 ### Atmospheric lapse rate

The lapse rate of the atmosphere is usually defined as the rate of change of temperature with altitude. As shown in the analysis below, the lapse rate is easily determined by calculating the change in pressure with altitude using conventional thermodynamic relationships. If the atmosphere can be approximated as a dry ideal gas (no water vapor in the mixture), the relationship of the pressure of the fluid microelement in the gravitational field with height is given by:

dp=-ρgdz    (1.1)

where p is the atmospheric pressure, ρ is the atmospheric density, z is the altitude, and g is the local gravitational acceleration (assumed to be a constant).

The minus sign is because height is usually measured in the positive direction upward, so the pressure p falls in the positive direction of z.

For a closed system of ideal gas per unit mass, the first law of ideal gas thermodynamics can be written as:

dq=du+pdv=dh-vdp=cpdT-(1/ρ)dp    (1.2)

In the formula, T is the temperature, q is the heat transferred, u is the internal energy, h is the baking, is the specific volume, and cp is the constant pressure specific heat.

For an adiabatic process (no heat transfer) dg=0, equation (1.3) p= 101. 29 -(0. 011837)z+(4. 793×10-7)z2 becomes:

cpdT=(1/ρ)dp   (1.4)

Replace dp in equation (1.1) with the above formula and write:

If one assumes that the changes in g and cp with height are ignored, then under adiabatic conditions the temperature change is a constant. Substitute into g=9.81 m/s2 and cp=1.005 kJ/kgK to get:

Thus, for a system with no heat transfer, the lapse rate of temperature with increasing altitude is approximately 1°C per 100 meters. This is the so-called dry adiabatic lapse rate, which is defined by conventional notation as the negative value of the atmospheric temperature gradient. Therefore, the dry adiabatic lapse rate can be written as:

The dry adiabatic lapse rate is very important in meteorological research because comparing it with the actual lapse rate in the lower atmosphere is a measure of atmospheric stability. International standard atmospheric lapse rates based on meteorological data have been defined and used for this comparison. Specifically, for the mid-latitudes, the mean temperature decreases linearly with altitude until about 10 000 m (the defined target is 10.8 km), and the mean temperature at sea level is 288°K, and decreases all the way to 216.7°K at an altitude of 10.8 km. The standard temperature gradient thus given is: