If we climb to altitude we get closer to the sun. So why do we need warm clothes on top of Everest?

Imagine a parcel of air molecules with an imaginary boundary around them. Move this parcel upwards-it will experience less pressure and vice versa. If it expands (as the pressure decreases) work must be done as the molecules move apart, this work or energy is in the form of heat thus the parcel of air will cool. Thus we can see why we should expect parcels of air that have moved to higher altitudes (and thus expanded) to be cooler.

Under ideal conditions, on an overcast day with the air perfectly dry, the rate of cooling is around 1^oC per 100m of altitude rise. The change in temperature with altitude is called the lapse rate and the theoretical ideal we call the dry adiabatic lapse rate.

Let us have a brief physics lesson to consider why this is the case.

DETERMINING THE ADIABATIC LAPSE RATE

We need to establish an equation that describes the relationship between the temperature and pressure of an air parcel as it moves up and down in the atmosphere.

We make the assumption that any changes we make at any particular time are very small.

The ideal gas law and first law of thermodynamics can be combined to give the following equation:

Where

dQ is the quantity of heat added to parcel per unit mass of air (J/kg)

C_p is the specific heat of air (at constant pressure.) J/kg.^oC

dT is the incremental temperature change (K)

V is the volume per unit mass of air (m³/kg)

This gives us an indication of how atmospheric temperature changes with air pressure but what we need to know is how it changes with altitude.

Consider a static column of air with cross section A

A horizontal slice of air in this column with thickness dz and density , will have a mass A.dz.

If the pressure at the top of the slice = P_{(z + dz)}

Then pressure at the bottom of slice P_(z) will be P_{(z + dz)} plus the added weight per unit area of the slice itself.

Where g is the gravitational constant

We can write the incremental change in pressure with altitude dP as:

If we now express the rate of change in temperature with altitude dT/dz. as the product (dT/dP)(dP/dz) and substitute equations (2) and (4) into this product we see:

However

Since V = volume per unit mass and is mass per unit volume the product and thus

The negative sign indicates that temperature decreases with altitude.

If we substitute for g = 9.806 m/s²

and the specific heat for dry air at room temperature , C_p = 1005 J/kg^oC

and substitute for the joule in terms of kg m²/s² equation (6) tells us that :

dT/dz = (9.806 m/s² /1005 J/kg ^oC)x 1J/ kg m²/s² = -0.00976^oC/m

We normally get rid of the negative sign and express -dT/dz as 9.76^oC/km.

This is normally given the symbol , and defined as the dry adiabatic lapse rate.

In effect is about 1 ^oC/100m

REAL CONDITIONS

In our discussion on the dry adiabatic lapse rate we assumed that the air was dry.

The specific heat of the gas is clearly important in fixing the lapse rate and a specific correction for water vapour may be necessary if air is very wet. If air is saturated the situation is complicated by the release of latent heat when water condenses. The wet adiabatic lapse rate is not a constant but a good average is about 6^oC/km. However for our purposes we will continue to treat the air as dry.

WHY IS THE LAPSE RATE IMPORTANT TO US?

If lapse rate is greater than predicted by theory the air is unstable and mixing of pollutants takes place.

If the lapse rate is less than predicted the air is stable and pollutants are not dispersed.