M. R. Foster

Continuation or breakdown in tornado-like vortices

Laboratory experiments on swirling flows through tubes often exhibit a phenomenon called vortex breakdown, in which a bubble of reversed flow forms on the axis of swirl. Mager has identified breakdown with a discontinuity in solutions of the quasicylindrical flow equations. In this study we define a tornado-like vortex as one for which the axial velocity falls to zero for sufficiently large radius, and seek to clarify the conditions under which the solution of the quasi-cylindrical flow equations can be continued indefinitely or breaks down at a finite height. Vortex breakdown occurs as a dynamical process. Hence latent-heat effects, though doubtless important to the overall structure and maintenance of the tornado, are neglected here on the scale of the breakdown process. The results show that breakdown occurs when the effective axial momentum flux (flow force) is less than a critical value; for higher values of the flow force, the solution continues indefinitely, with Longs (1962) similarity solution as the terminal state. When applied to the conditions of the 1957 Dallas tornado, the computed breakdown location is in agreement with Hoeckers analysis of the observations.

The inviscid stability of a trailing line vortex

SummaryA finite difference method has been developed to study the inviscid stability of swirling flows to small non-axisymmetric disturbances. We apply the method to Batchelors trailing line vortex solution [3]. The method appears to be more efficient, and simpler to implement for this class of problem, than previously reported methods.ZusammenfassungZur Untersuchung der Stabilität von Wirbelströmungen bei nichtaxisymmetrischen kleinen Störungen ist eine Differenzemethode entwickelt worden. Sie wird auf Batchelors Lösung für die abgehende Wirbellinie angewendet. Die Methode erscheint für diese Art von Problemen gut geeignet und einfach in der Anwendung.

Analysis of the boundary conditions for a Hele--Shaw bubble

Effective boundary conditions are derived to be used with the classical Hele–Shaw equations in calculating the shape and motion of a Hele–Shaw bubble. The main assumptions of this analysis are that the displaced fluid wets the plates, and that the capillary number Ca and the ratio of gap width to characteristic bubble length e are both small. In a small region at the sides of the bubble, it is found that the thin‐film thickness scales with e2/5 Ca4/5, rather than the Ca2/3 scaling that is valid over most of the thin film above and below the bubble.

The inviscid stability of Long’s vortex

The inviscid stability of Long’s vortex, which is believed to model tornado‐like flows, is investigated using a finite‐difference method. Flows of this type are found to be generally unstable to small, short‐wave, helical disturbances, while no instabilities to axisymmetric modes have been found using this method. This instability provides a possible mechanism for the development of ’’suction vortices’’ in tornadoes.

The transient response of a contained rotating stratified fluid to impulsively started surface forcing

The transient response of a contained stratified rapidly rotating fluid to an impulsive surface stress has been studied theoretically and experimentally. The analysis predicts, and the experiments confirm, that for low values of the Burger number S the initial fluid adjustment within the E −½ Ω −1 timescale is characterized by a barotropic response in which the magnitude of the interior velocity is independent of depth. (Here E and Ω are the Ekman number and rotation rate respectively.) The period of the barotropic response decreases as S increases. For large S , the barotropic flow adjusts subsequently to a baroclinic flow within the E −½ Ω −1 scale, and during this later stage the excess and deficit in velocity in the lower and upper parts respectively of the fluid are removed. The baroclinic flow forced by the surface stress in these cases is thereby established in a timescale which is typically less than the spin-up time for a homogeneous fluid. The agreement between theory and experiment is shown to be qualitatively good, and the quantitative discrepancies observed between the predicted and measured interior velocities are discussed.

Spin-up of stratified rotating flows at large Schmidt number: experiment and theory

