Abstract
We present a new theoretical picture of magnetically dominated, decaying turbulence in the absence of a mean magnetic field. We demonstrate that such turbulence is governed by the reconnection of magnetic structures, and not by ideal dynamics, as has previously been assumed. We obtain predictions for the magnetic-energy decay laws by proposing that turbulence decays on reconnection timescales, while respecting the conservation of certain integral invariants representing topological constraints satisfied by the reconnecting magnetic field. As is well known, the magnetic helicity is such an invariant for initially helical field configurations, but does not constrain non-helical decay, where the volume-averaged magnetic-helicity density vanishes. For such a decay, we propose a new integral invariant, analogous to the Loitsyansky and Saffman invariants of hydrodynamic turbulence, that expresses the conservation of the random (scaling as $\mathrm{volume}^{1/2}$) magnetic helicity contained in any sufficiently large volume. Our treatment leads to novel predictions for the magnetic-energy decay laws: in particular, while we expect the canonical $t^{-2/3}$ power law for helical turbulence when reconnection is fast (i.e., plasmoid-dominated or stochastic), we find a shallower $t^{-4/7}$ decay in the slow `Sweet-Parker' reconnection regime, in better agreement with existing numerical simulations. For non-helical fields, for which there currently exists no definitive theory, we predict power laws of $t^{-10/9}$ and $t^{-20/17}$ in the fast- and slow-reconnection regimes, respectively. We formulate a general principle of decay of turbulent systems subject to conservation of Saffman-like invariants, and propose how it may be applied to MHD turbulence with a strong mean magnetic field and to isotropic MHD turbulence with initial equipartition between the magnetic and kinetic energies.
Highlights
The nature of the decay of magnetohydrodynamic (MHD) turbulence is an important outstanding problem in fluid dynamics, with far-reaching consequences in astrophysics, from the evolution of primordial magnetic fields in cosmology [1,2,3] to the dynamics of the solar wind [4]
Equation (8) states that the rate of change of magnetic helicity is smaller than the rate of the energy decay due to Ohmic heating by a factor equal to the ratio of the integral scale to the resistive dissipation length scale, which becomes arbitrarily small as ηn → 0þ
III, we proposed a way to impose the constraint of magnetic-helicity conservation on the decay of nonhelical MHD turbulence via the conservation of IH
Summary
The nature of the decay of magnetohydrodynamic (MHD) turbulence is an important outstanding problem in fluid dynamics, with far-reaching consequences in astrophysics, from the evolution of primordial magnetic fields in cosmology [1,2,3] to the dynamics of the solar wind [4]. III, we consider the decay of nonhelical MHD turbulence from a magnetically dominated state The mechanism controlling this type of decay has so far remained unknown: Because the mean helicity density vanishes, its conservation cannot be used to derive a scaling relation relating B, the characteristic magnetic field, to its characteristic length scale L. A Kolmogorov-style argument leads to a t−1 power law independently of whether the decay occurs on the ideal [21,24,33,41] or the Sweet-Parker [35,43,50] timescale (see the Appendix A) That both treatments predict the same power law is a coincidence related to the fact that anastrophy conservation implies a constant Lundquist number for n 1⁄4 2 resistive dissipation [50] (incidentally, the scaling argument is essentially this same point with the direction of implication reversed). We conjecture that the simultaneous conservation of both the Saffman-type cross-helicity invariant and the magnetic helicity might govern the initial period of decay of an MHD state starting with U ∼ B
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