Atmospheric Thermodynamics

Transcription

1 Atmosheric hermodynamics 3 he theory of thermodynamics is one of the cornerstones and crowning glories of classical hysics. It has alications not only in hysics, chemistry, and the Earth sciences, but in subjects as diverse as biology and economics. hermodynamics lays an imortant role in our quantitative understanding of atmosheric henomena ranging from the smallest cloud microhysical rocesses to the general circulation of the atmoshere. he urose of this chater is to introduce some fundamental ideas and relationshis in thermodynamics and to aly them to a number of simle, but imortant, atmosheric situations. Further alications of the concets develoed in this chater occur throughout this book. he first section considers the ideal gas equation and its alication to dry air, water vaor, and moist air. In Section 3.2 an imortant meteorological relationshi, known as the hydrostatic equation,is derived and interreted. he next section is concerned with the relationshi between the mechanical work done by a system and the heat the system receives, as exressed in the first law of thermodynamics. here follow several sections concerned with alications of the foregoing to the atmoshere. Finally, in Section 3.7, the second law of thermodynamics and the concet of entroy are introduced and used to derive some imortant relationshis for atmosheric science. 3.1 Gas Laws Laboratory exeriments show that the ressure, volume, and temerature of any material can be related by an equation of state over a wide range of conditions. All gases are found to follow aroximately the same equation of state, which is referred to as the ideal gas equation. For most uroses we may assume that atmosheric gases, whether considered individually or as a mixture, obey the ideal gas equation exactly. his section considers various forms of the ideal gas equation and its alication to dry and moist air. he ideal gas equation may be written as V mr (3.1) where, V, m, and are the ressure (Pa), volume (m 3 ), mass (kg), and absolute temerature (in kelvin, K, where K C ) of the gas, resectively, and R is a constant (called the gas constant) for 1 kg of a gas. he value of R deends on the articular gas under consideration. Because mv, where is the density of the gas, the ideal gas equation may also be written in the form R (3.2) For a unit mass (1 kg) of gas m 1 and we may write (3.1) as R (3.3) where 1 is the secific volume of the gas, i.e., the volume occuied by 1 kg of the gas at ressure and temerature. If the temerature is constant (3.1) reduces to Boyle s law, 1 which states if the temerature of a fixed mass of gas is held constant, the volume of the 1 he Hon. Sir Robert Boyle ( ) Fourteenth child of the first Earl of Cork. Physicist and chemist, often called the father of modern chemistry. Discovered the law named after him in Resonsible for the first sealed thermometer made in England. One of the founders of the Royal Society of London, Boyle declared: he Royal Society values no knowledge but as it has a tendency to use it! 63

2 64 Atmosheric hermodynamics gas is inversely roortional to its ressure.changes in the hysical state of a body that occur at constant temerature are termed isothermal. Also imlicit in (3.1) are Charles two laws. 2 he first of these laws states for a fixed mass of gas at constant ressure, the volume of the gas is directly roortional to its absolute temerature. he second of Charles laws states for a fixed mass of gas held within a fixed volume, the ressure of the gas is roortional to its absolute temerature. 3.1 Gas Laws and the Kinetic heory of Gases: Handball Anyone? he kinetic theory of gases ictures a gas as an assemblage of numerous identical articles (atoms or molecules) 3 that move in random directions with a variety of seeds. he articles are assumed to be very small comared to their average searation and are erfectly elastic (i.e., if one of the articles hits another, or a fixed wall, it rebounds, on average, with the same seed that it ossessed just rior to the collision). It is shown in the kinetic theory of gases that the mean kinetic energy of the articles is roortional to the temerature in degrees kelvin of the gas. Imagine now a handball court in a zero-gravity world in which the molecules of a gas are both the balls and the layers. A countless (but fixed) number of elastic balls, each of mass m and with mean velocity v,are moving randomly in all directions as they bounce back and forth between the walls. 7 he force exerted on a wall of the court by the bouncing of balls is equal to the momentum exchanged in a tyical collision (which is roortional to mv) multilied by the frequency with which the balls imact the wall. Consider the following thought exeriments. i. Let the volume of the court increase while holding v (and therefore the temerature of the gas) constant. he frequency of collisions will decrease in inverse roortion to the change in volume of the court, and the force (and therefore the ressure) on a wall will decrease similarly. his is Boyle s law. ii. Let v increase while holding the volume of the court constant. Both the frequency of collisions with a wall and the momentum exchanged in each collision of a ball with a wall will increase in linear roortion to v. herefore, the ressure on a wall will increase as mv 2,which is roortional to the mean kinetic energy of the molecules and therefore to their temerature in degrees kelvin. his is the second of Charles laws. It is left as an exercise for the reader to rove Charles first law, using the same analogy. 2 Jacques A. C. Charles ( ) French hysical chemist and inventor. Pioneer in the use of hydrogen in man-carrying balloons. When Benjamin Franklin s exeriments with lightning became known, Charles reeated them with his own innovations. Franklin visited Charles and congratulated him on his work. 3 he idea that a gas consists of atoms in random motion was first roosed by Lucretius. 4 his idea was revived by Bernouilli 5 in 1738 and was treated in mathematical detail by Maxwell. 6 4 itus Lucretius Carus (ca B.C.) Latin oet and hilosoher.building on the seculations of the Greek hilosohers Leucius and Democritus, Lucretius, in his oem On the Nature of hings, roounds an atomic theory of matter.lucretius basic theorem is nothing exists but atoms and voids. He assumed that the quantity of matter and motion in the world never changes, thereby anticiating by nearly 2000 years the statements of the conservation of mass and energy. 5 Daniel Bernouilli ( ) Member of a famous family of Swiss mathematicians and hysicists. Professor of botany, anatomy, and natural hilosohy (i.e., hysics) at University of Basel. His most famous work, Hydrodynamics (1738), deals with the behavior of fluids. 6 James Clark Maxwell ( ) Scottish hysicist. Made fundamental contributions to the theories of electricity and magnetism (showed that light is an electromagnetic wave), color vision (roduced one of the first color hotograhs), and the kinetic theory of gases. First Cavendish Professor of Physics at Cambridge University; designed the Cavendish Laboratory. 7 In the kinetic theory of gases, the aroriate velocity of the molecules is their root mean square velocity, which is a little less than the arithmetic mean of the molecular velocities.

3 3.1 Gas Laws 65 We define now a gram-molecular weight or a mole (abbreviated to mol) of any substance as the molecular weight, M, of the substance exressed in grams. 8 For examle, the molecular weight of water is ; therefore, 1 mol of water is g of water. he number of moles n in mass m (in grams) of a substance is given by Because the gas constant for N A molecules is R*, we have k R* N A (3.7) Hence, for a gas containing n 0 molecules er unit volume, the ideal gas equation is n m M (3.4) n 0 k (3.8) Because the masses contained in 1 mol of different substances bear the same ratios to each other as the molecular weights of the substances, 1 mol of any substance must contain the same number of molecules as 1 mol of any other substance. herefore, the number of molecules in 1 mol of any substance is a universal constant, called Avogadro s 9 number, N A. he value of N A is er mole. According to Avogadro s hyothesis, gases containing the same number of molecules occuy the same volumes at the same temerature and ressure. It follows from this hyothesis that rovided we take the same number of molecules of any gas, the constant R in (3.1) will be the same. However, 1 mol of any gas contains the same number of molecules as 1mol of any other gas.herefore,the constant R in (3.1) for 1 mol is the same for all gases; it is called the universal gas constant (R*). he magnitude of R* is J K 1 mol 1.he ideal gas equation for 1mol of any gas can be written as and for n moles of any gas as V R* (3.5) V nr* (3.6) he gas constant for one molecule of any gas is also a universal constant, known as Boltzmann s 10 constant, k. If the ressure and secific volume of dry air (i.e., the mixture of gases in air, excluding water vaor) are d and d,resectively,the ideal gas equation in the form of (3.3) becomes d d R d (3.9) where R d is the gas constant for 1 kg of dry air. By analogy with (3.4), we can define the aarent molecular weight M d of dry air as the total mass (in grams) of the constituent gases in dry air divided by the total number of moles of the constituent gases; that is, M d m i i (3.10) where m i and M i reresent the mass (in grams) and molecular weight, resectively, of the ith constituent in the mixture. he aarent molecular weight of dry air is Because R* is the gas constant for 1 mol of any substance, or for M d ( 28.97) grams of dry air, the gas constant for 1 g of dry air is R*M d,and for 1 kg of dry air it is R d 1000 R* M d J K1 kg 1 m i i M i (3.11) 8 In the first edition of this book we defined a kilogram-molecular weight (or kmole), which is 1000 moles. Although the kmole is more consistent with the SI system of units than the mole, it has not become widely used. For examle, the mole is used almost universally in chemistry. One consequence of the use of the mole, rather than kmole, is that a factor of 1000, which serves to convert kmoles to moles, aears in some relationshis [e.g. (3.11) and (3.13) shown later]. 9 Amedeo Avogadro, Count of Quaregna ( ) Practiced law before turning to science at age 23. Later in life became a rofessor of hysics at the University of urin. His famous hyothesis was ublished in 1811, but it was not generally acceted until a half century later. Introduced the term molecule. 10 Ludwig Boltzmann ( ) Austrian hysicist. Made fundamental contributions to the kinetic theory of gases. Adhered to the view that atoms and molecules are real at a time when these concets were in disute. Committed suicide.

