Yangang Liu

Complexity in the Air

Aerosol Droplet Turbulent Eddies Clouds Clusters GlobalMolecule

6 September 2012Self-similarity/scale-invariance/turbulence/chaos

Self-similarity;scale-invariance/turbulence/chaos

“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 …

Appetizer

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

Multiscale Climate Hierarchy

Aerosol

Droplet

Turbulent Eddy

Cloud

Cloud system

Global

MoleculeBottom-Up

Space Scale

Time Scale

Top-Down

Another Way to Look at Multiscale Climate Hierarchy: Climate

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.

ScaleInteractions&Dependence

EarthUroboros

n(r) (cm-3m-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 water droplets microscopically

Mean droplet radius~ 10 micrometer

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 representation of clouds.(Liu et al., AR, 1995; Liu & Hallett, QJ, 1998; JAS, 1998, 2002; Liu et al, 2002)

Conventional theory

Observation

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.

Droplet Population as a System

Kinetics failed to explain observed thermodynamics

Know equations For each droplet Knew Newton’s mechanics

for each molecule

Maxwell, Boltzmann, Gibbsintroduced statistical principles& established statistical mechanics

Uniform models failed to explain observed size distributions--

Establish the systemstheory

Most probable energy distribution

Molecular system, Gas Clouds

Most probablesize distribution

From 1800s to early 1900s Since mid-20th century

Analogy between molecular system and droplet system

x = Hamiltonian variable; X = total amount of xper unit volume; n(x) = droplet number distribution with respect to x; (x) = n(x)/N denotes the probability that a droplet of x occurs. Note the correspondence between x and the constraints.

Droplet System and Spectral Entropy

(1)

(2)

Consider the droplet system constrained by

(x)dx =1

Xxρ(x)dx =

NDroplet spectral entropy is defined as

(3) )]dxρ(x)ln[ρ(x-=H

Maximizing the spectral entropy subject to the two constraints given by Eqs. (1) and (2) yields the most probable distribution 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 “kBT”, or the mean energy per molecule.

Most Probable Distribution w.r.t. x

* 1 xρ x = exp -

α α(4)

* N xn x = exp -

α α(5)

The most probable distribution with respect to x is

Most Probable Droplet Size Distribution

Assume that the Hamiltonian variable x and droplet radius r follow a power-law relationship

bx = arSubstitution of the above equation into the exponential most probable distribution with respect to x yields the most probable droplet size distribution:

* b-1 b0

0

n r = N r exp -λr

N = ab/α;λ = a/α

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

This is a general Weibull distribution.

The systems theory works for other particle systems as well.

= e

ffect

ive

radi

us/v

olum

e-m

ean

radi

us

2/32

1/32

1 + 2εβ =

1 + ε

Entropy flux equals to internal entropy production at steady state.

“What an organism feeds upon is negative entropy.”

The same holds for the Earth system …

Schrodinger’s View on Life

The Earth System Exchanges Radiation with Space

Radiation Balance: Absorbed Incoming Solar Radiation Energy = Outgoing Emitted Longwave Radiation Energy

4Tσ4

S1E

R

S

(1-R)S

44 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.

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?

Entropy Production of Earth System

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

TS = 5800 KSun

TE = 280 KEarth

2.7 KUniverse

Entroy Production P = aE1/4(1-R)3/4 –b(1-R)

(See Wu and Liu, 2010a, b for more accurate expressions)

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 = E

R = 3.1E1.29

R = E

3/4γ/4F ~ R 1- R

Application of R = E leads to the negative entropy flux Eq:

MEP determines the optimal albedo and emissivity

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.

γ

* γE = α

3 + γ

* γR =

3 + γ

F* = f(S, , )

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)

Question 2: How to derive MEP principle?

LES

DNS

Increasing computer power

Microphysics

GC

M

CRM

WRF

GCRM

Model Grid Size

RCM

Model/Data Multiscale Hierarchy

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

Mod

el D

omai

n Si

ze

MMF

SCMParcel Model

NWP

Aerosol Droplet Turbulent EddiesS. Cu ClustersGlobalMolecule

200 km

Climate Model, Fast Physics and Parameterization

PhysicsFast Dynamicsdt

dX

21'2

'1

'2

'1

21

'2

'12121

x,xfxxzationParameteri

xxPhysics F

xxDynamics

xxxxxx

Parameterization is responsible for model uncertainty and significant resource consumption.

