  A theory of potential vorticity (PV) mixing in a disordered (tangled)
magnetic field is presented. The analysis is in the context of $\beta$-plane
MHD, with a special focus on the physics of momentum transport in the stably
stratified, quasi-2D solar tachocline. A physical picture of mean PV evolution
by vorticity advection and tilting of magnetic fields is proposed. In the case
of weak-field perturbations, quasi-linear theory predicts that the Reynolds and
magnetic stresses balance as turbulence Alfv\'enizes for a larger mean magnetic
field. Jet formation is explored quantitatively in the mean field-resistivity
parameter space. However, since even a modest mean magnetic field leads to
large magnetic perturbations for large magnetic Reynolds number, the physically
relevant case is that of a strong but disordered field. We show that numerical
calculations indicate that the Reynolds stress is modified well before
Alfv\'enization -- i.e. before fluid and magnetic energies balance. To
understand these trends, a double-average model of PV mixing in a stochastic
magnetic field is developed. Calculations indicate that mean-square fields
strongly modify Reynolds stress phase coherence and also induce a magnetic drag
on zonal flows. The physics of transport reduction by tangled fields is
elucidated and linked to the related quench of turbulent resistivity. We
propose a physical picture of the system as a resisto-elastic medium threaded
by a tangled magnetic network. Applications of the theory to momentum transport
in the tachocline and other systems are discussed in detail.
