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Statistical models of cloud-turbulence interactions Jeffery, Christopher A. M. 2001

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Statistical Models of Cloud-Turbulence Interactions by Christopher A . M . Jeffery M.Sc , University of British Columbia, 1996 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF Doctor of Philosophy in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Earth and Ocean Sciences) We accept this thesis as conforming to the required standard The University of British Columbia September 2001 © Christopher A . M . Jeffery, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The application of statistical turbulence theory to the study of atmospheric clouds has a long history that traces back to the pioneering work of L. F. Richardson in the 1920s. At a phenomenological level, both atmospheric clouds and turbulence are now well understood, but analytic theories with the power to predict as well as explain are still lacking. This deficiency is notable because the prediction of statistical cloud change in response to anthropogenic forcing is a preeminent scientific challenge in atmospheric science. In this dissertation, I apply the statistical rigor of new developments in passive scalar theory to problems in cloud physics at small scales (9(10 cm), where a white-in-time or (^-correlated closure is asymptotically exact, and at large scales 0(100 km) where a statistical approach towards unresolved cloud variability is essential. Using either the 5-correlated model or a self-consistent statistical approach I investigate (i) the preferential concentration or inertial clumping of cloud droplets; (ii) the effect of velocity field intermittency on clumping; (iii) the small-scale spatial statistics of condensed liquid water density and (iv) the large-scale parameterization of unresolved low-cloud physical and optical variability. My investigations, (i) to (iv), lead to the following conclusions: Preferent ial Concentrat ion: Inertial particles (droplets) preferentially concentrate at scales ranging from 6O77 at St « 0.2 to 877 at St 0.6, where 77 is the Kolmogorov length and St is the Stokes number. Clumping becomes significant at St « 0.3. Effect of Intermit tency: An effective Stokes number, Stefj = St(^ r /3) 1 / 2 where T is the longitudinal velocity-gradient flatness factor (kurtosis) explicitly incorporates velocity-gradient intermittency (i.e. non-Gaussian statistics) into the St-dependence of particle clumping. In the atmospheric boundary-layer, Steff « 2.7St. Intermit-tency effects significantly increase the degree of preferential concentration of large cloud droplets. C l o u d Spat ia l Scaling: Density fluctuations of an inert passive scalar are typically spa-tially homogeneous, whereas root-mean-square cloud liquid water (<&) fluctuations increase linearly with height above, cloud base. As a result, the qi spectral density is axisymmetric and complex. A model of low-cloud viscous-convective statistics where axisymmetric/non-homogeneous production of scalar covariance due to con-densation/evaporation is balanced by an axisymmetric rotation reproduces recent i i experimental measurements [Davis et al, 1999]. Low-c loud Opt i ca l Propert ies : The assumption of height-independence in unresolved saturation vapour density fluctuations (s) and the introduction of unresolved cloud-top height fluctuations (z't0 ) into a statistical cloud scheme couple parameterized subgrid low-cloud physical and optical variability. Analytic relationships between optical depth, cloud fraction and (s,z'top) provide a convenient framework for a G C M cloud parameterization that prognoses both the mean and variance of optical depth ii i Contents Abstract ii Contents iv List of Figures vii Acknowledgements viii Dedication ix 1 Introduction 1 1.1 A short anecdotal history of "turbulence" 3 1.2 Problems addressed in this dissertation 12 1.2.1 Cloud droplet number concentration inhomogeneities 12 1.2.2 Cloud liquid water density inhomogeneities 14 1.2.3 Unresolved low cloud optical properties 14 2 The ^-Correlated Model 16 2.1 Introduction 16 2.2 Hamiltonian Fluid Mechanics 17 2.2.1 Lagrangian formulation 18 2.2.2 Eulerian formulation 19 2.3 <5-Correlated Model 24 2.4 Mean Scalar Concentration 26 2.5 Mean Scalar Covariance 29 2.6 Summary: 1968-present 30 3 Spatial Statistics of Inertial Particles 35 3.1 Introduction 35 3.2 Correlation function 36 3.3 Spectral density 38 3.3.1 Small-scale solution (k ^> 77-1) 39 3.3.2 Large-scale solution (O.I77-1 < k < 39 3.4 Analysis 40 iv 3.5 Spectra and Discussion 42 3.6 Experimental verification 43 3.7 Summary 45 4 Intermittency arid Preferential Concentration 47 4.1 Introduction 47 4.2 St-ReA dependence 48 4.3 The Shaw Model and Vortex Tubes 50 4.4 Experimental verification 52 4.5 Summary 52 5 Spatial Statistics of Cloud Droplets 54 5.1 Introduction 54 5.2 Condensation/Evaporation Source Term 57 5.3 C E in the Batchelor limit 59 5.4 Axisymmetric Kraichnan Transfer 60 5.4.1 Viscous regime solution 60 5.4.2 Inertial-Convective regime solution 62 5.5 The axisymmetric source / 63 5.6 Determination of £ 65 5.7 Spectra and discussion 66 5.8 Experimental verification 69 5.9 Summary 70 6 Unresolved Variability of Low Cloud 72 6.1 Introduction 72 6.2 Statistical Cloud Schemes 74 6.3 Shortwave Optical Depth Formulation 75 6.4 Unresolved Low Cloud Optical Variability 78 6.5 Low Cloud Optical Properties 82 6.6 r e f f - r relationships in GCMs 82 6.7 Low Cloud Radiative Feedback 86 6.8 Experimental verification 88 6.9 Summary 89 7 Summary 91 Bibliography 96 Appendix A List of Principal Symbols 115 v Appendix B Triangle Distributions 118 B . l Smith [1990]'s triangle distribution 118 B. 2 Modified triangle distribution 118 Appendix C Calculation of low-cloud optical properties in Sec. 6.5 121 C l Calculation of R 121 C. 2 Calculation of e 122 vi List of Figures 2.1 Temperature and velocity spectra from Grant et al. [1968] 32 3.1 Accuracy of the fourth-order approximation Q{k) 41 3.2 Plot of the scale break kb and the self-excitation Xuc(°o, S t ) /x ; c — 1. . . . 44 3.3 Effect of particle inertia on the scalar spectrum 45 3.4 Characteristic scale of preferential concentration 46 5.1 ID cloud liquid-water scalar spectrum from S O C E X 56 5.2 Effect of condensation/evaporation on the scalar spectrum 67 5.3 Comparison of S O C E X data with the predicted ID scalar spectrum. . . . 68 6.1 Comparison of Ac vs u using Landsat data 81 6.2 Zonal accuracy of the P P H approximation 83 6.3 Effect-of model vertical resolution on the reff-r relationship 85 B . l The modified triangle distribution 120 vii Acknowledgements In the spring of 1993 I wandered into Phil Austin's office looking for employment for the summer. At the time, I had no noteworthy skills, little knowledge of computers, and only a cursory understanding of clouds. Moreover, I had plans to study biophysics in the fall. Atmospheric science, quite frankly, was not in my plans for the future. It was to my great fortune that Phil took pity on my impoverished state and offered me a programming job for the summer. Merely hours after formalizing my employment, he was quite surprised, I imagine, to learn that I had confused my knowledge of V M S with UNIX, and thus, my meager computer expertise was quite useless. But Phil persevered through constant interruptions and my intolerably slow progress, and thus began our friendship that eventually lead to his supervision of this thesis. To state that Phil has been an exemplary supervisor is an understatement; the knowl-edge, advice and support that I received from Phil over the last four years is the greatest fortune of my graduate career. His constant urgings that I should study the cloud param-eterization literature—particularly Barker [1996b] and Considine et al. [1997]—proved invaluable and led us to develop a new statistical treatment of cloud optical variability [Jeffery and Austin, 2001b]. For all that Phil has done for me I am sincerely grateful. Yet any success that I might have achieved alone with Phil would ring hollow were it not for the love and support of my wife, Nicole. Through the trials and tribulations of my graduate career she has been a pillar of both intellectual and emotional support; both her careful reading of my articles and this thesis, and the sacrifices she has made while pursuing her own Ph.D. were invaluable. Nothing has bolstered my spirit more over the last four years than the time spent with my wife and daughter. I am truly indebted. I thank my friends Brian, Andres, Joel and Tom for good times, and Vincent for the answers to all my questions about computers. I am most grateful to my parents, not only for beginnings, but for the steady support they always offer. I owe special thanks to my committee Douw Steyn and Roland Stull and to my colleagues Marcia Baker, Howard Barker, Anthony Davis, Wojtek Grabowski, Ray Shaw and Katepalli Sreenivasan. C H R I S T O P H E R A . M . J E F F E R Y The University of British Columbia September 2001 viii I dedicate this thesis to my daugher Sophia and to the late Lewis F. Richardson. Chapter 1 Introduction Big whorls have little whorls, Which feed on their velocity; And little whorls have lesser whorls, And so on to viscosity (in the molecular sense). [Richardson, 1922] Although the men whose work heralded the beginnings of turbulence theory un-doubtedly turned to the atmosphere for inspiration—Reynolds, Taylor, Prandtl and von Karman come to mind—it is Lewis Fry Richardson who first recognized the important and central role of statistical turbulence theory in atmospheric studies. Thus in 1922 when Richardson's book Weather Prediction by Numerical Process was first published, the marriage of theoretical turbulence and atmospheric science research effectively began. Initially criticized by his contemporaries for containing "too much heterogeneous mate-rial to be satisfactory" [Platzmann, 1967], Richardson's book now stands as a monument to the genius and prescience of its author and as a milestone introducing two fundamen-tal paradigms that are among the most important in atmospheric research: numerical weather prediction and the turbulent energy cascade. The latter he expressed poetically on page 66 of his book in an oft quoted imitation of Jonathan Swift (reproduced above). The renaissance of atmospheric turbulence, one can argue, occurred during the 1970s. In those days the efforts of atmospheric scientists like Leith, Lumley, Herring, Hi l l , Van Atta and Wyngaard, engineers like Corrsin and Lundgren, mathematicians like Man-delbrot and Rosenblatt and physicists like Burgers, Kraichnan and Lorentz crossed dis-ciplines: Kraichnan published in J. Atmos. Sci. [Leith and Kraichnan, 1972; Herring et ai, 1973; Kraichnan, 1976], repercussions from the Kansas and Minnesota experi-ments [Kaimal and Wyngaard, 1990] were felt in the turbulence research community at large [Monin and Yaglom, 1975], and a significant cross-fertilization of ideas occurred at meetings where members from all four groups were present [Rosenblatt and Van Atta, 1972]. Nowhere in atmospheric science did the influence of the "other" turbulence com-munities have a greater impact than in boundary layer meteorology (BLM). By the end 1 of the decade B L M was considered a success, along with its methodology: " . . . the pa-tient, systematic attack of fundamental problems through a combination of theory and experiment and the expression of the findings in simple, effective terms that make them useful in a wide range of applications" [Kaimal and Wyngaard, 1990]. In modern times the marriage of turbulence and atmospheric science research has faltered. The theoretical study of turbulence has become highly specialized, while the B L M community has turned its attention to less idealized problems that are considered too complex to yield to the analytic machinery of theoretical turbulence. The attitude of the atmospheric science community towards the prospect of analytic advances in tur-bulence theory is best summarized by a recent quote from John C. Wyngaard, one of the principal investigators in the Kansas and Minnesota experiments [Wyngaard, 1998a]: We are still searching for a solution to the "turbulence problem" . . . . Years ago, every time I saw him my father-in-law would ask me if I had solved the problem I was working on. He eventually stopped asking. The renormaliza-tion group, or R G , is the latest in a long line of failed attempts at a solution (Eyink, 1994). Although Wyngaard [1998a]'s comments are directed primarily at the "velocity problem"—determination of the statistical properties of a turbulent flow—they imply a similar pessimism towards the determination of the statistical properties of a pollutant in the flow: the "passive scalar problem". I believe we have some cause for optimism. Although a deep theoretical understanding of a turbulent velocity field remains elusive, the problem of predicting the statistical properties of a passive tracer or scalar in a turbulent flow has recently yielded to analytic attack. Scale renormalization leads to a white-in-time or "5-correlated" velocity field that is an exact statistical surrogate for the real field at the smallest turbulent scales and a reasonable approximation at larger scales. In this 5-correlated limit, analytic expressions for the intermittency correction to the normal or similarity scaling exponents of the passive scalar structure functions have been found [Shraiman and Siggia, 1994; Chertkov et al, 1995b; Gawedzki and Kupiainen, 1995; Balkovsky and Lebedev, 1998], whereas the theory of a turbulent velocity field has no such realistic yet simplified analytic model. Moreover, the prospect of extending 5-correlated theory in the near future to include a non-white temporal decorrelation is encouraging. In addition, the renormalization group method that Wyngaard [1998a] chastises for its failure with the velocity problem has had some success in the ^-correlated limit [Adzhemyan et al., 1998; Antonov, 1999; Adzhemyan et al, 2001]. Certainly, the velocity problem and the passive scalar problem are two very different creatures; the Navier-Stokes equation is non-linear w.r.t. the velocity while the advection-diffusion equation is linear w.r.t. the pollutant concentration. In this dissertation, I take a few small steps towards the goal of reappraising and advancing theoretical cloud physics using the analytic techniques that have proven suc-cessful with the inert passive scalar problem. While cloud liquid water is treated as a 2 reactive scalar, one may ignore, in a first-order iteration, the non-passive coupling of atmospheric turbulence with latent heating or cooling that accompanies phase change. From the outset I acknowledge that much work remains to be done to accomplish the stated goal. Certainly, say, an instanton for the cloud droplet number distribution or an expression for the tails of a stratiform cloud liquid water density function constitute significant advancements in theoretical cloud physics. I have done neither. What I have accomplished over the last four years is to identify specific problems in cloud physics that are particularly amenable to analytic attack and to develop models that either provide new information at small scales—on the order of centimeters—or better predictions at large scales—on the order of tens of kilometers. In this work the 5-correlated model takes center stage because it is both highly accurate at small scales and is also the paradigm for new theories of cloud physics that I hope to develop later in my career. Although my work on large scale cloud parameterizations does not explicitly involve the r5-correlated closure, I hope that the reader recognizes a certain "continuity of approach" when we move from an application of the, 5-correlated model in Chapter 5 to the statistical treat-ment of unresolved cloud optical variability in Chapter 6. The methodology and approach that I learned from the work of others in the passive scalar community motivated the development of my statistical large-scale parameterization. The rest of this chapter is organized as follows. First I present a short anecdotal history of "turbulence" in which I have assembled a collection of some of the more interesting and amusing thoughts of the major players in this drama from the last century. Although, a thorough account of the history of turbulence theory would be a worthy and difficult challenge, I will not rise to the occasion here, and rather, refer readers to the recent resource letter by Nelkin [2000]. In Sec. 1.2 I sketch the problems that are addressed in this work and the progress I have made towards their solution. 1.1 A short anecdotal history of "turbulence" A contemporary of Richardson, G. I. Taylor, described Richardson as "a very interesting and original character who seldom thought on the same lines as his contemporaries and often was not understood by them" [Taylor, 1959]. In the winter of 1917 Richardson was driving an ambulance for a French infantry division on the Western Front. Under these appalling conditions, he had the buoyancy of spirit to carry out one of the most remarkable and prodigious calculation feats in the history of weather prediction: the numerical calculation by hand of the change in pressure and wind for a six-hour interval in an area (2 cells, each 200 x 200 km 2) of central Europe. Although the calculation failed to predict the weather—his equations had the barometer rising fast enough to make one's ears pop—in an act of genius, Richardson did foresee the future of weather prediction. Amazingly, it would take over thirty years and the arrival of electronic computers before Charney, Fjortoft and von Neumann completed the first "successful" numerical weather 3 prediction [Charney et al., 1950]. Richardson would not be deterred by the failure of his initial attempt at numeri-cal weather prediction. Near the end of Weather Prediction by Numerical Process he describes a phantasmagorical vision of a "weather factor": Imagine a large hall like theater, except that the circles and galleries go right round through the space usually occupied by the stage. The walls of this chamber are painted to form a map of the globe. . . . Myriad computers are at work on the weather of the part of the map where each sits, but each computer attends to only one equation or part of an equa-tion. The work of each region is coordinated by an official of higher rank. . . . From the floor of the pit, a tall pillar rises to half the height of the hall. It carries a large pulpit on its top. In this sits the man in charge of the whole theater; he is surrounded by several assistants and messengers. One of his duties is to maintain a uniform speed of progress in all parts of the globe. In this respect, he is like the conductor of an orchestra in which the instruments are slide rules and calculating machines. But instead of waving a baton, he turns a beam of rosy light on any workers who are running ahead of the rest and a beam of blue light on those who are behind. . . . In a neighboring building, there is a research department where they invent improvements. But there is much experimenting on a small scale before any change is made in the complex routine of the computing theater. In a base-ment, an enthusiast is observing eddies in the liquid lining of a huge spinning bowl, but so far, the arithmetic proves the better way. . . . Outside are play-ing fields, houses, mountains, and lakes, for it was thought that those who compute the weather should breathe of it freely. The brilliance of Richardson's vision is highlighted by the anachronisms; many first-time modern readers may not realize that Richardson's "computers" are, in fact, human beings until the last line of the passage! Perhaps, we can catch a glimpse of a Message-Passing Interface (MPI) in Richardson's conductor The above passage also illustrates the important role that turbulence played in Rich-ardson's vision of weather prediction. The "enthusiast" observing eddies would no doubt be concerned with the cascade of energy from eddy to eddy and with the bulk dispersive properties of the flow in the bowl. Richardson's revolutionary understanding of the turbulent energy cascade is encapsulated by his famous poem (pp. 1) which indicates the direction of the cascade, from large scales to small, while highlighting that molecular viscosity (and not eddy viscosity) is the mechanism of energy dissipation that ends the cascade. Richardson's intuition was so powerful that in his paper of 1926 he was able to establish the famous "four-thirds law" that relates eddy-viscosity K and eddy-size I in a turbulent flow: K ~ I4/3, by purely empirical means [Richardson, 1926]. In 1941, when Kolmogorov and Obukhov formulated the general quantitative theory of inertial-4 range turbulence, Richardson's four-thirds law was actually the only empirical result which indicated the existence of simple general rules underlying the inertial cascade. Commenting on Richardson's work Taylor [1959] remarks that "(i)t is perhaps rather surprising that he did not take the step which Kolmogoroff (1941) and Obukhov took fifteen years later . . . " . Although no reference was made to Richardson in Kolmogorov [1941], later Kolmogorov would generously make up for this omission [see pp. 47]. Despite the success of Kolmogorov [1941]'s inertial range theory which predicted Richardson's four-thirds law K ~ e 1/ 3/ 4/ 3 and a velocity spectrum Eu(k) ~ e2^3k~5^3 from dimensional analysis using I or wave-vector k and the turbulent kinetic energy dis-sipation rate e, many researchers in the 1940's and 50's were skeptical about the prospect of developing a truly predictive theory of turbulence. Sir Horace Lamb is said to have remarked [Goldstein, 1969] I am an old man now, and when I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former, I am really rather optimistic." John von Neumann, who we already encountered for his work on numerical weather pre-diction [Charney et ai, 1950], advocated a computational approach towards turbulence in 1949 because of the prohibitive mathematical difficulties [von Neumann, 1963]: . . . a considerable mathematical effort towards a detailed understanding of the mechanism of turbulence is called for. The entire experience with the subject indicates that the purely analytical approach is beset with difficulties, which at this moment are still prohibitive. The reason for this is probably . . . (t)hat our intuitive relationship to the subject is still too loose—not having succeeded at anything like deep mathematical penetration in any part of the subject, we are still quite disoriented as to the relevant factors, and as to the proper analytical machinery to be used. Under these conditions there might be some hope to "break the deadlock" by extensive, but well-planned, computational efforts. A few years after von Neumann's comments, the first English textbook on B L M was published and the author, Oliver G. Sutton, warned future meteorologists that an un-derstanding of turbulence was both essential and challenging [Sutton, 1953]: This type of motion is called turbulent As yet, the mathematical theory of this type of motion is only partially established, and most of the difficulties experienced in the study of micrometeorology are to be traced, ultimately, to the great complexity of the motion of the air in the lower layers of the atmosphere. 5 Beginning in the early 1960s considerable attention was given to the statistical prop-erties of e. Kolmogorov [1962] and Obukhov [1962] suggested that fluctuations in e could lead to deviations from Richardson's four-thirds law and the other predictions of Kol-mogorov [1941]'s theory. The so-called "Vancouver group" [Monin and Yaglom, 1975, pp. 457] consisting of Burling, Pond, Stewart and Wilson here at U .B .C . played a partic-ularly active role in the development of this idea during the 1960s. Novikov and Stewart [1964] developed the first simple random cascade model for e(x) that, at a phenomeno-logical level, accounted for the intermittency of e and predicted a scaling spatial spectrum of the form E£(k) ~ k~^ and velocity spectrum Eu ~ &-5/3-/V3. p0nd and Stewart [1965] (See also Pond [1965]) made refined hot-wire measurements of EE over the ocean (Span-ish Banks) and found \i « 0.6 in agreement with the earlier and questionable results of Gurvich and Zubkovskii [1963] (See Monin and Yaglom [1975, Sec. 25.3]). Pond and Stewart [1965] predicted that the limits of applicability of the Novikov-Stewart model "appear to be so wide that models of this kind may represent a new important chapter in the theory of random processes." As it turned out, the chapter on fractal cascades was largely written by Benoit B. Mandelbrot who introduced the fractal (non-integer Hausdorff) dimension, D, a measure of the space-filling properties of the random cascade and related to the Novikov-Stewart model by D = 3 — \i. Mandelbrot [1976] generalized the Novikov-Stewart model which he termed "absolute curdling" and defined a more general process "weighted curdling" such that each moment (eq) has an associated fractal dimension Dq. Mandelbrot's weighted curdling is now typically referred to as "multifractal" as popularized by Frisch and Parisi [1985] and Benzi et al. [1984]. A dynamical version of the Novikov-Stewart model called the /3-model was introduced by Frisch et al. [1978]. Phenomenological cascade models, while providing valuable insights, were not im-mune from criticism. Kraichnan [1972] emphasized that "these hypotheses are purely conjectural, with no support from analysis using the Navier-Stokes equation" and Frisch et al. [1978] remarked that "Neither the /3-model of intermittency nor the lognormal model should be taken too seriously." Arguably the greatest contribution to the analytic theory of turbulence was made by Robert H. Kraichnan. Kraichnan was well trained in the techniques of theoretical physics having been Einstein's penultimate assistant in Princeton from 1949 to 1950. Kraichnan's Direct Interaction Approximation (DIA) and Lagrangian DIA theories have been described in detail in the book by Leslie [1973] who writes that "(t)here is now an increasing body of opinion which holds that, although his work is not conclusive, it does represent a great advance in our understanding, and that much in it is of permanent value." One of the more profound ideas to emerge from Kraichnan's work is the importance of the sweeping of small scales by the large, and of the difficulties associated with the removal of sweeping. The coupling of small and large scale motions can easily be seen mathematically in the non-linear term u • V u or physically in the motion of a small floating object entrained in the eddies of a river and swept along a complicated path by 6 the turbulent flow. Kraichnan [1972] writes: The basic questions about small-scale structure are highly resistant to res-olution by perturbation-related approaches But there is also another, almost unfair difficulty peculiar to the self-convection mechanism. The basic physical idea underlying Kolmogorov's theory is that sufficiently large scales should simply convect small scales, without distorting them and so without affecting the energy cascade process within the small scales. . . . Suppose that each flow system in the ensemble is subjected to a spatially constant transla-tion velocity — This can produce no distortion and therefore cannot affect the energy transfer, spectrum, or other statistics of a homogeneous ensemble. We may call this stochastic Galilean invariance. Kraichnan [1965] was able to correct for the lack of stochastic Galilean invariance in DIA by casting his field theoretic approach in terms of Lagrangian paths, hence La-grangian DIA. Kraichnan's approach was fundamentally correct, and gave rise to im-portant and influential insights into the description of turbulence, but did not provide a convenient technical way to consider all the orders of perturbation theory—only low order truncations were considered. Commenting on Lagrangian DIA, Kraichnan [1975] would later remark: "In higher orders, the term-by-term re-working is unacceptably clumsy, and it is to be hoped that some powerful functional technique will be discovered to re-place it." Further advances in the field theoretic description of turbulence were made by Belinicher and L'vov [1987] and are reviewed in L'vov and Procaccia [1997]. Stochastic Galilean invariance, though pernicious, is a technical concern that may be overcome, to a lessor or greater extent, by a Lagrangian enlargement of the phase space [Kraichnan, 1965; Belinicher and L'vov, 1987]. At a more fundamental level, one is apt to wonder why the renormalization group (RG) method that has proven so successful in the study of critical phenomena in equilibrium statistical mechanics has not yielded a solution to the turbulence problem. As first pointed out by Nelkin [1974, 1975] explicit analogies can be made between turbulence at high Reynolds number and critical phenom-ena provided that the roles of position and wavenumber space are interchanged. Loosely speaking, the R G idea is to look at the system from further and further away so that its microscopic details are eventually wiped out, while in turbulence we examine the sys-tem with a stronger and stronger magnifying glass until the large-scale details disappear. More precisely the lattice spacing a and the integral scale L are analogous and we look for homologous corrections to the spatial correlation function, either spin or velocity, of the form (r/a)^ or (r /L)^ in the limit (T — T c) —> 0 or v —>• 0, respectively. R G has had enormous success in predicting the critical exponent, C, of critical phenomena because these exponents are usually fractal—of the Novikov-Stewart type—while in turbulence a "multifractal" infinite hierarchy of exponents exist. This difference in scaling exponents is a manifestation of a more fundamental difference between the two systems: R G has succeeded in critical phenomena when the deviation from statistical equilibrium is small 7 while the essence of turbulence as a statistical-mechanical problem lies in the necessity to consider strong departures from absolute statistical equilibrium. The huge magnitude of the disequilibrium in turbulence is emphasized by Kraichnan [1975]: Interactions between wavenumbers that differ by as much as two octaves contribute strongly to the inertial-range energy transfer. This shows how strong is the disequilibrium A comparable disequilibrium in a gas of interacting particles would require that the temperature change by its own order of magnitude in a distance comparable to the range of the interaction potential. The analogy here is between Fourier modes of the turbulence and particles