Catling, D.C. (2015) Planetary atmospheres, in G. Schubert (ed.) Treatise on Geophysics, 2nd Ed., vol. 10, Oxford, Elsevier, p.429-472.
10.13 Planetary Atmospheres
DC Catling, University of Washington, Seattle, WA, USA
ã 2015 Elsevier B.V. All rights reserved.
10.13.1 Introduction and Scope 429 10.13.2 Composition and Vertical Structure 430
10.13.2.1 The Observed Compositional Diversity of Atmospheres 430 10.13.2.2 Nomenclature for the Vertical Structure of Atmospheres 432 10.13.2.3 The Equation of State, Atmospheric Composition, and the Hydrostatic Equation 434 10.13.2.4 Convection and Stability 435 10.13.2.5 Condensables and the Clausius–Clapeyron Equation 437
10.13.3 Radiation 438 10.13.3.1 Planetary Energy Balance, Blackbody Radiation, and the Greenhouse Effect 438 10.13.3.2 Radiative Transfer: Shortwave and Infrared 441 10.13.3.3 Optical Thickness and Thinness 444 10.13.3.4 Radiative–Convective Equilibrium 445
10.13.3.5 Absorption and Emission by Atmospheric Gases 450 10.13.3.5.1 Rotational transitions 451 10.13.3.5.2 Vibrational transitions 451 10.13.3.5.3 Collision-induced absorption 451 10.13.3.5.4 Broadening 451
10.13.3.5.5 Continuum 452
10.13.3.5.6 Numerical radiative transfer 452 10.13.4 Photochemistry 452 10.13.4.1 Basic Principles of Chemical Kinetics 452 10.13.4.2 Why Free Radicals Are So Important 453 10.13.4.3 Photolysis 453
10.13.4.4 Earth’s Atmospheric Composition and Chemistry 454 10.13.4.5 CO2 Atmospheres: Venus and Mars 455 10.13.4.6 Reducing Atmospheres: Giant Planets and Titan 455 10.13.5 Atmospheric Dynamics 455 10.13.5.1 Equations of Motion 455
10.13.5.2 Balance Regimes: Geostrophic, Cyclostrophic, and Thermodynamic 457 10.13.5.2.1 Geostrophic balance 457 10.13.5.2.2 Cyclostrophic balance 457 10.13.5.2.3 Thermodynamic balance 458 10.13.5.3 Zonal Mean Meridional Circulation: Hadley Cells and Thermally Driven Jets 458
10.13.5.4 Eddy-Driven Jet Streams 461 10.13.5.5 Planetary Waves and Barotropic and Baroclinic Instabilities 462 10.13.5.6 Deep and Shallow Atmospheres on Giant Planets 464 10.13.5.7 Buoyancy Waves and Thermal Tides 465 10.13.5.8 Superrotation 465
10.13.6 Outstanding Questions 466 Acknowledgments 467 References 467
10.13.1 Introduction and Scope a new frontier that should reveal a larger diversity of atmospheres (Seager, 2010; Seager and Deming, 2010; Seager et al., 2010). But The enormous topic of planetary atmospheres ranges from the at the moment, only a little can be said about exoplanet atmo- origin of atmospheres to the fluid dynamics of small weather spheres with any certainty. Currently, observational data from systems, so here, I restrict the scope to broad essentials at a level such atmospheres are arguably much sparser than data were for suitable for a PhD research student. I assume knowledge of atmospheres in our solar system before the Space Age. Rather calculus, plus basic physics and chemistry, but little else. The than speculate, I concentrate on key principles associated with audience could also include researchers in allied fields – for the planetary atmospheres that we know best. In fact, these are example, planetary geologists, geochemists, or astronomers. what will be needed to interpret exoplanet atmospheres correctly I focus on atmospheres in the solar system. Exoplanets represent (see Chapter 10.21).
Treatise on Geophysics, Second Edition http://dx.doi.org/10.1016/B978-0-444-53802-4.00185-8 429 Treatise on Geophysics, 2nd edition, (2015), vol. 10, pp. 429-472
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430 Planetary Atmospheres
Until the last 5 years or so, introductory syntheses about atmospheres (Moses et al., 2011) or early Mars (Zahnle et al., atmospheres were mostly confined to the Earth (e.g., 2008)), although we do not have an example in the modern
Houghton, 2002; Wallace and Hobbs, 2006), with a few excep- solar system, and such an atmosphere would likely contain tions (Chamberlain and Hunten, 1987; Yung and DeMore, considerable H2 as well. The atmosphere of the early Earth was 1999), but now, the coverage of planetary atmospheres is also reducing, albeit weakly, before its oxygenation at 2.4 Ga more available. A primer by Ingersoll (2013) is a good read (Ga giga anna billions of years ago) (Catling, 2014; Lyons ¼ ¼ for all levels, while Taylor (2010) provides a useful introduc- et al., 2014). In contrast, oxidizing atmospheres are relatively tion aimed at mid-level undergraduates. At the research level, poor in reducing hydrogen-bearing gases and include modern general texts are Pierrehumbert (2010), Sanchez-Lavega Earth and Mars (Table 2). Molecular O2 has a volume abun- (2010), and Catling and Kasting (in press). In a more special- dance of 21 and 0.15% on Earth and Mars, respectively, which ized area, Yung and Demore (1999) provided a detailed mono- greatly exceeds the amount of hydrogen-bearing reducing gases graph on photochemistry. Recent papers have also reviewed of 1.8 ppmv (parts per million by volume) CH on Earth 4 the probable atmospheric circulation regimes of exoplanets (Hartmann et al., 2013) and 15 5 ppmv H on Mars Æ 2 (Showman et al., 2010, 2013). (Krasnopolsky and Feldman, 2001). Venus is unusual. The The structure of this review is as follows. In order to discuss upper atmosphere of Venus is oxidizing, with a tenuous atmospheres, I start with an overview of composition and ozone (O3) layer at 90–105 km altitude on the nightside basic physical properties, including the layers of atmospheres. (Montmessin et al., 2011), while the atmosphere below
Thus, Section 10.13.2 describes structure and thermodynam- 20 km altitude is arguably weakly reducing because sulfur is ics. Atmospheric composition determines physical properties, converted into various reduced forms (Krasnopolsky and which in turn affects climate through the greenhouse effect. Pollack, 1994). Noting sulfur oxidation numbers in square Aplanet’sclimaterequiresagraspoftheradiativetransferof brackets, these sulfur species include sulfur dioxide (SO2) energy, which is discussed in Section 10.13.3.Abriefoverview [+4], carbonyl sulfide (OCS)[ 2], hydrogen sulfide (H2S) À of the photochemistry of atmospheres is then given in [ 2], and elemental sulfur allotropes of 1–8 atoms (S–S )[0]. À 8 Section 10.13.4 to provide an appreciation of the globally The OCS is sourced from weathering of sulfides on the surface, averaged bulk composition and vertical distribution of con- or volcanic outgassing, or the reduction of sulfur by carbon stituents. However, atmospheric motions transport heat and monoxide (e.g., S2 +CO), which is present at 20 ppmv below constituents, including condensable species, and are indis- 20 km (Fegley, 2014). Unlike Earth and Mars, O2 is absent in pensable for several aspects of climate and chemistry. Princi- Venus’ atmosphere with an upper limit of <0.3 ppmv. ples of atmospheric motion are described in Section 10.13.5. Broadly speaking, planetary formation theories from a Together, these various topics – thermodynamics, radiation, Sun-like nebula (reviewed by Chambers, 2014) favor more chemistry, and dynamics – provide the essentials of planetary hydrogen-rich atmospheres far from a star and more oxidizing atmospheres. Finally, in Section 10.13.6,Iverybrieflynote atmospheres nearby, but this can be confounded in an some key outstanding questions. extrasolar system if a planet’s orbital distance changes in plan-
One topic is missing: atmospheric origin and evolution. etary migration. Also, the very early rocky planets in our own This subject is of profound interest but involves consider- solar system may have had hydrogen-rich atmospheres able geochemistry and familiarity with all the topics of (Abe, 2011). this review, so the reader is referred to Catling and Kasting Redox also indicates the possible chemistry of aerosols, (in press) that comprehensively reviews this area (see which are fine particles found in atmospheres. In general,
Chapter 10.14). oxidizing atmospheres with sulfurous gases tend to produce
sulfate aerosols at altitude (McGouldrick et al., 2011), such as in the Earth’s stratosphere (Crutzen, 1976; Robock, 2000), on 10.13.2 Composition and Vertical Structure Venus (Thompson et al., 2011), or on Mars when it was volcanically active (Settle, 1979; Smith et al., 2014; Tian 10.13.2.1 The Observed Compositional Diversity et al., 2010). In contrast, reducing atmospheres generate of Atmospheres hydrocarbon aerosols, including the giant planets (Irwin, A physically meaningful classification of the composition of 2009; Moses et al., 2004; West et al., 1991, 2004, 2009), planetary atmospheres falls into two chemical categories: Titan (Krasnopolsky, 2010; Lavvas et al., 2008a,b; Wilson and reducing and oxidizing. A redox (reduction–oxidation) state Atreya, 2004), and probably the early Earth (Pavlov et al., indicates the chemical character of atmospheres, for example, 2001; Wolf and Toon, 2010; Zerkle et al., 2012). whether elements such as carbon are likely to be found in their In addition to the atmospheres discussed in the preceding reduced form (methane, CH4) or oxidized variety (carbon text, some smaller bodies possess very thin atmospheres. Exo- dioxide, CO2). spheres exist directly above the surfaces of several moons, The solar system contains several examples of oxidizing and which are defined as atmospheric layers of such low density reducing atmospheres. Reducing atmospheres are those that that they are virtually collisionless. We find the following thin are relatively rich in reducing gases, typically hydrogen-bearing atmospheres, where the surface pressure is given in bar gases, such as hydrogen itself (H ), methane (CH ), and pos- (1 bar 105 Pa) in parentheses: 2 4 ¼ sibly ammonia (NH3). Examples include all the giant planets and Saturn’s largest moon, Titan (Table 1). In principle, an (i) N2-rich atmospheres above surfaces covered in N2 ice: 6 atmosphere dominated by carbon monoxide (CO) would also Triton ( (14–20) 10À ), the largest moon of Neptune, Â 6 be reducing (perhaps applicable to some hot giant exoplanet and Pluto ( 15 10À ) Â
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Table 1 The composition of reducing atmospheres
Species Jupiter Saturn Uranus Neptune Titan
H 86.4 2.6% 88 2% 82.5 3.3% 80 3.2% 0.099% 2 Æ Æ Æ Æ He 13.6 0.3% 12 2% 15.2 3.3% 19.0 3.2% a Æ Æ Æ Æ CH 0.204 0.049% 0.47 0.2% 2.3% (below 1–2% (below 5.65% (1.48% in 4 Æ Æ CH4 clouds) CH4 clouds) stratosphere) 13 5 5 CH 1.9 0.1 10À 5.1 0.2 10À 0.062% 4 Æ Â Æ Â N2 – – – – 94.2% a 4 4 7 7 NH 5.74 0.22 10À 1.1 10À (p>1 bar, <1 10À 6 10À (at 6 bar) 3 Æ Â Â Â Â (8.9–11.7 bar, below NH3 below NH3 clouds) clouds) a 4 7 9 9 8 H O 4.23 1.38 10À (at 2.3 10À (upper 6–14 10À (at 1.7–4.1 10À (at <10À 2 Æ Â Â Â Â 17.6–20.9 bar) troposphere) p<0.03 mbar) p<0.6 mbar) a 5 6 7 6 H2S 6.7 0.4 10À (at p>16 bar) <0.4 10À <8 10À <3 10À – Æ Â5    520 Ne 1.99 10À <2 10À Ne  5  540 7 Ar 1.57 0.35 10À 3.4 10À Ar 2.1 10À Æ Â Â Â 36Ar 6 6 5 6 5 C H 5.8 1.5 10À (stratosphere) 7.0 1.5 10À 1.5 10À (at 1.5 10À 1 10À 2 6 Æ Â Æ Â Â Â Â (stratosphere) 0.5–1 mbar) (stratosphere) 6 6 7 PH 1.04 0.1 10À 5.9 0.2 10À <8.3 10À - 3 Æ Â Æ Â Â (p>0.5 bar) 8 7 8 +14.0 9 6 C2H2 4 10À (at 5 mbar) 3.0 0.2 10À (at 1.0 10À 6.0 4.0 10À 2 10À Â Æ Â Â À   1 mbar) (stratosphere) (stratosphere) 10 10 11 8 8 C H 3.5 10À (at p<10 mbar) 3.4 10À (at 4.0 0.5 10À 6 10À (at 10À (above stratospheric 2 2 Â Â Æ Â Â p<10 mbar) (at 0.1 mbar) p<5 mbar) condensation) 9 7 8 6 5 CO 1.6 0.3 10À (at 5–8 bar) <10À (stratosphere) <4 10À 6.5 3.5 10À 4.7 10À Æ Â Â Æ Â Â (stratosphere) (stratosphere) 5 +0.75 5 4 D/H 2.3–2.6 10À 1.7 0.45 10À 1.35 0.3 10À in H2  À  Æ+0.15 4 1.32 0.11 10À in CH4 À  13C/12C 0.0108 0.0005 0.011 0.01099 0.00017 in CH Æ Æ 4 aCondensable species, so levels are indicated.
Gases are given as volume mixing ratios. CH is condensable on Titan, Uranus, and Neptune. H O and NH are condensable on the giant planets, while H S reacts with NH to form 4 2 3 2 3 ammonium hydrosulfide (NH4SH) clouds. Source: Lodders K (2010) Atmospheric chemistry of the gas giant planets. Geochemical News 142: 1–11; Atreya SK, Mahaffy PR, Niemann HB, Wong MH, and Owen TC (2003) Composition and origin of the atmosphere of Jupiter – An update, and implications for the extrasolar giant planets. Planetary and Space Science 51: 105–112; Irwin P (2009) Giant Planets of Our Solar System: Atmospheres, Composition, and Structure, 2nd edn. Chichester, UK: Springer; Niemann HB, Atreya SK, Demick JE, Gautier D, Haberman JA, Harpold DN, Kasprzak WT, Lunine JI, et al. (2010) Composition of Titan’s lower atmosphere and simple surface volatiles as measured by the Cassini–Huygens probe gas chromatograph mass spectrometer experiment. Journal of Geophysical Research 115.
(ii) O2-rich exospheres above moons covered in water ice: come from magnetospheres, which are envelopes of ions around 12 13 Jupiter’s moons, Europa ( 10 –10 O2), (Smyth and planets controlled by the planetary magnetic fields. 12 Marconi, 2006), Ganymede ( 10À O2)(Hall et al., Volcanic small bodies have a different class of atmosphere. 12 1998), and Callisto ( 10À O2 possibly+CO2) The very thin SO2 atmosphere on Io, a rocky body, is sourced 5 1 (Carlson, 1999; Kliore et al., 2002; Liang et al., 2005), from high-temperature volcanism at a rate of 10 kg sÀ SO2 12 and Saturn’s moons, Rhea ( 10À O2 and CO2)(Teolis (Lellouch et al., 2003). The SO2 freezes to form frost away from 12 et al., 2010) and Dione ( 10À O2)(Tokar et al., 2012) the hot vents (Schmitt et al., 1994). Elemental sulfur is also 7 9 (iii) Volcanogenic atmospheres: SO2 on Io ( 10À –10À ), a volcanically outgassed, with SO2/S2 3–12 in the plume of the moon of Jupiter, and a water vapor–CO2 exosphere on volcano Pele (Spencer et al., 2000). In contrast to Io, Enceladus Enceladus, a moon of Saturn is covered in water ice and affected by cryovolcanism, that is, a
form of volcanism where ice turns to vapor or gets ejected as a The tenuous N2-rich atmospheres (with minor CH4) on Tri- result subsurface heating, unlike volcanism that involves mol- ton and Pluto arise from vapor equilibrium with N2 (and CH4) ten rock. Plumes at the south pole of Enceladus release icy 1 ice at prevailing surface temperatures. The O2-rich exospheres on particles at a rate of 200 30 kg sÀ (Hansen et al., 2011) Æ the moons of Jupiter and Saturn come from the breakdown of from parallel lineaments called ‘tiger stripes’ ( 500 m deep, water ice on the surfaces of these moons by radiolysis and 130 km long, 2 km wide) (Porco et al., 2006), causing a very sputtering ( Johnson et al., 2009). Radiolysis is the chemical asymmetrical exosphere (Tenishev et al., 2010). alteration of ices by charged particles, and sputtering is when The very thin atmospheres described in the preceding text charged particles collide and eject atoms or molecules. The faster are mentioned for completeness. Henceforth, I restrict most of escape of the hydrogen compared with that of the oxygen leaves the discussion to thicker atmospheres, the thinnest being that oxygen lingering behind in an exosphere. The charged particles of Mars.
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Table 2 The composition and properties of oxidizing atmospheres (noting that the lower atmosphere of Venus could be considered somewhat reducing)
Parameter Venus Earth Mars
Mean surface pressure (bar) 95.6 1.0 0.006 Mean surface temperature (K) 735 288 218
Mass relative to the Earth (5.97 1024 kg) 0.815 1.0 0.107 1/9 Â Mean radius relative to the Earth (6371 km) 0.950 1.0 0.532 1/2 a Key gases in atmosphere (by volume) CO2 96.5% N2 78.084% CO2 96.0 0.7% a Æ N2 3.5% O2 20.946% N2 1.89 0.01% c Æ SO2 H2O 0.1 ppm – 4% Ar 1.93 0.01% Æ 150 30 ppm (varies) O2 0.145 0.09% • Æ Æ (22–42 km) Ar 9340 ppm CO 0.06% 25 150 ppm CO a,b H O 0.03% • Æ 2 2 (12–22 km) 280 ppm (varies) • 40Ar 70 25 ppm (pre-industrial) He 10 ppm 36+38 Æ Ar 75 35 ppm • 396.5 ppm H2 15 5 ppm c Æ Æ H2O 30 15 ppm (5–45 km) (year 2013) Ne 2.5 ppm c Æ CO Ne 18.18 ppm Kr 0.3 ppm 4 • 45 10 ppm He 5.24 ppm O3 0–80 ppb Æ a (cloud top) CH4 1.7 ppm H2O2 0–40 ppb d • 17 1 ppm (12 km) Kr 1.14 ppm SO2 <0.3 ppb Æ a He 12 (+24/ 8)ppm H2 0.55 ppm À a Ne 7 3 ppm N2O 320 ppb c Æ a H 2.5 1 ppm CO 125 ppb 2 Æ (50–60 km) HCl 0.4 ppm (70 km) 84Kr 50 25 ppb Æ aBiologically cycled and influenced. bCurrently increasing about 2 ppm per year because of anthropogenic emissions. cAltitude-dependent. dIndicates lack of volcanic outgassing.
Source: Lodders K and Fegley B (1998) The Planetary Scientist’s Companion. New York: Oxford University Press; Pollack JB (1991) Kuiper prize lecture: Present and past climates of the terrestrial planets. Icarus 91: 173–198; Krasnopolsky VA and Lefevre F (2013) Chemistry of the atmospheres of Mars, Venus, and Titan. In: Mackwell SJ, Simon-Miller AA, Harder JW, and Bullock MA (eds.) Comparative Climatology of Terrestrial Planets, pp. 231–275. Tucson, AZ: University of Arizona Press; Mahaffy PR, Webster CR, Atreya SK, et al. (2013) Abundance and isotopic composition of gases in the martian atmosphere from the Curiosity Rover. Science 341: 263–266.
10.13.2.2 Nomenclature for the Vertical Structure (iv) The thermosphere from the mesopause to the thermopause of Atmospheres (or exobase) at that varies over altitudes of 250–500 km as Along with composition, a basic description of any atmospheres solar ultraviolet (UV) flux changes over cycles ranging from involves atmospheric structure. The term conventionally refers to daily to the 11-year solar cycle. (v) The exosphere, which blends the vertical temperature profile, which provides information into interplanetary space. about how chemical and physical processes change with altitude. Convection dominates lower tropospheres. Solar radiation
The average thermal profile is also the basis for naming atmo- heats the surface of rocky planets or is absorbed in the deeper spheric layers using a system developed for Earth in the early portions of tropospheres on giant planets. Consequently, warm 1900s (Lindemann and Dobson, 1923; Martyn and Pulley, buoyant air rises to altitudes of lower pressure where it expands 1936); thus, the nomenclature works with only some degree of and cools. Cool air parcels sink and warm. The net effect of such success for other bodies. New atmospheric layers begin where convection is a decline of temperature with altitude. On Earth, there is a temperature inflection (Figure 1). On Earth, the five the rate that temperature decreases, the lapse rate, averages 1 layers are the following: (i) the troposphere (tropos – Greek for 6 K kmÀ globally. In the upper troposphere where the air is the ‘turning’ of convection), where temperature declines from thinner, radiation transfers energy instead of convection above a the surface to the tropopause (pause – Greek for ‘stop’), which radiative–convective boundary. Higher still, emission of thermal lies at an average pressure of 0.16 bar (Sausen and Santer, 2003). infrared (IR) radiation from the air to space becomes so efficient The tropopause altitude is 0.2 bar (11 km) in the US Standard that a temperature minimum or tropopause is achieved. The
Atmosphere but varies in detail with latitude from 0.1 to 0.3 bar tropopause occurs near an average pressure of roughly 0.1 bar (tropics to pole) (Hoinka, 1998). (ii) The stratosphere (stratus– on Earth, Titan, and the giant planets. The similarity is because of Greek for ‘layered’), where temperature increases from the tro- a common scaling of broadband IR opacity and pressure. In popause to the stratopause at 50 km altitude where the air upper tropospheres, the IR opacity is pressure-dependent in the pressure is 100 Pa (1 mbar). (iii) The mesosphere from the same way – varying approximately with the square of the stratopause to the mesopause, at 85 km altitude, where pressure – while all these atmospheres are roughly similar in the air pressure is about 1–0.1 Pa (0.01–0.001 mbar). their strong opaqueness to IR at deeper pressures, such as 1 bar,
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500 Quiet Sun Active Sun
Exosphere 400
Thermopause
300
Heterosphere
Altitude (km) 200 Thermosphere
100 Mesopause Mesosphere Stratopause Stratosphere
Homosphere Tropopause Troposphere
0 500 1000 2000
Temperature (K)
Figure 1 The terminology for vertical layers of the Earth’s atmosphere, shown schematically. (Drawn by the author.)
despite the big differences in atmospheric composition and grav- molecules and atoms. In the tenuous gas, infrequent collisions ity (Robinson and Catling, 2014)(seeSection 10.13.3.4). between electrons and ions create an ionosphere, a shell of In stratospheres, the absorption of shortwave solar radiation electrons and ions that extends on Earth from the mesosphere causes a temperature inversion, defined as a temperature increase into the exosphere. The Earth’s hot thermosphere ( 1000 K at with altitude. On Earth, the ozone absorbs UV radiation, whereas 250 km altitude) reflects the slow rate of removing heat. Atoms on the giant planets and Titan, aerosol particles and methane and atomic ions do not have vibrational or rotational transi- absorb at wavelengths in the UV, visible, and near-IR. Mars and tions and so are poor radiators, unlike molecules. Instead, heat
Venus, which have predominantly CO2 atmospheres, do not is transferred inefficiently by conduction downward to the cold have strong shortwave absorbers above their tropospheres, so mesopause where radiative cooling is effective. Unlike the in the global average, they lack stratospheric inversions and Earth, the upper thermospheres on Mars and Venus are rela- tropopause minima (see Section 10.13.3.4). tively cold at 270 K for Mars and 250 K for Venus, because Above the Earth’s stratosphere, CO emission to space their 96% CO causes efficient radiative cooling to space 2 2 causes the mesosphere to have the lowest atmospheric temper- (Section 10.13.4.5). Finally, exospheres lie above thermo- atures. A cold mesopause forms the lower boundary of the spheres and are where atmospheres become collisionless. thermosphere where temperature increases with altitude. Chemically, we can divide atmospheres into two big Mesopauses occur around 1–0.1 Pa in planetary atmospheres layers: a homosphere where bulk gases are well mixed up to a in the solar system. They are cold because the air is so thin in homopause and a heterosphere above the homopause where upper mesospheres that energy-exchanging collisions between molecular diffusion becomes more important than turbulent molecules become less frequent than photon emission at a mixing. The homopause occurs where the diffusion separation level of radiative relaxation (Curtis and Goody, 1956; Lo´pez- time and turbulent mixing time of gases are equal. This tends to Puertas and Taylor, 2001). Consequently, excited molecules occur within a few orders of magnitude of 0.01 Pa (or lose energy by radiation rather than by collisions. 0.1 mbar). On Earth, the homopause is at 100 km height. In thermospheres, absorption of extreme ultraviolet radia- Above, at 250–1000 km altitude, diffusive separation causes tion (EUV) (with wavelength shorter than 120 nm) causes atomic oxygen (O) to become the dominant species, while temperature to increase. Molecules are broken apart by EUV higher up, helium dominates, and then, hydrogen above and photochemical reactions, and EUV and x-rays ionize both 2500 km height.