4 66 Atmosheric hermodynamics he ideal gas equation may be alied to the individual gaseous comonents of air. For examle, for water vaor (3.3) becomes e v R v (3.12) where e and v are, resectively, the ressure and secific volume of water vaor and R v is the gas constant for 1 kg of water vaor. Because the molecular weight of water is M w ( ) and the gas constant for M w grams of water vaor is R*, we have R v 1000 R* M w J K1 kg 1 From (3.11) and (3.13), R d R v M w M d (3.13) (3.14) Because air is a mixture of gases, it obeys Dalton s 11 law of artial ressures,which states the total ressure exerted by a mixture of gases that do not interact chemically is equal to the sum of the artial ressures of the gases. he artial ressure of a gas is the ressure it would exert at the same temerature as the mixture if it alone occuied all of the volume that the mixture occuies. Exercise 3.1 If at 0 C the density of dry air alone is kg m 3 and the density of water vaor alone is kg m 3, what is the total ressure exerted by a mixture of the dry air and water vaor at 0 C? Solution: From Dalton s law of artial ressures, the total ressure exerted by the mixture of dry air and water vaor is equal to the sum of their artial ressures. he artial ressure exerted by the dry air is, from (3.9), where d is the density of the dry air (1.275 kg m 3 at 273 K), R d is the gas constant for 1 kg of dry air (287.0 J K 1 kg 1 ), and is K. herefore, d Pa hpa Similarly, the artial ressure exerted by the water vaor is, from (3.12), e 1 R v vr v v where v is the density of the water vaor ( kg m 3 at 273 K), R v is the gas constant for 1 kg of water vaor (461.5 J K 1 kg 1 ), and is K. herefore, e Pa hpa Hence, the total ressure exerted by the mixture of dry air and water vaor is ( ) hpa or 1006 hpa Virtual emerature Moist air has a smaller aarent molecular weight than dry air. herefore, it follows from (3.11) that the gas constant for 1 kg of moist air is larger than that for 1 kg of dry air. However, rather than use a gas constant for moist air, the exact value of which would deend on the amount of water vaor in the air (which varies considerably), it is convenient to retain the gas constant for dry air and use a fictitious temerature (called the virtual temerature) in the ideal gas equation. We can derive an exression for the virtual temerature in the following way. Consider a volume V of moist air at temerature and total ressure that contains mass m d of dry air and mass m v of water vaor. he density of the moist air is given by d 1 d R d dr d m d m v V d v 11 John Dalton ( ) English chemist. Initiated modern atomic theory. In 1787 he commenced a meteorological diary that he continued all his life, recording 200,000 observations. Showed that the rain and dew deosited in England are equivalent to the quantity of water carried off by evaoration and by the rivers. his was an imortant contribution to the idea of a hydrological cycle. First to describe color blindness. He never found time to marry! His funeral in Manchester was attended by 40,000 mourners.

5 3.2 he Hydrostatic Equation 67 where d is the density that the same mass of dry air would have if it alone occuied all of the volume V and v is the density that the same mass of water vaor would have if it alone occuied all of the volume V. We may call these artial densities. Because d v, it might aear that the density of moist air is greater than that of dry air. However, this is not the case because the artial density v is less than the true density of dry air. 12 Alying the ideal gas equation in the form of (3.2) to the water vaor and dry air in turn, we have and where e and d are the artial ressures exerted by the water vaor and the dry air, resectively. Also, from Dalton s law of artial ressures, Combining the last four equations or where is defined by (3.14). he last equation may be written as where R d v e v R v d d R d d e e R d [1 e (1 )] R d v e R v 1 e (1 ) (3.15) (3.16) v is called the virtual temerature.if this fictitious temerature, rather than the actual temerature, is used for moist air, the total ressure and density of the moist air are related by a form of the ideal gas equation [namely, (3.15)], but with the gas constant the same as that for a unit mass of dry air (R d ) and the actual temerature relaced by the virtual temerature v.it follows that the virtual temerature is the temerature that dry air would need to attain in order to have the same density as the moist air at the same ressure. Because moist air is less dense than dry air at the same temerature and ressure, the virtual temerature is always greater than the actual temerature. However, even for very warm and moist air, the virtual temerature exceeds the actual temerature by only a few degrees (e.g., see Exercise 3.7 in Section 3.5). 3.2 he Hydrostatic Equation Air ressure at any height in the atmoshere is due to the force er unit area exerted by the weight of all of the air lying above that height. Consequently, atmosheric ressure decreases with increasing height above the ground (in the same way that the ressure at any level in a stack of foam mattresses deends on how many mattresses lie above that level). he net uward force acting on a thin horizontal slab of air, due to the decrease in atmosheric ressure with height, is generally very closely in balance with the downward force due to gravitational attraction that acts on the slab. If the net uward force on the slab is equal to the downward force on the slab, the atmoshere is said to be in hydrostatic balance. We will now derive an imortant equation for the atmoshere in hydrostatic balance. Consider a vertical column of air with unit horizontal cross-sectional area (Fig. 3.1). he mass of air between heights z and z z in the column is z, where is the density of the air at height z. he downward force acting on this slab of air due to the weight of the air is ɡz, where ɡ is the acceleration due to gravity at height z.now let us consider the net 12 he fact that moist air is less dense than dry air was first clearly stated by Sir Isaac Newton 13 in his Oticks (1717). However, the basis for this relationshi was not generally understood until the latter half of the 18th century. 13 Sir Isaac Newton ( ) Renowned English mathematician, hysicist, and astronomer. A osthumous, remature ( I could have been fitted into a quart mug at birth ), and only child. Discovered the laws of motion, the universal law of gravitation, calculus, the colored sectrum of white light, and constructed the first reflecting telescoe. He said of himself: I do not know what I may aear to the world, but to myself I seem to have been only like a boy laying on the seashore, and diverting myself in now and then finding a smoother ebble or a rettier shell than ordinary, while the great ocean of truth lay all undiscovered before me.

6 68 Atmosheric hermodynamics δz vertical force that acts on the slab of air between z and z z due to the ressure of the surrounding air. Let the change in ressure in going from height z to height z z be, as indicated in Fig Because we know that ressure decreases with height, must be a negative quantity, and the uward ressure on the lower face of the shaded block must be slightly greater than the downward ressure on the uer face of the block. herefore, the net vertical force on the block due to the vertical gradient of ressure is uward and given by the ositive quantity, as indicated in Fig.3.1.For an atmoshere in hydrostatic balance, the balance of forces in the vertical requires that or, in the limit as z : 0, z Ground δ gρδz ɡz Column with unit cross-sectional area Pressure = + δ Pressure = Fig. 3.1 Balance of vertical forces in an atmoshere in which there are no vertical accelerations (i.e., an atmoshere in hydrostatic balance). Small blue arrows indicate the downward force exerted on the air in the shaded slab due to the ressure of the air above the slab; longer blue arrows indicate the uward force exerted on the shaded slab due to the ressure of the air below the slab. Because the slab has a unit cross-sectional area, these two ressures have the same numerical values as forces. he net uward force due to these ressures () is indicated by the uward-ointing thick black arrow. Because the incremental ressure change is a negative quantity, is ositive. he downward-ointing thick black arrow is the force acting on the shaded slab due to the mass of the air in this slab. Equation (3.17) is the hydrostatic equation. 14 It should be noted that the negative sign in (3.17) ensures that the ressure decreases with increasing height. Because 1 (3.17) can be rearranged to give (3.18) If the ressure at height z is (z), we have, from (3.17), above a fixed oint on the Earth or, because () 0, (3.19) hat is, the ressure at height z is equal to the weight of the air in the vertical column of unit crosssectional area lying above that level. If the mass of the Earth s atmoshere were distributed uniformly over the globe, retaining the Earth s toograhy in its resent form, the ressure at sea level would be Pa, or 1013 hpa, which is referred to as 1 atmoshere (or 1 atm) Geootential he geootential at any oint in the Earth s atmoshere is defined as the work that must be done against the Earth s gravitational field to raise a mass of 1 kg from sea level to that oint. In other words, is the gravitational otential er unit mass. he units of geootential are J kg 1 or m 2 s 2. he force (in newtons) acting on 1 kg at height z above sea level is numerically equal to ɡ.he work (in joules) in raising 1 kg from z to z dz is ɡdz; therefore or, using (3.18), ɡdz d () (z) d (z) ɡdz z z d ɡdz ɡdz z ɡ (3.17) d ɡdz d (3.20) 14 In accordance with Eq. (1.3), the left-hand side of (3.17) is written in artial differential notation, i.e., z,because the variation of ressure with height is taken with other indeendent variables held constant.

7 3.2 he Hydrostatic Equation 69 he geootential (z) at height z is thus given by z (z) ɡdz (3.21) 0 where the geootential (0) at sea level (z 0) has, by convention, been taken as zero. he geootential at a articular oint in the atmoshere deends only on the height of that oint and not on the ath through which the unit mass is taken in reaching that oint. he work done in taking a mass of 1 kg from oint A with geootential A to oint B with geootential B is B A. We can also define a quantity called the geootential height Z as (3.22) where ɡ 0 is the globally averaged acceleration due to gravity at the Earth s surface (taken as 9.81 m s 2 ). Geootential height is used as the vertical coordinate in most atmosheric alications in which energy lays an imortant role (e.g., in large-scale atmosheric motions). It can be seen from able 3.1 that the values of z and Z are almost the same in the lower atmoshere where ɡ 0 ɡ. In meteorological ractice it is not convenient to deal with the density of a gas,,the value of which is generally not measured. By making use of (3.2) or (3.15) to eliminate in (3.17), we obtain Rearranging the last exression and using (3.20) yields d Z (z) ɡ0 1 z g g R R d v ɡ dz R d z ɡ00 ɡdz R d d v (3.23) able 3.1 Values of geootential height (Z) and acceleration due to gravity (ɡ) at 40 latitude for geometric height (z) z (km) Z (km) ɡ (m s 2 ) If we now integrate between ressure levels 1 and 2,with geootentials 1 and 2,resectively, or Dividing both sides of the last equation by ɡ 0 and reversing the limits of integration yields (3.24) his difference Z 2 Z 1 is referred to as the (geootential) thickness of the layer between ressure levels 1 and Scale Height and the Hysometric Equation For an isothermal atmoshere (i.e., temerature constant with height), if the virtual temerature correction is neglected, (3.24) becomes or where 2 1 d 2 1 R d 2 Z 2 Z 1 R d Z 2 Z 1 H ln( 1 2 ) 2 1 ex (Z 2 Z 1 ) H H R ɡ0 1 ɡ0 d v v d (3.25) (3.26) (3.27) H is the scale height as discussed in Section Because the atmoshere is well mixed below the turboause (about 105 km), the ressures and densities of the individual gases decrease with altitude at the same rate and with a scale height roortional to the gas constant R (and therefore inversely roortional to the aarent molecular weight of the mixture). If we take a value for v of 255 K (the aroximate mean value for the trooshere and stratoshere), the scale height H for air in the atmoshere is found from (3.27) to be about 7.5 km. 2 d R d v 1

