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

In this paper, we give a rigorous justification of the deviation 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 MHD (SHMHD) equations. Choosing an aspect ratio parameter $\varepsilon \in(0,\infty)$, we consider the case that if the horizontal and vertical viscous coefficients are of $\mu = O(1)$ and $\nu = O({\varepsilon ^\alpha })$, and the orders of magnetic diffusion coefficients $k$ and $\sigma$ are $k = O(1)$ and $\sigma = O({\varepsilon ^\alpha })$, with $\alpha > 2$, then the limiting system is the PEHM as $\varepsilon$ goes to zero. For ${H^1}$-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 $\varepsilon$ tends to zero. For ${H^1}$-initial data with additional regularity $({\partial _z}{\tilde A_0},{\partial _z}{\tilde B_0}) \in {L^p}(\Omega )(2<p<\infty)$, we slightly improve the well-posed result in \cite{2017-Cao-Li-Titi-Global} to extend the local-in-time strong convergences to the global-in-time one. For ${H^2}$-initial data, we show that the local-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as $\varepsilon$ goes to zero. Moreover, the rate of convergence is of the order $O({\varepsilon ^{\gamma /2}})$, where $\gamma = \min \{ 2,\alpha - 2\}$ with $\alpha \in (2,\infty )$. It should be noted that in contrast to the case $\alpha > 2$, the case $\alpha =2$ has been investigated by Du and Li in \cite{2023-Du-Li}, in which they consider the PEM and the rate of global-in-time convergences is of the order $O(\varepsilon)$.