of the gas. Perhaps Kraichnan's" greatest contribution to the "turbulence problem", broadly stated, is his work on passive scalars, in particular, the r5-correlated model which he first presented in 1968 [Kraichnan, 1968] and which is now sometimes referred to as the Kraichnan model. Historically the passive scalar problem has received less attention than the velocity problem. A k~5/3 spectrum for the scalar covariance was first predicted by Obukhov [1949] and Corrsin [1951] some ten years after the corresponding theory for the velocity was presented by Kolmogorov [1941], and the first Western experimental confir-mation of A ; - 5 / 3 Obukhov-Corrsin scaling was presented by Pond et al. [1966] (See also Pond [1965]) in agreement with the earlier Russian results from the Tsvang and Gurvich groups [Monin and Yaglom, 1975, Sec. 23.5]. Although Kraichnan originally developed his r5-correlated model to explain the k~l viscous-convective subrange, he would later return to a more general version in 1994 to derive an expression for the series of anoma-lous exponents that mark a departure from A ; - 5 / 3 Obukhov-Corrsin scaling [Kraichnan, 1994; Kraichnan et al, 1995]. As I mentioned previously, Kraichnan's 5-correlated model plays a central role in this dissertation, and a detailed presentation of its derivation and properties is given in Chapter 2. The passive scalar problem has always played an important role in B L M . One of the earliest and strongest statements concerning the importance of understanding boundary-layer transport was made by Priestley [1959]: One may perhaps therefore be excused in pressing the viewpoint herein that, while atmospheric turbulence may be of interest to the aerodynamicist in offering a new scale for his studies . . . its importance to the meteorologist rests largely in the knowledge of the transfers which it provides; that the study of profiles is to him primarily a means to this end and that of microstructure a tool for better understanding of the character of the transfer processes. The modern theory of B L M transport is based largely on semiempirical theories of turbu-lence which combine Reynolds decomposition, dimensional analysis and empirical mea-surements to parameterize B L M transport for various stability and energy considerations and boundary conditions. "Of course, from the viewpoint of "pure" theoretical physics, 8 all these theories must be considered as nonrigorous..." reminds Monin and Yaglom [1971, Sec. 5.9]. The class of semiempirical theories in which time-evolution equations play a secondary role are usually referred to as similarity theories. A comparison of the original texts on atmospheric similarity theory, e.g. Priestley [1959], Lumley and Panofsky [1964] and Monin and Yaglom [1971], with Stull [1988]'s more recent treatment reveals that the fundamental relations characterizing surface-layer (SL) and mixed-layer similarity developed in the 1950s and 1960s are largely used today. One exception might be unstable SL similarity in which three or four distinct approaches are still being debated [Kader and Yaglom, 1990; Stull, 1994; Zilitinkevich, 1994; Grachev et ai, 1997; Chang and Grossman, 1999]. Richardson [1920] first deduced the importance of the ratio of buoyant energy pro-duction to mechanical energy production in a thermally stratified fluid. Subsequently Obukhov [1946] parameterized Richardson's ratio (or number) in terms of the height-dependent ratio z/L where L is now known as the Obukhov length. In a regime of pure convection where mechanical energy production is negligible, e.g. z/L —>• —oo, SL similarity predicts that potential temperature asymptotes to a constant and the hori-zontal flux of potential temperature asymptotes to zero [Prandtl, 1932; Obukhov, 1946]. Experimental measurements refuting these predictions caused Monin and Yaglom [1971] to remark that "the motivation of the obvious discrepancy among these results is still unclear" [Sec. 8.2] and that "some problems exist which quite definitely require due consideration" [Sec. 8.5]. Motivated by the discrepancy between theory and measurement, Zilitinkevich [1971] and Betchov and Yaglom [1971] developed a three-sublayer model ofthe unstably strati-fied SL which includes a new free convection layer at approx. —1 < z/L < —0.1 between the usual mechanical layer and the pure convection layer described above. However, "the data available in 1971 were insufficient for the confirmation of the new theory" [Kader and Yaglom, 1990] and Betchov and Yaglom [1971] raised the further concern that "most of the existing observations on the unstably stratified surface layer of the atmosphere contain no data on the pure convection layer . . . if this layer exists at all in the atmo-sphere." Some twenty years after the Zilintinkevich-Betchov-Yaglom prediction, Kader and Yaglom [1990] presented a detailed study of the unstably stratified SL which con-firms the existence of both the free convection layer and the pure convection layer in the atmosphere. The Zilintinkevich-Betchov-Yaglom model assumes the existence of a mechanical sub-layer close to the ground where the friction velocity, u*, is finite. As early as 1959, Priestley pointed out that a comparison of a mechanical sublayer heat-flux, H ~ z~ll2, and the flux predicted by a convective plume model, H ~ z 1 / 2 , suggests that, in prac-tice, a mechanical sublayer with non-zero u* will exist as z —> 0 in a purely convective boundary-layer [Priestley, 1959, Chap. 6]: . . . the lowest layers will generally, in practice, obey a regime of forced convec-9 tion . . . , but it is of interest to note that in such an environment the solution for the plume will be of the form H ~ z 1 / 2 . . . whereas it follows . . . that H ~ z - 1 / 2 (in forced convection). The implication is that plumes can and are likely to exist in such a regime, while their relative role in heat flux will be negligible near the surface but will increase with increasing height. A mechanism for a non-zero u* in a purely convective boundary-layer was provided a few years later by Kraichnan [1962] who suggested that "turbulence of small spatial scale, generated locally in the boundary layers of the big eddies, can enhance heat convection near the boundary surfaces " The friction velocity was subsequently parameterized as a function of the free convective velocity scale by Businger [1973]. Recently, Stull [1994] has formulated a convective transport theory (CCT) for surfaces fluxes in which convective plume transport dominates mechanical transport as z —> 0. A distinguishing feature of C C T is that surface fluxes are independent of the surface roughness length. I conclude this short anecdotal history with a brief sample of some of the remarks that have been offered from time to time on the future of turbulence research. Frisch et al. [1978] are cautiously optimistic that a dynamical theory will be found: To go further, a genuine dynamical theory starting from the Navier-Stokes equations is needed. . . . So far no analytic tool has been found which allows these scaling laws to be determined theoretically although there have been many speculations that the renormalization-group theory developed for crit-ical phenomena . . . could be appropriate In the meantime, phenomeno-logical models can give important hints as to the necessary structure of an eventual dynamical theory. A year later, Liepmann [1979] chose to emphasize the slow progress made: The scientific study of turbulent flow spans approximately one hundred years; during this time some of the greatest names in physics, mechanics, and engi-neering have at one time or another taken a crack at the problem. Progress in many directions has been made, indeed significant progress. However, the "turbulence problem" as a whole-whatever that means-remains. In the 1980's, Tennekes [1985] suggested that the future of turbulence lies in a dynamical systems approach: I hope that the current research on modons will lead to a workable definition of individual eddies. From that point, we can venture in the direction of multiple interactions between eddies, and on toward chaotic behavior. Finally, we may arrive at a new theory of turbulence, in which coherent structures, strange attractors and a generalized concept of entropy are the cornerstones of understanding. 10 Nelkin [1992] sees little hope in statistical field theory: (T)urbulence . . . remains a fascinating unsolved problem. In the next several years I expect that new carefully controlled experiments, improvements in direct numerical simulation, and better mathematical understanding of the underlying Navier-Stokes equations will combine to increase our basic under-standing. I also hope that basic progress in statistical field theory will add essential contributions from theoretical physics, but nothing so far has given much substance to this hope. Two diametrically opposed assessments of turbulence theory were offered in 1997 and 1998. Tsinober [1998] offers these pessimistic remarks: Indeed the heaviest and the most ambitious armory from theoretical physics and mathematics was tried for more than fifty years, but without much success—fully developed turbulence, as a physical and mathematical problem remains unsolved. . . . turbulence remains among the fields with overproduc-tion of publications without any real breakthrough in understanding. . . . the number of predictions made in the field is limited by the number of fingers on one hand—the rest are correlations after experiments done, i.e. "postdic-tions". L'vov and Procaccia [1997], on the other hand, forecast a solution to the turbulence problem in a statement that is arguably the boldest and most optimistic prediction in the history of turbulence research. Commenting on Sir Horace Lamb's famous remarks (pp. 5), L'vov and Procaccia [1997] claim: Possibly Lamb's pessimism about turbulence was short-sighted. . . . The road ahead is not fully charted, but it seems that some of the conceptual diffi-culties have been surmounted We hope that the remaining 4 years of this century will suffice to achieve a proper understanding of the anomalous scaling exponents in turbulence. Progress . . . will bring the theory closer to the concern of the engineers. The marriage of physics and engineering will be the challenge of the 21st century. Ironically, though L'vov and Procaccia published 4 papers in 1995 and 7 papers in 1996 on their new theoretical approach paving the way for their bold prediction, the prediction itself would be their last contribution together in this endeavor. A more tempered opinion, lying somewhere between the remarks of Tsinober [1998] and L'vov and Procaccia [1997], is offered by Sreenivasan [1999]: Extrapolating from experience so far, future progress will take a zigzag path, and further order will be slow to emerge. . . . There is ground for optimism, and a meaningful interaction among theory, experiment, and computations must be able to take us far. It is a matter of time and persistence. 11 And finally, we return to Wyngaard [1998b] for a last assessment: In turbulence research one sees periods of feverish activity in what can be-come blind alleys, punctuated by occasional leaps forward. The long view reveals slow, steady progress in the physical understanding of the structure of turbulent flows in engineering and geophysics, accumulating evidence of the futility of head-on analytic attacks on turbulence, and repeated indications that experiment paces the growth in our understanding of turbulence 1.2 Problems addressed in this dissertation The difficulty in defining the turbulence problem reminds me of a cartoon in which a rather puzzled and dejected-looking researcher is introduced to a visitor: "After twenty years of research, Dr. Quimsey developed the answer and now he's forgotten the question!" [Liepmann, 1979] In this dissertation I address a number of topical questions that have been raised about the spatial structure of cloud number density and liquid water density at small scales O(10 cm) and about the prognosis of unresolved low cloud optical depth at large scales 0(100 km). Below I summarize the problems addressed and the results obtained in Chapters 3 through 6. Note that each chapter begins with a general introduction and ends with a summary so there is some redundancy in this presentation. The work presented in this dissertation also appears in the following articles: Chapter 3: Jeffery, C. A . , Effect of particle inertia on the viscous-convective subrange. Phys. Rev. E, 61(6), pp. 6578-6585, 2000. Chapter 4: Jeffery, C. A . , Investigating the small-scale structure of clouds using the r5-correlated closure. To appear, Atmos. Res., 2001. Chapter 5: Jeffery, C. A . , Effect of condensation and evaporation on the viscous-convective subrange. Phys. Fluids, 13(3), pp. 713-722, 2001. Chapter 6: Jeffery, C. A. and P. H. Austin, Unified treatment of the physical and optical variability of unresolved low clouds. J. Atmos. Sci., Submitted. Jeffery, C. A. , Parameterization of shortwave cloud properties in large-scale models: Is effective radius necessary? J. Atmos. Sci., Submitted. 1.2.1 C l o u d droplet number concentrat ion inhomogeneit ies Recently both Pinsky and Khain [1997] and Shaw et al. [1998] have suggested that cloud droplet inertia results in the formation of fine-scale concentration inhomogeneities that may lead to a modification of the droplet number spectrum. However, the mechanism by 12 which small scale variability of the droplet concentration impacts the number spectrum as suggested by these authors is distinctly different. Pinsky and Khain [1997] introduce "inertial drop mixing" whereby two small neighboring volumes of air exchange drops according to their velocity flux divergence under the assumption that the total number of droplets in the two volumes is conserved. As a result, two volumes with greatly differing number distributions at cloud base become homogenized as the volumes are lifted adiabatically, while two volumes that are identical at cloud base remain homogenized but will be stochastically different at cloud top. Results from a more refined model of inertial drop mixing are discussed in detail in Pinsky et al. [1999]. The Shaw et al. [1998] hypothesis, on the other hand, is distinctly different and much bolder. In their model of inertial effects, vortex tubes and thus the intermittency of large Reynolds number atmospheric flows plays a central role. Although a velocity intermittency effect had been hypothesized earlier [Tennekes and Woods, 1973; Cooper and Baumgardner, 1989], in the Shaw model the geometry of the fine-scale structure plays a key role for the first time. In contrast to Pinsky and Khain [1997]'s inertial mixing, Shaw et al. [1998] argue that the clumping or preferential concentration of cloud droplets leads to a greater segregation of droplets into regions with varying microphysical conditions resulting in droplet spectral broadening. Thus we have the Shaw model of "inertial drop segregation" in contradistinction to Pinsky and Khain [1997]'s inertial drop mixing. The Shaw model is based on two key assumptions that have been criticized [Grabowski and Vaillancourt, 1999; Vaillancourt and Yau, 2000]. The first assumption is that velocity intermittency leads to the clumping of cloud droplets which would not otherwise clump in a low Reynolds number flow. As pointed out by Grabowski and Vaillancourt [1999] the ratio of a cloud droplet's inertia to the velocity field inertia, known as the Stokes number, is too small for significant preferential concentration to occur in the absence of intermittency effects. The second assumption is that the vortex tube turn-over time is 0(10 s) such that individual drops become "trapped" in the tube and are prevented from mixing with the surrounding environment. However, this time-constant is 1-2 orders of magnitude larger than the Kolmogorov time. In Chapters 3 and 4 I address the following questions: (i) At what Stokes numbers and spatial scales does clumping occur? (ii) Does velocity intermittency lead to increased clumping? and (iii) If so, do vortex tubes (cylindrical vortices) play a special role in this increased clumping? To address these questions, I derive an analytic expression for the scalar spectrum of inertial particles using Kraichnan's r5-correlated closure in Chapter 3 and, in Chapter 4,1 introduce an effective Stokes number that incorporates intermittency effects. Using the effective Stokes number I am able to assess the preferential concentra-tion of cloud droplets at large Reynolds numbers in a quantitative manner for the first time. 13 1.2.2 C l o u d l iqu id water density inhomogeneit ies Recently, Davis et al. [1999] presented horizontal spectra of cloud liquid water density measured at an unprecedented resolution of 4 cm during the winter Southern Ocean Cloud Experiment (SOCEX). The spectra exhibit both an expected k~5/3 inertial-convective subrange and a k~l viscous-convective subrange. However, the scale-break between these two regimes is anomalous—typically the viscous-convective subrange begins near 10 cm in the atmosphere whereas in Davis et al. [1999]'s spectra the scale-break is greater than 1 m. This enhanced liquid variability at small-scales suggests that a source of cloud liquid water variance is present. In Chapter 5, I consider the effect of condensation and evaporation on the viscous-convective subrange—again using Kraichnan's 5-correlated closure—and I attempt to explain the presence of the observed enhanced liquid water variability. In particular, I argue that variability in cloud droplet mass due to condensation/evaporation results in a production subrange where the scalar dissipation rate increases with increasing k. This source of liquid water variability is contrasted with the effect of particle inertia in which a source of cloud number concentration variability is present at small scales. One of the more interesting ideas to emerge from my approach is the importance of the vertically non-homogeneous structure of liquid water fluctuations which leads to a liquid water spectral density that is complex (i.e. real and imaginary components). 1.2.3 Unreso lved low c loud opt ica l propert ies It has been known for some time that the plane-parallel-homogeneous assumption, where-by the mean solar reflectivity of a stratiform cloud layer is given by the reflectivity of the mean optical depth, leads to an overestimation of the prognosed reflectivity [McKee and Cox, 1974; Welch and Wielicki, 1984, 1985; Harshvardhan and Randall, 1985; Schertzer and Lovejoy, 1987; Cahalan et al., 1994; Kogan et al, 1995]. This follows trivially from the convexity of the function that relates reflectivity and optical depth. To reduce this bias many current global climate models (GCMs) multiply the prognosed mean optical depth by a constant factor x ~ 0-7 in the calculation of the average reflectivity [Cahalan et al, 1994]. Although x m a v be tuned in a particular G C M to reproduce the mea-sured radiative stream, this approach is ad-hoc in nature and may become increasingly inaccurate as the climate departs from its present state. A partial solution to this problem was pioneered by Barker [1996b] who approximated the distribution of optical depths by a 7-distribution specified by its mean and variance. This approximation is advantageous because a 7-distribution well represents satellite retrieved distributions of marine low cloud optical depth, and an analytic expression for the mean reflectivity (and emissivity) is available [Barker, 1996a; Barker et al., 1996]. However, it does have one drawback. A G C M must prognose both the variance and the mean of optical depth in the cloud field, a problem which Barker [1996b] left unsolved. 14 In Chapter 6 I offer a solution to this problem by coupling prognosed unresolved optical variability to the unresolved cloud physical variability predicted from a statistical cloud scheme. A key feature of my scheme is that the coupling of optical and physical variability is achieved through a simple linear model of cloud liquid water content in which the dynamically driven cloud-top height fluctuations and the thermodynamically driven temperature and moisture fluctuations are treated explicitly and separately. In Chapter 6 I also consider the effect of poor model vertical resolution on the prognosed cloud physical and optical variability, and I illustrate, using simple analytic response functions, that low resolution leads to an overestimation of the cloud fraction response. 15 Chapter 2 The ^-Correlated Model Does the Wind possess a Velocity? This question, at first sight foolish, improves on acquaintance. [Richardson, 1926] 2.1 Introduction In this chapter we consider the advection-diffusion equation for the concentration of a passive scalar in a velocity field. The velocity field is assumed to be turbulent which introduces difficulties but some simplifications. A turbulent velocity field is a random field in the sense that only the mean statistical properties are known; we do not have knowledge of any particular realization of our velocity field. On the other hand, a turbulent velocity field is distinctly different from a spatially-correlated Gaussian random field because there is structure in any given realization. In general, structure is apparent in the small scale statistics—particularly the velocity gradients—but absent in the larger scale statistics. In this chapter we ignore many of the statistical subtleties of a real turbulent velocity field and assume that our velocity field is Gaussian and that the two-point spatial correlation is known. In Chapter 4 we consider a particular implication of non-Gaussian velocity statistics in detail. The only real mathematical advantage afforded by a random turbulent velocity field is that the statistics of the velocity field at scales smaller than stirring scale are, to good measure, locally isotropic and locally homogeneous. Unfortunately, this does not imply local isotropy and homogeneity of the passive scalar statistics [Holzer and Siggia, 1994]. Our goal in this chapter is to derive equations for the equal-time 1-point and 2-point moments of a passive scalar in a 5-correlated in time velocity field. We could begin this derivation with the advection-diffusion (AD) equation, a well-known Eulerian statement of conservation of particle number (or mass) where the particle trajectories have a Brownian component. In fact, most reviews on passive scalar statistics in turbulent flows begin with A D equation [Pumir et al, 1999; Warhaft, 2000]. However, many ofthe tools used to solve the A D equation in the (^-correlated limit are the same tools needed 16 to derive the A D equation itself. Thus, instead, this chapter begins with a presentation of particle dynamics in a generalized Hamiltonian framework. Hopefully, the generality of the methodology and presentation will be of value to those readers familiar with Hamiltonian dynamics but with no specialized knowledge of the turbulence problem. Section 2.3 is a more formal discussion of the 5-correlated model using concepts taken from Piterbarg and Ostrovskii [1997]. In Sec. 2.4 we derive an equation for the mean scalar concentration in the 5-correlated limit using the methodology developed in Sec. 2.2, and in Sec. 2.5 we extend our derivation to the equal-time 2-point variance. Section 2.6 details the history and validation of the 5-correlated model and also summarizes some recent advances in this field. For clarity and brevity, we first introduce the notation and terminology. Vectors appear in bold print while scalars and norms of vectors appear in normal print. In this chapter components of a contravariant vector are denoted by a superscript, whereas a subscript is used for the components of a covariant vector. This convection will be dropped in later chapters where the contravariant/covariant distinction is not necessary. Vector components are labeled by Roman or Greek indices. Repeated Roman indices imply summation, but no summation is implied by repeated Greek indices. A centered dot denotes a scalar product, i.e. £ • £ = £ 2 . The subscript of a contravariant vector is a label. Thus given a contravariant vector £, £ Q = £a while £ { is simply £ labeled by i. The symbol (• • •) represents an ensemble average. Homogeneity refers to statistical invariance under spatial translation while isotropy refers to statistical rotational invariance. In particular, given a random function f(x, y) where x and y indicate spatial position, homogeneity of f implies (f(x, y)) = (f(y — x)) while homogeneity and isotropy demands that (f(x, y)) = (f(\y — x\)). A partial derivative with respect to some parameter, p, is denoted by d/dp. Partial differentiation w.r.t. time is also denoted by an over-dot, and w.r.t. space by V a = d/dxa. Brackets are used to denote the scope of a partial derivative if necessary, e.g. V afg = (Vaf)g + (V a c / ) / . Principal symbols are listed in Appendix A . 2.2 Hamiltonian Fluid Mechanics At first glance, Hamiltonian fluid mechanics is a strange starting point for a discussion of the turbulent mixing of a passive tracer because of the essential role that viscosity plays in turbulent dynamics. Certainly, dissipative systems are non-Hamiltonian. However, if we focus on a subset of the phase space, in this case the Lagrangian coordinates of our particles, then we can derive the A D equation within a Hamiltonian framework although kinematic viscosity does not appear explicitly in this formulation. With the notation and methodology in hand from this exercise, we are well prepared to tackle the closure of the A D equation in future sections. The theory of Hamiltonian dynamics and turbulent diffusion is presented in great detail in any number of excellent texts [Goldstein, 1980; Arnold, 1980; Dubrovin and Fomenko, 1992; Beris and Edwards, 1994] and reviews 17 [Morrison, 1982; Bennett, 1987; Salmon, 1988; Stull, 1993; Bennet, 1996; Klyatskin et al, 1996; Majda and Kramer, 1999]. The treatment here will thus be appropriately terse. 2.2.1 Lagrangian formulation Consider an individual parcel or element of a passive tracer in a fluid labeled by the parcel position x = a at time t = 0. Associated with this parcel is an initial scalar density 9(a,0). The parcel is shrunk to zero size so that there is one for each point in the fluid, while retaining a non-zero density 9. Although there are some conceptual difficulties with this Lagrangian formulation of zero-size finite-density tracer parcels, there are no mathematical difficulties because the infinite number of degrees of freedom in the corresponding continuous Eulerian formulation is preserved. The position of the a-labeled parcel at time t, £a(t) determines the Eulerian vector field a(x, t) by identifying x with £: X = ^ a ( x , t ) ( i ) ; or alternatively The labels a(x,t) are, by their very meaning, unchanged along the path of the parcel so that the labels obey the Liouville equation ~a(x,t) = 0 (2.1) where the total derivative D/Dt is given by D/Dt = d/dt + £ a • V . The parcel velocity • " p l ^ a W ^ ) = £ a W is> m general, different from the velocity of the fluid elements due to Brownian motion of the parcels. It is also different if the tracer particles (and hence parcels) have significant inertia. As a side note, one of the more subtle and significant ideas in classical mechanics is the connection between the conservation of parcel labels, Eq. (2.1), and potential vorticity. Ertel's theorem provides the way of translating the conserved parcel labels a into the conserved quantity # _ 1 ( V x up) • ( V a ) , i.e. potential vorticity. The symmetry corresponding to potential vorticity conservation is a parcel-relabeling symmetry since parcel-label variations leave the parcel density and entropy unchanged. One of the first connections between the parcel-relabeling symmetry property and the general vorticity conservation law was made here at U.B.C. by Calkin [1963]. In addition to £ we have the conjugate Lagrangian flux or tracer momenta ir f l(t)=i f l(t)0(o ,0). (2.2) The Lagrangian vector fields £,a(x,t) and 7r a(cc,i) describe the full phase space of the tracer. A variational action principle approach, provided the parcel velocity is inviscid, 18 leads to a Hamiltonian formalism for the dynamical evolution, 3F_ dt {F,H} (2-3) where {,} is the canonical (symplectic) Poisson bracket, H is the Hamiltonian, and F is any functional on the space of dynamical variables. For our purposes we restrict F to be £ a or the Eulerian scalar density 9(x,t). The canonical Poisson bracket is defined by [Poisson, 1809; Arnold, 1966] where 6 indicates a (Volterra) functional derivative. It follows from Eq. (2.4) that the Poisson bracket is antisymmetric w.r.t. functionals F and H, and therefore dH/dt = 0— energy is conserved. Equation (2.4) also gives {^,7r J '} = S1^ which is the classical analog ofthe well-known quantum-mechanical commutation relation [£ J ,7r J ] = ihS^. Restricting ourselves to the dynamics of a passive tracer, we require only the component of the Hamiltonian H(t) that contains the kinetic energy of the tracer: By definition a passive scalar does not contribute to the potential energy of the fluid. Using S£,%/5t;P = 5af}5(r—a) and dt;%/5irr = 0, a simple calculation recovers the definition of the canonical flux, Eq. (2.2), from Eqs. (2.3-2.5). There are a number of different ways of expressing conservation of (tracer) mass for the Eulerian scalar density 9(x,t). We will pursue these various derivations in this subsection, and in Sees. 2.4 and 2.5 we verify that our independent formulations of mass conservation lead to consistent expressions for scalar mean and covariance in the ^-correlated limit. Readers who object to this inherit redundancy may wish to skip to the following subsection. For the Eulerian specification it is convenient to label the parcels by their arrival at (x,t), i.e. £xt(i) = x. Associated with a tracer parcel is a volume element at some time t' given by d3£Xit(tf). Though the mass of the parcel must remain constant, its volume element may vary if the flow is compressible. Conservation of mass assumes the form (2.4) (2.5) 2.2.2 E u l e r i a n f o r m u l a t i o n mx,t(t),t)d%,t(t) = 9(£Xtt(t'),t')d%,t(t') which, rearranged, gives, 9{x,t) = g (* , ,«(*V) Mx,t) (2.6) 19 The Jacobian Jti(x,t) — \dxa / d£xt(t')\ defines the fractional change in parcel volume between £x>t{tf) and x. Taking the partial derivative of Eq. (2.6) with the limit t' —> t gives D9(T. i\ f)77l (cr. t\ (2.7) D e M =e(x,t)dJ^t{x't) Dt v ' ' dt Similarly evaluating d/dt' of the Jacobian gives dje(x,t) _ diitt(t') ^ ( dt' ~ da>t(t>)Jt'[X>t} with initial condition Jtt(x,t') = 1 and solution Jt'{x,t) = exp / V •up(£Xtt(e),o)do-Jv (2.8) Substituting Eq. (2.8) into Eqs. (2.6) and (2.7) we arrive at two equivalent expressions for the Eulerian density 6(x,t): the usual Markovian, Eulerian conservation law D9(x,t) Dt = -(V • up)9(x,t) and a non-Markovian path integral formulation 0(*,t) = 0(^,(0) exp Jv (2.9) (2.10) Equations (2.9) and (2.10) were known to Liouville [1838]. In general Eq. (2.10) is less tractable than Eq. (2.9) because of its explicit dependence on the Lagrangian, non-Markovian, parcel position ^Xtt(cr). To derive the A D equation from Eq. (2.9) we first separate the total parcel velocity up into advective and Brownian components: up(x, t) = u(x,t) + V2/cw(t) (2.11) where K is the molecular diffusivity and w is a centered Wiener process satisfying (wa{t)wp(s)) = 5al3mm(t,s), and the advective parcel velocity u is equal to the fluid velocity in the absence of inertial effects. Substituting Eq. (2.11) into Eq. (2.9) gives d9(x,t) dt + V • (u9(x, t)) + V2Kw(t) • V9(x, t) = 0. (2.12) A useful aid in our derivation of the A D equation is the Furutsu-Novikov formula [Furutsu, 1963; Novikov, 1964; Frisch, 1995; Klyatskin et al, 1996]. 