Point Process Analysis of Vortices in a Periodic Box

The motion of assemblies of point vortices in a periodic parallelogram can be described by the complex position $z_j(t)$ whose time derivative is given by the sum of the complex velocities induced by other vortices and the solid rotation centered at $z_j$. A numerical simulation up to 100 vortices in a square periodic box is performed with various initial conditions, including single and double rows, uniform spacing, checkered pattern, and complete spatial randomness. Point process theory in spatial ecology is applied in order to quantify clustering of the distribution of vortices. In many cases, clustering of the distribution persists after a long time if the initial condition is clustered. In the case of positive and negative vortices with the same absolute value of strength, the $L$ function becomes positive for both types of vortices. Scattering or recoupling of pairs of vortices by a third vortex is remarkable.

💡 Research Summary

The paper investigates the dynamics of point‑vortex assemblies confined to a periodic parallelogram, focusing on the special case of a square periodic box. Each vortex is represented by a complex position (z_j(t)=x_j(t)+i y_j(t)) whose time derivative is the sum of the complex velocities induced by all other vortices together with a solid‑body rotation term centered at the vortex position. The governing equation is

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