Complexity in the Air

1. Molecule Droplet Clusters Aerosol Turbulent Eddies Clouds Global Complexity in the Air Yangang Liu Self-similarity;scale-invariance/turbulence/chaos 6 September 2012 Self-similarity/scale-invariance/turbulence/chaos

2. Appetizer “I can see no other escape from this dilemma than that some of us should venture to embark on a synthesis of facts and theories, albeit with second-hand and incomplete knowledge of some of them – at the risk of making fools of ourselves.” I venture to take the risk of making fools of myself …

3. Complexity in the Air in History “Under the same sky in which the stars pursue their orbits, as symbols of the unchangeable lawfulness of Nature, we see the clouds towering, the rain pouring, and the wind changing, as symbols of just the opposite extreme, the most capricious of all processes in Nature, impossible to bring inside the fence of its Laws” Excerpt from a lecture titled “Cyclones and Thunderstorms” By the great German physicist Hermann von Helmholtz in 1875. 1821-1894

4. Multiscale Climate Hierarchy Molecule Aerosol Time Scale Top-Down Global Cloud system Cloud Turbulent Eddy Bottom-Up Droplet Space Scale

5. Another Way to Look at Multiscale Climate Hierarchy: Climate Uroboros Scale Interactions & Dependence Earth Uroboros Cosmic Uroboros was originated by Dr. Sheldon Glashow and popularized by Dr. Joel Primack. I am suggesting the concept of Earth Uroboros Uroboros is a legendary snake swallowing its own tail, representing hope for a unified theory linking the largest and smallest.

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

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

8. Droplet Population as a System Molecular system, Gas Clouds Knowequations For each droplet Knew Newton’s mechanics for each molecule Uniform models failed to explain observed size distributions-- Kinetics failed to explain observed thermodynamics Maxwell, Boltzmann, Gibbs introduced statistical principles & established statistical mechanics Establish the systems theory Most probable size distribution Most probable energy distribution Various fluctuations associated with turbulence and aerosols suggest considering droplet population as a system to obtain info on droplet size distributions without knowing details of individual droplets and their interactions. From 1800s to early 1900s Since mid-20th century Analogy between molecular system and droplet system

9. Droplet System and Spectral Entropy Consider the droplet system constrained by (1) (2) Droplet spectral entropy is defined as (3) x = Hamiltonian variable; X = total amount of x per unit volume; n(x) = droplet number distribution with respect to x; r(x) = n(x)/N denotes the probability that a droplet of x occurs. Note the correspondence between x and the constraints.

10. Most Probable Distribution w.r.t. x (4) Maximizing the spectral entropy subject to the two constraints given by Eqs. (1) and (2) yields the most probable distribution with respect to x: The most probable distribution with respect to x is (5) where a = 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 a is consistent with that of “kBT”, or the mean energy per molecule.

11. 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. Question one: What determines x in general, and a and b in particular?

12. The systems theory works for other particle systems as well. b= effective radius/volume-mean radius

13. Entropy flux equals to internal entropy production at steady state.

14. Schrodinger’s View on Life “What an organism feeds upon is negative entropy.” The same holds for the Earth system …

15. 4 4sET S (1-R)S The Earth System Exchanges Radiation with Space • Earth system is characterized by planetary albedo R for incident shortwave radiation energy, and planetary longwave radiation emissivity E; • R and E are normally assumed to be constant. Radiation Balance: Absorbed Incoming Solar Radiation Energy = Outgoing Emitted Longwave Radiation Energy

16. The Earth system as a whole is driven by negative radiation entropy flux (Wu and Liu, 2010, Rev of Geophys) Outgoing radiation entropy is about 20 times of incoming radiation entropy. What determines the radiation entropy flux?

17. Entropy Production of Earth System TE = 280 K Earth 2.7 K Universe TS = 5800 K Sun • The Earth system is primarily driven by solar energy, but proactively respond by adjusting its planetary albedo R and longwave emissivity E via a multitude of processes including clouds ----- an open system with an adaptive boundary. The total entropy production rate is given approximately by Cold scattered photons Hot concentrated photons Entroy Production P = aE1/4(1-R)3/4 –b(1-R) (See Wu and Liu, 2010a, b for more accurate expressions)

18. Relationship between Albedo and Emissivity Red dots are measurements for cirrus clouds taken from Platt et al. (1980). The black line represents an idealized case In general, assume a power-law relationship R = aEg R = 3.1E1.29 R = E Application of R = aEg leads to the negative entropy flux Eq:

19. MEP determines the optimal albedo and emissivity F* = f(S, a, g) The MEP state is determined by the relationship between shortwave albedo and longwave emissivity. A change in the albedo-emissivity relationship likely causes change in the climate state.

