Exercise 6

In deriving the 4/5'th law we obtain an expression, (Frisch, 6.51, p 85),

$$F(l)=(1+l\partial_l)(3+l\partial_l)(5+l\partial_l)\frac{S_3(l)}{l} =-12\epsilon,$$ (1)

which is valid in the inertial range. The inertial range is the range of scales so small that the forcing can be neglected (assumed only to act on the integral scales), but so large that the viscosity (acting on the internal scale) can be neglected as well. $\epsilon$ is the mean energy dissipation per unit mass.

Solve this differential equation with respect to S₃(l) and derive the 4/5'th law.

Hint: Use x=log(l) as independent variable and y=S₃(l)/l as dependent variable. Remember that S₃(l) must be finite for $l \rightarrow 0$.