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Species escape from the base of exospheres (the exobase) if Here, k is Boltzmann’s constant, m is the mean molecular mass, n their upward velocity exceeds the escape velocity v ,whichis is the number density or number of molecules per unit volume, esc given by a balance between gravitational energy and kinetic and r nm is the mass density. The universal gas constant R ¼ 1 1 energy (KE): (8.314 J KÀ molÀ ) is related to a specific gas constant (different 1 for each atmosphere) by R R=M K=m, where M (kg molÀ ) 1 GMm 2GM a3 ¼ ¼ mv2 v v a [1] is the mean molar mass. Dalton’s law states that the total pressure 2 esc ¼ r ) esc ¼ r ) esc / a / rffiffiffiffiffi is the sum of partial pressures pi, for each species i, p pi. rffiffiffiffiffiffiffiffiffiffiffi ¼ Applied to eqn [2], the mean molecular mass is if same density body of radius a P
n m Here, r is the radial distance, G is the universal|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl gravitational} i i i M 11 2 2 m [3] constant (6.672 10À Nm kgÀ ), M is the mass of the ¼ Xn ¼ NA  planet, a is the planet’s radius, and m is the molecular mass. where mi is the mass of the ith molecule and NA is Avogadro’s Today, hydrogen escapes efficiently from Earth at a small rate 1 number. On Mars, for example, using the data of Table 2,the- 3kgsÀ . Hydrogen atoms at the exobase in the tail of the 1 mean molar mass M 0:0435kgmolÀ and R R=M Maxwell–Boltzmann velocity distribution have enough ther- 1 1 ¼ ¼ 1 ¼ 191JkgÀ KÀ .FordryaironEarth,M 0:029kgmolÀ and mal velocity to evaporate and escape even though the root- 1 1 ¼ R R=M 287JKÀ kgÀ . mean-square speed (given by 1/2mv2 (3/2)kT, where ¼ ¼ 23 1 ¼ We usually describe atmospheric composition with the k 1.381 10À JKÀ is Boltzmann’s constant and T is exo- ¼  volume mixing ratio fi, for species i, which is also the mole base temperature) is less than the Earth’s escape velocity. Alter- fraction. Less commonly, we use a mass mixing ratio mi, which natively, the hydrogen is lost when chemical or charged is the mass of a particular gas divided by the total mass of a gas particle reactions impart speed to hydrogen atoms. The rate- mixture sample. For an ideal gas, limiting step is not a loss from the exobase itself but diffusion from the homopause to the exobase. Consequently, hydrogen pi Vi ni ri mi pi mi fi , mi fi [4] is said to be lost at an upper limit by diffusion-limited escape ¼ p ¼ V ¼ n ¼ r m p ¼ m (Hunten, 1973; Walker, 1977). Diffusion-limited escape of In all atmospheres, high total pressure at low altitudes hydrogen is an excellent theory given that data suggest that it pushes upward against the weight of overlying air, so that happens on Venus, Earth, Mars, and Titan. pressure declines with altitude. Air settles into hydrostatic For completeness, there are three ways that gases escape equilibrium. This equilibrium assumes no vertical motion – a from planets: (i) thermal escape, with end-member approxi- good approximation over large horizontal scales (e.g., a few mations of thermal evaporation (as mentioned in the preced- kilometers on the Earth) that average out updrafts and down- ing text) called Jean’s escape at one end and pressure-driven drafts. A column of air with unit area cross section and small hydrodynamic escape at the other (e.g., Catling and Zahnle, height Dz has a mass rDz, so that the pressure change going up 2009; Johnson et al., 2013; Walker, 1982), (ii) nonthermal in height Dz is Dp (rDz)g force/(unit area), where g is the escape where an atom or molecule is boosted to escape velocity ¼À ¼ gravitational acceleration. Written in calculus form, we get the because of a chemical reaction or ionic interaction, and hydrostatic equation relating pressure, density, and altitude: (iii) impact erosion where atmospheric gases are expelled as a result of the large body impacts. Helium escapes from the Earth @p 1 r z gz, pz r z gzdz [5] today through nonthermal escape. However, bodies can lose @z ¼À ðÞðÞ ðÞ¼ ðÞðÞ ðz their atmospheres in bulk. Theory demonstrates that the two For an ideal gas, the air density is r p=RT (eqn [2]), so that mechanisms that most efficiently strip away substantial atmo- ¼ we can rewrite eqn [5] as spheres are hydrodynamic escape driven by stellar irradiation (Lammer et al., 2008; Sekiya et al., 1980, 1981; Watson et al., @p gz @p @z À ðÞ p p 1981; Zahnle and Kasting, 1986; Zahnle et al., 1990) and @ z ¼ RzTz ) p ¼À H ) impact erosion (Griffith and Zahnle, 1995; Melosh and ðÞ ðÞ z dz z Vickery, 1989; Walker, 1986; Zahnle et al., 1992). Currently, psexp p psexp [6] ¼ À 0 H ) ÀH isothermal in the solar system, only Pluto’s atmosphere is believed to be ð undergoing hydrodynamic escape (Trafton et al., 1997). The Here, ps is the surface pressure and we introduce a vertical reader is referred to Catling and Kasting (in press) and Lammer length scale called the scale height, (2013) for more discussion of escape processes. H RT=g kT=mg [7] ¼ ¼ 10.13.2.3 The Equation of State, Atmospheric Composition, If we make the approximation that the temperature, gravity, and the Hydrostatic Equation and specific gas constant are constant with height, the integral of
eqn [6] can be evaluated to give the exponential shown in the last The most basic physics of atmospheres involves connecting the expression. Thus, we see the meaning of the scale height: the distributions of pressure, temperature, and density with (i) an vertical distance over which pressure drops by a factor of equation of state and (ii) a hydrostatic equation. 1/e 1/(2.7183) 1/3. In an isothermal atmosphere, density Except for deep within giant-planet atmospheres, the ideal ¼ also drops by 1/e over the same height, but the density scale gas law works well as for (i) height will be slightly different if there is any temperature change k with altitude. In the solar system, H 5–20 km because the ratio p nkT nm T p rRT [2] ¼ ¼ ðÞm ) ¼ T=mg in H kT=mg is similar given that smaller T (and larger g ¼ r for Jupiter) compensates for small m in H2-rich giant planets. |ffl{zffl}
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−5 10
−4 10 Mars Earth Uranus −3 10 Neptune
−2 10
(bar) Pressure −1 10 Jupiter Venus Titan 100 Saturn
101 50 100 150 200 250 300 350 Temperature (K)
Figure 2 Temperature–pressure profiles for planetary atmospheres in the solar system. The dashed line at 0.1 bar indicates that planets with stratospheric temperature inversions have tropopause minima near this pressure level. (Drawn by the author.)
The surface pressure measures an atmosphere’s mass 10.13.2.4 Convection and Stability because pressure is the weight of a column of air per unit Atmospheric temperatures vary with altitude (Figures 1 and 2) area. Ignoring lateral forces in the real 3-D spherical geometry primarily because of radiation and convection, although large- (which are a minor correction), we have scale atmospheric motion also plays a role (Section 10.13.5).
ps In lower tropospheres, warm air rises and cold air sinks so ps Mcolg Mcol [8] ¼ ) ¼ g that heat is transferred upward by convection. Air expands and becomes buoyant, that is, less dense than surroundings, where Mcol is the column mass. For example, Mcol (610 Pa)/ 2 2 4 2 ¼ through (i) radiative heating, (ii) contact with a planetary (3.72 m sÀ ) 164 kg mÀ on Mars, 10 kg mÀ on Earth, and 5 ¼2 surface warmed by sunlight, (iii) exothermic condensation of 2 10 kg mÀ on Titan. Equation [8] can also be applied to the  substances such as water vapor in the Earth’s atmosphere (see column mass above some reference pressure level for giant planets. Chapter 10.11), and (iv) heating from the interior of a planet, The hydrostatic equation combined with number density which is significant on giant planets but not usually on gives us another way to describe atmosphere composition, rocky ones. namely, column abundance N , which is the number of c Convection commences when the lapse rate exceeds a cer- molecules above a unit area. For example, the typical abun- 18 2 tain threshold. The lapse rate is denoted G (dT/dz) and dance of terrestrial ozone is NO3 8 10 molecules cmÀ .At 1 ¼À  usually given in K kmÀ . The minus sign is a convention to altitude z, the column abundance Nc(z) is defined by make G positive for typical tropospheric conditions. Convec- 1 tion is very efficient in transporting heat upward to the upper Nc z nzdz [9] ðÞ¼ z ðÞ troposphere. There, the overlying atmosphere is relatively ð transparent to thermal IR and energy is radiated to space. The column mass is Nc z m, so the pressure at altitude z,by ðÞ Also, in the upper troposphere, the lapse rate is no longer eqn [8],ispz Nc z mg. Using this, we obtain an expression ðÞ¼ ðÞ steep and convection ceases. In this way, a state of radiative– for the column abundance in terms of scale height H: convective equilibrium is achieved, in which outgoing IR pz nzkT z radiation from the warm planet balances the energy of incom- Nc z ðÞ ðÞ ðÞ Nc z nzHz [10] ing sunlight and sets a characteristic radiative–convective ðÞ¼mg ¼ mg ) ðÞ¼ ðÞ ðÞ temperature profile (see Section 10.13.3.4). In some planetary literature, column abundance is expressed The adiabatic lapse rate is the threshold lapse rate for as ‘cm-atm’ or ‘cm-amagat,’ which are equivalent. These units are convection. If the actual lapse rate is bigger or smaller, the the depth in centimeters that would be produced if a gas in a atmosphere convects or it does not. Adiabatic means no column of air were all extracted and compressed to 1 atm exchange of energy with surroundings. So if a vertically dis-
(101325 Pa) pressure and 0 C; they are given by the normaliza- placed air parcel expands or contracts adiabatically to adjust to 2 tion Nc(z)/n0,whereNc(z) is in units of molecules cmÀ and its surrounding pressure, the temperature change as a function 19 3 n 2.687 10 molecules cmÀ is the Loschmidt number, the 0 ¼ Â of height is readily calculable. Applying conservation of energy number density of an ideal gas at 1 atm and 273.15 K. Ozone to the parcel, we have column abundance is also expressed with the Dobson unit (DU), which is the depth in thousandths of centimeters of a column dq cvdT + pda ¼ of ozone taken to 1 atm and 0 C, that is, DUs are 1000NO3/n0. heat added=kg zero for adiabatic change in internal heat=kg work done=kg The Earth’s ozone column is typically 300 DU (0.3 cm-atm) ðÞ compared with a maximum of 6 DU on Mars. |{z} |ffl{zffl} |{z} [11]
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Here, dq is heat input per unit mass (zero if adiabatic) and cv is Our analysis is incomplete because condensable species the specific heat at constant volume. Work is ‘force incremental and aerosols strongly affect temperature profiles. For example,  distance’ or ‘pressure (p) incremental volume change.’ Because in the Earth’s troposphere, the condensation of water vapor  we are dealing with energy per unit mass, we define volume per releases heat making G smaller than the adiabatic value. Con- unit mass as a 1/density 1/r,sothatworkispda. Since densation of sulfuric acid on Venus has a similar thermody- ¼ ¼ a RT=p from eqn [2], the product rule of differentiation gives namic effect. We discuss a saturated adiabatic lapse rate (SALR) ¼ us da R=p dT RT=p2 dp.Substitutingineqn[11],wehave in Section 10.13.2.5. For deep atmospheres, such as on Venus, ¼ À one should also take account of the temperature dependence ÀÁ ÀÁ R RT RT of specific heat capacity cp so that Ga Ga(T). dq = c dT + p dT dp = (c + R)dT dp ¼ v 2 v A shorthand description of the stability of temperature p p p gradients uses potential temperature y, which is the tempera- ture that a parcel of air would have if it were taken to a dq = cp dT + gdz reference pressure p (e.g., 105 Pa for Earth) dry adiabatically. 12 ref ½ Potential temperature is an alternative thermodynamic vari- In the middle expression, we have inserted c c + R from able to temperature and, from its definition, is constant with p ¼ v kinetic theory, which relates specific heat at constant pressure altitude z if the lapse rate is adiabatic. Otherwise, an increase of y with z indicates convective stability, while negative @y/@z (cp) to specific heat at constant volume. We also substituted indicates instability. Using the adiabatic version of eqn [12] gdz for RT=p dp using the first expression in eqn [5], the À hydrostatic equation. For adiabatic changes, dq 0. Thus, from with dq 0, we define y as follows: ÀÁ ¼ ¼ the last expression in eqn [12], we obtain the adiabatic lapse dT R dp dp rate for dry air: k ; integrate y, T , pref , p T ¼ c p ¼ p ðÞðÞ p dT g k [13] T p pref Ga ln k ln y T [14] ¼À dz ¼ cp ) y ¼ pref ) ¼ p This simple but fundamental equation gives the temperature where k R=cp. The final expression in eqn [14] is Poisson’s ¼ change with altitude of a dry parcel of air moving up and down (adiabatic state) equation. We use eqn [14] later in through an atmosphere in hydrostatic equilibrium. Table 3 Section 10.13.3.4, and potential temperature comes up again gives the specific heat capacities, gravitational accelerations, in Section 10.13.5. and lapse rates Ga of various planetary atmospheres. In convectively stable atmospheres, a buoyancy force causes Comparison of the ambient lapse rate G with Ga indicates vertically displaced parcels to return to their original altitude, susceptibility to convection. When G
The resulting rapid mixing by convection of such unstable displaced upward dz from an altitude where the temperature atmospheres tends to adjust the temperature profile toward is T to a place where the environmental temperature is Tenv. The the adiabatic one. Thus, we find planetary atmospheres typi- parcel cools adiabatically to a new temperature Tp T Gadz, ¼ À cally with G Ga (ignoring local circumstances such as thun- while the environment cools less at the ambient lapse rate as derstorms) (Table 3). Tenv T Gdz. The difference in density between the parcel ¼ À
Table 3 Thermodynamic and physical properties of various planetary atmospheres
Ga g/cp ¼ Main Molar mass, M Specific gas constant Gravity g Specific heat capacity troposphere Mean G observed 1 1 1 2 1 1 1 1 Body gases (g molÀ ) JkgÀ KÀ (m sÀ ) (J kgÀ KÀ ) (K kmÀ ) (K kmÀ ) R( ) cp
Venus CO ,N 43.5 189 8.901 930 9.5 8.0 2 2 Earth N ,O 28.97 287 9.81 1004 9.8 6 2 2 Mars CO2,N2 43.5 191 3.72 850 4.4 2.5 a Jupiter H2, He 2.22 3745 24.25 10,988 2.1–2.45 1.9 a Saturn H2, He 2.14 3892 10.0 10,658 0.7–1.1 0.85 Titan N2, CH4 28.67 290 1.36 1044 1.3 1.0–1.4 a Uranus H2, He 2.3 3615 8.80 8643 0.7–1.1 0.75 a Neptune H2, He 2.3 3615 11.1 8643 0.85–1.34 0.95 Hot Jupiter H , He 2.0 4160 18.5 14300 1.3 0.2 (model)b 2 HD209458b
Ga is the tropospheric adiabatic lapse rate, while G is the observed tropospheric lapse rate. aOn giant planets, g varies from equator to pole. bMenou and Rauscher (2009).
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(rp) and air (re) drives a buoyancy force, which we can write as 20 Newton’s second law:
2 2 d ( z) d ( z) e (V ) = g( )V = g 1 15 p 2 p e 2 dt dt p mass buoyancy force acceleration 15 ½ Tenv = g 1 10 T p Substituting for temperatures T and T , we get p env
Vapor pressure (hPa) 5 d2 ( z) T z z z = g a 1 = g a g a z dt 2 T z T z T z
.. g 0 z+ ( ) z = 0 T a −20 −10 0 10 20 Temperature (ЊC) 16 ½ Figure 3 The saturation vapor pressure (SVP) curve between the vapor In the last expression of eqn [16], we recognize simple phase and condensed phase (liquid or ice) for water. The dashed harmonic motion, that is, the ‘x€+ o2x 0’ of elementary phys- line is the SVP with supercooled water. (Drawn by the author.) ¼ ics. Thus, the square of the angular (Brunt–Va¨isa¨la¨) frequency of oscillation is (Figure 3). The Clausius–Clapeyron equation defines this SVP
2 g g dT g g curve as a unique function of temperature. On Earth, we use a NB Ga G + S [17] T ðÞÀ T dz cp T Clausius–Clapeyron equation for water, but analogous equa- 1 tions apply for other condensables. When noncondensable Here, S(K kmÀ ) is static stability, which measures the gases are part of a gas mixture, the vapor pressure of the difference between the adiabatic lapse rate (eqn [13]) and actual condensable gas is a partial pressure, and if that partial pressure lapse rate ( dT/dz) and is usually positive. Equation [17] makes À reaches SVP, then the solid or liquid phase forms. intuitive sense: the more stable the atmosphere, the higher the The Clausius–Clapeyron equation comes from thermody- oscillation frequency, like stiffening an oscillating spring. namics. Two phases are in equilibrium along the SVP versus The effect of buoyancy oscillations can be observed. Vertical T line, so the Gibbs free energy change of the phase transition is deflection of horizontal air flow over an obstacle such as a moun- zero everywhere along the line. If e is the SVP, it can be shown tain on Earth or a large crater rim on Mars sometimes causes s by considering Gibbs free energy that the slope of e versus oscillations where the air swings up and down beyond the obsta- s temperature T is given by (e.g., Atkins and De Paula, 2010) cle at a frequency NB/2p,formingastandingbuoyancywave.At the cold crests of the waves, moisture can condense into a series of de ds entropy gained per kg from condensed to vapor s parallel clouds called lee wave clouds, given that they form in the dT ¼ da ¼ volume increase per kg from condensed to vapor lee of the obstacle. Roughly speaking, the distance between paral- lc lc lc lel clouds has wavelength, l (horizontal wind velocity/ [18] ¼ ¼ Tda ¼ T avapor acondensed Tavapor frequency) u/(NB/2p), which obviously can change with À ¼ ÀÁ altitude-dependent u and NB.Approximateterrestrialvaluesof where a is specific volume (a 1/r), which, in the last step, we 1 1 ¼ NB 0.01 sÀ and u 10 m sÀ give l 6km,consistentwith reasonably assumed, is far larger than for the vapor than for ¼ ¼ satellite imagery of lee wave cloud spacing of 3–20 km. condensed phase, that is, avapor acondensed.Wealsosubstituted lc Tds,wherelc is the latent heat or enthalpy, which is heat ¼ 10.13.2.5 Condensables and the Clausius–Clapeyron absorbed (or released) when a unit mass changes from one phase Equation to another at constant temperature and pressure. For different
phase transitions, this quantity can be ‘latent heat of Gases that condense – condensables – affect atmospheres vaporization’ for liquid–vapor, or latent heat of fusion for ice– through release or uptake of heat when they change into a liquid liquid, or latent heat of sublimation for ice–vapor. We can define or solid from the vapor phase, and vice versa. Condensables a specific gas constant for the condensable species vapor as include H2SO4 (SO3 +H2O) on Venus, H2OonEarth,H2Oand Rc R/Mc k/mc,whereMc and mc are its molar mass and molec- CO2 on Mars, H2O and NH3 on Jupiter and Saturn, and CH4 on ¼ ¼ ular mass, respectively. By substituting for avapor RcT/es from Titan, Uranus and Neptune, and HCN on Titan. Some condens- ¼ the ideal gas law, we obtain the Clausius–Clapeyron equation: ables also form rain (e.g., CH on Titan), snow (e.g., CO in 4 2 Mars’ polar atmosphere), and clouds (e.g., CH4 on Titan, des lces dlnes lc 2 ðÞ 2 es T Uranus, and Neptune). dT ¼ RcT ) dT ¼ RcT ) ðÞ The key concept governing condensable behavior is satura- T lc dT lc 1 1 tion vapor pressure (SVP), which is the vapor pressure exerted es T0 exp 2 es T0 exp [19] ¼ ðÞ Rc T ðÞ Rc T0 À T in equilibrium with a liquid or solid condensed phase. The ðT0 curve of SVP versus temperature for a particular substance Here, the last step makes the approximation that lc does not defines the transition between vapor and condensed phases change with temperature. Because exponents are additive,
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eqn [19] shows e exp( m l /kT). Thus, SVP is a Boltzmann disfavored by a steep compositional gradient in the envi- s / À c c equation, which makes sense: the numerator m l is the ronment due to condensation. Temperature profiles on Ura- c c energy to free a molecule from its neighbors, while the denom- nus and Neptune are superadiabatic where methane inator kT is the average molecular energy. For water vapor condenses, which suggests that convection may be inhibited above liquid water, we use T0 273.15K, es(T0) 611Pa, there (Guillot, 1995). 1 1 ¼ ¼ Rc 461JKÀ kgÀ , and latent heat of vaporization SVP also gives us a first-order understanding of clouds. ¼ 6 1 l 2.5 10 JkgÀ , finding that SVP roughly doubles for Clouds will form on average where the mean pressure– c ¼ Â every 10 K temperature rise. temperature conditions reach the SVP for various condensables On some bodies, SVP is essential for understanding the (Sanchez-Lavega et al., 2004)(Figure 4). bulk atmospheric gas. On Mars, for example, CO2 condenses at the cold poles. A temperature of 148 K corresponds to an SVP of 6 mbar ( 600 Pa) for CO , which is a typical air pres- 2 sure on the surface of Mars (Leighton and Murray, 1966). 10.13.3 Radiation Consequently, Mars’s CO ice caps are buffered at a tempera- 2 10.13.3.1 Planetary Energy Balance, Blackbody Radiation, ture near 148 K. On Triton and Pluto, the atmospheric pressure and the Greenhouse Effect is essentially the SVP of N2 over N2 ice at temperatures of 38 and 40 K, respectively (Brown and Ziegler, 1979; Lellouch Physical and chemical processes in planetary atmospheres are et al., 2011; Zalucha et al., 2011). driven by sources of free energy, meaning energy that can do The presence of even a minor condensable can change useful work. For most planets, light from the parent star the lapse rate considerably, as water vapor does on Earth. dominates free energy sources, although some giant planets Once condensation starts in a cooled, vertically displaced have an internal energy flux that leaks to the exterior and is parcel of air, further condensation (into liquid or ice) significant compared to the energy flux received from the reduces the rate of cooling by release of latent heat. An parent star.
