Review of Some Fundamental Thermodynamics

The first law of thermodynamics is a statement of the conservation of energy for thermodynamic systems.  It states that an increment of thermal energy  or heat (dq) absorbed by the system either raises the internal energy by an increment (du) or does and increment of work on the system (dw).  

The increment in internal enegy is proportional to an increment in temperature, where the proportionality depends on the substance (gas) and the process (constant pressure or constant volume) .

 For an adiabatic process, defined as one in which there is no heat exchanged between an air parcel and its surrounding (dq=0), we can equate any changes in internal energy with a change in pressure.   

This helps us define vertical motions in the atmosphere. Governed by the first law of thermodynamics, this expression tells us that small increases in temperature  are equated with small increases in pressure.  Or, if we recall the hydrostatic balance equation, we can substitute for dP as follows:

which gives:

This is the dry adiabatic lapse rate for the atmosphere.  This illustrates that the first law of thermodynamics, in the absence of external heating, determines the way air parcels respond to small perturbations in height. If a dry parcel experiences a small downward force, causing it to change elevation, its temperature must decrease at 9.8^o C/km, it must obey the first law.   If the same parcel experiences a small upward force, it temperature must increase at the same 9.8^o C/km.   This is an important concept for evaluating the stability of the atmosphere.  Of course, most of the time, the air parcel is not dry, but contains a finite amount of water vapor.  In this case, the temperature will decrease at the dry adiabatic lapse rate until the parcel has cooled to the point of saturation.  When saturation occurs, condensation of liquid water (or deposition of ice) from water vapor results in the release of  latent heat.  This addition of heat slows the rate at which the temperature decreases with height, it also means the process is no longer adiabatic.  However, if all the condensation products remain in the rising air parcel (ie., we form clouds, but not precipitation), then we can refer to the process as saturated-adiabatic.  If the condensation products are all removed, we say the process is thermodynamically irreversible, and not adiabatic, in meteorology, we refer to this as pseudoadiabatic.  Generally, the amount of heat carried out of the system by precipitation is small compared to the amount added to the air, and we find that the saturated-adiabatic lapse rate and the psuedoadiabatic lapse rate are nearly equal.

The first law can be written to take into account this known rate of heating:

We can rearrange this to get:

This gives us the moist adiabatic lapse rate relative to the dry adiabatic lapse rate, and we see that this quantity is not constant but depends on the pressure and temperature because the saturation mixing ratio varies with temperature and pressure.  In fact, the saturated adiabatic lapse rate ranges from very low lapse rates of ~4deg/km in warm humid surface air to effectively the same lapse rate as the dry adiabats in the upper troposphere, where there is very little moisture.  

The following paragraphs provide some reminders of how we express moisture content in the atmosphere.

Moisture Variables

Mixing Ratio is the amount of water vapor in a volume of dry air:

Mixing ratio is typically expressed in units of grams of water vapor per kilogram of dry air.  Vapor pressure and mixing ratio are related, where vapor pressure is just the partial pressure due to water vapor in the atmosphere

where epsilon is just the ratio of the molecular weights of water vapor and dry air respectively:

We know that water vapor, when considered as part of a mixture of gases, results in a lower molecular weight for moist air relative to dry air, and this translates into a lower density:  moist air is less dense than dry air.  This leads to the definition of another moisture variable used in meteorology, the virtual temperature. This is the temperature which a dry parcel of air must have in order to have the same density as a parcel of moist air at the same pressure.  Since the density of moist air is always lower, the virtual temperature is always greater than the actual temperature.  

At any given temperature, air can hold a finite amount of water vapor, we define this limit, the equilibrium condition between phases (water vapor in the atmosphere relative to liquid water or ice) as the saturation vapor pressure (with respect to ice or water).  

The saturation mixing ratio (with respect to water) is the ratio of the mass of water vapor in a parcel of air at saturation to the mass of dry air at the same temperature.   Relating the mass per unit volume (density) to the pressure throught the ideal gas law, we can rearrange and get an expression based on pressure and vapor pressure:

Relative humidity can be defined as a proximity to saturation, the ratio of the mixing ratio to the saturation mixing ratio (with respect to water) at the same temperature and pressure:

Finally we can define dew point, which is the temperature to which air must be cooled (at constant pressure) to reach saturation.  The dew point is the temperature where the saturation mixing ratio equals the actual mixing ratio.  It should be obvious that the dew point is always less than the actual temperature.

It is important to have this understanding of the vertical temperature/ moisture structure in order to understand radiative transfer through our atmosphere.  The following page illustrates this relationship.