Lagrangian Particle Dispersion Modeling of the Fumigation Process ...

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JOURNAL OF THE ATMOSPHERIC SCIENCES

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Lagrangian Particle Dispersion Modeling of the Fumigation Process Using Large-Eddy Simulation SI-WAN KIM

AND

CHIN-HOH MOENG

National Center for Atmospheric Research, Boulder, Colorado

JEFFREY C. WEIL CIRES, University of Colorado, Boulder, Colorado

MARY C. BARTH National Center for Atmospheric Research, Boulder, Colorado (Manuscript received 17 February 2004, in final form 1 October 2004) ABSTRACT A Lagrangian particle dispersion model (LPDM) is used to study fumigation of pollutants in and above the entrainment zone into a growing convective boundary layer. Probability density functions of particle location with height and time are calculated from particle trajectories driven by the sum of the resolvedscale velocity from a large-eddy simulation (LES) model and the stochastic subgrid-scale (SGS) velocity. The crosswind-integrated concentration (CWIC) fields show good agreement with water tank experimental data. A comparison of the LPDM output with an Eulerian diffusion model output based on the same LES flow shows qualitative agreement with each other except that a greater overshoot maximum of the groundlevel concentration occurs in the Eulerian model. The dimensionless CWICs near the surface for sources located above the entrainment zone collapse to a nearly universal curve provided that the profiles are time shifted, where the shift depends on the source heights. The dimensionless CWICs for sources located within the entrainment zone show a different behavior. Thus, fumigation from sources above the entrainment zone and within the entrainment zone should be treated separately. An examination of the application of Taylor’s translation hypothesis to the fumigation process showed the importance of using the mean boundary layer wind speed as a function of time rather than the initial mean boundary layer wind speed, because the mean boundary layer wind speed decreases as the simulation proceeds. The LPDM using LES is capable of accurately simulating fumigation of particles into the convective boundary layer. This technique provides more computationally efficient simulations than Eulerian models.

1. Introduction Fumigation is the phenomenon in which pollutants lying above the growing convective boundary layer are entrained into the boundary layer by penetrating thermal plumes. This process can increase the ground-level concentrations (GLCs) of pollutants significantly during daytime (e.g., Deardorff and Willis 1982). Fumigation is classified into two types depending on whether it is a temporal or spatial phenomenon. The former process, termed “nocturnal inversion breakup fumigation,” occurs when pollutants from an elevated stack or ozone in the residual layer are entrained into the growing convective boundary layer breaking the nocturnal inverCorresponding author address: Dr. Si-Wan Kim, MMM Division, NCAR, P.O. Box 3000, Boulder, CO 80307-3000. E-mail: [email protected]

© 2005 American Meteorological Society

JAS3435

sion in the morning. The spatial phenomenon, termed “shoreline fumigation,” occurs when a thermal internal boundary layer growing with downstream distance entrains pollutants from an elevated stack near a shoreline. In previous studies, both fumigation processes have been treated the same based on the Taylor’s hypothesis, which translates time to distance using the wind speed. The effects of the entrainment rate, initial plume location, and initial plume vertical depth on the vertical distributions of crosswind-integrated concentration (CWIC) and lateral and vertical dispersion characteristics have been the main topics of interest in the fumigation process. Water tank experiments of fumigation by Deardorff and Willis (1982, hereafter DW82) and more recently by Hibberd and Luhar (1996, hereafter HL96) have reported the topics mentioned above. These laboratory data have been utilized for the evalu-

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ation of various fumigation models. Early fumigation models applied the Gaussian model technique (Lyons and Cole 1973; Meroney et al. 1975; van Dop et al. 1979; Misra 1980a, b). However, DW82 showed that some assumptions of the Gaussian model were problematic. Specific problems were 1) the assumption of instantaneously perfect vertical mixing of entrained pollutants and 2) the model boundary layer height being represented as a smoothly varying function of distance rather than a locally fluctuating one. In the 1990s, a Lagrangian particle dispersion model (LPDM) based on parameterized turbulence statistics (Luhar and Britter 1990; Hurley and Physick 1991) and a simple probability density function approach (HL96) were used to study the fumigation process. These studies showed that vertical concentration profiles were quite sensitive to the parameterization of convective boundary layer turbulence (e.g., Hurley and Physick 1991). Recently, Cai and Luhar (2002, hereafter CL02) performed the first large-eddy simulation (LES) of fumigation by solving Eulerian diffusion equations for scalar tracers lying in and above the entrainment zone. Their LES of fumigation showed reasonable agreement with the water tank experiments of HL96. In this study, the fumigation phenomenon is studied by a Lagrangian particle dispersion model driven by LES-generated flows. Within the frame of LES, the LPDM approach has several advantages over the Eulerian diffusion model (EDM) approach. It uses the much simpler Lagrangian approach for tracking and finding the concentration field from the probability density function of particle position rather than solving the advection–diffusion equation for the entire LES domain. Another advantage is that the Lagrangian subgrid-scale (SGS) model is consistent with short and long time limits of Taylor’s (1921) theory, wherein the dispersion or plume width ␴z varies as ␴z ⬀ t (short time) and ␴z ⬀ t1/2 (long time). However, the Eulerian SGS model being an eddy-diffusivity approach only follows the long time behavior, and therefore, does not properly simulate the short-range dispersion from a small source. In addition, the SGS treatment in the LPDM combined with LES generally can account for inhomogeneous, non-Gaussian, and unsteady turbulence [following Thomson’s (1987) approach] although it was derived and applied here only for inhomogeneous, Gaussian, and weakly nonstationary turbulence. When large domains are considered, the LPDM is more computationally efficient than the EDM. Thus, more complex and detailed studies can be pursued with the LPDM once it has been evaluated. Our main objective in this study is to evaluate this new modeling approach for the fumigation process. We integrate our Lagrangian model for two different entrainment rates, various initial release heights, and two initial vertical plume depths, which were similar to those used by DW82, HL96, and CL02. Results are compared with water tank experiments (DW82; HL96),

the LES results of CL02, and our own Eulerian modeling results. Scaling issues and the importance of the LES vertical resolution are discussed. Finally, we discuss whether it is appropriate to treat both temporal and spatial fumigation phenomena in the same manner. The modeling tools are explained in section 2. The model experiments are described in section 3. Comparisons of LPDM results with water tank experiments and with Eulerian model results are shown in section 4, and a discussion follows in section 5. Section 6 provides summary and conclusions.

