A Velocity–Dissipation Lagrangian Stochastic Model for Turbulent Dispersion in Atmospheric Boundary-Layer and Canopy Flows

Abstract

An extended Lagrangian stochastic dispersion model that includes time variations of the turbulent kinetic energy dissipation rate is proposed. The instantaneous dissipation rate is described by a log-normal distribution to account for rare and intense bursts of dissipation occurring over short durations. This behaviour of the instantaneous dissipation rate is consistent with field measurements inside a pine forest and with published dissipation rate measurements in the atmospheric surface layer. The extended model is also shown to satisfy the well-mixed condition even for the highly inhomogeneous case of canopy flow. Application of this model to atmospheric boundary-layer and canopy flows reveals two types of motion that cannot be predicted by conventional dispersion models: a strong sweeping motion of particles towards the ground, and strong intermittent ejections of particles from the surface or canopy layer, which allows these particles to escape low-velocity regions to a high-velocity zone in the free air above. This ejective phenomenon increases the probability of marked fluid particles to reach far regions, creating a heavy tail in the mean concentration far from the scalar source.

Appendices

Appendix 1: Verification of the Well-Mixed Condition for the Log-Normal Model

All LS models used for turbulent dispersion must satisfy the so-called well-mixed condition (Thomson 1987). According to the well-mixed condition, particles of a tracer initially well-mixed in a turbulent flow must remain well-mixed at all times. The formulation of the drift and dispersion in the standard model by Thomson (1987) satisfies the well-mixed condition. In the log-normal model the well-mixed condition should be preserved by the construction of the model. However, this model is new and therefore the log-normal model must be first tested before being applied especially for the inhomogeneous flows considered here.

Here two checks are conducted—a theoretical one to assess whether the formulation of the log-normal model satisfies the well-mixed condition, and a numerical one where the log-normal model is tested for a case of strong inhomogeneous flow.

1.1 The Well-Mixed Condition: Formulation Analysis of the Log-Normal Model

By construction the model considered in the manuscript satisfies the well-mixed condition and has independent one-point one-time Eulerian PDFs of velocity and dissipation, i.e. the joint PDF of velocity and logarithm of the normalized dissipation $f_\mathrm{E}^{u \chi }$ can be simply written as $f_\mathrm{E}^{u \chi } = f_\mathrm{E}^u f_\mathrm{E}^{\chi }$.

These properties can be shown starting from a general model of the form,

$$\begin{aligned}&\displaystyle \mathrm{d}u_i = a_i(\mathbf x ,\mathbf u ,\epsilon )\mathrm{d}t+b_{ij}(\epsilon )\mathrm{d}W_j ,&\end{aligned}$$

(10)

$$\begin{aligned}&\displaystyle \mathrm{d}x_i = u_i \mathrm{d}t,&\end{aligned}$$

(11)

$$\begin{aligned}&\displaystyle \mathrm{d} \chi = M(\chi )\mathrm{d}t + D\mathrm{d}W,&\end{aligned}$$

(12)

where, $M(\chi )=-(\chi -\left\langle \chi \right\rangle )/T_\chi ,\; D=\sqrt{2 \sigma ^2 / T_\chi }$, and $\epsilon = \left\langle \epsilon \right\rangle$ exp$(\chi )$. For simplicity and convenience $\epsilon$ is used here for the random dissipation (instead of $\epsilon ^*$), and $\left\langle \epsilon \right\rangle$ stands for the averaged dissipation. Moreover, no special notation is used for the particle random variable appearing in the stochastic differential equation (SDE), although it must be understood that here the SDE refers to the Lagrangian particle quantity.

