Density gradient effects on the magnetorotational instability

The effects of the equilibrium density gradient on non-axisymmetric magnetorotational instability are investigated in a pure axial magnetic field for ideal incompressible plasmas. A second-order ordinary differential equation is employed to determine the magnetic field perturbation and the full dispersion relationship regarding the non-axisymmetric magnetohydrodynamic instability in the presence of gravitation and density gradient effects. By means of local linear analysis, the reduced dispersion relationship is derived with a small azimuthal wavenumber. Spatial variations in the radial field perturbation magnitude cannot be neglected in the calculation since this term has the same order of magnitude as LD, which is the scale length of radial density gradient. The analytical expression of the instability growth rate is presented. Our analysis shows that the instability criterion is modified by the density gradient which has a stabilizing effect when increasing outwards and conversely a destabilizing effect when decreasing outwards. The growth rate increases with LD when LD is small. For a sufficiently large LD, the growth rate decreases with increasing LD. The magnetic field exerts a similar effect on the growth rate and can totally quench the instability. The non-axisymmetric effect introduces a frequency shift and increases the growth rate but does not affect the instability criterion.