20 Furu tsu-Novikov formula. Let v(s) be a vector-valued centered Gaussian field and let F[v] be any differentiable functional. Then, assuming all averages exist, (v°(a)F[v]) = J ds' (j^m) Bm(s, s') (2.13) where Ba)3(s,s') = (va(s)vl3(s')). Equation (2.13) is just functional Gaussian integration by parts. Note that (5/5vlF) is an operator that is free to act on Bat. Using the Furutsu-Novikov formula, Eq. (2.12) becomes where (• • - ) w represents an average over the Wiener process w. To evaluate the functional derivative S9/6wa we return to Eq. (2.12): Swa(t') Jt, Swa(t') Jt, v ' dxa The first term of the r.h.s. goes to zero as t' —> t, while f*, daS(t' — a) = 1/2. Thus we arrive at the A D equation: + V •{ue(x,t)) = rzV29(x,t), 6(x,0) = dQ(x) (2.14) where #n are the initial conditions at t — 0, and we have dropped the (• • - ) w notation. Equation (2.14) although lacking an explicit w-dependence remains a it-dependent ran-dom process. Another useful Eulerian statement of conservation of mass can be derived using indi-cator functions. Consider the Lagrangian indicator function Rx^t(o) = 5(a — £ - 1 (a : ,£) ) which, like a(x,i), is unchanged along the path of the parcel so that DRXjt(a)/Dt = 0. The corresponding Eulerian indicator function is Ra(x,t) = 5(x - Ut)) = (2-15) Jo(x,t) The function Ra(x,t) is the evolution kernel for the A D equation because the scalar density 9(x,t) is, by definition, the convolution of Ra(x,t): 9(x,t) = Jd?r 9(r,0)Rr(x,t) (2.16) with initial conditions at t = 0. The equivalence of Eq. (2.16) with the other statements of mass conservation [Eqs. (2.9) and (2.10)] is immediately evident; substituting Eq. (2.15) into (2.16) recovers Eq. (2.6). Furthermore, averaging Ra(x, t) over the w-ensemble gives the Fokker-Planck or forward Kolmogorov equation dRa&'Q + V • (uRa(x, t)) = KV2Ra(x, t), Ra(x, 0) = 6{x - a) (2.17) 21 where again the (• • - ) w have been dropped. It is interesting to compare the A D equation for 9, (2.14), and Eq. (2.17) for Ra(x,t). The equations are the same but with different initial conditions. This reflects the different mathematical content of their solutions: Ra(x,t) describes the it-dependent probability density of a particle's position while 9 describes the tracer concentration in the Eulerian picture. However, their behaviour is identical if 9 is the Green's function solution for an initial tracer concentration that is a point source, i.e. 90(x) = 5(x — a). A path integral solution for Ra(x,t) can be obtained that is quite distinct from the path integral formulation for 9(x,t), Eq. (2.10). In this solution the most probable parcel trajectories minimize the Onsager-Machlup action [Onsager and Machlup, 1953]. The derivation presented here follows Risken [1996]. Consider the transition probability density of a particle at (a, 0) to a neighboring point X\ after a small time A i : Ra(xi, At). Integrating Eq. (2.17) in time from 0 to A t gives Ra(xi, At) = exp ( - V • uAt + nV2At) 5(xx - a). Writing the 5-function as a Fourier integral, we have 1 ( \ x i - a - uAt\2\ which is simply the Green's function solution for small X\ — a and A i . Following the general path-integral formalism of Feynman and Hibbs [1965] we then denote the path from (a = a?0,0 = to) to (x = xn,t = tn) by a series of intermediate steps (xi, ti), (x2, t 2 ) , ( x n - i , i„_i), which define a "path" with At = t/n. Since Ra(xi,ti) is a (it-dependent) probability, we have that Ra(x,t) = lim (T[R*.it.(xi+1,U+i)) (2.18) n—>oo \ -*- / \i=0 / w « . ( - , * ) = M m / - / n { [ 4 x K A r ^ ^ } e x p | _ g N + . - ^ . - ^ > ^ A f j where the integration is over the (tt-dependent) Wiener paths. To obtain the Onsager-Machlup solution we write Xi+\ — Xi — £ a ( i ; )A i and approximate the sum in the expo-nential by a time integral giving Ra(x,t)= / d [ £ o ] e x p { - — / d a [ U a ) - u ( U a ) , a ) n (2-19) where d[£a] stands for the measure on the set of paths {£ a(cr)} over which we integrate. Substitution of Eq. (2.19) into (2.16) gives another expression for 9(x,t). The utility of Eq. (2.19) comes from its Gaussian character—for Gaussian functions, integrating over its argument under given constraints gives, apart from a constant factor, the same result as maximizing the Gaussian under the same constraints. 22 The function [£ 0 — U] 2 / (4K) in the exponential on the r.h.s. of Eq. (2.19) is known as the Onsager-Machlup action; the dominant contribution to Ra(x,t) arises from those trajectories that minimize this action. Clearly, in the weak noise limit K —>• 0, the r.h.s. of Eq. (2.19) converges to S(x — £ a ) with £ a = u. For vanishing noise the Onsager-Machlup action can be evaluated by steepest descent, but in the strong-noise case a "classical" generalization of the quantum-mechanical Rayleigh-Ritz variational method is needed [Eyink, 1996]. The last exercise in this subsection is to demonstrate equivalence between the Hamil-tonian evolution equation, (2.3), and Eq. (2.9). To do so we must first determine the non-canonical Eulerian variables that correspond to our Lagrangian, canonical (in the in-viscid limit) vector fields £ and 7 r . The Eulerian variables are 9, u, entropy and potential vorticity. Only 9 and u are needed for our task. Traditionally, however, 9 and the cur-rent J = 9u are chosen instead because J and 9 form a Lie Algebra. The Hamiltonian, Eq. (2.5), in the Eulerian specification is J7w = / A s S + - (2-20) Using the chain rule we have that SH[9, J] _ /• 3 6H SJtfat) Siva ~ J X SJifat) 8ira where we have taken advantage of the fact that 59/5-n = 0. Using the Poisson bracket relation, Eq. (2.4), the bracket {9, H} becomes {9{x,t),H(x,t)} = J d?z Jdzr 89(x,t) 6H(x,t) SJtfat) 5£r SJ{{z,t) 8irr I d3z{9(x,t),r(z,t)}5^f) (2.21) in agreement with Eq. (6.3) in Salmon [1988]. From the definition of H, Eq. (2.20), it follows trivially that SH(X>*) - J a ^ 5 ( x - z) (2 22) 8J*(z,t) " 9(x,t) d [ X Z)- [ 2 - Z Z ) The last step is evaluation of {9, Ja} using Eq. (2.4). The interested reader is referred to Chapter 5.3 of Beris and Edwards [1994] for further details. The final result is {9(x,t),Ja(z,t)} = --^S(z - x)9(x,t) (2.23) where the corresponding quantum-mechanical analog [9(x), Ja{z)] contains an additional factor ih on the r.h.s. [Dashen and Sharp, 1968]. Inserting Eqs. (2.22) and (2.23) into Eq. (2.21) gives {9(x,t), H(x,t)} = - V • J(x,t) 23 and thus, along with Eq. (2.3), we have the usual mass conservation equation, (2.9). Unfortunately, our focus in this section on mass conservation has obscured the elegance of the Poisson bracket formalism and its generality; it applies to almost all areas of physics, from classical mechanics and electrodynamics to quantum mechanics and the new theories of subatomic particles, and it is applicable to nonlinear stability analyses. In fact, quantum mechanical commutators emerged from the Poisson bracket formulation of classical mechanics about three quarters of a century ago. The scalar density moments (6n) follow from Eqs. (2.10), (2.14) or (2.16) after integra-tion over the u-ensemble. Herein lies the difficulty. The turbulent velocity field u(x,t) has complex spatio-temporal correlations that make this integration, under general cir-cumstances, exceedingly difficult. We make no attempts to solve this problem in this dissertation. Rather, we restrict our attention to a specific passive scalar regime, called the viscous-convective subrange, where scale separation can be used to greatly simplify the temporal properties of u. Consider the statistics of 6(x,t) at some scale Ig. For the sake of simplicity, consider a homogeneous, isotropic velocity field with a single correlation time ru(lg) and standard deviation au(lg). In this case there are four independent time scales: • The observation time, t. • The Eulerian correlation time, rg = ru(lg). • The eddy turnover time, TT = le/o-u(lg). • The molecular diffusion time, To = IJ/K. If these four time scales have the same order of magnitude then this scale classification will not help us simplify the statistics of u—we are back to square one. However, if one or more of these scales is much smaller or bigger than the others, we can renormalize or rescale our velocity field such that this scale separation becomes infinite. Hopefully, the statistics of the renormalized velocity field will be more analytically tractable than the statistics of the original field. Formally, we accomplish this renormalization by defining 6 > 0 to be a small dimen-sionless parameter that modifies the relevant scales, and then we take the limit 5—^0. Consider the rescaled variables t' = a(6)t and where x is not rescaled and a(6) and (3(6) are some dimensionless functions. Our interest lies in a renormalized system where the temporal properties of the velocity field rapidly 2.3 (^-Correlated Model 24 decorrelate in time, but the renormalized scale-dependent eddy-diffusivity, K~s(lg), has a finite non-zero value. The last condition is crucial; if we carelessly modify the temporal properties of the velocity field we are apt to end up with a velocity field that transports mass infinitely fast or not at all. Dimensional analysis suggests Kg ~ o>(32, so that l i m ^ o Kg is well behaved if a(S) = [3~2(5). Taking (3 = S'1 and a = S2 we have that us(x,t') = 5~1u(x,^j . (2.24) Given our rescaling functions a and f3, we can now determine how the time scales TE, TT and TD diverge in the limit 5—^0. From the relations TE ~ 52, TT ~ 5 and rD ~ 1 it follows that [Piterbarg and Ostrovskii, 1997] r J 5 < r T < i - r D . (2.25) The simplification in the statistics of the renormalized velocity field defined by Eq. (2.24) is significant. Assuming that u is a Gaussian field completely determined by the space-time correlation tensor (ua(x + r, t + s)up(x, t)) = Fap(x + r,t + s;x,t), then under certain general conditions Us defined by Eq. (2.24) converges to a white noise process in the sense of distributions: \imFf{x + r,t + s-x,t) = 2r5(s)Faf}(x + r, x), (2.26) where r ~ TE is the renewal rate, r = Ig and Ig <§C lo where IQ is the integral length scale. The assumption that the velocity field has long-range spatial correlations but no memory as defined by Eq. (2.26) is known as the 5-correlated model or closure, the Kraichnan model or the Batchelor limit. It is important to emphasize that the renormalization defined by Eqs. (2.24) and (2.26) is not a uniformly valid theory in the large scale limit (Ig —>• l0) because of the strong infrared divergence in the A : - 5 / 3 Kolmogorov velocity spectrum. Therefore K(lg) obtained under this rescaling should not be interpreted as the bulk velocity-field eddy diffusivity K(l0). The correct renormalization in the large scale limit for the 5-correlated model with Kolmogorov velocity statistics can be found by computing a(S) such that the ensemble average of the rescaled passive scalar field (ds[x/8,t/a(5)]) has a non-trivial limit. This was done by Avellaneda and Majda [1994] who found the super-diffusive scaling a = 54/3 and concluded that K is unbounded satisfying K(l0) -> oo as Z0 —>• oo. Thus the second-order correlations are not well approximated by a simple Gaussian at large distances. Our last task in this section is to determine in what regime or subrange the scale separation given by Eq. (2.25) holds. Consider the regime where rjB < Ig < r\ where 25 n = ( z / 3 / ^ ) - 1 / 4 is the Kolmogorov length, r\B = Pr _ 1 / / 2?7 is the Batchelor length, e is the kinetic energy dissipation rate, u is the kinematic viscosity and Pr = U/K is the Prandtl number. Clearly, for T\B <C n we require Pr >^ 1. Since Ig < n the velocity field parameters are given by the Kolmogorov time T„ = (u/e)1^2 and the Kolmogorov velocity an = (z^e)1/4. Thus our time scales are TE ~ e~1 / /2, TT ~ e _ 1 / / 4 and TD ~ Pr. If we equate our renormalization parameter 8 with e~2 and take the large Reynolds number limit e —> co we recover our original time-scale separation given by Eq. (2.25). The regime defined by T\B < Ig < rj, Pr 3> 1 and the limit e —> oo is known as the viscous-convective subrange. In Chapters 3 and 5, the spatial covariance of 9 in the viscous-convective subrange is of primary interest. In contrast, in the inertial-convective subrange it is likely that TE ~ TD ~ e - 1 / 3 and therefore renormalization leads to homogenization [Piterbarg and Ostrovskii, 1997]. 2.4 Mean Scalar Concentration In this section we derive an equation for (9(x, t)) in the 5-correlated limit where the space-time correlation function of u is given by Eq. (2.26). For the sake of completeness and as a check of model consistency we offer two derivations: the first begins with Eq. (2.14). and the second from Eq. (2.10). Consider the A D equation, (2.14). Averaging over the u-ensemble, and using the Furutsu-Novikov formula, Eq. (2.13), we find d(9(x,t)) d / \ 3 „ / \ / 8 dt where (• • •) represents an average over the u-ensemble. Inserting the 5-correlated corre-lation function, Eq. (2.26), gives d(9(x,t)) + d f j 3 J 8 dt dx Returning to the A D equation, we calculate 89/8ua d9{x,a) d — 9{x, a) da ( K V 2 -+ /  ( 2 - V • u(x, cr)) dxa v ' 'dxa 89{x,a) 8ua(r,t')' The second term of the r.h.s. goes to zero as t' —> t, while J*t* da8(t' — a) = 1/2. Thus we arrive at [Piterbarg and Ostrovskii, 1997] 5 -9(x,t) = -l8(x-r)^-9(x,t). (2.28) 8ua(r,t) v ' ' 2 v 'dxc 26 Note that a closed form solution of 89(t)/8ua(t') is not available for t ^  t', hence the utility of the ^-correlated simplification. Inserting Eq. (2.28) into (2.27) gives m ^ t ) ] = VlTF^(x,x){e(x,t)) + KV2(e(x,t)). (2.29) Equation (2.29) can be simplified further if we note that FlJ(x,x) is constant in a ho-mogeneous velocity field. Defining an eddy-diffusivity Kafl = TFa(t(x,x) + K5a(>, which includes K, Eq. (2.29) becomes 'N K^VKd). (2.30) dt Thus (9) is described by an diffusion equation with an eddy-diffusivity K that is indepen-dent of the compressibility of the velocity field. This result should come as no surprise because in Sec. 2.3 we chose the rescaling parameters a and (3 such that the limit 5—^0 of Ks is finite. Following the procedure outlined in Elperin et al. [1998, Appendix A] we can also derive Eq. (2.30) from Eq. (2.10). This is possible because in the 5-correlated limit the non-Markovian, Lagrangian statistics of the path-integral formalism of Eq. (2.10) con-verge to Markovian, Eulerian statistics. Formally, this transformation from Lagrangian to Eulerian statistics is implemented by expanding the relevant parameters: 9, J', u and £ to small order in Ar . The concentration 9U at some near time t+At is given by (9(x,t+At)) = <Jf ^ t + A * ) ^ ^ * ) , * ) ) (2.31) where (• • •) represents an average over both the w and u ensembles. First we expand 9, the velocity compressibility b = V • u and u in Taylor series expansions: o(t.,«±t(t),t) = 0(x,t) + ^ ^ ^ X ! t + A t ( t ) - x r + l^^^^ - * ) m t e . W * ) - XT + • • • (2-32) Kt*M*t{°U) = K^t) + ^ ^ ( i X t t + M ( a ) - x r + --- (2.33) ua(Z»,*±t(<r),<r) ~ ua(x,t)-Aa(x,t)(a-t) + --- (2.34) where Aa = umVmua. Substituting Eq. (2.34) for u into the definition of (£ — x) gives /t+At ua(^t+At(a)^)da + V2rlwa(At) « -ua{x,t)At + Aa(x,t)^~ + \ / 2 W * ( A t ) . (2.35) 27 To calculate J 1 we have (t*,HAt(*) ~ *Y = ~ua(x, t)(t + A t - a) + • • • needed in Eq. (2.33) such that [Elperin et al, 1998, Eq. (A10)] ft+At ft+At db(x t) J K$Xjt{At(a),o-)da « J dab(x,t)-^^-um(x,t)(t + At-a) = b M A t - ^ ( x , ^ ^ ^ L . (2-36) Substituting Eq. (2.36) into Eq. (2.8) for J~l gives Jt-\x,t+At) * 1 - b(x,t)At + um(x,t)db^XJ) {Af + b2(x,t){-^. (2.37) Substituting Eqs. (2.32,2.35,2.37) into Eq. (2.31) and keeping terms up to (At ) 2 gives (0(t+At)) = ^ l - b A t + u m V m b ^ + b2^^J9(t) + (1 - bAt) (-umAt + A m ^ - + V2^wm) \ 2 / dxm + U-umAt + \ / W " ) ( - u n A t + \ / W M d 2 ^ ) \ . (2.38) 2 V y v ' dxmdxn / v y The next step is to demonstrate (umVmb) — —(b2) in a homogeneous flow: /u-(M«M = lim/U-(y,«)^ V \ v ' ' d x m / w->*\ v y y d x m / a 2 = l i m - — (um(y,t)un(x,t)) = -(b2{x,t)). Of course, we also have (Aa) = 0. Averaging over both the w and u ensembles gives (0(x, t+At)) = (0(x, t)) + \ (F™(At)2 + 2K8™At) • (2-39) To derive an expression for dO/dt from Eq. (2.39) it is necessary to evaluate the discrete derivate [9(t + At) — 9(t)]/At for small At . Although it is customary in non-stochastic calculus to take the limit A t —¥ 0, this is not appropriate here since our white-in-time velocity is only defined for A t = nr with n > 1. Forming the continuous derivative in the limit A t -+ 2r recovers Eq. (2.30). 28 2.5 Mean Scalar Covariance Following the methodology of Sec. 2.4 we derive the equal-time two-point covariance from both the A D equation (2.14) and from the path integral equation (2.10). Consider the correlation function $ = (6(a;)6(y)) where 0 = 9 — (9). Multiplying the A D equation by O(y), averaging over the u-ensemble and using the Furutsu-Novikov formula, Eq. (2.13), gives § + 2^/<i37 «fc( s s ^ J e(- . ' )e(»,t)>^(-, t : r , f f ) = 2«v»*. Inserting the 5-correlated correlation function, Eq. (2.26), gives ~dt + 2-^- f d3r(-^,G(x,t)e(y,t))2TFV(x,r) = 2KV2$ • (2.40) dx1 J \dw>(r,t) I which is the 2-point analog of Eq. (2.27). Using Eq. (2.28) we find that e(x,t)e(y,t) 5 -e(x,t)e(y,t) =-I 5ua (T*, i) 2 5 ( x - r ) i ^ + s(y-r)-d dxa v" 'dya which substituted into Eq. (2.40) gives (9<3> 82 d2 et = 2 a ^ T F " { x < ">* + 2 a ^ T F ' ^ »>* + 2 k V 2 * ' (2-41) in agreement with the 1-dimensional analogue found in Vergassola and Mazzino [1997, Eq. 4] and Chertkov et al. [1997, Eq. 5]. Assuming homogeneity of the velocity and scalar fields implies that Flj(x,y) = Flj(r), $(aj,y) = $(r) where r = y — x, and allows us to write Eq. (2.41) in the form ^ = - 2 [ ^ - r r F - ( 0 ) - r F ^ ( r ) ] ^ ^ + 2(r/3(a ;)%))$ - 4(Tu™(x)b(y))^. (2.42) Elperin et al. [1996] derived ^ = -2[K<5m" + r F - " ( 0 ) - r F m " ( r ) ] ^ ^ + 2(r6( : K)6(y))$ - A(rum(x)b(y))-^. (2.43) Unfortunately, the sign in the last term of Elperin et al. [1996]'s Eq. (2.43) is reversed. Elperin et al. [1998] solved Eq. (2.43) for $ in the viscous-convective subrange. I analyzed the scalar spectrum predicted by Eq. (2.43) in Jeffery [2000]. Recently, Elperin et al. [2001] present a new solution for $ in the viscous-convective subrange using the 29 correct Eq. (2.42). In Chapter 3 I will present the scalar spectrum predicted by Eq. (2.42) in the viscous-convective subrange. Equation (2.42) can also be derived from Eq. (2.10) using the path-integral formalism of Sec. 2.4. To do so, we must evaluate (0(x , t+At)0(y , t+At)> from Eq. (2.38). First we have the cross terms: 1 <92<E> (e(x,t+At)G(y,t)) = $ + -(Fmn(x,x)(At)2+ 2ri5mnAt) 2 V v ' /v / • / d x m d x n (Q(y,t+At)e(x,t)) = $ + ±(Fmn(y,y)(At)2 + 2K5mnAt)-^^ which follow immediately from Eq. (2.39). The mixed cross terms up to ( A i ) 2 from Eq. (2.38) are: b(x)+um(x)-^jAte(x) x[x^y}^ = (Atf where [/^ (a;)] x [a; —> y] = [^(x)] x [F(y)]. Using these results, assuming homogeneity (e.g. (b2(y)) = (b2(x))) and taking the continuous limit 3 $ = (Q(x,t+At)e(y,t+At)) - (e(x,t)e{y,t)) dt ~ At with the identification A i —> 2T recovers Eq. (2.42). 2.6 Summary: 1968-present The 5-correlated closure for the 2nd order correlation function in an incompressible ve-locity field was first introduced by Kraichnan [1968]. Using homogeneous and isotropic viscous-regime velocity correlation coefficients, Kraichnan [1968] derived the diffusion equation ^ = 2/ tV 2 $ + M (2r25mn - rmrn) n 9 ® (2.44) dt 3 v ' drmdrn v ' which is equivalent to Eq. (2.42) with b = 0 and r i™ = TF05^ - (|7|/6)(2r2<5m n - r m r " ) . A diffusion equation for $ in a rapidly fluctuating velocity field has been recovered earlier in a number of papers [Zel'dovich, 1937; Roberts, 1961; Kubo, 1963] with varying expressions for the effective.diffusivity. Following Batchelor [1959], |7| may be interpreted as the magnitude of the average value of the least principal rate of strain which scales with the Kolmogorov time: 7 = —(l/q)T~l where q is a universal constant for high Reynolds number flows. Recent numerical simulations suggest q « 5.5 [Bogucki et ai, 1997; Chasnov, 1998]. 30 Kraichnan [1968] derived an equation for the spectral density = (2ir) d J dr $(r) exp(-ik • r) where * € C : Re{\I>} € K+, ^(k) = by Fourier transforming Eq. (2.44): Although Kraichnan did not solve Eq. (2.45) he surmised the asymptotic behaviour \t ~ k~3 in the viscous-convective subrange, A; <C rig1, and \I> ~ exp(—A^A;) in the viscous-diffusive subrange, k ^> rig1, where XB = (QQ^^VB- Thus the 5-correlated model predicts a scalar spectrum Eg = 4irk2ty that scales as k"1 in the viscous-convective regime in agreement with the earlier prediction of Batchelor [1959]. Later, Mjolsness [1975] found the analytic form \I> ~ k~3(l + XBk) exp(—XBk) which solves Eq. (2.45). The first experimental validation of a k~l viscous-convective subrange was made by Grant et al. [1968] who analyzed temperature measurements taken from the Royal Canadian Navy's research vessel Oshawa in 1962. The data was collected in Discovery Passage near Campbell River, B.C. , an area that I am familiar with from fishing trips when I was a young boy. The measurements were made in the Southern end of the passage between Campbell River and Cape Mudge where the channel is fully turbulent and well mixed near the end of a flood tide. A typical Reynolds number based on the speed of the current and the depth of the water is 3 x 108. As an aside I would like to point out that Discovery Passage is known for more than k'1 spectra and Spring salmon. On Saturday, Apri l 5, 1958, 1,375 tons of explosives—the largest non-atomic blast at the time—lopped off 38 feet of a navigational hazard in the Seymour Narrows known as "Ripple Rock". Unfortunately, Grant et al. [1968]'s measurements were taken four years after the demise of Ripple Rock, and thus it is not known if the turbulent characteristics of the passage were irreversibly altered. Figure 7 in Grant et al. [1968] comparing one-dimensional temperature and velocity spectra has been reproduced in Fig. 2.1. Both temperature and velocity spectra have an inertial regime that is well described by a A ; - 5 / 3 power law. Furthermore, the temperature spectra also exhibit viscous-convective and viscous-diffusive regimes. Grant et al. [1968] fit the one-dimensional spectrum calculated from Batchelor [1959] 's three-dimensional analytic form for the viscous-convective subrange, E ~ A: - 1 exp(—A^A;2), to their temper-ature data and found good agreement for k < A ^ 1 . Both the temperature and velocity spectra calculated by Grant et al. [1968] and reproduced in Fig. 2.1 exhibit relatively little scatter and a clean decay at large k. The high quality of the data might be sur-prising to those readers who recall Grant et al. [1962]'s rather humorous account of the difficulties of ship-borne measurements: We have the advantage, working with a tidal current, of having a predictable steady current with a duration of several hours—but it does not look steady to a 200 ft. ship whose speed is limited to 2.5 knots. Frequently rather violent maneuvers are required to keep the ship from approaching too close to shallow (2.45) 31 water. Often we are carried into different regions of the current, where the turbulent intensity is quite different and the record must be interrupted by gain changes. Way has to be made for other shipping, which usually, observing our apparently irrational movements, treats us with some circumspection. logto *1 Figure 2.1: Temperature and velocity spectra from run 2, at a depth of 15 m near Cape Mudge. The velocity spectrum (bottom) exhibits a well-defined k~5/3 inertial subrange while the temperature spectrum (top) exhibits an inertial-convective subrange that evolves into a k~l viscous-convective regime. Reprinted from Grant et al. [1968] with the permission of Cambridge University Press. Recently Bogucki et al. [1997, Fig. 7] and Chasnov [1998, Fig. 3] compared the one-dimensional surrogates of the three-dimensional Kraichnan (E ~ A; - 1 (l-t-AsA;) exp(—XBk)) and Batchelor (E ~ k~l exp(—X2Bk2)) analytic forms with direct numerical simulation (DNS) for Pr > 1. They both observed excellent agreement between Kraichnan's spec-trum and the numerical results in the entire range of simulated wavenumbers, whereas, the Batchelor spectrum diverges from the numerical results for k > O^rjg1. Thus DNS provides numerical support for the time-scale based renormalization discussed in Sec. 2.3 where it was shown that the 5-correlated closure is an exact closure for the A D equation for Pr 1, k > n~l and the large Reynolds number limit e —> oo. During the mid 1990's several groups, working independently, realized that the 8-correlated model could be generalized and studied as a simplified model of turbulent diffusion [Chertkov et ai, 1994a, b; L'vov et ai, 1994; Kraichnan, 1994; Shraiman and 32 Siggia, 1994; Balkovsky et al, 1995; Chertkov et al, 1995a, b; Gawedzki and Kupiainen, 1995; Klyatskin and Woyczynski, 1995; Kraichnan et al, 1995; Shraiman and Siggia, 1995]. In particular, Kraichnan's closure by "linear ansatz" [Kraichnan, 1994; Kraich-nan et al, 1995] has motivated a large number of studies in this field [see Majda and Kramer, 1999, Sec. 4] because it exhibits anomalous scaling behaviour and provides a good testing ground for the capabilities of renormalization group methods [Fairhall et al, 1996; Chertkov et al, 1996; Adzhemyan et al, 1998; Belinicher et al, 1998] and renormal-ized perturbation theory [Gat et al, 1997; Shraiman and Siggia, 1998]. Kraichnan [1994] considered a 5-correlated velocity field with a generalized scaling Holder continuous spa-tial structure. Equation (2.42) with 6 = 0 reduces to $ = — M 2 $ where the Kraichnan operator M2 = K2in(C)V rmV and the diffusion coefficient K2(C) in the generalized homogeneous and isotropic velocity field u(C) is [See also Chertkov et al. [1995a] and Gawedzki and Kupiainen [1995]] where 0 < C < 2. Here ( is a fixed parameter that determines the scaling r2~^ of the 2nd-order velocity structure function. Assuming a smooth (viscous) velocity field, ( = 0, and setting K0 = recovers Eq. (2.44). The operator M2 is a 2nd-order, positive, elliptic differential operator. It is some-times called a "Hopf" operator [Shraiman and Siggia, 1995, 1996; Pumir et al, 1999] although Frisch et al. [1999] have remarked that there is a relatively weak association between Hopf's work and white-noise linear stochastic equations. In the limit C —> 2, M2 becomes proportional to the 3-dimensional Laplacian. For £ < 2 and K —> 0 the turbulent transport is super-diffusive with K2 ~ r 2 - ^ . As particles separate they diffuse faster and faster which results in the mean-square distance between particles growing as t2^ (e4 for C = 0). On the other hand, when the distance between particles is small diffusion is slow and particles which are close spend a relatively long time together. Some interesting and subtle points concerning u-averaged Lagrangian trajectories have recently been raised by Bernard et al. [1998]. Consider the conditional probability P(x, t\a, 0) = (Ra(x, t)) for K = 0 where (• • •) represents an average over the u-ensemble. A generalization of this one-parcel statistic is where X = (x±,..., xn) and A = (a\,..., an). It is natural to state that Pn{X, t\A, 0) is the joint probability density function (p.d.f.) of the differences of the endpoints of n Lagrangian trajectories. However, in making such a statement we have silently as-sumed that such trajectories are uniquely defined by their initial or final positions. A 33 straightforward consequence of such an assumption is the relation n Pn(X,t\A,0)\ a k = 0 * + l = ' " = a n Pk(X,t\A',0)Y[5(. ,Xj x n ) (2.46) where X' = (xi,..., Xk) and A' = ( o i , . . . , a*). Equation (2.46) expresses the elemen-tary property that the joint p.d.f. of coinciding random paths is concentrated at a single end-point. However, as explained in Bernard et al. [1998], Eq. (2.46) is only valid for C = 0, i.e. for a viscous regime 2nd-order velocity scaling of r 2 . For a Holder continuous velocity field C, > 0, the u-averaged trajectories behave as if there is a finite diffusivity K and Eq. (2.46) does not hold. This is exemplified with a one-dimensional model for the u-averaged separation p between two parcels in the inviscid limit [Frisch et al, 1999]: Equation (2.47) demonstrates that for £ > 0 the separation of nearby Lagrangian parcels becomes rapidly independent of the initial separation. This explosive separation at small it — s) can be contrasted with the behaviour of neighboring trajectories in a smooth flow (C = 0) where p(s) —> 0 as ii —>• 0 independent of s. With these insights in hand, we are now in a position to offer a qualified answer to the question "Does the Wind possess a Velocity?" that Richardson [1926] asks us on pp. 16. The wind that Richardson is referring to is the transport of air, leaves, broken umbrellas etc. that we see or experience daily. Watching a leaf on the ground at a tossed up into the air by the wind and landing at x after a turbulent flight lasting some time t, Richardson asks us if the trajectory of the leaf is uniquely associated with a velocity. The answer is no, provided that our leaf travels a distance greater than about 10 cm. For in the inertial regime where p2 is described by Richardson's law, p2 ~ et3, there is an inherent randomness to a leaf's trajectory analogous to the randomness of Brownian motion. Thus if we wait for an identical gust of wind to toss up an identical leaf, the trajectory of the leaf, this second time around, will be different. Although Richardson urged us to consider this matter seriously some 80 years ago, no doubt many of us in the atmospheric sciences have failed to heed Richardson's advice. In Chapter 3, we investigate the spectral solutions of Eq. (2.42) for inertial particles, i.e. b ^  0, and in Chapter 5 we consider solutions for cloud droplets where 6 = 0 but a condensation/evaporation source term is present. dp ds s < t with final separation p(t) = 77. The solution is simply (2.47) 34 Chapter 3 Spatial Statistics of Inertial Particles The same decade saw .. .the initial successes of the mixing-length concept, and, with the exception of a few striking and highly original contributions by L. F. Richardson, the empirical theories maintained their ascendancy until 1935 . . . In the field of micrometeorology, apart from the work of Richardson, which is in a class by itself, the only direct application of statistical concepts to the atmosphere during this period was made by Sutton in 1932 and 1934 . . . . [Sutton, 1953] 3.1 I n t r o d u c t i o n In this chapter we investigate the spatial and spectral scaling of inertial particle density in isotropic, homogeneous turbulence using Eq. (2.42). Particles with significant inertia have a velocity, u, that is different from the velocity, v, of the surrounding fluid. This results in the phenomena of "preferential concentration"—the accumulation of dense particles in regions of high strain and low vorticity in a turbulent flow—the theory of which has been developed largely in the engineering community and reviewed in Eaton and Fessler [1994]. Recently, Shaw et al. [1998] have suggested that preferential concentration is a mechanism for the broadening of cloud droplet spectra during condensational growth. They argue that a non-uniform droplet field implies a non-uniform supersaturation field which leads to a broader distribution of droplet growth rates. Some of the assumptions made by Shaw et al. [1998] have been questioned by Grabowski and Vaillancourt [1999]. The formal dependence of b = V • u—a measure of particle clumping—on the strain and vorticity of the surrounding fluid was first determined by Maxey [1987]. Let Vij = dvi/dxj be the velocity gradient tensor ofthe surrounding fluid. Maxey [1987] derived b ~ s 2 - tu2/2, where = (vij + Vj>i)/2 is the symmetric strain tensor, s 2 = trs 2 , Wij = (vtj — i>j,i)/2 is the vorticity tensor and ujk = ekijWij is the usual vorticity pseudovector. 