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Some characteristics of a plume from a point source based on analytical solution of the two-dimensional advection–diffusion equation
Atmospheric Environment 43 (2009) 2221–2227
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Atmospheric Environment journal homepage: www.elsevier.com/locate/atmosenv
Short communication
Some characteristics of a plume from a point source based on analytical solution of the two-dimensional advection–diffusion equation Tiziano Tirabassi a, *, Alessandro Tiesi a, Daniela Buske b, c, Marco T. Vilhena c, Davidson M. Moreira d a
Institute ISAC of CNR, 40129 Bologna, Italy Universidade Federal de Pelotas, DME, Pelotas, RS, Brazil c Universidade Federal do Rio Grande do Sul, PROMEC, Porto Alegre, RS, Brazil d Universidade UNIPAMPA, Bage´, RS, Brazil b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 July 2008 Received in revised form 13 January 2009 Accepted 13 January 2009
The moments of the concentration distribution obtained using a recent analytical solution of the steadystate two-dimensional advection–diffusion equation are presented. The solving methodology is the Generalized Integral Laplace Transform Technique, which allows obtaining a reliable solution of the advection–diffusion equation without any restrictive assumption about the eddy diffusivity coefficients and wind speed profiles. The first four moments and value and position of maximum ground level concentration are calculated. The concentration standard deviation is compared against the semiempirical ones used in operative Gaussian models. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Air pollution modelling Analytical solution Advection–diffusion equation Moments of concentration distribution
1. Introduction The vertical diffusion of a passive tracer release from a point source in unstable and stable atmosphere has been studied by an analytical solution of the advection–diffusion equation. Traditionally the operative modelling of the dispersion has been performed adopting a Gaussian approach that takes accounts of atmospheric turbulence assuming simple formulae for concentration distribution, in which the dispersion parameters depend simply on downwind distance and the meteorological state of boundary layer like the Pasquill–Gifford scheme (Arya, 1999). For the vertical dispersion the scheme performs adequately only for short horizontal distances and for near ground sources. Within this scheme the low source condition has the effect of the crude approximation of infinite height of the atmospheric boundary layer (ABL), then it turns out to overestimate the centroid (z) and the variance (s2z ) when the horizontal distance from the source approaches to the length scale of the ABL (Briggs, 1973). On the other hand the predicted ground level concentration, regardless of the ABL scenario, underestimates the experimental data (Irwin,
* Corresponding author. Tel.: þ39 051 639 9601; fax: þ39 051 639 9658. E-mail addresses: [email protected] (T. Tirabassi), [email protected] (A. Tiesi), [email protected] (D. Buske), [email protected] (M.T. Vilhena), davidson@ pq.cnpq.br (D.M. Moreira). 1352-2310/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2009.01.020
1983). In fact, although the ABL is assumed to have infinite height, its actual vertical limit affects the behaviour of all quantities. NonGaussian approaches are proved to perform more reliably, despite the more complicated parameterization of the ABL dynamics (Elliot, 1961; Malhotra and Cermak, 1964; Huang and Drake, 1977; Doran et al., 1978; Nieuwstadt and van Ulden, 1978; Tagliazucca et al., 1985; Hinrichson, 1986; Tirabassi et al., 1986; Brown et al., 1989; van Ulden, 1992; Sharan et al., 1996; Lin and Hildemann, 1997). One of the central equations to describe the evolution of pollutants in the ABL is the Advection–Diffusion Equation (ADE). The ADE is a second order partial differential equation, and in most cases it is solved numerically. Other approaches are based on the Lagrangian particle model (Seinfeld and Pandis, 1998). In this paper we present a study of the concentration distribution and its moments. The concentration Cðx; zÞ is obtained using a recent analytical solution of the steady-state twodimensional ADE (Moreira et al., 2005; Wortmann et al., 2005). The solution is obtained using the Generalized Integral Laplace Transform Technique (GILTT) (Moreira et al., 2005; Wortmann et al., 2005), without any approximation except for a round-off error. After solving the ADE, the maximum ground level concentration Cmax ðx; 0Þ is evaluated. Furthermore a complete analytical expression of the first four statistical symmetries (centroid z, variance s2z , skewness Sk and kurtosis Ku) of the
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vertical concentration distribution are presented. Special emphasis will be devoted to z and s2z , because of their great operative concern. We leave to a future reference the focus on the remaining higher order statistical descriptors, reminding that their importance is still a great issue.
m1 ¼
n X h ci ðxÞ h ðcosip 1Þ þ ; 2 c0 ðxÞp2 i ¼ 1 i2
(6)
m2 ¼
n 2h2 X ci ðxÞ h2 cosip þ z2 : 3 c0 ðxÞp2 i ¼ 1 i2
(7)
2. The GILTT solution
It is convenient to uniform the notation to the traditional one after assigning
The stationary ADE solved in two-spatial dimensions is the following:
z ¼ m1 ;
uðzÞ
vCðx; zÞ v vCðx; zÞ ¼ Kz ðzÞ vx vz vz
(1)
where C(x,z) is the y cross-wind integrated three-dimensional steady state concentration C(x,y,z), u(z) is the wind speed in the x direction and Kz(z) is the vertical eddy diffusivity which are both assumed depending on the vertical coordinate z. The equation is solved neglecting the longitudinal diffusion. We assume the boundary conditions of zero flux at the ground (z ¼ 0) and at the ABL height (z ¼ h); the source condition assumed is a point source at height hs , namely:
uðzÞcð0; zÞ ¼ Q dðz hs Þ at x ¼ 0;
N X
ci ðxÞji ðzÞ;
(3)
i¼0
where ci ðxÞ and ji ðzÞ are the solutions of the transformed equation and Sturm–Liouville problem respectively (for more details see the works of Wortmann et al. (2005) and Moreira et al. (2005)). The infinite series given in Eq. (3) can be truncated when the convergence is under a prefixed limit value. In our case we used 100 terms with an error of 0.5%.
