RiskWare  On-line Reference Manual

Release Level 1.9
Release Date 2000 06

Revision Level 1.1

Gaussian Puff Model

The Gaussian Puff Model in RiskWare is derived from the USEPA model INPUFF 2.4 (Petersen and Lavadas, 1986).

INPUFF is a Gaussian INtegrated PUFF model. It is designed to simulate dispersion from semi-instantaneous or continuous point sources over a spatially and temporally variable wind field. The algorithm is based upon Gaussian puff assumptions including a vertically uniform wind direction field and no chemical reactions.

The Gaussian puff diffusion equation is used to compute the contribution to the concentration at domain each grid cell from each puff every time step. Computations in INPUFF can be made for a single or multiple point sources.

The model has a a wide range of applications. The implied modeling scale is from tens of meters to tens of kilometers. The model is capable of addressing the accidental release of a substance over several minutes, or of modeling the more typical continuous plume from a stack.

In the default mode, the model assumes a homogeneous wind field. However, if a DEM is available for the domain and a wind field can be generated with DWM Diagnostic Wind Model, the wind field interpolated for each meteorological period will be used.

Three dispersion algorithms are utilized within INPUFF for dispersion downwind of the source. These include Pasquill's scheme as discussed by Turner (1970) and a dispersion algorithm discussed by Irwin (1983), which is a synthesis of Draxler's (1976) and Cramer's (1976) ideas. The third dispersion scheme is used for long travel times in which the growth of the puff becomes proportional to the square root of travel time. Removal is incorporated through deposition and gravitational settling algorithms.

Features of the INPUFF computer code include:

  • Optional stack downwash;
  • Optional buoyancy induced dispersion;
  • Wind speed extrapolated to release height;
  • Temporally variable source characteristics;
  • Temporally and spatially variable wind field use;
  • Consideration of terrain effects through externally generated wind field;
  • Consideration of moving source;
  • Alternative subroutines for estimating dispersion;
  • Optional alternative subroutine for estimating plume rise;
  • Removal through gravitational settling and deposition.

RiskWare will attempt to match model domain and DEM wherever possible. If a puff travels outside the modeling region, it is deleted from further consideration. If it travels outside the meteorological grid, but is still within the modeling region, the last wind experienced by the puff is used to advect it further.

Data Requirements

Information required on the source includes the following:
  • Location (km),
  • Emission rate (g/sec), for each simulation or meteorological period;
  • Physical stack height (m),
  • Stack gas temperature (K),
  • Stack diameter (m),
  • Stack gas velocity (m/sec),
  • Stack gas volume flow (m3/sec), and
  • Initial sr and sz (m),
  • Deposition and gravitational settling velocities (cm/sec).
Also the direction and speed of the source, if it is moving, must be provided as input.

The data for the source are automatically loaded through the RiskWare interface, set by the user through the embedded expert systems editing functions, are calculated with the release model.

The meteorological data needed for the computations are as follows:

  • Wind direction (deg),
  • Wind speed (m/sec),
  • Mixing height (m),
  • Stability class (dimensionless),
  • Standard deviation of elevation angle (radians),
  • Standard deviation of azimuth angle (radians),
  • Ambient air temperature (K), and
  • Anemometer height (m).

The meteorological information can vary for each meteorological time period. If dispersion is characterized by the on-site scheme, then the standard deviations of the azimuth and elevation angles are required.

Although INPUFF has several advantages over its continuous plume counterpart, it still retains several limitations, including:

  • Wind direction constant with height,
  • No consideration of chemical reactions, and
  • No explicit treatment of complex terrain
other then through the use of an externally generated distributed wind field.


In Gaussian-puff algorithms, source emissions are treated as a series of puffs emitted into the atmosphere. Constant conditions of wind and atmospheric stability are assumed during a time interval. The diffusion parameters are functions of travel time. During each time step, the puff centers are determined by the trajectory and the in-puff distributions are assumed to be Gaussian. Thus, each puff has a center and a volume which are determined separately by the mean wind, atmospheric stability, and travel time.


Plume rise is calculated using the methods of Briggs. Although plume rise from point sources is usually dominated by buoyancy, plume rise due to momentum is also considered. Building downwash, and gradual plume rise are not treated by INPUFF.

Stack-tip downwash (optional) can be considered using the methods of Briggs. In such an analysis, a height increment is deducted from the physical stack height before momentum or buoyancy rise is determined. Use of this option primarily affects computations from stacks having small ratios of exit velocity to wind speed.


Three dispersion algorithms are used within INPUFF for dispersion downwind of the source:

  • P-G scheme as discussed by Turner (1970),
  • On-site scheme formulated by Irwin (1983), and
  • Long travel time scheme.

The user has the option of choosing either the P-G or the on-site algorithm (for short travel time dispersion) and specifying when the long travel time dispersion parameters are to be implemented.

Dispersion downwind of a source, as characterized by the P-G scheme, is a function of stability class and downwind distance. Stability categories are commonly specified in terms of wind speed and solar radiation. The on-site dispersion algorithm is a synthesis of Draxler's (1976) and Cramer's (1976) ideas and requires specification of the variances of the vertical and lateral wind directions. The third dispersion scheme is used in conjunction with the other two and is for long travel times in which the growth of the puff is proportional to the square root of time.


Rao (1982) gave analytical solutions of a gradient-transfer model for dry deposition pollutants from a plume. His solutions treat gravitational settling and dry deposition of pollutants in a physically realistic manner, and are subject to the same basic assumptions and limitations associated with Gaussian plume models. His equations for deposition and settling were incorporated in several EPA air quality models including PALDS (Rao,1982). The equations used in INPUFF are the same as those used in PALDS except they are cast in terms of travel time instead of wind speed and downwind distance.


Three dispersion algorithms are incorporated within the model to account for initial dispersion, short travel time dispersion, and long travel time dispersion. The initial dispersion algorithm handles the finite size of the release through the use of initial dispersion parameters. Once the puff leaves the source its growth is determined by the short travel time dispersion algorithm. This algorithm has two schemes: the Pasquill-Gifford scheme which characterizes dispersion as a function of downwind distance and the on-site scheme which characterizes dispersion as a function of travel time. For long travel time, a dispersion algorithm that allows the puff to grow as a function of the square root of time is used.

Initial Dispersion

The initial dispersion of the plume at the source is modeled by specifying the initial horizontal and vertical dispersion parameters,sro and szo. For tall stacks these parameters, generally, have little influence on downwind concentrations. However, if the source is large enough or close enough to the ground, then initial size is important in determining ground level concentrations near the source. For a source near the ground, the initial horizontal dispersion can be calculated by dividing the initial horizontal dimension of the source by 4.3, and the initial vertical dispersion parameter is derived by dividing the initial height of the source by 2.15. This method of accounting for the initial size of near ground level release gives reasonable concentration estimates at downwind distances greater than about five times the initial horizontal dimension of the source.

Buoyancy Induced Dispersion

The buoyancy-induced dispersion feature is offered because emitted plumes undergo a certain amount of growth during the plume rise phase, due to the turbulent motions associated with the conditions of plume release and the turbulent entrainment of ambient air. Pasquill (1976) suggests that this induced dispersion, szo, can be approximated by DH/3.5, and the effective dispersion can be determined by adding variances, i.e.,

sze = SQRT(szo2 + sz2)

where sze is the effective dispersion and sz is the dispersion due to ambient turbulence levels. At the distance of final rise and beyond, szo is a constant using DH of final rise. At distances closer to the source, the DH used to determine szo is itself determined using gradual rise.

Since in the initial growth phases of release the plume is nearly symmetrical about its centerline, buoyancy-induced dispersion in the horizontal direction equal to that in the vertical is used, syo= DH/3.5. This expression is combined with that for dispersion due to ambient turbulence in the same manner as is shown above for the vertical.

In general, buoyancy-induced dispersion will have little effect upon maximum concentrations unless the stack height is small compared to the plume rise. Also, it is most effective in simulating concentrations near plume centerlines close to the source, where treating the emission as a point source confines the plume to a volume much smaller than the actual plume. It should be clarified here that the buoyancy-induced disperion close to the source is calculated using the gradual rise in INPUFF, even if the gradual plume rise option is not being used to determine the effective plume height.

Short Travel Time Dispersion

Dispersion downwind of the source can be characterized by the P-G scheme, which is a function of stability class and downwind distance, or by the on-site scheme, which is a function of travel time.

Pasquill-Gifford Scheme

The P-G values that appear as graphs in Turner (1970) are used in the model. However, for neutral atmospheric conditions two dispersion curves as suggested by Pasquill (1961) are incorporated into the model. Dispersion curves D1 and D2 are appropriate for adiabatic and subadiabatic conditions, respectively. The D2 curve is used in Turner (1970) for neutral conditions. From a practical point of view, since temperature soundings may not be available we refer to the D1 and D2 curves as D-day and D-night. P-G stability classes are numerical inputs to the puff model. Stability classes A through D-day are specified by 1-4, and classes D-night through F are specified by 5-7, respectively.

