Ekman Boundary Layer Research Articles

An intermediate model for large-scale ocean circulation studies

Abstract The design and purposes of an intermediate model are discussed along with fundamentals of the model and results of numerical experiments. The main purposes of the model are reconstructions of the schemes of the ocean large-scale circulation and paleocirculation. For these problems numerical effectiveness is the key factor. A novel feature is the parameterization of the side wall Ekman boundary layers which was introduced to enable the use of geostrophy for calculating baroclinic velocity. This approach of connecting the side frictional layers with a non-viscous interior is not model-specific and can be used in any model employing geostrophy in the interior. The method can facilitate the no-flux and no-slip boundary conditions at the side walls in such models. Preliminary numerical experiments with simple basin geometry and idealized forcing aimed at a comparison with primitive equation and planetary geostrophic models are carried out. A direct comparison with the Geophysical Fluid Dynamics Laboratory (GFDL) primitive equation model was performed for a quantitative test of the proposed model. The results of the experiments are discussed in the context of the applicability of intermediate models for studying ocean climate dynamics.

Three-Dimensional Particle Tracking Velocimetry With Laser-Light Sheet Scannings

This paper describes three-dimensional particle tracking velocimetry (3D PTV), which enables us to obtain remarkably larger number of velocity vectors than previous techniques. Instead of the usual stereoscopic image recordings, the present 3D PTV visualizes an entire three-dimensional flow with the scanning laser-light sheets generated from a pair of optical scanners and the images are taken by a high-speed video system synchronized with the scannings. The digital image analyses to derive velocity components are based on the numerical procedure (Ushijima and Tanaka, 1994), in which several improvements have been made on the extraction of particle images, the determination of their positions, the derivation of velocity components and others. The present 3D PTV was applied to the rotating fluids, accompanied by Ekman boundary layers, and their complicated secondary flow patterns, as well as the primary circulations, are quantitatively captured.

The Modulation Equations for the Asymptotic Suction Velocity Profile and the Ekman Boundary Layer

In this article two types of flows are considered, the asymptotic suction velocity profile, which is a nearly parallel flow, and the Ekman boundary layer, which is a nonparallel flow. The modified Orr‐Sommerfeld equation for the asymptotic suction velocity profile, which is the linearized stability equation for this flow, is analyzed and it is shown to have finitely many eigenvalues. In addition, the Ekman boundary layer is considered and the modulation equation for this nonparallel flow is derived for the first time.

Three-Dimensional Particle Tracking Velocimetry with Laser-Light Sheet Scanning System.

A three-dimensional particle tracking velocimetry (3D PTV) technique has been developed, which enables us to obtain markedly larger number of velocity vectors than can be obtained by previous techniques. Instead of the usual stereoscopic image recordings, the present 3D PTV technique visualizes an entire three-dimensional flow with scanning laser-light sheets generated from a pair of optical scanners and the images are taken by a high-speed video recording system synchronized with the scannings. Digital image analyses for the derivation of velocity components are based on a numerical procedure, in which several improvements have been made in terms of the extraction of particle images, the determination of their positions, and the derivation of velocity components. The present 3D PTV technique was applied to rotating fluids accompanied by Ekman boundary layers and their complicated secondary flow patterns as well as the primary circulations are quantitatively captured.

Approximate analytical solution of the nonlinear problem of a flow over a thermally inhomogeneous underlying surface

The problem of the thermal circulation over the underlying surface, has been studied analytically for the case when the temperature of the underlying surface depends linearly on one of the horizontal coordinates. A horizontal pressure gradient is specified at the upper boundary of the medium horizontal layer (that has been rotating around the vertical axis) being under consideration, and this fact provides the existence of the background horizontal flow. The problem is essentially nonlinear, since, first, the heat advection, second, the square friction and the heat exchange at the underlying surface are taken into account. The solution depends on three non-dimensional parameters that are determined by the absolute values of the specified horizontal temperature and pressure gradients and by the angle between these vectors. In dependence on the values of the above mentioned parameters the solution properties may be very different. When the horizontal temperature gradient is absent, the solution is a generalization of the Ekman boundary layer classical theory for a case of the nonlinear friction against the underlying surface. The temperature and pressure fields essentially depend on the existence or absence of the background motion velocity component in the direction of the temperature horizontal gradient.

