scholarly journals Erratum: “Libration driven elliptical instability” [Phys. Fluids 24, 061703 (2012)]

2017 ◽  
Vol 29 (5) ◽  
pp. 059901
Author(s):  
D. Cébron ◽  
M. Le Bars ◽  
J. Noir ◽  
J. M. Aurnou
Keyword(s):  
Elliptical Instability

Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

1999 ◽  
Vol 396 ◽  
pp. 73-108 ◽  
Author(s):  
D. M. MASON ◽  
R. R. KERSWELL
Keyword(s):  
Finite Amplitude ◽  
Complex Conjugate ◽  
Small Scale ◽  
Conjugate Pair ◽  
Weakly Nonlinear ◽  
Elliptical Instability ◽  
Elliptical Flow ◽  
Complex Conjugate Pair ◽  
Generic Instability ◽  
Secondary Instabilities

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

Tidal excitation of hydromagnetic waves and their damping in the Earth

1994 ◽  
Vol 274 ◽  
pp. 219-241 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Outer Core ◽  
Decay Rates ◽  
Wave Disturbance ◽  
Instability Mechanism ◽  
The Earth ◽  
Elliptical Instability ◽  
Hydromagnetic Waves

We examine the possibility that the Earth's outer core, as a tidally distorted fluid-filled rotating spheroid, may be the seat of an elliptical instability. The instability mechanism is described within the framework of a simple Earth-like model. The preferred forms of wave disturbance are explored and a likely growth rate supremum deduced. Estimates are made of the Ohmic and viscous decay rates of such hydromagnetic waves in the outer core. Rather than a conclusive disparity of scales, we find that typical elliptical growth rates, Ohmic decay rates and viscous decay rates all have the same order for plausible core fields and core-to-mantle conductivities. This study is all the more timely considering the recent realization that the Earth's precession may also drive similar instabilities at comparable strengths in the outer core.

scholarly journals Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers

2007 ◽  
Vol 585 ◽  
pp. 323-342 ◽  
Author(s):  
M. LE BARS ◽  
S. LE DIZÈS ◽  
P. LE GAL
Keyword(s):  
Growth Rates ◽  
Close Agreement ◽  
Unstable Mode ◽  
Quantitative Agreement ◽  
Fixed Rate ◽  
Elliptical Instability ◽  
Normal Mode Theory ◽  
Set Up ◽  
First Time ◽  
Selection Of

The effects of the Coriolis force on the elliptical instability are studied experimentally in cylindrical and spherical rotating containers placed on a table rotating at a fixed rate $\tilde{\Omega}^G$. For a given set-up, changing the ratio ΩG of global rotation $\tilde{\Omega}^G$ to flow rotation $\tilde{\Omega}^F$ leads to the selection of various unstable modes due to the presence of resonance bands, in close agreement with the normal-mode theory. No instability occurs when ΩG varies between −3/2 and −1/2 typically. On decreasing ΩG toward −1/2, resonance bands are first discretized for ΩG<0 and progressively overlap for −1/2 ≪ ΩG < 0. Simultaneously, the growth rates and wavenumbers of the prevalent stationary unstable mode significantly increase, in quantitative agreement with the viscous short-wavelength analysis. New complex resonances have been observed for the first time for the sphere, in addition to the standard spin-over. We argue that these results have significant implications in geo- and astrophysical contexts.

Elliptical instability of the Moore–Saffman model for a trailing wingtip vortex

2016 ◽  
Vol 803 ◽  
pp. 556-590 ◽  
Author(s):  
J. Feys ◽  
S. A. Maslowe
Keyword(s):  
Growth Rates ◽  
Stokes Equations ◽  
Normal Modes ◽  
Axial Flow ◽  
Velocity Profiles ◽  
Wing Loading ◽  
Axial Velocity Component ◽  
Important Conclusion ◽  
Elliptical Instability ◽  
Loading Parameter

