ONE. The Earth-atmosphere system CHAPTER
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1 CHAPTER ONE The Earth-atmoshere system 1.1 INTRODUCTION The Earth s atmoshere is the gaseous enveloe surrounding the lanet. Like other lanetary atmosheres, it figures centrally in transfers of energy between the sun, the Earth, and dee sace. It also figures in transfers of energy from one region of the globe to another. By maintaining thermal equilibrium, such transfers determine the Earth s climate. However, among neighboring lanets, the Earth s atmoshere is unique because it is related closely to ocean and surface rocesses that, together with the atmoshere, form the basis for life. Because it is a fluid system, the atmoshere is caable of suorting a wide sectrum of motions. These range from turbulent eddies of a few meters to circulations with dimensions of the Earth itself. By rearranging mass, air motion influences other atmosheric comonents such as water vaor, ozone, and cloud, which figure rominently in radiative and chemical rocesses. Such influence makes the atmosheric circulation a key ingredient of the global energy budget Descritions of atmosheric behavior The mobility of a fluid system makes its descrition comlex. Atmosheric motion redistributes mass and constituents into a variety of comlex configurations. Like any fluid system, the atmoshere is governed by the laws of continuum mechanics. They can be derived from the laws of mechanics and thermodynamics that govern a discrete fluid body by generalizing those laws to a continuum of such systems. In the atmoshere, the discrete system to which these laws aly is an infinitesimal fluid element, or air arcel, which is defined by a fixed collection of matter. 1
2 2 The Earth-atmoshere system Two frameworks are used to describe atmosheric behavior. The Eulerian descrition reresents atmosheric behavior in terms of field roerties, such as the instantaneous distributions of temerature, motion, and constituents. Governed by artial differential equations, the field descrition of atmosheric behavior is convenient for numerical alications. The Lagrangian descrition reresents atmosheric behavior in terms of the roerties of individual air arcels (e.g., in terms of their instantaneous ositions, temeratures, and constituent concentrations). Because it focuses on transformations of roerties within an air arcel and on interactions between that system and its environment, the Lagrangian descrition offers concetual as well as certain diagnostic advantages. For this reason, the basic rinciles governing atmosheric behavior are develoed in this text from a Lagrangian ersective. In the Lagrangian framework, the system considered is an individual air arcel moving through the circulation. Although it may change in form through deformation and in comosition through thermodynamic and chemical transformations, this system is uniquely identified by the matter comrising it initially. Mass can be transferred across the boundary of an air arcel through molecular diffusion and turbulent mixing. However, such transfers are slow enough to be ignored for many alications. An individual arcel can then change only through interaction with its environment and through internal transformations that alter its comosition and state Mechanisms influencing atmosheric behavior Of the factors influencing atmosheric behavior, gravity is the single most imortant. Even though it has no uer boundary, the atmoshere is contained by the gravitational field of the Earth, which revents atmosheric mass from escaing to sace. Because it is such a strong body force, gravity determines many atmosheric roerties. Most immediate is the geometry of the atmoshere. Atmosheric mass is concentrated in the lowest 10 km less than 1% of the Earth s radius. Gravitational attraction has comressed the atmoshere into a shallow layer above the Earth s surface in which mass and constituents are stratified vertically: They are layered. Through stratification of mass, gravity imoses a strong kinematic constraint on atmosheric motion. Circulations with dimensions greater than a few tens of kilometers are quasi-horizontal. Vertical dislacements of air are then much smaller than horizontal dislacements. Under these circumstances, constituents such as water vaor and ozone fan out in layers or strata. Vertical dislacements are comarable to horizontal dislacements only in small-scale circulations such as convective cells and fronts, which have horizontal dimensions comarable to the vertical scale of the mass distribution. The comressibility of air comlicates the descrition of atmosheric behavior by enabling the volume of a arcel to change as it exeriences changes in surrounding ressure. Therefore, concentrations of mass and constituents for the arcel can change, even though the number of molecules remains fixed. The concentration of a chemical constituent can also change through internal transformations, which alter the number of a articular tye of molecule. For examle, condensation decreases the abundance of water vaor in an air arcel that asses through a cloud system. Photodissociation of O 2 will increase the abundance of ozone in a arcel that asses through a region of sunlight. Exchanges of energy with its environment and transformations between one form of energy and another likewise alter the roerties of an air arcel. By exanding, an
