# Thermodynamics in a Nutshell

2.1 Overview

The atmospheres which are our principal objects of study are made of compressible gases. The compressibility has a profound effect on the vertical profile of temperature in these atmospheres. As things progress it will become clear that the vertical temperature variation in turn strongly influences the planet's climate. To deal with these effects it will be necessary to know some thermodynamics - though just a little. This chapter does not purport to be a complete course in thermodynamics. It can only provide a summary of the key thermodynamic concepts and formulae needed to treat the basic problems of planetary climate. It is assumed that the student has obtained (or will obtain) a more fundamental understanding of the general subject of thermodynamics elsewhere.

### 2.2 A few observations

The temperature profile in Figure 2.1, measured in the Earth's tropics introduces most of the features that are of interest in the study of general planetary atmospheres. It was obtained by releasing an instrumented balloon (radiosonde) which floats upwards from the ground, and sends back data on temperature and pressure as it rises. Pressure goes down monotonically with height, so the lower pressures represent greater altitudes. The units of pressure used in the figure are millibars (mb). One bar is very nearly the mean sea-level pressure on Earth, and there are 1000 mb in a bar.

Pressure is a very natural vertical coordinate to use. Many devices for measuring atmospheric profiles directly report pressure rather than altitude, since the former is generally easier to measure. More importantly, most problems in the physics of climate require knowledge only of the variation of temperature and other quantities with pressure; there are relatively few cases for which it is necessary to know the actual height corresponding to a given pressure. Pressure is also important because it is one of the fundamental thermodynamic variables determining the state of the gas making up the atmosphere. Atmospheres in essence present us with a thermodynamic diagram conveniently unfolded in height. Throughout, we will use pressure (or its logarithm) as our fundamental vertical coordinate.

1000

March 15 1993, 12Z 2.00 S 169.02 W 150 180 210 240 270 300 Temperature (Kelvin)

March 15 1993, 12Z 2.00 S 169.02 W

1000 0 6 12 18 24 Altitude (kilometers)

Figure 2.1: Left panel: Temperature profile measured at a point in the tropical Pacific. Right panel:The corresponding altitude. The measurements were obtained from a radiosonde ("weather balloon") launched at 12Z (an abbreviaton for Greenwich Mean Time) on March 15, 1993.

However, for various reasons one might nevertheless want to know at what altitude a given pressure level lies. By altitude tracking of the balloon, or using the methods to be described in Section 2.3, the height of the measurement can be obtained in terms of the pressure. The right panel of Figure 2.1 shows the relation between altitude and pressure for the sounding shown in Figure 2.1. One can see that the height is very nearly linearly related to the log of the pressure. This is the reason it is often convenient to plot quantities vs. pressure on a log plot. If po is representative of the largest pressure of interest, then — ln(p/po) is a nice height-like coordinate, since it is positive and increases with height.

We can now return to our discussion of the critical aspects of the temperature profile. The most striking feature of the temperature sounding is that the temperature goes down with altitude. This is a phenomenon familiar to those who have experienced weather in high mountains, but the sounding shows that the temperature drop continues to altitudes much higher than sampled at any mountain peak. This sounding was taken over the Pacific Ocean, so it also shows that the temperature drop has nothing to do with the presence of a mountain surface. The temperature drop continues until a critical height, known as the tropopause, and above that height (100mb, or 16 km in this sounding) begins to increase with height 1. The portion of the atmosphere below the tropopause is known as the troposphere, whereas the portion immediately above is the stratosphere. "Tropo" comes from the Greek root for "turning" (as in "turning over"), while "Strato.." refers to stratification. The reasons for this terminology will become clear shortly. The stratosphere was discovered in 1900 by Leon Phillipe Teisserenc de Bort, the French pioneer of instrumented balloon flights.

The sounding we have shown is typical. In fact, a similar pattern is encountered in the atmospheres of many other planets, as indicated in Figure 2.2 for Venus, Mars, Jupiter and Titan. In common with the Earth case, the lower portions of these atmospheres exhibits a sharp decrease of temperature with height, which gives way to a region of more gently decreasing, or even increasing, temperature at higher altitudes. In the case of Venus, it is striking that measurements taken with two completely different techniques, probing different locations of the atmosphere at different times of day and separated by a decade, nonetheless agree very well in the region of overlap of the measurements. This attests both to the accuracy of the measurement techniques, and the lack of what we would generally call "weather" on Venus, at least insofar as it is reflected in temperature variability. In the case of Mars, the temperature decrease shows most clearly in the summer afternoon when the surface is still warm from the Sun. As nignt approaches the upper level temperature decrease is still notable, but the lower atmosphere cools rapidly leading to a low level inversion, or region of temperature increase. In the Martian polar winter,the whole atmosphere cools markedly, and is much more isothermal than in the other cases.

