Reduction of the RACM scheme using Computational Singular Perturbation Analysis

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jd006743, 2006 Reduction of the RACM scheme using Computational Singular Perturbation Analysis T. Løvås, 1 E. Mastorakos, 1 and D. A. Goussis 2 Received 6 October 2005; revised 24 January 2006; accepted 15 February 2006; published 4 July [1] The modeling of dispersion of reacting pollutants in the atmosphere requires proper description of both the chemistry involved and the transport of pollutants. We present here highly reduced chemical schemes for rural and urban dispersion problems based on the Computational Singular Perturbation (CSP) technique. CSP identifies the steady-state species for removal, by resolving the fast and the slow chemistry components. Here, the CSP method is employed for the reduction of the Regional Atmospheric Chemistry Mechanism (RACM), which includes 77 species and 237 reactions. Three cases were tested: (1) a polluted scenario including ground emissions, (2) a relatively cleaner environment with 10% of the emission levels of case 1, and finally (3) rural conditions without any emissions, for which the effects of emissions on the reduction procedure were directly studied. It was found that reduced mechanisms with high degree of reduction, down to a 16 step mechanism, i.e., 56 species in steady state, produced excellent results for most species for all cases. These reduced mechanisms did not change much between urban or rural conditions and the inclusion of emissions did not affect significantly the selection of major species. This indicates the persistence of a well-defined fast subset of the mechanism, suggesting that the produced reduced mechanisms are valid for a wide range of conditions in the atmosphere. Citation: Løvås, T., E. Mastorakos, and D. A. Goussis (2006), Reduction of the RACM scheme using Computational Singular Perturbation Analysis, J. Geophys. Res., 111,, doi: /2005jd Introduction [2] The description of atmospheric chemistry is continually expanding in order to include new or trace pollutants resulting in huge chemical mechanisms, with thousands of reactions and hundreds or even thousands of species [Dodge, 2000; Jenkin et al., 1999]. If these extensive mechanisms are to be used in, for example, three-dimensional (3-D) Eulerian grid models for urban scale dispersion problems [Jacobson, 1999], then a prohibitively large number of equations need to be solved for each time step. However, the demand is growing for more detailed descriptions of the chemistry and hence the need for developing reduced chemical mechanisms to manageable sizes without loss of important information. [3] As the mechanisms become more complex, it is important to develop automated procedures that can reduce the mechanisms in a reliable manner with minimum input by the user. Over the last few years, several reduction techniques have been employed to atmospheric chemistries and presented in the literature. Often these techniques are used in combination, including sensitivity analysis [Heard 1 Department of Engineering, University of Cambridge, Cambridge, UK. 2 Institute of Chemical Engineering and High Temperature Processes, Rio-Patras, Greece. Copyright 2006 by the American Geophysical Union /06/2005JD et al., 1998; Djouad et al., 2003], lumping techniques [Atkinson et al., 1982; Aumont et al., 1996; Crassier et al., 2000; Fish, 2000; Sportisse and Djouad, 2000; Djouad and Sportisse, 2002], and timescale separation techniques [McRae et al., 1982; Lowe and Tomlin, 2000; Neophytou et al., 2004]. [4] Sensitivity analysis aims to define the elementary reactions in the mechanism that have a significant effect on the evolution of the system, and important or sensitive species are identified in an iteration procedure. Species that are not sensitive to the evolution can be removed from the system. The procedure is derived for a general chemical system by Turányi [1990] and has been widely used for combustion chemistries. Mauersberger [2005] presented a so-called iterative screening and structure analysis (ISSA) of a combined gas-phase and a liquid-phase chemical mechanism (RACM, discussed later, and CAPRAM2.4 [Ervens et al., 2003]) employing a reaction and sensitivity analysis in order to identify minor reaction paths not involving important species. [5] According to lumping techniques, chemical species that follow similar reaction paths are lumped together and are treated as one artificial species. Some widely used mechanisms for atmospheric chemistry were developed with these techniques, including the Carbon-Bond Mechanism (CBM-IV) by Gery et al. [1989], the Regional Acid Deposition Model (RADM2), and the Regional Atmospheric Chemistry Mechanism (RACM), both by Stockwell et al. [1990, 1997]. Sportisse and Djouad [2000] presented an 1of16

2 algorithm to build up lumped species in a systematic way based on a preprocessing step consisting of a timescale analysis of the chemical system. For example, slow reacting species (typically volatile organic compounds) can be lumped together according to this method. [6] Timescale separation techniques are based on the fact that chemical systems involve timescales that span a very wide range, the fastest of which are usually much smaller than the timescales characterizing the physical processes and the time range of interest. These fast chemical timescales can be removed, by applying the quasi-steady-state assumption (QSSA), without significant loss of accuracy or important information. With this approach, not only the size of the system is reduced, but also stiffness is removed, accelerating thus the numerical integration. For this reason the timescale separation methods are particularly appealing for reduction purposes. An analysis of the errors introduced by the QSSAs is used in itself as a selection criterion for reduction by Whitehouse et al. [2004], who employed the method to the Leeds Master Mechanism [Jenkin et al., 1997] for troposphere chemistry and achieved a factor of two reduction in the number of species. [7] There are several techniques on how to identify the fast component of the mechanism that gives rise to these fast timescales. One simple technique is to estimate