Transport of Infrared Atmospheric Radiation: Fundamentals of the Greenhouse Phenomenon 9783110628753, 9783110627657

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Boris M. Smirnov Transport of Infrared Atmospheric Radiation

Texts and Monographs in Theoretical Physics

Editor-in Chief Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA

Boris M. Smirnov

Transport of Infrared Atmospheric Radiation

Fundamentals of the Greenhouse Phenomenon

Physics and Astronomy Classification 2010 03.67.-a; 05.30.-d; 02.30.Tb; 02.50.-r Author Prof. Dr. Boris M. Smirnov Russian Academy of Sciences Joint Institute for High Temperatures Izhorskaya Str. 13/19 Moscow 125412 Russian Federation [email protected]

Despite careful production of our books, sometimes mistakes happen. Unfortunately, the frontmatter showed the wrong title in the original publication. This has been corrected. We apologize for the mistake. ISBN 978-3-11-062765-7 e-ISBN (PDF) 978-3-11-062875-3 e-ISBN (EPUB) 978-3-11-062903-3 ISSN 2627-3934 Library of Congress Control Number: 2019951211 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Newgen Knowledge Works (P) Ltd., Chennai, India Printing and binding: CPI books GmbH, Leck Cover image: agsandrew/iStock/Getty Images Plus www.degruyter.com

Preface Infrared emission of the atmosphere as a planar gaseous layer is determined mostly by radiative transitions between rotation and vibration-rotation transitions of H2 O and CO2 molecules, as well as by radiation of water microdrops which constitute clouds. We join general principles of spectroscopy with thermodynamics of the atmosphere and radiative field, including in consideration contemporary information from NASA atmosphere monitoring and data for rotation and vibration-rotation radiative transitions of molecules from the HITRAN database. Considering the Earth’s atmosphere as a nonuniform plain gaseous layer, we determine the atmospheric radiative flux to the Earth at each frequency of the infrared spectrum range with accounting for altitude distribution of atmospheric parameters together with the Mie theory for absorption of water microdrops. In this consideration photons formed in a certain atmosphere region are characterized by the temperature of this region. Summation of photons emitted from various atmosphere regions gives the radiative temperature T ω at a given frequency ω and defines an effective atmospheric layer which temperature determines the radiative temperature of the photon flux. Thus, this theory operates with specific variables, as the radiative temperature T ω for atmospheric molecules, the cloud temperature T cl or the temperature of water droplets which is independent of the frequency, the optical thickness u ω of the atmosphere at a given frequency, and the opaque factor g(ω) of the atmosphere below clouds. Reducing emission of a weakly nonuniform atmosphere to that of a constant temperature, we operate also with an effective radiating layer for each frequency which altitude h ω and temperature T ω determine the atmospheric radiative flux at a given frequency. The theory allows one to determine the radiative flux toward the Earth, as well as outgoing radiative flux, for each molecular component on the basis of its space distribution in the atmosphere and its temperature profile, whereas the cloud temperature is determined from the total radiative flux to the Earth or outside, as it follows from the atmospheric energetic balance. This theory allows one to formulate some aspects of atmospheric properties in a simple form and to remove the mistakes of climatological models which do not take into account the principal atmospheric behavior. The theory developed allows us to analyze various properties of the radiative atmosphere within the framework of the above terms. In particular, interaction of optically active atmospheric components is such, that an increase of the concentration of carbon dioxide causes an increasing radiative flux due to carbon dioxide molecules. But this is accompanied by a decrease of the radiative flux due to water molecules and water microdrops of clouds which almost compensate the decrease due to carbon dioxide. The change of the total radiative flux from the atmosphere has another order of magnitude than that due to carbon dioxide. This example exhibits that the results of this theory differ from those of climatological models based on empirical approach. In this book we use the theory of infrared atmospheric spectroscopy for determination the radiative fluxes from the standard atmosphere basing on parameters

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VI | Preface of about thousand spectral lines for radiative transitions which are taken from the HITRAN database. But this theory may be used for evaluation of the local atmospheric heat or radiative balance for regions of size of several kilometers, if the temperature profile and space distribution of emitting components are known. In particular, it may be used in meteorological evaluations. In addition note, that because throughout this book spectroscopy parameters are used, we infer that frequencies of radiative transitions, as well energies of molecular states, are expressed in the spectroscopy unit cm−1 . We have the following conversion this unit in other ones 1 cm−1 = 1.2398 ⋅ 10−4 eV = 1.8837 ⋅ 105 MHz = 1.9864 ⋅ 10−16 erg = 1.4388K. In the same manner, the temperature is measured in energetic units, so that 1K = 1.3807 ⋅ 10−16 erg = 8.6174 ⋅ 10−5 eV = 0.69504cm−1 = 1.9872 ⋅ 10−3 kcal/mol. This simplifies the analysis.

Abstract The theory of transport of infrared radiation is developed for a weakly nonuniform layer of molecular gas on the basis of general principles of interaction between the radiation field and atomic systems for local thermodynamic equilibrium between them. This theory operates with specific variables which include the radiative temperature T ω , the optical thickness of the layer u ω , a position of a layer h ω which is responsible for the radiative flux, and the opaque factor g(ω) of this layer. All quantities relate to a certain frequency ω. This theory is combined with the model of standard atmosphere and parameters of infrared radiative transitions for atmospheric molecules taking from the HITRAN database. On the basis of optical properties of liquid water and the Mie theory, emission of atmospheric water microdrops which constitute the clouds is described. The atmospheric infrared radiation toward the Earth is constructed from emission of clouds whose typical altitude follows from the energetic balance of the Earth, as well as from radiation of optically active molecules of atmospheric air located in the gap between the Earth and clouds. Various information about global atmospheric properties, mostly obtained in NASA programs for monitoring of atmospheric carbon dioxide and evolution of the global temperature, is used in evaluation of the atmosphere emission together with the model of standard atmosphere. Taking the contribution of water microdrops to the radiative flux toward the Earth on the basis of the energetic balance of the Earth and its atmosphere, one can determine this flux at each frequency together the contribution from each component. As a result, basic optically active components of the atmosphere, namely, H2 O, CO2 molecules and water microdrops from clouds, give the contribution to the total radiative flux from the standard atmosphere 51%, 19%, and 29% correspondingly, and the contribution of trace gases (methane and nitrogen dioxide) equals to 2%. In addition, approximately 95% of the radiative flux emitted by the Earth’s surface is absorbed by molecular atmospheric components below

Abstract |

VII

clouds at frequencies ω < 800cm−1 , whereas at larger frequencies the atmospheric molecules absorb 16% of this flux. Atmospheric optically active components are simultaneously radiators and absorbers, and this determines an interaction between them, if the concentration of one component varies. Then an increase of the radiative flux due to one component is compensated practically by a decrease of the flux due to other components. In particular, a doubling of the concentration of CO2 molecules for the standard atmosphere leads to an increase of the total radiative flux to the Earth by 1.3W/m2 , while the radiative flux to the Earth due to CO2 molecules increases by 7W/m2 approximately. The equilibrium climate sensitivity (ECS), i.e., the change of the global temperature as a result of doubling of the atmospheric carbon dioxide amount, is (0.6 ± 0.3)K in both cases under conditions where the content of other optically active components is unvaried. This gives that a contemporary anthropogenic increase of the global temperature as a result of combustion of fossil fuel is about 0.01K. For comparison, for a real atmosphere with a contemporary change the amount of atmospheric water and other components, ECS of (2.5 ± 0.4)K follows from NASA monitoring of the atmospheric carbon dioxide concentration and global temperature during last 150 years, if we assume that the contemporary change of the global temperature has an anthropogenic character. These results for atmospheric emission together with a study of the change of the global temperature in past are used for the analysis of evolution of the Earth’s thermal state. Conclusions differ from those of some climatological models. In particular, the contemporary concentration of atmospheric CO 2 molecules is larger compared to that in past. This is explained by disafforestation of our planet, rather than the combustion of fossil fuels.

Contents Preface | V Abstract | VI 1

Introduction | 1

2 2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2

Generals in statics and dynamics of global atmosphere | 7 Properties of global atmosphere | 7 Model of standard atmosphere | 7 Energetic balance for the Earth and its atmosphere | 10 Greenhouse molecular components in the atmosphere | 12 Atmospheric water vapor | 12 Atmospheric carbon dioxide | 16 Trace components in the atmosphere | 20 Evolution of global temperature | 22 Contemporary variation of global temperature | 22 Variation of global temperature in past | 25 Chemical equilibrium between the Earth and atmosphere | 27 Dynamics of atmospheric air with microparticles | 30 Convective motion of atmospheric air | 30 Transport of water microdrops in convective air | 33

3 3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 3.3.4

Water microdrops in atmospheric air | 37 Processes of water condensation in atmospheric air | 37 Behavior of water microdrops in tropospheric air | 37 Condition of water condensation in atmospheric air | 42 Mixing of air streams involving the condensed water phase | 44 Character of water condensation in atmospheric air | 45 Mechanisms of drop growth in air | 45 Growth of water drops due to coagulation and coalescence | 49 Gravitation mechanism of growth of water drops | 56 Water circulation between the Earth and atmosphere | 59 Character of water circulation through atmosphere | 59 Electric properties of water drops of cumulus cloud | 61 Processes of transformation of water drops in rain | 64 Kinetics of atmospheric electricity | 68

4 4.1 4.1.1 4.1.2

Thermodynamics of thermal atmospheric emission | 71 Radiation of flat layer | 71 Radiation in uniform gas | 71 Emission from flat gaseous layer | 73

X | Contents 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.3.3

Character of thermal atmospheric emission | 77 Emission of atmosphere as flat gaseous layer | 77 Model of frequency-independent absorption | 80 Atmospheric thermal radiation from thermodynamic standpoint | 82 Distribution over optical thicknesses | 85 Water microdrops in atmospheric emission | 86 Model of average atmospheric absorption | 86 Absorption by small water drops | 90 Water microdrops as atmospheric radiators | 96

5 5.1 5.1.1 5.1.2

Spectroscopy properties of radiative atmospheric molecules | 101 Radiative processes involving CO2 molecules | 101 Infrared spectroscopy of CO2 molecules | 101 Regular model for the absorption coefficient due to linear molecules | 106 Data of HITRAN bank in evaluation of absorption coefficient due to CO2 molecules | 112 Infrared resonant radiation of CO2 molecules | 116 Absorption by atmospheric water | 118 Absorption cross section due to atmospheric water molecules | 118

5.1.3 5.1.4 5.2 5.2.1

Part 2: Greenhouse phenomenon of standard atmosphere 6 6.1 6.1.1 6.1.2 6.1.3 6.2 6.2.1 6.2.2 7 7.1 7.1.1 7.1.2 7.1.3

Infrared radiative energetics of standard atmosphere | 125 Emission due to water molecules | 125 Models of infrared atmospheric emission | 125 Emission of atmospheric water molecules at low frequencies | 129 Water molecules as the main atmospheric radiator at low frequencies | 134 Infrared emission of the atmosphere with participation of CO2 molecules | 137 Atmospheric emission in the transition frequency range (580 − 700)cm−1 | 137 Atmospheric emission in the frequency range (700 − 800)cm−1 | 141 Shortwave infrared atmospheric emission | 149 Infrared atmospheric emission of water and carbon dioxide molecules at large frequencies | 149 Atmospheric emission in the frequency range (800 − 1000)cm−1 | 149 Atmospheric emission in the frequency range (1000 − 1100)cm−1 | 151 Model of single spectral lines | 152

Contents

7.1.4 7.2 7.2.1 7.2.2 7.2.3 8 8.1 8.1.1 8.1.2 8.1.3 8.2 8.2.1 8.2.2 8.2.3 8.3 8.3.1 8.3.2 8.3.3

9 9.1 9.1.1 9.1.2 9.1.3 9.2 9.2.1 9.2.2 9.2.3 9.3 9.3.1 9.3.2 9.3.3 9.3.4

| XI

Atmospheric emission due to resonant vibration transition of CO2 molecules | 156 Trace gases in atmospheric radiation | 159 Emission of atmospheric methane | 159 Emission of atmospheric nitrogen dioxide | 163 Infrared radiative fluxes for standard atmosphere | 169 Emission of a varied atmosphere | 173 Atmospheric emission at variation of greenhouse components | 173 Models of interaction between radiating components in a varied atmosphere | 173 Atmospheric emission at doubled concentration of carbon dioxide | 176 Radiative flux depending of concentration of radiating molecules | 182 Connection between global temperature and global energy fluxes | 185 Climate sensitivity due to radiation | 185 Climate sensitivity of the Earth and atmosphere | 187 Temporary changes of atmospheric fluxes | 192 Water condensed phase in atmospheric processes | 194 Power of outgoing atmospheric radiation | 194 Radiative water microdrops in the atmosphere | 196 Role of water microdrops in electrical and optical properties of the atmosphere | 200 Peculiarities of emission in atmospheric energetics | 203 Atmosphere as a media for transport of infrared radiation | 203 Model of standard atmosphere in its emission | 203 Weather of megalopolises | 205 Hydroelectric plants and greenhouse effect | 209 The problem of atmospheric carbon dioxide | 211 Correlation between carbon dioxide and global temperature | 211 Global deforestation | 212 World energetics in growth of atmospheric carbon dioxide | 215 Change of the Earth’s climate | 219 Solar variations related to climate | 219 Particle emission in climate change | 221 Cosmic rays and climate | 224 Paris agreements and climate change | 225

XII | Contents 10

Conclusion | 227

Bibliography | 229 Subject Index | 241 List of Figures | 243

1 Introduction The nature of the greenhouse phenomenon that was understood two centuries ago [1, 2], is explained in Fig.1.1. The flux J s of solar radiation is absorbed by a surface, so that the energetic balance of this surface is determined by absorption of solar radiation and emission in the infrared (IR) spectrum range. Assuming that this surface emits as a blackbody, we have the following energy balance equation for this surface J s = σT s4 ,

(1.0.1)

where σ = 5.67 ⋅ 10−8 W/m2 is the Stephan-Boltzmann constant, T s is the surface temperature. Let us now place this surface inside the greenhouse, which scheme is given in Fig.1.1. The partition of this greenhouse transmits solar radiation in the visible spectrum range freely, but it absorbs IR radiation and returns it to the surface. As a result, the absorbed flux increases together with an increasing surface temperature. Correspondingly, the total absorbed flux exceeds that of solar radiation. In particular, for the Earth and its atmosphere the ratio of the emitted IR radiative flux by the surface to the solar radiative flux absorbed by it is two, in the Venus case this ratio equals to hundred. As an example, one can use water as a greenhouse element. Fig.1.2 contains the absorption coefficient of liquid water. As it follows from this Figure, the mean free path for visible photons in water is 50m, whereas the mean free path of IR photons is 3μm. If the depth of the water layer in this partition ranges between these values, visible radiation passes freely through the partition, whereas IR radiation is absorbed by this water. The energy of absorbed radiation transfers later into heat which may be used for emission of this water. From this example it follows that the greenhouse partition is simultaneously the absorber of IR radiation and its radiator. This condition is fulfilled for the Earth’s atmosphere where absorbers and radiators are molecules as it was understood in the middle of 19 century [6–9]. Main greenhouse components of the Earth’s atmosphere are molecules of water and carbon dioxide, and also water microdrops.

Fig. 1.1: Scheme of the greenhouse. 1 - corpus of the greenhouse, 2 - separating partition, 3 - solar radiation, 4,6 - IR radiation of the partition inside the greenhouse and outside it, 5 - outgoing IR radiation of the surface.

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2 | 1 Introduction

Fig. 1.2: Absorption coefficient of liquid water under normal conditions [3, 4]. Arrows show the wavelength for solar radiation and IR one, a cross is a result of atmospheric measurements [5].

The greenhouse problem of the atmosphere is overlapped with adjacent atmospheric problems which therefore are the object of the consideration. We below analyze the energetic balance of the Earth and its atmosphere which includes their IR radiation. Hence, the energetic balance of the planet gives information for the greenhouse phenomenon. Evolution of the Earth local temperature in time may be restored on the basis of the isotope analysis of deposits and evolution of the global temperature follows from data of meteorological stations and satellites starting from 19 century. Because water and carbon dioxide are main greenhouse components, the balance is analyzed for these components which are in equilibrium between the Earth and atmosphere. Below this equilibrium is analyzed. Photon emission and absorption in the atmosphere is analyzed on the basis of the molecular spectroscopy. The absorption molecular spectrum in the IR spectral range consists of separate broaded spectral lines which corresponds to centers of certain rotation- and vibration-rotation transitions of molecules. Respectively, the absorption coefficient due to these molecules has the oscillation structure with maxima at corresponding rotation or vibration-rotation transitions. In construction the absorption coefficient of molecules, we are based on data of the HITRAN bank [10–12] for spectroscopy parameters of molecules. As a result, the absorption coefficient is expressed through information followed from the HITRAN data bank for radiative transitions in molecules. In evaluation the partial radiative flux from absorption in the atmosphere, we use the thermodynamic equilibrium between gaseous molecules and radiation in the course of atmospheric emission due to excited molecules formed in their collisions. This allows one to ignore reabsorption mechanisms, as well as radiation transport.

1 Introduction

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3

Second, because the Earth’s atmosphere is optically thick, this leads to separation of atmospheric layers which create the radiative fluxes directed to the Earth’s surface and outside. Third, though the atmosphere temperature varies with the altitude, the temperature gradient is low. This allows one to expand the radiative flux over a small parameter and reduce the partial radiative flux to that at a constant temperature. These facts allow us to understand the role of some factors in the radiative process and evaluate the yield radiative flux under real conditions. Specifics of atmospheric particles or microdrops (aerosols) in atmospheric emission is that their absorption spectrum occupies a wide spectral range [13]. Therefore atmospheric microdrops cover the spectral range which is transparent for atmospheric molecules. Next, according to Fig.1.2 data, the mean free path of visible photons in liquid water is about 50m, whereas in the case of IR photons this value is less than 10μm. Hence, if water microdrops are located in the atmosphere and their effective thickness is found between these values, visible photons pass freely through such an atmosphere, while IR photons are absorbed by these microdrops. When microdrops are formed in the atmosphere as a result of condensation of a water vapor, they consist of pure water. According to data of Fig.1.2, we hence encounter with such a situation where the sky is transparent for a visible spectrum range, i.e. for the man eye, and simultaneously these microdrops are effective radiators and absorbers in the IR spectrum range. Just this situation takes place in usual atmosphere, so that clouds are transparent in the visible spectrum range if they consist of pure water microdrops. Clouds become visible when absorb a small amount of admixtures (∼ 0.1% of its mass) in the form of salt molecules, aerosols or dust which absorb visible radiation (the Twomey effect [14, 15]). These conditions are fulfilled in a real atmosphere. Another atmospheric peculiarity of water microdrops is that they include of the order of 1% of atmospheric water, so that a shift of the equilibrium between water molecules and microdrops may influence dramatically on radiative atmospheric properties. Hence, it is necessary to analyze this problem in detail, and hence we include in consideration processes of formation and growth of water microdrops in the course of their growth. In this consideration it is essential that the absorption spectrum of CO2 molecules is overlapped with the spectrum of water molecules and water microdrops. In this case it is impossible to separate emission of carbon dioxide and water molecules. This means that the number density of carbon dioxide molecules increases and the radiative flux due to carbon dioxide molecules increases also, but a change of the total radiative flux is less than this one because a part of an increasing radiative flux is compensated by absorption of water molecules and water microdrops. This fact is taken into account in models of atmospheric emission under consideration. We set as a goal to compose the simple and reliable models for description the emission flux. The simplest model assumes the absorption coefficient of atmospheric components to be independent of the radiation frequency, and the final one accounts for the oscillation behavior of the absorption coefficient of H2 O and CO2 molecules. The series of simplest and transparent models for atmospheric emission which allows us to understand the character of the processes and determine their parameters.

4 | 1 Introduction Basing on general principles of infrared atmospheric radiation [16–23], we are guided by the standard atmospheric model [24], i.e. the atmosphere with average parameters over the globe and time. For this atmosphere we can evaluate the atmospheric radiative flux toward the Earth at certain concentrations of greenhouse components. This means the possibility to analyze the character of atmospheric emission depending on evolution of greenhouse components in time. In reality, the atmosphere is optically thick, and altitudes of atmospheric layers which are responsible for emission in a basic spectrum range are approximately in a region of about 3 km. From this it follows that this analysis is suitable for megalopolises which sizes exceed significantly the above value. Then depending on parameters of the atmosphere and the concentration of greenhouse components, one can evaluate the radiative flux toward the Earth. In the analysis of atmospheric emission, which is created due to vibration-rotation transitions of atmospheric water and carbon dioxide molecules, as well as water microdrops, we take into account the understanding of some aspects of this problem combined with general principles of the atmosphere [25–30]. In this consideration we are based on simple models which describe real atmospheric parameters, in the first place, a large optical thickness of the atmosphere. It is combined with thermodynamic equilibrium of vibrationally or rotationally excited water and carbon dioxide molecules with air molecules, and emitting excited molecules are formed in collisions with air molecules. This allows us to extract atmospheric layers which are responsible for atmospheric emission towards and outwards the Earth surface. Due to this character of radiation in the atmosphere, we ignore processes of reabsorption and the concept of atmospheric transparency. Infrared emission of the atmosphere is an important element of the energetic balance of the Earth and its atmosphere. We combine this problem with results of measurements and monitoring of some atmospheric parameters and their evolution which are fulfilled within the framework of some NASA programs. Summarizing these data with the analysis of the greenhouse atmospheric phenomenon, one can obtain a physical picture of atmospheric energetics. The problems under consideration are associated with the analysis of the Earth’s climate and its evolution in future [31–39]. Because various atmospheric problems attract the attention of many scientists during a long time, it is necessary to formulate the advantage of the methods and results under consideration compared to existing ones. We are guided by two aspects of the greenhouse problem, namely, infrared atmospheric emission, both at each frequency and the total ones summarized over frequencies, as well as the change of radiative fluxes and the global temperature as a result of variation of a greenhouse component. In this consideration it is used thermodynamic equilibrium in the atmosphere and a noncoherent character of radiation. Then the distribution of photons over frequencies which are created at a given point of the atmosphere, is characterized by the temperature of this point. Summation over these points leads to the concept of a layer that is responsible for radiation at a given frequency. This concept, on the one side, simplifies the analysis of atmospheric emission and, on the other side, allows one to extract the important mechanisms of this phenomenon. In particular, this analysis gives that the anthropogenic increase of the

1 Introduction

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global temperature from the end of 19 century up to now due to accumulation of carbon dioxide in the atmosphere is approximately 0.03K. But if we ignore the interaction between atmospheric water and carbon dioxide, this value will increase by an order of magnitude. The book recons on advanced students of atmospheric sciences, and also on professionallies and specialists in atmospheric physics.

