Verification of a One-Dimensional Model of $\hbox {CO}_{2}$ Atmospheric Transport Inside and Above a Forest Canopy Using Observations at the Norunda Research Station

Abstract

A model of $\hbox {CO}_{2}$ atmospheric transport in vegetated canopies is tested against measurements of the flow, as well as $\hbox {CO}_{2}$ concentrations at the Norunda research station located inside a mixed pine–spruce forest. We present the results of simulations of wind-speed profiles and $\hbox {CO}_{2}$ concentrations inside and above the forest canopy with a one-dimensional model of profiles of the turbulent diffusion coefficient above the canopy accounting for the influence of the roughness sub-layer on turbulent mixing according to Harman and Finnigan (Boundary-Layer Meteorol 129:323–351, 2008; hereafter HF08). Different modelling approaches are used to define the turbulent exchange coefficients for momentum and concentration inside the canopy: (1) the modified HF08 theory—numerical solution of the momentum and concentration equations with a non-constant distribution of leaf area per unit volume; (2) empirical parametrization of the turbulent diffusion coefficient using empirical data concerning the vertical profiles of the Lagrangian time scale and root-mean-square deviation of the vertical velocity component. For neutral, daytime conditions, the second-order turbulence model is also used. The flexibility of the empirical model enables the best fit of the simulated $\hbox {CO}_{2}$ concentrations inside the canopy to the observations, with the results of simulations for daytime conditions inside the canopy layer only successful provided the respiration fluxes are properly considered. The application of the developed model for radiocarbon atmospheric transport released in the form of $^{14}\hbox {CO}_{2}$ is presented and discussed.

Appendices

Appendix 1: Dependence of Frontal Leaf Area on LAI for the Case of a Coniferous Forest

From general considerations, the dependence of the average value of the ‘frontal area per unit volume’ (parameter a) should be related to the specific type of canopy and to the leaf-area index LAI. For the case of a pine forest, the total frontal leaf area FLAI is derived from LAI using the following considerations. According to the standard definition, LAI is the total one-sided area of leaves per unit area of ground. For the case of a pine forest, it is half of the total area of needles per unit area

$$\begin{aligned} {\textit{LAI}}=0.5\sum _i {A_i } =0.5\pi {\textit{Nby}}, \end{aligned}$$

(24)

where N is the number of needles per unit area, $A_i$ is surface area of an individual needle, b and y are the average width and length of a needle, respectively. Thus, if all needles are oriented vertically, then the total frontal leaf area is ${\textit{FLAI}}=\sum \nolimits _i {y_i} b_i =\left( {2/\pi }\right) {\textit{LAI}}$.

Assuming the directions of the needles are distributed randomly but homogenously, if the ends of all needles are brought to the same point, then their opposite ends will be uniformly distributed on a sphere of the radius y. We characterize the end of a needle by two angles $\phi , \theta$, where $\phi$ is the angle between the needle and the plane A, perpendicular to the wind direction, while $\theta$ is the angle between the needle and its projection on plane B, parallel to the ground. The vertical orientation of a needle corresponds to $\phi =0, \theta =\pi /2$. Then, the probability for the end of a needle to appear in the vicinity of an arbitrary point on the sphere is $P=\delta \phi \delta \theta /4\pi ^2$, where $\delta \phi , \delta \theta$ are small angles.

For the quadrant of a sphere $\phi =\left( {0, \pi /2} \right) , \theta =(0, \pi )$, the total projection of the surfaces of needles on plane A (frontal area) is equal to

$$\begin{aligned} \frac{Nyb}{4\pi ^{2}}\left| {\int \limits _0^\pi {\int \limits _0^{\pi /2} {\mathrm{d}\phi \mathrm{d}\theta \cos } } \phi } \right| =\frac{{\textit{Nyb}}}{4\pi }. \end{aligned}$$

(25)

As far as the frontal area is created by all needles, the total frontal area is greater than the value above by a factor of four, then

$$\begin{aligned} {\textit{FLAI}}=\frac{{\textit{ybN}}}{\pi }=\frac{2}{\pi ^{2}}{\textit{LAI}}. \end{aligned}$$

(26a)

The average value of the frontal area per unit volume is thus

$$\begin{aligned} \overline{{a}}=\frac{{\textit{FLAI}}}{h}=\frac{2}{\pi ^{2}h}{\textit{LAI}}. \end{aligned}$$

(26b)

