Research output: Contribution to journal › Article

**Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid.** / Billant, Paul; Dritschel, David Gerard; Chomaz, Jean-Marc.

Research output: Contribution to journal › Article

Billant, P, Dritschel, DG & Chomaz, J-M 2006, 'Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid' *Journal of Fluid Mechanics*, vol. 561, pp. 73-102. https://doi.org/10.1017/S0022112006000516

Billant, P., Dritschel, D. G., & Chomaz, J-M. (2006). Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid. *Journal of Fluid Mechanics*, *561*, 73-102. https://doi.org/10.1017/S0022112006000516

Billant P, Dritschel DG, Chomaz J-M. Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid. Journal of Fluid Mechanics. 2006 Aug 25;561:73-102. https://doi.org/10.1017/S0022112006000516

@article{81323ee742d346cbb9ad3440d1de56d3,

title = "Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid",

abstract = "In this paper, we investigate the three-dimensional stability of the Moore-Saffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes m = 1 (corresponding to a bending instability) and m = 2 (corresponding to a twisting instability).In the case of a quasi-geostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of long-wavelength instability of these two modes: the two-dimensional or three-dimensional nature of the instabilities is controlled only by the background strain rate gamma and by the rotation rate Omega of the principal axes of the elliptical vortex relative to the rotating frame of reference.For the m = 1 mode, it is shown that when Omega < -gamma, the vortex is stable to any long-wavelength disturbances, when -gamma < Omega less than or similar to 0, two-dimensional perturbations are most unstable, when 0 less than or similar to Omega < gamma, long-wavelength three-dimensional disturbances are the most unstable, and finally when gamma < Omega, short-wavelength three-dimensional perturbations are the most unstable. Similarly, the m = 2 instability is two-dimensional or three-dimensional depending only on gamma and Omega, independent of the Rossby number. This means that if a long-wavelength three-dimensional instability exists for a given elliptical vortex in a quasi-geostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids.The present results for the azimuthal mode m = 1 suggest that the vortex-bending instabilities observed previously in quasi-geostrophic fluids (tall-column instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.",

keywords = "Straight vortex filament, 3-dimensional stability, Field, Flows, Pair",

author = "Paul Billant and Dritschel, {David Gerard} and Jean-Marc Chomaz",

note = "This is a comprehensive analysis of the linear stability of columnar elliptical vortices subject to two-dimensional strain in a rotating, stratified fluid. It is the culmination of two lines of research, one started by Dritschel involving the tall-column instability, and another started by Billant and Chomaz involving the zigzag instability. Our joint work unifies these instabilities, and shows that they exist over a vast parameter space. This work represents over 7 years of collaborative effort.",

year = "2006",

month = "8",

day = "25",

doi = "10.1017/S0022112006000516",

language = "English",

volume = "561",

pages = "73--102",

journal = "Journal of Fluid Mechanics",

issn = "0022-1120",

publisher = "CAMBRIDGE UNIV PRESS",

}

TY - JOUR

T1 - Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid

AU - Billant, Paul

AU - Dritschel, David Gerard

AU - Chomaz, Jean-Marc

N1 - This is a comprehensive analysis of the linear stability of columnar elliptical vortices subject to two-dimensional strain in a rotating, stratified fluid. It is the culmination of two lines of research, one started by Dritschel involving the tall-column instability, and another started by Billant and Chomaz involving the zigzag instability. Our joint work unifies these instabilities, and shows that they exist over a vast parameter space. This work represents over 7 years of collaborative effort.

PY - 2006/8/25

Y1 - 2006/8/25

N2 - In this paper, we investigate the three-dimensional stability of the Moore-Saffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes m = 1 (corresponding to a bending instability) and m = 2 (corresponding to a twisting instability).In the case of a quasi-geostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of long-wavelength instability of these two modes: the two-dimensional or three-dimensional nature of the instabilities is controlled only by the background strain rate gamma and by the rotation rate Omega of the principal axes of the elliptical vortex relative to the rotating frame of reference.For the m = 1 mode, it is shown that when Omega < -gamma, the vortex is stable to any long-wavelength disturbances, when -gamma < Omega less than or similar to 0, two-dimensional perturbations are most unstable, when 0 less than or similar to Omega < gamma, long-wavelength three-dimensional disturbances are the most unstable, and finally when gamma < Omega, short-wavelength three-dimensional perturbations are the most unstable. Similarly, the m = 2 instability is two-dimensional or three-dimensional depending only on gamma and Omega, independent of the Rossby number. This means that if a long-wavelength three-dimensional instability exists for a given elliptical vortex in a quasi-geostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids.The present results for the azimuthal mode m = 1 suggest that the vortex-bending instabilities observed previously in quasi-geostrophic fluids (tall-column instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.

AB - In this paper, we investigate the three-dimensional stability of the Moore-Saffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes m = 1 (corresponding to a bending instability) and m = 2 (corresponding to a twisting instability).In the case of a quasi-geostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of long-wavelength instability of these two modes: the two-dimensional or three-dimensional nature of the instabilities is controlled only by the background strain rate gamma and by the rotation rate Omega of the principal axes of the elliptical vortex relative to the rotating frame of reference.For the m = 1 mode, it is shown that when Omega < -gamma, the vortex is stable to any long-wavelength disturbances, when -gamma < Omega less than or similar to 0, two-dimensional perturbations are most unstable, when 0 less than or similar to Omega < gamma, long-wavelength three-dimensional disturbances are the most unstable, and finally when gamma < Omega, short-wavelength three-dimensional perturbations are the most unstable. Similarly, the m = 2 instability is two-dimensional or three-dimensional depending only on gamma and Omega, independent of the Rossby number. This means that if a long-wavelength three-dimensional instability exists for a given elliptical vortex in a quasi-geostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids.The present results for the azimuthal mode m = 1 suggest that the vortex-bending instabilities observed previously in quasi-geostrophic fluids (tall-column instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.

KW - Straight vortex filament

KW - 3-dimensional stability

KW - Field

KW - Flows

KW - Pair

UR - http://www.scopus.com/inward/record.url?scp=33747040800&partnerID=8YFLogxK

U2 - 10.1017/S0022112006000516

DO - 10.1017/S0022112006000516

M3 - Article

VL - 561

SP - 73

EP - 102

JO - Journal of Fluid Mechanics

T2 - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -

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## Journal of Fluid Mechanics (Journal)

David Gerard Dritschel (Editor)2005 → …Activity: Publication peer-review and editorial work types › Editor of research journal

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ID: 329650