Theoretical analysis of the zigzag instability of a vertical co-rotating vortex pair in a strongly stratified fluid (original) (raw)
Journal of Fluid Mechanics, 2006
This paper investigates the three-dimensional stability of a pair of co-rotating vertical vortices in a rotating strongly stratified fluid. In a companion paper , we have shown that such a basic flow in a strongly stratified fluid is affected by a zigzag instability which bends the two vortices symmetrically. In the non-rotating flow, the most unstable wavelength of this instability scales as the buoyancy length and its growth rate scales as the external strain that each vortex induces on the other one. Here, we show that the zigzag instability remains active whatever the magnitude of the planetary rotation and is therefore connected to the tallcolumn instability in quasi-geostrophic fluids. Its growth rate is almost independent of the Rossby number. The most amplified wavelength follows the universal scaling λ = 2πF h b γ 1 /Ro 2 + γ 2 /Ro + γ 3 , where b is the separation distance between the two vortices, (γ 1 , γ 2 , γ 3 ) are constants, F h is the horizontal Froude number and Ro the Rossby number (F h = Γ /πa 2 N, Ro = Γ /πa 2 f , where Γ is the circulation of each vortex, a the vortex radius, N the Brunt-Väisälä frequency and f the Coriolis parameter). When Ro = ∞, the scaling λ ∝ F h b found in the companion paper Otheguy et al. is recovered. When Ro → 0, λ ∝ bf/N in agreement with the quasi-geostrophic theory. In contrast to previous results, the wavelength is found to depend on the separation distance between the two vortices b, and not on the vortex radius a.
Journal of Fluid Mechanics, 2000
This paper shows that a long vertical columnar vortex pair created by a double flap apparatus in a strongly stratified fluid is subjected to an instability distinct from the Crow and short-wavelength instabilities known to occur in homogeneous fluid. This new instability, which we name zigzag instability, is antisymmetric with respect to the plane separating the vortices. It is characterized by a vertically modulated twisting and bending of the whole vortex pair with almost no change of the dipole's cross- sectional structure. No saturation is observed and, ultimately, the vortex pair is sliced into thin horizontal layers of independent pancake dipoles. For the largest Brunt–Väisälä frequency N = 1.75 rad s−1 that may be achieved in the experiments, the zigzag instability is observed only in the range of Froude numbers: 0.13 < Fh0 < 0.21 (Fh0 = U0/NR, where U0 and R are the initial dipole travelling velocity and radius). When Fh0 > 0.21, the elliptic instability develop...
Three-dimensional stability of a vertical columnar vortex pair in a stratified fluid
Journal of Fluid Mechanics, 2000
This paper investigates the three-dimensional stability of a Lamb–Chaplygin columnar vertical vortex pair as a function of the vertical wavenumber kz, horizontal Froude number Fh, Reynolds number Re and Schmidt number Sc. The horizontal Froude number Fh (Fh = U/NR, where U is the dipole travelling velocity, R the dipole radius and N the Brunt–Väisälä frequency) is varied in the range [0.033, ∞[ and three set of Reynolds-Schmidt numbers are investigated: {Re = 10 000, Sc = 1}, Re = 1000, Sc = 1}, {Re = 200, Sc = 637}. In the whole range of Fh and Re, the dominant mode is always antisymmetric with respect to the middle plane between the vortices but its physical nature and properties change when Fh is varied. An elliptic instability prevails for Fh > 0.25, independently of the Reynolds number. It manifests itself by the bending of each vortex core in the opposite direction to the vortex periphery. The growth rate of the elliptic instability is reduced by stratification effects but ...
Evolution and instability of monopolar vortices in a stratified fluid
Physics of Fluids, 2003
The evolution of initially axisymmetric shielded pancake-like vortices in a nonrotating linearly stratified fluid has been investigated experimentally and numerically. The evolution process and the shape of tripoles in laboratory experiments depend on the experimental parameter values. In order to investigate this phenomenon we have considered the influence of the Reynolds ͑Re͒ and Froude ͑F͒ numbers on the tripole formation process. Also, the role of the ͑absolute͒ ratio ␥ between the vorticity values of the satellite vortices and the core vortex on the tripole evolution has been investigated. Additionally, a set of numerical simulations has been performed to enable an examination of the role of Reynolds numbers ͑up to Reϭ10 000) and Froude numbers (Fϭ0.1, 0.2, 0.4, and 0.8͒ outside the experimentally accessible range. The steepness parameter ␣ was varied between 2 and 8 in order to estimate the relative importance of the different modes constituting the perturbation. From this study we conclude that tripole formation and dipole splitting in a linearly stratified fluid can be well described in terms of the parameter set (Re,F,␣).
