## Appendix A Vertical Turbulent Mixing Scheme, Asymmetric Convective Model (ACM)  
The vertical coordinate system is associated with height, z, and it can be defined using \(M_{a}\), the column mass density of air (kg\(\cdot\)m\({}^{-2}\)), as \(M_{a}=\rho_{L}\Delta z\) where, \(\rho_{L}\) is the average air density \(\overline{\rho^{z}}\) (\(\cdot\), \(\rho_{L}=\overline{\rho^{z}}\)) and \(\Delta z\) is the difference between the height. Using the hydrostatic relation, \(dP=-\rho gdz\), the average air density is related to the height and the pressure \(\int dP=-g\int\rho dz\) or \(\Delta P=-g\overline{\rho^{z}}\Delta z\). The air density in the layer \(\rho_{L}\) can be expressed as \(\rho_{L}=-\frac{\Delta P}{g\Delta z}\). Since the height of each layer \(z_{k}\) can be determined as \(z_{k}=z_{0}+k\Delta z\) using \(\Delta z=(z_{0}+z_{i})/k_{h}\) for uniform vertical grid space of model layer when the surface height \(z_{0}=0\), the boundary layer height \(z_{i}=1000\) m, and the vertical grid \(k_{h}=64\). The time step \(\Delta t\) must be less than this critical time (\(0.5\times\mathrm{T}_{1}\)). The critical time for computational accuracy is the turnover time of air in the lowest layer \(T_{1}=(1/M_{u})(M_{a_{1}}/M_{a}S_{1})\) which is equivalent to an expression given by [32]. Using the vertical coordinate system defined as \(Ma\), definitions of upward (\(M_{u}\)) and downward (\(M_{d}\)) mixing rates are essential to understanding the tendency equations in the ACM, and how it conserves air mass. With this understanding, ACM can be used to mix any quantity that is conserved during air transport and changes during mixing in proportion to the masses of the parcels that mix. The upward mixing rate can be defined as the rate of fractional mass flux in kg air from the lowest layer (\(k=1\)) into any layer above per \(kg\) air in the upper layer per \(sec\) and estimated by convective velocity scale \(w^{*}\) and mixing height \(Z_{i}\) depending upon the sensible heat flux \(H_{0}\);  
\[M_{u}=\frac{w^{*}}{Z_{i}}\text{ with }w^{*}=\bigg{(}\frac{H_{0}gZ_{i}}{\rho c_{p}T _{0}}\bigg{)}^{\frac{1}{3}}.\] (A-1)  
Also, the downward mixing rate (\(M_{d}\)) can be defined as the rate of fractional mass flux in \(kg\) air from layer \(k\) per kg air in layer \(k\) down to layer \(k\)-\(1\) per second. \[M_{d_{k}}=M_{u}\bigg{[}\frac{MaS_{k}}{Ma_{k}}\bigg{]}\] (A-2)  
where, \(MaS_{k}\) is the sum from \(j=k\) to \(j=kh\) of \(Ma_{j}\), \(kh\) is the top layer of the model layer, \(Ma_{k}\) is the mass of air in layer \(k\), and \(z_{k}\) represents the thickness of layer \(k\), which is equal to \(z_{k}-z_{k\text{-}1}\). The upward mixing rate (\(M_{u}\)) in the entire levels of the CBL is constant with height and the downward mixing rate (\(M_{d}\)) increases from the top down in order to maintain mass continuity. Then, air mass conservation in the top layer (\(k=kh\)) requires \(M_{d_{kh}}=M_{u}\). At the lower layers down to and including layer 2, air mass conservation requires \(M_{d_{k}}=M_{u}+\frac{Ma_{k+1}}{Ma_{k}}M_{d_{k+1}}\). For the lowest layer (\(k=1\)), \(M_{d_{1}}=0\) but the mass flux out must equal the mass flux down from level \(2\), \(\Big{[}\frac{MaS_{2}}{Ma_{1}}\Big{]}M_{u}=\frac{Ma_{2}}{Ma_{1}}M_{d_{2}}\). It can be shown by substitution and induction that the value of \(M_{d_{k}}\) for all \(k\), except \(M_{d_{1}}=0\), is Equation (A-2).