Daniel Faraco (ICMAT-UAM), Sauli Lindberg (University of Helsinki) and László Székelyhidi Jr. (Max Planck Institute for Mathematics in the Sciences) are the authors of the paper Magnetic helicity, weak solutions and relaxation of ideal MHD, published in Communications on Pure and Applied Mathematics.

The unexpected solar flares that cause the aurora borealis seen at the Earth’s poles are caused by the behaviour of solar plasmas. Plasmas are fluids in which a magnetic field acts, which mathematicians study through the so-called magnetohydrodynamics (MHD) equations. These expressions determine the evolution of the velocity field of the fluid and the magnetic field acting on it. They are obtained by combining the classical equations of fluids (those of Euler and Navier-Stokes) with those of electromagnetism (those of Maxwell). To analyse them, physicists and mathematicians study the so-called conserved integral quantities, such as the total energy. In regular situations – i.e. when the fluid does not behave abruptly – they remain (approximately) constant over time.

However, in some hydrodynamic problems involving turbulence, the energy need not be conserved. Mathematically, in these situations, elements appear so irregular that it is not possible to apply the usual methods (i.e., derivative and integration) to study the integral quantities, but other approaches must be used. This was conjectured in the mid-20th century by the mathematician Andrei Kolmogorov and Lars Onsager (Nobel Prize in Chemistry in 1968).

Recently, Camillo De Lellis and László Székelyhidi Jr have developed a programme to give rigour to the ideas of Kolmogorov and Onsager, using a technique called convex integration. The methods have been used to study the equations of fluids in turbulent situations with success, but applying these ideas to the equations of magnetohydrodynamics is trickier, since other conserved integral quantities besides energy are involved, such as cross-helicity and magnetic helicity. The former measures how magnetic lines and velocity field flux lines intersect. The second describes the topological behaviour of the magnetic lines.

For turbulent plasmas, simulations and experiments suggested that both the total energy and the cross-helicity dissipate anomalously. However, magnetic helicity seems to behave differently: in very electrically conductive fluids, it is conserved. This statement was conjectured in 1974 by the physicist John Bryan Taylor and is the basis of his so-called relaxation theory.

In 2020, Daniel Faraco – professor at the Universidad Autónoma de Madrid (UAM) and member of the ICMAT – and Sauli Lindberg – University of Helsinki, proved Taylor’s conjecture from a mathematical point of view. In other words, they rigorously demonstrate that magnetic helicity is approximately conserved. And it is precisely because of the conservation of this property in such a turbulent regime as solar activity that flares are produced, which then cause auroras.

Recently Beckie, Bukcmaster and Vicol showed that there are integrable square solutions which do not preserve magnetic helicity. It was already known that, in these same solutions, if the velocity and magnetic field have spatial cubic integrability, the helicity is conserved.

Now, are there solutions that dissipate the magnetic helicity with integrability below the cubic integrability? Daniel Faraco (ICMAT-UAM), Sauli Lindberg (University of Helsinki) and László Székelyhidi Jr. (Max Planck Institute for Mathematics in the Sciences) have proved that there are. They also show that it is possible to construct solutions that exhibit anomalous energy dissipation but retain arbitrary magnetic helicity.

Their results are published in the journal Communications on Pure and Applied Mathematics.

Reference: Faraco, D., Lindberg, S. and Székelyhidi, L., Jr. (2024), Magnetic helicity, weak solutions and relaxation of ideal MHD. Comm. Pure Appl. Math., 77: 2387-2412. https://doi.org/10.1002/cpa.22168