Atmospheric chemical mechanisms, which are typically used in air quality research and forecasting codes, generally contain a large number of species and reactions. This poses a significant computational workload, which in some cases may account for more than 80 % of the total simulation time , even with the advent of modern hybrid computer architectures . These mechanisms describe an important set of processes in the troposphere; for example, the degradation of volatile organic compounds (VOCs) and the formation of ozone (O3), the latter being a major oxidant and pollutant. As a result, mechanisms with varying levels of complexity are included in regional and global atmospheric chemistry codes, the overall performance of which strongly depends on the choice of chemical mechanism.

Apart from the large number of species that require solving at every point in the computational domain and for every time step, there is a large disparity in the chemical timescales of the interacting species . This results in a stiff system of non-linear equations for the reaction rates, which is computationally expensive to integrate, and adds to the computational cost . Similar issues are encountered in the field of combustion research: detailed mechanisms describing the combustion of a fuel contain hundreds of species and thousands of reactions. However, from a practical point of view, one is usually only interested in a handful of important variables – in combustion this includes quantities such as ignition delay time, laminar flame speed etc., while in atmospheric chemistry this includes ozone/NOx mixing ratios and so on. In both cases, the sensitivity of quantities of interest to certain species and reactions, can be minor in relation to dominant species and reactions. As a result, solving for all species in the detailed chemical mechanism might not actually be required in order to obtain accurate estimates of the target quantities. To this end, chemistry reduction techniques have been developed to reduce the dimensionality of the problem. In turn, this results in a reduction of the computational requirements associated with detailed-chemistry simulations and an acceleration of the simulation. Even though this is common practice in combustion research using a variety of methods , chemistry reduction methods have seen limited use in atmospheric chemistry applications.

The usual reduction process of a detailed chemical mechanism begins with the identification of an accurate “skeletal” mechanism. The “skeletal” mechanism is a subset of the detailed mechanism, and is generated by eliminating unimportant species and reactions from the detailed mechanism for the problem at hand. Further reduction of the skeletal mechanism is also possible. This can be achieved by a variety of timescale analysis methods, which are applied to the skeletal mechanism, such as quasi-steady-state assumption (QSSA), computational singular perturbation (CSP) , intrinsic low dimensional manifolds (ILDM) etc. Timescale methods are employed for finding species which are approximately in steady state. Following this, a non-linear system of equations is solved for the steady-state species mixing ratios. As a result, timescale analysis methods are most efficient when applied to relatively small skeletal mechanisms rather than the full detailed mechanism. An approach such as this, using CSP, was utilized by in order to construct a reduced mechanism for the Carbon Bond mechanism (CBMIV) . In this study, our interest is in generating skeletal mechanisms, which is the first step in the reduction process, and can be used as a starting point for further reduction, in addition to being applied to more comprehensive chemistry codes.

Sensitivity analysis (SA) is perhaps the oldest and most straightforward of methods for identifying skeletal mechanisms . In SA, suitable sensitivity coefficients are defined which are usually reaction-based. The sensitivity of a species in each reaction is calculated for a particular configuration (reaction mode), and reactions that have sensitivity coefficients below a threshold value are identified as redundant and are removed from the detailed mechanism. An approach such as this was employed by to reduce the CBM-EX tropospheric chemical mechanism. The SA resulted in the elimination of a number of reactions from the detailed mechanism, and following steady-state assumptions this approach was further introduced to derive a reduced and computationally faster mechanism. A sensitivity analysis assisted tabulation method was also used by for accelerating the species integration. Furthermore, SA was employed by as a first reduction step for generating a skeletal mechanism from the Master Chemical Mechanism (MCM) . Reaction-based approaches such as sensitivity analysis and principal component analysis (PCA) , result in the removal of reactions from the detailed mechanism, but may not always significantly reduce the number of species which is the key factor controlling computational time in numerical simulations. To this end, a number of other techniques have been developed which are species-oriented rather than reaction-oriented. Species lumping is a popular approach in which a number of reacting species are combined into surrogate species; the net effect of these species on the system evolution remains approximately the same. Lumping has been used in the development of the Regional Acid Deposition Model (RADM2) , in the development of SAPRC , and for condensing the MCM . also used lumping for developing the Common Representatives Intermediates (CRI) mechanism from the MCM. Direct relation graph (DRG), is an alternative and efficient species-based method for the generation of skeletal mechanisms, originally proposed by . In DRG, a suitable species direct interaction coefficient (DIC) is defined. The DIC measures the importance a particular species has on a predefined set of target species. DRG eventually results in the removal of species, in contrast with classic reaction-based SA. In the original version of the DRG method, the target species set only included species appearing in the same reaction as the target species. However, species not interacting directly with the target species through a reaction, may still be indirectly important for a target species of interest. Therefore, an extension of the DRG method, namely DRG with Error Propagation (DRGEP) was proposed to address this issue . In DRGEP, the DIC is defined so that the effect of the reaction path is also taken into account during the reduction process. DRGEP has been extensively used to generate skeletal mechanisms for combustion applications, with good overall results, and many variants of the method have subsequently been developed using different DIC definitions and route-finding algorithms . DRGEP has been successfully used by , in combination with a number of other methods, to reduce the a-pinene oxidation subset of the MCM.

