the wall. We consider the Eq. (\[eq:total\_stress1\]) $$\label{eq:total_stress2}
{\kappa}^2 {y}^2( u'+ w')^2 I^{3/4}+(u'+w') = \frac{\tau_0}{\langle \tau_0 \rangle}(1-\frac{y}{Re\sqrt{f}}),$$ and first set $I=0$ in the viscous layer. Then $$\label{eq:u_o}
u = y - \frac{y^2}{2 Re\sqrt{f}}$$ by averaging and integration in $y$. Integrating Eq. (\[eq:total\_stress2\]) and subtracting $u$ gives, $$\label{eq:w_o}
w = \frac{\tau_0 - \langle \tau_0 \rangle}{\langle \tau_0 \rangle}\left( y - \frac{y^2}{2 Re\sqrt{f}}\right)$$ and $$\langle w^2 \rangle=\frac{\langle \tau_0^2 \rangle - \langle \tau_0 \rangle^2}{\langle \tau_0 \rangle^2}\left( y - \frac{y^2}{2 Re\sqrt{f}}\right)^2.$$

In the inertial layer $I=1$ and ignoring the small $O(1/y^4)$ term, we get that $$\begin{aligned}
u+w &=& \frac{1}{2\kappa^2 y} + 2\frac{\sqrt{\tau_0}}{\kappa \sqrt{\langle \tau_0 \rangle}}\sqrt{1-\frac{y}{2Re\sqrt{f}}} \nonumber\\
&-&2 \frac{\sqrt{\tau_0}}{\kappa \sqrt{\langle \tau_0 \rangle}} \tanh^{-1}\left(\sqrt{1-\frac{y}{2Re\sqrt{f}}}\right)+K, \end{aligned}$$ where $K$ is a constant. Then setting $w=0$, we get that $$\begin{aligned}
&&u = \frac{1}{2\kappa^2 y} + \frac{2}{\kappa}\sqrt{1-\frac{y}{2Re\sqrt{f}}}\nonumber\\
&-&\frac{2}{\kappa}\tanh^{-1}\left(\sqrt{1-\frac{y}{2Re\sqrt{f}}}\right)+K',\end{aligned}$$ where $K'$ is another constant, because $\tau_0$ becomes $\langle \tau_0 \rangle$. Subtracting, $u$ from $u+w$ we get $$\begin{aligned}
&&w =  2\frac{(\sqrt{\tau_0}-\sqrt{\langle \tau_0 \rangle})}{\kappa \sqrt{\langle \tau_0 \rangle}}\sqrt{1-\frac{y}{2Re\sqrt{f}}}{\nonumber}\\
&-&2 \frac{(\sqrt{\tau_0}-\sqrt{\langle \tau_0 \rangle})}{\kappa \sqrt{\langle \tau_0 \rangle}} \tanh^{-1}\left(\sqrt{1-\frac{y}{2Re\sqrt{f}}}\right)+C,\end{aligned}$$ where $C=K-K'$. Squaring $w$ and taking the average gives $$\begin{aligned}
&&\langle w^2 \rangle =  4C\frac{(\langle \sqrt{\tau_0} \rangle-\sqrt{\langle \tau_0 \rangle})}{\kappa \sqrt{\langle \tau_0 \rangle}}\sqrt{1-\frac{y}{2Re\sqrt{f}}} {\nonumber}\\
&-&4C \frac{(\langle \sqrt{\tau_0} \rangle-\sqrt{\langle \tau_0 \rangle})}{\kappa \sqrt{\langle \tau_0 \rangle}} \tanh^{-1}\left(\sqrt{1-\frac{y}{2Re\sqrt{f}}}\right){\nonumber}\\
&+&4\left[\frac{2(\langle\tau_0 \rangle-\sqrt{\langle \tau_0 \rangle}\langle \sqrt{\tau_0}\rangle)}{\kappa^2 \langle 