Based on the analytical expression of the concentration distribution obtained, it is a straightforward to analytically express the statistical descriptors moments characterizing the vertical concentration distribution. These are defined as follows:
Rh
zCðx; zÞdz
0
Cðx; zÞdz
;
It is also convenient to express the remaining moments m3 and m4 in dimensionless form defining the skewness and the kurtosis as
Sk ¼
m3 ; s3z
Ku ¼
m4 : s4z
(9)
Aware of these dimensionless definitions we have the expressions
Sk ¼
3 X n 3 h ci ðxÞ 2 2 p 1 cosði Þ þ c0 ðxÞp2 sz i ¼ 1 i2 i2 p2 i2 p2 3 1 h 3 3z z ; þ sz sz 4 sz
(10)
Ku ¼
4 X n 4 h ci ðxÞ 6 1 h 4 p 1 cosði Þ þ 5 sz c0 ðxÞp2 sz i ¼ 1 i2 i2 p2 2 4 4zSk z z 6 :
sz
sz
sz
ð11Þ
It is worthy of note to highlight the difference between the symmetries expressed in Eqs. (6), (7), (10) and (11) with those reported in Brown et al. (1997), where both wind and eddy diffusivities follow a power law. Brown et al. (1997) express the moments in terms of the gamma and the confluent hypergeometric functions. As far as concerns the expressions here reported, these are the exact solutions as the definitions (4) and (5) involve integrations by parts only due to the definition of the eigenfunction ji ðzÞ. In many operative air pollution models (i.e. Gaussian models), the variance is defined as (Zannetti, 1990)
s2s ¼
Rh
hs Þ2 Cðx; zÞdz ; Rh 0 Cðx; zÞdz
0 ðz
(12)
and the reason for such a definition lies on the possibility to relate the dispersion of the pollutant directly to the height source. Substituting Eq. (3) in Eq. (12) we obtain
3. Derivation of moments and ground-level concentration
m1 ¼ R0h
(8)
(2)
where h is the boundary layer height and Q is the pollutant emission rate. The GILTT method is used to solve the equation given in Eq. (1). This analytical method performs a series expansion of the concentration C(x,z) in terms of the eigenfunctions associated to a Sturm–Liouville problem. The resulting transformed equation is also solved, analytically, applying the Laplace Transform technique. According the works of Wortmann et al. (2005) and Moreira et al. (2005), the solution for problem (1) is written, either for the eddy diffusivity Kz ðzÞ and Kz ðx; zÞ, as:
Cðx; zÞ ¼
s2z ¼ m2 :
(4)
s2s ¼
n 2h2 X ci ðxÞ h2 cosip þ 2hs z þ h2s : 3 c0 ðxÞp2 i ¼ 1 i2
(13)
Compared to the s2z expressed in Eq. (7), s2s approaches to higher values, it is nonetheless easy to prove that s2s hs2z when the assignment hs /z is restored.
and the remaining 4. Turbulent parameterization
Rh
mm
ðz m Þm Cðx; zÞdz ¼ 0 Rh 1 ; m ¼ 2; 3; 4: 0 Cðx; zÞdz
(5)
After substituting Eq. (3) in Eqs. (4) and (5) we obtain the explicit expressions for the four moments. The first and the second moment are expressed in a dimensional form as
In the atmospheric diffusion problems the choice of a turbulent parameterization represents a fundamental decision for the pollutants dispersion modelling. The reliability of each model strongly depends on the way as turbulent parameters are determined and related to the current understanding of the PBL
T. Tirabassi et al. / Atmospheric Environment 43 (2009) 2221–2227
(Seinfeld and Pandis, 1998; Mangia et al., 2002). We adopted the parameterizations suggested in Degrazia et al. (1997, 2000). In terms of the convective scaling parameters the vertical eddy diffusivity can be formulated as (Degrazia et al., 1997)
z 1=3 z 1=3 h z Kz ¼ 0:22w* h 1 1 exp 4 h h h z i ; 0:0003exp 8 h
(14)
and for stable conditions as (Degrazia et al., 2000)
Kz ¼
0:3ð1 z=hÞu* z 1 þ 3:7z=L
(15)
where z is the height; h the thickness of the boundary layer, u* and w* are the velocity scales for the horizontal friction and vertical convection respectively. L is the local Monin–Obukhov length defined as L ¼ Lð1 z=hÞ5=4 where L is the Monin–Obukhov length. The main feature of the diffusivity Kz for convective and neutral scenarios stands in the quasi symmetric behaviour of it around 0:5h scanned versus z. The maximum value of Kz increases with the convection. Concerning stable regimes, Kz increases rapidly with z up to z0:1h. Over such a limit Kz decreases until the ABL top. The wind speed profile can be described by the power law expressed as follows (Panofsky and Dutton, 1988):
uz ¼ u1
n z ; z1
(16)
where uz and u1 are the mean wind velocity at the heights z and z1 , while n is an exponent that is related to the intensity of turbulence (Irwin, 1979a). In fact this empirical wind profile matches well similarity profile in the surface layer, and on the contrary is valid in all the ABL. It turns out that the exponent n depends on the Pasquill stability class, and it is shown in Table 1. It turns out that for the unstable classes A, B, and C, uz increases rapidly in the surface layer up to a value which remains nearly constant along the whole upper part of the ABL. Concerning the D, E, and F classes the wind uz increases moderately but constantly through the all ABL. The horizontal wind and diffusivity behaviours will affect the statistical symmetries to be discussed later. The vertical eddy diffusivities (Eqs. (14) and (15)) and the wind profile (Eq. (16)) will be substituted in the solution of the ADE (Eq. (3)), obtained with the GILTT method, to calculate the maximum ground-level concentration (Cmax ) and then to evaluate the statistical descriptors z, sz , Sk and Ku. 5. Statistical descriptors plots In Fig. 1 are reported the four vertical distribution descriptors for the single ABL regime C, the choice for such scenario lies in the possibility to represents a convective regime still very close to a neutral one. The class C is characterized by a Monin–Obukhov length L ¼ 30 m, and a horizontal wind speed u1 ¼ 4 m s1 at the fixed height z1 ¼ 10 m. Simulations have been performed
Table 1 Summary of quantities used to set the ABL stability regimes.