In comparison to SA, and lumping methods, DRGEP has seen limited use for reducing complex atmospheric chemical mechanisms, despite its large potential. In addition, the majority of studies in the literature (which used SA) focus on generating subsets of very detailed chemical mechanisms such as the MCM. As a result, the skeletal mechanisms generated from MCM are still of a prohibitive size for efficient forecasting purposes . Conversely, our focus in this study is on chemical mechanisms that are commonly used in atmospheric models. These mechanisms are already condensed mechanisms, which have typically been developed using a bottom-up approach, and include a large number of surrogate/lumped species. Thus, it is instructive to investigate whether DRGEP can be used for further reduction of these already condensed mechanisms, as a first step in the reduction process.

Another important point to note is that the majority of studies in the literature have only focused on a priori evaluation of the skeletal chemical mechanisms: their performance was only evaluated against the model scenario results, which usually involved 0-D box model simulations. However, in a practical air-quality forecast simulation, advection, diffusion, and the more refined calculation of photolysis rates, all affect the spatio-temporal concentration of the species. These effects, among others, lead to different mixing ratios in regions that have NOx-limited conditions compared with regions that have NOx-saturated conditions. This affects the species production/destruction rates and reduction process, and to date, at least to the knowledge of the authors, no a posteriori validation has been conducted using actual forecasting simulations.

In this study, DRGEP is employed in order to examine whether sufficiently accurate skeletal mechanisms can be generated for a detailed mechanism which is commonly used in forecasting codes, namely the Regional Atmospheric Chemistry Mechanism (RACM). This is an updated version of the Regional Acid Deposition Model (RADM2) mechanism by , and describes the degradation of a number of VOCs. It is a condensed chemical mechanism but it is relatively large, including 75 species and 237 reactions; therefore, it is a good candidate for reduction using DRGEP. The focus in particular, is on developing skeletal chemistry for application to ground-level ozone prediction in relatively polluted areas.

In the text which follows, Sect.  introduces the DRGEP method, and Sect.  lists the process for generating the skeletal mechanism as well as the a priori validation. Details of the a posteriori validation using a popular weather research and forecasting code are given in Sect. .

Direct relation graph (DRG) is a method for generating subsets of detailed chemical mechanisms by removing species that have a negligible effect on a predefined set of target species. In the original version of the DRG method developed by , the DIC rT,B between a target species T and a non-target species B is defined as rT,B=∑i=1Nr|w˙i,Tδi,B|∑i=1Nr|w˙i,T|, where w˙i,T is the net rate of species T from reaction i, and δi,B is an index specifying the existence of B in reaction i. The net rate of a species from a general reversible reaction is calculated using w˙i,T=νi,T⋅w˙ir, where νi,T is the difference in the stoichiometric coefficients of species T in reaction i, and w˙ir is the net rate of reaction i. The index δi,B is equal to 1 if B exists in reaction i, and 0 otherwise. Clearly, 0≤rT,B≤1, and rT,B is generally not equal to rB,T. A large value of rT,B implies that species B is important in the evaluation of the rate of T, while a low value implies that it is not as important. A threshold (ϵ) is introduced, and provided rT,B>ϵ, species B is added to the set of dependent species of T, otherwise it is deemed unimportant and removed. This relation is denoted as a direct path T→B. An example of a DRG involving four species A, B, C, and D, is given in Fig.  with the numbers indicating the values of the DICs for each pair. The process is repeated for all target species Ti, and the final set of species in the skeletal mechanism is constructed from the union of all target species sets. Species not included in the union set are eliminated, as are reactions involving any of the eliminated species.

