R. C. Kloosterziel

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.

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.

Emergence and evolution of triangular vortices

Laboratory observations and numerical simulations reveal that, in addition to monopoles, dipoles and tripoles, yet another stable coherent vortex may emerge from unstable isolated circular vortices. This new vortex is the finite-amplitude result of the growth of an azimuthal wavenumber-3 perturbation. It consists of a triangular core of single-signed vorticity surrounded by three semicircular satellites of oppositely signed vorticity. The stability of this triangular vortex is analysed through a series of high-resolution numerical simulations and by an investigation of point-vortex models. This new compound vortex rotates about its centre and is stable to small perturbations. If perturbed strongly enough, it undergoes an instability in which two of the outer satellites merge, resulting in the formation of an axisymmetric tripole, which subsequently breaks down into either a pair of dipoles or a dipole plus a monopole. The growth of higher-azimuthal-wavenumber perturbations leads to the formation of more intricate compound vortices with cores in the shape of squares, pentagons, etc. However, numerical simulations show that these vortices are unstable, which agrees with results from point-vortex models.

Inertial instability in rotating and stratified fluids: barotropic vortices

The unfolding of inertial instability in intially barotropic vortices in a uniformly rotating and stratified fluid is studied through numerical simulations. The vortex dynamics during the instability is examined in detail. We demonstrate that the instability is stabilized via redistribution of angular momentum in a way that produces a new equilibrated barotropic vortex with a stable velocity profile. Based on extrapolations from the results of a series of simulations in which the Reynolds number and strength of stratification are varied, we arrive at a construction based on angular momentum mixing that predicts the infinite-Reynolds-number form of the equilibrated vortex toward which inertial instability drives an unstable vortex. The essential constraint is conservation of total absolute angular momentum. The construction can be used to predict the total energy loss during the equilibration process. It also shows that the equilibration process can result in anticyclones that are more susceptible to horizontal shear instabilities than they were initially, a phenomenon previously observed in laboratory and numerical studies.

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.

On the large-time asymptotics of the diffusion equation on infinite domains

It is shown that expansions in similarity solutions provide a quick and economical method for assessing the large-time asymptotics of the diffusion equation on infinite and certain semi-infinite domains if Dirichlet or Neumann conditions are imposed. The similarity solutions are shown to form a basis for the Hilbert space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaCa% aaleqabaGaaGOmaaaakiaacIcaruWu5XMzHvMBaGqbbiaa-jfadaah% aaWcbeqaaiaad6gaaaGccaGGSaGaaeyzamaaCaaaleqabaWaa0baaW% qaaiaaikdaaeaacaaIXaaaaaaakmaaCaaaleqabaGaaeiFaiaa-Hha% caqG8bWaaWbaaWqabeaacaqGYaaaaaaakiaacMcaaaa!4582!\[L^2 (R^n ,{\text{e}}^{_2^1 } ^{{\text{|}}x{\text{|}}^{\text{2}} } )\]. This implies that initial conditions for the diffusion equation that are square integrable with respect to the exponentially-growing weight function % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzamaaCa% aaleqabaWaaSbaaWqaamaaDaaabaGaaeOmaaqaaiaabgdaaaGaaeiF% aerbtLhBMfwzUbacfeGaa8hEaiaabYhadaahaaqabeaacaqGYaaaaa% qabaaaaaaa!3F7E!\[{\text{e}}^{_{_{\text{2}}^{\text{1}} {\text{|}}x{\text{|}}^{\text{2}} } } \] can be expanded in a discrete, infinite sum of mutually orthogonal similarity solutions, each having a different rate of amplitude decay. This leads to a rapid, almost effortless recognition of the large-time asymptotic behaviour of the solution.

Three-dimensionally perturbed vortex tubes in a rotating flow

Numerical experiments are used to study the evolution of perturbed vortex tubes in a rotating environment in order to better understand the process of two-dimensionalization of unsteady rotating flows. We specically consider non-axisymmetric perturbations to columnar vortices aligned along the axis of rotation. The basic unperturbed vortex is chosen to have a Gaussian cross-sectional vorticity distribution. The experiments cover a parameter space in which both the strength of the initial perturbation and the Rossby number are varied. The Rossby number is dened here as the ratio of the maximum amplitude of vorticity in the Gaussian vorticity prole to twice the ambient rotation rate. For small perturbations and small Rossby numbers, both cyclones and anticyclones behave similarly, relaxing rapidly back toward two-dimensional columnar vortices. For large perturbations and small Rossby numbers, a rapid instability occurs for both cyclones and anticyclones in which antiparallel vorticity is created. The tubes break up and then re-form again into columnar vortices parallel to the rotation axis (i.e. into a quasi-two-dimensional flow) through nonlinear processes. For Rossby numbers greater than 1, even small perturbations result in the complete breakdown of the anticyclonic vortex through centrifugal instability, while cyclones remain stable. For a range of Rossby numbers greater than 1, after the breakdown of the anticyclone, a new weaker anticyclone forms, with a small-scale background vorticity of spectral shape given approximately by the 5=3 energy spectral law.

Propagation of barotropic dipoles over topography in a rotating tank

Abstract It is shown how symmetric dipolar vortices can be formed by the action of an impulsive jet in a homogeneous single layer of fluid in a rotating tank. These dipoles are allowed to interact with a constant topographic slope, which can model a β-plane or a continental shelf. A dipoles trajectory bends toward the right when climbing a slope and to the left when descending, as predicted by numerical simulations and analytical arguments. The maximum penetration of the dipoles over a slope, the adjustment to the slope, and formation of trailing lobes are compared with both numerical simulations and a two-point vortex model. The results suggest that Rossby wave radiation plays an important role in the interaction process.

Saturation of inertial instability in rotating planar shear flows

Inertial instability in a rotating shear flow redistributes absolute linear momentum in such a way as to neutralize the instability. In the absence of other instabilities, the final equilibrium can be predicted by a simple construction based on conservation of total momentum. Numerical simulations, invariant in the along-stream direction, suppress barotropic instability and allow only inertial instability to develop. Such simulations, at high Reynolds numbers, are used to test the theoretical prediction. Four representative examples are given: a jet, a wall-bounded jet, a mixing layer and a wall-bounded shear layer.