We demonstrate that, for the case of quasi-equipartition between the velocity and the magnetic field, the Lagrangian-averaged magnetohydrodynamics α−model (LAMHD) reproduces well both the large-scale and small-scale properties of turbulent flows; in particular, it displays no increased (super-filter) bottleneck effect with its ensuing enhanced energy spectrum at the onset of the sub-filter-scales. This is in contrast to the case of the neutral fluid in which the Lagrangian-averaged Navier-Stokes α−model is somewhat limited in its applications because of the formation of spatial regions with no internal degrees of freedom and subsequent contamination of super-filter-scale spectral properties. We argue that, as the Lorentz force breaks the conservation of circulation and enables spectrally non-local energy transfer (associated to Alfvén waves), it is responsible for the absence of a viscous bottleneck in MHD, as compared to the fluid case.As LAMHD preserves Alfvén waves and the circulation properties of MHD, there is also no (super-filter) bottleneck found in LAMHD, making this method capable of large reductions in required numerical degrees of freedom; specifically, we find a reduction factor of ≈ 200 when compared to a direct numerical simulation on a large grid of 1536 3 points at the same Reynolds number.