# Vertical decorrelation of a vortex by an external shear flow in a strongly stratified fluid

Abstract : This thesis investigates, theoretically and numerically, the vertical decorrelation of an initially vertical vortex by an ambient sinusoidal shear flow in a stratified fluid. It has been conjectured that such process should trigger the shear instability and, as such, contribute to the generation of small scales in strongly stratified turbulence. This type of turbulence is encountered in the atmosphere and the oceans in an intermediate range of scaleswhere Coriolis effects are negligible.The first part analyses the evolutions of the total energy and enstrophy of the vortex in Direct Numerical Simulations (DNS) as functions of the control parameters. This study reveals that the dynamics differs from freely decaying flows: because of the presence of the ambient shear flow, the balance between stretching and dissipation terms in the global enstrophy budget implies that the maximum enstrophy of the vortex scales as $Re^{2/3}$, where $Re$ is the Reynolds number, instead of simply $Re$. However, such simplified balance does not account for the observed effect of the stratification.In order to overcome this difficulty, the local dynamics of the vortex has been investigated by means of two asymptotic analyses, presented in the second part. A short-time analysis first proves that the initial response of the vortex is non-hydrostatic regardless of the stratification. A long-wavelength analysis provides governing equations for the evolution of the angular velocity of the vortex and the deformations of its axis. Internal waves are excited at the start-up of the motion, explaining the initial non-hydrostatic regime. The vortex is mostly advected in the direction of the shear flow but also perpendicularly owing to the self-induced motion. Its angular velocity decays because of dynamic and viscous effects. The former effect is due to the squeezing of the isopycnals in the vortex core which implies a decrease of the vertical vorticity to conserve potential vorticity.In a third part, the DNS show that the shear instability is only triggered if the Froude number is moderate and the wavelength of the shear small enough. The numerical results are compared to the asymptotic predictions. In particular, the evolutions of the vertical shear of horizontal velocity and of the vertical buoyancy gradient for small Froude number are comprehensively and finely captured by the long-wavelength asymptotic predictions. The minimum value of the asymptotic Richardson number almost never goes below the critical threshold $1/4$ necessary for the development of the shear instability. The saturation of the vertical shear is due to the decay of the vortex in the regions of high ambient shear. These results suggest that the shear instability is not easilytriggered by decorrelation processes in strongly stratified flows in contradiction with previous conjectures.
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### Citation

Julien Bonnici. Vertical decorrelation of a vortex by an external shear flow in a strongly stratified fluid. Mechanics of the fluids [physics.class-ph]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLX024⟩. ⟨tel-01806347⟩

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