Radial mixing in protoplanetary accretion disks VII. 2-dimensional   transport of tracers
  The detection of significant concentrations of crystalline silicates in comets indicates an extensive radial mixing in the primordial solar nebula. In studying the radial transport of matter within protoplanetary disks by numerical model calculations it is essential to resolve the vertical disk structure since matter is mixed radially inward and outward by a complex 2-dimensional flow pattern that is superposed on the global inward directed accretion flow. A numerical model calculation for a protoplanetary accretion disks with radial and vertical mixing is performed in the 1+1-dimensional approximation. The global 2D velocity field of the disk is calculated from an analytical solution for the meridional flow pattern, that exhibits an inward drift in the upper layers and an outward drift in the midplane in most parts of the disk. The disk model is based on the $\beta$-prescription of viscosity and considers vertical self-gravitation of the disk. The mixing processes are studied for the following species: amorphous silicate grains (forsterite, enstatite) that crystallise by annealing, and carbonaceous grains that are destroyed by surface reactions with OH molecules at elevated temperatures. Considerable fractions of crystallised silicates and methane (formed as a by-product of carbon combustion) are transported to the site of comet formation far from the protosun within a period of 10$^6$ yrs. The 2-dimensional transport of tracers in the solar nebula offers a natural explanation for the presence of crystalline silicates in comets and the significant portions of crystalline silicates observed in accretion disks around young stellar objects. 
Michael Wehrstedt and Hans-Peter Gail