Nonlinear PDE System:

Subgrid and Complex

(Resolved) (Unresolved)

Ludwig E. Boltzmann (1844-1906)

Statistical Physics as a Bridge between Kinetics and

Thermodynamics

James C. Maxwell (1831–1879) Josiah W. Gibbs (1839 – 1903)

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.

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.

Classical Diagram of Cloud Ensemble for Convection Parameterization (Arakawa and Schubert, 1974, JAS)

Droplet Ensemble DNS, Systems Theory

Molecule EnsembleKinetics, Statistical Physics, Thermodynamics

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.

Kinetics failed to explain observed thermodynamic properties

Know equations For each droplet Knew Newton’s mechanics

for each molecule

Maxwell, Boltzmann, Gibbsintroduced statistical principles

& established statistical mechanics

Uniform models failed to explain

observations

Establish the systemstheory

Most probable distribution

Molecular system, Gas Clouds

Most probabledistribution

Least probabledistribution

Difference between Droplet System and Molecular System

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)

Conventional theory

Observation

Most Probable and Least Probable

Most probable

Least probable

- 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 ifaveraging scale < Ls.

Scale-Dependence of Size Distribution

Question 3How to quantify the scale-dependence and link to fractal/scaling systems?

GCM/SCM

Data Fusion &

Assimilation

Observation

Theory

HRM

Parameter.

4M-1E Complexity and Components Involved

“4M-1E” Complexity and 6 Components:

• 4M scientific -- Multibody -- Multiscale -- Multitype -- Multi-dimension

• 1E engineering -- coordination among both investigators and components

Aerosol Droplet Turbulent Eddies Convection Clusters Global

Seminar Tomorrow in Environmenal Sciences Department Thanks!

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, …

However, the cloud-to-rain conversion is critical for understanding 2nd aerosol indirect effect and improving climate models. Existing representations need “surgery”.

Rain initiation has also been a persistent puzzle in cloud physics

We developed a new theory by considering rain initiation a statistical barrier-crossing process

(McGraw & Liu, Phys. Rev. Lett., 2003; Phys. Rev., 2004).

Despite understanding of collision/coalescence of drops fallingin 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, lackingclear physics and having tunable parameters.

Dr. Irving Langmuir

Newsday, August 3, 2004

Nobel prize winner & a pioneer in rain formation and weather modification in 1940s.

Dr. Irving Langmuir

Traditional Theory

The condensational equation of the uniform theory

dr S

=dt A + B r

The larger the droplet, the slower the growth.

Droplet population approaches a narrow droplet size distribution

Long-standing issue of spectral broadening

Systems theories for representing all aerosol/cloud/precipitation processes

Kohler theory

“Stable” state

Rain Initiation

KPT theory

Today’s talk focuses on the most probable size distribution for those “stable” state derived from the principle of maximum, INTEGRATION WITH KPT THEORY.

“Stable state”

“Stable” state

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!

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)

Ludwig E. Boltzmann (1844-1906)

Entropy as Statistical Measure of Disorder

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.

James C. Maxwell (1831–1879) Josiah W. Gibbs (1839 – 1903)

Information Entropy and Probability

Claude E. Shannon (1916-2001) (1903 – 1957)

John von Neumann Ralf Landauer(1927 – 1999)

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)

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)

Jaynes Formalism

Some considers/uses Janyes’ formalism as a common framework for both equilibrium and non-equilibrium systems.

Reflectivity of Monodisperse Clouds

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.

Neglect of dispersion effect significantly overestimates cloud

reflectivity

Green dashed line indicates the reflectivity error where overestimated cooling equals to the magnitude of warming by greenhouse gases.

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 -N relationship (Liu & Daum, Nature 2002; Liu et al. 2006, GRL).