35 Recently, the 5-correlated model has been used to assess the effect of compressibility of u [Elperin et al, 1995; Adzhemyan and Antonov, 1998] or, more specifically, particle inertia [Elperin et al, 1996, 1998; Jeffery, 2000] on the spatial statistics of the particle density 9. Elperin et al. [1996] use the 5-correlated model to assess the effect of particle inertia on spatial statistics and have found a mechanism for intermittency in particle concentrations (preferential concentration). They derive solutions for the nth-order cor-relation function <&„ of the form $„(£, r) = II" - $ 2 ( a ^ — exp[0.5n(n — 1)72*], where i 7^  j from which it follows that if the second moment of the particle concentration grows (72 > 0), then so do all higher-order correlation functions. A Reynolds number criterion for 72 > 0, when satisfied, implies self-excitation of fluctuations in particle con-centration without external pumping, and thus, intermittency. In a later work, Elperin et al. [1998] present a steady-state solution for $(r) = $2( r ) in the viscous-convective subrange (i.e. r < 77) and find that anomalous scaling appears when the degree of com-pressibility a > 1/27. I have investigated the spectral scaling of inertial particle density in the viscous-convective subrange and find a peak in the spectrum when the ratio of the energies in the compressible and the incompressible components of the particle's velocity is greater than 0.007 [Jeffery, 2000]. Unfortunately, the results presented in Elperin et al. [1998] and Jeffery [2000] begin with Eq. (2.43) derived in Elperin et al. [1996] which is incorrect. Recently, Elperin et al. [2001] present a corrected solution for $ 2 . In this chapter, I reanalyze the effect of particle inertia on $(r) and \I/(fc) following closely the methodology of Elperin et al. [1998] and Jeffery [2000] but beginning with Eq. (2.42). This chapter is organized as follows. In Sec. 3.2 Elperin et al. [2001]'s corrected solution for $ 2 is discussed and in Sec. 3.3 the corresponding spectral equation for \f/(fc) is presented and solved. Section 3.5 is a discussion of the effect of particle inertia on \t(fc); a number of figures are used for illustration and an explicit comparison with results from Wang and Maxey [1993] is made. Section 3.6 discusses experimental validation of the present model and Sec. 3.7 contains a summary. 3.2 Correlation function Particles with small but finite inertia have velocity U / D where v is the velocity of the surrounding fluid. Thus, in the case that v is divergenceless, homogeneous and isotropic, u is compressible, homogeneous and isotropic with correlation function [Elperin et al, where F' = dF/dr, F(0) = 1 —Fc(0), and DT = U\T jd where u$ is the characteristic veloc-ity of turbulent fluctuations with relaxation time r, and d is the spatial dimension. The 1995] (rum{x)un{x + r)) = DT { [F(r) + Fc(r)}5, mn 36 function F(r) describes the solenoidal (incompressible) component of the longitudinal correlation coefficient whereas Fc(r) describes the potential (compressible) component. Elperin et al. [1998] solved Eq. (2.43) using (3.1) for al = 3 in the viscous regime r < rj. In this regime the correlation coefficients can be approximated F(r) = (1 - c)[l - a(r/V)2}, Fc(r) = e[l - a(r/V)2}, (3.2) where a is a constant and e is a measure of the compressibility. Equation (2.43) with the above F(r), estimated from the familiar Batchelor parameterization for the second order structure function [Monin and Yaglom, 1975], is accurate in the range r < 5r?. However, Eq. (2.43) with the above Fc(r) is less accurate in the regime r ~ 0{v) because of the higher order derivatives F" and F"' that appear via (rb(x)b(y)). In Sec. 3.3 Eq. (3.2) for F(r) is used with a more accurate expression for Fc[r) such that $ is valid in the range r < bn. In order to recover the well-known viscous-diffusive regime in the limit e —r 0 it is necessary to write a = f(r,u0,rj) and r = g(\j\) where 7 = — (l/q)r~l is the average value of the least principal rate of strain, and q is a universal constant for high Reynolds number flows [Batchelor, 1959]. The choice r = |7| - 1 /(3 « rv and a — TJ2/(12T2UI) is consistent with both the well-known result a « rj2d/(30r2Uo) and q from recent numerical simulations [Bogucki et al, 1997; Chasnov, 1998] giving q = \/30 ~ 5.5. The solution of the incorrect Eq. (2.43) using (3.2) is [Elperin et ai, 1998, Eq. (12)] $(r) = ±(l + X2r/2S(X), (3.3) where S(X) = Re{4iP c " ( iX) + A2Q%(iX)}, P£(iX) and Q$(iX) are the Legendre func-tions with imaginary argument Z = iX, X = r/Xs, XB = [6K|7 | _ 1 (1 — 2<J/(1 + 3a)] 1/ 2 is a diffusive length scale (oc nB), n = 15a/(l + 3a), C(C + 1) = ^ 2 — 5/x + 2, and the parameter of compressibility a = e/(l — e). Note that for an incompressible velocity field, a = 0, the correlation function for r 3> A is $(r) = constant, corresponding to the well-known k~l viscous-convective scaling. Recently, Elperin et al. [2001, Eq. (34)] presented a solution for $ modeled after Eq. (3.3) but derived from the correct Eq. (2.42). Unfortunately, Eq. (34) in Elperin et al. [2001] contains an error, and furthermore, when corrected it may be simplified substantially. We begin this discussion by reproducing Eq. (34): $(r) = {1 + X2)XL(X)/X, (3.4) where L{X) = Re{AiP c " ( iX) + A2Q%{iX)}, p = 5a / ( l + 3a), C = p - 2 and A = a (a — 3)/2(l -I- 3a) 2. First, note that the definition of A is incorrect; in analogy with Eq. (3.3), A = —fi/2. Second, the above definition of L(X) does not hold when /x ± £ is an integer but \i itself is not an integer [Gradshteyn and Ryzhik, 1994, Sec. 8.707], as is the case here. In our case L(X) = Re{AP ;_ 2 (zX) + A2P^2(iX)}. (3.5) 37 Equations (3.4) and (3.5) correctly describe $(r) but a much simpler expression is available. First, note that a special-case recurrence relation is derivable from a more general result [Abramoutitz and Stegun, 1970, Eq. (8.5.3)]: Second, Eq. (8.2.1) in Abramowitz and Stegun [1970] states that P ^ _ 1 ( z ) = Pjf(z). Combining these relations gives P»_2(iX) = iXP^iX) = iXP^(iX). Inserting Abramowitz and Stegun [1970]'s Eq. (8.6.16), P^(z) ~ (z2 - l)~^l2 gives L(X) = AxX(l + X2)-*'2 + Re{A2p-^{iX)}. (3.6) In a similar fashion the second term in Eq. (3.6) can be shown to diverge as r —>• 0 which requires A2 = 0. Inserting Eq. (3.6) into (3.4) with A = —/i / 2 gives $(r) = A^l + X2)-11. (3.7) 3.3 Spectral density It is often useful to consider the spectral covariance density function \I/(fc) to gain some understanding of the relative contribution of individual wave-numbers to the 1-point scalar variance. Following Jeffery [2000] we first expand Fc to fourth-order, Fc(r) = e[l — a(r/rj)2 + [3a(r/r/)4], where j3 = 1/154 is estimated from Pinsky et al. [1999]'s Gaussian approximation for the correlation function B(r) = (b(x)b(y)). Using this fourth-order Fc, B becomes B(r) = e30| 7 | 2 1 UP fr" 2" (3.8) The fourth order term is ignored in (um(x)un(y)) as it makes only a minor contribution in this range. Thus the second-order expression for F(r), Eq. (3.2), and the fourth-order expression for Fc(r) produce results that are accurate in the r < 5n regime. Recall, Eq. (2.42): ^ = -2{K5- + TF-(0)-rF-{r)}^^ + 2(Tb{x)b{y))^ - 4(rum(x)b(y))d® o r A n analytic solution is not available over the whole range of interest, k > O.I77 x . Fol-lowing Jeffery [2000] we separate the solution to this equation over the full range of k 38 into small-scale and large-scale analytic solutions which match near k « n~l. We first derive a small-scale solution accurate in the range k S> r / - 1 where the effect of molecular diffusivity is important and Fc expanded to second-order, i.e. Eq. (3.2), is sufficiently accurate. We then obtain a large-scale solution accurate in the range O . l T y - 1 < k < rf~x where molecular diffusivity can be ignored, i.e. K = 0, and Fc is expanded to fourth-order. 3.3.1 Smal l -scale solut ion (A: >^ r ? - 1 ) Fourier transforming Eq. (2.42) [d/drj —> ikj,rj —> id/dkj] and using second-order isotropic, viscous regime velocity correlation coefficients (3.2) yields a Bessel type equa-tion accurate for k > rf1 0 = -\\kH + /fc V + A2W (3.9) valid for u < 0 where ^f(k) e R+, XB = [6« | 'y | - 1 ( l - 2a / ( l + 3a)] 1/ 2 is a diffusive length scale (oc r}B), A2 = 2[2 — 5a / ( l + 3a)], and the primes denote differentiation with respect to k. The spectral density ^(k) may be obtained by solving Eq. (3.9) [Abramowitz and Stegun, 1970] or by Fourier transforming Eq. (3.7) directly giving *(k) = WK^Xsk) + C^I^Xsk) (3.10) where I and K are modified Bessel functions, and \i = (1 — A2)/2 = —3/2 + 5a / ( l + 3a). Note that for an incompressible velocity field, a = 0, Eq. (3.10) reduces to the analytic form first derived in [Mjolsness, 1975]: \I>(A;) ~ k~z(l + XBk) exp(—XBk). 3.3.2 Large-scale solut ion (O.I77 - 1 < k < rj~l) Evaluating the contribution of the fourth-order term in Fc to the last two terms on the rhs of Eq. (2.42) and Fourier transforming gives, respectively, two new terms in Eq. (3.9): Writing \& ~ A; - 3 + Sf(k) in the viscous-convective regime (i.e. E ~ A; - 1 ) for some small parameter 5 and substituting above, we find that the second term is of order 5 compared to the first and can be ignored. Thus the dominant contribution of the fourth-order term in Fc is through B(r), and the large-scale equation for \I> has the form 0 = {k2 + A2)$>" + A2W, where Ax = [10/ l l7T 2 a/ ( l + 3a)] 1/ 2, A2 is defined as per Eq. (3.9) and the diffusion term —X2Bk2^ of (3.9) is ignored for the range k -C A ^ 1 . Integrating \&" twice gives [Gradshteyn and Ryzhik, 1994, Sec. 3.194] * = C3k2» 2 F i ( l / 2 - f z , - u , l - / i ; -Axk~2) + C 4 (3.12) 39 where 2^1 G C is a hypergeometric function. Neither Eq. (3.9) nor (3.12) are valid over the whole range of k; they are related through the asymptotic l i m ^ o [Eq. (3.9)] = l i m b e c * [Eq. (3.12)]. For Prandtl number Pr » 1 (i.e. A s <C 77 ) there is a well-defined region near k = r / - 1 where these two expressions for \I/ join smoothly. Thus we write r c^K^Xsk) k>km I C3k2»2Fi{l/2-n,-tji,l-K-A1k-2) k<km [ - 1 valid for p, < 0 where C3 = T(—//)/2(AB/2)' 2 and km is computed numerically from the intersection tf(fcm)[Eq. (3.9)] = *(fcm)[Eq. (3.12)]. 3.4 Analysis The time-evolution equation for the spherically integrated scalar covariance spectrum E{k) = 4irk2ty(k) may be written as dE(k) dx(k) O M t 2 dt dk -2Kk2E(k)+V(k), (3.14) where x(h) is the scalar dissipation rate and V(k) is the production spectrum of scalar variance. Solving for x m the steady state for the range k <C A ^ 1 gives x(k)~Xo~ / V(S)d£, (3.15) where x o = 2K J0°° k2E(k)dk. Equation (3.15) was used by Mjolsness [1975] with V — 0 to derive the constant of proportionality in the 6 = 0 solution of Eq. (2.42) [See Eq. (2.44)] which agrees well with numerical simulations [Bogucki et al, 1997; Chasnov, 1998]. Phillips [1965] used (3.15) where V is the spectrum ofthe conventional Reynolds stress term {bw)d(b)/dz and b is the buoyancy —g6/(9), along with the earlier results of Lumley [1964] to derive the buoyancy (temperature) spectrum in a stably stratified fluid. Phillips' derivation was corrected by Weinstock [1985] who showed that the Lumley-Phillips buoyancy subrange theory predicts the temperature spectrum is proportional to k~3 at small k—consistent with experiments. For inertial particles (a ^ 0) the terms in Eq. (2.42) corresponding to V(k) in (3.14) may be easily identified; the covariance production terms are the second term and one-half the third term on the r.h.s. of (2.42). Note that the third term is equal to the sum of the contribution from incompressible advection along compressible streamlines (i.e. spectral transfer) and the contribution from compressible advection along incom-pressible streamlines (i.e. spectral production). Using (3.1), (3.2) and Eq. (3.11), the scalar production spectrum is v(k) = ink2 1 0 | 7 | a l l n 2 (3.16) 40 n 1—i 1—i 1—r 0.01 0.05 0.50 5.00 Figure 3 . 1 : Comparison of the exact scalar production term 47TA;2(2T</> * (symbols) with the approximation Q(k) (lines). The cut-off used in Eq. ( 3 . 1 7 ) at k — 0 . 3 5 T ? _ 1 is shown as a dashed vertical line. Next we determine the accuracy of x(k) calculated from V{k) via Eq. ( 3 . 1 5 ) in the range k < ? 7 - 1 . A useful measure of the accuracy of B(r) and hence V(k) can be constructed from the spectral density of the velocity divergence <j>[k) = (b(0)b(k)). The Fourier transform of the scalar production term 2{rb(x)b(y))^ from Eq. ( 2 . 4 2 ) approximated to fourth-order using ( 3 . 8 ) is Q(k) = 4?rA; 2 {10|7 |cr/[3(l + a)][3* + ^"/{llrf)}} which can be compared with the exact expression 47T/S2(2T</> * \I>) where * is a spherical con-volution. This comparison is shown in Figure 3 . 1 with Xo s e t to unity, C\ calculated numerically from ( 3 . 1 5 ) , (j) from Jeffery [ 2 0 0 0 ] and \& given by ( 3 . 1 3 ) . Although Q(k) is not equal to the production spectrum V(k)—it lacks a contribution from the source term 2(Tum(x)b(y))d<&/dxm—the two functions are very similar. Both the approximate and exact source terms shown in Figure 3 . 1 decrease with decreasing k, but clearly Q(k) over-estimates the source term in the region k < 77-1. We are therefore required to introduce a cut-off kc in the evaluation of x(k) via Eq. ( 3 . 1 5 ) where kc is defined by / Q(k)= f 4irk2(2T(f>*^)dk= . 1 0 ^ g . / E(k)dk. Jkc Jo ( l + ^ i o For 0 < cr < 1 / 2 7 , kc = 0 . 3 5 ? 7 _ 1 suffices with reasonable accuracy and is shown as the dashed vertical line in Figure 3 . 1 . Note that this fc-space cut-off is consistent with the 4 1 previous statement that the velocity correlation coefficients are accurate in the range r < 5rj. Eq. (3.15) therefore becomes POO X{k) « Xo - / K = max(A;, O . 3 5 7 7 - 1 ) . (3.17) Eqs. (3.13-3.17) complete the determination of ^(k) as a function of the parameters Xo, h i . and a. 3.5 Spectra and Discussion Equation (2.42) for the correlation function $(r) reveals that the covariance of inertial particles is controlled by the competition between radial diffusion i f e f f V 2 $ with positive-definite diffusivity Kes, radial advection V^d^/dr with negative radial velocity Ves ~ (rur(0)b(r)) and production $ / r p with time constant T~1 ~ (rb(0)b(r)). Isotropy and/or parity invariance of the turbulent flow ensures that iv~eff — » K and —>• 0 at the smallest scales r —>• 0. Production is maximal at r = 0 but the accumulation/rarefaction of particles at such small scales is mitigated by molecular diffusion which transports the covariance to scales larger than A#. The compressible velocity Vefi which peaks near r = IO77 always contributes to the transport of particle covariance to smaller scales. At near inertial scales, however, iv"eff begins to dominate both production and ballistic transport, thereby preventing the build up of covariance, i.e. preferential concentration. We can therefore expect preferential concentration to peak near the inertial-viscous subrange transition and this is, in fact, what the present model predicts. It is unlikely that the production of scalar variance due to particle inertia has a large effect on the scaling of the inertial-convective subrange for a < 1, and, in fact, simulations of mass-less particles in low-Mach number compressible turbulence show little variation in scaling for a = 1 [Cai et al, 1998]. Numerical simulations [Bogucki et ai, 1997; Chasnov, 1998] of the viscous-convective subrange for incompressible flow a = 0 suggest that the scale break kb between the inertial-convective and viscous-convective subranges occurs, naturally, at the intersection between the inertial-convective spectrum Eic{k) = ClcXiC£~1/3k'5/3 and E(k): Eic(kb) = E(kb) where Cic « 3/4 is the Obukhov-Corrsin constant, e is the energy dissipation rate and Xic = x(h) = constant is the inertial-convective range scalar dissipation rate. For a = 0, Xic — Xo and the scale break is given by kb = (CiC/q)3/2r)~l w 0.05T? _ 1 where q is a function of |7|, whereas for cr ^ 0, Xic decreases with increasing a according to Eq. (3.17). However, for purposes of comparison, it is convenient to set Xic(c") to a constant reference value, i.e. unity. The scalar dissipation in the viscous-convective range is therefore Xvc(k, cr) = x(&, cr)/x{kb, a) which increases with increasing a. A n important parameter in most studies of preferential concentration in the literature is the Stokes number which plays a role similar to e (or a) in this work. The Stokes number, St, is usually defined as the ratio Tp/r^ where rv is the particle aerodynamic 42 time constant [Eaton and Fessler, 1994]. Comparing Eq. (3.8) to Pinsky et al. [1999]'s expression B(r) = (4/15)T 2 /T 4 + • • • gives e = -ist2. (3.18) 15 The scale break kb shown in Figure 3.2 has been computed numerically using model (3.13-3.17) and the inertial-convective range spectrum (above) for Pr in the range 500 to 8000. Also shown is a measure of the increase in scalar dissipation rate through the viscous-convective regime x t J C(co, o~)/xic ~ 1- The figure illustrates a number of robust trends that are worthy of discussion. First, kf, increases from « 0.05??-1 at St = 0 to fa 0.16?7—1 near St = 0.6; the viscous-convective regime gets pushed to smaller scales. Second, and as mentioned above, x„ c (oo) increases with increasing St. To aid in this discussion, x„ c (oo) has been fit to the approximate analytic form, X„c(oo) « Xic exp [ln(Pr)°-5 5(17cr - 35a2)] (3.19) depicted with • in Figure 3.2. This equation demonstrates, formally, the self-excitation (exponential growth) in the second moment of particle concentration first discussed in [Elperin et ai, 1996]. The weak Pr dependence results from a "longer" viscous-convective regime with increasing Pr and therefore more self-excitation. Self-excitation begins near St as 0.3 and, along with self-excitation of higher order moments, leads to preferential concentration. The scalar spectrum E, computed numerically using model (3.13-3.17) and the iner-tial-convective range spectrum, is shown in Figure 3.3 along with the change in scalar dis-sipation rate Xvc{k)/Xic- Both features discussed above—the suppression of the viscous-convective regime at larger scales and the self-excitation of the spectrum—are clearly visible. Beginning at St « 0.2 a peak at kp « 0.1?7—1 is visible in the spectrum and becomes more pronounced as St increases. The wave-number kp is a useful measure of the length scale at which preferential concentration occurs. The wave-number of the spectral peak kp defined by dE/dk\kp = 0 is plotted as a function of St in Figure 3.4 in the range 0.18 < St < 0.6. The figure reveals that the characteristic scale of preferential concentration increases from « 60?7 at St « 0.2 to ~ 877 at St « 0.6, consistent with the estimate in [Eaton and Fessler, 1994] that particles with a Stokes number of around 1 are concentrated at length scales 6 — 20 n. Note also that rapid growth in the viscous-convective regime for Stokes numbers greater than 0.2 shown in Figure 3.3 is consistent with the rapid growth in preferential concentration in the range 0.2 < St < 1 shown in Figure 18 of [Wang and Maxey, 1993]. 3.6 Experimental verification Unfortunately, experimental knowledge of the scalar spectrum of a continuous field of in-ertial particles is presently nonexistent. Investigations using direct numerical simulation 43 n I i I i i r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 St Figure 3.2: Plot showing both the scale break kb (right-axis) between the k~5/3 inertial-convective subrange and the viscous-convective subrange, and the measure of self-excitation Xvc(oo, St) jxic — 1 (left-axis) computed numerically (lines) and approximated analytically by Eq. (3.19) (•). These functions have been calculated using model (3.13-3.17) at four Prandtl numbers and appear such that the lines are, from top to bottom, Pr = 8000, 3000, 1200 and 500. (DNS) typically provide discrete particle pair statistics such as the radial distribution function which can not be directly related to the continuous field correlation function [Wang and Maxey, 1993; Sundaram and Collins, 1997, 1999; Wang et al, 2000]. Simu-lations that use a much larger number of particles, e.g. O(10 6), are capable of resolving a continuous particle field but are extremely limited in resolution [Wang and Maxey, 1993]. If we restrict our attention to the viscous-convective subrange, then the Monte Carlo Lagrangian methods recently proposed independently by Frisch et al. [1998] and Gat et al. [1998] could be perhaps modified to examine the scaling of inertial particle density (See also Celani et al. [1999]). However, it should be emphasized that this La-grangian technique involves the addition of a large scale forcing term to the A D equation which does not appear (explicitly) in viscous-convective theory. Nevertheless, qualitative comparisons between the results of this chapter and numeri-cal simulations can be drawn. The present theory predicts a rapid (exponential) increase in preferential concentration with increasing Stokes number for St < 1 which is qual-44 o o o d > 0.02 0.10 0.50 2.00 10.00 Figure 3.3: The scalar spectrum computed at various St using model (3.13-3.17) and the corresponding increase in scalar dissipation rate. Note that the increase in Xvc begins at k = 0.3577-1 as per Eq. (3.17). The parameter values are e = 0.01 and Pr = 1000. itatively consistent with results from numerical simulations. Wang and Maxey [1993] calculated the integrated variance of the probability of finding a discrete concentration, 6 j , in the their DNS simulations and they found that this global measure of local particle accumulation increased rapidly with increasing St for St < 1. In this chapter we have investigated the effect of particle inertia on the viscous-convective subrange in the small Stokes number regime, St < 0.6. Using the (^-correlated model, an analytic expression for the second-order spectral density of inertial particles is derived as a function of a Batchelor-type diffusive length scale, the rate of strain of Kolmogorov eddies, the scalar dissipation rate and the degree of compressibility (Stokes number). A rich spectral behaviour is observed in the viscous-convective regime that includes the following features: particle inertia suppresses small k viscous-convective scaling; the start of the viscous-convective regime is pushed to smaller scales with increasing St. Associated with increased compressibility is the emergence of a well-defined "bump" in 3.7 Summary 45 0.2 0.3 0.4 0.5 0.6 St Figure 3.4: Plot showing the characteristic scale of preferential concentration as a function of the Stokes number. The wavenumber kp is computed from dE/dk]^ = 0 using model (3.13-3.17). the spectrum beginning at St « 0.2. The bump represents the accumulation of inertial particles in regions of high-strain and low-vorticity in the flow and is a manifestation of intermittency in the spatial statistics. The characteristic scale of this preferential concentration ranges from around 6O77 at St = 0.2 to about 877 at St = 0.6. The relative height of the bump increases exponentially with increasing Stokes number. 46 Chapter 4 Intermittency and Preferential Concentration The hypothesis concerning the local structure of turbulence at high Reynolds number, developed in the years 1939-41 by myself and Oboukhov . . . were based physically on Richardson's idea of the existence in the turbulent flow of vortices on all possible scales I < r < L between the 'external scale' L and the 'internal scale' I and of a certain uniform mechanism of energy transfer from the coarser-scaled vortices to the finer. [Kolmogorov, 1962] 4.1 Introduction In this chapter we consider the effect of velocity field intermittency on preferential concen-tration in a quantitative manner. A Taylor microscale Reynolds number (Re>) dependent effective Stokes number is derived that is proportional to the square-root of the flatness factor of the longitudinal velocity derivative. We will also discuss implications for Shaw et al. [1998, 1999]'s hypothesis, that the fine-scale structure of atmospheric turbulence allows for the clumping of cloud droplets despite small Stokes numbers (St <C 0.15). In a recent paper, Shaw et al. [1998] present a mechanism for the broadening of cloud droplet spectra during condensational growth: cloud droplets preferentially accu-mulate in regions of high strain and low vorticity in a turbulent flow due to inertia. They further suggest that a non-uniform droplet field implies a non-uniform supersat-uration field which leads to a broader distribution of droplet growth rates. In a short comment, Grabowski and Vaillancourt [1999] question a number of Shaw et al. [1998]'s assumptions. In particular, Grabowski and Vaillancourt [1999] suggest that the Stokes number (St)—the ratio between the particle's inertial response time and the Kolmogorov time—is too small for significant preferential concentration to occur. They estimate that for typical atmospheric conditions and growing droplets (radius < 5 fim) St < 0.008, whereas laboratory experiments demonstrate that significant preferential concentration 47 occurs for St = 0(1). In response to Grabowski and Vaillancourt [1999], Shaw et al. [1999] argue that at high Reynolds numbers where intense vortex tubes are present, the Stokes number range for preferential concentration increases significantly. Support for this argument comes from direct numerical simulation (DNS) experiments at fixed St [Reade and Collins, 2000; Wang et al, 2000] that demonstrate an increase in particle clumping with increasing Reynolds number. This chapter is organized as follows. In Sec. 4.2, the effect of velocity field intermit-tency on preferential concentration is determined. A n effective Stokes number is pre-sented that is an explicit function of the Taylor microscale Reynolds number of the flow. Furthermore, we address Shaw et al. [1999]'s hypothesis that the fine-scale structure of atmospheric turbulence can account for the clumping of droplets at small St. In Sec. 4.3 we examine the trapping of cloud droplets in vortex tubes that is an essential feature of the Shaw model of droplet spectral broadening. We address the question "Are vortex tubes a statistically relevant feature of small scale turbulence?" Section 4.4 discusses experimental validation of the present hypothesis and Section 4.5 contains a summary. The results in this chapter also appear in Jeffery [2001c]. 