Turbulent mixing of a secularly unstable dust layer ($s\sim 2\pi ϵG{\sigma }_0|k|/\Omega -D{k}^2$)

Turbulence can raise dust random speeds, increase layer thickness and mix density enhancements by eddy diffusion, delaying secular gravitational instability. On sufficiently long wavelengths, weak-drag growth is proportional to $|k|$, whereas simple diffusive damping is proportional to $k^2$. Thus diffusion alone need not remove all long-wave growth in an infinite idealization, although finite size and lifetime can make it ineffective. Coherent turbulent concentration can also increase local surface density, so the net effect requires a stochastic transport model.