  We formulate a coarse-graining approach to the dynamics of
magnetohydrodynamic (MHD) fluids at a continuum of length-scales. In this
methodology, effective equations are derived for the observable velocity and
magnetic fields spatially-averaged at an arbitrary scale of resolution. The
microscopic equations for the bare velocity and magnetic fields are
renormalized by coarse-graining to yield macroscopic effective equations that
contain both a subscale stress and a subscale electromotive force (EMF)
generated by nonlinear interaction of eliminated fields and plasma motions. At
large coarse-graining length-scales, the direct dissipation of invariants by
microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is
shown to be negligible. The balance at large scales is dominated instead by the
subscale nonlinear terms, which can transfer invariants across scales, and are
interpreted in terms of work concepts for energy and in terms of topological
flux-linkage for the two helicities. An important application of this approach
is to MHD turbulence, where the coarse-graining length $\ell$ lies in the
inertial cascade range. We show that in the case of sufficiently rough velocity
and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and
can persist to arbitrarily small scales. Although closed expressions are not
available for subscale stress and subscale EMF, we derive rigorous upper bounds
on the effective dissipation they produce in terms of scaling exponents of the
velocity and magnetic fields. These bounds provide exact constraints on
phenomenological theories of MHD turbulence in order to allow the nonlinear
cascade of energy and cross-helicity. On the other hand, we show that the
forward cascade of magnetic helicity to asymptotically small scales is
impossible unless 3rd-order moments of either velocity or magnetic field become
infinite.