2. Description of models used in study a. Lagrangian particle dispersion model The LPDM adopted in this study is based on the model described in Weil et al. (2000, 2004). In this approach, particle position xp at time t ⫹ dt is given by xp共xos, t ⫹ dt兲 ⫽ xp共xos, t兲 ⫹ uL共xos, t兲 dt,

共1兲

where xos is the source or initial particle position, t is the time, and uL is the Lagrangian velocity of a particle. A bold character denotes a vector. We determine uL as the sum of the resolved-scale velocity component ur generated by LES and an SGS velocity component us: uL共xos, t兲 ⫽ ur关xp共xos, t兲, t兴 ⫹ us关xp共xos, t兲, t兴.

共2兲

The resolved-scale velocity component ur used LESgenerated flow fields, and the us is predicted from a stochastic differential equation [Eq. (3) below]. This approach is similar to that used by Lamb (1978), but our SGS velocity model (described in Weil et al. 2004) is somewhat more detailed based on Thomson’s (1987) Lagrangian stochastic theory. Here, we give a short description. The SGS turbulence is assumed to be Gaussian and isotropic. The increment in the ith component of the SGS velocity, usi over dt is given by the stochastic differential equation dusi ⫽ ⫺

fsC0⑀ usi 1 dt ⫹ 2 2 ␴s 2

冉

冊

⭸␴s2 1 d␴s2 u ⫹ dt si ⭸xi ␴s2 dt

⫹ 共 fsC0⑀兲1Ⲑ2d␰i,

共3兲

␴ 2s

where ⫽ 2es/3 is the isotropic stress component, es is the SGS turbulent kinetic energy, ⑀ is dissipation rate, d␰i is a Gaussian white noise process, and C0 is an assumed constant (⫽4 ⫾ 2; Thomson 1987). The dimensionless coefficient fs, which scales the first and last terms to the contribution from SGS turbulent kinetic energy is given by fs ⫽

具␴s2典 , 具␴a2␷典 ⫹ 具␴s2典

共4兲

where ␴ 2a␷ ⫽ (␴ 2ru ⫹ ␴ 2r␷ ⫹ ␴ 2rw)/3 is the resolved-scale velocity variance and the brackets denote a horizontal

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average. Both the SGS turbulent kinetic energy and the resolved-scale variance ␴ 2a␷ are output from the LES solutions that were generated from the National Center for Atmospheric Research (NCAR)-LES code (Moeng 1984; Sullivan et al. 1994). Weil et al. (2004) show that this LPDM approach applied to LES flow fields is not sensitive to the universal constant C0 over the range 2 ⱕ C0 ⱕ 6. ⌻his was contrary to a previous LPDM model study based on ensemble turbulence statistics (Du 1997), which found the best-fit value of C0 ⫽ 3 and poorer agreement with larger C0 values. The insensitivity of the current LPDM to C0 may be attributed to the model’s use only for the SGS velocities. At the surface, a perfect reflection condition is adopted, and at the top of the mixed-layer height, no constraint on particle position is applied.

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and the streamwise-integrated CWIC cxy(z, t) is given by c xy共z, t兲 ⫽ Qi

冕 冕 ⬁

⬁

⫺⬁

⫺⬁

p共x, y, z, t兲 dy dx.

共7兲

This equation can be written in the following dimensionless form: c xy共z, t兲zi ⲐQi ⫽ zi

冕 冕 ⬁

⬁

⫺⬁

⫺⬁

p共x, y, z, t兲 dy dx.

共8兲

In the water tank experiment by Willis and Deardorff (1976a), DW82, and HL96, a finite-line source of length Lx is adopted such that the CWIC averaged over Lx is x

c y 共z, t兲 ⫽

Qi Lx

冕冕 Lx

⬁

0

⫺⬁

p共x, y, z, t兲 dy dx.

共9兲

The dimensionless form of Eq. (9) is given by

b. Eulerian diffusion model

x

To make a clear comparison between the Lagrangian and Eulerian models, we apply the same LES flow field to these two models. For an Eulerian diffusion model, we added a conservation equation for a passive scalar into the NCAR-LES code with an area source located within or above the entrainment zone. A pseudospectral method is applied to solve the horizontal advection terms of the passive scalar, while a monotone scheme is used to solve its vertical advection. A radiation boundary condition (Klemp and Durran 1983) is used as the upper boundary condition, and a zero gradient method is used as the lower boundary condition.

c. Definitions of crosswind-integrated concentration for the laboratory experiments, Lagrangian, and Eulerian models We define the CWICs to compare results from the LPDM, which released point sources; the laboratory experiments, which released line sources; and the Eulerian diffusion model (EDM), which used an area source. In the LPDM, dispersion from an ensemble of instantaneous point sources above or within the entrainment zone is simulated. For an instantaneous point source of strength Qi (unit mass), the concentration at point x, y, z downwind of the source and at time t after the release is c共x, y, z, t兲 ⫽ Qi p共x, y, z, t兲,

共5兲

c y 共z, t兲zi ⲐQli ⫽ zi

冕冕 Lx

⬁

0

⫺⬁

p共x, y, z, t兲 dy dx,

共10兲

where Qli ⫽ Qi/Lx (mass per unit length), and this can be put in the dimensionless form used by DW82 for a continuous point source of strength Q (mass per unit time) as x

x

c y 共z, t兲zi ⲐQli ⫽ c y 共z, t兲Uzi ⲐQ,

共11兲

where Q ⫽ QliU ⫽ QiU/Lx. In making the above equivalence, we assume that all particles from the instantaneous point sources are translated at the constant wind speed U and t ⫽ x/U, that is, where streamwise wind fluctuations and dispersion are ignored. In the EDM, the concentration averaged over the horizontal domain (Lx ⫻ Ly) c xy is given by c xy共z, t兲 ⫽

Qi LxLy

冕冕 Lx

Ly

0

0

p共x, y, z, t兲 dy dx,

共12兲

for the total strength of source Qi (unit mass). The dimensionless form of Eq. (12) is given by c xy共z, t兲zi ⲐQai ⫽ zi

冕冕 Lx

Ly

0

0

p共x, y, z, t兲 dy dx,

共13兲

where Qai ⫽ Qi /(LxLy) (mass per unit area). The righthand side of Eq. (13) is the same as those of Eqs. (8) and (10) except that upper and lower limits of integrals differ with source type. Hence, we can assume the following equivalence: x

where p(x, y, z, t) is the particle position PDF at time t, which can be calculated from an ensemble of numerically computed particle trajectories. The CWIC cy(x, z, t) is given by cy共x, z, t兲 ⫽ Qi

冕

⬁

⫺⬁

p共x, y, z, t兲 dy,

共6兲

Cy共z, t兲 ⫽ c xy共z, t兲zi ⲐQi ⫽ c y 共z, t兲ziⲐQli ⫽ c xy共z, t兲ziⲐQai, 共14兲 y

where C (z, t) will be called hereafter the dimensionless CWIC that is not dependent on the source type. Equation (14) links the dimensionless CWICs from our LPDM (point source) results, water tank experiment data (line source), and EDM (area source) results. Be-

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KIM ET AL. TABLE 1. LES input parameters.

