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Douglas G. Fox


The motions resulting from the sudden release of a fixed amount of buoyancy in an incompressible fluid are simulated. Solutions are allowed to reach a steady state in the finite computed volume by the introduction of a dynamical stretching of the coordinate system. Fully nonlinear, transformed, and finite-differenced Navier-Stokes equations are integrated in time over a three-dimensional grid. It is shown that a steady-state solution to the transformed equations is equivalent to a self-preserving solution in real space. Physically realistic results are presented for a range of Reynolds numbers between 10 and 100. In a strongly diffusive regime the simulation agrees with an existing theoretical solution. Reynolds numbers of order 50 are sufficient to reproduce many of the features of laboratory experiments.

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Uncertainty in Air Quality Modeling

A Summary of the AMS Workshop on Quantifying and Communicating Model Uncertainty, Woods Hole, Mass., September 1982

Douglas G . Fox

Under the direction of the AMS Steering Committee for the EPA Cooperative Agreement on Air Quality Modeling, a small group of scientists convened to consider the question of uncertainty in air quality modeling. Because the group was particularly concerned with the regulatory use of models, its discussion focused on modeling tall stack, point source emissions.

The group agreed that air quality model results should be viewed as containing both reducible error and inherent uncertainty. Reducible error results from improper or inadequate meteorological and air quality data inputs, and from inadequacies in the models. Inherent uncertainty results from the basic stochastic nature of the turbulent atmospheric motions that are responsible for transport and diffusion of released materials. Modelers should acknowledge that all their predictions to date contain some associated uncertainty and strive also to quantify uncertainty.

How can the uncertainty be quantified? There was no consensus from the group as to precisely how uncertainty should be calculated. One subgroup, which addressed statistical procedures, suggested that uncertainty information could be obtained from comparisons of observations and predictions. Following recommendations from a previous AMS workshop on performance evaluation (Fox, 1981), the subgroup suggested construction of probability distribution functions from the differences between observations and predictions. Further, they recommended that relatively new computer-intensive statistical procedures be considered to improve the quality of uncertainty estimates for the extreme value statistics of interest in regulatory applications.

A second subgroup, which addressed the basic nature of uncertainty in a stochastic system, also recommended that uncertainty be quantified by consideration of the differences between observations and predictions. They suggested that the average of the difference squared was appropriate to isolate the inherent uncertainty that arises because individual realizations of the concentrations are different from the average concentrations. The average square difference allows quantification of this fact. Viewed in this framework, uncertainty is related to the variance of concentration fluctuations and the integral time scale of the turbulent flow.

How can the uncertainty be communicated to decision makers? There was concern expressed by a third subgroup that non-technical people would have little understanding of quantified uncertainty. This places an increased burden on modelers to ensure that their efforts are useful. Similarly, decision makers will need to educate themselves and accept the challenge of decision making with quantified uncertainty.

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Judging Air Quality Model Performance

A Summary of the AMS Workshop on Dispersion Model Performance, Woods Hole, Mass., 8–11 September 1980

Douglas G. Fox

Under the direction of the AMS Steering Committee for the EPA Cooperative Agreement on air quality modeling, a small group of scientists was convened to review and recommend procedures to evaluate the performance of air quality models. Particular attention was paid to the operational use of models in regulatory settings.

A number of general recommendations resulted from the workshop. The usefulness of models to aid decision makers in air quality management was reiterated. Concerns about the absolute, rather than the statistical, nature of air quality standards were raised, with a recommendation to reformulate standards accordingly. Model performance evaluation was suggested on the basis of differences between observed and predicted concentrations. The bias (average), the variance (noise), and the gross variability (gross error) of such differences were the recommended performance measures. In addition, correlation measures calculated in time, space, and jointly between observation and predictions were recommended. These measures and suggestions on how to apply them are outlined. Some qualification on the use of these measures for application to point sources using limited data sets was made.

The workshop did not recommend any specific numerical standards for acceptable model performance; there was insufficient information to do this. Rather, considerations for evaluating model performance using statistically constructed confidence intervals and comparison against EPA's currently used models were suggested.

Finally, areas in need of further research were identified. These include development and description of performance measures; application, especially to point source models; analysis of meteorological and air quality data, both existing and newly developed; and evaluation of diffusion modeling techniques.

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Morris H. McCutchan and Douglas G. Fox


Meteorological patterns around an isolated, conical mountain were analyzed for 2 consecutive years July through October. Nine remote automatic weather stations were deployed on nearly uniform slopes, all four aspects, and over an elevational range of from 2700 to 3300 m. The data were sorted according to light and strong winds, both night and day, in order to highlight influences due to airflow, surface heating and topography. Wind speeds grater than 5 m s−1 negate any slope, elevation or aspect differences present at low wind speeds. Detailed statistical analysis illustrated that the effects of aspect were much more pronounced than the effects of elevation on the wind components and, to a lesser degree on temperature.

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Morris H. McCutchan, Douglas G. Fox, and R. William Furman

The San Antonio Mountain Experiment (SAMEX) involves a 3325 m, conically shaped, isolated mountain in north-central New Mexico where hourly observations of temperature, relative humidity, wind speed, wind direction, and precipitation are being taken at nine locations over a three- to five-year period that began in 1980. The experiment is designed to isolate the effect of topography on these meteorological variables by using a geometric configuration sufficiently simple to lead to generalized results. One remote automatic weather station (RAWS) is located at the peak (3322 m); four are located at midslope (3033 m) on southwest, southeast, northeast, and northwest aspects; and four are at the base (2743 m) on southwest, southeast, northeast, and northwest aspects. The surface observations are supplemented by rawinsonde, pibal, tethersonde, and constant-level balloon observations at selected times during each year. The unique set of meteorological data collected in the experiment will be used to 1) determine the effect of elevation and aspect on the meteorological variables; 2) compare the temperature, humidity, and wind components on the mountain with observations and/or predictions of these variables in the free air nearby; and 3) validate temperature, humidity, and wind models in complex terrain.

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Douglas G. Fox and James E. Fairobent

The results of a 22-member expert panel on dispersion modeling, which was convened by the National Commission on Air Quality in 1979, are reviewed. The panel affirmed the validity of using models in support of air quality regulations. It also recognized the need to convey some of the uncertainty in modeling and recommended technical details for the commission to consider.

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