to the $k_x^{-1}$ with $a=0.9994$ in the buffer region. (b) Theoretical curve has a uniform $I$-integral with the $k_x^{-5/3}$ scaling present in buffer and inertial regions.[]{data-label="fig:mean velocity"}](Figure_2bis.pdf){width="45.00000%"}

The Spectral Theory of mean-square fluctuations {#sec:BW}
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In the above sections we have not used the spectral information in the integral $I$, in Eq. (\[eq:spectcont\]). We have just used the attached eddy hypothesis and set $I=0$ in the viscous layer and $I=1$ in the inertial layer. But following Ref. [@GGGC10], we can now use the spectral information through the integral $I$ to find the beginning of the buffer layer and the form of both the MVP $u$ and the fluctuation $w$ in the buffer layer and in the wake. This allows one to obtain the full functional form of both $u$ and $w$ as functions of the distance $y$ from the wall and compare it with the experimental data in the next section. By use of the energy Eq. (\[eq:energy\]) and the relation $\eta = \nu^{3/4}\epsilon^{-1/4}$ we can find an expression for $\eta/y$, the viscosity parameter that increases as we approach the wall $y\to 0$. If we set the fluctuation equal to zero, $$\eta/y = (\tilde u'(1-\tilde y/Re\sqrt{f})-(\tilde u')^2)^{-1/4}\tilde y^{-1}$$ and find 