Analogies and differences between the stability of an isolated pancake vortex and a columnar vortex in stratified fluid

Abstract

In order to understand the dynamics of pancake shaped vortices in stably stratified fluids, we perform a linear stability analysis of an axisymmetric vortex with Gaussian angular velocity in both the radial and axial directions with an aspect ratio of α. The results are compared to those for a columnar vortex (α = ∞) in order to identify the instabilities. Centrifugal instability occurs when R > c(m) where R = ReF 2h is the buoyancy Reynolds number, F h the Froude number, Re the Reynolds number and c(m) a constant which differs for the three unstable azimuthal wavenumbers m = 0, 1, 2. The maximum growth rate depends mostly on R and is almost independent of the aspect ratio α. For sufficiently large buoyancy Reynolds number, the axisymmetric mode is the most unstable centrifugal mode whereas for moderate R, the mode m = 1 is the most unstable. Shear instability for m = 2 develops only when F h 0.5α. By considering the characteristics of shear instability for a columnar vortex with the same parameters, this condition is shown to be such that the vortex is taller than the minimum wavelength of shear instability in the columnar case. For larger Froude number F h 1.5α, the isopycnals overturn and gravitational instability can operate. Just below this threshold, the azimuthal wavenumbers m = 1, 2, 3 are unstable to baroclinic instability. A simple model shows that baroclinic instability develops only above a critical vertical Froude number F h /α because of confinement effects.

Diffusive effects in local instabilities of a baroclinic axisymmetric vortex

“…(2021 a , b ), also establishes the local stability framework as an alternative approach to understand curvature effects. Finally, an insightful interpretation of (3.10) is that the square of the absolute circulation () decreases with radius when moving along iso-density surfaces (Emanuel 1979); this form of the symmetric instability criterion has been previously used to interpret instabilities in pancake vortices (Negretti & Billant 2013; Yim & Billant 2016; Yim, Billant & Ménesguen 2016).…”

“…Depending on the vorticity distribution, Reynolds number, Froude number, Burger number and Schmidt number, these pancake vortices can be in different instability regimes, including barotropic and baroclinic instabilities (Yim et al. 2016; Storer, Poulin & Ménesguen 2018), short-wavelength instabilities such as symmetric and double-diffusive instabilities (Negretti & Billant 2013; Yim & Billant 2016). Some of these instabilities are recognized as driving mechanisms for density layering observed in oceanic eddies (Hua et al.…”

“…2013). Of specific relevance to our study are the results of Yim & Billant (2016), who performed global stability analysis to investigate short-wavelength and other instabilities in Gaussian pancake vortices. Yim & Billant (2016) also presented results for , to show that oscillatory instability (McIntyre 1970) occurs at but not at .…”

Stable Vortices in a Continuously Stratified Ocean with Thin Active Layer

“…This result fits nicely with the concept of "uniform PV in the lower layer" advocated in [14,15], and, thus, appears to resolve the paradox of the longevity of oceanic eddies. On the other hand, non-uniform-PV vortices with continuous stratification turned out to be unstable [17][18][19].…”

Piecewise uniform potential vorticity pancake shielded vortices

“…These vortices appear to be unstable to small perturbations (see Sutyrin 2015 for a discussion). More recently, in the absence of background rotation, Yim and Billiant (2016) have studied the stability of a stratified vortex with a Gaussian angular velocity profile in cyclostrophic and hydrostatic balances for various values of height-to-width aspect ratio of the structure. The authors have shown that the vortex may be sensitive to centrifugal, shear, gravitational and baroclinic instabilities, depending in particular on the Reynolds and Froude numbers, and the vortex aspect ratio.…”