4. A Brief Review of Thermodynamics, Part 2

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1 ATMOSPHERE OCEAN INTERACTIONS :: LECTURE NOTES 4. A Brief Review of Thermodynamics, Part 2 J. S. Wright jswright@tsinghua.edu.cn 4.1 OVERVIEW This chater continues our review of the key thermodynamics concets and formulae needed for understanding the atmoshere ocean system. The vertical rofiles of temerature, ressure, and density in the atmoshere are revisited. The thermodynamics of dry atmosheres are resented, including the concets of buoyancy and stability and the nature of convection in dry atmosheres. The role of water in the atmoshere is described and the fundamental atmosheric thermodynamics are extended to moist atmosheres. We conclude with a brief look at radiative-convective equilibrium. 4.2 THERMODYNAMICS OF DRY ATMOSPHERES As fluids, the ocean and atmoshere share many of the same basic features, even though the roerties of the atmoshere and ocean are vastly different. The total mass kg) and secific heat 1004 J K 1 kg 1 ) of the atmoshere are much smaller than those of the ocean kg and 3994 J K 1 kg 1, resectively). Accordingly, the thermal inertia of the atmoshere is much smaller. The atmoshere resonds very quickly to changes in surface temerature, and its role in climate stability is rimarily radiative through the greenhouse effect) rather than thermal. Desite these fundamental differences, the vertical stability of the atmoshere can be understood in much the same context as that of the ocean VERTICAL STABILITY Assuming that it is well-mixed, the stability of a dry atmoshere can be understood in terms of the vertical variation of otential temerature alone. Suose a arcel of air is raised adiabatically from its initial osition with = old and θ = θ old to a new osition with environmental 1

2 Tz) θz) z 2 z 2 Height z 1 z 1 Temerature T 1 T 2 θ 1 θ 2 Potential temerature Figure 4.1: Schematic illustrations of dry convective adjustment to surface sensible heating relative to left) the environmental temerature rofile and right) the environmental otential temerature rofile. = new and θ = θ new. Potential temerature is conserved during adiabatic motions, so that we can rearrange the atmosheric equation of state to find the density of the arcel: ρ arcel = new R d θ old new 0 ) Rd /c 4.1) where 0 is the reference ressure as defined in section 3.3. The density of the environment is ρ env = new R d θ new new 0 ) Rd /c 4.2) The buoyancy of the uwardly dislaced arcel will be negative if ρ arcel > ρ env, which is only true if θ new > θ old. In a dry atmoshere, stability may therefore be defined in terms of the vertical gradient of otential temerature: the atmoshere is stable if otential temerature increases uward, neutrally stable if the change in otential temerature is zero, and unstable if otential temerature decreases uward. These features can also be exressed in terms of temerature. The atmoshere is unstable if the lase rate Eq. 2.8) is greater than the adiabatic lase rate, neutrally stable if the two are equal, and stable otherwise. The mean observed lase rate is aroximately 6.5 K km 1, while the mean adiabatic lase rate is 9.8 K km 1. The global mean atmoshere is therefore stable to dry convection. The diurnal cycle of solar radiation can create instabilities in the lower atmoshere by warming the surface, which then warms the atmoshere in contact with the surface. As an air arcel in contact with the surface warms, it becomes less dense than the air above it and begins to rise. As the air arcel rises it exands adiabatically, so that the temerature decreases 2

3 Tz) Tz) Height Height Temerature Temerature Figure 4.2: Schematic illustrations of the creation of temerature inversions by left) nighttime surface cooling and right) adiabatic warming during sinking from aloft and adiabatic cooling during rising from below. at the adiabatic lase rate. The air arcel continues to rise until its density is equal to the surrounding environmental density. The atmosheric equation of state shows that this occurs when the temerature of the arcel is equal to the environmental temerature, or where the adiabatic lase rate intersects with the mean environmental lase rate Fig. 4.1). This rocess tyically results in a well-mixed daytime surface boundary layer. The deth of this boundary layer deends on the environmental lase rate and the extent of surface warming. The oosite of this rocess occurs during calm winter nights, when radiative cooling of the surface leads to heat transfer from the atmoshere to the surface, gradually cooling the lowermost atmoshere. This rocess can create temerature inversion i.e., an increase in temerature with height, with Γ < 0; see Fig. 4.2). The atmoshere near the surface is then very stable, which allows ollutants to build u in the surface layer. Inversions are also common in the subtroics, where air descending from uer levels warms adiabatically. This adiabatic warming can create a semi-ermanent inversion where the descending air meets and mixes with air affected by conditions at the surface. Inversions may also be formed or intensified by the resence of nearby hills or mountains, which transmit the effects of daytime surface warming and nighttime surface cooling to higher levels in the atmoshere. This situation affects many cities, including Mexico City, Los Angeles, and Beijing. Inversions create a thermodynamic barrier that inhibits ventilation of the surface layer and allows ollution to build u near the surface. The buoyancy frequency for the atmoshere can be defined in terms of otential temerature: 3

