po po i.e. p p p(pi)( pi )R(pi)/cp(pi)-1( po )R(p2)/cP(p2)-1 < p(p2 ) (2.16) po p2

This yields the same criterion as the correct criterion given in Eq. 2.14 only if I/cp is constant.

The lack of a globally valid potential density complicates the precise analysis of the static stability of inhomogeneous atmospheres. Strictly speaking, one needs to examine potential density profiles for a range of different po covering the atmosphere. In practice, the potential density based on a single po can often provide a useful indication of static stability within a reasonably thick layer of the atmosphere because the variations of ñ/cp are usually slight unless the compositional variations are extreme.

An alternate approach to dealing with the effect of composition on buoyancy is to define a modified potential temperature based on a virtual temperature. The virtual temperature is the temperature at which the gas law for a standard composition (e.g. dry air) would yield the same density as the true gas law taking into account the effect of the actual composition on density (e.g. moist air). This approach is a common way of dealing with the effect of water vapor on buoyancy in Earth's atmosphere. It is equivalent to the potential density approach and shares the same limitations. Making use of a virtual temperature can be convenient for some purposes, but it can also be confusing in that one needs to keep track of the contexts in which one uses the virtual temperature and the ones in which one must use the actual temperature.

In planetary atmospheres, compositional variations often arise as a result of condensation, evaporation or sublimation of one of the atmospheric components. For example, the Martian atmosphere contains a moderate amount of Ar, and when CO2 condenses at the winter pole, it leaves an Ar-enriched layer near the surface. Because Ar has higher molecular weight than CO2, this layer tends to be statically stable. When CO2 sublimates from the polar glacier in the Spring, the pure CO2 layer resulting has enhanced buoyancy relative to the Ar-rich atmosphere above. Similarly, the evaporation of light from the surface of Titan makes the low level air positively buoyant with respect to the overlying N2, favoring upward mixing. Water vapor has a small but significant effect on buoyancy on the present Earth; the importance of the effect would increase sharply under warmer conditions in which the water vapor content of the atmosphere is greater. Condensation can also lead to compositional variations in the interior of atmospheres, and in some circumstances it may be necessary to take these into account when determining where the vertical mixing that creates the troposphere takes place. In principle, chemical reactions could also create compositional gradients strong enough to create buoyancy, though there do not at present appear to be any atmospheres where this is known to be a significant effect.

The hydrostatic relation relates pressure to altitude and the mass distribution of the atmosphere, and provides the chief reason that pressure is the most natural vertical coordinate to use in most atmospheric problems. Consider a column of any substance at rest, and suppose that the density of the substance as a function of height z is given by p(z). Suppose further that the range of altitudes being considered is small enough that the acceleration of gravity is essentially constant; The magnitude of this acceleration will be called g, and the force of gravity is taken to point along the direction of decreasing z. Now, consider a slice of the column with vertical thickness dz, having cross sectional area A in the horizontal direction. Since pressure is simply force per unit area, then the change in pressure from the base of this slice to the top of this slice is just the force exerted by the mass. By Newton's law, then, we have where dm is the increment of mass in the column per unit area. An immediate consequence of this relation is that

2.5 The hydrostatic relation

g which states that the amount of mass in a slab of atmosphere is proportional to the thickness of that slab, measured in pressure coordinates. A further consequence, upon dividing by dz is the relation

This differential equation expresses the hydrostatic relation. It is exact if the substance is at rest (hence the "static"), but if the material of the column is in motion, the relation is still approximately satisfied provided the acceleration is sufficiently small, compared to the acceleration of gravity. In practice, the hydrostatic relation is very accurate for most problems involving large scale motions in planetary atmospheres. It would not be a good approximation within small scale intense updrafts or downdrafts where the acceleration of the fluid may be large. Derivation of the precise conditions under which the hydrostatic approximation holds requires consideration of the equations of fluid motion, which will be taken up in a sequel to the present book.

An important consequence of the hydrostatic relation is that it enables us to determine the total mass of an atmosphere through measurements of pressure taken at the surface alone. Integrating Eqn 2.18 from the ground (p = ps) to space (p = 0) yields the relation m = ps (2.20)

where m is the total mass of the atmosphere located over a unit area of the planet's surface. Note that this relation presumes that the depth of the layer containing almost all the mass of the atmosphere is sufficiently shallow that gravity can be considered constant throughout the layer. Given that gravity decays inversely with the square of distance from the planet's center, this is equivalent to saying that the atmosphere must be shallow compared to the radius of the planet. For a well mixed substance A with mass-specific concentration ka relative to the whole atmosphere, the mass of substance A per square meter of the planet's surface is just mKA

Using the perfect gas law to eliminate p from Eqn 2.19 yields dp = —^Lp (2.21)

where R is the gas constant for the mixture making up the atmosphere. This has the solution

1 /• z p(z) = ps exp( — RTz), T(z) = (-J T-1dz)-1 (2.22)

Here, T(z) is the harmonic mean of temperature in the layer between the ground and altitude z. If temperature is constant, then pressure decays exponentially with scale height RT/g. Because temperature is measured relative to absolute zero, the mean temperature T(z) can be relatively constant despite fairly large variations of temperature within the layer. In consequence, pressure typically decays roughly exponentially with height even when temperature is altitude-dependent.

Exercise 2.5.1 Compute the mass of the Earth's atmosphere, assuming a mean surface pressure of 1000mb. (The Earth's radius is 6378km, and the acceleration of its gravity is 9.8m/s2). Compute the mass of the Martian atmosphere, assuming a mean surface pressure of 6mb. (Mars' radius is 3390km, and the acceleration of its gravity is 3.7m/s2.)

Note that the hydrostatic relation applies only to the total pressure of all constituents; it does not apply to partial pressures individually. However, in the special case in which the gases are well mixed, the total mass of each well-mixed component can still be determined from surface data alone. One simply multiplies the total mass obtained from surface pressure, by the appropriate (constant) mass-specific concentration.

