Analytical Derivation of Correlation Function for Vorticity-Depleted Regions Formed Around Strong Vortex

Background Two-dimensional turbulence exhibits distinct behavior from three-dimensional turbulence, characterized by an inverse energy cascade that transports energy from small to large scales, leading to the formation of large-scale vortices and organized structures. Onsager proposed the concept of negative absolute temperature states to explain this large-scale structure formation. Pure electron plasma experiments, where electron motion perpendicular to a strong axial magnetic field obeys the two-dimensional incompressible Euler equation, provide an ideal platform for studying such phenomena. In these experiments, the electron number density corresponds to vorticity, and the electrostatic potential to the stream function. Experiments at Kyoto University revealed the formation of depleted vorticity regions around strong vortices, which remained theoretically unexplained. Methods We employ linear response theory combined with mean-field approximation for point vortex systems with negative absolute temperature to investigate this depletion mechanism. Point vortices represent the vorticity field as a collection of delta-function singularities. We consider a delta-function vortex impulsively injected into an equilibrium state of uniform background vorticity and analytically derive the two-body correlation function that characterizes the system’s response to this perturbation. The mean-field approximation treats the continuous particle distribution emerging in the infinite particle limit. Results Our analysis demonstrates that the two-body correlation function exhibits negative values in the vicinity of the injected vortex. This negative correlation corresponds to a depleted vorticity region surrounding the strong vortex, consistent with experimental observations. Conclusions The analytical framework provides a qualitative explanation for the long-standing question of vorticity depletion formation around strong vortices in two-dimensional flows. However, our linear response treatment does not capture longer-time dynamics, and quantitative agreement with experiments requires further investigation through large-scale numerical simulations.