Semi-analytical Footprint Model Compliant with Arbitrary Atmospheric Stratification: Application to Monin–Obukhov Profiles

Abstract

A new model is proposed for the so-called scalar footprint and flux footprint in the atmospheric boundary layer. The underlying semi-analytical model allows computing the scalar concentration and flux fields related to turbulent diffusion of heat, water-vapor or to the dispersion of any scalar (e.g. passive pollutant) in the framework of K-theory. It offers improved capabilities regarding the representation of the gradual stratification in the boundary layer. In this model, the boundary layer is split in a series of sublayers in which the aerodynamic inertivity (a compound parameter aggregating wind-speed and eddy-diffusivity) is approximated by a sum of two power-law functions of a new vertical scale corresponding to the height-dependent downwind extension of the plume. This multilayer approach allows fitting with vanishing error any boundary-layer stratification, in particular those described by the Monin–Obukhov similarity theory (MOST) in the surface layer, while keeping the computation time of the footprint to low values. As a complement, a fully analytical surrogate model is presented for practical applications. For MOST profiles, the flux (resp. concentration) footprint is, to a RMS difference less than 1% (resp. 1.2%), equal (resp. equal to a constant multiplicative factor) to the inverse Gamma distribution. The optimal parameters of this distribution were evaluated for a broad range of atmospheric conditions and height. Regression formulas were also provided to compute the crosswind-integrated flux footprint distribution easily and with less than 1.6% RMS residual error. A comparison with the well-known footprint approximate model by Kormann and Meixner and the one by Hsieh, Katul and Chi has allowed quantifying their performances and limitations.

Appendices

Appendix 1: Power-Law Profiles

We discuss the Diffusion-Ascent-Associated Advection Distance (DAAAD), $\xi ^2$, and the aerodynamic inertivity b as functions of height z, for the two simplest cases: constant profiles and PL profiles. Assuming constant wind speed $u(z)=u_1$, constant eddy diffusivity $K(z)=K_1$, and a line source of strength Q at height $z_b=0$, the Liouville scale defined in Eq. 17, $\xi (z)$, which can be seen as the integrated (square root of) downwind advection distance of a cloud of particles while vertically diffusing up to height z, is in power 1 of height: $\xi (z)=\sqrt{u_1/K_1}z$. The DAAAD $\xi (z)^2$ is thus in power 2 of height. The aerodynamic inertivity of the atmosphere is constant: $b(z)=b_1=$$\sqrt{u_1K_1}$. The scalar field follows the classical Gaussian plume model:

$$\begin{aligned} \chi (x,z)=\frac{Q}{\sqrt{\pi K_1 u_1 x}}\exp \left( -\frac{u_1 z^2}{4K_1 x}\right) . \end{aligned}$$

(44)

The concentration at a given height z rises to a maximum at a downwind distance $x_{\chi max}(z)=\frac{u_1 z^2}{2K_1}$ and then decays continuously as distance x is further increased. Hence, the DAAAD $\xi (z)^2$ corresponds to two times $x_{\chi max}(z)$. At this distance the concentration has dropped by about 9.2% from the maximum concentration observed at height z.

When expressed in terms of aerodynamic inertivity and $\xi$, the 2D concentration field is:

$$\begin{aligned} \chi (x,\xi )=\frac{Q}{b_1\sqrt{\pi x}}\exp \left( -\frac{\xi ^2}{4x}\right) , \end{aligned}$$

(45)

where it can be seen that concentration is inversely proportional to the local aerodynamic inertivity $b(z)=b_1$, which motivated the choice of this name for this parameter.

Let us now assume the power-law profiles in Eq. 5 with again a line source at height $z_b=0$. The Liouville scale $\xi (z)$ and the aerodynamic inertivity b(z) are themselves PL functions of height:

$$\begin{aligned} \xi (z)= & {} \xi _1\left( \frac{z}{z_1}\right) ^{r/2}\quad ;\quad \xi _1 \equiv z_1\frac{2}{r}\sqrt{\frac{u_1}{K_1}}\quad ;\quad r \equiv m-n+2, \end{aligned}$$

(46)

$$\begin{aligned} b(z)= & {} b_1\left( \frac{z}{z_1}\right) ^{\frac{m+n}{2}} \quad ;\quad b_1 \equiv \sqrt{u_1 K_1}. \end{aligned}$$

(47)