The temperature decrease with height in the Earth's atmosphere has long been known from experience of mountain weather. It became a target of quantitative investigation not long after the invention of the thermometer, and was early recognized as a challenge to those seeking an understanding of the atmosphere. It was one of the central pre-occupations of the mountaineer and scientist Horace Benedict de Saussure (1740-1799). In the quest for an explanation, many false steps were taken, even by greats such as Fourier, before the correct answer was unveiled. As will be shown in the remainder of this chapter, some simple ideas based on thermodynamics and vertical mixing provide at least the core of an explanation for the temperature decrease with

1 Though the temperature minimum in vertical profiles of the Earth's atmosphere is the most obvious indicator of a transition between distinct layers, we'll eventually be able to provide a more dynamically based definition of the tropopause. In the definition we'll ultimately settle on, the tropopause need not be marked by a temperature minimum, though on Earth the temperature minimum lies only somewhat above the dynamically defined tropopause, and therefore serves as an approximate indicator of the tropopause location.

height. Towards the end of Chapter 3 we will introduce a theory of tropopause height that captures the essence of the problem; the theory of tropopause height will be revisited with increasing sophistication at various points in Chapters 4, 5 and 6. Nonetheless, some serious gaps remain in the state of understanding of the rate of decrease of temperature with height, and of the geographical distribution of tropopause height. In Chapters 3 and 4 we will see that the energy budget of a planet is crucially affected by the vertical structure of temperature; therefore, a thorough understanding of this feature is central to any theory of planetary climate.

2.3 Dry thermodynamics of an ideal gas 2.3.1 The equation of state for an ideal gas

The three thermodynamic variables with which we will mainly be concerned are: temperature (denoted by T), pressure ( denoted by p) and density (denoted by p). Temperature is proportional to the average amount of kinetic energy per molecule in the molecules making up the gas. We will always measure temperature in degrees Kelvin, which are the same as degrees Celsius (or Centigrade), except offset so that absolute zero - the temperature at which molecular motion ceases - occurs at zero Kelvin. In Celsius degrees, absolute zero occurs at about -273.15C. Pressure is defined as the force per unit area exerted on a surface in contact with the gas, in the direction perpendicular to the surface 2. It is independent of the orientation of the surface, and can be defined at a given location by making the surface increasingly small. In the mks units we employ throughout this book , pressure is measured in Pascals (Pa); 1 Pascal is 1 Newton of force per square meter of area, or equivalently 1 kg/(ms2). For historical reasons, atmospheric pressures are often measured in "bars" or "millibars." One bar, or equivalently 1000 millibars (mb) is approximately the mean sea-level pressure of the Earth's current atmosphere. We will often lapse into using mb as units of pressure, because the unit sounds comfortable to atmospheric scientists. For calculations, though, it is important to convert millibars to Pascals. This is easy, because 1 mb = 100 Pa. Hence, we should all learn to say "Hectopascal" in place of "millibar." It may take some time. When pressures are quoted in millibars or bars, one must make sure to convert them to Pascals before using the values in any thermodynamic calculations.

Density is simply the mass of the gas contained in a unit of volume. In mks units, it is measured in kg/m3.

For a perfect gas, the three thermodynamic variables are related by the perfect gas equation of state, which can be written p = knT (2.1)

where p is the pressure, n is the number of molecules per unit volume (which is proportional to density) and T is the temperature. k is the Boltzmann Thermodynamic Constant, a universal constant having dimensions of energy per unit temperature. Its value depends only on the units in which the thermodynamic quantities are measured. n is the particle number density of the gas. To relate n to mass density, we multiply it by the mass of a single molecule of the gas. Almost all of this mass comes from the protons and neutrons in the molecule, since electrons weigh next to nothing in comparison. Moreover, the mass of a neutron differs very little from the mass of a

2 Pressure can equivalently be defined as the amount of momentum per unit area per unit time which passes in both directions through a small hoop placed in the gas. This definition is equivalent to the exerted-force definition because when a molecule with velocity v and mass m bounces elastically off a surface, the momentum change is 2mv, but only half of the molecules are moving toward the surface at any given time. The momentum flux definition, in contrast, counts molecules going through the hoop in both directions