4 ) g θ N = θ z Buoyancy oscillations gravity waves) are ossible when the otential temerature increases with altitude stable conditions). Conversely, dry) convective instability results when the otential temerature decreases with altitude. One common tye of buoyancy oscillation in the atmoshere occurs when air flows over a mountain. The air is adiabatically lifted by the mountain, creating an oscillating density erturbation lee wave) that can extend as far as 100 km downstream. 4.3) 4.3 THERMODYNAMICS OF MOIST ATMOSPHERES Section 4.2 considers the thermodynamics and stability of a dry, well-mixed atmoshere. The major constituents of the atmoshere are indeed well-mixed Table 1.2); water vaor, by contrast, is not. The fractional amount of water vaor by volume in the trooshere varies from aroximately arts er million in the troical boundary layer) to aroximately one art er million near the troical trooause). Stability is not urely a function of otential temerature for an inhomogeneous atmoshere because the gas constant and secific heat vary by location. The molecular weight of water vaor is aroximately 60% of the mean molecular weight of dry air, so that air becomes less dense when its concentration of water vaor increases. Water can also change hase at atmosheric temeratures and ressures Fig. 4.3). As mentioned in section 3.4.1, water has a very high latent heat of vaorization. The resence of water in all its forms also creates satial variations in the secific heat and the absortion and/or reflection of radiation. Even though the amount of water in the atmoshere is small, its hase changes nonetheless have substantial effects on the energy budget and vertical stability of the atmoshere ATMOSPHERIC HUMIDITY The water vaor content of the atmoshere can be exressed in a variety of ways. The most fundamental is the vaor ressure e = ρ v R v T 4.4) where ρ v is the density of water vaor and R v is its gas constant J K 1 kg 1 ). Equation 4.4 is derived by alying the ideal gas law to water vaor. For a condensing substance such as water vaor on Earth), the maximum ossible vaor ressure is determined by temerature. The variation of this saturation vaor ressure with temerature can be calculated using the Clausius Claeyron equation: de dt = 1 T L v ρ 1 v ρ 1 c where L v is the latent heat of vaorization, ρ v is the density of the vaor hase, and ρ c is the density of the condensed hase liquid or ice). This equation is derived from fundamental 4.5) 4

5 10 5 Vaor ressure [Pa] ice suercooled liquid liquid Trile oint: 0.01 C, Pa 10 0 water vaor Temerature [ C] Figure 4.3: Phase diagram for water at a range of temeratures observed in the atmoshere. The y-axis shows the vaor ressure e Eq. 4.4). Equilibrium curves have been calculated using Eqs. 4.7 and 4.8. thermodynamic rinciles by calculating the work done in a reversible cycle of exansion and contraction across the condensation threshold see, e.g., Curry and Webster, 1999). Equation 4.5 can be simlified by alying the ideal gas law for water vaor and assuming that condensate is removed immediately: de dt = L ve R v T 2, 4.6) but it is often aroximated using emirical relationshis. Two emirical aroximations to Eq. 4.5 that are accurate at atmosheric temeratures are given by e = ex ) lnt ) 4.7) T e # = ex ) lnt ), 4.8) T where e is the saturation vaor ressure with resect to liquid water and e # is the saturation vaor ressure with resect to ice Emanuel, 1994). Direct measurements indicate that Equation 4.7 is accurate to within 0.006% between 0 C and 40 C and to within 0.3% for equilibrium between vaor and suercooled water down to 30 C), while Equation 4.8 is accurate to within 0.14% between 80 C and 0 C. 5

