A main result of this paper establishes the global stability of the
three-dimensional MHD equations near a background magnetic field with mixed
fractional partial dissipation with α,β∈(21​,1]. Namely,
the velocity equations involve dissipation (Λ12α​+Λ22α​+σΛ32α​)u with the case σ=1 and
σ=0. The magnetic equations without partial magnetic diffusion
Λi2β​bi​ but with the diffusion (−Δ)βb, where
Λis​(s>0) with i=1,2,3 are the directional fractional
operators. Then we focus on the vanishing vertical kinematic viscosity
coefficient limit of the MHD system with the case σ=1 to the case
σ=0. The convergent result is obtained in the sense of H1-norm