Transverse motion of a disk through a rotating viscous fluid

A thin rigid disk translates edgewise perpendicular to the rotation axis of an unbounded fluid undergoing solid-body rotation with angular velocity Ω. The disk face, with radius a, is perpendicular to the rotation axis. For arbitrary values of the Taylor number, [Tscr ] = Ωa2/ν, and in the limit of zero Reynolds number [Rscr ]e, the linearized viscous equations reduce to a complex-valued set of dual integral equations. The solution of these dual equations yields an exact representation for the velocity and pressure fields generated by the translating disk.For large rotation rates [Tscr ] [Gt ] 1, the O(1) disturbance velocity field is confined to a thin O([Tscr ]−1/2) boundary layer adjacent to the disk. Within this boundary layer, the flow field near the disk centre undergoes an Ekman spiral similar to that created by a nearly geostrophic flow adjacent to an infinite rigid plate. Additionally, flow within the boundary layer drives a weak O([Tscr ]−1/2) secondary flow which extends parallel to the rotation axis and into the far field. This flow consists of two counter-rotating columnar eddies, centred over the edge of the disk, which create a net in-plane flow at an angle of 45° to the translation direction of the disk. Fluid is transported axially toward/away from the disk within the core of these eddies. The hydrodynamic force (drag and lift) varies as O([Tscr ]1/2) for [Tscr ] [Gt ] 1; this scaling is consistent with the viscous stresses created in the Ekman boundary layer. Additionally, an approximate expression, suitable for all Taylor numbers, is given for the hydrodynamic force on a disk translating broadside along the rotation axis and edgewise transverse to the rotation axis.

Symmetry, sidewalls, and the transition to chaos in baroclinic systems

A high-resolution, quasi-geostrophic numerical model is utilized to examine two-layer baroclinic flow in a cylinder. Particular attention is given to the role of horizontal shear of the basic state induced by viscosity near the cylinder wall, and to the desymmetrization brought about by the cylindrical geometry, in the transition to baroclinic chaos. Solutions are computed for both f-plane and β-plane situations, and the results are compared to previous laboratory experiments. Agreement in the former case is found to be good, although the onset of chaos occurs at slightly lower forcing in the laboratory when its basic flow is prograde, and at higher forcing amplitude when the experimental basic azimuthal currents are retrograde. This suggests that the modest discrepancies may be attributable to computationally neglected ageostrophic effects in the interior fluid and Ekman boundary layers. When β ≠ 0, the numerical and laboratory results are in excellent agreement. The computational simulations indicate that the viscous sidewall boundary layer plays a pivotal role in the dynamics. Moreover, in contrast to previous studies performed in a periodic, rectilinear channel, the route to chaos is largely temporal and involves relatively few spatial modes. The reduction in symmetries upon going from f-plane channel to either f-plane or β-plane cylinder models leads to fewer secondary instabilities and fewer spatial modes that are active in the dynamics.

On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary layers

The Ekman boundary layer scalings associated with small oscillatory flows in a rapidly rotating system are well known to break down at the critical latitudes. The effect of these anomalous boundary regions on the interior flow has long been an issue of some concern. We argue, using analytical results derived in a model problem based on Stewartson's (1957) split-disk configuration, that these boundary regions spawn internal shear layers. These shear layers have their natural length scale of E1/3 and carry velocities only O(E1/6) smaller than the Rossby number – where E is the Ekman number – if the boundary is concave away from the fluid at the critical latitude. Otherwise, only a weaker shear layer is produced of length scale E1/5 containing velocities of O(E3/10) smaller than the Rossby number rather than the usual O(E1/5) value for viscous flows in the interior. We show how these layers can carry angular momentum by themselves or more significantly in combination with the interior inviscid flow, so as to influence the mean flow at leading order. These conclusions are then discussed in the context of a precessing oblate spheroid of fluid for which detailed experimental observations are available.

A search for free core nutation modes in VLBI nutation observations

In recent decades much effort has been devoted to searching observations for evidence of free core nutations predicted by theoretical models. Polar phototube data, tidal gravity observations and very long baseline nutation measurements have all been used. In this paper we report the first direct detection of the free core nutation in the frequency domain. A data set which consists of nutation series from 1100 very long baseline interferometry observing sessions from January 1984 to May 1992 is used in the spectral analysis. The results of the spectral analysis of the nutation series show that the period of the retrograde free core nutation mode is −431.0 (−425.5 to −436.7) solar days with an estimated amplitude of 0.19 ± 0.07 milliarcseconds. The recovered Q factor is 2000 ± 100. Based on an Ekman boundary layer on the surface of the core, this Q value implies a kinematic viscosity at the core-mantle boundary of 1.2 × 10 5 cm 2 s −1. A spectral peak at the predicted location of the prograde free core nutation is below the 95% confidence interval and cannot be considered significant.

