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Evolution of a vortex in a strongly stratified shear flow. Part 2. Numerical simulations
- Paul Billant, Julien Bonnici
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- Journal:
- Journal of Fluid Mechanics / Volume 893 / 25 June 2020
- Published online by Cambridge University Press:
- 22 April 2020, A18
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We conduct direct numerical simulations of an initially vertical Lamb–Oseen vortex in an ambient shear flow varying sinusoidally along the vertical in a stratified fluid. The Froude number $F_{h}$ and the Reynolds number $Re$, based on the circulation $\unicode[STIX]{x1D6E4}$ and radius $a_{0}$ of the vortex, have been varied in the ranges: $0.1\leqslant F_{h}\leqslant 0.5$ and $3000\leqslant Re\leqslant 10\,000$. The shear flow amplitude $\hat{U} _{S}$ and vertical wavenumber $\hat{k}_{z}$ lie in the ranges: $0.02\leqslant 2\unicode[STIX]{x03C0}a_{0}\hat{U} _{S}/\unicode[STIX]{x1D6E4}\leqslant 0.4$ and $0.1\leqslant \hat{k}_{z}a_{0}\leqslant 2\unicode[STIX]{x03C0}$. The results are analysed in the light of the asymptotic analyses performed in Part $1$. The vortex is mostly advected in the direction of the shear flow but also in the perpendicular direction owing to the self-induction. The decay of potential vorticity is strongly enhanced in the regions of high shear. The long-wavelength analysis for $\hat{k}_{z}a_{0}F_{h}\ll 1$ predicts very well the deformations of the vortex axis. The evolutions of the vertical shear of the horizontal velocity and of the vertical gradient of the buoyancy at the location of maximum shear are also in good agreement with the asymptotic predictions when $\hat{k}_{z}a_{0}F_{h}$ is sufficiently small. As predicted by the asymptotic analysis, the minimum Richardson number never goes below the critical value $1/4$ when $\hat{k}_{z}a_{0}F_{h}\ll 1$. The numerical simulations show that the shear instability is triggered only when $\hat{k}_{z}a_{0}F_{h}\gtrsim 1.6$ for sufficiently high buoyancy Reynolds number $ReF_{h}^{2}$. There is also a weak dependence of this threshold on the shear flow amplitude. In agreement with the numerical simulations, the long-wavelength analysis predicts that the minimum Richardson number goes below $1/4$ when $\hat{k}_{z}a_{0}F_{h}\gtrsim 1.7$ although this is beyond its expected range of validity.
Evolution of a vortex in a strongly stratified shear flow. Part 1. Asymptotic analysis
- Julien Bonnici, Paul Billant
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- Journal:
- Journal of Fluid Mechanics / Volume 893 / 25 June 2020
- Published online by Cambridge University Press:
- 22 April 2020, A17
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- Article
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In this paper, we investigate the dynamics of an initially vertical vortex embedded in a shear flow in a stratified fluid by means of a long-wavelength analysis. The main goal is to determine, whether or not, the Kelvin–Helmholtz instability can develop unconditionally as speculated by Lilly (J. Atmos. Sci., vol. 40, 1983, pp. 749–761). The analysis is performed in the case of the Lamb–Oseen vortex profile and a shear flow uniform in the horizontal and varying sinusoidally along the vertical using the assumption $\hat{k}_{z}a_{0}F_{h}\ll 1$, where $\hat{k}_{z}$ is the vertical wavenumber, $a_{0}$ the vortex radius and $F_{h}$ the horizontal Froude number based on the circulation of the vortex. The results show that the vortex axis is advected not only in the direction of the shear flow but also in the perpendicular direction owing to the self-induced motion of the vortex. In addition, internal waves are transiently excited at the beginning, generating an initial non-hydrostatic regime. Their relative effects on the displacements of the vortex axis are weak except initially. The angular velocity of the vortex decays owing to a dynamic effect and viscous effects related to the vertical shear. The former effect is due to the squeezing of isopycnals in the vortex core, which implies a decrease of the vertical vorticity to satisfy potential vorticity conservation. In addition, a horizontal velocity field with an azimuthal wavenumber $m=2$ is generated, meaning that the shape of the vortex becomes slightly elliptical. We further show that the minimum asymptotic Richardson number is bounded, $\min (Ri)>3.43$, when $\hat{k}_{z}a_{0}F_{h}\ll 1$ and therefore cannot go below the critical value $1/4$. This is because the growth of the vertical shear of the horizontal velocity of the vortex saturates owing to the decay of its angular velocity and because the squeezing of isopycnals increases the stratification strength. This suggests that the shear instability cannot always develop in strongly stratified flows, contrary to the conjecture of Lilly (as above). These predictions will be tested against direct numerical simulations in Part 2.