data.

Approximations to the MVP and mean square fluctuations, based on the formulas in Sec. \[sec:functional\] are given in Fig. \[fig:mean velocity\] and \[fig:mean variation\], respectively. To compare with experimental data one must solve the differential equations $$\label{eq:udiff}
u'=-\frac{1}{2 \kappa^2 I^{3/4}y^2} + \frac{1}{\kappa I^{3/8}y} \sqrt{1 - \frac{y}{Re\sqrt{f}}+\frac{1}{4\kappa^2 I^{3/4}y^2}}$$ with the initial condition $u = 4.17$ at the beginning of the buffer layer $y =4.17$. For the fluctuation we first have to solve the differential equation, ignoring term of order $O(1/y^3)$ and higher, $$\label{eq:wdiff}
w'=\frac{ \sqrt{\tau_0}-\sqrt{\langle\tau_0\rangle}}{ \kappa I^{3/8}y\sqrt{\langle \tau_0 \rangle}} \sqrt{1 - \frac{y}{Re\sqrt{f}}},$$ with the initial condition $w=\frac{ \tau_0-\langle\tau_0\rangle}{ \langle \tau_0 \rangle}\left(4.17-\frac{17.39}{2 Re\sqrt{f}}\right)$, from Eq. (\[eq:w\_o\]), at the beginning of the buffer layer. Here $I(y)$ is the integral in Eq. (\[eq:spectcont\]).

![image](Figure_3.pdf){width="80.00000%"}

In practice it is easier to vary the initial conditions than to change $\beta_d$ and $\beta_e$, thus we will let the initial condition $y_o$, of $w$, from Equation (\[eq:w\_o\]), vary slightly depending on the Reynolds number in the simulations below. The other initial condition $w_o$ is given by the formula $w_o=\frac{ \tau_0-\langle\tau_0\rangle}{ \langle \tau_0 \rangle}\left(y_o-\frac{y_o^2}{2 Re\sqrt{f}}\right)$.

Comparison with Experimental Data {#sec:data}
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The data we use to compare with the theory comes from the wind tunnel experiments at the 