Ekman Boundary Layer Research Articles

Weakening the effect of boundaries: “diffusion-free” boundary conditions as a “do least harm” alternative to Neumann

In this note, we discuss a poorly known alternative boundary condition to the usual Neumann or “stress-free” boundary condition typically used to weaken boundary layers when diffusion is present but very small. These “diffusion-free” boundary conditions were first developed (as far as the authors know) in 1995 (Sureshkumar and Beris J. Non-Newtonian Fluid Mech. 60, 53–80, 1995) in viscoelastic flow modelling but are worthy of general consideration in other research areas. To illustrate their use, we solve two simple ODE problems and then treat a PDE problem – the inertial wave eigenvalue problem in a rotating cylinder, sphere and spherical shell for small but non-zero Ekman number E. Where inviscid inertial waves exist (cylinder and sphere), the viscous flows in the Ekman boundary layer are O ( E 1 / 2 ) weaker than for the corresponding stress-free layer and fully O ( E ) weaker than in a non-slip layer. These diffusion-free boundary conditions can also be used with hyperdiffusion and provide a systematic way to generate as many further boundary conditions as required. The weakening effect of this boundary condition could allow precious numerical resources to focus on other areas of the flow and thereby make smaller, more realistic values of diffusion accessible to simulations.

Ice base slope effects on the turbulent ice shelf-ocean boundary current

Abstract Efforts to parameterize ice shelf basal melting within climate models are limited by an incomplete understanding of the influence of ice base slope on the turbulent ice shelf-ocean boundary current (ISOBC). Here we examine the relationship between ice base slope, boundary current dynamics, and melt rate using 3-D, turbulence-permitting large-eddy simulations (LES) of an idealized ice shelf-ocean boundary current forced solely by melt-induced buoyancy. The range of simulated slopes (3-10%) is appropriate to the grounding zone of small Antarctic ice shelves and to the flanks of relatively wide ice base channels, and the initial conditions are representative of warm-cavity ocean conditions. In line with previous studies, the simulations feature the development of an Ekman boundary layer adjacent to the ice, overlaying a broad pycnocline. The time-averaged flow within the pycnocline is in thermal wind balance, with a mean shear that is only weakly dependent on the ice base slope angle α, resulting in a mean gradient Richardson number 〈Rig〉 that decreases approximately linearly with sinα. Combining this inverse relationship with a linear approximation to the density profile, we derive formulations for the friction velocity, thermal forcing, and melt rate in terms of slope angle and total buoyancy input. This theory predicts that melt rate varies like the square root of slope, which is consistent with the LES results and differs from a previously proposed linear trend. The derived scalings provide a potential framework for incorporating slope-dependence into parameterizations of mixing and melting at the base of ice shelves.

Ekman layer of rotating stratified viscous Boussinesq equations in rotation-dominant limit

The asymptotics of weak solutions to the Boussinesq equations with no-slip boundary and moderately ill-prepared data is investigated in rotation-dominant limit regime as the Rossby number, the Froude number and the vertical viscosity tend to zero simultaneously. The new ingredient of this paper is to give a first proof of the three scale singular limit coupled with Ekman boundary layer by introducing an asymptotic profile to the original system.

Convective instability of Ekman boundary layer flow of a Newtonian fluid over a stretchable rotating disk

ABSTRACT This paper examines a linear convective instability of an incompressible, Newtonian, viscous fluid rotating over a stretchy spinning disk in an Ekman boundary layer flow. First time in the literature, a numerical analysis of impacts of the stretching mechanism on the stability of the Ekman flow in the presence of the Coriolis force has been carried out, where the bottom disk is allowed to uniformly extend in radial direction. The von Kármán similarity transformations have been used to convert the governing equations of motion into a system of non-linear coupled ordinary differential equations (ODEs) in non-dimensional variables. Then velocity profiles of the flow have been obtained by numerically solving these dimensionless ODEs. The neutral stability curves were then obtained by performing a linear stability analysis for the convective mechanism by using the Chebyshev collocation method. Surface stretching has a globally stabilizing influence on both Type-I and Type-II instability modes of the Ekman flow, according to the stability curves. Then, the result established from the stability curves has been verified by calculating total kinetic energy change due to induced perturbations in the flow system. y.