Expts

(d⌰/dz)fa (K m⫺1)

w␪sfc (K m s⫺1)

U (m s⫺1)

XL, YL (km)

ZL (km)

⌬x (m)

⌬y (m)

⌬z (m)

se fe fe_f se_u

0.020 0.006 0.006 0.020

0.05 0.05 0.05 0.05

0 0 0 3

1.84 1.84 1.84 1.84

0.72 0.80 0.80 0.72

10 10 10 10

10 10 10 10

5 5 2.5 5

cause each study used different scaling, we define Cyzi0, Cyzs, and Cyzi(t) representing Cy nondimensionalized by zi0 (boundary layer height at the moment of plume insertion), zs (initial source height), and zi(t) (boundary layer height as a function of time), respectively. For a continuous source in the presence of a mean wind, the CWIC can be obtained from another formulation that accounts for streamwise dispersion (Weil et al. 2004), which will be used to examine the Taylor’s translation hypothesis. In this formulation, the CWIC at point x, z downstream of the source is found by summing all particles that arrive at the point x, y, z at time t over the source emission history. This CWIC is given by c y共x, z兲 ⫽ Q

冕冕 ⬁

⬁

0

⫺⬁

p共x, y, z, t⬘兲 dy dt⬘,

共15兲

where t⬘ ⫽ t ⫺ te, and te is the particle emission time. In practice, the upper limit of the time integration in Eq. (15) is replaced by t⬘f ⫽ 1.5xe /U, where xe is the largest sampling distance of interest; this t⬘f was found to be sufficiently large for convergence of the integral (Weil et al. 2004). The dimensionless form of cy(x, z) is Cy共x, z兲 ⫽ c y共x, z兲Uzi ⲐQ.

共16兲

Results using Eq. (16), which is a function of x and z, will be compared to those using Eq. (8), which is a function of t and z, to examine the Taylor’s translation hypothesis.

are given in Table 1. The LES experiments representing slow and fast entrainments are labeled “se” and “fe,” respectively, and the stability of the free atmosphere is 0.02 and 0.006 K m⫺1, respectively. The surface heat flux is 0.05 K m s⫺1 in all cases. The case fe_f, where the vertical resolution is increased to 2.5 m, is designed to examine the effect of the vertical resolution on the CWIC. The free convection cases se, fe, and fe_f with initial mean wind U ⫽ 0 are used to compare with the water tank experiments. For comparison with laboratory experiments, the Coriolis force and large-scale subsidence are omitted in our LESs. To examine Taylor’s hypothesis, we ran the case se_u where the initial mean wind U ⫽ 3 m s⫺1. The periodicity of the flow field is applied in the horizontal direction so that particles reaching one end of the horizontal domain are cycled back to the other end of the domain. The LES output parameters (Table 2) relevant to the fumigation process are obtained at the simulation time of 5400 s after the simulated turbulence has reached quasi-equilibrium. At t ⫽ 5400 s when the particles are released, the mixed-layer height zi0 (height of the minimum heat flux) is 260 m for both entrainment-rate regimes. In both regimes, the convective velocity scale w 0 at this time (5400 s) is 0.76 m s⫺1, and the eddy * turnover time t 0 ⫽ zi0/w 0 is 342 s. Figure 1 shows the * * growth of the boundary layer heights for the two cases, where the dimensionless entrainment rates are about 0.014 and 0.039 for the se and fe cases, respectively. These values are similar to those of DW82.

3. Description of experiments a. LES experiments We performed two LESs, which had entrainment rates similar to those in DW82. In DW82, the dimensionless entrainment rate, we /w 0 of the slow entrain* ment case is 0.015 ⫾ 0.003, while that of the fast entrainment case is 0.042 ⫾ 0.003. These entrainment-rate conditions are also included in the experiments of HL96 and the LESs of CL02. The parameters used to generate LESs for the slow and fast entrainment cases TABLE 2. LES output parameters. Expts

we (m s⫺1)

zi0 (m)

se fe

0.011 0.030

260 260

w 0 (m s⫺1) * 0.76 0.76

t

(s) *0 342 342

we /w 0 * 0.014 0.039

FIG. 1. Time evolution of the boundary layer height (zi, defined by the flux minimum method) for cases se (dotted line) and fe (solid line).

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JOURNAL OF THE ATMOSPHERIC SCIENCES TABLE 3. Simulations performed by LPDM.

␴z0 /zi0

zs /zi0

1–14

we /w 0 * 0.014 (se)

0.036, 0.092

15–28

0.039 (fe)

0.036, 0.092

0.039 (fe_f) 0.014 (se_u)

0.036 0.036

1.30, 1.25, 1.19, 1.15, 1.09, 1.00, 0.91 1.30, 1.25, 1.19, 1.15, 1.09, 1.00, 0.91 1.19 1.19

Simulations

29 30

b. LPDM experiments The LPDM simulations are used to show the effects of entrainment rate, source height (within and above the entrainment zone), and vertical depth of the initial plume on the fumigation process. These conditions for the LPDM are given in Table 3. We selected seven dimensionless initial source heights, zs /zi0 of 1.30, 1.25, 1.19, 1.15, 1.09, 1.00, and 0.91, similar to those imposed in the Eulerian model runs by CL02. We categorized the zs /zi0 ⫽ 1.30, 1.25, 1.19, 1.15 as “above the entrainment zone,” and zs /zi0 ⫽ 1.09, 1.00, 0.91 as “within the entrainment zone.” This categorization is based on the heat flux profile (Fig. 2a) and the fraction of turbulent area at the source height (Fig. 2b). The fraction of turbulent area shown in Fig. 2b is defined as the percentage of grid points that are below the maximum ␪ gradient height zig(x, y) (for zig definition, see Sullivan et al. 1998) at t ⫽ 5400 s. At zs / zi0 ⱖ 1.15, the heat flux

