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Applications in Engineering: not just for climate science
9. September 2014
Von
Dan Hughes
- Veröffentlicht auf Amazon.com

Format: Kindle Edition
Verifizierter Kauf

The book is not to just for the atmospheric sciences, the material is useful for engineering applications, also. Note I have been verified by Amazon as a buyer of the book.

The book presents the synthesis and grand exposition of decades of detailed work by both the first author and the joint work of both authors. The focus of the authors' work is the atmospheric sciences, and both homogeneous and heterogeneous nucleation and growth are covered for both liquid and solid along with applications to modeling clouds and associated atmospheric problems. Additionally, the nitty-gritty details of the physical world that torture beautiful theories are considered, as they must always be. We seldom get to play in the purity of the theoretical.

The coverage is exhaustive in physical phenomena and processes and mathematical modeling details. The literature review and citations are also exhaustive including coverage from the earliest works to recent years. The engineering literature, of course, is not a focus area for the work. It might be a useful exercise to apply the models and methods developed in the book to the experimental data that are available in the engineering literature; especially that concerned with condensation of the liquid phase from its vapor or gaseous mixtures ( humid air, for example ).

As is the general case whenever two or more phases or fluids are involved, the nomenclature naturally leads to many sub-and super-scripts, and combinations of subs- and supers-, and many of these require combinations of mnemonic symbols. That being said, the nomenclature in the book is a model of clarity. The development of the mathematics is another model of clarity. Intermediate steps that are frequently omitted are instead included and detailed descriptions of the justifications of the equation development are spelled out step-by-step. The derivation of the van der Waals form of the equation of state, and its variations, developed in Chapter 3 are examples of the level of details throughout the text that I have read. And this chapter reviews the thermodynamics of pure components, phase transitions, and aqueous solutions.

The depth of development of concepts is generally not available in journal publications. I think the book will serve as a reference text and as a supplement to engineering texts and journal publications. Selected parts can be used to form university courses; the entire book likely cannot be covered in a single academic year.

I have not read the entire book, and I'm certain that digesting the contents of the entire book will be a major undertaking. I have taken an over-all scan of the major features and I have read chapters and sections that discuss those aspects that relate to engineering applications with which I am familiar. A few of these cross-cutting areas include: equation of state for water ( especially the meta-stable states ), condensation and growth of the liquid phase from its vapor phase or gaseous mixtures, effects of turbulence, effects of surfactants, effects of nuclei in heterogeneous nucleation and growth, particle drag, and experimental data for model validation.

I think the coverage of the equation of state for water will be immediately useful in engineering applications in which meta-stable states are encountered. How to handle thermodynamic state, transport, and thermal-physical properties for superheated liquid and subcooled vapor are always an issue in these problems. Generally it seems that maximum liquid superheating limits can approach the spinodal line but uncertainty is associated with the case of vapor subcooling limits and the spinodal line on that side of 'the doom'.

The equation of state for water developed by the International Association for the Properties of Water and Steam (IAPWS) is the gold standard in engineering. The IAPWS-95 version is a recent formulation and this formulation is discussed and its validation up to 2012 is mentioned in Chapter 4. The formulation is computationally expensive and many times the official version is seldom used whenever rapid transients are part of the problem. There are hundreds, maybe even thousands, of engineering-grade versions floating around. I doubt that the pure formulation can be used in GCMs so the approximations presented in Chapter 4 are important. GCMS of course do not consider rapid transients, but they do involve a massive number of grid cells and somewhat long time spans.

The many solid states of water are also discussed in Chapter 4. Ice Nine is mentioned.*

Nucleation and growth of liquid droplets from a vapor, or mixture of non-condensing and condensing gases, are important in several engineering areas such as steam flows in turbines, flows in supersonic nozzles such as rocket propulsion, and flows around airplanes in humid air, among others.

