A New Sensitivity Analysis and Solution Method for Scintillometer Measurements of Area-Averaged Turbulent Fluxes

Iteration, like friction, is likely to generate heat instead of progress. - George Eliot

Abstract

Scintillometer measurements of the turbulence inner-scale length $l_\mathrm{o }$ and refractive index structure function $C_n^2$ allow for the retrieval of large-scale area-averaged turbulent fluxes in the atmospheric surface layer. This retrieval involves the solution of the non-linear set of equations defined by the Monin–Obukhov similarity hypothesis. A new method that uses an analytic solution to the set of equations is presented, which leads to a stable and efficient numerical method of computation that has the potential of eliminating computational error. Mathematical expressions are derived that map out the sensitivity of the turbulent flux measurements to uncertainties in source measurements such as $l_\mathrm{o }$. These sensitivity functions differ from results in the previous literature; the reasons for the differences are explored.

Appendices

Appendix A: Relations Between $M$ and $\zeta$

$$\begin{aligned} M&= \frac{\zeta ^3}{(1+d(-\zeta )^{2/3})^3(1-b\zeta )},\end{aligned}$$

(54)

$$\begin{aligned} \left( \frac{\partial \zeta }{\partial M}\right)&= \left( \frac{(1\!-\!b\zeta )(1\!+\!d(-\zeta )^{2/3})^3}{3\zeta ^2\!+\!M[2d(1\!-\!b\zeta )(1 \!+\!d(-\zeta )^{2/3})^2(-\zeta )^{-1/3}\!+\!b(1\!+\!d(-\zeta )^{2/3})^3]}\right) , \end{aligned}$$

(55)

$$\begin{aligned} M\left( \frac{\partial \zeta }{\partial M}\right)&= \left( \frac{\zeta (1-b\zeta )(1+d(-\zeta )^{2/3})}{(3-2b\zeta )(1+d(-\zeta )^{2/3}) +2d\zeta (-\zeta )^{-1/3}(1-b\zeta )} \right) . \end{aligned}$$

(56)

Appendix B: Individual Terms in $S_{T_\star ,\epsilon }$ for Unstable Conditions ($\zeta <0$)

$$\begin{aligned} \left( \frac{\partial T_\star }{\partial \zeta }\right)&= T_\star \left( \frac{-b}{3(1-b\zeta )}\right) , \end{aligned}$$

(57)

$$\begin{aligned} \left( \frac{\partial M}{\partial \epsilon }\right)&= -2M/\epsilon . \end{aligned}$$

(58)

Appendix C: Individual Terms in $S_{T_\star ,z}$ for Unstable Conditions ($\zeta <0$)

$$\begin{aligned} \left( \frac{\partial T_\star }{\partial z}\right) _\zeta&= \frac{T_\star }{3z}, \end{aligned}$$

(59)

$$\begin{aligned} \left( \frac{\partial T_\star }{\partial \zeta }\right) _z&= T_\star \left( \frac{-b}{3(1-b\zeta )}\right) , \end{aligned}$$

(60)

$$\begin{aligned} \left( \frac{\partial M}{\partial z}\right)&= 2M/z. \end{aligned}$$

(61)

Appendix D: Individual Terms in $S_{u_\star ,\epsilon }$ for Unstable Conditions ($\zeta <0$)

$$\begin{aligned} \left( \frac{\partial u_\star }{\partial \epsilon }\right) _\zeta&= \frac{u_\star }{3\epsilon }, \end{aligned}$$

(62)

$$\begin{aligned} \left( \frac{\partial u_\star }{\partial \zeta }\right) _\epsilon&= u_\star \left( \frac{d}{3(1+d(-\zeta )^{2/3})(-\zeta )^{1/3}}\right) ,\end{aligned}$$

(63)

$$\begin{aligned} \left( \frac{\partial M}{\partial \epsilon }\right)&= -2M/\epsilon . \end{aligned}$$

(64)

Appendix E: Individual Terms in $S_{u_\star ,z}$ for Unstable Conditions ($\zeta <0$)

$$\begin{aligned} \left( \frac{\partial u_\star }{\partial z}\right) _\zeta&= \frac{u_\star }{3z}, \end{aligned}$$

(65)

$$\begin{aligned} \left( \frac{\partial u_\star }{\partial \zeta }\right) _z&= u_\star \left( \frac{d}{3(1+d(-\zeta )^{2/3})(-\zeta )^{1/3}}\right) , \end{aligned}$$

(66)

$$\begin{aligned} \left( \frac{\partial M}{\partial z}\right)&= 2M/z. \end{aligned}$$

