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Balanced ellipsoidal vortex equilibria in a background shear flow at finite Rossby number

Published online by Cambridge University Press:  15 September 2021

William J. McKiver
William J. McKiver*
Affiliation:
ISMAR-CNR, Arsenale - Tesa 104, Castello 2737/F, 30122Venice, Italy
*
†Email address for correspondence: [email protected]
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Abstract

We consider a uniform ellipsoid of potential vorticity (PV), where we exploit analytical solutions derived for a balanced model at the second order in the Rossby number, the next order to quasi-geostrophic (QG) theory, the so-called QG+1 model. We consider this vortex in the presence of an external background shear flow, acting as a proxy for the effect of external vortices. For the QG model the system depends on four parameters, the height-to-width aspect ratio of the vortex, $h/r$, as well as three parameters characterising the background flow, the strain rate, $\gamma$, the ratio of the background rotation rate to the strain, $\beta$, and the angle from which the flow is applied, $\theta$. However, the QG+1 model also depends on the PV, as well as the Prandtl ratio, $f/N$ ($f$ and $N$ are the Coriolis and buoyancy frequencies, respectively). For QG and QG+1 we determine equilibria for different values of the background flow parameters for increasing values of the imposed strain rate up to the critical strain rate, $\gamma _c$, beyond which equilibria do not exist. We also compute the linear stability of this vortex to second-order modes, determining the marginal strain $\gamma _m$ at which ellipsoidal instability erupts. The results show that for QG+1 the most resilient cyclonic ellipsoids are slightly prolate, while anticyclonic ellipsoids tend to be more oblate. The highest values of $\gamma _m$ occur as $\beta \to 1$. For large values of $f/N$, changes in the marginal strain rates occur, stabilising anticyclonic ellipsoids while destabilising cyclonic ellipsoids.

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 926 , 10 November 2021 , A38
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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