Elliptic and magneto-elliptic instabilities (original) (raw)

Elliptic instability of a co-rotating vortex pair

Journal of Fluid Mechanics, 2005

In this paper, we report experimental results concerning a three-dimensional shortwave instability observed in a pair of equal co-rotating vortices. The pair is generated in water by impulsively started plates, and is analysed through dye visualizations and detailed quantitative measurements using particle image velocimetry. The instability mode, which is found to be stationary in the rotating frame of reference of the two-vortex system, consists of internal deformations of the vortex cores, which are characteristic of the elliptic instability occurring in strained vortical flows. Measurements of the spatial structure, wavelengths and growth rates are presented, as functions of Reynolds number and non-dimensional core size. The self-induced rotation of the vortex pair, which is not a background rotation of the entire flow, is found to lead to a shift of the unstable wavelength band to higher values, as well as to higher growth rates. In addition, a dramatic increase in the width of the unstable bands for large values of the rescaled core radius is found. Comparisons with recent theoretical results by Le concerning elliptic instability of co-rotating vortices show very good agreement.

Spatio-temporal development of the long and short-wave vortex-pair instabilities

Physics of Fluids, 2000

We consider the spatio-temporal development of the long-wave and shortwave instabilities in a pair of counter-rotating vortices in the presence of a uniform axial advection velocity. The stability properties depend upon the aspect ratio a=b of the vortex pair, where a is the core radius of the vortices and b their separation, and upon W 0 =U 0 the ratio between the self-induced velocity of the pair and the axial advection velocity. For su ciently small W 0 =U 0 , the instabilities are convective, but an

Evolution of an Intense Vortex in a Periodic Sheared Flow

IUTAM Bookseries, 2008

An analytical theory is presented for the motion of a localized vortex in the presence of a zonal Rossby wave on the β-plane. In the framework of the equivalent-barotropic quasi-geostrophic model, the analytical method developed by Sutyrin and Flierl [Sutyrin G.G., Flierl G.R., 1994. Intense vortex motion on the β-plane: development of the beta gyres. J. Atmos. Sci. 51, 773-790] for intense vortices with piecewiseconstant potential vorticity is generalized to take into account a slowly propagating Rossby wave which modifies the background potential vorticity. The predictions of these asymptotic expansions are compared with the results of numerical simulations. The theory describes the vortex advection by the wave and the vortex drift due to the background potential vorticity gradient. The net vortex drift speed due to the wave is found to be smaller than the maximum wave velocity; this is due to the baroclinic ␤-effect and to the periodic structure of the background potential vorticity gradient. Besides known elliptical core deformations, triangular deformations are generated by the wave on the core boundary. Additionally, the planetary ␤-effect provides a predominantly westward vortex drift with nearly the same speed as the wave propagation speed. The asymptotic theory is shown to agree well the results of a numerical pseudo-spectral, high-resolution biperiodic model when the vortex velocity is much larger than the wave velocity. Both meridional and zonal vortex drifts are slightly overestimated when the wave velocity is comparable with the vortex velocity. Vortex size is shown to be more influential on vortex trajectory than the Rossby wave length. In particular, smaller vortices drift westward farther and faster than large ones. Vortex core deformations typically contain modes 2 and 3 with a stronger mode 3 component for more intense Rossby waves as predicted by theory.

Baroclinic instability of Kirchhoff's elliptic vortex

Journal of Fluid Mechanics, 1994

The linear instability of Kirchhoff's elliptic vortex in a vertically stratified rotating fluid is investigated using the quasi-geostrophic, f-plane approximation. Any elliptic vortex is shown to be unstable to baroclinic disturbances of azimuthal wavenumber m = 1 (bending mode) and m = 2 (elliptical deformation). The axial wavenumber of the unstable bending mode approaches Λc = 1.7046 in the limit of small ellipticity, indicating that it is a short-wave baroclinic instability. The instability occurs when the bending wave rotates around the vortex axis with angular velocity identical to the rotation rate of the undisturbed elliptic vortex. On the other hand, the wavenumber of the elliptical deformation mode approaches zero in the same limit, showing that it is a long-wave sideband instability.

