We consider a 2D incompressible and electrically conducting fluid in the domain $${\mathbb {T}}\times {\mathbb {R}}$$
T
×
R
. The aim is to quantify stability properties of the Couette flow (y, 0) with a constant homogenous magnetic field $$(\beta ,0)$$
(
β
,
0
)
when $$|\beta |>1/2$$
|
β
|
>
1
/
2
. The focus lies on the regime with small fluid viscosity $$\nu $$
ν
, magnetic resistivity $$\mu $$
μ
and we assume that the magnetic Prandtl number satisfies $$\mu ^2\lesssim \textrm{Pr}_{\textrm{m}}=\nu /\mu \le 1$$
μ
2
≲
Pr
m
=
ν
/
μ
≤
1
. We establish that small perturbations around this steady state remain close to it, provided their size is of order $$\varepsilon \ll \nu ^\frac{2}{3}$$
ε
≪
ν
2
3
in $$H^N$$
H
N
with N large enough. Additionally, the vorticity and current density experience a transient growth of order $$\nu ^{-\frac{1}{3}}$$
ν
-
1
3
while converging exponentially fast to an x-independent state after a time-scale of order $$\nu ^{-\frac{1}{3}}$$
ν
-
1
3
. The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system’s dynamic behavior.