Wake effects on turbulent transport in the convective boundary layer

The effect of the downstream propagation of a wake on the transport of momentum, energy and scalars (such as humidity) in the convective boundary layer (CBL) is studied using a direct numerical simulation. The incompressible Navier–Stokes and energy equations are integrated under neutral and unstable thermal stratification conditions in a rotating coordinate frame with the Ekman layer approximation. Wake effects are introduced by modifying the mean velocity field as an initial condition on a converged turbulent Ekman layer flow. With this initial velocity distribution, the governing equations are integrated in time to determine how turbulent transport in the CBL is affected by the wake. Through the use of Taylor’s hypothesis, temporal evolution of the flow field in a doubly periodic computational domain is transformed into a spatial evolution. The results clearly indicate an increase in the scalar flux at the surface for the neutrally stratified case. An increase in wall scalar and heat flux is also noted for the CBL under unstable stratification, though the effects are diminished given the enhanced buoyant mixing associated with the hot wall.     

On the basic structure of oceanic gravity currents

Results from numerical simulations of idealised, 2.5-dimensional Boussinesq, gravity currents on an inclined plane in a rotating frame are used to determine the qualitative and quantitative characteristics of such currents. The current is initially geostrophically adjusted. The Richardson number is varied between different experiments. The results demonstrate that the gravity current has a two-part structure consisting of: (1) the vein, the thick part that is governed by geostrophic dynamics with an Ekman layer at its bottom, and (2) a thin friction layer at the downslope side of the vein, the thin part of the gravity current. Water from the vein detrains into the friction layer via the bottom Ekman layer. A self consistent picture of the dynamics of a gravity current is obtained and some of the large-scale characteristics of a gravity current can be analytically calculated, for small Reynolds number flow, using linear Ekman layer theory. The evolution of the gravity current is shown to be governed by bottom friction. A minimal model for the vein dynamics, based on the heat equation, is derived and compares very well to the solutions of the 2.5-dimensional Boussinesq simulations. The heat equation is linear for a linear (Rayleigh) friction law and non-linear for a quadratic drag law. I demonstrate that the thickness of a gravity current cannot be modelled by a local parameterisation when bottom friction is relevant. The difference between the vein and the gravity current is of paramount importance as simplified (streamtube) models should model the dynamics of the vein rather than the dynamics of the total gravity current. In basin-wide numerical models of the ocean dynamics the friction layer has to be resolved to correctly represent gravity currents and, thus, the ocean dynamics.     

A 3-D coastal tidal rectification process: observations, theory and experiments

In homogeneous rotating fluid, when there is an oscillating forcing in the interior fluid with a period long enough for an Ekman layer to develop, there is an interaction between the oscillatory Ekman layer and the vertical wall, since the latter imposes an alternating adjustment flow confined near the wall. As a result, this coastal rectification process leads to a Lagrangian transport along the coast. The Ekman number, the Rossby number and the temporal Rossby number of the forcing flow are the governing parameters of that mechanism which can be described by a simplified analytical model taking into account both the vertical time-dependent structure of the current and the presence of the wall. The model shows that the residual (rectified) current flowing with the coast to its right results from the strong nonlinear interaction between along- and cross-shore tidal currents leading to asymmetrical momentum exchanges between the Ekman bottom layer and the coastal boundary layer. The model provides simple scaling laws for the maximum intensity and width of the residual current. The latter is significantly larger than the friction (Stokes) lateral boundary layer of the forcing flow. A comprehensive set of experiments is performed in the 13 m diameter rotating tank by oscillating an 8 m×2 m horizontal plate and vertical wall in a homogeneous fluid at rest in solid-body rotation and measuring the two horizontal components of the current at several locations and depths above the central part of the plate. The predicted and experimentally measured maximum intensity and width of the residual current are in very good agreement, within the range of validity of the model, i.e. when the Ekman number is sufficiently small. However experiments also show that the residual current still occurs when the Ekman layer thickness is of the same order as the fluid depth, but it is then confined to a narrower band along the vertical wall. The flow structure found experimentally is also correctly described by a numerical model developed by Zhang et al. (1994). Current measurements in the Eastern part of the English Channel near the French coast reveal a significant coastal residual current flowing Northward and the coastal rectification process described here may account for part of it.     

