Question

Consider potential radiative energy balance at the surface of the Earth, with the Sun directly overhead in the sky. Assume that the incident solar radiative flux is reduced passing through the atmosphere to 0.85 times the TOA value, and that 20% of this is reflected at the ground. Calculate the temperature that the surface (radiating as a blackbody) would need to be at in order to be in instantaneous radiative energy balance with the solar radiation. What would you say about such an energy balance ever holding on the surface of Earth? Explain. Assume some reasonable high surface temperature for the Earth, say 40 degrees C, and calculate the (instantaneous) energy imbalance between absorbed solar and emitted IR radiation for these conditions.

Answer 1

Solar Luminosity is

Therefore, solar flux density is

where, is the sun to earth distance. Putting the values, we get

85% of this amount can pass through, which means the amount of flux reaches earth is

Out of this 20% is reflected back. Therefore, the amount absorbed by earth surface is

Amount of energy absorbed is the earth surface projection area times this density, which is , where is earth radius.

The Stefan-Boltzmann law of blackbody radiation says that the amount of energy flux radiated is proportional to the temperature of the object raised to the fourth power.

is the flux of energy, is the temperature in Kelvin, is a constant =

Amount of energy radiated is total earth surface area times this flux which is

From energy balance, we can write

This energy corresponds to a temperature of (answer)

Now talking about the imbalance the amount of radiation by the blackbody (273+40=313 K) is

Energy imbalance is