Likewise nucleation and growth of vapor regions from its liquid phase as encountered primarily in boiling flows and to a lesser degree cavitation at lower pressures, are encountered in many engineered systems. Generation of electric power by use of water and its vapor has led to an enormous number of investigations into vapor-bubble nucleation, growth and the subsequent effects of these on the hydrodynamics of the flows in the engineering literature. The phenomena and processes discussed in the book are encountered in the general engineering areas of the multi-phase thermal-hydraulic sciences and their applications.

Many engineered systems involve the use of structures to bound the flows of interest and the interface between the fluids and structures is sometimes critically important relative to nucleation and growth of both vapor and liquid regions. The present book does not directly consider engineered equipment and the effects of fluid-solid interfaces due to equipment boundaries are not consider at all.

The fundamental concepts developed in the book, for both homogeneous and heterogeneous nucleation and growth of liquid from its vapor phase, are directly useful in many engineering applications. I am not aware of many cases for which nucleation and growth of the solid phase from the liquid phase is encountered in engineered equipment. That is not to say that such situations do not exist.

The Kindle edition is especially accessible as it takes maximum advantage of HTML with extensive use of links to equations, chapters, sections, and references. My experience so far indicates that all the HTML aspects are successfully handled in the Kindle version. The Kindle edition is also very useful for increasing the accessibility of the numerous equations in the text. And so far all the equations are correctly handled. Checking the mnemonic symbols in sub- and super-scripts is greatly facilitated by use of the Kindle utilities.

A personal note. It happens that I have both the hardcopy and Kindle versions. How and why this happened is a different story altogether. I recommend the Kindle version over the hardcopy. The hardcopy uses a small font size and these are small indeed by the time you get to the compound sub-scripts in long complicated equations. It's been hard to break the I have to hold a book syndrome, but the Kindle is converting me, even after some initial failures when it came to mathematical equations.

A second personal note. Engineers, even those USA converts to SI, will have to be aware of, and beware of, the units used in the book.

*Vonnegut Jr., Kurt

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Excellent reference for students and researchers in cloud physics
10. Februar 2015
Von
Vladimir V. Chukin
- Veröffentlicht auf Amazon.com

Format: Gebundene Ausgabe

REVIEW

by Dr. Vladimir V. Chukin, Associate Professor of the Russian State Hydrometeorological University, on February 10, 2015

The book is a good reference for researchers in cloud physics and remote sensing. The book can be especially useful for the students with good mathematical background. Students can use it as a manual studies and for numerical simulations of cloud microphysical and optical properties, droplet activation, ice nucleation, diffusion and coagulation growth, parameterization of the size spectra and fall velocities, and most other cloud processes. All the methods of calculations of these properties are described in detail, step-by-step, which makes reading easier. A valuable feature of the book is presentation of the modern concepts and theories developed in cloud physics over the last 2-3 decades and absent in the older books.

Indeed, the book combines simplicity and clarity of presenting the material. For example, a detailed description of Wegener-Bergeron-Findeisen process (page 158) is given, allowing to students to create a Python script for numerical simulation of the process in a few minutes.

The thermodynamics relations, used in cloud microphysics, are widely and in detail described. For example, the Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac statistics are presented. The authors show the boundaries of their usage (pages 56) and possible application for homogeneous and heterogeneous ice nucleation at low energy of ice germ formation (page 299). This detailed consideration of various statistical distributions allows more accurate calculations, extensions to lower temperatures, and shows that the critics expressed in one of the previous reviews by Paul Pukite is erroneous.

Since I am an expert in the field of supercooled droplets crystallization, I'm pleasantly surprised by the high level of presentation of the theory of the ice nucleation. The equations of classical nucleation theory (CNT) for the critical radius and energy of an ice germ (page 318) are substantially generalized, so that they depend on both the temperature and the saturation ratio, which allows to consider both dependencies simultaneously and with account for their mutual strong feedbacks.

Unfortunately, the book does not address the atmospheric electricity, in particular, the mechanisms of droplets and crystals electrification, leading to the appearance of strong electric fields in the clouds. This could be caused by the already large size of the book. It certainly does not reduce the importance of this excellent book.

