G. J. F. van Heijst

An experimental study of unstable barotropic vortices in a rotating fluid

Laboratory experiments on barotropic vortices in a rotating fluid revealed that the instability behaviour of cyclonic and anticyclonic vortices is remarkably different. Depending on its initial vorticity distribution, the cyclonic vortex has in a number of experiments been observed to be unstable to wavenumber-2 perturbations, leading to the gradual formation of a stable tripolar vortex structure. This tripole consists of an elongated cyclonic core vortex adjoined by two anticyclonic satellite vortices. In contrast, the anticyclonic vortex shows a rather explosive instability behaviour, in the sense that it is observed to immediately split up into two dipoles. Under somewhat different circumstances the higher-order mode-3 instability is observed, in which the anticyclonic core has a triangular shape, with three smaller cyclonic satellite vortices a t its sides. A modified version of Rayleigh’s instability criterion offers a qualitative explanation for this apparent difference between unstable cyclonic and anticyclonic vortices.

On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid

This paper describes exact solutions of two-dimensional vortex structures that were published by Chaplygin (1899, 1903) at the turn of the last century, which seem to have escaped the attention of later investigators in this field. Chaplygins solutions include that of an elliptical patch of uniform vorticity in an exterior field of pure shear and that of a (symmetric or non-symmetric) dipolar vortex with a continuous distribution of vorticity translating steadily along a straight path. In addition, a solution is presented for a non-symmetric vortex dipole moving along a circular trajectory. A concise account of Chaplygins solutions is given, complemented by a more detailed analysis of some of their relevant properties.

The flow between two finite rotating disks enclosed by a cylinder

The flow between two finite rotating disks enclosed by a cylinder is investigated both numerically and experimentally. For this finite geometry the full stationary Navier–Stokes equations are solved numerically without similarity assumptions. Experimental results are obtained by means of stereophotography of small tracer particles. The results are in good agreement with the numerical solution. Owing to the presence of the cylinder sidewall, the solution is found to be unique for all values of the parameters considered. When the disks rotate in opposite senses with counter-rotation above 15%, a stagnation point appears at the slower-rotating disk. This stagnation point is associated with a two-cell structure in the meridional plane and is experimentally observed as a ring of particles at the slower-rotating disk. Near the axis of rotation the solution is found to satisfy similarity demands; for weak counter-rotation the solution is of Batchelor type near the axis of rotation, but for strong counter-rotation a Stewartson profile is found to be more adequate for the description of the tangential velocity near the axis.

Laboratory experiments on the tripolar vortex in a rotating fluid

Within the framework of the study of coherent vortex structures as emerging in rotating, quasi-two-dimensional flows, the tripolar vortex is a relatively novel feature. It consists of a symmetric, linear arrangement of three patches of distributed vorticity of alternate signs, and the axis of this configuration rotates about the centre of the core vortex. This paper describes an experimental study of the formation of a tripole from an unstable axisymmetric vortex in a solidly rotating, homogeneous fluid. The flow is visualized by addition of dye, and is measured by streak photography of tracer particles. After digitization, the spatial distributions of the vorticity ω and the stream function ψ are calculated numerically, and scatter plots’ of ω versus ψ are presented for the various stages in the tripole formation process. Owing to viscous effects (spin-down by the bottom Ekman layer and lateral entrainment of ambient fluid) the tripole shows an exponential decay, both in its rotation speed and its internal, relative flow. The comparison of the observed flow characteristics with a simple point-vortex model shows reasonable quantitative agreement.

Propagation of barotropic vortices over topography in a rotating tank

A small-scale cyclonic vortex in a relatively broad valley tends to climb up and out of the valley in a cyclonic spiral about the centre, and when over a relatively broad hill it tends to climb toward the top in an anticyclonic spiral around the peak. This phenomenon is examined here through two-dimensional numerical simulations and rotating-tank experiments. The basic mechanism involved is shown to be the same as that which accounts for the northwest propagation of cyclones on a β-plane. This inviscid nonlinear effect is also shown to be responsible for the observed translationary motion of barotropic vortices in a free-surface rotating tank. The behaviour of isolated vortices is contrasted with that of vortices with non-vanishing circulation.