In the study of atmospheric dynamics, the hydrostatic equation is used to compute the pressure gradients which drive the great atmospheric circulations. Outside of dynamics, there are rather few problems in physics of climate that require one to know the altitude corresponding to a given pressure level. Our main use of the hydrostatic relation in this book will be in the form of Eqn 2.18, which tells us the mass between two pressure surfaces.

The hydrostatic relation also allows us to derive a useful alternate form of the heat budget, by re-writing the heat balance equation as follows:

¿Q = cpdT — p-1dp = cpdT — p-1 ^dz = d(cpT + gz) (2.23)

assuming cp to be constant. The quantity cpT + gz is known as the dry static energy. Dry static energy provides a more convenient basis for atmospheric energy budgets than entropy, since changes in dry static energy following an air parcel are equal to the net energy added to or removed from the parcel by heat sources such as solar radiation. For example, if there are no horizontal transports and if there is no net flux of energy between the atmosphere and the underlying planetary surface, then the rate of change of the net dry static energy in a (dry) atmospheric column is the difference between the rate at which solar energy flows into the top of the atmosphere and the rate at which infrared radiation leaves the top of the atmosphere; one needs to know nothing about how the heat is deposited within the atmosphere in order to determine how the net dry static energy changes. This is not the case for entropy.

Note that the dry static energy as defined above is actually the energy per unit mass of atmosphere. Thus, the total energy in a column of atmosphere, per unit surface area, is J(cpT + gz)pdz, which by the hydrostatic relation is equal to f (cpT + gz)dp/g if the pressure integral is taken in the direction of increasing pressure.

When a substance changes from one form to another (e.g. water vapor condensing into liquid water or gaseous carbon dioxide condensing into dry ice) energy is released or absorbed even if the temperature of the mass is unchanged after the transformation has taken place. This happens because the amount of energy stored in the form of intermolecular interactions is generally different from one form, or phase to another. The amount of energy released when a unit of mass of a substance changes from one phase to another, holding temperature constant, is known as the latent heat associated with that phase change. By convention, latent heats are stated as positive numbers, with the phase change going in the direction that releases energy. Phase changes are reversible. If one kilogram of matter releases L joules of energy in going from phase A to phase B, it will take the same L joules of energy to turn the mass back into phase A. The units of latent heat are energy per unit mass (Joules per kilogram in mks units).

Condensible substances play a central role in the atmospheres of many planets and satellites. On Earth, it is water that condenses, both into liquid water and ice. On Mars, CO2 condenses into dry ice in clouds and in the form of frost at the surface. On Jupiter and Saturn, not only water but ammonia (NH3) and a number of other substances condense. The thick clouds of Venus are composed of condensed sulfuric acid. On Titan it is methane, and on Neptune's moon Triton nitrogen itself condenses. Table 2.1 lists the latent heats for the liquid-vapor (evaporation), liquidsolid (fusion) and solid-vapor (sublimation) phase transitions are given for a number of common constituents of planetary atmospheres. Water has an unusually large latent heat; the condensation of 1 kg of water vapor into ice releases nearly five times as much energy as the condensation of 1kg of carbon dioxide gas into dry ice. This is why the relatively small amount of water vapor in Earth's present atmosphere can nonetheless have a great effect on atmospheric structure and dynamics. Ammonia also has an unusually large latent heat, though not so much so as water. In both cases, the anomalous latent heat arises from the considerable energy needed to break hydrogen bonds in the condensed phase.

Like most thermodynamic properties, latent heat varies somewhat with temperature. For example, the latent heat of vaporization of water is 2.5 • 106J/kg at 0C, but only 2.26 • 106 J/kg at 100C. For precise calculations, the variation of latent heat must be taken into account, but nonetheless for many purposes it will be sufficient to assume latent heat to be constant over fairly broad temperature ranges.

The three main phases of interest are solid, liquid and gas (also called vapor), though other phases can be important in exotic circumstances. There is generally a triple point in temperature-pressure space where all three phases can co-exist. Above the triple point temperature, the substance undergoes a vapor-liquid phase transition as temperature is decreased or pressure is increased; below the triple point temperature vapor condenses directly into solid, once thermo-dynamic equilibrium has been attained. For water, the triple point occurs at a temperature of 273.15K and pressure of 6.11mb (see Table 2.1 for other gases). Generally, the triple point temperature can also be taken as an approximation to the "freezing point" - the temperature at which a liquid becomes solid - because the freezing temperature varies only weakly with pressure until very large pressures are reached. Though we will generally take the freezing point to be identical to the triple point in our discussions, the effect of pressure on freezing of liquid can nonetheless be of great importance at the base of glaciers and in the interior of icy planets or moons, and perhaps also in very dense,cold atmospheres.

Typically, the solid phase is more dense than the liquid phase, but water again is exceptional. Water ice floats on liquid water, whereas carbon dioxide ice would sink in an ocean of liquid carbon dioxide, and methane ice would sink in a methane lake on Titan. This has profound consequences for the climates of planets with a water ocean such as Earth has, since ice formed in winter remains near the surface where it can be more readily melted when summer arrives.

Exercise 2.6.1 Per square meter, how many Joules of energy would be required to evaporate a puddle of Methane on Titan, having a depth of 20m?

Atmospheres can transport energy from one place to another by heating an air parcel by an amount ¿T, moving the parcel vertically or horizontally, and then cooling it down to its original temperature. This process moves an amount of heat cp^T per unit mass of the parcel. Latent heat provides an alternate way to transport energy, since energy can be used to evaporate liquid into an air parcel until its mixing ratio increases by ¿r, moving it and then condensing the substance until the mixing ratio returns to its original value. This process transports an amount of heat ¿¿r per unit mass of the planet's uncondensible air, and can be much more effective at transporting heat than inducing temperature fluctuations, especially when the latent heat is large. "Ordinary" heat - the kind that feels hot when you touch it, and which is stored in the form of the temperature increase of a substance - is known in atmospheric circles as "sensible" heat.