8 70 Atmosheric hermodynamics Above the turboause the vertical distribution of gases is largely controlled by molecular diffusion and a scale height may then be defined for each of the individual gases in air. Because for each gas the scale height is roortional to the gas constant for a unit mass of the gas, which varies inversely as the molecular weight of the gas [see, for examle (3.13)], the ressures (and densities) of heavier gases fall off more raidly with height above the turboause than those of lighter gases. Exercise 3.2 If the ratio of the number density of oxygen atoms to the number density of hydrogen atoms at a geootential height of 200 km above the Earth s surface is 10 5,calculate the ratio of the number densities of these two constituents at a geootential height of 1400 km. Assume an isothermal atmoshere between 200 and 1400 km with a temerature of 2000 K. Solution: At these altitudes, the distribution of the individual gases is determined by diffusion and therefore by (3.26). Also, at constant temerature, the ratio of the number densities of two gases is equal to the ratio of their ressures. From (3.26) ( 1400 km ) oxy ( 1400 km ) hyd From the definition of scale height (3.27) and analogous exressions to (3.11) for oxygen and hydrogen atoms and the fact that the atomic weights of oxygen and hydrogen are 16 and 1, resectively, we have at 2000 K and ( 200 km) oxy ex[1200 kmh oxy (km)] ( 200 km ) hyd ex[1200 kmh hyd (km)] 10 5 ex 1200 km 1 H oxy 1 H hyd H oxy 1000R* m H hyd 1000R* m m m m m 9.81 herefore, and ln ln 2 ln 1 Hence, the ratio of the number densities of oxygen to hydrogen atoms at a geootential height of 1400 km is 2.5. he temerature of the atmoshere generally varies with height and the virtual temerature correction cannot always be neglected. In this more general case (3.24) may be integrated if we define amean virtual temerature v with resect to as shown in Fig hat is, v # A B v Virtual temerature, v (K) H oxy H 6 m 1 hyd ( 1400 km ) oxy ( 1400 km ) hyd 10 5 ex (10.6) v d(ln ) 1 2 d(ln ) hen, from (3.24) and (3.28), km 1 Z 2 Z 1 H ln 1 2 R d v D (3.28) (3.29) Equation (3.29) is called the hysometric equation. Exercise 3.3 Calculate the geootential height of the 1000-hPa ressure surface when the ressure at sea level is 1014 hpa. he scale height of the atmoshere may be taken as 8 km. C E From radiosonde data Fig. 3.2 Vertical rofile, or sounding, of virtual temerature. If area ABC area CDE, v is the mean virtual temerature with resect to ln between the ressure levels 1 and 2. 1 d v 2 ln 1 2 ɡ0 ln 1 2

9 P Ch03.qxd 9/12/05 7:41 PM Page he Hydrostatic Equation 71 Solution: From the hysometric equation (3.29) Z 1000 hpa Z sea level H ln H ln H where 0 is the sea-level ressure and the relationshi ln (1 x) x for x 1 has been used. Substituting H 8000 into this exression, and recalling that Z sea level 0 (able 3.1), gives herefore, with hpa, the geootential height Z 1000 hpa of the 1000-hPa ressure surface is found to be 112 m above sea level hickness and Heights of Constant Pressure Surfaces Because ressure decreases monotonically with height, ressure surfaces (i.e., imaginary surfaces on which ressure is constant) never intersect. It can be seen from (3.29) that the thickness of the layer between any two ressure surfaces 2 and 1 is roortional to the mean virtual temerature of the layer, v. We can visualize that as v increases, the air between the two ressure levels exands and the layer becomes thicker. Exercise 3.4 Calculate the thickness of the layer between the and 500-hPa ressure surfaces (a) at a oint in the troics where the mean virtual temerature of the layer is 15 C and (b) at a oint in the olar regions where the corresonding mean virtual temerature is 40 C. Solution: From (3.29) Z 1000 hpa 8 ( ) Z Z 500 hpa Z 1000 hpa R d v herefore, for the troics with v 288 K, Z 5846 m. For olar regions with v 233 K, Z 4730 m.in oerational ractice,thickness is rounded to the nearest 10 m and is exressed in decameters (dam). Hence, answers for this exercise would normally be exressed as 585 and 473 dam, resectively. ɡ0 ln v m Before the advent of remote sensing of the atmoshere by satellite-borne radiometers, thickness was evaluated almost exclusively from radiosonde data, which rovide measurements of the ressure, temerature, and humidity at various levels in the atmoshere. he virtual temerature v at each level was calculated and mean values for various layers were estimated using the grahical method illustrated in Fig Using soundings from a network of stations, it was ossible to construct toograhical mas of the distribution of geootential height on selected ressure surfaces. hese calculations, which were first erformed by observers working on site, are now incororated into sohisticated data assimilation rotocols, as described in the Aendix of Chater 8 on the book Web site. In moving from a given ressure surface to another ressure surface located above or below it, the change in the geootential height is related geometrically to the thickness of the intervening layer, which, in turn, is directly roortional to the mean virtual temerature of the layer. herefore, if the three-dimensional distribution of virtual temerature is known, together with the distribution of geootential height on one ressure surface, it is ossible to infer the distribution of geootential height of any other ressure surface. he same hysometric relationshi between the three-dimensional temerature field and the shae of ressure surface can be used in a qualitative way to gain some useful insights into the three-dimensional structure of atmosheric disturbances, as illustrated by the following examles. Height (km) i. he air near the center of a hurricane is warmer than its surroundings. Consequently, the intensity of the storm (as measured by the deression of the isobaric surfaces) must decrease with height (Fig. 3.3a). he winds in such warm core lows 12 0 W A R M (a) Fig. 3.3 Cross sections in the longitude height lane. he solid lines indicate various constant ressure surfaces. he sections are drawn such that the thickness between adjacent ressure surfaces is smaller in the cold (blue) regions and larger in the warm (red) regions. W A R M C O L D (b)

10 72 Atmosheric hermodynamics always exhibit their greatest intensity near the ground and diminish with increasing height above the ground. ii. Some uer level lows do not extend downward to the ground, as indicated in Fig. 3.3b. It follows from the hysometric equation that these lows must be cold core below the level at which they achieve their greatest intensity and warm core above that level, as shown in Fig. 3.3b Reduction of Pressure to Sea Level In mountainous regions the difference in surface ressure from one observing station to another is largely due to differences in elevation. o isolate that art of the ressure field that is due to the assage of weather systems, it is necessary to reduce the ressures to a common reference level. For this urose, sea level is normally used. Let the subscrits g and 0 refer to conditions at the ground and at sea level (Z 0), resectively. hen, for the layer between the Earth s surface and sea level, the hysometric equation (3.29) assumes the form (3.30) which can be solved to obtain the sea-level ressure 0 ɡ ex Zɡ (3.31) If Z ɡ is small, the scale height H can be evaluated from the ground temerature. Also, if the exonential in (3.31) can be aroximated by 1 Zɡ H,in which case (3.31) becomes 0 ɡ Zɡ ɡ H ɡ R d v (3.32) Because and H 8000 m, the ressure correction (in hpa) is roughly equal to Z ɡ (in ɡ 1000 ha Zɡ H ln 0 ɡ H ɡ ex R d ɡ0Zɡ ɡ0Zɡ Zɡ H 1, meters) divided by 8. In other words, for altitudes u to a few hundred meters above (or below) sea level, the ressure decreases by about 1 hpa for every 8 m of vertical ascent. 3.3 he First Law of hermodynamics 15 In addition to the macroscoic kinetic and otential energy that a system as a whole may ossess, it also contains internal energy due to the kinetic and otential energy of its molecules or atoms. Increases in internal kinetic energy in the form of molecular motions are manifested as increases in the temerature of the system, whereas changes in the otential energy of the molecules are caused by changes in their relative ositions by virtue of any forces that act between the molecules. Let us suose that a closed system 16 of unit mass takes in a certain quantity of thermal energy q (measured in joules), which it can receive by thermal conduction andor radiation. As a result the system may do a certain amount of external work w (also measured in joules). he excess of the energy sulied to the body over and above the external work done by the body is q w. herefore,if there is no change in the macroscoic kinetic and otential energy of the body, it follows from the rincile of conservation of energy that the internal energy of the system must increase by q w.hat is, q w u 2 u 1 (3.33) where u 1 and u 2 are the internal energies of the system before and after the change. In differential form (3.33) becomes dq dw du (3.34) where dq is the differential increment of heat added to the system, dw is the differential element 15 he first law of thermodynamics is a statement of the conservation of energy, taking into account the conversions between the various forms that it can assume and the exchanges of energy between a system and its environment that can take lace through the transfer of heat and the erformance of mechanical work. A general formulation of the first law of thermodynamics is beyond the scoe of this text because it requires consideration of conservation laws, not only for energy, but also for momentum and mass. his section resents a simlified formulation that ignores the macroscoic kinetic and otential energy (i.e., the energy that air molecules ossess by virtue of their height above sea level and their organized fluid motions). As it turns out, the exression for the first law of thermodynamics that emerges in this simlified treatment is identical to the one recovered from a more comlete treatment of the conservation laws, as is done in J. R. Holton, Introduction to Dynamic Meteorology,4th Edition,Academic Press,New York,2004, A closed system is one in which the total amount of matter, which may be in the form of gas, liquid, solid or a mixture of these hases, is ket constant.

11 3.3 he First Law of hermodynamics 73 of work done by the system, and du is the differential increase in internal energy of the system. Equations (3.33) and (3.34) are statements of the first law of thermodynamics. In fact (3.34) rovides a definition of du.he change in internal energy du deends only on the initial and final states of the system and is therefore indeendent of the manner by which the system is transferred between these two states. Such arameters are referred to as functions of state. 17 o visualize the work term dw in (3.34) in a simle case, consider a substance, often called the working substance, contained in a cylinder of fixed cross-sectional area that is fitted with a movable, frictionless iston (Fig. 3.4). he volume of the substance is roortional to the distance from the base of the cylinder to the face of the iston and can be reresented on the horizontal axis of the grah shown in Fig he ressure of the substance in the cylinder can be reresented on the vertical axis of this grah. herefore, every state of the substance, corresonding to a given osition of the iston, is reresented by a oint on this ressure volume ( V) diagram. When the substance is in equilibrium at a state reresented by oint P on the grah, its ressure is and its volume is V (Fig. 3.4). If the iston moves outward through an incremental distance dx while its ressure remains essentially constant at, the work dw done by the substance in ushing the external force F through a distance dx is When the substance asses from state A with volume V 1 to state B with volume V 2 (Fig. 3.4), during which its ressure changes, the work W done by the material is equal to the area under the curve AB. hat is, W V 2 V 1 dv (3.36) Equations (3.35) and (3.36) are quite general and reresent work done by any substance (or system) due to a change in its volume. If V 2 V 1, W is ositive, indicating that the substance does work on its environment. If V 2 V 1, W is negative, which indicates that the environment does work on the substance. he V diagram shown in Fig. 3.4 is an examle of a thermodynamic diagram in which the hysical state of a substance is reresented by two thermodynamic variables. Such diagrams are very useful in meteorology; we will discuss other examles later in this chater. Working substance Cylinder Distance, x Piston F dw Fdx or, because F A where A is the cross-sectional area of the face of the iston, Pressure 1 2 A P Q B dw A dx dv (3.35) In other words, the work done by the substance when its volume increases by a small increment dv is equal to the ressure of the substance multilied by its increase in volume, which is equal to the blue-shaded area in the grah shown in Fig. 3.4; that is, it is equal to the area under the curve PQ. V 1 V V 2 dv Volume Fig. 3.4 Reresentation of the state of a working substance in a cylinder on a V diagram. he work done by the working substance in assing from P to Q is dv, which is equal to the blue-shaded area. [Rerinted from Atmosheric Science: An Introductory Survey, 1st Edition, J. M. Wallace and P. V. Hobbs,. 62, Coyright 1977, with ermission from Elsevier.] 17 Neither the heat q nor the work w are functions of state, since their values deend on how a system is transformed from one state to another. For examle, a system may or may not receive heat and it may or may not do external work as it undergoes transitions between different states.