A numerical study of detached shear layers in a rotating sliced cylinder

Abstract

The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ^¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the Rossby number arising from the analysis in the ageostrophic part of the detached shear layer.     

Ekman Layer in 3D-Model of the Geodynamo

The main subject of the paper is to resolve the Ekman layer analytically and to formulate an appropriate set of 3D-geodynamo equations. The equations are formulated in the mean field approximation where the mean values of magnetic field and velocity over azimuthal direction vanish. This approach should allow the numerical calculation to be performed for small Ekman numbers, down to 10^–12 , which are usually considered to be realistic in the geodynamo. The solution of the Ekman layer is also newly interpreted and consequently a new term appears in the usual expression for the geostrophic shear. The viscous terms are neglected in the main volume of the core and their leading role is assumed just in the thin Ekman layer. The inner core is not included in these considerations and no concrete calculations of a model are presented.     

Modeling high-order statistics in the turbulent Ekman layer

Results from a direct numerical simulation (DNS) of the neutral and unstable turbulent Ekman layer at a Reynolds number of 1000 were used to evaluate turbulence closure models. For the neutrally stratified Ekman layer, the higher-order moments of velocity were examined and the accuracy of a kurtosis model was assessed. For the unstable Ekman layer, the analysis of higher-order moments was extended to temperature-velocity correlations. Model coefficients were optimised using DNS data and it was shown that the optimised models accurately captured the distributions of all fourth-order moments. These low-Reynolds number results can be extrapolated to higher Reynolds numbers to parameterise turbulence in other flow fields with rotational effects such as the atmospheric boundary layer.     

Experimental study of the Ekman layer instability in steady or oscillating flows

To investigate the instabilities of steady and oscillating Ekman layers, an 8 m×2 m horizontal plate was moved at controlled speed in homogeneous water at rest in solid body rotation in the “Coriolis” 13 m diameter rotating tank. For a steady Ekman layer two distinct wave types were found, in agreement with previous experimental or numerical studies. Type I was stationary, was oriented positively with respect to the flow direction and had a wavelength of about 10 times the Ekman layer thickness. Type II was oriented negatively with respect to the flow direction and had a wavelength which was more than 20 times the Ekman layer thickness and a phase-speed between 0.3 and 0.5 the forcing interior velocity. The growth rates of both type I and type II waves for various Reynolds numbers Re (computed with the Ekman layer thickness) were estimated and their Re-variations qualitatively agree with previous numerical results. For an oscillating Ekman layer, experimental results depended strongly on Ro_t, the temporal Rossby number: only when Ro_t<1 was it possible to observe either type I or type II instabilities. Moreover, for all Ro_t and average to high Re, there was a noticeable upward turbulent transport occurring during each cycle between the flow maximum and the flow reversal. Such an upward turbulent transport is consistent with observations in the English Channel where maximum upward benthic movements and maximum turbidity were recorded at the flow reversal, hence Ekman layer instabilities and transition to turbulence are likely to occur in shallow tidal seas where they may be relevant for sediment resuspension and transport as well as for some biological processes.     

An analytical model for ice-edge upwelling

Abstract

This paper presents an analytical, two-dimensional model of the wind-induced homogeneous circulation near the edge of an ice pack floating on the ocean surface. It is shown that a vertical shear layer arises under the ice edge, by which the wind-driven geostrophic motion in the open ocean is matched to the flow region underneath the ice. As in coastal upwelling models, this shear layer consists of a thin E ^1/2-layer inside a thicker E ^1/4-layer (E being the Ekman number). Under certain conditions the shear layer produces a vertical mass flux from the bottom to the surface Ekman layer. Near the surface this upwelling flux is concentrated in the narrow E ^1/2-layer. Comparison with observations of upwelling at the edge of a polar ice pack shows good agreement.     

Improved bounds on linear instability of barotropic zonal flow within the shallow water equations

Here we develop mathematical results to describe the location of linear instability of a parallel mean flow within the framework of the shallow water equations; growth estimates of near neutral modes (for disturbances subcritical with respect to gravity wave speed) in the cases of non-rotating and rotating shallow water. The bottom topography is taken to be one-dimensional and the isobaths are parallel to the mean flow. In the case of a rotating fluid, the isobaths and the mean flow are assumed to be zonal. The flow is front-like: there is a monotonic increase of mean flow velocity. Our results show that for barotropic flows the location of instabilities will be a semi-ellipse region in the complex wave velocity plane, that is based on the wave-number, Froude number, and depth of the fluid layer. We also explore the instability region for the case of spatially unbounded mean velocity profiles for non-rotating shallow water.     