the chemical timescales from the inverse elements of the Jacobian of the chemical source term [McRae et al., 1982], which will be discussed more fully in section 4 of this paper. This definition of timescales can be supplemented by additional information such as sensitivity coefficients as proposed by Løvås et al. [2000] for combustion chemistries, to produce a more reliable selection of candidate species for the steady-state assumption. More rigorously, the Intrinsic Low Dimensional Manifold (ILDM) method proposed by Mass and Pope [1992, 1994] and Maas [1998] for combustion has also been applied to atmospheric chemistries by Lowe and Tomlin [2000]. Use of ILDM requires the construction of large look-up tables. This procedure demands significant computational memory and becomes cumbersome when the chemical mechanisms are complex and the number of steady-state species is large, typically the case for atmospheric chemistries. Lowe and Tomlin [2000] proposed to use repro-modeling to overcome this problem, where the species evolution is described using simple difference equations. [8] The present work utilizes a timescale separation method called Computational Singular Perturbation (CSP) [Lam and Gaussis, 1988, 1991, 1994; Lam, 1993]. This technique has been validated for a variety of problems, including flame and NO x chemistry [Goussis, 1996; Massias et al., 1999a, 1999b] and autoignition for engine simulations [Valorani and Goussis, 2001; Løvås et al., 2002]. CSP has also been successfully applied to atmospheric chemistry by Trevino and Mendez [1999] and Neophytou et al. [2004], where the CBM-IV mechanism of 28 species and 74 reactions was reduced down to 13 major species, i.e., 15 species put in steady state, and was applied to both urban and rural pollution conditions. [9] In this paper, we present CSP-derived reduced mechanisms for RACM, as described by Stockwell et al. [1997]. RACM is a gas-phase chemical mechanism containing 77 species and 237 reactions, 23 of which are photolysis reactions. RACM is an extended and substantially revised version of the popular RADM2 mechanism, incorporating modified reaction rates and providing improved accuracy in the prediction of species concentration evolution. RACM was validated for rural and polluted conditions and has been tested against environmental chamber data with satisfying results reported by Stockwell et al. [1997]. It is noted that RACM is significantly more detailed and demanding than the CBM-IV mechanism reduced previously, as it includes many stable inorganic species, inorganic intermediates, stable organic species, and organic intermediates. Hence RACM represents a greater challenge for the CSP method. [10] The paper is organized as follows: in the second section we discuss the reduction procedure and present the methodology of the CSP technique. In the third section we discuss the physical system toward which the reduced chemistry is validated. In the fourth section we discuss the associated timescales of the chemical mechanism and analyse the effect of changing conditions and ground emissions on the timescales and the subsequent reductions. We also present reduced mechanisms with increasing degree of reduction and discuss their limitations. Finally, we conclude and discuss future applications in the last section. 2. Formulation and Model Description 2.1. Overview of the CSP Methodology [11] The core of CSP is to resolve the set of differential equations governing the chemical system in a fast and a slow component. This decomposition is achieved by employing the CSP set of basis vectors. Here these vectors are approximated by their leading order expression, provided by an eigenvalue analysis of the system s Jacobian. The work on developing the CSP method can be studied in detail in a series of papers [Lam and Gaussis, 1988, 1991, 1994; Lam, 1993; Goussis, 1996; Massias et al., 1999a, 1999b]. The CSP procedure for producing reduced mechanisms has been described by Massias et al. [1999a, 1999b]. Given a detailed mechanism, the procedure consists of the following steps. First, a numerical solution of the time evolution of the system is obtained, using the detailed mechanism and a representative set of initial conditions describing the system, in our particular case a mechanism valid for dispersion problems in remote to polluted conditions from the Earth s surface to the upper troposphere. Second, the number of steps in the reduced mechanism is prescribed, so that the number of steady-state species is set. Third, the steady-state species are identified, by the so-called CSP-pointer, discussed next. Finally, the reduced chemical mechanism is constructed, via simple linear algebra manipulations, having the form of global reactions accompanied by their reaction rates being linear combinations of the elementary rates. [12] The procedure employed here involves only the first three steps above, so that the resulting reduced model consists of the steady-state relations and the original differential equations for the remaining species. It is straightforward to show that the reduced models created with this methodology provide equivalent accuracy with those produced by the algorithm of Massias et al. [1999a, 1999b]. 2of16

3 2.2. Local and Integrated CSP Pointers [13] The second and third steps of the CSP procedure will be outlined in more detail below. These involve setting the number of the desired global steps in the reduced mechanism, noted with an S, and determining the steady-state species by examining the local CSP pointer for each species. The number of steady-state species, say M, is therefore given by M = N E S where N is the total number of species in the full mechanism and E is the number of elements in the mechanism; in the case of RACM, N = 77 and E = 5 (C, O, H, S, and N). For example, for a choice of a 16 step reduced mechanism, the number of selected steady-state species M = = 56. [14] The set of balance equations that describes the evolution of the chemical system takes the form dy i dt ¼ g i ðy 1 ; Y 2 ; ::; Y N ; tþ ¼ P i þ L i Y i þ Emi i ; ð1þ where Y i is the species mass fraction of species i of total mass, g i is the chemical source term that is a function of all species concentrations