Appendix 2: Second-Order Closure Model

We used the second-order closure model presented by Katul and Albertson (1998, 1999), and briefly review this model and discuss its adaptation for the calculation of wind-speed and concentration profiles for the conditions of the Norunda station. The following system of equations is numerically solved for the Reynolds-averaged height-dependent along-wind horizontal velocity component $\overline{{u}}$, the momentum flux $\overline{u^{\prime }w^{\prime }}$, and variances of the along-wind, crosswind and vertical velocity components $\overline{u^{\prime 2}}, \overline{v^{\prime 2}} , \overline{w^{\prime 2}}$,

$$\begin{aligned}&\displaystyle -\,\frac{\mathrm{d}\overline{u^{\prime }w^{\prime }} }{\mathrm{d}z}-c_d a\left( z \right) \overline{{u}}^{2}=0, \end{aligned}$$

(27a)

$$\begin{aligned}&\displaystyle \left( {C_w q^{2}-\overline{w^{\prime 2}} } \right) \frac{\mathrm{d}\overline{{u}}}{\mathrm{d}z}+2\frac{\mathrm{d}}{\mathrm{d}z}\left( {q\lambda _1 \frac{\mathrm{d}\overline{u^{\prime }w^{\prime }} }{\mathrm{d}z}} \right) -\frac{q\overline{u^{\prime }w^{\prime }} }{3\lambda _2 }=0, \end{aligned}$$

(27b)

$$\begin{aligned}&\displaystyle -\,2\overline{u^{\prime }w^{\prime }} \frac{\mathrm{d}\overline{{u}}}{\mathrm{d}z}+\frac{\mathrm{d}}{\mathrm{d}z}\left( {q\lambda _1 \frac{\mathrm{d}\overline{u^{\prime 2}} }{\mathrm{d}z}} \right) +2c_d a\left( z \right) \overline{{u}}\left( z \right) ^{3}-\frac{q}{3\lambda _2 }\left( {\overline{u^{\prime 2}} -\frac{q^{2}}{3}} \right) -\frac{2}{3}\frac{q^{3}}{\lambda _3 }=0,\qquad \quad \end{aligned}$$

(27c)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}\left( {q\lambda _1 \frac{\mathrm{d}\overline{v^{\prime 2}} }{\mathrm{d}z}} \right) -\frac{q}{3\lambda _2 }\left( {\overline{v^{\prime 2}} -\frac{q^{2}}{3}} \right) -\frac{2}{3}\frac{q^{3}}{\lambda _3 }=0, \end{aligned}$$

(27d)

$$\begin{aligned}&\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}\left( {3q\lambda _1 \frac{\mathrm{d}\overline{w^{\prime 2}} }{\mathrm{d}z}} \right) -\frac{q}{3\lambda _2 }\left( {\overline{w^{\prime 2}} -\frac{q^{2}}{3}} \right) -\frac{2}{3}\frac{q^{3}}{\lambda _3 }=0. \end{aligned}$$

(27e)

In (27), the characteristic turbulent velocity $q=\sqrt{\overline{u^{\prime 2}} +\overline{v^{\prime 2}} +\overline{w^{\prime 2}} }$ and the length scales $\lambda _i =\gamma _i L\left( z \right)$ are proportional to a single length scale (Katul and Chang 1999),

$$\begin{aligned} L\left( z \right) = \max \left\{ {\begin{array}{ll} \kappa z\,\, {\textit{if}}\;{ z}-{h}<{d} \; \mathrm{or} \kappa \left( {z-d} \right) \;{\textit{if}}\,\, z-h\ge d \\ \displaystyle \frac{\varepsilon }{c_d a\left( z \right) }\hbox { with }\frac{{\text {d}}L}{{\text {d}}z}\le \kappa \\ \end{array}} \right. . \end{aligned}$$

(28)

At $z=0$, all variables are set to zero, $\overline{{u}}=0, \overline{u^{\prime }w^{\prime }} =0$, $\overline{u^{\prime 2}} =\overline{v^{\prime 2}} =\overline{w^{\prime 2}} =0$. At the top of the computational domain (reference height), the variables are fixed, i.e. at $z=z_{r}: \overline{u^{\prime }w^{\prime }} =-u_*^2$, $\overline{{u}}=A_m u_*$, $\overline{u^{\prime 2}} =A_u^2 u_*^2$, $\overline{v^{\prime 2}} =A_v^2 u_*^2, \overline{w^{\prime 2}} =A_w^2 u_*^2$.

