MHD instability associated with critical layers

Critical levels appear as singularities of waves propagating in shear flows. When magnetic field exists, critical levels are located where the phase velocity of the wave relative to the basic flow matches the velocity of Alfvén waves. Critical levels are known for locally strong wave amplitude in its vicinity, known as the critical layers. In this talk, I will demonstrate the situation where magnetic critical layers can contribute to the instability of the MHD flow.  We consider two different flow configurations. One is the shallow water flow, and the other is the 2D flow on a sphere. Asymptotic analysis has been used to explore deeper insights of the instability mechanism.