2D Simulations of the Line-Driven Instability in Hot-Star Winds: II. Approximations for the 2D Radiation Force

We present initial attempts to include the multi-dimensional nature of radiation transport in hydrodynamical simulations of the small-scale structure that arises from the line-driven instability in hot-star winds. Compared to previous 1D or 2D models that assume a purely radial radiation force, we seek additionally to treat the lateral momentum and transport of diffuse line-radiation, initially here within a 2D context. A key incentive is to study the damping effect of the associated diffuse line-drag on the dynamical properties of the flow, focusing particularly on whether this might prevent lateral break-up of shell structures at scales near the lateral Sobolev angle of ca. $1^{\rm o}$. We first explore nonlinear simulations that cast the lateral diffuse force in the simple, local form of a parallel viscosity. Second, to account for the lateral mixing of radiation associated with the radial driving, we next explore models in which the radial force is azimuthally smoothed over a chosen scale. Third, to account for both the lateral line-drag and the lateral mixing in a more self-consistent way, we explore further a method first proposed by Owocki (1999), which uses a restricted 3-ray approach that combines a radial ray with two oblique rays set to have an impact parameter $p < R_{\ast}$ within the stellar core. From numerical simulations, we find that, compared to equivalent 1-ray simulations, the high-resolution 3-ray models show systematically a much higher lateral coherence.... (Full abstract in paper)