Testing For Universality of Markov Solar Wind Turbulence at the Earth’s Magnetosphere on Kinetic Scales Based on the MMS Mission

Understanding turbulence in space and astrophysical plasmas is critical for advancing our comprehension of complex systems governed by nonlinear dynamics. This study extends the application of the Markovian framework in small-scale turbulence in the Earth’s magnetosphere, with a particular focus on the solar wind – magnetosphere interaction observed by NASA Magnetospheric Multiscale (MMS) mission. We benefit from the exceptional resolution of the Fluxgate Magnetometer instrument as well as the Fast Plasma Investigation instrument onboard the MMS. The high temporal resolution, coupled with recent machine learning methods, allows one to identify the turbulent regions and magnetic reconnection events with great accuracy. Hence, the data are analyzed across diverse magnetospheric regions, enabling insights into turbulence-driven energy transfers. With the obtained measurements we could analyze the magnetic field gradients, turbulence intensity, and the plasma parameters. 

By employing the multi-scale probabilistic approach, we explore the turbulent cascade using conditional Probability Density Functions (cPDFs) and the Markovian properties of fluctuations, revealing new insights into the dynamics of energy transfer at sub-ion scales. Our results confirm the Markovian necessary and sufficient properties of the turbulent cascade across kinetic scales, emphasizing the significance of the Einstein-Markov (EM) scale and the intermittent nature of energy transfer to smaller scales. The derived Fokker-Planck equation in scale governs the evolution of cPDFs through drift and diffusion coefficients, which have been directly calculated from the empirical data. This employed framework captures key features of turbulence, including its hierarchical structure, deviations from self-similarity, and the phenomenon of intermittency, evidenced by non-Gaussian statistics and broadened PDF tails. These findings provide a robust description of the cascade process, from large-scale energy input to dissipation at smaller scales.

By investigating turbulence in two electron diffusion regions, where magnetic reconnection may occur, the highlighted Markovian framework and Fokker-Planck methodology are interestingly still viable to describe the complexity of turbulence processes. This gives promising insight into understanding the stochastic nature of reconnection-driven turbulence.

Despite some limitations, including the simplifying assumptions inherent to the Markovian framework and second-order Fokker-Planck equations, our results demonstrate its effectiveness in capturing the essence of kinetic-scale turbulence. The connection between scale-dependent statistics and underlying physical processes, such as intermittency and energy cascades, highlights the framework’s utility for both theoretical and observational studies. 

This work bridges statistical physics and plasma turbulence for analyzing scale-dependent phenomena in magnetospheric plasmas. We hope that by elucidating the interplay of order and randomness in these systems, our findings support the idea of extending stochastic modeling to higher-dimensional problems.

Acknowledgments: This work has been supported by the National Science Centre, Poland (NCN), through grant No. 2021/41/B/ST10/00823.

[1] W. M. Macek, D. Wójcik, & J. L. Burch, 2023, Magnetospheric Multiscale Observations of Markov Turbulence on Kinetic Scales, Astrophys. J. 943:152, https://doi.org/10.3847/1538-4357/aca0a0.
[2] W. M. Macek & D. Wójcik, 2023, Statistical analysis of stochastic magnetic fluctuations in space plasma based on the MMS mission, MNRAS, 526, 5779–5790, https://doi.org/10.1093/mnras/stad2584.
[3] D. Wójcik & W. M. Macek 2024, Testing for Markovian character of transfer of fluctuations in solar wind turbulence on kinetic scales, Phys. Rev. E 110, 025203, https://doi.org/10.1103/PhysRevE.110.025203.