Jupiter

Jupiter 100 150 200 250 300 350 400 450 T

Venus

Venus 100 200 300 400 500 600 700 800 T Figure 2.2: The vertical profile of temperature for a portion of the atmosphere of Jupiter (upper left panel), for the atmosphere of Venus (upper right panel) , and for the atmosphere of Mars (lower left panel) and that of Titan (lower right panel). The Venus Magellan and Mars data derive from observations of radio transmission through the atmosphere, taken by the Magellan (late 1980's) and Global Surveyor orbiters, respectively. The information on the lower portion of the Venus atmosphere comes from one of the four 1978 Pioneer Venus probes (the others show a similar pattern). The Jupiter data derives from in situ deceleration measurements of the Galileo probe. The full Mars profile dataset reveals considerable seasonal and geographical variation. The profiles shown here were taken in the Southern Hemisphere by Mars Global Surveyor. The warmest one is in the late afternoon of 1998 in the Summer subtropics while the next warmest is at night-time under otherwise similar conditions. The coldest Mars sounding shown in the plot is from the winter South Polar region. 100 Pascal = 1 mb

Figure 2.2: The vertical profile of temperature for a portion of the atmosphere of Jupiter (upper left panel), for the atmosphere of Venus (upper right panel) , and for the atmosphere of Mars (lower left panel) and that of Titan (lower right panel). The Venus Magellan and Mars data derive from observations of radio transmission through the atmosphere, taken by the Magellan (late 1980's) and Global Surveyor orbiters, respectively. The information on the lower portion of the Venus atmosphere comes from one of the four 1978 Pioneer Venus probes (the others show a similar pattern). The Jupiter data derives from in situ deceleration measurements of the Galileo probe. The full Mars profile dataset reveals considerable seasonal and geographical variation. The profiles shown here were taken in the Southern Hemisphere by Mars Global Surveyor. The warmest one is in the late afternoon of 1998 in the Summer subtropics while the next warmest is at night-time under otherwise similar conditions. The coldest Mars sounding shown in the plot is from the winter South Polar region. 100 Pascal = 1 mb proton, so for our purposes the mass of the molecule can be taken to be M • y where y is the mass of a proton and M is the molecular weight - an integer giving the count of neutrons and protons in the molecule. (The equivalent count for an individual atom of an element is the atomic weight). The density is thus p = n • M • y). If we define the Universal Gas Constant as R* = k/y the perfect gas equation of state can be rewritten

In mks units, R* = 8314.5(m/s)2K-1 We can also define a gas constant R = R*/M particular to the gas in question. For example, dry Earth air has a mean molecular weight of 28.97, so Rdryair = 287 (m/s)2K-1, in mks units.

If y is measured in kilograms, then 1/y is the number of protons needed to make up a kilogram. This large number is known as a Mole, and is commonly used as a unit of measurement of numbers of molecules, just as one commonly counts eggs by the dozen. For any substance, a quantity of that substance whose mass in kilograms is equal to the molecular weight of the substance will contain one Mole of molecules. For example, 2 kg of H2 is a Mole of Hydrogen molecules, while 32 kg of the most common form of O2 is a Mole of molecular Oxygen. If n were measured in Moles/m3 instead of molecules per m3, then density would be p = n • M. One can also define the gram-mole (or mole for short), which is the number of protons needed to make a gram; this number is known as Avogadro's number, and is approximately 6.022 • 1023.

Generally speaking, a gas obeys the perfect gas law when it is tenuous enough that the energy stored in forces between the molecules making up the gas is negligible. Deviations from the perfect gas law can be important for the dense atmosphere of Venus, but for the purposes of the current atmosphere of Earth or Mars, or the upper part of the Jovian or Venusian atmosphere, the perfect gas law can be regarded as an accurate model of the thermodynamics.

An extension of the concept of a perfect gas is the law of partial pressures. This states that, in a mixture of gases in a given volume, each component gas behaves just as it would if it occupied the volume alone. The pressure due to one component gas is called the partial pressure of that gas. Consider a gas which is a mixture of substance A (with molecular weight Ma) and substance B (with molecular weight MB). The partial pressures of the two gases are

or equivalently,

where Ra = R*/Ma and RB = R*/MB. The same temperature appears in both equations, since thermodynamic equilibrium dictates that all components of the system have the same temperature. The ratio of partial pressures of any two components of a gas is a convenient way to describe the composition of the gas. From Eq. 2.3, pa/pb = «A/nB, so the ratio of partial pressure of A to that of B is also the ratio of number of molecules of A to the number of molecules of B. This ratio is called the molar mixing ratio. When we refer to a mixing ratio without qualification, we will generally mean the molar mixing ratio. Alternately, one can describe the composition in terms of the ratio of partial pressure of one component to total presssure of the gas (pa/(pa + pB) in the two-component example). Summing the two partial pressure equations in Eq. 2.3, we see that this is also the ratio of number of molecules of A to total number of molecules; hence we will use the term molar concentration for this ratio 3. If £a is the molar mixing ratio of A to B, then the

3The term volumetric mixing ratio or concentration is often used interchangeably with the term molar, as in "ppmv" for "parts per million by volume." The reason for this nomenclature is that the volume occupied by a given quantity of gas at a fixed temperature and pressure is proportional to the number of molecules of the gas contained molar concentration is £a/(1 + £a), from which we see that for the molar concentration and molar mixing ratio are nearly the same for substances which are very dilute (i.e. ^ 1). We will use the notation nA throughout the book to denote the molar concentration of substance A.

