Magnetic topology in fluids

Abstract

We study the evolution of turbulent magnetic fields from a topological point of view, invoking commonplace mathematical tools from general topology and dynamical systems theory which connect magnetic field evolution to time reversal invariance, entropy increase and the second law of thermodynamics. We show that in fact magnetic topology is well-defined only in the phase space corresponding to a dynamical system governed by the induction equation. Hence the field's topology and stochasticity can be studied in terms of the corresponding phase space trajectories rather than the field lines in real Euclidean space. In fact, our results suggest that magnetic field lines should not be taken too literally because their existence and uniqueness and more importantly continuity in time require strong mathematical conditions, hardly satisfied in astrophysical systems. As for magnetic topology change, it is shown that the phase space topology is preserved in time for a magnetic field which, besides satisfying few continuity conditions, solves a time reversal invariant induction equation. What breaks the time symmetry in the induction equation is the presence of non-ideal plasma effects at small scales such as resistivity, which results from random collisions between diffusing electrons and other particles. The small scale, stochastic disturbances produced thereby are super-linearly amplified by Richardson diffusion in the turbulent cascade, which are eventually manifested as large scale reconnection events, somehow similar to stretching quantum fluctuations during inflation to seed large scale cosmological structures. This suggests that reconnection is rooted in the second law of thermodynamics that dictates entropy increase which in turn breaks the time symmetry.