## Final Report: Buoyant Plume Dispersal in the Convective Boundary Layer: Analysis of Experimental Data and Lagrangian Modeling

**EPA Grant Number:**R826160

**Title:**Buoyant Plume Dispersal in the Convective Boundary Layer: Analysis of Experimental Data and Lagrangian Modeling

**Investigators:**Weil, Jeffrey C.

**Institution:**

**University of Colorado at Boulder**

**EPA Project Officer:**Shapiro, Paul

**Project Period:**February 3, 1998 through February 2, 2001 (Extended to February 2, 2003)

**Project Amount:**$244,000

**RFA:**Exploratory Research - Physics (1997) RFA Text | Recipients Lists

**Research Category:**Water , Land and Waste Management , Air , Engineering and Environmental Chemistry

### Objective:

The overall goal of this research program was to improve our understanding and predictive capability of dispersion in the convective boundary layer (CBL) with emphasis on highly buoyant plumes. The three key objectives were to:

1. Increase our knowledge of highly buoyant plumes that loft at the CBL top and resist downward mixing. The main goal was to develop and evaluate a gravity current model for the lofting plume.

2. Enhance the existing Lagrangian dispersion model by including the lofting model and a better treatment of the plume interaction with the inversion capping the CBL.

3. Develop further a probability density function (PDF) model for dispersion by incorporating the new lofting model. The PDF model is simpler in form than the Lagrangian model, which makes it of more immediate use in air quality applications.

### Summary/Accomplishments (Outputs/Outcomes):

The model developments, results, and their significance are presented below with a focus on the first two objectives. The Lagrangian model is a statistical or stochastic approach in which one follows plume "particles" given the environmental turbulence and source conditions. Plume material that reaches the surface either originates from the rising plume within the CBL mixed layer or from plume material lofting at the CBL top. There were two development efforts for these models, which we labeled as "initial" and "improved," and they are summarized in order.

The degree of plume buoyancy is characterized by the dimensionless buoyancy
flux *F*_{*}=*Fb* / (*Uw*^{2}_{*}*zi*),
where *Fb* is the source buoyancy flux, U is
the mean wind speed, *w*_{*} is the convective velocity scale,
and *zi* is the CBL depth. In the
CBL, the
rms turbulence velocities and the eddy length scale are proportional to *w*_{*}
and *zi*, respectively. The source buoyancy can
be divided into three ranges: low buoyancy for *F*_{*} 0.05,
where the plume behavior is similar to that for a nonbuoyant release; moderate
buoyancy for 0.05 *F*_{*} 0.1;
and high buoyancy for *F*_{*} > 0.1, which corresponds to
plume lofting.

__Initial Lofting and Lagrangian Models.__

The purpose of the lofting model is to predict the lateral and vertical plume dimensions, the plume density deficit, and most importantly the rate of removal or detrainment of plume material by the CBL turbulence. The removal rate determines how rapidly plume material is brought into the CBL and eventually to the ground. The initial model represented a change in our conceptual picture of a buoyant plume lofting at the CBL top and was based on analysis of laboratory dispersion data. In the new approach, the elevated plume was assumed to be embedded vertically within the entrainment layer that occupies the top 20 percent of the CBL. The plume was assumed to have a constant vertical depth, to spread laterally, and to lose buoyancy and pollutants due to entrainment by the CBL turbulence below it.

For the lateral spread, we adopted a gravity current model, which predicts
the horizontal advancement of one fluid into another due to the density difference
between them. For a plume with a conserved buoyancy flux, an equation for the
lateral spread led to a simple power law dependence of spread *y* on downwind
distance or travel time *t: y (Fb/U) ^{1/3}t^{2/3}*. For a lofting plume
that was slowly eroded by the CBL turbulence, the buoyancy loss resulted in
a slower
growth rate far downstream. Comparisons between the modeled dispersion and
laboratory data showed that the predicted spread followed the data trends quite
well, especially for cases with

*F*

_{*}0.2. The results with the buoyancy loss included were better than those with a constant buoyancy flux.

