  We consider a two-dimensional MHD model describing the evolution of viscous,
compressible and electrically conducting fluids under the action of vertical
magnetic field without resistivity. Existence of global weak solutions is
established for any adiabatic exponent \gamma >1. Inspired by the approximate
scheme proposed in [15], we consider a two-level approximate system with
artificial diffusion and pressure term. At the first level, we prove global
well-posedness of the regularized system and establish uniform-in-\epsilon
estimates to the regular solutions. At the second level, we show global
existence of weak solutions to the system with artificial pressure by sending
\epsilon to 0 and deriving uniform-in-\delta estimates. Then global weak
solution to the original system is constructed by vanishing \delta. The key
issue in the limit passage is the strong convergence of approximate sequence of
the density and magnetic field. This is accomplished by following the technique
developed in [15, 26] and using the new technique of variable reduction
developed by Vasseur et al. [33] in order to handle the cross terms.