“The Ekman Drain”: a conduit to the deep ocean for shelf material

A long (167 days) acoustic Doppler current profiler time series from the European continental slope west of Scotland has been analysed to investigate the influence of bathymetric steering on the slope current and the extent of down-slope transport in the bottom boundary layer. Within an interior region between the surface and bottom boundary layers, the direction of the flow is found to be remarkably consistent as required by the Taylor-Proudman theorem for geostrophic flow. The mean value of this interior flow direction is taken to be the effective direction of the bathymetry in controlling the geostrophic flow and so defines the rotation of coordinates required to determine along and cross-flow transports. Within a bottom boundary layer (BBL) of thickness ~100 m, the direction of the flow was deflected increasingly to the left with the mean veering angle ~12.5° at 12 mab and a down-slope speed of 2.6 cm s^?1. The corresponding integrated transport (the “Ekman drain”) had an average value of ~1.6 m² s^?1 over the full observation period. This down-slope flow was significantly correlated (at 0.1 % level), with the stress applied by the along-slope flow although with considerable scatter (r.m.s. ~1 m² s^?1) which suggests the influence of other forcing mechanisms. Combining the BBL volume transport with an estimate of the mean concentration of suspended particulate material indicates an annual down-slope flux of 3.0?±?0.6 tonnes m^?1 year^?1, of which ~0.36?±?0.1 tonnes m^?1 year^?1 is carbon. Biogeochemical measurements indicate that the carbon flux in the Ekman drain predominates over settlement of organic material through the water column over the slope and provides for relatively rapid delivery of material to deep water.     

Tides and mixing in the northwestern East China Sea Part I: Rotating and reversing tidal flows

Bottom-mounted ADV and ADCP instruments in combination with CTD profiling measurements taken along the Chinese coast of the East China Sea were used to study the vertical structure of temperature, salinity, and velocity in reversing tidal currents on a shallow inner shelf and in rotating tidal flows over a deeper sloping bottom of the outer shelf. These two regimes of barotropic tide affect small-scale dynamics in the lower part of the water column differently. The reversing flow was superimposed by seiches of ∼2.3 h period generated in semienclosed Jiaozhou Bay located nearby. As the tidal vector rotates over the sloping bottom, the height of the near-bottom logarithmic layer is subjected to tidal-induced variations. A maximum of horizontal velocity U_max appears at the upper boundary of the log layer during the first half of the current vector rotation from the minor to the major axis of tidal ellipse. In rotating tidal flow, vertical shear generated at the seafloor, propagated slowly to the water interior up to the height of U_max, with a phase speed of ∼5 m/h. The time-shifted shear inside the water column, relative to the shear at the bottom, was associated with periodically changing increases and decreases of the tidal velocity above the log layer toward the sea surface. In reversing flows, the shear generated near the bottom and the shear at the upper levels were almost in phase.     

风驱动下f-平面准地转三维海洋环流的形成及总质量守恒的应用

将理想化的南中国海海盆在垂直方向上划分为Ekman层、惯性层和摩擦层. Ekman层中的运动由大气风应力驱动,其底部的扰动压力将作为其下惯性层中运动的上边界条件. 惯性层中的运动是由f 平面三维非线性方程在准地转近似下位势涡度守恒控制,由此得到控制惯性层中运动关于扰动压力的三维椭圆型方程. 在惯性层以下考虑到深层的海盆水平尺度很小,由此引进带有底部摩擦的线性控制方程,方程的边界条件为惯性层和摩擦层交界面上的扰动压力连续,沿海盆边界假定海水与相邻的固壁间无热量交换,由此设在海盆边界上扰动温度为零. 在此基础上分别利用惯性层和摩擦层中的椭圆型控制方程计算了相应层次上冬、夏季的扰动压力和准地转流. 结果表明冬季各层上以气旋式环流为主,且随深度的增加流速减小;夏季各层上以反气旋式环流为主,流速也随深度增加而减小. 这在一定程度上和观测事实相符.     

