A unified linear theory of homogeneous and stratified rotating fluids

1967 ◽  
Vol 29 (3) ◽  
pp. 609-621 ◽  
Author(s):  
V. Barcilon ◽  
J. Pedlosky
Keyword(s):  
Side Wall ◽  
Extreme Case ◽  
Ekman Layer ◽  
Rotating Fluids ◽  
Homogeneous Fluid ◽  
Hybrid Features ◽  
Wall Boundary Layer ◽  
Ekman Layers ◽  
Buoyancy Forces ◽  
The One

A unified picture of the linear dynamics of rotating fluids with given arbitrary stratification is presented. The range of stratification which lies outside the region of validity of both the theories of homogeneous fluids, $\sigma S < E^{\frac{2}{3}}$ and the strongly stratified fluids, σS > E½, is studied, where σS = vαgΔT/κΩ2L and E =v/ΩL2. The transition from one dynamics to the other is elucidated by a detailed study of the intermediate region E2/3 < σS < E½. It is shown that, within this intermediate stratification range, the dynamics differs from that of either extreme case, except in the neighbourhood of horizontal boundaries where Ekman layers are present. In particular the side wall boundary layer exhibits a triple structure and is made up of (i) a buoyancy sublayer of thickness (σS)−1/4E½ in which the viscous and buoyancy forces balance, (ii) an intermediate hydrostatic, baroclinic layer of thickness (σS)½ and (iii) an outer E¼-layer which is analogous to the one occurring in a homogeneous fluid. In the interior, the dynamics is mainly controlled by Ekman-layer suction, but displays hybrid features; in particular the dynamical fields can be decomposed into a ‘homogeneous component’ which satisfies the Taylor-Proudman theorem, and into a ‘stratified component’ which is baroclinic and which satisfies a thermal wind relation. In all regions the structure of the flow is displayed in detail.

The spin-up of a homogeneous fluid bounded below by a permeable medium

1970 ◽  
Vol 40 (2) ◽  
pp. 225-239 ◽  
Author(s):  
John Kroll ◽  
George Veronis
Keyword(s):  
Side Wall ◽  
Ekman Layer ◽  
Maximum Difference ◽  
Rotating Fluid ◽  
Homogeneous Fluid ◽  
Wall Boundary Layer ◽  
Wall Boundary ◽  
Bounded Below ◽  
Unbounded Fluid ◽  
Theory And Experiment

The spin-up of a homogeneous rotating fluid bounded at the top and/or bottom by a permeable medium has been proposed by Bretherton & Spiegel (1968) as a model for the spin-up in natural flows where turbulent processes transmit the direct effect of the boundaries deeper into the fluid than does the laminar Ekman layer. The theoretical analysis for the spin-up of a laterally unbounded fluid bounded by a permeable medium below is presented here. In addition, an experimental study of the process is presented. Theory and experiment agree reasonably well with a maximum difference of about 8% in the predicted and measured spin-up times. The effects of the side-wall boundary have been studied theoretically by Howard (1969). Experimental observations in the side-wall boundary layer confirm qualitatively the results of Howard's theory.

Propagation of the Velocity Shear Front in Spin-up From Rest in a Cut-Cone

1995 ◽  
Vol 117 (1) ◽  
pp. 58-61 ◽  
Author(s):  
Jae Won Kim ◽  
Jae Min Hyun
Keyword(s):  
Side Wall ◽  
Azimuthal Velocity ◽  
Processing Technique ◽  
Velocity Shear ◽  
Image Processing Technique ◽  
Incline Angle ◽  
Homogeneous Fluid ◽  
Experimental Apparatus ◽  
Ekman Layers ◽  
Shear Front

The behavior of the dominant azimuthal velocity field during spin-up from rest of a homogeneous fluid in a cut-cone is investigated. The fundamental mechanism of spin-up process is recapitulated. In line with the classical flow model of Wedemeyer, the importance of the meridional circulation, driven by the Ekman layers, is stressed. The experimental apparatus, together with the image processing technique of the visualized flow data, is described. The reliability and accuracy of this experimental method are validated by performing parallel measurements using an LDV system. The experimental results clearly indicate that the azimuthal velocity shear front propagates faster as the incline angle of the side wall decreases. In the rotating zone of the interior, the azimuthal velocities are larger in magnitude in a cut-cone than in a circular cylinder of comparable size. Plausible physical explanations are offered, and the experimental observations are supportive of these physical arguments.

