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Poster 23 The new GIADMT approach to simulate the pollutant dispersion in the planetary boundary layer
Developments in Environmental Science, Volume 6 C. Borrego and E. Renner (Editors) Copyright r 2007 Elsevier Ltd. All rights reserved. ISSN: 1474-8177/DOI:10.1016/S1474-8177(07)06823-4
805
Poster 23 The new GIADMT approach to simulate the pollutant dispersion in the planetary boundary layer Camila Costa, Marco Tullio Vilhena, Davidson Moreira and Tiziano Tirabassi Abstract Analytical solutions of equations are of fundamental importance in understanding and describing the phenomenon of turbulent diffusion in the atmosphere. Focusing our attention in this direction, in this work we report a semi-analytical solution for the three-dimensional advection–diffusion equation in order to simulate pollutant dispersion in the atmosphere considering a vertically inhomogeneous PBL. This work relies on the semi-analytical solution for the three-dimensional advection–diffusion equation combining the ADMM (Advection Diffusion Multilayer Model) and GITT (Generalized Integral Transform Technique) methods. We coin this technique as GIADMT (Generalized Integral Advection Diffusion Multilayer Technique). 1. Model application and results
The advection–diffusion equation of air pollution in the atmosphere is written as @c @c @ @c @ @c @ @c þU ¼ Kx Ky Kz þ þ þS @t @x @x @x @y @y @z @z
(1)
where c denotes the average concentration, Kx, Ky and Kz are the Cartesian components of eddy diffusivity, U is the mean wind and S is the source term. The X-axis of the Cartesian coordinate system is aligned in the direction of the actual wind near the surface, the Y-axis is oriented in the horizontal crosswind direction, the Z-axis is chosen vertically upwards and t is the time.
Camila Costa et al.
806
Table 1. Statistical evaluation of model results with Copenhagen data set Model
nmse
r
fa2
fb
fs
GIADMT
0.10
0.90
0.96
0.06
0.22
After the application of the GIADMT method, we get the solution " ( Nj Nk 1 X X P¯ j cosðli yÞ X Pk pffiffiffiffiffiffi cðt; x; y; zÞ ¼ w wk ¯j t N i k¼1 x j¼1 i¼0 0 139 = An eRz þ Bn eRz þ A 5 ð2Þ @ Q ; þ 2Ra ðeRðzHsÞ eRðzH s Þ ÞHðz H s Þ n where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u 2 u 1 Pj Pk Pk Rin ¼ t K xn Un þ K yn l2i þ K zn x x t vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u pffiffiffiffiffiffi 2 ! P N i Pj u P Pk j k 2 tK zn þ K yn li þ U n K xn Ran ¼ t x x cosðli yo Þ t where the constants An and Bn are determined solving a linear system applying the (2N2) interface conditions, and wk, wj and Pk, Pj are the Gaussian Quadrature parameters tabulated (Stroud and Secrest, 1966), Nk is the number of inversions, Q is emission rate, Hs is the height source, H(zHs) is the Heaviside function Ni is the norm (see Costa et al., 2006) and li is the eigenvalue from the GITT approach. Table 1 shows the statistical analysis of the new model compared with the moderately unstable experiments of Copenhagen (Gryning and Lyck, 1984) for centreline concentrations. Analysing the statistical indices (Hanna, 1989) in Table 1, it is possible to notice that the models simulate satisfactorily the observed concentrations, with nmse (normalised mean square error), fb (fractional bias) and fs (fractional standard deviation) values relatively near to 0 and r (correlation coefficient) and fa2 (factor of two) relatively near to 1.
REFERENCES Costa, C.P., Vilhena, M.T., Moreira, D.M., Tirabassi, T., 2006. Semi-analytical solution of the steady three-dimensional advection–diffusion equation in the planetary boundary layer. Atmos. Environ. 40, 5659–5669.
Pollutant Dispersion in Planetary Boundary Layer
807
Gryning, S.E., Lyck, E., 1984. Atmospheric dispersion from elevated source in an urban area: Comparison between tracer experiments and model calculations. J. Climate Appl. Meteorol. 23, 651–654. Hanna, S.R., 1989. Confidence limit for air quality models as estimated by bootstrap and jacknife resampling methods. Atmos. Environ. 23, 1385–1395. Stroud, A.H., Secrest, D., 1966. Gaussian Quadrature Formulas. Englewood Cliffs, NJ, Prentice Hall.
