A comparison of numerical pseudodiffusion and atmospheric diffusion

Atmospheric Environment Vol. 19, No. 7, pp. 1065-1068, 1985

0004-6981/85 $3.00 + 0.00 © 1985 Pergamon Press Ltd.

Printed in Great Britain.

A COMPARISON OF NUMERICAL PSEUDODIFFUSION AND ATMOSPHERIC DIFFUSION C. M. SHEIH Environmental Research Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. and F. L. LUDWIG Atmospheric Science Center, SRI International, Menlo Park, CA 94025, U.S.A. (First received 13 April 1984 and in finalJbrm 26 November 1984)

Abstract--Some numerical pseudodiffusions reported in the literature were compared with atmospheric diffusion for urban to regional scale modeling. For grid sizes of 0.5 and 5 km, such as are used in urban and mesoscale modeling, most of the numerical schemes produced numerical diffusion smaller than natural diffusion. However, for 50 km grid sizes like those used in regional models, about half of the numerical schemes produced artificial diffusion larger than natural diffusion. Key word index: Pseudodiffusion, numerical diffusion, diffusion, atmospheric diffusion.

INTRODUCTION

It is known that finite-difference approximations of the pollutant transport equation introduce artificial numerical diffusion or pseudodiffusion which can be quite large and might significantly alter the calculated distribution of pollutant concentration. Although the importance of the problem has long been recognized (e.g. Molenkamp, 1968; Sheih, 1978), studies of pseudodiffusion have been comparing the results of various numerical methods (e.g. Long and Pepper, 1976) rather than evaluating the relative importance of pseudodiffusion and natural diffusion. Obviously, assessments of numerical errors with respect to that of natural diffusion will have great practical value. This paper examines the relative importance of natural horizontal diffusion, as computed from a semiempirical formula, and the corresponding numerical pseudodiffusion reported by many investigators.

EVALUATION OF NUMERICAL

PSEUDODIFFUSION

Cp(to)'

Ra

O'2(to ) o.2 (t) ,

(2)

where a(t) isdispersioncoefficientat time t.The initial dispersion coefficienta(to) is simply the initialhalf width of the pollutant puff. The following semiempiricalformula reported by Gifford (1982)isused in the present study to estimatethe horizontaldispersion coefficients, a2(t) = 2 K t + (Vo/fl) 2 (1 - exp( - f i t ) ) 2

Numerical experiments testing the effects of the pseudodiffusion produced by a numerical method generally employ a conical concentration pattern (or sometimes a mound with a cosine profile) in a field of solid rotation (e.g. Molenkamp, 1968). Since these studies normally report the ratio of peak concentrations at the final and the initial time steps, this ratio has been chosen as a basis for evaluation. The ratio or the attenuation factor can be expressed by Rn = C p ( t )

where R n is the numerical attenuation factor, C_(t) is • , P the peak concentration at time t and t o is the mitial time. To compute the attenuation factor for atmospheric diffusion, it is more convenient to use dispersion coefficients because they are easier to obtain and more widely reported than other parameters. In a twodimensional field of isotropic turbulence, the ratio of the final to the initial peak concentration is equivalent to the reciprocal of the ratio of the squares of the corresponding dispersion coefficients. Thus, the attenuation factor for atmospheric diffusion can be expressed by

(1)

- ( K / f l ) (3 - 4 exp( - fit) + exp( - 2fit)),

(3)

where K = 5 x 1 0 4 m 2 s - 1 , Vo=0.15ms-t and fl = 10 -4 s - ~. This formula is fairly general because it was derived from theoretical considerations and the values of the constants were determined by fitting the formula to field data reported by many investigators. To compute horizontal dispersion coefficients from (3), the travel time t has to be estimated. The travel time depends upon initial dispersion coefficient, the grid size and advection wind velocity. The procedure for the computation is as follows.

