  We prove the global well-posedness of the 2D incompressible non-resistive MHD
equations with a velocity damping term near the non-zero constant background
magnetic field. To this end, we newly design a normal mode method of
effectively leveraging the anisotropy of the linear propagator that encodes
both the partially dissipative nature of the non-resistive MHD system and the
stabilizing mechanism of the underlying magnetic field. Isolating new key
quantities and estimating them with themselves in an entangling way via the
eigenvalue analysis based on Duhamel's formulation, we establish the global
well-posedness for any initial data $(v_0,B_0)$ that is sufficiently small in a
space rougher than $H^{4}\cap L^1$. This improves the recent work in SIAM J.
Math. Anal. 47, 2630-2656 (2015) where the similar result was obtained provided
that $(v_0,B_0)$ was small enough in a space strictly embedded in $H^{20}\cap
W^{6,1}$.
