This results of this chapter are pertinent to a planet with a distinct surface, which may defined as an interface across which the density increases substantially and discontinously. The typical interface would be between a gaseous atmosphere and a solid or liquid surface. In the Solar system, there are only four examples of bodies having both a distinct surface and a thick enough atmosphere to significantly affect the surface temperature. These are Venus, Earth, Titan and Mars; among these, the present Martian atmosphere is so thin that it only marginally affects the surface temperature, though this situation was probably different early in the planet's history when the atmosphere may have been thicker. Although thin atmospheres have little effect on the surface temperature, the atmosphere itself can still have interesting behavior, and the flux of energy from the surface to the atmosphere provides a crucial part of the forcing which drives the atmospheric circulation. This is the case for example, for the thin Nitrogen atmosphere of Neptune's moon Triton. Apart from the examples we know, it is worth thinking of the surface balance in general terms, because of the light it sheds on the possible nature of the climates of extrasolar planets already detected or awaiting discovery.
The exchange of energy between the surface and the overlying atmosphere determines the surface temperature relative to the air temperature. It also turns out that it determines the exchange of mass between the surface and the atmosphere (as in sublimation from a glacier or evaporation from an ocean, lake or swamp). Because outer space is essentially a vacuum, the only energy exchange terms at the top of the atmosphere are radiative. At the surface, energy can be exchanged by means of fluid motions as well as by radiation.
The atmospheric gas in direct contact with the surface must have the same velocity as the surface; because the surface material is so much denser (and in the case of a solid so much more rigid) than the atmosphere, the atmospheric flow must typically adjust to the presence of the surface over a rather short distance. The resulting strong shears lead to random-seeming complex turbulent motions sustained by the kinetic energy of the shear flow near the boundary. We may subdivide the atmosphere into the free atmosphere - which is sufficiently far above the surface to be little affected by turbulence stirred up at the surface, and the planetary boundary layer (PBL, for short) where the transfer of heat, chemical substances, and momentum is strongly affected by surface-driven turbulence. We may further identify the surface layer, which is the thin portion of the PBL near the ground within which all the vertical fluxes may be considered independent of height.
Given that the whole troposphere is created by convection - which is a form of buoyancy-driven turbulence - it is not at once clear why the PBL should exist as a distinct entity from the troposphere in general. The main reason one can typically distinguish the PBL is that mechanically driven turbulence is more trapped near the surface than is buoyancy driven turbulence, and also has distinct time and space scales. On the present Earth, the effect of moisture is also important in maintaining the distinction, since moisture gives deep convection an intermittent character: most of the troposphere-forming mixing takes place in rare convective events, while most of the troposphere remains quiescent most of the time. Because dry (i.e. noncondensing) convection is typically shallower than moist convection, in planets which have both forms the dry convection can often be treated as part of the boundary layer. This is the case for Earth, and likely for other planets with a surface and an atmosphere in which latent heat release is important (Titan and perhaps Early Mars being the only other known examples so far). For planets like Present Mars or Venus, where dry convection is the only form of convection, it is less clear that the PBL can be productively distinguished from the troposphere in general. Even in such cases, though, one can identify a constant-flux surface layer; the depth of the surface layer typically range from a few meters to a few tens of meters.
As in previous chapters, we let Tg be the temperature of the planet's surface. Previously, we used Tsa to denote the temperature of the air in immediate contact with the ground, but now we modify the definition somewhat, and allow Tsa to be the temperature at the top of the surface layer, assuming the air at the bottom of the surface layer (which is in contact with the ground) has the same temperature as the ground itself. A model of the PBL is necessary to connect Tsa to the temperature of the lowest part of the free troposphere. For many purposes, we can dispense with the PBL and patch the surface layer directly to the free troposphere. We shall adhere to this expedient in most of the following discussion.
Now let's discuss, in general terms, how the surface budget affects the climate. The state of the atmosphere and the ground must adjust so that the top-of-atmosphere and surface budgets are simultaneously satisfied. If the atmosphere is optically thick in the longwave spectrum, the top-of-atmosphere budget becomes decoupled from the surface budget, since radiation from the ground and lower portions of the atmosphere is absorbed before it escapes to space. In this case, the determination of Tg can be decomposed into two stages carried out in sequence. First one determines Tsa by adjusting this temperature until the top of atmosphere balance is satisfied, assuming that the rest of the troposphere is related to Tsa through the appropriate dry or moist adiabat. Then, once Tsa is known, one makes use of a model of the surface flux terms to determine the value of Tg which balances the surface budget with Tsa fixed at the previously determined value. This can be done without reference to the top-of-atmosphere budget, since the OLR is independent of Tg in the optically thick limit.