12 74 Atmosheric hermodynamics If we are dealing with a unit mass of a substance, the volume V is relaced by the secific volume. herefore, the work dw that is done when the secific volume increases by d is Combination of (3.34) and (3.37) yields (3.37) (3.38) which is an alternative statement of the first law of thermodynamics Joule s Law dw d dq du d Following a series of laboratory exeriments on air, Joule 19 concluded in 1848 that when a gas exands without doing external work, by exanding into a chamber that has been evacuated, and without taking in or giving out heat, the temerature of the gas does not change. his statement, which is known as Joule s law, is strictly true only for an ideal gas,but air (and many other gases) behaves very similarly to an ideal gas over a wide range of conditions. Joule s law leads to an imortant conclusion concerning the internal energy of an ideal gas. If a gas neither does external work nor takes in or gives out heat, dw 0 and dq 0 in (3.38), so that du 0. Also, according to Joule s law, under these conditions the temerature of the gas does not change, which imlies that the kinetic energy of the molecules remains constant. herefore, because the total internal energy of the gas is constant, that art of the internal energy due to the otential energy must also remain unchanged, even though the volume of the gas changes. In other words, the internal energy of an ideal gas is indeendent of its volume if the temerature is ket constant. his can be the case only if the molecules of an ideal gas do not exert forces on each other. In this case, the internal energy of an ideal gas will deend only on its temerature More Handball? Box 3.1. showed that the gas laws can be illustrated by icturing the molecules of a gas as elastic balls bouncing around randomly in a handball court. Suose now that the walls of the court are ermitted to move outward when subjected to a force. he force on the walls is sulied by the imact of the balls, and the work required to move the walls outward comes from a decrease in the kinetic energy of the balls that rebound from the walls with lower velocities than they struck them. his decrease in kinetic energy is in accordance with the first law of thermodynamics under adiabatic conditions. he work done by the system by ushing the walls outward is equal to the decrease in the internal energy of the system [see (3.38)]. Of course, if the outside of the walls of the court are bombarded by balls in a similar manner to the inside walls, there will be no net force on the walls and no work will be done. 18 We have assumed here that the only work done by or on a system is due to a change in the volume of the system. However, there are other ways in which a system may do work, e.g., by the creation of new surface area between two hases (such as between liquid and air when a soa film is formed). Unless stated otherwise, we will assume that the work done by or on a system is due entirely to changes in the volume of the system. 19 James Prescott Joule ( ) Son of a wealthy English brewer; one of the great exerimentalists of the 19th century. He started his scientific work (carried out in laboratories in his home and at his own exense) at age 19. He measured the mechanical equivalent of heat, recognized the dynamical nature of heat, and develoed the rincile of conservation of energy. 20 Subsequent exeriments carried out by Lord Kelvin 21 revealed the existence of small forces between the molecules of a gas. 21 Lord Kelvin 1st Baron (William homson) ( ) Scottish mathematician and hysicist. Entered Glasgow University at age 11. At 22 became Professor of Natural Philosohy at the same university. Carried out incomarable work in thermodynamics, electricity, and hydrodynamics.

13 3.3 he First Law of hermodynamics Secific Heats Suose a small quantity of heat dq is given to a unit mass of a material and, as a consequence, the temerature of the material increases from to d without any changes in hase occurring within the material. he ratio dqd is called the secific heat of the material. he secific heat defined in this way could have any number of values, deending on how the material changes as it receives the heat. If the volume of the material is ket constant, a secific heat at constant volume c v is defined where the material is allowed to exand as heat is added to it and its temerature rises, but its ressure remains constant. In this case, a certain amount of the heat added to the material will have to be exended to do work as the system exands against the constant ressure of its environment. herefore, a larger quantity of heat must be added to the material to raise its temerature by a given amount than if the volume of the material were ket constant. For the case of an ideal gas, this inequality can be seen mathematically as follows. Equation (3.41) can be rewritten in the form c v dq d v const (3.39) dq c v d d() d (3.43) However, if the volume of the material is constant (3.38) becomes dq du.herefore c v du d v const For an ideal gas, Joule s law alies and therefore u deends only on temerature. herefore, regardless of whether the volume of a gas changes, we may write c v du d (3.40) From (3.38) and (3.40), the first law of thermodynamics for an ideal gas can be written in the form 22 dq c v d d (3.41) Because u is a function of state, no matter how the material changes from state 1 to state 2, the change in its internal energy is, from (3.40), u 2 u c v d We can also define a secific heat at constant ressure c c dq d const (3.42) From the ideal gas equation (3.3), d() Rd. herefore (3.43) becomes (3.44) At constant ressure, the last term in (3.44) vanishes; therefore, from (3.42) and (3.44), (3.45) he secific heats at constant volume and at constant ressure for dry air are 717 and 1004 J K 1 kg 1, resectively, and the difference between them is 287 J K 1 kg 1,which is the gas constant for dry air. It can be shown that for ideal monatomic gases c :c v :R 5:3:2, and for ideal diatomic gases c :c v :R 7:5:2. By combining (3.44) and (3.45) we obtain an alternate form of the first law of thermodynamics: Enthaly dq (c v R)d d c c v R dq c d d (3.46) If heat is added to a material at constant ressure so that the secific volume of the material increases from 1 to 2,the work done by a unit mass of the material is ( 2 1 ). herefore, from (3.38), the finite quantity of heat q added to 22 he term dq is sometimes called the diabatic (or nonadiabatic) heating or cooling, where diabatic means involving the transfer of heat. he term diabatic would be redundant if heating and cooling were always taken to mean the addition or removal of heat. However, heating and cooling are often used in the sense of to raise or lower the temerature of, in which case it is meaningful to distinguish between that art of the temerature change d due to diabatic effects (dq) and that art due to adiabatic effects (d).

14 76 Atmosheric hermodynamics a unit mass of the material at constant ressure is given by where u 1 and u 2 are, resectively, the initial and final internal energies for a unit mass of the material. herefore, at constant ressure, where h is the enthaly of a unit mass of the material, which is defined by (3.47) Because u,,and are functions of state, h is a function of state. Differentiating (3.47), we obtain Substituting for du from (3.40) and combining with (3.43), we obtain (3.48) which is yet another form of the first law of thermodynamics. By comaring (3.46) and (3.48) we see that or, in integrated form, q (u 2 u 1 ) (2 1) (u 2 2) (u 1 1) q h 2 h 1 h # u dh du d() dq dh d dh c d h c (3.49) (3.50) where h is taken as zero when 0. In view of (3.50), h corresonds to the heat required to raise the temerature of a material from 0 to K at constant ressure. When a layer of air that is at rest and in hydrostatic balance is heated, for examle, by radiative transfer, the weight of the overlying air ressing down on it remains constant. Hence, the heating is at constant ressure. he energy added to the air is realized in the form of an increase in enthaly (or sensible heat, as atmosheric scientists commonly refer to it) and dq dh c d he air within the layer exands as it warms, doing work on the overlying air by lifting it against the Earth s gravitational attraction. Of the energy er unit mass imarted to the air by the heating, we see from (3.40) and (3.41) that du c v d is reflected in an increase in internal energy and d Rd is exended doing work on the overlying air. Because the Earth s atmoshere is made u mainly of the diatomic gases N 2 and O 2,the energy added by the heating dq is artitioned between the increase in internal energy du and the exansion work d in the ratio 5:2. We can write a more general exression that is alicable to a moving air arcel, the ressure of which changes as it rises or sinks relative to the surrounding air. By combining (3.20), (3.48), and (3.50) we obtain dq d(h ) d(c ) (3.51) Hence, if the material is a arcel of air with a fixed mass that is moving about in an hydrostatic atmoshere, the quantity (h ), which is called the dry static energy, is constant rovided the arcel neither gains nor loses heat (i.e., dq 0) Adiabatic Processes If a material undergoes a change in its hysical state (e.g., its ressure, volume, or temerature) without any heat being added to it or withdrawn from it, the change is said to be adiabatic. Suose that the initial state of a material is reresented by the oint A on the V diagram in Fig. 3.5 and that when the material undergoes an isothermal transformation it moves along the line AB. If the same material underwent a similar change in volume but under adiabatic conditions, the transformation would 23 Strictly seaking, Eq. (3.51) holds only for an atmoshere in which there are no fluid motions. However, it is correct to within a few ercent for the Earth s atmoshere where the kinetic energy of fluid motions reresents only a very small fraction of the total energy. An exact relationshi can be obtained by using Newton s second law of motion and the continuity equation in lace of Eq. (3.20) in the derivation. See J. R. Holton, An Introduction to Dynamic Meteorology,4th ed.,academic Press, (2004).

