An analytical two-layer model used to explore linear stability characteristics indicates that bottom topography (two meridional ridges) is important for the Kuroshio stability characteristics in the Luzon Strait. In the two-layer model with the two ridges, the flow is stabilized for the long-wave mode but destabilized for the short-wave mode (due to increasing vertical shear in the horizontal velocity). The analytical model produces wavelengths and phase speeds for the most unstable mode which is similar to the observation, but the growth rate is underestimated. However, a spectral numerical model applied with a more realistic stratification and velocity structure does obtain faster growth rates comparable to the observations. Parameter sensitivity tests were conducted using the analytical model. The characteristics of the most unstable mode are most sensitive to the surface front location relative to the bottom topography but not sensitive to varying the density difference and thickness of the upper layer.     

Three-dimensional structure of tidal current in the East China Sea and the Yellow Sea

A three-dimensional tidal current model is developed and applied to the East China Sea (ECS), the Yellow Sea and the Bohai Sea. The model well reproduces the major four tides, namely M₂, S₂, K₁ and O₁ tides, and their currents. The horizontal distributions of the major four tidal currents are the same as those calculated by the horizontal two-dimensional models. With its high resolutions in the horizontal (12.5 km) and the vertical (20 layers), the model is used to investigate the vertical distributions of tidal current. Four vertical eddy viscosity models are used in the numerical experiments. As the tidal current becomes strong, its vertical shear becomes large and its vertical profile becomes sensitive to the vertical eddy viscosity. As a conclusion, the HU (a) model (Davieset al., 1997), which relates the vertical eddy viscosity to the water depth and depth mean velocity, gives the closest results to the observed data. The reproduction of the amphidromic point of M₂ tide in Liaodong Bay is discussed and it is concluded that it depends on the bottom friction stress. The model reproduces a unique vertical profile of tidal current in the Yellow Sea, which is also found in the observed data. The reason for the reproduction of such a unique profile in the model is investigated.     

Inertial Instability of Arbitrarily Meandering Currents Governed by the Eccentrically Cyclogeostrophic Equation

Using natural coordinates, we have derived a criterion for the inertial instability of arbitrarily meandering currents. Such currents, governed by the eccentrically cyclogeostrophic equation, are adopted as the basic current field for the parcel method. We assume that any virtual displacement which is given to a water parcel moving in the basic field has no influence on this field. From the conservation of mechanical energy for a virtual displacement we derive an inertial instability frequency ω _m = [(f + 2u/r)Z]^0.5 for the eccentrically cyclogeostrophic current, where f is the Coriolis parameter, u the velocity (always positive), r the radius of curvature of a streamline (negative for an anticyclonic meander), and Z the vertical component of absolute vorticity. If ω _m ² is negative, the eccentrically cyclogeostrophic current becomes unstable. Although the conventional, centrifugal instability criterion, derived from the conservation of angular momentum in a circularly symmetric current field, has a certain meaning for a monopolar vortex, it contains a radial shear vorticity that is difficult to use in arbitrarily meandering currents. The new criterion ω _m ² contains a lateral shear vorticity that is applicable to arbitrarily meandering currents. Examining instabilities of concentric rings with radii of 50–100 km, we consider reasons why the anticyclonic supersolid rotation has been very much less frequently observed than the cyclonic supersolid rotation, despite a prediction of some common stability and a rapid change in radial velocity gradient for the former. Classifying eccentric streamlines into the large and small curvature-gradient types, we point out that the large-gradient curvature in anticyclonic rings is apt to be unstable. This revised version was published online in July 2006 with corrections to the Cover Date.