  In this paper, the infinite limit of the Prandtl number is justified for the
two-dimensional incompressible magneto-convection, which describes the
nonlinear interaction between the Rayleigh-B$\rm\acute{e}$nard convection and
an externally magnetic field. Both the convergence rates and the thickness of
initial layer are obtained. Moreover, based on the method of formal asymptotic
expansions, an effective dynamics is constructed to simulate the motion within
the initial layer.