I would like to express my gratitude to the authors for their work. Special thanks to the head of the library of RSHU Mrs. Elena Astafeva for her assistance in the purchase of the book for the university.

COMMENTS

on the wrong and misleading review by Paul Pukite

As I dealt for many years with cloud particles nucleation and parameterization of their size spectra, I have a few comments on the review by Paul Pukite, which seems to me completely wrong, misleading and based on poor knowledge of both quantum statistics and methods used in cloud physics. The review at Amazon.com site of Paul Pukite of September 5 2014 for the book by Khvorostyanov and Curry contains two main statements, which, he thinks, are wrong in the book.

1) The first statement is "The authors apply Bose-Einstein statistics to the formation of both liquid and ice nucleation. There is no precedent for using B-E statistics in anything other than very-low, near absolute zero, physical behaviors".

The transition from the Bose-Einstein to Boltzmann statistics occurs not at a temperature close to zero K, as Pukite erroneously assumes, but is determined by another condition. Both statistical distribution include the exponent exp(-E/kT). When E ≤ kT, then the more complete Bose-Einstein statistics should be used. Under another condition, E >> kT, the Bose-Einstein statistics is simplified and converts into the Boltzmann statistics. Thus, not proximity to zero K, but the ratio E/kT is the criterion of transition from Bose-Einstein to Boltzmann statistics.

There are several different derivations of the Bose-Einstein statistics in theoretical physics but none of these derivations requires that the temperature should be near absolute zero (e.g., Landau and Lifshitz, v. 5, Statistical Physics, 1958b; Born, 1963; Atkins, 1982), etc. In contrast to Pukite's claim, there are many precedents of using Bose-Einstein statistics at any temperature (including very high) for the particles with integer spin, like, e.g., photons, gluons (a component of protons and neutrons, which compose all the matter that surrounds us in everyday life at any temperature), or Higgs's bosons that give inertia to any particles at any temperature, and to all the other more than 70 bosons- at any temperature. P.W. Atkins in his classical book "Physical Chemistry" (2nd edition, 1982, page 688) writes: "When the particles are ordinary molecules they obey a type of statistics known as Bose-Einstein statistics which can be developed as follows". Then Atkins derives the BE statistics applicable for molecules and other particles with integer spin at any temperature. Then Atkins gives a list of 12 books, where various statistical distributions are derived (it would be useful to Paul Pukite to read some of these or similar books). Then Atkins considers on several next pages applications of various statistics (including BE) for various molecules at various temperatures including 100 K, 298 K, 5000 K, etc. Thus, Bose-Einstein statistics is applied at high temperatures and not near "absolute zero" as Pukite erroneously assumes.

Another example of application of BE statistics is the Planck function, which is simply BE distribution multiplied by the second power of radiation frequency as described on page 55 of the book by Khvorostyanov and Curry (2014, hereafter, KC-2014). Integration of the Planck function (proportional to BE distribution) over all frequencies gives the integral flux of the black-body radiation (Stefan-Boltzmann law), i.e., the integral flux that the body emits at a given temperature T. Well-known calculations show that the approximate emission temperature of the Earth is ~300 K, the emission temperature of the sun is ~6000 K and both temperatures are evaluated based on BE statistics. Is it close to "absolute zero" as Pukite assumes? So, Bose-Einstein statistics is valid far above from absolute zero, and understanding of Paul Pukite of this BE statistics is completely wrong.

Yet another "precedents" of using BE statistics far above zero K are Einstein's (1906) and Debye's (1912) theories of heat capacity of solids. E.g., the heat capacity for ice was calculated with BE statistics for harmonic oscillators in various models with the Debye's characteristic temperature as 192 K or 318 K, so that BE statistics was applied far above zero K (e.g., Hobbs, 1974; Landau and Lifshitz, v. 5, 1958b; Born, 1963; see also pages 114-116 in Khvorostyanov and Curry, 2014), in conflict with illiterate Pukite's statement that it should be near zero K.