On-site Meteorology Scheme

The sigma-curves of the P-G scheme above are based on data of near-ground level releases and short-range dispersion studies. These data are used to extrapolate the P-G curves to high release heights and far receptor distances. In view of this, INPUFF has an option of using on-site meteorological data to estimate dispersion. This scheme is a result of the recommendations of the American Meteorological Society's workshop on stability classification schemes and sigma curves (Hanna et al., 1977). Irwin (1983) proposed characterizing sy and sz in a manner similar to Cramer (1976) and Draxler (1976). The standard deviation of the crosswind concentration distribution, sy, is

sy = svtfy   (7)
where sv is the standard deviation of the horizontal crosswind component of the wind, t is the downstream travel time of the pollutant, and fy is a nondimensional function of travel time. The standard deviation of the vertical concentration distribution, sz, for an elevated source, when sz is less than the source height, is
sz = swtfz   (8)
where sw the standard deviation of the vertical component of the wind, and fzis a nondimensional function, primarily dependent upon travel time. The nondimensional functions fy and fz were characterized by Irwin (1983).

Besides the P-G stability class, the scheme requires sv and sw, which are assumed to be typical of conditions at final plume height. For small angles, sv = sau and sw =seu where u is the wind speed at measurement height and sa and se are the standard deviations of the horizontal and vertical wind angle, respectively. The puff model requires sa and se as data input and computes sv and sw.

Long Travel Time Dispersion

That the dispersion parameters used in INPUFF satisfy the diffusion theory developed by Taylor (1921) is desirable. Taylor showed that while the growth of the puff is linear with time near the source, the growth becomes proportional to the square root of time at large distances. In the model, after the puff has attained a specified horizontal dimension, the algorithm automatically goes to a long travel time growth rate proportional to the square root of time. The size of the puff at that time is specified by the user. For example, the user may decide that when sr for the puff is greater than 1000 meters the long travel time dispersion parameters should by utilized. A very large SYMAX value results in the long travel time code not being executed.


Depending on the stack height, plume rise, and height of the mixing layer, the puffs can be above or below the mixed depth layer, L. If the puffs are above L then there are two cases that govern their growth. Initially the puffs are allowed to grow according to the P-G, F curve, or if the on-site scheme is used, the puffs are restricted to a vertical growth rate characterized by sw=0.01m/sec. After the puffs attain a given size of sr (not actual puff size) specified by the user, the growth rate is specified by the Mt.

When the puffs are below L, then there are four cases that must be considered. Cases one and two are puffs which are not well mixed vertically and whose growth rates are characterized by the short travel time sigmas or by Mt. Cases three and four are puffs that are well mixed vertically and whose growth for sr is for short travel times or according to Mt. During the modeling simulation, every puff is given a key to indicate whether it is above or below L and whether its growth rate is characterized by the short travel time sigmas or by Mt.

In the modeling design, puffs are allowed to change their dispersion keys. When the height of L becomes greater than the puff height, the puffs are allowed to grow at the rate characterized by surface measurements. Normally this is a neutral or unstable situation. This transition period is likely to occur in the morning hours. In the afternoon, despite the decay of active mixing, a puff remains well mixed through the maximum mixing lid as shown in Figure 2. The maximum height of L is stored for each puff and is never allowed to decrease. This method assures that concentration does not increase with downwind distance or travel time, so as to violate the second law of thermodynamics.


As discussed earlier, short travel time dispersion can be characterized by two schemes, the P-G scheme and the on-site scheme. The P-G scheme uses the empirical P-G curves and stability classification to estimate dispersion coefficients (Turner, 1970), whereas the on-site scheme relates diffusion directly to turbulence. If on-site meteorological data are not available, only the widely used P-G scheme can be adopted. If on-site meteorological data are available, either scheme can be used.

INPUFF's on-site scheme adopts Irwin's algorithm (1983) in characterizing sy and sz. This scheme essentially requires information on the standard deviations of horizontal (sa) and vertical (se) wind fluctuations and wind speed at measurement height. Stability is classified as stable or unstable from the near-surface data for temperature difference, Richardson Number, or stability parameter.


The analytical solutions for atmospheric concentration of a gaseous or suspended particulate pollutant, incorporating dry deposition and gravitational settling were given by Rao (1982). That document provides a review of deposition models and the details of the derivation of the equations used in INPUFF. The equations used in INPUFF reduce to the Gaussian puff equations for Vd and W = 0.


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The Theory of Homogeneous Turbulence. Cambridge University Press, London.
Briggs, G. A. 1969.
Plume Rise. USAEC Critical Review Series. TID-25075, National Technical Information Service, Springfield, VA. 81 pp.
Briggs, G. A. 1971.
Some Recent Analyses of Plume Rise Observation. In: Proceedings of the Second International Clean Air Congress, H. M. Englund and W. T. Beery, eds., Academic Press, New York. pp. 1029-1032.
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Diffusion Estimation for Small Emissions. NOAA Atmospheric Turbulence and Diffusion Laboratory, Contribution File No. (Draft) 79. Oak Ridge, TN. 59 pp.
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Plume Rise Predictions. In: Lectures on Air Pollution and Environmental Impact Analysis, D. A. Haugen, ed. Am. Meteorol. Soc., Boston, MA. pp. 59-111.
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