and is given by 1/sec. The source term can be split into a production term, P i, and a consumption term that is a product of the consumption rate, L i, and the species mass fraction. In the present work ground emissions are also included and are represented through an extra term in the balance equation, Emi i, set independently for some selected species. This term is independent of the species concentrations and, for simplicity, is held constant throughout the time integration. [15] Application of the CSP involves the introduction of a set of NN-dimensional vectors [Massias et al., 1999b], a,so that equation (1) is cast as: dy dt ¼ a mh m þ a s h s ; where a m is a (N M)-dimensional matrix for the M fast basis vectors and a s is a (N (N M))-dimensional matrix for the remaining N M slow basis vectors. h m and h s are the socalled fast and slow amplitude vectors, being the projections of the source term vector g along the vectors in a m and a s, respectively; i.e., h m = b m g and h s = b s g, where b m and b s form the set of dual basis vectors and satisfy the relations b m a m = I, b s a m = 0, b m a s = 0, and b s a s = I. The CSP basis vectors in a m and a s are approximated by the eigenvectors of the Jacobian of g; the ones related to the M largest eigenvalues in a m and the ones related to the N M slowest ones in a s. The first term on the right-hand side in equation (2) is insignificant when M steady-state assumptions are valid, i.e., h m 0, resulting in the following expression: dy dt a sh s : [16] The CSP pointer, employed for the identification of the steady-state species, is defined by the diagonal elements of the dyadic a m b m : ð2þ ð3þ D ¼ diag 1 M a 1b 1 þ a 2 b 2 þ...þ a M b M : ð4þ The local CSP pointer as given in equation (4) is a function of time and indicates the influence of the M fastest chemical timescales on species i [Valorani et al., 2003]. The pointer takes the value between zero and one. When D i is close to unity, species i is fully influenced by the M fastest timescales and is a candidate for being set to steady state. The opposite occurs when D i becomes close to zero, the species is weakly influenced by the M fastest chemical timescales and can not be characterized as steady state. The M elements of D that are closest to unity are pointed out as the best steady-state candidates. However, a pointer may change value during the course of the simulation and hence a global selection parameter needs to be defined for reduced mechanisms that aim to be valid in the nonlocal sense. [17] Massias et al. [1999b] propose two alternative integration procedures in order to achieve a global measure for the overall influence of fast and slow chemistry. One procedure involves only the integration of the pointer, D i, itself. The second procedure includes a weighting factor in the integration, accounting for the net production rate and the species concentration. Here, a variation of the second approach is employed, also used by Neophytou et al. [2004], defined as I i ¼ Z T2 T 1 D i jg i ðþ t j dt; ð5þ j j g max i where T 1 and T 2 are integration limits; in this case the starting time and the end time of the simulation. Equation (5) defines the integrated CSP pointer. Since the integrated pointer gives an overall measure of how much the species is influenced by the fast timescales compared to the other species, the M species with the highest value of integrated pointer, I i, are those automatically selected by the CSP method as the best candidates for the steady-state approximation. [18] There is certain flexibility in the selection of steadystate species when the values of their integrated CSP pointer are not too different. From experience, it is well known that in many applications and/or a wide range of operation conditions some species are important for the overall accuracy of the simulation, e.g., the ground state oxygen atom O( 3 P) is important for the production of O 3. In such cases, it is possible to keep these species as non-steady-state even when their CSP pointers suggest them as good steadystate candidates. This makes the user of the procedure able to optimize the resulting reduced mechanism. This will be further discussed in section Physical System 3.1. Box Model [19] The model employed in the present work is a zerodimensional box model, i.e., we assume that the species concentrations are uniform in space and their evolution is governed by equation (1). Advanced grid models for atmospheric dispersion problems often use an operator splitting technique, where chemical and physical processes are treated separately for each time step. Hence each grid point is considered to be an independent box at a given time step. For the purpose of studying the applicability of 3of16

4 Table 1. List of All RACM Species, Their Abbreviations, and Definition (From Stockwell et al. [1997]) Species Symbol Species Name and Definition Stable inorganic compounds 1. O 3 ozone 2. H 2 O 2 hydrogen peroxide 3. NO nitric monoxide 4. NO 2 nitric dioxide 5. NO 3 nitric trioxide 6. N 2 O 5 dinitrogen pentoxide 7. HONO nitrous acid 8. HNO 3 nitric acid 9. HNO 4 pernitric acid 10. SO 2 sulfur dioxide 11. SULF sulfuric acid 12. CO carbon monoxide 13. CO 2 carbon dioxide Abundant Stable Species 14. N 2 nitrogen 15. O 2 oxygen 16. H 2 O water 17. H 2 hydrogen Inorganic Short-Lived Intermediates 18. O( 3 P) ground state oxygen atom 19. O( 1 D) excited state oxygen atom 20. HO hydroxyl radical 21. HO 2 hydroperoxy radical Stable Organic Compounds 22. CH 4 methane 23. ETH ethane 24. HC3 alkanes, alcohols, esters and alkynes (low HO rate const.) 25. HC5 alkanes, alcohols, esters and alkynes (mediate HO rate const.) 26. HC8 alkanes, alcohols, esters and alkynes (high HO rate const.) 