As a consequence, $\sqrt{b}$ is itself a PL function of $\xi$ with power $\mu -1/2$ where $\mu \equiv (m+1)/(m-n+2)$. Notice that it is therefore a particular case of the $\sqrt{b}$-EPL profiles in Eq. 27 with $\nu =\mu -1$. The scalar field (Robert’s solution) expresses as:

$$\begin{aligned} \chi (x,z)=\frac{rQ}{z_1 u_1 \varGamma (\mu )}\left( \frac{u_1 z_1^2}{K_1 r^2 x}\right) ^{\mu }\exp \left( -\frac{u_1 z_1^2}{K_1 r^2 x}\left( \frac{z}{z_1}\right) ^r\right) . \end{aligned}$$

(48)

Upon introducing $\xi (z)$ expressed in Eq. 46 we obtain Eq. 7. This expression corresponds to Eq. 20 in Kormann and Meixner (2001), knowing their “$\xi$” variable is equivalent to present $\xi ^2/4$. Further introducing the aerodynamic inertivity b(z) expressed in Eq. 47 we obtain:

$$\begin{aligned} \chi (x,\xi )=\frac{2Q}{b \xi \varGamma (\mu )} \left( \frac{\xi ^2}{4x}\right) ^{\mu } \exp \left( -\frac{\xi ^2}{4x}\right) , \end{aligned}$$

(49)

which shows again that the concentration is inversely proportional to the local aerodynamic inertivity $b(\xi (z))$. The inferred concentration footprint can be nondimensionalized by multiplying it by $b \xi$. This gives:

$$\begin{aligned} \left\langle \overline{f_{\chi }}^y\right\rangle \equiv \overline{f_{\chi }}^y{b \xi }=\frac{2}{\varGamma (\mu )} \left( \frac{\xi ^2}{4x}\right) ^{\mu } \exp \left( -\frac{\xi ^2}{4x}\right) , \end{aligned}$$

(50)

It is easy to see that the DAAAD $\xi (z)^2$ and the downwind distance $x_{\chi max}(z)$ at which the concentration footprint reaches its maximum are now related through $\xi (z)^2=4\mu x_{\chi max}(z)$ which implies that $x_{\chi max}(z)$ increases to the power r of height. Similarly, the flux footprint in Eq. 11 can be nondimensionalized by multiplying it by $\xi ^2$, which gives:

$$\begin{aligned} \left\langle \overline{f_{\varphi }}^y\right\rangle \equiv \overline{f_{\varphi }}^y{\xi ^2 }=\frac{4}{\varGamma (\mu )}\left( \frac{\xi ^2}{4x}\right) ^{\mu +1} \exp \left( -\frac{\xi ^2}{4x}\right) . \end{aligned}$$

(51)

The DAAAD $\xi (z)^2$ and the downwind distance $x_{\varphi max}(z)$ at which the flux footprint reaches its maximum are related through $\xi (z)^2=4(\mu +1) x_{\varphi max}(z)$. The downwind distances at which the concentration and flux footprints reach their maximum are in a constant ratio $(\mu +1)/\mu$.

The inverse Gamma distribution is defined by:

$$\begin{aligned} f\left( x,\mu ,\beta \right) =\frac{1}{\beta \varGamma (\mu )}\left( \frac{\beta }{x}\right) ^{\mu +1} \exp \left( -\frac{\beta }{x}\right) , \end{aligned}$$

(52)

with $\mu >0$, the shape parameter. The flux footprint in Eq. 11 can thus be expressed vs the inverse Gamma distribution as:

$$\begin{aligned} \overline{f_{\varphi }}^y=f\left( x,\mu ,\frac{\xi ^2}{4}\right) , \end{aligned}$$

(53)

and the cumulative flux footprint can be expressed vs the regularized Gamma function $F\left( x,\mu ,\beta \right)$ as:

$$\begin{aligned} \overline{F_{\varphi }}^y=F\left( x,\mu ,\frac{\xi ^2}{4}\right) \equiv \frac{\varGamma \left( \mu ,\frac{\xi ^2}{4x}\right) }{\varGamma (\mu )}, \end{aligned}$$

(54)

where $\varGamma \left( \mu ,t\right)$ is the upper incomplete Gamma function defined as $\varGamma \left( \mu ,t\right) =\int _{t}^{\infty }w^{\mu -1}e^{-w}dw$. The limit of the cumulative flux footprint for $x \rightarrow \infty$, is 1, as expected. The condition $\mu >0$ must be fullfilled, which is the case for PL profiles fitting realistic atmospheric profiles.