After reading Chapters 8 and 9 of the book, I could understand as it is clearly described in these sections of the book that all calculations of the nucleation rates and particles concentrations in this book were done with the traditional Boltzmann's statistics as it is usually done in classical nucleation theory (CNT) (page 293, eq. (8.2.1) and subsequent equations in Chapter 8; page 397, eq. (9.2.6) and page 411, eq. (9.6.2) and subsequent equations in Chapter 9). Possible application and testing of Bose-Einstein statistics is just briefly outlined on half a page as a possible generalization of Boltzmann's statistics at sufficiently low temperatures when critical energy of a germ formation F-cr may become comparable to (kT) (page 299 in Chapter 8). This regime may occur not at very low temperatures close to zero Kelvin, as Pukite erroneously assumes, but at intermediate temperatures due to low surface tension or other parameters of CNT decreasing with temperature (e.g., around 200 K). But Bose-Einstein statistics was never applied for any calculations in this book. Thus, the statement in Pukite's review that "The authors apply Bose-Einstein statistics to the formation of both liquid and ice nucleation" is completely wrong falsification of what has done in this book. Boltzmann's statistics is applied in all calculations in this book, not B-E statistics.

The explanations above show that Pukite's claim that BE statistics is applicable only near zero K is based on nothing (where did he take it from?) and is conflict with all the modern views of physics. Note however that if nucleation calculations were performed in KC-2014 book not with Boltzmann but with Bose-Einstein distribution, they would be valid and could be extended to lower temperatures since BE statistics includes Boltzmann statistics as a particular case. It would be interesting to carry out ice nucleation calculations with Bose-Einstein distribution when F-cr ≤ kT and Boltzmann statistics becomes invalid. This may occur at intermediate temperatures, e.g., 180-200 K, still well above zero K.

2) The second statement in Pukite's review is "Other sections talk about "the older power law and newer lognormal parameterizations of aerosol size spectra" -- these are already out-of-date as modern uncertainty approaches always favor power-law distributions".

This again illustrates Pukite's fundamental unawareness in cloud microphysics. All major types of drop, crystal and aerosol size spectra are considered in the book, including power laws, exponential and gamma distributions, and the quantitative relations among them are established.

However, everybody who deals with cloud microphysics now and has even a minimum knowledge here, from the undergraduate students, to the most experienced experts in this area, knows that the size spectra usually used now for parameterization of drop, crystal and aerosol distributions are lognormal or gamma distributions (see reviews in Chapters 2, 6, 13, 14 in this book KC-2014; Pruppacher and Klett, 1997, and Seinfeld and Pandis, 1998 and many other books and papers). Vice versa, the power law distributions suggested for aerosol at the beginning of 1950's, are very rarely used in the modern literature.

Thus, the claim of Pukite that "lognormal parameterizations are already out-of-date as modern uncertainty approaches always favor power-law distributions" are in conflict with all the modern parameterizations of the size spectra, and shows his complete unawareness about the current literature on cloud physics and parameterizations used in the cloud models of all types- from the parcel models, to cloud-scale models, and to weather and climate models. For example, an Intercomparison of simulations of the recent MPACE (Mixed-Phase-Arctic Clouds Experiment), where several tens leading modelers from all over the world participated, used lognormal size spectra for aerosol and gamma distributions for drops and crystals (Klein et al., 2009; Morrison et al., 2009). Similar parameterizations were used for simulations of ISDAC and CRYSTAL (2004) campaigns (big series of papers of the last decade in JGR and JAS). And no power laws as Pukite again erroneously assumes for the "modern approaches".

