Geostrophic tripolar vortices in a two-layer fluid: Linear stability and nonlinear evolution of equilibria

Reinaud, Jean Noel; Sokolovskiy, Mikhail; Carton, Xavier

doi:10.1063/1.4978806

Cited by 9 publications

(7 citation statements)

References 69 publications

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“…Thomson's seminal work also inspired further studies, including Dritschel (1985) where the point vortex configurations were generalised to arrays of two-dimensional finite area patches of uniform vorticity. Other configurations of finite area vortex equilibria with m−fold symmetries have been sought for two-dimensional vortices by Burbea (1982); Wu et al (1984); Crowdy (2002Crowdy ( , 2003; Kizner & Khvoles (2004a,b); Xue et al (2017) and for geophysical vortices by Kizner et al (2007); Shteinbuch-Fridman et al 2015; Kizner et al (2017); Shteinbuch-Fridman et al 2017; Reinaud et al (2017) to name but a few studies.…”

Section: Introductionmentioning

confidence: 99%

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

Reinaud

2019

J. Fluid Mech.

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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.

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Section: Introductionmentioning

confidence: 99%

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

Reinaud

2019

J. Fluid Mech.

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“…This means that the minimal distance from the fixed vortex to either of the moving ones is equal to half the initial distance between pair's vortices. are performed using the contour dynamic techniques as in the work 38 . Detailed analysis of the configuration using the finite-size vortex model is reported in the accompanying paper 44 .…”

Section: Symmetric Casementioning

confidence: 99%

“…Since the model allows for vorticity redistribution through the deformation of the vortices and generally more complicated dynamics, many authors look for steady or periodic solutions that ensure that finite area vortices engaged into the interactions remain coherent and retain their initial shape for a reasonably long time. Approaches into finding such solutions include the ones based on contour dynamics 10,[35][36][37][38][39] or analytical ones using complex analysis 40 . In particular, recent studies 38,41 attest that finite area vortex systems can manifest dynamical regimes very similar to their analogous point-vortex counterparts.…”

Section: Introductionmentioning

confidence: 99%

“…Approaches into finding such solutions include the ones based on contour dynamics 10,[35][36][37][38][39] or analytical ones using complex analysis 40 . In particular, recent studies 38,41 attest that finite area vortex systems can manifest dynamical regimes very similar to their analogous point-vortex counterparts. Nevertheless, the problem of the stability of the finite area vortex systems needs a separate thorough consideration.…”

Section: Introductionmentioning

confidence: 99%

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Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices

et al. 2018

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The problem of a pair of point vortices impinging on a fixed point vortex of arbitrary strengths [E. Ryzhov and K. Koshel, EPL 102, 44004 (2013)] is revisited and investigated comprehensively. Although the motion of pair of point vortices is established to be regular, the model presents a plethora of possible bounded and unbounded solutions with complicated vortex trajectories. The initial classification [E. Ryzhov and K. Koshel, EPL 102, 44004 (2013)] revealed that pair could be compelled to perform bounded or unbounded motion without giving a full classification of either of those dynamical regimes. The present work capitalizes upon the previous results and introduces a finer classification with a multitude of possible regimes of motion. Regimes of bounded motion for the vortex pair entrapped near the fixed vortex or of unbounded motion, when the vortex pair moves away from the fixed vortex, can be categorized by varying the two governing parameters: (i) the ratio of the distances between pair's vortices and the fixed vortex, and (ii) the ratio of the strengths of the vortices of the pair and the strength of the fixed vortex. In particular, a bounded motion regime where one of pair's vortices does not rotate about the fixed vortex is revealed. In this case, only one of pair's vortices rotates about the fixed vortex, while the other one oscillates at a certain distance. Extending the results obtained with the point-vortex model to an equivalent model of finite size vortices is the focus of a second paper.

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“…35 Apart from these 2D vortex equilibria, geophysical tripolar vortex equilibria in a two-layer fluid have been found numerically and studied recently. [36][37][38] A further, new, class of multipolar vortex equilibria are computed here: finite-area vortex equilibria consisting of two sets of nested polygonally arranged vortex patches, i.e., a structure comprising two sets of identical vortex patches arranged at the vertices of polygons of different sizes but sharing the same centre. There exist two distinct cases: when the two sets of polygonal patches are aligned with each other and when they are staggered.…”

Section: Introductionmentioning

confidence: 99%

New families of vortex patch equilibria for the two-dimensional Euler equations

2017

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Various modified forms of contour dynamics are used to compute multipolar vortex equilibria, i.e., configurations of constant vorticity patches which are invariant in a steady rotating frame. There are two distinct solution families for “N + 1” point vortex-vortex patch equilibria in which a finite-area central patch is surrounded by N identical point vortices: one with the central patch having opposite-signed vorticity and the other having same-signed vorticity to the satellite vortices. Each solution family exhibits limiting states beyond which no equilibria can be found. At the limiting state, the central patch of a same-signed equilibrium acquires N corners on its boundary. The limiting states of the opposite-signed equilibria have cusp-like behaviour on the boundary of the central patch. Linear stability analysis reveals that the central patch is most linearly unstable as it approaches the limiting states. For equilibria comprising a central patch surrounded by N identical finite-area satellite patches, again two distinct families of solutions exist: one with the central patch and satellite patches having the same-signed vorticity and the other in which they are opposite-signed. In each family, there are two limiting behaviours in which either the central patch or the satellite patches develop corners or cusps. Streamline plots and time-dependent simulations indicate that opposite-signed multipolar equilibria are robust structures and same-signed equilibria are generally less stable. Streamlines also reveal stable and unstable (saddle point) stagnation points, indicating the existence of new equilibria in which additional patches of vorticity are “grown” at the stagnation points. Examples of such equilibria are computed, and a general numerical routine is briefly described for finding even more complex finite-area equilibria. Finally, new nested polygonal vortex equilibria consisting of two sets of polygonally arranged vortex patches (named “N + N” equilibria here) are computed for two distinct cases: one with the corners of the polygons aligned with each other and the other when they are staggered. Various limiting states are computed for these equilibria. Time-dependent simulations reveal that the aligned equilibria are susceptible to instability, while the staggered equilibria survive a relatively long time. In some parameter regimes, following instability, these structures evolve into known structures such as “N + 1” multipolar vortex equilibria and N-polygon co-rotating equilibria.

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Geostrophic tripolar vortices in a two-layer fluid: Linear stability and nonlinear evolution of equilibria

Cited by 9 publications

References 69 publications

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

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

Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices

New families of vortex patch equilibria for the two-dimensional Euler equations

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