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Decay of Magnetohydrodynamic Turbulence

Decaying isotropic hydrodynamic turbulence is a well-developed subject, thanks to the work of Kolmogorov, Landau, Batchelor, Saffman and others. However, the decay of magnetohydrodynamic turbulence, i.e., of chaotic magnetic fields in a conducting fluid, is less well understood, with considerable discrepancy between theory and modern numerical simulations. In this talk, I will show how the decay of magnetic energy can be predicted by assuming a reconnection-mediated dynamics that respects the conservation of integral invariants representing topological constraints. It is well known that magnetic helicity is an invariant of this sort, although it has been believed that it does not constrain the evolution of parity-symmetric fields. I propose that such fields respect an integral invariant that is analogous to the Loitsyansky and Saffman invariants of hydrodynamic turbulence, but that expresses the conservation of mean square fluctuations of magnetic helicity arising in large volumes. I briefly explain how these ideas might be important for solving an outstanding problem in astrophysics: the origin of magnetic fields in cosmic voids.