size $R$), respectively. $\kappa_\epsilon$ is a dimensionless parameter, $\epsilon$ is the turbulent energy dissipation rate, $\eta=\nu^{3/4}\epsilon^{-1/4}$ is the viscous length scale and $R$ is the largest length scale in the flow. The dissipative correction function is typically an exponential cutoff function $c_d(\eta k) =\exp(-\beta_d \eta k)$, and the energetic-range (wake) correction function is $c_e(Rk)=(1+(\beta_e/(Rk))^2)^{-17/4}$, which is the form that was proposed by von Kármán. $\beta_d$ and $\beta_e$ are non-negative fitting parameters that can be adjusted to data. By the change of variables $\xi = sk$, we recast Eq. (\[eq:vs\]) as $$v_s^2 = (\kappa_\epsilon \epsilon s)^{2/3}I\left(\frac{\eta}{s},\frac{s}{R}\right),$$ where the spectral function $I$ is given by the formula [@GGGC10] $$\begin{aligned}
\label{eq:spectcont}
&I\left(\frac{\eta}{s},\frac{s}{R}\right)=  \nonumber \\
&\frac{2}{3} \int_1^\infty e^{-\xi \beta_d \eta/s}\xi^{-5/3}\left(1+\left(\frac{\beta_e s}{R\xi}\right)^2\right)^{-17/6} d\xi.\end{aligned}$$ The integral sums the energies of all eddies of a smaller radius than $s$, and computes their contribution to the energy of the eddy of radius $s$. This is the energy (or spectral) formulation of the attached eddy hypothesis of Townsend [@T76]. The $I$-function correctly captures the buffer layer, as the transition from the viscous to the inertial layer, and the asymptotic of the MVP in the energetic wake. The asymptotic values are such that in the inertial layer $I=1$ and in 