u* (m s1) u (m s1) L1 (m s1) n
A
B
C
D
E
F
0.1 1.5 0.14 0.07
0.17 2.5 0.09 0.07
0.25 4 0.03 0.1
0.26 4.5 0 0.15
0.16 3.5 0.03 0.35
0.09 2.5 0.14 0.55
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adopting six values of the source height, in order hs =h ¼ 0.05, 0.1, 0.2, 0.3, 0.4, 0.5. Distribution statistics are plotted versus the dimensionless horizontal distance x=h (Observe that the definition of dimensionless distance is not the canonical one which is usually set in such a way to summarise the ABL regime. The reason for such a simpler definition stands in the intention to put emphasis to distances relatively to the ABL height h, to better appreciate the validity of our results which are valid for limited ABL height, and for long horizontal distances.) For each plot the long distance value of the moment approaches to a common asymptotic value, regardless the source height. For each plume source height the centroid starts from the respective source height and slowly approaches to the asymptotic value of 0:5h. The plots referring to the sz =h are obtained using the definition of Eq. (5), all tending to the asymptotic value z0:3h. The highest source emissions show a nearly Gaussian distribution, this is particularly manifest in the Sk and the Ku plots already for short distances. The kurtosis plots are strongly depending on the source heights. In all plots it is anyway very easy to highlight the terrain influence on the lowest sources. Kurtosis plots reveal a poorly peaked vertical distribution tendency in respect of the traditional Gaussian behaviour, which is still widely used for operative purposes. The power law vertical behaviour of the horizontal wind profile ensures the wind to increase rapidly in the lower part of the ABL. For the convective cases, above the heightzz0:2h the wind remains nearly constant. Because of such behaviour the highest sources releases seems to be poorly affected by the vertical structure of the horizontal wind. Furthermore they reach earlier than lower source release cases the tendency to approach to the asymptotic limit. Concerning the second moment sz =h, in Fig. 1 is reported the plot using the definition of Eq. (5). The first three plots behave homogeneously. It is also worth of note to remind the symmetric shape of the diffusivity Kz, for such a reason the highest sources released distributions tend to have a Gaussian shaped skewness rather soon after emission. Regarding the kurtosis plots two different behaviours are highlighted at short horizontal distances from the release point. Because of the turbulent parameterization mentioned earlier, the nearly symmetric scenario for the highest sourced distributions makes possible for Ku to decrease with the distance. On the other hand, the lower sourced distribution approach to a maximum, which means a narrow sharp distribution (leptokurtic). For long horizontal distribution then all kurtosis decrease to an asymptotic value z1:5 which represents a widely spread distribution (platykurtic). In Fig. 2 we reported the comparison of the above first four moments with the four ones, obtained with a solution that accept power low profile of wind and eddy diffusivity coefficient only (Brown et al., 1997). The diffusion coefficient used in our solution is symmetric and realistically go to 0 at the top of ABL, while in the solution adopted in Brown et al. (1997) Kz profiles do not go to 0 at the top of ABL. Moreover, in the approach of Brown et al. (1997) an infinite ABL is considered, so we limited the comparison for the two lowest sources (i.e. 0.05h and 0.1h) and for a maximum distance from the source x ¼ 10h in order to minimize the influence of the ABL top. The meteorological condition are the same of Fig. 1, but, in the case of the Brown et al. (1997) approach, Kz follows a power low profile with an exponent equal to 0.9 and matching the profile from Degrazia et al. (1997) at z ¼ 0.01h. The moments obtained with the GILTT approach increase more rapidly than the ones presented in Brown et al. (1997), but, in the case of skewness and kurtosis, these last ones reach a bigger maximum, showing a concentration vertical distribution more asymmetric and more ‘‘peakedness’’.
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Fig. 1. Plots of the four moments for the vertical distribution of the concentration with stability class C (L ¼ 30 m; u ¼ 4 m s1). The curves refer to the six emission heights hS =h ¼ 0.05, 0.1, 0.2, 0.3, 0.4, 0.5.
6. Position and value of the maximum ground level concentration Through the main advantages in using analytic expressions for the concentration field is the ability to evaluate the value of C(x,z) straightforwardly. This feature sets a significant difference with numerical models, whose evaluation of a concentration field in a single point involves the evaluation of the field in the grid points surrounding the source. This point assumes a strategic importance in the environmental management. By this point, to find the position and the value of the maximum ground level concentration Cmax ðx; 0Þ, it is enough to calculate its first derivative and then work out the horizontal distance xmax at which Cmax ðxmax ; 0Þ occurs. In order to show positions and values of the maximum ground level concentration in several different ABL scenarios, we adopted the wind velocity u1 and the Monin–Obukhov length L relative to the six Pasquill stability classes (Pasquill and Smith, 1984) using the definition of the classes and the semi empirical relation among ABL parameters and the empirical Pasquill classes proposed by Golder (1972) and fitted in Irwin (1979b). This allowed us to identify for each class the wind u, the Monin–Obukhov length L and the friction velocity u*. The set of meteorological parameters used for the six stability classes are shown in Table 1. In Table 2 the value of xmax and Cmax for the ABL scenarios corresponding to the six different stability classes and for six source heights hS =h varying from 0.001 to 0.5 are shown. According to Table 2, xmax increases with the source height, while the concentration
value decreases as the turbulence decreases in unstable cases (although an important role is due to the wind velocity too which increases as the turbulence decreases). Concerning the highest sources the results for the stable classes E and F are not reported because xmax turns out to be over 50 km far from the source, so they are out of range of reliability for a steady-state model. It is possible to see the dependence of concentration maximum value on the source height in Fig. 3. The two curves ðhS =hÞð1Þ and ðhS =hÞð2Þ are also shown to compare the qualitative behaviour of our results with the Brown et al. (1997) ones showing that a Cmax fh S with 1 a 2. 7. Plume standard deviation The most diffused operative models nowadays rely on the empiric estimates of the plume vertical standard deviation sz , the so-called sigmas (Pasquill and Smith, 1984). Although empirical Gaussian models have been proved to be not completely reliable they still enjoy a large extent of popularity in the environmental management and the air quality control. Then it is useful to compare such empirical expressions with those finally available based on the exact solution of the ADE. However, the accuracy of results still depends by the choice of the ABL parameterization. Despite such aspects, it is still significant to evaluate the reliability of the methodology adopting physically sounding theoretical parameterization. In our case we adopted the ABL parameterization mentioned in Section 4 which has been proved to perform quite
T. Tirabassi et al. / Atmospheric Environment 43 (2009) 2221–2227
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Fig. 2. Comparison of the first four moments obtained with the GILTT approach against the ones obtained with the approach described in Brown et al. (1997).