The values of the following constants are taken according to Katul and Albertson (1999): $\gamma _1 =0.392, \gamma _2 =0.85, \gamma _3 =16.58, C_w =0.0771, \varepsilon =0.13$, while the value of the friction velocity in the simulation is set to $u_*=1\hbox { m s}^{-1}$. The reference height is set according to the simulation coinciding with one of the measurement levels of the wind speed at the Norunda station, $z_r =73\hbox { m}$, and the friction coefficient is set according to the simulations above, $c_d =0.25$. The values of $A_m,A_u,A_v,A_w$ are defined by averaging the measurements at the Norunda station in neutral stratification through the same period, with the following values obtained: $A_m =8.015,A_u=1.943,A_v =1.61,A_w =1.29$. Equation 27 was solved using the numerical procedure as described by Wilson (1988).

For the calculation of the concentration for the particular date, the equation for the turbulent concentration flux was first solved

$$\begin{aligned} \frac{\mathrm{d}\overline{w^{\prime }C^{\prime }} }{\mathrm{d}z}=S_\chi \left( z \right) , \end{aligned}$$

(29)

where the vertical profile of the source term $S_\chi$ is defined in the same way regarding the right-hand side of (2) as described above. Katul and Albertson (1999) presented the concentration transport equation in a form suitable for the calculation of the sources and sinks from the available concentration measurements. From their equations, we derived the concentration transport equation in the following form suitable for the calculation of concentrations given the estimated distributions of sources and sinks,

$$\begin{aligned}&\overline{w^{\prime 2}} \frac{\mathrm{d}C}{\mathrm{d}z}+\frac{3}{c_8 }\frac{\mathrm{d}}{\mathrm{d}z}\left( {\left( {\lambda _3 \lambda _1 \frac{\mathrm{d}\overline{w^{\prime 2}} }{\mathrm{d}z}} \right) \frac{\mathrm{d}C}{\mathrm{d}z}} \right) =\frac{\mathrm{d}}{\mathrm{d}z}\left( {\frac{\tau }{c_8 }\overline{w^{\prime }C^{\prime }} \frac{\mathrm{d}\overline{w^{\prime 2}} }{\mathrm{d}z}} \right) \nonumber \\&\quad +\,2\frac{\mathrm{d}}{\mathrm{d}z}\left( {\frac{\tau }{c_8 }\overline{w^{\prime 2}} \frac{\mathrm{d}\overline{w^{\prime }C^{\prime }} }{\mathrm{d}z}} \right) -c_4 \frac{\overline{w^{\prime }C^{\prime }} }{\tau }, \end{aligned}$$

(30)

where $\tau =\lambda _3/q$, with the boundary conditions $C\left( {z=z_r}\right) =C_0$ and $\mathrm{d}C/\mathrm{d}z\left( {z=0} \right) =0$. All respiration fluxes originating from the soil are represented as a source term $S_\chi$ in the bottom 2-m layer of the canopy. The values of constants entering Eq.30 are taken according to Katul and Albertson (1999), and are equal to $c_8 =9, c_4 =5$. The variance of the vertical velocity component in Eq. 30 is obtained by multiplication of the corresponding solution of the system Eq. 27 for $u_*=1\hbox { m}\hbox { s}^{-1}$ by the value of $u_*$ observed at the particular date and time. Thus, Eq. 30 was solved for neutral, daytime conditions for all dates described above in the main text.

Appendix 3: Set-Up of the Model of Siqueira and Katul (2010) for the Simulation of the Vertical Profile of $^{14}\hbox {CO}_{2}$

Siqueira and Katul (2010) (hereafter SK10) analytically solved Eq. 2 describing $\hbox {CO}_{2}$ turbulent transport assuming neutral stability conditions, a uniform leaf area profile inside the canopy, and a definition of the diffusion coefficient inside canopy similar to the HF08 relationship (9). However, in contrast to HF08 theory, the SK10 solution was provided for a canopy of finite depth, and allowed for the flux from the soil surface. Here, we briefly summarize the relationships from SK10 and their relevance for our work. We preserve the use of the previously defined notation and coordinate system. Hence, the relationships from SK10 are adapted accordingly.

