  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.