2 Generals in statics and dynamics of global atmosphere Abstract: General principles of the global atmosphere which are the basis for study the atmospheric greenhouse phenomena, include the energetic balance of the Earth and its atmosphere, the equilibrium between atmospheric and Earth’s water, as well the equilibrium between atmospheric carbon dioxide and solid carbon bonded in carbon compounds at the Earth’s surface. The NASA programs and palaeontological studies give the evolution of the global temperature, i.e. the temperature averaged over coordinates of the Earth’s surface and time. The results are based on data of meteorological stations and satellites during last 150 years, as well as on the isotope analysis of deposits. The Pauling concept based on the thermodynamic equilibrium between free atmospheric CO2 molecules in the atmosphere and bound CO2 molecules in oceans, creates the feedback in the connection between the global temperature and concentration of atmospheric carbon dioxide. Under the assumption that an observed evolution of the global temperature is determined by the concentration change of atmospheric carbon dioxide, the doubling of the concentration of CO2 molecules leads to an increase of ∆T = (2.5 ± 0.4)K of the global temperature according to NASA measurements. The convective character of air motion determine transport processes in the atmosphere.

2.1 Properties of global atmosphere 2.1.1 Model of standard atmosphere Our goal is to analyze the greenhouse phenomenon in the Earth’s atmosphere which is determined by the state of atmospheric air. But parameters of the Earth’s atmosphere which determine this phenomenon vary in time and depend on a geographic point. In order to simplify the analysis, we will be based on the model of standard atmosphere [24] which averages atmospheric parameters over time and relates roughly to the USA atmosphere. This model gives a certain altitude distribution for atmospheric parameters. Let us consider first the altitude distribution of the number density of air molecules and temperature. Though air consists of nitrogen (79%) and oxygen (20%), we below assume air molecules to be identical with the molecular weight m = 29a.m.u.. Ignoring at the first stage of the analysis the convective motion of air molecules in the atmosphere, we use the Boltzmann law for the altitude distribution of air molecules T dh N = N o exp (− ∫ , (2.1.1) ), Λ = Λ mg where h is the altitude, T is the air temperature, g is the free fall acceleration. In particular, at the Earth surface (T = 288K) we have Λ = 8.4km.

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8 | 2 Generals in statics and dynamics of global atmosphere

Fig. 2.1: Scaling parameter for the pressure scaling parameter according to formula (2.1.3) for the model of standard atmosphere [24].

In order to account for air convection in the altitude distribution of the number density of air molecules, one can use the data for standard atmosphere, approximating the number density N of air molecules as h N(h) = N(0) exp (− ) Λ

(2.1.2)

The altitude dependence for the scaling parameter Λ follows from the model of standard atmosphere. In the same manner one can represent the altitude dependence for the air pressure p(h) according to formula p(h) = p o exp (−

h ), Λp

(2.1.3)

where p o is the average pressure near the Earth’s surface, and Λ p the parameter of pressure scaling for atmospheric air. The altitude dependence for the pressure scaling parameter is given in Fig.2.1. Fig.2.2 contains the altitude dependence of the atmosphere temperature. Being guided by the greenhouse effect, we are restricted by the troposphere and lower stratosphere only. As it follows from Fig.2.2, the temperature gradient in the troposphere is equal to dT/dh = −6.5K/km. Let us compare this with the adiabatic law of air expansion as a result of air transition to higher layers. Take an air element contained n air molecules and displace it to another altitude. Basing on the adiabatic law, we assume the energy of air molecules to be conserved in this element. We below determine the temperature gradient for atmospheric layers near the Earth’s surface under such conditions.

2.1 Properties of global atmosphere

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Fig. 2.2: Altitude dependence of the atmosphere temperature in the troposphere for the model of standard atmosphere [24].

Let us consider displacement of an air element contained n air molecules to a layer of another altitude as an adiabatic process [40–43]. We have the following equation for the internal energy E of this air element as a result of its displacement to another atmospheric layer dE = c v ndT + pdV = 0 (2.1.4) Here c V is the heat capacity per molecule, p is the gas pressure, V is the volume of this air element. Considering diatomic molecules of nitrogen and oxygen as air molecules, we have that at room temperature rotation degrees of freedom are strongly excited for these molecules, whereas the vibration states are not excited. This lead to five degrees of freedom for an individual molecule, i.e. c V = 5/2. Introducing the number density N of air molecules, we have V = n/N, and using the state equation for air as an ideal gas p = NT, transform equation (2.1.4) to the form ln (

T ) = const N 2/5

(2.1.5)

From this on the basis of formula (2.1.2) it follows dT 0.4T =− = −14K/km dh Λ

(2.1.6)

As is seen, the adiabatic character for variation of the the atmosphere temperature as an altitude function does not correspond to the standard atmosphere model. Indeed, this model postulates dT/dh = const in the troposphere, whereas this is valid, until the temperature variation is low. In addition, the temperature gradient according to formula (2.1.5) is twice compared to that of the standard atmosphere model. One can expect that accounting for the condensation of atmospheric water may decrease the temperature gradient in the troposphere. Below we check this possibility. One can note that the temperature gradient causes a heat flux q from the Earth’s surface to atmosphere according to the standard formula q=−

dT Pc = −λ c , S dh

(2.1.7)

10 | 2 Generals in statics and dynamics of global atmosphere where P c = 2.4 ⋅ 1016 W is the power due to convective transfer of heat, S = 5.1 ⋅ 1018 cm2 is the area of the Earth’s surface, λ c is the effective coefficient of thermal conductivity which is determined by air convection. As it follows from formula (2.1.7), the effective coefficient of thermal conductivity is equal λ c = 72(W/cm ⋅ K), and this exceeds that of motionless atmosphere by several orders of magnitude. On the basis of the state equation p = NT for atmospheric air one can connected the scales of atmospheric pressure and density given by formulas (2.1.2) and (2.1.3). We have h dT ln Λ p = ln Λ N − ln (1 − (2.1.8) ), T E dh where T E is the temperature at the Earth’s surface. The second term of the right hand of formula (2.1.8) is relatively small. For example, at the altitude h = 3km we have Λ p = 1.07Λ N . Hence, with an accuracy of 10% one can consider the scaling parameters Λ p and Λ N to be identical.

2.1.2 Energetic balance for the Earth and its atmosphere Our goal is to analyze the long-term change of the Earth’s energetic balance that is a part of the problem of climate evolution. The later is discussed widely including monographs [23, 28, 30, 32–36]. We below consider physical aspects of this problem by summation the data for measured atmospheric parameters within the framework of this problem and then use them in simple, but reliable models for radiative properties of the atmosphere. This allows us to construct a general physical picture of atmospheric processes which are of importance for the greenhouse atmospheric phenomenon. Basing on the model of standard atmosphere [24, 44, 45], where any parameter depends on an altitude over the sea level only, we characterize the thermal state of the Earth by only one parameter - the global temperature which is the temperature of the Earth’s surface averaged over time of day and season during a year and also over the globe. Variation of the global temperature during a long time characterizes the climate evolution. In this consideration, the energetic balance of the Earth and its atmosphere is of importance for understanding the character of atmospheric energetic processes including the greenhouse phenomenon in the Earth’s atmosphere. Parameters of this energetic balance given in Fig.2.3 are taken from the author’s books [46–48] and are borrowed from the NASA report (evidently, from [31]). Subsequently, these data were represented in the form of specific powers [49–54] which result from dividing of powers given in Fig.2.3 by the area S = 5.1 ⋅ 1014 m2 of the globe surface. These values obtained from data of Fig.2.3 are represented in Fig.2.4. They include the indicated data and are given in atmospheric books [28, 37, 39] with coincidence within the limits of a few percent for basic channels. Evidently, the reason of this coincidence is that the data are taken from the same source - the data of NASA [31]. As it follows from Fig.2.3, the basis of the Earth energetics is solar radiation which is converted partially in infrared radiation as a result of the greenhouse effect, i.e.due

2.1 Properties of global atmosphere

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Fig. 2.3: Expressed in 1016 W the powers of indicated processes which are obtained or lost by the Earth as a whole as well as by its atmosphere. Absorbed powers are given inside corresponding rectangulars, consumed powers are indicated near arrows.

Fig. 2.4: Average energy fluxes in W/m2 for indicated channels followed from data of Fig.2.3 for the energetic balance of the Earth and its atmosphere [30].

to emission and absorption of infrared radiation by the Earth and its atmosphere. Note that the Earth emits infrared radiation almost as a blackbody. Indeed, if we apply the Stephan-Boltzmann law for radiation of the Earth’s surface with the power given in Fig.2.3, one can obtain the Earth temperature as T = 287K, whereas the standard atmosphere model contains the global Earth temperature T = 288K. We note also that the atmosphere removal leads to the global temperature T = 278K, if the Earth absorbs solar radiation completely and emits as a blackbody. This is confirmed by optical properties in the infrared spectrum range for objects located onto the Earth’s surface which are confirmed by the data of Fig.2.5. This allows one to consider emission of the Earth’s surface similar to that of blackbody, i.e. the radiative flux J E from the Earth surface in the infrared spectral range is given by the Stephan-Boltzmann law J E = σT E4 ,

(2.1.9)

12 | 2 Generals in statics and dynamics of global atmosphere

Fig. 2.5: Emissivity of various surfaces via the photon wavelength [22].

where σ = 5.67 ⋅ 10−8 W/m2 is the Stephan-Boltzmann constant, and T E is the global temperature. In this consideration we include advantages of atmospheric physics during last decades. The basic ones from them is monitoring of the carbon dioxide concentration in Mauna Loa observatory [55, 56] and evolution of the global temperature of the Earth from the end of 19 century up to now on the basis of treatment of information from thousands meteorological stations [57–59]. Both ones were a result of some NASA programs. Note that a huge work in obtaining the corresponding results requires a careful analysis which is made there. We will use some results of measurements and its analysis which are of interest for the greenhouse phenomenon.

2.2 Greenhouse molecular components in the atmosphere 2.2.1 Atmospheric water vapor There are three basic greenhouse atmospheric components which are responsible for this phenomenon and include free water molecules, carbon dioxide molecules, and water microdrops. Therefore we consider below the behavior of these components in the atmosphere. The main greenhouse atmospheric component is a water vapor. We collect below the data related to the amount of an atmospheric water vapor and rates of processes which determine its behavior. The mean amount of atmospheric water is 1.3 ⋅ 1019 g [60–63] that corresponds to the mean water density in the atmosphere as 3g/m3 . One can compare the water mass in the atmosphere with that of atmospheric air 5.1⋅1021 g which relates to nitrogen and oxygen. This corresponds to the average concentration of water molecules in atmospheric air as approximately 0.4%, whereas near the Earth’s surface the average concentration of water molecules in air is equal 1.7%. The total rate of water evaporation

2.2 Greenhouse molecular components in the atmosphere

| 13

Fig. 2.6: Rates of exchange by water between land, oceans and atmosphere which are expressed in 1018 g/yr and are given near arrows. Water amount is expressed in 1015 g (billion ton) and is indicated inside a corresponding rectangle [60, 64–67].

from the Earth’s surface is 4.8 ⋅ 1020 g/yr [60, 64–67] or 1.5 ⋅ 1013 g/s = 1.5 ⋅ 107 ton/s (only 1.0 ⋅ 1018 g/yr in the form of snow), and the same rate relates to water returning to the Earth’s surface. As it follows from this, the power of the evaporation process is equal 2.9 ⋅ 1016 W (see Fig.2.6) which is returned as a result of water condensation in the atmosphere. In this manner, the above power is transferred from the Earth to the atmosphere. From the amount of atmospheric water and the rate of its formation it follows that an average time of residence of water molecules in the atmosphere is approximately 9 days [67]. Fig.2.6 gives the water balance between the land, ocean and atmosphere. Precipitation of atmospheric water on the Earth’s surface and uniform distribution over it gives a layer of liquid water of a thickness 2.5 cm [68]. The height of a precipitated layer may be used as a unit for amount of a water vapor in the atmosphere as it made in Fig.2.6, Fig.2.7. As it follows from Fig.2.7, approximately 80% of atmospheric water is located at altitudes below 3km. According to data of Fig.2.8, the rate of water precipitation on the Earth’s surface is higher for oceans, whereas season variations of this rate are more for land.

Fig. 2.7: Evolution of the average amount of atmospheric water expressed in heights of precipitated liquid water [69, 70].

14 | 2 Generals in statics and dynamics of global atmosphere

Fig. 2.8: Evolution of the water moisture averaged over a globe and expressed through the relative mass of atmospheric water molecules [71–73].

Atmospheric water contains a small part of Earth’s water which mass is 1.4 ⋅ 1024 g. If Earth’s water would be distributed over the Earth’s surface uniformly, the layer thickness will be 2.7km. From this it follows that the most part of this water is located underground. Note that 96% of Earth’s water is salty. In addition, open water located on the Earth’s surface is a source of atmospheric water and is found in equilibrium with it. Near the Earth’s surface, the average partial pressure of water is about 12 Torr, whereas the saturated water pressure is 13 Torr at its temperature [74–76]. Therefore atmospheric water is found there in the form of free molecules, as well as in all the atmosphere, at least, at altitudes below 3km. Water aerosols (microdrops) are formed and exist at larger altitudes in the form of clouds. Processes involving water particles or water aerosols [25, 77–84] are of principle for radiative, electric and other atmospheric properties. Formation of water microdrops may proceed if the partial pressure of water exceeds the saturated vapor pressure at a current atmosphere temperature which is taken from [75]. The ratio of these pressures is the air moisture. There are reliable methods of measurements of the global moisture (for example, [85, 86]), and Fig.2.8 gives evolution of the moisture of atmospheric air at some altitudes in time. The altitude dependence of the number density of water molecules, as well as this dependence for the air moisture, allows one to find the parameters for the altitude dependence of the number density of water molecules. In particular, from Fig.2.8 it follows that the concentration of water molecules in atmospheric air is 1.7%, 0.34% and 0.045% at the Earth’s surface and at altitudes of 4.2km and 9km correspondingly according to averaging of data in 1950 − 1960, and the above values according to averaging for 2000 − 2010 are correspondingly 1.7%, 0.33%, and 0.039%. Let us approximate the number density of water molecules as an altitude function N(H2 O) as N(H 2 O) = N o exp(−h/λ),

(2.2.1)

2.2 Greenhouse molecular components in the atmosphere

| 15

where h is the altitude. Then from 1950−1960 data one can find the following parameters of formula (2.2.1) as N o = 4.2 ⋅ 1017 cm−3 , λ = (2.0 ± 0.1)km, and for 2000 − 2010 we have N o = 4.3 ⋅ 1017 cm−3 , λ = (1.9 ± 0.1)km. From this one can find the total amount of atmospheric water, and the mass of atmospheric water is equal M w = N o λSm w , (2.2.2) where S = 5.1 ⋅ 1018 cm2 is the area of the Earth’s surface, m w = 3.0 ⋅ 10−23 g is the mass of an individual water molecule. From this formula it follows M w = 1.3 ⋅ 1019 g that coincides with the above value. Thus, with an accuracy of a few percent the total amount of atmospheric water is identical, being obtained in different ways. Thus, the average number density of water molecules near the Earth’s surface is N(H2 O) = 4.3 ⋅ 1017 cm−3 , the number density per unit vertical column is equal n(H2 O) = 1.4 ⋅ 1023 cm−2 , and its variation from the end of 19 century is ∆n(H2 O) = 5.7 ⋅ 1021 cm−2 , if we take in accordance with [87] that an increase of the amount of atmospheric water vapor is 4%. As is seen, variations of an amount of atmospheric water exceeds several times that of carbon dioxide. Let us consider the altitude dependence for the average atmospheric temperature accounting for evaporation of water from the Earth’s surface and its condensation in the atmosphere that leads to energy release. Then equation (2.1.3) is transformed to dE = c v ndT + pdV + ∆εdn w ,

(2.2.3)

where n w is a number of water molecules in a given air element, and ∆ε is the energy which is released as a result of transition of a free water molecule in a bound state of a drop. Because n w = n ⋅ c, where c is the concentration of water molecules in atmospheric air, we have dn w = ndc in the last term of equation (2.2.3). This equation shows also that the altitude dependence of the temperature has not a simple form as it is used in the model of standard atmosphere (dT/dh = −6.5K/km). Let us estimate the contribution of the water condensation process in the troposphere heat balance. Taking the integral from equation (2.2.3) over the troposphere, one can introduce the ratio ∆ε ⋅ c ξ= (2.2.4) 2.5∆T One can obtain for the standard atmosphere ξ ≈ 0.4. This means that condensation of atmospheric water influences remarkably on the atmospheric temperature profile. Note that the total amount of atmospheric water grows in time. This equals to 0.07g/kg per decade (gram of water per kilogram of air) [88–90] that corresponds to variation of the concentration of atmospheric water as d ln c = 7 ⋅ 10−3 yr−1 dt

(2.2.5)

From the end of 19 century an increase of the air moisture near the Earth’s surface is equal approximately 4% [87]. Season variations of the global air moisture which

16 | 2 Generals in statics and dynamics of global atmosphere are given in Fig.2.8, follow from different rates of season water evaporation from the Earth’s surface and the opposite process. One can expect that because the tropospheric temperature decreases with altitude, the average moisture of atmospheric air as an altitude function increases. Then at a certain altitude the partial pressure of water equalizes to the saturated vapor pressure, and formation of microdrops proceeds at higher altitudes. Fig.2.8 shows that this is not realized. One can formulate the model which describes the behavior of atmospheric water in the atmosphere. In absence of condensation of atmospheric water that takes place near the Earth’s surface, the atmospheric temperature decreases with an altitude increase because of air expansion. In accordance with formula (2.1.6), the temperature gradient is dT/dh = −14K/km until water condensation is absent. In any case, condensation of an atmospheric water vapor is absent, until its partial pressure is lower than the saturated vapor pressure at a current atmosphere temperature. Taking the temperature gradient according to formula (2.1.6) and the partial water pressure near the Earth’s surface to be 1.7kPa for the model of standard atmosphere, we obtain that the condensation process proceeds at altitudes above 1km where the temperature is 277K. Within the framework of the standard atmosphere model, the number density of free water molecules is N o = 4.3 ⋅ 1017 cm−3 that at the temperature T = 288K corresponds to the partial pressure of an atomic water vapor p = 13Torr and the average moisture of air of 82%. From this analysis it follows that a nonuniform distribution of atmospheric water is caused by a heightened rate of condensation in regions with aerosols, and this leads to location of clouds in a restricted region. One can take the average cloudness to be 2/3, i.e. clouds cover 2/3 of the Earth’s surface.

2.2.2 Atmospheric carbon dioxide Another molecular greenhouse component of the atmosphere is carbon dioxide, and we below consider its behavior there. Fig.2.10 contains basic channels of carbon dioxide transport to the Earth’s atmosphere and from it. One can see that the basic channel of transformation of atmospheric carbon dioxide into a solid carbon at the Earth’s surface is the photosynthesis process, and formation of atmospheric carbon dioxide results from oxidation of solid carbon at the Earth’s surface. Though the accuracy of data of Fig.2.9 is restricted, this Figure shows that the photosynthesis process proceeds more effectively at land, than in oceans. Oxidation of organic compounds at the Earth’s surface results from breathing of living organisms and plants, and also from decomposition and putrefaction of dead organisms and plants. As it follows from this Figure, the human activity gives the contribution of about 5% to carbon flux in the atmosphere. Nevertheless, excluding of the contribution of the industrial production of atmospheric carbon dioxide leads to violation of the balance of carbon dioxide in the atmosphere, that leads to a decrease of its amount in the atmosphere. Note that in contrast to equilibrium between the Earth’s surface and atmosphere for water, in the case of carbon the chemical state of transferring molecule changes as

2.2 Greenhouse molecular components in the atmosphere

| 17

Fig. 2.9: Carbon fluxes through the Earth, atmosphere in 109 ton per year [91–95].

a result of this transition. Transformation of atmospheric carbon dioxide in solid carbon at the Earth’s surface proceeds under the action of solar radiation as a result of the photosynthesis process where chlorophyll is a catalysis [96] for conversion of gaseous carbon dioxide in solid carbon. This process depends on many factors which include the plant type, the phase of plant growth and external conditions. But in general, the photosynthesis process proceeds according to the scheme 6H2 O + 6CO2 + ω → C6 H12 O6 + 6O2 ,

(2.2.6)

so that as a result of this multistage process, glucose C6 H12 O6 and oxygen are formed. Subsequently glucose is transformed in various forms of solid carbon [97, 98]. The theoretical maximum efficiency for transformation of the solar energy into the chemical one of bound carbon is approximately 11%. The photosynthetic efficiency is estimated as (3 − 6)% [99, 100]. As the chemical process, photoionization consists of many subsequent stages [97, 101–105] and has a complex character. Plants are divided in C3 and C4 groups according to the process character depending on the number of carbon atoms participated in an elementary chemical process [106]. In the case of plants of the group C3 which includes wheat, rice, beans, the process proceeds through collisions between the CO2 molecule and some intermediate products, while in the case of plants of the group C4 carbon dioxide molecules are captured by the cell, and the chemical process involves molecules in the bound state. As a result, the dependence of the photosynthesis rate on the concentration of CO2 molecules in atmospheric air are different for plants of these groups [107]. Along with the photosynthesis process, removal of carbon dioxide from the atmosphere may result in dissolving of carbon dioxide in oceans, that leads to formation of carbonates such as CaCO3 [108]. Destruction of carbonates leads to transition of carbon dioxide into the atmosphere. Other channels of formation of atmospheric carbon dioxide are processes of plant rot, and also processes of breathing of plants and microbes. Note the principal difference between photosynthesis processes at land and in oceans. Forests and the agriculture production are responsible for the photosynthesis

18 | 2 Generals in statics and dynamics of global atmosphere process at land, and its annual result may be determined on the basis of an increase of the mass of carbon-contained products. The photosynthesis process in oceans is connected with the phytoplankton that is an analog of plants at land [109, 110]. Indeed, microparticles of phytoplankton absorb atmospheric carbon dioxide and transforms it into bonded solid carbon under the action solar radiation. Subsequently phytoplankton is used by fishes as a nutrient medium [111]. In this way a biomass of oceans grows. For its measurement, the isotope method is used on the basis of the radioactive 14 C isotope [112, 113]. Along with it, the satellite method is used by measurements of the chlorophyll concentration in oceans [114]. As a result, the total rate for the biomass accumulated in oceans is estimated as 50GtC/yr [115], that is 5⋅1016 g/yr of carbon which constitute this biomass. Combining products of photosynthesis at land and in oceans, one can determine the total mass of solid carbon from the photosynthesis process. This rate determined on the basis of isotopic analysis of plants, ranges from 150GtC/yr up to 175GtC/yr [116]. This value exceeds the old estimation [117] according to which the indicated value is taken as 105GtC and is less than the value 217GtC/yr of Fig.2.9. Comparison of these values testifies about their accuracy. Detailed information about atmospheric carbon dioxide follows from monitoring of atmospheric CO2 that is made from 1959 in the Mauna Loa observatory (Hawaii, USA) [55, 56, 119, 120]. This observatory is located at altitude 3400m above the sea level and far from sources or absorbers of carbon dioxide. Some results of this monitoring are given in Fig.2.10. As is seen, the concentration of carbon dioxide molecules in atmospheric air increases from 316ppm in 1959 up to 409ppm in 2017. In addition, the rate of an increase of the carbon dioxide concentration grows in time from 0.7ppm/yr in 1959 up to approximately 2.1ppm in 2017. Note also that in 1750 the concentration of CO2 molecules was equal (277 ± 3)ppm, and in 1870 this value was (288 ± 3)ppm [118]. Season oscillations of this concentration is represented in Fig.2.10c, where the season character of the photosynthesis process is given. The Mauna Loa observatory is located in the northern hemisphere close to the equator, and carbon dioxide from the southern hemisphere reaches this observatory that leads to small season oscillations. But because carbon dioxide from the northern hemisphere dominates, a season decrease of the carbon dioxide concentration proceeds in the period from May to September. Note that season variations of the carbon dioxide concentration in atmospheric air in regions far from the equator are larger than those at the Mauna Loa observatory. Let us construct the month distribution for the rate of transport of carbon dioxide from the Earth’s surface in the atmosphere and vice versa. One can use the average rates during the year according to Fig.2.10 data and take into account that the photosynthesis is absent from October to March. Requiring the average rate during year for change of the carbon dioxide concentration in the atmosphere to be zero in scales of these fluxes, one can find the month distribution for fluxes of atmospheric carbon dioxide that is given in Fig.2.10b for standard atmosphere. As is seen, oscillations of the carbon

2.2 Greenhouse molecular components in the atmosphere

| 19

(a)

(b)

(c)

Fig. 2.10: Concentration of CO2 molecules in atmospheric air during the last half century (a) and for the last five years (b) according to [55, 56]; open circles correspond to an average during a month, and filled squares relate to averaged data for year (one half year before and after an indicated data). c) Month oscillations of the concentration of CO2 molecules in atmospheric air obtained by averaging of this month data wth respect to an average value of this year for 30 years [121].