SALR, or moist adiabat, denoted Gsa,occurswhenchilled Consequently, for planetary atmospheres, the solar flux at moist air is held at saturation by the warming associated the top of the atmosphere is a key parameter. Satellites measure 2 with continual condensation of vapor. This lapse rate is the Earth’s solar constant as S 1360.8 0.5 W mÀ (Kopp 0 ¼ Æ called the pseudoadiabatic lapse rate if condensates are and Lean, 2011), consistent with ground-based inferences assumed to fall out of the parcel because then the process (Chapman et al., 2012). As a result, the solar flux at the top of is not reversible and adiabatic. However, saturated and the atmosphere averaged over the area of the whole Earth and 2 pseudoadiabats tend to be similar because condensation over time is S0/4 340 W mÀ . We divide S0 by 2 to account for products carry relatively little heat out of a parcel of air. glancing angles on a spherical planet, and we divide it by 2 The SALR can be derived as a function of temperature and again because only one side of the Earth is in daylight. Alterna- saturation mass mixing ratio m mc=m fc E es=p , tively, another way to understand the factor of ¼ is that the cms ¼ ðÞ¼ ðÞ where fc is the volume mixing ratio of the condensable, p solar flux intercepted by a planet of radius R is equivalent to that 2 is the total pressure, and E is the ratio of molecular masses. on a projected disk of area pR . So, the ¼ factor represents In this way, the SALR can be expressed as a modification of the ratio of received flux on a disk to the whole planetary area 2 2 2 the adiabatic lapse rate, that is, Gsa Ga f(T,mcms)(e.g.,see (pR /4pR ) to get a global average in W mÀ . ¼ Â Andrews (2010),pp.37–39).FormulationsoftheSALRfor Afractionofsunlight’sincidentpowerisreflectedand major constituent condensation are also possible (Ingersoll, scattered back to space, which, over all wavelengths, is 1969; Kasting, 1988, 1991). called the Bond albedo A (or planetary albedo). For the Under saturated conditions, static stability is controlled by solar system planets with atmospheres, A ranges from 0.25 comparing the actual lapse rate to the saturated lapse rate, Gsa. for Mars to 0.76 for Venus (Moroz et al., 1985), the 1 Air is stable if G
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Venus Earth Mars 0.01 0.1 0.001
0.1 0.01
H2O
1 0.1 P (bar) P (bar) H2O H SO P (mbar) 2 4 10 CO 1 2
100 1 200 400 600 800 200 220 240 260 280 300 90 120 150 180 210 240 T (K) T (K) T (K)
Jupiter Saturn Titan 0.001 0.001 0.001 1: NH3 1: NH3
0.01 2: NH4SH 0.01 2: NH4SH C2H6 3: H2O 3: H2O 0.01 CH 4 0.1 0.1
P (bar) P (bar) P (bar) 0.1 1
1 1 1 2 2 3 3 1 10 3 10 0 100 200 300 400 0 100 200 300 400 40 80 120 160 200 T (K) T (K) T (K)
Uranus Neptune HD209458b 0.01 0.01 0.1 1:CH4 1:CH4 0.1 2:SH2 0.1 2:SH2 3: NH3 3: NH3 4: NH4SH 4: NH4SH 1 5: H O 1 5: H O 1 2 2 1 1 2 P (bar) P (bar) P (bar) 10 2 10 3 3 MgSiO3 Fe 100 4 100 4
5 5 1000 1000 10 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400
T (K) T (K) T (K) Figure 4 Use of the vapor pressure and saturation vapor pressure (SVP) for various condensables to predict cloud formation in solar system atmospheres and hot Jupiter HD209458b. The solid line is the vertical profile of pressure ( p) versus temperature (T). For Mars, an annual average p-T profile is shown as well as a colder profile. The dashed lines are the saturation vapor pressure (SVP) curves for various condensable species, as indicated, assuming a fixed volume mixing ratio of condensable. It is assumed that particles condense when the partial pressure (e) reaches the saturation vapor pressure (e ), that is, where e f pT"e T , where f is the volume mixing ratio of the condensable. Thus, the base of clouds is marked where s ¼ c ðÞsðÞ c a dashed line crosses the solid pressure–temperature profile. Below this level, the phase is vapor because the ambient air is too hot. Various mixing ratios are assumed to calculate the dashed curves as follows: Venus 2 ppm and 2000 ppm H2SO4 (where liquid clouds form); the Earth 250 ppm and 1.5% H2O, giving rise to ice and liquid clouds, respectively; Mars 0.95 CO2 and 300 ppm H2O giving ice clouds; Jupiter 200 ppm NH3, 36 pppm NH4SH, 50 ppm H2O, 0.17% H2O giving solid phase clouds except for the higher water level; Saturn 200 ppm NH3, 36 pppm NH4SH, 0.17% H2O, giving solid particles; Titan 5% CH4, 10 ppm C2H6, giving solid particles; Uranus and Neptune 2% CH4, 37 ppm SH2, 200 ppm NH3, 36 pppm NH4SH, 0.17% H2O, giving solid particles; and HD209458b 75 ppm MgSiO3 and 68 ppm Fe, giving solid particles. Reprinted from Sanchez-Lavega A, Perez-Hoyos S, and Hueso R (2004) Clouds in planetary atmospheres: A useful application of the Clausius–Clapeyron equation. American Journal of Physics 72: 767–774.
3 2 34 2hu du 2hc dl Here, h 6.626 10À J s is Planck’s constant and c is the Bu du , Bl dl , ¼ Â ¼ c2 ehv=kT 1 ¼ l5 ehc=lkT 1 speed of light, which is related to the frequency (u) and wave- ðÞÀ ðÞÀ length (l) of radiation by c ul. The third expression in 2 1 1 2 1 1 ¼ Bu in WmÀ HzÀ srÀ Bl in WmÀ mÀ srÀ eqn [20] is in terms of wavenumber u 1=l, as used by some ¼ 3 spectroscopists. Thus, each form of the Planck function in the |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl~u du~ {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} B~d~u 2hc [20] u ¼ ehc~u=kT 1 preceding text applies when the spectrume is specified in units ðÞÀ of frequency (Hz), wavelength (m), or wavenumber (usually 1 2 1 1 1 cmÀ ), respectively. In eqn [20], blackbody radiation B (T)isa B in WmÀ cm À sr u u~ ðÞÀ À |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Treatise on Geophysics, 2nd edition, (2015), vol. 10, pp. 429-472
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2 radiance (or intensity), which is the power (W) per unit et al., 1990)), which is puzzling. For the Earth, Fi 0.09 W m 2 frequency (or wavelength or wave number) interval crossing can be neglected, Sp 1361 W mÀ , and Bond albedo A 0.3. 2 1 1 ¼ ¼ unit area (mÀ ) in unit solid angle (steradianÀ or srÀ )ina Solving eqn [23] gives equilibrium temperature T 255 K, eq ¼ given direction. which is also the Earth’s effective temperature. Clearly, this A flux density or irradiance (often shortened to just ‘flux’) temperature is not achieved at the surface, which has a mean 2 1 2 1 2 1 has units of WmÀ HzÀ (or WmÀ mÀ or WmÀ cmÀ ) and temperature of 288 K. For planets with atmospheres, the dif- is the intensity integrated over all directions. In spherical polar ference between the surface temperature and effective temper- coordinates, an elemental solid angle is sin ydydf where y is ature arises because of the greenhouse effect described in the the angle from the polar axis and f is the azimuthal angle. If succeeding text. the intensity Bu is isotropic (i.e., independent of y), the flux An idealized atmospheric level where Earth achieves its over an entire hemisphere is p times Bu, as follows: effective temperature can be estimated. The Earth’s average 1 tropospheric lapse rate is 6KkmÀ , so given the surface f 2p y 2p x 1 ¼ ¼ ¼ temperature of 288 K, the air temperature will be at 255 K at F B cosy siny dydf 2p B xdx u ¼ u ¼ u 5 km altitude. This altitude is called the emission level, ð0 ð0 ð0 x although it is somewhat fictitious because in reality, the atmo- 1 x2 sphere radiates to space from different levels as a function of 2pB pB |ffl{zffl} [21] ¼ u 2 ¼ u wavelength, depending upon the IR transparency of the over- 0 lying air at that wavelength. Nonetheless, the Earth cannot 2 The flux over all frequencies (W mÀ ) is found by substitut- buck conservation of energy, and in equilibrium, it must radi- ing for Bu (eqn [20]) in eqn [21] and integrating ate a total flux back to space equivalent to a 255 K blackbody of the Stefan–Boltzmann law. 2 1 2ph 1 u du F B d The greenhouse effect is when an atmosphere warms up p u u 2 hv=kT ¼ 0 ¼ c 0 e 1 and provides a flux of thermal IR radiation to the surface (or ð ð À 4 4 3ÀÁ5 4 lower layers of giant planets) that adds to the shortwave solar 2pk T 1 x dx 2p k 4 4 2 3 x 2 3 T F sT [22] flux. The surface of a rocky planet with an atmosphere is ¼ c h 0 e 1 ¼ 15c h ) ¼ ð ðÞÀ warmer than it would be without an atmosphere because the
standard integral p4=15 surface receives energy from both the Sun and the heated ¼ atmosphere. For specificity, the fraction of the total flux reach- |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Here, we substituted x hu/kT to obtain a standard integral. The ing the surface due to IR from the atmosphere is about 99.9%, ¼ final expression in eqn [22] is the Stefan–Boltzmann law with 2/3, 1/3, and 9/10 for Venus, Earth, Mars, and Titan, respec- 8 2 4 the Stefan–Boltzmann constant s 5.67 10À WmÀ KÀ . tively (Courtin et al., 1992). For rocky planets, the difference ¼ Â Blackbodies are ideal absorbers and emitters, and because between Teq and surface temperature Ts is another way to planets are made of materials that absorb IR radiation strongly express the size of the greenhouse effect: and reach thermal equilibrium, the spectrum that planets emit can be represented by a blackbody. Essentially, a planet DTg Ts Teq [24] À absorbs visible sunlight, warms up, and radiates like a black- body in the IR. When outgoing radiation balances incoming For the Earth, DTg 288 255 33 K. Of course, this defi- ¼ À ¼ radiation along with an additional internal heat flux F (as nition assumes that the albedo would be the same where the air i removed – which it certainly would not – so T should be occurs on the giant planets), the planet achieves an effective D g temperature Te, that is, the equivalent blackbody temperature viewed judiciously as a useful metric, a way to express the flux 2 that characterizes the total flux output: difference of about 150 W mÀ between blackbodies at 255 and 288 K. Of course, slowly rotating airless bodies can have large Sp 4 day–night temperature differences, and so, the mean tempe- 1 A + Fi sTe , ðÞÀ 4 ¼ rature will be different from the equilibrium temperature, as is
internal heat flux outgoing IR radiation well known (Pierrehumbert, 2010). absorbed solar radiation |{z} There are various greenhouse gases. In the Earth’s modern |fflfflfflfflfflffl{zfflfflfflfflfflffl} |{z} Sp 4 Sp clear sky atmosphere, water vapor contributes 60% of DTg, or 1 A sT if Fi 1 A [23] ðÞÀ 4 ¼ eq ðÞÀ 4 CO 25%, and ozone 8%, with the rest from CH ,NO, 4 2 anthropogenic chlorofluorocarbons (CFCs), and other trace outgoing IR radiation gases (Karl and Trenberth, 2003; Stephens and Tjemkes, |ffl{zffl} Here, Sp is the solar flux at the orbital distance of the planet 1993). Importantly, water vapor condenses and exits the atmo- from the Sun. When the internal heat flux is negligible, the sphere as rain or snow, whereas the Earth’s other main green- effective temperature is called the equilibrium temperature, house gas, CO2, does not condense. H2O is effectively ‘a slave’
Teq, because there is equilibrium between incoming sunlight to the forcing by the concentration of CO2. By forcing, we mean 2 and outgoing IR. On Jupiter, Fi is 5.4 W mÀ compared with a persistent disturbance to the climate system that changes the 2 8.1 W mÀ of absorbed sunlight (Hanel et al., 1981). For giant energy balance. If CO2 levels goes up, more water vapor enters a planets, sources of internal energy include gravitational energy warmer atmosphere and increases the greenhouse effect. Thus, released from formation and contraction as well as subsequent the climatic role of water vapor is a feedback rather than a internal differentiation. However, unlike the other gas giants, forcing (Lacis et al., 2010, 2013). The bulk background gases 2 Uranus has little internal heat flow (42 47 mW mÀ (Pearl that absorb very little IR, such as N and O , are themselves Æ 2 2
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critical for the greenhouse effect because collisions of green- The Sun behaves like a blackbody because the ions and house gases (CO ,HO, etc.) with N or O take away or add electrons inside the Sun are in thermal equilibrium. So, solar 2 2 2 2 energy to the greenhouse gas molecules, which allows them to radiation has an effective temperature of 5780 K. Wien’s law absorb over a much wider range of photon wavelengths. This indicates peak emission at lmax (0.0029mK)/(5780K) ¼ ¼ process of pressure broadening (see Section 10.13.3.5) is fun- 0.5mm, in the visible. The Earth is roughly 20 times cooler damental to the greenhouse effect in thick atmospheres. On than the Sun’s surface, and so, its Wien peak is 20 times larger other bodies, rather than just trace gases, the bulk atmospheric at 10 mm. Given typical temperatures, planets generally have gases are greenhouse gases. On all the giant planets, H2 is the their Wien peak at thermal IR wavelengths, which is where main source of IR opacity (see Section 10.13.3.5), on Titan, N2, many atmospheric gases and aerosol particles absorb strongly. H2, and CH4 are important (McKay et al., 1991; Samuelson Several atmospheres have windows, which are spectral inter- et al., 1981), while CO2 is the key greenhouse gas on Venus vals that are relatively transparent to thermal IR compared to (Bullock and Grinspoon, 2001; Titov et al., 2007) and Mars. high opacity at other wavelengths. A window can be a key
Clouds can cool a planet because of their contribution to the means for a planet to radiate IR energy to space and cool. On albedo, but they can also warm through adding to the green- Titan, a window at 16.5–25 mm is close to Titan’s Wien peak of house effect. For example, on Earth, icy cirrus clouds contribute 31 mm. H , which is a minor constituent ( 0.1%), absorbs 2 to greenhouse warming since they are largely transparent to weakly in this window (McKay et al., 1989). On Earth, the key solar radiation but opaque to thermal IR, while low-lying atmospheric window of 8–12 mm encompasses the Wien peak marine stratus predominantly contributes to shortwave albedo. at 10 mm. Again, absorption by a minor constituent, water If we take clouds into account in our apportioning of vapor, modulates the outgoing IR flux through the window. On the greenhouse effect on Earth, water vapor is responsible Venus, dense CO2 and clouds leaves only a few narrow win- for 50%, clouds 25%, CO2 20%, and other gases the dows in the near-IR. The significant ones are 1–1.2 mm in which remainder (Schmidt et al., 2010). Regarding their cooling only H2O absorbs, 1.7 mm where CO and HCl absorb, and effect, clouds contribute about half of the Earth’s albedo of 2.2–2.5 mm where CO and SO are seen (Crisp and Titov, 2 0.3. According to satellite measurements, clouds reduce the 1997; Titov et al., 2007). These are wavelengths to peer relatively 2 Earth’s absorbed radiation by 48 W mÀ , but enhance deep into Venus’ atmosphere, which is otherwise opaque. À2 the greenhouse effect by +30 W mÀ . So, overall, clouds cool the Earth by 18 W m2 (Ramanathan and Inamdar, 2006). À A hypothetical Earth without clouds would be far warmer 10.13.3.2 Radiative Transfer: Shortwave and Infrared (Goldblatt et al., 2013). Like water vapor, clouds are a feedback and change in Atmospheres like the Earth’s are transparent to incoming solar radiation but not entirely. The ozone in the Earth’s stratosphere response to levels of CO2 and other greenhouse gases. Feed- backs can alter the mean temperature of a planet in positive absorbs solar UV (0.2–0.4 mm) in the Hartley and Huggins feedback (like the aforementioned water vapor feedback) or bands, and ozone in both the stratosphere and troposphere absorbs visible light in the 0.4–0.85 mmChappuisband.The stabilize it in negative feedback. Another important positive Earth’s tropospheric water vapor absorbs solar near-IR (1–4 m feedback is the ice-albedo feedback. An increase in surface m temperature causes a decrease in albedo from diminished region). A more extreme case is Titan’s stratospheric methane snow and ice cover, which causes further warming, or the and haze of aerosols, which absorb in the UV, visible, and near- other way, cooling increases albedo with more snow and ice, IR (for methane) to such an extent that the reduction of solar causing further cooling. The ice-albedo feedback is crucial for flux reaching Titan’s surface is called an antigreenhouse effect
(McKay et al., 1989, 1991). Temporary antigreenhouse effects understanding the Earth’s ice ages during the last few million years (Lorius et al., 1990). Negative feedbacks include an are also possible from debris raised in asteroid impacts (Pollack increase in outgoing thermal IR as a planet warms up, operat- et al., 1983) or, on Earth, smoke created in nuclear war (Robock ing on short timescales of weeks. Over geologic timescales, et al., 2007; Turco et al., 1983). geochemical cycles act as negative feedbacks on the Earth’s Radiative transfer restricted to the visible and UV is simpler than in the IR because the Planck function (eqn [20])indicates climate. The important one for Earth is the carbonate–silicate negative feedback, which acts over a timescale of 0.5 My that we can neglect visible or UV emission at temperatures typical (Berner, 2004; Walker et al., 1981). If the climate warms, of planetary surfaces and atmospheres. This is not true for IR calculations. The starting point in radiative transfer is the greater rainfall and temperatures consume atmospheric CO2 in weathering reactions on the continents, which cools the extinction law (or Beer–Lambert–Bouguer law). Experiments show that the loss of intensity of a beam of light of frequency v Earth. If the climate cools, the consumption of CO2 from dry passing through a substance depends linearly on the incident air is slowed, and CO2 accumulates due to volcanic emissions, which increases the greenhouse effect. On long times, negative intensity Iv and the elemental mass of absorbing or scattering feedbacks have evidently ‘won’ overall because the Earth has material expressed as ra ds,wherera is the density of the absorber 3 remained habitable since at least 3.5 Ga (Buick, 2007). or scatterer (kg mÀ )andds is the elemental path length:
One further importance of blackbody theory is that it tells s tv us the wavelength where most of the radiative energy is emitted dIv kvIvr ds Iv Iv0exp k r ds0 Iv0eÀ [26] ¼À a ) ¼ À u a ¼ from planets and stars. The Planck function is maximal where ð0
@Bl/@l 0, which gives Wien’s law: Here, the proportionality constant kv is the mass extinction ¼ 2 1 coefficient (m kgÀ ) or opacity. In eqn [26], I is the intensity 3 v0 lmaxT 2:897 10À meters K [25] at distance s 0 and tv is a dimensionless optical path or ¼ Â ¼
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optical thickness along the beam direction. We usually define atmosphere can instead create ions in photoionization or our optical path as increasing from the top of the atmosphere break molecules in photodissociation. downward. In this geometry, we call tv the optical depth. In the In contrast to absorption, scattering is when radiation is case of a pencil beam inclined at zenith angle y (from the redirected out of the original direction of propagation as a result vertical), the elemental path along the beam ds is related to of interaction with aerosols or when radiative decay of a mol- that along the vertical path dz by geometry as cos y dz/ds. ecule emits a photon in a random direction, attenuating the ¼ Replacing ds in eqn [26] by dz/cos y gives direct beam. The type of scattering can be diagnosed from the
size of a scatterer using a dimensionless size parameter, where rp tv =cosy tv =cosy I I eÀ transmissivity T eÀ [27] v ¼ v0 ) v ¼ is the scatterer radius and l is the wavelength of radiation,
Here, the transmissivity (or transmittance) Tv is the ratio of 2prp light getting through (I ) to the incident intensity (I ). x [29] v v0 ¼ l The visible optical depth can be large in planetary atmo- For gas molecules or particles with x 1, the scattering of spheres. On Mars, the broadband visible optical depth in global dust storms reaches 5(Colburn et al., 1989; Kahn et al., solar radiation is Rayleigh scattering, which is isotropic with a 4 1992), which means that direct-beam sunlight is attenuated scattering efficiency Qs ∝lÀ , a strong wavelength dependence 5 by a factor of eÀ 1/148. On Titan’s surface at 0.5 mm visible that causes the Earth’s blue sky when convolved with the solar wavelength, the extinction optical depth is 10 due to atmo- spectrum. Aerosols (dust, smoke, and cloud droplets) with spheric haze (Tomasko and West, 2009). However, these sur- 0.1
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Wavelength (µm)
20 15 12 11 10 9 8 7
) − 1 ) − 1 150
(cm 320 K − 1 H2O sr − 2 O 280 K 3 100 W m
− 3 CO 2 240 K 50 CH4
Radiance (10 200 K H2O
0 400 600 800 1000 1200 1400
−1 (a) Wavenumber (cm ) s (215 K)4
Tropopause
s (265 K)4
s (320 K)4
Ground 15 µm 11 µm 20 µm CO2 window H2O (b) band band
Figure 5 (a) The Planck function for noon over a vegetated region of the Niger Valley in N. Africa where the surface temperature is 320 K. Dashed lines show blackbody curves for particular temperatures (Adapted from Hanel RA, Conrath BJ, Kunde VG, et al. (1972) The Nimbus 4 infrared spectroscopy. Experiment, I: calibrated thermal emission spectra. Journal of Geophysical Research 77: 2629–2641). (b) A schematic diagram showing the meaning of parts of the curve in (a). The arrows indicate the layers in the atmosphere where blackbody fluxes originate, according to the Stefan–Boltzmann law. (Drawn by the author.)