An existing Lagrangian dispersion model (Weil, 1994, 1998) was modified to: (1) predict dispersion by the motion of buoyant particles rather than by a "meandering" plume used earlier, (2) account for environmental turbulence effects on plumes through detrainment of plume material by the ambient turbulence, and (3) incorporate the new gravity current/lofting model. The treatment of the plume by a large number of buoyant particles was intended to improve the modeling of the plume interaction with the elevated inversion.

In the new model (Weil, 2000), particles were tracked by superimposing the plume rise velocity and the local ambient turbulence velocity, which was treated stochastically. The variation of the plume properties with time was obtained from equations for mass, momentum, and energy conservation, but included the detrainment of those quantities. Detrainment was parameterized similarly to entrainment except that the detrained flux had the opposite sign. The parameterizations differed for particles originating from the rising plume in the CBL and the lofted plume at the CBL top. A particle was assumed to behave as "passive" material—without source buoyancy or momentum—once it had been removed from the plume.

The Lagrangian model was evaluated with dispersion data from convection tank
experiments (Weil, et al., 2002), which covered cases with *F*_{*} = 0 to 0.4. Comparisons
of
the predicted mean plume height, vertical dispersion, and surface values
of the crosswind-integrated concentration (CWIC) agreed well with the data
both as a function of distance and plume buoyancy. One of the improvements
of the new model over the earlier approach was the shape of the vertical
CWIC profile. The profile showed a peak CWIC near the CBL top with the peak
maintained over a considerable distance downstream in agreement with the
laboratory data. Thus, the lofted plume acted as an elevated reservoir of
pollutants. The vertical profile of CWIC in the mixed layer also was in approximate
agreement with the data for moderate distances. However, for large distances,
the model overestimated the mixed layer concentrations while underestimating
the concentrations in the entrainment zone aloft. This was attributed to
a deficiency in the entrainment velocity formulation in the lofting plume
(Weil, 2000).

__Improved Lofting and Lagrangian Models__

The "initial" model deficiencies resulted in lofted plume material being mixed too rapidly into the CBL. Further analysis revealed a conceptual problem in a lofted plume with a constant vertical depth, although this was suggested by laboratory data. The apparent constant depth was caused by vertical meandering of the instantaneous (lofted) plume by fluctuations in the CBL height. The analysis prompted a reexamination of the lofting model and a consideration of the effects of CBL height fluctuations on the mean concentration profiles.

In the "improved" lofting model, both the vertical (*rz*)
and lateral (*ry*) plume dimensions varied with
downstream distance or travel time (i.e.,
*rz* was not a constant). An entrainment formulation
was adopted to model the plume mass and buoyancy fluxes; the change of the
lateral plume dimension with
time was given by the gravity current model. Thus, there were three equations
for the three variables *ry*, *rz*,
and the plume density deficit .
A partial solution showed that: (1) the vertical half-width *rz* decreased
with time, (2) the plume density deficit remained constant, and (3) the species
concentration
also was constant. The second point is quite important because the local density
deficit enables the plume to remain aloft and resist detrainment. Thus, for
a constant rather than a decreasing as in the initial model, more of the
plume should remain aloft for a longer time, consistent with the laboratory
data.

A numerical solution to the equations showed that for a zero entrainment velocity,
*ry* *t ^{2/3}* consistent
with the “initial” model, but

*rz*decreased as

*t*. With detrainment included, the

^{-2/3}*ry*followed the

*t*behavior for some time and then increased less rapidly. Likewise, the

^{2/3}*rz*varied as

*t*but then decreased more rapidly as material was removed from the lofted plume by the CBL turbulence. The reduction in

^{-2/3}*rz*was consistent with visual observations of plumes in laboratory experiments (Snyder, et al., 2002; Weil, et al., 2002). The predicted fraction of material remaining in the lofted plume was computed as a function of time or distance and found to agree reasonably well with measured fractions for most distances.