27. ETE ethene 28. OLT terminal alkenes 29. OLI internal alkenes 30. DIEN butadien and other anthropogenic diens 31. ISO isoprene 32. API a pinene and other cyclic terpenes with one double bond 33. LIM d-limonene and other cyclic diene-terpenes 34. TOL toluene and less reactive aromatics 35. XYL xylene and more reactive aromatics 36. CSL cresol and other hydroxyl substituted aromatics 37. HCHO formaldehyde 38. ALD acetaldehyde and higher aldehydes 39. KET ketones 40. GLY glyoxal 41. MGLY methylglyoxal and other a-carbonyl aldehydes 42. DCB unsaturated dicarbonyls 43. MACR methacrolein and other unsaturated monoaldehydes 44. UDD unsaturated dihydroxy dicarbonyl 45. HKET hydroxyl ketone 46. ONIT organic nitrate 47. PAN peroxyacetyl nitrate and higher saturated PANs 48. TPAN unsaturated PANs 49. OP1 methyl hydrogen peroxide 50. OP2 higher organic peroxides 51. PAA peroxyacetic acid and higher analogs 52. ORA1 formic acid 53. ORA2 acetic acid and higher acids Organic Short-Lived Intermediates 54. MO2 methyl peroxy radical 55. ETHP peroxy radical formed from ETH 56. HC3P peroxy radical formed from HC3 57. HC5P peroxy radical formed from HC5 58. HC8P peroxy radical formed from HC8 59. ETEP peroxy radical formed from ETE 60. OLTP peroxy radical formed from OLT 61. OLIP peroxy radical formed from OLI 62. ISOP peroxy radical formed from ISO and DIEN 63. APIP peroxy radical formed from API 64. LIMP peroxy radical formed from LIM 65. PHO phenoxy radical and similar radicals 66. ADDT aromatic-ho adduct from TOL 67. ADDX aromatic-ho adduct from XYL 68. ADDC aromatic-ho adduct from CSL 4of16

5 Table 1. (continued) Species Symbol Species Name and Definition 69. TOLP peroxy radicals formed from TOL 70. XYLP peroxy radicals formed from XYL 71. CSLP peroxy radicals formed from CSL 72. ACO 3 acetyl peroxy and higher saturated acyl peroxy radicals 73. TCO 3 unsaturated acyl peroxy radicals 74. KETP peroxy radicals formed from KET 75. OLNN NO3-alkane adduct reacting to form carbonitrates + HO2 76. OLND NO3-alkane adduct reacting via decomposition 77. XO 2 Accounts for additional NO and NO2 conversions. reduced mechanisms and their suitability to predict the time evolution of the chemical species, a simple box model is the natural choice as it does not contain the influence of diffusion and convection terms. [20] The model is supplemented by a set of initial conditions and operating parameters needed in the chemical reaction rates such as the zenith angle. The numerical solver employed here to solve the remaining differential equations is CHEMEQ2, proposed by Mott et al. [2000], a predictorcorrector type solver suitable for stiff ordinary differential equation (ODE) systems. The scheme avoids some of the start-up costs associated with more accurate backward differential (BDF) methods like LSODE [Hindmarsh, 1983], VODPK [Brown et al., 1989], and SMVGEAR [Jacobson and Turco, 1994]. (For comparison of stiff system solvers for atmospheric chemistries, see Sandu et al. [1997].) The scheme is therefore particularly useful in situations requiring operator splitting, such as multidimensional reacting flow problems, using reduced chemistry. This solver is applicable to any reaction kinetics and significant experience has been gained with the solver for combustion problems [Wright et al., 2005]. For a typical box model calculation with the present RACM chemistry, CHEMEQ2 predicted concentrations within 4% of those calculated by VODPK for O 3, while the other species were predicted within a single percentage of the solution given by VODPK. [21] The RACM reaction rates are given by Stockwell et al. [1997], and a list of species is reproduced here as reference for the reader in Table 1. The mechanism consists of 214 Arrhenius-type reactions where the reaction coefficient is given by k = AT n exp[ (E/R)/T], and A is the Arrhenius coefficient, T is the temperature with an experimentally defined exponential factor n, E is the activation energy, and R is the universal gas constant. In addition, the mechanism contains 23 photolysis reactions where the reaction rate is a function of the solar zenith angle. The photolysis frequencies are calculated independently and accessed by the box model through a look-up procedure for the given zenith solar angle. The photolytic rates are based on the Tropospheric Ultraviolet-Visible Model (TUV) by Madronich and Flocke [1998] (see also Stockwell et al. [1997] and references therein). [22] Results are presented for three different cases, two of which are similar to cases 19 and 20 discussed by Stockwell et al. [1997]. These cases are a polluted atmosphere with significant emissions, and a less polluted case with only 10% of the initial concentrations and emissions levels from the previous case. We have included a third case, which is identical to the second cleaner atmosphere, but without any emissions, in order to explore whether the external timescale imposed by the emissions may affect the selection procedure. The initial conditions for these three cases are listed in Table 2 together with corresponding emission rates and will hereafter be denoted as cases 1, 2, and 3. [23] The calculations are started at 1200 (noon) on 21 June located at a longitude of E0.0 and latitude of N40.0 and ended at 1200 (noon) the next day. Results are reported every 3 min, but CHEMEQ2 uses internally a much smaller adaptive time step. The cloud coverage is assumed constant throughout the calculations. Also, no equation is solved for the temperature that is thus kept constant at K Solving for Steady-State Species [24] For the reference solution, the full set of differential equations as given by equation (1) for the 77 species is solved by the ODE solver. However, when steady-state species are identified, their concentration is calculated as follows: dy i dt ð ¼ P i þ L i Y i þ Emi i ¼ 0 ) Y i;ss ¼ P i þ Emi i Þ : ð6þ L i [25] The remaining species concentrations are affected by the steady-state species concentrations because the latter appear in the reaction rates. Equation (6) is a nonlinear algebraic system whose solution is the set of concentrations of the steady-state species. In the present work, a simple fixpoint inner iteration is employed to solve equation (6) and is fully incorporated into the subroutines calculating the righthand side of equation (1) for the non-steady-state species that is supplied to CHEMEQ2. The code is written so as to handle any set of steady states. [26] The inner iteration loop can converge slowly when the set becomes large; the steady-state species being strongly coupled. In some situations, it is found that the inner iteration is as time consuming as the solution of the detailed mechanism (i.e., equation (1) for all species, without any steady-state assumptions). The implications of this are discussed later in the paper. It is noted here that the inner iteration is observed to be slow when certain pairs of species are not in the same state, i.e., that one is steady state and another is not. For example, when PAN is in steady state, the inner iteration converges faster when also ALD is in steady state. Identification of such pairs may be made through a sensitivity analysis and an exploration of the 5of16