DRG has been applied with good overall results in a number of studies in combustion research, yet the simple definition of the interaction coefficient in Eq. () has some important limitations. Consider the model situation depicted in Fig. . If A is the target species and C is the species in question, then with an example threshold value (ϵ) of 0.1, C would be added to the dependent set of A. However, it is clear that “stronger”, i.e., more important paths from A to C exist, for example, A → B → C. Thus, the notion of “path” becomes important, and a suitable DIC definition is required able to describe this. In addition, there are also alternative paths, e.g., A → D → C or A → D → B → C.

The DRGEP method aims to account for the above points by using an improved DIC definition and reduction strategy. In DRGEP , the DIC is first defined using rT,B=|∑i=1Nrw˙i,Tδi,B|max⁡(PT,CT), where the production PT and consumption terms CT of T are defined as PT=∑i=1Nrmax⁡(0,w˙i,T) and CT=∑i=1Nrmax⁡(0,-w˙i,T). The DIC, as defined above, is calculated for all species. The path interaction coefficient (PIC) rT,Bp for a given path p connecting target species T and B is then defined as rT,Bp=∏i=1n-1rSiSi+1|p, i.e., it is the product of all DICs along that path. The PIC is calculated for all possible paths connecting T to B, and an overall path interaction coefficient (OIC), rT,Bo is then calculated using rT,Bo=max⁡∏i=1n-1rSiSi+1p=max⁡rT,Bp, i.e., the strongest path from T to B is identified based on the product of the DICs of connected nodes across all paths linking T and B. In the example in Fig. , the strongest path is A → B → C. This is due to the fact that for this path rA–Co=0.9⋅0.7=0.63, which is the largest value, and both B and C are included in the set of A. The process is repeated for all target species of interest, and species with overall interaction coefficients less than a predefined threshold value are removed.

The identification of the strongest path is a common problem in computational science, and a number of different route-finding algorithms have been developed for this task. In this study, we employ a classic algorithm for searching through the connected nodes and obtaining rT,Bo, which equates to the “strongest” path . An in-house code, namely REDCHEM_ v0.0 was specifically developed for the DRGEP method, and for all associated functions including the route-finding subroutines.

Reduction methods require input data, and these data should be representative of the actual reaction scenario. This translates to using actual weather-forecast simulation data as input for the DRGEP method. However, this is hardly ever done in practice, as there is a large computational overload for conducting these kinds of simulations in the first place. As a result, a computationally more efficient initial-value problem (box model) is used as a model scenario for the reduction, which is common practice in chemical mechanism reduction studies . The species mixing ratios Ci evolve according to dCidt=Gi(Ck), where Gi are non-linear functions of the species rates, with initial conditions Ci0 for the species mixing ratios, and T0 for temperature. The pressure is kept fixed at 1.0 atm, and the temperature at 298.0 K. The kinetic pre-processor library (KPP) is used for the numerical integration of Eq. (). The KPP library includes a number of different solvers, and in this study a 5-stage Runge–Kutta method was used from the package (Radau5). This method is stiffly accurate and robust, and is often used for benchmarking purposes.

Six different initial scenarios are considered. The species mixing ratios for each model scenario are given in Table . These model scenarios were used by for reducing the CBMEX mechanism using sensitivity analysis. In this work, the same mixing ratios are used as per Table 1 of , but are adapted for the chemical mechanism used in this study. In this paper we group important alkane, alkene, and aromatic species for each mechanism, and initialize their mixing ratios based on the relevant alkane, alkene, and aromatic species found in the study of . Cases A–C correspond to increasing VOC/NOx ratios not including isoprene, while cases D–F correspond to about the same VOC/NOx ratios with isoprene included. The VOC/NOx ratios are relatively high, and this ensures that a large number of VOC-relevant reactions are activated; thus, in this process a relatively large region of the composition space is covered. At the same time we are interested in ozone production in relatively highly polluted areas where such conditions are typically found.

