the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# 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 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 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
entrainment-limited 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 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 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 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 first-order 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
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 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 first-order 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 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 approach
for bimolecular reactions.

Bimolecular reactions, $A+B\to \text{products}$, typically follow
second-order 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 gas-phase reactant concentrations within clouds, designated
*c*_{A,i} and *c*_{B,i}. These concentrations are related to the
grid-average 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 pseudo-first-order 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 gas-phase,
in-cloud reaction rates for bimolecular reactions:

The system of Eqs. (2) and (3) can be solved by root-finding methods or
fixed-point 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 entrainment-limited bimolecular
reaction rate coefficient. The grid-cell 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 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 (${k}_{A,i}\gg {k}_{\text{c}}$ or ${k}_{B,i}\gg {k}_{\text{c}})$, the grid-scale losses of *A* and *B*
are determined by the rate at which the limiting reactant is entrained into
clouds:

In the limit where in-cloud 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 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 second-order rate coefficient:

Although Eq. (7a) is finite and well defined for all values of *f*_{c},
numerical overflow could occur with finite-precision arithmetic when *f*_{c}
approaches zero or one. To improve stability and accuracy, numerical calculations
can use the equivalent expression

This approximate entrainment-limited bimolecular reaction rate coefficient Eq. (7a or b) can be used in Eq. (4a).

The accuracy of entrainment-limited 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 gas-phase 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, gas-phase 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,
gas-phase 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 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.

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
${k}_{AB}{c}_{B}/{k}_{\text{c}}$ values in regions that are about half cloudy, and these
errors do not exceed 4 %. By the same metric, the approximate
entrainment-limited bimolecular algorithm has up to 10 %–30 % error (Fig. 2). The thin-cloud method has much larger errors than either of the
entrainment-limited methods over most of the parameter space in Fig. 2.
These thin-cloud errors exceed 1000 % when cloud fractions are small and
in-cloud reactions are fast. As *f*_{c} approaches one, however, the
thin-cloud method has increasingly good accuracy, with errors under 0.1 %
for *f*_{c}≥0.97. Numerical codes can, therefore, use thin-cloud instead
of entrainment-limited 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 entrainment-limited method is about 15
times faster than the exact method, and the thin-cloud 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 entrainment-limited method is about 2.3 times faster than the
exact method. The thin-cloud method, meanwhile, is only about 25 % faster
than the exact entrainment-limited 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
entrainment-limited reaction kinetics.

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 *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 *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 entrainment-limited 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 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 ≪1 % error after 1 h and always <4 %. Entrainment-limited kinetics has already been applied to numerous first-order reactions, and the extension here to bimolecular reactions should further expand its applicability and usefulness in atmospheric chemistry modeling.

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

The supplement related to this article is available online at: https://doi.org/10.5194/acp-22-9011-2022-supplement.

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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