  The 4/5 and 2/3 laws of turbulence can emerge from a theory of 'engineered'
random vector fields $\mathcal{X}_{i}(x,t)
=X_{i}(x,t)+\tfrac{\theta}{\sqrt{d(d+2)}} X_{i}(x,t)\psi(x)$ existing within
$\mathbf{D}\subset\mathbf{R}^{d}$. Here, $X_{i}(x,t)$ is a smooth deterministic
vector field obeying a nonlinear PDE for all
$(x,t)\in\mathbf{D}\times\mathbf{R}^{+}$, and $\theta$ is a small parameter.
The field $\psi(x)$ is a regulated and differentiable Gaussian random field
with expectation $\mathbb{E}[\psi(x)]=0$, but having an antisymmetric
covariance kernel
$\mathscr{K}(x,y)=\mathbb{E}[\psi(x)\psi(y)]=f(x,y)K(\|x-y\|;\lambda)$ with
$f(x,y)=-f(y,x)=1,f(x,x)=f(y,y)=0$ and with $K(\|x-y\|;\lambda)$ a standard
stationary symmetric kernel. For $0\le\ell\le \lambda<L$ with
$X_{i}(x,t)=X_{i}=(0,0,X)$ and $\theta=1$ then for $d=3$, the third-order
structure function is \begin{align}
S_{3}[\ell]=\mathbb{E}\left[|\mathcal{X}_{i}(x+\ell,t)-\mathcal{X}(x,t)|^{3}\right]=-\frac{4}{5}\|X_{i}\|^{3}=-\frac{4}{5}X^{3}\nonumber
\end{align} and $S_{2}[\ell]=CX^{2}$. The classical 4/5 and 2/3-scaling laws
then emerge if one identifies the random field $\mathcal{X}_{i}(x,t)$ with a
turbulent fluid flow $\mathcal{U}_{i}(x,t)$ or velocity, with mean flow
$\mathbb{E}[\mathcal{U}_{i}(x,t)]=U_{i}(x,t)=U_{i}$ being a trivial solution of
Burger's equation. Assuming constant dissipation rate $\epsilon$, small
constant viscosity $\nu$, corresponding to high Reynolds number, and the
standard energy balance law, then for a range $\eta\le\ell\ll \lambda<L$
\begin{align}
S_{3}[\ell]=\mathbb{E}\left[|\mathcal{U}_{i}(x+\ell,t)-\mathcal{U}(x,t)|^{3}\right]=-\frac{4}{5}\epsilon\ell\nonumber
\end{align} where $\eta=(\nu^{3/4}\epsilon)^{-1/4}$. For the second-order
structure function, the 2/3-law emerges as
$S_{2}[\ell]=C\epsilon^{2/3}\ell^{2/3}$.
