Exercises. Exercises 145

Transcription

1 Exercises 145 cations for the global energy balance, as discussed in Section It is notable that over some of the world s hottest desert regions, the outgoing longwave radiation exceeds absorbed solar radiation. The exclusive focus on radiative fluxes in this section is justified by the fact that radiative transfer is the only process capable of exchanging energy between the Earth and the rest of the universe. The energy balance at the Earth s surface is more complicated because conduction of latent and sensible heat across the Earth s surface also plays important roles, as discussed in Chapter 9. Exercises Net Radiation W m 2 Fig Global distribution of the net imbalance between the annual-mean net incoming solar radiation and the outgoing longwave radiation. Positive values indicate a downward flux. [Based on data from the NASA Earth Radiation Budget Experiment. Courtesy of Dennis L. Hartmann.] 4.11 Explain or interpret the following in terms of the principles discussed in this chapter. (a) Species of plants that require a relatively cool, moist environment tend to grow on poleward-facing slopes. (b) On a clear day a snow surface is much brighter when the sun is nearly overhead than when it is just above the horizon. (c) Light-colored clothing is often worn in hot climates. (d) The stars twinkle, whereas the planets do not. (e) The radiation emitted by the sun is isotropic, yet solar radiation incident on the Earth s atmosphere may be regarded as parallel beam. (f) The colors of the stars are related to their temperatures, whereas the colors of the planets are not. (g) The color temperature of the sun is slightly different from its equivalent blackbody temperature. (h) The equivalent blackbody temperature of Venus is lower than that of Earth even though Venus is closer to the sun. (i) The equivalent blackbody temperature of the Earth is lower than the global-mean surface temperature by 34 C. (j) Frost may form on the ground when temperatures just above the ground are above freezing. (k) At night if air temperatures are uniform, frost on highways is likely to form first on bridges and hilltops. (l) Aerosols in the atmosphere (or on windows) are more clearly visible when viewed looking toward a light source than away from it. (m) Clouds behave as blackbodies in the infrared region of the spectrum, but are relatively transparent in the microwave region. (n) Sunlit objects appear reddish around sunrise and sunset on days when the air is relatively free of aerosols. (o) A given mass of liquid water would produce more scattering if it were distributed among a large number of small cloud droplets than among a smaller number of larger droplets (assume that both sizes of droplets fall within the geometric optics regime). (p) The upper layers of the atmosphere in Fig appear blue, whereas the lower layers appear red. (q) Smoke particles with a radius of 0.5 mm appear bluish when viewed against a dark background but reddish when viewed against a light background (e.g., the sky). (r) The disk of the full moon appears uniformly bright all the way out to the edge. (s) The absorption coefficient of a gas is a function of temperature and pressure. (t) The absorptivity of greenhouse gases in the troposphere is enhanced by the presence of high concentrations of N 2 and O 2.

2 146 Radiative Transfer (z) Low clouds are not visible in satellite imagery in the water vapor channel. (aa) Ground fog is more clearly evident in visible than in infrared satellite imagery. (bb) The fraction of the incoming solar radiation that is backscattered to space by clouds is higher when the sun is low in the sky than when it is overhead. (cc) Under what conditions can the flux density of solar radiation at the Earth s surface (locally) be greater than that at the top of the atmosphere? 4.12 Remote sensing in the microwave part of the spectrum relies on radiation emitted by oxygen molecules at frequencies near 55 GHz. Calculate the wavelength and wave number of this radiation The spectrum of monochromatic intensity can be defined either in terms of wavelength or wave number such that the area under the spectrum, plotted as a linear function of or,isproportional to intensity.show that I 2 I A body is emitting radiation with the following idealized spectrum of monochromatic flux density. Fig In these views of the limb of the Earth from space, the upper atmosphere shows up blue, while the lower atmosphere exhibits an orange hue. The lower photo, taken 2 months after the eruption of Mt. Pinatubo, shows layers of sulfate aerosols in the lower stratosphere. [Photographs courtesy of NASA.] 0.35 m 0.50 m 0.70 m 0.35 m 0.50 m 0.70 m 1.00 m 1.00 m F 0 F F 1.0 W m F 0.5 W m F 0.2 W m 0 2 m 1 2 m 1 2 m 1 (u) Low clouds emit more infrared radiation than high clouds of comparable optical thickness. (v) The presence of cloud cover tends to favor lower daytime surface temperatures and higher nighttime surface temperatures. (w) Temperature inversions tend to form at night immediately above the tops of cloud layers. (x) Convection cells are often observed within cloud layers. (y) At night, when a surface based inversion is present, surface temperatures tend to rise when a deck of low clouds moves overhead. Calculate the flux density of the radiation An opaque surface with the following absorption spectrum is subjected to the radiation described in the previous exercise: 0.70 m 0.70 m 0 1 How much of the radiation is absorbed? How much is reflected? 4.16 Calculate the ratios of the incident solar radiation at noon on north- and south-facing 5 slopes (relative to the horizon) in seasons in which the solar zenith angle is (a) 30 and (b) 60.