All gases are condensible at low enough temperatures or high enough pressures. On Earth (in the present climate) CO2 is not a condensible substance, but on Mars it is. The ability of a gas to condense is characterized by the saturation vapor pressure, psat of that gas, which may be a function of any number of thermodynamic variables. When the partial pressure pa of gas

A is below psat,A, more of the gas can be added, raising the partial pressure, without causing condensation. However, once the partial pressure reaches psat,A, any further addition of A will condense out. The state pa = psat,A is referred to as "saturated" with regard to substance A. Each condensed state (e.g. liquid or solid) will have its own distinct saturation vapor pressure. Rather remarkably, for a mixture of perfect gases, the saturation vapor pressure of each component is independent of the presence of the other gases. Water vapor mixed with 1000 mb worth of dry air at a temperature of 300K will condense when it reaches a partial pressure of 38mb; a box of pure water vapor at 300K condenses at precisely the same 38mb. If a substance "A" has partial pressure pa that is below the saturation vapor pressure, it is said to be "subsaturated," or "unsaturated." The degree of subsaturation is measured by the saturation ratio pa/psatiA, which is often stated as a percent. Applied to water vapor, this ratio is called the relative humidity, and one often speaks of the relative humidity of other substances, e.g. "methane relative humidity" instead of saturation ratios. Note that the relative humidity is also equal to the mixing ratio of the substance A in a given mixture to the mixing ratio the air would have if the substance were saturated. This is different from the ratio of specific humidity to saturation specific humidity, or the ratio of molar concentration to saturation molar concentration except when the mixing ratio is small.

It is intuitively plausible that the saturation vapor pressure should increase with increasing temperature, as molecules move faster at higher temperatures, making it harder for them to stick together to form condensate. The temperature dependence of saturation vapor pressure is expressed by a remarkable thermodynamic relation known as the Clausius-Clapeyron equation. It is derived from very general thermodynamic principles, via a detailed accounting of the work done in an reversible expansion-contraction cycle crossing the condensation threshold, and requires neither approximation nor detailed knowledge of the nature of the substance condensing. The relation reads djpTf = T -=TL--T (2.24)

dT T P- — -c where pv is the density of the less condensed phase, pc is the density of the more condensed phase, and L is the latent heat associated with the transformation to the more condensed phase. For vapor to liquid or solid transitions, pc ^ pv, enabling one to ignore the second term in the denominator of Eqn 2.24. Further, upon substituting for density from the perfect gas law, one obtains the simplified form dpsat _ L

where Ra is the gas constant for the substance which is condensing. If we make the approximation that L is constant, then Eqn 2.24 can be integrated analytically, resulting in

where To is some reference temperature. This equation shows that saturation water vapor content is very sensitive to temperature, decaying rapidly to zero as temperature is reduced and increasing rapidly as temperature is increased. The rate at which the change occurs is determined by the characteristic temperature appearing in the exponential. For the transition of water vapor to liquid, it has the value 5420K at temperatures near 300K. For CO2 gas to dry ice, it is 3138K, and for methane gas to liquid methane it is 1031K. Equation 2.25 seems to imply that the psat asymptotes to a constant value when T ^ L/Ra. This is a spurious limit, though, since the assumption of constant L invariably breaks down over such large temperature ranges. In fact, L typically approaches zero at some critical temperature, where the distinction between the two phases disappears. For water vapor, this critical point occurs at a temperature and pressure of 647.1K and 221bars. For carbon dioxide, the critical point occurs for the vapor-liquid transition,

Very High ]

Temperature (Kelvin)

Figure 2.4: The general form of a phase diagram showing the regions of temperature-pressure space where a substance exists in solid, liquid or gaseous forms. The triple point is marked with a black circle while the critical point is marked with a grey circle. The solid-liquid phase boundary for a "normal" substance (whose solid phase is denser than its liquid phase) is shown as a solid curve, whereas the phase boundary for water (ice less dense than liquid) is shown as a dashed curve. The critical point pressure is typically several orders of magnitude above the triple point pressure, while the critical point temperature is generally only a factor of two or three above the triple point temperature. Therefore, the pressure axis on this diagram should be thought of as logarithmic, while the temperature axis should be thought of as linear. This choice of axes also reflects the fact that the pressure must typically be changed by an order of magnitude or more to cause a significant change in the temperature of the solid/liquid phase transition.

Very Low

^High emperature

Pressure

Temperature (Kelvin)

Figure 2.4: The general form of a phase diagram showing the regions of temperature-pressure space where a substance exists in solid, liquid or gaseous forms. The triple point is marked with a black circle while the critical point is marked with a grey circle. The solid-liquid phase boundary for a "normal" substance (whose solid phase is denser than its liquid phase) is shown as a solid curve, whereas the phase boundary for water (ice less dense than liquid) is shown as a dashed curve. The critical point pressure is typically several orders of magnitude above the triple point pressure, while the critical point temperature is generally only a factor of two or three above the triple point temperature. Therefore, the pressure axis on this diagram should be thought of as logarithmic, while the temperature axis should be thought of as linear. This choice of axes also reflects the fact that the pressure must typically be changed by an order of magnitude or more to cause a significant change in the temperature of the solid/liquid phase transition.

at 304.2K and 73.825 bars. Critical points for other atmospheric gases are shown in Table 2.1. At high pressures, the solid/liquid phase boundary does not typically terminate in a critical point, but instead gives way to a bewildering variety of distinct solid phases distinguished primarily by crystal structure.

Exercise 2.6.2 Show that the slope d lnpsat/dT becomes infinite as T ^ 0. Show that it decreases monotonically with T provided the latent heat decreases or stays constant as T increases. Show that the curve psat(T) is infinitely flat near T = 0, in the sense that all the derivitives dnpsat/dTn vanish there. In Fig. 2.4 why is the curvature of the phase boundary sketched the way it is at low temperature?