Diurnally Varying Ekman Layer in a Rossby Wave

Abstract Weak but persistent synoptic-scale ascent may play a role in the initiation or maintenance of nocturnal convection over the central United States. An analytical model is used to explore the nocturnal low-level jets (NLLJ) and ascent that develop in an idealized diurnally varying frictional (Ekman) boundary layer in a neutrally stratified barotropic environment when the flow aloft is a zonally propagating Rossby wave. Steady-periodic solutions are obtained of the linearized Reynolds-averaged Boussinesq-approximated equations of motion on a beta plane with an eddy viscosity that is specified to increase abruptly at sunrise and decrease abruptly at sunset. Rayleigh damping terms are used to parameterize momentum loss due to radiation of inertia–gravity waves. The model-predicted vertical velocity is (approximately) proportional to the wavenumber and wave amplitude. There are two main modes of ascent in midlatitudes, an afternoon mode and a nocturnal mode. The latter arises as a gentle but persistent surge induced by the decrease of turbulence at sunset, the same mechanism that triggers inertial oscillations in the Blackadar theory of NLLJs. If the Rayleigh damping terms are omitted, the boundary layer depth becomes infinite at three critical latitudes, and the vertical velocity becomes infinite far above the ground at two of those latitudes. With the damping terms retained, the solution is well behaved. Peak daytime ascent in the model occurs progressively later in the afternoon at more southern locations (in the Northern Hemisphere) until the first (most northern) critical latitude is reached; south of that latitude the nocturnal mode is dominant.

Direct numerical simulation of sand particles transporting in the atmospheric Ekman boundary layer

Abstract This study presents a direct numerical simulation that investigates the transport mechanisms of sand particles in the neutrally stable atmospheric Ekman boundary layer (AEBL). The simulation models the AEBL in a half-channel flow, taking into account the Earth's rotation and adding a Coriolis term to the Navier–Stokes equations. The Lagrangian point-particle method with one-way coupling is used to track the sand particles. The main objective is to examine the impact of gravity and particle Stokes number on the sand particle dynamics. The results indicate that gravity has a significant effect on large-size sand particles, as seen in the mean and root-mean-square sand velocity profiles. The slip velocity profiles of sand particles relative to the surrounding air show that larger particles experience higher drag forces in the viscous sublayer, which hinders their forward movement. This effect is also amplified by gravity. Furthermore, the mean profiles of the streamwise and spanwise slip velocities exhibit distinct demarcations of the viscous sublayer, buffer layer and log-law region. The spatial and temporal Voronoï analysis reveals that gravity increases the clustering level of sand particles in the entire turbulent Ekman layer and reduces the time for the change of the Voronoï volume, particularly for large-size particles.

Bathtub vortex effect on Torricelli's law

When emptying a water filled container through a bottom hole, a ``bathtub vortex'' may appear and deform the free surface. Torricelli's law, which predicts the flow rate as a function of the water height, has been known for 400 years, but deviations are observed in the presence of such a vortex. This study focuses on the impact of the bathtub vortex on the emptying velocity. Through an experiment on the unsteady draining flow in a rotating tank, we show that Torricelli's law is modified as a function of the surface deformation, and that the draining time is mainly determined by a nondimensional parameter corresponding to the ratio of the outlet size to the Ekman boundary layer thickness.

Analysis of a stable bathtub vortex in a rotating container

Rotating flows with free-surface vortices can be found in many engineering applications, such as pump and turbine intakes, vessels, and nuclear reactors. The need to address rather different flow regions existing in such flows, such as Ekman and Stewartson layers and the line vortex zone, in a coupled manner, makes modeling of free-surface rotating flows very challenging. In this work, the flow field of a free-surface vortex, created in a rotating cylinder with a drain hole in its bottom, is investigated numerically and analytically. Above the drain hole of the cylinder, a free-surface vortex, accompanied by axial velocity, is created. This axial velocity profile is governed by the Ekman boundary layer far from the axis and by the drainage in its proximity. The experiments of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)] on the so-called bathtub vortex are numerically modeled with the finite volume method. The simulations are validated with the available measurements from the experiments. Using the simulation results, self-similar and non-self-similar models, describing the velocity fields in the Ekman boundary layer, are compared and tested. It is shown that the self-similar model is more accurate than the non-self-similar model. It is also demonstrated that the analytical model of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)], when modified as suggested in the present study, is capable of predicting the free-surface profile for low rotation rates. However, for high rotation rates, only the numerical simulation can predict the relation between the flow field within the liquid and the free-surface profile.