Exercise 2.3.1 Show that a mixture of gases with molar concentrations nA = nA/(nA + nB) and = nB/(nA + nB) behaves like a perfect gas with mean molecular weight M = nAMA + nBMB . (i.e. derive the expression relating total pressure pA + pB to total density pA + pB and identify the effective gas constant). Compute the mean molecular weight of dry Earth air. (Dry Earth air consists primarily of 78.084% N2, 20.947% O2, and .934% Ar, by count of molecules.)

The mass mixing ratio is the ratio of the mass of substance A to that of substance B in a given parcel of gas, i.e. pa/pb . From Eq. 2.4 it is related to the molar mixing ratio by pa ma PA

Pb Mb pb

Throughout this book,we will use the symbol r to denote mass mixing ratios and n for molar mixing ratios, with subscripts added as necessary to distinguish the species involved. Yet another measure of composition is specific concentration, defined as the ratio of the mass of a given substance to the total mass of the parcel (e.g. pa/(pa + pB) in the two-component case). We'll use the symbol q, with subscripts as necessary, to denote the specific concentration of a substance. Using the law of partial pressures, the specific concentration of substance A in a mixture is related to the molar concentration by pa ma pa ma , .

Ptot M ptot M

where M is the mean molecular weight of the mixture, with the mean computed using weighting according to molar concentrations of the species, as in Exercise 2.3.1.

All of the ratios we have just defined are convenient to use because, unlike densities, they remain unchanged as a parcel of air expands or contracts, provided the constituents under consideration do not undergo condensation, chemical reaction or other forms of internal sources or sinks. Hence, for a compressible gas, two components A and B are well-mixed relative to each other if the mixing ratio between them is independant of position.

Constituents will tend to become well mixed over a great depth of the atmosphere if they are created or destroyed slowly, if at all, relative to the characteristic time required for mixing. In the Earth's atmosphere, the mixing ratio of oxygen to nitrogen is virtually constant up to about 80km above the surface. The mixing ratio of carbon dioxide in air can vary considerably in the vicinity of sources at the surface, such as urban areas where much fuel is burned, or under forest canopies when photosynthesis is active. Away from the surface, however, the carbon dioxide mixing ratio varies little. Variations of a few parts per million can be detected in the relatively slowly mixed stratosphere, associated with the industrial-era upward trend in fossil fuel carbon dioxide emissions. Small seasonal and interhemispheric fluctuations in the tropospheric mixing ratio, associated with variations in the surface sources, can also be detected. For most purposes, though, carbon dioxide can be regarded as well mixed throughout the atmosphere. In contrast, water vapor has a strong in that quantity. To see this, write n = N/V, where N is the number of molecules and V is the volume they occupy. Then, the ideal gas law can be written in the alternate form V = (kT/p)N. Hence the ratio of standardized volumes is equal to the molar mixing ratio, and so forth. Abbreviations like "ppmv" for molar mixing ratios are common and convenient, because the "v" can unambiguously remind us that we are talking about a volumetric (i.e. molar) mixing ratio or concentration, whereas in an abbreviation like "ppmm" one is left wondering whether the second "m" means "mass" or "molar."

internal sink in Earth's atmosphere, because it is condensible there; hence its mixing ratio shows considerable vertical and horizontal variations. Carbon dioxide, methane and ammonia are not condensible on Earth at present, but their condensation can become significant in colder planetary atmospheres.

Exercise 2.3.2 (a) In the year 2000, the concentration of CO2 in the atmosphere was about 370 parts per million molar. What is the ratio pco2fptot? Estimate pco2 in mb at sea level. Does the molar concentration differ significantly from the molar mixing ratio? What is the mass mixing ratio of CO2 in air? What is the mass mixing ratio of carbon (in the form of CO2) in air - i.e. how many kilograms of carbon would have to be burned into CO2 in order to produce the CO2 in 1 kg of air? Note: The mean molecular weight of air is about 29. (b) The molar concentration of O2 in Earth air is about 20%. How many grams of O2 does a 1 liter breath of air contain at sea level (1000mb)? At the top of Qomolangma (a.k.a. "Mt. Everest," about 300mb)? Does the temperature of the air (within reasonable limits) affect your answer much?