Mechanisms of bottom boundary fluxes in a numerical model of the Shetland shelf

Across-slope bottom boundary layer (BBL) fluxes on the shelf-edge connect this region to deeper waters. Two proposed ways in which across-slope BBL fluxes can occur, in regions that have a slope current aligned to the bathymetry, are the frictional veering of bottom currents termed the ‘Ekman drain’ and through local wind-forced downwelling (wind-driven surface Ekman flow with an associated bottom flow). We investigate the variability, magnitude and spatial scale of BBL fluxes on the Shetland shelf, which has a prominent slope current, using a high-resolution (～2 km) configuration of the MITgcm model. Fluxes are analysed in the BBL at the shelf break near the 200 m isobath and are found to have a seasonal variability with high/low volume transport in winter/summer respectively. By using a multivariate regression approach, we find that the locally wind-driven Ekman transport plays no explicit role in explaining daily bottom fluxes. We can better explain the variability of the across-slope BBL flux as a linear function of the speed and across-slope component of the interior flow, corresponding to an Ekman plus mean-flow flux. We estimate that the mean-flow is a greater contributor than the Ekman flux to the BBL flux. The spatial heterogeneity of the BBL fluxes can be attributed to the mean-flow, which has a much shorter decorrelation length compared to the Ekman flux. We conclude that both the speed and direction of the interior current determines the daily BBL flux. The wind does not explicitly contribute through local downwelling, but may influence the interior current and therefore implicitly the BBL fluxes on longer timescales.     

Frontal upwelling in a rotating two-layer fluid

Abstract

This paper describes the source-sink driven flow in a two-layer fluid confined in a rotating annulus. Light fluid is injected at the inner wall, while denser fluid is withdrawn at the outer wall. The interface between the immiscible fluids intersects the bottom and thus produces a front. The net transport from the source to the sink is carried by Ekman layers at the bottom and at the interface, and by Stewartson layers at the side walls. A detached Stewartson layer arises at the front, leading to a pronounced upwelling circulation.     

Wind and melt driven circulation in a marginal sea ice edge frontal system: a numerical model

A seasonal ice edge zone is a unique frontal system with an air-ice-sea interface. This paper is a report on the numerical results from a quasi-three dimensional, time dependent, non-linear numerical model of circulation at a continental shelf-seasonal ice edge zone. The purpose of the experiments is to model the hydrography and circulation, including upwelling, baroclinic geostrophic flow, and inertial oscillations, at the ice edge with emphasis on examining the driving forces of wind and melting ice. It is suggested that the non-linear acceleration terms and vertical density diffusion terms are negligible and that the horizontal density diffusion terms are of secondary importance within the time and space scales of the experiments. The vertical eddy viscosity terms are important in a spin-up time scale and for Ekman transport and a bottom Ekman layer. The effects of the horizontal eddy viscosity terms are observable (a long-ice jet is diffused away from the ice edge) by the end (72 h) of the model runs.Model results are compared with available oceanographic and meteorological data for verification. The observed and modeled features of melt water induced water column stability, frontal structure, and ice edge upwelling are briefly discussed relative to observed ice edge primary production. Because the model is relatively general in nature, it is readily applicable to other seasonal or marginal ice edge zones in either hemisphere.     

Flows within the Ekman layer during spin-up of a thermally stratified fluid

Abstract

Finite-difference numerical solutions were obtained to present the flow and temperature field details within the transient Ekman layer during spin-up of a thermally stratified fluid in a cylinder. This complements the earlier studies on stratified spin-up which examined the flows in the interior core region. As the stratification increases, the following changes in the flow field are noticeable. The radial velocity in the Ekman layer decreases in magnitude. The azimuthal flows adjust smoothly from the interior region to the endwall boundary, and the Ekman layer in the azimuthal flow field fades. Vertical motions are inhibited, resulting in a weakened Ekman pumping. The axial vorticity field behaves similarly to the azimuthal flows. The temperature deviation from the equilibrium profile decreases, and the heat transfer flux from the endwall to the fluid decreases. The thickness of the thermal layer is larger than the velocity layer thickness. Illustrative comparisons of the relative sizes of the terms in the governing equations are conducted in order to assess the stratification effect in the adjustment process of the fluid.     

