Magnetohydrodynamic Shearing Waves

I consider the nonaxisymmetric linear theory of an isothermal magnetohydrodynamic (MHD) shear flow. The analysis is performed in the shearing box, a local model appropriate for a thin disk geometry. Linear perturbations in this model can be decomposed in terms of shearing waves (shwaves), which appear spatially as plane waves in a frame comoving with the shear. The time dependence of these waves cannot in general be expressed in terms of a frequency eigenvalue as in a normal mode decomposition, and numerical integration of a set of first-order amplitude equations is required for a complete characterization of their behavior. Their generic time dependence, however, is oscillatory with slowly-varying frequency and amplitude, and one can construct accurate analytic solutions by applying the Wentzel-Kramers-Brillouin method to the full set of amplitude equations. For the bulk of wavenumber space, therefore, the shwaves are well-approximated as modes with time-dependent frequencies and amplitudes. The incompressive shwaves, which require a sub-thermal mean field, include the nonaxisymmetric extension of magneto-rotationally unstable modes. The compressive shwaves are the extension of the fast and slow magnetosonic modes to a shear flow; the validity of the latter requires a super-thermal mean field. The oscillatory nature of the incompressive shwaves breaks down for very weak fields, in which case their amplitudes can be expressed in terms of hypergeometric functions. The complete set of solutions constitutes a comprehensive linear test suite for numerical MHD algorithms that incorporate a background shear flow. I conclude with a brief discussion of possible astrophysical applications. Subject headings: accretion, accretion disks — MHD