## 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 qandΔ prepresent changes in_{s}qandpdue to evaporation and condensation, respectively. The pre-adjusted values are indicated by ^ and post-adjusted values are expressed as_{s}pand_{s}q.Then, the relationship

holds. Ground surface vapor flux can be expressed as

Thus,

Δ qcaused by evaporation can be expressed

In addition,

Δ qcaused by condensation can be formulated as

Based on the above relationships,

pwill be adjusted by_{s}

Because the value of

pis adjusted, the value of_{s}qalso needs to be adjusted accordingly. This adjustment needs to be made by conserving the water vapor mass in the course of adjusting the value ofp. This condition can be expressed as_{s}

From this condition, the following relationship holds:

Thus,