The previous system of SDE satisfies a Fokker–Planck equation for the Lagrangian joint PDF $f_\mathrm{L}^{u \chi }$ and, by the relation between Lagrangian and Eulerian PDFs (e.g. Novikov 1969, 1986; Pope and Chen 1990; Pope 2000; Thomson 1987), the one-point one-time Eulerian PDF $f_\mathrm{E}^{u \chi }$ satisfies as well a Fokker–Plank equation of the form,

$$\begin{aligned} \frac{\partial f_\mathrm{E}^{u \chi }}{\partial t} + \frac{\partial u_i f_\mathrm{E}^{u \chi }}{\partial x_i} + \frac{\partial a_i f_\mathrm{E}^{u \chi }}{\partial u_i} - \frac{1}{2} b_{ik} b_{kj} \frac{\partial ^2 f_\mathrm{E}^{u \chi }}{\partial u_i \partial u_j} + \frac{\partial M f_\mathrm{E}^{u \chi }}{\partial \chi } . - \frac{1}{2} \frac{\partial ^2 D^2 f_\mathrm{E}^{u \chi }}{\partial \chi \partial \chi } = 0 \end{aligned}$$

(13)

This equation is general and does not assume that the two PDFs are independent. Assuming independency $f_\mathrm{E}^{u \chi } = f_\mathrm{E}^u f_\mathrm{E}^{\chi }$ and integrating over the whole space of the variable $\chi$ (under some regularity conditions, see e.g. Pope (2000), p. 466 and Thomson (1987), p.534, it is possible to obtain an equation for the marginal PDF $f_\mathrm{E}^u$,

$$\begin{aligned} \frac{\partial f_\mathrm{E}^{u}}{\partial t} + \frac{\partial u_i f_\mathrm{E}^{u}}{\partial x_i} + \frac{\partial \left\langle a_i \right\rangle ^{\chi } f_\mathrm{E}^{u}}{\partial u_i} , \; - \frac{1}{2} \left\langle b_{ik} b_{kj} \right\rangle ^{\chi } \frac{\partial ^2 f_\mathrm{E}^{u}}{\partial u_i \partial u_j} = 0 \end{aligned}$$

(14)

where the notation $\left\langle \ \right\rangle ^{\chi }\left( \equiv \int _{-\infty } ^{\infty } f_\mathrm{E}^{\chi } \mathrm{d} \chi \right)$ indicates that the average is taken only over the space of $\chi$. With the choice of $b_{ij}(\epsilon )=\sqrt{C_0 \epsilon } \delta _{ij}$ used here, and by noting that $\left\langle \epsilon \right\rangle ^\chi = \left\langle \left\langle \epsilon \right\rangle \text {exp}(\chi ) \right\rangle ^\chi = \left\langle \epsilon \right\rangle \left\langle \text {exp}(\chi ) \right\rangle ^\chi = \left\langle \epsilon \right\rangle$, since $\chi$ is normalized so that $\left\langle \text {exp}(\chi ) \right\rangle ^\chi$ is unity (Pope and Chen 1990), we have,

$$\begin{aligned} \frac{\partial f_\mathrm{E}^{u}}{\partial t} + \frac{\partial u_i f_\mathrm{E}^{u}}{\partial x_i} + \frac{\partial \left\langle a_i \right\rangle ^{\chi } f_\mathrm{E}^{u}}{\partial u_i} - \frac{1}{2} C_0 \left\langle \epsilon \right\rangle \frac{\partial ^2 f_\mathrm{E}^{u}}{\partial u_i \partial u_i} = 0 . \end{aligned}$$

(15)

If we assume a pre-defined form of $f_\mathrm{E}^u$ we obtain from the previous equation that

$$\begin{aligned} \left\langle a_i \right\rangle ^\chi = \frac{1}{f_\mathrm{E}^u} \left( \frac{1}{2} C_0 \left\langle \epsilon \right\rangle \frac{\partial f_\mathrm{E}^{u}}{\partial u_i} + \varPhi _i(u_i)\right) , \end{aligned}$$