The J values for the photolysis rate coefficients are based on parameterizations as developed in the MCM . These are given by J=Icos⁡M(θ)e-Nsec⁡(θ), where θ is the solar azimuth angle, and I, M, and N are reaction-specific constants. A 48 h run is conducted for each scenario, which results in a total of 496 datasets. These scenarios are used as input for DRGEP and the reduction process is done on a dataset basis. Important species are retained for each dataset, and the process is repeated for the next dataset. Any new species not already included in the dataset is added. Once all datasets are considered, a species union set is formed, and any reactions involving species other than those included in the species union set are removed. The target species is set to be O3, which is an important pollutant of interest. In addition, O3 is eventually produced by the degradation of the VOCs through a large number of reaction pathways; therefore, it is a good target for the reduction.

As an example, Table  lists the OIC values for target O3 as obtained for scenario A at midday (t=12 h) and midnight (t=24 h). Clearly, there is a difference in the OIC values for each species since the rate constants depend on the solar azimuth angle which determines the rate constants of the photolytic reactions. Top scoring species for O3 include third-body species M, and oxygen species such as O3P (ground-state oxygen atom) and O1D (excited state oxygen atom). This is expected since these species readily react to produce O3 through the reactions O3P+M⇒O3 and O1D+M⇒O3. Nitrogen oxides NO and NO2 also score high as they too react both directly and indirectly with ozone. Direct paths include the reactions O3+NO⇒NO2+O2 and O3+NO2⇒NO3+O2, while indirect paths include reactions with O3P. The methyl peroxy radical MO2 ranks high as it is involved in numerous reactions with VOCs. The hydroxyl radical HO also scores high since it is the main oxidation path of the VOCs which eventually end up producing ozone. With a threshold of 9.0×10-3 (which results in a 54-species subset as explained later in the text), 18 species will be included in the set for t=12 h, and 11 species will be included for t=24 h. The process is then repeated for all other datasets to form the overall species union set for the target O3.

In order to quantify the quality of the skeletal mechanisms generated using DRGEP, a percentage error is defined based on the target species of interest, i.e., ozone. For a mixing ratio obtained using the skeletal mechanism Cis, and a mixing ratio using the detailed mechanism Cid, this is defined as e=1N∑i=1N1T∫100⋅|Cis(t)-Cid(t)||Cid(t)|dt, where N is the total number of cases. Note that zero values are not taken into account for the error calculation.

Figure  shows this error against the number of species in the skeletal mechanism. As expected, the error increases as more species are removed. With about 10 species removed (75 to 65) the error is negligibly small. The error then increases once more than about 10 species are removed; however, the error remains small down to about 55 species (20 species removed) at less than 10 %. The huge spike in the error below about 55 species results from the removal of important intermediates.

Figure  shows the percentage speed-up (CPU time) gained, against the number of species in the skeletal mechanisms. This is done for case A, and similar results were observed for the rest of the cases. The speed-up is calculated for both the total simulation time, and for integrating Eq. () alone. The threshold line where the error in O3 prediction spikes as per the results in Fig.  is also shown. Clearly, as the number of species is reduced, the computational time drops and there is an increase in both speed-ups. It is interesting to note that for a decrease from 64 to 58 species there is no increase in integration speed-up. This implies that the stiffness of the remaining species and equations is unchanged. Nevertheless, there is an increase in the total speed-up of almost 10 % for a decrease from 64 to 58 species, which is due to simulation overheads alone and is quite significant. The average speed-up due to overhead computations is found to be 6.6 %. At the threshold error (e) of 10 %, the smallest possible skeletal mechanism contains 54 species and 150 reactions (the threshold for OIC at this point is 9.0×10-3), which is significantly smaller than the detailed mechanism.

