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.     

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.     

The bottom Ekman layer and the apparent violation of the maximum principle

The solution for the bottom Ekman layer has a somewhat counter intuitive character, which seems to violate the maximum principle: at a certain level the velocity within the Ekman layer is higher than the velocity in the geostrophic layer above. I explain this character by looking at an analogous problem in an inertial frame of reference and show that it is the result of observing the flow from a rotating frame of reference (i.e. within a system that is not in steady state). The flow in the bottom Ekman layer is a superposition of the flow that results from the force exerted on the fluid by the rotating Earth and of the flow that results from the pressure-gradient term. Therefore, at a certain level the speed is higher than the speed of the geostrophic layer above which results from the pressure gradient alone.     

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.     

低纬磁层边界区的可压缩性Kelvin-Helmholtz不稳定波

本文用“三层模式”探讨了低纬磁层边界区的可压缩Kelvin-Helmholtz不稳定波的特性。低纬边界层的存在降低了不稳定的阈值,几乎整个低纬边界区都是不稳定的。从内边界向磁层传输的能流和动量流都比双层模式增大,扰动磁场是椭圆偏振的。还讨论了低纬边界区该不稳定性的动力学效应及其与P_c4-5脉动的联系。     

森林冠层和森林边界层大涡模拟

在采用各向异性湍流动能闭合方案和3阶Runge Kutta时间积分方案的大涡模式中,引入由森林冠层粗糙元造成的动量拖曳项、热量输入项和TKE耗散项,以模拟森林冠层和森林边界层的气象场. 通过中性和不稳定层结条件下不同叶面积指数算例的模拟分析及与已有观测结果的比较表明,本文所建大涡模式对森林冠层和森林边界层有较好的模拟效果. 进一步研究表明：不稳定层结条件下较稠密的森林冠层中特有的Kinking & Pairing湍涡结构与森林边界层中湍流的大涡运动相互作用,形成了森林冠层附近的温度斜坡型结构.     

Instability of a geostrophic front and its energetics

Abstract

The instability of a current with a geostrophic surface density front is investigated by means of a reduced gravity model having a velocity profile with nearly uniform potential vorticity. It is shown that currents are unstable when the mean potential vorticity decreases toward the surface front at the critical point of the frontal trapped waves investigated by Paldor (1983). This instability is identical with that demonstrated by Killworth (1983) in the longwave limit.

The cross-stream component of mass flux and the rates of energy conversions among the five energy forms defined by Orlanski (1968) are also calculated. The main results are as follows, (a) The mass flux toward the surface front is positive near the front and negative around the critical point. The positive mass flux near the front does not vanish at the position of the undisturbed surface front, so that the mean position of the front moves outward and the region of the strong current spreads. (b) The potential energy of the mean flow integrated over the fluid is released through the work done by the force of the pressure gradient of the mean flow on the fluid, and is converted into the kinetic energy of the mean flow. (c) In the critical layer, the mean flow is rapidly accelerated with the growth of the unstable wave. This acceleration is caused by the rapid phase shift of the unstable wave in the critical layer.     

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.     

Experimental investigation of stratorotational instability using a thermally stratified system: instability,waves and associated momentum flux

Stratorotational instability (SRI) has been proposed as a mechanism for outward angular momentum transport in Keplerian accretion disks. A particular designed Taylor–Couette laboratory experiment with axial stratification is suitable for studying the instability. Bottom endplate is cooled and top endplate is heated to achieve axial stratification. Due to constructive constraints, endplates are visually unamenable and quantitative measurement techniques in the co-rotating frame can only be done by looking through the outer cylinder. For this purpose, we built a co-rotating mini-PIV (Particle Image Velocimetry) system with a camera having a tilted viewing angle regarding the horizontal laser sheet. The aim of this study is (i) to quantify the uncertainty of the mini-PIV together with the used calibration technique and (ii) to compare experimental findings on SRI with theoretical predictions.