On coupling between the Poincaré equation and the heat equation: non-slip boundary condition

In contrast to the well-known columnar convection mode in rapidly rotating spherical fluid systems, the viscous dissipation of the preferred convection mode at sufficiently small Prandtl numberPrtakes place only in the Ekman boundary layer. It follows that different types of velocity boundary condition lead to totally different forms of the asymptotic relationship between the Rayleigh numberRand the Ekman numberEfor the onset of convection. We extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition. Complete analytical solutions – the critical parameters for the onset of convection and the corresponding flow and temperature structure – are obtained and a new asymptotic relation betweenRandEis derived. While an explicit solution of the Ekman boundary-layer problem can be avoided by constructing a proper surface integral in the case of the stress-free boundary problem, an explicit solution of the spherical Ekman boundary layer is required and then obtained to derive the solvability condition for the present problem. In the corresponding numerical analysis, velocity and temperature are expanded in terms of spherical harmonics and Chebychev functions. Accurate numerical solutions are obtained in the asymptotic regime of smallEandPr, and comparison between the analytical and numerical solutions is then made to demonstrate that a satisfactory quantitative agreement between the analytical and numerical analyses is reached.

Ekman Boundary Layers and Energy Dissipation inRotating Drums During Spin‐Down Process

The Laser Doppler Velocimetry (LDV) is employed to investigate energy dissipation during a spin‐down process inside a rotating drum. The tracer/light sheet method is applied to observe flow patterns in the entire flow field from which the instantaneous, two‐dimensional velocity distribution and the formation and subsequent time wise variation of the Ekman boundary layer are determined. Results are synthesized to find the relationship between the Ekman boundary layer and the redistribution of secondary‐flow induced angular momentum. The fluid viscosity, drum size and speed of rotation are varied to determine their effects on both the Ekman boundary layer and energy dissipation during spin‐down process. The role of Ekman boundary layer in the reduction of rotating fluid motion is determined. Results from the study may be used to develop a method to achieve uniform mixing in an enclosed vessel.

On the Spatial Instability Modes of the Laminar Ekman Boundary Layer

On the Spatial Instability Modes of the Laminar Ekman Boundary Layer

Spin-up of a rapidly rotating heavy gas in a thermally insulated annulus

The linear spin-up problem for a rapidly rotating viscous diffusive ideal gas is considered in the limit of vanishing Ekman number E. Particular attention is given to gases having a large molecular weight. The gas is enclosed in a cylindrical annulus, with flat top and bottom walls, which is rotating around its axis of symmetry with rotation rate Ω. The walls of the container are adiabatic. In a rotating gas (of any molecular weight), the Ekman layers on adiabatic walls are weak, which implies that there is no distinct non-diffusive response of the gas outside the Ekman and Stewartson boundary layers on the timescale E−1/2Ω−1 for spin-up of a homogeneous fluid. For the case of adiabatic walls, it is shown that the spin-up mechanisms due to viscous diffusion and Ekman suction are, from a formal point of view, equally strong. Therefore, the gas will adjust to the increased rotation rate of the container on the diffusive timescale E−1Ω−1. However, if E1/3 [Lt ] γ – 1 [Lt ] 1 and M [siml ] 1, which characterizes rapidly rotating heavy gases (where γ is the ratio of specific heats of the gas and M the Mach number), it is shown that the gas spins up mainly by Ekman suction on the shorter timescale (γ–1)2E−1Ω−1. In such cases, the interior motion splits up into a non-diffusive part of geostrophic character and diffusive boundary layers of thickness (γ – 1) outside the Ekman and Stewartson layers. The motion approaches the steady state of rigid rotation algebraically instead of exponentially as is usually the case for spin-up.

Quantitative Visualization of Surface Flows on Rotating Disks

A novel scheme for the visualization of surface flows is developed. It utilizes the strong adhesion forces between micrometer-sized particles and solid surfaces to register the surface streaklines, or equivalently, streamlines for steady flows. Fluorescent particles are used to allow the spectral separation of particle fluorescence emission from morphology-related elastic light scattering from the surface. This scheme was employed to investigate the surface flow on rotating disks in a disk-drive-like environment. Trajectories of the streaklines were digitized and quantitatively analyzed using image processing for orientation and spatial distribution. The surface streaklines provide information about the boundary layers on the disk while the spreading angle of the jets in the self-pumped through-flow reveal details about the bulk flow outside the boundary layer. The spiral angle of the streaklines over a major portion of the disk surface was found to be in good agreement with the theory for laminar Ekman boundary layers. The spreading of the streaklines, reflecting the width of the self-pumped jets emanating from holes in the hub, was found to increase linearly with radius.

On coupling between the Poincaré equation and the heat equation

It has been suggested that in a rapidly rotating fluid sphere, convection would be in the form of slowly drifting columnar rolls with small azimuthal scale (Roberts 1968; Busse 1970). The results in this paper show that there are two alternative convection modes which are preferred at small Prandtl numbers. The two new convection modes are, at leading order, essentially those inertial oscillation modes of the Poincaré equation with the simplest structure along the axis of rotation and equatorial symmetry: one propagates in the eastward direction and the other propagates in the westward direction; both are trapped in the equatorial region. Buoyancy forces appear at next order to drive the oscillation against the weak effects of viscous damping. On the basis of the perturbation of solutions of the Poincaré equation, and taking into account the effects of the Ekman boundary layer, complete analytical convection solutions are obtained for the first time in rotating spherical fluid systems. The condition of an inner sphere exerts an insignificant influence on equatorially trapped convection. Full numerical analysis of the problem demonstrates a quantitative agreement between the analytical and numerical analyses.