ANALYTIC SOLUTIONS FOR EKMAN BOUNDARY LAYER ON A POROUS PLATE

This study tends to explore a model of Ekman boundary layer on a porous Plate is considered and the governing differential equations are introduced. In this research, the researcher tries to find out steady state solutions for the velocity and temperature distributions are obtained. In this regard, the effect of buoyancy forces on velocity and temperature distributions are studied. The results of this study have indicated that the effects of buoyancy forces have significant contribution to the field of profiles.

Evaluation for the models of turbulent-diffusion term due to pressure-vorticity correlation in the Ekman boundary layer

Evaluation for the models of turbulent-diffusion term due to pressure-vorticity correlation in the Ekman boundary layer

Eddy viscosity profiles and currents characterization of the bottom Ekman layer for the Western Gulf of Mexico continental shelf

By analyzing a 4-year time series of currents and temperature from a set of moorings deployed over the Western Gulf of Mexico shelf, typical bottom Ekman layer parameters as the buoyancy frequency and bottom stress were estimated. Mean vertical structure and variance of the near-bottom currents are presented and the near-neutral stratification assumption is discussed considering what is expected from a turbulent bottom Ekman layer. Constant values for eddy viscosity (K) are obtained here from measured bottom Ekman spirals, which are further used to reproduce theoretical K profiles and the expected variability at each location. The results offer the first regional characterization of the bottom Ekman boundary layer, and are intended to serve in the calculation of mesoscale cross-shelf transport, as well as a reference to the turbulent parameters necessary in the development of benthic biogeochemical models.

Exploring stratification effects in stable Ekman boundary layers using a stochastic one-dimensional turbulence model

Abstract. Small-scale processes in atmospheric boundary layers are typically not resolved due to cost constraints but modeled based on physical relations with the resolved scales, neglecting expensive backscatter. This lack in modeling is addressed in the present study with the aid of the one-dimensional turbulence (ODT) model. ODT is applied as stand-alone column model to numerically investigate stratification effects in long-lived transient Ekman flows as canonical example of polar boundary layers by resolving turbulent winds and fluctuating temperature profiles on all relevant scales of the flow. We first calibrate the adjustable model parameters for neutral cases based on the surface drag law which yields slightly different optimal model set-ups for finite low and moderate Reynolds numbers. For the stably stratified cases, previously calibrated parameters are kept fixed and the model predictions are compared with various reference numerical simulations and also observations by an exploitation of boundary layer similarity. ODT reasonably captures the temporally developing flow for various prescribed stratification profiles, but fails to fully capture the near-surface laminarization by remaining longer in a fully developed turbulent state, which suggests preferential applicability to high-Reynolds-number flow regimes. Nevertheless, the model suggests that large near-surface turbulence scales are primarily affected by the developing stratification due to scale-selective buoyancy damping which agrees with the literature. The variability of the wind-turning angle represented by the ensemble of stratified cases simulated covers a wider range than reference reanalysis data. The present study suggests that the vertical-column ODT formulation that is highly resolved in space and time can help to accurately represent multi-physics boundary-layer and subgrid-scale processes, offering new opportunities for analysis of very stable polar boundary layer and atmospheric chemistry applications.

Retraction Statement: True Gravity in Atmospheric Ekman Layer Dynamics

Chu, P. C. True gravity in atmospheric Ekman layer dynamics. Journal of Geophysical Research: Atmospheres, 126, e2021JD035293. https://doi.org/10.1029/2021JD035293. The above article from Journal of Geophysical Research Atmospheres, published online on 29 September 2021, has been retracted by agreement between the journal Editor‐in‐Chief, Minghua Zhang; AGU; and Wiley Periodicals LLC. The retraction has been agreed because the conclusions of the paper were found to be incorrect, since they depend on the choice of the coordinate system that does not apply to practical application of the theory of atmospheric Ekman boundary layer. The author disagrees with this assessment.