correctly when referred to ABL regimes whose experimental ground level concentration are recorded (Degrazia et al., 1997, 2000). For these reasons, we will use the sigma calculated in respect to the source height hS as defined in Eq. (12) to be able to compare the model performances with the empirical s2s . It is Table 2 Table of xmax and Cmax divided by source height and scanned through stability classes of the ABL. Stability hS ¼ 0:001 h
A B C D E F
xmax h
3 2 C 10 (s m ) xmax h Q
w0.001 w0.001 w0.001 w0.001 w0.001 w0.001
208 126 84 82 145 266
hS ¼ 0:1 h xmax h A B C D E F
hS ¼ 0:02 h
1.06 1.31 1.94 6.52 83.00 –
hS ¼ 0:2 h C xmax 104 (s m2) Q h 20.867 12.520 7.521 6.872 8.080 –
hS ¼ 0:05 h
C 103 (s m2) xmax Q h
0.23 11.517 0.28 6.910 0.40 4.348 0.88 4.329 2.47 6.595 5.67 9.336
0.56 0.69 1.01 2.79 16.80 53.80
C 103 (s m2) Q 4.233 2.540 1.558 1.501 1.996 2.313
hS ¼ 0:5 h
C xmax 104 (s m2) Q h
2.08 10.652 2.57 6.391 3.88 3.755 15.80 3.212 – – – –
worthy of note that the empirical s2s have been drawn from experiments with near-ground source heights, and involved horizontal distances up to few kilometres from the source (Pasquill and Smith, 1984; Arya, 1999). Despite such a local validity restriction, their use is usually extended to higher source heights as well as longer horizontal distances. As an example, the U.S. Environmental Protection Agency (EPA) admits their adoption even as far as 50 km away from the source.
w7 w10 w23 w73 – –
C 104 (s m2) Q 5.069 3.067 1.736 1.342 – –
Fig. 3. Plots of cmax ðx; 0Þ,Q 1 versus the dimensionless distance hs =h. The plot shows the two curves proportional to ðhs =hÞð2Þ and ðhs =hÞð1Þ .
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T. Tirabassi et al. / Atmospheric Environment 43 (2009) 2221–2227
Fig. 4. Plots of sz =h for the six stability classes. For each class the sz is evaluated for source height hs =h ¼ 0.01 (empty squares), 0.1 (black squares), 0.2 (empty triangles) and 0.5 (black triangles) using Eq. (5) and hs =h ¼ 0.01 (empty stars), 0.1 (black stars), 0.2 (empty circles) and 0.5 (black circles) using Eq. (12).
Through the most diffused empirical sigmas, it is best remembered those proposed by Briggs (Arya, 1999), whose applicability in both rural and urban versions provides an interpolation scheme agreeing with those proposed by Pasquill and Gifford, except for the vertical sigma in a convective scenario where they approximate the Brookhaven curves (Arya, 1999). In our work we have compared the values of ss coming from Eq. (13) with those proposed by Briggs for rural terrain, setting
the meteorology scenarios where they are defined. In order to allow comparison we set u, u* and L according to the six Pasquill stability classes as we did previously in Section 4 introducing Table 1. In Fig. 4 the empirical standard deviations and those derived by our solution are compared for four different source heights hS =h ¼ 0.05, 0.1, 0.2, 0.5. We can see that for stable classes E and F the empirical sigma overestimate the analytical ones through the
T. Tirabassi et al. / Atmospheric Environment 43 (2009) 2221–2227
all range of horizontal distances, while, for the unstable cases A, B and C, there is a qualitative agreement near the source although such agreement occurs with a large uncertainty. Far from the source the functions show a rather different behaviour. In particular, the empirical ss approach to larger values than the analytical functions. The reason for such a great discrepancy stands on the crude approximation of an unlimited ABL. Nonetheless it is reasonable to adopt such an approximation when near-ground sources are considered. In fact the neutral case seems to better reproduce agreement between the different sigmas. It is known that the sigma functions are depending on the source height though the relationship between them is still an open issue. With the aim to better highlight such an aspect, Fig 4 shows a comparison between the two sigmas for unstable, neutral and stable scenarios and relative to four different source heights hs =h ¼ 0.05, 0.1, 0.2, 0.5. The plots show that at short horizontal distances from the source the empirical curves match the theoretical ones, then for large distances these reaches far higher values than the theoretical ones. For the neutral case the agreement reaches relatively the top in agreement. As a demonstration of the fact that the empirical ss are based on real low source patterns, in case D the agreement is maximum. On the other hand, the empirical curves for the neutral regime underestimate for all distances the pattern for the highest sources. Moreover, in the Fig 4 the sigmas based on Eq. (5) are plotted also. It is easy to see that for unstable regime they magnify the behaviour of sigmas using Eq. (12). In stable regimes they show the same behaviour while in the neutral one they underestimate empirical sigma in the case of low sources. Based on the results shown, it is necessary to bear in mind that finding any relationship between the theoretical curves and the effective source height is an important issue. Furthermore, it is also necessary to stress the point that the use of the traditional sigma schemes can lead to unreliable results in trying to reproduce the ground level concentration C(x,0). 8. Conclusions Using an analytical solution of the two-dimensional steadystate ADE for a point source release, without any restriction on the vertical profiles of wind and eddy exchange coefficients, we have derived formulae for the first four moments of the vertical concentrations for both surface and elevated releases. Moreover, it is possible to easily evaluate the position and value of maximum ground concentrations. The behaviour of the plume vertical standard deviation was outlined and compared with some very popular empirical ones, used in many operative air pollution models. It was outlined that discrepancies occur for tall sources and for long distances. It is evident that empirical formulae for ss need to take into account the height of sources and of ABL. The formulae presented here can be useful for operative evaluation of atmospheric dispersion and a better understanding of advection–diffusion phenomena. Acknowledgements The authors thank the Brazilian CNPq and CAPES, the Italian CNR and project ‘‘Laboratorio LaRIA’’ for the partial financial support of this work.