805
Poster 23 The new GIADMT approach to simulate the pollutant dispersion in the planetary boundary layer Camila Costa, Marco Tullio Vilhena, Davidson Moreira and Tiziano Tirabassi Abstract Analytical solutions of equations are of fundamental importance in understanding and describing the phenomenon of turbulent diffusion in the atmosphere. Focusing our attention in this direction, in this work we report a semi-analytical solution for the three-dimensional advection–diffusion equation in order to simulate pollutant dispersion in the atmosphere considering a vertically inhomogeneous PBL. This work relies on the semi-analytical solution for the three-dimensional advection–diffusion equation combining the ADMM (Advection Diffusion Multilayer Model) and GITT (Generalized Integral Transform Technique) methods. We coin this technique as GIADMT (Generalized Integral Advection Diffusion Multilayer Technique). 1. Model application and results
The advection–diffusion equation of air pollution in the atmosphere is written as @c @c @ @c @ @c @ @c þU ¼ Kx Ky Kz þ þ þS @t @x @x @x @y @y @z @z
(1)
where c denotes the average concentration, Kx, Ky and Kz are the Cartesian components of eddy diffusivity, U is the mean wind and S is the source term. The X-axis of the Cartesian coordinate system is aligned in the direction of the actual wind near the surface, the Y-axis is oriented in the horizontal crosswind direction, the Z-axis is chosen vertically upwards and t is the time.
Camila Costa et al.
806
Table 1. Statistical evaluation of model results with Copenhagen data set Model
nmse
r
fa2
fb
fs
GIADMT
0.10
0.90
0.96
0.06
0.22
After the application of the GIADMT method, we get the solution " ( Nj Nk 1 X X P¯ j cosðli yÞ X Pk pffiffiffiffiffiffi cðt; x; y; zÞ ¼ w wk ¯j t N i k¼1 x j¼1 i¼0 0 139 = An eRz þ Bn eRz þ A 5 ð2Þ @ Q ; þ 2Ra ðeRðzHsÞ eRðzH s Þ ÞHðz H s Þ n where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u 2 u 1 Pj Pk Pk Rin ¼ t K xn Un þ K yn l2i þ K zn x x t vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u pffiffiffiffiffiffi 2 ! P N i Pj u P Pk j k 2 tK zn þ K yn li þ U n K xn Ran ¼ t x x cosðli yo Þ t where the constants An and Bn are determined solving a linear system applying the (2N2) interface conditions, and wk, wj and Pk, Pj are the Gaussian Quadrature parameters tabulated (Stroud and Secrest, 1966), Nk is the number of inversions, Q is emission rate, Hs is the height source, H(zHs) is the Heaviside function Ni is the norm (see Costa et al., 2006) and li is the eigenvalue from the GITT approach. Table 1 shows the statistical analysis of the new model compared with the moderately unstable experiments of Copenhagen (Gryning and Lyck, 1984) for centreline concentrations. Analysing the statistical indices (Hanna, 1989) in Table 1, it is possible to notice that the models simulate satisfactorily the observed concentrations, with nmse (normalised mean square error), fb (fractional bias) and fs (fractional standard deviation) values relatively near to 0 and r (correlation coefficient) and fa2 (factor of two) relatively near to 1.
REFERENCES Costa, C.P., Vilhena, M.T., Moreira, D.M., Tirabassi, T., 2006. Semi-analytical solution of the steady three-dimensional advection–diffusion equation in the planetary boundary layer. Atmos. Environ. 40, 5659–5669.
Pollutant Dispersion in Planetary Boundary Layer
807
Gryning, S.E., Lyck, E., 1984. Atmospheric dispersion from elevated source in an urban area: Comparison between tracer experiments and model calculations. J. Climate Appl. Meteorol. 23, 651–654. Hanna, S.R., 1989. Confidence limit for air quality models as estimated by bootstrap and jacknife resampling methods. Atmos. Environ. 23, 1385–1395. Stroud, A.H., Secrest, D., 1966. Gaussian Quadrature Formulas. Englewood Cliffs, NJ, Prentice Hall.