In this paper, we are concerned with 2D ideal MHD equations with magnetic damping term in domain \(\mathbb {T}\times \mathbb {R}\) . Stability and exponential decay are obtained. Our analysis is based on some key observations and ideas to maximize the hidden dissipation in the equations, such as the Poincaré’s inequality in one direction. Without the coupling with the magnetic field, the velocity of the fluid is governed by the 2D incompressible Euler equations and can grow rather rapidly in time. The results obtained here indicate the stabilization effect of the magnetic field for conductive fluids. As a by-product, stability and exponential decay for 2D MHD equations with only a velocity damping term are also shown.
