The generation of spiral density waves by MRI in accretion discs
We investigate the linear dynamics of non-axisymmetric perturbations in Keplerian discs subject to a weak uniform vertical magnetic field in the shearing box approximation. Perturbations are decomposed into shearing waves and evolved by numerically integrating the linearized ideal MHD equations. The disc flow supports three basic perturbation modes: two incompressible modes - magnetic mode that undergoes magnetorotational instability (MRI) and inertia-magnetic waves - and compressible spiral density waves. The magnetic mode and inertia-magnetic waves have a low frequency of the order of Alfv\'en and orbital frequencies, respectively, while density waves have high-frequency. We introduce mode eigenfunctions and governing modal equations to analyze the dynamics of individual modes. For non-axisymmetric modes, the modal equations are coupled due to the shear of the Keplerian rotation of the disc, giving rise to a new shear-induced linear mode coupling process, which is rooted in the non-self-adjoint nature of shear flows. We focus on the generation of density waves by the dominant MRI-unstable magnetic mode. We show that initially imposed magnetic mode undergoes MRI growth and abruptly excites density waves when its radial wavenumber crosses zero. The density wave-MRI coupling is most efficient when the azimuthal and vertical wavelengths of perturbations are comparable to the disc scale height. Since density waves are compressible, whereas MRI is incompressible, this wave excitation process can also be regarded as a linear mechanism generating compressible motions via MRI-driven incompressible ones. Its implications for compressible nonzero net vertical field MRI-turbulence are also discussed.