The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are and , and the orders of magnetic diffusion coefficients κ and σ are and , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For -initial data with additional regularity , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal.272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order , where with . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order .