### 2.3.2 Specific heat and conservation of energy

Conservation of energy is one of the three great pillars upon which the edifice of thermodynamics rests. When expressed in terms of changes in the state of matter, it is known as the First Law of Thermodynamics. When a gas expands or contracts, it does work by pushing against the environment as its boundaries move. Since pressure is force per unit area, and work is force times distance, the work done in the course of an expansion of volume dV is pdV. This is the amount of energy that must be added to the parcel of gas to allow the increase in volume to take place. For atmospheric purposes, it is more convenient to do write all thermodynamic relations on a per unit mass basis. Dividing V by the mass contained in the volume yields p-1, whence the work per unit mass is pdp-1. This is not the end of energy accounting. Changing the temperature of a unit mass of the substance while holding volume fixed changes the energy stored in the various motions of the molecules by an amount cv dT, where cv is a proportionality factor known as the specific heat at constant volume. For example it takes about 720 Joules of energy to raise the temperature of 1kg of air by 1K while holding the volume fixed. For ideal gases, the specific heat can depend on temperature, though the dependence is typically weak. For non-ideal gases, specific heat can depend on pressure as well.

Exercise 2.3.3 There are 20 students and one professor in a well-insulated classroom measuring 20 meters by 20 meters by 3 meters. Each person in the classroom puts out energy at a rate of 100 Watts (1 Watt = 1 Joule/second). The classroom is dark, except for a computer and LCD projector which together consume power at a rate of 200 Watts. The classroom is filled with air at a pressure of 1000mb (no extra charge). The room is sealed so no air can enter or leave, and has an initial temperature of 290K. How much does the temperature of the classroom rise during the course of a 1 hour lecture?

Combining the two contributions to energy change we find the expression for the amount of energy that must be added per unit mass in order to accomplish a change of both temperature and volume:

Using the perfect gas law, the heat balance can be re-written in the form

SQ = cvdT + d(pp-1) - p-1dp = (cv + R)dT - p-1dp (2.8)

From this relation, we can identify the specific heat at constant pressure, cp = cv + R, which is the amount of energy needed to warm a unit mass by 1K while allowing it to expand enough to keep pressure constant.

The units in which we measure temperature are an artifact of the marks one researcher or other once decided to put on some device that responded to heat and cold. Since temperature is proportional to the energy per molecule of a substance, it would make sense to set the proportionality constant to unity and simply use energy as the measure of temperature. This not being common practice, one has occasion to make use of the Boltzmann thermodynamic constant, k, which expresses the proportionality between temperature and energy. More precisely, each degree of freedom in a system with temperature T has a mean energy 1 kT. For example, a gas made of rigid spherical atoms has three degrees of freedom per atom (one for each direction it can move), and therefore each atom has energy |kT on average; a molecule which could store energy in the form of rotation or vibration would have more degrees of freedom, and therefore each molecule would have more energy at any given temperature. The energy-temperature relation is made possible by an important thermodynamic principle, the equipartion principle, which states that in equilibrium, each degree of freedom accessible to a system gets an equal share of the total energy of the system. In constrast to physical constants like the speed of light, the Boltzmann constant should not be considered a fundamental constant of the Universe. It is just a unit conversion factor.

2.3.3 Entropy, reversibility and Potential temperature; The Second Law

One cannot use Eqn 2.8 to define a "heat content" Q of a state (p, T) relative to a reference state (po,To), because the amount of heat needed to go from one state to another depends on the path in pressure-temperature space taken to get there; the right hand side of Eqn 2.8 is not an exact differential. However, it can be made into an exact differential by dividing the equation by T and using the perfect gas law as follows:

assuming cp to be constant. This equation defines the entropy, s = cp ln(Tp-R/Cp). Entropy is a nice quantity to work with because it is a state variable - its change between two states is independent of the path taken to get from one to the other. A process affecting a parcel of matter is said to be adiabatic if it occurs without addition or loss of heat from the parcel. By definition, 5Q = 0 for adiabatic processes. In consequence, adiabatic processes leave entropy unchanged, provided that the changes in state of the system are slow enough that the system remains close to thermodynamic equilibrium at all times. The latter condition is satisfied for all atmospheric phenomena that will concern us, but could be violated, for example, in the case of explosive adiabatic expansion of a formerly-confined gas into a vacuum. Entropy can also be defined for gases whose specific heat depends on temperature and pressure, for inhomogeneous mixtures of gases, and for non-ideal gases.

The Second Law of Thermodynamics states that entropy never decreases for energetically closed systems - systems to which energy is neither added nor subtracted in the course of their evolution. The formal derivation of the law from the microscopic properties of molecular interactions is in many ways an unfinished work of science, but the tendency towards an increase in entropy - an increase in disorder - seems to be a nearly universal property of systems consisting of a great many interacting components. A process during which the entropy remains constant is reversible, since it can be run both ways.