(67)

Appendix F: Total Differential Expansion as in Andreas (1992) for Unstable Conditions ($\zeta <0$)

Here we reproduce the analysis of Andreas (1992). Subscripts indicate the equation that is being differentiated locally. The coupled equations are

$$\begin{aligned} \zeta&= \frac{zgk}{u_\star ^2 T}\left( T_\star +\frac{0.61T}{\rho +0.61q}q_\star \right) , \end{aligned}$$

(68)

$$\begin{aligned} \epsilon&= \frac{u_\star ^3}{\kappa z}\phi (\zeta ) = \frac{u_\star ^3}{\kappa z}(1+d(-\zeta )^{2/3})^{3/2} , \end{aligned}$$

(69)

$$\begin{aligned} T_\star&= \frac{(1-b\zeta )^{1/3}z^{1/3}}{\sqrt{(}a)}\left( \frac{\text{ sign }_1\sqrt{C_{n_1}^2}B_2-\text{ sign }_2 \sqrt{C_{n_2}^2}B_1}{A_1B_2-A_2B_1}\right) , \end{aligned}$$

(70)

$$\begin{aligned} q_\star&= \frac{(1-b\zeta )^{1/3}z^{1/3}}{\sqrt{(}a)}\left( \frac{\text{ sign }_2\sqrt{C_{n_2}^2}A_1-\text{ sign }_1 \sqrt{C_{n_1}^2}A_2}{A_1B_2-A_2B_1}\right) . \end{aligned}$$

(71)

We expand Eqs. 68 and 69 as

$$\begin{aligned} d\zeta&= \left( \frac{\partial \zeta }{\partial z}\right) _{68}dz+\left( \frac{\partial \zeta }{\partial T_\star }\right) _{68}dT_\star +\left( \frac{\partial \zeta }{\partial q_\star }\right) _{68}dq_\star ,\end{aligned}$$

(72)

$$\begin{aligned} d\epsilon&= \left( \frac{\partial \epsilon }{\partial u_\star }\right) _{69}du_\star +\left( \frac{\partial \epsilon }{\partial z}\right) _{69}dz+\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}d\zeta . \end{aligned}$$

(73)

Combining these, we obtain

$$\begin{aligned} d\epsilon&= \left[ \left( \frac{\partial \epsilon }{\partial u_\star }\right) _{69}+\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial u_\star }\right) _{68}\right] du_\star \nonumber \\&+\left[ \left( \frac{\partial \epsilon }{\partial z}\right) _{69}+\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial z}\right) _{68}\right] dz \nonumber \\&+\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial T_\star }\right) _{68}dT_\star \nonumber \\&+\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial q_\star }\right) _{68}dT_\star , \end{aligned}$$

(74)

$$\begin{aligned} \frac{d\epsilon }{\epsilon }&= \frac{u_\star }{\epsilon } \frac{du_\star }{u_\star } \left[ \left( \frac{\partial \epsilon }{\partial u_\star }\right) _{69}+\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial u_\star }\right) _{68}\right] \nonumber \\&+\frac{z}{\epsilon }\frac{dz}{z}\left[ \left( \frac{\partial \epsilon }{\partial z}\right) _{69}+\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial z}\right) _{68}\right] \nonumber \\&+\frac{T_\star }{\epsilon }\frac{dT_\star }{T_\star }\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial T_\star }\right) _{68} \nonumber \\&+\frac{q_\star }{\epsilon }\frac{dq_\star }{q_\star }\left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}\left( \frac{\partial \zeta }{\partial q_\star }\right) _{68} , \end{aligned}$$

(75)

where the local derivatives are given by

$$\begin{aligned} \left( \frac{\partial \epsilon }{\partial u_\star }\right) _{69}&= \frac{3\epsilon }{u_\star },\end{aligned}$$

(76)

$$\begin{aligned} \left( \frac{\partial \zeta }{\partial u_\star }\right) _{68}&= \frac{-2\zeta }{u_\star },\end{aligned}$$

(77)

$$\begin{aligned} \left( \frac{\partial \epsilon }{\partial \zeta }\right) _{69}&= \frac{\epsilon }{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta ), \end{aligned}$$

(78)

$$\begin{aligned} \left( \frac{\partial \epsilon }{\partial z}\right) _{69}&= -\frac{\epsilon }{z}, \end{aligned}$$

(79)

$$\begin{aligned} \left( \frac{\partial \zeta }{\partial z}\right) _{68}&= \frac{\zeta }{z} ,\end{aligned}$$