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verified purchase of hardcover A physically adequate description of clouds and hydrometeors is one of the greatest challenges in
3. Dezember 2014
Von
Dr. Olaf Hellmuth
- Veröffentlicht auf Amazon.com

Format: Gebundene Ausgabe

Vitaly I. Khvorostyanov and Judith A. Curry

Thermodynamics, Kinetics, and Microphysics of Clouds

Cambridge University Press, First Edition, ISBN 978-1-107-01603-3 Hardback

Book Review by Olaf Hellmuth, TROPOS Leibniz Institute for Tropospheric Research, Leipzig, Germany

on November 23, 2014, verified purchase of hardcover

A physically adequate description of clouds and hydrometeors is one of the greatest challenges in numerical weather prediction (NWP) and climate modelling. Cloud modeling is a multiscale problem, including processes ranging from nanometric to planetary scales. Owing to the scientific and practical importance of cloud physics, the number of related scientific publications is very large and continues to grow. The amount of obtained knowledge is very difficult to percept already for experienced people, even more for students and polymaths. In view of the rapid development of cloud physics, any attempt to either ‘condense’ the state-of-the art knowledge and/or to compile the pieces of the puzzle of in-depth knowledge in favour of a broader, more universal perspective is very helpful, even now indispensable for both beginners and advanced learners. Considering the number of instructive books on cloud physics available on the market, one might ask: is there a need for a ‘new book’ on cloud physics? My answer to this question is a clear ‘Yes’: the differing ways of reader’s perception require differing styles of writer’s presentation. Moreover, if any new book contains new approaches, methods and results, then the book fills a gap. The “Khvorostyanov/Curry” is, without any doubt, such a book! The book contains scientific ‘rootwork’, useful for beginners but not exclusively and even not primarily aiming at this target group. The authors are evicted experts in the field of cloud physics; they present both phenomenological findings and ‘fine arts’ of theory as the title promises: thermodynamics, kinetics and microphysics of clouds. I appreciate this inasmuch as for me as a meteorologist the theoretical understanding of cloud physics is - as ever - a very challenging and exhaustive endeavour. The authors present a key for this ragged terrain, the conquest of which, however, deserves some hard days work by the willing and inquisitorial reader especially regarding the theoretical parts. The presented material in the book is very ‘nutritive’: it contains practical solutions for sophisticated microphysical questions of interest. For students it is very recommendable to have this book on hand, for cloud physicists and modellers it is a ‘must’, especially for the ones who already appreciate the classic books on this topic!

By a quick look the book can be subdivided into two parts: a general part with phenomenology and basic notions of cloud physics, and a special part addressing the genesis of clouds and presentation of advanced theories of cloud physics.

The general part (Chapters 1-4) presents the general description of clouds: cloud classification, global cloud distribution, role of clouds in the general circulation of the atmosphere and ocean, conditions of cloud formation, microphysical and optical properties and their parameterisations. This part of the book also contains a detailed description of the thermodynamics of the bulk and discrete media, solutions and gases, major statistical distributions, the general equilibrium (entropy) equation and of the major equations used in cloud physics, phase rules, dry and wet adiabatic processes. The modern theories of the various states of water, solid and amorphous ices at low temperatures and high pressures, stability conjectures and possible transformations of various forms of water and ice are described. It is very nice that the authors also adequately considered very recent developments, such as the new Thermodynamic Equation of Seawater TEOS-10 for the determination of the thermodynamic properties of water in its vapourised and condensed (liquid, solid) forms.

