the viscous layer $I=0$. The $I$-function combines the Kolmogorov-Obukhov theory with the observed spectrum in the viscous layer, the inertial layer and the wake and is thus able to capture the transition from one layer to the next. In Ref [@GGGC10], it was used to give the details of the MVP. In this paper, we will use it to capture the profile of mean-square fluctuations.

In the buffer layer a different scaling of the attached eddies comes into play, this is the $k_x^{-1}$ scaling of the spectrum that has been debated in literature, but clearly shows up in recent simulations and experiments in the middle of the buffer layer, see Figure 9 (a) in Ref. [@LM15] and Figure 12 (b) in Ref. [@Sa18]. In the spectral theory, corresponding $I$-function for this scaling regime is $$\begin{aligned}
\label{eq:spectcont1}
&I_b\left(\frac{\eta}{s},\frac{s}{R}\right)= \nonumber\\
&\frac{2}{3}s^{-\frac{2}{3}} \int_1^\infty e^{-\xi \beta_d \frac{\eta}{s}}\xi^{-1}\left(1+\left(\frac{\beta_e s}{R\xi}\right)^2\right)^{-\frac{17}{6}} d\xi, \end{aligned}$$ where the subscript $b$ stands for “buffer”. The mean velocity is primarily influenced by the $I$-function, whereas the variation (fluctuation squared) is greatly influenced by the $I_b$-function in the buffer layer. $I$ is associated with the Kolmogorov-Obukhov energy cascade $k_x^{-5/3}$, in the inertial layer, whereas $I_b$ is associated with the $k_x^{-1}$ scaling in the 