20 | 2 Generals in statics and dynamics of global atmosphere

Fig. 2.11: Season distribution for carbon dioxide fluxes for standard atmosphere [122].

dioxide concentration for standard atmosphere with accounting for various latitudes is larger than that observed at the Mauna Loa observatory. Thus, we have that in the first approximation atmospheric carbon dioxide is mixed with air and the concentration of atmospheric carbon dioxide is constant. If we take into account the season character of formation and loss of carbon dioxide in the atmosphere, one can obtain in the following approximation remarkable variations of its concentration in time which according to data of Fig.2.10c are of the order of 20%. The data of Fig.2.10a give the contemporary rate of change of a current annual concentration as d ln c = 0.006yr−1 , (2.2.7) dt with an accuracy approximately 10%. As it follows from Fig.2.10b, the amplitude of season oscillations of the carbon dioxide exceeds the annual change of the carbon dioxide concentration that relates to the latitude of the Mauna Loa observatory. The season oscillations are stronger for larger latitudes, as it exhibits in Fig.2.11. This shows also the natural character of variations of the atmospheric CO2 amount. In addition, the average number density of carbon dioxide molecules near the Earth’s surface is N(CO2 ) = 1.0 ⋅ 1016 cm−3 ; the amount of carbon dioxide molecules in a vertical column is n(CO2 ) = 8 ⋅ 1021 cm−2 , and its variation from the end of 19 century equals approximately ∆n(CO2 ) = 2.5 ⋅ 1021 cm−2 .

2.2.3 Trace components in the atmosphere We now give a cursory glance on trace atmospheric molecular gases which include ozone, methane, and nitrogen dioxide. Stratospheric ozone is concentrated mostly at altitudes from 18 km to 40 km and the amplitude of day and season oscillations of its atmospheric amount is compared with its total amount. Basing on data [26–28, 123, 124], we take an average ozone amount in a vertical atmospheric column to be 2 ⋅ 1019 cm−2 or approximately 700DU (this unit is equal 1DU(Dobson) =

2.2 Greenhouse molecular components in the atmosphere

(a)

| 21

(b)

Fig. 2.12: Methane in the Earth’s atmosphere.

2.69 ⋅ 1016 cm−2 [125]). Stratospheric ozone is concentrated in the altitude region between 18 km and 30 km [126], where the temperature of the standard atmosphere ranges between 217K and 227K. Because of a narrow temperature range, we below consider the ozone layer to have an indentical temperature to be T = 220K. Oxygen atoms of the ozone molecule form an isosceles triangle with an obtuse angle of 116.8o and absorption of stratospheric ozone takes place near the vibration transition of frequency 1042cm−1 (λ = 9.6μm). Atmospheric methane is another greenhouse component (see Fig.2.12. Basing on data [127–129] for atmospheric methane, one can observe the growth of the amount of methane molecules in the atmosphere with a high rate of season oscillations, as it is demonstrated in Fig.2.12b. The concentration of atmospheric methane molecules varies from 0.72ppm in the pre-industrial epoch to 1.9ppm now [127]. From this we take the atmospheric concentration to be approximately 2ppm and the number density of methane molecules to be N v = 5⋅1013 cm−3 at the Earth surface. We below focus on the main radiative infrared transition in the atmosphere related to an upper state of the methane molecule which corresponds to a triply degenerate deformation vibration with the transition frequency ω o = 1306 cm−1 , and one can take parameters of vibration-rotation transitions from [130–133]. The radiative lifetime of the excited vibrational state with the excitation energy ω o = 1306 cm−1 is τ v = 0.4s. Due to the tetrahedral structure, the methane molecule as a rotator is characterized by three identical rotational constants B = 5.2cm−1 . Note that the number density of methane molecules in a vertical atmosphere column (approximately 4 ⋅ 1019 cm−2 ) is higher than that for the ozone case, but the greenhouse effect is weaker in the methane case. This is explained by a different rotation constants which characterize an energetic distance between neighboring spectral lines. One more greenhouse component of atmospheric air is nitrous oxide (N2 O). Though the concentration of N2 O molecules in the atmosphere is less by an order of magnitude than that for methane molecules, more favorable spectroscopic parameters may provide the existence of an absorption band even at such concentration. We

22 | 2 Generals in statics and dynamics of global atmosphere take the concentration of atmospheric N2 O molecules to be 0.3ppm according to measurements [134, 135], and this concentration grows slightly in time being 0.27ppm in 18 century, 0.28ppm in 19 century, and 0.29ppm in 20 century [136]. We also use below spectroscopic parameters of N2 O molecule [130–133, 137, 138] which include the rotation constant of this linear molecule B o = 0.419cm−1 . The radiative time τ r = 5ms corresponds to the vibration frequency ν1 = 2224cm−1 , and the radiative time τ r = 80ms relates to the vibration frequency ν3 = 1285cm−1 .

2.3 Evolution of global temperature 2.3.1 Contemporary variation of global temperature Let us introduce the global Earth’s temperature as a parameter of the Earth’s energetic balance, i.e. the temperature of the Earth’s surface averaged over all the globe surface. Variations of the global temperature in time characterize the climate evolution, and our task is to determine the rate of change of the global temperature dT/dt in a certain range of time. The problem is that chaotic variations of the global temperature, as well as different temperature changes at different geographical points, both daily and season variations, reach tens degrees of Kelvin. Therefore fluctuations of the global temperatures averaged over year may attain several Kelvin. But variations of the global temperature for a century do not exceed 1K [139]. The method [57] for determination the variation of the global temperature allows one to overcome the indicated trouble and to decrease the fluctuations of the global temperature by one order of magnitude. Within this method, the temperatures are compared for the same geographical point, but at the same daily and season time, and the difference of these temperatures ∆T is constructed for one or several years. At the next step, this temperature difference is averaged over daily and season times during the year, as well as over geographical points. In this manner one can determine variation ∆T of the global temperature in time and the rate d∆T/dt of its variation. Fluctuations of variation of the global temperature in this method are estimated as 0.1K. Realization of this method requires large information and a huge work. Nevertheless, this information follows from data of meteorological stations. A number of such meteorological stations was about 6 thousands in the second half of 19 century and now their number decreases in three times, whereas the basic information follows from satellite measurements. Treatment of this information is made within the framework of NASA programs [Gottard Institute for Space Studies - GISS], and results of this treatment are given in [59, 140–143]. Comparison of the global temperature change in a summer and winter, as well as in a day time and night time or in North and South hemispheres gives that indicated values of ∆T during a year does not exceed 0.2K [59, 141]. Fig.2.13 describes evolution of the global temperature with averaging over a year, five years and fifteen years. One can see that the larger time of averaging, the smoother

2.3 Evolution of global temperature

(a)

|

23

(b)

Fig. 2.13: Variation of the global temperature with averaging for one and fife years (a) [140], and for five and fifteen years (b) [59, 141]. Arrows indicate years when the global temperature does not vary in average and is taken as a basis ones.

this dependence. From Fig.2.13 it follows that year fluctuations of the global temperature is of the order of 0.1K. One can see a non-monotonic evolution of the global temperature in time. Indeed, during 1880-1910 a weak cooling was observed that was changed by a weak heating in 1910-1940. Subsequent cooling was observed during 1940-1950, during next 1950-1980 the global temperature did not vary within the limit of its accuracy, and the Earth heating takes place after 1980. Basing on this, we give in Fig.2.14 evolution of the global temperature after 1985 when the global temperature was increased monotonically. An approximation of these data by the linear time dependence describes evolution of the average change of the global temperature by the equation d∆T = 0.018K/yr, (2.3.1) dt and the fluctuation (standard deviation) for data of Fig.2.14 is equal ∆ = 0.09K [144]. The latter means that a smooth (linear) time dependence for the global temperature change ∆T is valid at time intervals above 5 years. Evolution of a local temperature of the Earth’s surface has a more complicated character compared with the global temperature during a long period. For example, measurements at the Mauna-Loa observatory during 1977-2006 [145] show warming in night time (from 22.00 pm up to 6.00 am) with the rate 0.040o C/yr, whereas a slight cooling with the rate −0.014o C/yr is observed at noon (12.00 am). The average temperature change averaged over all times is 0.021o C/yr that is close to evolution of the global temperature according to formula (2.3.1). As it follows from this example, the global temperature is the atmospheric characteristic which contains information about a thermal state of the Earth. Thus, from treatment of data of thousands meteorological stations it follows a warming during last decades in accordance with formula (2.3.1). In addition, the global temperature increase for last 130 years is equal (0.8 ± 0.1)o C [139]. One can refer to these data twofold. If we consider this as a steady warming, one can expect a subsequent increase of the global temperature in future decades. From another

24 | 2 Generals in statics and dynamics of global atmosphere

Fig. 2.14: Evolution of global temperature during last years [59]. The temperature variation is counted from the average global temperature in 1950-1980.

standpoint, one can consider this warming as a random fluctuation. In this case prediction of the subsequent warming is not correct. To answer this question, let us analyze evolution of the Earth’s temperature and variation of the Earth’s climate in past [36, 146–148]. Our position is that a contemporary warming is a long fluctuation of the global temperature, and hence this warming may be changed by a cooling through a certain time. Within the framework of the Arrenius concept [149], one can take the concentration of CO2 molecules as a measure for the atmosphere state, because carbon dioxide is an important atmospheric component with a large lifetime in the atmosphere. Let us introduce the equilibrium climate sensitivity ECS [150] as a change of the global Earth temperature at the doubling of the atmospheric carbon dioxide concentration, i.e. ECS = ∆T

ln 2 , ln(c2 /c1 )

(2.3.2)

where c1 and c2 are concentrations of CO2 molecules at the initial and final times, and ∆T is the global temperature variation for this time interval. On the basis of formulas (2.3.1) and (2.2.7) one can find on the basis of formula (2.3.2) δ = (2.5 ± 0.4)o C

(2.3.3)

One can note that the value (2.3.3) results from the totality of measurements, as well as their careful treatment rather than estimates. In addition, it does not prove that an observed temperature change results from the greenhouse effect.

2.3 Evolution of global temperature

|

25

Fig. 2.15: Evolution of the Earth’s temperature which is determined by isotopic methods at various points of the planet surface [151].

2.3.2 Variation of global temperature in past In considering the Earth’s temperature in past, note that in this case it is possible to determine the local temperature only. To confirm this position, we give in Fig.2.15 evolution of a typical Earth temperature in past, as it follows from the isotopic analysis of deposits inside the Earth [151]. In contrast to the global temperature that results from averaging over various geographical points on the Earth’s surface, data of this Figure relate to random points of the globe, and hence their analysis has a qualitative character. One can see an irregular form of the time dependence for the temperature. In addition, the more is a fluctuation time, the more is the fluctuation amplitude. As it follows from Fig.2.15, the maximum amplitude of temperature variation reaches 14K. Probably, large fluctuations of the Earth temperature are connected with glacial epochs. The period of a glacial epoch is of the order of 100kyr and the nature of existence of the glacial epochs is explained according to the Milankovich theory [152, 153] due to character of Earth motion in its rotation around the Sun along elliptic orbits. Probably, fluctuations in the Earth temperature with a typical time of the order of a century are determined by variations in the solar activity [154]. Evidently, it was the reason of the temperature lowering during so called Maunder minimum that lasted from 1645 to 1715 and was characterized by a lower solar irradiance [155]. One can connect the solar activity with some processes inside the Sun [156, 157], that in turn influences on the Earth temperature. In particular, the amplitude of the temperature change during the Maunder minimum is approximately 1K [154] that corresponds to variation of the global temperature in the last century. One can give one more example of the last millennium that allows us to consider an Earth heating of the last century as a random fluctuation of the global temperature. Indeed, during 11 -14 centuries a warm period was observed in England [158], when the grape was ripen and vikings occupy Greenland. It was changed by a cold period, when the river Thames was frozen each year and vikings leave Greenland [158]. Thus our experience for the temperature variation in some regions of the globe, both in the last millennium and during longer periods, testifies about a non-regular evolution of the Earth temperature in time, and temperature fluctuations are the

26 | 2 Generals in statics and dynamics of global atmosphere

Fig. 2.16: Evolution of the concentration of carbon dioxide molecules in the Earth’s atmosphere and of the Earth’s temperature in past [162, 163].

stronger, the more long period is considered. Note that in this consideration we assume the contribution of human activity to be relatively small, and we ignore this. Hence, we consider this problem in greater detail. In the case of the equilibrium between atmospheric carbon dioxide and carbon bonded in organic compounds at the Earth surface, there is a correlation between a change of the global temperature and the amount of atmospheric carbon dioxide. The most experience in study of this correlation follows from the analysis of deposits on the basis of isotope methods [159–161]. Then the temperature of the object follows from the ratio of concentrations of stable isotopes. In particular, the ratio of oxygen stable isotopes 18 O and 16 O with their occurrence in the Earth’s crust 0.2% and 99.76% correspondingly. In addition, stable carbon isotopes 13 C and 12 C with the natural occurrence 1.10% and 98.90%, stable argon isotopes 38 Ar and 40 Ar with the natural occurrence 0.06% and 99.60%, as well as stable hydrogen isotopes, deuteron 2 H and proton 1 H with the natural occurrence 0.015% and 98.985% are used for determination of the temperature when this object is formed. Simultaneously with determination the temperature change, the time of isotope location inside an object under consideration follows from measurements of an amount of radioactive isotopes [161]. As a result, this gives information about the climate at certain time in past that is the object of study of paleontology (for example, [164–169])). Along with the temperature, the concentration of atmospheric gases at that time follows from the analysis of gaseous bubbles located inside deposits in the form of ice or clay. As an example of such a research, Fig.2.16 contains evolution of the concentration of carbon dioxide and Earth’s temperature [162, 163] on the basis of the deposit analysis taken at the meteorological station "Vostok" in Antarctica. One can see the correlation between changes of the temperature and carbon dioxide concentration during time up to 400kyr ago. In addition, the carbon concentration at this period ranges from 172 to 300 ppm. This corresponds to results of the bottom analysis of Atlantic ocean west of Sierra Leone which gave the range between 213 ppm and

2.3 Evolution of global temperature

|

27

283 ppm during the period from 900 kyr to 2.1 million years ago [163]. This shows that the amount of carbon dioxide in the atmosphere in past is not constant and fluctuates between 200 and 300ppm. Contemporary content of atmospheric carbon dioxide is approximately 400ppm that is higher than that in past. As it follows from Fig.2.16, maxima of the temperature are observed with a period of approximately 100kyr in accordance with the Milankovich theory [152, 153], i.e. variation of the temperature in this period is connected mostly with variations of the Earth’s orbit with respect to the Sun. From this standpoint, the contemporary global temperature is higher approximately 3K than the average one, i.e. we are found now in a warm period. Next, as it follows from Fig.2.16, the amplitude of variations for the concentration of CO2 molecules is approximately 70ppm, and the amplitude of the temperature variation during several ice periods exceeds 10K. One can estimate from this the equilibrium climate sensitivity on the basis of formula (2.3.2), that is equal to δ ≈ 18K. As is seen, this value exceeds by one order of magnitude than the contemporary value. From this it follows a small contribution of carbon dioxide to the planet heat balance. Thus, the experience of the temperature evolution in past exhibits about oscillations of global parameters with the period approximately 100 kyr. The amplitude of the temperature change exceeds 10o C and the amplitude of oscillations for the concentration of CO2 molecules is approximately 100 ppm, while the maximum concentration of CO2 molecules does not exceed 300ppm. Because the human activity does not influence on these values, one can consider them to describe the behavior of the atmospheric state in a preindustrial period.

2.3.3 Chemical equilibrium between the Earth and atmosphere The concept of ECS (Equilibrium Climate Sensitivity) was introduced on the basis of the Arrenius paper in 1896 [149] and this quantity is the change of the global temperature ∆T in the case of doubling of the concentration of atmospheric CO2 molecules in accordance with formula (2.3.2). In past this quantity may be evaluated by comparison of the rate d∆T/dt to d ln c/dt, say, on the basis of data of Fig.2.16 that gives [170] ECS = (3.0 ± 1.5)o C

(2.3.4)

This result follows from averaging over many evaluations (for example, [171–178])), where the assumption is used that the change of the radiative flux due to atmospheric CO2 molecules was equated to the change of the total radiative flux toward the Earth, whereas these values differ by several times. In order to evaluate the ECS for present time, we take into account that the global temperature increases from the end of 19 century up to now by ∆T = (0.8 ± 0.1)o C [139], whereas the concentration of atmospheric CO2 molecules increases from 280ppm [118] at the end of 18 century up to 410ppm now. From this it follows ECS = (1.9 ± 0.4)o C,

(2.3.5)

28 | 2 Generals in statics and dynamics of global atmosphere that is in accordance with this value (2.3.3) where we use data of last 30 years. Note we take here that the concentration of CO2 molecules here is the parameter described the atmosphere state, and the change of the global temperature results for other reasons, rather than due to CO2 molecules. We also check the validity of the Pauling concept [179, 180] according to which an increase of the carbon dioxide concentration in the atmosphere results from an increase of the global temperature, but not vise versa, as it is assumed in the standard consideration. In this case an atmospheric amount of carbon dioxide follows from equilibrium with ocean carbon dioxide which is dissolved in oceans and is in the bound state inside some compounds of the carbon acid. For definiteness, we consider the following equilibrium between atmospheric and oceanic carbon dioxide CO2 + CaO ←→ CaCO3 ,

(2.3.6)

where the enthalpy of this transition is ∆H = 178kJ/mol [108]. Because an amount of atmospheric carbon dioxide is less approximately in 60 times compared to that of oceanic carbon dioxide [180], the concentration c(CO2 ) of atmospheric carbon dioxide depends on the temperature as c(CO2 ) ∼ exp (−

∆H ) T

(2.3.7)

This leads to the following relation between the concentration change of atmospheric carbon dioxide and the temperature change ∆T of the Earth’s surface ∆ ln c(CO2 ) ∆H = 2 d∆T T

(2.3.8)

If we assume the concentration of atmospheric carbon dioxide to be determined by the global temperature only, one can obtain d ln c(CO2 ) ∆H d∆T = 2 dt T dt Taking d∆T/dt = 0.018K/yr in accordance with formula (2.2.7) and using the above value of the enthalpy change ∆H, one can reduce this equation to the form d ln c(CO2 ) = 0.0046yr−1 dt

(2.3.9)

One can see that the Pauling concept of equilibrium between the carbon dioxide concentration in the atmosphere and the global temperature (2.2.7) gives the main contribution the observed variation (2.2.7) of evolution of atmospheric carbon dioxide concentration. One can expect that another part of the change of the carbon dioxide concentration is determined by variation of the rate of the photosynthesis process in accordance with Fig.2.10, evidently, as a result of decrease of the photosynthesis rate because of deforestation. We above operate with the global temperature while the rates under consideration are determined by local temperatures, i.e. an average is required after determination a

2.3 Evolution of global temperature

|

29

certain value, rather than before it, and this leads to an error. This error is proportional to the difference between the global and local temperatures at a certain time and geographical coordinate. In order to estimate this error, let us derive equation (2.3.8) with accounting for an indicated difference. Then we obtain instead of equation (2.3.8) d ln c(CO2 ) ∆Hd∆T 1 = ⟨ 2⟩⋅ , dt dt T

(2.3.10)

where T(r, t) is the local temperature, and brackets mean an average over the globe and time. Since usually a local temperature differs from the global one less than by 10%, using the global temperature in equation (2.3.8) leads to an error below 10%. Let us consider also the chemical equilibrium between atmospheric water vapor and bound water located in water reservoirs at the Earth’s surface. One can introduce the concentration c(H 2 O) = N w /N a of water molecules, where N a is the number density of air molecules, N w is the number density of atmospheric water molecules. Assuming the residence time of molecules in the atmosphere to be independent of the evaporation rate, one can obtain the following temperature dependence for the concentration of water molecules c(H2 O) ∼ exp (−

ϵb ), T

(2.3.11)

where ϵ b = 0.43eV [75] is the binding energy of water molecules in liquid water. By analogy with equation (2.3.8), one can obtain in this case d ln c(H 2 O) ϵ b dT = 2⋅ dt dt T

(2.3.12)

From this on the basis of formula (2.3.1) it follows d ln c/dt = 1.1 ⋅ 10−3 yr−1 compared to 7 ⋅ 10−3 yr−1 for CO2 molecules according to formula (2.3.12). Formula (2.3.12) gives the connection between a variation of the global temperature ∆T and that of the logarithm of the concentration of atmospheric water molecules which variation is ∆ ln c T2 ∆T = ∆ ln c ⋅ (2.3.13) ϵb In particular, from the end of 19 century the global temperature increases by ∆T = 0.8K [139], that according to formula (2.3.13) corresponds to relative change of the water concentration in the atmosphere by approximately 5%. The latter is in accord with data [87] that gives 4%. In addition, one can obtain from this that an increase of the global temperature ∆T that causes by an increase of the concentration of atmospheric water molecules ∆c(H2 O) is given by formula ϵb ln ∆c = 2 = 0.06K −1 ∆T T

(2.3.14)