Turbulence phenomena for viscous fluids: Vortices and instability

Applied Mathematics and Computation, 2020

Through the Ginzburg-Landau and the Navier-Stokes equations, we study turbulence phenomena for viscous incompressible and compressible fluids by a second order phase transition. For this model, the velocity is defined by the sum of classical and whirling components. Moreover, the laminar-turbulent transition is controlled by rotational effects of the fluid. Hence, the thermodynamic compatibility of the differential system is proved. The same model is used to understand the origins of tornadoes and their behavior and the birth of the vortices resulting from the fall of water in a vertical tube. Finally, we demonstrate how the weak Coriolis force is able to change the rotation direction of the vortices by modifying the minima of the Ginzburg-Landau equation. Hence, we conclude the paper with the differential system describing the water vorticity and its thermodynamic compatibility.

Three-dimensional stability of a vortex pair

2001

This paper investigates the three-dimensional stability of the Lamb-Chaplygin vortex pair. Short-wavelength instabilities, both symmetric and antisymmetric, are found. The antisymmetric mode possesses the largest growth rate and is indeed the one reported in a recent experimental study ͓J. Fluid Mech. 360, 85 ͑1998͔͒. The growth rates, wave numbers of maximum amplification, and spatial eigenmodes of these short-wavelength instabilities are in good agreement with the predictions from elliptic instability theory. A long-wavelength symmetric instability similar to the Crow instability of a pair of vortex filaments is also recovered. Oscillatory bulging instabilities, both symmetric and antisymmetric, are identified albeit their growth rates are lower than for the short-wavelength instabilities. Their behavior and eigenmodes resemble those of the oscillatory bulging instability occurring in the mixing layer.

Instability of strongly nonlinear waves in vortex flows

Journal of Fluid Mechanics, 1994

Weakly nonlinear descriptions of axisymmetric cnoidal and solitary waves in vortices recently have been shown to have strongly nonlinear counterparts. The linear stability of these strongly nonlinear waves to three-dimensional perturbations is studied, motivated by the problem of vortex breakdown in open flows. The basic axisymmetric flow varies both radially and axially, and the linear stability problem is therefore nonseparable. To regularize the generalization of a critical layer, viscosity is introduced in the perturbation problem. In the absence of the waves, the vortex flows are linearly stable. As the amplitude of a wave constituting the basic flow increases owing to variation in the level of swirl, stability is first lost to non-axisymmetric 'bending' modes. This instability occurs when the wave amplitude exceeds a critical value, provided that the Reynolds number is larger enough. The critical wave amplitudes for instability typically are large, but not large enough to create regions of closed streamlines. Examination of the most-amplified eigenvectors shows that the perturbations tend to be concentrated downstream of the maximum streamline displacement in the wave, in a position consistent with the observed three-dimensional perturbations in the interior of a bubble type of vortex breakdown.

The three-dimensional instability of elliptical vortices in a viscoelastic fluid

Journal of Non-newtonian Fluid Mechanics, 1993

A linear stability analysis is presented for an upper-convected Maxwell fluid undergoing unbounded two-dimensional flows with elliptical streamlines and uniform vorticity. The flows are found to exhibit a Floquet-type instability to a plane-wave disturbance whose wave vector is periodically distorted in time. The elasticity of the fluid is found to exert a destabilizing influence.

Influence of an elliptic instability on the merging of a co-rotating vortex pair

2007

We study the nonlinear evolution of the elliptic instability and its influence on the merging process of two corotating Batchelor vortices using a spectral DNS approach. First, we analyse the nonlinear saturation of the elliptic instability for a single strained vortex, with and without axial jet, for moderate Reynolds numbers (Re = Γ/ν ≈ 12500, where Γ is the circulation and ν the kinematic viscosity). We show that the vortex deformation induced by the instability remains limited to the vortex core region. The second part of our work focuses on the influence of the elliptic instability on the merging process. We compare three cases : no instability (2D), elliptic instability without axial jet, and elliptic instability with axial jet, the latter case being relevant to aircraft wakes. Qualitative and quantitative differences between the three different cases are pointed out and discussed in the context of aircraft vortices.