Formation of Precipitation and Relationship with Thermodynamic Concept

Saturation is typically quantified by relative humidity.

The relative humidity (RH) is defined as:

The actual vapor pressure is dependent on the water vapor concentration (or density p_v) of the

parcel of air. These variables are related through the ideal gas law, given by:

 

where: p_v is the water vapor density, R_d is the gas constant of dry air (=287.04 Joules kg^-1 K^-1)

T is the absolute temperature (K)

NOTE: the 0.622 = 18/29 which is the ratio of the molecular weights of water and air.
 

As noted earlier, the saturation vapor pressure is defined as the pressure at which the parcel of air is saturated with water vapor. That is - you cannot add more water vapor to this parcel of air without changing its temperature. The saturation vapor pressure varies only with temperature as what is defined by the Claussius-Clapeyron equation.

 

where T_c is the air temperature (degrees C).

 

The relationship between e_s and T_c is shown in the below graph. We can evaluate from the graph that RH →100% if ρ_v increases (for a fixed temperature) or if temperature decreases. It can be seen that decreasing temperature decreases exponentially e_s  thereby forcing e_s = e_a .

One key mechanism for decreasing temperature is LIFTING the parcel of air to higher elevation.

As a parcel of air is lifted to higher elevations – it becomes cooler, and its e_s  drops. The height or

elevation at which e_s = e_a, clouds will form. The term LIFTING CONDENSATION LEVEL (LCL) is often used to describe this elevation or state. Roughly speaking, the air cools by about 10K per 1000 m (or 1 Km). This quantity is known is the DRY ADIABATIC LAPSE RATE (Γ_d ).

By definition,

 

where g is the gravitational acceleration (=9.8 ms^-2) and C_p is the specific heat capacity of dry air at constant pressure (=1005 Joules Kg^-1 K^-1).