The effect of planetary rotation on the zigzag instability of co-rotating vortices in a stratified fluid

Othéguy, Pantxika; Billant, Paul; Chomaz, Jean‐Marc

doi:10.1017/s0022112005008050

Cited by 15 publications

(16 citation statements)

References 22 publications

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“…The growth rate of the symmetric mode (3.13) is plotted in figure 13 as a function of the rescaled vertical wavenumberkbF h for various Rossby numbers forb =6.7 and a fixed Froude number F h =0.5. As found by Otheguy et al (2006a), we see that the symmetric mode is unstable for all the Ro values while it is known to be stable in homogeneous fluids (Jimenez 1975). The antisymmetric mode (3.12) always remains stable in stratified-rotating fluids like in homogeneous fluids.…”

Section: Effect Of the Rossby Number For A Strongly Stratified Fluidsupporting

confidence: 70%

“…In the case of co-rotating vortex pairs, the zigzag instability is symmetric (Otheguy et al 2006b) whereas no such bending instability occurs in homogeneous fluids (Jimenez 1975;Meunier & Leweke 2005). The zigzag instability has also been shown to remain active in strongly stratified and rotating fluids whatever the magnitude of the planetary rotation in the cases of an equal strength co-rotating vertical vortex pair (Otheguy, Billant & Chomaz 2006a) and of an elliptic vortex (Billant, Dritschel & Chomaz 2006). Thereby, the zigzag instability is of the same nature as the tall-column instability observed by Dritschel & de la Torre Juárez (1996) in quasi-geostrophic fluids (strongly stratified and rapidly rotating fluids).…”

Section: Introductionmentioning

confidence: 99%

“…The elliptic instability becomes the most dangerous instability only when the Froude number is much larger: F h & 5. In the presence of a background rotation, the growth rate of the zigzag instability in the strongly stratified regime, F h < 1, is almost independent of the Rossby number Ro = Γ/(4πR 2 Ω b ), where Ω b is the planetary rotation rate (Otheguy et al 2006a). However, the most amplified wavelength varies with the Rossby number like F h b/f (Ro), where the function f (Ro) is such that f (Ro) → constant for Ro →∞ and f (Ro) → Ro for Ro → 0.…”

Section: Introductionmentioning

confidence: 99%

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Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses.

et al. 2010

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The three-dimensional stability of vertical vortex pairs in stratified and rotating fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-rotating and counter-rotating vortex pairs are most unstable to the zigzag instability when the Froude number Fh = Γ/(2πR2N) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Väisälä frequency) is lower than unity independently of the Rossby number Ro = Γ/(4πR2Ωb) (Ωb is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2πb2) (b is the separation distance between the vortices) and is almost independent of Fh and Ro. The most amplified wavelength scales like Fhb when the Rossby number is large and like Fhb/|Ro| when |Ro| ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects.When Fh is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous fluids. However, its growth rate is lower than in homogeneous fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution.Physically, the different stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.

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Section: Effect Of the Rossby Number For A Strongly Stratified Fluidsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses.

et al. 2010

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“…For Froude numbers F h ≈ 0.01-0.1 typical of these flows, the ratios λ/D and H/D are therefore of the same order, meaning that at least one wavelength of the zigzag instability could fit in the vortex height. A similar discussion for the case of oceanic vortices can be found in Otheguy, Billant & Chomaz (2006a).…”