3 Exercises Compute the daily insolation at the North Pole at the time of the summer solstice when the Earth sun distance is km. The tilt of the Earth s axis is Compute the daily insolation at the top of the atmosphere at the equator at the time of the equinox (a) by integrating the flux density over a 24-h period and (b) by simple geometric considerations. Compare your result with the value in the previous exercise and with Fig Orbitally induced variations in solar flux density incident in the top of the atmosphere during summer at high latitudes of the northern hemisphere play a central role in the orbital theory of the ice ages discussed in Section By what factor does the flux density at noon, at 55 N on the day of the summer solstice, vary between the extremes of the orbital cycles? 4.20 What fraction of the flux of energy emitted by the sun does the Earth intercept? 4.21 Show that for small perturbations in the Earth s radiation balance T E 1 F E T E 4 F E where T E is the Earth s equivalent blackbody temperature and F E is the flux density of radiation emitted from the top of its atmosphere. [Hint:Take the logarithm of the Stefan Boltzmann law (4.12) and then take the differential.] Use this relationship to estimate the change in equivalent blackbody temperature that would occur in response to (a) the seasonal variations in the sun Earth distance due to the eccentricity of the Earth s orbit (presently 3.5%) and (b) an increase in the Earth s albedo from to Show that the flux density of incident solar radiation on any planet in the solar system is 1368 W m 2 r 2, where r is the planet sun distance, expressed in astronomical units Estimate the flux density of the radiation emitted from the solar photosphere using two different approaches: (a) starting with the intensity ( W m 2 sr 1 ) and making use of the results in Exercise 4.3 and (b) making use of the relationship derived in the previous exercise. (c) Estimate the output of the sun in watts By differentiating the Planck function (4.10), derive Wien s displacement law. [Hint:in the wavelength range of interest, the exponential term in the denominator of (4.10) is much larger than 1.] 4.25 Show that for radiation with very long wavelengths, the Planck monochromatic intensity B (T) is linearly proportional to absolute temperature. This is referred to as the Rayleigh Jeans limit Use the relationship derived in Exercise 4.22 to check the numerical values of T E in Table The observed equivalent blackbody temperature of Jupiter is 125 K, 20 K higher than the value in Table 4.1. Assuming that the temperature of Jupiter is in a steady state, estimate the flux density of radiation emitted from the top of its atmosphere that is generated internally by processes on the planet If the moon subtends the same arc of solid angle in the sky that the sun does and it is directly overhead, prove that the flux density of moonlight on a horizontal surface on Earth is given by F s a(r s d) 2, where F s is the flux density of solar radiation intercepted by the Earth, a is the moon s albedo, R s is the radius of the sun, and d is the Earth sun distance. Estimate the flux density of moonlight under these conditions, assuming a lunar albedo of Suppose that the sun s emission or the Earth s albedo were to change abruptly by a small increment. Show that the radiative relaxation rate for the atmosphere (i.e., the initial rate at which the Earth s equivalent blackbody temperature would respond to the change, assuming that the atmosphere is thermally isolated from the other components of the Earth system) is given by dt 4 T E T 3 E dt c p p sɡ 1 where T E is the initial departure of the equivalent blackbody temperature from radiative equilibrium, is the Stefan Boltzmann constant, T E is the equivalent