Stable and unstable monopolar vortices in a stratified fluid
Journal of Fluid Mechanics, 1996
This paper presents experiments on planar monopolar vortex structures generated in a non-rotating, stratifi ed fluid. In order to study the dynamics of such planar vortices in the laboratory, angular momentum was generated in a specifi c horizontal layer of the stratifi ed fl uid, by using three different generation mechanisms. The lens-shaped monopolar vortices thus created were in some cases stable and conserved their circular symmetry, while in other cases they appeared to be unstable, leading to the formation of a multipoled vortex with a different topology. Characteristics such as cross-sectional profiles (angular velocity and vorticity) and vorticity-stream function scatter plots have been measured experimentally by using digital image processing techniques. The characteristics of the monopolar vortices are compared with analytical vortex models known from literature. Simple models, based on vertical diffusion of vorticity, are proposed to describe the monopolar vortex decay ; they show reasonable agreement with the experimental results. From the multipolar structures, the tripolar vortex and a specific case of a triangular vortex, neither having been observed before in a stratified fluid, are studied in detail. A comparison with point-vortex models yields good agreement. Although these multipolar vortices appear to persist for a long while, they are found eventually to be unstable and to transform into a monopolar vortex.
Pairing of two vertical columnar vortices in a stratified fluid
European Journal of Mechanics - B/Fluids, 2015
We present three-dimensional (3D) numerical simulations of the pairing of two vertical columnar vortices in a stably stratified fluid. Whereas in two dimensions, merging of two isolated vortices occurs on a diffusion time scale, in the three-dimensional stratified case we show that merging is a much faster process that occurs over an inertial time scale. The sequence of dynamical processes that leads to this accelerated pairing involves first a linear stage where the zigzag instability develops displacing vortices alternately closer and farther with a vertical periodicity scaling on the buoyancy length scale L B = F h b, where F h is the horizontal Froude number (F h = Γ /πa 2 N with a the core size of the vortices, Γ their circulation and N the Brunt-Väisälä frequency) and b is the separation distance between the vortices. In layers where the vortices have started to move closer, their distance decreases exponentially with the growth rate of the zigzag instability. Non-linearities do not seem to affect this process and the decrease only stops when the pairing is completed in that layer. At the same time, enstrophy that has also grown exponentially reaches a magnitude of the order of the Reynolds number Re = Γ /(π ν) (where ν is the kinematic viscosity of the fluid) if the Reynolds number is not too large, meaning that energy is then dissipated on the inertial time scale. This dissipation occurs in thin layers and the vortices that were originally moving away in the intermediate layer start slowing down and rapidly merge.
Numerical study of the evolution of vortices in a linerly stratified fluid
Il Nuovo Cimento C, 1999
This paper presents a numerical study in which the evolution of vortices in a stratified fluid is compared to the evolution of two-dimensional vortices. The influence of the Reynolds number and the Froude number are investigated, for the evolution of axisymmetric vortices, for their azimuthal instability and for the subsequent formation of tripoles. It is found that due to radial diffusion axisymmetric vortices with various initial vorticity profiles all evolve towards the same profile. This evolution reduces the growth of azimuthal instabilities which may lead to the formation of a tripole. For vortices in a stratified fluid the effect of the ambient stratification on the evolution of the vortices is investigated. It is found that a process of vortex stretching, which becomes more pronounced for increasing Froude numbers, leads to a weaker tripole formation.
Stabilization of Isolated Vortices in a Rotating Stratified Fluid
Fluids, 2016
The key element of Geophysical Fluid Dynamics-reorganization of potential vorticity (PV) by nonlinear processes-is studied numerically for isolated vortices in a uniform environment. Many theoretical studies and laboratory experiments suggest that axisymmetric vortices with a Gaussian shape are not able to remain circular owing to the growth of small perturbations in the typical parameter range of abundant long-lived vortices. An example of vortex destabilization and the eventual formation of more intense self-propagating structures is presented using a 3D rotating stratified Boussinesq numerical model. The peak vorticity growth found during the stages of strong elongation and fragmentation is related to the transfer of available potential energy into kinetic energy of vortices. In order to develop a theoretical model of a stable circular vortex with a small Burger number compatible with observations, we suggest a simple stabilizing procedure involving the modification of peripheral PV gradients. The results have important implications for better understanding of real-ocean eddies.