The Second Law is perhaps more intuitive when restated in the following way: In an energetically closed system, heat flows from a hotter part of the system to a colder part of the system, causing the system to evolve toward a state of uniform temperature. To see that this statement is equivalent to the entropy-increase principle, consider a thermally insulated box of gas having uniform pressure, but within which the left half of the mass is at temperature Ti and the right half of the mass is at temperature T2 < Ti. Now suppose that we transfer an amount of heat ¿Q from the left half of the box to the right half. This transfer leaves the net energy unchanged, but it changes the entropy. Specifically, according to Eq. 2.9, the entropy change summed over the two halves of the gas is ds = (— i )£Q. Since T2 < Ti, this change is positive only if > 0, representing a transfer from the hotter to the colder portion of the gas. Entropy can be increased by further heat transfers until Ti = T2, at which point the maximum entropy state has been attained.

The Second Law endows the Universe with an arrow of time. If one watches a movie of a closed system and sees that the system starts with large fluctuations of temperature (low entropy) and proceeds to a state of uniform temperature (high entropy), one knows that time is running forward. If one sees a thermally homogeneous object spontaneously generate large temperature inhomogeneities, then one knows that the movie is being run backwards. Note that the Second Law applies only to closed systems. The entropy of a subcomponent can decrease, if it exchanges energy with the outside world and increases the entropy of the rest of the Universe. This is how a refrigerator works.

Entropy is a very general concept, of which we have seen only the most basic instance. For a homogeneous ideal gas near thermodynamic equilibrium, the notions of reversible (i.e. isentropic) and adiabatic processes are equivalent, but caution must be exercised when extending this picture to more complex systems involving mixtures of gases. For example, if a box of gas at uniform temperature T contains pure N2 in its left half and pure O2 in its right half, then entropy will increase when the two gases spontaneously mix, even if no energy is let into the box. The entropy can still be defined in terms of changes in ¿Q/T, but it requires careful attention to what precisely is meant by ¿Q and to which subsystems the heat changes are being applied. The references given in the Further Readings section of this chapter provide a deeper and more general understanding of the use of entropy in solving thermodynamic problems.

Now let's get back to basics. Entropy can be used to determine how the temperature of an air parcel changes when it is compressed or expanded adiabatically. This is important because it tells us what happens to temperature is a bit of the atmosphere is lifted from low altitudes (where the pressure is high) to higher altitudes (where the pressure is lower), provided the lifting occurs so fast that the air parcel has little time to exchange heat with its surroundings. If the initial temperature and pressure are (T, p), then conservation of entropy tells us that the temperature To found upon adiabatically compressing or expanding to pressure po is given by Tp-R/Cp = T0p0 R/Cp This leads us to define the potential temperature d = T (P )-R/cp (2.10)

Po which is simply the temperature an air parcel would have if reduced adiabatically to a reference pressure po. Like entropy, potential temperature is conserved for adiabatic processes.

To understand why the presence of cold air above warm air in the sounding of Figure 2.1 does not succumb immediately to instability, we need only look at the corresponding profile of potential temperature, shown in Figure 2.3. This figure shows that potential temperature increases monotonically with height. This profile tells us that the air aloft is cold, but that if it were pushed down to lower altitudes, compression would warm it to the point that it is warmer than the

March 15 1993, 12Z Figure 2.3: The dry potential temperature profile for the sounding in Figure 2.1

surrounding air, and thus being positively buoyant, will tend to float back up to its original level rather than continuing its descent. We see also where the stratosphere gets its name: potential temperature increases very strongly with height there, so air parcels are very resistant to vertical displacement. This part of the atmosphere is therefore strongly stratified.

The troposphere is stable, but has much weaker gradients of 0. In a compressible atmosphere, a well-stirred layer would have constant 0 rather than constant T, since it is the former that is conserved for adiabatic processes such as would be caused by rapid vertical displacements. This is the essence of the explanation for why temperature decreases with height: turbulent stirring relaxes the troposphere towards constant 0, yielding the dry adiabat

In this formula, 0 has the constant value T(po).

On the dry adiabat, the slope dlnT/dlnp has the value R/cp. From the first equality in Eq. 2.9,it can be seen that this result remains valid even if R/cp depends on temperature and pressure. In general, the slope d ln T/d lnp provides a convenient measure of how sharply temperature decreases in the vertical; positive values correspond to decrease of temperature with height, since pressure decreases with altitude. The dry adiabatic slope is then R/cp. In atmospheric science, it is common to characterize the temperature structure by -dT/dz - the lapse rate - but there are very few circumstances in which the altitude z is a convenient coordinate for climate calculations. Unless noted otherwise, we will (somewhat unconventionally) use the term lapse rate to refer to the slope d ln T/d ln p. With this definition,the dry adiabatic lapse rate would be R/cp. (see Problem ?? for an exploration of the more conventional use of the term.)

