# American Institute of Mathematical Sciences

July  2022, 21(7): 2383-2397. doi: 10.3934/cpaa.2022076

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

 Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, Telangana 502285, India

Received  February 2022 Revised  March 2022 Published  July 2022 Early access  April 2022

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.

Citation: Vikas S. Krishnamurthy. Liouville links and chains on the plane and associated stationary point vortex equilibria. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2383-2397. doi: 10.3934/cpaa.2022076
##### References:
 [1] M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-deVries equation, Commun. Math. Phys., 30 (1978), 1-30. [2] A. Adriani, A. Mura, G. Orton and et al., Clusters of cyclones encircling Jupiter's poles, Nature, 555 (2018), 216-219. [3] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [4] A. Constantin, D. G. Crowdy, V. S. Krishnamurthy and M. H. Wheeler, Stuart-type polar vortices on a rotating sphere, Discrete Contin. Dyn. Syst. A, 41 (2021), 201-215.  doi: 10.3934/dcds.2020263. [5] A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. R. Soc. A, 477 (2021), 20200424. [6] A. Constantin and V. S. Krishnamurthy, Stuart-type vortices on a rotating sphere, J. Fluid Mech., 865 (2019), 1072-1084.  doi: 10.1017/jfm.2019.109. [7] D. Crowdy, Polygonal N-vortex arrays: A Stuart model, Phys. Fluids, 15 (2003), 3710-3717.  doi: 10.1063/1.1623766. [8] D. G. Crowdy, General solutions to the 2D Liouville equation, Int. J. Eng. Sci., 35 (1997), 141-149.  doi: 10.1016/S0020-7225(96)00080-8. [9] D. G. Crowdy, Stuart vortices on a sphere, J. Fluid Mech., 498 (2004), 381-402.  doi: 10.1017/S0022112003007043. [10] A. E. Gill, Atmosphere-Ocean Dynamics: An Introductory Text, International geophysics series, Academic Press, 1982. [11] V. S. Krishnamurthy, M. H. Wheeler, D. G. Crowdy and A. Constantin, Steady point vortex pair in a field of Stuart-type vorticity, J. Fluid Mech., 874 (2019), 11 pp. doi: 10.1017/jfm.2019.502. [12] V. S. Krishnamurthy, M. H. Wheeler, D. G. Crowdy and A. Constantin, A transformation between stationary point vortex equilibria, Proc. R. Soc. A, 476 (2020), 20200310. [13] V. S. Krishnamurthy, M. H. Wheeler, D. G. Crowdy and A. Constantin, Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation, J. Fluid Mech., 921 (2021), 35 pp. doi: 10.1017/jfm.2021.285. [14] C. Li, A. P. Ingersoll, A. P. Klipfel and H. Brettle, Modeling the stability of polygonal patterns of vortices at the poles of Jupiter as revealed by the Juno spacecraft, Proceed. Nation. Acad. Sci., 117 (2020), 24082-24087. [15] I. Loutsenko, Equilibrium of charges and differential equations solved by polynomials, J. Phys. A: Math. Gen., 37 (2004), 1309-1321.  doi: 10.1088/0305-4470/37/4/017. [16] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. [17] K. Marynets, Stuart-type vortices modeling the antarctic circumpolar current, Monatshefte für Mathematik, 191 (2019), 749-759.  doi: 10.1007/s00605-019-01319-0. [18] A. Mura, A. Adriani and A. Bracco, et al., Oscillations and stability of the Jupiter polar cyclones, Geophys. Res. Lett., 48 (2021), 8 pp. [19] P. K. Newton, The $N$-vortex problem: Analytical techniques, in Applied Mathematical Sciences, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3. [20] P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992. [21] J. T. Stuart, On finite amplitude oscillations in laminar mixing layers, J. Fluid Mech., 29 (1967), 417-440. [22] A. Tur and V. Yanovsky, Point vortices with a rational necklace: New exact stationary solutions of the two-dimensional Euler equation, Phys. Fluids, 16 (2004), 2877-2885.  doi: 10.1063/1.1760772.

