When a plasma, which rotates differentially about a fixed center, is subjected to a magnetic field, perpendicular to the plane of rotation of the fluid, the coupling of the current with the field (the Lorentz force density) may be disruptive if the angular velocity of the gas decreases with the increase of the radial coordinate. This phenomenon is commonly referred to as the magnetorotational instability (MRI). For a perfectly conducting, inviscid plasma (ideal approximation), the problem can be treated analytically. Such an approach is particularly useful for the description of the dynamic evolution of accretion disks, astrophysical structures consisting of ionized gases which rotate about compact objects (black holes, neutron stars). In this case, the flow is assumed to exhibit a Keplerian profile, thereby leading to a dispersion relation which is biquadratic in the growth rate of the MRI. However, the associated instability condition implies the perturbative wavelength increases with the increase of the radial coordinate. This means that, as the radial coordinate decreases, the perturbative frequency increases with no limit. Recently, there has been some progress towards an analytical formulation of the MRI by including dissipative effects for the rotating plasma. In this work, by introducing both finite resistivity and viscosity for a Keplerian accretion disk, it is found that the growth rate of the instability may satisfy a quadratic equation and become suppressed by a term which depends on the magnetic Prandtl number. It is also shown that, when resistive effects dominate, the perturbative wavelength saturates asymptotically to a minimum value which does not depend on the radial coordinate.
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