30 | 2 Generals in statics and dynamics of global atmosphere

2.4 Dynamics of atmospheric air with microparticles 2.4.1 Convective motion of atmospheric air Transport of air with some additives as well as heat transport in tropospheric layers results from air convection which is accompanied by processes of water condensation with formation of water microdrops [181]. Simultaneously solid water consisting of ice or snow may be formed in the middle and high troposphere. Atmospheric convection is developed in the troposphere because of large atmosphere sizes and high atmosphere pressures. Convective motion in atmospheric air is of importance not only for transport of air and water in the atmosphere, but also for atmospheric electric phenomena. In the first place, transport of atmospheric air and water is of importance for the weather. In this case information about atmospheric parameters at a given time and that for surrounding geographical points allows one to describe the weather for certain time intervals at a given place on the basis of solution of the corresponding transport equations. This concept of the weather prediction as a result of numerical solution of transport equations is developed during a century (for example, [182]) and the accuracy of such a prediction improves with time. The maximum time of weather prediction in this manner is estimated as one-two weeks, that is a typical time during which an air perturbation encircles the globe. The uncertainty in the local weather forecast results from the random character of dynamics of atmospheric transport processes [183–186], and a turbulent character of motion results from collision of air fluxes [187, 188]. This establishes the random nature of transport of atmospheric air and its components, as well as electric atmospheric phenomena [189–194, 194, 195]. One can consider the convective air motion as a sum of individual vortices which arise in atmospheric air as a result of the Rayleigh-Taylor instability [196–200] under the action of the Earth’s gravitation field and temperature gradient. If conditions of the convective instability are fulfilled in a region between two heated plates, some space structures, Bennard cells [201–203], are formed above the threshold of this instability. If vortices are formed in a space of large sizes, they are mixed, and a random convective air motion develops in the form of vortices of different sizes. Let us estimate parameters of air convective motion on the basis of the criterion of development of the Rayleigh-Taylor instability, that takes place if the Rayleigh number is Ra ∼ 1000 [196, 197, 204]. The Rayleigh number for a vortex of size l, that is a size of the elementary air cell, is given by [196] gl4 dT gl4 = (2.4.1) Ra = Tdh νχ Λ T νχ Here g = 980cm/s2 is the free fall acceleration at the Earth’s surface, ν = η/ρ = 0.3cm 2 /s is the air kinematic viscosity, ρ = 1.3kg/m3 is the air density, Λ T = Tdh/dT ≈ 44km is a typical size for variation of the tropospheric temperature variation and we are guided by altitudes near the Earth’s surface, where the kinematic viscosity equals ν = 0.15cm2 /s, and the air thermal diffusivity is equal

2.4 Dynamics of atmospheric air with microparticles

| 31

χ = κ/Nc p = 0.2cm2 /s. Here N ≈ 2.5 ⋅ 1019 cm−3 is the number density of air molecules near the Earth’s surface, c p = 7/2 is the heat capacity per air molecule. This gives for a typical vortex size l≈(

Ra ⋅ νχΛ T 1/4 ) g

(2.4.2)

From this one can estimate the minimal vortex size in the atmosphere that is equal l ≈ 20cm at altitudes near the Earth’s surface. A typical vortex size l depends on the number density of air molecules N as l ∼ 1/√N. One can estimate the velocity v l of motion for a vortex of a size l from the NavierStokes equation vl ∂2 v (2.4.3) η 2 ∼ η 2 ∼ PN 󸀠 , ∂h l where η is the air viscosity, P = mg is the molecule weight (m is the molecule mass, g is the free fall acceleration), N 󸀠 ∼ Nl/Λ T is the difference of the molecule number density at a distance l between two air layers, N is the number density of air molecules (N 󸀠 ≪ N). From this we have for the vortex velocity v l of a size l vl ∼

mgNl3 , ΛT η

(2.4.4)

and the air diffusion coefficient due to transport of vortices of a size L is estimated as Dl ≈

v l l mgNl4 ∼ 3 3Λ T η

(2.4.5)

Applying these results for the troposphere near the Earth surface, one can obtain v l ≈ 10cm/s, and the diffusion coefficient for air transport through vortices of such sizes equals D l ∼ 100cm2 /s. For such vortex sizes the Reynolds number is equal Re = v l l/ν ∼ 103 that has the same order of magnitude for gases as the Rayleigh number Re ∼ Ra [200]. Air transport on large distances proceeds through large vortices of sizes L which significantly exceed a typical vortex size l. The largest elementary vortex leads to turbulent motion. Transition to the turbulent character of motion corresponds to the critical Reynolds number that is Re cr = v L L/ν = 2 ⋅ 105 [196]. From this one can obtain for the diffusion coefficient of atmospheric air through large vortices DL ≈

Re cr ν ≈ 3 ⋅ 104 cm2 /s 3

(2.4.6)

In addition, formulas (2.4.4) and (2.4.5) allow one to estimate a size of large vortices in accordance with 3ΛηD L 1/4 (2.4.7) L≈( ) mgN Applying these formulas to the troposphere, as early, we obtain L ∼ 70cm, v L ∼ 103 cm/s. Thus, we obtain roughly for the ratio of sizes for large and small vortices in

32 | 2 Generals in statics and dynamics of global atmosphere

Fig. 2.17: Altitude dependence for the coefficient of turbulent diffusion [205, 206].

the course of air convection in the troposphere L/l ≈ 3. One can see that transport of atmospheric air and its components for large distances is realized through vortices of a maximum size, and the appropriate diffusion coefficient for tropospheric air is estimated as D L ≈ 3 ⋅ 104 cm2 /s. Thus, mass and energy transport in atmospheric air on large distances has a random character [184–186] that is created by collisions of air fluxes resulted from mixing of air layers [187, 188]. Because of the random character of the transport process, one can describe this transport by the diffusion coefficient (2.4.6). This diffusion coefficient varies with an altitude, and Fig.2.17 gives the convective diffusion coefficient for air of the standard atmosphere, where observational data for air transport are treated on the basis of the diffusion character of its motion.The convective diffusion coefficient for motion at large distances which follows from treatment of data of Fig.2.17 with averaging for day and night is equal D L = 4 ⋅ 104 cm2 /s,

(2.4.8)

that is in accordance with previous estimations. In the regime which is realized in the troposphere we have the following dependence of parameters of largest vortices on the number density N of air molecules L∼

1 1 DL 1 , DL ∼ , vL ∼ ∼ N L √N √N

(2.4.9)

This similarity law holds true until the velocity v L of vortices is small compared to a thermal velocity of air molecules or the sound speed c s . This takes place in the low atmosphere (2.4.10) N ≫ No , where N o ∼ 1016 cm−3 that corresponds to altitudes h < 55 km, i.e. the regime under consideration is fulfilled in the mesosphere and stratosphere. In considering the air transport at higher altitudes, one can assume the velocity of the largest vortices to be equal to the sound speed v L ≈ c s ≈ 3 ⋅ 104 cm/s. Since

2.4 Dynamics of atmospheric air with microparticles

| 33

the critical value of the Reynolds number Re cr = v L L/ν = 2 ⋅ 105 [196] corresponds to largest vortices, one can obtain the similarity law instead of (2.4.9) L∼

1 1 , DL ∼ , vL = cs , N N

(2.4.11)

Re cr ν cs

(2.4.12)

and the largest vortex size is L=

In particular, for the altitude h = 100km this formula gives (ν = 30m2 /s [24]) that corresponds to L ∼ 10km which follows from treatment of observational data. Next, formula (2.4.6) Re cr ν (2.4.13) DL ≈ 3 gives at the altitude h = 100km D L ∼ 2⋅106 cm2 /s. This value is less than that followed from treatment of observational data D L ∼ 1 ⋅ 107 cm2 /s [207]. Note that in this case L ∼ Λ T , i.e. the diffusion character of air transport for large distances is violated.

2.4.2 Transport of water microdrops in convective air Convective motion of air is accompanied by drift of molecules and particles of air under the action of gravitation forces. We now analyze this drift of atmospheric molecules and particles. In the above consideration the convection is created by air molecules, i.e. nitrogen and oxygen molecules. Therefore the average mass of these molecules (m = 29 in units of atomic masses) is included in formulas (2.4.4) and (2.4.5). Other components and admixtures do not contribute to the convective air motion. Collisions between molecules are of importance, for example, the mean free path of air molecules at the altitude of 100km is 14cm [24] that is small compared with values L and Λ T which are responsible for convection. Therefore components of air are captured by its motion and are mixed with air. As a result, the concentration of air admixture does not depend on the altitude h up to h ∼ 100km. One can add to this that transport of air in this consideration consists of two parts, namely, diffusion motion of air under the action of convection and drift motion of molecules due to gravitation forces. In this consideration displacement of molecules owing to gravitation forces is weak, so that one can neglect the action of the molecule weight on the convection process. Let us introduce an effective drift velocity w L of a given component due to convection, that follows from the expression for a flux j of this component j = −D L ∇N = w L N,

(2.4.14)

where N is the number density of particles for this component, and its drift velocity is wL ≈

DL , Λ

(2.4.15)

34 | 2 Generals in statics and dynamics of global atmosphere where Λ ≈ 8km is the scale size according to formula (2.1.2) and, as it follows from formula (2.4.15), w L ≈ 0.01cm/s. From this one can find the criterion of the convective motion for a given component. Indeed, this component partakes in the convective motion of air molecules, if this drift velocity w L is large compared to the drift velocity w g under the action of the particle weight. An indicated drift velocity follows from the Einstein relation [208–210], that is given by wg =

Dmg T

(2.4.16)

Here m is the mass of an admixture particle, g ≈ 980cm/s2 is the free fall acceleration, T is the air temperature expressed in energetic units, and D is the diffusion coefficient of a given atomic particle in air. This diffusion coefficient is equal within the framework of the Chapman-Enskog approximation [211–213] and the hard-sphere model for colliding atoms and molecules [213, 214] D=

3√πT , 8√2μNσ g

(2.4.17)

where μ = m1 m2 /(m1 + m2 ) is the reduced mass of a colliding particle and air molecule which masses are m1 and m2 correspondingly, N is the number density of air molecules, σ g is the gas-kinetic cross section for colliding atoms or molecules. The drift velocity of air molecules, where the gas-kinetic cross section at room temperature is σ g = 38 A˚ 2 [214], we have on the basis of formulas (2.4.10) and (2.4.11) at room temperature (2.4.18) N a w g = 6 ⋅ 1012 cm−2 s−1 , and w g ≈ 2 ⋅ 10−7 cm/s near the Earth’s surface. From this we have for the troposphere w L ∼ 0.03cm/s, whereas at the boundary altitude h ∼ 100km for convective motion we have for air molecules v L ∼ 3 ⋅ 104 cm/s, and w g ∼ 1cm/s. Therefore, the criterion wL ≫ wg ,

(2.4.19)

that allows us to neglect the action of gravitation forces on the convective character of motion of air molecules in the Earth atmosphere. This criterion holds true for air molecules in all the region of air convective motion. We now analyze the behavior of a macroscopic particle in air which transport proceeds in the form of convective motion. For definiteness, we take a uniform spherical particle of a radius r with a mass density ρ of its material. Locating in the troposphere, this particle falls down with the velocity w g [196] w g = gτ rel =

2ρgr2 , 9η

(2.4.20)

and for water microdrops under consideration this formula has the form w g /r2 = 1.2 ⋅ 106 s−1 cm−1 . From this one can find that w L = w g , if r ≈ 1μm. Hence the criterion

2.4 Dynamics of atmospheric air with microparticles

| 35

Fig. 2.18: Concentration of free water molecules (the ratio of the number density of water molecules to the number density of air molecules) as an altitude function according to NOAA (NASA) 30 May 2019 at Hilo, Hawaii [215]. For comparison, the contemporary concentration of CH4 molecules is marked by an arrow.

(2.4.19) is fulfilled for particles of a size r ≪ 1μm, and then microdrops are captured by air fluxes. In this case the altitude distribution (2.1.2) for microdrops is fulfilled. Let us introduce the altitude distribution of the number density of microdrops N(h) near an altitude h o in the form N(h) = N o ⋅ exp (−

h − ho ), λa

(2.4.21)

where N o = N(h o ). Evidently, in the limiting case w L ≪ w g we have λ a ∼ Λw L /w g . One can approximate a typical altitude variation λ a for the particle number density λa =

Λ 1 + w g /w L

(2.4.22)

This formula takes into account that, starting from a certain size, formed water microdrops cannot rise up because of the action of the gravitation force. In addition, we give in Fig.2.18 a typical distribution of the concentration of water molecules over altitudes that follows from balloon measurements. In this case, the measurements of the water content are fulfilled during the balloon rise, and then it goes down, so that the water density in the form of free molecules is fixed in the course of the altitude change. As is seen, the concentration of water molecules decreases with an increasing altitude because of the condensation process with formation the water microdrops. Regions of strong condensation are characterized by horizontal lines. From this one can represent the distribution of condensed water in the atmosphere. The condensation process proceeds in some regions where wet air streams are mixed with cold air. Hence, regions with a supersaturated water vapor are distributed randomly in the atmosphere and condensed water is located in the form of random streams or spots. It should be noted also that water condensation in the atmosphere leads to a sharper decrease of the number density of water molecules at an increasing altitude in accordance with formula (2.2.1).

3 Water microdrops in atmospheric air Abstract: Processes involving water microdrops in the atmosphere are considered. The mechanisms of formation and growth of water microdrops in the atmosphere include the processes of attachment of free water molecules to microdrops, coagulation and coalescence processes, and gravitation growth of drops which fall down under the action of their weight. Circulation of water in the atmosphere proceeds through water evaporation from the Earth’s surface, conversion of atmospheric water molecules into water microdrops at altitudes of several km and fall of the drops to the Earth’s surface. The model of standard atmosphere which deals with average atmospheric parameters does not admit the condensation process because the average number density of atmospheric water molecules at any altitude is lower than that at the saturated vapor pressure. Hence, the condensation process in the atmosphere is determined by fluctuations and takes place in the course of mixing of air streams which have different temperature and moisture. Kinetics of growth of water microdrops in clouds usually result from competition between the coalescence and gravitation growth character. Processes of growth of water microdrops in clouds are connected with atmospheric electricity.

3.1 Processes of water condensation in atmospheric air 3.1.1 Behavior of water microdrops in tropospheric air It should be noted that the main part of atmospheric condensed water is contained in clouds. Starting from L.Howard [216, 217], the classification of cloud types is produced. Different types of atmospheric clouds which are represented in Fig.3.1 are characterized by different parameters, as well their contribution to water transport through the

Fig. 3.1: Types of clouds [218].

https://doi.org/10.1515/9783110628753-003

38 | 3 Water microdrops in atmospheric air atmosphere is different [219]. One can extract three types of clouds according to their shape, namely, cirrus, cumulus, and stratus ones [220]. Cirrus means a curl or hair, and these clouds contain rare condensed water. Cirrus clouds are observed at high altitudes above 6 km. Cumulus clouds are more dense, and stratus clouds usually cover all or most of the sky with some layer. The prefix "alto" relates to corresponding layers located at high altitudes. The suffix "nimbus" corresponds to dense clouds and means "rain". In particular, towering clouds with flat tops, called as cumulonimbus clouds, create thunderstorms, as well as nimbostratus clouds may be connected with rain. Two last cloud types give the main contribution to formation and transport of atmospheric condensed water and, analyzing the behavior of water microdrops in the atmosphere, we are guided by these clouds. In addition, fog clouds are formed near the ground, usually at night, when air cools near the ground and the number density of water molecules exceeds that at the saturated water pressure, and then the water excess is transformed in microdrops. Since our interest is connected with clouds contained the basic mass of drops, and we are guided by these rain-caused clouds, which are cumulonimbus and nimbostratus ones. These clouds grow under specific conditions where warm wet air rises up and cooling of this air at higher altitudes leads to transformation of water molecules contained in it into microdrops. Formation of water drops is accompanied by air heating. If wet air rises up as a whole, a towered cumulus cloud is formed. As a result, it occupies a wide range of altitudes that facilitates transition to rain. Then water microdrops fall down under the action of their weight and, collecting all the drops of a smaller size on its way, they are converted in drops of millimeter sizes. In this manner water microdrops create rain. This takes place mostly in towered cumulus clouds which example is given in Fig.3.2 [221, 222]. Therefore we will be guided basically by such clouds. According to the above analysis, water microdrops which radius exceeds 1μm cannot rise as a result of convective motion. We below consider the behavior of water microdrops in atmospheric air, being guided by clouds which consist of such microdrops. In estimations we are based on parameters of water microdrops located in a typical mature cumulus cloud, which are characterized by following parameters on average [223–226] r = 8μm, N d = 103 cm−3 , (3.1.1) where we assume a microdrop to have a spherical shape, r is a drop radius, N d is the number density of drops. A number of bound water molecules of a microdrop and its radius r are connected by the relation n=(

r rW

3

) ,

(3.1.2)

˚ where r W is the Wigner-Seitz radius [227, 228] which is equal r W = 1.92 A[229] for liquid water. According to this formula, the number of water molecules in a water drop of a radius r = 8μm is equal n = 7 ⋅ 1013 , and the average number density of bound

3.1 Processes of water condensation in atmospheric air |

39

Fig. 3.2: A cumulus cloud [221, 222].

water molecules in microdrops is nN d = 7 ⋅ 1016 cm−3 . As is seen, the concentration of bound water molecules in cumulus clouds is close to the average concentration of water molecules in the atmosphere, as well as to the number density of free water molecules in the atmosphere at the saturated vapor pressure. Note that in this consideration a microdrop is large compared to the mean free path of air molecules in atmospheric air, that is equal 0.1μm. In addition, for a drop size (3.1.1) we have for the Reynolds number of atmospheric air located near an individual microdrop vl r ∼ 0.01 (3.1.3) ν We below consider the behavior of an individual microdrop in atmospheric air, starting from the case, where a motionless microdrop is inserted in an air flux moving with the velocity vo . Under these conditions, the motion equation for the microdrop of a mass M and current velocity v has the form Re =

M

M(v − vo ) dv = F = 4πrη(v − vo ) = − , dt τ rel

where F is the Stokes force [196], and τ rel is relaxation time in accordance with [230] that is given by τ rel =

2ρr2o 9η

(3.1.4)

40 | 3 Water microdrops in atmospheric air Here ρ = 1g/cm3 is the mass water density, and η = 1.85g/(cm ⋅ s) is the air viscosity coefficient at room temperature. If a microdrop consists of liquid water, we have from this formula τ rel /r2 = 1.2 ⋅ 103 s/cm2 . For typical parameters of liquid microdrops according to formula (3.1.1) we have τ rel ≈ 7 ⋅ 10−4 s. Let us compare this relaxation time with a typical time τ l of velocity variation in vortices of a size which is estimated as τ l ∼ l/v l . Restricting ourselves by large vortices of a size L which determine the air transport due to convection, we have τ L ∼ L/v L ∼ 0.06s for atmospheric air near the Earth surface. Requiring τ rel ≪ τ L , we have that fast establishment of equilibrium for water microdrops in atmospheric air according to this criterion takes place at sizes r ≪ 70μm. The range of microdrop sizes r ∼ 1−10μm is of interest now, such microdrops are not captured by air vortices in the troposphere. Another type of relaxation processes results in establishment of the equilibrium thermal regime. Let the temperature T d of a water microdrop differs from the temperature of surrounding air T a . Then the heat flux q occurs that equalizes the drop temperature and the air one. Since q = −κ

dT , dR

where κ is the air thermal conductivity, and R is a distance from the drop center. The power Q which is transfered from air to the drop or vice versa is equal P = 4πR2 κ

dT dR

(3.1.5)

Solving equation Q = const with the boundary conditions T(r o ) = T d , T(∞) = T o , we have for the current temperature T(R) T(R) = T o +

Td − To R

(3.1.6)

This distribution is established fast, and the heat power from a heated microdrop to surrounding air is given by P = 4πrκ∆T, ∆T = |T d − T o |

(3.1.7)

As a result of this process, the temperature T d tends to the temperature T o of surrounding air. This process is described by equation Cp

d∆T = −P = −4πrκ∆T, dt

(3.1.8)

where the drop heat capacity is Cp = cp ⋅

4πr3 ρ 3

Here the specific heat capacity for liquid water is c p = 1cal/(g ⋅ K). Representing equation (3.1.8) as

3.1 Processes of water condensation in atmospheric air |

Cp

r2 ρc p d∆T ∆T = −P = −4πrκ , , τr = dt τr 3κ

41

(3.1.9)

where τ r /r2 is the relaxation time for heat. In particular, for water microdrops in atmospheric air we have τ r /r2 = 5.5⋅103 s/cm2 . For a size of a water microdrop r = 8μm according to formula (3.1.1) we have τ r = 0.9ms. It is essential that the processes of molecule attachment to microdrops and evaporation of molecules are accompanied by heat processes. This may lead to a change of the character of these processes. Let us analyze these processes near a microdrop of a radius r, taking c o as the concentration of water molecules, and taking c∗ as the water concentration at the saturated vapor pressure for a given temperature. For a nonsaturated vapor c o < c∗ a microdrop evaporates. For the diffusion regime of this process the molecule flux i from the microdrop surface is equal at a distance R from the particle surface dc i = −DN w , (3.1.10) dR where D is the diffusion coefficient for a water molecule in atmospheric air, which is equal under normal conditions D = 0.22cm2 /s [214, 231], and N w is the number density of water molecules. A number of water molecules which intersect a sphere of a radius R j = 4πR2 ⋅ i (3.1.11) One can consider this relation as an equation for the concentration c(R) of water molecules because the flux j is independent of R. Solving this equation, one can obtain [229] r∆c c(R) = c o + , j = 4πDN w r∆c, (3.1.12) R where ∆c = c∗ − c o . Note that evaporation and growth of water microdrops is accompanied by processes of the energy absorption and release. The power of the total process is equal Q = ε o j, where j is the flux of molecules, and ε o = 0.43 eV [75] is the binding energy per water molecule. From this we have in accordance with formula (3.1.6) for the temperature change [229] ∆T DN m ε o =− , (3.1.13) ∆c κ In particular, near the Earth’s surface within the framework of the model of standard atmosphere we have T = 288K, κ = 2.5 ⋅ 10−4 W/(cm ⋅ K), c∗ = 0.017. From this it follows ∆T/∆c = 1.4 ⋅ 103 K, and ∆T/c∗ = 23K. As is seen, the microdrop temperature differs from that of surrounding air, and this influences on the rates of evaporation and growth, as well as on the drop thermal balance. We also compare the heat flux due to thermal conductivity P and that due to radiation of the microdrop J. Assuming a drop radius r to be large compared to the wavelength of radiation and considering water as a blackbody, we have for exchange by radiation between the microdrop and surrounding air that is optically thick for radiation. Then we have J = 4σT 3 ∆T ⋅ πr2 , (3.1.14)

42 | 3 Water microdrops in atmospheric air where σ is the Stephan-Boltzmann constant, πr2 is the absorption cross section for a blackbody. From this we obtain for the ratio of the radiative power J and the power P of heat transport J 4σT 3 r ξ= = (3.1.15) P κ In particular, for the drop size in a cumulus cloud (3.1.1) and room temperature we have ξ = 4 ⋅ 10−4 . From this one can conclude that the radiative transfer is not important for the behavior of an individual micron-sized drop. We also determine the temperature change ∆T if a particle does not absorb radiation from an environment. Then, comparing the emission power P = σT 4 from a drop which emits as a blackbody, and the power (3.1.7) of drop heating due to thermal conductivity, we obtain for a decrease ∆T of the drop temperature with respect to surrounding air ∆T =

σT 4 ⋅ r κ

(3.1.16)

In particular, this gives near the Earth surface (T = 288K, κ = 2.5 ⋅ 10−4 W/(cm ⋅ K), r = 8μm, and ∆T = 0.12K. One can see that drop emission is a slow process.