no particular coordinate system or assumption about whether dIv s ðÞ Iv s Bv s [32] the source function behaves like a blackbody or not. dtv s1, s ¼ ðÞÀ ðÞ ðÞ Next, we impose a coordinate system and assume that the where the elemental optical path dt is between distance s and medium is a blackbody, that is, a purely absorbing, non- some distance s1. This is sometimes called Schwarzschild’s scattering medium in thermal equilibrium with source func- equation after Karl Schwarzschild who first considered such a tion Jv Bv the Planck function and kv the mass absorption radiative transfer equation for the Sun’s atmosphere in 1914. ¼ ¼ coefficient. Minus signs are confusing in radiative transfer due With a little work, a solution to eqn [32] can be found by tv s1, s to different geometric conventions, so let us be clear that we multiplying by an integrating factor, eÀ ðÞ, integrating now define our IR optical depth as measured downward along between 0 and s1, and rewriting as a path s, that is, t is zero at an observer at s and increases s1 ¼1 tv s1, 0 tv s1, 0 backward along the beam to some value t inside the atmo- Iv s1 Iv0eÀ ðÞ+ Bv s eÀ ðÞkur ds [33] ðÞ¼ ðÞ a sphere. Thus, ð0 The first term is just the exponential attenuation of the t 0 1 ¼ emission from s 0tos s1. The second is the sum of the t kvrads dt,orifs is altitude : ¼ ¼ ¼ s ¼ t t À contributions from each point s along the path 0 to s1, each ð ð ¼ t 0 exponentially attenuated by the path length to the observer. 1 ¼ t kvra dz dt [31] Another useful form of the radiative transfer equation makes ¼ z ¼ t t À ð ð ¼ the plane-parallel approximation, that is, where pressure, With these assumptions, the equation of radiative transfer temperature, and composition are assumed to be functions of becomes height only. Then, if a beam of radiance Iv passes upward
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through a layer of depth dz at an angle y to the vertical, particular UV or visible wavelength. As a result, UV heating or geometry gives dz ds cos y (ds)m, defining the cosine of ionization can occur at this level. ¼ ¼ zenith angle as m cos y. Given our definition of optical depth It is possible to justify the concept of absorption near tv 1 ¼ (eqn [31]), the equation for radiative transfer in a plane-parallel quantitatively for both solar and IR radiation. For solar radia- atmosphere is a slight modification of eqn [32], as follows, tion, we work with fluxes directly because of the typical lack of where we return to generality by using Jv: emission in atmospheres at solar frequencies. Within a plane- tary atmosphere at height z, the vertical component of direct dIv m Iv Jv [34] flux crossing a surface parallel to the ground for a solar beam at dtv ¼ À zenith angle y and frequency v can be deduced from the If we imagine looking down on a planet from some level in extinction law (eqn [27])as the atmosphere, we integrate eqn [34] from that level where the tu=m Fu mFu eÀ [38] optical depth is t to the surface where t t*. Given t as a limit of ¼ ¼ integration, we use a dummy variable t0 t in the integral for 3 1 À The volume heating rate Q [W mÀ HzÀ ] is related to how path lengths. To obtain the upward intensity at any level where this flux changes with altitude due to absorption by the expres- the optical depth is t, we multiply by an integrating factor sion, Q (dFnet,v/dz), where Fnet,v is the net flux, that is, the t0 t =m ¼À eÀðÞÀ and solve. Assuming that a u (frequency) subscript upward minus the downward flux. With no upward solar flux, on t is implicit, the solution for upward intensity is the heating rate is t = * = dt0 t* t m t0 t m dF dF dt 1 Iv t " Iv t* eÀ À + Jv t0 eÀðÞÀ 1>m > 0 u u u tu=m ðÞ Q mFu eÀ kur Q ðÞ ¼ ðÞ t ðÞ m ¼ dz ¼ dt dz ¼Àm ðÞÀ u ) ð [35] k r À u u where the arrow superscript indicates upward. If we are looking tu=m kuruFu e|{z}À [39] down from space, we simply put t 0 in the limit of integra- ¼ ðÞ ¼ tion, and the intensity observed at the satellite instrument is Initially, the value of Q in eqn [39] increases downward as the term kvrv in the curved brackets increases because tv is t* d t*=m t0=m t0 Iv t 0 " Iv t* eÀ + Jv t0 eÀ [36] small at the top of the atmosphere and the exponential factor is ðÞ¼ ¼ ðÞ 0 ðÞ m ð near 1. Eventually, the optical depth becomes significant, so surface contribution the term exp ( t /m) begins to dominate over the downward contributions along the path À u |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} increase of kvrv, and Q begins to decrease exponentially Here, the first term is the extinction|fflfflfflfflfflfflfflfflfflfflfflfflffl along{zfflfflfflfflfflfflfflfflfflfflfflfflffl the path} from the downward toward zero. We deduce that Q has a single maxi- ground to the spacecraft where Iv(t*)istheupwardintensityatthe mum at some altitude in between. ground. The second term is the upward emission contribution We find the maximum in Q by differentiating w.r.t. z, using from points along the path to the spacecraft. Here, Jv(t0)corre- the product rule and setting dQ/dz 0 ¼ sponds to emission in the upward direction. The same expression can also be written in altitude coordinates but is more complicated: dQ −−/µ ()k /µ d()k = 0 = ()k F e uuuu + F e uu z dz uu uuµ dz 1 m kv z0 ra z0 dz0 40 À ðÞ ðÞ 2 2 ½ Iv z " Iv z 0 e ð0 2 ðÞ ¼ ðÞ¼ d ()kuuu ()ku d v 1 d v z [37] = = 1 2 z k dz00 dz µ dz µ dz m vra dz0 + k z z J z eÀ 0 v 0 ra 0 v 0 ð 0 ðÞ ðÞðÞ m In the last step, we use dt /dz k r from the definition of ð u ¼À u u optical depth (eqn [31]). Because the density of absorbing gases We derive a hemispheric monochromatic flux Fv from the usually drops exponentially with increasing height and the measured intensity Iv at a particular frequency v by integrating over 2 ahemisphere.Then,wederiveatotalflux(WmÀ )byintegrating absorption coefficient is either constant or an approximate power law relationship with pressure, optical depth should vary over all frequencies. We will use eqn [37] in the succeeding text. approximately exponentially with altitude, that is, t t 0 exp u ¼ u (z/Ht)whereHt is a scale height for the optical depth. If this 10.13.3.3 Optical Thickness and Thinness expression for tu is inserted into eqn [40], after a little manipula- tion, an important result is obtained that dQ/dz 0 occurs where Vertical sections of atmospheres are described as optically thick ¼ (tv 1) or optically thin (tv 1) to radiation of a particular tusecy 1, or tu 1 when zenith angle y 0 [41] frequency. ‘Optically thick’ just means opaque to a particular ¼ ¼ ¼ frequency or frequency band of radiation, and ‘optically thin’ Thus, at any frequency, the maximum volume heating rate means nearly transparent. This is a useful concept in looking at Q, or related effect such as dissociation or ionization, occurs spectra such as Figure 5(a). The IR photons can be thought of where the optical depth is unity for normal incidence radiation. as ‘escaping’ from the level where the IR optical depth drops The altitude is higher by factor sec y for solar radiation incident below 1 and then passing up through overlying ‘optically at zenith angle y. Also, the characteristic width of activity is H . t thin’ or IR-transparent air to space. We can also apply the Such a vertical structure is called a ‘Chapman layer’. idea to incoming solar flux. Solar photons are mainly absorbed A similar insight about the importance of unity optical in the atmosphere near the level of unity optical depth for a depth applies in the thermal IR for atmospheres that are
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optically thick at depth, that is, when we cannot see the surface a characterization of an atmosphere that is useful for semi- emission looking downward from space. In our expression for quantitative comparison of planetary atmospheres. upward intensity from an atmosphere (eqn [37]), we neglect • The second is that the diffuse approximation is implemen- the first term due to surface radiation, extend the integral to the ted. This replaces radiances I and B in Schwarzschild’s whole atmosphere, and assume blackbody emission. Then, radiative transfer equation with fluxes, with the optical depth scaled by a factor D, called the diffusivity factor to 1 1 Iv secy kvr Bvexp secy kvr dz0 dz account for the integration of the radiances over a hemi- ðÞ1 " a À a ð0 ðz sphere. A value of D 1.66 is close to numerical results 1 ¼ Iv secy BvWv dz [42] (Armstrong, 1968; Rodgers and Walshaw, 1966) and equiv- ) ðÞ1 "¼ 0 ð alent to assuming an average zenith angle of 53, that is, where W is a weighting function, defined by 1/m secy 1.66, in eqn [34]. Others set D 3/2 (Goody v ¼ ¼ ¼ and Yung, 1989; Weaver and Ramanathan, 1995)orD 2. ¼ 1 The third simplification is to handle thermal IR with a two- Wv z kv z ra z exp secy kv z0 ra z0 dz0 [43] • ðÞ ðÞ ðÞ À ðÞ ðÞ + ðz stream approximation, in which upward (F ) and down- Equation [42] says that the radiance to space is a weighted ward (FÀ) fluxes are treated separately. vertical integral of the Planck function. The weighting function Radiative–convective equations are easiest to formulate W is mathematically similar to heating rate Q in eqn [39]. Just u with optical depth t as the vertical coordinate but can be like Q, function W is zero at the bottom of the atmosphere u converted into physical space using a relationship between t where the exponential reaches zero, Wu is zero at the top of the and pressure. In a plane-parallel atmosphere, Schwarzschild’s atmosphere where density is zero, and Wu has a maximum in equation (eqn [34]) for the upwelling and downwelling ther- between. As a result, the monochromatic IR radiance comes + mal radiative fluxes (F and FÀ, respectively) can be written as from a layer with characteristic thickness that is a scale height + in optical depth or smaller and centered where optical depth dF + dFÀ DF pB , DFÀ pB ; d d has exactly the same tu 1 condition as eqn [41]. For frequen- t ¼ ðÞÀ t ¼À ðÞÀ 4 cies with large k ra that are highly absorbing, the t 1 peak of where pB t sT t [44] u u ðÞ¼ ðÞ the weighting function W will be high up (large z), given that u Equation [44] allows calculation of upwelling and downwelling t is the integral of k ra from z to . This is seen in a planetary u u 1 thermal fluxes if T(t) is specified. The net thermal flux is IR spectrum such as Figure 5(a) in dips where greenhouse + Fnet F FÀ. Subtracting dFÀ/dt from dF*/dt in eqn [44],weget gases absorb. The dependence of Iv( ) on gas composition ¼ À 1 " and wavelength in eqn [42] is also how we deduce the temper- dFnet + DF + FÀ 2pB [45] ature and chemical structure of planetary atmospheres using dt ¼ ðÞÀ remote sensing, for example, from orbiters around other and differentiating w. r. t. t gives planets or ground-based telescopes.
2 + 2 d Fnet 0d F + FÀ dB1 d Fnet 2 dB D ðÞ2p D Fnet 2pD dt2 ¼ dt À dt ) dt2 À ¼À dt 10.13.3.4 Radiative–Convective Equilibrium B C B C B C B DFnet C Combining the theory of convection (Section 10.13.2.4)and @ A radiation (Sections 10.13.3.1–10.13.3.3)allowsustounderstand |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} [46] + the average vertical structure of planetary atmospheres quantita- In the last step, addition of dFÀ/d and dF /d from t t tively given that it is primarily controlled by these processes. + eqn [44] gives d(F +FÀ)/dt DFnet. Equation [46] can be ¼ A hypothetical atmospheric structure determined only by solved for blackbody radiance B(t) and the atmospheric tem- radiation balance (ignoring convection) will generate an unre- perature profile using the Stefan–Boltzmann law if Fnet(t) and alistic temperature profile in an IR-opaque troposphere with a a boundary condition are known. steep temperature gradient that is actually unstable to convec- Boundary conditions involve the solar flux. In the solar tion. Conceptually, a temperature profile for purely radiative system, atmospheres absorb solar radiation in the troposphere transfer for a stratosphere and upper troposphere has to be or at the surface and partly in stratospheres. The latter often leads joined to a convective profile below with both continuities of to stratospheric temperature inversions (Figure 2), while absorp- temperature and flux. How do we calculate this? For analytic tion in tropospheres can stabilize upper tropospheres against calculations, three approximations are made, as follows: convection, as happens on Titan. Exponential attenuation is a
• The first is a gray approximation, where the thermal opacity reasonable approximation for solar radiation, so the net absorbed of the atmosphere is assumed to be independent of wave- stellar flux Fnet can be written as (Robinson and Catling, 2012)
length and represented by a single, broadband value. ‘Gray ktropot kstratiot Fnet t Fnet eÀ + Fstrato eÀ opacity’ should be viewed as a fitting parameter to match an ðÞ¼ observed thermal structure. It rarely makes sense to labori- tropospheric and surface solar absorption stratospheric solar absorption ously compute gray opacities from spectral averaging, and it ðÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}[47] is a misconception to consider gray opacities as ‘real’ prop-
erties of atmospheres, given that the real opacity is Here, Fstrato and Ftropo are the top-of-atmosphere net absorbed wavelength-sensitive. Instead, a gray optical property gives stellar fluxes in the stratosphere and troposphere (including
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4 the surface for planets with surfaces), respectively. Parameters rule), eqn [54] reduces to sT (t) (Fnet (0)/2)(1+Dt), which ¼ kstrato tsws/t and ktropo tswt/t control the attenuation of solar is a common form for simple radiative models in textbooks ¼ ¼ flux in two channels, where tsws and tswt are shortwave optical (e.g., Andrews (2010), p. 85; Goody and Yung (1989), p. 392). 4 depths and t is the gray thermal IR optical depth. At the top of Using sT (t) in eqn [54], the downwelling flux can also be the atmosphere, where t 0, the net absorbed stellar flux must deduced from eqn [53] and the upwelling flux from a similar ¼ be (eqn [23]) expression. We can now ask: what is required for an atmosphere to have F F t 0 1 A [48] a strong stratospheric temperature inversion and thus a netðÞ¼¼ ðÞÀ 4 well-developed tropopause minimum? The minimum occurs where A is the planet’s Bond albedo and F is the solar flux in the radiative regime, and we can simplify eqn [54] by incident at the top of the atmosphere. The overall energy omitting terms in tropospheric shortwave attenuation, that is, balance at any atmospheric level is between the net thermal k 1, which is the case for real atmospheres. Then, setting tropo flux with the absorbed stellar flux and any internal energy flux, the derivative w.r.t. t of the resulting expression to zero, the Fi from a planet’s interior (relevant for giant planets): optical depth of the tropopause minimum is (Robinson and Catling, 2014) Fnet t F t + Fi [49] ðÞ¼ netðÞ
By combining eqn [47] with eqn [49] and inserting into 2 1 F k eqn [46], we obtain ln2 strato stato 1 3 [55] ttp 2 ¼ kstrato Ftropo Fi D À dsT4 1 6 þ 7 2 2 ktropot 1=100 6 3 7 D ktropo Ftropo eÀ 6 1=10 10 7 dt ¼ 2D À À 6 7 4 5 h |fflffl{zfflffl} |5fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} 2 2 kstrato t 2 |fflfflfflfflfflffl{zfflfflfflfflfflffl} + D kstrato Fstrato eÀ + D Fi [50] À |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} i The numbers given under parts of eqn [55] are typical which integrates to ÀÁ averages for the Earth, Titan, and the giant planets, which all 2 2 1 D ktropo have stratospheres and tropopause minima (Table 4). Using T4 T4 0 + À 1 e ktropot F s t s À tropo ðÞ¼ ðÞ 2D" ktropo À these numbers, the gray IR optical depth where the tropopause
ÀÁ minimum occurs is ttp 0.05. This optical depth depends on D2 k2 Fstrato /(Ftropo +Fi), which is the ratio of the stratospheric strato kstratot 2 + À 1 eÀ F + D Fit [51] k À strato absorbed flux relative to that from below and kstratio, which is strato ÀÁ the ratio of shortwave optical depth to IR optical depth in the The flux at the top of the atmosphere in this equation stratosphere. Equation [55] only makes mathematical sense if (sT4(0)) can be eliminated by applying the boundary condi- tion that there is no downwelling thermal radiation at the top 2 2 of the atmosphere where t 0. We first use the definition kstrato > D 1+ Ftropo + Fi =Fstrato [56] + ¼ F F FÀ to rewrite eqn [45] as hi net ¼ À which sets a threshold condition on kstrato for a tropopause dFnet 4 DFnet +2FÀ 2sT [52] minimum or, alternatively, the condition for a stratospheric dt ¼ À temperature inversion. It tells us that to have a stratospheric ÀÁ4 and, second, we solve for sT using eqns [49] and [47]: inversion, an atmosphere has to have a large kstrato, that is, a strong shortwave absorber with a shortwave optical depth that 4 sT t FÀ t ðÞ¼ ðÞ dominates over the IR opacity; otherwise, cooling to space by D + k 1 tropo ktropot D + kstrato kstratot the IR-active molecules cancels shortwave heating of a strato- + Ftropo eÀ + Fstrato eÀ + Fi 2 D D sphere. Typical flux ratios in the planets of the solar system in 2 [53] eqn [56] mean that kstrato must be 10 for a well-developed tropopause minimum. Third, inserting t 0, and noting that FÀ (0) is zero, gives ¼ The condition for a stratospheric inversion (eqn [56])is T4(0). Inserting T4(0) into eqn [51] gives a final expression s s satisfied by some planets but not by others depending on for the temperature profile in the radiative part of a gray stratospheric composition and the balance of shortwave atmosphere that sits above a radiative–convective boundary absorption to IR cooling (Table 5). The Earth’s stratospheric and extends into the stratosphere: ozone absorbs strongly in the UV, so kstrato is large and heating is not overwhelmed by emission by CO at 15 mm, water F D k D 2 4 strato strato kstratot sT t 1+ + eÀ vapor, or ozone at 9.6 mm. Consequently, the Earth has a ðÞ¼ 2 k D À k strato strato stratospheric inversion. In contrast, CO in the atmospheres 2 F D k D F of Mars and Venus does not absorb strongly in the visible or tropo tropo ktropot i + 1+ + eÀ + 1+Dt 2 k D À k 2 ðÞsolar near-IR; instead, in the upper atmosphere, the 15 mm tropo tropo [54] band of CO2 radiates heat to space efficiently. Thus, kstrato is too small for stratospheric inversions in the global average on If there are no internal heat (F 0) and no solar atte- Mars and Venus. On the giant planets, dark stratospheric aero- i ¼ nuation (the limit k k 0, applying l’Hoˆpital’s sols absorb in the UV and visible, while methane has many strato ¼ tropo !
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Table 4 Solar absorption and infrared properties of gray radiative–convective parameters for the Earth, Titan, and the giant planets
Parameter World
Earth Jupiter Saturn Titan Uranus Neptune
Pref [bar] 1 1 1 1.5 1 1 T [K] 288 166 135 94 76 72 0 ac 0.6 0.85 0.94 0.77 0.83 0.87 2 Fstrato [W mÀ ] 7 1.3 0.41 1.49 0.24 0.09 2 Ftropo [W mÀ ] 233 7.0 2.04 1.12 0.41 0.18 2 F [W mÀ ] Minor, 0.087 5.4 2.01 Minor Minor, 0.042 0.047 0.43 i Æ kstrato 90 90 180 120 220 580 ktropo 0.16 0.06 0.03 0.20 0.08 0.20 trc 0.15 0.34 0.44 4.4 0.62 0.41 a t0 1.9 6.3 9.2 5.6 8.7 3.0 ttp 0.050 0.064 0.040 0.077 0.042 0.017 ptp [bar] (model) 0.16 0.10 0.066 0.18 0.070 0.075 ptp [bar] (observed) 0.16 0.14 0.08 0.2 0.11 0.1 a For Titan, t0 2.5 at p 1 bar. ¼ ¼ Parameters inferred from measurements include a tropospheric reference pressure (P ) and temperature (T ), an empirical adjustment factor applied to the dry adiabat to account ref ref for volatile condensation (ac) (eqn [59]), an internal energy flux (Fi), and the solar flux absorbed in the stratosphere (Fstrato ) and troposphere (Ftropo ). For rocky bodies, Ftropo 2 includes surface absorption. For example, the Earth has F + F 240W/m , the global mean absorbed solar flux. Parameters that control solar energy attenuation strato tropo ¼ in the stratosphere and troposphere are Kstrato tsws/t and Ktropo tswt/t, where tsws and tswt are shortwave optical depths and t is the gray thermal IR optical depth at ¼ ¼ any atmospheric level (eqn [47]). The gray IR optical depths at the reference pressure, radiative–convective boundary, and tropopause are t0, trc and ttp, respectively. A comparison of t0 values shows that IR optical depth is somewhat larger on the giant planets than the Earth or Titan. Also, Titan has the largest fraction of solar radiation absorbed in its stratosphere. Source: Robinson TD and Catling DC (2014) Common 0.1 bar tropopause in thick atmospheres set by pressure-dependent infrared transparency Nature Geoscience 7: 12–15.
Table 5 Molecules that dominate stratospheric shortwave heating and infrared cooling in atmospheres of the solar system
Planet Stratospheric heating Stratospheric cooling
Oxidizing stratospheres Venus Unknown absorber (UV), CO2, sulfur-bearing aerosols CO2, sulfur-bearing aerosols Earth Ozone CO2, water vapor, ozone Mars CO2 CO2 Reducing stratospheres
Jupiter Aerosols (UV/vis), CH4 (NIR) H2–H2 and H2–He CIA continuum, acetylene (C2H2), ethane (C2H6), methane Saturn Aerosols (UV/vis), CH (NIR) H –H and H –He CIA continuum, acetylene (C H ), ethane (C H ), 4 2 2 2 2 2 2 6 methane
Titan Haze, CH4 (NIR) Acetylene (C2H2), ethane (C2H6), haze Uranus Aerosols (UV/vis), CH4 (NIR) H2–H2 and H2–He CIA continuum, acetylene (C2H2), ethane (C2H6), methane Neptune Aerosols (UV/vis), CH4 (NIR) H2–H2 and H2–He CIA continuum, acetylene (C2H2), ethane (C2H6), methane
CIA, collision-induced absorption.
