Ra. Jarvis et Aw. Woods, THE NUCLEATION, GROWTH AND SETTLING OF CRYSTALS FROM A TURBULENTLY CONVECTING FLUID, Journal of Fluid Mechanics, 273, 1994, pp. 83-107
We present a new model to describe the thermal and compositional evolu
tion of a binary alloy which is cooled from above. Explicit account is
taken of the nucleation of crystals in the cold upper thermal boundar
y layer, the growth of crystals in the turbulently convecting interior
, and their subsequent gravitational settling to the floor of the cham
ber. The crystallization of one solid phase only is considered. When t
he residence time of a typical crystal within the convecting bulk is s
hort compared with the overall cooling time of the fluid, the crystal
size distribution loses memory of earlier conditions in the fluid and
the number density simply decays exponentially with the cube of the cr
ystal size. A quasi-steady state exists in which the rate of crystal p
roduction is balanced by the rate of sedimentation at the floor, allow
ing the volume fraction of suspended crystals to remain small until co
nvection ceases to be vigorous. We focus on the situation in which the
latent heat released by solidification would far exceed the heat flux
extracted through convection if the melt undercooling were maintained
equal to the initial temperature difference applied at the cold upper
boundary. In this case, either the growth or nucleation of crystals m
ust be limited in order that the fluid continues to cool. Both the gro
wth-limited and nucleation-limited regimes may develop during the cool
ing of an individual fluid body, depending upon the thermal boundary c
ondition at the upper boundary of the convecting portion of the fluid.
We calculate how the mean crystal size within the sedimented crystal
pile evolves as the fluid cools. During the growth-limited regime, the
mean crystal size in the crystal pile typically decreases with height
, owing to the decrease in the extracted heat flux and the greater eff
iciency of crystal settling as the fluid layer becomes shallower. In c
ontrast, during the nucleation-limited regime, the fluid undercooling
may increase significantly as the fluid cools, and inverse grading (la
rge crystals over small) is possible. We discuss the possible applicat
ion of our theory to the cooling of large igneous intrusions.