If the atmosphere is very optically thin in the longwave spectrum, the OLR is determined entirely by the ground temperature and ground emissivity. Further, since an optically thin atmosphere radiates very little, the only way the atmosphere itself loses energy is through turbulent exchange with the surface. Suppose first that the atmosphere is transparent to solar radiation. In that case, in equilibrium the net turbulent exchange between atmosphere and surface must vanish, since otherwise the atmospheric temperature would rise or fall, there being nothing to balance a net exchange. In consequence, the ground temperature will be just what it would have been without an atmosphere despite the presence of turbulence. In this case, one determines the ground temperature as if the planet were in a vacuum, the top of atmosphere budget is automatically satisfied, and then, once Tg is known, the surface budget is used to determine Tsa, and (via an adiabat) the rest of the atmospheric structure. It is exactly the inverse of the process used in the optically thick case. In fact, the basic picture is little altered even if the atmosphere absorbs solar radiation. In that case, the requirement that the atmosphere be in equilibrium implies that any solar radiation absorbed in the atmosphere be passed on to the surface by turbulent fluxes. The result is much the same as if the solar radiation were absorbed directly by the surface; one does the ground temperature calculation as before, but simply remembers to add the atmospheric absorption to the solar energy directly absorbed by the ground. It should be kept in mind that these considerations apply only in equilibrium. Even an optically thin atmosphere can affect the transient behavior of the surface (e.g. in the diurnal or seasonal cycle), as will be discussed in Chapter 7.
In the intermediate case, where the atmosphere is neither optically thick nor thin, one must solve for Tsa and Tg simultaneously, so as to find the values that satisfy both the top-of-atmosphere and surface energy budgets. We'll do this crudely in the present chapter through the introduction of atmospheric transparency factors. Generally speaking, though, when the atmosphere is not too optically thin, the surface budget will have some effect on the temperature of the ground. For Earth this temperature is of interest because the ground is where people live and where much of the biosphere resides as well; for a broad range of planets actual or hypothetical the ground temperature also affects chemical processes which determine atmospheric composition, as well as the melting of ices at the surface. We shall see, however, that it is a fairly common circumstance that the surface fluxes effectively constrain the ground temperature to be nearly equal to the overlying air temperature, so that the climate can be determined without detailed reference to how the surface balance works out.
The surface receives radiant energy in the form of shortwave (solar) and longwave (thermal infrared) flux. The shortwave flux incident on the surface is equal to the shortwave flux incident at the top of the atmosphere, diminished by whatever proportion is absorbed in the atmosphere or scattered back to space. We will call the shortwave flux incident on the ground Sg. The shortwave flux absorbed at the surface is then (1 — ag)Sg, where ag is the albedo of the ground. Sg is affected by clouds, atmospheric absorption and atmospheric Rayleigh scattering.
The longwave radiation striking the surface is the infrared back radiation emitted by the atmosphere, which was discussed in Chapter 4. The back radiation depends on both the greenhouse gas content of the atmosphere - which determines its emissivity - and the temperature profile. When the atmosphere is optically thick in the infrared, most of the back radiation comes from the portions of the atmosphere near the ground, whereas in an optically thinner atmosphere the back radiation comes from higher - and generally colder - parts of the atmosphere, and is correspondingly weaker. If the atmosphere is very optically thin, the back radiation will be weak regardless of the atmospheric temperature profile, simply because an optically thin atmosphere radiates very little. As in Chapter 4, I-s will denote the back radiation integrated over all longwave frequencies. The absorbed infrared flux is then egI-s, where eg is the longwave emissivity of the ground. The ground loses energy by upward radiation at a rate eguT4. Thus, the net infrared cooling of the
Surface Cooling Factor, Moist Case
Surface Cooling Factor, Moist Case
Surface Cooling Factor, Dry Case
Surface Cooling Factor, Dry Case
Figure 6.1: Surface cooling factor e* for a 1bar Nitrogen-Oxygen atmosphere with water vapor and CO2. The surface gravity is that for Earth. In the left panel, the calculations were done with free tropospheric relative humidity set to 50%, and low-level relative humidity set to 80%. Results in the right panel are for a dry atmosphere (zero relative humidity, but with the temperature profile kept the same as in the moist case). In both cases, the numbers on the curves indicate the partial pressure of CO2 in mb.
