Exercise 3.

Burgers equation,

$$\delta_t u + u\delta_xu=\nu \delta_{xx}u$$ (1)

is a one dimensional toy-model of turbulence (u=u(x,t)). We shall return to Burgers equation later in the course.

The inviscid case, $\nu=0$, is just an advection equation in 1 dimension,

$$\frac{Du}{Dt}=0.$$ (2)

Derive dynamical equations to fourth order for the ensamble averages, $\langle u \rangle$. Assume that u is a gaussian random variable and apply the gaussian closure of the system of equations. Derive an equation for $\langle u^2 \rangle$.

The gaussian closure is (from Wick's theorem):

$$\langle u^4 \rangle = 3 \langle u^2 \rangle^2.$$ (3)

Show this.