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(Fig. 2a) is vanishingly small and the fraction of turbulent area (Fig. 2b) is less than about 10%. Hence, those source heights are classified as above the entrainment zone. It is noteworthy that at zs / zi0 ⱕ 0.91, the fraction of turbulent area increases to nearly 100%, indicating that strong turbulence exists within the entrainment zone (although the heat flux is negative). Figure 3 shows the positions of source heights relative to the local boundary layer height zig. For zs /zi0 ⱖ 1.15, the source is initially located mostly above the top of the turbulent layer; for zs /zi0 ⱕ 0.91, the source is located mostly inside the turbulent layer; and for zs between those limits, source comes from both inside and outside of the turbulent layer, which strictly speaking is a mixed process between dispersion from a elevated source within CBL and fumigation. The initial plumes are assumed to have a Gaussian shape with a root-mean-square vertical spread of ␴z0 centered at the source height zs. We examined two initial vertical spreads of the plume, ␴z0 /zi0 ⫽ 0.036 and 0.092, which were the same values used by CL02. (Note that the smallest source used in this study results in a source diameter of 0.072 to 0.14 zi0, whereas the vertical grid ⌬z/zi0 ranges only from 0.01 to 0.02. Thus, the source was several times the grid size in this fumigation study.) The ␴z0 /zi0 in the water tank experiments of HL96 was less than 0.040 for entrainment rates comparable to our cases se and fe. In the DW82 experiment, the initial vertical spread of the plume was not given,

FIG. 2. Profiles of (a) heat flux normalized by surface heat flux Qs and (b) fraction of turbulent area (percentage of area below zig, zig is the boundary layer height defined by the maximum ␪ gradient method) at t ⫽ 5400 s. Solid (filled circle) and dotted (open circle) lines represent the slow and the fast entrainment cases, respectively. Symbols in (b) represent selected source heights.

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c. Eulerian diffusion model experiments Similar to the LPDM experiments, we performed eight Eulerian diffusion model experiments for each entrainment regime (Table 4). An area source at height zs covering the whole LES horizontal domain with a Gaussian distribution in the vertical direction was inserted at t ⫽ 5400 s of each LES run; the dimensionless source heights and initial vertical dispersion were: zs /zi0 ⫽ 1.19, 1.09, 1.00, 0.91, and ␴z0 /zi0 ⫽ 0.036, 0.092. The CWIC as functions of z and t are calculated from the LES scalar concentration over 3 (2) h for slow (fast) entrainment, using Eqs. (12) and (13).

4. Comparison of LPDM output with water tank experiments and Eulerian model a. The definition of the height scaling parameter and the starting time of fumigation

FIG. 3. Cross section of local boundary layer height (zig, thick solid lines) and the location of initial source heights for the (a) slow and (b) fast entrainment cases. Solid, dotted, dashed, dash– dotted, and dash–triple dotted lines represent zs /zi0 ⫽ 0.91, 1.00, 1.09, 1.15, and 1.19, respectively.

The definitions of the height scaling parameter and the starting time of fumigation are different in the analyses of DW82, HL96, and CL02. DW82 employed the time-varying mixed-layer height zi(t) as the height scaling parameter (see Luhar and Britter 1990); HL96 used the initial plume height zs; and CL02 used zi0, the mixed-layer height defined by the heat flux minimum at the moment of plume insertion. The definitions of the

thus we must estimate ␴z0 /zi0. In their experiment, the mixed-layer depth was in the vicinity of 0.2 to 0.3 m, and the initial fumigant ribbon was subject to local gravity wave undulations of amplitude about ⫾1 cm before fumigation began. If we assume ␴z0 ⫽ 1 cm in DW82 experiment, then ␴z0 /zi0 is about 0.033 to 0.05. Hence, our LPDM output for ␴z0 /zi0 ⫽ 0.036 cases can be compared to the experimental data of the DW82 and HL96. For the LPDM runs, the LES data files are applied at a time interval of 20 s. For times between two successive LES files, the resolved-scale velocities, subgridscale energy, and resolved-scale velocity variances are linearly interpolated with time. A total time period of 3 (2) h of the LES flow fields is used for the slow (fast) entrainment case, where the slow entrainment case takes longer to complete the fumigation. Initially, a total of about 100 000 particles are released from 400 different source locations covering most of the LES horizontal domain. PDFs of the particle height are calculated as a function of time for LPDM runs. For case se_u, PDFs of particle height are computed both as a function of time t [Eqs. (7) and (8)] and distance x [Eqs. (15) and (16)] to check Taylor’s hypothesis. TABLE 4. EDM simulations. Simulations 1–8 9–16

we /w 0 * 0.014 (se) 0.039 (fe)

␴z0 /zi0

zs /zi0

0.036, 0.092 0.036, 0.092

1.19, 1.09, 1.00, 0.91 1.19, 1.09, 1.00, 0.91

FIG. 4. Contours of the dimensionless CWIC (Cyzi0) for the slow entrainment case se with zs /zi0 of (a) 1.25 and (b) 1.19.

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FIG. 5. The same as in Fig. 4 except for fast entrainment case fe.

starting time of fumigation ts also differ in the three studies. DW82 set ts equal to the earliest time at which the initial dye ribbon was observed to be subjected to downward diffusion motions. In HL96, ts is taken to be the earliest time at which downward diffusion was observed to occur along a significant length (about 0.5 zs) of the initial fumigant ribbon; the horizontal length of fumigant was 1.5 to 2.0 m and 0.5 zs was 0.113 to 0.165 m. Hence ts in HL96 is the time at which approximately 6% to 11% of the total length of fumigant exhibits downward diffusion. While the starting times of fumigation defined by DW82 and HL96 are based on the plume location, the ts in CL02 is defined as the plume insertion time. It has been debatable whether the GLC distributions in time are simply time shifted with respect to different initial heights of inserted plumes in the fumigation process; that is, the GLC distribution with time is shifted to a greater time with an increase in zs. When particles are released within the entrainment zone, fumigation begins rapidly (CL02), thus allowing the starting time of fumigation to be independent of release height. CL02 did not discuss the starting time of fumigation for particles released above the entrainment zone. Most likely the starting time of fumigation will be delayed. Here we will quantify the starting time of fumigation for particles released above the entrainment zone. Figure 4 shows the contour plots of dimensionless

FIG. 6. Variation of dimensionless CWIC (Cyzs) near the surface (z/zi0 ⫽ 0.2) with time for (a) slow and (b) fast entrainment cases; the time origin is taken at t0 ⫽ 0. Solid lines represent results for zs /zi0 ⫽ 1.30, dotted lines for zs /zi0 ⫽ 1.25, dashed lines for zs /zi0 ⫽ 1.19, dash–dotted lines for zs /zi0 ⫽ 1.15, dash–triple dotted lines for zs /zi0 ⫽ 1.09, and long dashed lines for zs /zi0 ⫽ 1.00. Symbols represent the HL96 data.

