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Langevin equation

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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

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