ALBERT

Description: This paper describes an experimental and theoretical study of the complicated disturbance (Taylor column) due to the slow relative motion between a spherical, or short cylindrical, rigid object and an incompressible fluid of low viscosity in which the object is immersed, when the motion of the object is that of steady revolution with angular speed Ω rad/see about an axis (the Z-axis) whose perpendicular distance from the centre of the object,$overline{R}$, is much greater than a typical linear dimension of the object,L, and the undisturbed fluid motion is one of steady rotation about the same axis with angular speed (Ω+Uϑ/R) and zero relative vorticity (i.e.d(UϑR)/dR= 0). It extends earlier experimental work on Taylor columns to systems of sufficiently large axial dimensions for Z variations in the disturbance pattern to be perceptible. Over the ranges of Rossby and Ekman numbers (based onL) covered by the experiments, namely ε = 1·89 × 10−3to 2·36 ×10−1and γ = 1·30 × 10−3to 2·03 × 10−2respectively, the axis of the Taylor column is found to trail in the downstream direction at a small angle ϕ = tan−1(Kε) to the line parallel to the Z-axis through the centre of the object, whereK= (1·54 ± 0·04) for a sphere. The variation with Z of the amplitude of the disturbance is roughly linear and the scale-length of this variation, Zc, is close toL/γ¼over the limited range of γ covered by the experiments.The experimental value ofKis remarkably close to the theoretical value derived by Prof. Lighthill in the appendix, where he applies his general linear theory of waves generated in a dispersive system by travelling forcing effects to the problem of describing a Taylor column at large distances from the moving object when the fluid is inviscid and unbounded.

Description: The dispersion relationship for plane hydromagnetic waves in a stratified rotating fluid (α) indicates that the well-known analogy between rotating fluids and stratified fluids in regard to their hydrodynamic behaviour does not extend to magnetohydrodynamic behaviour, and (b) lends credence to a certain conjecture made in a previous paper, namely that effects due to density stratification can be neglected when considering the dispersion relationship for free hydromagnetic oscillations of the Earth's core if the Brunt—Väisälä frequency is much less than twice the angular speed of the Earth's rotation.

Description: The properties of edge waves confined by the interaction of buoyancy and Coriolis forces to the vicinity of a rigid plane boundary in a rotating, stratified, electrically conducting fluid pervaded by a magnetic field are established in some simple cases. The background shear is taken to be zero, the basic Alfvén velocity V and Brunt–Väisälä frequency N are assumed uniform, and all dissipative effects are taken to be vanishingly small. It is shown that waves trapped against the bounding wall can occur only if V is parallel to the wall. When the basic rotation vector Ω is also parallel to the wall, the hydromagnetic edge waves have a higher frequency and smaller spatial extent perpendicular to the wall than their non-hydromagnetic counterparts, but more complex behaviour is found when Ω possesses a component normal to the wall. There are conditions under which edge waves may exist even when the basic density stratification is top-heavy (i.e. when N2 〈 0). © 1975, Cambridge University Press. All rights reserved.

Description: The structure transport properties and regimes of flow exhibited in a rotating fluid annulus, subject to internal heating and sidewall cooling, are studied both in the laboratory and in numerical simulations. The performance of the numerical model is verified quantitatively to within a few per cent in several cases by direct comparison with measurements in the laboratory of temperature and horizontal velocity fields in the axisymmetric and regular wave regimes. The basic azimuthal mean flow produced by this distribution of heat sources and sinks leads to strips of potential vorticity in which the radial gradient of potential vorticity changes sign in both the vertical and horizontal directions. From diagnosis of the energy budget of numerical simulations, the principal instability of the flow is shown to be predominantly baroclinic in nature, though with a non-negligible contribution towards the maintenance of the non-axisymmetric flow components from the barotropic wave-zonal flow interaction. The structure of the regime diagram for the internally heated baroclinic waves is shown to have some aspects in common with conventional wall-heated annulus waves, but the former shows no evidence for time-dependence in the form of 'amplitude vacillation'. Internally heated flows instead evidently prefer to make transitions between wavenumbers in the regular regime via a form of vortex merging and/or splitting, indicating a mixed vortex/wave character to the non-axisymmetric flows in this system. The transition towards irregular flow occurs via a form of wavenumber vacillation, also involving vortex splitting and merging events. Baroclinic eddies are shown to develop from an initial axisymmetric flow via a mixed sinuous/varicose instability, leading to the formation of detached vortices of the same sign as the ambient axisymmetric potential vorticity at that level, in a manner which resembles recent simulations of atmospheric baroclinic frontal instability and varicose barotropic instabilities. Dye tracer experiments confirm the mixed wave/vortex character of the equilibrated instabilities, and exhibit chaotic advection in time-dependent flows.

