# Modelling atmospheric temperature as function of elevation

If I have a body that is exposed to light of intensity $$I$$, then its temperature should follow the ODE

$$\dot{T} = a I - b T^4$$

Mountains are covered with snow, yet $$I$$ should be the same as at sea level (or slightly higher becuase there is less stuff to pass through). Lower pressure means lower temperature, but that shouldn't affect the rock surface temperature.

1. How does temperature at elevation z depend on temperature at level z0. I think the classic graph showing decreasing temperatures show an average. I wonder what it looks like if you change sea level temperature

2. What effect causes the mountain top to cool off. Is it mainly wind? Note: lesser dense atmosphere should make the surface hotter because of higher isolation. I guess

$$\frac{\partial T}{\partial t } = D\nabla^2T$$

With boundary condition

$$\dot{T}_\text{surface} = a I - b T_\text{surface}^4$$

But $$D$$ is lower at lower air densities, thus the mountain surface should have a higher temperature.

• Define your terms and provide refs for your equations, please. Commented Mar 16, 2022 at 0:13
• Energy transport in the (thermally) optically thick part of the atmosphere is not dominated by radiation, but by convection. The temperature profile therefore remains close to adiabatic and you have to solve the adiabatic, hydrostatic structure equation instead of the radiation equilibrium that you wrote down. Temperature diffusion is also just radiation transport in the optically thick region of the atmosphere. Commented Mar 16, 2022 at 8:49

If I have a body that is exposed to light of intensity $$I$$, then its temperature should follow the ODE $$\dot{T} = a I - b T^4$$

You are implicitly assuming the Earth's surface is radiating into empty space. The Earth's surface doesn't "see" empty space in the thermal infrared. It instead "sees" an atmosphere that is much warmer than empty space in the thermal infrared, typically about 10 °C to 20 °C cooler than the surface. You are failing to account for the back radiation from the atmosphere.

Lower pressure means lower temperature, but that shouldn't affect the rock surface temperature.

That atmospheric temperature in the troposphere decreases with increasing altitude does affect rock surface temperature. This is the primary reason mountaintops and high elevation plateaus are cool.

However, consider the Tibetan Plateau. The daily mean temperature in Lhasa (altitude = 3656 meters) is 8.8 °C. A naive application of the environmental lapse rate of 6.5 °C per kilometer would suggest Lhasa's mean temperature should be about 24 °C cooler than that of locales at the same latitude but at sea level. Lhasa is indeed cooler than those corresponding sea level locales, but not by 24 °C. Lhasa instead is only about 11 to 13 °C cooler than those corresponding sea level locales. In fact, the Tibetan Plateau acts as a significant heat source for the upper troposphere.

Lower pressure means lower temperature, but that shouldn't affect the rock surface temperature

This is ignoring the fact that the rock surface exchanges temperature with the atmosphere constantly, so they should be in an approximate equilibrium on an annual time scale.