20. Principle of Maximum Entropy Production • A nonlinear, far-from equilibrium system maintained at steady state via exchanges of energy/matter with its environment tend to generate entropy at a maximum rate possible. • For a nonlinear, far-from equilibrium system maintained at steady state via exchanges, the state of maximum entropy production occurs with the largest probability. Question 2: How to derive MEP principle? (Quarterly J of Met. Soc., 104, 1978)

21. GCM GCRM Molecule Droplet Clusters Aerosol Turbulent Eddies S. Cu Global Increasing computer power WRF CRM Microphysics LES DNS DNS = Direct Numerical Simulation LES = Large Eddy Simulation CRM = Cloud-Resolving Model WRF = Weather Research and Forecast Model GCM = Global Climate Model RCM = Regional Climate Model GCRM = Global CRM NWP = Numerical Weather Forecasting SCM = Single Column Model MMF Model/Data Multiscale Hierarchy NWP RCM SCM Model Domain Size Parcel Model Model Grid Size

22. 200 km (Unresolved) (Resolved) Climate Model, Fast Physics and Parameterization Subgrid and Complex Nonlinear PDE System: Parameterization is responsible for model uncertainty and significant resource consumption.

23. Statistical Physics as a Bridge between Kinetics and Thermodynamics Ludwig E. Boltzmann (1844-1906) Josiah W. Gibbs (1839 – 1903) James C. Maxwell (1831–1879) Gibbs coined the term "statistical mechanics" to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of large ensembles of particles.

24. Fast Physics Parameterization as Generalized Statistical Physics • “Statistical physics“ is to account for the observed thermodynamic properties of systems in terms of the statistics of large ensembles of particles. • “Parameterization” is to account for collective effects of many smaller scale processes on large scales. Molecule Ensemble Kinetics, Statistical Physics, Thermodynamics Classical Diagram of Cloud Ensemble for Convection Parameterization (Arakawa and Schubert, 1974, JAS) Droplet Ensemble DNS, Systems Theory

25. Fluctuations lead us to assume that droplet size distributions occur with different probabilities, and info on size distributions can be obtained without knowing details of individual droplets. Difference between Droplet System and Molecular System Molecular system, Gas Clouds Knowequations For each droplet Knew Newton’s mechanics for each molecule Uniform models failed to explain observations Kinetics failed to explain observed thermodynamic properties Maxwell, Boltzmann, Gibbs introduced statistical principles & established statistical mechanics Establish the systems theory Most probable distribution Most probable distribution Least probable distribution

26. Observation Conventional theory Most Probable and Least Probable Most probable Least probable We have developed a systems theory that describes the most and least probable distribution; the most and least probable distribution approximately correspond to observed and traditional theory-predicted ones.(Liu et al., JAS, 1998, 2002; Liu et al, 2002)

27. Scale-Dependence of Size Distribution - Fluctuations increases from level 1 to 3. • Saturation scale Ls is defined as the averaging scale beyond which distributions do not change. • Distributions are scale-dependent and ill-defined if averaging scale < Ls. Question 3 How to quantify the scale-dependence and link to fractal/scaling systems?

28. 4M-1E Complexity and Components Involved GCM/SCM Droplet Clusters Aerosol Turbulent Eddies Convection Global HRM Data Fusion & Assimilation Parameter. Observation Theory • “4M-1E” Complexity and 6 Components: • 4Mscientific • -- Multibody • -- Multiscale • -- Multitype • -- Multi-dimension • 1E engineering • -- coordination among both investigators and components Seminar Tomorrow in Environmenal Sciences Department Thanks!

29. My Summary • Isolated system favors the state of maximum entropy. • Near equilibrium system approaches the maximum entropy state along the path of minimum entropy production. • Far-from equilibrium system approaches the maximum entropy state along the path of maximum entropy production. • A unifying theme: nature favors the most probable state. • Knowing the most probable state is not enough for “small” system; scale-dependence is another challenge for “small” systems. • Other definitions/applications: relative entropy, mutual entropy, Renyi entropy, Tasali entropy, …

30. Rain initiation has also been a persistent puzzle in cloud physics Dr. Irving Langmuir Nobel prize winner & a pioneer in rain formation and weather modification in 1940s. Dr. Irving Langmuir However, the cloud-to-rain conversion is critical for understanding 2nd aerosol indirect effect and improving climate models. Existing representations need “surgery”. Newsday, August 3, 2004 Despite understanding of collision/coalescence of drops falling in clouds, cloud-to-rain conversion has been another persistent puzzle in cloud physics. And representation of this process has been either intuitive or empirical in climate models, lacking clear physics and having tunable parameters. We developed a new theory by considering rain initiation a statistical barrier-crossing process (McGraw & Liu, Phys. Rev. Lett., 2003; Phys. Rev., 2004).

31. 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

32. Rain Initiation Systems theories for representing all aerosol/cloud/precipitation processes Kohler theory KPT theory “Stable” state “Stable state” “Stable” state Today’s talk focuses on the most probable size distribution for those “stable” state derived from the principle of maximum, INTEGRATION WITH KPT THEORY.