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##### References:
 [1] M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-deVries equation, Commun. Math. Phys., 30 (1978), 1-30. [2] A. Adriani, A. Mura, G. Orton and et al., Clusters of cyclones encircling Jupiter's poles, Nature, 555 (2018), 216-219. [3] A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019. [4] A. Constantin, D. G. Crowdy, V. S. Krishnamurthy and M. H. Wheeler, Stuart-type polar vortices on a rotating sphere, Discrete Contin. Dyn. Syst. A, 41 (2021), 201-215.  doi: 10.3934/dcds.2020263. [5] A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. R. Soc. A, 477 (2021), 20200424. [6] A. Constantin and V. S. Krishnamurthy, Stuart-type vortices on a rotating sphere, J. Fluid Mech., 865 (2019), 1072-1084.  doi: 10.1017/jfm.2019.109. [7] D. Crowdy, Polygonal N-vortex arrays: A Stuart model, Phys. Fluids, 15 (2003), 3710-3717.  doi: 10.1063/1.1623766. [8] D. G. Crowdy, General solutions to the 2D Liouville equation, Int. J. Eng. Sci., 35 (1997), 141-149.  doi: 10.1016/S0020-7225(96)00080-8. [9] D. G. Crowdy, Stuart vortices on a sphere, J. Fluid Mech., 498 (2004), 381-402.  doi: 10.1017/S0022112003007043. [10] A. E. Gill, Atmosphere-Ocean Dynamics: An Introductory Text, International geophysics series, Academic Press, 1982. [11] V. S. Krishnamurthy, M. H. Wheeler, D. G. Crowdy and A. Constantin, Steady point vortex pair in a field of Stuart-type vorticity, J. Fluid Mech., 874 (2019), 11 pp. doi: 10.1017/jfm.2019.502. [12] V. S. Krishnamurthy, M. H. Wheeler, D. G. Crowdy and A. Constantin, A transformation between stationary point vortex equilibria, Proc. R. Soc. A, 476 (2020), 20200310. [13] V. S. Krishnamurthy, M. H. Wheeler, D. G. Crowdy and A. Constantin, Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation, J. Fluid Mech., 921 (2021), 35 pp. doi: 10.1017/jfm.2021.285. [14] C. Li, A. P. Ingersoll, A. P. Klipfel and H. Brettle, Modeling the stability of polygonal patterns of vortices at the poles of Jupiter as revealed by the Juno spacecraft, Proceed. Nation. Acad. Sci., 117 (2020), 24082-24087. [15] I. Loutsenko, Equilibrium of charges and differential equations solved by polynomials, J. Phys. A: Math. Gen., 37 (2004), 1309-1321.  doi: 10.1088/0305-4470/37/4/017. [16] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. [17] K. Marynets, Stuart-type vortices modeling the antarctic circumpolar current, Monatshefte für Mathematik, 191 (2019), 749-759.  doi: 10.1007/s00605-019-01319-0. [18] A. Mura, A. Adriani and A. Bracco, et al., Oscillations and stability of the Jupiter polar cyclones, Geophys. Res. Lett., 48 (2021), 8 pp. [19] P. K. Newton, The $N$-vortex problem: Analytical techniques, in Applied Mathematical Sciences, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3. [20] P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992. [21] J. T. Stuart, On finite amplitude oscillations in laminar mixing layers, J. Fluid Mech., 29 (1967), 417-440. [22] A. Tur and V. Yanovsky, Point vortices with a rational necklace: New exact stationary solutions of the two-dimensional Euler equation, Phys. Fluids, 16 (2004), 2877-2885.  doi: 10.1063/1.1760772.
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
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
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
Collapse configurations for AM, $\text{L}_1$ and $\text{L}_2$ with the choice of constants $C_n = -g_n(1+i)$ for $n\geq 1$
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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