Similarly, the concentration footprint can be expressed vs the inverse Gamma distribution as:

$$\begin{aligned} \overline{f_{\chi }}^y=\frac{\xi }{2b}\frac{\varGamma (\mu -1)}{\varGamma (\mu )}f\left( x,\mu -1,\frac{\xi ^2}{4}\right) , \end{aligned}$$

(55)

and the cumulative concentration footprint can be expressed vs the regularized Gamma function $F\left( x,\mu ,\beta \right)$ as:

$$\begin{aligned} \overline{F_{\chi }}^y=\frac{\xi }{2b}\frac{\varGamma (\mu -1)}{\varGamma (\mu )}F\left( x,\mu -1,\frac{\xi ^2}{4}\right) = \frac{\xi }{2b}\frac{\varGamma \left( \mu -1,\frac{\xi ^2}{4x}\right) }{\varGamma (\mu )}. \end{aligned}$$

(56)

The limit of the cumulative concentration footprint for $x \rightarrow \infty$ is $\frac{\xi }{2b}\frac{\varGamma (\mu -1)}{\varGamma (\mu )}$ provided $\mu >1$, otherwise the function is unbounded.

Appendix 2: Quadrupole Method

The four entries A, B, C, D of the quadrupole related to a graded layer of $\sqrt{b}$-EPL type, Eqs. 27 and 28, are expressed as:

$$\begin{aligned} A= & {} \frac{{{\check{\xi }}}_{1}h_{1}\sqrt{p}}{h_{0}}\left( I_{\nu ,0}K_{\nu +1,1}+I_{\nu +1,1}K_{\nu ,0}\right) -\frac{2\nu A_{D}{{\check{\xi }}}_{1}^{-\nu }}{h_{0}} \left( I_{\nu ,0}K_{\nu ,1}-I_{\nu ,1}K_{\nu ,0}\right) , \end{aligned}$$

(57)

$$\begin{aligned} B= & {} -\frac{1}{h_{0}h_{1}}\left( I_{\nu ,0}K_{\nu ,1}-I_{\nu ,1}K_{\nu ,0}\right) , \end{aligned}$$

(58)

$$\begin{aligned}{} & {} \begin{aligned} C&= -h_{0}h_{1}{{\check{\xi }}}_{0}{{\check{\xi }}}_{1}p \left( I_{\nu +1,0}K_{\nu +1,1}-I_{\nu +1,1}K_{\nu +1,0}\right) \\&\quad +\, 2\nu A_{D}h_{0}{{\check{\xi }}}_{0}{{\check{\xi }}}_{1}^{-\nu }\sqrt{p} \left( I_{\nu +1,0}K_{\nu ,1}+I_{\nu ,1}K_{\nu +1,0}\right) \\&\quad -\, 2\nu A_{D}h_{1}{{\check{\xi }}}_{0}^{-\nu }{{\check{\xi }}}_{1}\sqrt{p} \left( I_{\nu ,0}K_{\nu +1,1}+I_{\nu +1,1}K_{\nu ,0}\right) \\&\quad +\, 4\nu ^2 A_{D}^2{{\check{\xi }}}_{0}^{-\nu }{{\check{\xi }}}_{1}^{-\nu } \left( I_{\nu ,0}K_{\nu ,1}-I_{\nu ,1}K_{\nu ,0}\right) , \end{aligned} \end{aligned}$$

(59)

$$\begin{aligned} D= & {} \frac{{{\check{\xi }}}_{0}h_{0}\sqrt{p}}{h_{1}}\left( I_{\nu +1,0}K_{\nu ,1}+I_{\nu ,1}K_{\nu +1,0}\right) +\frac{2\nu A_{D}{{\check{\xi }}}_{0}^{-\nu }}{h_{1}} \left( I_{\nu ,0}K_{\nu ,1}-I_{\nu ,1}K_{\nu ,0}\right) . \end{aligned}$$

(60)

In the brackets present in Eqs. 57–60, the symbols $I_{\lambda ,k}$ and $K_{\lambda ,k}$ have been introduced, where $\lambda$ takes the values $\nu$ or $\nu +1$ and k takes the values 0 or 1. They represent respectively $I_{\lambda }(\sqrt{p}{{\check{\xi }}}_{k})$ and $K_{\lambda }(\sqrt{p}{{\check{\xi }}}_{k})$. Furthermore, ${{\check{\xi }}}$ and h with indices 0 and 1 refer to the values taken by these functions (see Eqs. 26, 27) at the low, resp. high boundary of the considered layer.