Figure 2.4 summarizes the features of a typical phase diagram. Over ranges of a few bars of pressure, the solid-liquid boundary can be considered nearly vertical. In fact the exact form of the Clausius-Clapeyron relation (Eq. 2.24) tells us why the boundary is nearly vertical and how it deviates from verticality. Because the difference in density between solid and liquid is typically quite small while the latent heat of fusion is comparatively large, Eq. 2.24 implies that the slope dp/dT is very large (i.e. nearly vertical. The equation also tells us that in the "normal" case where

Saturation Vapor Pressure for Water

Saturation Vapor Pressure for Water

Figure 2.5: Saturation vapor pressure for water, based on the constant-L form of the Clausius-Clapeyron relation. Curves are shown for vapor pressure based on the latent heat of vaporization, and (below freezing) for latent heat of sublimation. The latter is the appropriate curve for sub-freezing temperatures.

Figure 2.5: Saturation vapor pressure for water, based on the constant-L form of the Clausius-Clapeyron relation. Curves are shown for vapor pressure based on the latent heat of vaporization, and (below freezing) for latent heat of sublimation. The latter is the appropriate curve for sub-freezing temperatures.

ice is denser than liquid, the phase boundary tilts to the right, and so the freezing temperature increases with pressure; at fixed pressure, one can cause a cold liquid to freeze by squeezing it. The unusual lightness of water ice relative to the liquid phase implies that instead the phase boundary tilts to the left; one can melt solid ice by squeezing it. Substituting the difference in density between water ice and liquid water, and the latent heat of fusion, into Eq. 2.24, we estimate that 1006ars of pressure decreases the freezing point temperature by about .74K. This is roughly the pressure caused by about a kilometer of ice on Earth. The effect is small, but can nonetheless be significant at the base of thick glaciers.

Below the triple point temperature, the favored transition is gas/solid,and so the appropriate latent heat to use in the Clausius-Clapeyron relation is the latent heat of sublimation. Above the triple point, the favored transition is gas/liquid, whence one should use the latent heat of vaporization. The triple point (T,p) provides a convenient base for use with the simplified Clausius-Clapeyron solution in Eqn. 2.26, or indeed for a numerical integration of the relation with variable L. Results for water vapor are shown in Figure 2.5. These results were computed using the constant L approximation for sublimation and vaporization, but in fact a plot of the empirical results on a logarithmic plot of this type would not be distinguishable from the curves shown. The more exact result does differ from the constant L idealization by a few percent, which can be important in some applications. Be that as it may, the figure reveals the extreme sensitivity of vapor pressure to temperature. The vapor pressure ranges from about .1 Pascals at 200K (the tropical tropause temperature) to 35mb at a typical tropical surface temperature of 300K, rising further to 100mb at 320K. Over this span of temperatures, water ranges from a trace gas to a major constituent; at temperatures much above 320K, it rapidly becomes the dominant constituent of the atmosphere. Note also that the distinction between the ice and liquid phase transitions has a marked effect on the vapor pressure. Because the latent heat of sublimation is larger than the latent heat of vaporization, the vapor pressure over ice is lower than the vapor pressure over liquid would be, at subfreezing temperatures. At 200K, the ratio is nearly a factor of three.

Exercise 2.6.3 Let's consider once more the case of the airliner cruising at an altitude of 300mb, discussed in an earlier Exercise. Suppose that the ambient air at flight level has 100% relative humidity. What is the relative humidity once the air has been brought into the cabin, compressed to 1000mb, and chilled to a room temperature of 290K?

Once the saturation vapor pressure is known, one can compute the molar or mass mixing ratios with respect to the background non-condensible gas, if any, just as for any other pair of gases. The saturation vapor pressure is used in this calculation just like any other partial pressure. For example, the molar mixing ratio is just psat/pa, if pa is the partial pressure of the noncondensible background. Note that, while the saturation vapor pressure is independent of the pressure of the gas with which the condensible substance is mixed, the saturation mixing ratio is not.

Exercise 2.6.4 What is the saturation molar mixing ratio of water vapor in air at the ground in tropical conditions (1000mb and 300K)? What is the mass mixing ratio? What is the mass-specific humidity? What is the molar mixing ratio (in ppm) of water vapor in air at the tropical tropopause (100mb and 200K)?

When air is lifted, it cools by adiabatic expansion, and if it gets cold enough that one of the components of the atmosphere begins to condense, latent heat is released. This makes the lifted air parcel warmer than the dry adiabat would predict. Less commonly, condensation may occur as a result of subsidence and compression, since the increase of partial pressure of one of the compressed atmospheric components may overwhelm the increase in saturation vapor pressure resulting from adiabatic warming. Whichever direction leads to condensation, if we assume further that the condensation is efficient enough that it keeps the system at saturation, the resulting temperature profile will be referred to as the moist adiabat, regardless of whether the condensing substance is water vapor (as on Earth) or something else (CO2 on Mars or methane on Titan). We now proceed to make this quantitative.

The simplest case to consider is that of a single component atmosphere, which can attain cold enough temperatures to reach saturation and condense. This case is relevant to present Mars, which has an almost pure CO2 atmosphere that can condense in the cold Winter hemisphere and at upper levels at any time of year. A pure CO2 atmosphere with a surface pressure on the order of two or three bars is a commonly used model of the atmosphere of Early Mars, though the true atmospheric composition in that instance is largely a matter of speculation. Another important application of a single component condensible atmosphere is the pure steam (water vapor) atmosphere, which occurs when a planet with an ocean gets warm enough that the mass of water which evaporates into the atmosphere dominates the other gases that may be present. This case figures prominenently in the runaway greenhouse effect that will be studied in Chapter 4.

For a single component atmosphere, the partial pressure of the condensible substance is in fact the total atmospheric pressure. Therefore, at saturation, the pressure is related to the temperature by the Clausius-Clapeyron relation. To find the saturated moist adiabat, we simply solve for T in terms of psat in the Clausius-Clapeyron relation, and recall that p = psat because we are assuming the atmosphere to be saturated everywhere. Using the simplified form of Clausius-

Clapeyron given in Eqn 2.26, the saturated moist adiabat is thus

T T o where R is the gas constant for the substance making up the atmosphere. This equation is really just an alternate form of Clausius-Clapeyron. It can be thought of as a formula for the "dew-point" or "frost-point" temperature corresponding to pressure p: given a box of gas at fixed pressure p, condensation will occur when the temperature is made lower than the temperature T(p) given by Eq. 2.27.