The second part of the book (Chapters 5-14) is more specifically related to cloud physics. This part presents the methods and tools used in cloud, NWP and climate models, as well as in analyses of laboratory and fields experiments. Fortunately, the authors considered in detail the water vapour balance in ‘cloudy air’, which is of crucial importance for understanding of cloud microphysics. The concepts presented here go far beyond the widely employed ‘saturation adjustment’ schemes and include the derivation of equations for the differential and integral water and ice supersaturations in various forms, including the supersaturation (phase) relaxation times. The kinetic equations for the size spectra of regular condensation, deposition and the coagulation/accretion growth of drops and crystals are derived. A simplified analytical treatment of the glaciation process is given, which is of interest for the long time existence of of mixed-phase clouds and for cloud seeding experiments. The aerosol hygroscopic growth and activation of the cloud condensation nuclei (CCN) into cloud drops are described. This is followed by the extended classical nucleation theory (CNT) of homogeneous and heterogeneous nucleation of drops and crystals, and by the similar theory of deliquescence and efflorescence. A unified treatment of cloud particle fall velocities for liquid and crystalline cloud particles over the wide size range in the atmosphere is given as a quasi-power laws with coefficients as continuous functions by the Best and Reynolds numbers, which is convenient with respect to application. The authors also discussed applications of this theory for the other planets and other objects in various media. The explanation of the broadness of observed droplet size spectra is of crucial importance for any progress in cloud and rain prediction. In Chapters 13 and 14 various mechanisms to explain the broadening of droplet size spectra in clouds at all stages of their development are suggested and reviewed. The major attention is paid to the theories of stochastic condensation and coagulation and their analytical solutions.

Comparing the present book with previous classical books on cloud physics (e.g., Fletcher, 1962; Dufour and Defay, 1963; Defay et al, 1966; Pruppacher and Klett, 1997; Seinfeld and Pandis, 1998; the list is not claimed to be complete) I would like to emphasise a few specific features of the “Khvorostyanov/Curry”:

1) In Chapter 5, the authors presented equations for diffusion growth rates of drops and crystals, which are written as the product of the three major factors: supersaturation, kinetic, and psychrometric corrections. It is shown that this form is equivalent to the usually used in the literature more complicated equations, but is more convenient for the analytical derivations in the subsequent chapters.

2) In Chapter 6, solutions for the wet and critical CCN radii and the supersaturation are derived, which represent generalisations of Köhler’s classical theory with respect to three aspects: (i) allowance of an arbitrary soluble fraction, i.e., relinquishment of the usual restriction to strongly diluted aqueous solutions; (ii) treatment of the cases that the soluble fraction is proportional to the CCN volume or surface respectively (characterized by two hygroscopicity parameters); (iii) extension to the case of high water supersaturation. The authors presented the CCN spectra in a special algebraic form, which is equivalent to the lognormal distributions but allows more convenient calculation of hygroscopic growth and drop activation. Using this new algebraic spectrum, a fast numerical algorithm was developed for drop activation. The authors found 4 analytical limits of the mathematical solution, which can be directly applied in various models.

3) The parameters C and k appearing in the well-known and widely employed Twomey’s power law for the drop concentration, Nd (s) = Csk, as a function of supersaturation s are rather uncertain. These parameters are usually empirically adjusted to different airmass types. In Chapter 6 it is shown that both parameters are intrinsically not ‘constants’ but instead functions of supersaturation s, whereas C(s) and k(s) strongly decrease with increasing s. This finding agrees with laboratory and field experiments. The authors presented, to the best of my knowledge for the first time, the parameters C and k as analytical functions of supersaturation and of the CCN microphysical properties (size spectrum, soluble fraction). Retaining the basic form of Twomey’s power law, the degree of nonlinearity is enhanced by consideration of the dependence on CCN microstructure, shown in the modified expression Nd = C(s)sk(s).

4) For several decades, ice nucleation rates were parameterised on the base of empirical data either as functions of solely supercooling (in terms of temperature T) or as functions of solely water vapour supersaturation (in terms of saturation ratio Sw). Attempts to combine these dependencies were rare and not successful. Based on a series of their own previous papers, Khvorostyanov and Curry presented in their book a general theory of immersion freezing, called deliquescent-heterogeneous freezing, which was obtained from combination of the classical nucleation theory (CNT) with the Gibbs-Kelvin-Köhler theory of thermodynamic equilibrium between a solvent vapour and an aqueous electrolyte solution droplet. In this way, the authors derived new expressions for the radius of the critical ice embryo and the thermodynamic energy barrier for formation of the critical embryo as analytical functions of several observables: temperature, water vapour saturation ratio, pressure, radius of the particle, and misfit strain. Functional dependencies have been derived for submicron haze particles and for supermicron cloud drops. The outcome of the theory is important; it allows a simultaneous consideration of both liquid and vapour metastability in the determination of immersion freezing rates (in terms of supercooling and supersaturation, measurable thermophysical properties of supercooled solutions and hexagonal ice, and in terms of surface characteristics of the catalysers, which are – on principle – accessible from dedicated laboratory experiments).