15 3.4 Adiabatic Processes 77 Fig. 3.5 Pressure be reresented by a curve such as AC, which is called an adiabat.he reason why the adiabat AC is steeer than the isotherm AB on a V diagram can be seen as follows. During adiabatic comression, the internal energy increases [because dq 0 and d is negative in (3.38)] and therefore the temerature of the system rises. However, for isothermal comression, the temerature remains constant. Hence, C B and therefore C B Concet of an Air Parcel C B Isotherm Volume Adiabat An isotherm and an adiabat on a V diagram. In many fluid mechanics roblems, mixing is viewed as a result of the random motions of individual molecules. In the atmoshere, molecular mixing is imortant only within a centimeter of the Earth s surface and at levels above the turboause (105 km). At intermediate levels, virtually all mixing in the vertical is accomlished by the exchange of macroscale air arcels with horizontal dimensions ranging from millimeters to the scale of the Earth itself. o gain some insights into the nature of vertical mixing in the atmoshere, it is useful to consider the behavior of an air arcel of infinitesimal dimensions that is assumed to be i. thermally insulated from its environment so that its temerature changes adiabatically as it rises or sinks, always remaining at exactly the same ressure as the environmental air at the same level, 24 which is assumed to be in hydrostatic equilibrium; and ii. moving slowly enough that the macroscoic kinetic energy of the air arcel is a negligible fraction of its total energy. Although in the case of real air arcels one or more of these assumtions is nearly always violated A to some extent, this simle, idealized model is helful in understanding some of the hysical rocesses that influence the distribution of vertical motions and vertical mixing in the atmoshere he Dry Adiabatic Lase Rate We will now derive an exression for the rate of change of temerature with height of a arcel of dry air that moves about in the Earth s atmoshere while always satisfying the conditions listed at the end of Section Because the air arcel undergoes only adiabatic transformations (dq 0) and the atmoshere is in hydrostatic equilibrium, for a unit mass of air in the arcel we have, from (3.51), d(c ) 0 (3.52) Dividing through by dz and making use of (3.20) we obtain d dz dry arcel (3.53) where d is called the dry adiabatic lase rate.because an air arcel exands as it rises in the atmoshere, its temerature will decrease with height so that d defined by (3.53) is a ositive quantity. Substituting ɡ 9.81 m s 2 and c 1004 J K 1 kg 1 into (3.53) gives d K m 1 or 9.8 K km 1,which is the numerical value of the dry adiabatic lase rate. It should be emhasized again that d is the rate of change of temerature following a arcel of dry air that is being raised or lowered adiabatically in the atmoshere. he actual lase rate of temerature in a column of air, which we will indicate by z, as measured, for examle, by a radiosonde, averages 6 7 K km 1 in the trooshere, but it takes on a wide range of values at individual locations Potential emerature he otential temerature of an air arcel is defined as the temerature that the arcel of air would have if it were exanded or comressed adiabatically from its existing ressure and temerature to a standard ressure 0 (generally taken as 1000 hpa). ɡ c d 24 Any ressure differences between the arcel and its environment give rise to sound waves that roduce an almost instantaneous adjustment. emerature differences, however, are eliminated by much slower rocesses.

16 78 Atmosheric hermodynamics We can derive an exression for the otential temerature of an air arcel in terms of its ressure, temerature, and the standard ressure 0 as follows. For an adiabatic transformation (dq 0) (3.46) becomes Substituting from (3.3) into this exression yields Integrating uward from 0 (where, by definition, ) to,we obtain or aking the antilog of both sides or c d d 0 c d R d 0 c R c R ln d d 0 cr 0 0 ln 0 Rc (3.54) Equation (3.54) is called Poisson s 25 equation. It is usually assumed that R R d 287 J K 1 kg 1 and c c d 1004 J K 1 kg 1 ;therefore, R c Parameters that remain constant during certain transformations are said to be conserved. Potential temerature is a conserved quantity for an air arcel that moves around in the atmoshere under adiabatic conditions (see Exercise 3.36). Potential temerature is an extremely useful arameter in atmosheric thermodynamics, since atmosheric rocesses are often close to adiabatic, and therefore remains essentially constant, like density in an incomressible fluid hermodynamic Diagrams Poisson s equation may be conveniently solved in grahical form. If ressure is lotted on the ordinate on a distorted scale, in which the distance from the origin is roortional to R d c, or is used, regardless of whether air is dry or moist, and temerature (in K) is lotted on the abscissa, then (3.54) becomes (3.55) For a constant value of,eq.(3.55) is of the form y x where y 0.286, x,and the constant of roortionality is Each constant value of reresents adry adiabat,which is defined by a straight line with a articular sloe that asses through the oint 0, 0. If the ressure scale is inverted so that increases downward, the relation takes the form shown in Fig. 3.6, which is the basis for the seudoadiabatic chart that used to be widely used for meteorological comutations. he region of the chart of greatest interest in the atmoshere is the ortion shown within the dotted lines in Fig. 3.6, and this is generally the only ortion of the chart that is rinted. In the seudoadiabatic chart, isotherms are vertical and dry adiabats (constant ) are oriented at an acute angle relative to isotherms (Fig. 3.6). Because changes in temerature with height in the atmoshere generally lie between isothermal and dry adiabatic, most temerature soundings lie within a narrow range of angles when lotted on a seudoadiabatic chart. his restriction is overcome in the so-called skew ln chart, in which the ordinate (y) is ln (the minus sign ensures that lower ressure levels are located above higher ressure levels on the chart) and the abscissa (x) is Since, from (3.56), x (constant)y (constant) ln y x (constant) (3.56) and for an isotherm is constant, the relationshi between y and x for an isotherm is of the form 25 Simeon Denis Poisson ( ) French mathematician. Studied medicine but turned to alied mathematics and became the first rofessor of mechanics at the Sorbonne in Paris.

17 3.5 Water Vaor in Air θ = 100 K θ = 200 K θ = 300 K θ = 400 K θ = 500 K Pressure (hpa) θ = 253 K Dry Adiabat Isotherm = 20 C θ = 273 K θ = 293 K = 0 C = 20 C Pressure (hpa) Fig. 3.7 Schematic of a ortion of the skew ln chart. (An accurate reroduction of a larger ortion of the chart is available on the book web site that accomanies this book, from which it can be rinted and used for solving exercises.) emerature (K) 400 are more aarent than they are on the seudoadiabatic chart. Fig. 3.6 he comlete seudoadiabatic chart. Note that increases downward and is lotted on a distorted scale (reresenting ). Only the blue-shaded area is generally rinted for use in meteorological comutations. he sloing lines, each labeled with a value of the otential temerature, are dry adiabats. As required by the definition of, the actual temerature of the air (given on the abscissa) at 1000 hpa is equal to its otential temerature. y mx c, where m is the same for all isotherms and c is a different constant for each isotherm. herefore, on the skew ln chart, isotherms are straight arallel lines that sloe uward from left to right. he scale for the x axis is generally chosen to make the angle between the isotherms and the isobars about 45, as deicted schematically in Fig Note that the isotherms on a skew ln chart are intentionally skewed by about 45 from their vertical orientation in the seudoadiabatic chart (hence the name skew ln chart). From (3.55), the equation for a dry adiabat ( constant) is ln (constant) ln constant Hence, on a ln versus ln chart, dry adiabats would be straight lines. Since ln is the ordinate on the skew ln chart, but the abscissa is not ln, dry adiabats on this chart are slightly curved lines that run from the lower right to the uer left. he angle between the isotherms and the dry adiabats on a skew ln chart is aroximately 90 (Fig. 3.7). herefore, when atmosheric temerature soundings are lotted on this chart, small differences in sloe Exercise 3.5 A arcel of air has a temerature of 51 C at the 250-hPa level. What is its otential temerature? What temerature will the arcel have if it is brought into the cabin of a jet aircraft and comressed adiabatically to a cabin ressure of 850 hpa? Solution: his exercise can be solved using the skew ln chart. Locate the original state of the air arcel on the chart at ressure 250 hpa and temerature 51 C. he label on the dry adiabat that asses through this oint is 60 C, which is therefore the otential temerature of the air. he temerature acquired by the ambient air if it is comressed adiabatically to a ressure of 850 hpa can be found from the chart by following the dry adiabat that asses through the oint located by 250 hpa and 51 C down to a ressure of 850 hpa and reading off the temerature at that oint. It is 44.5 C. (Note that this suggests that ambient air brought into the cabin of a jet aircraft at cruise altitude has to be cooled by about 20 C to rovide a comfortable environment.) 3.5 Water Vaor in Air So far we have indicated the resence of water vaor in the air through the vaor ressure e that it exerts, and we have quantified its effect on the density of air by introducing the concet of virtual temerature. However, the amount of water vaor resent in a certain quantity of air may be exressed in many different ways, some of the more imortant

18 80 Atmosheric hermodynamics of which are resented later. We must also discuss what haens when water vaor condenses in air Moisture Parameters a. Mixing ratio and secific humidity he amount of water vaor in a certain volume of air may be defined as the ratio of the mass m v of water vaor to the mass of dry air; this is called the mixing ratio w.hat is w m v m d (3.57) he mixing ratio is usually exressed in grams of water vaor er kilogram of dry air (but in solving numerical exercises w must be exressed as a dimensionless number, e.g., as kg of water vaor er kg of dry air). In the atmoshere, the magnitude of w tyically ranges from a few grams er kilogram in middle latitudes to values of around 20 g kg 1 in the troics. If neither condensation nor evaoration takes lace, the mixing ratio of an air arcel is constant (i.e., it is a conserved quantity). he mass of water vaor m v in a unit mass of air (dry air lus water vaor) is called the secific humidity q,that is m v q w m v m d 1 w Because the magnitude of w is only a few ercent, it follows that the numerical values of w and q are nearly equivalent. Exercise 3.6 If air contains water vaor with a mixing ratio of 5.5 g kg 1 and the total ressure is hpa, calculate the vaor ressure e. Solution: he artial ressure exerted by any constituent in a mixture of gases is roortional to the number of moles of the constituent in the mixture. herefore, the ressure e due to water vaor in air is given by m v n v and n d are the number of moles of water vaor and dry air in the mixture, resectively, M w is the molecular weight of water, M d is the aarent molece n v M w n d n v m d m v M d M w (3.58) ular weight of dry air, and is the total ressure of the moist air. From (3.57) and (3.58) we obtain e w w (3.59) where is defined by (3.14). Substituting hpa and w kg kg 1 into (3.59), we obtain e 9.0 hpa. Exercise 3.7 Calculate the virtual temerature correction for moist air at 30 C that has a mixing ratio of 20 g kg 1. Solution: Substituting e from (3.59) into (3.16) and simlifying v w (1 w) Dividing the denominator into the numerator in this exression and neglecting terms in w 2 and higher orders of w,we obtain v 1 w or, substituting and rearranging, v (1 0.61w) (3.60) With 303 K and w kg kg 1, Eq. (3.60) gives v K. herefore, the virtual temerature correction is v 3.7 degrees (K or C). Note that (3.60) is a useful exression for obtaining v from and the moisture arameter w. b. Saturation vaor ressures Consider a small closed box, the floor of which is covered with ure water at temerature. Initially assume that the air is comletely dry. Water will begin to evaorate and, as it does, the number of water molecules in the box, and therefore the water vaor ressure, will increase. As the water vaor ressure increases, so will the rate at which the water molecules condense from the vaor hase back to the liquid hase. If the rate of condensation is less than the rate of evaoration, the box is said to be unsaturated at temerature (Fig. 3.8a). When the water vaor ressure in the box increases to the oint that the rate of condensation is equal to the rate of evaoration (Fig. 3.8b), the air is said to be saturated with resect