(80)

$$\begin{aligned} \zeta _T&\equiv \frac{zg\kappa }{u_\star ^2 T}T_\star ,\end{aligned}$$

(81)

$$\begin{aligned} \zeta _q&\equiv \frac{zg\kappa }{u_\star ^2 T}\left( \frac{0.61T}{\rho +0.61q}\right) q_\star ,\end{aligned}$$

(82)

$$\begin{aligned} \zeta&= \zeta _T+\zeta _q ,\end{aligned}$$

(83)

$$\begin{aligned}&\left( \frac{\partial \zeta }{\partial T_\star }\right) _{68}=\frac{\zeta _T}{T_\star },\end{aligned}$$

(84)

$$\begin{aligned}&\left( \frac{\partial \zeta }{\partial q_\star }\right) _{68}=\frac{\zeta _q}{q_\star }. \end{aligned}$$

(85)

Thus the expansion becomes

$$\begin{aligned} \frac{d\epsilon }{\epsilon }&= \frac{du_\star }{u_\star } \left( 3-\frac{2\zeta }{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta )\right) \nonumber \\&+\frac{dz}{z}\left( -1+\frac{\zeta }{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta )\right) \nonumber \\&+\frac{dT_\star }{T_\star }\frac{\zeta _T}{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta ) \nonumber \\&+\frac{dq_\star }{q_\star }\frac{\zeta _q}{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta ), \end{aligned}$$

(86)

where $dT_\star$ and $dq_\star$ have been expanded in Andreas (1989) as

$$\begin{aligned} \frac{dT_\star }{T_\star }&= S_z \frac{dz}{z}+S_{u_\star }\frac{du_\star }{u_\star }+S_{TC_{n_1}} \frac{dC_{n_1}}{C_{n_1}}+S_{TC_{n_2}}\frac{dC_{n_2}}{C_{n_2}}, \end{aligned}$$

(87)

$$\begin{aligned} \frac{dq_\star }{q_\star }&= S_z \frac{dz}{z}+S_{u_\star }\frac{du_\star }{u_\star }+S_{QC_{n_1}} \frac{dC_{n_1}}{C_{n_1}}+S_{QC_{n_2}}\frac{dC_{n_2}}{C_{n_2}}. \end{aligned}$$

(88)

Thus we have

$$\begin{aligned} \frac{d_\epsilon }{\epsilon }&= \frac{du_\star }{u_\star } \left( 3+\frac{\zeta }{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta )(S_{u_\star }-2)\right) \nonumber \\&+\frac{dz}{z}\left( -1+\frac{\zeta }{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta )(S_z+1)\right) +(\ldots )\frac{dC_{n_1}}{C_{n_1}} +(\ldots )\frac{dC_{n_2}}{C_{n_2}} , \end{aligned}$$

(89)

which gives us

$$\begin{aligned} S_{u_\star ,\epsilon }&= \frac{(1/3)}{\left( 1+\frac{1}{3} \frac{\zeta }{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta )(S_{u_\star }-2)\right) },\end{aligned}$$

(90)

$$\begin{aligned} S_{u_\star ,z}&= \frac{\frac{1}{3}\left( 1-\frac{\zeta }{\phi (\zeta )}\frac{\partial \phi }{\partial \zeta }(\zeta )(S_z+1)\right) }{\left( 1+\frac{1}{3}\frac{\zeta }{\phi (\zeta )} \frac{\partial \phi }{\partial \zeta }(\zeta )(S_{u_\star }-2)\right) } , \end{aligned}$$

(91)

where the terms $(S_{u_\star }-2)$ and $(S_z+1)$ are $(S_{u_\star }-4)$ and $(S_z+2)$ in Andreas (1992). Equations 90 and 91 reduce to Eqs. 44 and 46. Also from Andreas (1989) we have

$$\begin{aligned} S_{u_\star }=\frac{2b\zeta }{3-2b\zeta }, \end{aligned}$$

(92)

$$\begin{aligned} S_z=\frac{1-2b\zeta }{3-2b\zeta }, \end{aligned}$$

(93)

where $S_{u_\star }$ would be denoted here as $S_{T_\star ,u_\star }$ and $S_z$ would be written here as $S_{T_\star ,z}$ for a large-aperture scintillometer strategy not involving the derivation of $u_\star$ from Eq. 69. Equations 92 and 93 can be derived directly from the expressions in Andreas (1989) or they can be derived using the methodology outlined in this study. An alternative to using the results from Andreas (1989) in Eqs. 87 and 88 is to perform the total-differential expansion in Andreas (1992) from all the equations including an expansion of Eqs. 70 and 71, although the results are the same as here.