3.1.2 Condition of water condensation in atmospheric air Water microdrops are of interest for atmospheric phenomena, such as atmospheric electricity and atmospheric greenhouse effect. Therefore the nucleation processes involving water molecules are of importance for these phenomena and they are the object of this consideration. It is clear that the stable existence of water microdrops in atmospheric air is possible if the air temperature is below the dew point [232], that is the temperature at which the water partial pressure in air is equal to the saturated vapor pressure. Table 1 contains the saturated vapor pressure and corresponding number density of molecules depending on the air temperature. Since the water condensed phase exists in air stable, if the partial pressure of water molecules exceeds the saturated vapor pressure, the condition of the presence of the condensed phase in air has the form N > N sat ,

(3.1.17)

where N is the total number density of free and bound water molecules. For the model of standard atmosphere, the number density of water molecules near the Earth’s surface is N = 4.3 ⋅ 1017 cm−3 that corresponds to the saturated vapor number density at the global temperature according to data of Table 1. In this analysis it is convenient to introduce the atmospheric moisture as the ratio of the number density of free water molecules in the atmosphere to that at the saturated vapor pressure for a given temperature. One can see from the average number density of water molecules near the

3.1 Processes of water condensation in atmospheric air

|

43

Table 1: Parameters of a saturated water vapor in atmospheric air [74, 75]: p sat is the water saturated vapor pressure at an indicated temperature, N sat is the number density of water molecules in a saturated vapor, A s is the ratio of a mass of a saturated water vapor to an air mass located in a given volume at atmospheric pressure. T, K

p sat , Torr

N sat , 1017 cm−3

A s , g/kg

253 258 263 268 273 278 283 288 293 298

0.77 1.24 1.95 3.02 4.58 6.55 9.22 12.8 17.6 23.8

0.295 0.463 0.716 1.09 1.62 2.27 3.14 4.29 5.78 7.70

0.631 1.01 1.59 2.46 3.74 5.35 7.52 10.4 14.3 19.4

ground that the water vapor is saturated there. But the average moisture according to data of Fig.2.8 is approximately 82%, i.e. we deal with nonsaturated atmospheric water vapor. Moreover, as it follow from Fig.2.8, the air moisture decreases with an increasing altitude, i.e. the condensed water phase is absent in an averaged uniform atmosphere. Indeed, let us analyze the character of the moisture change with an altitude. Take the temperature dependence for the saturated number density of a water vapor as N sat (T) ∼ exp (−

εb ), T

where ε b = 0.43eV is the binding energy of a water molecule with a liquid surface. This leads to the following altitude dependence for the saturated number density of a water vapor N sat (T) ∼ exp (−

h T2 ≈ 2.6km ) , h ef = h ef ε b ⋅ |dT/dh|

(3.1.18)

Comparing this formula with formula (2.2.1), one can see that within the framework of the standard atmosphere model, the number density of free water molecules decreases sharper with an increasing altitude than the saturated number density of molecules. Hence, the criterion (3.1.17) is violated for the standard atmosphere model, and violation is the stronger, the higher altitude is. Thus, the model of standard atmosphere contradicts to observations according to which water condensation proceeds effectively at altitudes of several kilometers. The reason of this contradiction is that the atmosphere is nonuniform with respect to condensation processes, whereas the model of standard atmosphere deals with a uniform atmosphere. We obtain that the water condensed phase is not formed in atmospheric air under equilibrium conditions, and it results only from fluctuations or nonequilibrium processes.

44 | 3 Water microdrops in atmospheric air

Fig. 3.3: Long-scale winds in atmospheric air [233].

3.1.3 Mixing of air streams involving the condensed water phase From the above analysis it follows that the condensed phase of water is absent in a uniform atmosphere with averaged parameters since air is nonsaturated on average. Hence, condensation takes place as a result of air dynamics which leads to air cooling in some regions. This process consists in mixing of wet air from different regions under the action of winds which equalizise atmospheric parameters of these regions. Large-scale winds lead to displacements of air layers to large distances. In this case the energetic balance of the atmosphere given in Fig.2.3 depends on a latitude, so that the solar radiative flux is larger significantly at the equator compared with that in polar regions. This creates a force which causes the heat flux in the atmosphere from the equator. But these forces are different at various altitudes, and hence the horizontal force depends on an altitude. The character of long-scale winds as a result of the above reason is given in Fig.3.3. Along with these long-scale winds, mixing of atmospheric layers may be resulted from local air transport depending on meteorological conditions and landscape. In particular, a neighborhood of open water in the form of a lake or ocean with a mountain leads to transformation the horizontal winds in vertical ones that causes a mixing of atmospheric layers of different moisture and temperature. As a result of mixing of warm and cold air streams, the condensed water phase may be formed in atmospheric air. We below consider the example presented in Fig.3.4 with a mixing of air stream if near-surface air of the atmosphere penetrates to the altitude of 3 km and mixes there with air adiabatically. We take the moisture of the upper air layer in accordance with the model of standard atmosphere, i.e. its moisture is η = 65% [71–73]. We consider two versions for the moisture of near-surface air. In the first case we take it in accordance with the model of standard atmosphere η = 82% [71–73]. In the second case this air comes from the surface of a lake or ocean, and its moisture is η = 100%. As is seen, the above operation leads to formation of supersaturated air only in the second case. Then an excess of atmospheric water is transformed in liquid water microdrops, and the number density of free water molecules is equal to that at the

3.2 Character of water condensation in atmospheric air

|

45

Fig. 3.4: Mixing of atmospheric air volumes from altitudes of 0 km and 3 km in equal proportions. The average air moisture η = 65% at altitude of 3 km according to [71–73], as well as the moisture η = 82% for the case "a" for the lower layer. In the case "b" the lower air layer is taken over the surface of an ocean or lake.

saturation vapor pressure. This provides the equilibrium between bound and free water molecules in air. Another fact is that the number density of free water molecules exceeds that of bound molecules under such conditions of their formation. Below we use parameters of this example in subsequent estimations. In analyzing the growth of liquid microdrops as the condensed phase of water in the second example, note that the diffusion regime of this process (for example, [230]) is realized. This means that motion of associated particles has the diffusion character that takes place at drop sizes r ≫ 0.1μm. Fig.3.3 demonstrates the character of growth of atmospheric water drops as a result of mixing of atmospheric layers from different altitudes. As a result of mixing of air volumes from altitudes of 0 km and 3 km in equal proportions, a non-saturated water vapor is formed in atmospheric air in the case "a" of Fig.3.4, i.e. the condensed phase is absent there. If The Earth’s surface is a lake or ocean, its moisture is 100%, and its mixing with cold air leads to formation of a supersaturated water vapor, and its excess gives formation of the condensed phase. Note that such mixing takes place under the action of vertical winds. In addition, formula (3.1.1) for typical parameters of water microdrops in a cumulus cloud gives for the number density of bound water molecules in drops as N b ∼ 7 ⋅ 1016 cm−3 , i.e., this exceeds by several times that of Fig.3.4.

3.2 Character of water condensation in atmospheric air 3.2.1 Mechanisms of drop growth in air There are four mechanisms for formation and growth of microdrops which are given in Fig.3.5, and are important for water in the Earth’s atmosphere. The stronger process of

46 | 3 Water microdrops in atmospheric air

Fig. 3.5: Mechanisms of formation and growth of small water drops in atmospheric air [230, 234, 235].

Fig.3.5 includes the attachment of free water molecules to a drop, and due to this process and the inverse one of molecule evaporation from the drop surface an equilibrium establishes between free water molecules and microdrops. As a result of these processes, the number density of water molecules becomes equal to the saturated number density at a given temperature. Another mechanism involving water drops in atmospheric air is coagulation, that is joining of two microdrops as a result of their contact. The following process is coalescence or Ostwald ripening [236, 237] that results from interaction between microdrops and free water molecules. As a result of processes of attachment of molecules to the drop surface and molecule evaporation from the drop surface, large drops grow and small drops are evaporated. The total coalescence process leads to an increase of the average drop radius. The gravitation growth of water drops in the atmosphere is of importance for origin of rain and relates to large drop sizes. Then large microdrops fall faster than small ones. Overtaking small drops, large drops join with them. As a result, water microdrops are converted in millimeter drops of rain. Each of indicated mechanisms of growth of a condensed phase dominates under appropriate conditions. In particular, the classical theory of nucleation through formation of an embryo of a critical size is not realized in the real atmosphere, and water microdrops are nuclei of condensation in a region of a supersaturated water vapor. For this reason, the distribution of clouds in the atmosphere is non-uniform. On contrary, near the Earth’s surface in regions where winds are absent and nuclei of condensation are present, a mist is formed at nighttime, when the temperature changes sharply, as it takes place at autumn. In considering the growth of liquid drops in atmospheric air as a buffer gas, one can extract two inverse regime of this process. In the diffusion regime of drop growth or evaporation, the drop size exceeds significantly the mean free path λ of air molecules in atmospheric air r ≫ λ, (3.2.1) and the inverse criterion corresponds to the kinetic regime of drop growth. The kinetic and diffusion regimes of growth of water microdrops differ by the character of motion

3.2 Character of water condensation in atmospheric air |

47

for associated particles that leads to different expressions for the rate constants of the condensation process. It is clear that the first stage of the drop growth proceeds in the kinetic regime, whereas the diffusion regime of growth is realized when drops become large. In both cases the drop is taken as a macroscopic one, i.e. it consists of a large number n of molecules n ≫ 1. Let us consider the first stage of drop growth in the kinetic regime, if nuclei of condensation are present in atmospheric air. These nuclei of condensation are atmospheric ions, radicals and dust particles, so that water molecules form a bond as a result of collision with them. This drop grows after subsequent collisions with other water molecules. Assuming each collision of a water molecule with a growing drop leads to its attachment, i.e. the sticking probability is one, we have for the cross section of molecule attachment σ at in the kinetic regime σ at = πr2 = πr2W n2/3 ,

(3.2.2)

where r is the drop radius, r W is the Wigner-Seitz radius. Correspondingly, the rate constant of molecule attachment to a drop k at in the kinetic regime is equal k at = v T σ at = k o n2/3 , k o = √

8πT 2 r , m W

(3.2.3)

where v T is the average velocity of water molecules, and m is the mass of a water molecule. In particular, for water molecules at room temperature we have k o = 6.8∗ 10−11 cm3 /s. This allows one to analyze the first stage of nucleation if the condensation process proceeds due to nuclei of condensation. Denoting by N nc the number density of condensation nuclei to which water molecules attach, we have the balance equation for a current number n of water molecules in a growing drop dn = N nc k o n2/3 dt

(3.2.4)

A formal solution of this equation under the initial conditions n(0) = 1 gives for the growth time τ n of a drop up to a size n τn =

3n1/3 N nc k o

(3.2.5)

We now estimate the number density N nc of nuclei condensation which provides the total transformation of the excess of water molecules into drops. Denoting the number density of these excess molecules by N w , we have for the average number n of molecules in one drop when the all excess molecules become bound ones n=

Nw N nc

48 | 3 Water microdrops in atmospheric air From this we have for the number density of nuclei condensation N nc which provides an observed time τ ∼ 104 s of this transformation as 1/3

N nc =

3N w ko τ

(3.2.6)

Taking N w ∼ 1017 cm−3 , we have from this for the necessary number density of condensation nuclei N nc ∼ 1 ⋅ 109 cm−3 , and the drop radius of water drops in the end of this process is r ∼ 10nm. This shows that nuclei of condensation are not atmospheric ions which number density is lower than this estimation by several orders of magnitude. This rough estimation allows us to represent the character of condensation of atmospheric water vapor. Nuclei of condensation are necessary for growth of water drops. In reality, attachment of water molecules to nuclei of condensation is a chemical reaction between water molecules and radicals in the form of nitrogen oxides, ammonia and sulfur compounds [21, 27, 28, 30, 44]. Attachment of water molecules leads to formation of corresponding acids and their salts. When a size of drops becomes large, the character of the first stage of growth does not influence on a subsequent drop growth. At some sizes of growing drops, the diffusion regime of approach of attaching molecules to a drop is realized that results from the diffusion character of their motion. In this case an uniform space distribution of attaching molecules is violated near the drop because of the attachment process. Let us determine the effective rate constant of molecule attachment to a drop in the diffusion regime if one can ignore the evaporation process. Taking the number density of attaching molecule in a buffer gas to be N o far from the drop and N(R) at a distance R from the drop center, we have for the molecule flux toward the drop j = −D

dN , dR

where D is the diffusion coefficient for attaching molecules in a buffer gas. Because of absence of molecule absorption in a space, the total number of molecules attached to the drop is independent on R and is equal J at = 4πR2 j = −4πDR2

dN dR

Solving the equation J at (R) = const with the boundary condition for free molecules N(R) = 0, according to which each contact with the drop leads to molecule attachment, one can find for the number density of attaching molecules N(R) = N o (1 −

r ) R

(3.2.7)

This gives for the rate of molecule attachment J at = k at N o = 4πDrN o , k at = 4πDr

(3.2.8)

3.2 Character of water condensation in atmospheric air |

49

In particular, at room temperature of atmospheric air this formula gives for attachment of water molecules to a water drop in air as a buffer gas k at /r o = 2.8cm2 /s. For typical drop parameters in cumulus clouds (3.1.1) from this it follows k at = 2.3 ⋅ 10−3 cm3 /s. Above we consider conditions of attachment of water molecules to a water drop if the evaporation process is weak. Evidently, this takes place in a supersaturated vapor where the number density of attaching molecules N o is large, i.e. N o ≫ N sat , where N sat is the number density of molecules in the saturated vapor. Because these rates are equal, if N o = N sat , we have for the evaporation rate of molecules J ev from the drop surface J ev = k at N sat . Correspondingly, the total rate J of growth or evaporation of the drop is equal J = J at − J ev = k at (N o − N sat )

(3.2.9)

It should be noted that processes of drop growth and evaporation are accompanied by heat extraction or release. As a result, in a stable regime of the growth or evaporation process, the drop temperature differs from the temperature of surrounding air far from the drop. The power Q = ε b ⋅ J is extracted as a result of this process, where J is the molecule flux to the drop surface, and it is equal to the power P according to formula (3.1.9) that is transported as a result of the thermal conductivity process in a buffer gas. This gives for the difference of temperatures ∆T of the drop and air far from it that follows from the balance equation P = Q ∆T =

ε b k at (N o − N sat ) ε b k at = B(N o − N sat ), B = 4πκr 4πκr

(3.2.10)

In particular, under room temperature and atmospheric pressure of air where a water drop is located B = 6.1 ∗ 10−17 cm3 ⋅ K(r = 8μm).

3.2.2 Growth of water drops due to coagulation and coalescence Let us consider the coagulation process of growth of liquid drops in the diffusion regime. Assuming that each contact of two drops leads to drop association, we have for the rate constant of association k as for two drops of radii r1 and r2 according to the Smolukhowski formula [238] k as = 4π(D1 + D2 )(r1 + r2 ), where D1 and D2 are the diffusion coefficients for colliding microdrops in air. The rate constant of association averaged over aerosol sizes is [234] k as =

8T , 3η

(3.2.11)

where η is the viscosity coefficient for air. This formula gives at room temperature k as = 5.8 ⋅ 10−10 cm3 /s.

50 | 3 Water microdrops in atmospheric air

Fig. 3.6: Size distribution function of drops f(u) for the diffusion mechanism of coalescence in accordance with formula (3.2.1) [239, 242, 243].

We are based on equation for variation of the average drop size n for this growth mechanism [234], that has the form dn 1 1 = k as N b = k as N d n, dt 2 2

(3.2.12)

where N b = N d n is the total number density of molecules in drops, N d is the number density of drops. Assuming the total number density of bound molecules does not vary in time, we have from this equation n=

1 k as N b t, 2

(3.2.13)

Let us introduce the drop doubling time τ d during which a number of drop molecules is doubled, and the average drop radius becomes equal 21/3 r with respect to the initial average drop radius r. This time is given by τd =

2n 2 = k as N b k as N d

(3.2.14)

In particular, taking drop parameters as typical ones for cumulus clouds according to (3.1.1), one can obtain the doubling time for a drop size of a typical cumulus cloud τ d = 2/(k as N d ) = 40days. Since this time exceeds the lifetime of a cumulus cloud, one can conclude that the coagulation mechanism of drop growth is not of interest for evolution of drops in a cumulus cloud. We now consider the coalescence process using the automodel form of the size distribution function [239–242], where the reduced size u = n/n cr is the ratio of a number of drop molecules n to the critical number n cr . The classical theory of coalescence [239–243] relates to the diffusion mechanism of this process according to criterion (3.2.1), while if the criterion (3.2.1) is violated, the kinetic regime of coalescence [230, 244] is realized. In the diffusion regime of coalescence, the approach of an

3.2 Character of water condensation in atmospheric air

|

51

attaching molecule to the drop has the diffusion character, while in the kinetic regime of coalescence a straightforward trajectory of an attaching molecule corresponds to its approach to the drop. The diffusion mechanism of coalescence is realized if the criterion (3.2.1) holds true. Coalescence for small drop sizes subjects to the kinetic theory of coalescence. We give in Fig.3.6 the size distribution function f(u) of drops in the diffusion regime of the coalescence process which has the automodel character. As is seen, the size distribution function f(u) is nonzero only at u < 1.5. This means that large drops which may lead to divergence of the total flux of attaching molecules, are absent. Note that this distribution function is normalized to one. Next, though this distribution function relates to the diffusion regime of coalescence, its difference with the distribution function for the kinetic regime of coalescence is low [230]. In analyzing general principles of the coalescence process, we note its selfconsistent character. On the one hand, the total flux of molecules attached to large drops and evaporated from small drops, are identical. On the other hand, as a result of the total process, the average size of drops grows and their number density decreases. Simultaneously the number density of free molecules decreases slowly. Because a number of drop molecules is large n ≫ 1, one can use a small parameter n−1/3 and represent the total binding energy E n of bound molecules in the drop as an expansion over a small parameter in the form [245] E n = ε o n − An2/3

(3.2.15)

Here ε o is the average binding energy per one water molecule in a macroscopic drop, A is the specific surface energy, and these parameters are equal for water drops [75] ε o = 0.43eV, A = 0.21eV, where these parameters relate to the liquid aggregate state of drops at the temperature T = 293K. This gives the energy difference ∆ε(n) for water molecules in a drop consisting of n bound molecules compared to the limit of a large drop 2A dE n − εo = (3.2.16) ∆ε(n) = dn 3n1/3 As is seen, this change is determined by the water surface tension. We now consider an equilibrium between free and bound molecules for very large drops, if the criterion A ≪ T ⋅n1/3 holds true. Under equilibrium the number density of free molecules N w is equal to that of a saturated vapor N sat (T), i.e. N w = N sat (T). Taking for the saturated vapor pressure N sat (T) ∼ exp(−ε o /T), where ε o is the molecule binding energy in bulk water, we obtain for the critical drop size [229] N w = N sat (T) exp [

∆ε(n cr ) ], T

(3.2.17)

In analyzing the diffusion regime of coalescence, we take for simplicity that each contact of a free molecule with the drop surface leads to molecule attachment. Then one can obtain on the basis of the Smolukhowski formula [238] for the number of molecules attaching to a drop of a radius r per unit time J at = 4πDrN w ,

(3.2.18)

52 | 3 Water microdrops in atmospheric air

Fig. 3.7: Partial rate of molecule attachment to drops j(u) [239, 242].

where D is the diffusion coefficient for water molecules in air, N w is the number density of free water molecules in air. The total change of the number of bound molecules for a drop of a given size is J = J ev − J at = J o j(u), J o =

8πAr W DN w , u = (n/n cr )1/3 , n < n cr 3T

(3.2.19)

where J ev (n, T) is the evaporation rate, and this formula relate to drops whose size is below the critical one. Fig.3.7 contains the partial flux of molecules to the drop surface that is represented as j(u) = (1 − u)f(u), (3.2.20) and Fig.3.7 gives its size dependence. From the equilibrium it follows that numbers of evaporated and attached molecules are equal, i.e. 1

1.5

∫(1 − u)f(u)du = ∫ (u − 1)f(u)du 0

(3.2.21)

1

From data of Fig.3.7 we obtain for the total rate of molecule release for drops which size is below the critical one or for the rate of molecule attachment for drops of a larger size 1

1.5

J = J o ∫ j(u)du = −J o ∫ j(u)du = 0.085J o 0

(3.2.22)

1

From this we have equation for the average drop size n that is connected with the critical drop size by the relation n = 1.13n cr dn n J = = 0.056J o , dt n cr f(1)

(3.2.23)

This gives the doubling time of growth of an average drop size, if the average number of bound drop molecules varies from n to 2n [230, 244] τd =

36n 4.3n T = Jo Dr W N w A

(3.2.24)

3.2 Character of water condensation in atmospheric air |

53

Fig. 3.8: Thermal factor Φ(T) that characterizes the rate of growth of water drops which size exceeds the critical one under the equilibrium of these drops with the water vapor in atmospheric air. The growth process proceeds in the diffusion regime as a result of coalescence [230, 244].

It should be noted that processes of evaporation of molecules from the drop surface and attachment of molecules to drops are accompanied by heat processes. From this it follows that the drop temperature and that of surrounding air are different. This fact influences on the total rate of drop evaporation or molecule attachment to water drops. To characterize this process, we introduce the parameter Φ that is the ratio of the partial rate of drop growth without accounting for the heat effect J to that I which takes into account drop heating J , (3.2.25) Φ where the flux J is given by formulas (3.2.22) and (3.2.19). The temperature dependence for the factor Φ for growth of water drops in atmospheric air is given in Fig.3.8 with using parameters of water and its vapor [75], and the diffusion coefficient of water molecules in air at the temperature T = 300K and atmospheric pressure is given by D = 0.22cm 2 /s [214, 231]. Accounting for this effect leads to an increase of the doubling time of a drop size in Φ times compared to that according to formula (3.2.24). Fig.3.9 gives the temperature dependence for the doubling time of a drop size in a cumulus cloud that is equal according to formula (3.2.24) I=

τd =

4.3nΦ T Dr W N w A

(3.2.26)

The doubling time (3.2.26) may be used in the balance equation for the average drop size n, i.e. the average number of bound molecules of drops, that has the form dn ln 2 ⋅ n = dt τd It is convenient to rewrite this equation for the average drop radius r which is introduced according to formula (3.1.2). Then we obtain the growth equation in the form A r2 dr = K, K = 0.072Dr4W N w dt T

(3.2.27)

54 | 3 Water microdrops in atmospheric air

Fig. 3.9: Doubling time for an average number of drop molecules in a cumulus cloud with parameters (3.1.1) in atmospheric air if the drop growth is determined by the diffusion regime of coalescence [230, 244].

