Simulating Dispersion in the Evening-Transition Boundary Layer

Abstract

We investigate dispersion in the evening-transition boundary layer using large-eddy simulation (LES). In the LES, a particle model traces pollutant paths using a combination of the resolved flow velocities and a random displacement model to represent subgrid-scale motions. The LES is forced with both a sudden switch-off of the surface heat flux and also a more gradual observed evolution. The LES shows ‘lofting’ of plumes from near-surface releases in the pre-transition convective boundary layer; it also shows the subsequent ‘trapping’ of releases in the post-transition near-surface stable boundary layer and residual layer above. Given the paucity of observations for pollution dispersion in evening transitions, the LES proves a useful reference. We then use the LES to test and improve a one-dimensional Lagrangian Stochastic Model (LSM) such as is often used in practical dispersion studies. The LSM used here includes both time-varying and skewed turbulence statistics. It is forced with the vertical velocity variance, skewness and dissipation from the LES for particle releases at various heights and times in the evening transition. The LSM plume spreads are significantly larger than those from the LES in the post-transition stable boundary-layer trapping regime. The forcing from the LES was thus insufficient to constrain the plume evolution, and inclusion of the significant stratification effects was required. In the so-called modified LSM, a correction to the vertical velocity variance was included to represent the effect of stable stratification and the consequent presence of wave-like motions. The modified LSM shows improved trapping of particles in the post-transition stable boundary layer.

Appendix: LSM Including Skewness and Time Dependence

Here we outline the formulation of the LSM used in this study. The LSM differs from the normal formulation (Eqs. 3–6) by the inclusion of skewness and time dependence. We modify $a(z,w)$ and $b(z,w)$ in Eq. 3 to now be functions of time, i.e. $a=a(z,w,t)$ and $b=b(z,w,t)$. Equations 3 and 4 have a corresponding Fokker–Planck equation, and we require this to have a solution equal to the probability density function of the positions and velocities of all air parcels ($P(w,z,t)$), i.e. we require the ‘well-mixed condition’ (Thomson 1987) to be satisfied.

With the assumption that $b$ is given by Eq. 6, manipulation of the Fokker–Planck equation will allow the determination of $a(z,w,t)$ for a given form of $P(w, z, t)$. The Fokker–Planck equation can be written in terms of the probability flux in the $w$ direction, $\phi$, as

$$\begin{aligned} {\partial \phi \over \partial w}=-{\partial P \over \partial t}-{\partial \over \partial z}(wP) , \end{aligned}$$

(14)

where

$$\begin{aligned} \phi =aP - {\partial \over \partial w}\left( {C_0 \epsilon \over 2}P\right) , \end{aligned}$$

(15)

and the boundary condition of no flux at infinity is

$$\begin{aligned} \phi \rightarrow 0 \text{ as } \vert w \vert \rightarrow \infty . \end{aligned}$$

(16)

The procedure for deriving the LSM is to first prescribe a form for $P$, whether skewed, time-dependent or both, and then determine $\phi$ using Eqs. 14 and 16. Once $\phi$ is determined, $a(z,w,t)$ can be found from Eq. 15.

1.1 Including Skewness

In order to include skewness, $P$ is specified, following Baerentsen and Berkowicz (1984), as the weighted sum of two Gaussian distributions

$$\begin{aligned} P=F_1 P_{1}+ F_2 P_{2}, \end{aligned}$$

(17)

where

$$\begin{aligned} P_{1,2}={1 \over \sqrt{2\pi }\sigma _{1,2}}\exp \left[ -{1 \over 2}\left( {w - w_{1,2} \over \sigma _{1,2}}\right) ^{2}\right] , \end{aligned}$$

(18)

and $F_1$ and $F_2$ are the weights. At this stage $P$ is assumed time independent. We then follow Luhar and Britter (1989) and substitute Eqs. 17 and 18 into Eq. 14 and integrate with respect to $w$. The expression for $\phi$ for the time independent case ($\phi _s$) is then found to be

