the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Technical note: Entrainmentlimited kinetics of bimolecular reactions in clouds
The method of entrainmentlimited kinetics enables atmospheric chemistry models that do not resolve clouds to simulate heterogeneous (surface and multiphase) cloud chemistry more accurately and efficiently than previous numerical methods. The method, which was previously described for reactions with firstorder kinetics in clouds, incorporates cloud entrainment into the kinetic rate coefficient. This technical note shows how bimolecular reactions with secondorder kinetics in clouds can also be treated with entrainmentlimited kinetics, enabling efficient simulations of a wider range of cloud chemistry reactions. Accuracy is demonstrated using oxidation of SO_{2} to S(VI) – a key step in the formation of acid rain – as an example. Over a large range of reaction rates, cloud fractions, and initial reactant concentrations, the numerical errors in the entrainmentlimited bimolecular reaction rates are typically ≪1 % and always <4 %; thus, they are far smaller than the errors found in several commonly used methods of simulating cloud chemistry with fractional cloud cover.
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Aqueous reactions in clouds play an important role in atmospheric chemistry, with the production of acid rain from SO_{2} being a prominent example (Seinfeld and Pandis, 2016). Rapid heterogeneous (surface and multiphase) reactions can consume reactants within clouds, making the overall reaction rate dependent on entrainment to supply additional reactants from the surrounding air. As clouds are subgridscale features in many largescale regional and global atmospheric models, accounting for these processes in chemical transport models is challenging. To address these challenges, Holmes et al. (2019) introduced entrainmentlimited uptake, an algorithm to accurately and efficiently account for cloud chemistry occurring in just a fraction of a grid cell. The method incorporates cloud fraction and entrainment into the kinetic rate expression, enabling calculation of concentrations in a partly cloudy model grid cell with very little computational effort. The original paper applied entrainmentlimited uptake to firstorder loss of nitrogen oxide compounds (NO_{2}, NO_{3}, N_{2}O_{5}) and showed that clouds are a globally significant sink for these gases (Holmes et al., 2019). The method has since been applied to nitrogen oxide isotopes (Alexander et al., 2020), nitrate in urban haze (Chan et al., 2021), dimethyl sulfide oxidation products (Novak et al., 2021; Jernigan et al., 2022), mercury (Shah et al., 2021), and reactive halogens (Wang et al., 2021), all of which also involved firstorder loss reactions in clouds. This note derives entrainmentlimited reaction kinetics for bimolecular reactions with secondorder kinetics so that the entrainmentlimited method can be applied to a wider range of chemical systems that are important in the atmosphere.
The computational challenge of cloud chemistry in a fractionally cloudy grid cell is that explicitly calculating reactant concentrations in the cloudy and clear fractions would increase the model's variables and computational effort. For cloud reactions with firstorder kinetics, however, Holmes et al. (2019) showed that explicitly calculating concentrations within clouds can be avoided. For a reaction with loss frequency k_{i} in clouds, the reaction rate in a partly cloudy grid cell is
where c is the reactant concentration in the grid cell (averaged over cloudy and clear fractions), $x/\left(\mathrm{1}+x\right)$ is the fraction of reactant inside cloud, and
Here, the cloud fraction is f_{c}, and $\mathrm{1}/{k}_{\text{c}}$ is the mean residence time of air in clouds. The expression is exact for steady decay in which concentrations in and out of clouds decline at the same fractional rate. The overall idea is that kinetics governing the gridcell concentration follows the usual firstorder form (Eq. 1a) with rate coefficients that depend on entrainment as well as chemical kinetics. We will follow a similar approach for bimolecular reactions.