As it follows from this equation, growth of drops becomes slower for large drops. Next, since of the growth process for the coalescence mechanism is determined by processes of molecule attachment to the drop surface and evaporation of water molecules from it, its rate is proportional to the number density N w of free water molecules that, in turn, is equal to the number density of a saturated vapor at a given temperature. Hence, the constant K varies in a narrow range for the coalescence growth in atmospheric air. Taking the number density of free water molecules in accordance with Fig.3.3 as N w = 2.3 ⋅ 1017 cm−3 and K = 4.3 ⋅ 10−15 cm3 /s, we obtain under these conditions that the drop growth time up to size r = 8μm according to (3.1.1) proceeds through time t ≈ 11hours. In analyzing the coalescence process, we assume a low time of equilibrium establishment between water drops and free water molecules compared to a coalescence time. Let us determine a time of establishment of this equilibrium in the limit of large drops where the equilibrium number density of molecules near a drop N o is equal to their number density N sat in the saturated vapor. The number density of bound water molecules N b in drops is given by Nb = Nd (

r rW

3

) ,

(3.2.28)

where N d is the drop number density, r is the average drop radius. Taking N b ≫ N sat , we obtain from the equilibrium for the number density variations δN sat = −δN b , so that violation of the equilibrium does not influence on the drop parameters. Then vapor relaxation is described by equation dN w = 4πDr(N sat − N w ), dt

(3.2.29)

3.2 Character of water condensation in atmospheric air |

55

Fig. 3.10: Hierarchy of typical times in a system consisting of water drops and water vapor which are located in atmospheric air at the temperature T = 0o C and pressure 1 atm, if typical drop parameters correspond to a cumulus cloud (3.1.1). 1 - equilibrium between free water molecules and drops, 2 - gravitation growth of neutral drops, 3 - gravitation growth of charged drops, 4 - coalescence, 5 coagulation for neutral drops [230, 235].

and the solution of this equation gives for the number density of free molecules N w = N sat − (N sat − N o ) exp(−t/τ),

(3.2.30)

As is seen, a relaxation time or a time of establishment of the equilibrium between free molecules and drops is equal 1 τ eq = (3.2.31) 4πDrN d This time depends weakly on the air temperature and in the above consideration it is lower than a growth time for drops. In particular, for parameters of a typical cumulus cloud (3.1.1) in air at atmospheric pressure and the air temperature in the range 260 − 300K this time ranges 0.5 − 0.4s according to formula (3.2.31). Fig.3.10 contains typical times of drop growth, as well as a time of equilibrium establishment between drops and saturated water vapor in atmospheric air for parameters of a typical cumulus cloud (3.1.1) and the air temperature 0o C. Note that the number densities of free and bound water molecules are compared under these conditions. In particular, at the pressure 0.5atm that corresponds to an altitude of standard atmosphere approximately 6km, the specific water content in drops is 3.3g/kg, while the amount of free water molecules in the saturated vapor is equal 7.5g/kg. Let us estimate a typical time of growth or evaporation of a drop as a result of interaction with a vapor of free molecules. The balance equation for a drop size in the diffusion regime of these processes has the form [230] dn = 4πDr(N w − N sat ), dt

(3.2.32)

where n is a number of bound molecules in a drop, r is its radius, so that n = (r/r W )3 , r W = 0.19nm is the Wigner-Seits radius of a liquid water drop, D is the diffusion coefficient of water molecules in air, N w is a current number density of water molecules, and N sat is the number density at the saturated vapor pressure. This equation gives

56 | 3 Water microdrops in atmospheric air that at N w > N sat a drop grows, while at N w < N sat the drop evaporates. On the basis of formula (3.1.2) one can rewrite this equation as dr2 Nw = α( − 1) , α = 1.5r3W DN sat dt N sat

(3.2.33)

In particular, taking the diffusion coefficient of water molecules in atmospheric air D m = 0.22cm2 /s [214, 231] at the temperature T = 273K, where N sat = 1.6⋅1017 cm−3 , we have α = 3.8 ⋅ 10−4 cm2 /s. In particular, one can determine from this the evaporation time in air where the number density of free water molecules is small compared to that N sat at the saturated vapor pressure. Under these conditions, evolution of the number n of drop molecules in the course of drop vaporization is described by equation (3.2.32) that now has the form dn = −4πDrN sat (3.2.34) dt Taking the number n of molecules in a drop of a radius r as n = (r/r W )3 , where r W is the Wigner-Seits radius, and solving equation (3.2.34), one can obtain for the evaporation time τ ev 3r2 τ ev = (3.2.35) 8πDN sat r3W In particular, for a water drop with an average radius in cumulus clouds r = 8μm at room temperature, where N sat = 4.3 ⋅ 1017 cm−3 , D = 0.22cm2 /s [214, 231], formula (3.2.35) gives for the evaporation time of this drop in the case of absence of free water molecules in atmospheric air τ ev = 0.1s.

3.2.3 Gravitation mechanism of growth of water drops We now consider the gravitation mechanism of drop growth (see Fig.3.4) which is significant at large sizes of associated drops, i.e. in this case the diffusion growth regime is realized. We first estimate the rate constant of gravitational association of drops as k as ∼ ∆v ⋅ σ, where ∆v is the difference of velocities of two drops, σ = π(r1 + r2 )2 is the cross section of contact between two drops of radii r1 and r2 . As early, we assume that each contact of two liquid drops leads to their association. As a result, taking the fall velocity (2.2.2), one can estimate the rate constant of drop association as [230] k as ∼

ρgr4 , η

(3.2.36)

where r is the average drop radius. Denoting by r the average drop radius, we have the associative rate constant k as as [230] k as =

2ρgr4 , η

(3.2.37)

3.2 Character of water condensation in atmospheric air

|

57

It is convenient to represent formula (3.2.37) in the form r 4 k as = k o ( ) , a where k o = 1.1 ⋅ 10−5 cm3 /s and a = 10μm. In the course of growth of water liquid drops in atmospheric air, the coalescence mechanism is replaced by the gravitation mechanism as a drop size increases. Note that the rate of the coalescence growth mechanism decreases with an increasing drop size, whereas the rate of the gravitation mechanism increases under these conditions. Hence, the size distribution function of drops has the maximum where the rates for these mechanisms are compared. Under conditions of Fig.3.3, it takes place at a drop radius r ∼ 50μm, where the doubling time is τ d ∼ 1min. Note that formula (3.2.36) requires the fulfilment of the criterion wg ≫ √

T , m

(3.2.38)

where w g is the falling velocity, m is the drop mass, T is the gas temperature expressed in energetic units. This criterion gives r≫(

η2 T ) ρ3 g2

1/7

(3.2.39)

In particular, for water aerosols located in atmospheric air at room temperature this criterion has the form r ≫ 2μm, and this consideration relates to micron-size particles. Let us write the balance equation for the average number of drop molecules n. If two molecules consisting of n1 and n2 molecules are joined as a result of gravitation association, the number of drop molecules is n = n1 + n2 , whereas before the joining the average number of drop molecules is (n1 + n2 )/2, and the average change of the number of drop molecules is n/2. Taking N b /n as the average number density of drops with N b as the number density of bound molecules, we obtain the balance equation for the average number n of drop molecules dn 1 = k as N b dt 2

(3.2.40)

Correspondingly, equation for the average drop radius r takes the form ρgN b r3W dr = b, b = 3η r2 dt

(3.2.41)

For example given in Fig.3.4 we have b = 0.56cm−1 s−1 , whereas for parameters (3.1.1) of an average cumulus cloud this parameter is equal b = 1.7cm−1 s−1 . In order to account for the coalescence and gravitation mechanisms of drop growth simultaneously, it is necessary to join the balance equations for a drop size

58 | 3 Water microdrops in atmospheric air

Fig. 3.11: Dependence of the doubling time which is given by formula (3.2.44) on the microdrop radius with using of the following growth parameters : 1 - an example of Fig.3.4, 2 -a cumulus cloud (3.1.1).

due to coalescence (3.2.27) and gravitation fall (3.2.41). As a result, one can obtain dt =

r2 dr K + br4

(3.2.42)

From this one can determine the total time of drop growth t gr that is equal t gr

∞

∞

0

0

r2 dr x2 dx −1/4 −3/4 =∫ = K b = 1.1K −1/4 b−3/4 ∫ K + br4 1 + x4

(3.2.43)

We obtain for the example of Fig.3.4 that the parameters of this formula are K = 4.3 ⋅ 10−3 μm3 /s and b = 5.6 ⋅ 10−5 μm−1 s−1 . This gives t gr = 19min. Using the parameters (3.1.1) for the cumulus cloud, we have b = 1.7 ⋅ 10−4 μm−1 m−1 . In addition, the parameter K is proportional to the number density of free water molecules and is equal to that at the saturated vapor pressure. Therefore, one can take the parameter K to be identical in both cases, i.e., K = 4.3 ⋅ 10−3 μm3 /s. As a result, under parameters (3.1.1) of a cumulus cloud we obtain for the growth time of water drops t gr = 8min. Note that these results relate to neutral water drops. One can introduce the time of doubling of water microdrops τ d during which a number of bound water molecules in a drop is doubled. Then the average drop radius increases from r to r ⋅ 21/3 , or representing it in the symmetric form, we have for the radius change ∆r ∆r = 21/6 r − 2−1/6 r = 0.23r As a result of we have for the doubling time in the case under consideration τd =

0.23r3 K + br4

(3.2.44)

Fig.3.11 contains the doubling time of a microdrop according to formula (3.2.44) for the growth cases under consideration. The maximum of the doubling time corresponds to

3.3 Water circulation between the Earth and atmosphere

| 59

the radius microdrop r m = (3k)1/4 b−1/4 and its value τ max is equal τ max =

0.23r3m 0.13 = 1/4 3/4 4K K b

(3.2.45)

Comparing formulas (3.2.43) and (3.2.45), one can obtain t gr = 8.4 τ max

(3.2.46)

This means that the main contribution to growth of a microdrop follows from large sizes compared to those corresponded to τ max . In particular, for the growth parameters of Fig.3.4 this typical size is ≈ 80nm.

3.3 Water circulation between the Earth and atmosphere 3.3.1 Character of water circulation through atmosphere Water penetrates into the atmosphere as a result of evaporation from open reservoirs in the form of free water molecules. Then water is condensed in the atmosphere, and water drops fall on the Earth’s surface in the form of rain. As a result, circulation of water through the atmosphere [246] takes place. Fig.3.12 contains the scheme of basic processes in the course of water circulation in the atmosphere, and below we analyze the sequence of these processes. Above we convince that the model of standard atmosphere cannot describe water condensation in it, and this process is determined by a nonuniform atmosphere structure. Note that condensation processes proceed in cold atmospheric regions, evidently, with the temperature below 0o C where water particles may be formed in a solid state (snow or ice). Then the phase transition due to heat resulting from attachment of additional water molecules may prevent a growing water particle from evaporation. One more peculiarity of formation and growth of water drops and particles is connected with a low rate of condensation for the classical theory of nucleation [247–249] in a uniform system where the condensation process proceeds through formation of an embryo of the critical size. The weakness of this mechanism follows from the large binding energy of the forming embryo compared with a thermal energy [250, 251]. Nuclei of condensation in the form of ions and radicals initiate the nucleation process in a real atmosphere. Subsequently growing water drops are nuclei of condensation, so that water molecules from neighboring regions penetrate in the region of condensation and attach to the drops. This determines a nonuniform character of condensation in the atmosphere. The alternative with respect to this scheme of water circulation in the atmosphere is transport of water molecules as a result of the atmospheric diffusion. Indeed, due to the diffusion process during residence in the atmosphere, water molecules move on a distance h c that is given by h c = √2D L t r , (3.3.1)

60 | 3 Water microdrops in atmospheric air

Fig. 3.12: Sequence of processes in the course of water circulation in the Earth’s atmosphere. 1- evaporation of water molecules from the Earth’s surface, 2 - convective diffusion of water molecules to upper layers of the atmosphere, 3,4 - formation of water microdrops from free water molecules in cold atmospheric layers and subsequent transformation of water microdrops into drops of rain, 5 - falling of water drops to the Earth’s surface in the form of rain.

where the diffusion coefficient in the atmosphere is D L = 4 ⋅ 104 cm2 /s according to formula (2.4.8), and t r = 9 days is the residence time of a water molecule in the atmosphere. We obtain from this h c ≈ 3km that coincides with the scale (2.2.1) for the altitude distribution function of water molecules in the atmosphere. Hence, this mechanism for transport of water molecules in the atmosphere deserves attention. In this case water molecules partake in the convection motion together with air molecules, and their altitude distribution function for water and air molecules must be identical. Because of the difference of these distributions for air and water molecules which are described by formulas (2.1.2) and (2.2.1), we will keep below the scheme of Fig.3.12 for water circulation in the atmosphere. Nevertheless, transport of water molecules as a result of convection gives a certain contribution to the behavior of atmospheric water. Keeping to the scheme of Fig.3.12 for water transport in the atmosphere in the course of its circulation, we have the relation between the amount of condensed atmospheric water and that in the form free water molecules. Indeed, because evaporated water in this scheme passes through stages of free water molecules and condensed water, the relation between the number density of free atmospheric molecules N m and bound molecules N b in water microdrops follows from the balance equation Nm Nb =χ , tr t gr

(3.3.2)

3.3 Water circulation between the Earth and atmosphere

| 61

where τ c is the residence time of atmospheric water molecules in the condensed phase, χ is the probability to locate for water molecules in regions of a supersaturated water vapor. Basing on the example of Fig.3.4, where N m /N b = 16, and assuming water microdrops to be neutral, we have for the growth time of water microdrops t gr = 19min on the basis of formula (3.2.43). From this one can find for the probability of regions where water condensation is possible as χ ∼ 0.01. We cannot accept or disprove this value, but the analysis of its deriving causes some contradictions. First, a typical size r of growing water microdrops which corresponds to the maximum of the size distribution function of microdrops is r ≈ 1μm. In reality we observe large sizes of water drops, as, for example, for cumulus clouds according to (3.1.1). Second, a typical growth time of drops or the observed lifetime is measured in hours and exceeds remarkably that followed from formula (3.2.43). This means that the model under consideration for gravitation growth of neutral microdrops is not realized. It is necessary to take into account a charge of growing water drops and to combine the processes of drop growth with those of atmosphere electricity.

3.3.2 Electric properties of water drops of cumulus cloud The basis of atmospheric electricity is formation of charge microdrops as a result of collision of water microdrops located in different aggregate states [252]. Subsequently, charges are separated as a result of faster falling of more heavy microdrops, and the falling velocity is approximately 1cm/s for a drop radius of r = 10μm and is proportional to r2 . As is seen, separation of atmospheric charge is possible at not low sizes of falling drops. In the analysis of the character of growth processes for charged drops, we will be guided by those taken place in cumulus clouds. We are based on a general scheme of electrical processes in the atmosphere [30]. Then drop charging results from collisions of water particles in different aggregate states [252–254], mostly in collision of graupels (snow and water particles) and ice particles. Falling of charged particles leads to charge separation because of different velocities of this process. As a result, clouds contain water particles of a certain type with larger velocities of falling. Evidently, these water particles are charged negatively, because finally they transfer the negative charge to the Earth. We now estimate the charge of falling drops assuming that they are found in the liquid aggregate state. Charge separation results from fast falling of negatively charged water particles which form finally negatively charged clouds, and their charge will be increased when new portions of water drops go to this cloud. In the end, electric breakdown from this cloud to the Earth transfers the cloud electric charge to the Earth’s surface. Atmospheric ions formed by atmosphere ionization under the action of cosmic rays, provide discharging the Earth. Keeping this scheme, we obtain an altitude range where the electric current to the Earth is created by falling of charged water particles. Assume the Earth’s electric current to be distributed uniformly over its surface, i.e. the electric current density i is

62 | 3 Water microdrops in atmospheric air equal to its average value, that is i = 3.3 ⋅ 10−16 A/cm2 = 2 ⋅ 103 e/cm2 ∗ s. For definiteness, we assume falling water particles to be liquid and their charging takes place at the altitude of h = 6km. Because the electric atmosphere phenomenon is a secondary phenomenon of water circulation, this allows us to determine the specific charge that is carried by water liquid drops. Because the average flux of water molecules at the altitude 6 km is equal 5 ⋅ 1015 cm−2 s−1 , the average charge transferred by one molecule equals to 4 ⋅ 10−13 e, i.e. the drop of a radius about 3μm carries one electron charge on average. At lower altitudes neutral water molecules adjoin to charged falling drops. Taking the association rate due to the gravitation mechanism for joining of water drops according to formula (3.2.37), we have for a cumulus cloud with an average drop radius r = 8μm for the association rate k as = 4.3 ⋅ 10−6 cm3 /s. If drops are neutral, the lifetime of a cumulus cloud with respect to doubling of a drop size is τ ≈ 8min, that is less compared to real lifetimes of a cumulus cloud which is measured in hours. From this it follows that water drops of a cumulus cloud are slightly charged that prevents the drops from association partially. Assuming drop radii and charges to be identical in a cumulus cloud, one can find for the rate constant of drop association [230] k as =

Z 2 e2 2r4 ρg exp (− ) η 2rT

(3.3.3)

instead of formula (3.2.37) for association of neutral drops. In this case electrostatic repulsion of charged drops prevents them from the contact. One can represent formula (3.2.44) for liquid water drops in the form [30] k as = r4 exp(−0.06Z 2 /r),

(3.3.4)

where the rate constant of drop association k as is measured in 10−9 cm3 /s, and an average drop radius r is expressed in μm. Note that a typical drop charge Z ≈ 27 creates a weak field on its surface that under given conditions is equal E = Ze/r2 ≈ 6V/cm and corresponds to the electric potential with respect to surrounding air U = Ze/r ≈ 5mV. Let us determine the size doubling time of water drops in a cumulus cloud due to the gravitation association mechanism, where the rate constant of drop association is given by formula (3.3.3). Assuming that the total number density of bound water molecules N b does not vary in time, as well as the specific drop charge Z, we obtain for the doubling time τ d on the basis of formula (3.2.44) for the rate constant of drop association n2

r1

1

2

2dn Z2 3η 1 exp = d τd = ∫ ( ) = τ o F[Z 2 /(2rT)], ∫ k as N b ρgN b r3W r 2rT n r

(3.3.5)

where n1 , n2 are the initial and final average numbers of bound molecules in a microdrop, r1 , r2 are the average radii, and the parameters of formula (3.3.3) are τo =

3η ρgN b r3W r

, F(x) = x [exp(x) − exp(x/21/3 )]

(3.3.6)

3.3 Water circulation between the Earth and atmosphere

| 63

Fig. 3.13: Function F(x) in accordance with formula (3.3.6).

Fig. 3.14: Average microdrop charge Z for a typical cumulus cloud depending on a doubling time for a number of water molecules in a drop [244].

On the basis of typical parameters of water drops (3.1.1) in a cumulus cloud, we have for a doubling time τ o = 24min, and the function F(x) is given in Fig.3.13. If we take a typical lifetime of a cumulus cloud between 3 and 12 hours, one can obtain on the basis of formula (3.3.6) for the corresponding drop charge [30, 235] Z = 27 ± 2

(3.3.7)

According to data of Fig.3.14, the drop charge depends weakly on a typical time of association of water microdrops. The repulsion electrostatic potential for drops of the average charge Z = 27 and radius r = 8μm is equal Z 2 e2 /(2r o ) = 760K, that exceeds a thermal molecule energy. In particular, at room temperature Z 2 e2 /(2r o T) = 2.6 and exp[Z 2 e2 /(2rT)] = 13. Charges of Fig.3.14 confirm a weak dependence of the drop charge Z on a growth time, and an increase of the cloud lifetime by the order of magnitude leads to an increase of the drop charge by 15%. This fact characterizes the reliability of determination of the drop charge and confirms the existence of the drop stage of water circulation in the atmosphere. In addition, Ze2 /r = 56K, i.e., the drop charge does not influence of bound excess electrons of the drop.

64 | 3 Water microdrops in atmospheric air Let us return to the concept where the electricity transport in the atmosphere is a secondary phenomenon with respect to water transport, and the ratio of the transferred charge to the mass of transferred water is 1.3⋅10−10 C/g [30, 235]. If water transport in the atmospheric water circulation is accompanied a channel by drop charging in a cumulus cloud, we obtain the ratio of a transported drop charge Ze to the water mass m in drops for this channel Ze = 2 ⋅ 10−9 C/g, m

(3.3.8)

and this ratio is larger than that for the atmosphere on average. The average ratio of the electric current (I = 1700A) of Earth charging under the action of lightning to the water mass per unit time (dM/dt = 1.5 ⋅ 1013 g/s) is equal 1.3 ⋅ 10−10 C/g [30, 235] that is the ratio of a transferred charge through the atmosphere to transported water as a result of water circulation in the atmosphere. Let us determine a typical thickness of a charged layer of a cumulus cloud. A typical electric field strength is determined by a charge per unit square of a cloud. On the other hand, formulas (3.1.1) and (3.3.7) give the charge number density for a cloud, and the combination of these quantities allows one to determine the thickness of the cloud lower edge where a charge is located. Basing on the observational value of the cloud voltage U = 20−100MV [255], we take for estimations the average voltage U = 60MV, that at an altitude of the lower cloud edge L = 3km gives for the average electric field strength E = 200V/cm. From this we have for the charge density per unit cloud square e E σ= ≈ 1 ⋅ 108 ≈ 2 ⋅ 10−11 C/cm2 (3.3.9) 4π cm2 The charge density according to formulas (3.1.1) and (3.3.9) is equal ZeN d = 4 ⋅ 10−15 C/cm3 . The ratio of the surface charge density to its volume density gives for the thickness of a charged cloud layer l ∼ 100m [230] that is less than a cloud size.

