Chapter 1: Why non-grey?

Almost every climate-modelling course starts with a radiating planet at one temperature seen through one atmosphere at one opacity. The grey atmosphere. From it you get Milne’s solution, a lapse rate, and a qualitative sense of the greenhouse effect. You do not, however, get stratospheric cooling. You do not get the CO₂ forcing. You do not get anything that a satellite sees when it looks down.

This chapter shows why.

The mean of an exponential is not the exponential of the mean

Consider a single spectral interval of width $\Delta\nu$ containing many absorption lines. The monochromatic transmission through a column of path length $L$ at wavenumber $\nu$ is

\[T_\nu(L) = \exp\!\left(-\sigma(\nu)\, L\right).\]

The band-averaged transmission is

\[\langle T(L)\rangle = \frac{1}{\Delta\nu}\int T_\nu(L)\,d\nu = \left\langle e^{-\sigma L}\right\rangle.\]

A grey model replaces this with $e^{-\langle\sigma\rangle L}$. These two are equal only when $\sigma(\nu)$ is constant across the band. For a realistic band — a few strong lines sitting on top of weak continuum — they are wildly different.

Figure 1: **Figure 1.1** — Band-averaged transmission (true) vs. grey approximation for a toy band with 10 % strong lines and 90 % weak continuum. The grey model is wrong at every path length.

The physical interpretation: most of the transmitted energy comes through the weak part of the band; most of the absorption happens in the strong part. Averaging $\sigma$ first smears this separation out.

If we want a radiation scheme that is cheap but not wrong, we need to represent $\langle e^{-\sigma L}\rangle$ directly — not $e^{-\langle\sigma\rangle L}$. That leads to the k-distribution (Chapter 3).

Non-grey phenomena you get for free

Once you have proper band-averaged transmission you recover:

Stratospheric cooling. The window (8–12 μm) is nearly transparent; the CO₂ band at 15 μm is opaque. High in the stratosphere the atmosphere emits in both regions, but only absorbs significantly in the CO₂ band. Net result: cooling. A grey model cannot do this — it has only one opacity and cannot separate emission from absorption at different wavelengths.

CO₂ forcing. Doubling CO₂ moves the effective emission level in the 15 μm band upward to colder temperatures, reducing OLR from that band. The window band is unaffected. Broadband OLR drops. A grey model cannot separate these — it either doubles all opacity (enormous forcing) or nothing (zero forcing).

Solar heating of the stratosphere. UV is absorbed by O₃; the visible is largely transparent. A grey shortwave model cannot separate these.

The picket-fence scheme you will implement in the remaining chapters is the minimal non-grey model that captures all three phenomena.

Try it yourself

Open examples/spectral_radiation_anatomy.ipynb and run the first three cells. You will see the window/CO₂-band contrast directly from CorkLongwaveRadiation’s per-band flux diagnostics.

References

Goody, R. M., and Y. L. Yung. 1989. Atmospheric Radiation: Theoretical Basis. 2nd ed. Oxford University Press. https://academic.oup.com/book/40894.

Pierrehumbert, R. T. 2010. Principles of Planetary Climate. Cambridge University Press. https://doi.org/10.1017/CBO9780511780783.

--- title: "Chapter 1: Why non-grey?" bibliography: ../../references.bib --- Almost every climate-modelling course starts with a radiating planet at one temperature seen through one atmosphere at one opacity. The grey atmosphere. From it you get Milne's solution, a lapse rate, and a qualitative sense of the greenhouse effect. You do not, however, get stratospheric cooling. You do not get the CO₂ forcing. You do not get anything that a satellite sees when it looks down. This chapter shows why. ## The mean of an exponential is not the exponential of the mean Consider a single spectral interval of width $\Delta\nu$ containing many absorption lines. The monochromatic transmission through a column of path length $L$ at wavenumber $\nu$ is $$T_\nu(L) = \exp\!\left(-\sigma(\nu)\, L\right).$$ The band-averaged transmission is $$\langle T(L)\rangle = \frac{1}{\Delta\nu}\int T_\nu(L)\,d\nu = \left\langle e^{-\sigma L}\right\rangle.$$ A grey model replaces this with $e^{-\langle\sigma\rangle L}$. These two are equal only when $\sigma(\nu)$ is constant across the band. For a realistic band — a few strong lines sitting on top of weak continuum — they are wildly different. ![**Figure 1.1** — Band-averaged transmission (true) vs. grey approximation for a toy band with 10 % strong lines and 90 % weak continuum. The grey model is wrong at every path length.](_artifacts/01_mean_of_exp.png){#fig-mean-of-exp} The physical interpretation: most of the transmitted energy comes through the *weak* part of the band; most of the absorption happens in the *strong* part. Averaging $\sigma$ first smears this separation out. If we want a radiation scheme that is cheap but not wrong, we need to represent $\langle e^{-\sigma L}\rangle$ directly — not $e^{-\langle\sigma\rangle L}$. That leads to the k-distribution (Chapter 3). ## Non-grey phenomena you get for free Once you have proper band-averaged transmission you recover: **Stratospheric cooling.** The window (8–12 μm) is nearly transparent; the CO₂ band at 15 μm is opaque. High in the stratosphere the atmosphere emits in both regions, but only absorbs significantly in the CO₂ band. Net result: cooling. A grey model cannot do this — it has only one opacity and cannot separate emission from absorption at different wavelengths. **CO₂ forcing.** Doubling CO₂ moves the effective emission level in the 15 μm band upward to colder temperatures, reducing OLR from that band. The window band is unaffected. Broadband OLR drops. A grey model cannot separate these — it either doubles all opacity (enormous forcing) or nothing (zero forcing). **Solar heating of the stratosphere.** UV is absorbed by O₃; the visible is largely transparent. A grey shortwave model cannot separate these. The picket-fence scheme you will implement in the remaining chapters is the minimal non-grey model that captures all three phenomena. ::: {.callout-tip} ## Try it yourself Open `examples/spectral_radiation_anatomy.ipynb` and run the first three cells. You will see the window/CO₂-band contrast directly from `CorkLongwaveRadiation`'s per-band flux diagnostics. ::: ## Further reading - @goody1989, Chapter 4 — the definitive treatment of band models and the limits of the grey approximation. - @pierrehumbert2010, Chapter 4 — a modern planetary-science perspective on why non-grey matters.