Eq. (\[eq:velocity\]) in the sense that if $\sqrt{\tau_0}=\langle \tau_0\rangle^{1/2}$, then $\tilde w =0$ and we recover Eq. (\[eq:velocity\]). Thus squaring Eq. (\[eq:velocity1\]) gives that $${\tilde u}^2+2\tilde u \tilde w +{\tilde w}^2= \frac{\tau_0}{\langle \tau_0\rangle \tilde \kappa^2} (\log(\hat y))^2+2\frac{\sqrt{\tau_0}}{\tilde \kappa \sqrt{\langle \tau_0 \rangle}} C\log(\tilde y) +C^2.$$ Taking the average, using that $\langle \tilde w \rangle = 0$ and Eq. (\[eq:velocity\]), we get that $$\langle \tilde w^2 \rangle = \frac{2C\langle \sqrt{\tau_0}\rangle-2D\sqrt{\langle\tau_0\rangle}}{\tilde \kappa \sqrt{\langle \tau_0 \rangle}}\log(\tilde y) +C^2-D^2.$$ By comparing this with the generalised log-law in Eq. (\[eq:l-lfluct\]), for the fluctuations squared, we obtain $$\label{eq:gloglaw}
\langle \tilde w^2 \rangle = -A \log(\tilde y)+B,$$ where $A = - \frac{2C\langle \sqrt{\tau_0}\rangle-2D\sqrt{\langle\tau_0\rangle}}{\tilde \kappa \sqrt{\langle \tau_0 \rangle}}$ and $B = C^2-D^2$ are the Townsend-Perry constants. The full formulas in next section show that Eq. (\[eq:gloglaw\]) is the leading term and $A = - 2C(\frac{\langle \sqrt{\tau_0}\rangle-\sqrt{\langle\tau_0\rangle}}{\tilde \kappa \sqrt{\langle \tau_0 \rangle}})$, with $C=D$.

To simplify the notation, we will now drop the tilde’s from all the variable with the dimensionless units implicitly assumed, unless otherwise stated.

The functional form of the Townsend-Perry law {#sec:functional}
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We will now use Eq. (\[eq:total\_stress1\]) to find the general form of the average of the fluctuations squared as a function of the distance to 