A Simplified Mathematical Description of Cloud Formation

Let us assume that the air parcel has mass m and air density p (excluding the liquid water). The velocity of the air parcel will be the result of buoyancy forces and the gravitational force due to liquid water. The buoyancy force is proportional to the volume of the air parcel m/p and the density difference between the air parcel and its surroundings, 

FIGURE 17.12 Schematic description of the cloud formation mathematical framework. 

As the air parcel is moving, it causes the acceleration of surrounding airmasses, resulting in a decelerating force on the air parcel. The deceleration force is proportional to the mass of the displaced air, m, and the corresponding deceleration, -dW/dt. Pruppacher and Klett (1997) show that this effect is actually equivalent to an acceleration of an induced mass m/2 and therefore a term - m dW/dt should be added on the right-hand side of (17.52). Using the ideal-gas law (p - p)/p = (T - T )/T, and the modified (17.52) can be rewritten as 

The rate of change of temperature can be calculated using (17.51), noting that dT/dt = WdT/dz, and also that wvs should be replaced by wv to allow the creation of supersaturations. The final result is 

The condensed water is related to wv through the water mass balance for the entraining parcel. If airmass dm enters the parcel from the outside, then the water vapor and liquid water mixing ratios will change according to 

---