## 2 Model Description
### Model Equations  
The TKM describes the reactions which occur over a substantial range of chemical time scales in a dry CBL with turbulence driven by constant surface heat flux without cumulus convection when the growth of the CBL is ignorable during the afternoon in the summer months as shown in Figure 1a. Previously, the TKM has conducted a simple NO\({}_{x}\)-O\({}_{3}\) photochemistry study to quantitatively account for turbulence-induced incomplete mixing of two interacting chemical species [10]. In this study, the NO and O\({}_{3}\) fluxes are not opposite in sign, which means they both have net upward transfers, so they are diluted in the upper CBL due to entrainment of 'clean air' from the free troposphere. Governing equations in the TKM are described using a concentration field splitting method (CSM) for an effective chemical reaction rate [33; 34] and a non-local closure asymmetric convective model (ACM) for the vertical mixing scheme [30]. As a computationally simple scheme, as shown in Figure 1b, the ACM provides realistic simulation of vertical mixing in the CBL as mentioned in the Appendix A. The effective reaction rate related to the intensity of segregation \(I_{s}\) can be calculated using the concept of the CSM which explains the possible states of mixing of the two species considering the concentration fields (the reactive species and their inert surrogates) in position and time. Based on the CSM considering the binary reaction between \(A\) and \(B\), the hypothetical concentrations of inert species \(A^{I}\) and \(B^{I}\) are present in the same position and time with the reactive concentration of the reactive species \(A\) and \(B\) because the species \(A^{I}\), \(B^{I}\), \(A\) and \(B\) transport in the same convective flow of the CBL. Also, the concept of a local phenomenal extent reaction \(\xi\) incorporates both stoichiometry and diffusion effects simultaneously in space and time. To reduce the number of partial differential equations needed to describe the evolution of multimolecular systems in the turbulent (incompletely mixed) field, the stoichiometric relations are combined with the concept of inert surrogate concentrations (see References [10; 30; 31] for details). Therefore, the mean concentration of the reactive species \(A\) and \(B\), \(\overline{C}_{A}\) and \(\overline{C}_{B}\), can be calculated using the inert surrogate mean concentration, \(\overline{C}_{A}^{I}\) and \(\overline{C}_{B}^{I}\)and the phenomenal extent of reaction \(\xi\) in space and time \((\overrightarrow{x},t)\): \(\overline{\mathcal{C}}_{A}\left(\overrightarrow{x},t\right)=\overline{ \mathcal{C}}_{A}^{I}\left(\overrightarrow{x},t\right)-\overline{\xi}\left( \overrightarrow{x},t\right)\) and \(\overline{\mathcal{C}}_{B}\left(\overrightarrow{x},t\right)=\overline{ \mathcal{C}}_{B}^{I}\left(\overrightarrow{x},t\right)-\overline{\xi}\left( \overrightarrow{x},t\right)\) for two reactants and \(\overline{\mathcal{C}}_{P}\left(\overrightarrow{x},t\right)=\overline{ \mathcal{C}}_{P}^{I}\left(\overrightarrow{x},t\right)+\overline{\xi}\left( \overrightarrow{x},t\right)\) for product \(P\). The governing equations of variables in the TKM for inert species, for reactive species, and for phenomenal extent of reaction \(\xi\) are time-dependent, nonlinear and coupled partial differential equations, which are expressed in a fixed Eulerian framework as given below (see Reference [10] for details):  
\[