Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations.

International audienceIn stratified and rotating fluids, pairs of columnar vertical vortices are subjected to three-dimensional bending instabilities known as the zigzag instability or as the tall-column instability in the quasi-geostrophic limit. This paper presents a general asymptotic theory for these instabilities. The equations governing the interactions between the strain and the slow bending waves of each vortex column in stratified and rotating fluids are derived for long vertical wavelength and when the two vortices are well separated, i.e. when the radii R of the vortex cores are small compared to the vortex separation distance b. These equations have the same form as those obtained for vortex filaments in homogeneous fluids except that the expressions of the mutual-induction and self-induction functions are different. A key difference is that the sign of the self-induction function is reversed compared to homogeneous fluids when the fluid is strongly stratified: vertical bar(Omega) over cap (max)vertical bar (max)vertical bar the maximum angular velocity of the vortex) for any vortex profile and magnitude of the planetary rotation. Physically, this means that slow bending waves of a vortex rotate in the same direction as the flow inside the vortex when the fluid is stratified-rotating in contrast to homogeneous fluids. When the stratification is weaker, i.e. vertical bar(Omega) over cap (max)vertical bar > N, the self-induction function is complex because the bending waves are damped by a viscous critical layer at the radial location where the angular velocity of the vortex is equal to the Brunt Vaisala frequency. In contrast to previous theories, which apply only to strongly stratified non-rotating fluids, the present theory is valid for any planetary rotation rate and when the strain is smaller than the Brunt-Vaisala frequency: Gamma/(2 pi b(2)) << N, where Gamma is the vortex circulation. Since the strain is small, this condition is met across a wide range of stratification: from weakly to strongly stratified fluids. The theory is further generalized formally to any basic flow made of an arbitrary number of vortices in stratified and rotating fluids. Viscous and diffusive effects are also taken into account at leading order in Reynolds number when there is no critical layer. In Part 2 (Billant et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000282X), the stability of vortex pairs will be investigated using the present theory and the predictions will be shown to be in very good agreement with the results of direct numerical stability analyses. The existence of the zigzag instability and the distinctive stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids will be demonstrated to originate from the sign reversal of the self-induction function

The stably stratified Taylor-Couette flow is always unstable except for solid-body rotation

International audienceThe stability of the flow between two concentric cylinders is studied numerically and analytically when the fluid is stably stratified. We show that such flow is unstable when the angular velocity Omega(r) increases along the radial direction, a regime never explored before. The instability is highly non-axisymmetric and involves the resonance of two families of inertia-gravity waves like for the strato-rotational instability. The growth rate is maximum when only the outer cylinder is rotating and goes to zero when Omega(r) is constant. The sufficient condition for linear, inviscid instability derived previously, d Omega(2)/dr < 0, is therefore extended to d Omega(2)/dr not equal 0, meaning that only the regime of solid-body rotation is stable in stratified fluids. A Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) analysis in the inviscid limit, confirmed by the numerical results, shows that the instability occurs only when the Froude number is below a critical value and only for a particular band of azimuthal wavenumbers. It is also demonstrated that the instability originates from a reversal of the radial group velocity of the waves, or equivalently from a wave over-reflection phenomenon. The instability persists in the presence of viscous effects

Radiative instability of an anticyclonic vortex in a stratified rotating fluid

International audienceIn strongly stratified fluids, an axisymmetric vertical columnar vortex is unstable because of a spontaneous radiation of internal waves. The growth rate of this radiative instability is strongly reduced in the presence of a cyclonic background rotation f/2 and is smaller than the growth rate of the centrifugal instability for anticyclonic rotation, so it is generally expected to affect vortices in geophysical flows only if the Rossby number Ro = 2 Omega/f is large (where Omega is the angular velocity of the vortex). However, we show here that an anticyclonic Rankine vortex with low Rossby number in the range -1 <= Ro < 0, which is centrifugally stable, is unstable to the radiative instability when the azimuthal wavenumber vertical bar m vertical bar is larger than 2. Its growth rate for Ro = -1 is comparable to the values reported in non-rotating stratified fluids. In the case of continuous vortex profiles, this new radiative instability is shown to occur if the potential vorticity of the base flow has a sufficiently steep radial profile. The most unstable azimuthal wavenumber is inversely proportional to the steepness of the vorticity jump. The properties and mechanism of the instability are explained by asymptotic analyses for large wavenumbers

A unified criterion for the centrifugal instabilities of vortices and swirling jets