6 Table 2. Initial Concentrations and Emission Levels for the Three Cases Case 1, Case 2, and Case 3 a Case 1 Polluted Case 2 Rural Case 3 Rural Initial Conditions, ppb H 2 O (1%) (1%) (1%) O NO NO HNO CO CH H 2 O HCHO O (20.9%) (20.9%) (20.9%) N (78.1%) (78.1%) (78.1%) Emissions, ppb/min Acet-aldehyde CO Ethene Ethane HC HC HC HCHO Ketone NO Internal alkanes Terminal alkanes SO Toulene Xylene a The remaining species are set to an initial concentration of ppb. QSSA errors (see section 4.1), but such a systematic effort is not employed here. 4. Results and Discussion [27] In this section we analyze and discuss the reduced mechanisms constructed with the CSP method outlined in the previous section. This involves discussion of the timescales derived from the CSP analysis, the selection of steady-state species according to the integrated CSP pointers, and the general performance of the reduced mechanism as compared to the reference solution Chemical Timescales [28] If the set of species mass fractions at time t 0 is given by the vector Y 0, the mass fractions given by Y =[Y 1, Y 2,...,Y N ] T after a time increment Dt will be described by the following [Neophytou et al., 2004]: dy dt ¼ g, d ð Y 0 þ DYÞ g dt 0 þ JDY, dðdyþ g dt 0 þ JDY; ð7þ since Y 0 is constant in time. Here J is the Jacobian of the source term vector g 0 (i.e., J ij i /@Y j ). When assuming no second-order reactions like A + A! B, and the terms P i and L i in equation (1) are constant, J is a diagonal matrix with L i as diagonal elements. From equation (7) we have that DY i / expð L i DtÞ: ð8þ This implies that the species relaxes to a local equilibrium according to a timescale of 1/L i. The inverse of the diagonal elements of the Jacobian are commonly used as definitions of the chemical timescales, t i = jj ii 1 j = j(@g i /@Y i ) 1 j. This method was employed for atmospheric chemistry by McRae et al. [1982] and studied in great detail for combustion chemistries by Turányi [1990] and Tomlin et al. [1997]. The method gives a reliable first hand indication about the characteristic timescale for each species. According to this approach, no off-diagonal elements are considered. [29] Løvås et al. [2000] argued that to justify setting a species in steady state, not only must the characteristic lifetime be small, but also the errors introduced to the selected steady-state species concentrations must be small. These two criteria do not necessarily coincide. For example, in combustion the hydrogen atom is very reactive and has a short characteristic lifetime, but it may create significant errors to the system if set to steady state. This can be mathematically described through a classic Taylor series expansion of the species concentration equation for a species j, which is a function of the concentration of species i, i.e. c j = F(c i ). The expansion of F(c i ) will take the form Fc ð i þ Dc i Þ ¼ Fc ð i ð iþ Dc i þ...; i representing the new concentration F(c i + Dc i ) for species j after a change Dc i. Thus the error introduced to concentration c j, f j = F(c i + Dc i ) F(c i ), can be expressed as the following (ignoring second-order terms): ð Þ f j ¼ Fc ð i þ Dc i Þ Fc ð i i Dc i ) f i f i : ð10þ 6of16

7 This shows that the errors introduced are proportional to the species sensitivity parameter expressed by S ji j /@c i, leading to the level of importance parameter introduced by Løvås et al. [2000] that combines the chemical characteristic lifetime as given by t j with a species sensitivity parameter S ji toward a chosen i. [30] The level of importance includes the diagonal elements of the Jacobian and species sensitivity but does not account for the nonlinearity in the source terms themselves. The CSP pointers as derived above do account for the offdiagonal elements and are thus a more reliable parameter for defining the timescales of the species for highly nonlinear chemical systems. Bell et al. [2003] discuss in detail the implications of the presence of these couplings for the proper analysis of experimental data from the atmosphere. [31] Figure 1 shows a direct comparison between the CSP pointers (to the left) and the corresponding timescales as given by t i (to the right) for selected species as function of time. The values correspond to the urban case 1 from Table 2. First, let us focus on the differences in the CSP pointers between the species. The two top plots, Figures 1a 1b, show values for species that are all short lived intermediates apart from OP2, which is a higher organic peroxide and a stable compound. The species that are candidates for steady state are the species with a CSP pointer close to unity. It is clear from the two top left plots that for many species, the CSP pointer takes a clear value close to 0 or 1 throughout the simulation. Figure 1c shows that abundant stable species, such as N 2,O 2,H 2 O, and H 2, have all CSP pointers very close to 0 throughout the simulation, as expected. [32] For some species, the value of the CSP pointer changes during the course of the simulation. In Figure 1a, HC8P, a peroxy radical from alkanes, is shown as an example of a species whose CSP pointer dips for short periods. This happens just as the photolysis reactions slow down at around 20 hours in the evening and restart at around 03 hours in the morning driven by the sunset and sunrise. NO and NO 2, can take intermediate values, as shown in Figure 1d, indicating the importance of offdiagonal elements on these species. These species are involved in the photolysis reactions and will be affected by the rapidly changing photolysis rates that go to zero at night. In Figure 1d, the pointer for O 3 always has low values, indicating that it should not be put in steady state. [33] The right-hand plots, Figures 