CWIC for the slow entrainment case with zs /zi0 of 1.25 and 1.19. The time at which the contour of 0.1 reaches the ground is about 10 dimensionless time units for zs /zi0 ⫽ 1.25, and about 6 dimensionless time units for zs /zi0 ⫽ 1.19, demonstrating the dependence of fumigation on the source height. The dimensionless CWIC for the fast entrainment case also exhibits a positive time shift in the GLC distribution as zs /zi0 increases from 1.19 and 1.25 (Fig. 5). The time shift between Fig. 4a (zs /zi0 ⫽ 1.25) and Fig. 4b (zs /zi0 ⫽ 1.19) is about 4 dimensionless time units, while that between Fig. 5a (zs /zi0 ⫽ 1.25) and Fig. 5b (zs /zi0 ⫽ 1.19) is about 2. We define three starting times of fumigation for the LPDM simulations. The first starting time (t0) corresponds to the time of the insertion of the plume. The second and third starting times are set to the time at which 5% (ts5%) and 10% (ts10%) of released particles begin to reside within the well-mixed layer below 0.85 zs. This definition is similar to that of HL96: the time at which a significant amount of fumigant (6% to 11% of the total fumigant) shows a downward motion. Similar to HL96, the CWICs calculated from our

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FIG. 7. The same as in Fig. 6 except that the time origin is taken at t0 ⫽ ts5%.

FIG. 8. The same as in Fig. 6 except that the time origin is taken at t0 ⫽ ts10%.

LPDM results at z/zi0 ⫽ 0.2 are normalized by zs. When plotted with t0 equal to the plume insertion time (Fig. 6), the near-surface CWICs clearly reach maximum values at different times, indicating a time shift with CWIC maxima for low zs /zi0 occurring sooner than those with higher source heights. The modeled CWICs agree with the laboratory data for only certain heights: zs /zi0 ⫽ 1.15 for the slow entrainment case and zs /zi0 ⫽ 1.15 and 1.19 for the fast entrainment case. When t0 ⫽ ts5% is adopted, the CWICs for zs ⱖ 1.15 zi0 collapse nearly to a single curve as shown in Fig. 7 and show good agreement with the HL96 data for pure fumigation cases (i.e., cases with zs /zi0 ⱖ 1.15) for both slow and fast entrainment cases. The CWICs with t0 ⫽ ts10% (Fig. 8) also collapse to a single curve, but agree less well with the laboratory data, especially at the early stage of fumigation, suggesting that the 10% criterion is not appropriate. Figure 7 shows that the curves that do not collapse to the laboratory data are the ones with a plume release height within the entrainment zone. As shown in section 3b, for source heights above the entrainment zone (zs /zi0 ⱖ 1.15) particles are almost all above the turbulent layer, while for source heights within the entrainment zone (zs /zi0 ⬍ 1.15) some particles originate inside and some outside of the turbulent layer. This leads to a different behavior of concentration fields for

sources above the entrainment zone and within the entrainment zone. Thus, CWICs from source heights originating above the entrainment zone depend on the definition of the fumigation starting time, while CWICs from source heights originating within the entrainment zone do not depend on the fumigation starting time.

b. The vertical distribution of crosswind-integrated concentration Contours of the distributions CWIC (z, t) computed from Eq. (8) from our LPDM are compared with those from the HL96 tank experiment in Fig. 9 for the source height zs /zi0 ⫽ 1.25 and the initial vertical spread ␴z0 /zi0 ⫽ 0.036. The dimensionless entrainment rates in our LPDM and the experiments are identical or close, 0.014 (0.039) and 0.014 (0.038), for the slow (fast) entrainment cases, respectively. We select zs /zi0 ⫽ 1.25 case to compare with the HL96 experiments, even though the information of zs /zi0 is not available from HL96. In section 4a, we showed that the results should be insensitive to zs /zi0 if zs /zi0 is above the entrainment zone and if the starting time of fumigation is properly defined. The CWIC distributions from our model and the experiments show good agreement for both the slow and fast entrainment cases. In both cases, the local

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FIG. 9. Contours of the dimensionless CWICs (Cyzs) for the slow entrainment case from (a) LPDM (we /w 0 ⫽ 0.014) and from (b) * HL96 (we /w*0 ⫽ 0.014) and for fast entrainment case from (c) LPDM (we /w*0 ⫽ 0.039) and from (d) HL96 (we /w*0 ⫽ 0.038). LPDM plots are for the source of zs /zi0 ⫽ 1.25 and ␴z0 /zi0 ⫽ 0.036.

maximum near-surface CWIC occurs much sooner for fast entrainment (at dimensionless t of 7) than for slow entrainment (at dimensionless t of 15).

c. Near-ground-level crosswind-integrated concentration The modeled CWICs near ground level for the source height zs /zi0 ⫽ 1.25 and the initial vertical spread ␴z0 /zi0 ⫽ 0.036 are compared with those from the DW82 (Fig. 10a) and HL96 experiments (Fig. 10b), where the height scale in Fig. 10a uses zi(t) according to DW82 and in Fig. 10b uses zs according to HL96. We calculate the crosswind-integrated GLCs of DW82 using their centerline GLCs and ␴y assuming a Gaussian distribution [see Luhar and Britter (1990) and HL96]. Our LPDM results are quite close to both the DW82 and HL96 data. Next we compare the near-surface CWICs normalized by zi(t) between the LPDM and EDM results (for various initial source heights) in Fig. 11. The EDM ex-

hibits much greater CWIC maximum values compared to the LPDM for the releases within the entrainment zone, zs /zi0 ⫽ 0.91; for this case, the surface CWIC increases rapidly right after the release with an overshoot maximum before dropping to an equilibrium level. This characteristic is common for both entrainment-rate regimes and both cases of initial vertical spread (not shown in the figure). A possible reason for this overshoot maximum for zs /zi0 ⫽ 0.91 is based on previous studies of particle dispersion within the CBL. For particle release within the CBL, such as zs /zi0 ⫽ 0.07, 0.24, and 0.5, the overshoot maximum is also greater than one, that is, the dimensionless CWIC at the well-mixed state (Willis and Deardorff 1976a, 1978, 1981). For source height of zs /zi0 ⫽ 0.91, most particles are initially located within the turbulent layer (section 3b, Figs. 2 and 3), resulting in concentration fields that behave like those from sources within the CBL. A greater maximum in EDM than that in LPDM might be due to the too-diffusive nature of SGS eddy diffusivity and advection scheme in EDM.