The axisymmetric convective regime for a rigidly bounded rotating annulus

1968 ◽  
Vol 32 (4) ◽  
pp. 625-655 ◽  
Author(s):  
Michael E. Mcintyre
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Axisymmetric Flow ◽  
Side Wall ◽  
Boundary Layer Equations ◽  
Wall Boundary Layer ◽  
Ekman Layers ◽  
Interior Flow ◽  
Side Walls ◽  
Annular Container

The axisymmetric flow of liquid in a rigidly bounded annular container of heightH, rotating with angular velocity Ω and subjected to a temperature difference ΔTbetween its vertical cylindrical perfectly conducting side walls, whose distance apart isL, is analysed in the boundary-layer approximation for small Ekman numberv/2ΩL2, withgαΔTHv/4Ω2L2K∼ 1. The heat transfer across the annulus is then convection-dominated, as is characteristic of the experimentally observed ‘upper symmetric regime’. The Prandtl numberv/kis assumed large, andHis restricted to be less than about 2L. The side wall boundary-layer equations are the same as in (non-rotating) convection in a rectangular cavity. The horizontal boundary layers are Ekman layers and the four boundary layers, together with certain spatialaveragesin the interior, are determined independently of the interior flow details. The determination of the latter comprises a ‘secondary’ problem in which viscosity and heat conduction are important throughout the interior; the meridional streamlines are not necessarily parallel to the isotherms. The secondary problem is discussed qualitatively but not solved. The theory agrees fairly well with an available numerical experiment in the upper symmetric regime, forv/k[bumpe ] 7, after finite-Ekmannumber effects such as finite boundary-layer thickness are allowed for heuris-tically.

Linear theory of rotating stratified fluid motions

1967 ◽  
Vol 29 (1) ◽  
pp. 1-16 ◽  
Author(s):  
V. Barcilon ◽  
J. Pedlosky
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Linear Theory ◽  
Stable Stratification ◽  
Ekman Layer ◽  
Stratified Fluid ◽  
Rotating Fluids ◽  
Ekman Layers ◽  
Steady Motions ◽  
Rotating Stratified Fluid

A linear theory for steady motions in a rotating stratified fluid is presented, valid under the assumption that ε < E, where ε and E are respectively the Rossby and Ekman numbers. The fact that the stable stratification inhibits vertical motions has important consequences and many features of the dynamics of homogeneous rotating fluids are no longer present. For instance, in addition to the absence of the Taylor-Proudman constraint, it is found that Ekman layer suction no longer controls the interior dynamics. In fact, the Ekman layers themselves are frequently absent. Furthermore, the vertical Stewartson boundary layers are replaced by a new kind of boundary layer whose structure is characteristic of rotating stratified fluids. The interior dynamics are found to be controlled by dissipative processes.

scholarly journals Reviving the genitive. Prescription and practice in the Netherlands (1770–1840)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Andreas Krogull ◽  
Gijsbert Rutten
Keyword(s):  
Language Use ◽  
Extreme Case ◽  
Spoken Language ◽  
Interesting Case ◽  
Research Literature ◽  
National Language ◽  
Prepositional Phrases ◽  
Historical Sociolinguistics ◽  
The One ◽  
Genitive Case

AbstractHistorical metalinguistic discourse is known to often prescribe linguistic variants that are not very frequent in actual language use, and to proscribe frequent variants. Infrequent variants that are promoted through prescription can be innovations, but they can also be conservative forms that have already largely vanished from the spoken language and are now also disappearing in writing. An extreme case in point is the genitive case in Dutch. This has been in decline in usage from at least the thirteenth century onwards, gradually giving way to analytical alternatives such as prepositional phrases. In the grammatical tradition, however, a preference for the genitive case was maintained for centuries. When ‘standard’ Dutch is officially codified in 1805 in the context of a national language policy, the genitive case is again strongly preferred, still aiming to ‘revive’ the synthetic forms. The striking discrepancy between metalinguistic discourse on the one hand, and developments in language use on the other, make the genitive case in Dutch an interesting case for historical sociolinguistics. In this paper, we tackle various issues raised by the research literature, such as the importance of genre differences as well as variation within particular genres, through a detailed corpus-based analysis of the influence of prescription on language practices in eighteenth- and nineteenth-century Dutch.

Stratified Ekman layers evolving under a finite-time stabilizing buoyancy flux

2018 ◽  
Vol 840 ◽  
pp. 266-290 ◽  
Author(s):  
S. M. Iman Gohari ◽  
Sutanu Sarkar
Keyword(s):  
Finite Time ◽  
Richardson Number ◽  
Ekman Layer ◽  
Buoyancy Flux ◽  
Late Time ◽  
Near Surface ◽  
Obukhov Length ◽  
Low Level ◽  
Ekman Layers ◽  
Low Level Jet