The source term $S_\chi$ was represented by SK10 by

$$\begin{aligned} S_\chi \left( z \right) =a\left( z \right) g_s \left( {C_i -C\left( z \right) } \right) =a\left( z \right) g_s C\left( z \right) \left( {\chi -1} \right) , \end{aligned}$$

(31)

with $C_i$ being the mean intercellular $\hbox {CO}_{2}$ concentration, $g_s$ is the total conductance, and $\chi =C_i{/}C$ is assumed constant in the canopy layer. The total conductance $g_s$ is assumed to decrease inside the canopy with decreasing height similar to photosynthetically active radiation as in (22). The value of $g_s$ at the top of the canopy is denoted as $g_{\max } =g_s \left( {z=h} \right)$. Providing those assumptions, SK10 represented the concentration transport equation Eq. 2 inside the canopy and the boundary conditions in non-dimensional form as

$$\begin{aligned}&\displaystyle \frac{\partial ^{2}C}{\partial \hat{{z}}^{2}}+\beta \frac{\partial C}{\partial \hat{{z}}}+\Lambda \exp \left( {\nu \hat{{z}}} \right) =0, \end{aligned}$$

(32a)

$$\begin{aligned}&\displaystyle \hat{{z}}=\left( {z-h} \right) /l_m , \end{aligned}$$

(32b)

$$\begin{aligned}&\displaystyle \Lambda =S_{cc} \frac{l_m }{h}\frac{\chi \,{\textit{LAI}}}{\beta }\frac{g_{\max } }{U_h }, \end{aligned}$$

(32c)

$$\begin{aligned}&\displaystyle \nu =\frac{l_m }{h}\,{\textit{LAI}}\, k_e -\beta \end{aligned}$$

(32d)

and

$$\begin{aligned}&\displaystyle C=C_h\, {\textit{at}}\, \hat{{z}}=0, \end{aligned}$$

(33a)

$$\begin{aligned}&\displaystyle \frac{\partial C}{\partial \hat{{z}}}=\hat{{R}}=-\frac{R_0 S_{cc} }{u_*}\exp \left( {\beta h/l_m } \right) \, {\textit{at}}\, \hat{{z}}=-\frac{h}{l_m }, \end{aligned}$$

(33b)

where $R_0$ is part of the flux originating from the soil surface. The solution to $C\left( {\hat{{z}}} \right)$ was represented by SK10 in the form

$$\begin{aligned} C\left( {\hat{{z}}} \right) =\hat{{R}}f_R \left( {\hat{{z}}} \right) +C_h f_{{\textit{Ch}}} \left( {\hat{{z}}} \right) . \end{aligned}$$

(34)

The following relationships are easily proved for the functions $f_R \left( {\hat{{z}}} \right) ,f_{{\textit{Ch}}} \left( {\hat{{z}}} \right)$, since relationship (34) should satisfy boundary conditions (33) for arbitrary $C_h$ and $\hat{{R}}$,

$$\begin{aligned}&\displaystyle \left. {f_R } \right| _{\hat{{z}}=0} =0, \end{aligned}$$

(35a)

$$\begin{aligned}&\displaystyle \left. { \frac{{\text {d}}f_R }{\mathrm{d}\hat{{z}}}} \right| _{\hat{{z}}=-h/l_m } =1, \end{aligned}$$

(35b)

$$\begin{aligned}&\displaystyle \left. {f_{{\textit{Ch}}} } \right| _{\hat{{z}}=0} =1, \end{aligned}$$

(35c)

$$\begin{aligned}&\displaystyle \left. { \frac{{\text {d}}f_{{\textit{Ch}}} }{\mathrm{d}\hat{{z}}}} \right| _{\hat{{z}}=-h/l_m } =0. \end{aligned}$$

(35d)

We define the analytical form of the functions $f_R \left( {\hat{{z}}} \right) ,f_{{\textit{Ch}}} \left( {\hat{{z}}} \right)$ by solving Eq. 32 with the Matlab symbolic toolbox for the boundary conditions $C_h =0, \hat{{R}}=1$ for $f_R$, and $C_h =1, \hat{{R}}=0$ for $f_{{\textit{Ch}}}$.

In SK10, the solution $C\left( z \right)$ above the canopy for neutral stratification was represented following the relationships (7) and (10) for the diffusion coefficient,

$$\begin{aligned}&\displaystyle C\left( z \right) =\frac{c_*}{\kappa }\left\{ {C\left( {z_R } \right) +\ln \left( {\frac{z-h+d_c }{z_r -h+d_c }} \right) +\Psi _c \left( z \right) } \right\} , \end{aligned}$$

(36a)

$$\begin{aligned}&\displaystyle \Psi _c \left( z \right) =\int \limits _{z-h}^{z_r -h} {\frac{1-\hat{{\varphi }}_c \left( {z^{\prime }} \right) }{z^{\prime }+d_c }} \mathrm{d}z^{\prime }. \end{aligned}$$