R=cp cp cv =cp absorption bands in solar near-IR. A collision-induced p p ðÞÀ T Tref T Tref T absorption (CIA) continuum of H2–H2 and H2–He and ther- ¼ pref ) ¼ pref ) g 1 =g mal IR emission bands of acetylene (C2H2, 13.7 mm) and p ðÞÀ Tref [57] ethane (C2H6, 12.1 mm) and methane (7.6 mm) cool these ¼ p ref stratospheres inefficiently (Yelle et al., 2001; Zhang et al., Here, T is a reference temperature at a reference pressure p 2013). Overall, kstrato is large, and all the giant planets have ref ref stratospheric inversions. On Titan, a thick stratospheric haze (e.g., 1 bar), while g is the ratio of specific heats cp/cv. Kinetic absorbs visible and UV radiation, along with stratospheric theory relates g to the degrees of freedom, Ndof, for the primary atmospheric constituent(s), where absorption in the near-IR by methane. The results are a large kstrato and a pronounced stratospheric inversion. 2 The aforementioned is radiation only; the other part of the g 1+ [58] ¼ Ndof radiative–convective problem to consider is the convective regime. Equation [14] describes a dry adiabat, which we can For atmospheres of linear diatomic gases, such as N2–O2 rewrite as temperature as a function of pressure, (Earth), H2 (giant planets), or N2 (Titan), the molecules have
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3 translational and 2 rotational degrees of freedom, so that atmosphere is complicated, with a 8–12 mm window, a steep
Ndof 5 and g 7/5 1.4. For CO2-dominated atmospheres (np 4–5) t p scaling for the 6.3 mm water absorption band ¼ ¼ ¼ ¼ À (Venus and Mars), there are 3 translational, 2 rotational, and water bands beyond about 20 mm, and an n 2 scaling p ¼ and 0.3 vibrational degrees of freedom, so Ndof 6.3 and for the 15 mmCO2 band. But it turns out that np 2 is a good ¼ ¼ g 1.3 (Bent, 1965). With these g values, a tropospheric tem- approximation for tropospheres and lower stratosphere for all ¼ perature T is proportional to p0.3 and p0.2 for diatomic and the planets with thick atmospheres in the solar system, vali- triatomic planetary atmospheres, respectively. However, no dated by comparison of numerical calculations (Robinson and observed troposphere follows a dry adiabat because condens- Catling, 2012, 2014). ables release latent heat and lower the lapse rate. Following An n 2 power law in eqn [62] also has a physical basis. p ¼ Robinson and Catling (2012), a semiempirical approximation The primary absorption processes that cause IR opacity in adjusts the convective tropospheric temperature structure as thick tropospheres are pressure broadening and CIA (see follows: Section 10.13.3.5). Because both processes rely on collisions,
they tend to have an absorption coefficient ki p,soby ac g 1 =g 2 / p ðÞÀ integration of dt p, it follows that t p . T Tref [59] / / ¼ pref With an opacity–pressure power law scaling of eqn [62], we can rewrite our tropospheric convective profile with opacity where ac is a dimensionless constant, 0.6–0.9 for planets in as the vertical coordinate instead of pressure: the solar system, representing the average ratio of the observed lapse rate to dry adiabatic lapse rate in the planet’s convective ac g 1 =npg b=np t ðÞÀ t regime. T Tref Tref [63] ¼ t0 ¼ t0 Equation [59] gives temperature as a function of pressure in the convective regime, whereas eqn [54] gives temperature as a This equation can be inserted into Schwarzschild’s differ- function of optical depth in the radiative regime, so how is ential equation that describes the upwelling thermal radiative optical depth related to pressure? The gray differential optical flux (eqn [44]) to give a differential equation in dF+/dt, which depth, dt, across an elemental layer of vertical thickness dz is can be solved (see the Appendix of Robinson and Catling the sum of various contributions from N sources of opacity, (2012)) to give an analytic solution for the convective upwell- a such as absorbing gases: ing flux below the radiative–convective boundary in a gray troposphere: Na N N dp dt ki ri dz kimi rdz dt kimi + 4 D t0 t ¼À ¼À ) ¼ g F t sTref eÀ ðÞÀ i 1 i 1 i 1 ðÞ¼ X¼ X¼ X¼ mir dp=dz gr ¼À attenuated upwelling flux from t0 level N ai + bi |ffl{zffl} p |{z}dp ki pref mi pref [60] T4 |efflfflfflfflfflfflfflfflfflfflDt {zfflfflfflfflfflfflfflfflfflffl} 4 4 ¼ ðÞðÞpref g s ref b b i 1 + Gu 1+ ,Dt Gu 1+ ,Dt0 ¼ 4b=np X D np À np t0 Here, k is the mass absorption coefficient for absorber i, which ðÞ i generally depends on pressure and temperature, while ri mir flux contribution from atmosphere above the reference level ¼ is the corresponding density of the ith absorber, where mi is the |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}[64] mass mixing ratio (eqn [4]). In the second step, we substitute for the hydrostatic eqn [5]. In the last step, we use the fact that Here, Gu is an upper incomplete gamma function, which is a the profiles of and k are generally power laws of pressure: mathematical function defined by mi i
ai bi 1 a 1 t p p Gu a, x t À eÀ dt [65] m p m p , ki p ki p [61] ðÞ iðÞ¼ iðÞref p ðÞ¼ ðÞref p ðx ref ref Although this looks complicated, from a practical perspec- Water vapor is the main source of opacity in the Earth’s tive, the incomplete gamma function is evaluated as simply as atmosphere, whereas in other atmospheres, gases have similar a sine or cosine in computer programs, such as MATLAB, IDL, pressure dependence of opacity and mass mixing ratio, such as or Python. Thus, the thermal flux in eqn [64] can be instantly N2 and CH4 on Titan. Furthermore, the sum over kimi in computed as a function of optical depth, if input parameters eqn [60] could be replaced with a single weighted-mean (np, b) and variables (T0, t0) are specified. absorption coefficient k. Thus, we can integrate and normalize In the preceding text, we have written analytic equations eqn [60] to get a power law dependence of optical depth on that govern radiative–convective equilibrium in planetary pressure, where the power is np a+b+1, ¼ atmospheres with the gray approximation. The temperature
p a + b p a + b +1 p np follows a convective profile according to eqn [63] up to a dt dp t t0 ,ort t0 [62] radiative–convective boundary, where the temperature then 1 pref ) ¼ pref ¼ pref obeys eqn [54] according to radiative equilibrium. There
Here, t0 is the gray IR optical depth at reference pressure pref, will be a tropopause temperature minimum above the such as 1 bar. In tropospheres and lower stratospheres, often a radiative–convective boundary at an IR optical depth given well-mixed gas (a 0) with an absorption coefficient linearly by eqn [55] if the stratospheric shortwave absorption is ¼ dependent on pressure (b 1) occurs, such as H in giant- large compared to IR opacity, that is, the condition of ¼ 2 planet atmospheres. In that case, n 2. The Earth’s large k in eqn [56]. Figure 6(a) is a schematic profile p ¼ strato
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that shows a purely radiative–convective equilibrium for an F F F 4 tropo+ strato + i 4 (absorbed solar flux) atmosphere without a stratospheric shortwave absorber. The Tskin = Tskin = 2 2 temperature asymptotes to an isothermal stratosphere with 4 aso-calledskintemperature.Theskintemperatureisgiven T T e T e by eqn [53] with t 0, no downward thermal flux, and no = skin = 1/4 ¼ 2 2 solar atmospheric absorption. If we also neglect the internal 66 heat flux, we get ½
1/4 Skin temperature, Tskin= Te/2
Radiative–convective Boundary
Convective profile
R a d i a
Altitude t i v e
p r o f il e
Radiative surface Temperature
(a) Temperature
Strong stratospheric inversion if short-wave optical depth >> infrared optical depth
pressure
r tp r tp, ~ 0.1 bar where scaled infrared optical depth, Dt tp, ~ 0.1
Radiactive–convective boundary where scaled r rc infrared optical depth, Dt rc, ~ 1 (or greater)
Reference infrared r 0 optical depth t ~2− 9 0
(b) Temperature
Figure 6 (a) A schematic profile of a radiative–convective equilibrium for an atmosphere without a shortwave absorber. A purely radiative profile (solid line) is unphysical. It is unstable to convection and has a discontinuity at the surface, so turbulence ensues and an adiabatic (i.e., convective) profile (dashed line) is established. The convective profile intersects the radiative profile at a level called a radiative–convective boundary with continuity of net flux and temperature. Without a stratospheric inversion, there is no tropopause minimum. In the absence of significant internal heat from the planet, the temperature asymptotes at altitude to a skin temperature that is a simple function of the effective temperature Te, as shown. In our solar system, this schematic radiative–convective profile is roughly representative of the global mean profile of Venus. (b) A more general radiative– convective equilibrium profile where a strong shortwave stratospheric absorber is present. The conditions for the stratospheric inversion, tropopause minimum, and typical IR optical depths are shown (applicable for the Earth, Titan, and the giant planets). D is the diffusivity factor (see text). Reproduced from Robinson TD and Catling DC (2014) Common 0.1 bar tropopause in thick atmospheres set by pressure-dependent infrared transparency. Nature Geoscience 7: 12–15.
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where Te is the effective temperature. Thus, the skin tempera- (described as optically or radiatively active gases), whereas ture of an Earth-like world without the properties that give it a others do not (optically inactive). Why this is so involves an 1/4 stratospheric inversion would be 255 K/2 , which is understanding of molecular spectroscopy. 214 K. Although simple, the skin temperature actually gives a In a spectrum from a planetary atmosphere, we see discrete rough estimate for the tropopause minimum temperature; on lines, groups of lines called bands, windows of weak spectral Earth, the observed global mean tropopause temperature is absorption, and a continuum of smoother varying absorption. 208 K (Han et al., 2011). Figure 6(b) shows a schematic for Our starting point recognizes that the total internal energy of a the more general case for an atmosphere with a stratospheric gas molecule can be written as inversion. E E + E + E + E [67] Of course, gray radiation is an approximation, and numerical total ¼ electronic vibrational rotational translational nongray calculationsarerequiredtoreplicatethedetailsof where Eelectronic is the energy associated electron orbits and atmospheres, for example, Goody and Yung (1989), other terms represent vibrational energy, rotational KE, and p. 404–407; Thomas and Stamnes (1999), p. 450–453; and translational KE, respectively. Molecular collisions depend on Satoh (2004),p.380–383.Suchmodelswerefirstpioneeredfor temperature through Etranslational and redistribute energy the Earth by Manabe and coworkers (Figure 7)(Manabe and between the various forms (if the air is dense enough) to Strickler, 1964; Manabe and Wetherald, 1967). Since then, radia- establish local thermodynamic equilibrium where a local vol- tive and radiative–convective models have been applied to Venus ume behaves like a blackbody. This was the assumption in the
(Bullock and Grinspoon, 2001; Crisp, 1986, 1989; Meadows and radiative transfer discussed in Sections 10.13.3.3 and Crisp, 1996), Mars (Kasting, 1991; Marinova et al., 2005), Jupiter 10.13.3.4. and Saturn (Appleby and Hogan, 1984), Uranus and Neptune A photon can be absorbed if its energy is close to a quantized (Appleby, 1986), Titan (McKay et al., 1991, 1999), and exoplanets energy jump within a molecule or it can be scattered by chang- (Heng et al., 2012; Robinson and Catling, 2012). ing phase or direction. Transitions in rotational speed require
less energy than vibrational ones, while electronic transitions require the most. Planck’s law for photon energy, E hv hc/l, ¼ ¼ 10.13.3.5 Absorption and Emission by Atmospheric Gases indicates that types of transition correspond to general wave- length ranges (Table 6). However, absorption often occurs as In the previous subsection, we implicitly assumed that some combinations of transitions, such as rotational and vibrational. gases absorb or scatter radiation at certain wavelengths An electric dipole moment due to separated positive charge and negative charge determines whether a molecule undergoes 2.3 certain rotational or vibrational transitions. For example, the
(40) slight positive charge on hydrogen relative to oxygen in H2O produces an electric dipole moment. The electric field in elec-
tromagnetic radiation torques the electric dipole and induces
rotation. Isolated homoatomic molecules such as H2 do not possess electric dipoles but dipoles can be induced by 10 (30) collisions. CH4 and CO2 also have no electric dipoles, but their bonds can bend, allowing a temporary electric dipole
with associated rotational–vibrational transitions.
Some molecules possess a magnetic dipole moment. The (20) magnetic field in an electromagnetic wave can torque a molecule with a magnetic dipole moment into rotation, analogous to a Altitude (km) (mb) Pressure 100 Pure radiative equil. torqued needle on a compass. Many molecules do not have Dry adiabatic adj. significant magnetic moments because electrons of opposite
spin pair up. O notably has two unpaired electrons and a mag- 6.5 °Ckm−1 adj. (10) 2 netic dipole, which allows rotational bands in the microwave. In fact, satellite sensing of microwave emission from O2 measures the Earth’s atmospheric temperature over broad altitudes accor- ding to instrument weighting functions like those discussed in 1000 (0) Section 10.13.3.2 (Shine et al., 2008; Yang et al., 2013). 180 220 260 300 340 Temperature (K) Table 6 Typical wavelength ranges of molecular transitions
Figure 7 Results from radiative transfer modeling for the Earth that Dominant transition Wavelength range Band includes the effects of stratospheric ozone that causes a temperature inversion above a tropopause minimum. Three models are shown: purely Electronic <1 mm X-ray to UV to visible radiative equilibrium, radiative–convective equilibrium with a dry adiabat, Vibration 1–20 mm Near-IR to far-IR and radiative–convective equilibrium with a typical adiabat modified by Rotation >20 mm Far-IR to microwavea water vapor. Reprinted from Manabe S and Strickler RF (1964) Thermal a equilibrium of the atmosphere with a convective adjustment. Journal H2 can have rotational transitions at shorter wavelengths in warm, dense atmospheres of Atmospheric Sciences 21: 361–385, with permission from the (Wordsworth and Pierrehumbert, 2013). American Meteorological Society. IR, infrared; UV, ultraviolet.
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For IR opacity and climate, the most important transitions displacements of atoms in molecules corresponding to are vibrational and rotational, so we discuss these next. Dv 2, 3,...produce less intense overtones. ¼Æ Æ The frequency of combined vibrational and rotational tran- 10.13.3.5.1 Rotational transitions sitions is slightly lower or higher than the pure vibration. Rotating molecules have a rotational frequency that is quan- Consequently, rotational–vibrational spectra have lines on either side of a pure vibrational frequency. The P branch is the tized analogous to a household fan spinning at set speeds. 1 2 low-frequency side corresponding to changes DJ 1, while A molecule’s rotational KE is =2Io , where I is the moment ¼À the R branch is the higher-frequency side with DJ +1. The of inertia. In general, there are three principal moments of ¼ central frequency line with DJ 0 is called the Q branch, inertia about three axes of symmetry through the center of ¼ mass of a molecule, denoted l ,l , and l from smallest to which may not exist if quantum mechanics requires DJ 0. A B C 6¼ largest. However, in many simple molecules, some moments Lines in the P and R branches are labeled according to the initial rotational quantum number J, that is, R(0), R(1),..., of inertia are equivalent or zero. For example, the axis through the atoms of CO , a linear molecule (O C O), has l 0, and P(0), P(1), etc. Superposed DJ 2 or +2 electric quadru- 2 ¼ ¼ A ¼À while the moments of inertia around two perpendicular axes pole transitions can also occur. Instead of a PQR branch struc- are equal and can be designated as l. In quantum theory, the ture, there are the O branch (DJ 2) and S branch (DJ +2). ¼À ¼ quantized total rotational angular momentum L, associated Giant-planet atmospheres have electric quadrupole absorption
lines of H2 in the far-IR, for example. with a so-called linear rotor such as CO2,is Vibrational bands of CO are important on Venus, Earth, 2 h and Mars. The bending of CO produces a 15 m rotation– L JJ+1 [68] 2 m ¼ 2p ðÞ vibration band – actually a PQR branch structure but individual pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where J 0, 1, 2, ... is the rotational quantum number. The lines are hidden by pressure broadening (see the succeeding
¼ text). The 4.3 mm asymmetrical stretch CO2 band is important rotational KE for number J is on Venus for absorbing a large amount of outgoing IR at depth 2 2 2 1 2 Io L h because of Venus’s high temperature. A vibration–rotation EJ Io ðÞ JJ+1 hB J J +1 where B ¼ 2 ¼ 2I ¼ 2I ¼ 8p2IðÞ¼ðÞ ðÞðÞ band of methane at 7.6 mm contributes IR opacity in the Earth’s h atmosphere but appears in emission in spectra of the giant 2 rotational constant ¼ 8p I ¼ planets and Titan due to warm CH4-containing stratospheres.
Only transitions between adjacent states J and J+1 are 10.13.3.5.3 Collision-induced absorption allowed, giving an energy change, Homoatomic molecules such as H2 or N2 lack permanent
DE hB J +1 J +2 JJ+1 hB J2 +3J +2 J2 J electric dipoles, but collisions at high-pressure distort the elec- ¼ ½ðÞðÞÀðÞ¼ À À tronic structure and produce a temporary electric dipole, allow- 2hB J +1 Âà ¼ ðÞ ing vibrational and rotational transitions (Trafton, 1966; Thus, the photon frequency needed to cause a rotational Trafton et al., 1998). CIA in various pairs of molecules creates transition in a linear rotor is significant IR continuum opacity in several planetary atmo-
spheres: H2–H2 and H2–He on the giant planets, N2–N2 and DE N –CH in Titan’s atmosphere (McKay et al., 1989), and CO – v 2BJ+1 [69] 2 4 2 ¼ h ¼ ðÞ CO2 on Venus (Moskalenko et al., 1979) or the early Earth Consequently, a pure rotational absorption spectrum has a (Kasting and Ackerman, 1986; Wordsworth et al., 2010). As series of lines separated by frequency differences Dv 2B.Asa first noted by Sagan (1977)H2–N2 and H2–CO2 CIA may have ¼ molecule rotates with increasing J, centrifugal force causes been important on prebiotic Earth (Wordsworth and Pierrehumbert, 2013) and early Mars (Ramirez et al., 2014). bonds to stretch, which requires a more complex treatment. Also, similar expressions can be derived for more molecules At Earth-like temperatures, H2 is a strong greenhouse gas with many moments of inertia. because its small moment of inertia causes widely spaced rotational energy levels (eqn [69]) that absorb across the thermal IR. 10.13.3.5.2 Vibrational transitions Molecular bonds behave like springs, so a diatomic molecule 10.13.3.5.4 Broadening has a classical simple harmonic resonant frequency v0. In quan- tum mechanics, however, the oscillation frequency and associ- Broadening processes are critical for planetary climates because ated energy levels of a diatomic molecule are quantized as otherwise, IR absorption lines would be delta-functions with negligible climatic effect. The three processes are the following:
1 1 v v+ v0, Ev hv0 v + Natural broadening, in which a finite time for absorption ¼ 2 ¼ 2 • or emission causes energy and frequencies to spread by where the letter ‘vee,’ v 0, 1, 2,..., is a vibrational quantum Heisenberg’s uncertainty principle. Apart from the UV, ¼ number. The v 0 state has a finite zero point energy, this broadening is tiny compared to the other two processes ¼ E hv /2, and a transition from this state to the next highest, in the succeeding text. 0 ¼ 0 or vice versa, is the fundamental, which typically produces • Doppler broadening, where random motion of molecules the strongest line because at atmospheric temperatures and relative to the radiation source causes absorption and emis- pressures, most molecules are in the ground state. Large sion at frequencies that are Doppler-shifted relative to the
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rest position of a line. This broadening tends to dominate 10.13.4 Photochemistry in upper stratospheres and higher. 10.13.4.1 Basic Principles of Chemical Kinetics • Pressure (or collisional) broadening, where collisions add or remove energy during radiative transitions allowing emis- Clearly, through molecular properties like those discussed in sion or absorption over a broader range of photon energies. the preceding text, atmospheric composition sets and controls Pressure broadening is a key source of IR opacity in tropo- planetary climates and atmospheric structures. But what sets spheres (Section 10.13.3.4). composition? The concentration of any gas is determined by chemical kinetics, that is, a competition between source and Pressure broadening produces a characteristic line shape sink fluxes, which depends on reaction rates. In general, atmo- called the Lorenz profile allowing absorption and emission far spheres are never in chemical equilibrium because of the free from the line center (Petty, 2006). A key property is that energy from sunlight (or starlight for exoplanets), as well as pressure-broadened lines have a width linearly proportional interior or tidal energy fluxes. However, giant-planet atmo- to pressure (i.e., the mass absorption coefficient relationship kv spheres that convect air to pressurized, hot interiors, where p, as assumed in Section 10.13.3.4 for tropospheres) unlike / reactions are efficient, are far closer to chemical equilibrium Doppler broadened lines, but both Doppler line broadening than shallower atmospheres on rocky planets such as the Earth and pressure line broadening depend on temperature. or Mars. One can think of chemical equilibrium as telling you
what ‘should happen’ if everything fully reacted, whereas kinet-
ics determines what actually happens according to the speed of 10.13.3.5.5 Continuum reactions. Thus, the starting point for atmospheric chemistry is A CIA continuum for giant planets and Titan was mentioned in to consider reaction rates. Then, we consider other principles the preceding text. For Earth, weak continuum absorption before discussing specific planetary atmospheres. from 12.5 to 8 mm is mainly due to water vapor with absorp- A general two body reaction can be written as tion coefficient proportional to the square of the water vapor bB+cC gG+hH [70] density. This absorption may be caused by the distant wings of ! The number of reactions per unit volume and unit time is very many lines in the far-IR or by dimers (H2O∙H2O), trimers, given by the reaction rate, and polymers of H2O molecules, but the problem is not fully resolved. At shorter wavelengths, continuum absorption arises m n Rr krnB nc [71] from well-known processes, including photoionization and ¼ 3 photodissociation. where ni is number density (molecules cmÀ )ofspeciesi.Here, the exponents are related to the order of the reaction m+n and are determined experimentally. A rate coefficient or rate constant kr is established in the lab by mixing reactants and measuring reaction 10.13.3.5.6 Numerical radiative transfer rates. The units of kr depend on the order of the reaction and the Although gray models like those in Section 10.13.3.4 are indis- concentration units. If m 1 and n 0, the reaction would be first pensable for intuitive understanding, numerical radiative trans- ¼ 1 ¼ order with kr in units of sÀ ;ifm n 1, the reaction would be fer calculations are needed for realism, which deal with hundreds ¼3 ¼ 1 1 second order with kr in units of cm moleculeÀ sÀ ; and so on. of lines that often overlap. Line-by-line calculations explicitly The rates of change of species are related: sum the effects of all the contributions of individual lines. This accurate method is very computationally demanding for climate dnB dnG b rate bRr, gRr , etc:,so models. Consequently, two more practical approaches have dt ¼À  ¼À dt ¼ [72] 1dnB 1dnC 1dnG 1dnH been developed (Goody and Yung, 1989; Liou, 2002; Petty, b dt c dt g dt h dt 2006; Thomas and Stamnes, 1999; Vardavas and Taylor, 2007): À ¼À ¼ ¼ We can consider an example, oxidation of methane by a • Band models use an analytic form for the positions and strengths of lines. Over a spectral interval, lines can be hydroxyl (OH) species: represented by a mean line strength and spread in a random OH + CH4 CH3 +H2O [73] ! distribution or by some other spacing that is amenable to 5 The Earth’s troposphere has concentrations nOH 8 10 analytic formulation. 3 13 3  molecules cmÀ and n 4:3 10 molecules cmÀ , while The correlated k-distribution method is a statistical representa- CH4  • an average tropospheric rate constant for eqn [73] is tion of the apparently haphazard variation of absorption 15 3 1 1 kr CH4 3:6 10À cm moleculeÀ sÀ . The units indicate a coefficients kv over spectral interval Dv with a smooth ¼  first-order reaction. By applying eqn [71], we can estimate the cumulative probability function of the occurrence of kv. chemical lifetime of CH4, that is, the average time a CH4 Within Dv, the Planck function can be considered a func- molecule spends in the atmosphere before destruction: tion of temperature without frequency dependence. The order in which the k ’s are summed then does not matter. v n CH4 n In this way, integration can be done using bigger spectral CH4 tCH4 intervals than in the detailed line-by-line spectrum. ¼ destruction rate ¼ krCH4 nOHn CH4 Line-by-line calculations validate the aforementioned two 1 s 15 3 1 5 3 methods and constrain the region of parameter space where ¼ 3:6 10 À cm molecules 8 10 cm  À À  À the inherent assumptions are valid. 12years ÀÁÀÁ
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Thus, methane would be removed rapidly without resupply. (Section 10.13.4.3), creating radicals that transmit the free Methane attains a nonnegligible abundance of 1.8 ppmv in energy. Reactions of radicals with nonradical molecules make the Earth’s present atmosphere because of a significant source further radicals because electrons do not pair up, that is, flux: 80–90% is from the Earth’s microbial biosphere, and the radical + nonradical radical + nonradical rest is from geothermal sources (Kirschke et al., 2013). On Mars !