3 1.2 Comosition and structure 3 air arcel exchanges energy mechanically with its environment through work that it erforms on the surroundings. Heat transfer, as occurs through absortion of radiant energy and conduction with the Earth s surface, reresents a thermal exchange of energy with a arcel s environment. Absortion of water vaor by an air arcel (e.g., through contact with a warm ocean surface) has a similar effect. When the vaor condenses, latent heat of vaorization carried by the vaor is released to the surrounding molecules of dry air that comrise the arcel. If the condensed water then reciitates back to the Earth s surface, this rocess leads to a net transfer of heat from the Earth s surface to the arcel. The Earth s rotation, like gravity, exerts an imortant influence on atmosheric motion and, hence, on distributions of atmosheric roerties. Because the Earth is a noninertial reference frame, the conventional laws of mechanics do not hold; they must be modified to account for its acceleration. Aarent forces introduced by the Earth s rotation are resonsible for roerties of the large-scale circulation, in articular, the flow of air around centers of low and high ressure. Those forces also inhibit meridional, e.g., NS (North-South) motion. Consequently, they inhibit transfers of heat and constituents between the equator and oles. For this reason, rotation tends to stratify roerties meridionally, just as gravity tends to stratify them vertically. The hysical rocesses described in the receding aragrahs do not oerate indeendently. They are interwoven in a comlex fabric of radiation, chemistry, and dynamics that govern the Earth-atmoshere system. Interactions among these can be just as imortant as the individual rocesses themselves. For instance, radiative transfer controls the thermal structure of the atmoshere, which determines the circulation. Transort by the circulation, in turn, influences the distributions of radiatively active comonents such as water vaor, ozone, and cloud. In view of their interdeendence, understanding how one of these rocesses influences behavior requires an understanding of how that rocess interacts with others. This feature makes the study of the Earth-atmoshere system an eclectic one, involving the integration of many different hysical rinciles. This book develos the most fundamental of these. 1.2 COMPOSITION AND STRUCTURE The Earth s atmoshere consists of a mixture of gases, mostly molecular nitrogen (78% by volume) and molecular oxygen (21% by volume); see Table 1.1. Water vaor, carbon dioxide, and ozone, along with other minor constituents, comrise the remaining 1% of the atmoshere. Although resent in very small abundances, trace secies such as water vaor and ozone lay a key role in the energy balance of the Earth through their involvement in radiative rocesses. Because they are created and destroyed in articular regions and are linked to the circulation through transort, these and other minor secies are highly variable. For this reason, trace secies are treated searately from the rimary atmosheric constituents, which are referred to simly as dry air Descrition of air The starting oint for describing atmosheric behavior is the ideal gas law V = nr T = m M R T (1.1) = mrt,
4 4 The Earth-atmoshere system Table 1.1. Atmosheric Comosition. Constituents are listed with volume mixing ratios reresentative of the Trooshere or Stratoshere, how the latter are distributed vertically, and controlling rocesses Troosheric Vertical Distribution Constituent Mixing Ratio (Mixing Ratio) Controlling Processes N Homogeneous Vertical Mixing O Homogeneous Vertical Mixing H 2 O Decreases sharly in Trooshere Evaoration, Condensation, Increases in Stratoshere Transort Highly Variable Production by CH 4 Oxidation Ar.0093 Homogeneous Vertical Mixing CO mv Homogeneous Vertical Mixing Production by Surface and Anthroogenic Processes O 3 10 mv $ Increases sharly in Stratoshere Photochemical Production Highly Variable in Stratoshere; secondarily through ollution in trooshere Destruction at Surface Transort CH mv Homogeneous in Trooshere Production by Surface Processes Decreases in Middle Atmoshere Oxidation Produces H 2 O N 2 O 320 bv Homogeneous in Trooshere Production by Surface and Decreases in Middle Atmoshere Anthroogenic Processes Dissociation in Middle Atmoshere Produces NO Transort CO 70 bv Decreases in Trooshere Production Anthroogenically Increases in Stratoshere and by Oxidation of CH 4 Transort NO 0.1 bv $ Increases Vertically Production by Dissociation of N 2 O Catalytic Destruction of O 3 CFC bv Homogeneous in Trooshere Industrial Production CFC bv Decreases in Stratoshere Mixing in Trooshere HFC 134A 30 t Photo-dissociation in Stratoshere Radiatively active $ Stratosheric value which constitutes the equation of state for a ure (single-comonent) gas. In (1.1),, T,andM denote the ressure, temerature, and molar weight of the gas, and V, m, and n = m refer to the volume, mass, and molar abundance of a fixed collection of M matter (e.g., an air arcel). The secific gas constant R is related to the universal gas constant R through R = R M. (1.2) Equivalent forms of the ideal gas law that do not deend on the dimension of the system are = ρrt v = RT, (1.3) where ρ and v = 1 ρ (also denoted α) are the density and secific volume of the gas.