6 Pressure [hpa] S 60 S 30 S 0 30 N 60 N 90 N Zonal mean relative humidity Pressure [hpa] Zonal mean secific humidity S 60 S 30 S 0 30 N 60 N 90 N Relative humidity [%] Secific humidity [g kg 1 ] Figure 4.4: Zonal mean distributions of relative humidity RH; left) and secific humidity q; right) in the atmoshere. Data from the Jaanese 55-year Reanalysis. Vaor ressure is not the most hysically intuitive reresentation of water vaor content, and several other reresentations are used in a variety of contexts. For examle, relative humidity is defined as the ratio of the vaor ressure to the saturation vaor ressure RH = e e 4.9) so that RH = 1 when e = e i.e., when the air is saturated). Relative humidity can also be calculated with resect to ice: RH i = e e # 4.10) The saturation vaor ressure over ice is uniformly less than the saturation vaor ressure over liquid water Fig. 4.3), so that relative humidity with resect to ice is always greater than relative humidity with resect to liquid water. The distribution of relative humidity in the atmoshere is shown in the left anel of Fig The mass mixing ratio r is defined as the ratio of the mass of water vaor to the mass of dry air i.e., ρv /ρ d). Using the ideal gas law, r can be related to e: r = e /R v T d/rd T = R d R v e e = ɛ e e 4.11) where ɛ is the ratio of the mean molecular weight of water vaor to the mean molecular weight of dry air Rd /R v = Mv /M d /28.97 = 0.622). Observations of water vaor are often reorted as mass mixing ratios in units of g kg 1. The secific humidity q is defined as the ratio of the mass of water vaor to the total mass of the arcel i.e., ρv /ρ v + ρ d )), and can be calculated as e q = ɛ 1 ɛ)e = r 1 + r 4.12) 6

7 Since r and q are both generally less than 0.04 in the atmoshere, r q. The distribution of secific humidity in the atmoshere is shown in the right anel of Fig THERMODYNAMIC EFFECTS OF WATER VAPOR The resence of water vaor modifies the secific heat and the gas constant of air. Consider again our exression of the first law of thermodynamics in Eq. δq = c dt + ρ 1 d). The change in heat content δq in a moist but unsaturated air arcel at constant ressure can be exressed as m d + m v )dq = c d m d + c v m v )dt, 4.13) where m d and m v are the masses of dry air and water vaor inside the arcel, resectively, c d 1004 J K 1 kg 1 is the secific heat of dry air at constant ressure and c v 1870 J K 1 kg 1 is the secific heat of water vaor at constant ressure. The secific heat is defined as the change in heat content for each 1 K change in temerature: ) Q T = c d m d + c v m v m d + m v = c d + c v mv /md) 1 + mv /m d) = c d + c v r 1 + r where r is in units of kg kg 1. We can simlify this exression even further by using the fact that r is small throughout the atmoshere: c m = c d 1 + c v /cd)r 1 + r ) [ )] cv c d 1 + r 1. c d Here, c m reresents the secific heat at constant ressure for moist air. Substituting aroximate values for c d and c v, we get c m c d r ). We can use a similar aroach and c vv 1410 J K 1 kg 1 ) to show that c vm c vd r ). The density of an unsaturated arcel is also affected by the resence of water vaor. Using the ideal gas law and the definition of r, ρ = ρ d + ρ v = d R d T + e R v T = d R d T 1 + r ) = R d T We can exress this in the familiar form of the ideal gas law: ρ = d ɛ d + e) 1 + r ) =, 1 + r. R d T 1 + r /ɛ R m T, R m R d 1 + r /ɛ 1 + r. 4.14) The arameter ɛ is less than 1, therefore R m > R d. This means that the density of a moist air arcel is less than the density of a dry air arcel at the same ressure and temerature. For adiabatic rocesses, ds = c dt T R d = 0, so that and the otential temerature of moist unsaturated air is dlnt ) = R m c m dln ) 4.15) 7

8 ) Rm /cm ) R d 1+ r /ɛ ) c d 1+r cv /cd ) R d r ) c d θ m = T = T T ) Temerature will change less raidly during adiabatic dislacements of moist air θ m < θ d ) because of the higher heat caacity of water vaor relative to dry air; however, this effect is relatively small less than 1%), and otential temerature is therefore generally calculated using Eq As shown by Eq. 4.14, moist air is less dense than dry air at the same temerature and ressure. The deendence of density on humidity is often exressed by the use of the virtual temerature. The virtual temerature of an air arcel is the temerature dry air would have if it had the same density and ressure as the arcel, and is defined such that R d T v = R m T. Substituting the exression of R m in Eq. 4.14, the virtual temerature can be calculated as 1 + r ) /ɛ T v T T r ). 4.17) 1 + r Because r is ositive definite, the virtual temerature T v is always larger than T. The virtual otential temerature is then defined as θ v T v 0 ) Rd c d. 4.18) Virtual otential temerature is directly related to density, and is therefore a useful measure of the relative density of unsaturated air arcels. An unsaturated atmoshere is stable if θ v increases with height and unstable if θ v decreases with height. Both θ d and r are conserved during adiabatic rocesses in unsaturated air, so that θ v is also effectively conserved THERMODYNAMIC EFFECTS OF SATURATION A arcel of air cools by adiabatic exansion as it is lifted. If the arcel contains water vaor, some of that water vaor will condense once the arcel cools enough that the saturation vaor ressure becomes less than the vaor ressure i.e., RH 1). The condensation of that water vaor releases latent heat, which warms the arcel relative to the dry adiabat and increases the otential temerature. Condensation and evaoration are diabatic rocesses, and neither otential temerature nor virtual otential temerature are conserved when they occur. Our understanding of cloud rocesses therefore relies on a different set of conserved variables. Neglecting ice, the secific entroy entroy er unit mass) of an air arcel that includes water in both the vaor and liquid hases is equal to the sum of the secific entroies for dry air, water vaor and liquid water in the arcel: s = s d + r s v + r l s l = s d + r + r l )s l + r s v s l ) Substituting the Clausius Claeyron equation as L v = T sv s l ), where sv entroy of water vaor in equilibrium with the liquid water, we have is the secific 8