– with another oxidizing atmosphere – the lifetime of methane HCN, formation of 0e:g:,CN+ CH CH + 1 is 300 years (Lefevre and Forget, 2009; Zahnle et al., 2011), so 4 ! 3 hydrogen cyanide on Titan B C methane sources are very limited or nonexistent given in situ B nitrile radical C B methyl radical C measurements of <1 ppbv CH (Webster et al., 2013). @ A 4 |{z} |ffl{zffl} [77] Generally, we deduce that a lifetime is related to chemical sinks as follows: However, radical–radical reactions remove radicals in two ways: disproportionation into nonradicals kr1 X+Y1 ! radical + radical nonradical + nonradical ! kr2 1 X+Y products, chemical lifetime of X 2 tx a reaction at ! ) ¼ ¼ krinYi 0 1 e:g:,C2H3 +C2H3 C2H4 +C2H2 , 200 km i ! B C X B altitude on TitanC k B vinyl or ethenyl radical ethylene acetylene C ri B ðÞ C etc: [74] @ A ! |fflffl{zfflffl} |fflffl{zfflffl} |fflffl{zfflffl} [78] Three-body (termolecular) reactions are also important in and recombination with the formation of a single molecule planetary atmospheres, for example, the formation of ozone
(O3) in the Earth’s stratosphere from O+O2 +a third body. radical + radical + M nonradical + M ! When two gases, A and B, combine, bond formation releases e:g:,NO + OH + M HNO + M, formation of nitric acid on ð 2 ! 3 energy that must be removed to prevent the adduct (the form Earth [79] AB) from splitting. A third body ‘M’ takes away the energy, Þ where ‘M’ represents any atmospheric gas, usually the most By removing radicals, a chain of photochemical reactions abundant one: initiated by sunlight is terminated. The net effect of all reac- tions is polymerization in reducing atmospheres, which can
A+B+M AB + M [75] produce particles of hydrocarbons, sulfur, phosphorus, and so ! Because a third body is needed, termolecular reactions are on, depending on the gas mixture, while in oxidizing pressure-dependent up to a high-pressure limit, where the atmospheres, the outcome is analogous to combustion, for exact concentration of M no longer matters. Also, termolecular example, full oxidation of CH4 to water and CO2 or oxidation of sulfur gases to sulfate aerosols (Hu et al., 2012, 2013). reactions are usually faster at lower temperatures because the Certain gas species can also be lost to planetary surfaces. molecules approach each other at slower speeds. More generally, rate constants have an Arrhenius depen- ‘Dry deposition’ is interception with the surface and loss with- dence on thermal energy kT,as out the aid of precipitation, although the surface could be wet or dry. ‘Wet deposition’ is when raindrops, snow, or cloud or k Aexp E =kT [76] r ¼ ðÞÀ a fog particles scavenge gases to the surface. For example, nitric acid formed in eqn [79] is dry-deposited onto the driest desert where a pre-exponential factor A and activation energy Ea are determined experimentally. A few useful rules are the follow- on Earth, the Atacama Desert (Catling et al., 2010; Michalski ing: (i) Reactions between molecules and radicals (species with et al., 2004), whereas elsewhere, it usually dissolves in rain that unpaired electrons, see Section 10.13.4.2) usually have posi- falls to the surface. Deposition is how atmospheric chemistry can alter the geochemistry of a planetary surface. For example, tive Ea so that kr increases with temperature. (ii) Fast reactions the surface of Mars has accumulated sulfate salts that formed in between two radicals generally have negative activation energy its atmosphere from the oxidation of volcanic sulfur (Settle, Ea so that kr decreases with temperature. (iii) Reactions between nonradical molecules are normally very slow unless 1979; Smith et al., 2014). they react on particle surfaces. (iv) Favored reactions tend to be exothermic. 10.13.4.3 Photolysis
10.13.4.2 Why Free Radicals Are So Important If species A is photolyzed, its number density nA (molecules 3 cmÀ ) declines with a photodissociation rate, Appreciably fast atmospheric reactions, with few exceptions, involve at least one radical species. A free radical has an @nA A+hv products, rate jAnA [80] unpaired electron in its outer shell (e.g., atomic chlorine, Cl) ! ¼ @t ¼À and reacts strongly to obtain more stable bonding. An excep- where jA is the photolysis rate (or photodissociation) coeffi- tion is ground-state atomic oxygen (O), a biradical that has 1 cient (sÀ ). The probability Ci per second for a molecule i to eight electrons of which two are unpaired. Some literature absorb radiation is related to ji, as follows: indicates radicals with dots, for example, HO . Á Photochemistry is initiated by the Sun, loosely analogous 1 1 Ci si I dl sÀ , ji qi si I dl sÀ [81] ¼ l l ¼ l l l to the spark plugs of an engine. Photons break up molecules ðl ðl
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Here, sil is the absorption cross section for species i (usually NOx from turning reactive chlorine back into unreactive chlo- 2 1 2 cm moleculeÀ ), and l dl is the photon flux crossing 1 cm in rine nitrate via l spectral interval l to l+dl. Absorption of a photon may some- ClO + NO2 +M ClONO2 + M [88] times lead to a different process (collisional deactivation, ! radiation, or breaking of a different bond), so qil is a dimen- Nitric oxide (NO) is naturally present because biogenically sionless quantum yield for the photodissociation of interest. produced nitrous oxide (N2O) reacts with an energetic oxygen atom (denoted O(1D)) to generate two NO radicals.
In Antarctica, springtime sunlight releases Cl from parti- 2 10.13.4.4 Earth’s Atmospheric Composition and Chemistry cles, which photolyzes into Cl radicals, causing ozone deple- tion via eqn [86]. The process is encouraged by a springtime With the principles in the preceding text, we can understand stratospheric vortex of eastward winds that prevents inflow the Earth’s atmosphere. The two key constituents responsible of new odd nitrogen or ozone from lower latitudes. Unlike for its chemical character are very large amounts of O2 and Antarctica, buoyancy waves associated with north–south
H2O vapor compared with any other known atmosphere. The mountain ranges disrupt the development of a coherent Arctic oxidizing chemistry caused by radicals associated with O and 2 vortex, so the Arctic lacks an equivalent ozone hole. H2O is critical for radiative transfer because it renders the One aspect of the Earth’s stratosphere that bears some Earth’s troposphere largely transparent to visible sunlight. At commonality to other oxidizing atmospheres (e.g., Venus and the same time, O2 causes the creation of the ozone layer, early Mars) is that sulfur gases (sulfur dioxide, SO , and car- 2 shielding the Earth’s surface from biologically harmful UV bonyl sulfide, OCS) are oxidized into sulfate aerosols that radiation with wavelengths less than about 300 nm. make a thin haze called the Junge layer (at 20–25 km alti- In the stratosphere, the basic scheme for making O3 from tude). The Junge layer scatters sunlight back to space and cools the O atoms of O2 was deduced by Sydney Chapman in 1930 the Earth. The Junge layer is a photochemical haze of sulfates, with the so-called Chapman reactions: analogous to Venus’ clouds, although far thinner because the
Earth’s sulfur is mostly not gaseous but tied up in the form of O2 + hv l 242nm O + O [82] ðÞ ! sulfate and sulfide minerals on or below the Earth’s surface. O+O +M O +M [83] 2 ! 3 However, it is a general feature of oxidizing atmospheres with sulfur gases to produce sulfate hazes. O3 + hv l 1140nm O + O [84] ðÞ ! We now turn to the Earth’s troposphere. All major gases
O3 +O 2O2 [85] (except Ar) in the Earth’s atmosphere are mediated by biolog- ! Energy absorbed in ozone photolysis (eqn [84]) heats the ical cycles, including O2,N2, and CO2. In addition, various stratosphere, causing its temperature inversion. O atoms trace gases have significant biological sources, such as CH4 and bigger hydrocarbons, N2O, OCS, dimethyl sulfide (CH3SCH3 derived from ozone photolysis cycle around to reform ozone and DMS), and carbon disulfide (CS2). via eqn [83]. But O3 and O (the so-called odd oxygen species) A key factor for the Earth’s tropospheric chemistry is the return to stable O2 molecules via eqn [85]. Chapman’s scheme overpredicted the abundance of strato- role of the dominant oxidizing radical, OH (Levy, 1971; Prinn, 2014). This species arises because of the presence of large spheric O3 compared to observations. In the 1970s, it was realized that minor constituents catalyze the removal of odd amounts of water vapor and O2, where the latter leads to the generation of tropospheric ozone ( 10% of the ozone column oxygen, abundance) in pathways involving NO . In the stratosphere, x X+O XO + O NO destroys ozone, but in the troposphere, NO makes ozone 3 ! 2 x x XO + O X+O [86] (Crutzen, 1979). To create ozone, O2 and O need to be com- ! 2 O3 +O O2 +O2 bined, so O atoms are required. These derive from NO2, whose ! progenitor NO comes from lightning, soils, or transport from the stratosphere: where X is a radical: NO, OH, Cl, Br, or H.
In the 1980s, it was further shown that heterogeneous NMHCS + NO + hv NO2 + products a net reaction reactions (gas–solid) on particles in polar stratospheres could ! ½ of many deplete the ozone, as opposed to the homogeneous reactions
(gas–gas) reactions such as eqns [82]–[86]. In Antarctica, an NO2 + hv l 410 nm NO + O [89] ðÞ ! ozone hole of low ozone column abundance appears each NMHCs are nonmethane reactive hydrocarbons (C2–C5 com- southern spring. Chlorine radicals released from the UV pounds plus isoprene (C5H8), e.g., butane (C4H10) and ethylene decomposition of anthropogenic CFCs are responsible (C2H4)), which have biogenic sources such as marine plankton (Russell et al., 1996). It was deduced that heterogeneous reac- and vegetation. The O atoms from eqn [89] react with O2 to form tions convert unreactive forms of chlorine, such as HCl and ozone (eqn [83]). Oxidation of methane and carbon monoxide, chlorine nitrate into Cl2 on polar stratospheric cloud particles: catalyzed by NOx, can also produce ozone. Then, ozone photol-
ClONO2 + HCl Cl2 + HNO3 [87] ysis with relatively energetic photons generates exited O atoms ! that react with water vapor to generate OH radicals: chlorine nitrate nitric acid 1 O3 + hv l 310 nm O2 +O D [90] This reaction|fflfflfflfflffl sequesters{zfflfflfflfflffl} odd nitrogen|fflfflffl{z (NOfflfflffl} NO+NO , ðÞ ! x ¼ 2 pronounced ‘nox’) into unreactive nitric acid, which prevents H O+O 1D OH + OHÀÁ [91] 2 ! ÀÁ
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OH is the ‘detergent of the atmosphere’ because it helps agreaterconcentrationofOatomsonVenusversusMarsbecause oxidize sulfur-containing gases (SO ,H S, DMS, etc.) into sul- of more photodissociation of CO and a smaller-scale height 2 2 2 fates, removes hydrocarbon gases such as CH4 and NMHCs by (eqn [7])forthesametemperature,giventhedependenceongra- initiating their oxidation, and converts NOx into nitric acid. vitational acceleration (Krasnopolsky, 2011). On both planets, Without significant OH, the Earth’s atmosphere would accu- chemistry produces vibrationally excited CO2 (denoted by an mulate hydrocarbons and sulfurous gases and be very different. asterisk),
O+CO2 CO2* +O [93] ! which decays and radiates 15 mm photons through an optically 10.13.4.5 CO2 Atmospheres: Venus and Mars thin path to space, causing cooling. Venus and Mars both have CO2-dominated atmospheres and so have some common behavior (Krasnopolsky and Lefevre, 10.13.4.6 Reducing Atmospheres: Giant Planets and Titan 2013). For example, CO does not significantly decompose 2 Much could be said about the photochemistry on Titan and the into CO by photolysis on either planet because catalytic reac- tions recombine CO2 from CO and O. Also, both planets have giant planets (e.g., Yung and Demore, 1999), but here, I focus cold thermospheres caused by CO2 radiative cooling, but on one general aspect – the fate of hydrogenated gases. Venus’ thermosphere is colder, which is counterintuitive given Although Titan and the giant planets have reducing atmo- Venus’ location. Photochemistry lies behind this situation. spheres, hydrogen escapes to space on Titan but not from the giant planets where H is the major constituent. This changes Of course, the recombination of CO2 relies on radicals. On 2 Mars, OH radicals (derived from water vapor or hydrogen the fate of carbon species on Titan compared to the giant peroxide photolysis) catalyze the recombination of CO and planets and is also relevant for the processing of nitrogen, O in cycles (McElroy and Donahue, 1972; Parkinson and phosphorus, and sulfur species on the giant planets. Hunten, 1972), for example, Methane is the most abundant carbon gas in the cold reducing atmospheres of the solar system (although CO is OH + CO CO2 +H ! thermodynamically favored in hot Jupiters). Unlike the rapid H+O2 +M HO2 +M conversion to CO2 and water vapor that occurs in oxidizing ! atmospheres (Section 10.13.4.1), methane is destroyed when HO2 +O OH + O2 ! UV photons with wavelength of <145 nm break C–H bonds. Net : CO + O CO2 The radicals that are released combine to make bigger hydro- ! carbons in the following overall schematic net reaction: Unlike Mars, where the chemistry is largely restricted to C–O–H, seven principal elements are involved on Venus: CH4 molecules + hv radicals + hydrogen O–C–S–H–Cl–N–F. Chlorine atoms (derived from HCl pho- ðÞ! large hydrocarbons + hydrogen tolysis) play an important role in CO2 recombination on ! ðÞ [94] Venus, along with various catalysts:
Cl + CO + M ClCO + M On Titan, for example, the lifetime of atmospheric methane ! against complete photolysis loss is 30–100 My (Yung et al., 1984). ClCO + O2 +M ClCO3 +M ! The hydrocarbons can be as simple as ethane (C2H6)orascom- ClCO +X Cl + CO + XO plex as polyaromatic hydrocarbons, which are molecules consist- 3 ! 2 ing of linked benzene rings. On Titan, organic molecules also XO + O X+O2 ! incorporate nitrogen to form nitriles (organics with C Ngroups) Net : CO + O CO2 (Atreya et al., 2009). But on Titan, hydrogen escapes, so the ! organics accumulate on the surface as solids or liquids, and meth- where X H, Cl, O, SO, or SO . ¼ 2 A fundamental difference of Venus compared with Mars (and ane is destroyed (Figure 8(a)). However, on the giant planets, Earth) is a total cloud cover. Venus’ clouds of sulfuric acid are hydrogen does not escape and is the main gas. There, hydrocar- bons eventually get mixed down to deep, hot ( 1000 K) layers photochemical, originating from the photolysis of SO2 and CO2: where they are hydrogenated and converted back into methane.
SO2 + hv SO + O Similar cycles occur on the giant planets with nitrogen, phos- ! phorus, and sulfur species. Ammonia (NH3), phosphine (PH3), CO2 + hv CO + O ! [92] and hydrogen sulfide (H2S) are photolyzed to make hydrazine, SO +O+M SO +M 2 ! 3 elemental phosphorus, and sulfur-containing solids. However,
SO3 +H2O H2SO4 sedimentation and mixing to depth cyclically reconstitute ! ammonia, phosphine, and hydrogen sulfide (Figure 8(b)). Unlike the Earth’s clouds that require updrafts and cooling for condensation, photochemical haze occurs globally on Venus, analogous to Titan’s haze. Sulfur on Venus also disproportionates, 10.13.5 Atmospheric Dynamics that is, some gets oxidized, while the rest is reduced. Molecules of 10.13.5.1 Equations of Motion sulfur atoms and sulfane (sulfur–hydrogen) absorb visible and UV radiation and may be responsible for cloud coloring. Atmospheric dynamics concerns the study of atmospheric Venus’ upper thermosphere is colder than Mar’s ( 250 K vs. motions and their relationship to heating and forces (Holton, 270 K) because O atoms are 10 times more abundant. There is 2004; Houghton, 2002; Mak, 2011; Vallis, 2006). Winds are
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Hydrogen escape a +2V v + V V r ðÞ ½Â ðÞÂ
acceleration on co-rotating frame coriolis acceleration centripetal acceleration Hydrocarbons Fg F |{z} + |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} CH4 (C H , (C H ) , PAHs ¼ m m Photolysis + 2 6 2 2 n chemistry force of gravity all other forces Sedimentation |{z} |{z} [95]
(a) Surface where V is the planet’s angular velocity vector, the acceleration and velocity are measured by a corotating observer, and the last term includes pressure gradient forces, frictional forces, and CH4 Hydrocarbons (C H , C H , etc.) 2 6 2 2 external tidal forces caused by gravitational interaction with PH3 Phosphorus, P4 Hydrazine, N H the Sun or parent star, from a moon, or from a planet being NH3 Photolysis + 2 4 Sulfur solids (S , H S (s),(NH ) S (s) orbited if we are considering the atmosphere of a large moon. H S chemistry 8 x y 4 x y 2 Usually, the centripetal acceleration is taken to the right-hand side, treated as ‘centrifugal force per kg’ and, combined with Sedimentation gravity to form an effective, measured gravity g. A modified g + diffusion vector is valid because rotating planets deform into oblate +H2 reactions surfaces as a result of the centrifugal force pulling outward at (b) Hydrogenation at hot depths the equator. Simplifying eqn [95] by neglecting tidal torques, Figure 8 (a) Net hydrocarbon production on Titan, associated with the we get the usual basic form of the momentum equation in escape of hydrogen. PAHs are polyaromatic hydrocarbons. (b) The atmospheric dynamics, closed cycles of hydrogen-bearing species on giant planets. (Drawn by the author.) dv 1 À p 2V v gk + Fvise [96] dt ¼ r r À ðÞÂ À
Coriolis force=kg preesure gradient force=kg |fflfflfflfflffl{zfflfflfflfflffl} |{z} critical for climate because they transport heat (e.g., from where Fvisc is the frictional or viscous force per unit mass, v tropics to cold poles) and affect the formation of clouds by is the wind vector, and k is an upward unit vector. It is carrying condensables. Winds also redistribute non- possible that tidal forces should not be neglected for long- condensables (e.g., O3 in the Earth’s stratosphere), modify term climate models rather than weather models. But sur- surfaces (e.g., dunes on Mars or Titan), and can drive secondary prisingly little work has been done to examine this issue circulations (e.g., ocean currents on Earth). (Houben, 2013). Atmospheric motions are usually considered from the Equation [96] can be expanded into local Cartesian (x, y, z) perspective of an observer corotating with a planet. coordinates on a sphere with equations in three velocity A rotating frame is accelerating, and in accelerating frames, components (e.g., see Holton (2004), Chapter 2): a zonal apparent forces arise in the opposite direction to real accel- wind u in the west to east or ‘zonal’ x direction, a meridional erations. For example, scales in an upwardly accelerating wind v in the south-to-north y direction, and a vertical wind elevator would incorrectly measure weight without account- w upward in the z direction, that is, v ui + vj + wk. The ¼ ing for a downward apparent force. In a rotating frame, resulting momentum equations simplify for large-scale centrifugal and Coriolis apparent forces arise. Centrifugal motions, if we ignore friction and vertical motion (given that forces are known from everyday experience, but the Coriolis vertical winds are usually much smaller than horizontal ones) force is less familiar and, consequently, often illustrated by a (e.g., see Andrews (2010) or Holton (2004)): ball thrown from the center of a merry-go-round to a catcher @p at its edge. If the merry-go-round does not rotate, the ball Vertical hydrostatic equation : gr [97] reaches the catcher in a straight line, but with counter- ðÞ@z ¼À clockwise rotation, the catcher rotates around and the ball du utanf 1 @p Zonal momentum : f + v [98] misses him; the ball appears to be deflected to the right of its dt À a ¼Àr @x motion from the viewpoint of the corotating catcher and Coroilis term Centrifugal term thrower. The analogy is imperfect because the ball also expe- dv Meridional momentum : riences a radially outward centrifugal force. But the basic dt utanf 1 @p picture is analogous to looking down upon a planet spinning + f + u [99] a ¼Àr @y west to east: the Coriolis force acts to the right of motion Coroilis term Centrifugal term (e.g., a wind vector) such as in the Earth’s northern hemi- sphere. Viewed from below, the merry-go-round’s rotation is Here, a is the planetary radius at the local x, y, z origin, the clockwise and the Coriolis force acts to the left of the latitude is f, and f is the Coriolis parameter defined by motion, akin to the effect on winds in the Earth’s southern f 2Osinf [100] hemisphere. ¼
Newton’s second law for motion on a rotating planet (e.g., where O is the planet’s angular velocity. Specifically, O 2p/td ¼ see Taylor, 2005)is where td is the rotation period ( a sidereal day), for example, ¼
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5 1 O 2p/(86164 s) 7.29 10À rad sÀ for Earth. If put on the Equation [103] uses a scale analysis. The variables U and Lh ¼ ¼ Â right-hand side, terms fv and fu are x and y components of the are characteristic wind speeds and horizontal length scales, À 1 Coriolis force per kg that act to the right of the wind in Earth’s respectively. For example, Jupiter has U 100 m sÀ , 8 4 1 northern hemisphere. Terms (u tan f/a)v and +(u tan f/a)u Lh 100000 km 10 m, and f 1.8 10À sÀ at 45N. À ¼ Â are x and y components of cyclostrophic acceleration, which are Thus, Ro 0.01, so rotation dominates and geostrophy applies. important on slowly rotating planets (see Section 10.13.5.2). Similarly, Ro 1 for other giant planets, Earth, and Mars. But In eqns [98] and [99], the derivatives in v and u are sub- at the equator, f 0 and f 0, so Ro cannot be small. The ¼ ! stantial or material derivatives. The dX/dt form is the result is that circulations on rotating planets are more (Lagrangian) rate of change of flow-field quantity X with symmetrical with latitude close to the equator but rotation respect to time following a fluid parcel, unlike dX/dt, the affects flows strongly at higher latitudes, producing big eddies. (Eulerian) rate of change at a fixed point. They are related by Geostrophy predicts that zonal winds should typically increase in speed with altitude. Usually, the distribution of sun- dX @X @X @X @X + u + v + w light creates a temperature decrease from equator to pole, so dt @t @x @y @z substantial derivative local derivative columns of air on average shrink toward the pole because they are cooler, which must create an increasing meridional pressure advective terms @X gradient at higher altitudes. Since @p/@y grows with altitude, the + v X |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} [101] ¼ @t Ár zonal wind ug will grow too by eqn [102].Thislinkbetween
horizontal temperature gradients and the wind shear – that is, Here, v is the velocity vector and we remind ourselves that change of wind speed and direction with height – is captured in a b a1b1 +a2b2 +a3b3, while the gradient operator is @/@x Á ¼ r¼ the thermal wind equation, which is derived from eqn [102] by +@/@y+@/@z. For example, if 2-D spatial contours of tempera- substituting for r p=RT from the ideal gas law (eqn [2]), dif- ture T are fixed in time (@T/@t 0) and an observer moves ¼ ¼ ferentiating w.r.t. z and then substituting the hydrostatic eqn [6]: eastward with velocity u in time. Dt between 250 and 270 K contours, then eqn [101] gives dT/dt u T (Dx/Dt) ¼ r ¼ Â vg 1 p g RT 1 g T (DT/Dx) 20 K/Dt. Alternatively, if the 250 K contour moves f RT = = T ¼ 2 with the wind speed and replaces the 270 K downstream con- z x p z R x T T x
tour, dT/dt 0 because the local change @T/@t ( 20K/Dt)is g/RT ¼ À balanced by advection of temperature, u T. ‘Advection’ (Latin g T r = ‘carry to’) is the transport of an atmospheric property by the T x velocity field. The difference between advection and convection is that advection usually refers to the predominantly horizontal, An approximation sign is used for ignoring vertical variations
large-scale atmospheric motions, while convection describes in T. With analogous manipulation of the zonal geostrophic the mostly vertical, locally induced motions. wind equation, the thermal wind equations in ug and vg are
@ug g @T @vg g @T f , f [104] @z ÀT @y @z T @x 10.13.5.2 Balance Regimes: Geostrophic, Cyclostrophic, and Thermodynamic These equations (or slightly higher approximations) allow calculation of winds on other planets (e.g., Smith et al., 2001). 10.13.5.2.1 Geostrophic balance After deriving temperatures from radiances measured by an
