Extended Self-Similarity in Turbulent Systems: An Analytically Soluble Example

In turbulent flows the nth order structure functions $S_n\left(R\right)$ scale like $R^{{\zeta }_n}$ when $R$ is in the “inertial range.” Extended self-similarity refers to the substantial increase in the range of power law behavior of the $S_n\left(R\right)$ when they are plotted as a function of $S_2\left(R\right)$ or $S_3\left(R\right)$. Here we demonstrate this phenomenon analytically in the context of the “multiscaling” turbulent advection of a passive scalar. This model gives rise to a series of differential equations for the structure functions $S_n\left(R\right)$ which can be solved and shown to exhibit extended self-similarity. The phenomenon is understood by comparing the equations for $S_n\left(R\right)$ to those for $S_n\left(S_2\right)$.