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 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

Aqueous reactions in clouds play an important role in atmospheric chemistry,
with the production of acid rain from SO

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 first-order kinetics, however, Holmes et
al. (2019) showed that explicitly calculating concentrations within clouds
can be avoided. For a reaction with loss frequency

Bimolecular reactions,

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 (

Combining these limits gives an approximation of the
entrainment-limited bimolecular loss rates, expressed as a grid-scale
second-order rate coefficient:

Although Eq. (7a) is finite and well defined for all values of

The accuracy of entrainment-limited bimolecular reaction rates will now be
demonstrated using oxidation of S(IV) by aqueous

Comparison of numerical solutions for reaction of dissolved
SO

Figure 1 shows that the exact entrainment-limited algorithm (Eq. 4) is nearly identical to a reference solution in a two-box model that explicitly represents concentrations inside clouds and entrainment mixing with clear air. The approximate entrainment-limited solution (Eq. 7) also resembles the exact entrainment-limited 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 thin-cloud approximation, in which loss is computed for the entire grid cell using the grid-average 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 entrainment-limited method by large amounts.

Accuracy of (

Figure 2 shows accumulated error in the entrainment-limited 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 two-box model, after 1 h of integration. Over most of the
parameter space, the errors in the entrainment-limited calculations are much
less than 1 %. The largest errors occur over a narrow range of

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

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

The results here and in the earlier work of Holmes et al. (2019) show that
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
entrainment into the rate coefficient, the usual first- and second-order
rate expressions are retained, allowing the entrainment-limited kinetics to
be easily implemented in numerical codes. The entrainment-limited approach
provides far greater accuracy than other methods currently in use; typical
errors for bimolecular reactions are

Python code for implementing the entrainment-limited bimolecular kinetics is provided in the Supplement.

The supplement related to this article is available online at:

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.