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Some characteristics of a plume from a point source based on analytical solution of the two-dimensional advection–diffusion equation Tiziano Tirabassi a, *, Alessandro Tiesi a, Daniela Buske b, c, Marco T. Vilhena c, Davidson M. Moreira d a
Institute ISAC of CNR, 40129 Bologna, Italy Universidade Federal de Pelotas, DME, Pelotas, RS, Brazil c Universidade Federal do Rio Grande do Sul, PROMEC, Porto Alegre, RS, Brazil d Universidade UNIPAMPA, Bage´, RS, Brazil b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 July 2008 Received in revised form 13 January 2009 Accepted 13 January 2009
The moments of the concentration distribution obtained using a recent analytical solution of the steadystate two-dimensional advection–diffusion equation are presented. The solving methodology is the Generalized Integral Laplace Transform Technique, which allows obtaining a reliable solution of the advection–diffusion equation without any restrictive assumption about the eddy diffusivity coefficients and wind speed profiles. The first four moments and value and position of maximum ground level concentration are calculated. The concentration standard deviation is compared against the semiempirical ones used in operative Gaussian models. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Air pollution modelling Analytical solution Advection–diffusion equation Moments of concentration distribution
1. Introduction The vertical diffusion of a passive tracer release from a point source in unstable and stable atmosphere has been studied by an analytical solution of the advection–diffusion equation. Traditionally the operative modelling of the dispersion has been performed adopting a Gaussian approach that takes accounts of atmospheric turbulence assuming simple formulae for concentration distribution, in which the dispersion parameters depend simply on downwind distance and the meteorological state of boundary layer like the Pasquill–Gifford scheme (Arya, 1999). For the vertical dispersion the scheme performs adequately only for short horizontal distances and for near ground sources. Within this scheme the low source condition has the effect of the crude approximation of infinite height of the atmospheric boundary layer (ABL), then it turns out to overestimate the centroid (z) and the variance (s2z ) when the horizontal distance from the source approaches to the length scale of the ABL (Briggs, 1973). On the other hand the predicted ground level concentration, regardless of the ABL scenario, underestimates the experimental data (Irwin,
* Corresponding author. Tel.: þ39 051 639 9601; fax: þ39 051 639 9658. E-mail addresses: [email protected] (T. Tirabassi), [email protected] (A. Tiesi), [email protected] (D. Buske), [email protected] (M.T. Vilhena), davidson@ pq.cnpq.br (D.M. Moreira). 1352-2310/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2009.01.020
1983). In fact, although the ABL is assumed to have infinite height, its actual vertical limit affects the behaviour of all quantities. NonGaussian approaches are proved to perform more reliably, despite the more complicated parameterization of the ABL dynamics (Elliot, 1961; Malhotra and Cermak, 1964; Huang and Drake, 1977; Doran et al., 1978; Nieuwstadt and van Ulden, 1978; Tagliazucca et al., 1985; Hinrichson, 1986; Tirabassi et al., 1986; Brown et al., 1989; van Ulden, 1992; Sharan et al., 1996; Lin and Hildemann, 1997). One of the central equations to describe the evolution of pollutants in the ABL is the Advection–Diffusion Equation (ADE). The ADE is a second order partial differential equation, and in most cases it is solved numerically. Other approaches are based on the Lagrangian particle model (Seinfeld and Pandis, 1998). In this paper we present a study of the concentration distribution and its moments. The concentration Cðx; zÞ is obtained using a recent analytical solution of the steady-state twodimensional ADE (Moreira et al., 2005; Wortmann et al., 2005). The solution is obtained using the Generalized Integral Laplace Transform Technique (GILTT) (Moreira et al., 2005; Wortmann et al., 2005), without any approximation except for a round-off error. After solving the ADE, the maximum ground level concentration Cmax ðx; 0Þ is evaluated. Furthermore a complete analytical expression of the first four statistical symmetries (centroid z, variance s2z , skewness Sk and kurtosis Ku) of the
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vertical concentration distribution are presented. Special emphasis will be devoted to z and s2z , because of their great operative concern. We leave to a future reference the focus on the remaining higher order statistical descriptors, reminding that their importance is still a great issue.
m1 ¼
n X h ci ðxÞ h ðcosip 1Þ þ ; 2 c0 ðxÞp2 i ¼ 1 i2
(6)
m2 ¼
n 2h2 X ci ðxÞ h2 cosip þ z2 : 3 c0 ðxÞp2 i ¼ 1 i2
(7)
2. The GILTT solution
It is convenient to uniform the notation to the traditional one after assigning
The stationary ADE solved in two-spatial dimensions is the following:
z ¼ m1 ;
uðzÞ
vCðx; zÞ v vCðx; zÞ ¼ Kz ðzÞ vx vz vz
(1)
where C(x,z) is the y cross-wind integrated three-dimensional steady state concentration C(x,y,z), u(z) is the wind speed in the x direction and Kz(z) is the vertical eddy diffusivity which are both assumed depending on the vertical coordinate z. The equation is solved neglecting the longitudinal diffusion. We assume the boundary conditions of zero flux at the ground (z ¼ 0) and at the ABL height (z ¼ h); the source condition assumed is a point source at height hs , namely:
uðzÞcð0; zÞ ¼ Q dðz hs Þ at x ¼ 0;
N X
ci ðxÞji ðzÞ;
(3)
i¼0
where ci ðxÞ and ji ðzÞ are the solutions of the transformed equation and Sturm–Liouville problem respectively (for more details see the works of Wortmann et al. (2005) and Moreira et al. (2005)). The infinite series given in Eq. (3) can be truncated when the convergence is under a prefixed limit value. In our case we used 100 terms with an error of 0.5%.