5 1.2 Comosition and structure 5 Because it is a mixture of gases, air obeys similar relationshis. So do its individual comonents. The artial ressure i of the ith comonent is that ressure the ith comonent would exert in isolation at the same volume and temerature as the mixture. It satisfies the equation of state i V = m i R i T, (1.4.1) where R i is the secific gas constant of the ith comonent. Similarly, the artial volume V i is that volume the ith comonent would occuy in isolation at the same ressure and temerature as the mixture. It satisfies the equation of state V i = m i R i T. (1.4.2) Dalton s law asserts that the ressure of a mixture of gases equals the sum of their artial ressures = i i. (1.5) Likewise, the volume of the mixture equals the sum of the artial volumes 1 V = i V i. (1.6) The equation of state for the mixture can be obtained by summing (1.4) over all of the comonents V = T i m i R i. Then, defining the mean secific gas constant i R = m i R i m yields the equation of state for the mixture (1.7) The mean molar weight of the mixture is defined by V = mrt. (1.8) M = m n. (1.9) Because the molar abundance of the mixture is equal to the sum of the molar abundances of the individual comonents, n = i m i M i, (1.9) may be exressed M = R m ( ). i m R i M i 1 These are among several consequences of the Gibbs-Dalton law, which relates the roerties of a mixture to roerties of the individual comonents (e.g., Keenan, 1970).
6 6 The Earth-atmoshere system Then alying (1.2) for the ith comonent together with (1.7) leads to R = R M. (1.10) Equation (1.10) is analogous to (1.2) for a single-comonent gas. Because of their involvement in radiative and chemical rocesses, variable comonents of air must be quantified. The absolute concentration of the ith secies is measured by its density ρ i or, alternatively, by its number density [i] = ( NA M i ) ρ i (1.11) (also denoted n i ), where N A is Avogadro s number and M i is the molar weight of the secies. Partial ressure i and artial volume V i are other measures of absolute concentration. The comressibility of air makes absolute concentration an ambiguous measure of a constituent s abundance. Even if a constituent is assive, namely if the number of molecules inside an individual arcel is fixed, its absolute concentration can change through changes of volume. For this reason, a constituent s abundance is more faithfully described by the relative concentration, which is referenced to the overall abundance of air or simly dry air. The relative concentration of the ith secies is measured by the molar fraction N i = n i n. (1.12) Dividing (1.4) by (1.8) and alying (1.2) for the ith comonent leads to N i = i = V i V. (1.13) Molar fraction uses as a reference the molar abundance of the mixture, which can vary through changes of individual secies. A more convenient measure of relative concentration is mixing ratio. The mass mixing ratio of the ith secies r i = m i m d, (1.14) where the subscrit d refers to dry air, is dimensionless. It is exressed in g kg 1 for troosheric water vaor and in arts er million by mass (mm, or simly m) for stratosheric ozone. Unlike molar abundance, the reference mass m d is constant for an individual air arcel. If the ith secies is assive, namely if it does not undergo a transformation of hase or a chemical reaction, its mass m i is also constant. The mixing ratio r i is then fixed for an individual air arcel. For a trace secies, such as water vaor or ozone, the mixing ratio is closely related to the molar fraction N i = r i ɛ i, (1.15.1)
7 1.2 Comosition and structure 7 where ɛ i = M i M d, (1.15.2) because the mass of air in the resence of such secies is nearly identical to that of dry air. The volume mixing ratio rovides similar information. It is distinguished from mass mixing ratio by dimensions such as arts er million by volume (mv) for stratosheric ozone (Probs. 1.2, 1.3). From (1.13) and (1.12), it follows that the volume mixing ratio is aroximately equal to the molar fraction. Each measures the relative abundance of molecules of the ith secies. As noted, the mixing ratio of a assive secies is fixed for an individual air arcel. A roerty that is invariant for individual arcels is said to be conserved. Although constant for individual air arcels, a conserved roerty is generally not constant in sace and time. Unless that roerty haens to be homogeneous, its distribution must vary satially and temorally, as arcels with different values exchange ositions. A conserved roerty is a material tracer because articular values track the motion of individual air arcels. Thus, tracking articular values of r i rovides a descrition of how air is rearranged by the circulation and, therefore, of how all conserved secies are redistributed Stratification of mass By confining mass to a shallow layer above the Earth s surface, gravity exerts a rofound influence on atmosheric behavior. If vertical accelerations are ignored, then Newton s second law of motion alied to the column of air between