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FIG. 10. Near ground-level dimensionless CWICs as a function of time compared to (a) DW82 (zm/zi0 ⫽ 0.07), and (b) HL96 (zm/zi0 ⫽ 0.2); where zm is the measurement height; LPDM results are plotted at the measurement heights of the corresponding experiments. Symbols represent the laboratory data and lines the LPDM results. Filled (DW82 dye expt; HL96 expt 5) and open (DW82 oil expt; HL96 expt 7) circles and dotted lines apply to the fast entrainment case, and triangles (DW82 dye expt; HL96 expt 4) and solid lines to the slow entrainment case. LPDM plots are for the source height of zs /zi0 ⫽ 1.25 and plume spread of ␴z0 /zi0 ⫽ 0.036.

For the fast entrainment case (Fig. 11b), the equilibrium (far field) GLCs from our LPDM (tw 0 /zi0 ⬎ 10) * are greater than those from the EDM because of particle accumulation near the ground in the LPDM solution, which will be discussed in section 5a. Here, the equilibrium GLCs indicate ground-level concentrations corresponding to an appropriately well-mixed vertical distribution that occurs after fumigation is completed. The equilibrium GLC should be 1 for Cyzi(t). The surface CWICs from our EDM results are also compared with the LES results from CL02 (Fig. 12). Now the CWICs are normalized by zi0, as used by CL02. Our EDM (thick lines) and CL02 (thin lines) results are qualitatively similar. However, our EDM results exhibit a faster onset of fumigation for most initial plume heights except for zs /zi0 ⫽ 1.19. The equilibrium CWICs agree well with each other for thin plumes (Fig. 12a), but the equilibrium CWICs from CL02 are slightly lower than our EDM results for thick

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FIG. 11. The dimensionless CWIC (Cyzi(t)) near the surface (z/zi0 ⫽ 0.1) from the LPDM (thick lines) and from the Eulerian model (thin lines) with ␴z0 /zi0 ⫽ 0.036 for (a) slow entrainment and (b) fast entrainment cases. Solid lines correspond to zs /zi0 ⫽ 1.19, dotted lines to zs /zi0 ⫽ 1.09, dashed lines to zs /zi0 ⫽ 1.00, and dash–dotted lines to zs /zi0 ⫽ 0.91.

plumes (Fig. 12b). The difference between CL02 and our EDM may be due to the different numerical schemes used in the LES codes: CL02 used a finite difference method, while we used a pseudospectral representation in x and y.

d. Dependence of GLCs on the initial plume thickness The surface CWIC distributions from our LPDM are compared for a thin (thin lines) and thick (thick lines) initial plume in Fig. 13. For the cases with zs /zi0 ⱖ 1.09, the thicker plume resulted in larger near-ground CWICs at the earlier stage of fumigation and then showed less CWICs at the later time, compared to the thinner plume release. For both entrainment cases and both initial plume thicknesses, overshoot maxima appear with zs /zi0 ⫽ 0.91. This overshoot maximum is slightly larger for the thinner plume in both entrainment cases. However, this difference in overshoot maximum in LPDM is much smaller than that in EDM (not shown).

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FIG. 12. Comparison of near-surface dimensionless CWICs (Cyzi0) from the NCAR Eulerian model (thick lines, z/zi0 ⫽ 0.1) and CL02 (thin lines) for (a) ␴z0 /zi0 ⫽ 0.036 and (b) ␴z0 /zi0 ⫽ 0.092 for slow entrainment case. Solid lines correspond to zs /zi0 ⫽ 1.19, dotted lines to zs /zi0 ⫽ 1.09, dashed lines for zs /zi0 ⫽ 1.00, and dash–dotted lines for zs /zi0 ⫽ 0.91.

FIG. 13. The dimensionless CWICs (Cyzi(t)) near the surface (z/zi0 ⫽ 0.1) from the LPDM results for ␴z0 /zi0 ⫽ 0.036 (thin lines) and ␴z0 /zi0 ⫽ 0.092 (thick lines); (a) for slow entrainment and (b) for fast entrainment. Solid lines correspond to zs /zi0 ⫽ 1.19, dotted lines to zs /zi0 ⫽ 1.09, dashed lines to zs /zi0 ⫽ 1.00, and dash– dotted lines to zs /zi0 ⫽ 0.91.

5. Discussion

equilibrium state. One possible reason for the accumulation is a too-coarse vertical resolution in the LES. To see if the resolution affects the concentration near the surface, we perform a simulation with a finer vertical resolution of 2.5 m in case fe_f (Fig. 14). The accumulation of CWIC near the surface is slightly reduced with the finer vertical resolution, but the accumulation still remains. The maximum contour value decreases from 1.2 (Fig. 14a; ⌬z ⫽ 5 m) to 1.1 (Fig. 14b; ⌬z ⫽ 2.5 m). Another possible source for this accumulation is an incomplete formulation of the SGS velocity us for the fast entraining CBL. The derivation for us (Weil et al. 2004) includes the nonstationary SGS turbulence term ⳵␴ 2s /⳵t in Eq. (3), but it neglects the change in zi with time, which could contribute to us. Note that the purpose of this term is to maintain an initially well-mixed distribution of particles as well mixed; this is Thomson’s (1987) “well mixed” condition and one of the cornerstone criteria governing the formulation of Lagrangian stochastic models. One would expect that the omission of this term would manifest itself in the fast entrainment case as indeed was found in this study. A formulation for the random (Lagrangian) vertical velocity wL in an one-

Results from the LPDM show good agreement with previous laboratory and LES studies of fumigation, indicating that LPDM can be an accurate and efficient tool to further examine fumigation. The LPDM did not agree well with previous studies at later times in the fast entrainment simulation near the surface when particles accumulated. Here we will discuss reasons for the particle accumulation. Then we will address the use of Taylor’s hypothesis for fumigation studies.

a. Accumulation of particles near the surface in the fe case The LPDM results from the fast entrainment show an accumulation of particles after the plume centerline reaches the ground (Figs. 5, 11b, and 13b), which is in contrast to the expected well-mixed vertical profile in the equilibrium state. When our LPDM was applied to the dispersion from sources within a slowly entraining CBL [this study and Weil et al. (2004)], there was no accumulation of concentration near the surface in the

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x ⫺ xs ⫽ U共t ⫺ ts兲.

