The merger of geophysical vortices at finite Rossby and Froude number

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.

Section: Discussionsupporting

confidence: 86%

Section: Discussionmentioning

confidence: 98%

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

McKiver

2021

“…We determine the equilibria for the m-vortex problem for 2 m 7. It should be noted that the equilibria for m = 2 were originally obtained at lower resolution by Reinaud & Dritschel (2002) and at high resolution in Reinaud & Dritschel (2018a).…”

Section: Finite Volume Vorticesmentioning

confidence: 99%

Three-dimensional quasi-geostrophic vortex equilibria with -fold symmetry

Reinaud

2019

Self Cite

We investigate arrays of $m$ three-dimensional, unit-Burger-number, quasi-geostrophic vortices in mutual equilibrium whose centroids lie on a horizontal circular ring; or$m+1$ vortices where the additional vortex lies on the vertical ‘central’ axis passing through the centre of the array. We first analyse the linear stability of circular point vortex arrays. Three distinct categories of vortex arrays are considered. In the first category, the $m$ identical point vortices are equally spaced on a circular ring and no vortex is located on the vertical central axis. In the other two categories, a ‘central’ vortex is added. The latter two categories differ by the sign of the central vortex. We next turn our attention to finite-volume vortices for the same three categories. The vortices consist of finite volumes of uniform potential vorticity, and the equilibrium vortex arrays have an (imposed) $m$-fold symmetry. For simplicity, all vortices have the same volume and the same potential vorticity, in absolute value. For such finite-volume vortex arrays, we determine families of equilibria which are spanned by the ratio of a distance separating the vortices and the array centre to the vortices’ mean radius. We determine numerically the shape of the equilibria for $m=2$ up to $m=7$, for each three categories, and we address their linear stability. For the $m$-vortex circular arrays, all configurations with $m\geqslant 6$ are unstable. Point vortex arrays are linearly stable for $m<6$. Finite-volume vortices may, however, be sensitive to instabilities deforming the vortices for $m<6$ if the ratio of the distance separating the vortices to their mean radius is smaller than a threshold depending on $m$. Adding a vortex on the central axis modifies the overall stability properties of the vortex arrays. For $m=2$, a central vortex tends to destabilise the vortex array unless the central vortex has opposite sign and is intense. For $m>2$, the unstable regime can be obtained if the strength of the central vortex is larger in magnitude than a threshold depending on the number of vortices. This is true whether the central vortex has the same sign as or the opposite sign to the peripheral vortices. A moderate-strength like-signed central vortex tends, however, to stabilise the vortex array when located near the plane containing the array. On the contrary, most of the vortex arrays with an opposite-signed central vortex are unstable.

“…Tsang & Dritschel 2015; McKiver & Dritschel 2016; Mahdinia et al. 2017; Reinaud & Dritschel 2018; McKiver 2020; Sokolovskiy, Carton & Filyushkin 2020, and references therein).…”

Section: Introductionmentioning

confidence: 99%

Point-vortex dynamics in three-dimensional ageostrophic balanced flows

Weiss

2022

Geophysical turbulent flows, characterized by rapid rotation, quantified by small Rossby number, and stable stratification, often self-organize into a collection of coherent vortices, referred to as a vortex gas. The lowest-order asymptotic expansion in Rossby number is quasigeostrophy, which has purely horizontal velocities and cyclone–anticyclone antisymmetry. Ageostrophic effects are important components of many geophysical flows and, as such, these phenomena are not well modelled by quasigeostrophy. The next-order correction in Rossby number, which includes ageostrophic effects, is the so-called balanced dynamics. Balanced dynamics includes ageostrophic vertical velocity and breaks the geostrophic cyclone–anticyclone antisymmetry. Point-vortex solutions are well known in two-dimensional and quasigeostrophic dynamics and are useful for studying the vortex-gas regime of geophysical turbulence. Here, we find point-vortex solutions in fully three-dimensional continuously stratified QG $^{+1}$ dynamics, a particular formulation of balanced dynamics. Simulations of QG $^{+1}$ point vortices show several interesting features not captured by quasigeostrophic point vortices including significant vertical transport on long time scales. The ageostrophic component of QG $^{+1}$ point vortex point-vortex dynamics renders them useful in modelling flows where quasigeostrophy filters out important physical processes.