We consider the nonlinear spin-up/down of a rotating stratified fluid in a coni- cal container. An analysis of axisymmetric similarity-type solutions to the relevant boundary-layer problem, Duck, Foster & Hewitt (1997), has revealed three types of behaviour for this geometry. In general, the boundary layer evolves to either a steady state, or a gradually thickening boundary layer, or a finite-time singularity depending on the Schmidt number, the ratio of initial to final rotation rates, and the relative importance of rotation and stratification. In this paper we emphasize the experimental aspects of an investigation into the initial readjustment process. We make comparisons with the previously presented boundary-layer theory, showing good quantitative agreement for positive changes in the rotation rate of the container (relative to the initial rotation sense). The boundary-layer analysis is shown to be less successful in predicting the flow evolution for nonlinear decelerations of the container. We discuss the qualitative features of the spin-down experiments, which, in general, are dominated by non-axisymmetric effects. The experiments are conducted using salt-stratified solutions, which have a Schmidt number of approximately 700. The latter sections of the paper present some stability results for the steady boundary-layer states. A high degree of non-uniqueness is possible for the system of steady governing equations; however the experimental results are repeatable and stability calculations suggest that �higher branch� solutions are, in general, unstable. The eigenvalue spectrum arising from the linear stability analysis is shown to have both continuous and discrete components. Some analytical results concerning the continuous spectrum are presented in an appendix. A brief appendix completes the previous analysis of Duck, Foster & Hewitt (1997), presenting numerical evidence of a different form of finite-time singularity available for a more general boundary-layer problem.

Delayed separation in eastward, rotating flow on a β-plane

We consider the small-Rossby-number flow of a fluid past an obstacle in a coordinate frame in which the rotation rate varies linearly in the direction normal to the flow in a manner that models the variation of the Coriolis force for midlatitude planetary motions. The eastward flow is characterized by strong upstream divergence of the streamlines like that noted by Davies & Boyer (1982), and a similarly severe streamline convergence in the lee of the obstacle. Such a structure occurs for small values of the β-parameter that measures the importance of the lateral angular-velocity variation. In this parameter range, Rossby waves occur, but are confined to a narrow region in the lee of the object. The presence of these waves modifies the edge velocity ‘seen’ by the Stewartson quarter layer in such a way as to delay the onset of separation beyond what one might expect based on the analysis of Walker & Stewartson (1974) for a flow without beta-effect.

Instabilities in the spin-up of a rotating, stratified fluid

Theoretical analyses and laboratory experiments have been performed on the stability of a flow generated by the differential cyclonic corotation of a flat, rigid disk in a uniformly rotating, linearly stratified fluid contained within a cylindrical tank. The undisturbed fluid is stably stratified with salt (Schmidt number σ≈670) and the (vertical) axes of rotation of the disk and the fluid container are coincident. The theoretical analysis shows that when the interior flow satisfies gradient wind balance (or, alternatively, thermal wind balance), it is destabilized by the action of viscosity. In the experiments, the manifestation of the viscous overturning instability is seen to be the formation of steplike internal microstructures in the density field, observed as regularly spaced, curved ring-shaped sheets with associated localized sharp, vertical density gradients. A stability analysis of the flow shows that the instability criterion is dependent on local values of the vertical and radial gradients of ...

The flow caused by the differential rotation of a right circular cylindrical depression in one of two rapidly rotating parallel planes

The flow induced by the differential rotation of a cylindrical depression of radius a in one of two parallel rigid planes rapidly rotating about their common normal at speed Q is studied. A Taylor column bounded by the usual Stewartson layers arises, but the shear-layer structure is rather different from any previously studied. The Ei-layers (E = v/ωa 2 ) smooth the discontinuity in the geostrophic flow, but the way in which this is accomplished is related to the possible singu-larities of the E1/3-layer solutions. The fact that the 1/4-layer is partially free and partially attached to a vertical boundary accounts for the new joining conditions for the 1/4-layer. The drag on a right circular cylindrical bump in uniform flow is given in addition to some general comments on the applicability of these joining conditions to the motion of an axisymmetric object of quite general shape.

Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone

We consider a set of nonlinear boundary-layer equations that have been derived by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified fluid in a conical container. In contrast to the unsteady analysis of DFH, we re- strict attention to steady solutions and extend the previous discussion further by allowing the container to both co-rotate and counter-rotate relative to the contained swirling fluid. The system is governed by three parameters, which are essentially non- dimensional measures of the rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; however, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to this system proves to be rather complicated and shows many interesting features, including non-uniqueness, singular solutions and bifurcation phenomena. We present a broad description of the steady states with particular emphasis on boundaries in parameter space beyond which steady states cannot be continued. A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to non-axisymmetric boundary-layer states appearing as bifurcations of the axisymmetric solutions. In an Appendix we give details of an exact, non-axisymmetric solution to the Navier� Stokes equations (with axisymmetric boundary conditions) corresponding to the flow of homogeneous fluid above a rotating disk.