Description: An incompressible fluid fills a container of fixed shape and size and of uniform cross-section in the (x, y)-plane, themrigid side walls and the two rigid end walls being in contact with the fluid. Here (x, y, z) are the Cartesian co-ordinates of a general point in the frame of reference in which the container is stationary. Fluid is withdrawn from the container atQcm3/sec via certain permeable parts of the side walls and replaced at the same steady rate via other permeable parts of the side walls. As, by hypothesis, the vorticity of the entering and leaving fluid relative to the container is zero, the concomitant fluid motion within the container, Eulerian velocityu= −∇ϕ − ∇ ×A, is irrotational when the container is stationary in an inertial frame. The present paper is concerned with the effects onuof uniform rotation of the whole system with angular velocity Ω about thez-axis when the normal component ofuon the side walls is independent ofz.In the simplest conceivable case,D≡zu−zlis infinite (butD/Qremains finite). End effects are then negligible anduis everywhere independent ofz.The solenoidal component ofu, − ∇ ×A, corresponds tojgyres, one for each of thejirreducible sets of circuits across which the net flow of fluid does not vanish that can be drawn within them-ply connected region bounded by the side walls. While ∇ϕ, which satisfies ∇2ϕ = 0, depends onQbut not on Ω,jandv(the coefficient of kinematic viscosity), ∇ ×Adepends on all these quantities but vanishes identically whenjΩ = 0. WhenjΩ ≠ 0 butv→ 0, ∇2A+ 2Ω, the absolute vorticity, tends to zero everywhere except in certain singular regions near the bounding surfaces, where boundary layers form.End effects cannot be ignored whenDis finite. WhenDis independent ofxandyand equal toD0(say) and Ω is sufficiently large for the boundary layers on the end walls to be of the Ekman type, 95% thickness δ = 3(v/Ω)½(δ [Lt ]D0), the end effects that then arise are only confined to these boundary layers whenj= 0. Whenj≠ 0 boundary-layer suction influences the flow everywhere; thus ∇2Aand ∇ϕ (but not ∇ ×A) are reduced to zero in the main body of the fluid, the regions of non-zero ∇ϕ and ∇2Abeing the Ekman boundary layers on the end walls and boundary layers of another type, 95% thickness δs(typically greater than δ), on the side walls. A theoretical analysis of the structure of these boundary layers shows that non-linear effects, though unimportant in the end-wall boundary layers, can be significant and even dominant in the side-wall boundary layers. The analysis of an axisymmetric system, whose side walls are two coaxial cylinders, suggests an approximate expression for Δs. WhenDis not everywhere independent ofxandy,non-viscous end effects arise which produce relative vorticity in the main body of the fluid even whenj= 0.Experiments using a variety of source-sink distribution generally confirm the results of the theory, show that instabilities of various kinds may occur under certain circumstances, and suggest several promising lines for future work.

Description: A general theoretical relationship between the (three-dimensional) vcLacity field and density field is established for geostrophic flow of a non-homogeneous fluid. The practical value of the relationship – which reduces to the celebrated two-dimensional theorem due to Proudman and Taylor when the fluid is homogeneous – is illustrated by means of three examples of flows in rapidly rotating fluids, namely (i) baroclinic waves in laboratory systems and in the atmosphere, (ii) the gyroscopic ‘steering’ processes in Jupiter's atmosphere that are implied by the ‘Taylor column’ theory of the Great Red Spot, and (iii) certain striking properties of ocean currents revealed by recent observations in the Atlantic and Mediterranean. © 1971, Cambridge University Press. All rights reserved.

Description: The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω0 rad/s about a vertical axis, and stirred by rotating about the same axis at Ω1 rad/s a disk of radius a cm and thickness b’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω0, Ω1 and a, ranges of Rossby number, the modulus of ε ≡ (Ω1 + Ω0)/½ (Ω1 + Ω0), from 0·01 to 0·3, and Ekman number, E ≡ 2v/a2(Ω1 + Ω0), from 10−5 to 5 × 10−4 were attained. Although the apparatus was axisymmetric, only when |ε| did not exceed a certain critical value, |εT|, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |ε| 〉 |εT|, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when ε was positive, and of a blunt ellipse when ε was negative.The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when ε is positive, depend in relatively simple ways on ε and E. The dependence of|εT| on E can be expressed by the empirical relationship |εT| = AEn, where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing ε.