1065

1066

C . M . SHEIH and F. L. LUDWIG

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A comparison of numerical pseudodiffusion and atmospheric diffusion (1) The initial radius of the advected diffusing blob for a given numerical experiment is used for a (to) and the corresponding initial time t o is obtained by solving (3) with the Newton-Raphson method (e.g. McCormick and Salvadori, 1965). (2) The elapsed time of a numerical integration is computed from reported values of total angle of rotation, total number of time steps and radius of rotation and assumed values of grid size and advection speed. The center of the advected diffusing blob is assumed to move at a wind speed 5, 10 and 20 m s - t, and the grid size is either 0.5, 5 or 50 kin. (3) The elapsed time for each numerical experiment (and each assumed grid size) was added to the corresponding value for t o to obtain the end time t which was used to calculate a2(t). Once trz(t) is obtained, the attenuation factor for atmospheric diffusion can be computed from (2). Since integration times differ from one numerical experiment to another, relative values of R, and R aare easier to interpret. Thus, the relative dilution factor, defined as ( 1 - R n ) / ( 1 - R a ) , is used in evaluating various numerical schemes. The factor represents the ratio of concentration reductions due to numerical and atmospheric diffusion. For pseudodiffusion to be smaller than atmospheric diffusion, the relative dilution factor has to be smaller than unity. Ideally, it should be zero, and a criterion for evaluating a numerical scheme is that the relative dilution factor be as small as possible. RESULTS AND DISCUSSIONS The results of evaluation for various numerical methods are shown in Table 1. To ensure a fair comparison free from the effects of initial spatial resolution of numerical integration, only those simulations with the same ratio of the initial radius of the diffusing blob to the finite-difference grid size are included. The value of the ratio used here is 4 because it appears to be the most widely used one. The table shows the relative dilution factors for various combinations of wind speeds 5, 10 and 20 m s - 1 (a range normally encountered in the atmosphere) and grid sizes 0.5, 5 and 50 km (usually used in urban, meso and regional scale modeling, respectively). The results indicate that increases in wind speed appear to make the numerical diffusion more important with respect to natural diffusion. However, the magnitude of this effect depends strongly upon the numerical method being used. For example, at the grid size 50 km and with wind speed increasing from 5 to 20 m s - t , the relative dilution factor increases substantially from 5.04 to 18.54 for the mass-in-cell method, but experiences much smaller changes from 0.03 to 0.07 for the Galerkin-Chapeau function method. The ranges of the relative dilution factors shown in the table are 0.02-4.72 for the urban scale, 0.02-3.50 for the mesoscale and 0.03-18.54 for the regional scale. Although there is a slight decrease in the relative dilution factor

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from urban scale to mesoscale, the difference is insignificant. However, much larger values of the relative dilution factor are obtained for the regional scales. For urban and meso scales, the effect of pseudodiffusion appears to be smaller than that of the atmospheric diffusion for most of the numerical schemes except the methods of upstream difference, Lax-Wendroff, mass in cell, and the second moment. For regional scales, the table indicates that the majority of the numerical schemes tested produce larger pseudodiffusion than natural diffusion. The worst case is the mass in cell method tested by Pedersen and Prahm (1974), where the dilution of concentration due to numerical diffusion reaches 18.54 times that of natural diffusion, when the wind speed is 20 m s - t and the grid size is 50 km. The overall ranking shown in the Table indicates that the Galerkin-Chapeau function method is the first, while the mass in cell method appears to be the last.

CONCLUSION For the grid sizes tested, the pseudodiffusion of the upstream differencing method always produces larger dispersion than natural diffusion, while the numerical pseudodiffusions for finite-element, Crank-Nicolson, pseudospectral, and pseudospectral associated methods produce numerical dispersion that is comparable to or smaller than natural diffusion. For grid sizes of 0.5 and 5 km, as often used in urban and mesoscale modeling, most of the reported numerical schemes produce smaller numerical than natural diffusion. However, for the grid size of 50 km frequently used in regional-scale models, the majority of the numerical schemes tested produces larger numerical than natural dispersion. Acknowledoement--The results reported here were obtained from work supported by the Electric Power Research Institute. REFERENCES

Anderson D. and Fattahi B. (1974) A comparison of numerical solutions to the advective equation. J. atmos. Sci. 31, 1500-1506. Christenscn O. and Prahm L. P. (1976) A pseudo spectral model for dispersion of atmospheric pollutants. J. appl. Met. 15, 1284-1294. Gifford F. A. (1982) Horizontal diffusion in the atmosphere: a Lagrangian-dynamic theory. Atmospheric Environment 16, 505-512. Lee H. N. and Meyers R. E. (1979) On time dependent multigrid numerical technique. Paper presented at TICOM Meeting, University of Texas, 26 March 1979. Long P. E. and Pepper D. W. (1976) A comparison of six numerical schemes for calculating the advection of atmospheric pollution. Preprints, Third Symposium on Atmospheric Turbulence Diffusion and Air Quality, Am. Met. Soc., 181-187. McCormick J. M. and Salvadori M. G. (1965) Numerical Methods in FORTRAN, pp. 61--64. Prentice-Hall, Englewood Cliffs, New Jersey.

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McRae G. J., Goodin W. R. and Seinfeid J. H. (1982) Numerical solution of the atmospheric diffusion equation for chemically reacting flows. J. Computation Phys. 45, 1--42. Molenkamp C. R. (1968) Accuracy of finite-difference methods applied to the advection equation. J. appl. Met. 7, 160-167. Orszag P. E. and Pepper D. W. (1971) Numerical simulation of incompressible flows within simple boundaries: ac-

curacy. J. Fluid Mech. 49, 75-112. Pedersen L. B. and Prahm L. P. (1974) A method for numerical solution of the advection equation. Tellus 26, 594-602. Shannon J. D. (1979) A Gaussian moment-conservation diffusion model. J. appl. Met. 18, 1406-1414. Sheih C. M. (1978) A puff-on-cell model for computing pollutant transport and diffusion. J. appl. Met. 17, 141-147.