  We investigate the behavior of flows, including turbulent flows, driven by a
horizontal body-force and subject to a vertical magnetic field, with the
following question in mind: for very strong applied magnetic field, is the flow
mostly two-dimensional, with remaining weak three-dimensional fluctuations, or
does it become exactly 2D, with no dependence along the vertical?
  We first focus on the quasi-static approximation, i.e. the asymptotic limit
of vanishing magnetic Reynolds number Rm << 1: we prove that the flow becomes
exactly 2D asymptotically in time, regardless of the initial condition and
provided the interaction parameter N is larger than a threshold value. We call
this property "absolute two-dimensionalization": the attractor of the system is
necessarily a (possibly turbulent) 2D flow.
  We then consider the full-magnetohydrodynamic equations and we prove that,
for low enough Rm and large enough N, the flow becomes exactly two-dimensional
in the long-time limit provided the initial vertically-dependent perturbations
are infinitesimal. We call this phenomenon "linear two-dimensionalization": the
(possibly turbulent) 2D flow is an attractor of the dynamics, but it is not
necessarily the only attractor of the system. Some 3D attractors may also exist
and be attained for strong enough initial 3D perturbations.
  These results shed some light on the existence of a dissipation anomaly for
magnetohydrodynamic flows subject to a strong external magnetic field.