The influence of the land–sea breeze on coastal upwelling systems: locally forced vs internal wave vertical mixing and implications for thermal fronts

Land–sea breeze forcing near a land boundary drives both a locally forced response and an associated offshore propagating internal wave response, the effects of which can be difficult to separate. These processes enhance vertical mixing near the critical latitude for diurnal-inertial resonance (30° N/S), and are a feature of all four major eastern boundary upwelling systems. Here, we employ 1D- and 2D-vertical model configurations forced by a land–sea breeze to quantify the relative contributions of the locally forced and internal wave responses to surface currents and vertical mixing, and test sensitivity to latitude and bottom slope. We further include a sub-inertial alongshore wind to consider the role of the land–sea breeze in the context of upwelling systems. At the critical latitude, the internal waves generated via thermocline pumping near the land boundary are evanescent (in agreement with theory) and largely absent ∼50 km offshore. The internal waves are shown to contribute to vertical mixing, which can be ∼20% greater than that due to the forced response alone, further deepening the surface Ekman boundary layer. This deepening reduces the sub-inertial offshore advection of surface waters, thereby retaining the upwelling front closer to the land boundary and driving a net warming of the nearshore surface waters. Cross-shore horizontal oscillations of the upwelling front generated by the land–sea breeze drive strong diurnal variability in sea surface temperature, in agreement with observations from a cross-shore mooring array in the southern Benguela (∼32.3° S).

Anisotropy and stratification effects in the dynamics of fast rotating compressible fluids

The primary goal of this paper is to develop robust methods to handle two ubiquitous features appearing in the modeling of geophysical flows: (i) the anisotropy of the viscous stress tensor and (ii) stratification effects.We focus on the barotropic Navier–Stokes equations with Coriolis and gravitational forces. Two results are the main contributions of the paper. Firstly, we establish a local well-posedness result for finite-energy solutions, via a maximal regularity approach. This method allows us to circumvent the use of the effective viscous flux, which plays a key role in the weak solutions theories of Lions–Feireisl and Hoff, but seems to be restricted to isotropic viscous stress tensors. Moreover, our approach is sturdy enough to take into account nonconstant reference density states; this is crucial when dealing with stratification effects. Secondly, we study the structure of the solutions to the previous model in the regime when the Rossby, Mach and Froude numbers are of the same order of magnitude. We prove an error estimate on the relative entropy between actual solutions and their approximation by a large-scale quasi-geostrophic flow supplemented with Ekman boundary layers. Our analysis holds for a large class of barotropic pressure laws.

Stability of conducting fluid flow between coaxial cylinders under thermal gradient and axial magnetic field

Numerical simulations were performed to investigate the bifurcation in swirling flow between two coaxial vertical cylinders produced by the thermal gradient. The suppressed effects of an axial magnetic field on both vortex breakdown and fluid layers are analyzed. The governing Navier-Stokes, temperature, and potential equations are solved by using the finite-volume method. A conducting fluid is placed in the gap between two coaxial cylinders characterized by a small Prandtl number (Pr = 0.032). Three annular gaps were R = 0.7, 0.8, and 0.9 compared in terms of flow stability, and heat transfer rates. The combination of aspect ratio =1.5 and Reynolds number, Re=1500 is the detailed case in this study. In the hydrodynamic case, vortex breakdown takes place near the inner cylinder due to the increased pumping action of the Ekman boundary layer. In addition, the competition between buoyancy and viscous forces develops a fluid layered structure. It is shown that the onset of the oscillatory instability set in by increasing Reynolds number to the critical value. The results show that with an intensified magnetic field, the vortex breakdown disappears, the number of fluid layers will be reduced and the onset of the oscillatory instability will be retarded. Stability diagrams corresponding to the limits of transition from the multiple fluid layers to the one fluid layer are obtained.

Stochastic modeling of transient neutral and stably‐stratified Ekman boundary layers

AbstractNeutral and stably‐stratified Ekman boundary layers (EBLs) are numerically investigated with a stochastic one‐dimensional turbulence (ODT) model. EBLs achieve the bulk‐surface coupling in Earth's atmosphere. They are numerically challenging due to transient and non‐universal turbulence properties even at small scales. ODT addresses this problem by distinguishing turbulent‐advective from molecular‐diffusive transport processes for a vertical column along which all relevant scales of the flow are resolved. We demonstrate the model's capabilities for economical, accurate, and stratification regime independent simulation of EBLs for the wind‐turning angle. ODT reproduces and extrapolates reference direct numerical simulation results consistent with observations. We conclude that ODT may be useful for modeling of atmospheric surface layers.