Barotropic instability of a boundary jet on a sloping bottom

Abstract

The stability of a shear flow on a sloping bottom in a homogeneous, rotating system was investigated by means of a laboratory experiment.

The basic flow was driven near a vertical wall of a circular container by a ring-shaped plate that contacted with a free surface of the working fluid and rotated relative to the fluid container. The velocity profile was asymmetric in the radial direction and had only one inflection point. The velocity profile was well expressed by a linear theory for the vertical shear layer.

The effect of the circular geometry was checked by comparing experimental results obtained in two fluid systems in which only the sign of the curvature was opposite and it was confirmed that circular geometry was not essential for the shear flow on the sloping bottom in this experiment.

It was found that the sloping bottom stabilizes the basic flow only when the drift direction of the topographic Rossby wave is opposite to that of the basic flow. The viscous dissipation in both the Ekman layer and the interior region was also important in determining the critical Rossby number.

The eddy fields caused by the instability can be classified into two types: One is the stationary eddy field in which a row of eddies moves along the basic flow without changing form. The other is the flow pattern in which eddies have finite life times and their configuration is not well organized. When the sloping bottom does not stabilize the basic flow, the former flow pattern is realized, otherwise the latter flow pattern appears.

The wave numbers of the eddies in the regular flow pattern were observed as a function of the Rossby number. The relation did not fit to linear preferred modes predicted by an eigenvalue problem.     

Three-layer model for vertical velocity distribution in open channel flow with submerged rigid vegetation

Based on the detailed laboratory experiments and theoretical analysis, a new three-layer model is proposed to predict the vertical velocity distribution in an open channel flow with submerged vegetation. The time averaged velocity and turbulence behaviour of a steady uniform flow with fully submerged artificial rigid vegetation was measured using a 3D Micro ADV, and the vertical distribution of velocity and Reynolds shear stress at different vegetation height, vegetation density and measuring positions were obtained. The results show that the velocity profile consists of three hydrodynamic regimes (i.e. the upper non-vegetated layer, the outer and bottom layer within vegetation); accordingly different methods had been adopted to describe the vertical velocity distribution. For the upper non-vegetated layer, a modified mixing length theory combined with the concept of ‘the new vegetation boundary layer’ was adopted, and an analytical model was presented to predict the vertical velocity distribution in this region. For the bottom layer within vegetation, the depth average velocity was obtained by numerically solving the momentum equations. For the upper layer within vegetation, the analytical solution was presented by expressing the shear stress as a formula fitted to the experimental data. Finally, the analytical predictions of the vertical velocity over the whole flow depth were compared with the results obtained by other researchers, and the good agreement proved that the three-layer model can be used to predict the velocity distribution of the open channel flow with submerged rigid vegetation.     

Rotating flow past a short,sliced cylinder at finite Rossby number

Abstract

Flow past a short obstacle in a rotating reference frame generates a wake that is crucial to the overall flow structure if the Rossby number is of the order of the quarter power of the Ekman number. We present here a theory for such flows for the case when the obstacle's top is an oblique, planar surface. The results arise from a combination of asymptotic analysis and numerical computation, and show that even weak asymmetry generates significant global effect on the entire flow-field. Comparisons with the experiments reported by Foster and Davies (1996) are generally good when the high edge is at 90° to the oncoming flow.     

Wave energy input into the Ekman layer

This paper is concerned with the wave energy input into the Ekman layer, based on 3 observational facts that surface waves could significantly affect the profile of the Ekman layer. Under the assumption of constant vertical diffusivity, the analytical form of wave energy input into the Ekman layer is derived. Analysis of the energy balance shows that the energy input to the Ekman layer through the wind stress and the interaction of the Stokes-drift with planetary vorticity can be divided into two kinds. One is the wind energy input, and the other is the wave energy input which is dependent on wind speed, wave characteristics and the wind direction relative to the wave direction. Estimates of wave energy input show that wave energy input can be up to 10% in high-latitude and high-wind speed areas and higher than 20% in the Antarctic Circumpolar Current, compared with the wind energy input into the classical Ekman layer. Results of this paper are of significance to the study of wave-induced large scale effects.