Volume 26, Number 5
Volume 26, Number 5, 2021
Special Issue: 200th birthday of Hermann von Helmholtz
Dritschel D. G., Sokolovskiy M. A., Stremler M. A.
Abstract

Dritschel D. G.
Abstract
This paper examines the stability and nonlinear evolution of configurations of equalstrength point vortices equally spaced on a ring of constant radius, with or without a central vortex, in the threedimensional quasigeostrophic compressible atmosphere model. While the ring lies at constant height, the central vortex can be at a different height and also have a different strength to the vortices on the ring. All such configurations are relative equilibria, in the sense that they steadily rotate about the $z$ axis. Here, the domains of stability for two or more vortices on a ring with an additional central vortex are determined. For a compressible atmosphere, the problem also depends on the density scale height $H$, the vertical scale over which the background density varies by a factor $e$. Decreasing $H$ while holding other parameters fixed generally stabilises a configuration. Nonlinear simulations of the dynamics verify the linear analysis and reveal potentially chaotic dynamics for configurations having four or more vortices in total. The simulations also reveal the existence of staggered ring configurations, and oscillations between single and double ring configurations. The results are consistent with the observations of ring configurations of polar vortices seen at both of Jupiter’s poles [1].

Stremler M. A.
Abstract
The classic problem of three point vortex motion on the plane is revisited by using the interior angles of the vortex triangle, $\theta_{j}$, $j=1,2,3$, as the key system variables instead of the lengths of the triangle sides, $s_j$, as has been used classically. Similar to the classic approach, the relative vortex motion can be represented in a phase space, with the topology of the level curves characterizing the motion. In contrast to the classic approach, the alternate formulation gives a compact, consistent phase space representation and facilitates comparisons of vortex motion in a comoving frame. This alternate formulation is used to explore the vortex behavior in the two canonical cases of equal vortex strength magnitudes, $\Gamma_{1} = \Gamma_{2} = \Gamma_{3}$ and $\Gamma_{1} = \Gamma_{2} = \Gamma_{3}$.

Reinaud J. N.
Abstract
We determine and characterise relative equilibria for arrays of point vortices in a
threedimensional quasigeostrophic flow. The vortices are equally spaced along two horizontal
rings whose centre lies on the same vertical axis. An additional vortex may be placed along
this vertical axis. Depending on the parameters defining the array, the vortices on the two rings
are of equal or opposite sign. We address the linear stability of the point vortex arrays. We
find both stable equilibria and unstable equilibria, depending on the geometry of the array. For
unstable arrays, the instability may lead to the quasiregular or to the chaotic motion of the
point vortices. The linear stability of the vortex arrays depends on the number of vortices in
the array, on the radius ratio between the two rings, on the vertical offset between the rings
and on the vertical offset between the rings and the central vortex, when the latter is present.
In this case the linear stability also depends on the strength of the central vortex. The nonlinear
evolution of a selection of unstable cases is presented exhibiting examples of quasiregular
motion and of chaotic motion.

Kurakin L. G., Ostrovskaya I. V.
Abstract
A system of four point vortices on a plane is considered. Its motion is described by
the Kirchhoff equations. Three vortices have unit intensity and one vortex has arbitrary intensity $\varkappa$. We study the stability problem for the stationary rotation of a vortex quadrupole consisting of three identical vortices located uniformly on a circle around a fourth vortex.
It is known that for $ \varkappa> 1 $ the regime under study is unstable,
and in the case of $ \varkappa <3 $ and $ 0 <\varkappa <1 $ the orbital stability takes place. New results are obtained for $ 3 <\varkappa <0 $. It is found that, for all values of $ \varkappa $ in the
stability problem, there is a resonance $1:1$ (diagonalizable case). Some other resonances
through order four are found and investigated: double zero resonance
(diagonalizable case), resonances 1:2 and 1:3, occurring with isolated values of $\varkappa $.
The stability of the equilibrium of the system reduced by one degree of freedom with the involvement of the
terms in the Hamiltonian through degree four is proved for all $ \varkappa \in (3,0) $.

de Marez C., Carton X.
Abstract
Coastal upwellings, due to offshore Ekman transport, are more energetic at
the western boundaries of the oceans, where they are intensified by incoming
Rossby waves, than at the eastern boundaries. Western boundary upwellings
are often accompanied by a local vortex field. The instability of a developed
upwelling front and its interaction with an external vortex field is studied
here with a threedimensional numerical model of the hydrostatic rotating NavierStokes
equations (the primitive equations). The baroclinic instability of the front
leads to the growth of meanders with 100200 km wavelength, in the absence of
external vortex. On the $f$plane, these waves can break into a row of vortices
when the instability is intense. The $\beta$effect is stabilizing and strongly
decreases the amplitude of meanders. Simulations are then performed with a front
initially accompanied by one or several external vortices. The evolutions in
this case are compared with those of the unstable jet alone. On the $f$plane,
when an external vortex is close to the front, this latter sheds a long filament
which wraps up around the vortex. This occurs over a period similar to that of
the instability of the isolated front. Cyclones are more efficient in tearing
such filaments offshore than anticyclones. On the $\beta$plane, the filaments
are short and turbulence is confined to the vicinity of the front. At long times,
waves propagate along the front, thus extending turbulence alongshore. The
initial presence of a vortex alley leads to a stronger destabilization of the
front and to a larger crossshore flux than for a single vortex, with many
filaments and small vortices pushed far offshore. In the ocean, this crossshore
exchange has important consequences on the local biological activity.

Riccardi G., Dritschel D. G.
Abstract
The paper deals with the calculation of the internal singularities of the Schwarz
function corresponding to the boundary of a planar vortex patch during its selfinduced motion
in an inviscid, isochoric fluid. The vortex boundary is approximated by a simple, timedependent
map onto the unit circle, whose coefficients are obtained by fitting to the boundary computed in
a contour dynamics numerical simulation of the motion. At any given time, the branch points of
the Schwarz function are calculated, and from them, the generally curved shape of the internal
branch cut, together with the jump of the Schwarz function across it. The knowledge of the
internal singularities enables the calculation of the Schwarz function at any point inside the
vortex, so that it is possible to check the validity of the map during the motion by comparing
left and right hand sides of the evolution equation of the Schwarz function. Our procedure
yields explicit functional forms of the analytic continuations of the velocity and its conjugate
on the vortex boundary. It also opens a new way to understand the relation between the time
evolution of the shape of a vortex patch during its motion, and the corresponding changes in
the singular set of its Schwarz function.
