C. Details of the Model

d. Change in Surface Pressure by Evaporation and Condensation

If water vapor is removed from the atmosphere at a certain height due to condensation, the effect of the change of mass at that height should be taken into consideration, thus should be reflected in changes in ps and vertical wind. However, implementing such treatment in a σ-coordinate system is quite intricate. Here, only the change in the total mass within a certain column (vertically-integrated values of the evaporation and condensation) will be considered, and this change will be adjusted at the end of each time step. At a particular time step, Δ q and Δ p_s represent changes in q and p_s due to evaporation and condensation, respectively. The pre-adjusted values are indicated by ^ and post-adjusted values are expressed as p_s and q.

Then, the relationship

holds.

Ground surface vapor flux can be expressed as

$\begin{displaymath} F_q \sim L C (q^{\ast} - qs) \rho\end{displaymath}$

Thus, Δ q caused by evaporation can be expressed

In addition, Δ q caused by condensation can be formulated as

$\begin{displaymath} \Delta q = S^{cond}_{q} \times \Delta t\end{displaymath}$

Based on the above relationships, p_s will be adjusted by
$\begin{displaymath} \Delta p_s = F_q(k=1) \frac{g}{L} \Delta t + \int S^{cond}_{q} \Delta t dp\end{displaymath}$

Because the value of p_s is adjusted, the value of q also needs to be adjusted accordingly. This adjustment needs to be made by conserving the water vapor mass in the course of adjusting the value of p_s. This condition can be expressed as

$\begin{displaymath} \hat{\rho} \hat{q} = \rho q\end{displaymath}$

From this condition, the following relationship holds:

$\begin{displaymath} \frac{\hat{p_s} \sigma}{RT} \hat{q} = \frac{P_s \sigma}{RT} q = \frac{(\hat{p_s} + \Delta p_s) \sigma}{RT} q\end{displaymath}$

Thus,

$\begin{displaymath} q = \frac{\hat{p_s}}{\hat{p_s} + \Delta p_S}.\end{displaymath}$

C.d. Change in Surface Pressure by Evaporation and Condensation