According to Eq. 4.21, I-s approaches uT^ when the atmosphere is optically thick throughout the infrared. In order to characterize the optical thickness of the atmosphere, we introduce the effective low level atmospheric emissivity ea, defined so that I-s = eaaT'44a. ea depends on the temperature profile as well as the optical thickness, as illustrated by Eq. 4.21 in the optically thick limit. When Tg = Tsa the surface cooling becomes eg • (1 — ea)uT4, which vanishes in the optically thick limit where ea ^ 1. Let e* = (1 — ea); this is the effective emissivity of the ground when the air temperature equals the ground temperature. If the air temperature is not too different from the ground temperature, we may linearize the term uT^ about Tg = Tsa, which results in
Fgir = eg • e*uTS4a + (4uTg3eg)(Tg — Tsa) (6.2)
From this equation we can define the infrared coupling coefficient, bir = 4aTg^eg. When bir is large, a small temperature difference leads to a large radiative imbalance, and it is correspondingly hard for the ground temperature to differ much from the overlying air temperature. Later, we will derive analogous coupling coefficients for the turbulent transfers.
Figure 6.1 shows how e* varies with temperature for an Earthlike atmosphere in which the only greenhouse gases are water vapor and CO2, with the water vapor relative humidity held fixed as temperature is changed. In the moist case (left panel), e* rapidly approaches zero as the temperature increases; this is because of the increasing optical thickness caused by the increase of water vapor content with temperature (owing to the fixed relative humidity). Increasing the CO2 content also increases the optical thickness, correspondingly reducing e*. At low temperatures, the CO2 effect dominates, because there is little water in the atmosphere. However, by the time Earthlike tropical temperatures (300K) are reached, water vapor is sufficient to make e* essentially zero all on its own without any help from CO2. To underscore the relative role of CO2 and water vapor, results for a dry atmosphere are given in the right hand panel of Figure 6.1. e* still goes down with temperature, because temperature affects the opacity of CO2; however the decline is much less pronounced than it is in the moist case. Even with 100mb of CO2 in the atmosphere, e* falls only to about .4 at 320K, and significant infrared cooling of the surface is possible. In sum, CO2 by itself is relatively ineffective at limiting surface cooling, but the opacity of water vapor can practically eliminate surface infrared cooling at temperatures above 300K, unless the ground temperature significantly exceeds the air temperature.
Though the results of Fig. 6.1 were were computed for Earth conditions, they give a fair indication of the extent of surface radiative cooling on other planets whose atmospheres consist of an infrared-transparent background gas mixed with CO2 and with water vapor fed through exchange with a condensed reservoir. Through the hydrostatic relation, the surface gravity g affects the mass of greenhouse gas represented by a given partial pressure; the lower the g the greater the mass (and hence the greater the optical thickness), and conversely. This is especially important in the case of water vapor, since in that case the partial pressure is set by temperature, through the Clausius-Clapeyron relation. Thus, for a "large Earth" with high g, it takes a higher temperature to make the lower atmosphere optically thick. For example, calculations of the sort used to make Figure 6.1 show that with 1mb of CO2 in a moist atmosphere having temperature 280K, increasing g to 100m/s2 increases e* to .507 (vs .303 for g = 10m/s2). In the same atmospheric conditions, e* falls to .102 for a "mini-Earth" with g = 1m/s2. Increasing the pressure of the transparent background gas makes the greenhouse gases more optically thick through pressure broadening. With g = 10m/s2, increasing the background air pressure to 10bar has a very profound effect, lowering e* to .094. Reducing the air pressure below 1000mb should in principle increase e*, but in fact it is found to very slightly reduce it, to .299. It appears that the reduction in opacity from less pressure broadening is offset by the changes in the moist adiabat that occur when the air pressure is reduced: the latent heat of condensation is spread over less background gas, so the temperature aloft is greater and hence the air aloft contains more water.
