The excitation of spiral density waves through turbulent fluctuations in accretion discs I: WKBJ theory

Abstract

We study and elucidate the mechanism of spiral density wave excitation in a differentially rotating turbulent flow. We formulate a set of wave equations with sources that are only non-zero in the presence of turbulent fluctuations. We solve these in a shearing box domain using a WKBJ method. It is found that, for a particular azimuthal wave length, the wave excitation occurs through a sequence of regularly spaced swings during which the wave changes from leading to trailing form. This is a generic process that is expected to occur in shearing discs with turbulence. Trailing waves of equal amplitude propagating in opposite directions are produced, both of which produce an outward angular momentum flux that we give expressions for as functions of the disc parameters and azimuthal wave length. By solving the wave amplitude equations numerically we justify the WKBJ approach for a Keplerian rotation law for all parameter regimes of interest. In order to quantify the wave excitation completely the important wave source terms need to be specified. Assuming conditions of weak nonlinearity, these can be identified and are associated with a quantity related to the potential vorticity, being the only survivors in the linear regime. Under the additional assumption that the source has a flat power spectrum at long azimuthal wave lengths, the optimal azimuthal wave length produced is found to be determined solely by the WKBJ response and is estimated to be 2 pi H, with H being the nominal disc scale height.