International audienceSwirling jets and vortices can both be unstable to the centrifugal instability but with a different wavenumber selection: the most unstable perturbations for swirling jets in inviscid fluids have an infinite azimuthal wavenumber, whereas, for vortices, they are axisymmetric but with an infinite axial wavenumber. Accordingly, sufficient condition for instability in inviscid fluids have been derived asymptotically in the limits of large azimuthal wavenumber m for swirling jets (Leibovich and Stewartson, J. Fluid Mech. vol. 126, 1983, pp. 335-356) and large dimensionless axial wavenumber k for vortices (Billant and Gallaire, J. Fluid Mech., vol. 542, 2005, pp. 365-379). In this paper, we derive a unified criterion valid whatever the magnitude of the axial flow by performing an asymptotic analysis for large total wavenumber root k(2) + m(2). The new criterion recovers the criterion of Billant and Gallaire when the axial flow is small and the Leibovich and Stewartson criterion when the axial flow is finite and its profile sufficiently different from the angular velocity profile. When the latter condition is not satisfied, it is shown that the accuracy of the Leibovich and Stewartson asymptotics is strongly reduced. The unified criterion is validated by comparisons with numerical stability analyses of various classes of swirling jet profiles. In the case of the Batchelor vortex, it provides accurate predictions over a wider range of axial wavenumbers than the Leibovich-Stewartson criterion. The criterion shows also that a whole range of azimuthal wavenumbers are destabilized as soon as a small axial velocity component is present in a centrifugally unstable vortex

Instabilities and waves on a columnar vortex in a strongly stratified and rotating fluid

International audienceThis paper investigates the effect of the background rotation on the radiative instability of a columnar Rankine vortex in a strongly stratified fluid. We show that a cyclonic background rotation strongly stabilizes the radiative instability. The modes become neutral when the Rossby number Ro is below a critical value which depends on the azimuthal wavenumber of the wave. In the limit of small Rossby number, there exist fast neutral waves that are not captured by the quasi-geostrophic theory. In the presence of anticyclonic background rotation, the centrifugal instability dominates the radiative instability only when -400 less than or similar to Ro < -1. The numerical stability analysis is completed by asymptotic analyses for large wavenumbers which explain the properties and mechanisms of the waves and the instabilities. The stability of a continuous smoothed Rankine vortex is also investigated. The most amplified azimuthal wavenumber is then finite instead of infinite for the Rankine vortex. (C) 2013 AIP Publishing LLC

On the mechanism of the Gent–McWilliams instability of a columnar vortex in stratified rotating fluids

International audienceIn stably stratified and rotating fluids, an axisymmetric columnar vortex can be unstable to a special instability with an azimuthal wavenumber m = 1 which bends and slices the vortex into pancake vortices (Gent & McWilliams Geophys. Astrophys. Fluid Dyn., vol. 35 (1-4), 1986, pp. 209-233). This bending instability, called the 'Gent-McWilliams instability' herein, is distinct from the shear, centrifugal or radiative instabilities. The goals of the paper are to better understand the origin and properties of this instability and to explain why it operates only in stratified rotating fluids. Both numerical and asymptotic stability analyses of several velocity profiles have been performed for wide ranges of Froude number Fr-h = Omega(0)/N and Rossby number R-0 = 2 Omega(0)/f, where Omega(0) is the angular velocity on the vortex axis, N the Brunt-Vaisala frequency and f the Coriolis parameter. Numerical analyses restricted to the centrifugally stable range show that the maximum growth rate of the Gent-McWilliams instability increases with R-0 and is independent of Frh for Fr-h 1, the maximum growth rate decreases dramatically with Fr-h. Long axial wavelength asymptotic analyses for isolated vortices prove that the Gent-McWilliams instability is due to the destabilization of the long-wavelength bending mode by a critical layer at the radius r(c) where the angular velocity Omega is equal to the frequency omega : Omega(r(c)) = omega. A necessary and sufficient instability condition valid for long wavelengths, finite Rossby number and Fr-h 0. Such a critical layer r(c) exists for finite Rossby and Froude numbers because the real part of the frequency of the long-wavelength bending mode is positive instead of being negative as in a homogeneous non-rotating fluid (R-0 = Fr-h = infinity). When Fr-h > 1, the instability condition zeta' (r(c)) > 0 is necessary but not sufficient because the destabilizing effect of the critical layer r(c) is strongly reduced by a second stabilizing critical layer r(c2) existing at the radius where the angular velocity is equal to the Brunt-Vaisala frequency. For non-isolated vortices, numerical results show that only finite axial wavenumbers are unstable to the Gent-McWilliams instability

