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## ERQ · 12 marks · Topics: B.2 Greenhouse effect + C.2 Wave model · Archetype: data_response
**Integration:** primary=B.2 Greenhouse effect, secondary=C.2 Wave model (strength: supporting)
**Stem.** A simplified two-stream model of Earth's energy balance treats the atmosphere as a single thin layer that absorbs a fraction α of the long-wavelength infrared (IR) radiation emitted upward by the surface and re-radiates equally upward and downward as a black body. Solar shortwave radiation passes through the atmosphere with negligible absorption. The following data are extracted from satellite measurements:
| Quantity | Symbol | Value |
|---|---|---|
| Mean solar irradiance at top of atmosphere | S | 1361 W m⁻² |
| Planetary (Bond) albedo | a | 0.30 |
| Measured mean surface temperature | T_s | 288 K |
| Stefan–Boltzmann constant | σ | 5.67 × 10⁻⁸ W m⁻² K⁻⁴ |
| Peak wavelength of surface IR emission | λ_peak | 10.1 μm |
| Spectral half-width of surface IR emission band | Δλ | 7.0 μm |
The figure (not shown) presents the modelled energy fluxes: incoming solar S(1−a)/4 absorbed by the surface, surface emission σT_s⁴, atmospheric absorption fraction α, and atmospheric back-radiation reaching the surface.
### Part (a) State [2 marks] · AO1 · Topic: B.2
State two greenhouse gases present in Earth's atmosphere that absorb radiation in the wavelength band centred on λ_peak, and state one molecular property that allows them to absorb in this band.
### Part (b)(i) Calculate [3 marks] · AO2 · Topic: B.2
Show that the surface emits long-wavelength radiation at approximately 390 W m⁻² and calculate the effective emission temperature T_e that the Earth would have in the absence of the atmospheric layer.
### Part (b)(ii) Determine [3 marks] · AO2 · Topic: B.2
By writing the energy-balance equation for the surface (incoming absorbed solar flux plus atmospheric back-radiation equals upward surface emission), determine the fraction α of surface IR absorbed by the atmospheric layer that is consistent with the measured T_s = 288 K. Use the Show-that value 390 W m⁻² from (b)(i).
### Part (c) Explain [2 marks] · AO3 · Topic: B.2
Recent satellite data show that, while α is increasing due to rising CO₂, the upward IR flux measured at the top of the atmosphere has remained nearly constant over decades. Explain how this observation is consistent with a rising surface temperature T_s.
### Part (d) Evaluate [2 marks] · AO3 · Topic: B.2 + C.2 · ASSUMPTIONS DISCRIMINATOR
The two-stream model treats atmospheric back-radiation as an incoherent thermal flux whose power simply adds to the absorbed solar flux. A student proposes refining the model by representing the upward surface emission and the downward atmospheric re-emission as two coherent monochromatic plane waves at λ_peak that could interfere, claiming this would change α. Using the spectral half-width Δλ = 7.0 μm given in the data table, estimate the coherence length L_c ≈ λ_peak²/Δλ of the thermal IR radiation and evaluate whether the student's proposed refinement is justified.
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## Mark Scheme
### Part (a) [2 marks]
- M1: Names any two of: H₂O / water vapour, CO₂ / carbon dioxide, CH₄ / methane, N₂O, O₃ [no ECF]
- M2: Molecules possess a vibrational (bending/stretching) mode that produces a changing electric dipole moment / resonant frequency matching IR photon energy [no ECF]
### Part (b)(i) [3 marks] — Show that + Calculate
- M1: Substitution σT_s⁴ = (5.67 × 10⁻⁸)(288)⁴ giving 390.1 W m⁻² (Show-that target: 390.1 W m⁻²; accept 389–391) [no ECF]
- M2: Energy-balance for bare Earth: σT_e⁴ = S(1−a)/4 = (1361)(0.70)/4 = 238 W m⁻² [no ECF]
- M3: T_e = (238/5.67 × 10⁻⁸)^(1/4) = 255 K (accept 254–256 K) [ECF from M2]
### Part (b)(ii) [3 marks] — Determine
- M1: Surface balance equation written as S(1−a)/4 + α·σT_s⁴/2 = σT_s⁴ (or equivalent two-stream form with atmospheric layer emitting ασT_s⁴/2 downward) [no ECF]
- M2: Substitution of numerical values: 238 + α(390/2) = 390, i.e. 238 + 195α = 390 [ECF from (b)(i) M1 and (b)(i) M2]
- M3: Rearrangement α = (390 − 238)/195 = 152/195 and final value α ≈ 0.78 (accept 0.77–0.79) [ECF from M2]
### Part (c) [2 marks] — Explain (causal chain per §4.4.1)
- M1: A larger α reduces the fraction of surface IR transmitted directly to space, so for fixed solar input the surface must warm to restore radiative balance [observation + mechanism]
- M2: **Therefore** T_s rises until the increased σT_s⁴ compensates the increased atmospheric trapping, leaving the top-of-atmosphere upward flux unchanged at S(1−a)/4 ≈ 238 W m⁻² [causal link to observation]
### Part (d) [2 marks] — Evaluate (position + support + limitation per §4.4.1)
- M1: **Coherence-length estimate using C.2 wave physics:** L_c ≈ λ_peak²/Δλ = (10.1 × 10⁻⁶)²/(7.0 × 10⁻⁶) ≈ 1.5 × 10⁻⁵ m (≈ 15 μm), which is vastly smaller than the atmospheric scale height (~10⁴ m) over which back-radiation is generated [quantitative wave-model result]
- M2: **Position + justification:** The refinement is NOT justified — because L_c is many orders of magnitude shorter than the path differences between independently emitting molecules, the two fluxes are mutually incoherent, so time-averaged intensities (not amplitudes) add and no stable interference modifies α [evaluative judgement tied to M1]
### Marker notes
- Show-that target in (b)(i): 390.1 W m⁻² given to 4 sig figs; student-derived 389–391 W m⁻² acceptable. Subsequent parts MUST use 390 W m⁻².
- Alternative method accepted for (b)(ii): top-of-atmosphere balance σT_s⁴(1−α) + α·σT_s⁴/2 = S(1−a)/4 yields the same α ≈ 0.78.
- (a) accept any other IB-syllabus greenhouse gas; CFCs accepted. Reject O₂, N₂ (no IR-active dipole transitions).
- (d) accept equivalent reasoning via: (i) broad Planck spectrum implies superposition of many uncorrelated frequencies; (ii) random thermal phases of independent molecular emitters; (iii) explicit comparison L_c ≪ atmospheric path lengths. Full marks require BOTH a numerical coherence length AND an explicit evaluative position.
- ECF: a wrong α in (b)(ii) does not affect (d). A wrong L_c in (d) M1 still allows M2 if the student's evaluative reasoning is internally consistent.
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