1e 1h, show the characteristic timescales taken from the inverse of the diagonal elements of the Jacobian matrix, i.e., the conventional definition of timescale. Species that are candidates for steady-state selection are those with short chemical lifetime. Compared to the CSP pointers, the species with low values of chemical timescales do indeed match those with high values of pointers, and those species with long timescales correspond to those with low values of the CSP pointer, as also observed from a similar comparison for the CBM-IV mechanism by Neophytou et al. [2004]. The timescales span many orders of magnitude and change significantly during the integration domain. This makes their use for QSSA selection cumbersome, while the CSP pointers produce a more consistent steady-state species selection because most species are either close to unity or close to zero, with few species taking intermediate values. [34] The major difference between urban and rural conditions is the ratio of volatile organic compounds, VOC, to nitric oxides, NO X (VOC/NO X = 1.43 for the urban scenario as opposed to VOC/NO X = 14.3 for the rural case). When ground emissions are included in the simulations, their emission rate is added to the production rate in the source term for the species. In order to study this effect, we have compared two rural cases with identical initial conditions, one including the emissions and one case without emissions. The CSP pointers as function of time for a selected set of species are plotted in Figure 2 for rural conditions. For consistency with Figure 1, we have included the corresponding timescale as given by t i. Regarding the comparison between CSP pointer and timescales, the same trend as seen in Figure 1 is evident also in Figure 2, i.e., that the fast timescales correspond to CSP pointers close to unity and that very few CSP pointers have intermediate values, in contrast to the timescales. The data show that the effect of emissions does not alter either the timescales or the CSP pointers. Hence the emissions do not affect the inner dynamics of the system, nor the manner in which the species are affected by the fast and slow timescales, and hence the steady-state species selection is affected by the presence of emissions. This robustness of the QSSA s selection was also observed by Neophytou et al. [2004] in the CBM-IV mechanism by examination of timeseries of concentrations provided by an Eulerian Grid code including transport. Therefore the CSP reduced mechanisms described here should be applicable to a wide range of atmospheric pollutant dispersion problems Selection of Steady-State Species Based on CSP [35] Table 3 lists three sets of values of integrated pointers for a 20 step mechanism (M = 52) resulting from three different initial conditions. For brevity, only 40 of the 77 species pointers are included. The chemistry will be affected by the initial conditions and thus also the CSP pointers are expected to be different when the atmospheric conditions are changed. This is reflected in the different ordering of the species according to integrated CSP pointers in the three scenarios given in Table 3. Although the ordering is different between the three cases, most of the species retained in the mechanism as non-steady-state remain the same. The species that do differ between the urban case 1 and the rural case 2 have CSP pointers close to the cut-off limit, for example HCHO and ONIT. The same trend is found for the integrated CSP pointers resulting from Case 3 for rural conditions without emissions. The species retained in the mechanism as non-steady-state species are to a large extent the same as for the two previous cases, and the species that do differ are found to all lie close to the cutoff limit. For case 2 with emissions, MO 2,NO 2, and HO 2 are replaced as steady-state species by TOL, ETH, and OP2 in Case 3. The remaining 22 species are the same. This further extends the previous conclusion that the species affected by the fast chemistry have little sensitivity to the external timescales imposed by the emissions. For all three cases, abundant stable species like H 2,O 2,N 2, and H 2 O are retained in the mechanism, while O 3 is kept together with important pollutants like the acid HNO 3 and sulfuric acid, HSO 3, and organic 7of16

8 Figure 1. Local pointers (left) and timescales defined from t j = jj 1 jj j (right) as function of time for selected RACM species. All values results from analysis from urban conditions, case 1. 8of16

9 Figure 2. Local pointers (left) and timescales defined from t j = jj 1 jj j (right) as function of time for selected species for rural conditions, cases 2 and 3. acids ORA1 and ORA2. Important intermediates for photolysis reactions like ketones and atomic oxygen O( 3 P) are also retained in all three cases. Thus we may use a single set of reduced species for both scenarios. This is consistent with the findings of Neophytou et al. [2004] who also concluded that the reduction did not change considerably dependent on whether nighttime or daytime chemistry was considered for the CBM-IV mechanism. [36] The CSP pointers may vary by altering the value of M, the required number of steady-state species. Selected steady-state species for the urban polluted condition, Case 1, are listed in Table 4 for increasing degree of reduction. The first column lists the steady-state species for a 30 step mechanism for M = 42 and the second column lists the steady-state species for the 20 step mechanism discussed above where M = 52. This indicates that 10 more species would be identified as steady state for the 20 step mecha- 9of16