Based on the analytical expression of the concentration distribution obtained, it is a straightforward to analytically express the statistical descriptors moments characterizing the vertical concentration distribution. These are defined as follows:
Rh
zCðx; zÞdz
0
Cðx; zÞdz
;
It is also convenient to express the remaining moments m3 and m4 in dimensionless form defining the skewness and the kurtosis as
Sk ¼
m3 ; s3z
Ku ¼
m4 : s4z
(9)
Aware of these dimensionless definitions we have the expressions
Sk ¼
3 X n 3 h ci ðxÞ 2 2 p 1 cosði Þ þ c0 ðxÞp2 sz i ¼ 1 i2 i2 p2 i2 p2 3 1 h 3 3z z ; þ sz sz 4 sz
(10)
Ku ¼
4 X n 4 h ci ðxÞ 6 1 h 4 p 1 cosði Þ þ 5 sz c0 ðxÞp2 sz i ¼ 1 i2 i2 p2 2 4 4zSk z z 6 :
sz
sz
sz
ð11Þ
It is worthy of note to highlight the difference between the symmetries expressed in Eqs. (6), (7), (10) and (11) with those reported in Brown et al. (1997), where both wind and eddy diffusivities follow a power law. Brown et al. (1997) express the moments in terms of the gamma and the confluent hypergeometric functions. As far as concerns the expressions here reported, these are the exact solutions as the definitions (4) and (5) involve integrations by parts only due to the definition of the eigenfunction ji ðzÞ. In many operative air pollution models (i.e. Gaussian models), the variance is defined as (Zannetti, 1990)
s2s ¼
Rh
hs Þ2 Cðx; zÞdz ; Rh 0 Cðx; zÞdz
0 ðz
(12)
and the reason for such a definition lies on the possibility to relate the dispersion of the pollutant directly to the height source. Substituting Eq. (3) in Eq. (12) we obtain
3. Derivation of moments and ground-level concentration
m1 ¼ R0h
(8)
(2)
where h is the boundary layer height and Q is the pollutant emission rate. The GILTT method is used to solve the equation given in Eq. (1). This analytical method performs a series expansion of the concentration C(x,z) in terms of the eigenfunctions associated to a Sturm–Liouville problem. The resulting transformed equation is also solved, analytically, applying the Laplace Transform technique. According the works of Wortmann et al. (2005) and Moreira et al. (2005), the solution for problem (1) is written, either for the eddy diffusivity Kz ðzÞ and Kz ðx; zÞ, as:
Cðx; zÞ ¼
s2z ¼ m2 :
(4)
s2s ¼
n 2h2 X ci ðxÞ h2 cosip þ 2hs z þ h2s : 3 c0 ðxÞp2 i ¼ 1 i2
(13)
Compared to the s2z expressed in Eq. (7), s2s approaches to higher values, it is nonetheless easy to prove that s2s hs2z when the assignment hs /z is restored.
and the remaining 4. Turbulent parameterization
Rh
mm
ðz m Þm Cðx; zÞdz ¼ 0 Rh 1 ; m ¼ 2; 3; 4: 0 Cðx; zÞdz
(5)
After substituting Eq. (3) in Eqs. (4) and (5) we obtain the explicit expressions for the four moments. The first and the second moment are expressed in a dimensional form as
In the atmospheric diffusion problems the choice of a turbulent parameterization represents a fundamental decision for the pollutants dispersion modelling. The reliability of each model strongly depends on the way as turbulent parameters are determined and related to the current understanding of the PBL
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(Seinfeld and Pandis, 1998; Mangia et al., 2002). We adopted the parameterizations suggested in Degrazia et al. (1997, 2000). In terms of the convective scaling parameters the vertical eddy diffusivity can be formulated as (Degrazia et al., 1997)
z 1=3 z 1=3 h z Kz ¼ 0:22w* h 1 1 exp 4 h h h z i ; 0:0003exp 8 h
(14)
and for stable conditions as (Degrazia et al., 2000)
Kz ¼
0:3ð1 z=hÞu* z 1 þ 3:7z=L
(15)
where z is the height; h the thickness of the boundary layer, u* and w* are the velocity scales for the horizontal friction and vertical convection respectively. L is the local Monin–Obukhov length defined as L ¼ Lð1 z=hÞ5=4 where L is the Monin–Obukhov length. The main feature of the diffusivity Kz for convective and neutral scenarios stands in the quasi symmetric behaviour of it around 0:5h scanned versus z. The maximum value of Kz increases with the convection. Concerning stable regimes, Kz increases rapidly with z up to z0:1h. Over such a limit Kz decreases until the ABL top. The wind speed profile can be described by the power law expressed as follows (Panofsky and Dutton, 1988):
uz ¼ u1
n z ; z1
(16)
where uz and u1 are the mean wind velocity at the heights z and z1 , while n is an exponent that is related to the intensity of turbulence (Irwin, 1979a). In fact this empirical wind profile matches well similarity profile in the surface layer, and on the contrary is valid in all the ABL. It turns out that the exponent n depends on the Pasquill stability class, and it is shown in Table 1. It turns out that for the unstable classes A, B, and C, uz increases rapidly in the surface layer up to a value which remains nearly constant along the whole upper part of the ABL. Concerning the D, E, and F classes the wind uz increases moderately but constantly through the all ABL. The horizontal wind and diffusivity behaviours will affect the statistical symmetries to be discussed later. The vertical eddy diffusivities (Eqs. (14) and (15)) and the wind profile (Eq. (16)) will be substituted in the solution of the ADE (Eq. (3)), obtained with the GILTT method, to calculate the maximum ground-level concentration (Cmax ) and then to evaluate the statistical descriptors z, sz , Sk and Ku. 5. Statistical descriptors plots In Fig. 1 are reported the four vertical distribution descriptors for the single ABL regime C, the choice for such scenario lies in the possibility to represents a convective regime still very close to a neutral one. The class C is characterized by a Monin–Obukhov length L ¼ 30 m, and a horizontal wind speed u1 ¼ 4 m s1 at the fixed height z1 ¼ 10 m. Simulations have been performed
Table 1 Summary of quantities used to set the ABL stability regimes.
