Kinetics of growth process controlled by convective fluctuations

A model of the spherical (compact) growth process controlled by a fluctuating local convective velocity field of the fluid particles is introduced. It is assumed that the particle velocity fluctuations are purely noisy, Gaussian, of zero mean, and of various correlations: Dirac delta, exponential, and algebraic (power law). It is shown that for a large class of the velocity fluctuations, the long-time asymptotics of the growth kinetics is universal (i.e., it does not depend on the details of the statistics of fluctuations) and displays the power-law time dependence with the classical exponent 1/2 resembling the diffusion limited growth. For very slow decay of algebraic correlations of fluctuations asymptotically like t(-gamma), gamma in (0,1]), kinetics is anomalous and depends strongly on the exponent gamma. For the averaged radius of the crystal approximately t(1-gamma/2) for 0 approximately (t ln t)1/2 for gamma=1.