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Yangang Liu 1 Robert McGraw 1, Peter Daum 1, and John Hallett 2 Systems Approach to Cloud Droplet Size Distributions: Analogy with Statistical Physical and Kinetics Workshop on Clouds and Turbulence, London, March 2009 (1= Brookhaven National Laboratory; 2 = Desert Research Institute)

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Rich History of Cloud Physics Here “Progress in cloud physics has been hindered by a poor appreciation of these interactions between processes ranging from nucleation phenomena on the molecular scale to the [turbulent] dynamics of extensive cloud systems on the scale of hundreds of thousands of kilometers” (Quote from 1 st ed. preface, 1957) B. J. Mason Formerly Prof. of Cloud Physics Imperial College of Science and Technology (1948 – 1965)

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n(r) (cm -3 m -1 ) Macroscopic view of clouds is an optical manifestation of cloud particles Microscopic Zoom-in A central task of cloud physics is to predict the cloud droplet size distribution, n(r). Clouds are systems of water droplets Mean droplet radius ~ 10 micrometer

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Traditional Theory The condensational equation of the uniform theory The larger the droplet, the slower the growth. Droplet population approaches a narrow droplet size distribution Long-standing issue of spectral broadening

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Spectral broadening is a long-standing, unsolved problem in cloud physics “…., it appears unlikely that internal turbulence can cause deviations greater than about 1 m from the sizes predicted by the theory of condensation in a steady updraft.” (Quote from Mason, 2 nd ed, P145, 1971). But, turbulence remains to be a key to this day. Regular theory Observation Conventional theory Droplet radius Concentration

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Commonly Used Size Distribution Functions (Most already summarized in “The Physics of Clouds” by B. J. Mason 1957) Problem is still at large: New developments in stochastic condensation, entrainment-mixing, modeling activities (from other speakers), and SYSTEMS THEORY (MY TALK).

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Various fluctuations associated with turbulence and aerosols suggest considering droplet population as a system to obtain information on droplet size distributions without knowing details of individual droplets and their interactions. Droplet Population as a System Boltzmann equation Droplets & equations for each droplet (DNS) Molecules & Newton’s mechanics for each molecule Maxwell, Boltzmann & Gibbs introduced statistical principles & established statistical mechanics Various kinetic equations (e.g., stochastic condensation) Systems theory Most probable energy distribution Molecular system (gas) Cloud Most probable size distribution Least probable size distribution We developed a systems theory (Liu & Hallett, QJ, 1998; Liu et al., AR, 1995, JAS, 1998, 2002a, b). Today mainly on MPSD based on the maximum entropy principle.

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x = Hamiltonian variable, X = total amount of per unit volume, n(x) = droplet number distribution with respect to x, (x) = n(x)/N = probability that a droplet of x occurs. Droplet System (1) (2) Consider the droplet system constrained by

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Liu et al. (1992, 1995, 2000), Liu (1995), Liu & Hallett (1997, 1998) Note the correspondence between the Hamiltonian variable x and the constraint Droplet spectral entropy is defined as Droplet Spectral Entropy (3)

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Maximizing the spectral entropy subject to the two constraints given by Eqs. (1) and (2) yields the most probable PDF with respect to x: where = X/N represents the mean amount of x per droplet. Note that the Boltzman energy distribution becomes special of Eq. (5) when x = molecular energy. The physical meaning of is consistent with that of “k B T”, or the mean energy per molecule. Most Probable Distribution w.r.t. x (4) (5) The most probable distribution with respect to x is

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Most Probable Droplet Size Distribution Assume that the Hamiltonian variable x and droplet radius r follow a power-law relationship Substitution of the above equation into the exponential most probable distribution with respect to x yields the most probable droplet size distribution: This is a general Weibull distribution!

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This figure demonstrates that the Weibull distribution from the systems theory well describes observed droplet size distributions. Observational Verification Data from marine and continental clouds = Standard deviation/mean Boltzmann

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Mixing-Dominated Horizontal Regime h = 260 m Relative Dispersion Mean-Volume Radius ( m) Condensation-Dominated Vertical Regime July 27, 2005 Relative Dispersion Droplet Concentration N (cm -3 ) Difference between Gas and Droplet Systems Note the opposite relationships of mean-volume radius to relative dispersion ! A striking difference between molecular and droplet systems is that the relative dispersion of the Boltzmann energy distribution is constant whereas the relative dispersion of the droplet size distribution varies (with turbulence properties).