Without loss of generality, we may suppose that To is taken to be the surface temperature, so that psat(To) is the surface pressure ps. Since the logarithm is negative, the temperature decreases with altitude (recalling that lower pressure corresponds to higher altitude). Further, the factor multiplying the logarithm is the ratio of the surface temperature to the characteristic temperature L/R. Since the characteristic temperature is large, the prefactor is small, and as a result the temperature of saturated adiabat for a one-component atmosphere varies very little over a great range of pressures. For example, in the case of the CO2 vapor-ice transition, an atmospheric surface pressure of 7mb (similar to that of present Mars) would be in equilibrium with a surface dry-ice glacier at a temperature of 149K; at .07mb - one one-hundredth of the surface pressure -the temperature on the saturated adiabat would only fall to 122K.

Exercise 2.7.1 In the above example, what would the temperature aloft have been if there were no condensation and the parcel were lifted along the dry adiabat?

The criterion determining whether condensation occurs on ascent or descent for an arbitrary one-component atmosphere is derived in Problem ??. Note that for a single-component saturated atmosphere, the supposition that the atmosphere is saturated is sufficient to determine the temperature profile, regardless of the means by which the saturation is maintained. Thus, it does not actually matter to the result whether the saturation is maintained by condensation due to vertical displacement of air parcels, or some other physical mechanism such as radiative cooling of the atmosphere to the point that condensation occurs.

Unless there is a reservoir of condensate at the surface to maintain saturation, it would be rare for an atmosphere to be saturated all the way to the ground. Suppose now that a one-component atmosphere has warm enough surface temperature that the surface pressure is lower than the saturation vapor pressure computed at the surface temperature. In this case, when a parcel is lifted by convection, its temperature will follow the dry or noncondensing adiabat, until the temperature falls so much that the gas becomes saturated. The level at which this occurs is called the lifted condensation level. Above the lifted condensation level, ascent causes condensation and the parcel follows the saturated adiabat. Since the temperature curve along the saturated adiabat falls with altitude so much less steeply than the dry adiabat, it is very easy for the two curves to intersect provided the surface temperature is not exceedingly large. An example for present Summer Martian conditions (specifically, like the warmest sounding in 2.2) is shown in Figure 2.6. A comparison with the Martian profiles in Figure 2.2 indicates that something interesting is going on in the Martian atmosphere. For the warm sounding, whose surface temperature is close to 255K, the entire atmosphere aloft is considerably warmer than the adiabat, and the temperature nowhere comes close to the condensation threshold, though the very lowest portion of the observed atmosphere, below the 200 Pa level, does follow the dry adiabat quite closely. Clearly, something we haven't taken into account is warming up the atmosphere. A likely candidate for the missing piece is the absorption of solar energy by CO2 and dust.

Figure 2.6: The adiabatic profile for a pure CO2 atmosphere with a surface pressure of 450 Pa (4.5mb) and a surface temperature of 255K. The conditions are similar to those encountered on the warmer portions of present-day Mars.

Although results like Figure 2.6 show a region of weak temperature dependence aloft which bears a superficial resemblance to the stratosphere seen in Earth soundings (and also at the top of the Venus, Jupiter and Titan soundings), one should not jump to the conclusion that the stratosphere is caused by condensation. This is not generally the case, and there are other explanations for the upper atmospheric temperature structure, which will be taken up in the next few chapters.

2.7.2 Mixtures of condensible with noncondensible gases

As a final step up on the ladder of generality, let's consider a mixture of a condensible substance with a substance that doesn't condense under the range of temperatures encountered in the atmosphere under consideration. This might be a mixture of condensible methane on Titan with non-condensible nitrogen, or condensible carbon dioxide on Early Mars with non-condensible nitrogen, or water vapor on Earth with a non-condensible mixture of oxygen and nitrogen. Whatever the substance, we distinguish the properties of the condensible substance with the subscript "c," and those of the non-condensible substance by the subscript "a" (for "air"). We now need to do the energy budget for a parcel of the mixture, assuming that it is initially saturated, and that it is displaced in such a way that condensation (releasing latent heat) must occur in order to keep the parcel from becoming supersaturated. We further introduce the assumption that essentially all of the condensate is immediately removed from the system, so that the heat storage in whatever mass of condensate is left in suspension may be neglected. This is a reasonable approximation for water or ice clouds on Earth, but even in that case the slight effect of the mass of retained condensate on buoyancy can be significant in some circumstances. In other planetary atmospheres the effect of retained condensate could be of greater importance. The temperature profile obtained by assuming condensate is removed from the system is called a pseudoadiabat, because the process is not truly reversible. One cannot return to the original saturated state, because the condensate is lost. At the opposite extreme, if all condensate is retained, it can be re-evaporated when the parcel is compressed, allowing for true reversibility.

Let the partial pressure, density, molecular weight, gas constant and specific heat of the noncondensible substance be pa, pa,Ma, Ra, and cpa, and similarly for the condensible substance. Further, let L be the latent heat of the phase transition between the vapor and condensed phase of the condensible substance, and let pc,sat(T) be the saturation vapor pressure of this substance, as determined by the Clausius-Clapeyron relation. The assumption of saturation amounts to saying that pc = pc,sat(T); if the parcel weren't at saturation, there would be no condensation and we could simply use the dry adiabat based on a noncondensing mixture of substance "a" and "c."

Now consider a parcel consisting of a mass ma of noncondensible gas with an initial mass mc of condensible gas occupying a volume V. Suppose that the temperature is changed by an amount dT, the partial pressure of noncondensible gas is changed by an amount dpa, and the partial pressure of the condensible gas is changed by an amount dpc. The mechanical work term for the mixture is Vd(pa + pc), but by the definition of density V = ma/pa = mc/pc. Thus, the total heat budget of the parcel can be written in the form mc

(ma + mc)£Q = macpadT--dpa + mccpcdT--dpc + Ldmc (2.28)

Pa Pc where mc is the mass of condensible substance in the vapor phase. Thus, when some vapor is condensed out, dmc is negative, yielding a negative contribution to ¿Q when L is positive. Since condensation releases latent heat, this sign may seem counter-intuitive. Recall, however, that condensation puts the system in a lower energy state, which is why there is heat available to be "released." If the system is energetically closed (¿Q = 0), then the negative contribution of the latent heat term is offset by an increase in the remaining terms - e.g. an increase in temperature.

