Appendix A
Supercritical-Channel-Flow Hydraulic Theory Uncertainty Estimation
The uncertainties in the calculations presented in section 3.3 and Table 2 are estimated as follows. The total uncertainty, \(\text{δq},\ \)of a function \(q\left(x,\ \ldots,\ z\right)\) for which the uncertainties in \(x,\ \ldots,\ z\) are \(\delta x,\ \ldots,\ \delta z\), respectively, is estimated via Taylor (1997) as
\begin{equation} \delta q=\sqrt{\left(\frac{\partial q}{\partial x}\text{δx}\right)^{2}+\ldots+\left(\frac{\partial q}{\partial z}\text{δz}\right)^{2}}\nonumber \\ \end{equation}
In the supercritical-channel-flow cases analyzed in Section 3, upstream wind speed and PBL height, lower-troposphere static stability, and the angular change in the flow direction due to the bending boundary (Figs. 6, 8, 10, and 12) are obtained from visual inspection of the respective fields and determine  Fr and , along with their corresponding uncertainties. For instance, using Eq. (A1), the total uncertainty in the Froude Number (\(\delta Fr)\) upstream of the Guiana Highland expansion fan is given by
\begin{equation} \delta Fr=\sqrt{\left(\frac{\partial Fr}{\partial U}\text{δU}\right)^{2}+\left(\frac{\partial Fr}{\partial g^{\prime}}\delta g^{\prime}\right)^{2}+\left(\frac{\partial Fr}{\partial h}\text{δh}\right)^{2}}\nonumber \\ \end{equation}
\(\delta Fr=0.22\)
so that \(Fr=1.32\pm 0.22\).
The measured values and uncertainties of Fr and propagate through the subsequent calculations in Eqs. (4)–(7). The total uncertainty for every new variable is obtained using Eq. (A1), with the exception of δβ in Eq. (4), where \(\beta\) is a function of Fr only, for which a simpler formula can be used:
\begin{equation} \delta\beta=\left|\frac{\text{dβ}}{\text{dFr}}\right|\text{δFr}\nonumber \\ \end{equation}
Table 2 summarizes the predicted final wind speeds, PBL heights, and the corresponding uncertainties for each expansion fan.
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