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We investigate aspects of low-magnetic-Reynolds-number flow between twoparallel, perfectly insulating walls, in the presence of an imposed magneticfield parallel to the bounding walls. We find a functional basis to describethe flow, well adapted to the problem of finding the attractor dimension, andwhich is also used in subsequent direct numerical simulation of these flows.For given Reynolds and Hartmann numbers, we obtain an upper bound for thedimension of the attractor by means of known bounds on the nonlinear inertialterm and this functional basis for the flow. Three distinct flow regimesemerge: a quasi-isotropic 3D flow, a non-isotropic three-dimensional (3D) flow,and a 2D flow. We find the transition curves between these regimes in the spaceparameterized by Hartmann number Ha and attractor dimension $d_\text{att}$. Wefind how the attractor dimension scales as a function of Reynolds and Hartmannnumbers (Re and Ha) in each regime. We also investigate the thickness of theboundary layer along the bounding wall, and find that in all regimes thisscales as 1/Re, independently of the value of Ha, unlike Hartmann boundarylayers found when the field is normal to the channel. The structure of the setof least dissipative modes is indeed quite different between these two casesbut the properties of turbulence far from the walls (smallest scales and numberof degrees of freedom) are found to be very similar.

Bounds on the attractor dimension for magnetohydrodynamic channel flow with parallel magnetic field at low magnetic Reynolds number