On planets that rotate fairly rapidly, such as Earth, Mars, and orbital instrument, eqn [104] implies the geostrophic winds. the giant planets, the Coriolis force is important. In the Earth’s Often, no surface wind is assumed as a boundary condition at northern hemisphere (NH), the Coriolis force acting to the the expense of some accuracy. right of large-scale flow is frequently approximately balanced by low pressure to the left and high pressure to the right, that is, 10.13.5.2.2 Cyclostrophic balance a pressure gradient force. Thus, on NH weather maps, winds The analysis in the preceding text applies to rapidly rotating blow counterclockwise around a ‘low’ cyclonic system or clock- planets, but for slowly rotating planets, the balance is different. wise around a ‘high’ anticyclonic system. The balance of Cor- 6 1 For example, on Venus, Lh 1000 km 10 m, U 10 m sÀ , iolis and pressure gradient forces is called geostrophic balance. 7 1 ¼ 7 1 and f 2O sin f 2(2.98 10À sÀ )(sin 45) 4.2 10À sÀ Because f u/tan f, eqns [98] and [99] provide this balance as ¼ ¼  ¼  at 45N, so the Ro 10 from eqn [103] and geostrophy is 1 @p 1 @p invalid. If the atmosphere itself rotates sufficiently rapidly ug , vg [102] ¼Àf r @y ¼ f r @x over some altitude range, the substantial derivatives and f z z terms are negligible in eqns [98] and [99], and a cyclostrophic The z subscript indicates a partial derivative evaluated balance and a cyclostrophic thermal wind apply, as follows: along an altitude surface, while the g subscript denotes geo-
strophic winds. In deriving eqn [102], we assumed d(u or u 2 tan 1 p ; v)/dt f to eliminate d/dt terms. This condition is true if the = a y dimensionless Rossby number (Ro), which is the ratio of the (a) cyclostrophic balance Lagrangian acceleration to Coriolis acceleration, is <0.1 or 105 2 ½ so, where tan (u ) 1 p R T g T g T RT = = Relative acceleration U2=L U a z y p z R T 2 y T y Rossby number, Ro h [103] ¼ Coriolis acceleration ¼ fU ¼ fLh (b) cyclostrophic thermal wind equation
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W equilibrium temperature. The radiative time constant can be Pressure estimated as
force 2 acosf f 2 heat change per m pecp DTe 1 A Sp u /( acosf) trad ðÞ= ðÞÀ f ¼ absorbed flux per m2 ¼ g 4 a u2tanf/a u f E quat 4pecp DTe or ðÞ [108] ¼ 1 A S g Horizontal ðÞÀ p centrifugal where p is the pressure at the emission level, g is the gravita- force e tional acceleration, DTe is the temperature change forced by the Figure 9 The geometry of cyclostrophic balance and the balance of absorption of solar radiation ( 2 K for Earth or 80 K for Mars), forces on a slowly rotating planet such as Venus or Titan. Here, f is cp is the specific heat capacity, A is the Bond albedo, and Sp is latitude, a is planetary radius, u is the zonal wind vector, and O is a the top-of-atmosphere solar flux. This uses eqns [8] and [23]. planetary rotation vector. (Drawn by author.) We can see the effect of eqn [107] in latitude-averaged
temperature structures of planetary atmospheres. Static stabil-
Equation [105](a) is a balance between centrifugal force per unit ity (G G) is generally positive, so if we measure T
10.13.5.2.3 Thermodynamic balance 10.13.5.3 Zonal Mean Meridional Circulation: Hadley Cells Large-scale ascent and descent produce expansion and compres- and Thermally Driven Jets sion, which can change temperatures from those expected with simple radiative–convective equilibrium (Section 10.13.3.4). Jet streams or jets are significant features of the circulations of The physics derives from the first law of thermodynamics, writ- planetary atmospheres, defined as zonal wind maxima, whether ten dq cpdT adp,wherea 1/r,fromeqn[12].Dividingbydt eastward or westward. The style of atmospheric circulations is ¼ À ¼ and cp and taking the limit dt 0, closely connected with jet stream structure, which varies greatly ! between planets. Jets can be single or multiple in each hemi- dT a dp Q dT0 @T g + w + sphere. They may be strong or weak, meandering in latitude or dt À c dt ¼ c ) dt @z c p p p straight. They are generally associated with transient eddies or waves whose zonal wave numbers (the number of complete grw time variable component workdone À Q @T g Q waves around a latitude circle) vary between planets from wave |fflffl{zfflffl} w |{z} + |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} [106] number one on Venus to wave number >40 on Jupiter. ¼ cp ) @z cp cp It is convenient to think of two types of jets: those driven
heat gain static stability heat gain directly by the equator-to-pole temperature gradient (thermally driven jets) and those driven indirectly by eddies (eddy-driven 1 1 Here, Q dq/dt|{z}[J kgÀ |sfflfflfflfflfflfflfflfflfflÀ ]{z isfflfflfflfflfflfflfflfflffl the} diabatic|{z} heating per kg ¼ jets), respectively (Lee and Kim, 2003). In large-scale atmo- caused by radiation, convection, or conduction. In the middle spheric dynamics, an eddy means a departure from a steady part of eqn [106], we used eqn [101] to define T0 as the and uniform east–west flow, which results in variations in horizontal (x, y) and time variable component of temperature pressure with longitude. We consider thermally driven jets first. and T as a horizontal and time average temperature. Since the The subtropical jet that occurs in the Earth’s atmosphere at term in T’ is usually small, we neglect it for the final expression, 30–40 latitude is thermally driven and associated with the which is a thermodynamic energy equation. This describes descending branch of a large overturning of the tropical atmo- 1 1 a rate of temperature change (K sÀ or K dayÀ ) for a parcel sphere called the Hadley circulation. This circulation is a large of air with vertical velocity and diabatic heating. We can loop of airflow where air warms and rises in the tropics, cools replace the diabatic heating term with a Newtonian cooling and sinks at higher latitudes, and returns to the tropics near approximation, in which temperature relaxes back to an equi- the surface to complete the loop, all under the influence of librium reference temperature field described by Teqm as planetary rotation (Figure 11). George Hadley (1685–1786) Q/cp aN(T Teq), where aN 1/trad where trad is a radiative ¼À À ¼ invented the concept to explain tropical trade winds (Hadley, time constant. Recognizing static stability (eqn [17])in 1735). Equatorial-moving air is deflected by the Coriolis force eqn [106], we can write thermodynamic balance as toward the west and produces persistent westward tropical winds. In the subtropics, sinking causes high-pressure zones w Ga G aN T Teqm [107] ðÞÀ ¼À À of dry air that also explain deserts such as the Sahara. This equation indicates that heatingÀÁ or cooling from vertical Hadley imagined a circulation symmetrical about the equa- motion is balanced by radiative relaxation back to an tor, but we know now that the Earth’s Hadley circulation is
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150 0.01
180
160
0.1 180 170
170 150 160 160 150 170
(mbar) Pressure 1 190 200 210 220 230 240
10
−90 −60 −30 0 30 60 90 Latitude
Figure 10 Contours of the zonal mean meridional temperatures (in K) at northern hemisphere autumnal equinox on Mars derived from measurements by the Thermal Emission Spectrometer (TES) instrument on NASA’s Mars Global Surveyor orbiter. The solid line at the lower boundary of the temperature contours is the zonal mean atmospheric pressure at the surface. Adapted from Smith MD, Pearl JC, Conrath BJ, and Christensen PR (2001) Thermal Emission Spectrometer results: Mars atmospheric thermal structure and aerosol distribution. Journal of Geophysical Research 106: 23929–23945 with permission.
North where low-level southwestward trade winds from the northern Pole hemisphere converge with northwestward trade winds from the south. The ITCZ is generally located somewhat off the
L equator (Figure 12). The descending branches of the Hadley L Tropospheric circulation are also not uniform in longitude. Subsidence takes J subtropical jet place preferentially in high-pressure areas over the subtropical H H H Atlantic, Pacific, and Indian Oceans. Monsoons are associated with the Hadley circulation and Trade Hadley exist on Earth and Mars. A monsoon is a seasonal reversal of winds cells winds associated with latitudinal movement of the ITCZ, and
the monsoon behaves as an intensification of the Hadley circu-
H H H lation in particular longitude ranges (e.g., Schneider (2006)). J On Earth, the ITCZ is offset from the equator because of plan- L L etary obliquity and the distribution of oceans and continents. Surface winds curve across the equator and turn from westward (a) to eastward due to the Coriolis force and become a monsoon
Figure 11 Idealized symmetrical Hadley cells on Earth. Moist air rises circulation. The Martian orbit has a relatively high eccentricity near the equator tropics causing a belt of clouds. The air aloft travels with more incoming solar radiation near the southern summer poleward, acquiring a fast zonal speed by conservation of angular winter solstice, and there is asymmetrical topography between momentum. This results in a subtropical jet near the descending branch of the Hadley cell. At the surface, air returns to complete the loop, causing hemispheres, causing the Hadley circulation to be stronger than during southern winter solstice (Haberle et al., 1993b; westward trade winds in the tropics. In each hemisphere, a different circulation is found beyond the maximum latitudinal extend of the Hadley Richardson and Wilson, 2002). The lower Hadley branch can circulation. In midlatitudes, there are waves that are found in the eastward be channeled by large-scale topography to form monsoon-like winds. (Adapted from Wallace JM and Hobbs PV (2006) Atmospheric circulations ( Joshi et al., 1994, 1995). Science: An Introductory Survey, 2nd edn. Burlington, MA: Elsevier A key physical concept behind the Hadley circulation is Academic Press, International geophysics series. Used with permission.) conservation of angular momentum. In the simplest model,
one assumes that angular momentum per unit mass is conserved in the upper branch of the Hadley circulation. At the equator, symmetrical in neither latitude nor longitude (Cook, 2004; assuming no zonal wind, the angular momentum per unit mass Dima and Wallace, 2003; Lindzen and Hou, 1988). Rising air is Oa2,wherea is the planetary radius. If angular momentum per occurs as relatively small-scale convection associated with unit mass is conserved in transport of air parcels from the equa- heavy rainfall in the intertropical convergence zone (ITCZ) tor to latitude f, a zonal wind u is acquired, as follows:
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90 ЊN
60 ЊN
July 30 ЊN
0Њ
January 30 ЊS
60 ЊS
(b) (a) 90 ЊS 180 ЊS 120 ЊS60ЊS0ЊS60ЊN 120 ЊN 180 ЊN
Figure 12 (a) The intertropical convergence zone (ITCZ) of the trade winds is visible on Earth as a band of cloud across the eastern Pacific from west to east (Courtesy: NOAA/NASA GOES project). (b) The latitude of the ITCZ changes seasonally and the annual average latitude of the ITCZ is north of the equator unlike an idealized symmetrical Hadley circulation.
W the first equality, and we solve for u. Imagine air in the upper Angular momentum at latitude f branch of the Hadley circulation carried from the equator, =( a cos + u) a cos W f f where u 0, to latitude 30. Equation [109] predicts ¼ 1 u 134 m sÀ at 30 Thus, we expect strong subtropical east- ¼ ward winds at altitude.
a cos f The subtropical jet core in the upper branch of the 1 Earth’s Hadley circulation seasonally reaches 30–70 m sÀ u –nowhereashighaspredictedintheprecedingtextbecause a eddies from the midlatitudes provide westward angular f momentum that decelerates the upper branch of the Hadley
cell (Schneider, 2006). Of course, friction causes slow sur- face winds, so that winds must increase with altitude near the jet. We can look at the thermal wind eqn [104] in ‘reverse’: a large wind shear must have a correspondingly big meridional temperature gradient @T/@y in the vicinity
of the jet, unlike the smooth variation of equilibrium tem-
perature with latitude that we might expect. Large-scale ascent is associated with T
Oa2 Oacosf + u acosf ucosf models suggest that midlatitude eddies confine the Hadley ¼ ðÞ) ang:momentum@equator circulation. Possibly, the Hadley circulation winds and tem- angular momentum@latitude f peratures become unstable, which prevents penetration into |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} sin 2f midlatitudes. Energy carried from the tropics is transferred a 1 cos 2 u a [109] O f O from the Hadley cells to eddies and waves characteristic of ¼ À ) ¼ ðÞcosf ÀÁ midlatitudes. Using such ideas, Held (2000) derived a weaker 2 sin f 0.5 0.5 dependence on rotation of fH (g /(Oa)) . Earth-like 1 A point on the planet’s|fflfflfflfflfflfflfflffl{z surfacefflfflfflfflfflfflfflffl} at latitude f moves with general circulation model simulations confirm that the Hadley tangential velocity Oa cos f, to which we add the zonal wind circulation contracts in latitude range with increasing rotation speed, u (Figure 13). The radius to the planetary rotation axis is rate (Del Genio and Suozzo, 1987; Navarra and Boccaletti, a cos f, so that angular momentum conservation is given by 2002; Showman et al., 2013). With slow rotation, if the
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Ro≳10, a planet behaves as an ‘all tropics’ world because Hadley cells extend to high latitudes (Showman et al., 2013). 50 As slowly rotating bodies, Venus and Titan should have Hadley circulations extending to high latitude, but relatively 0 2 little evidence proves this. On Venus, 20 W mÀ of sunlight reaches the surface (Titov et al., 2007), so the Hadley circula- Latitude (deg) −50 tion must be weak. Surface wind markings from Magellan radar have an equatorward direction at higher latitudes and (a) −100 −50 0 50 100 150 200 an east–west direction nearer the equator consistent with a large-scale Hadley circulation (Greeley et al., 1997). Venus 2 absorbs 100 W mÀ near its cloud tops ( 60–70 km), which 50 should drive a second Hadley-like circulation above the clouds
(Baker and Leovy, 1987). A strong temperature gradient 0 and weak maximum in the zonal wind speed at 55 latitude in both hemispheres indicates this circulation’s probable Latitude (deg) −50 extent. On Titan, low-latitude dunes indicate eastward flow, opposite to the expected trade winds ( Jaumann et al., 2009). (b) −100 0 100 200 300 400 500 Possibly, the dunes form during periods when equatorial zonal winds reverse and strengthen (Tokano, 2009, 2010) (see Chapter 10.12). 50
0 10.13.5.4 Eddy-Driven Jet Streams
In terrestrial midlatitudes, we find an eddy-driven polar jet Latitude (deg) −50 stream associated with systematically structured eddies super- imposed on the zonal mean flow. A familiar example of such (c) −100 0 100 200 300 eddies are traveling cyclones and anticyclones. There is also a stationary wave structure due to the distribution of oceans and continents. On the giant planets, multiple jets are probably eddy- 50 driven also (Ingersoll, 1990; Williams, 1978)(Figure 14). On Mars and Venus, the single, primarily thermally driven jet in 0 each hemisphere is also altered by large-scale stationary and
transient eddies. Latitude (deg) −50 To understand eddy-driven jets, we introduce concepts of rotation in fluid dynamics. Rather than quantify spin in terms (d) −400 −200 0 200 400 of velocity and distance to a center of rotation, vorticity is used, Eastward wind speed (m s−1) a single quantity that describes the spin of a fluid element at a point, equivalent to the curl of the wind vector, curl v v. Figure 14 Jet streams on the giant planets are shown in zonal mean ¼r In 2-D flows, ignoring the velocity dependence on altitude and zonal winds from cloud tracking on (a) Jupiter, (b) Saturn, (c) Uranus, and (d) Neptune. The peaks, whether eastward or westward, correspond the vertical velocity, relative vorticity z is a measure of spin to jets. Jupiter and Saturn have 20 east–west jets, with a strong about a vertical axis on a planet’s surface: prograde jet over the equator, while Uranus and Neptune appear to have 3 broad jets, with strong retrograde jet over the equator. Adapted from @ @ @u @v z h vh , u, v z + [110] Showman AP, Cho JYK, and Menou K (2010) Atmospheric circulation of ¼r  ¼ @x @y  ðÞ)¼À@y @x extrasolar planets. In: Seager S (ed.) Exoplanets, pp. 471–516. Tucson, AZ: University of Arizona Press. The sign of z follows a right-hand rule: with fingers in the direction of curved flow, ‘thumb up’ is positive. Vorticity mea- sures both change of velocity transverse to the flow (horizontal An element of fluid has a component of the planet’s spin as shear) and curvature of the flow. In solid-body rotation, well as any relative vorticity. The solid planet has a vorticity of
2O, and its vertical component of 2Osinf at latitude f is the relative vorticity is 2o (where o is angular velocity): a shear contribution of the change in tangential velocity with radius, planetary vorticity, equivalent to the Coriolis parameter, f @V/@r @(o r)/@r o, adds to curvature from the change in (eqn [100]). If we add relative vorticity (z) and planetary ¼ ¼ the tangential velocity with angle, V(@(angle)/@(path)) vorticity (f ), we obtain the absolute vorticity : ¼ V(2p/2pr) V/r o. Cyclonic relative vorticity is always in ¼ ¼ @u @v the same direction as the component of the planet’s rotation, z + f + + f [111] ¼ ¼À@y @x and a cyclonic vortex is always a low-pressure zone. In the Earth’s northern hemisphere, cyclonic flow is counterclockwise In adiabatic, inviscid flow in a thin homogeneous layer but would be clockwise in the southern hemisphere. Anticy- absolute vorticity, , is conserved, that is, stays constant with clonic relative vorticity is in the opposite direction to the time, if there is no friction or stirring. We denote the latter by a component of planetary rotation. vorticity forcing term S:
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d The result means the following: a poleward vorticity flux by a no friction or stirring 0, ðÞ dt ¼ eddies that increases the average vorticity [] in the domical
d region poleward of f1 also increases the mean zonal wind b with friction or stirring S [112] ðÞ dt ¼ u at f1. Conversely, if eddies drift poleward to another latitude f2 Here, the substantial derivative is d/dt @/@t +u@/@x+v@/@y. ¼ where there is dissipation, then S is negative. At latitude f2, Equation [112](b) can be used to understand how jets are 0 0 the eddy vorticity flux will be equatorward (negative), and a forced by eddies. A common technique in fluid dynamics is line- retrograde jet will be forced via eqn [115]. Thus, if eddies prop- arization in which variables are represented by mean values and agate north or south, prograde and retrograde jets can be gener- perturbations and inserted into equations; then certain terms are ated and multiple jets could form on rapidly rotating planets. eliminated to obtain equations that can be solved analytically. Another way to look at eddy-driven jets is to consider the Consider a uniform zonal mean flow and an eddy fluctuation north–south flux of momentum from the eddies. The north- deviation that is primed, u u + u0,alongwithasmallmeridional 2 1 ¼ ward eddy momentum across a latitude circle is rv’ (kg m sÀ ) perturbation v v’. Primed variables represent parts that vary in ¼ so that the poleward flux of eastward eddy momentum is longitude, associated with eddies. The average of the product of fluctuation components, for example, u v ,isthecovarianceandis u rv dA where the overbar is a zonal average and dA is an 0 0 0ðÞ0 nonzero only when u’andv’ are correlated, that is, in phase. Thus, ð element of area of an imaginary great wall around the latitude applying a perturbation to eqn [112](b), we have circle. If the northward eddy wind and eastward eddy wind are
@ @ @ correlated, u0v0 > 0, and there is northward flux of zonal + u + u0 + v0 + 0 S0 @t ðÞ@x @y ðÞ¼ momentum per unit mass. Negative u0v0 corresponds to a southward flux. From vector addition of the primed velocities, @ @ @ + u 0 + v0 S 0 [113] eddy wind component covariance corresponds to a tendency ) @t @x @y ¼ for tilted eddies. Where u0v0 > 0, there is SW to NE tilt of the Here, has no dependence on t or x, and we eliminate small eddies. Conversely, u0v0 < 0 corresponds to systematic NW to terms. If we multiply eqn [113] by ’, take the zonal average, SE tilt. So, air parcels traveling along a SW–NE trajectory and and delete terms with a zonal mean of zero, we get an then back feed momentum to an eastward jet to the north.
More generally, if the shear of the zonal wind along a meridian, ' ' du=dy, and u0v0 have the same sign, eddies accelerate jets. ' + u ' + v' ' = S ' ' v' ' t x y Eddies are said to transport momentum upgradient (Held, poleward vorticity 1975; Thompson, 1971). This appears to be true on Jupiter 1 ( ')2 transport by eddies 2 t 114 (Salyk et al., 2006). On Earth, we find a departure from the 2 ½ ( ') / t S ' ' symmetrical sinusoid form of waves and a systematic eastward = + 2 / y / y tilt in eddies in midlatitudes so that southward motions have a smaller eastward component than northward motions (Starr, vorticity variance term eddy forcing term 1948), which causes net angular momentum transport
Examine eqn [114]. Because f increases with latitude and @f/@y (Figure 15). is positive, the meridional gradient of absolute vorticity @=@y is generally >0. So, the generation of vorticity variance by eddy 10.13.5.5 Planetary Waves and Barotropic and Baroclinic stirring is associated with poleward flux of vorticity across Instabilities some latitude circle f1 by the eddies represented by v00 at Jets meander in latitude because of waves in the circulation. latitude f1. This increases vorticity in the whole area of the The most important class of waves in this regard are Rossby planet poleward of f1. waves, which are oscillations that arise primarily in the hori- It turns out that an increase of vorticity above latitude f1 can zontal because conservation of absolute vorticity acts to restore drive a prograde jet at f1 (here, the word prograde means in the same direction as the rotation of the planet, e.g., eastward for Earth; flows displaced meridionally to the latitude of origin. Consider retrograde means the opposite direction). The vorticity (rotation at an air parcel moving eastward with zero relative vorticity (z 0) that is perturbed northward, perhaps because of flow apoint)isrelatedtothecirculation(rotationoveranarea)around ¼ over topography. Planetary vorticity f increases northward aclosedcurveenclosingthatareabyStokes’theorem.Writing down this theorem and applying it to the whole domical region
(a spherical cap) north of latitude f1,wehave Cold polar air, slow eastward momentum
G V ds dA C ¼ s ¼ L L þ ððarea A circulation m2s 1 velocityalong arc ds ½À H 2 acos u |{z} p |{z}f1 ) 2pðÞp=2 Warm air, fast eastward momentum a2 cosfdfdl ¼ l 0 f f ¼ ¼ 1 Figure 15 Tilts in the eastward winds cause momentum to be ð ð p=2 2p a2 cosfdf [115] transported from the equator to poles. (These tilts were also drawn on ¼ ½ f f Figure 11 as dashed lines). (Drawn by the author.) ð ¼ 1
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(eqn [100]), so relative vorticity z must become negative to Here, k 2p/lx and n 2p/ly are zonal and meridional wave ¼ ¼ keep absolute vorticity constant ( z+f ). Remembering the numbers, where lx and ly are zonal and meridional wave- ¼ right-hand grip rule, negative z is anticyclonic curvature (clock- lengths, respectively. In the simple case where n 0, the ¼ wise in the Earth’s northern hemisphere), which sends the waves propagate eastward relative to the planet and westward parcel southward, back to its initial latitude (Figure 16). relative to the zonal wind, at phase speeds that increase with Then, an overshoot beyond the initial latitude will induce increasing wavelength or smaller k. Setting n 0 and c 0in ¼ ¼ cyclonic curvature and send the parcel back northward. The eqn [117] gives k2 b=u for stationary waves with a west–east ¼ 0:5 air parcel sinusoidally oscillates as it moves downstream mak- wavelength of lx 2p u=b . For example, in the Earth’s mid- ¼ ðÞ 11 1 1 ing a Rossby wave. Meanders in the Earth’s midlatitude jet dle troposphere at 45N, b 1.6 10À mÀ sÀ , the station- Â stream with typical wave numbers of 4 to 6 are guided by ary wavelength is 7000 km if u 20ms 1, and four such ¼ À Rossby waves. Low-pressure cyclones tend to form in the wavelengths would encircle the Earth. troughs of the waves, and high-pressure anticyclones tend to Strictly horizontal flows with no temperature variations form in the crests. Similar Rossby waves have been observed on (parallel isobars and isotherms) are called barotropic and Mars in midlatitudes in spring and winter with wave numbers such flows are stable when b @2u=@y2"0. However, if the À 1–2 (Hollingsworth and Barnes, 1996; Read and Lewis, 2003; second term – the curvature of the zonal wind profile with Wilson et al., 2002). latitude – becomes bigger than b, then instability is produced The restoring effect due to the gradient in planetary vorticity called the barotropic instability or Rayleigh–Kuo instability.