4 148 Radiative Transfer blackbody temperature in K, c p is the specific heat of air, p s is the global-mean surface pressure, and g is the gravitational acceleration. The time T E (dt/dt) 1 required for the atmosphere to fully adjust to the change in radiative forcing, if this initial time rate of change of temperature were maintained until the new equilibrium was established, is called the radiative relaxation time.estimate the radiative relaxation time for the Earth s atmosphere A small, perfectly black, spherical satellite is in orbit around the Earth at an altitude of 2000 km as depicted in Fig What angle does the Earth subtend when viewed from the satellite? 4.31 If the Earth radiates as a blackbody at an equivalent blackbody temperature T E 255 K, calculate the radiative equilibrium temperature of the satellite when it is in the Earth s shadow. [Hint:Let de be the amount of radiation flux imparted to the satellite by the flux density de received within the infinitesimal element of solid angle d.] Then, de r 2 Id where r is the radius of the satellite and I is the intensity of the radiation emitted by the Earth, i.e., the flux density of blackbody radiation, as given by (4.12), divided by.integrate the above expression over the arc of solid angle subtended by the Earth, as computed in the previous exercise, noting that the radiation is isotropic, to obtain the total energy absorbed by the satellite per unit time Q 2.21r 2 T 4 E Finally, show that the temperature of the satellite is given by T s T E Show that the approach in Exercise 4.5 in the text, when applied to the previous exercise, yields a temperature of T s T E K Explain why this approach underestimates the temperature of the satellite. Show that the answer obtained with this approach converges to the exact solution in the previous exercise as the distance between the satellite and the center of the Earth becomes large in comparison to the radius of the Earth R E.[Hint:show that as d R E :,the arc of solid angle subtended by the Earth approaches R 2 E d2.] 4.33 Calculate the radiative equilibrium temperature of the satellite immediately after it emerges from the Earth s shadow (i.e., when the satellite is sunlit but the Earth, as viewed from the satellite, is still entirely in shadow) The satellite has a mass of 100 kg, a radius of 1 m, and a specific heat of 10 3 J kg 1 K 1. Calculate the rate at which the satellite heats up immediately after it (instantaneously) emerges from the Earth s shadow Consider two opaque walls facing one another. One of the walls is a blackbody and the other wall is gray (i.e., independent of ).The walls are initially at the same temperature T and, apart from the exchange of radiation between them, they are thermally insulated from their surroundings. If and are the absorptivity and emissivity of the gray wall, prove that. Solution: The flux emitted by the black wall is F T 4 and the flux absorbed by the gray wall is T 4.The flux emitted by the gray wall is T 4. The rate at which the gray wall gains or loses energy from the exchange of radiation with the black wall is Earth satellite H T 4 T 4 ( ) T 4 Fig Geometric setting for Exercises If H is not zero, then the temperature of the wall must change in response to the energy

