Energy Balance Problems (HL IB Physics)

The diagram below shows a simple energy balance climate model in which the atmosphere and the Earth’s surface are treated as two bodies.

The Earth’s surface receives both solar radiation and radiation emitted from the atmosphere.

At current atmospheric greenhouse gas concentrations, the temperature of Earth’s atmosphere is set to increase by 6 K.

Data for this model:

• Current mean temperature of the Earth’s atmosphere = 242 K
• Current mean temperature of the Earth’s surface = 288 K
• Solar intensity per unit area at top of the atmosphere = 344 W m^–2
• Emissivity of the atmosphere, e = 0.720
• Albedo of the atmosphere, a = 0.280

Use this data to estimate the increase in temperature of the Earth’s surface.

Answer:

Step 1: List the known quantities

• Solar intensity above atmosphere, I_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
• Intensity absorbed at the Earth’s surface = I_s
• New temperature of Earth’s surface = T_s

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

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

I_s = e × I_a

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

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

• Since intensity = power per unit area

I = eσT⁴

Step 4: Calculate the new intensity radiated by the atmosphere

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

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

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

New intensity, I_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

I_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