We perform measurements of the azimuthal and radial component of the velocity in axial stably stratified Taylor–Couette flows, consider velocity profiles and do frequency-filtering and flow decomposition. The absolute error of the mini-PIV system is 2% and we realised that stratified Taylor–Couette flows have smaller Ekman endwall effects than homogeneous ones. Still, Ekman pumping has an impact of the flow and might be responsible for differences between the data and theoretical models ignoring the endwalls. Here we focus on the flow structure during transition to SRI, the drift rate of SRI modes and the radial momentum flux as a function of the Reynolds number. Whereas the structure in form of trapped boundary Kelvin modes and the drift rate corresponds well with earlier predictions, the momentum flux shows a nonlinear dependency with respect to the Reynolds number. Away from the region of transition, theoretical models show a linear relationship. Several possible reasons for the mismatch between the experimental and theoretical models are discussed. Most important, we experimentally demonstrated that in the Rayleigh stable flow regime the SRI can provide a significant amount of outward momentum flux which makes this instability interesting in the context of accretion disks and also of atmospheric vortices where rotation and stratification also play a significant role.     

On total turbulent energy and the passive and active role of buoyancy in turbulent momentum and mass transfer

Measurements of turbulent fluctuations of horizontal and vertical components of velocity, salinity and suspended particulate matter are presented. Turbulent Prandtl numbers are found to increase with stratification and to become larger than 1. Consequently, the vertical turbulent mass transport is suppressed by buoyancy forces, before the turbulent kinetic energy (TKE) and vertical turbulent momentum exchange are inhibited. With increasing stratification, the buoyancy fluxes do not cease, instead they become countergradient. We find that buoyantly driven motions play an active role in the transfer of mass. This is in agreement with trends derived from Monin–Obukhov scaling. For positive Richardson flux numbers (Ri _f ), the log velocity profile in the near-bed layer requires correction with a drag reduction. For negative Ri _f , the log velocity profile should be corrected with a drag increase, with increasing |Ri _f |. This highlights the active role played by buoyancy in momentum transfer and the production of TKE. However, the data do not appear to entirely follow Monin–Obukhov scaling. This is consistent with the notion that the turbulence field is not in equilibrium. The large stratification results in the decay of turbulence and countergradient buoyancy fluxes act to restore equilibrium in the energy budget. This implies that there is a finite adjustment timescale of the turbulence field to changes in velocity shear and density stratification. The energy transfers associated with the source and sink function of the buoyancy flux can be modeled with the concept of total turbulent energy.     

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.     

Boundary layer dissipation and the nonlinear interaction of equatorial baroclinic and barotropic rossby waves

Two-layer equatorial primitive equations for the free troposphere in the presence of a thin atmospheric boundary layer and thermal dissipation are developed here. An asymptotic theory for the resonant nonlinear interaction of long equatorial baroclinic and barotropic Rossby waves is derived in the presence of such dissipation. In this model, a self-consistent asymptotic derivation establishes that boundary layer flows are generated by meridional pressure gradients in the lower troposphere and give rise to degenerate equatorial Ekman friction. That is to say, the asymptotic model has the property that the dissipation matrix has one eigenvalue which is nearly zero: therefore the dynamics rapidly dissipates flows with pressure at the base of the troposphere and creates barotropic/baroclinic spin up/spin down. The simplified asymptotic equations for the amplitudes of the dissipative equatorial barotropic and baroclinic waves are studied by linear theory and integrated numerically. The results indicate that although the dissipation slightly weakens the tropics to midlatitude connection, strong localized wave packets are nonetheless able to exchange energy between barotropic and baroclinic waves on intraseasonal timescales in the presence of baroclinic mean shear. Interesting dissipation balanced wave-mean flow states are discovered through numerical simulations. In general, the boundary layer dissipation is very efficient for flows in which the barotropic and baroclinic components are of the same sign at the base of the free troposphere whereas the boundary layer dissipation is less efficient for flows whose barotropic and baroclinic components are of opposite sign at the base of the free troposphere.     

Inertial effects in an oceanic circulation model

Abstract

The flow in a mechanically driven thin barotropic rotating fluid system is analysed. The linear theory of Baker and Robinson (1969) is modified and extended into the non-linear regime.

An internal parameter, the “local Rossby number”, is indicative of the onset of nonlinear effects. If this parameter is 0(1) then inertial effects are as important as Coriolis accelerations in the interior of the transport-turning western boundary layer and both of its Ekman layers. The inertial effects in the Ekman layers, ignored in previous explorations of non-linear wind driven oceanic circulation, are retained here and calculated using an approximation of the Oseen type. The circulation problem is reduced to a system of scalar equations in only two independent variables; the system is valid for non-small local Rossby number provided only that the approximate total vorticity is positive.

