A more accurate treatment of the radiative transfer, originally worked out by Sir Arthur Eddington, for the atmospheres of stars, modifies our result to
At the top of the troposphere, t = 0 (all photons escape upward). This means that T = 0.84Te. If we use Te = 246, then T = 207 K. The infrared optical depth from the bottom to the top of the troposphere is about 2, giving T = 1.2Te, or about 293 K.
Convection also plays an important role in the energy transport in the lower atmosphere. As the air near the ground is heated, it expands. The buoyant force on the air will make it rise until it reaches air of its own density. The more rapidly the temperature falls with altitude, the more rapidly the pressure falls with altitude. A faster pressure fall-off means a larger pressure difference between the top and bottom of any parcel of air. The larger pressure difference provides a larger buoyant force, and the air rises more quickly, making convection more important. Therefore, convection becomes more important when the fall-off in temperature with altitude is larger.
As the air rises, it encounters lower pressure air and expands. The gas does work in the expansion. It takes no energy from its surroundings, so it cools. A process in which no energy is exchanged is called adiabatic. The convection process itself will modify the temperature gradient, dT/dz, to the value appropriate for an adiabatic process. If the temperature gradient by radiation is less than the adiabatic gradient, the temperature gradient is not enough to drive convection. However, if the temperature gradient is greater than the adiabatic gradient, convection will set in. Remembering that dT/dz is negative, we can say that the condition for significant convection is that
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