The bottom layer, whose lower end is fixed at a height correponding to the roughness length, $z=z_0$, requires particular attention (notice that $z_b$ in the definition of the Liouville scale in Eq. 17 is then set to $z_0$). As a matter of fact, wind speed vanishes at $z=z_0$, hence the aerodynamic inertivity vanishes as well. The $\sqrt{b}$-EPL profile in Eq. 27 should thus be chosen so that $b(\xi )$ vanishes at $\xi =0$. Setting the constant $\xi _{c}$ to 0 allows achieving this. One should then keep only those power-functions (from the two) that have a positive exponent ($1/2+\nu$ and $1/2-\nu$). Assuming that the MO similarity theory is valid down to $z_0$, the effusivity profile follows asymptotically the neutral profile, i.e. $b(z)\propto z ln(z/z_0)$. While fitting a $\sqrt{b}$-EPL profile to the neutral profile (after expressing it in terms of $\xi$) we checked that both power-law functions of exponent $1/2+\nu$ and $1/2-\nu$ in Eq. 27 can be kept (indeed, the optimal value for $\nu$ is about 0.3). As a consequence, we have $h_{0}=0$ and $h_{1}=A_{B}\xi _{1}^{\nu }+A_{D}\xi _{1}^{-\nu }$ for the bottom layer. However, several terms in Eqs. 57–60 are now unbounded because involving $\check{\xi _0}=\xi _0=0$, in particular the terms $\check{\xi _0}^{-\nu }$ and $K_{\nu ,0}$ and $K_{\nu +1,0}$ which correspond here to $K_{\nu }(0)$ and $K_{\nu +1}(0)$. Expansion in power series of the Bessel functions is required to solve the singularities. Notice that only the two entries C and D prove to be necessary when concentration and flux are to be calculated at heights not lower than the top of the bottom layer, which will be the case (refer to Eq. 34). After some manipulations (see the Supplementary Material Sect. S1–2) we find the following expressions for the entries C and D of the bottom quadrupole:

$$\begin{aligned}{} & {} \begin{aligned} C&=A_{B}h_{1}{{\check{\xi }}}_{1}\sqrt{p}^{\;-\nu +1}2^{\nu }\varGamma (\nu +1)I_{\nu +1,1}\\&\quad +\, \nu A_{B}A_{D}{{\check{\xi }}}_{1}^{-\nu }\sqrt{p}^{\;-\nu }2^{\nu +1}\varGamma (\nu +1)I_{\nu ,1}\\&\quad -\, \nu A_{D}h_{1}{{\check{\xi }}}_{1}\sqrt{p}^{\nu +1} \left[ 2^{-\nu +1}/\varGamma (\nu +1)K_{\nu +1,1}+2^{-\nu }\varGamma (-\nu )I_{\nu +1,1}\right] \\&\quad +\, \nu ^2 A_{D}^2{{\check{\xi }}}_{1}^{-\nu }\sqrt{p}^{\nu } \left[ 2^{-\nu +2}/\varGamma (\nu +1)K_{\nu ,1}-2^{-\nu +1}\varGamma (-\nu )I_{\nu ,1}\right] , \end{aligned} \end{aligned}$$

(61)

$$\begin{aligned}{} & {} \begin{aligned}&D=A_{B}h_{1}^{-1}\sqrt{p}^{\;-\nu }2^{\nu }\varGamma \left( \nu +1 \right) I_{\nu ,1}\\&\quad \quad +\nu A_{D}h_{1}^{-1}\sqrt{p}^{\nu }\left[ 2^{-\nu +1}/\varGamma \left( \nu +1\right) K_{\nu ,1}-2^{-\nu }\varGamma \left( -\nu \right) I_{\nu ,1}\right] . \end{aligned} \end{aligned}$$

(62)

We now develop the expression of the admittance at the bottom of a semi-infinite layer having a $\sqrt{b}$-EPL profile (third option for the closure of the multilayer problem discussed in Sect. 2.5). From the definition of the admittance we obtain, for a generally graded layer (Krapez 2016, Eq. 19):

$$\begin{aligned} Y=\sqrt{b}\left( \sqrt{b}'\psi -\sqrt{b}\psi '\right) \psi ^{-1}. \end{aligned}$$

(63)

On the other hand, from Eqs. 23, 27 and 28, the general form for the concentration is:

$$\begin{aligned} X(p,\xi )=\frac{1}{h({\check{\xi }})}\left[ A_I I_{\nu }(\sqrt{p}{\check{\xi }})+A_K K_{\nu }(\sqrt{p}{\check{\xi }})\right] . \end{aligned}$$

(64)

The asymptotic expansions of the modified Bessel functions for $\vert t\vert \rightarrow +\infty$ are:

$$\begin{aligned} I_{\nu }(t)\approx \frac{1}{\sqrt{2\pi t}} \exp {\left( t\right) }\quad ;\quad K_{\nu }(t)\approx \sqrt{\frac{\pi }{2t}} \exp {\left( -t\right) }, \end{aligned}$$

(65)

which shows that the particular solution involving $I_{\nu }$ in Eq. 64 is unbounded for $\xi \rightarrow \infty$ and should thus be discarded. The admittance thus simplifies as:

$$\begin{aligned} Y =\frac{h{{\check{\xi }}}\left( h'K_{\nu }-hK_{\nu }'\right) }{K_{\nu }}. \end{aligned}$$

(66)

The admittance at $\xi =\xi _0$, the bottom of the topmost (“sacrificial”) layer, finally reads:

$$\begin{aligned} Y_{top}=h_{0}\left( h_{0}{{\check{\xi }}}_{0}\sqrt{p}\frac{K_{\nu +1,0}}{K_{\nu ,0}} -2\nu A_{D}{{\check{\xi }}}_{0}^{-\nu }\right) . \end{aligned}$$

(67)

Appendix 3: Verification of the Semi-analytical Model

Matlab R2021a on a PC equipped with an Intel core Xeon W-2123 @ 3.6GHz CPU and 8GB RAM has been employed for the computations and evaluation of the computation time. The first verification test was performed with the classical uK-PL(z) profiles (the expressions of related concentration and flux are in Appendix 1, Eqs. 50 and 51). The whole process, from step 1 to step 8, was applied (see Sect. 2.8). Notice that since any pair of uK-PL(z) profiles leads, in the Liouville space, to a b-PL$(\xi )$ profile, this one being a special case of the $\sqrt{b}$-EPL$({\check{\xi }})$ profiles (see Appendix 1), the fitting in step 3 was thus formally trivial. In practice however, small (negligible) errors were observed at the end of step 4, which actually came from the quadrature computation to evaluate $\xi (z)$ in step 2. In the end, the comparison with the analytical solution in Eq. 51 was very satisfactory, since the relative error was less than $10^{-9}$.

We didn’t find in the literature other expressions for graded profiles u(z) and K(z) that are analytically solvable, namely leading to an exact and closed form analytical solution of the stationary advection–diffusion problem. Nevertheless, the “thermal” solutions developed in Krapez (2016) can offer a starting point to build such solvable profiles. Among the numerous possibilities, we selected the following inertivity profiles, which were called $\tanh ^2$-type profiles:

$$\begin{aligned} b(\xi )=b_{\infty }\left( \tanh \left( \xi /\xi _{c}\right) \right) ^2, \end{aligned}$$

(68)

where $b_{\infty }$ is the asymptotic inertivity for increasing height and $\xi _{c}$ a scale for the Liouville variable. Among the infinite number of pairs of profiles u(z) and K(z) from which these inertivity profiles $b(\xi )$ can derive, we will consider, by way of illustration, the simplest one, i.e. the one embodying the same spatial variability in u(z) and K(z), thus ensuring a proportionality between z and $\xi$:

$$\begin{aligned}&u(z)=u_{\infty }\left( \tanh \left( \left( z-z_0\right) /z_{c}\right) \right) ^2, \end{aligned}$$

(69a)

$$\begin{aligned}&K(z)=K_{\infty }\left( \tanh \left( \left( z-z_0\right) /z_{c}\right) \right) ^2, \end{aligned}$$

