Technical note: Entrainment-limited kinetics of bimolecular reactions in clouds

. The method of entrainment-limited 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 10 reactions with first-order kinetics in clouds, incorporates cloud entrainment into the kinetic rate coefficient. This technical note shows how bimolecular reactions with second-order kinetics in clouds can also be treated with entrainment-limited 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 formation of acid rain—as an example. Over a large range of reaction rates, cloud fractions, and initial 15 reactant concentrations, the numerical errors in the entrainment-limited bimolecular reaction rates are typically << 1 % and always < 4 %, which is far smaller than the errors found in several commonly used methods of simulating cloud chemistry with fractional cloud cover.


Introduction
Aqueous reactions in clouds play an important role in atmospheric chemistry, production of acid rain 20 from SO2 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. Since clouds are sub-grid-scale features in many large-scale regional and global atmospheric models, accounting for these processes in chemical transport models is challenging. To address these challenges, Holmes et al. (2019) introduced 25 entrainment-limited 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 https://doi.org/10.5194/acp-2021-752 Preprint. Discussion started: 24 September 2021 c Author(s) 2021. CC BY 4.0 License. the kinetic rate expression, enabling calculation of concentrations in a partly cloudy model grid cell with very little computational effort. The original paper applied entrainment-limited uptake to firstorder loss of nitrogen oxide compounds (NO2, NO3, N2O5) and showed that clouds are a globally 30 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 (Chen et al., 2021), dimethyl sulfide oxidation (Novak et al., 2020), and reactive halogens (Wang et al., 2021), all of which also involved first-order loss reactions in clouds. This note derives entrainment-limited reaction kinetics for bimolecular reactions with second-order kinetics so that the entrainment-limited method can be applied 35 to a wider range of chemical systems that are important in the atmosphere.

Derivation
The computational challenge of cloud chemistry in a fractionally cloudy grid cell is that explicitly (1 + ′2 + ′2 + 2 ′ + 2 ′ − 2 ′ ′ ) 1/2 , ′ ≡ , ′ ≡ 1 − . 2 The cloud fraction is and 1/ 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 grid-cell concentration follows the usual first-order form (Eq. 1a) with rate coefficients that depend on entrainment as well as chemical kinetics. We will follow a similar 50 approach for bimolecular reactions.
The system of equations 2 and 3 can be solved by root finding methods or fixed-point iteration. After evaluating and , 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 entrainment-limited bimolecular reaction rate coefficient.
We can also derive an approximation to the entrainment-limited bimolecular rate coefficient that does not require iteration to solve. In the limit where the in-cloud reaction is much faster than entrainment 70 ( , ≫ or , ≫ ), the grid-scale losses of A and B are determined by the rate at which the limiting reactant is entrained into clouds:

≈ ′ min([A], [B]) . 5
In the limit where in-cloud reactions are slow ( , ≪ and , ≪ ), the losses follow second-order kinetics determined by the grid-scale mean concentrations: Combining these limits gives an approximation of the entrainment-limited bimolecular loss rates, expressed as a grid-scale 2 nd order rate coefficient

Evaluation
The accuracy of entrainment-limited bimolecular reaction rates will now be demonstrated using oxidation of S(IV) by aqueous H2O2, which is a prominent step in the formation of S(VI) and acid rain, as an example (Chameides, 1984 incorporating the solubility and dissociation equilibria, cloud liquid water content, and aqueous kinetics into the effective rate coefficient (e.g., Park et al., 2004). For a cloud with 1 g m -3 liquid water at pH 5, 284 K, and 800 hPa, the effective, gas-phase bimolecular rate coefficient is = 3.7 ×  The entrainment-limited 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 115 B are consumed in only one reaction. While additional in-cloud reactions and reactants can be incorporated into the pseudo-first order loss rates (Eq. 3), to account for their effects on and , 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 entrainment-limited equations. 120 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 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.

Conclusion
The results here and in the earlier work of Holmes et al. (2019) show that the entrainment-limited 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 errors for bimolecular reactions are << 1 % error after 1 hour and always < 4 %. Entrainment-limited kinetics have already been applied to numerous first-order reactions and the extension here to 135 bimolecular reactions should further expand its applicability and usefulness in atmospheric chemistry modeling.

Code availability
Python code implementing the entrainment-limited bimolecular kinetics is provided in the supplement. and its global implications, Geophys.
Res. Lett., 46, 4980-4990,  4a and 7, bottom row). Accuracy is shown as the percent difference (%) in the cumulative loss of reactants after 1 hour relative to a reference two-box model.