## Ro t im

Hence show that it has a spectrum, TmTm, where T*m is the complex conjugate, given by Eq. 12-2. Graph the spectrum using a log-log plot and hence convince yourself that fluctuations with a frequency greater than 1/yO are damped.

3. For the one-layer ''leaky greenhouse'' model considered in Fig. 2.8 of Chapter 2, suppose that, all else being fixed, the atmospheric absorption depends linearly on atmospheric CO2 concentration as e _ £0 + [CO2] e\, where [CO2] is CO2 concentration (in ppm), £0 _ 0.734, and ex _ 1.0 x 10-4 (ppm)-1. Calculate, for this model, the surface temperature:

(a) for the present atmosphere, with [CO2] _ 380 ppm (see Table 1.2);

(b) in pre-industrial times, with [CO2] _ 280 ppm; and

(c) in a future atmosphere with [CO2] doubled from its present value.

### 4. Faint early Sun paradox

The emission temperature of the Earth at the present time in its history is 255 K. Way back in the early history of the solar system, the radiative output of the Sun was thought to be 25% less than it is now. Assuming all else (Earth-Sun distance, Earth albedo, atmospheric concentration of greenhouse gases, etc.) has remained fixed, use the one-layer ''leaky greenhouse'' model explored in Problem 3 to:

(a) determine the emission temperature of the Earth at that time if greenhouse forcing then was the same as it is now. Hence deduce that the Earth must have been completely frozen over.

(b) if the early Earth were not frozen over because of the presence of elevated levels of CO2, use your answer to Problem 3 to estimate how much CO2 would have had to have been present. Comment on your answer in view of Fig. 12.14.

### 5. Bolide impact

There is strong evidence that a large meteorite or comet hit the Earth about 65M y ago near the Yucatan Peninsula, extinguishing perhaps 75% of all life on Earth—the K-T extinction marking the end of the Cretaceous (K) (see Fig. 12.12). It is speculated that the smoke and fine dust generated by the resulting fires would have resulted in intense radiative heating of the midtroposphere with substantial surface cooling (by as much as 20°C) which could interrupt plant photosynthesis and thus destroy much of the Earth's vegetation and animal life.

A slight generalization of the one-dimensional problems considered in Chapter 2 provide insights in to the problem.

By assuming that a fraction f ' of the incoming solar radiation in Fig. 2.8 is absorbed by a dust layer and that, as before, a fraction ' e of terrestrial wavelengths emitted from the ground is absorbed in the layer, show that:

T,, where Te is the given by Eq. 2-4. [Hint: write down expressions for the radiative equilibrium of the dust layer and the ground.]

Investigate the extreme case where the dust layer is so black that it has zero albedo (no radiation reflected, ap = 0) and is completely absorbing f = 1) at solar wavelengths.

6. Assuming that the land ice over the North American continent at the Last Glacial Maximum shown in Fig. 12.17 had an average thickness of 2 km, estimate the freshwater flux into the adjacent oceans (in Sv) that would have occurred if it had completely melted in 10 y, 100 y, 1000 y. Compare your estimates to the observed freshwater meridional flux in the ocean, Fig. 11.32. Another useful comparative measure is the flux of the Amazon river, 0.2 Sv.

Appendices

A.1. Derivations

### A.1.1. The Planck function

A.1.2. Computation of available potential energy A.1.3. Internal energy for a compressible atmosphere A.2. Mathematical definitions and notation A.2.1. Taylor expansion A.2.2. Vector identities A.2.3. Polar and spherical coordinates A.3. Use of foraminifera shells in paleo climate A.4. Laboratory experiments A.4.1. Rotating tables A.4.2. List of laboratory experiments A.5. Figures and access to data over the web

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