For this mechanism, the total speed-ups and integration speed-ups are 54.4 % and 43.7 %, respectively. In general, the CPU time scales with the total number of species in the system due to the evaluation of the Jacobian when dealing with stiff systems. For integration of the system alone, the expected speed-up percentage scales as speed-up (%)=100⋅|(Nsp/Nsp,det)2-1|, and this is shown as a blue line in Fig. . Clearly, there is a good qualitative agreement with the theoretical result.

In order to visualize the errors in the ozone mixing ratio more clearly, Fig.  shows the solution profiles for scenarios A–C for the 48 h simulations, using the detailed and worst performing skeletal mechanism (which has 54 species). It is clear that the agreement with the detailed mechanism is very good for all six scenarios. In the isoprene scenarios D–F, similarly good results were obtained, although these results are not presented here. It is also important to note that the agreement is particularly good both early in the simulation and at later times for the box model problem. In a forecasting simulation, the situation is somewhat different. Typical time steps used in forecasting simulations are of the order of minutes, and a new initial-value problem is solved at every time step for the species mixing ratios, using the previous time step mixing ratios as the new initial condition. Furthermore, an operator splitting approach is used in the majority of codes for integrating the species mixing ratios, and this process is equivalent to filtering/smoothing the mixing ratio fields, which reduces the stiffness of the system. From this point of view, integration in forecasting codes is less stringent than integration in box model runs. In this sense, the skeletal mechanism need only be accurate enough over the integration time step, before the next initial-value problem is solved.

It is also important to note that the skeletal mechanism was developed with relatively polluted conditions in mind, for predicting ozone, which explains the relatively large VOC/NOx ratios used for the reduction in Table . However, the method is general and can be tailored for modelling conditions of specific interest, e.g., low VOC/NOx conditions. In order to examine the performance of the smallest skeletal mechanism for such conditions, additional box model runs were conducted for significantly lower VOC/NOx ratios. In particular, the NO and NO2 mixing ratios in Table  were increased by a factor of 6 for each species, leading to initial conditions with VOC/NOx ratios of 0.7, 1.8, and 3.2 for modified scenarios A′–C′, respectively. The results for the worst-performing skeletal mechanism (which has 54 species) are shown in Fig. . The agreement with the detailed chemical mechanism is particularly good for all three cases, both at early times and for longer times. Even though the chemistry for low VOC/NOx conditions is somewhat different, the results in Fig.  indicate that the reduction process is not as sensitive to the VOC/NOx ratio – the most important parameter was instead found to be variation in sunlight intensity due to the activation/deactivation of photosensitive reactions.

The results of the previous section constitute an a priori validation. In an actual simulation, the photolysis rates and the species mixing ratios due to the effects of advection, diffusion, and so on, can be substantially different from the conditions in a box model. As a result, the species rates will also differ which affects the reduction process. The aim of this section is to examine the performance of the skeletal mechanism generated in the previous section, by implementing it in an actual atmospheric-chemistry simulation, and comparing it with results using the detailed chemical mechanism.

A direct relation graph approach for generating skeletal chemical mechanisms from more detailed mechanisms was employed, in order to produce a more computationally efficient mechanism for accelerated atmospheric chemistry simulations. A code was developed for the task, and the method was applied to a commonly used mechanism, namely RACM, with the target species being ozone, which is a major pollutant.

The skeletal mechanism was developed using input from a 0-D initial-value problem, and was validated both a priori against the 0-D problem results and a posteriori. The a posteriori validation involved implementing both the detailed and the skeletal mechanisms in an actual air-quality forecasting code, namely WRF-Chem, and running simulations to compare the spatio-temporal ozone mixing ratio profiles. The skeletal mechanism was found to perform well, with relatively low percentage errors. A speed-up of 24.6 % was achieved for the total simulation time, which does not yet include any speed-up due to overheads such as input/output computations.

The method is general, and can be applied to any chemical mechanism in the WRF-Chem package or other chemistry–transport codes, for producing computationally more efficient air quality and climate simulations. Since a significant speed-up has been achieved with the already optimized chemical mechanism used in this study, it is expected that future application to more comprehensive chemistry mechanisms may lead to significant gains in computational efficiency.