共17兲

Equation (17) can be written in the dimensionless form X ⫽ 关共x ⫺ xs兲Ⲑzi0兴共w 0 ⲐU兲 ⫽ T, *

FIG. 14. Dimensionless CWIC (Cyzi(t)) contours from (a) case fe and from (b) case fe_f. (c) Near ground-level concentration (z/zi0 ⫽ 0.1) as a function of time for fe_f (solid) and fe (dotted) case. Plots are for the source height of zs /zi0 ⫽ 1.19 and plume spread of ␴z0 /zi0 ⫽ 0.036.

dimensional ensemble-mean dispersion model for a rapidly growing CBL developed by Weil (1992) includes this ⳵zi/⳵t effect. The derivation for the corresponding term in the case of the SGS velocity requires careful consideration and further work.

b. Examination of Taylor’s hypothesis in the fumigation problem Previous studies have stated that Taylor’s translation hypothesis can be applied to relate the diffusion travel time t and the downstream distance x as

共18兲

where T ⫽ (t ⫺ ts)w 0 /zi0. By this hypothesis, shoreline * fumigation has been regarded as the same as nocturnal inversion breakup fumigation, but the applicability of this hypothesis under realistic atmospheric conditions has not been verified. Taylor’s hypothesis has been recommended for application to the CBL when the mean wind is in the range between 1.2w ⱕ U ⱕ 6w (Willis * * and Deardorff 1976a, b), where the lower limit ensures the negligibility of streamwise diffusion and the upper limit ensures the neglect of shear effects. The mean mixed-layer wind (1.9 m s⫺1) in case se_u falls within this range. Willis and Deardorff (1976b) have also shown that Taylor’s hypothesis starts to break down for ␴U /U ⬎ 0.5. In case se_u, ␴U /U increases from 0.19 to 0.29 during the simulation time. Thus, the value of ␴U /U is within the range of condition where Taylor’s hypothesis was considered as applicable. Since case se_u has a nonzero mean transport wind, we can calculate the CWIC both as function of time t [Eqs. (7) and (8)] and distance x [Eqs. (15) and (16)]. By plotting the dimensionless CWICs as a function of time and distance (Fig. 15), it is shown that the CWICs calculated as a function of T differ from those as a function of X. The GLC reaches a value of 0.1 at times 8 and 12 for T and X, respectively. CWIC differences between the two scalings also appear at dimensionless height greater than 1 because of the vertical variation of wind within the mixed layer, the entrainment zone, and above the entrainment zone (Fig. 16). In addition, the decrease of the mean wind speed in the mixed layer with time (⳵U/⳵t ⫽ ⫺⳵uw/⳵z ⬍ 0; uw is the turbulent momentum flux in x direction) would produce a modified relationship between X and T than given by Eq. (18). Taylor’s hypothesis was further examined for the various source heights and using different normalization wind speeds. In Fig. 17, we adopted two normalization wind speeds; one is the mean boundary layer wind speed U0 at the time of particle release (thin solid lines) and the other is the mean boundary layer wind speed Ut as a function of time (thick solid lines). With zs /zi0 ⫽ 1.19, the near-surface dimensionless CWIC with T is higher than that with X normalized by either U0 or Ut (Fig. 17a). At zs /zi0 of 1.00 and 0.80, differences in CWIC with T and X are smaller than those for zs /zi0 ⫽ 1.19 (Figs. 17b, c). The large differences for zs /zi0 ⫽ 1.19 is due to the fact that the actual wind at the source height is larger than the normalized wind speed (see Fig. 16). With zs /zi0 ⫽ 1.00 and 0.80, CWICs with X normalized by Ut agree better with those with T, compared to those normalized by U0. The difference in near source height CWICs with X and T is also large for zs /zi0 ⫽ 1.19, compared to zs /zi0 ⫽ 1.00 and 0.80 (Figs.

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FIG. 16. Time evolution of vertical profile of U (m s⫺1) at time intervals of 3600 s (solid, dotted, dashed, dash–dot lines with increasing time).

6. Summary and conclusions

FIG. 15. Dimensionless CWIC (Cyzi0) fields computed as a function of (a) dimensionless time and (b) dimensionless distance for case se_u. (c) Near ground-level concentration (z/zi0 ⫽ 0.1) as a function of X (solid line, normalized by boundary layer wind speed U0 at the time of particle release) and as a function of T (dotted line). Plots are for the source height of zs /zi0 ⫽ 1.19 and plume spread of ␴z0 /zi0 ⫽ 0.036.

17d,e,f). When the source lies in the upper entrainment zone or above it (zs /zi0 ⫽ 1.09 to 1.25, not shown here), actual particle transport speed before being entrained is higher than mean boundary layer wind speed. The use of a smaller wind speed (U0 or Ut) rather than the actual transporting wind speed at the source height reduces dimensionless CWIC values following Eq. (16). When the source lies at and below zs /zi0 ⫽ 1.00, using Ut as a normalization wind speed makes Taylor’s translation hypothesis more appropriate.

This study shows that the Lagrangian modeling approach driven by LES-generated flow fields is capable of reproducing results from the water tank experiments of DW82 and HL96. The dependence of ground-level concentration (GLC) on the entrainment rate and the initial vertical thickness of plume is consistent with the findings from those laboratory experiments and a previous LES study by CL02. The near-surface crosswind-integrated concentration (CWIC) depends on whether the initial plume height lies within or above the entrainment zone. For initial plume heights within the entrainment zone, some particles originate within the turbulent layer and therefore behave like an elevated source within the CBL. Only for source heights above the entrainment zone does the dispersion behave like pure fumigation. The nearsurface CWIC distributions collapse to a universal curve once an appropriate starting time shift is applied. We recommend a proper starting time shift based on a significant fraction (5%) of the released particles reaching below 85% of the initial plume height. The differences between Lagrangian (LPDM) and Eulerian (EDM) approaches for modeling fumigation

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FIG. 17. Dimensionless CWIC (Cyzi0) near the surface and at the source height computed as a function of dimensionless time (dotted line) or distance (solid lines) with zs /zi0 ⫽ (a), (d) 1.19; (b), (e) 1.00; and (c), (f) 0.80 for case se_u. CWIC and X in thin solid line are normalized by boundary layer wind speed U0 at the time of particle release, and those in thick solid line are normalized by boundary layer wind speed Ut as a function of time.

are compared. Qualitatively both approaches based on the same LES flow field have a similar dependence of the CWICs on the entrainment rate, initial plume height, and initial plume vertical spread. The EDM tends to predict a higher overshoot of GLC than the LPDM. One aspect of the LPDM results is an undesired accumulation of particle concentration near the surface during the equilibrium state for the fast entraining CBL. Future improvement to the SGS velocity model is needed for rapidly growing boundary layer.