(f ) is called the beta effect where beta b @f/@y is the merid- Then, growing unstable eddies derive their energy from the ¼ ¼ ional gradient of planetary vorticity. If we assume a zonal mean KE of the zonal mean flow. In prograde jets, the zonal wind zonal wind and differentiate absolute vorticity (eqn [111]) decreases to the right of the flow where @2u=@y2 can be large. w.r.t. y, we get Barotropic instability produces unstable eddies that propagate to latitudes where they decay and accelerate retrograde zonal @ @2u @f @ 2Osinf b , where b ðÞ mean flow. Numerical simulations of jets forced by the eddies @y ¼ À @y2 @y ¼ @y indicate a meridional length scale associated with this instabil- ity, characteristic of the width of the jets, called the Rhines dy adf, spherical geometry ¼ 1/2 radius, rRh (2U/b) , where U is the typical speed of the jet 1 df 2Ocosf |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} [116] (Rhines, 1975). On giant planets, vortices stop growing at this ¼ a df ¼ a size in the meridional directional and stretch out east–west,
Usually, b is much larger than @2u=@y2 so @=@y b À around a circle of latitude. Thus, the number of bands on the provides the restoring effect. face of a gas giant is roughly a semicircle meridian divided by It is possible to derive sinusoidal functions in x, y, and t the Rhines radius, (pa)/rRh (Menou et al., 2003). À that represent Rossby waves. The procedure is to rewrite the In real atmospheres, we must consider vertical variations. vorticity eqn [113] in terms of a single variable, a stream General flows with vertical wind variations, nonzero vertical function, c, which is a scalar field for a fluid flow with contour velocity, and horizontal temperature gradients (thus, mutually lines that are streamlines of the flow, defined as u @c/@ ¼À inclined isobars and isotherms) are called ‘baroclinic.’ Instead y and v @c/@x (e.g., see Holton (2004), pp. 213–217). ¼À of vorticity, an analogous quantity, potential vorticity (PV), is The Rossby wave solutions in c are found to have a dispersion introduced to allow for the vertical stretching or shrinking of relation, meaning how the eastward phase speed c of the fluid columns. Shrinking of the radius of a column of spinning
Rossby wave varies with wavelength, as follows: fluid increases the spin rate, like an ice skater drawing in their
arms. PV is conserved for a fluid element in the absence of o b c u mixing with other air parcels or frictional loss of KE and, in a ¼ k ¼ À k2 + n2 ðÞ constant density fluid, can be defined as phase speed relative to planet’s surface [117] ¼ f + z 1 1 P ðÞmÀ sÀ [118] v ¼ h ¼ h ÀÁ where h is the height of a short segment of a fluid column (e.g., z < 0 z < 0 Wallace and Hobbs, 2006, p. 289). Here, the word potential crops up again. In potential temperature (Section 10.13.2.4), Latitude, f1 Relative vorticity, z =0 we had a quantity remaining the same in adiabatic expansion,
although the actual temperature changed. Potential vorticity is
conserved for a fluid element, but the element’s relative vortic- z > 0 ity changes with changing latitude or column height. When density r varies with height, a more general version of PV is Figure 16 Illustration of the mechanism of Rossby waves in the x-y Ertel’s potential vorticity QP: plane: a restoring effect caused by the gradient of absolute vorticity. A fluid parcel in the northern hemisphere moves eastward through the @y QP [119] wave pattern. As it crosses latitude f1, the relative vorticity z 0. @z ¼ ¼ r North of f1, conservation of absolute vorticity requires z< 0, so the trajectory curves to the south anticyclonically. When it shoots over where y is potential temperature and the sign of @y/@z indicates latitude f1, to conserve absolute vorticity z > 0 and the trajectory static stability (Section 10.13.2.4). In adiabatic flow, if static curves northward cyclonically. (Drawn by the author.) stability @y/@z increases, shrinking and horizontal expansion
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of fluid columns causes the absolute vorticity n to decrease rapidly rotating rocky planets at the appropriate latitudes and according to eqn [119],ifQ stays constant. The converse occurs on Mars, making associated baroclinic waves (Banfield P also applies. Since QP involves potential temperature, it can et al., 2004; Collins and James, 1995; Hinson et al., 2012; Read be generated and dissipated by diabatic heating, stirring, or and Lewis, 2003; Wilson et al., 2002). The net thermal effect of friction. 3-D baroclinic eddies is to transport warm low-latitude air Ertel’s PV can be calculated from temperature and pressure poleward and up and cold high-latitude air equatorward and fields, but analytic approaches often use quasigeostrophic (QG) down. For this reason, the process has been called stantwise (or potential vorticity qg, which is an approximate form of PV valid sloping) convection (Hide and Mason, 1975). As described in when the Ro is small and flows are approximately geostrophic. Section 10.13.5.4, eddies can drive jet streams. However, bar- Equations based around QGPV are helpful in studying large-scale oclinic instability should be less important on slowly rotating motions characteristic of atmospheric circulations. An equation planets with small horizontal temperature gradients. analogous to eqn [113] can be derived but with absolute vorticity The horizontal length scale of the most unstable eddies
replaced by QGPV, qg, which again can be written in terms of a produced by baroclinic instability is the Rossby radius of single stream function and solved to give 3-D planetary wave deformation: solutions (e.g., see Andrews, 2010). The key difference is that NBH such waves have an amplitude perturbation c’ that depends on rRo [122] 1 2 ¼ f0 height. Analogous to =2 mass velocity , the wave KE den- 2 ðÞÂðÞ sity rc 0 is independent of height in the absence of dissipation. where H is the scale height, N the Brunt–Vaisala frequency, B So, for this quantity to remain constant, the amplitude grows as and f0 a reference Coriolis parameter. The Rossby radius can be 0:5 c’ r À , where r0 is the background density. Such a relation- understood as follows. Imagine air rising vertically through a 0 ship is used in photochemical models of planetary atmospheres height H and then spreading out perhaps when it reaches the where turbulence (at least to certain altitudes) is often parame- tropopause. Initially, the spreading gas will not ‘feel’ the effects 0.5 0.5 terized proportional to À .Thesame À dependence also r r of the planetary rotation, but after a certain distance, its lateral applies to the vertical amplitude growth of buoyancy waves spread will be converted to rotational motion around a core. (Section 10.13.5.7). This distance is roughly the Rossby radius. In the Earth’s midlat- 4 1 The meridional gradient of zonal mean QGPV is somewhat itude troposphere, rRo 700 km, using f0 10 sÀ , H 7km, 2 1 À analogous to eqn [116] but has an extra term that depends on and N 10 sÀ . Perhaps as a consequence, only one eddy- B the vertical shear of the zonal mean zonal wind (e.g., see Salby driven jet can fit in the Earth’s midlatitudes (Schneider and
(2012), p. 465, or Andrews (2010), p. 128 for derivation of Walker, 2006). On Jupiter, rRo [(0.003–0.02) 27]/[2 (2p/   QGPV equations): 35730) sin(45)] 325–2200km using a range of NB values  (Magalhaes et al., 2002; Shetty and Marcus, 2010), which is @ 2 2 qg @ u 1 @ f0 @u small compared to Jupiter’s vast size. b + r [120] @y ¼ À @y2 r @z 0 N2 @z 0 B
Here, f0 is the value of the Coriolis parameter f at a reference 10.13.5.6 Deep and Shallow Atmospheres on Giant Planets latitude (f f0 +by), NB is the Brunt–Va¨isa¨la¨ frequency ¼ (Section 10.13.2.4), and r0 is a height-dependent zonal and The pattern of strong eastward jets at the equator on Jupiter time-mean density, that is, rs exp( z/H), where H is density and Saturn, with alternatively eastward and westward jets at À scale height and rs a reference density. Just as barotropic insta- higher latitudes is a puzzle of atmospheric dynamics. Histori- bility was caused by a change of sign of @/@y in eqn [116],so cally, there have been two approaches to explain the jet struc- instability arises when @qs=@y changes sign. When the third ture (Vasavada and Showman, 2005): term 1=r @=@z r B@u=@z dominates, a different kind of ðÞ0 ðÞ0 instability emerges. This is baroclinic instability, which is (i) The observed wind systems are occurring in a shallow when flow is driven by the extraction of potential energy layer overlying a horizontally uniform but convective from a latitudinal temperature gradient, causing large-scale deep interior (Williams, 1978). (ii) The observed winds somehow reflect the dynamic activity eddies. If we use the barometric law r0 rs exp( z/H) and ¼ À of an active deep atmosphere, and the jets probably rep- differentiate the last term of eqn [120] w.r.t. z using the product rule, we see that this term depends on the vertical wind shear: resent very deeply rooted systems (Busse, 1976, 1983). In the first scenario, shallow-layer models can produce 1 u B u 2 u 1 u 0 eddies (Cho and Polvani, 1996; Huang and Robinson, 1998) ()0 B + 0 2 z z H z z H z 0 0 – sometimes associated with the moist convection (Lian and 121 Showman, 2010) – that force alternating zonal jets, though ½ many models produce westward jets at the equator. In the Thus, baroclinic instability is most commonly found near second scenario, circulations extend deeply into the interior. the regions of strong vertical shear of the mean zonal wind. On The deep interior opacities of giant planets should lead to 1 1 Earth, it occurs when @u=@z exceeds 1msÀ kmÀ . This nearly adiabatic and convective conditions (Nutzman et al., threshold is small and less than the wind shear predicted 2009). The jets themselves could perhaps result from the sur- from the thermal wind equation with a typical horizontal face manifestation of large circulating cells in systematic align- gradient induced by the spread of sunlight over midlatitudes. ment with the jets. Galileo probe measurements of a vertical Consequently, baroclinic instability is generally predicted on wind profile on Jupiter show nearly uniform strong zonal
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winds to great depth from 5 to 20 bar, but consistent with below the breaking altitude and winds diminish upward temperature measurements interpreted in the light of the ther- from the core of a jet. This process is important for shaping mal wind equation, these measurements show decay of wind jets in general, acts as a drag on the zonal mean flow, and strength upward in the uppermost troposphere (Atkinson makes upper atmospheres ‘feel’ the rotation of the underlying et al., 1997). This wind profile supports the deep circulation planet. Thermal wind balance (eqn [104]) applied to the effect camp, but because the profile may not be representative, it of waves closing off the jets and declining wind speed with does not prove it. The probe descended in a ‘hot spot’ of high altitude is associated with a reverse equator-to-pole tempera-
IR emission at 5 mm, which emerges from deeper, unusually ture gradient in the Earth’s stratosphere. In turn, through ther- cloud-free parts of the troposphere (Ortiz et al., 1998). modynamic balance (eqn [107]), this is connected with ascent On Uranus and Neptune (the ice giants), a broad band of and descent, where tropospheric air rises into the tropical strong retrograde winds exist at low latitudes with a single stratosphere and subsides at the poles – the Brewer–Dobson high-latitude prograde jet in each hemisphere (Figure 14). circulation (Brewer, 1949; Dobson, 1956) – which carries 1 Moreover, Neptune’s retrograde jet speed exceeds 500 m sÀ , stratospheric ozone from equator to poles. the fastest in the solar system. It is unclear why there are such An important and special class of large-scale buoyancy significant circulation differences between the gas giants and waves are atmospheric thermal tides, which are global oscilla- Uranus and Neptune. Possibly, convection is weaker on the ice tions in pressure and wind fields caused by the heating and giants and unable to drive the eddies needed for the kind of jet cooling associated with the apparent motion of the Sun over a structure seen on the giant planets or only operates to drive planet (Chapman and Lindzen, 1970). The wave periods are prograde jets at high latitudes. Perhaps, a surprising result is fractions of a solar day, so that the wave with a solar day length that the jet speeds increase going outward in the solar system, is a diurnal tide, while that with half a day length is a semidi- which is counterintuitive if one assumes that incident sunlight urnal tide. Furthermore, the tides divide into migrating tides is important. This puzzle has been called Suomi’s paradox after that move around the planet with the Sun’s apparent motion
Vern Suomi, who first pointed it out and is not resolved. and nonmigrating tides that do not, which are modes generated
by migrating tides interacting with planet-scale topography. A key theoretical result is that global-scale thermal tidal 10.13.5.7 Buoyancy Waves and Thermal Tides waves only propagate vertically if their angular frequency o Besides planetary waves, another class of atmospheric waves exceeds the Coriolis parameter, that is, o>f (Lindzen, 1971). that are very important for circulation are propagating buoy- A diurnal tide with o O cannot propagate vertically beyond ¼ ancy waves (also called internal gravity waves), which corre- 30 latitude because f 2Osin(30) O at that latitude. Thus, ¼ ¼ spond to oscillations primarily in the vertical plane when the on Earth and Mars, the diurnal tide acts as a buoyancy wave 2 square of the Brunt–Va¨isa¨la¨ frequency exceeds zero, NB >0. only between latitudes 30 where it propagates vertically and 2 Æ Given that NB has the same sign as static stability (eqn [17]), its amplitude increases with height, whereas the semidiurnal this situation is common in atmospheres. As discussed in tide is a global buoyancy wave from pole to pole with very large
Section 10.13.2.4, the restoring force of the mean vertical vertical wavelength 100 km. Migrating and nonmigrating buoyancy gradient acts on vertically displaced fluid parcels to tides have been detected on Earth and Mars, and migrating cause oscillations. It is possible to derive a wave equation for tides have been observed on Venus (Migliorini et al., 2012; buoyancy waves called the Taylor–Goldstein equation and find Pechmann and Ingersoll, 1984; Peralta et al., 2012; Schofield solutions (e.g., Holton, 2004, pp. 196–201; Nappo, 2002). In and Taylor, 1983). In the case of Venus, which rotates every the x-z plane, wave perturbations in density, vertical velocity, 243 Earth days, the thermal tides in the middle atmosphere (z/2H) i(kx ot) pressure, and potential temperature vary as e e À , reflect atmospheric rotation, clearly not the rotation of the where the factor exp(z/2H) accounts for the vertical variation solid planet. The atmosphere near the cloud tops at 65– 1 of atmospheric density, where H is the density scale height. 70 km altitude rotates at 110 m sÀ past the subsolar point, Such waves are generated mainly by flow over topography establishing waves that migrate with the Sun but move oppo- and large-scale flow associated with convective cells. Wave site to the direction of the mean flow (Gierasch et al., 1997). 1/2 amplitude varies with density as rÀ , so amplitudes grow Tides are important in atmospheres. On Mars, feedback upward as density decreases, causing some waves to break between tidal surface winds and lifting of dust high in the aloft, in crude analogy to ocean waves breaking on a beach. atmosphere may contribute to the generation of planet encir- Theory shows that buoyancy waves can also be absorbed at cling dust storms (Leovy et al., 1973), while breaking of tides lower levels, the so-called critical levels, if the phase speed of may be important for mixing atmospheric constituents. On the wave c equals the mean zonal wind, that is, c u (Forbes, Venus, tides may be implicated in superrotation, which we ¼ 2002; Nappo, 2002). Otherwise, the waves continue to prop- discuss next. agate vertically and break well above these critical levels. Thus, for example, if an eastward jet exists at altitude, buoyancy 10.13.5.8 Superrotation waves propagating upward with eastward phase speeds will be absorbed and are said to be filtered out, whereas waves Superrotation is where the equatorial zonal wind speed is with westward or near-zero relative speed (the speed of the persistently greater than the speed of the underlying rotating underlying surface) will penetrate higher, break, and dump planet, which applies to Venus (Belton et al., 1991; Peralta their westward momentum (Forbes, 2002). The result is that et al., 2007), Titan (Bird et al., 2005; Kostiuk et al., 2006), these breaking waves diminish eastward zonal winds aloft, Jupiter (Ingersoll et al., 2004), and Saturn (Del Genio et al., which is called closing off the winds – that is, a jet exists 2009). The angular momentum of a planet’s surface is maximal
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at the equator, and so, the superrotation cannot be maintained 2007; Lebonnois et al., 2010; Yamamoto and Takahashi, by some simple transport of angular momentum (Hide, 1969, 2009). The waves near the superrotating level on Venus may
1970), as, for example, occurs from equator to high latitudes in be atmospheric tides (Newman and Leovy, 1992). It is unlikely Hadley cells. Also, since viscosity should tend to dissipate high to be a coincidence that the peak winds occur near a maximum equatorial winds, superrotation is puzzling. in the vertical profile of solar heating maximum (Figure 17) Generally, it has been proposed that equatorial superrota- (Gierasch et al., 1997; Leovy, 1987; Pechmann and Ingersoll, tion might be maintained by eddies with the kind of 1984; Schofield and Taylor, 1983). However, the underlying
‘upgradient’ transport discussed in Section 10.13.5.4 (e.g., physics of equatorial superrotation remains an area of research Gierasch, 1975) (although gravitational torque acting on the (Mitchell and Vallis, 2010). atmosphere is an alternative idea (Gold and Soter, 1971; Schubert, 1983)). Eddies could carry prograde momentum systematically toward the jet region, either by motions in the 10.13.6 Outstanding Questions vertical plane with upward and zonal velocities correlated below the maximum wind level and downward and zonal Although the Space Age has greatly improved our knowledge of velocities correlated above the maximum or by motions in the atmospheres of the solar system, including the Earth, a long the horizontal plane with equatorward and prograde winds list of unsolved problems remains. Planetary atmospheres as a correlated (Schneider and Liu, 2009). If the former case whole are very undersampled, with no in situ measurements applies, buoyancy waves may be implicated; in the latter case, at all for Saturn, Uranus, and Neptune, a possibly unrepresen-
Rossby waves. Either way, generation of wave activity by some tative descent profile from Jupiter, poor knowledge of the process near the equatorial wind maximum would be required. atmosphere of Venus below the clouds, and barely any mea- Attempts to model superrotation on Venus have been made surements of winds in all extraterrestrial atmospheres. Even on with varying degrees of success, and models that work have Earth, matters of basic enquiry remain. These include, for exam- generally assumed background superrotation below the cloud ple, how the Hadley circulation interacts with eddies (Levine base (e.g., Del Genio and Zhou, 1996; Hollingsworth et al., and Schneider, 2011), how stratospheric circulation and
180 185 110 190 185 170 180 190 5 185 180 105 175 180 170 175 175 100
4 170 170 170 165 175 95
180 90 175 185 3 190 180
195 ht (km)
185 200 85 g 190 205 (Pressure) (atm) (Pressure)
Hei
10 195 200 210 80 205 − log 2 210 215 230 215 220 75 220 225 225 230 235 230 70 235 225 235 240 240 245 220 1 245 250 250 65 260 260 270 270 60
290 290 310 310 55 330 330 0 0 90 180 270 360 −10 0 10 20 30 40 50 60 70 80 90 Solar longitude Latitude
Figure 17 Cross sections of temperature (K) in the atmosphere of Venus above 55 km altitude (Schofield and Taylor, 1983). On the left, the temperature field is averaged between 0 and30N and shown in Sun-following coordinates where zero longitude is noon and 180 is the antisolar point. There is a minimum temperature of 170 K at 95 km altitude, above which the equator-to-pole temperature gradient is reversed. On the right, temperatures are shown as a zonal mean cross section. From above the cloud tops (70 km) to 95 km, a semidiurnal variation is revealed by waves in the temperature contours with a phase that moves in an eastward direction at higher altitude (left). The phase in the solar longitude plot on the left suggests a Sun-following eastward-propagating oscillation forced at lower altitudes. Gierasch PJ, Goody RM, Young RE, et al. (1997) The general circulation of the Venus atmosphere: An assessment. In: Bougher SW, Hunten DM, and Phillips RJ (eds.) Venus II, pp. 459–500. Tucson, AZ: University of Arizona Press.
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chemistry affects the troposphere (Thompson et al., 2011), and and NASA Astrobiology Institute’s Virtual Planetary Laboratory the response of cloud formation (Boucher et al., 2013) and the award under solicitation no. NNH05ZDA001C. The author stability of ice sheets to climate forcing (McNeall et al., 2011). also thanks the late Conway Leovy for helping improve his After the Earth, Mars has the most observed atmosphere. knowledge of planetary atmospheres and Tyler Robinson for a On Mars, global dust storms occur in some Martian years and helpful review. not others (Martin and Zurek, 1993). But we do not know why. What are the mechanisms for storm initiation, growth, and decay and how are these storms connected to circulation fea- References tures such as tides and planetary waves? Also, the role of clouds and surface exchange in the water cycle on Mars and its inter- Abe Y (2011) Protoatmospheres and surface environment of protoplanets. Earth, Moon, action with the dust cycle remains to be elucidated. and Planets 108: 9–14. On Venus, winds and temperature fields in the lower atmo- Andrews DG (2010) An Introduction to Atmospheric Physics, 2nd edn. New York: Cambridge University Press. sphere are essential for understanding the circulation and yet Appleby JF (1986) Radiative convective equilibrium models of Uranus and Neptune. are poorly measured or understood. Clouds play such an Icarus 65: 383–405. important role for Venus’ climate that further elucidation of Appleby JF and Hogan JS (1984) Radiative-convective equilibrium models of Jupiter their size distribution, shapes, and composition and the distri- and Saturn. Icarus 59: 336–366. bution of minor sulfur and chlorine species would also help Armstrong BH (1968) Theory of the diffusivity factor for atmospheric radiation. Journal of Quantitative Spectroscopy and Radiative Transfer 8: 1577–1599. define chemistry, climate, and dynamics. Also, many reactions Atkins PW and De Paula J (2010) Physical Chemistry, 9th edn. New York: W.H. relevant to chemistry on Venus that involve elemental sulfur Freeman. lack laboratory data. Atkinson DH, Ingersoll AP, and Seiff A (1997) Deep winds on Jupiter as measured by We do not have a fundamental understanding of the differ- the Galileo probe. Nature 388: 649–650. Atreya SK, Lorenz RD, and Waite JH (2009) Volatile origin and cycles: Nitrogen and ences between the giant-planet atmospheric circulations. There methane. In: Brown RH, Lebreton J-P, and Waite JH (eds.) Titan from Cassini- are two main puzzles. The first is the different jet structures on Huygens, pp. 77–99. New York: Springer. Uranus and Neptune compared with those on Jupiter and Atreya SK, Mahaffy PR, Niemann HB, Wong MH, and Owen TC (2003) Composition and Saturn. The latter have several jets in each hemisphere origin of the atmosphere of Jupiter – An update, and implications for the extrasolar (Jupiter has 5–6 eastward jets and Saturn has 4), whereas the giant planets. Planetary and Space Science 51: 105–112. Baker NL and Leovy CB (1987) Zonal winds near Venus cloud top level: A model study ice giants have only one jet (Figure 14). Whereas the equatorial of the interaction between the zonal mean circulation and the semidiurnal tide. jets are retrograde on the ice giants, they are prograde on the Icarus 69: 202–220.
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Treatise on Geophysics, 2nd edition, (2015), vol. 10, pp. 429-472