Without water vapor, it takes an enormous amount of CO2 to make the lower atmosphere optically thick. This case is relevant to Venus and Venus-like planets, which may be defined to be planets having a dry rocky surface and a thick, dry CO2 atmosphere. The near surface radiative properties can be determined using the homebrew exponential-sum radiation code; for better accuracy, we did this based on exponential-sum tables computed for the surface temperature and pressure conditions under consideration, so as to minimize errors due to pressure and temperature scaling of absorption coefficients. Line parameters in the HITRAN database were used to compute the absorption coefficients. For a 1 bar pure CO2 atmosphere on a planet with the gravity of Venus, this calculation yields e* = .43 when the surface temperature is 300K, falling further to under 10-6 for pressures of 10 bar or more. The sharp decline in surface cooling between 1 bar and 10 bar arises from line-broadening, which fills in the window regions in the CO2 absorption spectrum. At the 727K surface temperature of Venus, the surface emission shifts toward higher wavenumbers where CO2 doesn't absorb as well, but the high temperature increases the line strengths while the high pressure causes the absorption to further spill over into the windows. Hence, at 92 bar and 727K the calculation still yields a value of e* that is under 10-5, even assuming CO2 to be completely transparent for wavenumbers higher than 10000 cm-1. The estimates of e* at high pressure should be viewed with some caution, however. At high pressures, the contribution of each line to spectral distances far removed from the line center is considerable, and there is much uncertainty about the appropriate form of line shape to be used in computing this far-field contribution. We'll see shortly that it actually makes some difference to the climate of Venus whether e* is zero or 0.05 .
In the opposite extreme, atmospheres like the thin Martian atmosphere have very little effect on the surface radiative cooling. For a Martian CO2 atmosphere on the dry adiabat with 7mb of surface pressure, e* = .9 at 220K, falling only modestly to .86 at 280K. Recall that, per square meter of surface, Mars actually has vastly more CO2 in it's atmosphere than the Earth has at present; allowing for the difference in gravity, a 7mb pure CO2 atmosphere on Mars has as much CO2 per unit area as an Earth atmosphere with a CO2 partial pressure of 18.5mb at the ground. In comparison, the present Earth's atmosphere has a partial CO2 pressure of a mere .38mb (in 2006). The weak emission of the Martian atmosphere is due to the low total pressure, which yields little collisional broadening of the emission lines. If the same amount of CO2 on Mars at present were mixed into a 1bar atmosphere of N2, the effective surface emissivity e* falls to .75 at 230K and .69 at 280K.
Among common greenhouse gases, water vapor appears unique in its ability to make the lower atmosphere nearly opaque to infrared, even at concentrations as low as a few percent.
Clouds made of an infrared-absorbing substance such as water act just like a very effective greenhouse gas in making the lower atmosphere optically thick (making ea close to unity). It takes very little cloud water to make the lower atmosphere act essentially like a blackbody. Infrared-scattering clouds in the surface layer, like those made of methane or CO2, have a very different effect on the back-radiation. First, they shield the surface from back-radiation coming down from the upper atmosphere by reflecting it, rather than absorbing it; hence the shielding is accomplished without the cloud layer heating up in response to absorption. More importantly, the downwelling radiation from a reflective cloud is determined by the upwelling ground radiation incident upon it; the resulting back radiation is then determined by the ground temperature, and is independent of the cloud temperature. As a result, the surface cannot increase its longwave cooling by warming up until it is substantially warmer than the atmosphere. This gives a scattering cloud great potency to increase the ground temperature, if it allows sufficient solar radiation to get through to the ground. Either IR-reflecting or absorbing clouds are different from a greenhouse gas, in that they also strongly increase the shortwave albedo.
6.2.3 Radiatively driven ground-air temperature difference
Now we consider the equilibrium temperature difference between the ground and the overlying air that would be attained in the absence of turbulent heat exchange. This temperature difference is important in determining the extent to which convection is driven from below, by positive buoyancy generated near the ground. We have already discussed this issue for the case in which the atmosphere itself is in pure radiative equilibrium (See 3.6,4.3.4 and 4.7). Our concern now is with what happens once convection has set in and altered the atmospheric temperature profile.
If the only heat exchange is radiative, the surface budget reads
Since the second term on the left hand side is positive, the infrared back-radiation always drives Tg to exceed its no-atmosphere value. However, this value might be more or less than Tsa. To examine this difference, we linearize the surface radiation budget about Tsa, which results in
The linearized form can be immediately solved for the ground-air temperature difference. Substi tuting the expression for bir, we find
4 Tsa where To is the no-atmosphere ground temperature, which satisfies aegTO4 = (1 — ag)Sg. For planets with an optically thick lower atmosphere, the ground temperature can get extraordinarily hot relative to the air temperature if there are no turbulent fluxes to help carry away the heat. The first term on the right hand side of Eq. 6.5 is large in tropical Earth conditions. For (1 — ag)Sg = 300W/m2 and Tsa = 300K with eg = 1, it has the value 49K. But in tropical Earth conditions, e* is on the order of .1, so the second term subtracts little (15K for Tsa = 300K). Thus, the ground temperature is 34K warmer than the overlying air temperature, or 334K. In reality, the sea surface temperature hardly ever gets more than a few degrees warmer than the free-air temperature in the Earth's tropics.
