Bottleneck Effects in Turbulence: Scaling Phenomena in $\mathit{r}$ versus $\mathit{p}$ Space

We (analytically) calculate the energy spectrum corresponding to various experimental and numerical turbulence data analyzed by Benzi et al. We find two bottleneck phenomena: While the local scaling exponent ${\zeta }_r\left(r\right)$ of the structure function decreases monotonically, the local scaling exponent ${\zeta }_p\left(p\right)$ of the corresponding spectrum has a minimum of ${\zeta }_p\left(p_{\min }\right)\approx 0.45$ at $p_{\min }\approx (10\eta {)}^{-1\mathrm{}}$ and a maximum of ${\zeta }_p\left(p_{\max }\right)\approx 0.77$ at $p_{\max }\approx {8L}^{-1\mathrm{}}$. A physical argument starting from the constant energy flux in $p$ space reveals the general mechanism underlying the energy pileups at both ends of the $p$-space scaling range. In the case studied here, they are induced by viscous dissipation and the reduced spectral strength on the scale of the system size, respectively.