Probing Small Scale Solar Wind Turbulence: Markovian Analysis and Scale Interactions from Inertial to Kinetic Regimes

The solar wind flowing from the solar corona at supersonic and super-Alfvénic speeds is the subject of intense research at present. Numerous studies focused on temporal variability of plasma parameters, crucial to define solar wind plasma, showed that spectral distributions exhibit Power-law dependence. Additionally, examinations of its fluctuations revealed a distinctive evolution in shape of PDF’s shifting from Gaussian (Maxwellian) to peaked and heavy-tailed distributions, towards smaller scales. 

Turbulence stands as a complex, nonlinear, and multiscale phenomenon, based on a sophisticated cascade theory. While its complete description remains an unsolved challenge, various statistical methods such as structural functions or power spectra offer partial insights. Although the well-established Kolmogorov theory (K41) holds in the inertial region of magnetized plasma [10.1103/PhysRevE.78.026414], its applicability at smaller scales or higher frequencies is known to be an exception. Thus, alternative methods, such as a kinetic treatment, should be considered. 

Common approaches rely on various plasma parameters and processes, limiting their applicability in highly dynamic turbulence like the solar wind. Consequently, an alternative approach based on the framework of stochastic processes theory, particularly Markov processes, has been introduced to characterize energy transfer across the turbulent cascade. Statistical evidence suggests that turbulence has Markov properties. Furthermore, the differential equation of the Markov process can be extracted directly from data. Estimation of the Kramers-Moyal coefficients plays a pivotal role in discerning the form of the Fokker-Planck (or equivalently Langevin) equation that governs the evolution of the PDF with scale for the increments. Models based on a drift force and diffusion strength depending on scale have been emerging as a viable approach for elucidating the dynamics of solar wind turbulence, hence this method can be considered as a junction between the statistical and dynamical analysis.

Based on the data collected by the Magnetospheric Multiscale (MMS) mission’s satellites, we delve into the subject of turbulence on inertial, sub-ion, and kinetic scales. Building upon prior Markovian analysis of turbulence of the transfer of magnetic-to-magnetic field fluctuations in the near-Earth space environment [10.1093/mnras/stad2584, 10.3847/1538-4357/aca0a0], we also extend our investigation to ion velocity-to-velocity and magnetic-to-velocity cases. However, we direct our focus towards the purer statistical facet of the analysis, joint with the elements of dynamical approach. We analyze whether the transfer of increments exhibits `local` or `non-local` character, which in this context, they describe the scales involved in interactions that lead to the turbulent cascade. Additionally, we observe a global scale-invariance in relation to the Fokker-Planck equation, for a magnetic field case. 

Finally, we briefly discuss a potential non-parametric approach, namely a stochastic dynamical jump-diffusion model, or alternatively a multi-fractal approach, which can be useful to describe the underlying process accurately. We believe that such a comparative approach spanning diverse conditions is meaningful, as it aims to unveil any underlying universality within the statistical properties of the near-Earth solar wind space plasma at the intricate kinetic and sub-ion scales.

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