Kinetic Phase Space Coherent Structures in Quasi-stationary Electron Holes

Electrostatic phase space coherent structures, present in electron and ion holes, are ubiquitous features in laboratory and space plasmas. In this work, we focus on the two-stream instability, a generator of electron holes, in a Vlasov-Poisson (VP) system with a self-consistent evolution of the electric field that leads to Bernstein Green Kruskal (BGK)-like structures and its self-organization process. Unlike stationary BGK modes, these structures, verified by Gkyell simulations, exhibit persistent time-dependent fluctuations. We find that numerical dynamical system tools, like Lagrangian coherent structures (LCS) and finite time Lyapunov exponents (FTLE), which are regularly used in cutting edge research in fluid mechanics, provide key insight into phase space hole structure, enabling us to surpass the traditional time independent treatment. For analytic progress, we apply techniques in asymptotic methods and ergodic theory to classify and predict LCS growth in real and Fourier space. In real space, we find that LCS dynamics promote a chaos quenching process, and in Fourier space, we report a new stable conic LCS. These persistent LCS generate intermittency in phase space turbulence and their dynamic formation reveals new insights on time-dependent plasma self-organization.