Stratified flow in nocturnal boundary layers is studied using direct numerical simulation (DNS) of the Ekman layer, a model problem that is useful to understand atmospheric boundary-layer (ABL) turbulence. A stabilizing buoyancy flux is applied for a finite time to a neutral Ekman layer. Based on previous studies and the simulations conducted here, the choice of $L_{\mathit{cri}}^{+}=Lu_{\ast }/\unicode[STIX]{x1D708}\approx 700$ ($L$ is the Obukhov length scale and $u_{\ast }$ is the friction velocity) provides a cooling flux that is sufficiently strong to cause the initial collapse of turbulence. The turbulent kinetic energy decays over a time scale of $4.06L/u_{\ast }$ during the collapse. The simulations suggest that imposing $L_{\mathit{cri}}^{+}\approx 700$ on the neutral Ekman layer results in turbulence collapse during the initial transient, independent of Reynolds number, $Re_{\ast }$. However, the long-time state of the flow, i.e. turbulent with spatial intermittency or non-turbulent, is found to depend on the initial value of $Re_{\ast }$ since the cooling flux and resultant stratification increase with $Re_{\ast }$ for a given $L^{+}$. The lower-$Re_{\ast }$ cases have sustained turbulence with shear and stratification profiles that evolve in a manner such that the gradient Richardson number, $Ri_{g}$, in the near-surface layer, including the low-level jet, remains subcritical. The highest $Re_{\ast }$ case has supercritical $Ri_{g}$ in the low-level jet and turbulence does not recover. A theoretical discussion is performed to infer that the bulk Richardson number, $Ri_{b}$, is more suitable than $L^{+}$ to determine the fate of stratified Ekman layers at late time. DNS results support the implications of $Ri_{b}$ for the effect of initial $Re_{\ast }$ and $L^{+}$ on the flow.

Coriolis-force attenuation of blocking in a stratified flow

1979 ◽  
Vol 94 (4) ◽  
pp. 711-727 ◽  
Author(s):  
M. R. Foster
Keyword(s):  
Boundary Layers ◽  
Detailed Examination ◽  
Ekman Layer ◽  
Ekman Pumping ◽  
Ekman Number ◽  
Coriolis Forces ◽  
Ekman Layers ◽  
The Face ◽  
Vorticity Generation ◽  
Upstream Flow

Even very small Coriolis forces are shown to alter significantly the nature of the upstream wake of an object in slow (small Froude number) translation through a non-diffusive stratified fluid. If the Ekman number is of order one, the far upstream extent of the wake is reduced. If the fluid rotation is sufficient to make the Ekman number small, the contraction of the wake is much greater. We study a particular case in detail; the Ekman number is small enough to make horizontal boundary layers Ekman layers. In this case, the wake is confined to the vicinity of the object, the upstream flow arising from a combination of Ekman pumping and baroclinic vorticity generation. The upstream flow is described by an eigenfunction whose amplitude is dependent on object geometry. If the object is a semi-infinite rectangular parallelepiped, that amplitude is determined by detailed examination of the shear layer at the face of the parallelepiped and its interaction with the Ekman layer on the top surface of the object

scholarly journals An Inertial Model of the Interaction of Ekman Layers and Planetary Islands

2013 ◽  
Vol 43 (7) ◽  
pp. 1398-1406
Author(s):  
Joseph Pedlosky
Keyword(s):  
Nonlinear Model ◽  
Model Problem ◽  
Three Dimensional ◽  
Ekman Layer ◽  
Ekman Transport ◽  
Complex Flow ◽  
Ekman Layers ◽  
Principal Effect ◽  
Quasigeostrophic Model ◽  
Dissipative Model

Abstract An adiabatic, inertial, and quasigeostrophic model is used to discuss the interaction of surface Ekman transport with an island. The theory extends the recent work of Spall and Pedlosky to include an analytical and nonlinear model for the interaction. The presence of an island that interrupts a uniform Ekman layer transport raises interesting questions about the resulting circulation. The consequential upwelling around the island can lead to a local intake of fluid from the geostrophic region beneath the Ekman layer or to a more complex flow around the island in which the fluid entering the Ekman layer on one portion of the island's perimeter is replaced by a flow along the island's boundary from a downwelling region located elsewhere on the island. This becomes especially pertinent when the flow is quasigeostrophic and adiabatic. The oncoming geostrophic flow that balances the offshore Ekman flux is largely diverted around the island, and the Ekman flux is fed by a transfer of fluid from the western to the eastern side of the island. As opposed to the linear, dissipative model described earlier, this transfer takes place even in the absence of a topographic skirt around the island. The principal effect of topography in the inertial model is to introduce an asymmetry between the circulation on the northern and southern sides of the island. The quasigeostrophic model allows a simple solution to the model problem with topography and yet the resulting three-dimensional circulation is surprisingly complex with streamlines connecting each side of the island.

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