Problems

1. At present the emission temperature of the Earth is 255 K, and its albedo is 30%. How would the emission temperature change if:

(a) the albedo was reduced to 10% (and all else were held fixed)?

(b) the infrared absorptivity of the atmosphere (e in Fig. 2.8) was doubled, but albedo remained fixed at 30%?

2. Suppose that the Earth is, after all, flat. Specifically, consider it to be a thin circular disk (of radius 6370 km), orbiting the Sun at the same distance as the Earth; the planetary albedo is 30%. The vector normal to one face of this disk always points directly towards the Sun, and the disk is made of perfectly conducting material, so both faces of the disk are at the same temperature. Calculate the emission temperature of this disk, and compare with Eq. 2-4 for a spherical Earth.

3. Consider the thermal balance of

Jupiter.

(a) Assuming a balance between incoming and outgoing radiation, and given the data in Table 2.1, calculate the emission temperature for Jupiter.

(b) In fact, Jupiter has an internal heat source resulting from its gravitational collapse. The measured emission temperature Te defined by

(outgoing flux of 0T4, = planetary radiation per unit surface area)

is 130 K. Comment in view of your theoretical prediction in part (a). Modify your expression for emission temperature for the case where a planet has an internal heat source giving a surface heat flux Q per unit area. Calculate the magnitude of Jupiter's internal heat source.

(c) It is believed that the source of Q on Jupiter is the release of gravitational potential energy by a slow contraction of the planet. On the simplest assumption that Jupiter is of uniform density and remains so as it contracts, calculate the annual change in its radius ajup required to produce your value of Q. (Only one half of the released gravitational energy is convertible to heat, the remainder appearing as the additional kinetic energy required to preserve the angular momentum of the planet.)

[A uniform sphere of mass M and radius a has a gravitational potential energy of - § G Mr, where G is the gravitational constant = 6.7 x 10-11 kg-1 m3 s-2. The mass of Jupiter is 2 x 1027 kg and its radius is ajup = 7.1 x 107 m.]

4. Consider the "two-slab" greenhouse model illustrated in Fig. 2.9, in which the atmosphere is represented by two perfectly absorbing layers of temperature Ta and Tb.

Determine Ta, Tb, and the surface temperature Ts in terms of the emission temperature Te.

5. Consider an atmosphere that is completely transparent to shortwave (solar) radiation, but very opaque to infrared (IR) terrestrial radiation. Specifically, assume that it can be represented by N slabs of atmosphere, each of which is completely absorbing of IR, as depicted in Fig. 2.12 (not all layers are shown).

(a) By considering the radiative equilibrium of the surface, show that the surface must be warmer than the lowest atmospheric layer.

(b) By considering the radiative equilibrium of the nth layer, show that, in equilibrium,

2T„4 = Th + Th, where Tn is the temperature of the nth layer, for n> 1. Hence argue that the equilibrium surface temperature is

Ts = (N + 1)1/4 Te, where Te is the planetary emission temperature. [Hint: Use your answer to part (a); determine T1 and use Eq. 2-16 to get a relationship for temperature differences between adjacent layers.]

6. Determine the emission temperature of the planet Venus. You may assume the following: the mean radius of Venus' orbit is 0.72 times that of the Earth's orbit; the solar flux So decreases as the square of the distance from the Sun and has a value of 1367 W m - 2 at the mean Earth orbit; Venus' planetary albedo = 0.77.

The observed mean surface temperature of the planet Venus is about 750 K (see Table 2.1). How many layers of the N-layer model considered in Problem 5 would be required to achieve this degree of warming? Comment.

FIGURE 2.12. An atmosphere made up of N slabs, each of which is completely absorbing in the IR spectrum.

7. Climate feedback due to

Stefan-Boltzmann.

(a) Show that the globally averaged incident solar flux at the ground is 1 (1 - ap)So.

(b) If the outgoing longwave radiation from the Earth's surface were governed by the Stefan-Boltzmann law, then we showed in Eq. 2-15 that for every 1 Wm-2 increase in the forcing of the surface energy balance, the surface temperature will increase by about a quarter of a degree. Use your answer to (a) to estimate by how much one would have to increase the solar constant to achieve a 1°C increase in surface temperature? You may assume that the albedo of Earth is 0.3 and does not change.

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