(69b)

where $u_{\infty }$ and $K_{\infty }$ are the limiting values of wind speed and diffusivity for increasing height and $z_{c}$ is a scale for height. In this case $b_{\infty }=\sqrt{u_{\infty }K_{\infty }}$, $\xi =(z-z_0)\sqrt{u_{\infty }/K_{\infty }}$, and $\xi _{c}=z_{c}\sqrt{u_{\infty }/K_{\infty }}$. As a result of a source of strength Q at $z=z_{0}$ (i.e. $\xi =0$), the concentration is:

$$\begin{aligned} \chi (x,\xi )=\frac{Q}{b_{\infty }\sqrt{\pi x}}\left( 1+\frac{\xi \xi _{c}}{2x\tanh \left( \xi /\xi _{c}\right) }\right) \exp \left( -\frac{\xi ^2}{4x}\right) , \end{aligned}$$

(70)

and the flux is:

$$\begin{aligned} \varphi (x,\xi )=\frac{4Q}{\sqrt{\pi }\xi ^2} \left[ \left( 1-\frac{\xi _{c}}{\xi }\sqrt{\frac{b}{b_{\infty }}}\right) \gamma ^{3/2}+ 2\frac{\xi _{c}}{\xi }\sqrt{\frac{b}{b_{\infty }}}\gamma ^{5/2} \right] \exp (-\gamma ), \end{aligned}$$

(71)

with $\gamma =\xi ^2/(4x)$ (see Supplementary material, Sect. S2). Even though the $\tanh ^2$-type profiles in Eq. 69 are quite far from MOST profiles, they present the mathematical advantage of leading to simple, closed-form analytical expressions of the footprints; in addition, compared to uK-PL(z) profiles, they present the characteristic of being bounded. Hence, they have an interest for benchmarking of semi-analytical or numerical solutions.

The semi-analytical model was tested with the $\tanh ^2$-type profiles in Eq. 69 by comparing the computed flux to the theoretical values obtained from Eq. 71. Starting with one EPL layer on each side of the assumed measurement level, the layers were subdivided until the fitting (relative) error was everywhere lower than a prescribed value ranging from 0.7 down to $3\times 10^{-5}$. The number of required layers ranged from 3 to 37. The maximum (relative) error with respect to the theoretical flux in Eq. 71 decreased from 0.07 to $10^{-6}$ (the relative error was obtained by dividing the difference in footprint values by the maximum of the theoretical footprint). Discretizing in 11 layers was enough to reach a relative error lower than about $2\times 10^{-3}$ in the aerodynamic inertivity and $10^{-4}$ in the flux-footprint. In this case, the CPU time was about 0.66 s (0.6 s for the aerodynamic-inertivity profile definition and fitting - step 1 to 4, and 0.06 s for the quadrupole manipulations and Laplace inversion for 100 points along x-axis—step 5 to 8). More information on the verification task, in particular a detailed analysis on the interplay between the fitting-error constraint, the number of discretized layers, and the footprint error can be found in the Supplementary Material Sect. S2.

Appendix 4: Monin–Obukhov Profiles

Over a horizontally homogeneous surface in steady-state conditions the vertical fluxes are nearly constant with height in the lower part of the boundary layer. The surface layer is commonly defined to be the layer where the vertical fluxes vary by less than 10% of their magnitude (Stull 2012). Among them are the momentum flux ($-\overline{w'u'}$) the sensible heat flux ($-\overline{w'\theta '}$) and the trace gas flux ($-\overline{w'\chi '}$) (w is the vertical wind speed, $\theta$ the temperature, the overbar denotes Reynolds average and each primed variable represents the deviation from its average). This allows defining in the surface layer scales for velocity (friction velocity $u_*=\sqrt{|\overline{w'u'}|}$), temperature ($\theta _*=-\overline{w'\theta '}/u_*$) and tracer-concentration ($\chi _*=-\overline{w'\chi '}/u_*$). The Obukhov length is defined as:

$$\begin{aligned} L=-\frac{\theta _v u_*^3}{\kappa g \overline{w'\theta _v'}}, \end{aligned}$$

(72)

where $\theta _v$ and $\overline{w'\theta _v'}$ are the virtual potential temperature and buoyancy flux, respectively, g is the gravitational acceleration and $\kappa$ is the von Karman constant (commonly accepted value is $\kappa =0.4$). The Obukhov length can be used to classify atmospheric stability conditions: L is negative when buoyancy flux is positive, namely upwards (unstable stratification), L is positive when when buoyancy flux is negative (stable), $\vert L\vert \rightarrow \infty$ when buoyancy flux vanishes (neutral). The MO theory states that when properly scaled with $u_*$, $\theta _*$ and $\chi _*$, the wind-shear, temperature and tracer-concentration gradients are functions (namely universal similarity functions) that depend only on the stability parameter $\zeta \equiv z/L$, where L is the Obukhov length scale:

$$\begin{aligned}&\frac{\kappa z}{u_*}\frac{du}{dz}=\phi _m(\zeta ), \end{aligned}$$