Bimolecular reactions, $A+B\to \text{products}$, typically follow secondorder kinetic rate expressions of the form R=k_{AB}c_{A}c_{B}, where k_{AB} is the rate coefficient. For reactions within clouds, the rate depends on gasphase reactant concentrations within clouds, designated c_{A,i} and c_{B,i}. These concentrations are related to the gridaverage concentration via ${c}_{A,i}/{c}_{A}={x}_{A}/{f}_{\text{c}}\left(\mathrm{1}+{x}_{A}\right)$, where x_{A} is defined by Eq. (2) using the loss frequency for A within cloud. ${c}_{B,i}/{c}_{B}$ and x_{B} are defined similarly. The loss frequency for A within cloud is the pseudofirstorder rate ${k}_{A,i}={k}_{AB}{c}_{B,i}$, and ${k}_{B,i}={k}_{AB}{c}_{A,i}$ is the analogous loss for B. This forms a system of equations that collectively define gasphase, incloud reaction rates for bimolecular reactions:
The system of Eqs. (2) and (3) can be solved by rootfinding methods or fixedpoint iteration. After evaluating x_{A} and x_{B}, the overall reaction rate in a partly cloudy grid cell is found by substituting Eq. (3a) into Eq. (1):
Equation (4b) is the exact form of the entrainmentlimited bimolecular reaction rate coefficient. The gridcell concentrations c_{A} and c_{B} typically have units of molecules per cubic centimeter (molec. cm^{−3}), and the bimolecular rate coefficients k_{2} and k_{AB} typically have units of cubic centimeters per molecule per second (cm^{3} molec.^{−1} s^{−1}).
We can also derive an approximation to the entrainmentlimited bimolecular rate coefficient that does not require iteration to solve. In the limit where the incloud reaction is much faster than entrainment (${k}_{A,i}\gg {k}_{\text{c}}$ or ${k}_{B,i}\gg {k}_{\text{c}})$, the gridscale losses of A and B are determined by the rate at which the limiting reactant is entrained into clouds:
In the limit where incloud reactions are slow (${k}_{A,i}\ll {k}_{\text{c}}\phantom{\rule{0.125em}{0ex}}\mathrm{and}\phantom{\rule{0.125em}{0ex}}{k}_{B,i}\ll {k}_{\text{c}})$ or the cloud fraction approaches one, the losses follow secondorder kinetics determined by the gridscale mean concentrations:
Combining these limits gives an approximation of the entrainmentlimited bimolecular loss rates, expressed as a gridscale secondorder rate coefficient:
Although Eq. (7a) is finite and well defined for all values of f_{c}, numerical overflow could occur with finiteprecision arithmetic when f_{c} approaches zero or one. To improve stability and accuracy, numerical calculations can use the equivalent expression
This approximate entrainmentlimited bimolecular reaction rate coefficient Eq. (7a or b) can be used in Eq. (4a).
The accuracy of entrainmentlimited bimolecular reaction rates will now be demonstrated using oxidation of S(IV) by aqueous H_{2}O_{2}, which is a prominent step in the formation of S(VI) and acid rain, as an example (Chameides, 1984). One key aqueous reaction is ${\mathrm{HSO}}_{\mathrm{3}}^{}$ + H_{2}O_{2} + H^{+} → ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}}$ + 2H^{+} + H_{2}O, where the reactants are dissolved forms of gaseous SO_{2} and H_{2}O_{2}. While the reaction occurs in cloud droplets, the reaction rate can be expressed in terms of the gasphase concentrations of SO_{2} and H_{2}O_{2} by incorporating the solubility and dissociation equilibria, cloud liquid water content, and aqueous kinetics into the effective, gasphase rate coefficient (e.g., Park et al., 2004). For a cloud with 1 g m^{−3} liquid water at a pH of 5, 284 K, and 800 hPa, the effective, gasphase bimolecular rate coefficient is ${k}_{\mathrm{eff}}=\mathrm{3.7}\times {\mathrm{10}}^{\mathrm{14}}$ cm^{3} molec.^{−1} s^{−1}, which will be used in examples below. A similar approach can be applied to other bimolecular aqueous reactions.
Figure 1 shows that the exact entrainmentlimited algorithm (Eq. 4) is nearly identical to a reference solution in a twobox model that explicitly represents concentrations inside clouds and entrainment mixing with clear air. The approximate entrainmentlimited solution (Eq. 7) also resembles the exact entrainmentlimited and reference solutions, but remaining reactant concentrations diverge by 3 % after 1 h and 10 % after 4 h. Two other cloud chemistry methods that are used in current atmospheric chemistry models are also shown in Fig. 1: the thincloud approximation, in which loss is computed for the entire grid cell using the gridaverage liquid water content, and the cloud partitioning method, in which only reactants within the cloudy fraction can react, but the concentrations are homogenized across cloudy and clear regions each time step of the chemical solver. Holmes et al. (2019) describe these other methods in greater detail. Both of the other methods diverge from the reference solution and entrainmentlimited method by large amounts.