4.2 St-Re^ dependence In response to Grabowski and Vaillancourt [1999] 's contention that the Stokes number of cloud droplets is too small for significant preferential concentration to occur, Shaw et al. [1999] argue that the Stokes number range for preferential concentration is significantly broader in clouds because of the large Reynolds number of atmospheric turbulence. The fine-scale structure of large Reynolds number flow is highly intermittent [Sreenivasan and Antonia, 1997] and small-scale coherent structures, i.e. vortex tubes, may influence the clumping process. Support for this claim comes from recent DNS experiments [Reade and Collins, 2000; Wang et ai, 2000] that demonstrate an increase in the radial distribution function with increasing Taylor micro-scale Reynolds number, Re^, at fixed St. Inspection of Eq. (2.42) reveals that the variance of the particle velocity flux divergence (b2) controls the degree of preferential concentration in the (^-correlated limit. Below we estimate the effect of velocity field intermittency on (b2). Recall from Sec. 3.1 that Maxey [1987] first derived the relationship between the compressibility, b = V • u, of an inertial particle's trajectory and the velocity, v, of the surrounding fluid for St < 1: b = -Xg^ijVjJ = -AjV-"72) where Xg = g/Vt, g is the acceleration of gravity, Vt is the terminal fall velocity of a particle and it is assumed pf <C pp where pf and pp are the densities of the fluid and particles, respectively. Using the zero-fourth-cumulant (Gaussian) hypothesis and 48 assuming isotropy and incompressibility, Pinsky et al. [1999] related ((vijVjti)2) to the longitudinal velocity derivative (v2^) giving (b2) = 60X;2(vl)2. (4.1) The St dependence for a particle falling at Stokes terminal velocity in a large Re^ velocity field is (b2) = T ^ S t 2 , (4.2) where the relation v2^ = r~ 2 /15 was determined in Sec. 3.5 [See Eq. (3.18)]. The Gaussian approximation used by Pinsky et al. [1999] to derive Eq. (4.1) is appro-priate for fourth-order moments of the velocity field which are approximately Gaussian, but it is unlikely to be accurate for fourth-order velocity gradients; large Re> flows are highly intermittent. In fact, the flatness factor T of viti has a strong Re> dependence and reaches 20-25 in the atmosphere where Re^ = O(104) [Sreenivasan and Antonia, 1997, Fig. 6]. This suggests that (b2) in the atmosphere may be a factor of 7-8 larger than the prediction of Eq. (4.1). On the other hand (v4^) is just one term in the expansion of (b2), and, therefore, more information is required to determine the Re^-dependence of (b2). Consider the symmetric strain tensor Sij = (vij+Vj^)/2 and the vorticity pseudovector w*; = ekijWij where the vorticity tensor Wij = (vij—Vj^/2 and e^- is the alternating (Levi-Civita) symbol. Twenty-years ago Siggia [1981] demonstrated that fourth-order velocity derivative moments can be expressed as a linear combination of four irreducible scalar invariants: Ii = (s 4), I2 = (s2u>2), I3 — (uiiSijSjkOJk) and J 4 = (to4), where s2 = trs 2 . These four invariants are functions of familiar dynamical quantities [Pedlosky, 1987]: 2i^s2 is the kinetic energy dissipation rate, co2 is the enstrophy and WjSij is the vortex stretching term in the vorticity equation. Writing vitj in terms of s^ - and u>k and substituting into Eq. (4.1) gives (b2) = \;2(I1-I2 + IA/4). '• (4.3) The Re^-dependence of (b2) follows from knowledge of i i , I2 and I4. Furthermore, even if these three invariants have a simple power-law dependence on Re>, the resulting Re^-dependence of (b2) might be quite complex. Unfortunately, measurement of Ia at large Reynolds numbers typical of the atmo-spheric boundary layer has not yet been made. However, the task at hand simplifies if we assume that (b2) scales with Ii ~ (vii)- This approximation is justified at low R e A by the recent analysis of the ratios A = I2/I1 and C = J 4 / J 1 by Zhou and Antonia [2000]. Figure 18 in Zhou and Antonia [2000] reveals that A and C have no discernible Re^-dependence for Re> < 100. Combining Eqs. (4.2) and (4.3) with the approximation h/h ~ h/h ~ constant, we incorporate the Reynolds number dependence of preferen-tial concentration into St by defining an effective Stokes number Steff: Steff = S t ^ / 3 ) 1 / 2 , (4.4) 49 where T = (vfti)/(v2tl)2 is the longitudinal flatness factor. Deviations of T above the Gaussian value of 3 are a manifestation of intermittency in the turbulence fine-scale structure. In the atmospheric boundary-layer where F « 22, Steff is around 2.7 St. Shaw et al. [1999, Table 1] estimate that St < 0.02 for kinetic energy dissipation rate e = 0.01 m 2s~ 3 and St < 0.064 for e = 0.1 m 2s~ 3 where r < 8 pm. Including the Re^ dependence these numbers become Steff < (0.054,0.17), respectively. In Sec. 3.5 we found that clumping begins at St « 0.2 but does not become significant until St > 0.3. Thus for r < 8 / t r a , i.e. during most of a droplet's growth, Steff is still too small for significant preferential concentration to occur. On the other hand, for larger drops with r > 15 /xm the Stokes number are St eg > (0.19,0.60), respectively. Thus these large drops, particularly in high e regions of a cloud, are preferentially concentrated. It should be emphasized that the criterion St > 0.3 does not reflect the smallest St at which preferential concentration is statistically detectable. Rather it is a measure of the smallest St at which the generation of small-scale covariance due to clumping at some scale k~l becomes comparable in magnitude to the cascade of covariance from fc-1 to smaller scales. Thus for St > 0.3 small-scale generation of covariance dominates the usual cascade from large to small scales. The main assumption in the derivation of Eq. (4.4) is I\ ~ I2 ~ h which has been verified for laboratory grid turbulence at Re> < 100. Obviously, this approximation should also be verified (or modified) from atmospheric data where Re^ = O(10 4). 4.3 The Shaw Model and Vortex Tubes At the heart of the arguments put forth by Shaw et al. [1998, 1999] is a requirement that the lifetime of a vortex tube increase with increasing Re>. For large Re> atmospheric flows, Shaw et al. [1998] chose a vortex lifetime r s of 5, 10, or 15 s. This value can be compared with the eddy turn-over time used in Sec. 2.3: le 0.01 m TT = 7 T 7 7 7 ~ r = 0.5 s (ue)1^ 0.02 m/s while the Eulerian correlation time rg ~ rv « 0.05 s. As pointed out by Grabowski and Vaillancourt [1999], the vortex in the Shaw model has to survive many tens of its turn-over time to generate a strong effect on the cloud spectrum. The importance of TS (or TT) is that long lifetimes are required to develop significant differences in the supersaturation between the droplet-free vortex and the environment. Thus the parameter that is most relevant here is the particle separation p introduced in Sec. 2.6. Let us recall the behaviour of p2 as given by Eq. (2.47): • Viscous-Diffusive Subrange: p2 = Kt. • Viscous-Convective Subrange: p2 ~ exp(t/r,,). 50 • Inertial-Convective Subrange: p2 fa O.let3. Using these relations we can estimate p2 after 10 s. Of the three regimes, it is in fact the viscous-convective subrange that has greatest potential of large p2 for t/r^ = 0(100) considered by Shaw et al. [1998]. As a result, particles in close proximity will rapidly separate until p fa 1 cm at the end of the viscous-convective subrange. For p > 1 cm the inertial-convective scaling takes over and after 10 seconds or so p2 ~ 1 m 2 . Thus, due to the exponential growth of p in the viscous-convective subrange, neighboring particles will almost certainly have separated after 10 seconds. This behaviour is in contradistinction to the Shaw model where p2 remains small for particles trapped in a vortex tube. How can we reconcile the large p2 predicted by the ^-correlated model with the large eddy-trapping times predicted by Shaw et al. [1998]'s vortex model? I believe that any contradictions are resolved by the following claim: Vortex tubes are not statistically relevant players in the dynamics of two-point passive scalar moments and/or pair separation at small-scales and large Re^. Here I present only a sketch of the argument which will be investigated further in a forthcoming publication using a numerical vortex model. Let us return to Eq. (4.3) which includes the effect of intermittency on particle clumping. Consider the following question: "How might the ratios A = I2/I1 and C = J4 / J1 change in a velocity field dominated by cylindrical vortices?". Zhou and Antonia [2000]'s experimental measurements indicate that while F increases from 3 to 4.5 as Re A —¥ 100, A and C remain constant near their Gaussian ratios of 1.43 and 4.76, respectively. The invariant I2 is constrained by the incompressibility/homogeneity condition (a;2) = 2(s 2). Thus it is reasonable to expect I2 and i i to scale with Re^ such that A remains constant. On the other hand, there is no reason to expect the ratio C to remain near its Gaussian value in a velocity field dominated by cylindrical vortices which increase in intensity and persistence with increasing R e A -In fact, a maximization of the ratio of the spatially averaged mean-square enstrophy to mean-square dissipation around a cylindrical vortex tube has been discussed by He et al. [1998]. They find Cvort ~ 10.65, spatially averaged over a single Burgers vortex and independent of the circulation which can be compared with the Gaussian value C g a U s s ~ 4.76. As a result the increase in (b2) due to the presence of a Burgers vortex at fixed i i ( R e A ) is l - A + Cyprt/4 1 - A + C g a u s s / 4 ~ This, albeit approximate, analysis supports Shaw et al. [1998]'s claim that vortex tubes lead to increased particle clumping. On the other hand, Zhou and Antonia [2000]'s experimental measurement of C = 5 ± 1 does not support such a large value of C , albeit 51 at Re> < 100. More to the point, their data does not indicate an increase in C with increasing Re^ despite the increase in intensity and persistence of vortex tubes in this Re^ regime. This brings us full circle to the comments of Grabowski and Vaillancourt [1999] who argue that the volume fraction of vortex tubes used by Shaw et al. [1998], 50%, is much too high. Grabowski and Vaillancourt [1999] suggest that 1% is more realistic. Indeed, Zhou and Antonia [2000]'s measured value of C = 5 ± 1 and a strong Re^ dependence of I\ is consistent with a relatively small volume fraction of vortex tubes superimposed on a highly non-Gaussian background velocity-gradient field. Thus we conclude that it is the general intermittency of the velocity-gradient field as a whole, and not the presence of vortex tubes in particular, that could potentially lead to increased preferential concentration in atmospheric clouds. 4.4 Experimental verification Both Wang et al. [2000] and Reade and Collins [2000] have observed a Re^ dependence in the radial distribution function of inertial particles. Wang et al. [2000] approximate this dependence as a linear function of ReA. Reade and Collins [2000] point out that while the linear dependence proposed by Wang et al. [2000] is plausible, it might also represent the first term in a Taylor series expansion in which higher-order terms become significant at larger Re>. Note that Wang et al. [2000]'s measurements span a relatively small range of ReA from 45 to 75. A first test of the effective Stokes number defined by Eq. (4.4) is to compare Wang et al. [2000] 's linear dependence with the Steff dependence calculated from the DNS velocity field. Comparisons of the DNS velocity field with the experimental measurements of Zhou and Antonia [2000] is also valuable because they could potentially reveal some deficiencies in the DNS fine-scale statistics. As mentioned previously, the other important experimental measurements that need to be made in the atmospheric surface layer are the determination of three invariants I\, I2 and I4. Conceivably, aircraft measurements of a non-Poissonian distribution of cloud droplets could provide indirect experimental support for an effective Stokes number effect in large ReA atmospheric flows. However, the analysis of small-scale aircraft measurements of droplet spacing is controversial because (i) high frequency aircraft data are easily biased or contaminated and (ii) the size of the drops and hence their inertia is not accurately known. A recent analysis by Chaumat and Brenguier [2001] supports little droplet clumping. 4.5 Summary The debate over the existence of a preferential concentration of cloud droplets at small scales is fundamentally linked with the fine-scale structure of atmospheric turbulence. At the heart of the arguments put forth by Shaw et al. [1998, 1999] for a significant 52 clumping of droplets is an explicit Reynolds number dependence of the relevant small-scale parameters. In particular, the lifetime of a vortex tube is assumed to increase linearly with Re> [Shaw et al, 1999, pp. 1439]. Although this time-scale is paramount for a Lagrangian analysis of individual droplet interactions with a single vortex tube, the relevant parameter in the Eulerian analysis of small-scale droplet number density is the variance of the scalar VijVjj averaged over both the vortex and background fields. This scalar variance is decomposed into three invariant contributions which, in turn, are explicit functions of symmetric strain and (antisymmetric) vorticity. Based on the recent experimental results of Zhou and Antonia [2000], an effective Stokes number is derived in this chapter that is proportional to the square-root of the flatness of the longitudinal velocity derivative, thereby explicitly incorporating the intermittency of the velocity field and a ReA-dependence into the theoretical framework developed using the 5-correlated closure. Using this effective Stokes number and an atmospheric flatness factor of 20-25, it is determined that intermittency may lead to appreciable clumping for very large drops with diameters greater than 25 /am under general atmospheric conditions. This finding supports the arguments made by Shaw et al. [1998, 1999] that intermittency enhances preferential concentration at small St. On the other hand, the Shaw model of droplet spectral broadening is contingent on the interaction of cloud droplets and vortex tubes. Shaw et al. [1998, 1999] argue that the eddy-trapping time at large Re A is 0(10 s) which is approximately two orders of magnitude greater than the Kolmogorov time. Their arguments justifying such a large value are: (i) for a particle trapped in a cylindrical vortex the eddy-trapping time is equal to the vortex lifetime; (ii) vortex tubes are the (statistically) dominant small scale feature at large Re A; (iii) vortex lifetimes may increase rapidly with increasing Re A-However, the suggestion of a statistically dominant role played by cylindrical vortices as Re^ increases is inconsistent with experimental measurements which demonstrate that the ratio of mean-square enstrophy to mean-square dissipation is independent of Re\ [Zhou and Antonia, 2000] and close to the Gaussian value. Further arguments supporting the bulk statistical insignificance of vortex tubes are made by Tsinober [1998]. In addition we saw in Sec. 4.3 that the mean square displacement of non-inertial particles increases exponentially in time in the viscous-convective subrange which is suggestive of strong droplet mixing. A huge increase in the eddy-trapping time resulting from the interaction of inertial droplets and vortex tubes is an essential component of the Shaw model of droplet spectral broadening, because spectral broadening does not occur if the droplets rapidly sample a range of supersaturations. Thus, in the absence of a dominant particle-vortex interaction, particle inertia could actually lead to spectral narrowing due to increased inertial mixing as pointed out by Pinsky and Khain [1997] and Pinsky et al. [1999]. 53 Chapter 5 Spatial Statistics of Cloud Droplets For uniform stratus it is customary to make a very rough eye estimate of the water-content in such words as "thin, heavy, dark, ugly, lowering." Can photometric measurements yield a corresponding numerical quantity? [Richardson, 1919] 5.1 Introduction In this chapter we consider the effect of condensation and evaporation on the viscous-convective subrange using a general mean-field approximation that is consistent with the non-homogeneous vertical structure of the condensate's first and second moments and experimental observations of mean vertical flux in a condensation cloud. Our goal is to reproduce the spectral behaviour of new atmospheric measurements presented by Davis et al. [1999] that exhibit anomalous scaling of cloud liquid water in the near inertial-convective regime. Davis et al. [1999] presented horizontal spectra (j){kx) of cloud liquid water content (LWC) measured at an unprecedented resolution of 4 cm during the winter Southern Ocean Cloud Experiment (SOCEX). The scalar spectrum from the ensemble-average ofthe flight segments shown in Fig. 5.1 (•) exhibits two distinct scaling regimes: Kolmogorov scaling (—5/3) is evident at larger scales and viscous-convective like scaling (—1) is visible at the smallest scales. Although these spectral scalings are of no surprise the scale break between the inertial-convective and viscous-convective regimes, estimated by Davis et al. [1999] to occur at 2-5 meters (kb « 0.002?7_1, rj = Kolmogorov length) is anomalous. Normal viscous-convective scaling, also shown in the figure (-), intersects the inertial-convective subrange at kb fa 0.05r7_1 which corresponds to an r-space transition of around 10 cm in the atmosphere. Thus the observed scale break occurs at scales one order of magnitude larger than the standard theory predicts. What is particularly intriguing about these new observations are the implications for the scalar dissipation rate x ; with the new scale break, x m the viscous-convective regime is a factor of 14 larger than the inertial-convective x , suggesting that a source of scalar variance is present on scales of tens of centimeters. 54 Marshak et al. [1998] suggest that the strong variability shown in Fig. 5.1 on scales of 4 cm to 4 m is consistent with Shaw et al. [1998] 's discussion of a strong preferential concentration—the accumulation of inertial cloud droplets in regions of high strain and low vorticity in a turbulent flow. However, we saw in Chapter 3 that increased clumping of particles is associated with the suppression of viscous-convective scaling at near inertial-convective scales, i.e. the movement of kb to smaller scales. Thus the data of Fig. 5.1 and the predictions shown in Fig. 3.3 are clearly at odds. Gerber et al. [2001] suggests that the enhanced LWC variance at small scales is related to the small-scale entrainment features generated at cloud boundaries. However, as they admit, the spectral density distribution of entrainment scales and the in-cloud volume affected by entrainment and mixing are not known. Mazin [1999] proposes that the non-inertial-convective scaling is caused by the temporal relaxation of the supersaturation to its steady-state value with e-folding time, rp. Mazin argues that for updrafts with decorrelation time K T p the time is too short for a significant amount of phase change to occur and the turbulent laws for an inert scalar apply, whereas for t >^ TP the supersaturation is close to its steady-state value and the cloud LWC behaves like an inert scalar with a vertical mean gradient. For time scales close to rp the -5/3 law is violated. However, two aspects of Mazin's hypothesis are questionable. First, the linear increase of LWC variability with height above cloud base demonstrates that a condensation cloud is fundamentally distinct from an inert scalar with an imposed mean gradient as discussed below, and that this distinction is present over a wide range of scales. Thus, the scaling for times t 3> rv is just as likely to be anomalous as for times t ~ 0(rp). Second, it is not at all clear whether a change in the Lagrangian spectrum of supersaturation at temporal scales of 0(rp) will, in fact, lead to changes in the Eulerian spatial spectrum. In this chapter, I suggest that the anomalous scale break is caused by the effect of condensation and evaporation on scalar variance. Unlike other theories of conden-sation/evaporation effects on cloud microphysics, e.g. phase relaxation time [Mazin, 1999] or buoyancy reversal [Grabowski, 1993], the model proposed here does not invoke non-stationary, non-equilibrium, discrete or non-local phenomena such as sedimentation, buoyancy, entrainment or a non-continuous droplet field. Rather, the present model is fundamentally a mean-field approximation that relates the complex process of conden-sation/evaporation to the mean vertical structure of liquid water in the cloud. Thus the present model is akin to Lagrangian parcel models where condensation/evaporation is largely dictated by the vertical velocity and the average environmental conditions inside the parcel. The present model decouples LWC production from the vapour and tem-perature fields; therefore, water vapour and temperature are represented by only their first moments through the equilibrium vertical liquid water structure. In fact, it should come as no surprise that anomalous viscous-convective scaling is observed in clouds if one considers that condensation/evaporation is an asymmetric internal pumping coupled to a large Reynolds number ( R e A ) , inertial velocity field that exhibits a continuous range of scales. As a result of this coupling, production of LWC occurs over a wide range of 55 o 0.00001 0.001 0.1 M Figure 5.1: Ensemble-averaged ID scalar spectrum for cloud LWC data measured during the S O C E X field program and presented in Marshak et al. [1998] and Davis et al. [1999]. A typical atmospheric value of 0.76 mm is assumed for the Kolmogorov length r). Also shown is the usual ID inertial-convective/viscous-convective scaling calculated using q = 5.5 (Sec. 5.4) and (3 = 3/4 (Sec. 5.6). The observed spectrum is a factor of 14 greater than the normal spectrum in the viscous-convective regime. scales. Conceptually, it is not hard to see how condensation through lifting creates liquid water variance. Consider a fluctuating (mean zero) variance Q(x)Q(y) where the vertical velocities u3(x) and u3(y) are both positive. As the parcel rises, © at both x and y increases through condensation, and the variance grows. Thus condensation/evaporation coupled to vertical advection leads to a self-excitation of LWC variance; in Sec. 5.3 we derive an advection-type source term for the advection-diffusion equation of the form: source = velocity x d(Q(x)Q(y))/dx3. The above example illustrates an important distinction between homogeneous, iso-tropic, incompressible mixing of a passive scalar in a reacting system (condensation cloud) and in an inert system with an initially imposed scalar gradient. The density fluctuations in the latter are stationary, anisotropic and homogeneous—properties that follow immediately from the incompressible advection-diffusion equation. Furthermore, the initial mean scalar gradient is maintained. In contrast, the density fluctuations in 56 the former are stationary and anisotropic, but not homogeneous—mean-square density fluctuations increase in the direction of increasing mean density. Thus, although the mean density profiles of the two systems may be identical the statistical properties of the density fluctuations are not. The present model, although limited to wavenumbers k > kb, predicts that the LWC correlation function has an important non-homogeneous, vertical contribution from a term linear in r 3 . This general behaviour agrees well with aircraft measurements [Stephens and Piatt, 1987; Noonkester, 1984; Vali et al., 1998] and numerical simulations [Kogan et ai, 1995; Stevens et al., 1996; Khairoutdinov and Kogan, 1999] which demonstrate that both the mean and root-mean-square LWC in atmospheric clouds increase linearly with height. This chapter is organized as follows. The source term representing condensation/eva-poration is introduced in Sec. 5.2, and in Sec. 5.3 the resulting equation for the correlation function in the Batchelor limit is derived. Also in Sec. 5.3 I present an approximate analytic form for the correlation function that illustrates the general anisotropic and non-homogeneous properties of the full solution whose derivation follows in two parts. In Sec. 5.4,1 derive a general axisymmetric solution for the spectral density without the new source term for both the viscous and inertial-convective subranges while the contribution from the new source term is determined in Sec. 5.5. In Sec. 5.6 the magnitude of the axisymmetric contributions to the spectral density are determined using the new data shown in Fig. 5.1. Section 5.7 is a discussion of the predicted spectra, and experimental validation of the present model is discussed in Sec. 5.8. Section 5.9 contains a summary. The results in this chapter also appear in Jeffery [2001b]. 5.2 Condensation/Evaporation Source Term The density of a condensate ipc in an incompressible velocity field is described by the advection-diffusion equation, Eq. (2.14), ^ + u . V i = K V V c + C E ( i H i ) , (5.1) where C E is a source term that models condensation (tpv —>• ipc) and evaporation (ipc —>• ipv) between the condensate (ipc) and its vapour (ipv). Condensation/evaporation occurs as a result of imposed vertical gradients (z = x3 = e3) in the temperature, pressure and vapour fields. In general, C E is a function of ipc(x, t), tpv(x,t) and the macroscopic tem-perature field T(x, t) as well as a host of microscopic parameters including the saturation vapour-pressure, the diffusivity of heat and vapour and the latent heat of evaporation [Pruppacher and Klett, 1997]. Furthermore, in a closed system C E is non-stationary be-cause it is coupled through T(x, t) to irreversible thermodynamic processes, while, on the other hand, in an open system the spatial structure of C E has a non-trivial dependence on the thermal boundary conditions. To remove some of this complexity, we consider 57 a simplified model for C E that is decoupled from both ipv and T and hence stationary, i.e. thermodynamics are reversible. The model is based on the following deterministic equation for the vertical structure of ibc: a* . = (5.2) oz z where z(xi,x2) is the height above cloud base and p G R is a constant. Eq. (5.2) states that ibc(x + Az) is related to its neighbouring density ipc(x) through condensation {p > 0) or evaporation (p < 0), processes controlled by the vertical dependence of T and ipv which are assumed non-stochastic, i.e. T(x) = (T(z)). Thus Eq. (5.2) is a mean-field approximation, and as such, ignores non-local effects including entrainment of "non-cloud" environmental air at the system boundaries. The resulting vertical structure from (5.2), (ibc(z)), ~ zp, can be compared to experimental measurements of the system in question to determine the sign and magnitude of p. Using d/dz = d/dx3 = u^d/dt gives C E = ^ V c - (5.3) Note that the dependence dibc/dt ~ u3ipc/z of (5.3) is also exhibited by Lagrangian parcel models of diffusional growth of water drops in clouds where dibc/dt ~ (ipc/a)da/dt ~ ipc/t ~ wipc/z, a is the radius of the drop, and w is the vertical velocity of the parcel [Pruppacher and Klett, 1997]. There are a number of experimental and numerical studies that report the vertical distribution of LWC inside stratus and stratocumulus clouds [Stephens and Piatt, 1987; Noonkester, 1984; Nicholls and Leighton, 1986; Austin et ai, 1995; Duynkerke et ai, 1995; Kogan et al., 1995; Stevens et al, 1996; Vali et al., 1998; Khairoutdinov and Kogan, 1999]. A l l these observations and model predictions show that the mean cloud liquid water increases nearly linearly with height from cloud base corresponding to p = 1 in Eq. (5.2), and, therefore, the advection-diffusion equation for this system is ^ + « ' V ^ = KATPC + ^ V c - (5.4) dt z In this work we examine the second-order ensemble mean moments of ibc as described by Eq. (5.4). The ensemble-averaged advection-diffusion equation (5.1) with C E given by (5.3) does not predict (ipc(z)); the connection between C E and the mean vertical structure of ibc follows from deterministic Eq. (5.2). However, using Eqs. (5.1) and (5.3) and assuming stationarity and horizontal homogeneity, we find that the vertical flux of condensate obeys (u^ipc) ~ zp. Thus with an appropriate choice of p, the mean-field source term for condensation and evaporation (5.3) reproduces the experimentally observed vertical flux of condensate in the system of interest. Observational [Nicholls and Leighton, 1986; Duynkerke et al., 1995; Khairoutdinov and Kogan, 1999] and numerical studies [Moeng, 1986; Khairoutdinov and Kogan, 1999; Wang and Wang, 1999] of atmospheric clouds demonstrate that the mean vertical flux of LWC is approximately linear in z, i.e. p = 1 58 as above. The vertical dependence of (u3ifjc) is consistent with the discussion in Sec. 5.1 in that density fluctuations in a condensation cloud are non-homogeneous. 5 . 3 C E in the Batchelor limit In this section we evaluate C E using the 5-correlated closure. The diffusion equation for the second-order correlation function, <& = (0(aj)©(y)), from Eq. (2.42) with b = 0 is — = -2{K5mn + rFmn(0) - rFmn{r)]dx + J , (5.5) where O = ipc — (tbc) and I is the contribution from any source terms. Note that the con-ventional Reynolds stresses (Qun)d(ipc)/dxn that normally couple 0 to mean-gradients in the passive scalar field do not contribute in the 5-correlated model. Following the procedure outlined in Sec. 2.5 the condensation-evaporation source term I from Eq. (5.3) is evaluated as follows: ) = - 2 T Z LF3n(x,x)- 2rz lF3n{x,y) — dxn dyn = 2[rF3n(0)-rF3n(r)]z-1^- (5.6) oyn which has the form of an advection term. It should be noted that in the modeling and closure of the condensation/evaporation term pursued here, the closure of the advection term is independent of / resulting in Eq. (5.5). An alternate approach is to explicitly in-corporate I into the closure ofthe advection term. A comparison of these two approaches will be presented in a future study. It is illustrative to compare the velocity VCE = — 2[F 3 n(0) — F3n(r)]z~l from Eq. (5.6) with the velocity Vpj — 4dFmn(r)/drn caused by particle inertia in Eq. (2.42). Ignoring the anisotropic nature of the former, the two velocities scale according to: VCE ~ 7-2 and Vpi ~ r in the viscous regime, and VCE ~ r2/3 and Vpj ~ r - 1 / 3 in the inertial regime. Thus evaporation/condensation is a source of scalar variance that increases with increasing r (infrared divergence), whereas the effect of particle inertia is limited to scales 0(n). The general behaviour of Eq. (5.5) with / given by (5.6) is worthy of some discussion. First, note that the viscous-convective scaling $ = constant is the trivial solution of (5.5) with or without the source term I. Thus a normal viscous-convective subrange is one prediction of the present model. However, a cloud with a vertical mean-gradient is axisymmetric about the e3 axis, and therefore we expect $ to contain contributions from odd-order terms in r 3 . Furthermore, the anomalous ID (horizontal) scalar spectrum shown in Fig. 5.1 suggests that the non-homogeneous vertical component of $ which 59 does not directly contribute to the horizontal scalar spectrum may be disrupting normal viscous-convective scaling. Thus we assume that there are other non-trivial contributions to $ in the viscous-convective regime. Second, note that since r < z(x\,X2) can be treated as a constant parameter independent of r. If we assume that $ has a term $ ' = C i 2 T 3 with c i e > 0 which is consistent with aircraft measurements [Stephens and Piatt, 1987; Noonkester, 1984; Vali et ai, 1998] and numerical simulations [Kogan et al., 1995; Stevens et al., 1996; Khairoutdinov and Kogan, 1999] that show increasing LWC fluctuations with increasing height, then /($ ') becomes which is a positive source that increases with increasing r. The general form of the solution of Eq. (5.5) keeping terms greater than z - 1 and ignoring molecular diffusion then becomes where c 0 > 0 and the signs of c 2 and c 3 have yet to be determined. By definition the horizontal correlation function ^(ri ,^) as well as the horizontal viscous-convective spectral scaling is independent of r 3 ; only the effects of the first and last terms in Eq. (5.7) are evident in horizontal measurements. It is important to emphasize that Eq. (5.7) is not the solution of Eq. (5.5) but only illustrates the general r-z or in Fourier space k-z scaling that appears later when more rigorous methods are used. However, the picture that emerges from this analysis is robust—a non-homogeneous component ~ r 3 of $ that is z/r larger than the homogeneous components changes the normal k^l horizontal viscous-convective spectral scaling. 