Journal of Biological Physics 18: 297-306, 1992. (Q 1992 Kluwer Academic Publishers. Printed in the Netherlands. 297 The Effect of Turbulent Motion on the Diffusion of Microorganisms TREVOR LIPSCOMBE Princeton University Press, 41 William Street, Princeton, NJ 08.540, U.S.A. (Received: 6 July 1992) Abstract. The effect of turbulent fluid motion on the diffusion of simple organisms is discussed. The net reproduction rate and the turbulent flow are assumed to be Gaussian-correlated random variables. For homogeneous isotropic turbulence, simple equations for the average concentration of the organisms are derived in terms of the energy density of the fluid. It is shown that the effective diffusivity generated by the motion is positive-definite, and is independent of the helicity of the flow. Key words: biological fluid dynamics, diffusion, turbulence Introduction One important problem in epidemiology and ecologyis to determine the effect of fluid motion on the spread of airborne or waterborne organisms. This system can be modelled by the simple mathematical expression wx, t> ‘& + [v(x, t) * V] C(x, t) = D v2 C(x, t) + R(x, t) C(x, t) . (1) Here C(x, t) is the concentration of the organisms at a point x at time t. The velocity of the fluid v(x, t) is assumed to be incompressible, so that V . v = 0 and an equation of state neednot be specified.The incompressibility condition holds for a wide class of fluids, particularly liquids. The organismsare assumed to be small enoughso that they do not disturb the flow pattern. D is a constant - the diffusion rate of the speciesin the particular fluid at rest. R(x, t) is the net reproductionrate of the organisms, the differencebetween the birth rate and the death rate. Reproduction here is assumed to be independent of the organism population andto be asexual. The generalization to sexualreproduction is simple: The concentration of the two sexes canbe written as Ci (with i = 1 or 2), and the production ratethen becomes a four-component tensor,R+. Given thesedefinitions, Equation (1) is a consequence of the conservation of the number of organisms in a given fluid element.Equation (1) should be a good model for describingthe spread of bacteria or other micro-organisms in air, or for the motion of planktonin the ocean. In what follows, it is assumed that the statisticsof the turbulent velocity field and the random reproduction rate are known, that the averages of R and v are