Three-dimensional stability of a vortex pair

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 1998]. 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

Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid

International audienceWe present a three-dimensional linear stability analysis of a couple of co-rotating vertical vortices in a stratified fluid. When the fluid is non-stratified, the two vortices are unstable to the elliptic instability owing to the elliptic deformation of their core. These elliptic instability modes persist for weakly stratified flow: Fh > 10, where Fh is the horizontal Froude number (Fh = Gh/pab2 N where Gh is the circulation of the vortices, ab their core radius and N the Brunt-Väisälä frequency). For strong stratification (Fh < 2.85), a new zigzag instability is found that bends each vortex symmetrically with almost no internal deformation of the basic vortices. This instability may modify the vortex merging since at every half-wavelength along the vertical, the vortices are alternatively brought closer, accelerating the merging, and moved apart, delaying the merging. The most unstable vertical wavelength ?m of this new instability is shown to be proportional to Fhbb, where bb is the distance between the vortices, implying that ?m decreases with increasing stratification. The maximum growth rate, however, is independent of the stratification and proportional to the strain S = Gb/2pbb2. These scaling laws and the bending motion induced by this instability are similar to those of the zigzag instability of a counter-rotating vortex pair in a stratified fluid. © 2006 Cambridge University Press

Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

The well-known Rayleigh criterion is a necessary and sufficient condition for inviscid centrifugal instability of axisymmetric perturbations. We have generalized this criterion to disturbances of any azimuthal wavenumber m by means of large-axial-wavenumber WKB asymptotics. A sufficient condition for a free axisymmetric vortex with angular velocity Ω(r) to be unstable to a three-dimensional perturbation of azimuthal wavenumber m is that the real part of the growth rate...is positive at the complex radius r=r0 where ∂σ(r)/∂r=0, i.e. where ϕ=(1/r3)∂r4Ω2/∂r is the Rayleigh discriminant, provided that some a posteriori checks are satisfied. The application of this new criterion to various classes of vortex profiles shows that the growth rate of non-axisymmetric disturbances decreases as m increases until a cutoff is reached. The criterion is in excellent agreement with numerical stability analyses of the Carton & McWilliams (1989) vortices and allows one to analyse the competition between the centrifugal instability and the shear instability. The generalized criterion is also valid for a vertical vortex in a stably stratified and rotating fluid, except that φ becomes \phi{=}(1/r^3)\partial{r^4(\Omega+\Omega_b)^2/\partial r, where Ωb is the background rotation about the vertical axis. The stratification is found to have no effect. For the Taylor-Couette flow between two coaxial cylinders, the same criterion applies except that r0 is real and equal to the inner cylinder radius. In sharp contrast, the maximum growth rate of non-axisymmetric disturbances is then independent of m

Stability of a Gaussian pancake vortex in a stratified fluid

International audienceVortices in stably stratified fluids generally have a pancake shape with a small vertical thickness compared with their horizontal size. In order to understand what mechanism determines their minimum thickness, the linear stability of an axisymmetric pancake vortex is investigated as a function of its aspect ratio alpha, the horizontal Froude number F-h, the Reynolds number Re and the Schmidt number Sc. The vertical vorticity profile of the base state is chosen to be Gaussian in both radial and vertical directions. The vortex is unstable when the aspect ratio is below a critical value, which scales with the Froude number: alpha(c) similar to 1.1F(h) for sufficiently large Reynolds numbers. The most unstable perturbation has an azimuthal wavenumber either m = 0, vertical bar m vertical bar = 1 or vertical bar m vertical bar = 2 depending on the control parameters. We show that the threshold corresponds to the appearance of gravitationally unstable regions in the vortex core due to the thermal wind balance. The Richardson criterion for shear instability based on the vertical shear is never satisfied alone. The dominance of the gravitational instability over the shear instability is shown to hold for a general class of pancake vortices with angular velocity of the form (Omega) over tilde (r, z) = Omega(r)f(z) provided that r partial derivative Omega/partial derivative r < 3 Omega everywhere. Finally, the growth rate and azimuthal wavenumber selection of the gravitational instability are accounted well by considering an unstably stratified viscous and diffusive layer in solid body rotation with a parabolic density gradient