5.4 Axisymmetric Kraichnan Transfer We begin the derivation of the spectral covariance density function \&(k) by considering the axisymmetric solutions of (5.5) without the source term / . 5.4.1 Viscous regime solution The equation for $ in the viscous-convective subrange was first derived by Kraichnan [1968] and is given by Eq. (2.44). Since $ and hence ^ in a condensation cloud is axisym-metric, we are interested in the generalization of Eq. (2.45) to axisymmetric variables. The Fourier transform of (2.44) is easily found by using d/dr3 —> ikj, r3 —> id/dk3 and then converting to axisymmetric variables where 6 = cos_ 1(fc • e3/|fc|) is the angle between the wave vector k and the vertical axis: /($ ') = 2 C l [ F 3 3 ( 0 ) - F 3 3 ( r ) ] $(r, r3, z) « c0 + cizr 3 + c 2 Cir 3 + c 3 cir 2 , (5.7) dt ~ ' T(tf) = k2 (5.8) (5.9) 60 The isotropic solution of Eqs. (5.8) and (5.9) is discussed in Sec. 2.6 and reproduced below: X A . 1 fc - 3 [ l + A BA;]exp(-A BA;), (5.10) 47r|7| XB = (6/.I7I-1)1/2, (5.11) where K is a modified Bessel function, x(z) ~ z2 is the non-homogeneous scalar dis-sipation rate and A^ is a diffusive length scale that is proportional to the Batchelor length. For an isotropic scalar field the corresponding scalar spectrum E{k) is defined as E = Airk2^. Therefore in the range k <C A ^ 1 , E(k) = x M - 1 ^ - 1 which is the usual A ; - 1 viscous-convective scaling. The general solution of Eqs. (5.8) and (5.9) is obtained using the method of separation of variables: oo 0 = {1~^)li9'~2^+j{j + l)Pj' (5>12) X*k2Bi = k 2 ^ + Ak^-2j{j + l)Bj> ( 5 - 1 3 ) where \i — cos 9 and c3- G C is an arbitrary constant. Immediately we identity P3 as a Legendre polynomial since Eq. (5.12) is the familiar Legendre equation [Abramowitz and Stegun, 1970]. The Fourier space symmetry relation ^f(k) = ty*(—k) restricts the Cj ' s such that for even j, Re{cj} G K+ and Im{cj} = 0, whereas for odd j, Re{cj} = 0. Thus the odd terms represent the vertically non-homogeneous component of the spectrum, whereas the even terms are homogeneous contributions. Note that for a passive scalar field in homogeneous turbulence with an imposed mean gradient the odd terms are identically zero [Herr et al, 1996]. Eq. (5.13) for B3 is a Bessel type equation with solution [Abramowitz and Stegun, 1970] B3 = k-zl2Kv{3)(\Bk) where u(j) = [9 + Sj(j + l ) ] 1 / 2 /2 . Note that the P / s satisfy f^dp Pj(fj) = 25(j) so that only the j = 0 term contributes to the spheri-cally averaged spectrum. The expansion of the scalar spectrum in terms of Legendre polynomials was first suggested by Herring [1974] who derived an equation for \P in ax-isymmetric turbulence using Kraichnan's direct interaction approximation (DIA). The ^-correlated model can be formally recovered from DIA in the limit that the Greens' function G(x,t\t;y,t\t0)—the scalar amplitude at (x,t) arising from a 5-function source at to located in the fluid element that arrives at (y,t)—becomes 53(x — y) [Kraichnan, 1968]. Thus Eqs. (5.12) and (5.13) are a special case of the more general results in Herring [1974]. 61 As discussed in Sec. 5.3 we are interested in the solution of the correlation function up to approximately second-order in r 3 which corresponds to expanding \I/ to j = 2. The axisymmetric (j = 1,2) contribution to the spectral density, \ & a x t : can be written = * S T ( * > + A * ) , (5-14) = 4 T ^ f > ) ^ ^ ( A ^ ) P 2 ( / i ) ' « T ( ^ ) = 3 ( 2 J ) 3 / 2 fc,/A ^ ( A B ^ P X M , (5.15) such that l i m ^ o ^ r ^ , " ) = ^ X / ^ T ^ A ^ A T 3 / 2 - ^ ^ ) and l i m ^ o M) = Cx/(4vr|7|)zA^2/<;-4P1(/ii), and where v = y/57/2 « 3.775. The constant C in Eq. (5.15) is of fundamental importance in what follows and plays the role of the Kolmogorov constant for the non-homogeneous (imaginary) component of the spectral density. As discussed in Sec. 5.3 the non-homogeneous component v&J11 is assumed to scale as zk times the homogeneous components W° or \&3fl as illustrated in Eq. (5.15).. The resulting r-space scalings predicted by Eq. (5.14): r and r 2 2 7 5 is close to the r and r 2 scaling estimated in Sec. 5.3. 5.4.2 Inertial-Convective regime solution A solution for \& in the inertial-convective subrange r > 77 is facilitated by the fact that molecular diffusivity can be ignored. Evaluating Eq. (5.5) with (3.1) and velocity correlation coefficient F(r) = 1 — a2r2^ where a2 is an arbitrary constant gives 82<t> \4r2S - r r l = 0 OTmOTn or, when Fourier transformed, n T o , 2 ^ 2 * 4cos0dtf d 2tf n 12* + 3A; 2 -— + 16A; — + ——^^ + 4 — ^ = 0. aAr <9A; smf> 08 082 Expanding the solution in terms of Legendre polynomials gives 00 3=0 0 = sfc^ + iefc^-iw + i)-!^-, with solution P j = A r 1 3 / 6 ^ - ^ + d3kv^} where = {169 + 12[4j"(j + 1) - 12]} x/ 2/6, and C j and d j are arbitrary constants. In general both c3- and d j are non-zero; however, in the small k viscous-convective regime, k~"^ ^> ku^\ Therefore, without loss of generality, we can set d j = 0. Note that the scaling of the isotropic (j = 0) solution 62 vT/"° ~ k~3 is invariant under a change in the velocity spatial correlation, a manifestation of the fact that $ = constant is the trivial solution of Eq. (5.5) independent of the effective diffusivity. In addition, the scaling of the (j = 1) solution \I/"XJ ~ k~A also remains invariant. The scaling of typ*1 (j = 2) changes only slightly from a viscous scaling of « —5.275 to an inertial scaling of —(13 + v /313)/6 « —5.115. Because of the steeper spectral decay of the axisymmetric contribution, \I>"0 3> for k > and therefore, only the k~5115 scaling makes a significant contribution to the overall spectral density. Using the approximation AT 5 - 1 1 5 m k~5, v p ^ 1 1 can be written nx\k,ri = ^\-B2k-5PM (5.16) which corresponds exactly with the r-space scaling of the last two terms in Eq. (5.7). Eqs. (5.15) and (5.16) are used in the rest of this work to represent axisymmetric viscous-convective scaling. 5.5 The axisymmetric source / The contribution of the axisymmetric, real term \I>5fJ to the overall spectral density in the absence of the condensation/evaporation source I is fundamentally limited by the restric-tion Re{^} > 0. Using P2(A t) = (3/j2 —1)/2 and an inertial-convective/viscous-convective boundary at k = kb this restriction becomes ^""(fct) > tyjfl(kb, /_i = 0, ±1) . For example, using the spectra \&" 0 = k~3, the maximum allowed anisotropy ^"g1 = B2(k)P2(/j,) where B2(k) = 2k~5kb, and the identities for the ID horizontal spectra (j)lS0{kx) — AnJ^kdk Wso(k) and (j)axi(kx) = TT / ~ kdk (1 - 3k2x/k2)B2{k) gives <j)iso(kx) = Ink'1 and (f)axi(kx) = — 8iv/15kx3kl. Thus the maximum possible change in the horizontal spectrum which occurs at the boundary k = kb is only 2/15 or about 13 percent! The physical interpretation of these results is straightforward. The solution tyRxl represents a conservative transfer or rotation of scalar variance along the e 3 axis, and thus in the absence of a source, little rotation is possible before the variance becomes depleted at \i = 0 or ± 1 . The source term / , therefore, plays a crucial role in balancing this anisotropic conservative spectral transfer. The equation for \& in the viscous regime (2.44) including the source term (5.6) is % = 2 ^ ^ [ 2 r 2 5 m n - r m r n ] ^ dt ' 3 L m " m n j drmdrn hi r 2 r 2x _ , , , ] — or, Fourier transformed as per Eq. (5.8), ^ = - 2 ^ + M r w _ i b l ^ w , 63 sm9 d2V 2d2m + cos 9 89dk + k~dW 2 cos 9 sin 9 k sin 9 k cos 9 ~d9 where T($) is given by (5.9). The equation for the spectral contribution \I>src from the source / is —— = -2Kk2^src + ^ T ( * s r c ) - -^TOi^Y1). (5.17) Using (5.15) the source term becomes nliV?') = i [Kci^f1) c o s 2 0 + TZs(*?xi) sin 2 9] , A 5 / 2 f" A 3 / / 2 Note that for small k, TZC < ?es and fts(#jxi) = 5 C X / ( 4 T T | 7 | ) A ^ 2 A ; - 5 . Therefore the small k, steady-state equation analogous to (5.17) is T(Vsrc) + ^ 7K2k-5 sin 2 9 = 0. (5.18) 47r|7| Assuming a solution ofthe form \T/ s r c = (A + Bp2)k~5 and evaluating (5.18) gives — 2B = -10A - 45 = 5 C X / ( 4 T T | 7 | ) A 5 2 or * ' r e ( * ^ ) = - 4 ^ A 5 2 A : ' 8 ^ ( 5 ' * 2 - 1 ) -Not surprisingly, this solution is invariant under the transformation from inertial to viscous velocity correlations (not shown). Assuming that the sum of the axisymmetric, real terms \&^x l (5.16) and \T/ s r c is approximately isotropic gives c 2 = (5/3)£, and the resulting spectral density is Thus axisymmetric production of scalar variance (\I/src) is balanced by a conservative axisymmetric transfer of scalar variance (^^X l). An assumption of perfect isotropy is not necessary to prevent the spectral density from becoming negative. However, a small degree of anisotropy in typ1 + \& s r c has little effect on the results and conclusions in the following sections. Combining Eqs. (5.10) and (5.19) gives the resultant spectral density * ( J f c , / i ) = ¥so{k) + [^axi + ysrc]{k) + i^aIxi(k,iJ,X), X I--3M i \ \ i„\ X C \ - 2 i „ - 5 47r|7|*" [l + *Bk]exp(-\Bk)-^-\B k-+ ^r(fc)jtx,c), 64 where is given by (5.15). The spectrum E(k) = 2irk2 d\x *(A;,/i) is therefore E{k) = ^k~l [1 + XBk] exp(-XBk) - ^\-2k~3. (5-20) Comparing Eq. (5.20) with (5.7) we find that c 2 = 0 as a result of the isotropic assumption (above). The Kolmogorov-like constant ( first introduced in Sec. 5.4.1 is determined in the next section. 5.6 Determination of £ The only free parameter in the present model is the fundamental constant £ defined by Eq. (5.15). Like the Kolmogorov constant, ( should asymptote to a well-defined value in the large ReA limit. Since independent information on ( is not yet available in the literature, its value is chosen to best reproduce the experimental data shown in Fig. 5.1. Following the methodology of Sec. 3.4 we evaluate C from Eq. (3.15). The determi-nation of the production spectrum of scalar variance, V, in this study is complicated by the fact that an expression is needed that is accurate in both the viscous k > rfx and inertial k < rf1 regimes. The viscous regime form, Vv, follows from Eq. (5.18): r7T 7 si JO Vv(k) = 2ivk2 I sin6d6 ^ A T / A T 5 sin 2 0, 10CX A R 2 A; 9 ^ The scale break between the viscous and inertial regimes k{ is usually taken to be around O.lry - 1. Thus for k < ki we can expect the inertial scaling Vi(k) ~ fc~5/3 where Vi(ki) = Vv(ki). The resulting expression for x using Eq. (3.15) is Xo - - § 7 4 7 3 [* _ 2 / 3 - ( 2 / W 2 / 3 ] * < k, *(*) = < 3 5 A ^ • (5-21) X o ~ 9 A p ^ k > k i The unknown constant £ can be determined from Eq. (5.21) in principle using the new liquid water data in Davis et al. [1999]. One source of uncertainty, however, is the magnitude of the Obukhov-Corrsin constant (3 in the inertial-convective regime parame-terization Eic(k) = feT1/3AT5/3, (5.22) where e is the energy dissipation rate and xtc — constant is the inertial-convective range scalar dissipation rate. Since the change in (3 due to condensation/evaporation is un-known, (3 is assigned its inert passive scalar value of ~ 3/4. Using the data in Davis et al. [1999] I estimate that x%c = x(h) = Xo/14 which gives C = |A|fcp[ifc 6- 2/ 3-(2/3)fcr 2 / 3]- 1, (5.23) 65 where kb is the wave-number of the scale break between the inertial-convective and viscous-convective regimes. Substituting Eq. (5.23) into (5.20) with the identification X ->• Xo gives E(k) = r^\k~l [1 + XBk] exp(—XBk) — n\ ^ ^ 4 / 3 [ V 2 / 3 - ( 2 / 3 ) f c - 2 / 3 ] - ^ - 3 . (5.24) The final step in the specification of E is the determination of kb- Numerical simulations [Bogucki et al, 1997; Chasnov, 1998] of the viscous-convective subrange with V = 0 suggest that the scale break kb occurs, naturally, at the intersection EiC(kb) = E(kb) which can be calculated numerically from (5.22) and (5.24). Using /3 = 3/4, ki = O.lry - 1 and recalling from Sec. 5.4.1 that 7 = —(l/c/)r~x where q = 5.5 produces kb = 0.0477-1. Thus the predicted scale break between the inertial-convective and viscous-convective regimes is at somewhat larger scales than the usual break at kb = (P/q)3^2^'1 ~ O.O577-1. This extension to larger scales is in contrast with the effect of particle inertia which suppresses near-inertial viscous-convective scaling as discussed in Sec. 3.5. In the atmosphere where 77 ~ 0(1 mm), the predicted r-space scale break of the present model occurs around 25 cm which is an order of magnitude smaller than the transition estimated by Davis et al. [1999] to occur at 2-5 meters. This apparent discrepancy is discussed further in the next section. Eqs. (5.21-5.24) complete the determination of E(k) as a function of the parameters xo> |7|> <\B> ki and kb. 5.7 Spectra and discussion Before embarking on a discussion of the predictions of the present model, it should be emphasized that these predictions are highly dependent on the value of the Kolmogorov-like parameter ( [Eq. (5.15)]. In particular, in the limit £ —> 0 normal A ; - 1 viscous convective scaling is recovered. Despite this deficiency, the present model provides an appealing analytical framework within which the anomalous scaling of cloud LWC can be explained. In the region (kb — 0.0477-1) < k < 0.35?7_1 the scalar dissipation rate increases with increasing k according to Eq. (5.21). A typical atmospheric value for 77 is w 1 mm, and therefore, this "production subrange" corresponds to scales of about 3 cm up to 30 cm. At smaller scales (r < 3cm) a normal viscous-convective subrange exists associated with a constant scalar dissipation rate xo and at larger scales (r > 30cm) the cascade of variance from larger to smaller scales dominates the dynamics. In the production subrange the spectral scaling changes from a negatively sloped A; - 1 scaling to a positively sloped A; - 3 scaling [Eq. (5.24)], reflecting the production of scalar variance in the vicinity of kb. The scalar spectrum given by Eqs. (5.21-5.24) is shown in Fig. 5.2 along with the change in scalar dissipation rate x{k)/Xic- The increase in variance beginning at k = kb 66 0.005 0.020 0.100 0.500 2.000 kr\ Figure 5.2: Plot showing the inertial-convective, production and viscous-convective sub-ranges predicted by the present model [Eqs. (5.21-5.24)]. The production subrange begins at kb = 0.04T7 - 1, ends near 0.4r7_ 1 and is associated with increasing scalar production and dissipation. The increase in x by a factor of 14 presented at the top of the figure is cho-sen to reproduce the ID LWC spectrum measured during S O C E X and shown in Fig. 5.3. Normal inertial-convective/viscous-convective scaling is also shown for comparison. The figure is generated using e = 0.01 m 2 s~3. is associated with a corresponding increase in the scalar dissipation rate. Outside of the production subrange normal inertial-convective and viscous-convective behaviour is evident. The bump in the scalar spectrum in the production subrange—a reflection of increased variance in this regime—is superficially similar to the spectral bump caused by particle inertia shown in Fig. 3.3. The location of the spectral peak at kp « 0.033r?_1 in the present model represents increased variance at scales one order of magnitude larger than preferential concentration (kp « 0.3?7_1), a relationship mirrored by the behaviour of the condensation/evaporation induced velocity {ru(0)u(r))/z [Eq. (5.6)] compared to the particle inertia induced velocity (ru(O)V-u(r)) [Eq. (2.42)]. Despite some similarities between Fig. 5.2 and Fig. 3.3, the physics of condensation/evaporation and preferential concentration is distinctly different. Preferential concentration is an accumulation or clumping of inertial particles in regions of high strain and low vorticity in a turbulent flow and therefore, by definition, is a manifestation of a non-uniform particle distribution. 67 In contrast, the increased variance exhibited by the present model arises from variable particle mass due to condensation and evaporation despite a uniform particle number distribution. The I D horizontal spectrum defined by (j)(kx) = Jfc°° k~ldk E{k) is shown Figure 5.3: Comparison of the ensemble-averaged I D LWC scalar spectrum mea-sured during S O C E X [Marshak et al, 1998; Davis et al, 1999] and the present model [Eqs. (5.21-5.24)]. The factor of 14 increase in x{k = kx) in the production subrange is chosen to produce good agreement between (f)(kx) and the data at large kx. The dis-crepancy in the modeled and observed spectra near kx = 0.008?7_1 may be a result of the unnaturally sharp transition between the inertial-convective and production regimes shown in Fig. 5.2 and used in the generation of 4>{kx). in Fig. 5.3 along with the experimental data from Davis et al [1999]. The good agree-ment between the modeled and observed spectra for k « 0.04r?_1 is not fortuitous—the relation Xic — Xo/14 used in Sec. 5.6 to determine the unknown constant C, was chosen to produce a close correspondence between the two spectra in this region. For k in the range 0.00277-1 < k < 0.04?7_1 the modeled spectrum falls some-what below the experimental data. The plateau in the modeled spectrum near k = k^ is associated with the sharply defined local minimum exhibited by E in the same region and shown in Fig. 5.2. It is very likely that the real transition between the inertial-convective and production regimes is much smoother than the prediction ofthe present model, i.e. Fig. 5.2, which may explain the discrepancy between the I D spectra shown in Fig. 5.3. The plateau in the modeled 68 spectrum near k = kb may also explain the discrepancy between the 25 centimeter scale break (kb = 0.04r7_1) predicted by the present model and the break estimated by Davis et al. [1999] from experimental data (Fig. 5.3) to occur at 2-5 meters (kb « 0.002ry_1). Certainly, since <f)(kx) is a projection of the actual 3D spectral density \J/(fc), an abrupt change in the scaling of * (or E) appears smooth and gradual when projected onto kx. Thus, the appearance of (f>(kx) is not necessarily a reliable indicator of the behaviour of E(k). Overall, the experimental data in Davis et al. [1999] does support the existence of a production subrange predicted by the present model (5.21-5.24). The key assumption in the derivation of the production subrange is the existence of the imaginary spectral density ^fjxt (5.15) that goes as zk~4/j, for small k. Because scales with an integer exponent the r-space contribution cannot be calculated without knowledge of a transition from the k~4 scaling to a different (non-integer) scaling regime. Clearly, more information on the spectral density of liquid water in clouds from numerical simulations is needed to ascertain the validity of the scaling and magnitude of used in the present model. 5.8 Experimental verification The basic principle of the present model is that variability in cloud liquid water density due to condensation/evaporation leads to increased variance at small scales. This hy-pothesis could be verified by aircraft measurements of cloud droplet number density at a comparable resolution of about 4 cm. If contemporaneous measurements of a normal number density spectrum and an anomalous water density spectrum are observed then this would provide strong support for the mean field model of this chapter. However, these measurements would not rule out Gerber et al. [2001]'s entrainment hypothesis if cloud-top entrainment were to affect droplet size but not droplet number. Unfortunately, the resolution of state-of-the-art cloud number density measurements is currently much greater than 4 cm. Experimental determination of the location of the scale-break, kb, between the inertial-convective and viscous-convective regimes under a variety of cloud-top conditions could potentially provide further information on the relative effect of entrainment on small-scale liquid water density. In the present model kb is a fitted parameter and therefore the relevant large-scale cloud features that impact kb are unknown. Another source of uncertainty in the present model is the value of the Obukhov-Corrsin constant (3 in the inertial-convective regime described by Eq. (5.22). In principle (3 could be determined from cloud liquid water density measurements that extend to scales smaller-than the Kolmogorov length. However, such high frequency measurements are currently not available. Hopefully this will improve in the near future. 69 5.9 Summary In this chapter a mean-field model for the effect of condensation and evaporation on passive scalar statistics is developed that relates the phase change of the condensate to the vertical structure of its first and second moments in the cloud. Unlike inert scalar statistics with an initially imposed scalar gradient, the new model predicts non-homogeneous vertical density fluctuations—in good agreement with atmospheric mea-surements [Stephens and Piatt, 1 9 8 7 ; Noonkester, 1 9 8 4 ; Vali et al, 1 9 9 8 ] and numerical simulations [Kogan et al, 1 9 9 5 ; Stevens et al, 1 9 9 6 ; Khairoutdinov and Kogan, 1 9 9 9 ] that show increasing liquid water fluctuations with increasing height in clouds. As a first step towards understanding the effect of condensation/evaporation on passive scalar statis-tics, an equation for the spectral density \I> is derived in the viscous-convective regime where an exact closure is available. The derivation proceeds in two parts: the axisym-metric spectral contribution in the Batchelor limit, derived for both viscous and inertial velocity correlations, is written as an infinite sum of Legendre polynomials of /J as first suggested by Herring [ 1 9 7 4 ] ; the first-order contribution from condensation/evaporation is also derived assuming that the imaginary (non-homogeneous) part of \I/ is significantly large. In the absence of condensation/evaporation axisymmetric Kraichnan transfer of scalar variance is virtually forbidden because of the restriction that the real part of \& be positive. However, in the presence of condensation/evaporation the possibility of axisym-metric transfer balancing axisymmetric production of variance to produce an isotropic, homogeneous contribution E{k) ~ A; - 3 exists and is explored. Under the assumption of spectral balance, an expression for \& is derived that re-produces the spectral behaviour of new experimental data of cloud liquid water density [Davis et al, 1 9 9 9 ] which exhibits anomalous viscous-convective scaling. The modeled spectrum has one adjustable constant reflecting the magnitude of the imaginary (non-homogeneous) part of the spectrum; the value of this constant is chosen judiciously so that good agreement is obtained between the observed and modeled horizontal spectra. The present model predicts a production subrange, 0 . 0 4 T / _ 1 < k < 0 . 3 5 T 7 - 1 , where the scalar dissipation rate increases with increasing k. Associated with increased dissipation is a change in the spectral scaling from the usual negatively-sloped A; - 1 viscous-convective scaling to an anomalous positively sloped A; - 3 regime. The resulting scalar spectrum in the production subrange has a well defined bump reflecting increased variance due to con-densation and evaporation, similar to the behaviour exhibited in the spectrum of inertial particles as discussed in Chapter 3 . The scale break between the inertial-convective and production (viscous-convective) subrange occurs at 0 .04?7 _ 1 —s l ight ly smaller than the usual transition near 0 . 0 5 7 7 - 1 for an inert scalar—although the break in the ID horizon-tal spectrum remains consistent with data and Davis et al. [ 1 9 9 9 ] ' s somewhat larger-scale estimate. Despite some uncertainty in the vicinity of the inertial-convective/production subrange transition, the present model provides a convenient analytic framework within which the non-homogeneous, anisotropic behaviour of condensation cloud spectral scaling 70 may be explored. Chapter 6 Unresolved Variability of Low Cloud When sufficient measurements of reflectivity have been accumulated they are likely to be used along with other data in answering such questions as :— (i) Would flooding the Sahara cause more solar radiation to be lost to space? (ii) When primaeval forests have been cut down and replaced by wheat fields, as in the United States of America and Canada, does the country entrap more or less solar energy? (iii) During geological times, as Ice Ages and tropical forests came and went, did the reflectivity of the vegetation have an influence on the extension or contraction of its field? [Richardson, 1930] 6.1 Introduction In this chapter we consider the unresolved physical and optical variability of low clouds using a statistical cloud scheme and the linear cloud model from Sec. 5.2. Although we do not use the (^-correlated closure per se, we continue to adhere to the statistical methodology that is developed in Chapters 2 through 5. As we shall see shortly, a careful and self-consistent treatment of the moments of unresolved cloud amount is at the heart of statistical cloud schemes whereby unresolved variability of temperature and moisture in a model cell is assumed to obey a known distribution, often a Gaussian. At first sight, this assumption may appear odd to those readers who are familiar with the theoretical passive scalar literature; certainly, much effort is currently being devoted to the theoretical determination of passive scalar probability density functions which are generally not Gaussian and not known. However, the behaviour of statistical cloud schemes is distinctly different from inert passive scalar statistics because of the influence of condensation and evaporation. In fact, at large cloud fractions, the modeled response of grid-averaged cloud amount to small changes in temperature and moisture depends 72 strongly on the condensation/evaporation parameterization but is relatively insensitive to the assumed shape of the unresolved fluctuations. Clouds of varying distribution, character, water content, and altitude are an intrinsic feature of our climate system. Of all the climate constituents, clouds are unique in their ability to significantly affect both the shortwave and longwave radiative streams that are the primary components of the Earth's energy budget. There is little debate that clouds—playing the part of an umbrella more than a greenhouse—currently act to cool our climate. However, the role clouds may play in future climate change, as important modifiers of the radiation budget or as integral elements of feedback loops which enhance or buffer the climate's dynamic response, is a question of considerable interest. Although the radiative impact of clouds on climate is understood in principle, the dependence of clouds on the variables of the climate system is only understood in iso-lated areas under rather limited conditions. Observational studies have provided corre-lation coefficients between various microphysical parameters, e.g. cloud liquid water qi and temperature T [Somerville and Remer, 1984], which are used to infer sensitivities, e.g. dqi j dT, assuming all other relevant parameters remain fixed. Cloud sensitivities can then be inserted into radiative transfer models of varying sophistication to determine the sign and magnitude of cloud feedback. Cloud feedback is a measure of the net radiative impact of clouds on the global energy balance; the precise definition of cloud feedback varies from study to study. Global climate models (GCMs) typically allow changes in cloud amount and optical thickness to feed back into the dynamics of the climate system. Thus in these "closed-loop" studies, cloud feedback is typically defined as the difference in equilibrium surface temperature between a model integration where the clouds re-spond to climate forcing and where cloud properties remain fixed. On the other hand, in "open-loop" studies, e.g. ID radiative-convective models, where global dynamics are not represented, cloud feedback is typically defined as the net (positive downward) radiative flux at the top of the atmosphere due to the cloud response. In both definitions cloud feedback is defined w.r.t. a positive temperature perturbation; a positive cloud feedback implies that cloud response enhances warming while a negative cloud feedback implies a relative cooling. Closed-loop studies of global cloud feedback are perhaps only possible through the use of GCMs. Comparative studies of GCMs have long shown a marked sensitivity of the predicted climate to the parameterization of cloud properties. Early attempts to quantify this sensitivity were hampered by cloud diagnostic schemes which were inherently biased towards the contemporary climate. Recently, prognostic cloud schemes based on an assumed statistical distribution of subgrid variability have replaced the older diagnostic schemes in many models. Although the relationship between unresolved variability and mean cloud amount is known in principle, a corresponding relationship between ice-free low cloud physical and optical properties is lacking. In this chapter we consider a unified theory of low cloud physical and optical variabil-ity that links mean reflectivity and emissivity to the underlying distribution of unresolved 73 fluctuations. In Sec. 6.2, statistical cloud schemes are introduced and a stochastic sub-grid variable s that represents unresolved fluctuations in the temperature and moisture fields is defined. A relationship between optical depth, cloud liquid water density and cloud number density is derived in Sec. 6.3. In Sec. 6.4 we bridge a gap between physical and optical variability in these schemes by first restricting s and hence Ps to be height independent in low clouds. At the same time, we consider a distribution of cloud top height fluctuations (z[0 ) that is orthogonal to a height independent Ps. Using our de-rived coupling of fluctuations s and z'top and an idealized climatology, we consider the unresolved variability and radiative properties of low clouds in Sec. 6.5. The relationship between unresolved variability and radiative impact is complicated by the fact that the total reflectivity (_Rtot = ACR) of a cloud layer is a non-linear function of cloud fraction, Ac—the fraction of sky covered by cloud when viewed from below—and cloud optical depth, r, a measure of the "opacity" of the cloud and related to the vertical integral of qi(z) from cloud base to cloud top. Thus, for example, an increase in qi (or r) caused by increasing temperatures does not necessarily imply an increase in i ? t o t if Ac decreases producing an optically thicker cloud field with smaller cloud fraction. In this chapter we also consider the impact of low model vertical resolution on G C M predictions of cloud optical properties. In Sec. 6.6 the work of Lohmann et al. [2000] is used as a case-study to demonstrate that effective radius, reff, at cloud top is a problematic diagnostic quantity in large-scale model studies. Two new diagnostic quantities that are not explicit functions of reff, fractional liquid water path and fractional column droplet concentration, are introduced and shown to be well-suited quantities for the diagnosis of cloud-radiation interactions in models with low vertical-resolution. In Sec. 6.7, we derive analytic expressions for two response functions that characterize two potential low cloud feedback scenarios in a warming climate. We find that low model vertical resolution can cause a significant overestimation of the unresolved low cloud Ac response by a factor of around 2.5. Section 6.8 discusses experimental validation of the present model and Sec. 6.9 contains a summary. The results in this chapter also appear in Jeffery and Austin [2001b] and Jeffery [2001d]. 