The translation of time to distance coordinates, or Taylor’s hypothesis, should be applied with caution for fumigation studies. When a mean boundary layer wind at the time of plume insertion is used for normalization, we find the distribution of CWIC as a function of dimensionless distance to be quite different from the CWIC distribution as a function of dimensionless time. When the sources lie above zs /zi0 ⫽ 1, height variation in the boundary layer winds leads to an earlier onset of fumigation when a time coordinate is used compared to that when a translated distance coordinate is used.

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Thus, when the duration of the fumigation episode is long (i.e., when some of the pollutants remain in or above the entrainment zone long after the onset of fumigation), the use of the mean boundary layer wind to translate time into distance is inappropriate if there is a big jump of wind speed across the entrainment zone. When the sources lie at or below zs /zi0 ⫽ 1, the use of mean boundary layer wind as a function of time make the application of Taylor’s translation hypothesis more appropriate. Acknowledgments. Ours thanks go to Drs. Luhar and Cai for giving us information about scaling of DW82 and definition of starting time of fumigation. We thank Edward (Ned) Patton and Peter Sullivan for their help with computing at NCAR. Helpful comments from G.-H. Kim are appreciated. This work was supported by the NCAR visitor program. NCAR is sponsored by the National Science Foundation. J. C. Weil has been supported by a grant from the U.S. Environmental Protection Agency’s Science to Achieve Results (STAR) program. Although this research has been funded partially by the U.S. EPA’s STAR program, it has not been subjected to any EPA review and does not necessarily reflect the views of the Agency, and no official endorsement should be inferred. REFERENCES Cai, X. M., and A. K. Luhar, 2002: Fumigation of pollutants in and above the entrainment zone into a growing convective boundary layer: A large-eddy simulation. Atmos. Environ., 36, 2997–3008. Deardorff, J. W., and G. E. Willis, 1982: Ground-level concentrations due to fumigation into an entraining mixed layer. Atmos. Environ., 16, 1159–1170. Du, S., 1997: Universality of the Lagrangian velocity structure function constant (C0) across different kinds of turbulence. Bound.-Layer Meteor., 83, 207–219. Hibberd, M. F., and A. K. Luhar, 1996: A laboratory study and improved PDF model of fumigation into a growing convective boundary layer. Atmos. Environ., 30, 3633–3649. Hurley, P., and W. Physick, 1991: A Lagrangian particle model of fumigation by breakdown of the nocturnal inversion. Atmos. Environ., 25A, 1313–1325. Klemp, J. B., and D. R. Durran, 1983: An upper boundary condition permitting internal gravity wave radiation in numerical mesoscale models. Mon. Wea. Rev., 111, 430–444. Lamb, R. G., 1978: A numerical simulation of dispersion from an

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elevated point source in the convective planetary boundary layer. Atmos. Environ., 12, 1297–1304. Luhar, A. K., and R. E. Britter, 1990: An application of Lagrangian stochastic modeling to dispersion during shoreline fumigation. Atmos. Environ., 24A, 871–881. Lyons, W. A., and H. S. Cole, 1973: Fumigation and trapping on the shores of Lake Michigan during stable onshore flow. J. Appl. Meteor., 12, 494–510. Meroney, R. N., J. E. Cermak, and B. T. Yang, 1975: Modelling of atmospheric transport and fumigation at shoreline site. Bound.-Layer Meteor., 9, 69–90. Misra, P. K., 1980a: Dispersion from tall stacks into a shoreline environment. Atmos. Environ., 14, 397–400. ——, 1980b: Verification of a shore-line dispersion model for continuous fumigation. Bound.-Layer Meteor., 19, 501–507. Moeng, C.-H., 1984: A large-eddy simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci., 41, 2052–2062. Sullivan, P. P., J. C. McWilliams, and C.-H. Moeng, 1994: A subgrid-scale model for large-eddy simulation of planetary boundary-layer flows. Bound.-Layer Meteor., 71, 247–276. ——, C.-H. Moeng, B. Stevens, D. H. Lenschow, and S. D. Mayor, 1998: Structure of the entrainment zone capping the convective atmospheric boundary layer. J. Atmos. Sci., 55, 3042– 3064. Taylor, G. I., 1921: Diffusion by continuous movements. Proc. London Math. Soc., 20, 196–212. Thomson, D. J., 1987: Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech., 180, 529–556. van Dop, H., R. Steenkist, and E. T. M. Nieustadt, 1979: Revised estimates for continuous shoreline fumigation. J. Appl. Meteor., 18, 133–137. Weil, J. C., 1992: Dispersion in a rapidly evolving convective boundary layer. Preprints, 10th Symp. on Turbulence and Diffusion, Portland, OR, Amer. Meteor. Soc., 204–207. ——, P. P. Sullivan, and C.-H. Moeng, 2000: Lagrangian modeling of dispersion in the convective boundary layer over a range of stability. Preprints, 11th Joint Conf. on the Applications of Air Pollution Meteorology with the A&WMA, Long Beach, CA, Amer. Meteor. Soc., 30–34. ——, ——, and ——, 2004: The use of large-eddy simulations in Lagrangian particle dispersion models. J. Atmos. Sci., 61, 2877–2887. Willis, G. E., and J. W. Deardorff, 1976a: A laboratory model of diffusion into the convective planetary boundary layer. Quart. J. Roy. Meteor. Soc., 102, 427–445. ——, and ——, 1976b: On the use of Taylor’s translation hypothesis for diffusion in the mixed layer. Quart. J. Roy. Meteor. Soc., 102, 817–822. ——, and ——, 1978: A laboratory study of dispersion from an elevated source within a modeled convective planetary boundary layer. Atmos. Environ., 12, 1305–1311. ——, and ——, 1981: A laboratory study of dispersion from a source in the middle of the convectively mixed layer. Atmos. Environ., 15, 109–117.