Entanglement and topological linking of magnetic and stream lines


 Helicity fluctuation, generation of linking number and effect on resistivity

This subject is motivated by the particular importance of the magnetic stochasticity in fusion plasma and in astrophysics. In tokamak the equilibrium configuration consists of toroidally nested magnetic surfaces. They are generated by magnetic field lines produced by the external (main) magnetic field and the externally-induced current flowing in the plasma. The lines are helical and can be closed by periodicity (the set has the power of the set of rational numbers) or they effectively create a 2D toroidal surface by ergodicity ( = the line can come arbitrarily close to any point of this surface). The magnetic perturbations create islands and when two islands come into contact it occurs the exponential instability (= from two arbitrarily close initial points the magnetic lines evolve arbitrarily far, after a distance which is the inverse of the Lyapunov exponent). The exponential instability is the definition of chaos. Since plasma moves mainly along the magnetic field line, a chaotic field line transports plasma quantities from one region to regions that can be at large distances, which means that the parameters are equalized over the region of chaoticity. This is very inconvenient for confinement.

The magnetic field lines are linked and the quasi-ideal nature of the plasma tends to preserve the relative positions in space of the lines. This means that any topological quantity that characterizes the entanglement of the lines is a good invariant and only the resistivity is able to change it. Breaking-up of magnetic lines and reconection with change of the topology is possible only when a real dissipative mechanism is present.

We discuss the linking of lines in space and in particular we look for a possiblity to quantify the change in the linking, which happens when there is reconnection.

Here is a text written in 2005 and presented in 2006 at a conference in Exeter. It is about the fluctuation of the topological linking.

Helicity fluctuation, generation of linking number and effect on resistivity

The following work is on arxiv.org

It has not been followed by further developments. The direction has changed toward mapping of 2D magnetic and flow structures, etc.


 Statistical analysis of the linking number in stochastic magnetic fields

In the volume where the magnetic field lines are stochastic, random reconnection events mediated by dissipative resistivity produce a continuous change of the topological content. The system consisting of the ensemble of magnetic lines must be characterized by a topological quantity, like the density of link, of writhe or twist. The topological content is fluctuating and a statistical analysis should be made. We present here a path-integral formulation of the problem: we define the generating function of the correlations of the linking number of an ensemble of lines in space and calculate the two-point correlation.

This work is an application of the theory of statistical analysis of polymers. The reference is: F. Tanaka, Progress of Theoretical Physics, 68, 148 and 164 (1982).

The work has been published in Romanian Journal of Physics.

There is a need for a further development of the "replica method" application.


 The current profile in tokamak: between topology and turbulence

The radial profile of the current density in tokamak is determined by the effect of two very different factors : topology and turbulence. I made a presentation on this subject at the workshop of Isaac Newton Institute of Mathematical Science.

The work that has been presented at the meeting is: The ideal versus the reality: topology and turbulence in the current density in tokamak , authors F. Spineanu and M. Vlad

The work has been included in the electronic Proceeding of the meeting.

Much work has been done after this, in seeking for a Baeklund map between solutions of the Painleve III equations. More difficult for us was to understand why we have insisted so much on this mapping, while the exact solution of sinh-Poisson must be doubly periodic, which excludes the Painleve approximation.


 A MHD invariant and the current/vorticity/density profiles in tokamak

There is a forzen-in invariant that extends to MHD the Ertel's Lagrangian invariant. The latter is an important constraint acting on the dynamical processes in 2D fluids, in planetary atmosphere and in plasma confined in strong magnetic field. The MHD invariant has been derived by Sagdeev, Moiseev Tur and Yanovskii.

The work A MHD invariant and the confinement regimes in Tokamak , authors F. Spineanu and M. Vlad, has been first presented in a workshop on Stochasticity in Fusion Plasma (Bad Honef 2015).

The work has been published in Nuclear Fusion.

(In the text published in NF symbols of integrals on closed loops have been erroneously replaced by symbols of double integral; this comes from the limited adaptation of the Latex package amssymb to the IOP style.)

The applications can be extremely important. It is a constraint that associates high current density to high vorticity, leading to a particular perspective on Internal Transport Barrier, or the edge layer of rotation at H-mode. Etc. Still to work to clarify the change from Lagrangian to quasi-Eulerian invariance. Possibly the approach of Robert and Sommeria for the weakly dissipative 2D fluid can help.