3.3.3 Processes of transformation of water drops in rain Above we consider the behavior of an individual microdrop, where a drop charge does not influence on surrounding air which can contain a water vapor also. In this case thermal processes involving a drop do not act on surrounding air far from the drop. We now consider the behavior of microdrops in a cloud consisting of microdrops, if a stream of humid air penetrates inside a cloud. As a result, a part of stream water attaches to microdrops, while other part remains inside a cloud. We assume that the stream capacity does not give a contribution to the thermal balance of this mixture, whereas processes of drop evaporation and vapor condensation are determined the heat balance of air with a cloud. Our task is to find which part of water molecules of the stream attaches to microdrops. Evidently, water microdrops of a cloud are in equilibrium with a water vapor inside it, and then the number density of free water molecules is that of the saturated

3.3 Water circulation between the Earth and atmosphere

| 65

Fig. 3.15: Temperature dependence for a moisture part ξ(T) which transfers from the humid air stream into microdrops according to formula (3.3.12).

vapor in accordance with data of Table 1. Subsequent attachment of water molecules to microdrops is accompanied by energy extraction, that shifts the equilibrium between free and bound water molecules. This is the basis of these evaluations. On the other hand, this process leads to a nonuniform distribution of atmospheric water in a space and determines the cloud origin in a nonuniform atmosphere. In this evaluation we take the initial air temperature to be T and the number density of free water molecules to be corresponded to a saturated vapor in accordance with Table 1. After addition a vapor portion ∆A to air, a part of this vapor will attach to drops, whereas another part ∆A s will be located in the vapor which amount increases because of the temperature growth due to vapor condensation. Let us consider equation the thermal balance in this case. On the one hand, according to the Clasius-Clayperon equation [256, 257] we have for variation of the saturated vapor ε o ∆T ∆A s = A s 2 (3.3.10) T On the other hand, air heating results from the condensation process according to equation ∆T = (∆A − ∆A s )C p , (3.3.11) where C p is the air heat capacity. From this we have for the part of water flux which is transformed into drops after mixing of a moist air flux with air containing water microdrops ε o A s (T)C p 1 ∆A − ∆A s = , y= (3.3.12) ξ= ∆A 1+y T2 The dependence ξ(T) is given in Fig.3.15. Thus, we obtain that water microdrops and clouds which consist of them may be formed under unequilibrium conditions, if fluxes of humid air are mixed with

66 | 3 Water microdrops in atmospheric air clouds consisting of water microdrops. This leads to an increase of cloud water in a forming mixture. Therefore, though the average number density of atmospheric water molecules is less than that for a saturated water pressure, in some atmospheric regions a large amount of water is gathered. As a result, a water disperse phase is formed that is of importance for electric and radiative atmospheric processes. Note that on the first stage of growth, a size of water microdrops is small (below 1μm) and they are kept in vortices of convection motion of atmospheric air. In addition, condensation in a supersaturated water vapor proceeds more effectively at low temperatures. From the above analysis it follows that the number density of atmospheric water molecules is less compared with that at the saturated vapor pressure both for the standard atmosphere model [24] and according to averaging of measurements (Fig.2.8). Hence, formation of the condensed water phase in the atmosphere may be result from fluctuations. This takes place if a flux of warm saturated air penetrates in a cold region. We below consider an example where vapor from a region near the Earth’s surface is transported to the altitude of 3 km, where it is mixed with air which parameters correspond to standard consitions. We assume in this example that the numbers of air molecules from two altitudes are identical and the water moisture η = 48% at the altitude 3 km corresponds to Fig.2.8. If we take from this Figure the moisture near the Earth to be η = 82%, a non-saturated vapor is formed (see Fig.3.4a). Another version is considered in Fig.3.4b, where displaced air is taken from open water (a lake or ocean) and its moisture is η = 100%. As a result of mixing, a supersaturated vapor is formed and an excess of a water vapor is transformed into the condensed phase. Along with equation of drop growth, we use the equation of drop motion under the action of its weight dh = wg , (3.3.13) dt where h is the way which a microdrop passes during its falling, w g is the falling velocity given by formula (2.2.2). One can combine this equation with the growth equation (3.2.12), and assuming the total number density of bound molecules N b to be independent of time, one can obtain dh = const = 1.4 ⋅ 106 , dr

(3.3.14)

where const is determined by initial conditions of this equation in accordance with parameters (3.1.1) of a cumulus cloud. In particular, if a fall height inside the cumulus thunder is h = 2km, the final drop radius is r = 1.4mm, and a typical time of drop fall is τ f ≈ 25min. Note that we leave aside the size distribution of water microdrops [265, 266] and for estimation assume all microdrop at a given time to have an identical size. According to observations [258], a radius of rain drops near the Earth’s surface is found in a range 0.1 − 9mm. From this it follows that a typical thickness of a cloud is h = 0.1 − 10km for typical parameters (3.1.1) of microdrops in a cumulus cloud. Next, formula (3.2.36) is violated at large drop sizes, and the falling velocities for drops of

3.3 Water circulation between the Earth and atmosphere

| 67

Fig. 3.16: Character of transformation of cumulus cloud in rain with lightning [181, 264].

radii 0.5mm and 5mm are equal correspondingly 2m/s and 9m/s, whereas according to formula (3.2.36) these velocities are 2m/s and 200m/s respectively. Indeed, at large Reynolds numbers the Stokes formula (3.2.36) is violated and the spherical drop shape is lost [258], i.e., the Stokes formula which is the basis of formula (2.2.2) may be used as an estimate. In addition, if drops of a cumulus cloud with parameters (3.1.1) are transformed in water at the Earth’s surface, the thickness of a water layer at the Earth surface ∆ is proportional to to the thickness L of a cumulus cloud, and ∆ = 2mm/km L

(3.3.15)

The rate of water precipitation for moderate rain ranges from 2.5mm/hour up to 10mm/hour [258, 259]. From this it follows that the air moisture contributes to the process where wet air is mixed with a cumulus cloud. Hence, convection of wet air and its interaction of jets of wet air with drops of a cumulus cloud is of importance for rain origin [260–262]. Note that the thunderstorm weather is accompanied by a high air moisture exceeded 7g/kg [263] that corresponds to the saturated vapor pressure at the temperature 8o C (see Table 1). As a result of this analysis, we obtain that the growth of drops inside a cumulus cloud becomes slower because water drops are charged. But this charge does not act on excess bound electrons of drops. Since the energy of repulsion U o of colliding drops grows as ln U ∼ r5 , the growth of identically charged drops in a cumulus cloud is stoped at an increasing drop size. Hence, for a subsequent growth of drops, it is necessary to remove their charge or to realize this growth by adding water molecules. Evidently, the second mechanism is realized. Fig.3.16 demonstrates this process.

68 | 3 Water microdrops in atmospheric air

Fig. 3.17: Evolution of the charged drop size in the course of drop growth as a result of their joining if the specific charge in accordance with formula (3.3.16) is equal 1.3 ⋅ 10−10 C/g (1) and at the beginning the growth parameters are taken for a typical cumulus cloud according to formulas (3.1.1) and (3.3.8) (2).

3.3.4 Kinetics of atmospheric electricity We above consider some aspects of evolution of neutral and charged water microdrops in the atmosphere. In the case of development of charged microdrops which are connected with atmospheric electricity, we are guided by some processes in cumulus clouds. We now analyze the kinetics of charged water microdrops taking into account the specific charge (3.3.8) in cumulus clouds, as well as the average specific charge for atmospheric electricity which is equal to 1.3 ⋅ 10−10 C/g. We thus have the character of drop growth in a cumulus cloud with formation of lightning and rain that is represented in Fig.3.16. We start from a cumulus cloud with drop parameters according to formula (3.1.1). In addition, drops have a charge of the same sign, and the average charge in units of electron charged is given by formula (3.3.7) Z = 27. Electrostatic repulsion of colliding drops prevent them from the contact, i.e. drop growth is ceased practically at these parameters. Attaching of water molecules to drops in accordance with Fig.3.17 leads to charge dilute and drop growth. But this dilution is restricted, because the specific charge of falling drops cannot be below the specific atmospheric current of the Earth charging, that is dq I = = 1.3 ⋅ 10−10 C/g (3.3.16) dM dM/dt Here q is the charge transferred by water microdrops through the atmosphere, I = dq/dt is the electric current passed through the atmosphere, dM/dt is the mass of atmospheric water which is transferred through the atmosphere per unit time. This formula shows that the atmospheric electric current is the secondary phenomenon of water transport through the atmosphere. When the drop charge is diluted by water in microdrops, they grow until their charge locks the subsequent growth of drops. But at a new level, the electric potential of the drop compels some negative ions to leave a microdrop. Appearance of negative ions in air may cause electric breakdown of atmospheric air that leads to development of lightning. Simultaneously, drops become

3.3 Water circulation between the Earth and atmosphere

| 69

free from the charge and are grown as a result of the gravitation joining, so that finally rain is formed. Thus, within the framework of this rough scheme, rain accompanies lightning in the course of development of the drop growth process. Below we consider the kinetics of drop growth at the stage before lightning and rain formation. In consideration the growth of charged microdrops, we assume their charge to be conserved in drops in the course of their growth. Replacing the rate of drop association of neutral drops according to formula (3.2.37) by that (3.3.3) for identically charged drops, we have the growth equation in the form dt =

r2 dr K+

br4 exp(−Z 2 e2 /(2rT)

(3.3.17)

instead of equation (3.2.42). Because the drop charge Z is proportional to its mass, i.e., to r3 , where r is the drop radius, one can represent equation (3.3.17) as dt =

r2 dr , K + br4 exp(−(r/r o )5

(3.3.18)

where r o = 8μm for parameters of a cumulus cloud, and r o = 23μm for parameters of an example of Fig.3.4b and the specific drop charge according to (3.3.16). Note that in this consideration we assume identical sizes of microdrops, i.e. the size distribution function has the form of delta-function, rather than its real form (for example, [265, 266]). Hence, the results of this analysis have the qualitative character. Fig.3.16 contains the dependence of the doubling time on the drop radius in accordance with equation (3.3.18), where the specific charge of growing drops is conserved in the course of drop growth and corresponds to formula (3.3.8) for a cumulus cloud, as well as to formula (3.3.16) for an example of Fig.3.4b. On a certain stage of the growth process, repulsion interaction of joining charged drops prevents them from association. Correspondingly, the doubling time for drops increases sharply with an increasing time. In particular, under indicated conditions the doubling time is τ d = 2hours at drop sizes r = 30μm in the first case and r = 10μm for parameters of a cumulus cloud, and their charges in units of the electron charge equals to Z = 95 and Z = 42 in these cases. The energy of electrostatic repulsion U defined as U=

Ze2 , 2r

is equal 0.22eV and 0.13eV in these cases. Under these conditions, the electric potential acted on an electron charge located at the drop surface is less. In particular, in the first case we have Ze2 /r ≈ 5meV. In conclusion of the analysis of atmospheric electricity, we describe schematically the stage before air electric breakdown where molecular ions are formed in a space of location of charged microdrops. We construct this physical picture based on observed data which is represented particular in Fig.3.16 where the growth process proceeds under the action of moist air fluxes. In this case the flux of warm moist air rises up and leads to growth of charged microdrops. When the cloud becomes ripen and charged

70 | 3 Water microdrops in atmospheric air microdrops grow up to a large size, they lost a charge, so that negative molecular ions are formed in surrounding air. This leads simultaneously to electric breakdown of air in the form of lightning and transformation of microdrops in rain. One can consider charged water microdrops as water drops with diluted negative ions O−2 inside them. These ions cannot leave the drop under the action of the field of its charge because it is relatively small. On contrary, the electrostatic interaction becomes significant during collisions of charged microdrops and can give a contribution to the binding energy of an excessive electron. In addition, in the course of approach of charged drops, the charge distribution inside them varies, i.e. bound negative ions move inside the drop. One can expect, this may lead to a release of a negative ion, that facilitates the electric breakdown.

4 Thermodynamics of thermal atmospheric emission Abstract: Radiation of a weakly nonuniform plain gaseous layer is analyzed in the case of thermodynamic equilibrium between radiation and gas molecules within the framework of the line-by-line model. The basic parameters of photon-molecule interaction and radiative transport are the layer optical thickness, the opaque factor of the layer, the radiative temperature, as well as the effective sublayer responsible for layer emission, and all these parameters are taken at a certain frequency. These parameters allow one to determine the radiative fluxes in both sides of an infinite layer whose parameters depend only on a distance from its sides. The equilibrium is considered between emission and absorption processes, according to which the rates of these radiative processes subject to the principle of detailed balance or the Kirchhoff law. The Mie theory is used for determination the absorption cross section of a liquid water microdrop in the infrared spectrum range on the basis of optical properties of bulk liquid water in this spectral range. Applying these principles to the Earth’s atmosphere, we formulated the realistic model of atmospheric emission where water and carbon dioxide molecules, as well as water microdrops, are radiators.

4.1 Radiation of flat layer 4.1.1 Radiation in uniform gas In considering radiation of the atmosphere, we analyze first propagation of radiation in a uniform gas system. The basic parameter for description of this and other processes is the absorption coefficient K ω for photons of a given frequency ω, so that 1/K ω is the mean free path for these photon in the gas. Note that the atmosphere as a radiating system may be represented as a flat layer located above the Earth’s surface, because the atmosphere thickness is several orders of magnitude below the Earth’s radius. A flat gaseous layer which properties depend on an altitude over the Earth’s surface only, is convenient to be described by the optical thickness u ω of the layer, which is defined as u ω = ∫ K ω dh,

(4.1.1)

where the integral is taken over the all thickness of this layer. We define i ω as a flux of photons which propagate inside the gas layer, and j ω is the flux of photons on the boundary of the gaseous layer which is directed outside it. The boundary of a layer is taken such that outside it absorption of photons is absent. In the case of a uniform layer in the transverse direction, as we model the Earth’s atmosphere, one can connect the photon fluxes if we have a sharp boundary. In this case 1 iω (4.1.2) j ω = ∫ i ω cos θd cos θ = , 2 4 https://doi.org/10.1515/9783110628753-004

72 | 4 Thermodynamics of thermal atmospheric emission where the factor 1/2 takes into account that only photons directed to this boundary determine the radiative flux, and the factor cos θ takes into account that the total photon flux is directed perpendicular to the Earth’s surface due to the symmetry. One can define also the energy radiative flux J ω which we call below as a radiative flux (4.1.3) J ω = ωj ω , If a photon flux passes through a gas, its intensity varies in the course of propagation according to equation (the Beer-Lambert law) [267, 268] dJ ω = −k ω J ω dz

(4.1.4)

This equation describes the flux of a low intensity if it does not change the properties of an absorbed media. From this it follows that the probability of survival of a photon P ω of a given frequency ω, if it passes a certain distance L from a point of the surface, is (4.1.5) P ω = exp (− ∫ k ω dl) , where dl is the element of the photon trajectory from the surface. Basing on the case under consideration, where the absorption coefficient of a gas k ω depends only on a distance from the surface, we have for this probability L

P ω = exp (− ∫

k ω dz uω ) = exp (− ) cos θ cos θ)

(4.1.6)

0

Here the z axis is directed perpendicular to the surface. Let us determine the radiative flux under thermodynamic equilibrium for the gaseous system and radiation field. Thermodynamic equilibrium inside the gaseous system means that the number densities of molecules in two states o and i are connected by the Boltzmann formula Ni = No

gi ∆E exp (− ) , go T

(4.1.7)

where N o , N i are the number density of molecules in these states, g o , g i are the statistical weights of these states, ∆E is the transition energy between states o and i, and T is the temperature. Using the thermodynamic equilibrium for the radiation field, we have that the probability to excite n photons for a given photon state according to the Boltzmann formula is proportional to exp(−ω/T), where ω is the photon energy. Hence the average number of photons in a given state n ω is equal ∑ n exp(−nω/T) nω =

n

∑ exp(−nω/T)

= [exp (

−1 ω ) − 1] T

(4.1.8)

n

Let us determine the equilibrium radiative flux on the basis of this formula. The equilibrium photon flux inside and equilibrium gaseous system is equal according to the

4.1 Radiation of flat layer | 73

Planck formula [269, 270] iω = 2

d∫k (2π)3

nω c =

ω2 ω2 nω = 2 2 π c π 2 c2 [exp ( ω T ) − 1]

(4.1.9)

Here the factor 2 accounts for two photon polarizations, k is the photon wave vector, c is the light speed, the integral is taken in one direction, λ = 2π/k = 2πc/ω is the wavelength for photons, and we use the dispersion equation for photons as electromagnetic waves ω = kc. From this one can find the energy flux of outgoing radiation for an equilibrium gaseous system which optical thickness at a given frequency is large (o)

J ω = ω

−1 ω ω3 iω = − 1] [exp ( ) 4 T 4π2 c2

(4.1.10)

This is the radiative flux of a blackbody.

4.1.2 Emission from flat gaseous layer Emission of the atmosphere is created by the troposphere, a low atmospheric layer of thickness approximately 10 km. Because the troposphere thickness is small compared to horizontal lengths of the Earth’s surface where its parameters vary remarkably, one can describe the atmospheric emission as that of a flat gaseous layer which parameters do not vary in the horizontal direction. We below consider atmospheric emission in the IR spectral range, being guided by the model of standard atmosphere and taking into account that atmospheric parameters depend on the altitude also. Correspondingly, we reduce the problem of atmospheric emission to that for a plain gaseous layer. In addition, absorbed and emitted photons are found in the thermodynamic equilibrium with atmospheric air. We also consider the case of optically thick atmosphere, so that photons formed in higher atmospheric layers and directed to the Earth’s surface are absorbed mostly on the way and do not reach the surface. It should be noted also that an excited molecule formed as a result of photon absorption, is quenched later in collisions with air molecules. Due to thermodynamic equilibrium, photons are formed in emission of excited molecules which in turn result from collisions of nonexcited molecules. We first consider the case of a constant temperature of air located in a layer. Then the equilibrium flux i ω propagates inside the layer, and the mean free path λ ω of photons with a given frequency is small compared to the layer thickness L, i.e. λ ω ∼ 1/k ω ≪ L, where k ω is the the absorption coefficient for atmospheric air. Because photons are found in thermodynamic equilibrium with air, the equilibrium flux of photons is equal according to formula (4.1.9) ω2 (4.1.11) iω = π 2 c2 [exp ( ω T ) − 1] We are based on the isotropic distribution of emitted photons, so that the probability of photon propagation in a direction that forms an angle θ with the perpendicular to

74 | 4 Thermodynamics of thermal atmospheric emission

Fig. 4.1: Geometry for an outgoing photon in the course of photon emission from a plain gas layer where the gas temperature depends on a distance from the boundary only. Here t is the trajectory of a propagated photon, p is the plain boundary of an emitted gas, 1 is the origin, 2 is a point of its intersection with a line of photon motion, 3 is the projection of this point onto the boundary plane direction [271].

the layer is d cos θ. Correspondingly, the projection of the photon flux dJ 󸀠ω for a given range of propagation angles is equal dj󸀠ω =

i ω cos θd cos θ 1

= i ω cos θd cos θ/2

(4.1.12)

∫ d cos θ

−1

Note that because the total flux at a given frequency is directed perpendicularly to the layer, di ω is the flux projection onto the direction which is perpendicular to the layer boundary. In this analysis, we use the geometry of the radiation flux given in Fig.4.1. Let us extract a think layer of a thickness dz and find the number of photons dj ω for a given frequency range per unit time and unit square. This flux propagates in an angle range d cos θ, is protected on the surface direction and is emitted by a layer of a thickness dz. This quantity is given by the expression dj ω = di ω

k ω dz = i ω k ω dzd cos θ/2, cos θ

(4.1.13)

where dz/ cos θ is the way passed through a test layer, and the factor k ω dz/ cos θ is the probability for this photon to be absorbed inside a test layer. We account for the photon flux i ω inside the layer is isotropic. Including absorption of photons before the boundary of a gaseous system according to formula (4.1.6), one can obtain from this for the energy flux J ω of radiation which intersects the boundary J ω = ω ∫ dj ω = ω

u

1

0

0

ωi ω iω uω g(u ω ) )= ∫ du ω ∫ d cos θ exp (− 2 cosθ 4

4.1 Radiation of flat layer | 75

One can define the opaque factor as 1

g(u ω ) = 2 ∫ cos θd cos θ [1 − exp (− 0

uω )] , cosθ

(4.1.14)

The opaque factor g(u ω ) is the part of the radiative flux compared to that of a blackbody at a given frequency and temperature. Here u ω is the optical thickness of the total layer at a given frequency, u ω = ∫ k ω dz. Let us take the case where the total optical thickness u ω of the layer at a given frequency is large u ω ≫ 1 and the temperature is independent of the altitude, i.e. the flux i ω does not depend on the altitude also. In this limiting case one can obtain from formula (4.1.14) the expression (4.1.10) for the radiative flux J ω which is given by formula (4.1.10) J ω = ω

−1 ω ω3 iω = [exp ( ) − 1] 2 2 4 T 4π c

One can obtain this connection in a simpler manner, but formula (4.1.14) for the outgoing photon flux is useful if the above assumptions are violated. We now consider the case of the local thermodynamic equilibrium if this equilibrium takes place for each point of a gaseous system, but its temperature varies in a space. Being guided by the atmosphere, we focus on the case where the gaseous system is a plain gaseous layer, and the gaseous temperature depends on the altitude only, i.e. on the coordinate which directed perpendicular to the layer. In addition, we consider this layer to be weakly nonuniform. Our task to determine the radiative flux outside this layer. Basing on the nature of this process, one can determine this flux as an equilibrium flux (4.1.10) at the radiative temperature T ω that is the temperature of an effective layer. This is a layer located at a distance h ω for which the optical thickness is of the order of one u ω (h ω ) ∼ 1 (4.1.15) This relation uses that the main contribution to the radiative flux relates to regions from which emitting photons reach the boundary. We below obtain this relation more accurately basing on a small parameter which takes into consideration a weak nonuniformity of a flat layer. This small parameter contains the temperature gradient near the boundary, and we represent it below. Taking into account that generation of photons at each point corresponds to the gaseous temperature at this point, one can represent formulas (4.1.10) and (4.1.14) in the form 1

u

0

0

−1 ω3 uω ω Jω = − 1] F(u cos θd(cos θ) du exp ), F(u ) = , ) ) (− [exp ( ∫ ∫ ω ω ω cos θ Tω 2π2 c2

(4.1.16) and we assume the function F(u ω ) to be weakly dependent of the temperature.

76 | 4 Thermodynamics of thermal atmospheric emission Expanding this value over a small parameter, we determine the photon energy flux which goes outside the layer under the condition of a large total optical thickness u at a given frequency, i.e. u ≫ 1. In evaluation this integral, one can take into account that the main contribution to this integral is determined by optical thicknesses u ∼ 1. Expanding the function F(u ω ) over a small parameter, we have F(u ω ) = F(u o ) + (u ω − u o )F 󸀠 (u o ) +

1 (u ω − u o )2 F 󸀠󸀠 (u o ) 2

(4.1.17)

We have the following values of the integrals in the course of expansion over a small parameter ∞

1

∫ cos θd cos θ ∫ du ω exp (− 0 1

0 ∞

∫ cos θd cos θ ∫ u ω du ω exp (− 0 1

1 uω )= , cos θ 3

0 ∞

∫ cos θd cos θ ∫ u2ω du ω exp (− 0

1 uω )= , cos θ 2

1 uω )= cos θ 4

0

Let us take the parameter u o in the expansion (4.1.17) such, that the second term of expansion is zero. This gives for the optical thickness of the effective layer which temperature coincides with the radiative one [272, 273] u o (T ω ) = 2/3,

(4.1.18)

and leads to the following expression for the radiation energy flux J ω = J ω ef (1 − α), α =

5F 󸀠󸀠 (u o ) , 18F(u o )

(4.1.19)

where α is a small parameter of expansion. In particular, in the limit ω ≫ T this small parameter is given by α=

ω dT 2 5 5 ω dT 2 ⋅ ( 2 ⋅ ) = ( 2 ) 18 T du 18 T k ω dh

(4.1.20)

This small parameter is realized under the condition kω ≫ (

ω dT ⋅ ) 2T 2 dh

(4.1.21)

because dT/du = dT/k ω dh. Taking the frequency at the maximum of the photon energy flux in accordance with the Wien law ω ≈ 3T [274] and the temperature gradient for the standard atmosphere model dT/dh = −6.5K/km, one can obtain the criterion α ≪ 1 in the following form for the absorption coefficient k ω and the layer optical thickness u ω k ω ≫ 3 ⋅ 10−7 cm−1 , u ω ≫ 0.2 (4.1.22)

4.2 Character of thermal atmospheric emission

| 77

4.2 Character of thermal atmospheric emission 4.2.1 Emission of atmosphere as flat gaseous layer We now apply the above results to the Earth’s atmosphere within the framework of the model of standard atmosphere. The radiative flux from the atmosphere toward the Earth J↓ and outside J↑ , as it follows from the energetic balance of the atmosphere, are equal on average J↓ = 327W/m2 , J↑ = 200W/m2 (4.2.1) From this one can obtain the temperatures T↓ and T↑ for the atmosphere layers which are responsible for atmospheric emission towards the Earth and outside it on the basis of the Stephan-Boltzmann law J↓ = σT↓4 , J↑ = σT↑4 ,

(4.2.2)

From this one can obtain the above temperatures [30, 275] T↓ = 276K, T↑ = 244K

(4.2.3)

The following step in this operation is the determination of the altitudes h↓ and h↑ for effective layers which are responsible for this radiation. These altitudes follow from equations T↓ = T[h↓ ] and T↑ = T([h↑ ]) and radiation towards the Earth, where T(h) is the temperature at an altitude h. The connection between the corresponding altitude and temperature has the form T↓ = T E − h↓

dT dT , T↑ = T E − h↑ , dh dh

(4.2.4)

where within the framework the standard atmosphere model the average temperature gradient is dT/dh = 6.5K/km, and T E = 288K is the temperature of the Earth’s surface. Equations (4.2.3) and (4.2.4) lead to the following values of altitudes which are responsible for the atmosphere emission toward the Earth and outside h↓ = 1.9km, h↑ = 6.8km

(4.2.5)

The method of determination of the effective altitude is presented in Fig.4.2. The above results relate to a high optical thickness of the atmosphere. We below generalize them to the case where the criterion u ω ≫ 1 may be violated, but the absorption coefficient k ω depends as early on an altitude only. According to the geometry of photon propagation toward the boundary, given in Fig.4.1, the total radiative flux j ω of the layer with an identical temperature T is given by [271] 1

uω

(o)

j ω = J ω g(u ω ), g(u ω ) = 2 ∫ d cos θ ∫ dx exp (− 0 1

= 2 ∫ cos θd cos θ [1 − exp (− 0

x ) cos θ

0

uω )] , cos θ

(4.2.6)

78 | 4 Thermodynamics of thermal atmospheric emission

Fig. 4.2: Method of determination of the altitude of the atmospheric layer which is responsible for emission of infrared atmospheric radiation toward the Earth and outside it [144].