10 Table 3. Integrated CSP Pointers for a 20 Step Mechanism (M = 52) a Case 1 Polluted Case 2 Rural Case 3 Rural Steady-State Species HNO E+06 GLY E+07 GLY E+18 ALD E+06 PAN E+07 MGLY E+18 GLY E+06 ALD E+07 O1D E+15 HC E+05 HC E+07 OP E+15 ETE E+05 N2O E+06 ONIT E+13 XYL E+05 XYL E+06 N2O E+07 PAN E+05 ETE E+06 HONO E+07 HO E+05 NO E+06 HO E+07 HCHO E+05 NO E+06 ETH E+07 NO E+05 MO E+05 HNO E+07 N2O E+05 NO E+05 TOL E+07 NO E+05 HO E+05 NO E+06 NO E+04 ONIT E+05 NO E+06 Non-Steady-State Species ONIT E+04 HCHO E+05 NO E+06 HKET E+03 HC E+04 MO E+05 TOL E+03 HKET E+04 HO E+05 HC E+03 HNO E+04 HCHO E+05 HNO E+03 OP E+04 HNO E+04 OP E+03 TOL E+03 HKET E+04 HC E+02 KET E+03 PAA E+04 OP E+02 OP E+03 OP E+03 O E+02 PAA E+03 HC E+03 KET E+02 O E+03 H2O E+02 H2O E+02 HC E+02 O E+02 PAA E+02 ETH E+02 H E+00 ETH E+01 H2O E+01 CO E-02 CO E+01 CO E+01 CH E-02 SO E+00 H E+00 ORA E-05 H E-02 SO E-01 ORA E-07 CH E-03 CH E-02 H2O E-08 H2O E-12 H2O E-12 O E-14 ORA E-12 ORA E-13 CO E-17 ORA E-13 CO E-13 SULF E-18 SULF E-13 SULF E-14 O3P E-60 O E-14 ORA E-14 SO E+00 CO E-14 O E-15 N E+00 O3P E-60 O3P E-60 HC E+00 N E+00 N E+00 KET E+00 a Only 40 of the 77 species with the smallest pointer value for the three atmospheric conditions (see Table 1) are shown. nisms. These are NO, NO 2,N 2 O 5,HC 8, ETE, XYL, HCHO, ALD, GLY, MGLY. Short-lived peroxy radicals from alkanes (MO 2, ETHP, HC 3 P, HC 5 P, HC 8 P) and inorganic radicals such as HO and HO 2, together with the nitrogen-containing radicals such as NO 3 and XO 2 are identified as steady state in both cases. A cursory examination of the timescales in Figures 1 and 2 indicates that these species are fast and hence they would probably be selected as steady state by the conventional reduction techniques. As the mechanism is reduced even further to an 18 step mechanism, M = 54, two more species are to be set to steady state. However, three new steady-state species are identified, TOL, OP2 and PAA, one of them replacing NO 2 that is no longer in steady state in the 18 step mechanism. These are the four species underlined in Table 4. Apparently, in a 20-step mechanism (M = 52) NO 2 is less affected by the 52 fast timescales than are TOL, OP2, and PAA. In contrast, in an 18-step mechanism (M = 54) TOL, OP2, and PAA are less affected by the 54 fast timesscales than is NO 2. [37] The above results demonstrate QSSA selections using the integrated CSP pointer provides steady-state species consistent with our expectations and with simplified timescale analysis and sets of species that do not change significantly in the presence of emissions or between different pollution scenarios. Therefore reduced mechanisms of a wide validity range are expected to arise from employing these steady-state assumptions. The accuracy of these reduced mechanisms is discussed next Reduced Mechanism Simulations [38] The error introduced by the steady-state assumptions will not only affect the steady-state species concentrations but also the evolution of the species not in steady state, since the former appear in the net reaction rates of the latter. Therefore the accuracy of the reduced mechanisms must be quantified. In this subsection, we study the performance of the reduced mechanisms and the degree of reduction after which significant errors are becoming evident. [39] We have performed reduction based on the three atmospheric conditions presented in Table 2, cases 1 3, and have compared in each case the performance of reduced mechanisms with increasing degree of reduction achieved by reducing the number of global steps (i.e., increasing the number of steady-state species). Some species profiles calculated using the full chemical system for case 2 and case 3 are plotted in Figure 3 to emphasize the effects of emissions on the evolution of some selected species. Despite the fact that the polluted case 1 has significantly higher concentrations than case 2, case 1 is not included in Figure 3 as the qualitatively picture is the same as for case 2. For Table 4. Steady-State Species for Urban Case, Case 1, With Emissions for Mechanisms With Increasing Degree of Reduction, Ranging From 30 Steps Down to 18 Steps 30 Step (M = 42) 20 Step (M = 52) 18 Step (M = 54) NO3 HONO NO3 HONO NO3 HONO HNO4 O1D HNO4 O1D HNO4 O1D HO HO2 HO HO2 HO HO2 OLT OLI OLT OLI OLT OLI DIEN ISO DIEN ISO DIEN ISO API LIM API LIM API LIM CSL DCB CSL DCB CSL DCB MACR UDD MACR UDD MACR UDD PAN TPAN PAN TPAN PAN TPAN MO2 ETHP MO2 ETHP MO2 ETHP HC3P HC5P HC3P HC5P HC3P HC5P HC8P ETEP HC8P ETEP HC8P ETEP OLTP OLIP OLTP OLIP OLTP OLIP ISOP APIP ISOP APIP ISOP APIP LIMP PHO LIMP PHO LIMP PHO ADDT ADDX ADDT ADDX ADDT ADDX ADDC TOLP ADDC TOLP ADDC TOLP XYLP CSLP XYLP CSLP XYLP CSLP ACO3 TCO3 ACO3 TCO3 ACO3 TCO3 KETP OLNN KETP OLNN KETP OLNN OLND XO2 OLND XO2 OLND XO2 NO NO NO2 OP2 N2O5 N2O5 HC8 HC8 ETE ETE XYL XYL HCHO HCHO ALD ALD GLY GLY MGLY MGLY TOL PAA 10 of 16

11 Figure 3. Concentration profiles for rural conditions as function of time comparing results from simulations with and without emissions. the species NO 2 (Figure 3a), O 3 (Figure 3b), NO (Figure 3c), and CO (Figure 3d) the concentrations are higher for the case including emissions, although only NO and CO have finite emissions (see Table 2). For NO 2,O 3, and NO, the emissions cause their concentrations to build up over time, which is consistent with the results presented by Stockwell et al. [1997]. However, without emissions their concentrations decrease over a 24-hour period. Only CO has a decreasing trend in both cases. [40] It is of interest to include the solution from the photostationary-state relationship for O 3, which is often employed for rural conditions with low concentrations of organic compounds [e.g., Jacobson, 1999]. The photostationary-state relationship is derived assuming that NO 2 is in steady state. In addition, the relationship is only used during daytime where the photolysis rates are nonzero. For conditions both with and without emissions, we find that the photostationary-state relationship over predicts the O 3 concentrations by up to 30%. This is partly due to the fact that as will be discussed later, NO 2 is found to introduce significant errors to the overall solution if set to steady state and should thus be retained in the mechanism. [41] In order to study the performance of reduced mechanisms with increasing degree of reduction, the box model was run for each case as presented in Table 2, first with the full set of species and solving the full set of ODEs, then with the reduced mechanisms where the steady-state species were treated separately employing algebraic equations. In Figure 4 the time evolution for a selected set of species with primary interest for air-quality modelers (e.g., NO 2, NO, CO, O 3, ETH, and PAN) are plotted resulting from the series of runs for the urban case 1. The