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.     

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.     

Inertial waves in spherical shells at low Ekman numbers

Inertial waves as oscillatory motions in rotating fluids generate internal shear layers at critical latitudes. We investigated the nonlinear interaction of inertial waves for global flows (3D flows) in dependence on the Ekman number. When the value of the Ekman number decreases, the influence of the Ekman layers to the flow pattern increases. Critical latitudes, the attractor flow pattern and certainly internal shear layers are observable mainly at greater values of the Ekman number. Although, with decreasing the Ekman number smaller flow structures become visible, nonlinear interactions in shear layers drive an axisymmetric flow whose amplitude diverges at the limit of the vanishing Ekman number. We show that this conclusion is valid not only for zonal wind driven by inertial modes but also for similarly driven global flows.     

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.     

Under-ice seiches

A closed, ice-covered water body containing water with homogeneous density distribution is considered. No-slip conditions are specified for flow velocity at the lower ice boundary and on the bed. Two variants of boundary conditions are considered on the side boundary: the boundary is either a solid vertical wall with a finite liquid depth or the liquid wedges out to zero depth values. Ice either is attached fast to the shore or is separated from it by an open-water zone. A basic fourth order equation is derived for the amplitude of ice oscillations. The introduction of friction results in the appearance of reduced depth. Eigenvalue problem is considered for evaluating seiche periods. For the case when the liquid wedges out at the shore, the basic equation becomes singular at water body boundaries. A lake with a longitudinal depth profile, which can be approximated by a parabola, is considered. The solution is found by the method of matched asymptotic expansions. In the inner domain, beyond the boundary layers, the equation is reduced to Legendre equation, which yields a new relationship for the spectrum of seiche oscillations both in the presence of ice and in an open lake with varying depth. Boundary layers appear at the margins of the lake, where the liquid wedges out; a solution is found for these layers.     

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.     

Ekman motion in shallow open sea in the presence of time-harmonic variation of water depth

The time variation of the wind-induced flow in a homogeneous unbounded sea region has been analytically investigated. The time-dependent Ekman solution in a homogeneous, shallow open sea has been further extended by taking into account the time variation in water depth which might be caused by either tidal motion or change in the mixed layer thickness. The solution approach taken in this study is based on the Galerkin-eigenfunction method originally developed by Heaps [1972. On the numerical solution of three-dimensional hydrodynamical equations for tides and storm surges. Memoires de la Societe Royale des Sciences de Liege Serie 6(2), 143–180]. A series of calculations have been made with emphasis on the influence of the time variation in water depth upon the build-up of Ekman spirals in the presence of oscillatory variations in water depth. It has been found that two oscillations contribute to the wind drift current, the inertial oscillation and the depth-variation-induced oscillation; the inertial oscillation decays with time, but the depth-variation-induced oscillation remains undamped despite the presence of bottom friction. The presence of time-harmonic variation produces peculiar forms of hodograph with a curled or circular pattern according to the angular frequency of the water depth variation.     

太湖风生流的三维数值模拟

建立了太湖三维风生流数值模型,并用差分法求解;垂直方向上采用了坐标变换技术,把任一节点的水深转换成无量纲水深,从而有效地消除了因风力作用造成的自由水面波动和湖底不规则的影响;水平面上采用锯齿网格处理,对于四周以闭边界为主的湖泊水域,显得比较合理。计算结果表明,湖泊风生流沿垂直、水平方向都有较大变化,流向上下、水平也并不一致,这是湖泊水流区别于其它水域水流之所在。计算模拟显示所建模型的有效性和可操作性。     

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.     

