Nonlocal transport of passive scalars in turbulent penetrative convection

We present a Green's function approach for quantifying the transport of a passive scalar (tracer) field in three-dimensional simulations of turbulent convection. Nonlocal, nondiffusive behavior is described by a transilient matrix (the discretized Green's function), whose elements contain the fractional tracer concentrations moving from one subvolume to another as a function of time. The approach was originally developed for and applied to geophysical flows, but here we extend the formalism and apply it in an astrophysical context to three-dimensional simulations of turbulent compressible convection with overshoot into convectively stable bounding regions. We introduce a novel technique to compute this matrix in a single simulation by advecting labeled particles rather than solving the passive scalar equation for a large number of different initial conditions. The transilient matrices thus computed are used as a diagnostic tool to quantitatively describe nonlocal transport via matrix moments and transport coefficients in a generalized, multiorder diffusion equation. Results indicate that transport in both the vertical and horizontal directions is strongly influenced by the presence of coherent velocity structures, generally resembling ballistic advection more than diffusion. The transport of a small fraction of tracer particles deep into the underlying stable region is reasonably efficient, a result which has possible implications for the problem of light-element depletion in late-type stars.