In this paper, we investigate the elliptical instability exhibited by two counter-rotating trailing vortices. This type of instability can be viewed as a resonance between two normal modes of a vortex and an external strain field. Recent numerical investigations have extended earlier results that ignored axial flow to include models with a simple wake-like axial flow such as the similarity solution found by Batchelor (J. Fluid Mech., vol. 20, 1964, pp. 645–658). We present herein growth rates of elliptical instability for a family of velocity profiles found by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 333, 1973, pp. 491–508). These profiles have a parameter $n$ that depends on the wing loading. As a result, unlike the Batchelor vortex, they are capable of modelling both the jet-like and the wake-like axial flow present in a trailing vortex at short and intermediate distances behind a wingtip. Direct numerical simulations of the linearized Navier–Stokes equations are performed using an efficient spectral method in cylindrical coordinates developed by Matsushima & Marcus (J. Comput. Phys., vol. 53, 1997, pp. 321–345). We compare our results with those for the Batchelor vortex, whose velocity profiles are closely approximated as the wing loading parameter $n$ approaches 1. An important conclusion of our investigation is that the stability characteristics vary considerably with $n$ and $W_{0}$, a parameter measuring the strength of the mean axial velocity component. In the case of an elliptically loaded wing ($n=0.50$), we find that the instability growth rates are up to 50 % greater than those for the Batchelor vortex. Our results demonstrate the significant effect of the distribution and intensity of the axial flow on the elliptical instability of a trailing vortex.

Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity

1999 ◽  
Vol 387 ◽  
pp. 205-226 ◽  
Author(s):  
P. G. POTYLITSIN ◽  
W. R. PELTIER
Keyword(s):  
Cross Sections ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Steady State Solution ◽  
Vorticity Distribution ◽  
Columnar Vortex ◽  
Elliptical Instability ◽  
The Stability ◽  
Dimensional Instability ◽  
Stuart Vortices

We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor–Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the ‘elliptical’ instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin & Peltier (1998) and Leblanc & Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third ‘hyperbolic’ mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.

scholarly journals Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability

2016 ◽  
Vol 459 (1) ◽  
pp. 924-938 ◽  
Author(s):  
Adrian J. Barker ◽  
Harry J. Braviner ◽  
Gordon I. Ogilvie
Keyword(s):  
Elliptical Instability ◽  
Non Linear ◽  
Global Modes

scholarly journals Elliptical instability of a rapidly rotating, strongly stratified fluid

2009 ◽  
Vol 21 (7) ◽  
pp. 074104 ◽  
Author(s):  
J. M. Aspden ◽  
J. Vanneste
Keyword(s):  
Stratified Fluid ◽  
Elliptical Instability

scholarly journals An experimental investigation on elliptical instability of a strongly asymmetric vortex pair in a stable density stratification

2006 ◽  
Vol 13 (6) ◽  
pp. 641-649 ◽  
Author(s):  
B. Cariteau ◽  
J.-B. Flór
Keyword(s):  
Froude Number ◽  
Vortex Pair ◽  
Azimuthal Velocity ◽  
Buoyancy Frequency ◽  
Nonlinear Interactions ◽  
Secondary Vortex ◽  
Asymmetric Vortex ◽  
Elliptical Instability ◽  
Acceleration And Deceleration ◽  
Instability Growth

Abstract. We investigate the elliptical instability of a strongly asymmetric vortex pair in a stratified fluid, generated by the acceleration and deceleration of the rotation of a single flap. The dominant parameter is the Froude number, Fr=U/(NR), based on the maximum azimuthal velocity, U, and corresponding radius, R, of the strongest vortex, i.e. the principal vortex, and buoyancy frequency N. For Fr>1, both vortices are elliptically unstable while the instability is suppressed for Fr<1. In an asymmetric vortex pair, the principal vortex is less – and the secondary vortex more – elliptical than the vortices in an equivalent symmetric dipolar vortex. The far more unstable secondary vortex interacts with the principal vortex and increases the strain on the latter, thus increasing its ellipticity and its instability growth rate. The nonlinear interactions render the elliptical instability more relevant. An asymmetric dipole can be more unstable than an equivalent symmetric dipole. Further, the wavelength of the instability is shown to be a function of the Froude number for strong stratifications corresponding to small Froude numbers, whereas it remains constant in the limit of a homogenous fluid.

scholarly journals Elliptical instability of the flow in a rotating shell

2005 ◽  
Vol 151 (3-4) ◽  
pp. 194-205 ◽  
Author(s):  
Laurent Lacaze ◽  
Patrice Le Gal ◽  
Stéphane Le Dizès
Keyword(s):  
Elliptical Instability ◽  
Rotating Shell
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