Abstract: A thin Hartmann boundary layer will be formed near the wall when the conducting fluid is passing through a vertical magnetic field.The property of Hartmann boundary layers is vital to design and operation of equipments in electromagnetic metallurgy and thermonuclear fusion cooling system.This stability problem is investigated by non-modal stability analysis method.Through solving the governing equations of disturbances and adjoint field variables iteratively, the amplification and spatial distributions of primary perturbations are obtained.The effects of magnetic field on the optimal perturbation amplification G_max, spanwise wavenumber β_opt and time t_opt are analyzed, and the interaction between two opposite Hartmann boundary layers is considered as well.Results indicate that the optimal initial perturbations are in the form of streamwise vortices, which is symmetric or antisymmetric with respect of the normal direction.When the Hartmann number Ha is larger(Ha>10), the initial perturbations of symmetric and antisymmetric vortices are amplified equally, and the two opposite Hartmann boundary layers can be considered as independent from each other.In this case, the dependence of optimal perturbation amplification G_max on the square of local Reynolds number R is obtained, and the corresponding optimal spanwise wavenumber β_opt and time t_opt are proportional to the Hartmann number Ha.When the Hartmann number Ha is smaller(Ha<10), the antisymmetric vortices are more unstable, and the perturbation amplifications G_max is larger than that of symmetric vortices.There is still a kind of interaction between the two opposite Hartmann boundary layers to influence the stability of the flow field.