Gradient Tensor Invariants in Space Plasmas: Evolution Equations and Statistics

This Ph.D thesis work discusses some topological features of streamlines and magnetic field lines in the framework of Magneto-Hydrodynamic (MHD) theory, i.e., the theoretical framework to describe the dynamics of fully ionized matter, the plasma, which constitutes most of the 98% of the baryonic matter of the universe. In particular, I derive a mathematical/theoretical model to describe the evolution of magnetic field and plasma structures in terms of geometrical SO(3) invariants of gradient tensors of the magnetic and velocity field. This theoretical model is then applied to characterize the different topological and dynamical properties of the structures of magnetic and velocity fields in turbulent plasma environments, such as the solar wind and the circum-terrestrial space. This work moves from the previous works by Villeifosse (1984) and Cantwell (1991, 1993) who introduced some invariant quantities (named P, Q, R) under rotations related to the characteristic polynomial of the matrix of the gradient tensor of the velocity field in the framework of incompressible fluids, capable of describing the topological features of the fluid streamlines, and extend the theoretical description of the dynamics of the gradient tensor invariants to the case of the MHD by deriving the dynamical equations for the Lagrangian evolution of these quantities. Differently from the fluid case, the dynamical equations of the gradient tensor invariants contain some specific quantities that describe the effects of the magnetic field on plasma motion topology and structures, so that an evaluation of the relative weights of these terms in respect to the terms appearing in the purely fluid situation would allow to characterize the dynamic regime of the considered plasma environment. Here, I present an analysis of the relative weight of the magnetic field effect on velocity field structure by considering data from both numerical simulations and in situ space plasma measurements. In particular, by considering observations and measurements from the NASA-Magnetospheric Multiscale (MMS) mission in the solar wind and in the Earth’s magnetosheath, the potentiality of the approach based on the gradient tensor invariants in characterizing the dynamical regime of these turbulent plasma environments is discussed, showing the different structural character of the observed plasma turbulence in spite of the similar spectral features of the magnetic and velocity field fluctuations. The same kind of analysis is also performed in the case of direct numerical simulations (DNSs) of decaying MHD turbulence. As a last point, I attempt a modelling of the evolution equations of the gradient tensor invariants via a dynamical system approach, concentrating my attention on the velocity gradient tensor invariants and on the role that the magnetic field plays in the dynamics of the topology of the streamlines. The results of this PhD thesis work are particularly interesting in the field of interplanetary and magnetospheric plasmas providing a different way to unveil the dynamical properties of turbulent plasmas in the different heliospheric regions and new insights and elements to model the topology of magnetic field and plasma structures and the interaction between different plasma regions. In other words, the approach, here presented, could lead to a different point of view to study the dynamics of turbulent space plasma and its role in the description of several Space Weather related phenomena.