The Rayleigh shearing instability limit of the magnetorotational instability

We use the geometric optics approximation to derive the stability criteria for the Rayleigh shearing instability and the magnetorotational instability. We examine the cases where each criterion is relevant by looking into the magnitude of the magnetic field using a small dimensionless parameter. Examining all the orders of this parameter in the characteristic equation we show that configurations with sufficiently small magnetic field are characterised by the Rayleigh shearing instability criterion rather than that of the magnetorotational instability.