19 3.5 Water Vaor in Air Can Air Be Saturated with Water Vaor? 26 It is common to use hrases such as the air is saturated with water vaor, the air can hold no more water vaor, and warm air can hold more water vaor than cold air. hese hrases, which suggest that air absorbs water vaor, rather like a songe, are misleading. We have seen that the total ressure exerted by a mixture of gases is equal to the sum of the ressures that each gas would exert if it alone occuied the total volume of the mixture of gases (Dalton s law of artial ressures). Hence, the exchange of water molecules between its liquid and vaor hases is (essentially) indeendent of the resence of air. Strictly seaking, the ressure exerted by water vaor that is in equilibrium with water at a given temerature is referred more aroriately to as equilibrium vaor ressure rather than saturation vaor ressure at that temerature. However, the latter term, and the terms unsaturated air and saturated air, rovide a convenient shorthand and are so deely rooted that they will aear in this book. to a lane surface of ure water at temerature, and the ressure e s that is then exerted by the water vaor is called the saturation vaor ressure over a lane surface of ure water at temerature. Similarly, if the water in Fig. 3.8 were relaced by a lane surface of ure ice at temerature and the rate of condensation of water vaor were equal to the rate of evaoration of the ice, the ressure e si exerted by the water vaor would be the saturation vaor ressure over a lane surface of ure ice at. Because, at any given temerature, the rate of evaoration from ice is less than from water, e s () e si (). he rate at which water molecules evaorate from either water or ice increases with increasing temerature. 27 Consequently, both e s and e si increase with increasing temerature, and their magnitudes Water (a) Unsaturated, e, e s Water (b) Saturated Fig. 3.8 A box (a) unsaturated and (b) saturated with resect to a lane surface of ure water at temerature. Dots reresent water molecules. Lengths of the arrows reresent the relative rates of evaoration and condensation. he saturated (i.e., equilibrium) vaor ressure over a lane surface of ure water at temerature is e s as indicated in (b). deend only on temerature. he variations with temerature of e s and e s e si are shown in Fig. 3.9, where it can be seen that the magnitude of e s e si reaches a eak value at about 12 C. It follows that if an ice article is in water-saturated air it will grow due to the deosition of water vaor uon it. In Section it is shown that this henomenon Saturation vaor ressure e s over ure water (hpa) e s e si emerature ( C) e s Fig. 3.9 Variations with temerature of the saturation (i.e., equilibrium) vaor ressure e s over a lane surface of ure water (red line, scale at left) and the difference between e s and the saturation vaor ressure over a lane surface of ice e si (blue line, scale at right). e s e si (hpa) 26 For further discussion of this and some other common misconcetions related to meteorology see C. F. Bohren s Clouds in a Glass of Beer,Wiley and Sons,New York, As a rough rule of thumb, it is useful to bear in mind that the saturation vaor ressure roughly doubles for a 10 C increase in temerature.

20 82 Atmosheric hermodynamics lays a role in the initial growth of reciitable articles in some clouds. c. Saturation mixing ratios he saturation mixing ratio w s with resect to water is defined as the ratio of the mass m vs of water vaor in a given volume of air that is saturated with resect to a lane surface of ure water to the mass m d of the dry air. hat is (3.61) Because water vaor and dry air both obey the ideal gas equation w s w s m vs m d vs e s d (R v ) ( e s) (R d ) (3.62) where vs is the artial density of water vaor required to saturate air with resect to water at temerature, d is the artial density of the dry air (see Section 3.1.1), and is the total ressure. Combining (3.62) with (3.14), we obtain w s e s For the range of temeratures observed in the Earth s atmoshere, e s ;therefore w s e s (3.63) Hence, at a given temerature, the saturation mixing ratio is inversely roortional to the total ressure. Because e s deends only on temerature, it follows from (3.63) that w s is a function of temerature and ressure. Lines of constant saturation mixing ratio are rinted as dashed green lines on the skew ln chart and are labeled with the value of w s in grams of water vaor er kilogram of dry air. It is aarent from the sloe of these lines that at constant ressure w s increases with increasing temerature, and at constant temerature w s increases with decreasing ressure. d. Relative humidity; dew oint and frost oint he relative humidity (RH) with resect to water is the ratio (exressed as a ercentage) of the actual e s mixing ratio w of the air to the saturation mixing ratio w s with resect to a lane surface of ure water at the same temerature and ressure. hat is RH 100 w w s 100 e e s (3.64) he dew oint d is the temerature to which air must be cooled at constant ressure for it to become saturated with resect to a lane surface of ure water. In other words, the dew oint is the temerature at which the saturation mixing ratio w s with resect to liquid water becomes equal to the actual mixing ratio w. It follows that the relative humidity at temerature and ressure is given by RH 100 w s (at temerature d and ressure ) w s (at temerature and ressure ) (3.65) A simle rule of thumb for converting RH to a dew oint deression ( d ) for moist air (RH 50%) is that d decreases by 1 C for every 5% decrease in RH (starting at d dry bulb temerature (), where RH 100%). For examle, if the RH is 85%, d and the dew oint 5 deression is d 3 C. he frost oint is defined as the temerature to which air must be cooled at constant ressure to saturate it with resect to a lane surface of ure ice. Saturation mixing ratios and relative humidities with resect to ice may be defined in analogous ways to their definitions with resect to liquid water. When the terms mixing ratio and relative humidity are used without qualification they are with resect to liquid water. Exercise 3.8 Air at 1000 hpa and 18 C has a mixing ratio of 6 g kg 1.What are the relative humidity and dew oint of the air? Solution: his exercise may be solved using a skew ln chart. he students should dulicate the following stes. First locate the oint with ressure 1000 hpa and temerature 18 C. We see from the chart that the saturation mixing ratio for this state is 13 g kg 1.Since the air secified in the roblem has a mixing ratio of only 6 g kg 1,it is unsaturated and its relative humidity is, from (3.64), %. o find the dew oint we move from right to

21 3.5 Water Vaor in Air 83 left along the 1000-hPa ordinate until we intercet the saturation mixing ratio line of magnitude 6 g kg 1 ;this occurs at a temerature of about 6.5 C. herefore, if the air is cooled at constant ressure, the water vaor it contains will just saturate the air with resect to water at a temerature of 6.5 C. herefore, by definition, the dew oint of the air is 6.5 C. At the Earth s surface, the ressure tyically varies by only a few ercent from lace to lace and from time to time. herefore, the dew oint is a good indicator of the moisture content of the air. In warm, humid weather the dew oint is also a convenient indicator of the level of human discomfort. For examle, most eole begin to feel uncomfortable when the dew oint rises above 20 C, and air with a dew oint above about 22 C is generally regarded as extremely humid or sticky. Fortunately, dew oints much above this temerature are rarely observed even in the troics. In contrast to the dew oint, relative humidity deends as much uon the temerature of the air as uon its moisture content. On a sunny day the relative humidity may dro by as much as 50% from morning to afternoon, just because of a rise in air temerature. Neither is relative humidity a good indicator of the level of human discomfort. For examle, a relative humidity of 70% may feel quite comfortable at a temerature of 20 C, but it would cause considerable discomfort to most eole at a temerature of 30 C. he highest dew oints occur over warm bodies of water or vegetated surfaces from which water is evaorating. In the absence of vertical mixing, the air just above these surfaces would become saturated with water vaor, at which oint the dew oint would be the same as the temerature of the underlying surface. Comlete saturation is rarely achieved over hot surfaces, but dew oints in excess of 25 C are sometimes observed over the warmest regions of the oceans. e. Lifting condensation level he lifting condensation level (LCL) is defined as the level to which an unsaturated (but moist) arcel of air can be lifted adiabatically before it becomes saturated with resect to a lane surface of ure water. During lifting the mixing ratio w and otential temerature of the air arcel remain constant, but the saturation mixing ratio w s decreases until it becomes equal to w at the LCL. herefore, the LCL is located at the intersection of the otential temerature line assing through the temerature and ressure of the air arcel, and the w s line that asses through the ressure and dew oint d of the arcel (Fig. 3.10). Since the dew oint and LCL are related in the manner indicated in Fig. 3.10, knowledge of either one is sufficient to determine the other. Similarly, a knowledge of,,and any one moisture arameter is sufficient to determine all the other moisture arameters we have defined. f. Wet-bulb temerature he wet-bulb temerature is measured with a thermometer, the glass bulb of which is covered with a moist cloth over which ambient air is drawn. he heat required to evaorate water from the moist cloth to saturate the ambient air is sulied by the air as it comes into contact with the cloth. When the difference between the temeratures of the bulb and the ambient air is steady and sufficient to suly the heat needed to evaorate the water, the thermometer will read a steady temerature, which is called the wet-bulb temerature. If a raindro falls through a layer of air that has a constant wet-bulb temerature, the raindro will eventually reach a temerature equal to the wet-bulb temerature of the air. he definition of wet-bulb temerature and dew oint both involve cooling a hyothetical air arcel to saturation, but there is a distinct difference. If the unsaturated air aroaching the wet bulb has a mixing ratio w,the dew oint d is the temerature to which the air must be cooled at constant ressure Pressure (hpa) Lifting condensation level for air at A w s constant C θ constant B A (, d, w s (A)) (,, w) Fig he lifting condensation level of a arcel of air at A, with ressure, temerature, and dew oint d, is at C on the skew ln chart.