(73a)

$$\begin{aligned}&\frac{\kappa z}{\theta _*}\frac{d\theta }{dz}=Pr_n\cdot \phi _{h}(\zeta ), \end{aligned}$$

(73b)

$$\begin{aligned}&\frac{\kappa z}{\chi _*}\frac{d\chi }{dz}=Sc_n\cdot \phi _{\chi }(\zeta ). \end{aligned}$$

(73c)

In this representation $\phi _{h,\chi }\rightarrow 1$ for $|\zeta |\rightarrow 0$ (neutral limit) whereas $Pr_n$ and $Sc_n$ are the turbulent Prandtl and Schmidt numbers in the neutral limit, respectively (see e.g. Leclerc and Foken (2014), Wilson (2015)).

The wind profile u(z) is obtained by integrating Eq. 73a which gives Eq. 38. On the other hand, upon applying the K-theory which expresses the turbulent flux $\overline{w'\chi '}$ as $-K$ times the gradient of concentration, where K is the eddy diffusivity related to tracer concentration, Eq. 73c leads to the expression of the eddy diffusivity in Eq. 39.

We considered the universal similarity functions initially proposed by Businger et al. (1971) and later revised by Högström (1988) (BH parameterizations) while taking $\kappa =0.4$.

For the unstable case ($L<0$):

$$\begin{aligned}&\phi _m(\zeta )=\left( 1-19.3\zeta \right) ^{-1/4}, \end{aligned}$$

(74a)

$$\begin{aligned}&Pr_n\cdot \phi _h(\zeta )=Pr_n\cdot \left( 1-11.6\zeta \right) ^{-1/2}, \end{aligned}$$

(74b)

while, for the stable case ($L>0$):

$$\begin{aligned}&\phi _m(\zeta )=1+6\zeta , \end{aligned}$$

(75a)

$$\begin{aligned}&Pr_n\cdot \phi _h(\zeta )=Pr_n\cdot \left( 1+8.21\zeta \right) , \end{aligned}$$

(75b)

with $Pr_n=0.95$. For the similarity function related to tracer concentration, we assumed both $\phi _{\chi }(\zeta )=\phi _h(\zeta )$ and $Sc_n=Pr_n=0.95$ although other values for $Sc_n$, the neutral turbulent Schmidt number, could be considered as well.

Based on the expressions in Eqs. 74a and 75a, the integral in Eq. 38 takes the form proposed by Paulson (1970):

$$\begin{aligned} u(z)=\frac{u_*}{\kappa } \left[ \ln \left( \frac{z}{z_0}\right) -\psi _m(\zeta )+\psi _m(\zeta _0)\right] , \end{aligned}$$

(76)

where $\zeta _0\equiv z_0/L$ and the integrated similarity function $\psi _m$ is given by, if $L<0$:

$$\begin{aligned} \psi _m(\zeta )= 2\ln \left( \frac{1+\phi _m^{-1}}{2}\right) +\ln \left( \frac{1+\phi _m^{-2}}{2}\right) -2\arctan (\phi _m^{-1})+\frac{\pi }{2}, \end{aligned}$$

(77)

where $\phi _m(\zeta )$ is taken from Eq. 74a, while, if $L>0$:

$$\begin{aligned} \psi _m(\zeta )=-6\zeta . \end{aligned}$$

(78)

In case the individual roughness elements are packed very closely together (e.g. forest canopy with close trees or cities with houses packed close together) a displacement distance d has to be introduced which is fraction of the mean height of the elements. The absolute height z appearing in Eqs. 38, 39, 74a–78 should then be replaced by the relative height $z-d$.

Typical profiles of the wind speed and eddy diffusivity according to MOST while taking into account the parameterisations shown before have been plotted in Fig. 4 in Supplementary Material Sect. S3.

Appendix 5: Parameterization of the Surrogate Model for the Flux Footprint

The surrogate model for the (crosswind-integrated) flux footprint is described in Eq. 40. The two parameters $\mu _s$ and $\xi _s^2$ on which this model is based were then fitted with respect to the scaled height $z_m/z_0$ and and the stability parameter $z_0/L$ (a scaling of $\xi _s^2$ with $\sqrt{z_0}$ was chosen to provide dimensionless results; in Eq. 40, the upwind distance x will thus have to be identically scaled).