The final expressions for the thermodynamic and kinetic energy barriers and nucleation rates are given in a straightforward analytical form. These expressions are not ‘empirical’ but represent exact solutions of a set of thermodynamic and kinetic constraints.

I would like to emphasise that the whole theory is an extension of CNT, which allows a theoretical prediction of nucleation rates (and on its base the determination of freezing temperature and freezing saturation ratio).

The simple form of the nucleation rate expression as function of temperature and saturation ratio allows a straightforward application in cloud models for interpretation of field and laboratory studies. It should be mentioned that the new theory also sets thermodynamic constraints for the applicability of previous empirical parameterisations. The new theory yields a number of useful expressions for practical applications: a quantitative relation between the solution and pressure effects in nucleation; analytical decompositions of nucleation rates and crystal concentrations as products of functions of solely temperature and saturation ratio; parameterisations of crystal concentrations for direct application in general circulation models (GCMs) and climate models.

5) In their Chapter 11 the authors applied the extended CNT to the problem of deliquescence and efflorescence, which allows the evaluation of several useful properties of these processes: the temperature dependence of solubility, dissolution heats, threshold relative humidities, location of the eutectic points on the phase diagram, and to plot a unified phase diagram for freezing, melting, deliquescence and efflorescence.

6) For many decades the gamma − distribution is used as an empirical parameterisation for cloud drop size spectra. In their Chapter 13, the authors demonstrated that the empiricism of this spectral shape can be traced back to fundamental cloud processes: the analytical solutions of the stochastic condensation equation for the small-size fraction of drop and crystal spectra (obtained in Chapters 13 and 14) were shown to have a form similar to the gamma − or the generalised gamma −distributions, depending on the involved processes. The parameters of the gamma −distributions, determining the width of the spectra and the rates of precipitation formation, are expressed in terms of atmospheric observables and fundamental constants.

7) In Chapter 14 the authors derived analytical solutions to the stochastic kinetic equation of coagulation for the large-size fractions of the drop and crystal spectra for various cases. These solutions were found to be similar to the gamma −distribution, exponential Marshal-Palmer and inverse power law size spectra that have been determined experimentally. Also the parameters of these spectra can be expressed in terms of cloud properties and fundamental constants. In summary, the mathematical solutions of the kinetic equations in Chapters 13 and 14 provide an a-posteriori justification and explanation for the most frequently observed size spectra and their parameterisations. Finally, based on the integral Chapman-Kolmogorov equation for stochastic processes, in Chapters 13 and 14 general integral stochastic kinetic equations for condensation and coagulation are formulated. These equations await further evaluation and solution in future studies.

The scientific content of the book is a great séance: it covers major topics and ‘hot spots’ of cloud physics and aims at a broad target group: undergraduate and graduate students, researchers working with cloud, weather and climate models, dealing with air pollution modelling, cloud seeding, remote sensing and other applications of cloud physics, not to forget environmental engineering. In view of the 782 pages of the book I would like to add that authors did a very diligent and expensive work to provide helpful solutions of pending problems. I am sure that the book gives many stimuli for further research and sophistication of the theoretical description of clouds. This book also shows that cloud physics is not a ‘terminated science’ but remains a heavy terrain and rich field for future research.

The layout, print quality and manufacturing of the hardback is excellent (e.g., despite of the 782 pages one can open the book on any page and it remains open without external power against the ‘clapping forces’!). If there is something to criticise then I would name the ‘small’ letter size, which is not a gift for exhausted eyes. A book like this can carry additional pages in favour of a somewhat larger letter size …

Summing up, I say ‘well done!’ and give five stars. A very good investment for root workers in cloud physics.