5 Exercises 149 imbalance, in which case, heat is being transferred from a colder body to a warmer body, in violation of the second law of thermodynamics. It follows that 4.36 (a) Extend the proof in the previous exercise to the case in which absorptivity and emissivity are wavelength dependent. Let one of the walls be black, as in the previous exercise, and let the other wall also be black, except within a very narrow wavelength range of width,centered at 1 where 1. [Hint:Because a 1 blackbody radiation is isotropic, it follows the blackbody flux in the interval is B( 1, T).Using this relationship,consider the energy balance as in the previous exercise and proceed to show that.] (b) Indicate how this result could be extended to prove that Consider a closed spherical cavity in which the walls are opaque and all at the same temperature. The surfaces on the top hemisphere are black and the surfaces on the bottom hemisphere reflect all the incident radiation at all angles. Prove that in all directions I B (a) Consider the situation described in Exercise 4.35, except the both plates are gray, one with absorptivity 1 and the other with absorptivity 2.Prove that F 1 1 F 2 2 where F 1 and F 2 are the flux densities of the radiation emitted from the two plates. Make use of the fact that the two plates are in radiative equilibrium at the same temperature but do not make use of Kirchhoff s law. [Hint:Consider the total flux densities F 1 from plate 1 to plate 2 and F 2 from plate 2 to plate 1. The problem can be worked without dealing explicitly with the multiple reflections between the plates.] 4.39 Consider the radiation balance of an atmosphere with a large number of layers, each of which is isothermal, transparent to solar radiation, and absorbs the fraction of the longwave radiation incident on it from above or below. (a) Show that the flux density of the radiation emitted by the topmost layer is F (2 ) where F is the flux density of the planetary radiation emitted to space. By applying the Stefan Boltzmann law (4.12) to an infinitesimally thin topmost layer, show that the radiative equilibrium temperature at the top of the atmosphere, sometimes referred to as the skin temperature,is given by T* TE (Were it not for the presence of stratospheric ozone, the temperature of the 20- to 80-km layer in the Earth s atmosphere would be close to the skin temperature.) 4.40 Consider an idealized aerosol consisting of spherical particles of radius r with a refractive index of 1.5. Using Fig. 4.13, estimate the smallest radius for which the particles would impart a bluish cast to transmitted white light, as in the rarely observed blue moon Consider an idealized cloud consisting of spherical droplets with a uniform radius of 20 m and concentrations of 1 cm 3.How long a path through such a cloud would be required to deplete a beam of visible radiation by a factor of e due to scattering alone? (Assume that none of the scattered radiation is subsequently scattered back into the path of the beam.) 4.42 Consider solar radiation with a zenith angle of 0 that is incident on a layer of aerosols with a single scattering albedo , an asymmetry factor g 0.7, and an optical thickness 0.1 averaged over the shortwave part of the spectrum. The albedo of the underlying surface is R s (a) Estimate the fraction of the incident radiation that is backscattered by the aerosol layer in its downward passage through the atmosphere. (b) Estimate the fraction of the incident radiation that is absorbed by the aerosol layer in its downward passage through the atmosphere. (c) Estimate the consequent corresponding impact of the aerosol layer upon the local albedo. Neglect multiple scattering. For simplicity, assume that the radiation

6 150 Radiative Transfer back-scattered from the earth s surface and clouds is parallel beam and oriented at 0 or 180 Zenith angle. (In reality it is isotropic.) [Hint:Show that the fraction of the radiation that is backscattered in its passage through the layer is (1 ɡ) b 0(1 e ) 2 and the fraction that is transmitted through the layer is and from the hypsometric equation applied to an isothermal layer Substituting for T and in (4.61) we obtain di T e 0e z H dz I (k r 0)e z H e t e (1 ɡ) 0(1 e ) 2 Then show that the total upward reflection from the top of the atmosphere is From (4.32) (k r 0) H(k r 0)e z H z e z H dz (4.62) b R s t 2 (1 br s b 2 R 2 s ) which can be rewritten in the form 4.43 Consider radiation with wavelength and zero zenith angle passing through a gas with an absorption coefficient k of 0.01 m 2 kg 1.What fraction of the beam is absorbed in passing through a layer containing 1 kg m 2 of the gas? What mass of gas would the layer have to contain in order to absorb half the incident radiation? 4.44 Show that for overhead parallel beam radiation incident on an isothermal atmosphere in which the r,the mixing ratio of the absorbing gas,and k,the volume absorption coefficient,are both independent of height, the strongest absorption per unit volume (i.e., di dz) is strongest at the level of unit optical depth. Solution: From (4.17) if r and k are both independent of height, di b R st 2 (1 br s ) dz I T (k r) (4.61) where I is the intensity of the radiation incident on the top of the atmosphere, T I I is the transmissivity of the overlying layer, and is the density of the ambient air. From (4.33) Using (4.62) to express e z H in the aforementioned equation in terms of optical depth, we obtain Now at the level where the absorption is strongest, d dz di dz di dz I H Performing the indicated differentiation, we obtain e I H d e d dz ( e ) 0 (1 dz ) 0 from which it follows that 1. Although this result is strictly applicable only to an isothermal atmosphere in which k and r are independent of height, it is qualitatively representative of conditions in planetary atmospheres in which the mixing ratios of the principal absorbing constituents do not change rapidly with height. It was originally developed by Chapman 28 for understanding radiative and photochemical processes related to the stratospheric ozone layer.

