# Langevin equation

The stochastic differential equation (SDE) for velocity component $U(t)$, the Langevin equation is

$dU(t) = - U(t) \frac{dt}{\tau} \frac{2 u'}{\tau}^{1/2} dW(t)$

where $dW(t)$ is a Wiener process. $u'$ is the turbulence intensity and $\tau$ a Lagrangian time-scale.

Th finite difference approximation of the above equation is

$U(t \Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}$

where $\mathcal{N}$ is a standardized Gaussian random variable with 0 mean an unity variance which is independent of $U$ on all other time steps (Pope 1994). The Wiener process can be understood as Gaussian random variable with 0 mean and variance $dt$