To complete the solution for small Rossby number a boundary condition for the inertially induced transport is needed. It is found by examining the dynamics controlling this additional transport from the western boundary layer as the transport recirculates through the rest of the ocean basin. The strong constraint of total recirculation within the western boundary layer (zero net inertial transport) is derived.

The calculated primary inertial effects are in agreement with the observations of the laboratory model of Baker and Robinson (1969).

The analysis indicates the extent to which three-dimensional non-linear circulation can be reduced to a two dimensional problem.     

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.     

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.     

Natural integration of scalar fluxes from complex terrain

Large eddy simulations of turbulent flow and transport in the atmospheric boundary layer were conducted over heterogeneous sources of heat and water vapor to identify the blending properties of the turbulent mixing in an unstably stratified boundary layer. The numerical simulations show that the concept of blending in the ABL is in fact a useful one, even under convective conditions, for a range of surface conditions. Since the transport eddies that are responsible for the blending have sizes that are constrained by the boundary layer depth, and since the vertical motion is so important under the unstable density stratification studied here, we see that when the length scales of the source variability on the land surface become significantly greater than the ABL depth the blending is lost. In this case the source fields remain relatively uncoupled by the important eddy motion. However, for smaller surface length scales, the dynamic eddy motion couples the surface patches. Hence, there is good reason that the land surface exchange phenomenon would not be scale invariant over the entire range of scales. Because of the active role of temperature the effects of inhomogeneous surface sources of sensible heat persist higher into the ABL than do the effects of surface sources from more passive scalars, such as water vapor. Moreover, the mean fields of potential temperature and specific humidity blend at much lower heights than do the vertical turbulent flux fields of these two scalars. We propose a useful measure of blending efficiency for simulation studies and show how this bridges from the dynamics responsible for the blending to the horizontal homogeneity of scalar flux fields at measurement heights in the ABL.     

INTERNATIONALE GLAZIOLOGISCHE GRÖNLANDEXPEDITION, 1957–1959

Abstract

Equations are derived for vapor transport within a constant flux layer of the atmosphere. The physical model is based on the Reynolds analogy for fully turbulent flow, so that vapor and momentum flux are assumed similar. The shear stress is deduced from analysis of the wind profile by means of equations available for non-neutral conditions. Empirical expressions are proposed using the friction velocity rather than the wind velocity at a given level in order to take account of the stability of the air.     

On the magnitude of upwelling fluxes in shelf-break canyons

A hydrodynamic model is employed to derive the magnitude of on-shelf fluxes through a shelf-break canyon for a wide range of canyon sizes and ambient oceanic conditions. Predicted canyon-upwelling fluxes are of the order of 0.05–0.1 Sv (1 Sv=1 million m³/s), being several orders of magnitude greater than upslope fluxes in the bottom Ekman layer on the ambient continental slope. On the basis of ∼150 simulations conducted, a bulk formula of upwelling flux in a submarine canyon is derived. For typical conditions, the upwelling flux varies quadratically with forcing strength (speed of incident flow), linearly with canyon depth, and is inversely proportional to the buoyancy frequency of the density stratification inside the canyon. Other parameters such as density stratification above shelf-break depth and bottom friction are found to have minor influences on the resultant canyon-upwelling flux.     

A note on the heat transfer by the axisymmetric thermal convection in a rotating fluid annulus

Abstract

Some new measurements are presented of the axisymmetric heat transport in a differentially heated rotating fluid annulus. Both rigid and free upper surface cases are studied, for Prandtl numbers of 7 and 45, from low to high rotation rates. The rigid lid case is extended to high rotation rates by suppressing the baroclinic waves, that would normally develop at some intermediate rotation rate, with the use of sloping endwalls.

A parameter P is defined as the square of the ratio of the (non-rotating) thermal sidewall layer thickness to the Ekman layer thickness. For small P the heat transport remains unaffected by the rotation, but as P increases to order unity the Ekman layer becomes thin enough to inhibit the radial mass transport, and hence the heat flux. No explicit Prandtl number dependence is observed. Also this scaling allows the identification of the region in which the azimuthal velocity reaches its maximum. Direct comparisons are drawn with previous experimental and numerical results, which show what can be interpreted as an inhibiting effect of increasing curvature on the heat transport.