overall performance is good for all mechanisms down to an 18 step mechanism. Such a highly reduced mechanism, approximately 60% reduction, is however starting to show problems reproducing the correct trends. [42] It was found that some species should be retained in the mechanism in order to achieve a reliable and converging result. These species included O( 3 P) and NO 2.O( 3 P) was found to have a high sensitivity on the predicted O 3 through the following reaction O 3! O 3 P þ O2 that has a high photolytic reaction rate [Stockwell et al., 1997]. Also, O( 3 P) with NO 2 was found very sensitive to the photolytic reaction rates through NO 2! O 3 P þ NO; in particular at the onset and offset of the photolysis in the morning and evening. Such sensitivity toward the photolytic reaction rates was also reported by Djouad et al. [2003] who performed an extensive sensitivity analysis on the similar RADM2 mechanism. Therefore NO 2 was deliberately retained in the 20 step mechanism in favor of OP2, which 11 of 16

12 Figure 4. case 1. Results comparing reduced chemistry with the detailed mechanism for urban conditions, becomes steady state in the 18 step mechanism. From Table 3 it is clear that NO 2 is just above the cut-off limit and little error will be introduced by exchanging NO 2 with one of the other species just below the cut-off limit. [43] Figure 4a and Figure 4b show very good agreement for O 3 and ETH, respectively, between the reduced and detailed mechanisms. Neither of these species is treated as steady state in any of the reduced mechanisms. Figure 4c and Figure 4d for NO and CO respectively reveal some deviations. NO is set to steady state in the 20 and 18 step mechanisms (see Table 4) and a certain deviation is thus expected in these two cases. However, the plot shows clearly that the species even when set to steady state are reproduced through the algebraic relations in good agreement with the exact solution. Regarding CO, the 18 step mechanism did not at first produce reliable predictions. However, retaining TOL in the mechanism in favor of OP1, which is just below the cut-off limit for the 18 step mechanism, turned out to be advantageous for an improved prediction of CO. TOL is an important aromatic compound mainly decomposed by an addition of HO to its aromatic ring [Jacobson, 1999]. This produces an HO 2, which in turn is important for the balance of CO. [44] The 20 and the 18 step mechanisms both contain many organic compounds in steady state, such as HCHO, GLY, and MGLY, which are thought to cause the overprediction of CO toward the morning hours. Figure 4e shows the predictions of NO 2 for the various reduced mechanisms. Note that the 20 and 18 step mechanisms retain deliberately NO 2 as major species. For both mechanisms, the correct trend is not well reproduced: the overall evening production of NO 2 is overpredicted and the peak in 12 of 16

13 Figure 5. Results comparing reduced chemistry with the detailed mechanism with emissions, case 2. morning hours is underpredicted. This is also reflected in Figure 4f showing the evolution of PAN. For the 18 step mechanism the sensitivity toward the offset and onsets of the photolytic reactions are evident causing the concentration profiles to peak at these two points. However, PAN goes into steady state already at the level of 30 steps and a certain deviation is expected. [45] We performed the same study for the rural cases, case 2 and 3, and the results are shown in Figures 5 and 6. In Figure 5 the same species concentration evolutions as in Figure 4 are plotted against time resulting from reduced mechanisms with increasing degree of reduction. This case represents a cleaner environment than the previous with only 10% of the initial concentrations and emission levels. Thus the trends between Figure 4 and Figure 5 are very similar, only with concentrations an order of magnitude lower in the latter. [46] Figures 5a 5c show excellent agreement between the reduced mechanisms and the reference solution, even down to a 16 step mechanism. This includes the species O 3 (Figure 5a), ETH (Figure 5b), and NO (Figure 5c). Note that NO goes into steady state already at the level of 30 steps, but its concentration is still well-predicted. The 16 step reduced mechanism did not reproduce reliable prediction for CO (Figure 5d). The 18 step mechanism including also HCHO and HC5 was found to give reasonable results apart from a steep unexpected increase toward the end of the simulation. As in the case of urban environment, retaining 13 of 16

14 Figure 6. Results comparing reduced chemistry with the detailed mechanism without emissions, case 3. TOL in the mechanism improved the CO predictions considerably throughout the simulation and this is the solution plotted in Figure 5d. This time, TOL was retained in favor of HCHO, which is just below the cut-off limit for the 18 step mechanism (see also Table 2 for the 20 step, case 2). Regarding NO 2 (Figure 5e) and PAN (Figure 5f), the same behavior was encountered as shown in Figure 4. NO 2 was fixed as non-steady-state for the two mostly reduced mechanisms, but is still overpredicted during night as seen in Figure 5e in Figure 5. The 20 step and 16 step reduced mechanisms show the same peaks for PAN during the offset and onset of the photolysis as for the urban case, and thus over predicting the concentrations during night time. However, PAN is steady-state even in the 30 step mechanism and steady state errors are evident. It is interesting to note that a stronger reduction was possible for the rural case than what was experienced for the urban case. Apart from CO (Figure 5d), all species concentrations were well reproduced using a 16 step mechanism. [47] We finally discuss the performance of reduced mechanisms for a rural atmosphere without the effect of emissions. The results plotted in Figure 6 are for the same set of species as in the previous cases. Once again it is shown that the overall performance is good for mechanisms down to 18 steps. However, some deviations are found particularly for species with low concentrations. In the case of no emissions, species like ETH (Figure 6b) remain low as there are few reactions producing these species, and most of 14 of 16