9 Pressure [hpa] Temerature T) Potential temerature θ) Virtual otential temerature θv) Equivalent otential temerature θe) Temerature [K] Figure 4.5: Profiles of temerature T ), dry otential temerature θ; Eq. 3.5), virtual otential temerature θ v ; Eq. 4.18) and equivalent otential temerature θ e ; Eq. 4.19) in the trooshere averaged along the equator. Data from the Jaanese 55-year Reanalysis. s = s d + r t s l + r L v T + r s v sv ) = [ ) r Lv c d lnt R d ln d + rt c l lnt ) + T + r c v lnt R v lne c v lnt + R v lne ) ] = c d + r t c l )lnt R d ln d + r L v T r R v ln e /e ) with r t = r + r l the total water mixing ratio and c l = 4190 J K 1 kg 1 the secific heat of liquid water. Differentiating and setting ds = 0, we have the following conservation exression for moist adiabatic rocesses: c d + r t c l )dlnt ) = R d dln d ) + r R v d ln e ) e r L v T. We can then define the equivalent otential temerature θ e as θ e = T d 0 ) Rd /c d +r T c l ) e e ) r Rv/c L v r ex c d + r T c l )T ) 4.19) Equivalent otential temerature is conserved during both moist and dry adiabatic rocesses. If the air is comletely dry r = 0), θ e reduces to θ. In the absence of diabatic heating i.e., if we neglect radiative transfer and neither condensation nor evaoration occur), r is unchanged and both θ and θ e are conserved. If condensation or evaoration occur, then θ e is conserved but θ is not. In the case of condensation, RH = e /e ) is close to one, so that θ changes by a factor 9

10 Figure 4.6: Thermal equilibrium temerature rofiles for a lase rate of 6.5 K km 1 dashed), a dry adiabatic lase rate of 10 K km 1 dotted), and ure radiative equilibrium solid) from Manabe and Strickler, 1964). of aroximately ex L v r c +r T c l )T ). Note that in this exression r is ositive for condensation because condensation increases θ). In the case of evaoration, which reduces θ, e /e < 1 and the term involving RH becomes more influential. Just as decreases of θ with height indicate vertical instabilities in the dry atmoshere, decreases of θ e with height indicate vertical instabilities in the moist atmoshere. These vertical instabilities are mixed out by the occurrence of moist convection. Figure 4.5 shows that θ and θ v increase with height in the troics, indicating that the mean state of the troical atmoshere is stable with resect to dry convection. By contrast, the gradient of θ e is aroximately neutral below 300 hpa and is even unstable in the lower atmoshere although this instability can only be triggered if condensation occurs). This difference highlights the critical imortance of water to convective mixing in the troics. A moist adiabat is defined as the temerature rofile corresonding to ascent with constant equivalent otential temerature. The equivalent otential temerature is a useful variable for describing cloud rocess, but it is not conserved in all cases. In articular, θ e is not conserved for radiative heating and cooling, nor is it conserved for sensible and latent heat fluxes from the surface to the atmoshere. Equivalent otential temerature is not even conserved for all cloud-related rocesses: for examle, the evaoration of falling raindros into unsaturated air results in an irreversible increase in entroy. We defined the virtual temerature to account for variations in density due to the resence of water vaor; however, T v does not account for variations in density due to the resence of condensed water. The density of a arcel containing condensed water can be exressed as ρ = m d + m v + m l + m i V a +V l +V i 10

