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Liouville links and chains on the plane and associated stationary point vortex equilibria

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  • Liouville links and chains are exact steady solutions of the Euler equation for two-dimensional, incompressible, homogeneous and planar fluid flow, uncovered recently in [11,12,13]. These solutions consist of a set of stationary point vortices embedded in a smooth non-zero and non-uniform background vorticity described by a Liouville-type partial differential equation. The solutions contain several arbitrary parameters and possess a rich structure. The background vorticity can be varied with one of the parameters, resulting in two limiting cases where it concentrates into some point vortex equilibrium configuration in one limit and another distinct point vortex equilibrium in the other limit. By a simple scaling of the point vortex strengths at a limit, a new steady solution can be constructed, and the procedure iterated indefinitely in some cases. The resulting sequence of solutions has been called a Liouville chain [13]. A transformation exists that can produce the limiting point vortex equilibria from a given seed equilibrium. In this paper, we collect together all these results in a review and present selected new examples corresponding to special sequences of 'collapse configurations.' The final section discusses possible applications to different geophysical flow scenarios.

    Mathematics Subject Classification: Primary: 76B47, 35Q35.


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  • Figure 1.  Collapse configurations for the Adler–Moser (AM, column 1) and the two Loutsenko hierarchies ($ \text{L}_1 $ and $ \text{L}_2 $, columns 2 and 3) shown at the stages $ n = 1, 3, 5, 10 $. Size of the markers is proportional to the strengths of the positive (blue diamond) and negative (black disc) point vortices

    Figure 2.  A single link in each Liouville chain shown corresponding to AM, $ \text{L}_1 $ and $ \text{L}_2 $. The parameter $ A $ varies increases from $ 0 $ towards $ \infty $ as we move down the columns. Similar links exist between every consecutive pair of SPVE in each column of figure 1

    Figure 3.  Collapse configurations for AM, $ \text{L}_1 $ and $ \text{L}_2 $ with the choice of constants $ C_n = -g_n(-2) $ for all $ n\geq 1 $. Selected equilibria, corresponding to $ g_5' $, $ g_{10}' $, $ g_{15}' $ and $ g_{20}' $, are shown for each hierarchy. The chosen collapse point $ -2 $ is marked by a red triangle

    Figure 4.  Collapse configurations for AM, $ \text{L}_1 $ and $ \text{L}_2 $ with the choice of constants $ C_n = -g_n(1+i) $ for $ n\geq 1 $

    Figure 5.  Collapse configurations $ g_{35}' $ in the AM, $ \text{L}_1 $ and $ \text{L}_2 $ hierarchies, with the choices of constants $ C_0 = -\frac{1}{2\varGamma+1} $ and $ C_n = -g_n(1) $ or $ C_n = -g_n(-2) $ for $ n\geq 1 $

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