$$\begin{aligned} \phi _s=&-{1 \over 2}\left( 1+\text{ erf }{v_{1} \over \sqrt{2}}\right) {\partial \over \partial z}\left( F_1 w_{1}\right) -{1 \over 2}\left( 1+\text{ erf }{v_{2} \over \sqrt{2}}\right) {\partial \over \partial z}\left( F_2 w_{2}\right) \nonumber \\&+P_{1}\sigma _{1}\left\{ \left( {\partial \over \partial z}\left( F_1 \sigma _{1}\right) + {F_1 w_{1} \over \sigma _{1}}{\partial w_{1} \over \partial z}\right) +\left( F_1{\partial w_{1} \over \partial z}+{F_1 w_{1} \over \sigma _{1}}{\partial \sigma _{1} \over \partial z}\right) v_{1}+F_1 {\partial \sigma _{1} \over \partial z}v_{1}^{2} \right\} \nonumber \\&+P_{2}\sigma _{2}\left\{ \left( {\partial \over \partial z}\left( F_2 \sigma _{2}\right) + {F_2 w_{2} \over \sigma _{2}}{\partial w_{2} \over \partial z}\right) +\left( F_2{\partial w_{2} \over \partial z}+{F_2 w_{2} \over \sigma _{2}}{\partial \sigma _{2} \over \partial z}\right) v_{2}+F_2{\partial \sigma _{2} \over \partial z}v_{2}^{2} \right\} , \end{aligned}$$

(19)

where

$$\begin{aligned} v_{1}={w-w_{1} \over \sigma _{1}} \end{aligned}$$

(20)

and

$$\begin{aligned} v_{2}={w-w_{2} \over \sigma _{2}}. \end{aligned}$$

(21)

Following Hudson and Thomson (1994), the values of $F_1$, $F_2$, $w_1$, $w_2$, $\sigma _1$ and $\sigma _2$ are set by ensuring that the variance ($\sigma _w^2$) and skewness ($S=\overline{w^3}/\sigma _w^3$) of $P$ match the values from the LES, by imposing the constraints that the integral of P is one and the mean of $w$ is zero, and by making the choice $w_1/\sigma _1 = - w_2/\sigma _2 = S^{1/3}$. This yields

$$\begin{aligned} w_1&= \alpha \sigma _1,\end{aligned}$$

(22)

$$\begin{aligned} w_2&= -\alpha \sigma _2,\end{aligned}$$

(23)

$$\begin{aligned} F_1&= \sigma _2/(\sigma _1 +\sigma _2),\end{aligned}$$

(24)

$$\begin{aligned} F_2&= \sigma _1/(\sigma _1 +\sigma _2),\end{aligned}$$

(25)

$$\begin{aligned} \sigma _1&= \sigma _2 + \gamma /\beta , \end{aligned}$$

(26)

and

$$\begin{aligned} \sigma _2=\frac{1}{2} \{ \sqrt{\gamma ^2/\beta ^2+ 4 \beta } - \gamma /\beta \}, \end{aligned}$$

(27)

where $\alpha =S^{1/3}$, $\beta =\sigma _w^2/(1+ \alpha ^2)$ and $\gamma =\overline{w^3}/(3\alpha + \alpha ^3)$. The choice $w_1/\sigma _1 = - w_2/\sigma _2 = S^{1/3}$ ensures $P$ is Gaussian for $S=0$.

1.2 Including Time Dependence

The above derivation can now be repeated, but without assuming that $P$ is time independent. The integral of $P$ with respect to $w$ can be written as

$$\begin{aligned} \int _{-\infty }^{w} \! P(w',z,t) \, dw'=T_{1}+T_{2}, \end{aligned}$$

(28)

where $T_1$ and $T_2$ are the contributions from the Gaussian distributions $P_1$ and $P_2$ and $w'$ is a dummy variable. Using Eq. 14, the expression for $\phi$ including both skewness and time-dependence is

$$\begin{aligned} \phi =-{\partial \over \partial t} \left( T_{1} + T_{2} \right) +\phi _s \end{aligned}$$

(29)

where the tendencies of $T_1$ and $T_2$ are given by

$$\begin{aligned} {\partial \over \partial t} T_{1}&={1 \over 2}\left( \text{ erf }{v_{1} \over \sqrt{2}}+1\right) {\partial F_1 \over \partial t}-P_{1}\sigma _{1}\left[ {F_1 \over \sigma _{1}}{\partial w_{1} \over \partial t}+{F_1 \over \sigma _{1}}{\partial \sigma _{1} \over \partial t}v_{1}\right] ,\end{aligned}$$

(30)

$$\begin{aligned} {\partial \over \partial t} T_{2}&={1 \over 2}\left( \text{ erf }{v_{2} \over \sqrt{2}}+1\right) {\partial F_2 \over \partial t}-P_{2}\sigma _{2}\left[ {F_2 \over \sigma _{2}}{\partial w_{2} \over \partial t}+{F_2 \over \sigma _{2}}{\partial \sigma _{2} \over \partial t}v_{2}\right] . \end{aligned}$$

(31)