Investigating structures through gradient tensors in turbulent space plasmas: invariants’ evolution equations and Schur decomposition

The availability of multi-point in situ data from space missions orbiting in solar wind and near-Earth environments offers valuable insights into fundamental physical phenomena such as shocks, magnetic reconnection, turbulence, waves, jets and so on. All these processes are related to dynamical evolving plasma structures in both space and time. In this context, invariant quantities derived from the gradient tensor method allows us to study the evolution of topological structures in velocity and magnetic fields across various regions of interplanetary space at different scales. The use of gradient tensors is primarily based on the availability of multi-point data from missions involving at least four satellites arranged in a tetrahedral formation.

Here we present some theoretical and observational results based on the analysis of gradient tensor invariants. We derive equations governing the temporal evolution of these quantities to get insights into the topological and morphological changes of these structures in time. These evolution equations also allow us to identify the dominant physical terms driving the observed changes. A preliminary analysis, based on MMS multi-point observations, suggests that the plasma in the near-Earth solar wind predominantly behaves like a fluid, whereas velocity and magnetic field interactions play a more significant role in the magnetosheath region.
We further introduce a novel approach for studying gradient tensor characteristics using the Schur transformation. This technique decomposes the velocity and magnetic field gradient tensors into a matrix representing eigenvalue contributions and another term associated with pressure and dissipative effects. This decomposition enables the identification of regions where dissipative effects are more prominent. These studies are of critical importance for future space missions which will extend the current multi-point paradigm, based on a single tetrahedron constellation, to multi-scale multiple tetrahedra configurations such as the NASA mission HelioSwarm (in the solar wind) and the ESA Phase A Plasma Observatory (in the near-Earth plasma).