(16)

where $\varPhi _i$ is as defined by Thomson (1987) Eq. 9b (for more details see Rodean 1996, Chap. 8). Satisfying this condition ensures that the model has independent PDFs $(f_\mathrm{E}^{u \chi } = f_\mathrm{E}^u f_\mathrm{E}^{\chi })$ and that the Eulerian velocity PDF is a solution to Eq. 15. However, this is a necessary but not sufficient condition for the well-mixed condition since it does not define the drift coefficient $a_i$ in Eq. 10 but only its averaged value. Indeed, looking at Eqs. 10 and 11 seperately from Eq. 12 we obtain another Fokker-Plank transport equation that the Eulerian velocity PDF $f_\mathrm{E}^u$ must satisfy (given $b_{ij}(\epsilon )=\sqrt{C_0 \epsilon } \delta _{ij}$),

$$\begin{aligned} \frac{\partial f_\mathrm{E}^{u}}{\partial t} + \frac{\partial u_i f_\mathrm{E}^{u}}{\partial x_i} + \frac{\partial a_i f_\mathrm{E}^{u}}{\partial u_i} - \frac{1}{2} C_0 \epsilon \frac{\partial ^2 f_\mathrm{E}^{u}}{\partial u_i \partial u_i} = 0 . \end{aligned}$$

(17)

Satisfying the well-mixed condition (i.e. the fact that a given $f_\mathrm{E}^u$ is a solution of Eq. 17) brings to the definition of the drift coefficient used here,

$$\begin{aligned} a_i = \frac{1}{f_\mathrm{E}^u} \left( \frac{1}{2} C_0 \epsilon \frac{\partial f_\mathrm{E}^{u}}{\partial u_i} + \varPhi _i(u_i)\right) , \end{aligned}$$

(18)

where $\varPhi _i$ is again as defined by Thomson (1987). Comparing the definition in Eqs. 18 and 16, we see that Eq. 18 respects Eq. 16 since upon ensemble averaging of Eq. 18 over the space of $\chi$, Eq. 16 is retrieved. Therefore, Eq. 18 is more restrictive, and its definition of the drift coefficient is necessary and sufficient for the well-mixed condition to be satisfied. Also, since Eq. 18 contains Eq. 16 it also turns out that $f_\mathrm{E}^u$ and $f_\mathrm{E}^\chi$ are independent.

1.2 The Well-Mixed Test

For the well-mixed test, a highly inhomogeneous canopy flow dataset was selected from an open-channel experiment, where laser Doppler anemometry measurements were available for all flow statistics (Poggi et al. 2006). For this type of flow, all the terms of the drift coefficients $a_u$ and $a_w$ are included (Eqs. 3, 4), so the most general form of the model can be examined. In all simulations, the particles that reach the ground or the water level, which extends to $5 h_\mathrm{c}$, are reflected in the vertical direction and the sign of their velocity fluctuations is also reversed. Since the canopy domain and the free flow domain above the canopy are of the same order of magnitude and have a distinct differences in the flow statistics, the test is presenting an extreme flow environment to the LS model in terms of flow inhomogeneity.

The flow conditions are described in Poggi et al. (2006) and the flow statistics are shown in their Fig. 2. In each test $5 \times 10^6$ particles were released uniformly across all $z$ at $x=0$. The trajectories were computed using the LS equations (Eqs. 1–7) for some 150 s—sufficiently enough to test whether they truly stay well-mixed over time. The log-normal model parameters were set to $\sigma = 1$ or 2.5, and $C_{\chi } = 0.5, 1.6$ or 3. For details about the choice of $\sigma$ and $C_{\chi }$, see Sects. 3 and 4 respectively. At the end of the simulation, the domain was divided into 50 vertical layers, and the number of particles in each layer was computed and then divided by the expected well-mixed value. a perfectly well-mixed distribution of the horizontally integrated concentration should be unity in each layer.

The test results clearly show that the well-mixed condition is satisfied for all the simulations. Figure 8 shows the normalized concentration at the end of the well-mixed test simulations for the log-normal model with $\sigma = 1$ and 2.5 for $C_{\chi } = 1.6$, compared with the results for the standard model. The particle distribution is equal to the expected well-mixed concentration at all heights with an error of less than $2\,\%$. All the other tests (for $C_{\chi } = 0.5$ and 3) gave similarly good results and are not shown here. These results verify that the log-normal model satisfies the well-mixed condition, and may be used to perform correct dispersion simulations in inhomogeneous atmospheric flows.

