Stochastic models for tracer experiments in the circulation: II. Serial random walks

The theory of the simple one-dimensional random walk has been extended to the case of variable diffusion coefficient D( x). An approximate solution of the diffusion equation with drift has been obtained which is applicable for the case of a quasi-normal spatial distribution C( x). This includes cases where D( x) is a slowly varying function and particularly cases early in time following initial introduction of a delta function distribution into the system. A relation is derived through which higher moments of C( x) can be obtained from lower moments and their integrals. A numerical iteration method is described by which the evolution of C( x) can be characterized through the evolution of its moments. The approximate effect of an upstream reflecting barrier on the distribution is discussed. The approximate effect of diffuse rather than discrete absorption at the terminus of the system is also estimated.