22 84 Atmosheric hermodynamics to become saturated. he air that leaves the wet bulb has a mixing ratio w that saturates it at temerature w.if the air aroaching the wet bulbis unsaturated, w is greater than w; therefore, d w, where the equality signs aly only to air saturated with resect to a lane surface of ure water. Usually w is close to the arithmetic mean of and d Latent Heats If heat is sulied to a system under certain conditions it may roduce a change in hase rather than a change in temerature. In this case, the increase in internal energy is associated entirely with a change in molecular configurations in the resence of intermolecular forces rather than an increase in the kinetic energy of the molecules (and therefore the temerature of the system). For examle, if heat is sulied to ice at 1 atm and 0 C, the temerature remains constant until all of the ice has melted. he latent heat of melting (L m ) is defined as the heat that has to be given to a unit mass of a material to convert it from the solid to the liquid hase without a change in temerature. he temerature at which this hase change occurs is called the melting oint. At 1 atm and 0 C the latent heat of melting of the water substance is J kg 1.he latent heat of freezing has the same numerical value as the latent heat of melting, but heat is released as a result of the change in hase from liquid to solid. Similarly, the latent heat of vaorization or evaoration (L v ) is the heat that has to be given to a unit mass of material to convert it from the liquid to the vaor hase without a change in temerature. For the water substance at 1 atm and 100 C (the boiling oint of water at 1 atm), the latent heat of vaorization is J kg 1. he latent heat of condensation has the same value as the latent heat of vaorization, but heat is released in the change in hase from vaor to liquid. 28 As will be shown in Section 3.7.3, the melting oint (and boiling oint) of a material deends on ressure Saturated Adiabatic and Pseudoadiabatic Processes When an air arcel rises in the atmoshere its temerature decreases with altitude at the dry adiabatic lase rate (see Section 3.4.2) until it becomes saturated with water vaor. Further lifting results in the condensation of liquid water (or the deosition of ice), which releases latent heat. Consequently, the rate of decrease in the temerature of the rising arcel is reduced. If all of the condensation roducts remain in the rising arcel, the rocess may still be considered to be adiabatic (and reversible), even though latent heat is released in the system, rovided that heat does not ass through the boundaries of the arcel. he air arcel is then said to undergo a saturated adiabatic rocess. However,if all of the condensation roducts immediately fall out of the air arcel, the rocess is irreversible, and not strictly adiabatic, because the condensation roducts carry some heat. he air arcel is then said to undergo a seudoadiabatic rocess.as the reader is invited to verify in Exercise 3.44, the amount of heat carried by condensation roducts is small comared to that carried by the air itself. herefore, the saturated-adiabatic and the seudoadiabatic lase rates are virtually identical he Saturated Adiabatic Lase Rate In contrast to the dry adiabatic lase rate d,which is constant, the numerical value of the saturated adiabatic lase rate s varies with ressure and temerature. (he reader is invited to derive an exression for s in Exercise 3.50; see the book Web site.) Because water vaor condenses when a saturated air arcel rises, it follows that s d. Actual values of s range from about 4 K km 1 near the ground in warm, humid air masses to tyical values of 67K km 1 in the middle trooshere. For tyical temeratures near the trooause, s is only slightly less than d because the saturation vaor ressure of the air is so small that the effect of condensation is negligible. 29 Lines that show the rate of decrease in 28 Normally, when heat is given to a substance, the temerature of the substance increases. his is called sensible heat.however,when heat is given to a substance that is melting or boiling, the temerature of the substance does not change until all of the substance is melted or vaorized. In this case, the heat aears to be latent (i.e., hidden). Hence the terms latent heat of melting and latent heat of vaorization. 29 William homson (later Lord Kelvin) was the first (in 1862) to derive quantitative estimates of the dry and saturated adiabatic lase rates based on theoretical arguments. For an interesting account of the contributions of other 19th-century scientists to the realization of the imortance of latent heat in the atmoshere, see W. E. K. Middleton, A History of the heories of Rain,Franklin Watts,Inc.,New York, 1965, Chater 8.

23 3.5 Water Vaor in Air 85 temerature with height of a arcel of air that is rising or sinking in the atmoshere under saturated adiabatic (or seudoadiabatic) conditions are called saturated adiabats (or seudoadiabats). On the skew ln chart these are the curved green lines that diverge uward and tend to become arallel to the dry adiabats. saturated adiabatic ascent or descent. Substituting (3.3) into (3.46) gives dq c d R d (3.66) From (3.54) the otential temerature is given by Exercise 3.9 A arcel of air with an initial temerature of 15 C and dew oint 2 C is lifted adiabatically from the 1000-hPa level. Determine its LCL and temerature at that level. If the air arcel is lifted a further 200 hpa above its LCL, what is its final temerature and how much liquid water is condensed during this rise? or, differentiating, ln ln R c ln constant c d c d R d (3.67) Solution: he student should dulicate the following stes on the skew ln chart (see the book Web site). First locate the initial state of the air on the chart at the intersection of the 15 C isotherm with the 1000-hPa isobar. Because the dew oint of the air is 2 C, the magnitude of the saturation mixing ratio line that asses through the 1000-hPa ressure level at 2 C is the actual mixing ratio of the air at 15 C and 1000 hpa. From the chart this is found to be about 4.4 g kg 1. Because the saturation mixing ratio at 1000 hpa and 15 C is about 10.7 g kg 1,the air is initially unsaturated. herefore, when it is lifted it will follow a dry adiabat (i.e., a line of constant otential temerature) until it intercets the saturation mixing ratio line of magnitude 4.4 g kg 1.Following uward along the dry adiabat ( 288 K) that asses through 1000 hpa and 15 C isotherm, the saturation mixing ratio line of 4.4 g kg 1 is interceted at about the 820-hPa level. his is the LCL of the air arcel. he temerature of the air at this oint is about 0.7 C. For lifting above this level the air arcel will follow a saturated adiabat. Following the saturated adiabat that asses through 820 hpa and 0.7 C u to the 620-hPa level, the final temerature of the air is found to be about 15 C. he saturation mixing ratio at 620 hpa and 15 C is 1.9 g kg 1.herefore,about g of water must have condensed out of each kilogram of air during the rise from 820 to 620 hpa Equivalent Potential emerature and Wet-Bulb Potential emerature We will now derive an equation that describes how temerature varies with ressure under conditions of Combining (3.66) and (3.67) and substituting dq L v dw s,we obtain In Exercise 3.52 we show that From (3.68) and (3.69) his last exression can be integrated to give (3.68) (3.69) (3.70) We will define the constant of integration in (3.70) by requiring that at low temeratures, as w s : 0,.hen : or e L v c dw s d L v c dw s d L v w s c d L v w s d c L vw s c ln constant L vw s c ln e ex L v w s c (3.71) he quantity e given by (3.71) is called the equivalent otential temerature.it can be seen that e is the e

24 86 Atmosheric hermodynamics otential temerature of a arcel of air when all the water vaor has condensed so that its saturation mixing ratio w s is zero. Hence, recalling the definition of, the equivalent otential temerature of an air arcel may be found as follows. he air is exanded (i.e., lifted) seudoadiabatically until all the vaor has condensed, released its latent heat, and fallen out. he air is then comressed dry adiabatically to the standard ressure of 1000 hpa, at which oint it will attain the temerature e.(if the air is initially unsaturated, w s and are the saturation mixing ratio and temerature at the oint where the air first becomes saturated after being lifted dry adiabatically.) We have seen in Section that otential temerature is a conserved quantity for adiabatic transformations. he equivalent otential temerature is conserved during both dry and saturated adiabatic rocesses. If the line of constant equivalent otential temerature (i.e., the seudoadiabat) that asses through the wet-bulb temerature of a arcel of air is traced back on a skew ln chart to the oint where it intersects the 1000-hPa isobar, the temerature at this intersection is called the wet-bulb otential temerature w of the air arcel. Like the equivalent otential temerature, the wet-bulb otential temerature is conserved during both dry and saturated adiabatic rocesses. On skew ln charts, seudoadiabats are labeled (along the 200-hPa isobar) with the wet-bulb otential temerature w (in C) and the equivalent otential temerature e (in K) of air that rises or sinks along that seudoadiabat. Both w and e rovide equivalent information and are valuable as tracers of air arcels. When height, rather than ressure, is used as the indeendent variable, the conserved quantity during adiabatic or seudoadiabatic ascent or descent with water undergoing transitions between liquid and vaor hases is the moist static energy (MSE) 30 MSE c L v q (3.72) where is the temerature of the air arcel, is the geootential, and q v is the secific humidity (nearly the same as w). he first term on the right side of (3.72) is the enthaly er unit mass of air. he second term is the otential energy, and the third term is the latent heat content. he first two terms, which also aear in (3.51), are the dry static energy.when air is lifted dry adiabatically, enthaly is converted into otential energy and the latent heat content remains unchanged. In saturated adiabatic ascent, energy is exchanged among all three terms on the right side of (3.72): otential energy increases, while the enthaly and latent heat content both decrease. However, the sum of the three terms remains constant Normand s Rule Many of the relationshis discussed in this section are embodied in the following theorem, known as Normand s 31 rule,which is extremely helful in many comutations involving the skew ln chart. Normand s rule states that on a skew ln chart the lifting condensation level of an air arcel is located at the intersection of the otential temerature line that asses through the oint located by the temerature and ressure of the air arcel, the equivalent otential temerature line (i.e., the seudoadiabat) that asses through the oint located by the wet-bulb temerature and ressure of the air arcel, and the saturation mixing ratio line that asses through the oint determined by the dew oint and ressure of the air. his rule is illustrated in Fig for the case of an air arcel with temerature,ressure, dew oint d,and wet-bulb temerature w. It can be seen that if,,and d are known, w may be readily determined using Normand s rule. Also, by extraolating the e line that asses through w to the 1000-hPa level, the wet-bulb otential temerature w may be found (Fig. 3.11) Net Effects of Ascent Followed by Descent When a arcel of air is lifted above its LCL so that condensation occurs and if the roducts of the condensation fall out as reciitation, the latent heat gained by the air during this rocess will be retained by the air if the arcel returns to its original level. 30 he word static derives from the fact that the kinetic energy associated with macroscale fluid motions is not included. he reader is invited to show that the kinetic energy er unit mass is much smaller than the other terms on the right side of (3.72), rovided that the wind seed is small in comarison to the seed of sound. 31 Sir Charles William Blyth Normand ( ) British meteorologist. Director-General of Indian Meteorological Service, A founding member of the National Science Academy of India. Imroved methods for measuring atmosheric ozone.

