Dynamics of pancake-like vortices in a stratified fluid: experiments, model and numerical simulations

Abstract

The dynamics and the three-dimensional structure of vortices in a linearly stratified, non-rotating fluid are investigated by means of laboratory experiments, an analytical model and through numerical simulations. The laboratory experiments show that such vortices have a thin pancake-like appearance. Due to vertical diffusion of momentum the strength of these vortices decreases rapidly and their thickness increases in time. Also it is found that inside a vortex the linear ambient density profile becomes perturbed, resulting in a local steepening of the density gradient. Based on the assumption of a quasi-two-dimensional axisymmetric flow (i.e. with zero vertical velocity) a model is derived from the Boussinesq equations that illustrates that the velocity field of the vortex decays due to diffusion and that the vortex is in so-called cyclostrophic balance. This means that the centrifugal force inside the vortex is balanced by a pressure gradient force that is provided by a perturbation of the density profile in a way that is observed in the experiments. Numerical simulations are performed, using a finite difference method in a cylindrical coordinate system. As an initial condition the three-dimensional vorticity and density structure of the vortex, found with the diffusion model, are used. The influence of the Froude number, Schmidt number and Reynolds number, as well as the initial thickness of the vortex, on the evolution of the flow are investigated. For a specific combination of flow parameters it is found that during the decay of the vortex the relaxation of the isopycnals back to their undisturbed positions can result in a stretching of the vortex. Potential energy of the perturbed isopycnals is then converted into kinetic energy of the vortex. However, when the stratification is strong enough (i.e. for small Froude numbers), the evolution of the vortex can be described almost perfectly by the diffusion model alone.

Evolution and instability of monopolar vortices in a stratified fluid

“…Although these values are rather inaccurate, they indicate that ␣ I Ͼ␣ II . The typical length scale L of the flow is defined as the radius where the vorticity changes sign, 12 and it can be estimated that L I ϭ6Ϯ1 cm and L II ϭ5Ϯ1 cm. The buoyancy frequency is fairly easy to determine from a vertical density profile measured during the experiments: Nϭ1.8 rad s Ϫ1 .…”

“…All the equations are nondimensionalized by a typical length scale L ͑the radius of the vortex͒, a characteristic velocity scale Vϭͱ2 max ͑with the vortex thickness and max the extremum vorticity value͒ and a time scale L/V. 12,13 The density perturbation is scaled by the density difference ⌬ϭN 2 L 0 /g. The perturbation pressure p is scaled by 0 V 2 .…”

“…In Sec. III, the evolution of the 3-D vortices with a similar initial velocity and density distribution as used by Beckers et al 12 for ␣ϭ2, but now for larger values of the steepness parameter ␣, is studied by numerical simulations. The influence of the Froude number on the tripole formation is investigated and the 3-D structure of a full-grown tripole is briefly discussed.…”

Long-range interaction and elastic collisions of isolated vortices

“…Oceanographic measurements indicate that the typical velocity distribution of coherent vortices is rather compact and that the relative vorticity changes sign in the vortex interior (Olson 1980;Olson et al 1985;Joyce & McDougall 1992;Cornillon & Park 2001). The laboratory experiments with decaying monopolar vortices also revealed a tendency for the vortices to evolve towards the shielded configuration (Trieling & van Heijst 1998;Beckers et al 2001). It is the focus on the completely shielded eddies that distinguishes our study from the earlier vortex-vortex interaction models.…”

Internal Waves Radiation by a Turbulent Jet Flow in a Stratified Fluid

“…Now it is well known that the fluid density variations in a pancake vortex arise as a result of the cyclostrophic balance, where the centrifugal force inside the vortex is balanced by a vertical pressure gradient force that is provided by a perturbation of the local density (cf. Beckers et al (2001)). This balance inside the individual vortex deflects isopycnals toward the vortex center and thus creates a two-layer distribution of the local density perturbation.…”