We propose the following parameterizations, which, for ease, are given in reference to the neutral case (with a superscript “$\textit{n}$”) according to:

$$\begin{aligned}&\mu _s\left( \frac{z_0}{L},\frac{z_m}{z_0}\right) =\mu _s^{n} \cdot R_{\mu }\left( \frac{z_0}{L},\frac{z_m}{L}\right) , \end{aligned}$$

(79a)

$$\begin{aligned}&\frac{\xi _s}{\sqrt{z_0}}\left( \frac{z_0}{L},\frac{z_m}{z_0}\right) =\frac{\xi _s^{n}}{\sqrt{z_0}} \cdot R_{\xi } \left( \frac{z_0}{L},\frac{z_m}{L}\right) . \end{aligned}$$

(79b)

The parameters for the neutral case are approximated by:

$$\begin{aligned}&\mu _s^n=0.7577+0.2423 \cdot \left( \tanh \left( 1.217\cdot X^{0.4135} \right) \right) ^{16.38}, \end{aligned}$$

(80a)

$$\begin{aligned}&\frac{\xi _s^n}{\sqrt{z_0}}=1.735 \cdot X \cdot Y^p, \end{aligned}$$

(80b)

where

$$\begin{aligned} \begin{aligned}&p=0.5164-6.604\times 10^{-3}\ln (Y)+2.578\times 10^{-4}\ln (Y)^2,\\&X=\ln \left( \frac{z_m}{z_0}\right) ,\\&Y=\frac{z_m}{z_0}-1. \end{aligned} \end{aligned}$$

(81)

The functions $R_{\mu }$ and $R_{\xi }$ were parameterized as described in the following expressions with possibly different forms depending on whether the atmosphere is stable or unstable (subscript “-” for unstable and “+” for stable):

$$\begin{aligned} R_{\mu }^{-}=&1+A_1 \tanh \left[ A_2\ln \left( A_3 Z_{\mu }+1\right) \right] , \end{aligned}$$

(82a)

$$\begin{aligned} R_{\mu }^{+}=&\left\{ 1+A_1 \tanh \left[ A_2\ln \left( A_3 Z_{\mu }+1\right) \right] \exp \left( A_4 Z_{\mu }^{1/4} \right) \right\} ^{-1}, \end{aligned}$$

(82b)

$$\begin{aligned} R_{\xi }^{-},\;R_{\xi }^{+}=&\left( 1+A_1 Z_{\xi }^{A_2}\right) ^{A_3}, \end{aligned}$$

(82c)

with

$$\begin{aligned}&Z_{\mu }= \frac{z_0}{\left| L\right| } \left( \frac{z_m}{z_0} -1 \right) , \end{aligned}$$

(83a)

$$\begin{aligned}&Z_{\xi }=\frac{z_m}{\left| L\right| }. \end{aligned}$$

(83b)

The coefficients $A_1$ to $A_5$ are expressed as functions of the parameter $z_0/L$ of the following type:

$$\begin{aligned} A_{i}=B_1+B_2 \left| \frac{z_0}{L} \right| ^{B_3},\quad i=1 \ldots 5, \end{aligned}$$

(84)

where the coefficients $B_1$ to $B_3$ have the optimal values reported in Table 3, 4, 5 and 6.

Table 3 Values of the coefficients $B_1$ to $B_3$ necessary to compute the coefficients $A_1$ to $A_3$ involved in the computation of $R_{\mu }^{-}$

Table 4 Values of the coefficients $B_1$ to $B_3$ necessary to compute the coefficients $A_1$ to $A_4$ involved in the computation of $R_{\mu }^{+}$

Table 5 Values of the coefficients $B_1$ to $B_3$ necessary to compute the coefficients $A_1$ to $A_3$ involved in the computation of $R_{\xi }^{-}$

Table 6 Values of the coefficients $B_1$ to $B_3$ necessary to compute the coefficients $A_1$ to $A_3$ involved in the computation of $R_{\xi }^{+}$

The surrogate model in Eq. 40, when fed with the parameters $\mu _s$ and $\xi _s^2/\sqrt{z_0}$ obtained from the empirical relations described from Eqs. 79a to 84, allows reaching the performances described in Fig. 11 in terms of residual errors on the flux footprint.