  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
$O(\varepsilon)$.