Exercise 2.7.2 Show that when there is no condensation (dmc = 0) the adiabat T(p) obtained by setting SQ = 0 has the same form as the usual dry adiabat, but with R and cp taking on the appropriate mean values for the mixture.

The heat budget does not contain any term corresponding to heat storage in the condensed phase, since it is assumed that all condensate disappears from the parcel by precipitation. The usual way to change dpa would be by lifting, causing expansion and reduction of pressure. Now, we divide by mcT, make use of the perfect gas law to substitute for pa and pc, and make use of the fact that mc/ma = (Mc/Ma)(pc/pa), since mc/ma is just the mass mixing ratio, denoted henceforth by rc. This yields n . JQ dT dPa} , dT dPc, , L

(1 + rc)^F = [Cpa^ - Ra-] + [cpcrc ~T - rcRc-] + T^c (2.29)

The first two bracketed groups of terms on the right hand side can be recognized as the contribution of the two substances to the noncondensing entropy of the mixture, weighted according to the relative abundance of each species. If there is no condensation, the mixing ratio is conserved as the parcel is displaced to a new pressure, drc = 0, and the expression reduces to the equivalent of Eqn. 2.9, leading to the dry adiabat for a mixture. At this point, we introduce the saturation assumption, which actually consists of two parts: First, we assume that the air parcel is initially saturated, so that before being displaced, pc = pc,sat(T) and rc = rsat = epc,sat(T)/pa, where e is the ratio of molecular weights Mc/Ma and pc,sat(T) is determined by the Clausius-Clapeyron relation. Second, we assume that a displacement conserving rc would cause supersaturation, so that condensation would occur and bring the partial pressure pc back to the saturation vapor pressure corresponding to the new value of T. Usually, this would occur as a result of ascent and cooling, since cooling strongly decreases the saturation vapor pressure. Typically (though not inevitably), the effect of compressional warming on saturation vapor pressure dominates the effect of increasing partial pressure, so that subsidence of initially saturated air follows the dry adiabat.

Assuming that the displacement causes condensation, we may replace pc by pc,sat(T) and rc by rsat everywhere in Eqn. 2.29. Next, we use Clausius-Clapeyron to re-write dpc,sat, observing that dpc,sat ,, d lnpc,sat ,rp /n on\

pc,sat dT

and drsat = ed c,sat = e — d ln — = rsat • (d ln pc,sat - d ln pa) (2.31)

pa pa pa

Upon substituting into Equation 2.29 and collecting terms in d ln T and d ln pa we find

(1 + rsat) 5T = (cpa + (cpc + ( rRLt - 1) ^ >sat)d ln T - (1 + R^^Rad ln pa (2.32)

To obtain the adiabat, we set SQ = 0, which leads to the following diferential equation defining ln T as a function of ln pa:

d ln T = Ra 1 + RLt rsat d ln pa Cpa 1 + ( + ( RLt - 1) cpLT )rsat )

Note that this expression reduces to the dry adiabat, as it should, when rsat ^ 0.

Exercise 2.7.3 What would the the slope d lnT/dlnpa be for a noncondensing mixture of the two gases? (Hint: lnp = lnpa + const. in this case). Why doesn't Eq. 2.33 reduce to this value as L ^ 0? (Hint: Think about the way Clausius-Clapeyron has been used in deriving the moist adiabat, and what it implies for variations of pc.)

A displaced parcel of atmosphere will follow the moist adiabat if the condensible substance condenses in the course of the displacement, but it will follow the dry adiabat if the displacement causes the condensible substance to become subsaturated. Does condensation occur on ascent (which lowers the total pressure) or descent (which raises the total pressure)? To answer this question, we must compare the moist adiabatic slope d ln T/d lnp computed from Eq. 2.33 with the dry adiabatic slope R/cp, with R and cp computed as the appropriate weighted average for the mixture. When the moist adiabatic slope is lower than R/cp,then lifting a parcel adiabatically creates enough cooling that the parcel becomes supersaturated, and condensation occurs. Conversely, if the moist adiabatic slope is less than R/cp, condensation occurs on descent instead.

The only problem with applying this criterion is that Eq. 2.33 gives us d ln T/d lnpa whereas we need d ln T/d ln p. Here, we will restrict attention to the dilute case, where there is little enough condensible substance present that p « pa. In this case d ln T/d lnp « d ln T/d lnpa and R/cp « Ra/cp a. The opposite limit of the condensible-dominated atmosphere is done in Problem ??, and the general case is done in Problem ??.

An examination of the properties of gases indicates that cpc/cpa is typically of order unity, whereas L/(RcT) is typically very large, so long as the temperature is not exceedingly great. If one drops the smaller terms from the denominator of Eq. 2.33, one finds that the temperature gradient along the moist adiabat is weaker than that along the dry adiabat provided eL/cpaT > 1, which is typically the case. In this typical case, condensation occurs on ascent and warms the ascending parcel through the release of latent heat. This behavior can fail when the latent heat is weak or the noncondensible specific heat is very large, whereupon the heat added by condensation has little effect on temperature. It is in this regime that condensation happens on descent rather than ascent. Given the thermodynamic properties of common atmospheric constituents, it is not an easy regime to get into. Even the case of H2O condensation in an H2 Jovian or Saturnian atmosphere yields condensation on ascent, despite the high specific heat of H2. In that case, the high specific heat of H2 is canceled out by the low value of e, which is typical behavior since specific heat tends to scale inversely with molecular weight - substances with low molecular weight have more degrees of freedom per kilogram. The quantity eL/cpaT decreases with temperature, especially since L approaches zero as the critical point is approached, and this suggests that condensation might occur during descent at very high temperatures. A proper evaluation of the possibility would require taking into account non-ideal gas effects, the temperature dependence of specific heat, the neglected condensate density term in Clausius-Clapeyron, and probably also non-dilute effects.