Fig. 8

The verification of the well-mixed condition: final particle distribution normalized by the expected well-mixed value. The vertical lines stand for a perfectly well-mixed distribution (at 1), and an error of ${\pm }5\,\%$

Appendix 2: Atmospheric Boundary-Layer Flow Statistics (MOST)

The profiles of the Eulerian flow statistics needed in the ABL case study are presented here. These include the mean velocity, its SD, the Reynolds stress, and the mean dissipation rate (or integral time scale). Using Monin–Obukhov similarity theory, the following profiles were employed for neutral, stable and unstable conditions (Kaimal and Finnigan 1994).

The mean wind speed $\overline{u}(z)$ is calculated as

$$\begin{aligned} \overline{u}(z) = \frac{k}{u_*} \left( \ln \left( \frac{z}{z_0}\right) -\psi \right) , \end{aligned}$$

(19)

where $k=0.4$ is the von Karman constant, $u_*$ is the friction velocity, and $z_0$ is the aerodynamic roughness length. The values of the last two were set to $u_* = 0.4 \,\hbox {m s}^{-1}$ and $z_0 = 1.7$ mm in all ABL simulations.

$\psi$ is the stability correction function, which is expressed by,

$$\begin{aligned} \begin{array}{ll} \psi (z) = -5z/L, &{} \quad \mathrm{for}\; z/L\ge 0 \\ \psi (z) = 2 \text {ln}\left( \frac{1+\phi }{2}\right) + \text {ln}\left( \frac{1+\phi ^2}{2}\right) -2 \text {tan}^{-1}(\phi )+\frac{\pi }{2} &{} \quad \mathrm{for}\; z/L<0, \end{array} \end{aligned}$$

(20)

where the Obukhov length $L$ was chosen to be 200 for stable conditions, $-10$ for unstable conditions, and $\phi$ is given by: $\phi (z) = (1-16z/L)^{1/4}$. For neutral conditions $L\rightarrow \infty$, and therefore $\psi =0$.

The velocity SD profiles are expressed as,

$$\begin{aligned} \begin{array}{lclcl} \sigma _u = 2.5 u_*, &{} \; &{} \sigma _w = 1.25 u_*, &{} \; &{} \text {for}\; z/L\ge 0 \\ \sigma _u = 2.5 u_*\left( 1-3\frac{z}{L}\right) ^{1/3}, &{} \; &{} \sigma _w = 1.25 u_*\left( 1-3\frac{z}{L}\right) ^{1/3} &{} \; &{} \text {for}\; z/L<0. \end{array} \end{aligned}$$

(21)

The Reynolds stress is constant for all stability conditions: $\overline{u'w'}=-u_*^2$.

Finally, the Lagrangian time scale is estimated using the diffusion coefficient according to K theory as $T_\mathrm{L}=K/\sigma _w^2$ (Rodean 1996), and the mean dissipation rate is calculated from the consistency with the Kolmogorov’s similarity theory for locally isotropic turbulence as $\epsilon =2\sigma _w^2/C_0 T_\mathrm{L}$. $C_0$ is a phenomenological constant, taken to be 3.125, based on matching of the Lagrangian time scale to similarity theory (Li and Taylor 2005), and the diffusion coefficient is estimated by $K(z)=k z u_* / \phi _h$ , with $\phi _h$ given by (Hsieh et al. 2000),

$$\begin{aligned} \begin{array}{ll} \phi _h(z) = 1+5\frac{z}{L}, &{} \quad \mathrm{for}\;z/L\ge 0 \\ \phi _h(z) = 0.032\left( 0.037-\frac{z}{L}\right) ^{-1/3}. &{} \quad \mathrm{for}\; z/L<0 \end{array} \end{aligned}$$

(22)

For stable conditions ($z/L>0$) all the correction functions are stretched to the top of the ABL (taken as 300 m for this case). For the unstable conditions ($z/L<0$), the corrections are used only for the surface layer, which is estimated as 20 % of the entire height of the ABL (200 m of the 1 km height of the ABL). Above this height, in the mixed layer, all the statistics are taken as constants for the unstable case.