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Statistical Models of Cloud-Turbulence Interactions

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Statistical Models of Cloud-Turbulence Interactions Statistical Models of Cloud-Turbulence Interactions by Christopher A. M. Jeffery M.Sc, University of British Columbia, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE 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: Preferential Concentration: 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. r 1 2 Effect of Intermittency: An effective Stokes number, Stefj = St(^ /3) / 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. Cloud Spatial 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 ii experimental measurements [Davis et al, 1999]. Low-cloud Optical Properties: 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 GCM cloud parameterization that prognoses both the mean and variance of optical depth iii 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 CE 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 reff-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 Cl 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, St)/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 SOCEX 56 5.2 Effect of condensation/evaporation on the scalar spectrum 67 5.3 Comparison of SOCEX 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 PPH 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 VMS 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. CHRISTOPHER A. M. JEFFERY 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).
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