Everything on the right hand side of Eqn 2.33 is either a thermodynamic constant, or can be computed in terms of ln T and ln pa. Therefore, the equation defines a first order ordinary differential equation which can be integrated (usually numerically) to obtain T as a function of pa. Usually one wants the temperature as a function of total pressure, rather than partial pressure of the noncondensible substance. This is no problem. Once T(pa) is known, the corresponding total pressure at the same point is obtained by computing p = pa + pc,saí(T(pa)). To make a plot, or a table, one treats the problem parametrically: computing both T and p as functions of pa. When the condensible substance is dilute, then pc,sat << pa, and p « pa, so Eqn 2.33 gives the desired result directly.

Figure 2.7 shows a family of solutions to Eqn 2.33, for the case of water vapor in Earth air. When the surface temperature is 250K, there is so little moisture in the atmosphere that the

Moist adiabat, water vapor in air

1111111111111111111111111111111111

10° ^ llllllllllllllllllllllllllllllllE

0 50 100 150 200 250 300 350 T

Molar concentration of water vapor

w1000 ft

0.0001 0.001 0.01 0.1 1 q (parts water per part total)

Figure 2.7: The moist adiabat for saturated water vapor mixed with Earth air having a partial pressure of 1 bar at the surface. Results are shown for various values of surface temperature, ranging from 250K to 350K. The left panel shows the temperature profile, while the right shows the profile of molar concentration of water vapor. A concentration value of .1 would mean that one molecule in 10 of the atmosphere is water vapor.

profile looks like the dry adiabat right to the ground. As temperature is increased, a region of weak gradients appears near the ground, representing the effect of latent heat on temperature. This layer gets progressively deeper as temperature increases and the moisture content of the atmosphere increases. When the surface temperature is 350K, so much moisture has entered the atmosphere that the surface pressure has actually increased to over 1300mb. Moreover, the moisture-dominated region extends all the way to 10 Pa (.1mb) , and even at 100 Pa (1mb) the atmosphere is 10% water by volume. Thus, for moderate surface temperatures, there is little water high up in the atmosphere. When the surface temperature approaches or exceeds 350K, though, the "cold trap" is lost, and a great deal of water is found aloft, where it is exposed to the destructive ultraviolet light of the sun and the possibility of thermal escape to space. In subsequent chapters, it will be seen that this phenomenon plays a major role in the life cycle of planets, and probably accounts for the present hot, dry state of Venus.

The mixing ratio of the condensible component varies along the moist adiabat, so the reader may wonder whether these variations can lead to compositionally induced buoyancy along the lines discussed in Section 2.4. Suppose a parcel starts on the moist adiabat at pressure p, with temperature T(p), and that the parcel is saturated. For the sake of definiteness, let's suppose also that condensation occurs on ascent, so ascending displacements are saturated and remain on the moist adiabat. Since there is a unique moist adiabat going through the starting point, and since saturation also determines the composition uniquely once temperature and pressure are known, then when the parcel is lifted to pressure pi it has the same temperature, composition and density as the ambient air there. Composition doesn't induce buoyancy, because the variations of R/cp are constrained by the saturation assumption, and have been taken into account in the computation of the moist adiabat. In this sense, the moist adiabat is neutrally stable against condensing ascent. On the other hand, a subsiding air parcel will become subsaturated, and descending along the dry adiabat will become warmer than the surrounding air, causing it to be positively buoyant. In this sense, the moist adiabat is stable against subsidence 4.

Saturated air can go up without expenditure of energy, but it takes energy to push it down. However, when an air parcel is initially subsaturated and must be lifted some distance before it becomes saturated and follows the moist adiabat, then the full range of issues discussed in Section 2.4 come into play. In that case, the compositional variations in the subsaturated layer can either favor or inhibit the triggering of vertical mixing. On a related note, suppose that the temperature profile T(p) happens to follow the moist adiabat, but that the air is unsaturated; this is in fact the case over most of the Earth's tropics, for dynamical reasons that are beyond the scope of this book. In that case, if a saturated air parcel from the ocean surface is lifted through the dry surroundings, it will follow the moist adiabat but because its composition will be different from the unsaturated surroundings it will in fact be positively buoyant if the condensible has lower molecular weight than the noncondensible background, as is the case for water vapor in Earth air. If the condensible has higher moleculer weight, the parcel will be negatively buoyant and vertical mixing will be choked off. Real atmospheres consist of a mix of saturated and unsaturated regions, and the representation of vertical mixing in these cases pose a considerable challenge. Compositional effects on Earth at its present temperature are slight, but for atmospheres in which the condensible is nondilute, these effects become more and more important. With regard to the modelling of convection, this represents largely unexplored territory. It is probably important for the case of methane on Titan, but would also be important on Earthlike planets having temperatures some tens of degrees or more warmer than the present Earth.

The mass of retained condensate - cloud mass - can also affect the buoyancy of lifted or subsiding parcels, since condensate alters the density of an air parcel. In such cases one also needs to consider cooling due to evaporation of condensate when the air parcel subsides, warms and becomes subsaturated. In quiescent conditions retained condensate mass is expected to be a small fraction of the total mass of an air parcel, since a large condensate loading usually leads to coalescence of cloud droplets and subsequent removal by precipitation. The effects of condensate loading are small, but significant, to convection in the present Earth's atmosphere, but the possibility should be kept in mind that condensate loading may play a more prominent role in planetary atmospheres that have not been as extensively studied as Earth's.

When the condensible substance is dilute, it is easy to define a moist static energy which is a generalization of the dry static energy defined in Eq. 2.23. This is accomplished by multiplying Eq. 2.29 by T to obtain the heat budget, dropping the terms proportional to rc (which are small in the dilute limit), and making use of RaT/pa = 1/pa « 1/p. The hydrostatic relation is used to rewrite the pressure work term in precisely the same way as was done for dry static energy. Then, if we further assume that the temperature range of interest is small enough that L may be regarded as constant, we obtain

4 Compositional effects will alter the buoyancy of subsiding parcels, and in principle could make subsiding parcels negatively buoyant in extreme cases. For typical atmospheric gases the compositional effect is too small to overwhelm the positive buoyancy due to compressional heating, but if one could find a situation where composition rendered subsiding parcels unstable to descent, the resulting convection would be very interesting indeed to study.