some level at ressure and a level incrementally higher at ressure + d (Fig. 1.1) reduces to a balance between the weight of that column and the net ressure force acting on it da ( + d)da = ρgdv, where g denotes the acceleration of gravity, or d = ρg. (1.16) dz This simle form of mechanical equilibrium is known as hydrostatic balance. It is a good aroximation even if the atmoshere is in motion because, for large-scale circulations, vertical dislacements of air are small. This feature renders vertical acceleration two to three orders of magnitude smaller than the forces in (1.16). Alying the same analysis between the ressure and the to of the atmoshere (where vanishes) illustrates the origin of atmosheric ressure: The ressure at any level must equal the weight of the atmosheric column of unit cross-sectional area above that level. Owing to the comressibility of air, the density in (1.16) is not constant. It deends on the air s ressure through the gas law. Eliminating ρ with (1.3) and integrating from the surface to an altitude z yields = e z dz z H (z ) s, (1.17.1) s
8 8 The Earth-atmoshere system Figure 1.1 Hydrostatic balance for an incremental atmosheric column of cross-sectional area da and height dz, bounded vertically by isobaric surfaces at ressures and + d. where H (z) = RT (z) g (1.17.2) is the ressure scale height and s is the surface ressure. The scale height reresents the characteristic vertical dimension of the mass distribution. A function of altitude, it varies from about 8 km near the Earth s surface to 6 km in very cold regions of the atmoshere. As illustrated by Fig. 1.2, global-mean ressure and density decrease with altitude aroximately exonentially. Pressure decreases from about 1000 hpa or 10 5 Pascals (Pa) at the surface to only 10% of that value at an altitude of 15 km (two scale heights). 2 According to hydrostatic balance, 90% of the atmoshere s mass then lies beneath this level. Pressure decreases by another factor of 10 for each additional 15 km of altitude. Density decreases with altitude at about the same rate, from a surface value of about 1.2 kg m 3. The shar uward decrease of ressure imlies that isobaric surfaces, along which = const, are quasi-horizontal. Deflections of those surfaces introduce comaratively small horizontal variations of ressure that drive atmosheric motion. 2 The historical unit of ressure, millibar (mb), has been relaced by its equivalent in the Standard International (SI) system of units, the hectopascal (hpa), where 1 hpa = 100 Pa = 1mb.See Aendix A for conversions between the SI system of units and others.
9 1.2 Comosition and structure Temerature (K) Altitude (km) Pressure Density Temerature Pressure (hpa) Density (kg m -3 ) Figure 1.2 Global-mean ressure (solid), density (dashed), and temerature (dotted), as functions of altitude. Source: U.S. Standard Atmoshere (1976). Above 100 km, ressure and density also decrease exonentially (Fig. 1.3), but at a rate which differs from that below and which varies gradually with altitude. The distinct change of behavior near 100 km marks a transition in the rocesses that control the stratification of mass and the comosition of air. The mean free ath of molecules is determined by the frequency of collisions. It varies inversely with air density. Consequently, the mean free ath increases exonentially with altitude, from about 10 7 m at the surface to of order 1 m at 100 km. Because it controls molecular diffusion, the mean free ath determines roerties of air such as viscosity and thermal conductivity. Diffusion of momentum and heat suorted by those roerties dissiate atmosheric motion by destroying gradients of velocity and temerature. Below 100 km, the mean free ath is short enough for turbulent eddies in the circulation to be only weakly damed by molecular diffusion. At those altitudes, bulk transort by turbulent air motion dominates diffusive transort of atmosheric constituents. Turbulence stirs different gases with equal efficiency. Mixing ratios of
10 Altitude (km) O 2 O Cambridge University Press 10 The Earth-atmoshere system 300 Temerature (K) M O T Normal Solar 200 He N 2 M T Active Solar O 2 M N He O O 2 M N 2 T He O Pressure (hpa) Number Density (m -3 ) Molar Weight Figure 1.3 Global-mean ressure (bold), temerature (stiled), mean molar weight (solid), and number densities of atmosheric constituents as functions of altitude. Source: U.S. Standard Atmoshere (1976). assive constituents are therefore homogeneous in this region. Those constituents are said to be well mixed. The densities of assive constituents then all decrease with altitude at the same exonential rate. This gives air a homogeneous comosition, with constant mixing ratios r N2 = 0.78, ro2 = 0.21, and the constant gas roerties 3 M d = gmol 1 (1.18.1) N 2 R d = Jkg 1 K 1. (1.18.2) The well-mixed region below 100 km is known as the homoshere. 3 Proerties of dry air are tabulated in Aendix B, along with other thermodynamic constants.
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