Section: Resultssupporting

confidence: 56%

Three-dimensional stability of vortex arrays in a stratified and rotating fluid

2011

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International audienceThis paper investigates numerically and through an asymptotic approach the three-dimensional stability of steady vertical vortex arrays in a stratified and rotating fluid. Three classical vortex arrays are studied: the KÃ¡rmÃ¡n vortex street, the symmetric double row and the single row of co-rotating vortices. The asymptotic analysis assumes well-separated vortices and long-wavelength bending perturbations following Billant (J. Fluid Mech., vol. 660, 2010, p. 354) and Robinson & Saffman (J. Fluid Mech., vol. 125, 1982, p. 411). Very good agreement with the numerical stability analysis is found even for finite wavelength and relatively close vortices. For a horizontal Froude number Fh = 1 and for a non-rotating fluid, it is found that the Kármán vortex street for a street spacing ratio (the distance h between the rows divided by the distance b between vortices in the same row) ? = 0.41 and the symmetric double row for any spacing ratio are most unstable to a three-dimensional instability of zigzag type that vertically bends the vortices. The most amplified vertical wavenumber scales like 1/(bFh) and the growth rate scales with the strain (2pb2), where is the vortex circulation. For the Kármán vortex street, the zigzag instability is symmetric with respect to the midplane between the two rows while it is antisymmetric for the symmetric double row. For the Kármán vortex street with well-separated vortex rows ? > 0.41 and the single row, the dominant instability is two-dimensional and corresponds to a pairing of adjacent vortices of the same row. The main differences between stratified and homogeneous fluids are the opposite symmetry of the dominant three-dimensional instabilities and the scaling of their most amplified wavenumber. When Fh > 1, three-dimensional instabilities are damped by a viscous critical layer. In the presence of background rotation in addition to the stratification, symmetric and antisymmetric modes no longer decouple and cyclonic vortices are less bent than anticyclonic vortices. However, the dominant instability remains qualitatively the same for the three vortex arrays, i.e. quasi-symmetric or quasi-antisymmetric and three-dimensional or two-dimensional. The growth rate continues to scale with the strain but the most unstable wavenumber of three-dimensional instabilities decreases with rotation and scales like Ro/(bFh) for small Rossby number Ro, in agreement with quasi-geostrophic scaling laws. © 2011 Cambridge University Press

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“…Perturbations changed from antisymmetric to symmetric when vortices were co-rotating, and their wavelength depended on the separation distance rather than vortex radius as was the case for counter-rotating vortices. The effect of planetary rotation on the corotating zigzag instability was investigated by Otheguy, Billant & Chomaz (2006b). Anticyclonic rotation with Ro < −3.67 was found to decrease the vertical length scale associated with zigzag instability, while weaker anticyclonic rotation rates increased the length scale.…”

Section: Introductionmentioning

confidence: 99%

Evolution of a stratified rotating shear layer with horizontal shear. Part 2. Nonlinear evolution

Arobone

Sarkar

2013

J. Fluid Mech.

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Direct numerical simulation is used to investigate the nonlinear evolution of a horizontally oriented mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important dimensional parameters governing inviscid dynamics are maximum shear S(t), buoyancy frequency N, angular velocity of rotation Ω and characteristic shear thickness L(t). The effect of rotation rate, Ω, on the development of fluctuations in the shear layer is systematically studied in a regime of strong stratification. An instability mechanism, qualitatively distinct from the inertial instability, is found to deform columnar vortex cores in vertical planes for a strongly stratified rotating mixing layer. This mechanism emerges when centreline absolute vertical vorticity, ω 3 (t) + 2Ω, is nearly zero as predicted by the linear stability analysis in Part 1 (J. Fluid. Mech., vol. 703, 2012, pp. 29-48). When the initial rotation rate is moderately anticyclonic, strong destabilization and a cascade to small scales is observed, consistent with prior studies involving horizontally sheared flow in the presence of rotation. Examination of enstrophy budgets in cases which are initially inertially unstable reveal the importance of baroclinic torque in maintaining lateral enstrophy fluctuations substantially beyond the time when the flow becomes inertially stable. The cyclonic stratified cases show weak nonlinearity in vortex dynamics. At high Reynolds number, despite the strong stratification, the flow exhibits three-dimensional, nonlinear dynamics and significant vertical mixing except for cases where the rotation is stabilizing.

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The effect of planetary rotation on the zigzag instability of co-rotating vortices in a stratified fluid

Cited by 15 publications

References 22 publications

Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses.

Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses.

Three-dimensional stability of vortex arrays in a stratified and rotating fluid

Evolution of a stratified rotating shear layer with horizontal shear. Part 2. Nonlinear evolution

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