7 Exercises For incident parallel beam solar radiation in an isothermal atmosphere in which k is independent of height (a) show that optical depth is linearly proportional to pressure and (b) show that the absorption per unit mass (and consequently the heating rate) is strongest, not at the level of unit optical depth but near the top of the atmosphere, where the incident radiation is virtually undepleted Consider a hypothetical planetary atmosphere comprised entirely of the gas in Exercise The atmospheric pressure at the surface of the planet is 1000 hpa, the lapse rate is isothermal, the scale height is 10 km, and the gravitational acceleration is 10 m s 2.Estimate the height and pressure of the level of unit normal optical depth (a) What percentage of the incident monochromatic intensity with wavelength and zero zenith angle is absorbed in passing through the layer of the atmosphere extending from an optical depth 0.2 to 4.0? (b) What percentage of the outgoing monochromatic intensity to space with wavelength and zero zenith angle is emitted from the layer of the atmosphere extending from an optical depth 0.2 to 4.0? (c) In an isothermal atmosphere, through how many scale heights would the layer in (a) and (b) extend? 4.48 For the atmosphere in Exercise 4.46, estimate the levels and pressures of unit (slant path) optical depth for downward parallel beam radiation with zenith angles of 30 and Prove that the optical thickness of a layer is equal to ( 1) times the natural logarithm of the transmissivity of the layer Prove that the fraction of the flux density of overhead solar radiation that is backscattered to space in its first encounter with a particle in the atmosphere is given by b 1 2 ɡ where ɡ is the asymmetry factor defined in (4.35). [Hint:The intensity of the scattered radiation must be integrated over zenith angle.] 4.51 Show that Schwarzschild s equation in the form that it appears in (4.41) in the text can be integrated along a path extending from 0to s 1 in Fig to obtain an expression for I (s 1 ) Prove that as the optical thickness of a layer approaches zero, the flux transmissivity, as defined in Eq. (4.45), becomes equal to the intensity transmissivity at a zenith angle of 60 degrees. over solid angle A thin, isothermal layer of air in thermal equilibrium at temperature T 0 is perturbed about that equilibrium value (e.g., by absorption of a burst of ultraviolet radiation emitted by the sun during a short-lived solar flare) by the temperature increment T.Using the cooling to space approximation (4.56) show that where di ( I B (T ))k rds 4.53 Integrate the expression for the heating rate, as approximated by cooling to space that appears as Eq. (4.55) of the text, namely dt 2 1 (4.55) dt c p 0 dt dt k rb (z)e d T k r e c p db dt T0 (4.63) (4.64) This formulation, in which cooling to space acts to bring the temperature back toward radiative equilibrium, is known as Newtonian cooling or radiative relaxation.it is widely used in 28 Sydney Chapman ( ) English geophysicist. Made important contributions to a wide range of geophysical problems, including geomagnetism, space physics, photochemistry, and diffusion and convection in the atmosphere.

8 152 Radiative Transfer parameterizing the effects of longwave radiative transfer in the middle atmosphere Prove that weighting function w i used in remote sensing, as defined in (4.59), can also be expressed as the vertical derivative of the transmittance of the overlying layer The annual mean surface air temperature ranges from roughly 23 C in the tropics to 25 C in the polar cap regions. On the basis of the Stefan Boltzmann law, estimate the ratio of the flux density of the emitted longwave radiation in the tropics to that in the polar cap region.