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Dynamics of a stratified vortex under the complete Coriolis force: two-dimensional three-components evolution
Published online by Cambridge University Press: 24 October 2022
Abstract
We study the dynamics of an initially axisymmetric and vertical Lamb–Oseen vortex in a stratified-rotating fluid under the complete Coriolis force on the 
$f$ plane, i.e. in the presence of a background rotation both along the vertical and horizontal directions. By a combination of direct numerical simulations and asymptotic analyses for small horizontal background rotation, we show that a critical layer appears at the radius where the angular velocity of the vortex is equal to the buoyancy frequency when the Froude number is larger than unity. This critical layer generates a vertical velocity that is invariant along the vertical and which first increases linearly with time and then saturates at an amplitude scaling like 
$Re^{1/3}/\widetilde{Ro}$, where 
$Re$ is the Reynolds number and 
$\widetilde {Ro}$ the non-traditional Rossby number based on the horizontal component of the background rotation. In turn, a quasi-axisymmetric anomaly of vertical vorticity is produced at the critical radius through the non-traditional Coriolis force. Below a critical 
$\widetilde {Ro}$ depending on 
$Re$, the Rayleigh's inflectional criterion is satisfied and a shear instability is subsequently triggered rendering the vertical vorticity fully non-axisymmetric. The decay of the angular velocity is then enhanced until it is everywhere lower than the buoyancy frequency. A theoretical criterion derived from the Rayleigh condition predicts well the instability. It shows that this phenomenon can occur even for a large non-traditional Rossby number 
$\widetilde {Ro}$ for large 
$Re$. Hence, the non-traditional Coriolis force might have much more impact on geophysical vortices than anticipated by considering the order of magnitude of 
$\widetilde {Ro}$.
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- © The Author(s), 2022. Published by Cambridge University Press
References
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Toghraei and Billant supplementary movie 1
Evolution of the vertical velocity (left) and vertical vorticity (right) for $Re=2000$, $F_h=10$, $Ro=23.1$, $\phi=60^{\circ}$ ($\widetilde{Ro}=40$).
Video
2.8 MB
Toghraei and Billant supplementary movie 2
Evolution of the vertical velocity (left) and vertical vorticity (right) for $Re=2000$, $F_h=2$, $Ro=23.1$, $\phi=60^{\circ}$ ($\widetilde{Ro}=40$).
Video
1.9 MB
Toghraei and Billant supplementary movie 3
Evolution of the vertical velocity (left) and vertical vorticity (right) for $Re=2000$, $F_h=2$, $Ro=20.3$, $\phi=80^{\circ}$ ($\widetilde{Ro}=115.2$).
Video
1.4 MB
Toghraei and Billant supplementary movie 4
Evolution of the vertical velocity (left) and vertical vorticity (right) for $Re=10000$, $F_h=2$, $Ro=20.3$, $\phi=80^{\circ}$ ($\widetilde{Ro}=115.5$).
Video
2.2 MB