25 3.5 Water Vaor in Air 87 Pressure (hpa) θ d w s θ e LCL he effects of the saturated ascent couled with the adiabatic descent are: i. net increases in the temerature and otential temerature of the arcel; ii. a decrease in moisture content (as indicated by changes in the mixing ratio, relative humidity, dew oint, or wet-bulb temerature); and, iii. no change in the equivalent otential temerature or wet-bulb otential temerature, which are conserved quantities for air arcels undergoing both dry and saturated rocesses. he following exercise illustrates these oints. Exercise 3.10 An air arcel at 950 hpa has a temerature of 14 C and a mixing ratio of 8 g kg 1. What is the wet-bulb otential temerature of the air? he air arcel is lifted to the 700-hPa level by assing over a mountain, and 70% of the water vaor that is condensed out by the ascent is removed by reciitation. Determine the temerature, otential temerature, mixing ratio, and wetbulb otential temerature of the air arcel after it has descended to the 950-hPa level on the other side of the mountain. w s w θw θ Fig Illustration of Normand s rule on the skew ln chart. he orange lines are isotherms. he method for determining the wet-bulb temerature ( w ) and the wet-bulb otential temerature ( w ) of an air arcel with temerature and dew oint d at ressure is illustrated. LCL denotes the lifting condensation level of this air arcel. Solution: On a skew ln chart (see the book Web site), locate the initial state of the air at 950 hpa and 14 C. he saturation mixing ratio for an air arcel with temerature and ressure is found from the chart to be 10.6 g kg 1.herefore,because the air has a mixing ratio of only 8 g kg 1,it is unsaturated.he wetbulb otential temerature ( w ) can be determined using the method indicated schematically in Fig. 3.11, which is as follows. race the constant otential temerature line that asses through the initial state of the air arcel u to the oint where it intersects the saturation mixing ratio line with value 8 g kg 1.his occurs at a ressure of about 890 hpa, which is the LCL of the air arcel. Now follow the equivalent otential temerature line that asses through this oint back down to the 1000-hPa level and read off the temerature on the abscissa it is 14 C. his is in the wet-bulb otential temerature of the air. When the air is lifted over the mountain, its temerature and ressure u to the LCL at 890 hpa are given by oints on the otential temerature line that asses through the oint 950 hpa and 14 C. With further ascent of the air arcel to the 700-hPa level, the air follows the saturated adiabat that asses through the LCL. his saturated adiabat intersects the 700-hPa level at a oint where the saturation mixing ratio is 4.7 g kg 1.herefore, g kg 1 of water vaor has to condense out between the LCL and the 700-hPa level, and 70% of this, or 2.3 g kg 1,is reciitated out. herefore, at the 700-hPa level 1 g kg 1 of liquid water remains in the air. he air arcel descends on the other side of the mountain at the saturated adiabatic lase rate until it evaorates all of its liquid water, at which oint the saturation mixing ratio will have risen to g kg 1.he air arcel is now at a ressure of 760 hpa and a temerature of 1.8 C. hereafter, the air arcel descends along a dry adiabat to the 950-hPa level, where its temerature is 20 C and the mixing ratio is still 5.7 g kg 1.If the method indicated in Fig is alied again, the wetbulb otential temerature of the air arcel will be found to be unchanged at 14 C. (he heating of air during its assage over a mountain, 6 C in this examle, is resonsible for the remarkable warmth of Föhn or Chinook winds, which often blow downward along the lee side of mountain ranges. 32 ) 32 he erson who first exlained the Föhn wind in this way aears to have been J. von Hann 33 in his classic book Lehrbuch der Meteorologie,Willibald Keller,Leizig, Julius F. von Hann ( ) Austrian meteorologist. Introduced thermodynamic rinciles into meteorology. Develoed theories for mountain and valley winds. Published the first comrehensive treatise on climatology (1883).

26 88 Atmosheric hermodynamics 3.6 Static Stability Unsaturated Air Consider a layer of the atmoshere in which the actual temerature lase rate (as measured, for examle, by a radiosonde) is less than the dry adiabatic lase rate d (Fig. 3.12a). If a arcel of unsaturated air originally located at level O is raised to the height defined by oints A and B, its temerature will fall to A,which is lower than the ambient temerature B at this level. Because the arcel immediately adjusts to the ressure of the ambient air, it is clear from the ideal gas equation that the colder arcel of air must be denser than the warmer ambient air. herefore, if left to itself, the arcel will tend to return to its original level. If the arcel is dislaced downward from O it becomes warmer than the ambient air and, if left to itself, the arcel will tend to rise back to its original level. In both cases, the arcel of air encounters a restoring force after being dislaced, which inhibits vertical mixing. hus, the condition d corresonds to a stable stratification (or ositive static stability) for unsaturated air arcels. In general, the larger the difference d,the greater the restoring force for a given dislacement and the greater the static stability. 34 Exercise 3.11 An unsaturated arcel of air has density and temerature, and the density and temerature of the ambient air are and. Derive an exression for the uward acceleration of the air arcel in terms of,,and ɡ. Solution: he situation is deicted in Fig If we consider a unit volume of the air arcel, its mass is. herefore,the downward force acting on unit volume of the arcel is ɡ. From the Archimedes 35 rincile we know that the uward force acting on the arcel is equal in magnitude to the gravitational force that acts on the ambient air that is dislaced by the air arcel. Because a unit volume of ambient air of density is dislaced by the air arcel, the magnitude of the uward force acting on the air arcel is ɡ. herefore,the net uward force (F) acting on a unit volume of the arcel is F ( ) ɡ ρg ρ, ρ, Height A B O A B emerature (a) Γ Γ d Height B A O B A emerature (b) Fig Conditions for (a) ositive static stability ( d ) and (b) negative static instability ( d ) for the dislacement of unsaturated air arcels. Γ d Γ ρ g Surface Fig he box reresents an air arcel of unit volume with its center of mass at height z above the Earth s surface. he density and temerature of the air arcel are and, resectively, and the density and temerature of the ambient air are and. he vertical forces acting on the air arcel are indicated by the thicker arrows. z 34 A more general method for determing static stability is given in Section Archimedes ( B.C.) he greatest of Greek scientists. He invented engines of war and the water screw and he derived the rincile of buoyancy named after him. When Syracuse was sacked by Rome, a soldier came uon the aged Archimedes absorbed in studying figures he had traced in the sand: Do not disturb my circles said Archimedes, but was killed instantly by the soldier. Unfortunately, right does not always conquer over might.

27 3.6 Static Stability 89 Because the mass of a unit volume of the air arcel is,the uward acceleration of the arcel is d 2 z dt 2 F ɡ herefore 0 ( d ) z ( d ) z where z is the height of the air arcel. he ressure of the air arcel is the same as that of the ambient air, since they are at the same height in the atmoshere. herefore, from the gas equation in the form of (3.2), the densities of the air arcel and the ambient air are inversely roortional to their temeratures. Hence, or 1 d 2 z dt d 2 z dt 2 ɡ (3.73) Strictly seaking, virtual temerature v should be used in lace of in all exressions relating to static stability. However, the virtual temerature correction is usually neglected excet in certain calculations relating to the boundary layer. Exercise 3.12 he air arcel in Fig. 3.12a is dislaced uward from its equilibrium level at z0 by a distance z to a new level where the ambient temerature is.he air arcel is then released.derive an exression that describes the subsequent vertical dislacement of the air arcel as a function of time in terms of, the lase rate of the ambient air (), and the dry adiabatic lase rate ( d ). Solution: Let z z 0 be the equilibrium level of the air arcel and zzz 0 be the vertical disalcement of the air arcel from its equilibrium level. Let 0 be the environmental air temerature at z z 0.If the air arcel is lifted dry adiabatically through a distance z from its equilibrium level, its temerature will be ɡ Substituting this last exression into (3.73), we obtain which may be written in the form where d 2 z dt 2 d 2 z dt 2 ɡ ( d ) z N2 z0 N ɡ ( d ) 1/2 (3.74) (3.75) N is referred to as the Brunt 36 Väisälä 37 frequency. Equation (3.74) is a second order ordinary differential equation. If the layer in question is stably stratified (that is to say, if d ), then we can be assured that N is real, N 2 is ositive, and the solution of (3.74) is z A cos Nt B sin Nt Making use of the conditions at the oint of maximum dislacement at time t 0, namely that zz(0) and dzdt 0 at t 0, it follows that z(t) z(0) cos Nt hat is to say, the arcel executes a buoyancy oscillation about its equilibrium level z with amlitude equal to its initial dislacement z(0), and frequency N (in units of radians er second). he Brunt Väisälä frequency is thus a measure of the static stability: the higher the frequency, the greater the ambient stability. Air arcels undergo buoyancy oscillations in association with gravity waves, a widesread henomenon in lanetary atmosheres, as illustrated in Fig Gravity waves may be excited by flow over 36 Sir David Brunt ( ) English meteorologist. First full-time rofessor of meteorology at Imerial College ( ). His textbook Physical and Dynamical Meteorology,ublished in the 1930s,was one of the first modern unifying accounts of meteorology. 37 Vilho Väisälä ( ) Finnish meteorologist. Develoed a number of meteorological instruments, including a version of the radiosonde in which readings of temerature, ressure, and moisture are telemetered in terms of radio frequencies. he modern counterart of this instrument is one of Finland s successful exorts.

28 90 Atmosheric hermodynamics Solution: For the eriod of the orograhic forcing to match the eriod of the buoyancy oscillation, it is required that where L is the sacing between the ridges. Hence, from this last exression and (3.75), or, in SI units, U 104 U LN m s 1 L U 2 N L 2ɡ ( d ) 1/ ( ) 103 1/2 Fig Gravity waves, as revealed by cloud atterns. he uer hotograh, based on NOAA GOES 8 visible satellite imagery, shows a wave attern in west to east (right to left) airflow over the north south-oriented mountain ranges of the Aalachians in the northeastern United States. he waves are transverse to the flow and their horizontal wavelength is 20 km. he atmosheric wave attern is more regular and widesread than the undulations in the terrain. he bottom hotograh, based on imagery from NASA s multiangle imaging sectro-radiometer (MISR), shows an even more regular wave attern in a thin layer of clouds over the Indian Ocean. Layers of air with negative lase rates (i.e., temeratures increasing with height) are called inversions. It is clear from the aforementioned discussion that these layers are marked by very strong static stability. A low-level inversion can act as a lid that tras ollution-laden air beneath it (Fig. 3.15). he layered structure of the stratoshere derives from the fact that it reresents an inversion in the vertical temerature rofile. If d (Fig. 3.12b), a arcel of unsaturated air dislaced uward from O will arrive at A with a temerature greater than that of its environment. herefore, it will be less dense than the ambient air mountainous terrain, as shown in the to hotograh in Fig or by an intense local disturbance, as shown in the bottom hotograh. he following exercise illustrates how buoyancy oscillations can be excited by flow over a mountain range. Exercise 3.13 A layer of unsaturated air flows over mountainous terrain in which the ridges are 10 km aart in the direction of the flow. he lase rate is 5 C km 1 and the temerature is 20 C. For what value of the wind seed U will the eriod of the orograhic (i.e., terrain-induced) forcing match the eriod of a buoyancy oscillation? Fig Looking down onto widesread haze over southern Africa during the biomass-burning season. he haze is confined below a temerature inversion. Above the inversion, the air is remarkably clean and the visibility is excellent. (Photo: P. V. Hobbs.)