u* (m s1) u (m s1) L1 (m s1) n
A
B
C
D
E
F
0.1 1.5 0.14 0.07
0.17 2.5 0.09 0.07
0.25 4 0.03 0.1
0.26 4.5 0 0.15
0.16 3.5 0.03 0.35
0.09 2.5 0.14 0.55
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adopting six values of the source height, in order hs =h ¼ 0.05, 0.1, 0.2, 0.3, 0.4, 0.5. Distribution statistics are plotted versus the dimensionless horizontal distance x=h (Observe that the definition of dimensionless distance is not the canonical one which is usually set in such a way to summarise the ABL regime. The reason for such a simpler definition stands in the intention to put emphasis to distances relatively to the ABL height h, to better appreciate the validity of our results which are valid for limited ABL height, and for long horizontal distances.) For each plot the long distance value of the moment approaches to a common asymptotic value, regardless the source height. For each plume source height the centroid starts from the respective source height and slowly approaches to the asymptotic value of 0:5h. The plots referring to the sz =h are obtained using the definition of Eq. (5), all tending to the asymptotic value z0:3h. The highest source emissions show a nearly Gaussian distribution, this is particularly manifest in the Sk and the Ku plots already for short distances. The kurtosis plots are strongly depending on the source heights. In all plots it is anyway very easy to highlight the terrain influence on the lowest sources. Kurtosis plots reveal a poorly peaked vertical distribution tendency in respect of the traditional Gaussian behaviour, which is still widely used for operative purposes. The power law vertical behaviour of the horizontal wind profile ensures the wind to increase rapidly in the lower part of the ABL. For the convective cases, above the heightzz0:2h the wind remains nearly constant. Because of such behaviour the highest sources releases seems to be poorly affected by the vertical structure of the horizontal wind. Furthermore they reach earlier than lower source release cases the tendency to approach to the asymptotic limit. Concerning the second moment sz =h, in Fig. 1 is reported the plot using the definition of Eq. (5). The first three plots behave homogeneously. It is also worth of note to remind the symmetric shape of the diffusivity Kz, for such a reason the highest sources released distributions tend to have a Gaussian shaped skewness rather soon after emission. Regarding the kurtosis plots two different behaviours are highlighted at short horizontal distances from the release point. Because of the turbulent parameterization mentioned earlier, the nearly symmetric scenario for the highest sourced distributions makes possible for Ku to decrease with the distance. On the other hand, the lower sourced distribution approach to a maximum, which means a narrow sharp distribution (leptokurtic). For long horizontal distribution then all kurtosis decrease to an asymptotic value z1:5 which represents a widely spread distribution (platykurtic). In Fig. 2 we reported the comparison of the above first four moments with the four ones, obtained with a solution that accept power low profile of wind and eddy diffusivity coefficient only (Brown et al., 1997). The diffusion coefficient used in our solution is symmetric and realistically go to 0 at the top of ABL, while in the solution adopted in Brown et al. (1997) Kz profiles do not go to 0 at the top of ABL. Moreover, in the approach of Brown et al. (1997) an infinite ABL is considered, so we limited the comparison for the two lowest sources (i.e. 0.05h and 0.1h) and for a maximum distance from the source x ¼ 10h in order to minimize the influence of the ABL top. The meteorological condition are the same of Fig. 1, but, in the case of the Brown et al. (1997) approach, Kz follows a power low profile with an exponent equal to 0.9 and matching the profile from Degrazia et al. (1997) at z ¼ 0.01h. The moments obtained with the GILTT approach increase more rapidly than the ones presented in Brown et al. (1997), but, in the case of skewness and kurtosis, these last ones reach a bigger maximum, showing a concentration vertical distribution more asymmetric and more ‘‘peakedness’’.
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Fig. 1. Plots of the four moments for the vertical distribution of the concentration with stability class C (L ¼ 30 m; u ¼ 4 m s1). The curves refer to the six emission heights hS =h ¼ 0.05, 0.1, 0.2, 0.3, 0.4, 0.5.
6. Position and value of the maximum ground level concentration Through the main advantages in using analytic expressions for the concentration field is the ability to evaluate the value of C(x,z) straightforwardly. This feature sets a significant difference with numerical models, whose evaluation of a concentration field in a single point involves the evaluation of the field in the grid points surrounding the source. This point assumes a strategic importance in the environmental management. By this point, to find the position and the value of the maximum ground level concentration Cmax ðx; 0Þ, it is enough to calculate its first derivative and then work out the horizontal distance xmax at which Cmax ðxmax ; 0Þ occurs. In order to show positions and values of the maximum ground level concentration in several different ABL scenarios, we adopted the wind velocity u1 and the Monin–Obukhov length L relative to the six Pasquill stability classes (Pasquill and Smith, 1984) using the definition of the classes and the semi empirical relation among ABL parameters and the empirical Pasquill classes proposed by Golder (1972) and fitted in Irwin (1979b). This allowed us to identify for each class the wind u, the Monin–Obukhov length L and the friction velocity u*. The set of meteorological parameters used for the six stability classes are shown in Table 1. In Table 2 the value of xmax and Cmax for the ABL scenarios corresponding to the six different stability classes and for six source heights hS =h varying from 0.001 to 0.5 are shown. According to Table 2, xmax increases with the source height, while the concentration
value decreases as the turbulence decreases in unstable cases (although an important role is due to the wind velocity too which increases as the turbulence decreases). Concerning the highest sources the results for the stable classes E and F are not reported because xmax turns out to be over 50 km far from the source, so they are out of range of reliability for a steady-state model. It is possible to see the dependence of concentration maximum value on the source height in Fig. 3. The two curves ðhS =hÞð1Þ and ðhS =hÞð2Þ are also shown to compare the qualitative behaviour of our results with the Brown et al. (1997) ones showing that a Cmax fh S with 1 a 2. 7. Plume standard deviation The most diffused operative models nowadays rely on the empiric estimates of the plume vertical standard deviation sz , the so-called sigmas (Pasquill and Smith, 1984). Although empirical Gaussian models have been proved to be not completely reliable they still enjoy a large extent of popularity in the environmental management and the air quality control. Then it is useful to compare such empirical expressions with those finally available based on the exact solution of the ADE. However, the accuracy of results still depends by the choice of the ABL parameterization. Despite such aspects, it is still significant to evaluate the reliability of the methodology adopting physically sounding theoretical parameterization. In our case we adopted the ABL parameterization mentioned in Section 4 which has been proved to perform quite
T. Tirabassi et al. / Atmospheric Environment 43 (2009) 2221–2227
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Fig. 2. Comparison of the first four moments obtained with the GILTT approach against the ones obtained with the approach described in Brown et al. (1997).