6.2 Statistical Cloud Schemes Statistical cloud schemes have a long history that dates back to the pioneering work of Sommeria and Deardorff [1977] and Mellor [1977]. Large-scale atmospheric models typically contain temperature, pressure and total water (vapour+liquid) fields that evolve according to prescribed dynamical and thermodynamical equations. Traditionally these numerical models would assign, for each field, a single average value to an individual grid cell, thereby ignoring any variability within the cell. The relative importance of this neglected variability is, not surprisingly, scale-dependent—for large-scale climate models with grid spacings 250 km or greater the unresolved variability can be a substantial 74 fraction of the mean value. Furthermore, the relative importance of subgrid variability is magnified ten-fold by the presence of condensation which is a small difference in two relatively large scalar quantities: the saturation vapour density, qa, and the cell's total vapour (or water) density, qv, prior to condensation. Early climate modelers were well aware that the use of "all-or-nothing" condensation schemes, whereby an individual grid cell is either completely clear or completely cloudy depending on the difference qv — qs, is a particularly acute problem. The Sommeria-Deardorff-Mellor (SDM) statistical cloud scheme introduces a stochas-tic subgrid variable s that represents unresolved fluctuations in qv — qs and is assumed to be normally distributed. The variance of s, cr2, in more sophisticated schemes can be diagnosed from a turbulence model [Ricard and Royer, 1993] or from neighbouring cells [Levkov et al., 1998; Cusack et al., 1999] but, in practice, is often taken as a prescribed fraction [Smith, 1990] of q2. A key assumption in the SDM scheme is that each grid cell is assumed to contain a complete ensemble of s from which the statistics of unresolved cloud are calculated, regardless of the size of the grid or the time-step of the model. For example, the mean liquid water to some power p, qf, in the cloudy region of a cell is given by • where Ps is the probability distribution function (e.g. Gaussian) of s, the cloud density, Ad, is the fraction of grid cell occupied by cloud, and < 1 is a parameter that accounts for latent heating of the cloud. For the small Ad of a typical G C M grid cell, aL is reduced from its adiabatic value due to detrainment of heat out of the cloud. Assuming a constant value ar, m 0.75 in the boundary-layer is consistent with a broad range of temperatures and sub-adiabaticities. In what follows an overbar is reserved to represent an average over the cloudy fraction of a cell, and unresolved variability in each cell is assumed to be centered (i.e. have zero mean). 6.3 S h o r t w a v e O p t i c a l D e p t h F o r m u l a t i o n Over 30 years ago, Hansen and Pollack [1970] made the important discovery that visible and near-infrared single-scattering optical properties of cloud droplets depend mainly on the mean droplet radius for extinction, r e x t , given by where r is the radius of the droplet, n(r)dr is the number per unit volume with radius [r, r + dr], Qext(r, A) is the extinction efficiency, and A is the wavelength of light. Hansen (6.1) Fact (A) = J0°° r3Qext(r, X)n(r)dr f0°° r2Qext(r, X)n(r)dr' 75 eliminated the wavelength dependence with the approximation Qext(r, A) = constant, which he found to be quite accurate for typical atmospheric cloud droplet size distribu-tions, and in the following year he introduced the effective radius [Hansen, 1971] (6.2) where (• • •) = Jjj30 • • -n(r)dr. The implications of Hansen's work are manifold: model-ing studies of radiative-transfer through cloudy atmospheres require knowledge of only second and third moments of n(r); details ofthe cloud condensation nuclei population of entraining air can be ignored because the largest droplets dominate reg [Choularton and Bower, 1993]; and the complicated dependence of n(r) on cloud type can be cat-egorized and quantified in terms of the single parameter re^. Recently, Damiano and Chylek [1994] have verified that reg is the single most suitable parameter for character-izing the single-scattering albedo UJ0, the asymmetry parameter g, and the normalized volume scattering (extinction) coefficient (3sca./qi (Pext/qi) where qi is the volume-averaged liquid water content. The shortwave optical properties of a cloud layer—reflectance, transmittance, absorp-tion—are largely a function of the cloud optical depth, r , defined as the integral of the volume extinction coefficient through the cloud layer. Using the approximation Qext{r, A) = 2, T is given by where pw is the density of water, Zbot is cloud base height and ztop is cloud top height. Stephens [1978] made the further assumption that reff(z) in Eq. (6.3) is independent of where L W P is the liquid water path. Stephen's parameterization of r in terms of LWP and reff is not a unique relationship, but it has emerged, in various forms [Slingo, 1989; Savijarvi et al, 1997; Chou et al, 1998], as a common parameterization in large-scale climate models. The advantages of using LWP to describe r were twofold: (i) LWP was already needed to calculate the longwave emissivities and (ii) global information on L W P was available at the time from satellite microwave radiometry. The removal of reg(z) from the vertical integral in Eq. (6.3) is an unnecessary approx-imation—the sum of the ratio (qi/reg)i can be evaluated in a discrete atmospheric model with no greater difficulty than the discrete evaluation of LWP. For low clouds where the liquid water content increases linearly above cloud base, the approximation reff = r eff(ztop) introduces an error in r of up to r /5 or 20% [Brenguier et al, 2000], a deviation which is much greater than the tolerances of the newer parameterized shortwave schemes [See the discussion in Brenguier et al. [2000]]. Yet, Stephen's approximation [Eq. (6.4)] is still (6.3) z so that r = 3 LWP 2p«, r e f f (6.4) 76 used to evaluate r in state-of-the-art GCMs where reff is evaluated at cloud top [Jones and Slingo, 1996; Rotstayn, 1997; Kiehl et al, 2000]. Errors associated with the vertical structure of res are also discussed by Pontikis [1995] and Fomin and Mazin [1998]. Below we derive a more accurate approximation for r due to Pontikis [1993]. Consider the optical depth of a vertically inhomogeneous cloud layer given by Eq. (6.3) with reff defined by Eq. (6.2): fZtop r = 2TT / dz (r2)(z), (6.5) J z b o t where again we have used Qext(f, A) = 2. We assume that n(r,z) is unknown. To determine r we must prognose or diagnose (r2) from the dynamical and thermodynamical variables available in our large-scale model. Assuming that (r 2) can be written as a function of two other moments, dimensional analysis gives (r 2) - (ry (r^y^y-K (6.6) Clearly qi is available to us if we are going to have any success at modeling clouds, so we set i = 3. The question then becomes what other moment of n(r) is available to us in addition to (r3)? One possibility is the cloud droplet concentration N = (r°) which is related to the density of subcloud condensation nuclei (CCN). The C C N density is a scalar quantity obeying an advection-diffusion equation and can, in principle, be carried by a large-scale model. Using j = 2/3 gives (r2) ~ (r 3 ) 2 / 3 ^ 1 / 3 , (6.7) which is the approximation first used by Twomey [1977]. There is a much more compelling reason, however, why Eq. (6.7) is the best approx-imation for (r 2) even if N is largely unknown. Unlike higher order moments of n(r), N in non-precipitating clouds is relatively insensitive to condensation-evaporation and, therefore, to first-order subgrid fluctuations. Furthermore, any unresolved variability in N that does exist is heavily damped by the 1/3 exponent. On the other hand, we will see shortly that a careful evaluation of the effect of subgrid fluctuations on ( r 3 ) 2 / 3 is essential to accurately represent G C M grid-cell optical properties. Substituting Eq. (6.7) into (6.5) and using, by definition, qi = 4 /37Tp w ( r 3 ) , gives where klJz is the constant of proportionality in Eq. (6.7), written for correspondence with Martin et al. [1994]'s reff parameterization. Equation (6.8) appears to have been first written down by Pontikis [1993] although it is anticipated by the approximation due to Twomey [1977]. 77 6.4 Unresolved Low Cloud Optical Variability Currently, state-of-the-art GCMs lack the methodology to include unresolved variability in the calculation of cloud reflectivity (R) or emissivity (e). Through the use of a statis-tical cloud scheme {e.g. SDM, Eq (6.1)}, many modern GCMs couple changes in cloud properties ( A A c , A r ) in a changing climate to the distribution of the subgrid variability s. But they also use the plane-parallel homogeneous (PPH) assumption jRpph = R(r) which assumes that r is uniformly distributed throughout each grid cell. Thus the P P H approximation decouples the optical properties R and e from the underlying physical cloud variability. The convexity of the functions relating R and e to r insures that the optical bias incurred from the P P H assumption is positive, i.e. R and e are overestimated. To reduce this bias many current GCMs use an effective optical depth reff = C T where C ~ 0.7 to calculate Rpph [Cahalan et al, 1994]. Although C may be tuned in a particu-lar G C M to reproduce the measured radiative stream, this approach is ad-hoc in nature and becomes increasingly inaccurate as the climate departs from its present state. Below we present a unified treatment of the physical and optical variability of boundary-layer clouds based on the SDM scheme (See also Jeffery and Austin [2001b]). Our model of boundary-layer cloud optical variability is based on two assumptions: (i) horizontal subgrid variability in the boundary-layer of large scale models exceeds vertical variability; and (ii) cloud liquid water increases linearly with height above cloud base, i.e. qs(z) = q0 — Twz. Assumption (i) is accurate for large-scale temperature and moisture fluctuations because the horizontal length of a grid cell in a climate model is much greater than the boundary layer height. However, it does not hold near cloud top where the vertical dependence of s at the cloud boundary is complex. We overcome this deficiency by introducing a distribution of unresolved cloud-top height fluctuations, Pz't , that is distinct from a ^-independent Ps, although s and ^ o p may be correlated. Our second assumption (ii) was also used in Sec. 5.2 and is well established both numerically and experimentally in the literature. Given assumptions (i) and (ii) we are now in a position to calculate the optical statis-tics of low clouds. For clarity and brevity, we first introduce the notation.. The variable dependence (x) labels unresolved horizontal variability, whereas (z) indicates a vertical dependence which by assumption (i) is non-stochastic, i.e. resolved. Furthermore, (x) represents unresolved variability in a single cell whereas (z) is a continuous dependence that may extend through a column of cells. Consider the integral of the qf(z,x) from cloud base zhot(x) to cloud top ztop(x) = ztop + z[ (x): (6.9) where s*(x) = s(x) (6.10) 78 and we have used z b o t = T~1(q0-qv + s) from (ii). The key feature of Eqs. (6.9) and (6.10) is that fluctuations in Zbot are defined by inverting qi(zb0t, s) = 0, whereas the unresolved cloud-top height fluctuations z't0 are absorbed into the new subgrid variability s*. Thus our distribution Pz> (z'top) can, in principle, be combined with Ps to give PSt(s*). This is advantageous because unresolved variability of qv, qs and ztop is contained in the single parameter s* and Ps* is ^ -independent. In analogy with the S D M scheme we assume that s* is centered and P s„ is known. The r.h.s. of Eq. (6.9) should not be confused with qf+l(zt0p, x)aJi1T~1 / (p+1) since the statistics of s*, generally, differ from the statistics of s. Formulation of the shortwave and longwave optical depths follows from Eq. (6.9) given the appropriate functional relation r ~ J dz func(<&,...). At this point, it is convenient to introduce the cloud thickness h(x) = ziop(x) — Z\>^{x) so that Eq. (6.9) is simply / dzqf(z,x) = ^ f h ^ ( x ) . (6.11) The longwave optical depth, which depends primarily on the liquid water path (LWP), is strictly given by the uh2 model" T(X) ~ h2(x) 2T„ {qv + r w z t o p - q0 - s*(x)} . (6.12) In the previous section we derived Eq. (6.8) for shortwave optical depth which corresponds to Eq. (6.11) with p = 2/3 and hence a " / i 5 / 3 model": 3 a 2 / 3 r{x) ~ ^ { q v + Twztop - q 0 - s*(x)}5/3. (6.13) O I VJ The moments of r can be calculated from P S t in analogy with Eq. (6.1) while Ac is given by /qv-qs(ztop) ds*Ps,(s*). (6.14) -oo Intuitively, we might expect the maximum cloud overlap assumption, Ac = max^^) = Ad(ztop), to hold for a model of cloud variability that ignores vertical variations in s. However, comparing Eqs. (6.1) and (6.14) we find that it does not hold since Ac is calculated from PStt and not Ps. The failure of the maximum cloud overlap assumption is due to the independence (orthogonality) of z'top and s in our approach. Note, the usual cell averaged quantities, e.g. qi(z), are independent of z[op and should be calculated with Knowledge of both Ps and Pz'top, and hence P s*, is extremely limited. Observational [Klein and Hartmann, 1993; Oreopoulos and Davies, 1993; Norris and Leovy, 1994; Klein et al, 1995; Bony et al., 1997] of the marine boundary-layer suggest that s is largely a 79 function of boundary-layer temperature, T, while ztop is largely controlled by the dif-ference in potential temperature (A0) between the top of the boundary layer and the surface. As mentioned previously as(z) is usually assumed to be proportional to qs(z), i.e. a? = constant. Recent observational studies [Norris, 1998a, b; Bajuk and Leovy, 1998; Chen et ai, 2000] have indicated the importance of cloud type in the analysis of cloud properties; in subsequent work we hope to parameterize the ratio o-s*/°~s a s a func-tion of cloud type diagnosed from various stability and potential energy considerations. On the other hand, in consideration of the poor boundary layer vertical resolution of typical GCMs and in the absence of knowledge of o~s*/o~s, we follow current G C M pa-rameterizations and assume <7S* ~ as ~ qs in Sec. 6.7. Note that aZtop is coupled to qs through Tw in Eq. (6.10). Since s* is ^-independent by definition in our scheme we will use qs(z) at the surface (i.e. qo(T)) to evaluate the temperature dependence of as* in Sec. 6.7. The great utility of Eqs. (6.12) and (6.13) is not in the calculation of r directly but, rather, in providing a methodology to include subgrid variability in the reflectivity and emissivity. We do this by calculating P t ( T ) from P S T using the equations above and a change of variable; then by definition X = J dr X(T)Pt(T) (6.15) where X = R or e and PT is the distribution of r in the cloudy part of the column. This approach is discussed in Considine et al. [1997] albeit not in the context of a generalized framework of unresolved variability. Unfortunately, the analytic expressions for R and e are sufficiently unwieldy to prevent an analytic evaluation of Eq. (6.15). Another approach, pioneered by Barker [1996b], is to assume an analytically friendly form for P T that is both sufficiently general to approximate P T over a wide range of conditions and that allows a closed-form expression for X. Barker et al. [1996] analyzed satellite data of marine low clouds and found that a generalized 7-distribution, P 7 ( r ) , closely approximates the observed distribution and allows Eq. (6.15) to be integrated analytically. In an article in preparation [Jeffery and Austin, 2001a] I show that using P 7 (T ) in place of P t ( T ) calculated from Eq. (6.13) and a Gaussian P S * are essentially equivalent (see also Considine et al. [1997]). The last step in our treatment of unresolved optical variability is to relate P 7 to Eqs. (6.12) or (6.13) and P , , . The shape of P 7 is controlled by the parameter v = T 2 / C T 2 which is a measure of the width ofthe distribution relative to its mean. Barker [1996b], who introduced P7(u, r ) , did not relate v, and in particular a2, to the unresolved physical variability s. Using the framework we have presented thus far, we calculate r and r 2 (and thus v) using Eqs. (6.12) or (6.13), and P S , d la Eq. (6.1). Analytic expressions for R(y,r) and e(u,r) used in our analysis in Sec. 6.5 are given in Barker [1996b] and Barker and Wielicki [1997], respectively, and reproduced in App. C. Our scheme provides a one-to-one relationship between v and Ac for a given P S „ that 80 can be tested against the Landsat satellite data compiled by Barker et al. [1996]. To make such a comparison we need to specify the distribution PSt. The triangle distribu-tion first used by Smith [1990] is a computationally efficient surrogate for the Gaussian distribution, but it does not accurately mimic a Gaussian at small Ac. We therefore introduce a modified triangle distribution (See App. B) that is similar to Smith's scheme (or a Gaussian) at large Ac but better reproduces Gaussian behaviour at small Ac. A comparison of Ac vs u is shown in Figure 6.1 for the h2 and / i 5 / 3 models calculated using Eqs. (6.12-6.14), and our new triangle distribution. Also shown is a subset, v G (0,6.5), of the Landsat data tabulated in Table 2 of Barker et al. [1996]. Agreement between the data and both theoretical models is good despite uncertainties in the retrieval of Ac from the satellite scenes [Barker et al, 1996]. It should be noted that v G (6.5,20) has a significant impact on the calculation of R and e and is in close agreement with the model, while the selected data shown in Fig. 6.1 emphasizes the region v G (0.5,6.5). However it is encouraging that the predictions of our model exhibit the same asymptotic behavior near v = 0.5 shown by the data. A comparison of Ac vs v for various Ps, will be presented in a future study [Jeffery and Austin, 2001a]. Figure 6.1: Plot of Ac vs v for the h2 (—) and / i 5 / 3 (- -) models calculated using Eqs. (6.12-6.14), and PSt from Sec. B.2. Landsat data (•) in the range v < 6.5 from Table 2 of Barker et al. [1996] is also shown for comparison. 81 6.5 Low Cloud Optical Properties As a demonstration of the behaviour of our statistical cloud scheme, we now consider the complicated coupling of unresolved variability and cloud optical properties in the model of Sec. 6.4 using a zonally-averaged climatology. While we acknowledge that our model neglects the meridional structure of cloud amount caused by atmospheric dynamics, e.g. the storm tracks, and meridional variations in surface properties or as [Rotstayn, 1997], we incorporate what we consider to be the major latitudinal dependencies: solar zenith angle, saturation vapour density and surface albedo. First consider the base-state climatology. Our zonally averaged model extends over latitudes <f> = —60 °S to 60 °N where our "grid cells" encompass one latitudinal band. We assume that our new triangle distribution (See Sec. B.2) defines the shape of the distribution PSt which represents meridional fluctuations in "subgrid" variability, and that this form is independent of <j> and T. Our base-state climatology is constructed from the Ac(4>) measurements of Warren et al. [1988] (See Ramaswamy and Chen [1993] and Kogan et al. [1997] for a similar approach). Mean optical depth T(4>) G (3.5,6.2) is estimated from the satellite measurements presented in Hatzianastassiou and Vardavas [1999]. Further details may be found in App. C. Note that qv and ztop are not determined independently, but rather, by specifying Ac, r, T(<f>) and qs(T), we solve for the two unknowns aSt and qv + r w ^ t o p . We use the hbl3 model for r (h2 model for longwave r) . The mean reflectivity and emissivity of our base-state climatology, averaged over the diurnal cycle at Equinox, is shown in Fig. 6.2 along with the corresponding val-ues predicted by the plane-parallel homogeneous approximation used by GCMs. The well-documented plane-parallel albedo (reflectivity) bias [Cahalan et al, 1994] of RpPh, roughly 0.06 in our model, is visible in the lower half of the figure, but it is overshadowed by the much larger bias of which averages near 0.3. Early studies of r feedback [Tempkin et ai, 1975; Somerville and Remer, 1984] assumed that longwave optical prop-erties of clouds are saturated (e = 1 ~ epTph) and as a result, changes in r only affect the cloud's shortwave properties. Although, as shown in Fig. 6.2] 1 is not saturated at global scales, assumptions concerning the behaviour of I are not significant for low cloud radiative forcing calculations since the longwave forcing is very small. Figure 6.2 reiterates that a coupled treatment of physical and optical variability can substantially impact the predicted absolute magnitude of the global radiative stream in a G C M cloud parameterization. 6.6 r e f F - r relationships in G C M s In this section we consider the effect of low model resolution on the r eff-r relationship predicted by a G C M [Lohmann et al., 2000]. The h5?3 model for shortwave r derived in Sec. 6.4 involves the analytic integration of q^3 from cloud-base to cloud-top assuming a 82 —^I 0 0 0 °oooooooooo o 0 ooooooooo 00 d " IW las 3 o 1—1 d " o d " • B • • i 1 1 1 1 1 r - 6 0 ° - 4 0 ° - 2 0 ° 0° 20° 40° 60° Latitude Figure 6.2: Comparison of R (—) and I (- -) with plane-parallel homogeneous values -Rpph = R { T ) (•) and eTpph" = e(r) (o). The convexity of R and e insures that the P P H approximation overestimates R and e. Shortwave f [Eq. (6.13)], longwave r [Eq. (6.12)], and AC [Eq. (6.14)] are all calculated using PSt from Sec. B.2. Expressions for R and e are from Barker [1996b] and Barker and Wielicki [1997], respectively. Reflectivities are diurnally averaged at Equinox. See App. C for further information. linear cloud liquid-water density profile. In a G C M radiation scheme this integral is done numerically. Thus at coarse vertical resolution the statistical properties of the analytic model, Eq. (6.13), and the discrete integration implemented in a G C M are generally different. As we shall see shortly this difference is especially prevalent in the predicted relationship between reff(z top) and r. A discussion of other errors associated with G C M optical depth parameterizations can be found in Jeffery [2001d]. In the absence of unresolved variability, integration of the Pontikis equation, (6.8), 1 /3 with qi = Tw(z — z\)0t), N(z) = N and reff ~ qx' gives relationships between reff(ztop), N, Fw, and LWP or r: r e f f (z t o p ) ~ ( r ^ A ^ L W P ) 1 / 6 [Pontikis, 1996] and r e f f(z top) ~ ( T t u A r _ 2 r ) 1 / 5 [Szczodrak et al, 2001]. Thus a scatter plot of high-resolution measurements of reff (^top) v s T o r LWP can be used to determine the relative scatter of the ratio Tw/N2 in the region of interest. For areas with large spatial variations in the anthropogenic production of C C N (N) and small temperature and moisture fluctuations (i.e. Tw(x) = Tw), the relative scatter provides some measure of the indirect cloud albedo effect in the 83 region [Szczodrak et ai, 2001]. Using high resolution (1 km) A V H R R data from a region off the coast of California (24°N to 35°N, 115°W to 136°W), Szczodrak et al. [2001] found r e f f(z to P) ~ rllb for 40% of their retrieved scenes, indicating little spatial variability in the ratio Tw/N2. This same region was analyzed by Lohmann et al. [2000] who used the E C H A M G C M to further investigate the reg-r relationship over the course of a year using data extracted twice daily. The Lohmann et al. [2000] study, however, is distinctly different from the study by Szczodrak et al. [2001] because their G C M was run in low horizontal (400 km) and vertical (19 levels) resolution. Surprisingly, Lohmann et al. [2000] found r e f f (2 top) ~ T 1 / 6 for a subset of their clouds (r < 10 or alternatively non-precipitating clouds) which is consistent with the analytic result r e f f ( z t o p ) ~ r 1 / 5 and little variability in Tw/N2. A significant source of error in the Lohmann et al. [2000] study is the coarse model vertical resolution. The E C H A M 4 uses a hybrid sigma-pressure coordinate system with 5 vertical levels in the boundary layer [Roeckner et al, 1996]. Over the sub-tropical ocean the level boundaries are approximately at Z i _ i / 2 = (0, 70, 235, 560,1080,1810) meters. With such poor resolution at the top of the boundary layer it is likely that the 5th model level z5 makes a dominant contribution to the discrete optical depth. In this case T = ~ of,^{zb)-l a n d we find that r M ~ ( r /A0 1 / 2 (6.16) where F eff ~ ql1^3 by definition [See Lohmann et al. [1999, Eq. 22]]. The relationship res(z5) ~ r 1 / 2 predicted by Eq. (6.16) is consistent with Twomey [1977]'s original formu-lation of the indirect albedo effect and is quite different from the r 1 / 5 result expected at high resolution. To verify that the ECHAM4's vertical resolution is too coarse to reproduce the relation ^eff(ztop) ~ T^1/5 seen by Szczodrak et al. [2001], reft(zc,) was calculated at fixed Tw and N € (10,100) c m - 3 using the linear cloud model introduced in Sec. 6.4 and ECHAM4' s first five model levels for Ac € (0.35,0.95). As in Sec. 6.5 the two unknown parameters Qv — Qs(ztop) and aSf are determined from the equations for r, Eq. (6.13), and Ac, (6.14). Note that Tw is only a relevant parameter in this analysis if Eq. (6.16), which is r„,-independent, does not hold. The results, shown in Fig. 6.3, demonstrate that r^z*,) ~ r 1 / 2 at fixed A^ as predicted by Eq. (6.16). The straight lines in Fig. 6.3 are, of course, distinctly different from the contour plots in Lohmann et al. [2000] which represent an average over a distribution of N; the key idea here is that Lohmann et al. [2000] should not expect to see reft(z5) ~ r 1 / 5 at fixed N. See the figure caption for parameter values. Equation (6.16) is only asymptotically exact as the number of levels occupied by cloud approaches one. Only minor deviations from a straight line are evident in Fig. 6.3 at large r and small N. A slight Ac dependence is seen in this regime. Following Lohmann et al. [2000] a line at constant LWP (150 g m - 2 ) is also shown in the figure. This particular r eff-r comparison is independent of model resolution and obeys r ~ f~^. As discussed in Lohmann et al. [2000], the negative reff-r correlation seen for r > 10 is probably the result 84 N = 100 cm - 3 5 10 15 20 25 x Figure 6.3: Log-log plot of reff(z5) vs r calculated using a linear cloud profile (Sec. 6.4) with N e (10,100) cm- 3 , Ac = 0.35 (—), Ac = 0.95 (- -) and ECHAM4's 5 lowest model levels Zi = (35,152.5,397.5, 820,1445) meters. A reference line res ~ r 1 / 2 has been added for comparison. Optical depth and Ac are calculated from Eqs. (6.13) and (6.14) using P s , from Sec. B.2 with parameter values Tw = 0.0016 g m - 4 estimated from Szczodrak et al. [2001], CLL = 0.75, pw = 1000 kg m - 3 and kd = 1. Effective radius is defined by Eq. 22 in Lohmann et al. [1999]. of an upper bound on LWP caused by ECHAM4's parameterization of the autoconversion of cloud droplets to precipitating rain drops which increases with increasing qi [Lohmann and Roeckner, 1996]. We conclude from Eq. (6.16) and Fig. 6.3 that the close agreement between the relation r eff(z t o p) ~ r 1 / 5 seen in A V H R R satellite retrievals [Szczodrak et al., 2001] and the relation reff(.2top) ~ ^ 1 / / 6 seen by Lohmann et al. [2000] in small r, non-precipitating clouds is fortuitous; the vertical resolution of E C H A M 4 is too course to reproduce the correct dependencies. In fact, Fig. 6.3 reveals that the large values of f eff (> 10 fim) prognosed by E C H A M 4 at small f (< 15) are consistent with N & 10 c m - 3 , whereas available low cloud data in the region indicates that N averaged over 1600 km 2 is unlikely to decrease below 40 c m - 3 [Albrecht, 1989; Hudson and Svensson, 1995; Twohy et ai, 1995; Szczodrak et al., 2001]. Possible causes for the large variability in reff seen by Lohmann et al. [2000] include the model prognosis of discrete jumps in cloud top height 85 coupled with a sharp decrease in N above the boundary layer and the inclusion of mid-level clouds in the statistical analysis [Lohmann, 2001]. The above discussion of the Lohmann et al. [2000] G C M study leads us to ask "Is the r e f f - r relationship really fundamental to the understanding of shortwave cloud-radiation interactions?". After all, our "reinterpretation" of the Lohmann et al. [2000] study reveals that this relationship is highly sensitive to both model resolution and N. We suggest that the two fundamental quantities that characterize shortwave r follow immediately from inspection of the Pontikis equation, (6.8). These two quantities are closely related to L W P and column droplet concentration [Han et al, 1998; Jeffery, 2001a]. In analogy with fractional Brownian motion we define the quantities i) fractional liquid water path (FLWP): r z t o P F L W P = / dz qf/3, J Zbot and ii) fractional column droplet concentration (A//c) fZtop jVfc = dz TV 1 / 3 . J ^bot Together the triad r — F L W P — A / / c can be used to characterize quantities of interest. For example effective liquid water content eff CiT 3/2 effective droplet concentration A W = fc F L W P 3 and cloud thickness , A/} C FLWP h = —1 CiT where ci = ( l / 3 p t l J ) 2 / 3 / ( l / 2 A ; ( i 7 r ) 1 / 3 from Eq. (6.8). In addition, the impact of precipita-tion-induced Qi limitations on r is best diagnosed from a G C M simulation by a contour plot of F L W P and r . Note that a F L W P - r comparison does not suffer from the same low vertical resolution difficulties that Lohmann et al. [2000]'s F eff-^ comparison suffers because both F L W P and r are vertically integrated quantities. 