Spin‐up in a rectangular container

The spin‐up from rest of (i) a homogeneous and (ii) a linearly stratified fluid in a rectangular container has been examined in the laboratory. In the spin‐up process leading to the ultimate state of rigid‐body rotation, three main stages can be discerned, these being (1) the starting flow, characterized by zero absolute vorticity, (2) flow separation due to cyclonic vorticity generation at the lateral tank walls, and (3) a subsequent organization of the flow into a regular array of alternately cyclonic and anticyclonic cells. During the final stage the flow in these cells gradually decays due to the spin‐down/spin‐up mechanism provided by the Ekman boundary layer present at the bottom of each cell. Experiments have been performed with free‐surface and rigid‐lid upper boundary conditions, and the organization of the flow in these cases was observed to be essentially different. In particular, it was noted that the central cell in the free‐surface case is always cyclonic. A model for this behavior is advanced, in terms of the tendency of cyclonic vortices to move toward the rotation axis in the free‐surface configuration.

Experimental study of dipolar vortices on a topographic /?-plane

The behaviour of dipolar vortices in a rotating fluid with a sloping bottom (simulating the variation of the Coriolis parameter on the Earth, with the direction of steepest bottom slope corresponding with the northern direction) has been investigated in the laboratory. Dipoles were generated by moving a vertical cylinder through the fluid. Dye photographs provided qualitative information, whereas quantitative information about the evolving flow field was obtained by streak photography. Dipoles initially directed under a certain angle relative to the west-east axis showed meandering or cycloid-like trajectories. Some asymmetries between east-travelling dipoles (ETD’s) and west-travelling dipoles (WTD’s) were observed. ETD’s are stable in the trajectory sense: a small deviation from zonal motion results in small oscillations around the equilibrium latitude. WTD’s are unstable : small initial deviations produce large displacements in northern or southern directions. This asymmetry arises because the vorticity of a dipole moving westward is anticorrelated with the ambient vorticity, while the vorticities are correlated when the dipole moves eastward. ETD’s increase in size and eventually split into two independent monopoles, the rate of growth depending on the gradient of planetary vorticity. WTD’s are initially more compact but owing to the large displacements in the meridional direction strong asymmetries in the circulation of the two halves are produced, resulting in a large deformation of the weaker part. The experimental observations show good qualitative agreement with analytical and numerical results obtained using a modulated point-vortex model.

The evolution of stable barotropic vortices in a rotating free-surface fluid

Laboratory experiments reveal that, for increasing time, barotropic cyclones typically show an increasing steepness in their flow profiles. This implies that such vortices become barotropically more unstable. This has been confirmed by observations which are further discussed Kloosterziel & van Heijst. We discuss the evolutionary process.

Vortical motion in the head of an axisymmetric gravity current

A series of experiments that examine the initial development of an axisymmetric gravity current have been carried out. The experiments highlight the growth of a ring vortex that dominates the dynamics of the gravity current’s early time propagation. In particular, the experiments show three distinct stages of early time development that have previously been described as the “initial phase” of a gravity current. The first phase of the early time development is dependent on the fractional depth of the lock release, followed by a secondary phase wherein the frontal speed is approximately constant and a third phase of reducing speed. The second phase of the gravity current’s propagation comes to an abrupt end with the breakdown of the ring vortex at a clearly defined position. All of the experimental results show the development of a complex flow field where the generation and collapse of a ring vortex dominate the gravity current’s early time propagation. The complexity of the flow field and the dependence of the propagation speed on the presence of the ring vortex in the head of the gravity current highlights the unsuitability of shallow-water modeling for axisymmetric lock releases at early times.

Interaction of two unequal corotating vortices

Previous high-resolution contour dynamics calculations [Dritschel and Waugh, Phys. Fluids A 4, 1737 (1992)] have shown that in two-dimensional inviscid flow the interaction of two unequal corotating vortices with uniform vorticity is not always associated with vortex growth and may lead to vortices smaller than the original vortices. In the present study, we investigate whether these results also hold for two-dimensional vortices with continuous vorticity distributions. Similar flow regimes are found as for uniform vorticity patches, but the variation of the flow regimes with the initial vortex radii and peak vorticities is more complicated and strongly dependent on the initial shape of the vorticity profile. It is found that the “halo” of low-value vorticity, which surrounds the cores of continuous vortices, significantly increases the critical distance at which the weaker vortex is destroyed. The halo also promotes the vortex cores to merge more efficiently, since it accounts for a substantial part of t...