### Atmospheric lapse rate

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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=c_{p}dT-(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})z^{2} becomes:

c_{p}dT=(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/s^{2} 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:

In this way, the standard lapse rate according to international practice is 0.66°C/100m.

Different temperature gradients produce different atmospheric steady states. Figure 1 shows the variation of the temperature profile from day to night due to differences in the heating of the Earth’s surface. Near the ground, the temperature profile (solid line) decreases with height before the sun rises, and the opposite is true after the sun rises (dashed line). The air near the ground is heated, and the temperature gradient near the ground increases with height, up to the so-called reversal height z_{i}. The air surface layer below z_{i} is called the troposphere or mixed layer. The temperature profile over z_{i} changes in the opposite direction.

The concept of atmospheric stability can be illustrated in this way. Consider an air cell moving up to a lower pressure, assuming that the standard lapse rate is 0.66℃/100m, and the considered ascending air cells cool with a dry adiabatic lapse rate (1℃ per 100m). If the air cells under consideration have the same temperature as the surrounding air at the beginning, then after rising 100 meters, they will be colder than the surrounding air, and the temperature will be 0.34℃ lower than them. The microelement becomes denser and tends to return to its original height position. This atmospheric state is called stable.

In general, atmospheres with dT/dz greater than (dT/dz)_{adiabatic} are stable. It should be noted that the international standard lapse rate rarely occurs in nature. This explains the need for daily balloon detection at major airports around the world to determine the actual lapse rate. Of course, a reversal state (temperature increases with altitude) is not necessary for the atmosphere to be stable. If it does exist, the atmosphere is more stable.