The equatorial dynamics of a deep homogeneous ocean

Abstract

The flow properties of an homogeneous fluid which is bounded by two concentric spheres and two meridional planes which intersect along a diameter of the spheres are investigated. The spheres rotate about this diameter with slightly different angular velocities. As in the axisymmetric case studied by Proudman (1956) and Stewartson (1966) the viscous terms in the equations of motion are important only in boundary layers on the spheres and on the cylinder C which circumscribes the inner sphere and which has generators parallel to the axis of rotation, provided the Ekman number E is small. In the inviscid region the velocities are independent of the coordinate measuring distance along the axis of rotation and are much weaker, by a factor 0(E ^½), than the velocities in the Ekman layer on the driving surface (outer sphere). (It is assumed that the reference frame is fixed in the slower rotating inner sphere.) If the separation of the spheres is small compared to their radii then the asymmetric circulation inside C is characterized by an intense jet along the western wall. Loss of fluid from this jet sustains the eastward and northward flow in the inviscid interior where motion is driven by the suction of the Ekman layer on the outer sphere. (Geophysical conventions have been adopted.) Outside C an intense current is present on the eastern, not western, wall while motion in the inviscid region is westward, and away from the axis of rotation. Though there is no transport across C in the inviscid region, the meridional transport of the Ekman layer on the outer sphere is continuous across C and increases, through suction, as the equator is approached until it drains into an eastward flowing equatorial current of width 0(E ^1/7). The eastern boundary current outside C and shear layers on C carry this fluid to the intersection of C and the western wall where it feeds the western boundary current inside C.

The relation between this study and the experiments of Baker and Robinson (1970) is discussed.     

Steady flow to a well near a stream with a leaky bed

We present an explicit analytic solution for steady, two-dimensional ground water flow to a well near a leaky streambed that penetrates the aquifer partially. Leakage from the stream is approximated as occurring along the centerline of the stream. The problem domain is infinite and pumping on one side of the stream induces flow on the other side. The solution includes the effects of uniform flow in the far field and a sloping hydraulic head in the stream. We use the solution to investigate the interaction between ground water and surface water in the stream, the effects of pumping on the opposite side of the stream, and the effects of the leaky streambed on the capture zone envelope of the well. We develop a relationship between parameters such that the pumping well will not capture water from the stream, or from the opposite side of the stream. When the discharge of the well is large enough to capture water from the stream, the shape of the capture zone envelope depends on flow conditions on the side of the stream opposite the well.     

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.     

Simulating a lake as a high-conductivity variably saturated porous medium

One approach for simulating ground water–lake interactions is to incorporate the lake into the ground water solution domain as a high-conductivity region. Previous studies have developed this approach using fully saturated models. This study extends this approach to variably saturated models, so that ground water–lake interactions may be more easily simulated with commonly used or public domain variably saturated codes that do not explicitly support coupled lake–water balance modeling. General guidelines are developed for the choices of saturated hydraulic conductivity and moisture retention and relative permeability curves for the lake region. When applied to an example ground water–lake system, model results are very similar to those from a model in which the lake is represented as a specified head boundary continuously updated by a lake mass balance. The high-conductivity region approach is most suitable for relatively simple geometries and lakes with slower and smaller fluctuations when the overall flow pattern and system fluxes, rather than the detailed flow pattern around the intersection of the lake and land surfaces, are of interest.     

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.     

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

We investigate, through both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. A particular emphasis is placed on a spherical-like cylinder whose diameter is nearly the same as its length. At this special aspect ratio, the strongest direct resonance occurs between the spatially simplest inertial mode and the precessional Poincaré forcing. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers. We also construct a nonlinear three-dimensional finite element model – which is validated against both the asymptotic solution and a constructed exact solution – for elucidating the nonlinear transition leading to disordered flow in the precessing spherical-like cylinder. Properties of both weakly and strongly precessing flows are investigated with the aid of a complete inertial-mode decomposition of the fully nonlinear solution. Despite a large effort being made, the well-known triadic resonance is not found in the precessing spherical-like cylinder. The energy contained in the precessionally forced inertial mode is primarily transferred, through nonlinear effects in the viscous boundary layers, to the geostrophic flow that becomes predominant when the precessional Poincaré force is sufficiently large. It is found that the nonlinear flow evolutes gradually and progressively from the laminar to disordered as the precessional force increases.     

Gravity currents down canyons: effects of rotation

The flow of dense water in a V-shaped laboratory-scale canyon is investigated by using a non-hydrostatic numerical ocean model with focus on the effects of rotation. By using a high-resolution model, a more detailed analysis of plumes investigated in the laboratory (Deep-Sea Res I 55:1021–1034 2008) for laminar flow is facilitated. The inflow rates are also increased to investigate plume structure for higher Reynolds numbers. With rotation, the plumes will lean to the side of the canyon, and there will be cross-canyon geostrophic currents and Ekman transports. In the present study, it is found that the cross-canyon velocities are approximately 5 % of the down-canyon velocities over the main body of the plume for the rotational case. With rotation, the flow of dense water through the body of the plume and into the plume head is reduced. The plume head becomes less developed, and the speed of advance of the head is reduced. Fluid parcels near the top of the plume will to a larger extent be left behind the faster flowing dense core of the plume in a rotating system. Near the top of the plume, the cross-canyon velocities change direction. Inside the plume, the cross-flow is up the side of the canyon, and above the interface to the ambient there is a compensating cross-flow down the side of the canyon. This means that parcels of fluid around the interface become separated. Parcels of fluid around the interface with small down-canyon velocity components and relative large cross-canyon components will follow a long helix-like path down the canyon. It is found that the entrainment coefficients often are larger in the rotational experiments than in corresponding experiments without rotation. The effects of rotation and higher inflow rates on the areal patterns of entrainment velocities are demonstrated. In particular, there are bands of higher entrainment velocities along the lateral edges of the plumes in the rotational cases.     

Ice cover,landscape setting,and geological framework of Lake Vostok,East Antarctica

Lake Vostok, located beneath more than 4 km of ice in the middle of East Antarctica, is a unique subglacial habitat and may contain microorganisms with distinct adaptations to such an extreme environment. Melting and freezing at the base of the ice sheet, which slowly flows across the lake, controls the flux of water, biota and sediment particles through the lake. The influx of thermal energy, however, is limited to contributions from below. Thus the geological origin of Lake Vostok is a critical boundary condition for the subglacial ecosystem. We present the first comprehensive maps of ice surface, ice thickness and subglacial topography around Lake Vostok. The ice flow across the lake and the landscape setting are closely linked to the geological origin of Lake Vostok. Our data show that Lake Vostok is located along a major geological boundary. Magnetic and gravity data are distinct east and west of the lake, as is the roughness of the subglacial topography. The physiographic setting of the lake has important consequences for the ice flow and thus the melting and freezing pattern and the lake’s circulation. Lake Vostok is a tectonically controlled subglacial lake. The tectonic processes provided the space for a unique habitat and recent minor tectonic activity could have the potential to introduce small, but significant amounts of thermal energy into the lake.     

太湖梅梁湾冬季湖流特征

2003年元月在盛行西北偏北风的情况下,对位于太湖北部的梅梁湾进行了面上湖流调查,发现梅梁湾湾口的湖流较为稳定,以向南流为主,且流速相对较大,最大达8cm/s,梅梁湾西岸有稳定的向北流,而从五里湖口至拖山附近的梅梁湾东线湖水由北向南流动,且在中部附近分为两支,一支向西,再流向北以补偿西岸的向北流,另一支扩散至整个梅梁湾南部,向南流经湾口进入太湖.在梅梁湾东北部,发现有弱辐合中心,该范围内Chl.a和TP、TN的含量明显高于周围水域.从所有点的垂直运动判断,梅梁湾水流以弱上升运动为主,大小为2cm/s以下.从水量平衡分析,以梅梁湾流入太湖为主要特征,水量补给主要来自于北部的五里湖和直湖港及武进港.     

A model study of spin-down and circulation in a cold water dome

A three-dimensional prognostic hydrodynamic model in cross sectional form is used to examine the influence of bottom friction, mixing and topography upon the spin-down and steady-state circulation in a cold water bottom-dome. Parameters characteristic of the Irish Sea or Yellow Sea cold water domes are used. In all calculations, motion is induced by specifying an initial temperature distribution characteristic of the dome, and an associated along frontal flow. The spin-down of the dome is found to be influenced by the coefficient of bottom friction, with a typical time scale of order 10 days, and in general to be independent of the chosen initial vertical profile of along frontal flow. However, in the case in which the along frontal flow is such that the near bed velocity is zero, then bottom stress is also zero, and there is no appreciable spin-down. Calculations showed that the formulation of viscosity and diffusivity had a greater effect upon the steady-state circulation than topography, suggesting that background mixing of tidal origin is important. The lack of topographic influence was due mainly to the formulation of the initial conditions which were taken to be independent of topography. The steady-state circulation was characterized by a cyclonic flow in the surface region, with an anti-cyclonic current near the bed, where frictional effects produced a bottom Ekman layer and an across frontal flow. This gave rise to vertical circulation cells in the frontal region of the dome with prevailing downwelling motion inside the dome. A detailed analysis of the dynamic balance of the various terms in the hydrodynamic equations yielded insight into the processes controlling the steady-state circulation in cold water domes. Responsible Editor: Phil Dyke