Figure 2 shows accumulated error in the entrainmentlimited kinetics over a wide range of initial reactant concentrations and cloud fractions. Results are presented as the error in total product formed, relative to the reference twobox model, after 1 h of integration. Over most of the parameter space, the errors in the entrainmentlimited calculations are much less than 1 %. The largest errors occur over a narrow range of ${k}_{AB}{c}_{B}/{k}_{\text{c}}$ values in regions that are about half cloudy, and these errors do not exceed 4 %. By the same metric, the approximate entrainmentlimited bimolecular algorithm has up to 10 %–30 % error (Fig. 2). The thincloud method has much larger errors than either of the entrainmentlimited methods over most of the parameter space in Fig. 2. These thincloud errors exceed 1000 % when cloud fractions are small and incloud reactions are fast. As f_{c} approaches one, however, the thincloud method has increasingly good accuracy, with errors under 0.1 % for f_{c}≥0.97. Numerical codes can, therefore, use thincloud instead of entrainmentlimited kinetics when ${f}_{\text{c}}>\sim \mathrm{0.97}$ for computational efficiency.
The relative computational performance of these cloud chemistry methods depends on numerous factors, such as reactant concentrations, cloud fraction, differential equation solver, error tolerances, optimizations, and programming language. Some general comparisons can be made, however, using the conditions of Fig. 1. (Code for timing tests is provided in the Supplement.) When evaluating the instantaneous reaction rate (e.g., at time t=0 in Fig. 1), the approximate entrainmentlimited method is about 15 times faster than the exact method, and the thincloud method is about 100 times faster. There is much less disparity in execution times when integrating the solution over time, however, because numerical solvers have many additional components. For the integration shown in Fig. 1, the approximate entrainmentlimited method is about 2.3 times faster than the exact method. The thincloud method, meanwhile, is only about 25 % faster than the exact entrainmentlimited solution, because the solver takes many more internal time steps as concentrations quickly decline. Speed differences between the methods would likely diminish further in a chemical mechanism with more compounds and reactions. Therefore, computational speed should not be a major impediment to adopting entrainmentlimited reaction kinetics.
The entrainmentlimited approach is best suited for applications and models that do not require highly detailed cloud and aqueous chemistry. For example, the derivation above assumes that reactants A and B are consumed in only one reaction. While additional incloud reactions and reactants can be incorporated into the pseudofirstorder loss rates (Eq. 3), to account for their effects on x_{A} and x_{B}, solving the system becomes more computationally intensive as more reactants are involved. For cloud reactions that depend on [H^{+}], the pH must be assumed or calculated via another method because it is infeasible to account for the relevant aqueous equilibria within the entrainmentlimited equations. Overcoming these limitations, however, requires explicit representation of reactant concentrations and entrainment in the cloudy fraction of a grid cell, along with the extra computational burden that this incurs. Despite the progression of atmospheric models to ever higher resolutions, fractional cloudiness is likely to remain a feature of many global and regional models for many years to come, necessitating some means of accounting for its effect on chemistry.
The results here and in the earlier work of Holmes et al. (2019) show that entrainmentlimited reaction kinetics can provide an efficient and accurate means of representing heterogeneous cloud chemistry in atmospheric models with fractional cloud cover. By incorporating cloud fraction and entrainment into the rate coefficient, the usual first and secondorder rate expressions are retained, allowing the entrainmentlimited kinetics to be easily implemented in numerical codes. The entrainmentlimited approach provides far greater accuracy than other methods currently in use; typical errors for bimolecular reactions are ≪1 % error after 1 h and always <4 %. Entrainmentlimited kinetics has already been applied to numerous firstorder reactions, and the extension here to bimolecular reactions should further expand its applicability and usefulness in atmospheric chemistry modeling.
Python code for implementing the entrainmentlimited bimolecular kinetics is provided in the Supplement.
The supplement related to this article is available online at: https://doi.org/10.5194/acp2290112022supplement.
The author has declared that there are no competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author is grateful to Daniel Jacob and Mike Long for helpful discussions.
This research has been supported by the National Aeronautics and Space Administration (New (Early Career) Investigator Program, grant no. NNX16AI57G).
This paper was edited by John Liggio and reviewed by two anonymous referees.
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