peak position is experimentally observed to be fixed, but its height shows a weak Reynolds number dependence $a = -3.06+0.99 \log(Re)$, see [@Sa18]. This relationship can be tested using our theory and this will be done in another publication, see also [@CS20]. Then, we integrate the differential equation from Eq. (\[eq:wdiff\]) for $w$ with the initial data described in last section from some point to the right of the peak, where above peak profile fits the initial condition, this give the profile of the fluctuations squared down to the flat part in the buffer layer. At the beginning of the flat part, $y \approx 60$, the second scaling from Section \[sec:MVP\] begins to dominate the fluctuations, modeling an inverse cascade of attached eddies in the buffer layer. Then we switch to the buffer $I$-function $I_ b$ in the integration and integrate with $I_b$ until we get into the inertial region where the Kolmogorov-Obukhov scaling dominates again and the attached eddies break up. This produces the curves in Fig. \[fig:mean variation\].

We can now compare the functional form of the fluctuations squared shown in Fig. \[fig:mean variation\] with the predictions of the stochastic closure theory (SCT) of turbulence, used in Refs. 