\sqrt{f}$, the attached eddies increase the viscosity (decrease the Reynolds number) and the MVP reaches its maximum increase at $y \approx 15$, independent of the Reynolds number. The energy transfer of the attached eddies is captured by the $I$-integral and we integrate the differential equation given by Eq. (\[eq:udiff\]), from $y=4.17$, with the initial condition $u=4.17$. This gives the MVP in Fig. \[fig:mean velocity\] (b). This was already done in Ref. [@GGGC10] and describes how the attached eddies transfer energy into the buffer and the inertial layer. However, we notice that in the predicted MVP over estimates the mean velocity in buffer region. This is because the $I$-function from Eq. (\[eq:spectcont\]) does not account for the formation of the attached eddies which reduce the net energy transfer in the direct cascade.

The curves for the fluctuations squared in Fig. \[fig:mean variation\] are obtained in a similar manner. The attached eddies fix the peak of $\langle w^2 \rangle$ at $y \approx 15$ and the peak profiles can be fitted by the viscous formula $\langle w^2 \rangle = a (y-\frac{y^2}{30})^2$ where $a \sim( \langle \tau _o^2\rangle -\langle \tau_o \rangle^2)/ \langle \tau_o \rangle^2$. This fit is shown in Fig. \[fig:mean variation\] (c). The 