Ironically, for planets which have such a strong greenhouse effect that the low level air temperature is much larger than the no-atmosphere value, Tg — Tsa can be quite small even if the lower atmosphere is optically thick enough to make e* « 0, and even in the absence of turbulent heat fluxes. This conclusion is readily deduced from the factor multiplying Tsa in the second line of Eq. 6.5. For example, Venus has a small TO because of the highly reflective clouds which keep sunlight from reaching the surface, yet has a high Tsa because of its strong greenhouse effect. In consequence, this factor is only .0024 for Venus in the limit e* = 0, whence Tg — Tsa « 1.8K. If some CO2 window region not reproduced by the procedure we used to estimate the surface radiative cooling on Venus allowed e* to increase modestly to 0.05, then the ground temperature would actually become slightly cooler than the overlying air temperature, leading to a low-level temperature inversion and cutting off near-surface convection. For planets like Venus, the surface radiation budget is dominated by infrared back-radiation, and the comparatively feeble sunlight has little power to drive the ground temperature to values much greater than the overlying air temperature. It is situations like the Earth's tropics, which combine an optically thick lower atmosphere (due to water vapor in our case) with a rather modest greenhouse effect, where the radiation budget tends to drive the ground temperature to large values relative to that of the overlying air.
When the lower atmosphere is optically thin, as in the case of present Mars, the ground-air temperature difference cannot be determined without considering the top-of-atmosphere balance simultaneously with the surface balance. For an optically thin atmosphere, Eq. 6.3 tells us that Tg is just slightly greater than its no-atmosphere value, but it does not by itself tell us how Tg relates to Tsa. The general idea for an optically thin atmosphere is that the ground temperature is close to what it would be without an atmosphere, while the atmosphere cools down until the energy it loses by emission is equal to the energy gained by absorption of infrared upwelling from the ground (plus atmospheric solar absorption, if there is any). This generally leaves the low level air temperature much colder than the ground, since the atmosphere loses energy by radiating out of both its top and its bottom. The most straightforward way to make this more precise is to consider the radiative energy budget of the atmosphere, which is the difference between top-of-atmosphere and surface energy budget.
The net infrared radiative flux into the bottom of the atmosphere is eguT^ — eaaT'44a, while the infrared flux out of the top of the atmosphere is the OLR. As discussed in Chapter 4, the OLR is the sum of the emission from the atmosphere itself and the portion of the upward emission from the ground which is transmitted by the atmosphere. Let a+ be the proportion of upward radiation from the ground which is absorbed by the full depth of the atmosphere, and express the upward atmospheric emission escaping the top of the atmosphere in the form ea)topuTs4a. Then OLR = ea,topaTS4a + (1 — a+)eguT^. Let's assume for the moment that the atmosphere does not absorb any solar radiation. Then, in the absence of turbulent heat fluxes the atmospheric energy budget reads
0 = OLR — (eg uTfl4 — eauT^) = a+eg uT^ — (ea,top + ea)uTS4a (6.6)
whence a e
Note that we have not yet made use of the assumption that the atmosphere is optically thin. For an optically thick atmosphere with a very strong greenhouse effect (like Venus), a+ « ea « 1 and ea,top ~ 0, and so we recover our previous result that Tsa « Tg for such an atmosphere,provided the emissivity of the ground is close to unity. For an optically thin atmosphere, a+, ea,top and ea are all small, so one needs to know precisely how small the absorption coefficient is relative to the two emission coefficients. For an isothermal atmosphere -whether grey or not- Eq. 4.9 implies ea,top = ea. For a grey atmosphere, it follows in addition that a+ = ea,top = ea. In this case Tsa = Tg/21/4, reproducing the result of Section 3.6. When the atmosphere is not grey, the absorption coefficient differs somewhat from atmospheric emission coefficient, because the spectrum of the upwelling radiation from the ground is different from that of the atmospheric emission (by virtue of the difference between ground temperature and air temperature). However, the deviation from the grey gas result is typically modest for an isothermal atmosphere. For example, a 7mb Marslike pure CO2 atmosphere with a uniform temperature of 230K has a+ = ea,top = ea « .14