correctly when referred to ABL regimes whose experimental ground level concentration are recorded (Degrazia et al., 1997, 2000). For these reasons, we will use the sigma calculated in respect to the source height hS as defined in Eq. (12) to be able to compare the model performances with the empirical s2s . It is Table 2 Table of xmax and Cmax divided by source height and scanned through stability classes of the ABL. Stability hS ¼ 0:001 h
A B C D E F
xmax h
3 2 C 10 (s m ) xmax h Q
w0.001 w0.001 w0.001 w0.001 w0.001 w0.001
208 126 84 82 145 266
hS ¼ 0:1 h xmax h A B C D E F
hS ¼ 0:02 h
1.06 1.31 1.94 6.52 83.00 –
hS ¼ 0:2 h C xmax 104 (s m2) Q h 20.867 12.520 7.521 6.872 8.080 –
hS ¼ 0:05 h
C 103 (s m2) xmax Q h
0.23 11.517 0.28 6.910 0.40 4.348 0.88 4.329 2.47 6.595 5.67 9.336
0.56 0.69 1.01 2.79 16.80 53.80
C 103 (s m2) Q 4.233 2.540 1.558 1.501 1.996 2.313
hS ¼ 0:5 h
C xmax 104 (s m2) Q h
2.08 10.652 2.57 6.391 3.88 3.755 15.80 3.212 – – – –
worthy of note that the empirical s2s have been drawn from experiments with near-ground source heights, and involved horizontal distances up to few kilometres from the source (Pasquill and Smith, 1984; Arya, 1999). Despite such a local validity restriction, their use is usually extended to higher source heights as well as longer horizontal distances. As an example, the U.S. Environmental Protection Agency (EPA) admits their adoption even as far as 50 km away from the source.
w7 w10 w23 w73 – –
C 104 (s m2) Q 5.069 3.067 1.736 1.342 – –
Fig. 3. Plots of cmax ðx; 0Þ,Q 1 versus the dimensionless distance hs =h. The plot shows the two curves proportional to ðhs =hÞð2Þ and ðhs =hÞð1Þ .
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Fig. 4. Plots of sz =h for the six stability classes. For each class the sz is evaluated for source height hs =h ¼ 0.01 (empty squares), 0.1 (black squares), 0.2 (empty triangles) and 0.5 (black triangles) using Eq. (5) and hs =h ¼ 0.01 (empty stars), 0.1 (black stars), 0.2 (empty circles) and 0.5 (black circles) using Eq. (12).
Through the most diffused empirical sigmas, it is best remembered those proposed by Briggs (Arya, 1999), whose applicability in both rural and urban versions provides an interpolation scheme agreeing with those proposed by Pasquill and Gifford, except for the vertical sigma in a convective scenario where they approximate the Brookhaven curves (Arya, 1999). In our work we have compared the values of ss coming from Eq. (13) with those proposed by Briggs for rural terrain, setting
the meteorology scenarios where they are defined. In order to allow comparison we set u, u* and L according to the six Pasquill stability classes as we did previously in Section 4 introducing Table 1. In Fig. 4 the empirical standard deviations and those derived by our solution are compared for four different source heights hS =h ¼ 0.05, 0.1, 0.2, 0.5. We can see that for stable classes E and F the empirical sigma overestimate the analytical ones through the
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all range of horizontal distances, while, for the unstable cases A, B and C, there is a qualitative agreement near the source although such agreement occurs with a large uncertainty. Far from the source the functions show a rather different behaviour. In particular, the empirical ss approach to larger values than the analytical functions. The reason for such a great discrepancy stands on the crude approximation of an unlimited ABL. Nonetheless it is reasonable to adopt such an approximation when near-ground sources are considered. In fact the neutral case seems to better reproduce agreement between the different sigmas. It is known that the sigma functions are depending on the source height though the relationship between them is still an open issue. With the aim to better highlight such an aspect, Fig 4 shows a comparison between the two sigmas for unstable, neutral and stable scenarios and relative to four different source heights hs =h ¼ 0.05, 0.1, 0.2, 0.5. The plots show that at short horizontal distances from the source the empirical curves match the theoretical ones, then for large distances these reaches far higher values than the theoretical ones. For the neutral case the agreement reaches relatively the top in agreement. As a demonstration of the fact that the empirical ss are based on real low source patterns, in case D the agreement is maximum. On the other hand, the empirical curves for the neutral regime underestimate for all distances the pattern for the highest sources. Moreover, in the Fig 4 the sigmas based on Eq. (5) are plotted also. It is easy to see that for unstable regime they magnify the behaviour of sigmas using Eq. (12). In stable regimes they show the same behaviour while in the neutral one they underestimate empirical sigma in the case of low sources. Based on the results shown, it is necessary to bear in mind that finding any relationship between the theoretical curves and the effective source height is an important issue. Furthermore, it is also necessary to stress the point that the use of the traditional sigma schemes can lead to unreliable results in trying to reproduce the ground level concentration C(x,0). 8. Conclusions Using an analytical solution of the two-dimensional steadystate ADE for a point source release, without any restriction on the vertical profiles of wind and eddy exchange coefficients, we have derived formulae for the first four moments of the vertical concentrations for both surface and elevated releases. Moreover, it is possible to easily evaluate the position and value of maximum ground concentrations. The behaviour of the plume vertical standard deviation was outlined and compared with some very popular empirical ones, used in many operative air pollution models. It was outlined that discrepancies occur for tall sources and for long distances. It is evident that empirical formulae for ss need to take into account the height of sources and of ABL. The formulae presented here can be useful for operative evaluation of atmospheric dispersion and a better understanding of advection–diffusion phenomena. Acknowledgements The authors thank the Brazilian CNPq and CAPES, the Italian CNR and project ‘‘Laboratorio LaRIA’’ for the partial financial support of this work.
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