33. Dependence of Entropy Production on Shortwave Albedo and Longwave Emissivity A maximum entropy production is expected if shortwave albedo and longwave emissivity are positively related to each other!

34. 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 (1796-1832)

35. Entropy as Statistical Measure of Disorder James C. Maxwell (1831–1879) Ludwig E. Boltzmann (1844-1906) Josiah W. Gibbs (1839 – 1903) In search of molecular root for thermodynamics, e.g, the 2nd law, Boltzmann, along with Maxwell and Gibbs etc, lay the foundation for kinetic theory and (equilibrium) statistical physics.

36. Information Entropy and Probability John von Neumann Ralf Landauer (1927 – 1999) Claude E. Shannon (1916-2001) (1903 – 1957) Information entropy “H” was chosen after Boltzmann’s H-theorem; Landauer principle further links information and thermodynamic entropy; application to model development remains to be explored. (Berut et al Nature 2012)

37. Jaynes Formalism Jaynes had further studied relationship between statistical physics, information theory, and probability theory since 1957. He argued that statistical physics can be seen as an application of a general formalism that seeks the most probable with limited information. Edwin T. Jaynes (1922–1998) Some considers/uses Janyes’ formalism as a common framework for both equilibrium and non-equilibrium systems.

38. Neglect of dispersion effect significantly overestimates cloud reflectivity Reflectivity of Monodisperse Clouds Green dashed line indicates the reflectivity error where overestimated cooling equals to the magnitude of warming by greenhouse gases. Neglecting dispersion can cause errors in cloud reflectivity, which further cause errors in temperature larger than warming by greenhouse gases. Dispersion may be a reason for overestimating cloud cooling effects by climate models.

39. Decreasing cloud reflectivity Increasing cloud reflectivity Aerosol-enhanced dispersion causes a warming effect on climate Enhanced dispersion has a warming effect that offsets the traditional 1st indirect effect by 10-80%, depending on the e-N relationship (Liu & Daum, Nature 2002; Liu et al. 2006, GRL).

40. New View Conventional Dispersion effect New View of Aerosol Indirect Effects New View: Indirect Effect = Number Effect + Dispersion Effect

41. Three Levels of Parameterization Stochastic parameterization Mean-field parameterization Unified parameterization Microphysics Resolved slaves subgrid Interacting subgrid processes Subgrid affects resolved Radiation Turbulence Convection PBL Process Surface-Process Fast Processes Resolved Grid Variables (self-consistency issues) Parameterization is not just practical necessity, but deep theoretical underpinning of scale-interactions within the multiscale system.

42. Outline • Background -- Recap -- Brief history • Particle Size Distribution Max. Entropy • Earth Climate Max. Entropy Production • Parameterization • Summary

43. Nonequilibrium Thermodynamics, Statistical Physics, and Dissipative Structure • Open non-equilibrium system exchanges energy/matter with its environment. • Principle of minimum entropy production (1947, near-equilibrium and linear). • Known examples of dissipative structure includes convection, hurricanes, …. • Self-organization, synergestics IIya Prigogine (1917 – 2003) 1977 Chemistry Nobel

44. Radiation Entropy and Blackbody Radiation Max Planck (1858–1947) Received the Nobel physics prize in 1918 for his contribution to quantum mechanics.

45. Kinetics, Statistical Physics and Thermodynamics With thermodynamics, one can calculate almost everything crudely; with kinetic theory, one can calculate few things, but more accurately; and with statistical mechanics one can calculate almost nothing exactly (Eugene Wigner) Eugene Paul Wigner (1902–1995 Shared the Nobel physics prize in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles“.

46. Brief History

47. Systems Theory onAtmospheric Particle Systems:Part I: Most Probable Size Distributions (Liu et al., AR, 1994, 1995; Liu & Hallett, QJ, 1998; JAS, 1998, 2002; Liu et al, 2002) Part II: On Rain Initiation and Autoconversion (McGraw and Liu, PRL, 2003, PRE, 2004; Liu et al., GRL, 2004, 2005, 2006, 2007, 2008)

48. Principle of Maximum Entropy Production • A nonlinear, far-from equilibrium system maintained at steady state via exchanges of energy/matter with its environment tend to generate entropy at a maximum rate possible. • For a nonlinear, far-from equilibrium system maintained at steady state via exchanges, the state of maximum entropy production occurs with the largest probability. (Quarterly J of Met. Soc., 104, 1978) Subsequent studies have applied the MEP principle to open material systems. What about the Earth system as a whole?

49. Droplet Spectral Entropy Droplet spectral entropy is defined as (3) Note the correspondence between the Hamiltonian variable x and the constraint defined in Eqs. (1) and (2).

50. Neglect of dispersion effect significantly overestimates cloud reflectivity Neglecting dispersion can cause errors in cloud reflectivity, which further cause errors in temperature larger than warming by greenhouse gases. Dispersion may be a reason for overestimating cloud cooling effects by climate models.