However, introduction of a vertical temperature gradient strongly affects the relative magnitude of the three coefficients. If we take the same Marslike atmosphere with the same ground temperature and pressure, but stipulate that the temperature is on the dry adiabat rather than isothermal, then a+ and ea are reduced slightly (to .116 and .106, respectively), but are still approximately equal. In contrast, ea,top is substantially reduced, to .043. In consequence, the temperature jump at the ground is Tsa = Tg/1.281/4 - substantially weaker than the isothermal case, but still quite unstable. Results for a dry Earth, with 300 ppmv of CO2 in a 1bar N2/O2 atmosphere having 300K surface temperature, are similar: a+ « ea « .14 while ea,top « .04. What is happening in both cases is that the atmosphere appears optically thin when averaged over all wavenumbers, but is really quite optically thick in a narrow band of wavenumbers near the principal CO2 absorption band. The optical thickness in this range introduces a strong asymmetry in the upward and downward radiation, and also weights the absorption towards the bottom of the atmosphere (which is also where a disproportionate amount of the infrared back-radiation is coming from. A rule of thumb for such cases is that a+ and ea will have similar magnitudes, while ea,top will be smaller (but, in the optically thin case, still non-negligible); it follows that the surface temperature jump is weaker than the isothermal case, but still unstable. For an optically thin grey gas the situation is different. In that case, ea — ea,top and both are less than a+; nonetheless, the relative magnitudes are such that an unstable temperature jump can generally be sustained at the surface even if the lower atmosphere is on a dry adiabat (see Problem ??).
The upshot of the preceding discussion is that, in the absence of atmospheric solar absorption, the radiative balance in an optically thin atmosphere almost always drives the surface to be notably warmer than the overlying atmosphere, even if convection has established an adiabat in the atmosphere. This provides a source of buoyancy that can maintain the convection which stirs the troposphere and maintains the adiabat. A moist adiabat is more isothermal than the dry adiabat, so our conclusion is even firmer in that case. Atmospheric solar absorption, on the other hand, would warm the atmosphere relative to the surface, weakening or even eliminating the unstable surface jump.
Moving on, let's consider the temperature the ground of a planet would have in radiative equilibrium at night-time, when Sg =0. In this case, there is little to be gained by linearizing the surface budget, as it reduces to simply aT4 = eaCTT4a, whence Tg/Tsa = (ea)1/4. For an optically thick lower atmosphere, the infrared back radiation keeps the ground temperature nearly equal to the air temperature. However, when the lower atmosphere is not optically thick, the ground temperature plummets at night, or would do so if it had time to reach equilibrium. Cold climates tend to be comparatively optically thin because they cannot hold much water vapor even in saturation. For example, using the moist case in Figure 6.1, we find that when Tsa = 240K, ea « .3 with .1mb of CO2 in the atmosphere. This implies that at night the ground temperature plunges toward the fearsomely cold value Tg = 177K. Liquid surfaces like oceans cannot generally cool down rapidly enough to approach the night-time equilibrium temperature, because turbulent motions in the fluid bring heat to the surface which keeps it warm. Solid surfaces like snow, ice, sand or rock can cool down very quickly, though, and do indeed plunge to very low temperatures at night. This situation applies to Snowball Earth and to the present-day Arctic and Antarctic. Very cold climates are of necessity dry, because of the limitations imposed by Clausius-Clapeyron. However, even relatively warm climates can be dry if the moisture source is lacking. This is why deserts can go from being unsurvivably hot in the daytime to uncomfortably cold at night. Turbulent fluxes can bring additional heat to the ground and moderate the night-time cooling somewhat, but these fluxes tend to be weak in the situation just described, because turbulent eddies must expend a great deal of energy to lift cold dense air from the ground to the outer edge of the surface layer (a matter taken up in more detail in Section 6.4)
The preceding discussion technically applies whether or not Tsa itself drops substantially at night, but is most meaningful in the situation where the atmosphere cools slowly enough that the atmosphere remains relatively warm as the night-time ground temperature drops. This is a fair description of the situation in the massive atmospheres of Titan, Earth and Venus, except to some extent during the long polar night on Titan and Earth. The tenuous atmosphere of present Mars, in contrast, cools substantially throughout its depth during the night, even at midlatitudes. In this situation, the relative temperature of air and ground at night is determined by the relative rates of cooling of the two media, rather than radiative equilibrium. We will take up the issue of thermal response time in detail in Chapter 7.
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