Note that this relation does not assume that the condensible mixing ratio rc is saturated. For adiabatic flow in the dilute case, this moist static energy will be conserved following an air parcel whether or not condensation occurs. If energy is added to or taken away from the parcel, then the change in moist static energy is equal to the net amount of energy added or taken away, per unit mass of the parcel.

When the condensate is nondilute, things are a bit more complicated. In this case significant amounts of mass can be lost from the atmosphere in the course of condensation, and in essence the precipitation of condensate can take away significant amounts of energy with it. In order to deal with the heat storage in condensate, one must make use of the specific heat of the condensed phase, which we'll refer to as cpc£ (regardless of whether the condensate is liquid or solid); the behavior of this specific heat is inextricably linked to the changes in latent heat with temperature through the thermodynamic relation dL

which is valid when the condensate density is much greater than the vapor density. This relation is essentially a consequence of energy conservation. Given this link, we will take into account the change of latent heat with temperature in carrying out the following analyis.

First let's analyze the energy budget per unit mass of the noncondensible substance. We'll write the pressure work term in Eq. 2.28 in the alternate form (ma/pa)dp. On dividing the equation by ma and using Eq. 2.35 and the hydrostatic relation we obtain

(1 + rc)£Q = d[(cpa + rcCpC£)T + (1 + rc)gz + L(T)r] - (c^T + gz)drc (2.36)

The quantity in square brackets is thus identified as the moist static energy per unit mass of noncondensible substance; it will be denoted by the symbol M. The second term on the right hand side, involving drc, represents the sink of moist static energy due to the heat and potential energy carried away by the condensate.

This form of moist static energy can be inconvenient to use, because the (1 + rc) weighting on the left hand side makes it hard to do the energy budget of a column of air knowing only the net input of energy into the column. The expression also becomes inconvenient when the atmosphere becomes dominated by the condensible substance, leading to very large values of rc. We can formulate the moist static energy per unit total mass of the gas by dividing Eq. 2.36 by (1 + rc). After carrying out a few basic manipulations, we find

¿Q = d[—-] - (cp^T + gz - —-)dln(1 + rc) (2.37)

The term M/(1 + rc) is the desired moist static energy per unit of total gaseous mass, and the second term is the corresponding sink due to precipitation.

Exercise 2.7.4 Rewrite the expression M/(1 + rc) in terms of the mass specific concentration q. What is the form of this expression when q « 1? What happens to the precipitation sink term in this limit?

An alternate approach to dealing with moist static energy in the nondilute case is to write an energy budget per unit total mass (condensate included) for an air parcel that retains its condensate. To allow for precipitation, one then deals explicitly with the energy loss occuring when some of the retained condensate is removed from the air parcel. This approach is especially useful in situations when the mass of retained condensate can be appreciable. It can be carried out using the moist entropy expressions derived in Emanuel (1994) (see Further Readings). Further modifications to the expression for moist static energy are required if both ice and liquid phases are present in the atmosphere, in order to account for the latent heat of the solid/liquid phase transition.

Because the change in moist static energy of an atmospheric column can be determined from the energy fluxes into the top of the column and the energy fluxes between the bottom of the column and the underlying surface, moist static energy provides a convenient basis for diagnostics of atmospheric energy flows. For the same reason, it provides a convenient basis for the formulation of simplified vertically-averaged energy balance models of climate. Such models are taken up briefly in Chapter 9. Moist static energy is also important to the formulation of simplified representations of buoyancy-driven vertical mixing an a statically unstable layer of the atmosphere. In such applications, when radiation or some other process creates a region of static instability, it is presumed that mixing will proceed until some layer of the atmosphere is reset to a state of neutral stability. The constraint that the moist static energy following mixing should be the same as that of the initial state (within a suitably chosen layer of atmosphere) plays a crucial role in determining what the temperature profile is following mixing.

2.8 For Further Reading

For a deeper discussion of thermodynamics, see

• Feynman RP, Leighton RB and Sands M 2005: The Feynman Lectures on Physics, Vol 1. Addison Wesley.

A physically based derivation of Clausius-Clapeyron is found in Section 45-3, and a good discussion of entropy is found in Section 44-6. A very thorough discussion of moist thermodynamics and of the representation of the effects of moist convection on the Earth's atmosphere can be found in

• Emanuel KA 1994: Atmospheric Convection. Oxford University Press.

An example of the computation of the dilute moist adiabat for a non-ideal noncondensible background gas can be found in the Appendix of

• Kasting JF 1991: CO2 Condensation and the Climate of Early Mars. Icarus 94 1-13.

The extension to the nondilute case would be straightforward were it not for the fact that the saturation vapor pressure can no longer be computed independently of the noncondensible gas state. This subject is under investigation for the N2/CH4 system on Titan, and may also be relevant for non-ideal mixed CO2/H2O mixtures on hot, wet planets such as Early Venus. The planetary implications of non-ideal non-dilute behavior are still at a very early stage of development.

A very nice analysis of the way the mixing of inhomogeneities increases entropy is given in

• Pauluis O and Held IM: Entropy budget of an atmosphere in radiative-convective equilibrium. Part II: Latent heat transport and moist processes. J. Atmos. Sci. 59 140-149.

This particular analysis applies to the mixing of moist and dry air, and shows that the mixing is a big part of what makes moist convection irreversible.

For thermodynamic properties of gases and associated phase transitions, the NIST Chemistry WebBook and the Air Liquide Gas Encyclopedia, found at

• http://encyclopedia.airliquide.com/encyclopedia.asp

are very useful. Properties for more exotic conditions can often be found in the Journal of Physical and Chemical Reference Data and results pertinent to planetary atmospheres are often reported in the journal Icarus.

Was this article helpful?

## Post a comment