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Determination of Relative Dispersion Adiabatic condensation: Done (e.g., Liu et al., GRL, 2006). Fokker-Planck and Langevin equations (recall Boltzmann equation, and go one step further): -- McGraw and Liu (2006, GRL): The steady-state droplet size distribution is maintained by the balance between diffusion (growth) and drift (depletion) coefficients. But this work is only for a special case of two constant coefficients. -- Latest research: The relative dispersion or b is determined by the dependence of the diffusion and drift coefficients on droplet radius. -- Future challenge: How to relate relative dispersion to turbulence properties? -- More challenging: Unify the blue and the white in sky ?

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Reflectivity of Monodisperse Clouds Neglecting dispersion can cause errors in cloud reflectivity, which further cause errors in temperature larger than warming by greenhouse gases (Liu et al., ERL, 2008) Neglect of dispersion effect significantly overestimates cloud reflectivity Green dashed line indicates the reflectivity error where overestimated cooling is equal to the magnitude of warming by greenhouse gases.

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Relative dispersion determines the threshold behavior of rain initiation AGU Highlight Combining the new rain initiation theory with theory for collision and coalescence of cloud drops leads to an analytical threshold function (Liu & Daum, JAS, 2004; Liu et al., GRL, 2004, 2005, 2006). Kessler scheme = Dispersion Note the importance of dispersion!

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Promising Future Boltzmann Laughing Buddha

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Entropic View of Earth System from Space The shortwave radiation absorbed by the Earth is emitted as infrared radiation (Energy balance). The Earth system enjoys a negative entropy flux F, which depends on the Earth’s planetary albedo R and emissivity : “Cold” photons + more entropy “Hot” photons T S = 5800 K Sun T E = 280 K Earth 2.7 K Universe How to relate E and R to clouds and further to MEP ?

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Relationship between Albedo and Emissivity Red dots are measurements for cirrus clouds taken from Platt et al. (1980). The black line represents an ideal stratiform cloud In general, assume a power-law relationship R = E R = 3.1E 1.29 R = E Application of R = E leads to the negative entropy flux Eq:

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MEP determines the optimal albedo and emissivity Cirrus Stratiform R* = 0.30 R* = 0.25 E* = 0.66 E* = 0.25 The earth tends to have maximum entropy production/maximum negative entropy flux; this MEP hypothesis offers an explanation for the stable earth albedo R ~ 0.29, or vise versa.

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Summary The principle of the maximum spectral entropy gives the most probable cloud droplet size distribution. The principle of MEP seems to keep the Earth’s climate stable by regulating cloud-related processes directly as well as indirectly (Note that a small change of the planetary albedo from 0.25 to 0.3 is sufficient to offset the warming caused by doubling CO2! This is a small number game). New challenge: How to quantify the relationship between the first and the second results? How to relate these results to climate change issues such as aerosol forcings? How to relate clouds to other components or processes such as biosphere? ….

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( = Standard Deviation/Mean Radius) The theoretical - expression can be used to examine the effect of DISPERSION on cloud reflectivity (R) via the known equation R = f Our systems theory well describes the ambient the cloud droplet size distribution

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Spectral broadening is a long-standing, unsolved problem in cloud physics We have developed a systems theory based on the maximum entropy principle, and applied it to derive a better representation of clouds. (Liu & Hallett, QJRS, 1998; Liu et al., AR, 1995, JAS, 1998, 2002; Liu & Daum GRL, 2000) Regular theory Observation Conventional theory Droplet radius Concentration

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Negative Entropy Flux Energy balance: “Cold” photons + more entropy “Hot” photons T S = 5800 K Sun T E = 280 K Earth 2.7 K Universe Negative entropy flux:

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Fundamentals of the traditional theory had been established by 1940’s “ The rapid progress of aerosol/cloud physics since the beginning 1940’s has not been characterized by numerous conceptual breakthroughs, but rather by a series of progressively more refined quantitative theoretical and experimental studies of previously identified microphysical processes (and ideas)” (Pruppacher and Klett, 1997). Further progress demands conceptual breakthrough!

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How do clouds respond to climate forcings? Low clouds High clouds Middle clouds

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Hansen & Travis (1974, Space Sci. Rev) introduced effective radius r e to describe light scattering by a cloud of particles Effective radius and Its Parameterization Cloud radiative properties are parameterized using liquid water (path) and r e, and very sensitive to r e (Slingo 1989, 1990). r e is further parameterized as The value of has been incorrectly assumed to be a constant!