Fig. 4.3: Comparison of dependencies on the optical thickness of a gaseous flat layer for the function g(u) defined by formulas (4.1.14), (4.2.6), and its approximation, given by formula (4.2.7). These functions are the ratio of the radiative flux from the gaseous layer of a given optical thickness to that for the layer of infinite optical thickness [276].

where the opaque factor g(u) is determined in accordance with formula (4.1.14) and is represented in Fig.4.3b, where u is the total optical thickness for a given frequency. Function g(u) tends to one in the limit u → ∞, and at u ∼ 1 it is approximated by the following dependence [276] G(u) = 1 − exp(−1.6u),

(4.2.7)

Both functions (4.2.6) and (4.2.7) are represented in Fig.4.3. Note that in the limit of low u formula (4.2.6) gives g(u) = 2u, whereas from formula (4.2.7) it follows g(u) = 1.6u. Hence, formula (4.2.7) is valid up to not small values u at u > 0.1. Indeed, the accurate value at u = 0.1 according to formula (4.2.6) is g(0.1) = 0.167, while formula (4.2.7) gives g(0.1) = 0.148, and from the asymptotic formula g(u) = 2u it follows g(0.1) = 0.2. One can introduce the ratio of the above functions

4.2 Character of thermal atmospheric emission

| 79

Fig. 4.4: Function F(u) = g(u)/G(u) defined by formula (4.2.8).

F(u) =

g(u) 1 − exp(−1.6u)

(4.2.8)

This function is given in Fig.4.4 and tends to one, if the approximation (4.2.7) coincides with (4.2.6). As is seen, this coincidence is fulfilled more or less at not low optical thicknesses of the layer. The function g(u) characterizes the probability to emit a photon of a given frequency, where this probability is unity for optically thick layer. In addition, the factor 1 − g(u) is the probability for a photon to reach the opposite boundary of the layer if it emits at another boundary. Hence the probability for the photon to survive in the course of propagation through the layer is equal 1 − g(u). One can determine on the basis of this formula the part of the radiative flux for an optically thick layer that is created below u ω = 2/3. We have g(2/3) = 0.66,

(4.2.9)

i.e. approximately 2/3 part of the radiative flux is formed at altitudes below the effective altitude (4.1.21). One can introduce the optical thickness u ef of an effective layer which temperature determines the radiative temperature of the radiative flux that intersects the boundary of a radiating layer. This value is given by formula (4.1.21) for an optically thick layer and allows one to determine the effective radiative temperature for a weakly nonuniform gas. Repeating derivation of formula (4.1.21), one can obtain the optical thickness U of an effective layer which characterizes the radiative temperature

80 | 4 Thermodynamics of thermal atmospheric emission

Fig. 4.5: The effective optical thickness of the layer u ef (u), which is responsible for emission of a weakly nonuniform gaseous layer and is determined by formula (4.2.10), and the approximation of this function U(u) according to formula formula (4.2.11).

1

u

U(ω) = 2 ∫ d cos θ ∫ xdx exp (− 0

x ), cos θ

(4.2.10)

0

where u(ω) is the total optical thickness of the layer at a given frequency. It is convenient to approximate this function by the dependence [276] U(u) =

u 2 exp(−u) + 1.5u

(4.2.11)

In the limit of optically thick gas u → ∞ formula (4.2.8) gives u ef = 2/3 in accordance with formula (4.1.21), whereas in another limiting case u ≪ 1 the effective layeris located in the middle of the optical thickness. Functions (4.2.10) and (4.2.11) are represented in Fig.4.5. Thus, the local thermodynamic equilibrium allows us to simplify the analysis of emission of a plain gaseous layer. In this simplified manner we introduce the radiative temperature as that of the layer which is responsible for emission of the total layer. In the case of the optically thick layer, the position of an effective layer which parameters relate to the yield radiative flux, that is given by formula (4.1.18). In a general case, the optical thickness u ω of the layer which parameters describe yield radiation, is determined by the relation (4.2.12) u ω = U(u t ), where u ω is the optical thickness of the effective layer, i.e., it is taken from the boundary up to the effective layer, and u t (ω) is the total optical thickness of the layer at a given frequency. 4.2.2 Model of frequency-independent absorption In considering the atmospheric emission, we move from a simple and rough model to more complex and realistic one. In this way we start from the simplest model

4.2 Character of thermal atmospheric emission

| 81

of frequency-uniform absorption [30, 275] where the absorption coefficient of the atmosphere is independent of the frequency [275]. This model was used for obtaining formula (4.2.2) because the temperature in formula (4.1.11) assumes to be identical for various frequencies. Along with determination the effective radiative temperatures (4.2.3) and altitudes (4.2.5) within the framework of this assumption, one can find the altitude dependence for the absorption coefficient k ω . Indeed, formula (4.1.21) leads to relations for altitudes which are responsible for atmospheric emission h=h↓

∞

h=0

h=h↑

2 2 ∫ du = , ∫ du = 3 3

(4.2.13)

These equations may be used for determination of the optical thickness of the standard atmosphere depending on the altitude. Evidently, the absorption coefficient k ω is a monotonic altitude function. Taking it to be independent of the radiation frequency ω, it is convenient to represent it in the simplest form as h du ω kω ≡ = A exp (− ) , (4.2.14) dh λ Substituting the dependence (4.2.14) into formula (4.2.13) with using formula (4.2.5), one can find parameters of the dependence (4.2.14) [30, 275] A = 0.41km−1 , λ = 5.5km, u t = Aλ = 2.3,

(4.2.15)

where u t is the total atmospheric optical thickness of the atmosphere for this model. Thus, the model under consideration [30, 275] proves a large optical thickness for the atmosphere in the infrared spectral range. This conclusion follows also from formula (4.2.3) according to which regions of emission toward the Earth and outside are separated. These parameters are connected with the radiative temperature which does not depend on frequency within the framework of this model and is given by first formula (4.2.4) ut dT Tω = TE − λ ⋅ ln ( (4.2.16) ) dh u t − 2/3 This formula is in accordance with the above atmospheric parameters (4.2.3) and (4.2.5) for the model of frequency-uniform absorption. Note that according to the energetic balance of the Earth and its atmosphere, we have that the average radiative flux from the Earth’s surface is J E = 386W/m2 , and the flux J p = 20W/m2 that passes through the atmosphere. Hence, the average probability to survive for a photon emitted by the Earth’s surface is approximately P = 1/20. This quantity is given by formula 1

P = ∫ cos θ exp (− 0

ut ) d cos θ, cos θ

(4.2.17)

82 | 4 Thermodynamics of thermal atmospheric emission which accounts for uniform emission of photons in all directions, and the projection of the photon flux emitted at angle θ to the surface normal is cos θ. Because the average probability P of photon surviving is 1/20, from this formula it follows for the layer optical thickness u t = 1.8. Comparison of this optical thickness of the atmosphere with that of formula (4.2.15) allows one to estimate the accuracy of parameters (4.2.15) within the framework of the model of frequency-uniform absorption. From this one can estimate its accuracy as 20%. Let us determine a change of the radiative flux ∆J↓ toward the Earth as a result of change of the concentration of atmospheric radiators by 10%. We have ∆J↓ = 4σT↓3 ∆T↓ =

4J↓ ∆T↓ T↓

(4.2.18)

On the other hand, a temperature change ∆T↓ for a layer which is located at an altitude h↓ and is responsible for atmospheric emission, is equal 󵄨󵄨 dT 󵄨󵄨 󵄨󵄨 dT 󵄨󵄨 ∆u 󵄨󵄨 󵄨󵄨 󵄨 󵄨 , ∆T↓ = 󵄨󵄨󵄨 󵄨󵄨 ∆h↓ = 󵄨󵄨󵄨 󵄨h 󵄨󵄨 dh 󵄨󵄨 󵄨󵄨 dh 󵄨󵄨󵄨 ↓ u

(4.2.19)

where the relative change of the atmospheric optical thickness is equal now ∆u/u = 0.1. From this we have in a given case ∆J↓ = 5.7W/m2 .

4.2.3 Atmospheric thermal radiation from thermodynamic standpoint The above results allow us to glance on the character of atmospheric radiation from general positions. Leaving aside the spectroscopy of atmospheric thermal radiation which determines parameters of this emission, we analyze the general character of this process from the thermodynamic standpoint. It should be noted that processes of emission and absorption are detailed inverse ones, and therefore their rates are connected by the Kirchhoff law [277]. We below consider this equilibrium if photons are emitted and absorbed as a result of transitions between o and i states of atomic particles which constitute an atomic system. Let us consider the equilibrium of emission and absorption processes in a gaseous system resulted from transitions between indicated states of atomic particles of this gas according to the scheme M o + ω 󴀘󴀯 M i , (4.2.20) where M is an atomic particle, and subscript indicates its state. From this the equality follows for rates of these processes which are taken as numbers of acts of these processes per unit volume and unit time. This gives Ni aω = No σω iω , τr

(4.2.21)

where N i is the number density of atomic particles in a state i which is connected by formula (4.1.7) with the number density N o of atomic particles in a lower transition

4.2 Character of thermal atmospheric emission

| 83

state, a ω is the photon distribution function, τ r is the radiative lifetime of an upper state i with respect to the lower state o, the partial photon flux i ω is given by formula (4.1.11). The photon distribution function is normalized by the relation ∫ a ω dω = 1

(4.2.22)

One can obtain from this on the basis of the Boltzmann law (4.1.7) the following expression for the absorption cross section σ ω in accordance with [278–280] σω =

ω Ni aω π2 c2 = Aa ω [1 − exp (− )] , No iω τr T ω2

(4.2.23)

where A is the first Einstein coefficient, g o and g i are the statistical weights for the lower and upper states of the transition. Note that spontaneous and stimulated radiation are included in this formula. Note that the balance equation (4.2.21) accounts for connection between spontaneous emission of excited atomic particles and absorption processes. Including in this scheme the stimulated radiation, we have the following relation between the absorption coefficient k ω due to a certain component and the absorption cross section σω kω = No σω (4.2.24) The connection (4.2.23) between parameters of the absorption and emission processes allows one to use parameters of the absorption process in the analysis of the emission one. One can add to this that the statistical description of radiative processes corresponds to the assumption that we deal with a noncoherent radiation. In this case an average is made over phases of individual photons, and the intensity from several radiative transitions is a sum of intensities of individual ones. Basing on this, we summarize powers of radiation of individual radiative transitions into the total one, that is used in formula (4.2.20). If radiation at a given frequency is determined by different components, the total power of radiation at a given frequency is the sum of those for each component. The above formulas will be applied to radiative vibration-rotation transitions between states of molecules which are located in atmospheric air. Being guided by radiative processes in the Earth’s atmosphere, we are based on the collision nature of the broadening of spectral lines. Then the distribution function of photons for a given transition has the form aω =

νj , 2π[(ω − ω j )2 + (ν j /2)2 ]

(4.2.25)

where ω j is the frequency of the spectral line center, ν j is the width of the spectral line. This allows one to determine the maximum of the absorption cross section σ max =

ω j π2 c2 2A j [1 − exp (− )] , 2 πν T ωj j

(4.2.26)

84 | 4 Thermodynamics of thermal atmospheric emission Under thermodynamic equilibrium, the radiative temperature T ω coincides with the temperature of a layer that is responsible for emission of a gaseous system. Let us consider the case where the altitude h ω which is responsible for emission at a given frequency is relatively small h ω ≪ Λ, λ, (4.2.27) where Λ and λ are the scales of the altitude variation for radiating components in accordance with formulas (2.1.2) and (2.2.1). On the basis of formula (4.1.18) for the altitude h ω in the case of an optically dense gaseous system hω =

2 3k ω

(4.2.28)

In this case the radiative temperature of the optically thick layer in accordance with formula (2.1.8) is given by 2 󵄨󵄨󵄨󵄨 dT 󵄨󵄨󵄨󵄨 Tω = TE − (4.2.29) 󵄨 󵄨, 3k ω 󵄨󵄨󵄨 dh 󵄨󵄨󵄨 where T E = 288K is the Earth’s temperature for the model of standard atmosphere. Correspondingly, the radiative flux in accordance with formula (4.1.10) Jω =

−1 ω ω3 − 1] [exp ( ) Tω 4π2 c2

(4.2.30)

Evidently, in a general case of an appropriate optical thickness of the atmospheric layer we have instead of formula (4.2.29) for the radiative temperature at a given frequency within the framework of the standard atmosphere model Tω = TE −

U[u t (ω)] 󵄨󵄨󵄨󵄨 dT 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨, 󵄨󵄨 dh 󵄨󵄨󵄨 kω

(4.2.31)

where U[u t (ω)] is the optical thickness of an effective layer is the solution of equations (4.2.11) and (4.2.12), u t (ω) is the total optical thickness of the atmospheric gap at a given frequency. Of course, this formula is transformed into formula (4.2.30) in the limit u t (ω) ≫ 1. Thus, considering the atmosphere as a plain gaseous layer, under local thermodynamic equilibrium in this gas one can reduce its emission at each frequency to that for the layer of a constant temperature. This temperature, or the radiative temperature of an emitted radiative flux, is the temperature of an effective layer. In this way, we reduce emission of the atmospheric gas to that of a gas with the constant temperature. Because of the local thermodynamic equilibrium, the radiative temperature at each frequency is the temperature of the effective layer in accordance with formula (4.2.29) at a large optical thickness of the gas layer. Thus, reducing emission of a weakly nonunform layer to that of a uniform layer, we deal with the concept of an effective layer or the radiative temperature at each frequency.

4.2 Character of thermal atmospheric emission

| 85

4.2.4 Distribution over optical thicknesses In considering emission of the atmosphere which temperature varies with an altitude, we reduce this problem to emission of a gaseous layer of a constant temperature and find the effective layer altitude that gives the main contribution to the radiative flux. Then the radiative temperature is the temperature of this effective layer which optical thickness is given by formula (4.2.12). We use the impact mechanism of the broadening of spectral lines for atmospheric molecules located in air, so that the photon distribution function a ω of a given spectral line has the Lorentz form and is described by formula (4.2.25) νj aω = ν 2π [(ω − ω j )2 + ( 2J )2 ] Let us introduce the distribution function f(u) over optical thicknesses of a layer under consideration. The value f(u)du is the probability that the atmospheric optical thickness ranges from u up to u+du, and we use a general principle of the statistical physics that this probability is proportional to a frequency interval dω, where this occurs. This relation may be obtained from the normalization condition f(u)du = a ω dω

(4.2.32)

Let us restrict by a frequency range not close to the center of a given spectral line where according to formula (3.1.15) aω ∼

νj , 2π [(ω − ω j )2 ]

(4.2.33)

On the other hand, in this frequency range of the action of neighboring spectral lines the photon distribution function is not strong, and therefore at wings of spectral lines uω ∼ aω ∼

1 (ω − ω j )2

(4.2.34)

This leads to the following distribution function over optical thicknesses f(u) ∼

1 √u

(4.2.35)

From this, summarizing the distribution function f(u) of those of individual spectral lines, one can find that in the basic frequency range the total distribution function has the form (4.2.35) under the above conditions. Formula (4.2.35) reflects the fact, that the optical thickness u of the atmosphere at a given frequency is proportional to the absorption cross section which, in turn, is proportional to the frequency distribution function a ω of photons. This distribution function is identical for various spectral lines at their wings, and the distribution function f(u) is identical due to various spectral lines. Let us introduce the maximum optical thickness u max that corresponds to an averaged optical density at centers of spectral lines, and in a similar way we introduce

86 | 4 Thermodynamics of thermal atmospheric emission the minimal optical thickness u min of the atmospheric layer, so that u max ≫ u min . Then from the normalization condition one can obtain the following formula for the distribution function over optical thicknesses f(u)du =

du 2√u max √u − u min

(4.2.36)

We now determine the mean radiative temperature on the basis of formulas (4.2.4) and (4.2.35). We have for the optically dense atmosphere 2 dT T ω = T E − h↓ 3 dh

√u max

∫ 0

dz u min + u o + z2

(4.2.37)

It was used the assumption that the absorption coefficient depends weakly on the altitude that takes place at low altitudes, and the condition of the optically dense atmosphere u max ≫ u o , where u o = κ ω h↓ = 2/3. In this limit we obtain Tω = TE −

π∆T 3√u min + 2/3

, ∆T = h↓

dT dh

(4.2.38)

As it follows from formula (4.2.38), the maximum amplitude of oscillations for the average frequency temperature is π∆T/√6, and this value decreases with an increasing optical thickness.

4.3 Water microdrops in atmospheric emission 4.3.1 Model of average atmospheric absorption The above analysis exhibits that a dense atmosphere emission as a whole is determined by the total absorption coefficient and a radiative temperature at a given frequency. This means that the radiative flux toward the Earth is described by the radiative temperature T ω that is the temperature of an effective layer, and for an optically dense atmosphere the radiative flux is given by formula (4.2.30). In considering the total atmospheric emission, we start from the simplest model of frequency-uniform absorption that is represented in section 4.2.2. Within the framework of this model, the radiative temperature is independent of the frequency. In spite of the roughness of this model, it allows one to estimate the values of radiative parameters. In particular, emission of the atmosphere is formed at altitude of the order of 3km and the radiative temperature for radiation toward the Earth is close to the temperature of the Earth’s surface. On the next stage, we take into account the character of this emission. Namely, the basic molecular components are water and carbon dioxide molecules which are responsible for atmospheric emission. In the range of frequencies inside the absorption bands of these molecules, the total absorption coefficient K ω of atmospheric air

4.3 Water microdrops in atmospheric emission

| 87

in the case of noncoherent radiation takes the form Kω = kω + κω ,

(4.3.1)

where k ω is the atmospheric absorption coefficient due to carbon dioxide molecules, κ ω is that owing to water molecules. From this it follows that parameters of emission of a gaseous system are determined by the total absorption coefficient K ω rather than (o) those due to a certain component. Hence, one can describe the radiative flux J ω at a given frequency ω by a common radiative temperature T ω . The relation (4.3.1) allows us to separate the contribution of individual components to the radiative flux from the gaseous system. Indeed, from relation (4.3.1) it follows for the probability ξ ω (CO2 ) that an emitting photon from a gaseous system at a given frequency results from emission of CO2 molecules as ξ ω (CO2 ) =

kω Kω

(4.3.2)

and the radiative flux due to emission of CO2 molecules J ω (CO2 ) is given by (o)

J ω (CO2 ) = ξ ω (CO2 )J ω ,

(4.3.3)

(o)

where J ω is given by formula (4.1.10). Correspondingly, the total radiative flux due to CO2 molecules is determined by formula J ω (CO2 ) = ∫ ξ ω (CO2 )

−1 ω ω3 dω − 1] [exp ( ) Tω 4π2 c2

(4.3.4)

Note that formula (4.3.2) includes parameters of an effective layer, rather than those at the Earth’s surface. One can analyze on this example also the reversibility of radiative processes which are determined by radiative transitions of atomic particles for thermodynamically equilibrium gaseous systems. In this case each component is simultaneously a radiator and absorber. In particular, if radiative processes proceed in accordance with scheme (4.2.20), atomic particles in a lower transition state o are absorbers and these atomic particles in an upper state i are radiators. From this it follows also that an increase of the number density of a certain component leads to an increase of the radiative flux due to this component and to a decrease of the radiative flux due to other components. Along with water and carbon dioxide molecules, water microdrops which constitute the clouds are an important optically active atmospheric components. The role of these components is proved by the Twomey concept [281] according to which some molecules may occupy a restricted spectrum range due to their discrete nature, and other spectrum range must be covered by particles as radiators with an infinite absorption band. Fig.4.6 demonstrates this concept for atmospheric air where the other spectrum part is occupied by radiation of water microdrops. From this Figure it follows that at low frequencies the radiative flux is determined by water and carbon dioxide

88 | 4 Thermodynamics of thermal atmospheric emission

Fig. 4.6: Spectrum of emission of CO2 molecules (up) and H2 O atmospheric molecules (down) [281, 282]. A dash-dotted line corresponds to thermal atmospheric emission.

molecules, whereas at large frequencies it is created mostly as a result of particle radiation. At low altitudes dust or a fume from volcano eruptions fulfill the role of such radiating particles. But the main contribution to thermal emission of the atmosphere due to particles gives water microparticles, mostly water microdrops. From this it follows the model of atmospheric emission according to which atmospheric radiators at low frequencies are water and carbon dioxide molecules, whereas at larger frequencies water microdrops fulfill the role of atmospheric radiators, and the boundary of two ranges we take as ω b = (750 − 800)cm−1 . In addition, we assume clouds consisting of water microdrops to be located at a certain altitude, and the optical thickness due to them is large. In addition, the optical thickness of the atmosphere at low frequencies is large, and hence the atmospheric radiative temperature at these frequencies is equal to the temperature of the Earth’s surface. Analogously to this, the optical thickness of the atmosphere in the absorption range of water microdrops is large. We assume it to be identical in all the absorption spectrum of water microdrops that corresponds to the assumption that the lower clouds are located in a narrow range of altitudes. Later we improve this model including radiation of water microdrops at low frequencies, and emission of water and carbon dioxide molecules in weak absorption bands, as well as radiation of trace gases in the atmosphere. Within the framework of this model, the behavior of radiative temperature is represented in Fig.4.7. This model is characterized by two radiative temperature, namely, by the Earth temperature which is T E = 288K according to the standard atmosphere model, and by the radiative temperature T cl of clouds. The latter is absent in the model of standard atmosphere as well water microdrops are not present in this model. Below

4.3 Water microdrops in atmospheric emission

| 89

Fig. 4.7: Radiative temperature within the framework of the model where frequency ranges of emission of molecules and water microdrops in the atmosphere are separated.

we evaluate this value on the basis of the Earth’s energetic balance according to Fig.2.3 and Fig.2.4. Under these conditions we have for the total radiative flux toward the Earth J↓ = J