6.7 Low Cloud Radiative Feedback In this section we derive simple analytic expressions that characterize the response of a prototypical G C M statistical cloud scheme to surface warming. We also use this approach to quantify the effect of poor model resolution on low cloud radiative feedback predicted 86 by such a model. Here we define low cloud feedback (LCF) as the change in the net (positive downward) shortwave radiative flux at the top of the boundary-layer. Longwave radiative feedback is negligible for low clouds because of the relatively small cloud/surface temperature difference in the boundary-layer. A more detailed treatment of L C F using the zonally-averaged climatology of Sec. 6.5 is given in Jeffery and Austin [2001b]. The relative tractability of Eqs. (6.13) and (6.14) presents us with the opportunity to transparently quantify cloud-temperature interactions for this subgrid-scale cloud pa-rameterization. Combining Eqs. (6.13) and (6.14), crs* « A^. , , Tw « A 2 g s and Smith's triangle distribution for subgrid variability (See Sec. B . l ) , we derive two analytic response functions for Ac < 0.5: where A = (Ai ,A2), Lv is the latent heat of vapourization, Ru is the gas constant for water vapour, and recall that qs must be evaluated at some fixed height (e.g. at the surface) since o\,* and Tw are z-independent by definition. Our use of the terminology "response function" is an analogy to response functions in the theory of thermodynamics, e.g. specific heat and adiabatic compressibility of an ideal gas. For Ac > 0.5 Eq. (6.18) remains valid but for Eq. (6.17) (d\nAc/dT)Ytx decays monotonically to zero as Ac —> 1. Overall, the simple \ow-Ac results given by Eqs. (6.17) and (6.18)—strictly valid for Ac < 0.5—are approximately valid for the range of cloud fractions, 0 < Ac < 0.65, typical of many large-scale boundary-layer low cloud fields. We can interpret Eqs. (6.17) and (6.18) as representing two potential low cloud shortwave feedback scenarios in a warming climate demarcated by df/dT = 0 and 8Ac/dT = 0, respectively. Let F < 0 be the net (positive downward) shortwave radiative flux reflected by the (unforced) low clouds and A T > 0 be the thermal forcing. Consider the small r approximation F ~ Acr. Then Eq. (6.17) implies L C F = - ( 4 / 5 ) F A T / T * , a positive cloud feedback, while for Eq. (6.18), L C F = (2/3)FAT/T*, a negative cloud feedback, where T* = RVT2/LV. Although we make no claims regarding the likelihood of the two scenarios described by Eqs. (6.17) and (6.18), the difference in sign between the equations leads to a non-trivial asymmetry between the (ylc,r)-response and L C F . Using Eq. (6.17) we find that (df/dT)x < 0 is a sufficient condition for a positive L C F while Eq. (6.18) implies that (dAc/dT)\ > 0 is a sufficient condition for a negative L C F . These relations follow from the naturally positive coupling between Ac and r , i.e. (dr / 8AC)T,X > 0. On the other hand, (df/dT)x > 0 and (dAc/dT)x < 0 do not uniquely specify the sign of the L C F . Thus our statistical approach formally links the observational evidence of a largely neg-ative r sensitivity [Tselioudis et al., 1993; Greenuiald et al., 1995; Bony et al., 1997] (6.17) (6.18) 87 with G C M simulations [Hansen et al., 1984; Wetherald and Manabe, 1986; Colman and McAvaney, 1997; Yao and Del Genio, 1999] that predict a positive L C F . We extend Eq. (6.18) to another useful form through the approximation (d\nr/dT)AC,\ « 2(d\nR/dT)Ac,x valid for r = 0(5) giving a i n ^ 1 K (6.19) dT ) A c X 3RVT* Since L C F is relatively insensitive to changes in e we can combine Eqs. (6.17) and (6.19): | ( L C F ) T i A | | ( L C F ) A c , A | 2.4, which illustrates that, in general, Ac feedback dominates the r feedback in this simplified model. We can also use our response functions to quantify the effect of low model vertical resolution on L C F . As we saw in Sec. 6.6 typical GCMs have only 4-6 model levels in the boundary layer (BL) and the vertical resolution of these levels usually decreases with height. As a result, the top model level in the B L dominates the discrete integration of q\ [Eq. (6.9)] and hence r. In this low resolution limit the h5?3 model for shortwave optical depth, Eq. (6.13), becomes [See Eq. (6.16)] T(X) ~ {qv + Ywztop - q 0 - s*(x)}2/3Az, where Az is the thickness of the model level centered at ztop. Computing the low vertical resolution response functions we find that the r response [Eq. (6.18)] remains unchanged while the Ac response becomes d In Ac \ „ Lv X 2 - ^ - , (6.20) dT J T M A z RvT2 independent of Tw. A comparison of Eqs. (6.17) and (6.20) reveals that the current implementation of statistical cloud schemes in low vertical resolution GCMs tends to overestimate the unresolved low cloud Ac response by a factor of 2.5, compared to the same statistical cloud scheme run at higher vertical resolution. 6.8 Experimental verification Experimental measurements supporting the unified treatment of unresolved physical and optical variability presented in this chapter are shown in Fig. 6.1. Although these initial results are encouraging a more comprehensive study is needed. In particular a detailed comparison of contemporaneous satellite retrievals of reflectivity and the reflectivity pre-dicted by a weather forecast model would be very enlightening and could potentially 88 lead to further improvements and refinements of the present model. The new Moderate-resolution Imaging Spectroradiometer (MODIS) aboard NASA's Terra and Aqua satel-lites will soon provide global retrievals of A C , r, reff and R which could be used to assess the validity of the (AC,T,R) relationships derived in this chapter. High-resolution in-situ nocturnal measurements of A C , longwave r and e in marine stratocumulus clouds from the Dynamics and Chemistry of Marine Stratocumulus Experiment (DYCOMS-II) scheduled for July, 2001 provide an additional opportunity to assess the accuracy of the linear cloud model derived in Sec. 6.4. 6.9 Summary Understanding the complex interaction of clouds and radiation is a formidable challenge. In this chapter the relationship between the physical cloud properties A C and r and the optical properties R and e is investigated within the context of a statistical cloud scheme. We restrict our attention to low clouds where the vertical profile of cloud liquid water is linear and where horizontal variability dominates. Assuming a known distribution of un-resolved variability that includes cloud-top height fluctuations, we derive a self-consistent and computationally efficient set of prognostic equations for A C and the moments of r, thereby incorporating subgrid optical fluctuations into the statistical cloud schemes first introduced in the 1970s [Sommeria and Deardorff, 1977; Mellor, 1977]. Our unified treat-ment of physical and optical variability is particularly appropriate for use in a G C M that incorporates a subgrid-scale turbulence scheme [Ricard and Royer, 1993]. In the future, we plan to extend the applicability of our coupled scheme by incorporating mixed-phase (water-ice) clouds and a more explicit treatment of cloud-top height fluctuations. Recent studies of the sensitivity of prognosed cloud properties to changes in vertical resolution [Bushell and Martin, 1999; Raisanen, 1999; Lane et ai, 2000] have highlighted difficulties associated with the accurate determination of low cloud physical properties in large-scale models with coarse vertical resolution. Using the recent GCM-based work of Lohmann et al. [2000] as a case study, we see that there are further difficulties involved in comparing cloud top quantities (e.g. reg) with vertically integrated quantities (e.g. r) in low resolution models. With respect to the cloud-radiation problem, this difficulty is avoidable as reg at cloud top is not essential to the determination of r. Rather, the fractional liquid water path and the fractional column droplet concentration are the two quantities which are most useful in diagnosing the interaction between N, qi and r in model prognosed clouds. In the future, increased vertical resolution and the determination of cloud top height by interpolation should further improve the prognosis of low cloud physical and optical properties in large-scale models. The model of unresolved cloud variability presented in this chapter is also used to probe the sensitivity of parameterized cloud fraction and optical depth to changes in temperature. The coupled ( A A c , A r ) global response of clouds to increasing temperature 89 is analogous to the response of an open thermodynamic system. Although the particu-lar thermodynamic trajectory that the system follows may be very sensitive to external forcing and boundary conditions, much can be learned by computing response functions where one of the thermodynamic coordinates is fixed along the trajectory. We derive an-alytic response functions in (Ac,f,T) space that demonstrate the overall dominance of the cloud fraction feedback in the model. In particular, our response functions indicate that global observational evidence of a largely negative optical depth sensitivity [Tselioudis et al, 1993; Greenwald et al., 1995; Bony et al, 1997] is indicative of a much stronger negative cloud fraction response and therefore a net positive low cloud feedback. Also we find that low model vertical resolution can cause a significant overestimation of the unresolved low cloud Ac response by a factor of around 2.5. It is hoped that these simple analytic results will be of use in future climate studies where they may provide guidance in the analysis of complicated cloud-climate interactions. 90 Chapter 7 Summary The root of the matter is that the greatest stimulus of scientific discovery are its practical applications. [Richardson, 1908] Turbulence, someone remarked, is too important to ignore, and too hard to solve [Pumir et al., 1999]. As early as 1922 Richardson was keenly aware of the important role that the statistical study of turbulence would play in atmospheric research. Over the last eight decades, significant progress has been made in our understanding of planetary boundary-layer (PBL) turbulence; in particular the fundamental stochastic quantities that describe the stability of the boundary layer and that characterize the fluxes of heat and mass are, for the most part, well-known. In contrast, much less attention has been given to the study of the statistical properties of clouds. Writes Wyngaard [1990] Thanks in part to the progress of the past few decades, we now have a good basis for dealing theoretically, numerically, and experimentally with conser-vative scalar fluxes in the idealized P B L Clouds . . . pose theoretical difficulties, particularly regarding their influence on the boundary layer, its coupling to the free atmosphere, and their transport properties. Theoretical difficulties also exist in the understanding of clouds at scales much smaller than the boundary-layer height. Recently, the cloud microphysics community has shown renewed interest in the fine-scale properties of turbulence due to concerns that fine-scale structure and/or fine-scale intermittency may influence droplet growth or coalescence in ways that are currently being ignored. A special session at the Conference on Cloud Physics, 17-21 August 1998 (Everett, WA) and a 3-day workshop organized by the Geo-physical Turbulence Program at N C A R , 9-11 November 2000 (Boulder, CO) on fine-scale turbulence and cloud microphysics have not resolved these concerns (See also Pinsky et al. [2000] and Vaillancourt and Yau [2000]). In 1968 a closed equation for the two-point equal-time cloud droplet number density covariance was derived by Kraichnan based upon the assumption that the temporal properties of the velocity rapidly decorrelate in time. The solution exhibits k~l scaling 91 for scales larger than the Batchelor length where molecular diffusion becomes important. This k~l regime has subsequently been verified experimentally [Grant et al, 1968] and numerically [Bogucki et ai, 1997; Chasnov, 1998] and is now known as the viscous-convective subrange. Kraichnan's ^-correlated model is thus the correct closure to use in the investigation of the spatial statistics of cloud droplets at viscous-convective scales. In this dissertation, I use Kraichnan's 5-correlated closure to investigate the clumping or preferential concentration of cloud droplets in the viscous-convective subrange. A n an-alytic expression for the second-order spectral density of inertial particles is derived that exhibits a rich behaviour. In particular, a well-defined bump appears in the spectrum as the inertia of the particles or droplets increases. The bump represents the accumula-tion of inertial particles in regions of high-strain and low-vorticity in the flow and is a manifestation of intermittency in the spatial statistics. The relative height of the bump increases exponentially with increasing Stokes number. Using a Gaussian assumption to link the compressibility of an inertial particle's ve-locity with the velocity statistics of the surrounding fluid, my analytic results indicate that very weak clumping occurs near Stokes numbers of 0.15 while significant clumping begins near St=0.3. In contrast the Stokes number of cloud droplets is typically less than 0.15. Thus these Gaussian results support the contention of Grabowski and Vaillancourt [1999] that the Stokes number of cloud droplets is too small for significant clumping to occur. However, it is well established that turbulent fine-scale statistics are highly non-Gaussian. In particular, the kurtosis of symmetric strain reaches 20-25 in the atmospheric boundary layer [Sreenivasan and Antonia, 1997]. It was first suggested by Tennekes and Woods [1973] that small-scale turbulent intermittency may effect cloud drop coalescence. More recently, Shaw et al. [1998] have argued that the interaction of cloud droplets with vortex tubes (i) increases clumping at fixed St and (ii) segregates droplets into widely varying microphysical environments due to long vortex-trapping times. Shaw et al. [1998] suggest that the combination of these two effects leads to droplet spectral broadening. In the (^-correlated limit, the equation for the spatial covariance has an explicit de-pendence on the mean-square inertial particle velocity divergence, (b2). Using the rela-tionship between b and the velocity of the surrounding fluid derived by Maxey [1987], my analysis reveals that clumping ((b2)) and hence Stokes number depends on a highly non-Gaussian fourth-order velocity derivative statistic which, in turn, is written as the sum of three scalar invariants introduced by Siggia [1981] for an isotropic, incompress-ible velocity field. Recent experimental data suggests that these three scalar invariants share the same Reynolds number (ReA) dependence [Zhou and Antonia, 2000]. Thus I introduce an "effective Stokes number", St eg defined by St e f f = S t (^ /3 ) 1 / 2 where T is the flatness (Kurtosis) of the symmetric strain (longitudinal velocity deriva-tive). Since T is a function of ReA, Steff may be used to explicitly incorporate Reynolds 92 number effects into the calculation of particle clumping. Using Steff I estimate that velocity field intermittency can account for the clumping of the largest drops with diameters greater than 25 /xm in atmospheric clouds. Thus my results provide some support for Shaw et al. [1998]'s claim that velocity intermittency leads to the preferential concentration of small St cloud droplets. On the other hand, and in contrast with the Shaw model, vortex tubes are not explicitly involved in this inter-mittency effect. Moreover, a comparison of the ratio of mean square enstrophy to mean square dissipation, which plays a central role in the Re^-dependence of Steff, between a synthetic vortex dominated velocity field [He et ai, 1998] and a real turbulent velocity field [Zhou and Antonia, 2000] suggests that, in fact, vortex tubes are not statistically relevant players in the clumping problem. This result is crucial because the Shaw model demands that vortex tubes dominate the statistics of particle-turbulence interactions. Therefore, my analysis casts serious doubt on the fundamental mechanism that Shaw et al. [1998] invoke in their model of droplet spectral broadening. In the future I hope to extend this work by comparing, in detail, the statistics of a field of Burgers vortices with corresponding Gaussian statistics as well as experimental data. In particular I hope to examine differences in (b(x)b(y)) between the vortex field and the Gaussian expression. In this dissertation I also use the (^-correlated model to investigate the effect of con-densation/evaporation (CE) on the viscous-convective subrange. Here, the cloud droplets are assumed to be non-inertial but their mass is variable. Condensation and evapora-tion is modeled by a simple mean-field expression that associates adiabatic lifting with condensation and sinking with evaporation. The resulting cloud statistical structure is non-homogeneous—the variance of cloud liquid water fluctuations increases linearly with height in the cloud. Vertical non-homogeneity introduces some rather interesting compli-cations; the spectral density is complex and the magnitude of the imaginary component, determined by a Kolmogorov like constant, cannot easily be determined. Despite these difficulties, I demonstrate that C E can explain the anomalous viscous-convective scaling recently found in boundary-layer clouds by Davis et al. [1999]. In the future I hope to extend this work to the inertial-convective subrange through the consideration of a velocity field with non-white temporal properties. In addition I plan to compare and contrast the closure used in this dissertation where advection and C E are decoupled with a closure which explicitly couples the two terms. Extension of this work will surely involve three-dimensional LES data; limited information is currently available from aircraft data which provides only a horizontal transect of the vertically inhomogeneous cloud liquid water field. Although the ^-correlated model which takes center stage in this thesis is only valid at very small scales 0(10 cm), the rigorous statistical approach embodied by modern turbulence theory is also valuable at very large scales 0(100 km). One problem that demands a more rigorous statistical methodology is the parameterization of unresolved cloud variability in global climate models (GCMs). In this dissertation I extend statistical 93 cloud schemes, pioneered by Sommeria and Deardorff [1977] and Mellor [1977], that link unresolved cloud physical variability to a distribution, Ps, of unresolved variability, s. Following Jeffery and Austin [2001b], my extension incorporates unresolved optical variability into these schemes by first restricting s and hence Ps to be height independent in low clouds. At the same time, I consider a distribution of cloud top height fluctuations (ztop) t n a t i s orthogonal to a height independent Ps. I then derive analytic expressions for the mean cloud layer optical properties—reflectivity and emissivity—as a function of a new subgrid variable s* which is related to s and z'top according to s* = s — Twziop where Tw is the lapse rate of cloud liquid water density. A key feature of my scheme is the prediction of a 1-to-l relationship between cloud fraction, Ac, and the normalized variance of optical depth, v, for a given PSm. A com-parison of Ac vs v calculated from a Gaussian-like PSit with satellite data is encouraging; the scheme is able to capture the essential features of the data including an asymptotic behaviour near v w 0.5 (See Fig. 6.1). Analysis of a zonally-averaged climatology reveals that a coupled treatment of physical and optical variability can substantially impact the absolute magnitude of the global radiative stream predicted by a G C M . Using my unified treatment of cloud physical and optical variability I also derive analytic expressions for two parameterized response functions that characterize two po-tential low cloud feedback scenarios in a warming climate. The key assumption used in the derivation of these expressions is a linear relationship between the temperature dependence of the standard deviation of s* and the saturation vapour density. This as-sumption is currently used by GCMs. Using my two analytic response functions I find that a negative optical depth response is a sufficient condition for a positive low cloud feedback while a positive cloud fraction response is a sufficient condition for a negative low cloud feedback. This non-trivial asymmetry may have important implications for the interpretation of experimentally or numerically determined cloud sensitivities. In the future I hope to improve the accuracy of my unified treatment of unresolved cloud variability through a more detailed treatment of cloud-top height fluctuations. In particular the ratio of the variances of s and z[op could potentially be parameterized as a function of cloud-type. A n investigation of this ratio would surely involve both cloud resolving model data and satellite data. In summary, a number of small steps have been taken in this dissertation in the application of the theoretical methodology of passive scalar turbulence to the study of clouds. At this stage, the predictions made and models presented in this dissertation are merely plausible. More detailed studies are needed to confirm or modify the various assumptions that appear in this work. However, at both very small and very large scales limited experimental data is available. At the end of each chapter in this dissertation I suggest experimental measurements that need to be made to shed further light on 94 the problems addressed. Unfortunately, the current potential for very accurate small-scale aircraft measurements in clouds and very accurate large-scale satellite retrievals of the relevant microphysical and dynamical variables that determine cloud-top height is extremely limited. On the other hand, considerable work still needs to be done to bring theoretical cloud physics up to the level of theoretical turbulence. It appears that there is a growing gap between the techniques used to tackle passive scalar problems in the turbulence community and the cloud physics community. Richardson would certainly not have approved of such a division. In the future I hope to work towards narrowing this perceived gap. I conclude with one final thought regarding the task at hand [Sreenivasan, 1999]: Since . . . (the) physical principles are still unclear, the task has an iterative character to it; thus each generation of students . . . has . . . made incremental progress. Progress has demanded that this grand problem be split into various sub-problems—some closer to basic physics and some to working practice. 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Phys., 7, 825-828, 1971. 114 Appendix A List of Principal Symbols a Tracer parcel position at t = 0 ar, Cloud latent heating parameter Ac Cloud fraction Ad(z) Cloud density b Inertial particle velocity divergence, V • u B Velocity divergence correlation function, (b(x)b(y)) C E Condensation/evaporation source term E Scalar spectrum FaP Parcel velocity correlation function, (ua(x)up(y)) F, Fc Solenoidal, potential correlation components T Longitudinal velocity-derivative flatness factor or kurtosis, (vl^/iv2^)2 2F1 Hypergeometric function F L W P Fractional liquid water path H Hamiltonian I Advection-diffusion equation source term Ia Fourth-order velocity gradient scalar invariant J Eulerian parcel current, 6u J Jacobian k Wave-vector kd Measure of droplet spectral shape, kd = (r2)3/((r3)2N) K Eddy-diffusivity Kfj, Modified Bessel function of order \x I Length scale Lv Latent heat of vapourization L C F Low cloud radiative feedback LWC Liquid-water content L W P Liquid-water path n(r) Droplet number distribution 115 N Droplet number density M Fractional column droplet concentration Qi Cloud liquid-water density Qv Cloud water-vapour density Qs{z) Saturation vapour density Qo Saturation vapour density at z = 0 Of order • • • Ps Distribution function of the random variable, s PM Legendre polynomial P 7 ( r ) 7-distribution P(x,t\a,0) Conditional probability of finding ( a , 0) labeled parcel at (x,t) V Production spectrum of scalar variance Pe Peclet number Pr Prandtl number, VJK r Radial vector, y — x reff Effective radius R Shortwave reflectivity Rx,t Lagrangian indicator function, 8(a — £~l(x,t)) Ra Eulerian indicator function, 5(x — £a(t)) Ry Water vapour gas constant Re A Taylor microscale Reynolds number s Unresolved variability of qv — qs s* Unresolved variability of qv — qs and z'top Sij Symmetric strain tensor, (vij + St Stokes number StefF Effective Stokes number t Time T Temperature u Eulerian contaminant parcel velocity V Eulerian fluid velocity vi,j Fluid velocity derivative, dvi/dxj w Wiener process w Eulerian vertical velocity, u3 X Eulerian position y Eulerian position z Vertical height, x3 z [Chap. 5] Height above cloud-base zbot Cloud base Ztop Cloud top l7l Magnitude of the average value of the least principal Rate of strain 116 Fw Low cloud liquid water lapse rate, dqi/dz 5 Small dimensionless parameter for renormalization S(- • •) Dirac 5-function 5aP Kronecker (5-function 5F(a, b,.. .)/5a Volterra functional derivative e [Chap. 3] Measure of the compressibility of u e [Chap. 6] Longwave emissivity e Energy dissipation rate 77 Kolmogorov length, (^ 3 / e ) - 1 / 4 T]B Batchelor length, P r - 1 / , 2 7 7 9 Passive scalar density © Passive scalar density fluctuations, 9 — (9) K Molecular diffusivity A B Diffusive decay scale, ~ TJB IX Angle cosine v Kinematic viscosity £ Lagrangian parcel position 7r Conjugate Lagrangian momenta p Particle separation pw Liquid water density a [Chap. 3] Measure of the compressibility of u, e/(l — e) as [Chap. 6] Standard deviation of random variable, s av Kolmogorov velocity, (ue)1^4 r 5-correlated velocity time-constant TD Molecular diffusion time TE Eulerian correlation time TT Eddy turn-over time TV Kolmogorov time, (v/e)1/2 T [Chap. 6] Optical depth cj) Latitude <b(kx) One-dimensional scalar spectrum $ Scalar correlation function, (Q(x)@(y)) X Scalar dissipation rate tpc Condensate density tbv Vapour density Scalar spectral density u)k Vorticity pseudovector, e ^ v y — fj,i)/2 117 Appendix B Triangle Distributions B . l Smith [1990]'s triangle dis t r ibut ion Smith [1990] introduces a symmetric triangle distribution for Ps in Eq. (6.1): 0o V ° o / where al = 6a2. Using Eqs. (6.1) and (B. l ) , cloud density Ad is [Smith, 1990, App. C] where QN = (e/„ — c/ 5)/a 0. The A-th moment of cloud liquid water follows in a similar manner: where F i = c r ( l + Q N ) X + 2 , F 2 = -2cTQXN+2, F 3 = cT(-l + Q N ) X + 2 and cT = ax[(X + l ) " 1 - ( A + 2)- 1]. Note that for Aa < 0.5, qx and Ad are related simply by qf ~ Aj . This feature of the triangle distribution facilitates the derivation of analytic response functions in Sec. 6.7. B . 2 Modi f i ed triangle dis t r ibut ion Smith [1990]'s triangle distribution, Eq. (B. l ) , is a computationally efficient surrogate for the Gaussian distribution, but it does not accurately mimic a Gaussian at small (B.l) Q N < - 1 (l + QN)2/2 -1<QN<0 1 - (1 - QN)2/2 0 < Q N < 1 1 1 < Q N 118 Ac- Jeffery and Austin [2001b] introduce a modified triangle distribution that exhibits Gaussian-like behaviour at small Ac. The modified triangle distribution is ^ ^ 2a0 ( 3cr0 l£l -o~o < S < O-Q, (B.2) where cr2 = (35/3)er2. Using Eqs. (6.1) and (B.2), cloud fraction Ac = Ad(ztop) is 0 Ar Q N < - 1 (l + QN)4(l-QN)/2 - K Q n < 0 1-(1-QN)\1 + QN)/2 0<QN<1 1 1 < Q N where QN = q*/a0 and q* = qv + T ^ Z t o p — %• The A-th moment of the cloud liquid water used in the calculation of r via Eqs. (6.12) or (6.13) follows in a similar manner: 0 A~1F1 A-\F1 + F2) Q N < - 1 - K Q N < 0 0 < Q N < 1 [ A-1(F1 + F2 + F3) 1<Q N where Fi = a%{l+QN)x+A / 3 - 5 Q N - 4 + 20QN A + l A + 2 | - 6 - 3 0 Q A T | 12+20Q^ 5 ( 1 + Q J V ) A + 3 A + 4 A + 5 Fo = ^ QXN+ L f -QN+3QN , Q N - 9 Q 3 N , 16 A + l 3 + N A + 2 A + 3 3 Q 3 A + 4 g £ ( - l + C M A + 4 [3+5Qjy - 4 - 2 0 ^ 2 \ A+l A + 2 | - 6 + 3 0 Q J V [ 12-20Qjy b(l-QN) A + 3 A + 4 A + 5 A comparison of the triangle distribution, Eq. (B. l ) , and the modified triangle distri-bution, (B.2), is shown in Fig. B . l . 119 Figure B . l : Comparison of Smith [1990]'s triangle distribution, Eq. (B. l ) , and the mod-ified triangle distribution, (B.2). The modified triangle distribution has a significantly larger density in the range 2.5 < |s|/<rs < 3.5. 120 Appendix C Calculation of low-cloud optical properties in Sec. 6.5 The following parameter values are used to generate Fig. 6.2: q0(T) = (1.826 x 109 g m" 3 ) exp{-Rv/(LvT)}, Rv = 461.5 J K - 1 k g _ 1 , Lv = 2.5 x 106 J kg" 1 , r„ = (4 x l fr 3 K m- 1 ){L^/( J R„T 2 )}g 0 (T) and aL = 0.75. C l Calcula t ion of R Low cloud reflectivity in Sec. 6.5 is calculated using Barker [1996b]'s 7-weighted 2-stream reflectivity Rica.{f, v) where v = T 2 / C T 2 is a measure of the shape of the 7-distribution P 7 ( T , T , */). _ The derivation of PicaC^, v) begins with following conservative scattering 2-stream reflectivity accurate for single scattering albedos near unity: R ( , = 7 i T ' + ( 7 3 - 7 i ^ o ) ( l - e - T ' / w ) 1 ' l + 7 ir ' The delta-Eddington solution gives the following parameter values: r ' = (1 — u>0g2)T, 71 = [7 - u'0(A + 3g')]/4, 73 = (2 - 3/W)/4, 74 = 1 - 73, UJ'0 = u0(l - g2)/(l - u0g2) and g' = g/(l + g) where /io is solar zenith angle, UQ = 1 is the single scattering albedo and g = 0.85 is the asymmetry parameter. Barker [1996b]'s 7-weighted 2-stream reflectivity is obtained when r has a 7-distri-bution, P 7 , in the cloudy region of a cell [Barker, 1996b, Eq. (7a)]: /•oo Pica(r, v) = / P1(T,T,V)R(T) dr Jo = 1 - ( ^ ) " [ ( W . + 7 , ) S ( 1 - , , ^ ) f \r f-t u^o + T' -(71/^0-73)y — 121 where Q(l — v,x) = exT(l — v, x) and r' = (1 — u>og2)r. Averaging of i? ; c a over the diurnal cycle at Equinox yields 2 /-COS <fi •^diurnal average ~ 7 / - ^ ( A O A * dp, COS (p J0 where <fi is latitude. The mean and variance of shortwave optical depth are calculated from Eqs. (6.13) and (6.8) as described in Sec. 6.5 with parameter values kd = 1, pw = 1 g c m - 3 and N = 200 x 106 m " 3 . C.2 Calcula t ion of e Low cloud emissivity in Sec. 6.5 is calculated using Barker and Wielicki [1997]'s 7-weighted emissivity ej c a(r, v). Low cloud emittance is accurately given by Beer's law, e = 1 — e~T^, where p is zenith angle. Barker and Wielicki [1997]'s 7-weighted emissivity, averaged over the hemisphere, is [Barker and Wielicki, 1997, Eq. (B3)] pao pi e i c a ( T , v) = 2 / / P1(T,T,v)e(T,p)pdpdr Jo Jo 1 _ r ^ ( r " ) { ( r / + ? ) \ V + T J v + 2 \ ^ + r /J The mean and variance of longwave optical depth are calculated from Eq. (6.12) as described in Sec. 6.5 where the constant of proportionality—the longwave absorption coefficient—is assigned the value 0.15 g _ 1 m 2 . r(„) - — r{u +1) V + T 122 

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