A statistical model was formulated to predict the effect of random fluctuations
in the CBL height on vertical profiles of the mean concentration. The model
consisted of: (1) an "instantaneous" concentration profile within
the CBL relative to the instantaneous CBL height *zi*, and (2) a statistical
distribution or PDF of the CBL height. The model was tested for predictions
of the mean temperature profiles in the CBL. The instantaneous temperature
was assumed to be uniform below *zi* and to have (1) a linearly increasing temperature
with height above *zi*, or (2) a temperature jump at *zi* followed by a linearly
increasing value. The PDF of *zi* was taken to be a Gaussian, which was parameterized
by the mean and standard deviation of *zi*.

The main effect of the fluctuations on the mean profile was found within plus or minus three standard deviations of the mean height. Beyond those ranges, the profile was either uniform below or linear with the height above. The predicted profiles agreed well with results obtained in two other studies: (1) convection tank measurements of mean temperature in a simulated CBL, and (2) large eddy simulations of the mean temperature profile. The model and results are being prepared for a research note (Weil, 2004a).

The improved Lagrangian dispersion model focused on the inclusion of the new
lofting model and an initial assessment of it on the dispersion predictions.
Preliminary runs were made for two of the source conditions explored earlier:
*F*_{*} = 0.1 and 0.4. The initial results showed promising
behavior. The main features were: (1) an elevated peak in the vertical profile
of the concentration at
the top of the CBL (i.e., at *zi*), and (2) a
nearly well-mixed concentration distribution for heights below about 0.7*zi*.
The predicted peak concentration at the inversion was maintained over a large
distance range, in agreement with
the laboratory data. Thus, the above features, especially the first, corrected
deficiencies in the initial Lagrangian dispersion model. The "improved" models
with the effects of the CBL height fluctuations included will be described
in a forthcoming paper (Weil, 2004b).

__PDF Dispersion Model__

The PDF model, including buoyancy effects, was developed under an earlier program and demonstrated generally good agreement with field observations. A somewhat deficient case was found for highly buoyant lofting plumes. However, due to the extensive efforts required for the development of the "improved" lofting and Lagrangian models, there was insufficient time to include the new lofting model in the PDF approach. This will be pursued in the future because an improved PDF model would be very worthwhile for applied dispersion modeling and regulatory applications.

__Significance and Use of Results__

The problem addressed above has been a longstanding scientific issue in dispersion modeling—how to deal with lofting plumes, which do not fit into the standard frameworks of most air quality models. The lofting situation also may be problematic for point sources in or near urban areas. Plumes from such sources can become trapped in shallow nocturnal boundary layers, driven by anthropogenic surface heating, and produce high surface concentrations. The model results fill an important void for the treatment of buoyant plumes interacting with elevated inversions in the CBL. The improved lofting model together with the Lagrangian dispersion model could be used by regulatory agencies (U.S. Environmental Protection Agency and state agencies) for dispersion predictions. Principal uses would be for determining the source emission limits for the protection of human health and welfare.

### Journal Articles:

No journal articles submitted with this report: View all 3 publications for this project### Supplemental Keywords:

*atmospheric dispersion, Lagrangian stochastic modeling, mean and fluctuating concentrations, peak concentration estimates, buoyant plume modeling, air, scientific discipline, atmospheric sciences, ecological risk assessment, engineering, chemistry, physics, Lagrangian modeling, air pollution modeling system, ambient emissions, atmospheric models, buoyant plume dispersal, convective boundary layers, ecological risk, hazardous air pollutants, mean and fugitive concentrations, probability density function air, ambient air, atmosphere, buoyant plumes, chemicals, exposure, mean concentrations, modeling, risk assessment, toxics.*, Scientific Discipline, Air, Physics, Atmospheric Sciences, Ecological Risk Assessment, Engineering, Chemistry, & Physics, risk assessment, hazardous air pollutants, air pollution modeling system, ambient emissions, buoyant plume dispersal, Langraian modeling, atmospheric dispersion, probablility density function, ecological risk, mean and fugitive concentrations, convective boundry layers