(36b)

Here, $\hat{{\varphi }}_c \left( {z^{\prime }} \right)$ is defined as in (10), and $d_c, d_m$ are the displacement heights for concentration and momentum, respectively. The following relationship for $\Psi _c \left( z \right)$ is

$$\begin{aligned} \Psi _c \left( z \right)= & {} \left( {1-\frac{\kappa S_{{\textit{cc}}} d_c }{2\beta d_m }} \right) \exp \left( {\frac{c_{2c} }{2}\frac{d_c }{d_m }} \right) \nonumber \\&\times \left[ {Ei\left( {\frac{-c_{2c} \left( {z_R -h+d_c } \right) }{2d_m }} \right) -Ei\left( {\frac{-c_{2c} \left( {z-h+d_c } \right) }{2d_m }} \right) } \right] , \end{aligned}$$

(37)

where Ei is the exponential integral. Finally, SK10 defined the parameters $C_h, c_*$, and $c_{2c}$ by fitting the concentration profile with the first and second derivatives at the top of the canopy derived from the above- and below-canopy profiles (36) and (34).

The described relationships of SK10 are directly applicable to the problem defined by (3) of $^{14}\hbox {CO}_{2}$ transport where the sink term of $^{14}\hbox {CO}_{2}, S_\chi ^{14}$ is assumed proportional to the $\hbox {CO}_{2}$ flux rate, $S_\chi ^{14}\left( z \right) =\left( {{C^{14}}{/}C} \right) S_\chi \left( z \right)$. After the substitution of (31), then $S_\chi ^{14} \left( z \right) =a\left( z \right) g_s C^{14}\left( z\right) \left( {\chi -1} \right)$. Hence, the relationships (32)–(34) are also valid for $C^{14}\left( z\right)$ provided that the boundary conditions are defined according to (5) for $C^{14}$.

For the application of the above equations for the simulation of the vertical profile of $^{14}\hbox {CO}_{2}$, we assume $d_c =d_m =d$ and $c_{2c}$ defined by (14) according to HF08. Hence, we obtain a concentration and turbulent scalar flux of $^{14}\hbox {CO}_{2}$ at the top of the canopy $C_h^{14} =C^{14}\left( h \right) , c_*^{14} =\overline{C^{{14'}} w^{{\prime }}} /u_*$ by fitting the concentration and its first derivative according to (34) and (36) at the top of the canopy. By considering $f_R \left( 0 \right) =0, f_{Ch} \left( 0 \right) =1, C^{14}\left( {z_R } \right) =0$, this gives the system of equations,

$$\begin{aligned}&\displaystyle C_h^{14} =\frac{c_*^{14} }{\kappa }\left[ {\ln \left( {\frac{d}{z_R -h+d}} \right) -\Psi _c \left( 0 \right) } \right] , \end{aligned}$$

(38a)

$$\begin{aligned}&\displaystyle \frac{\hat{{R}}^{14}}{l_m }f_R^{\prime } \left( 0 \right) +\frac{C_h^{14} }{l_m }f_{{\textit{Ch}}}^{\prime } \left( 0 \right) =\frac{c_*^{14} S_{{\textit{cc}}} }{2\beta d}, \end{aligned}$$

(38b)

where $\hat{{R}}^{14}$ is defined as $\hat{{R}}$ in (33), but with usage of the parameter $Q_0$ representing the $^{14}\hbox {CO}_{2}$ flux from the surface according to Eq. 5d instead of $R_0$. The system (38) is then solved against $C_h^{14}$ and $c_*^{14}$, with the solutions substituted into the relationships (34) and (36) to obtain the vertical profile $C^{14}\left( z \right)$.

In the application of the SK10 model to the simulation of $^{14}\hbox {CO}_{2}$ for the conditions of the Norunda station, we use the same parameter values as described above. The value of the ratio $C_i /C$ is consistent with SK10, where $C_i{/}C=0.2$ and hence $\chi =-\,0.8$. The parameter $g_{\max }$ is evaluated for this simulation by processing measurements from the Norunda station to obtain the estimation of the flux $S_\chi \left( h \right)$ according to the procedure described in Sect. 3.5. We then evaluate $g_{\max }$ by averaging the relationship $g_{\max } =S_\chi \left( h \right) /\left( {\overline{{a}}C\left( h \right) \chi } \right)$ through all the processed dates, which yields the estimate $g_{\max } =0.016$.