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Conventional Theory and Valley of Death Rain initiation has been a persistent puzzle in cloud physics; what are missing in this picture ? Fundamental difficulties: Spectral broadening Embryonic Raindrop Formation

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Earth-Atmosphere as a Heat Engine We must attribute to heat the great movements that we observe all about us on the Earth. Heat is the cause of currents in the atmosphere, of the rising motion of clouds, of the falling of rain and of other atmospheric phenomena (Sadi Carnot, 1824) Sadi Carnot

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Cloud optical depth is given by Substitution of r e = (L/N) 1/3 into the above equation yields With H, L, and N remaining the same, the relative difference in optical depth between “real” and the ideal monodisperse cloud ( 0 ) is Using = / 0 = (d) given by the Weibull size distribution, we can obtain the relative difference in optical depth (and therefore cloud albedo, and radiative forcing) as a function of d. Major Cloud Radiative Properties

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Fluctuations and interactions in turbulent clouds lead us to question the possibility of tracking individual droplets/drops and to consider droplets/drops as a system. Systems Approach as a New Paradigm Kinetics difficult to explain thermodynamic properties Knew Newton’s mechanics for each molecule Statistical mechanics; Phase Transition; Boltzmann equation Molecular system, Gas Know equations For each droplet Mainstream models difficult to explain size distributions Entropy principle; KPT; Fokker-Planck Equation Clouds

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Consider a droplet system constrained by Referred to Liu et al. (1992, 1995, 2000), Liu (1995), Liu & Hallett (1997, 1998) x = restriction variable, X = total amount of x per unit volume, n(x) = distribution with respect to x, (x) = n(x)/N = probability that a droplet of x occurs. Define spectral entropy * Maximizing H subject to Eq. (1) and Eq. (2) and using x = aD b, obtain the most probable size distribution n M (D) = N 0 D b-1 exp(- D b ) * Maximizing E subject to Eq. (1) and Eq. (2), obtain the least probably distribution n L (D) = N (D-D b ) The energy change to form a droplet with diameter D is Systems Theory

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Given cloud depth H, liquid water content L, and effective radius (r e ), optical depth is given by Substitution of r e = (L/N) 1/3 into the above equation yields With H, L, and N remaining the same, the relative difference in optical depth between “real” and the ideal monodisperse cloud ( 0 ) is Using = ( ) given by the Weibull size distribution, we can obtain the relative difference in optical depth (and therefore cloud albedo, and radiative forcing) as a function of . Dispersion Effect on Cloud Optical Depth

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Monodisperse cloud albedo is The difference in cloud albedo is Using = ( ) given by the Weibull size distribution, we can obtain the relative difference in cloud albedo as a function of . Dispersion Effect on Cloud Albedo Cloud albedo is given by The relative difference in cloud albedo is

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Cloud radiative forcing (CRF) is defined as Clear Sky F clear Clouds F cloud F = net downward radiative flux Cloud Radiative Forcing-Direvation A perturbation of cloud albedo DR will lead to a change in CRF: The neglect of relative dispersion will lead to errors in CRF:

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Derivation 1

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Derivation 2 Introduce a variable x such that: Substitution into the growth equation leads to Multiplication of Eq. (4) by x and averaging over droplet population leads to. (1) (2) (4) (3)

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Derivation 3 (5) (6) (7) Division of Eq. (5) by Eq. (1) gives., Substituting, we have.

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Derivation 4 Under the steady-state assumption, r 1 is given by (Cooper, 1989, JAS), Substituting into the expression for

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Relationship to CCN Properties Substituting into the expression for , we have Assuming a power-law CCN spectrum, Twomey (1959)

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Long Collection Kernel (10 m R 50 m) (R > 50 m) The general collection kernel is given by, and its general solution is too complicated to handle. Long (1978, J. Atmos. Sci.) gave a very accurate approximation: The (gravitational) collection kernel is negligible when R < 10 m.

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Continuous Collection Process R A drop of radius R fall through a polydisperse population of smaller droplets with size distribution n(r). The mass growth rate of the drop is (R 50 m) (R > 50 m) Application of the Long kernel yields the growth rate of the radius R: Unlike condensation, the collection growth rate of radius increases with the drop radius R.

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Hansen & Travis (1974, Space Sci. Rev) introduced effective radius r e to describe light scattering by a cloud of particles Effective radius and Its Parameterization Cloud radiative properties are parameterized using liquid water (path) and r e, and very sensitive to r e (Slingo 1989, 1990). r e is further parameterized as The value of has been incorrectly assumed to be a constant!

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* Slingo (1989, JAS) developed a scheme that uses liquid water path and r e to parameterize radiative properties of clouds. * Slingo (1990, Nature) found that the top-of-atmosphere forcing of doubling CO 2 could be offset by reducing r e by ~ 15 – 20%. * Kiehl (1994, JGR) reported diminishing known biases of the early NCAR CCM2 as a result of assigning difference values of r e to maritime and continental clouds. Introduction of r e into climate models has significantly improved cloud parameterizations in climate models and our capability to study indirect aerosol effect. Parameterization of Cloud Radiative Properties

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Two Key Equations Unlike condensation, the collection growth rate of radius increases with the drop radius R. According to the continuous collection process, we have The generalized mean value theorem for integrals is

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