A Note on Oscillatory Couette Flow in a Rotating System

An alternative solution is proposed for the oscillatory Ekman boundary layer flow bounded by two parallel plates in relative motion (Muzumder, 1991). The solution brings out among other things, the phenomenon of resonance which is of importance in rotating systems.

Experimental study in a rotating channel on the similarity law of tracer concentration distribution in the turbulent Ekman boundary layer

Concentration disrribution of airborne pollutants released into the atmosphere by large and elevated industrial sources or extended urban areas generally depends on the dynamical and thermal structure of the whole depth of the atmospheric planetary boundary layer (PBL). In order to be able to get meaningful predictions of pollutant dispersion from simple dispersion formulae using surface parameters as input data (like the Gaussian one), it is necessary to establish some reliable relationships between these surface parameters and the whole PBL's vertical structure. This paper presents the results of laboratory experiments in rotating hydraulic channel, aimed at investigating the influence of the Earth rotation on the dispersion of tracers which spread in a near-neutral Ekman turbulent PBL and at expressing the change of u and v components of mean wind with height in terms of rotation and turbulence scales typical of the atmospheric PBL. The results provide some evidence that in a near-neutral PBL the profiles of dilution ratio, adimensionalized by means of friction velocity u ∗ = √ u′w′ and Coriolis parameter f = 2Ω sinϕ , as a function of an adimensional height zf χ u ∗ , where χ is the von Karman constant, can be described by a universal function for a given Rossby number of the flow. This result confirms the theoretical expectation put forth by Wippermann and Yordanov in their study on the prediction of concentration patterns in a rotating turbulent PBL.

The influence of Ekman boundary layers on rotating convection

Abstract The influence of the boundary conditions on convective instability in rapidly (10−5 < E < 10−3) rotating spherical shells is investigated. For the purpose of comparison, both stress-free and rigid boundaries are studied at exactly the same parameter range of the problem. In the rigid boundary case an Ekman boundary layer is formed in middle latitudes where columnar convective rolls meet the outer spherical surface. There is also a boundary layer at the equatorial region of the inner sphere where the rolls are attached. When the Prandtl number Pr ≥ O(1), the Ekman boundary layer produced by the rigid boundaries has a destabilizing effect, the critical Reyleigh number being substantially reduced. The value of the critical azimuthal wavenumber and the drifting rate of the convection rolls are also significantly lower compared wilh the free boundary case. When the Prandtl number Pr < O(1), so thermal dissipation dominates over the viscous dissipation, the Ekman boundary layers exhibit a stabilizing e...

The connection between Ekman and Stewartson layers for a rotating disk

When a disk of finite radius and the surrounding medium rotate coaxially with slightly different angular velocities, a so-called Stewartson layer exists at the edge of the disk. The properties of this layer outside the boundary layer of the disk have been given in a previous publication. In the present paper it is shown how the radial flow of the Ekman boundary layer turns into the axial flow of the Stewartson layer. This happens in a region of which both the radial and axial dimensions are O(E1/2), where E is the Ekman number.

Model of two-dimensional vortex motion with an entrainment mechanism

A model for describing the vertically averaged vortex motions of an incompressible viscous fluid with an arbitrary vertical structure of the bottom Ekman boundary layer is proposed. An approach similar to that adopted in [1] is used: the second moments of the deviations from the average velocities required in order to close the vorticity equation are calculated by means of the Ekman solution for gradient flows, which makes it possible to take the integral bottom boundary layer effect into account. As a result, these terms lead to a specific form of nonlinear friction with a coefficient that depends on the vorticity of the average flow. In the particular case of a constant vertical turbulent transfer coefficient the inaccuracies of the model described in [1] can be eliminated. The generalized vorticity equation obtained has solutions of the vorticity spot type with a uniform internal vorticity distribution, which can be effectively investigated by means of appropriate algorithms [2]. The mechanism of entrainment at the vorticity front is illustrated with reference to the example of the evolution of vorticity spots. An exact solution of the problem of the evolution of an elliptic vortex (generalized Kirchhoff vortex), which in the case of fairly strong anticyclonic vorticity degenerates first into a line segment (vortex sheet) and then into a point vortex, is constructed. Equations describing the dynamics of an elliptic vorticity spot in an external field with a linear dependence of the velocity on the horizontal coordinates and generalizing the classical Chaplygin-Kida model [3, 4] are constructed.