The stochastic differential equation (SDE) for velocity component , the Langevin equation is
where is a Wiener process. is the turbulence intensity and a Lagrangian time-scale.
Th finite difference approximation of the above equation is
where is a standardized Gaussian random variable with 0 mean an unity variance which is independent of on all other time steps (Pope 1994). The Wiener process can be understood as Gaussian random variable with 0 mean and variance