Force balance in rapidly rotating Rayleigh-Bénard convection.

The force balance of rotating Rayleigh-Bénard convection regimes is investigated using direct numerical simulation on a laterally periodic domain, vertically bounded by no-slip walls. We provide a comprehensive view of the interplay between governing forces both in the bulk and near the walls. We observe, as in other prior studies, regimes of cells, convective Taylor columns, plumes, large-scale vortices (LSVs) and rotation-affected convection. Regimes of rapidly rotating convection are dominated by geostrophy, the balance between Coriolis and pressure-gradient forces. The higher-order interplay between inertial, viscous and buoyancy forces defines a subdominant balance that distinguishes the geostrophic states. It consists of viscous and buoyancy forces for cells and columns, inertial, viscous and buoyancy forces for plumes, and inertial forces for LSVs. In rotation-affected convection, inertial and pressure-gradient forces constitute the dominant balance; Coriolis, viscous and buoyancy forces form the subdominant balance. Near the walls, in geostrophic regimes, force magnitudes are larger than in the bulk; buoyancy contributes little to the subdominant balance of cells, columns and plumes. Increased force magnitudes denote increased ageostrophy near the walls. Nonetheless, the flow is geostrophic as the bulk. Inertia becomes increasingly more important compared to the bulk, and enters the subdominant balance of columns. As the bulk, the near-wall flow loses rotational constraint in rotation-affected convection. Consequently, kinetic boundary layers deviate from the expected behaviour from linear Ekman boundary layer theory. Our findings elucidate the dynamical balances of rotating thermal convection under realistic top/bottom boundary conditions, relevant to laboratory settings and large-scale natural flows.

Stability effect of an axial magnetic field on fluid flow bifurcation between coaxial cylinders

Numerical simulations aim to investigate the bifurcation caused by swirling flow between two coaxial vertical cylinders, and the fluid layers produced by the thermal gradient. The stability of both bifurcation and fluid layers by an axial magnetic field is analyzed. The finite-volume method is used to solve the governing Navier–Stokes, temperature and potential equations. A conducting viscous fluid characterized by a small Prandtl number [Formula: see text] is placed in the gap between two coaxial cylinders. The combination of aspect ratio, [Formula: see text] and Reynolds number, [Formula: see text] for three annular gaps ([Formula: see text] and [Formula: see text]) is compared in terms of flow stability, and heat transfer rates. Without a magnetic field, the vortex breakdown takes place near the inner cylinder due to the increased pumping action of the Ekman boundary layer. Fluid layered structures are developed by the competition between buoyancy and viscous forces. The increase in the magnitude of the magnetic field retarders the onset of the oscillatory instability caused by the disappearance of the vortex breakdown and reduces the number of fluid layers. The limits in which a vortex breakdown bubble manifests and the limits of transition from the multiple fluid layers to the single fluid layer are established.

Magnetohydrodynamic effect on vortex breakdown zones in coaxial cylinders

This research aims to investigate the behavior of vortex breakdown in vertical annuli filled with a conducting viscous fluid under the axial magnetic field. Three annular gaps and various ratios of annular height to the radius of the outer cylinder are considered. The pumping action of the Ekman boundary layer produced by rotation of the bottom disk sets up a secondary circulation along the meridional plane of the coaxial cylinders. For certain combinations of Reynolds numbers and aspect ratios, the vortex breakdown bubble occurred near to sidewall of the inner cylinder. The governing Navier–Stokes and potential equations are solved by using the finite-volume method, where a steady solution exists. The effects of the magnetic field, annular gaps, and aspect ratios on the three-dimensional structure of the recirculation zones and the position of vortex breakdown are developed. The results of the annular gap R=0.9 are compared with those Escudier’s stability diagram when R=1.0. The results obtained showed that the vortex breakdown is suppressed beyond the magnitude of the magnetic field exceeds a critical value and before the suppression of the electric vortex occurs. The stability diagrams (Re–H/Ro) corresponding to which a vortex breakdown bubble occurred near to the sidewall of the inner cylinder are obtained.