8.2.5 Earth's Surface–Atmosphere System

• It is useful to consider Earth’s energy balance in terms of how much incoming energy from the Sun is used and how much is returned to space
• If incoming and outgoing energy are in balance, the Earth’s temperature will remain constant
• This can be used to create models which can help climate scientists predict temperature fluctuations based on current and increased concentrations of greenhouse gases
  • At it’s simplest, the model involves a one-layer atmosphere above the Earth’s surface

Step 1: List the known quantities

• • Solar power above atmosphere, P_a = 344 W m^–2
  • Emissivity of the atmosphere, e = 0.720
  • Emissivity of the surface, e = 1
  • New temperature of Earth’s atmosphere, T_a = 242 + 6 = 248 K
  • Stefan-Boltzmann constant, σ = 5.67 × 10^–8 W m^–2 K^–4
  • Power absorbed at the Earth’s surface = P_s
  • New temperature of Earth’s surface = T_s

Step 2: Calculate the solar power absorbed at the Earth’s surface

• • This can be calculated using the emissivity and the solar power above the atmosphere

P_s = e × P_a

P_s = 0.720 × 344 = 247.68 = 248 W m^–2

Step 3: Write the equation for the power emitted by a body

P = eσT⁴

Step 4: Calculate the new power radiated by the atmosphere

P = 0.720 × (5.67 × 10^–8) × 248⁴ = 154.43 = 154 W m^–2

Step 5: Calculate the new power absorbed by the Earth’s surface

• • The power absorbed by the Earth’s surface is a sum of the solar radiation that reaches the surface plus the power radiated by the atmosphere

New power, P_s = 248 + 154 = 402 W m^–2

Step 6: Calculate the new temperature of the Earth’s surface

• • The Earth’s surface can be assumed to be a black body, hence e = 1

P_s = σT_s⁴

400 = (5.67 × 10^–8) × T_s⁴

T_s =  = 290 K

Step 7: Determine the increase in temperature

ΔT = 290 – 288 = 2 K