It is evident from Figure 2.3 that something prevents 6 from becoming completely well mixed. An equivalent way of seeing this is to compare the observed temperature profile with the dry adiabat. For example, if the air at 1000mb in Figure 2.3, having temperature 298K, were lifted dry-adiabatically to the tropopause, where the pressure is 100mb, then the temperature would be 298.()2/7, i.e. 154.3K (using the value R/cp = 2/7 for Earth air). This is much colder than the observed temperature, which is 188K. We will see shortly that in the Earth's atmosphere, condensation of water vapor is one of the factors in play, though it is not the only one affecting the tropospheric temperature profile. The question of what determines the tropospheric 6 gradient is at present still largely unsettled, particularly outside the Tropics.

It is no accident that the value of R/cp for air lies close to the ratio of two small integers. It is a consequence of the equipartition principle. Using methods of statistical thermodynamics, it can be shown that a gas made up of molecules with n degrees of freedom has R/cp = 2/(n + 2). Using the expression for the gas constant in terms of the specific heats, the adiabatic coefficient can also be written as R/cp = 1 — 1/y, where 7 = cp/cv; for exact equipartion, 7 = 1 + 2/n. The measured values of 7 for a few common atmospheric gases are shown in Table 2.1. Helium comes close to the theoretical value for a molecule with no internal degrees of freedom, underscoring that excitation of electron motions plays little role in heat storage for typical planetary temperatures. The diatomic molecules have values closest to the theoretical value for n = 5, one short of what one would expect from adding two rotational and one vibrational internal degrees of freedom. Among the triatomic molecules, water acts roughly as if it had n = 6 while carbon dioxide is closer to n = 7. The two most complex molecules, methane and ammonia, are also characterized by n = 7. The failure of thermodynamics to access all the degrees of freedom classically available to a molecule is a consequence of quantum theory. Since the energy stored in states of motion of a molecule in fact comes in discrete-sized chunks, or "quanta," one can have a situation where a molecule hardly ever gets enough energy from a collision to excite even a single vibrational degree of freedom, for example, leading to the phenomenon of partial excitation or even non-excitation of certain classical degrees of freedom. This is one of many ways that the quantum theory, operating on exceedingly tiny spatial scales, exerts a crucial control over macroscopic properties of matter that can effect the very habitability of the Universe. Generally speaking, the higher the temperature gets, the more easy it is to excite internal degrees of freedom, leading to a decrease in 7. This quantum effect is the chief reason that specific heats vary somewhat with temperature.

Exercise 2.3.4 (a) A commercial jet airliner cruises at an altitude of 300mb. The air outside has a temperature of 240K. To enable the passengers to breathe, the ambient air is compressed to a cabin pressure of 1000mb. What would the cabin temperature be if the air were compressed adiabatically? How do you think airlines deal with this problem? (b) Discuss whether the lower portion of the Venus temperature profile shown in Figure 2.2 is on the dry CO2 adiabat. Do the same for the Summer afternoon Mars sounding. (c) Assume that the Jupiter sounding is on a dry adiabat, and estimate the value of R/cp for the atmosphere. Based on your result, what is the dominant constituent of the Jovian atmosphere likely to be? What other gas might be mixed with the dominant one?

### 2.4 Static stability of inhomogeneous mixtures

An atmosphere is statically unstable if an air parcel displaced from its original position tends to continue rising or sinking instead of returning to its original position. Such a state will tend to mix itself until it becomes stable. Static stability is important to planetary climate because it affects the vertical mixing which creates a planet's troposphere.

 H2O CH4 CO2 N2 O2 H2 He NH3 Crit. point T 647.1 190.44 304.2 126.2 154.54 33.2 5.1 405.5 Crit. point p 221.e5 45.96e5 73.S25e5 34.0e5 50.43e5 12.9Se5 2.2Se5 112.S Triple point T 273.15 90.67 216.54 63.14 54.3 13.95 2.17 195.4 Triple point p 611. .117e5 5.1S5e5 .1253e5 .0015e5 .072e5 .0507e5 .061e5 L vap(b.p.) 22.55e5 5.1e5 - 1.9Se5 2.13e5 4.54e5 .203e5 13.71e5 L vap(t.p.) 24.93e5 5.36e5 3.97e5 2.1Se5 2.42e5 ?? ?? 16.5Se5 L fusion 3.34e5 .5S6Se5 1.96e5 .2573e5 .139e5 .5S2e5 ?? 3.314e5 L sublimation 2S.4e5 5.95e5 5.93e5 2.437e5 2.56e5 ?? ?? 19.S9e5 p liq(b.p.) 95S.4 450.2 1032. S0S.6 1141. 70.97 124.96 6S2. p liq(t.p.) 999.S7 ?? 1110. ?? 1307. ?? ?? 734.2 p solid 917. 509.3 1562. 1026. 1351. SS. 200. S22.6 cp (0C/1bar) 1S47. 2195. S20. 1037. 916. 14230. 5196. 2060. Y(cp/cv ) 1.331 1.305 1.294 1.403 1.393 1.3S4 1.664 1.309 Table 2.1: Thermodynamic properties of selected gases. Latent heats of vaporization are given