11 Dividing both the numerator and the denominator by m d yields so that 1 + r + r l + r i ρ = ρ d 1 + r T ) ρ d + r v ρ v + r l ρ l + r i ρ i ρ d R d T 1 + r T ) = R d T d ɛ d + e) 1 + r T ) = R d T We can then define the density temerature for which = ρr d T ρ : 1 + r T 1 + r /ɛ T ρ T 1 + r /ɛ 1 + r T, 4.20) which could in turn be used to define a density otential temerature. Unlike T v, T ρ may take values either smaller or larger than T, deending on the magnitude of the total water concentration. 4.4 RADIATIVE CONVECTIVE EQUILIBRIUM The temerature structure based urely on radiative equilibrium is unstable Fig. 4.6). In fact, this model suggests that the global mean surface temerature at ure radiative equilibrium would be more than 60 C! The temerature near the surface is much warmer than the temerature aloft. This rofile is unstable, and convection will occur. We can modify the simle energy balance models discussed in chater 1 to include convection, yielding yet another class of simle climate models called radiative convective equilibrium models. In most radiative convective equilibrium models such as that used to generate Fig. 4.6), the Earth s atmoshere is considered as a single vertical column just as in the ure radiative equilibrium energy balance model). Temerature at the surface and within the atmosheric column adjusts based on solar and long-wave radiative heating, long-wave radiative cooling, and convection. Such models may be configured to either include or exclude the thermodynamic effects of atmosheric moisture. The radiative equilibrium lase rate is larger than the equilibrium dry adiabatic lase rate, which means that the radiative equilibrium atmoshere is unstable to dry convection. Adjustment to the dry adiabatic rofile still results in a surface temerature of aroximately 35 C. Comarison with the thermal equilibrium for a global mean lase rate of 6.5 K km 1 indicates that the dry adiabatic rofile is further mixed by moist convection to yield the observed global mean vertical distribution of temerature in the trooshere. The imortance of radiative convective equilibrium in the trooshere is aarent in the zonal mean distribution of diabatic heating Fig 4.7). Diabatic heating is the sum of all nonadiabatic rocesses in the atmoshere, including radiative heating and cooling cf. Fig. 1.11), latent heating and cooling due to the formation and evaoration of clouds and reciitation e.g., the additional terms in Eq relative to Eq. 3.5), and heating due to turbulent mixing which we will discuss in more detail in lecture 6). Desite strong regional variations in diabatic heating and magnitudes of several Kelvins er day, the zonal mean isentroes in Fig. 4.7a do not show large deviations across the troics 11

12 Pressure [hpa] K a) Total heating rate 380K S 60 S 30 S 0 30 N 60 N 90 N 300K 280K 340K Pressure [hpa] K b) Radiative heating rate 380K S 60 S 30 S 0 30 N 60 N 90 N 300K 280K 340K Pressure [hpa] K c) Convective heating rate 380K S 60 S 30 S 0 30 N 60 N 90 N 300K 280K 340K Heating Rate [K d 1 ] Figure 4.7: Zonal mean a) total, b) radiative, and c) dee convective diabatic heating. Data from the Jaanese 55-year Reanalysis. 12

13 and inner subtroics. This lack of covariability suggests that, desite its name, diabatic heating does not strongly affect internal thermal) energy c T ) on long time scales. In fact, diabatic heating and cooling are rimarily converted to kinetic energy, with diabatic heating corresonding to uward motion across isentroes) in the atmoshere and diabatic cooling corresonding to downward motion across isentroes). In this sense, diabatic heating can be thought of as adding or removing otential energy g z). We will revisit these ideas later in the course. Figure 4.7 shows that both radiative and convective/moist rocesses are imortant in the trooshere. Accordingly, we may say that the trooshere is aroximately in radiative convective equilibrium. By contrast, the diabatic effects of moist rocesses and dry convection are aroximately zero above the trooause. This suggests that the stratoshere is nearly in ure radiative equilibrium. The stratoshere contains a temerature inversion due to the absortion of UV radiation by ozone Fig. 2.2), and is therefore very stable with a large buoyancy frequency N). Although convection is largely unimortant in the stratoshere, gravity waves and other oscillating disturbances lay a key role in stratosheric dynamics. REFERENCES Curry, J. A., and P. J. Webster 1999), Thermodynamics of Atmosheres and Oceans, 471., Academic Press, London, U.K. Emanuel, K. A. 1994), Atmosheric Convection, 580., Oxford University Press, Oxford, U.K. Manabe, S., and R. F. Strickler 1964), Thermal equilibrium of the atmoshere with a convective adjustment, J. Atmos. Sci., 21,