Characterizing the turbulent porosity of stellar-wind structure generated by the line-deshadowing instability

We analyze recent 2D simulations of the nonlinear evolution of the line-deshadowing instability (LDI) in hot-star winds, to quantify how the associated highly clumped density structure can lead to a "turbulent porosity" reduction in continuum absorption and/or scattering. The basic method is to examine the statistical variations of mass column as a function of path length, and fit these to analytic forms that lead to simple statistical scalings for the associated mean extinction. A key result is that one can characterize porosity effects on continuum transport in terms of a single "turbulent porosity length", found here to scale as $H \approx (f_{\rm cl} - 1) a$, where $f_{\rm cl} \equiv \left < \rho^2 \right >/\left < \rho \right >^2$ is the clumping factor in density $\rho$, and $a$ is the density autocorrelation length. For continuum absorption or scattering in an optically thick layer, we find the associated effective reduction in opacity scales as $\sim 1/\sqrt{1+\tau_{\rm H}}$, where $\tau_{\rm H} \equiv \kappa \rho H$ is the local optical thickness of this porosity length. For these LDI simulations, the inferred porosity lengths are small, only about a couple percent of the stellar radius, $H \approx 0.02 R_\ast$. For continuum processes like bound-free absorption of X-rays that are only marginally optically thick throughout the full stellar wind, this implies $\tau_{\rm H} \ll 1$, and thus that LDI-generated porosity should have little effect on X-ray transport in such winds. The formalism developed here could however be important for understanding the porous regulation of continuum-driven, super-Eddington outflows from luminous blue variables.