at both the boiling point (the point where saturation vapor pressure reaches 1bar) and the triple point. Liquid densities are given at the boiling point and the triple point. For CO2 the 'boiling point' is undefined, so the liquid density is given at 253K/20bar instead. Note that the maximum density of liquid water is 1000.00%/m3 and occurs at —4C. Densities of solids are given at or near the triple point. All units are mks, so pressures are quoted as Pa with the appropriate exponent. Thus, 1bar is written as 1e5 in the table.

For a well-mixed atmosphere, the potential temperature profile tells the whole story about static stability, since, according to the ideal gas law, the density of an air parcel with potential temperature 00 will be p0 = p1/(R00 • (p1/p0)R/Cp) upon being elevated to an altitude with pressure pi < po. The ambient density there is pi = p1/(R01 • (pi/po)R/Cp). The displaced parcel will be negatively buoyant and return toward its original position if p0 > p1, which is true if and only if 00 < 01, i.e. if the potential temperature increases with height.

For an inhomogeneous atmosphere, this is no longer the case, since the gas constant R depends on the mean molecular weight of the mixture, which varies from place to place. As an example, we may consider an atmosphere which has uniform 0 computed on the basis of a reference pressure p0, but which consists of pure N2 for p > p0 and pure CO2 for p < p0. One immediately notes that the system has an unstable density jump at po, since the density is po/RN2 0 just below the interface and po/Rco2 0 just above the interface. Since N2 has lower molecular weight (28) than CO2 (44), the gas constant for N2 is considerably greater than the gas constant for CO2. Given that RN2 > RCO2, the density is greater just above the interface than it is just below the interface. This is an unstable situation, and the N2 layer will tend to mix itself into the CO2, despite the constancy of 0.

The phenomenon is very familiar: it is why helium balloons rise in air, even when they are at the same temperature as their surroundings. The low molecular weight of helium makes it lighter (i.e. lower density) than air having the same temperature and pressure.

Exercise 2.4.1 Make sense of the following statement: "For the Earth's atmosphere, moist air is lighter than dry air." Would this still be true for a planet whose atmosphere is mainly H2?

Suppose now that a parcel of N2 is lifted from just below the interface to an altitude where the pressure is some lower pressure p1. Let the density of the parcel when it arrives at its destination be puft and the density of the ambient CO2 be pamb. The density difference is then

Pllft - pamb = ^ Rn2 (pi Jr^^ - Rco2 (p1/pal)-(R/cP)co2 ) (2.12)

The biggest effect in this equation comes from the fact that IN2 > ICO2, which assures that the N2 retains its buoyancy as it is lifted. A much weaker effect comes from the fact that I/cp is slightly greater for N2 (.286) than for CO2 (.230). This modulates the density difference as the parcel is lifted, but the effect is slight. Even lifting a great distance, so that pi/pa = .01, the pressure factor in the first term is only modestly greater than the pressure factor in the second term (3.74 vs. 2.89), yielding a modest reduction in the buoyancy of the lifted parcel. For other pairs of gases, the difference in I/cp could be more significant, and in principle it could also go in the opposite direction and increase buoyancy rather than reducing it.

For an arbitrary profile of composition and temperature, one can define a potential density, which is the density an air parcel would have if compressed or expanded adiabatically to a standard reference pressure po. Using the gas law, and the fact that mixing ratios are conserved (whence I/cp is conserved on adiabatic compression or expansion of the parcel), the potential density relative to reference level po is p(p|p0) = Ro

RT Po Po

The I and cp in this equation must be taken from the values prevailing at pressure p, since these values are determined by composition, which in this calculation is presumed to remain fixed as the parcel is displaced to the reference pressure po. When p(p|po) increases with p, then the system will be stable in the sense that a parcel from p < po will be positively buoyant when pushed down to po and thus tend to return to its original level; similarly a parcel lifted to po from some pressure p > po will be negatively buoyant and also tend to return to where it came from. When the atmosphere is homogeneous, I/cp is constant and one can determine whether a parcel at pressure pi will be buoyant when displaced adiabatically to p2 by adiabatically reducing both to the standard pressure po and comparing the densities there. In other words, for a homogeneous atmosphere, one can tell immediately from a single plot of potential density relative to po which regions are stable and which regions will tend to overturn. When the atmosphere is inhomogeneous, this is no longer the case. When a parcel with density p(pi) is displaced to p2, it will be positively buoyant if