Dec

reas

ing

clou

d re

flect

ivity

Increasing cloud reflectivity

New View of Aerosol Indirect Effects

New View:Indirect Effect = Number Effect + Dispersion Effect

Dispersion effect

Conventio

nal

New View

Three Levels of ParameterizationFast Processes

Microphysics

Convection

Radiation

Surface-Process

Turbulence

PBL Process

Mean-field parameterization

Resolved slaves subgrid

Subgrid affects resolved

Stochastic parameterization

Interacting subgrid processes

Unified parameterization

Resolved GridVariables

(self-consistency issues)

Parameterization is not just practical necessity, but deep theoretical underpinning of scale-interactions within the multiscale system.

Outline

Background-- Recap-- Brief history

Particle Size Distribution Max. Entropy

Earth Climate Max. Entropy Production

Parameterization

Summary

Nonequilibrium Thermodynamics, Statistical Physics, and Dissipative

Structure

IIya Prigogine(1917 – 2003)1977 Chemistry Nobel

• 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

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

Radiation Entropy and Blackbody Radiation

T

JJ

3

4

3

4F 4/34/1

1

κThν

exp

1

c

hνnI

2

30

ν

30

ν2

30

ν2

30

ν2

30

ν2

2

20

ν hνn

Icln

hνn

Ic

hνn

Ic1ln

hνn

Ic1

c

κνnL

4J T

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“.

Brief History

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)

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?

Note the correspondence between the Hamiltonian variable x and the constraint defined in Eqs. (1) and (2).

Droplet spectral entropy is defined as

Droplet Spectral Entropy

(3) )]dxρ(x)ln[ρ(x-=H

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.

Neglect of dispersion effect significantly overestimates cloud

reflectivity

Principle of Maximum Entropy Production

Slippery Concept of Entropy

Mesoscopic State

Statistic MechanicsKine

tics

Atmospheric Particle System

Molecular System

Systems Theory

Mesoscopic State

Systems Theory

Macroscopic State

Larger Scale

Macroscopic State

Thermodynamics

Microscopic State

Single Molecule

Mea

n-Fi

eld

Theo

ry

Microscopic State

Microscopic Theory

Maximum Entropy Principle and Most Proba

2/32

1/32

1+ 2εβ =

1+ ε

1/3 1/3

ew

3 Lr = β

4πρ N

(

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

Commonly Used Size Distribution Functions

Q Q 1τ = = =

R kQ k

Multimodal distributions are not listed here

Timeline of Entropy and 2nd Law of Thermodynamics

Emerging Period -- Late 1690s to early 1850s-- Heat powered engine -- Low efficiency (as low as 2%)-- Energy (caloric) loss-- Energy transformation -- Impossible perpetual motion (Lazare Carnot, 1803)-- Carnot cycle and thermodynamic reversibility (Sadi Carnot, 1823)

(25 January 1917 – 28 May 2003)

Timeline of Entropy and 2nd Law of Thermodynamics

Clausius 1854 to 1865Introduction of “entropy” & familiar form of 2nd law of thermodynamics:“I propose to name the quantity S the entropy of the system, after the Greek word “trope”, the transormation. I have deliberately chsen th word to be as similar as possible to the word energy: the two quanttities are closely related in physical signifance that a certain similarity in their names appears to be apropriate”.

Boltzmann 1854 to 1865Introduction of “entropy” & familiar form of 2nd law of thermodynmics:“I propose to name the quantity S the entropy of the system, after the Greek word “trope”, the transormation. I have deliberately chsen th word to be as similar as possible to the word energy: the two quanttities are closely related in physical signifance that a certain similarity in their names appears to be apropriate”.

Shannon entropy1854 to 1865Introduction of radiation entropy

Planck1854 to 1865Radiation entropy & blackbody formulation

Jaynes MaxInf formalism1854 to 1865Radiation entropy & blackbody formulation

Nonequlibrium thermodynamics And dissipitative structure, self-organization, …

Maximum entropy production(Paltriage 1975)

Particle spectral entropy(Liu 1992-2002)

Earth-Atmosphere as a Heat Engine

The atmosphere is molecules in motion. Atmospheric turbulence arises molecular dynamics via generation of vorticity in presence of anisotropies such as gravity, earth rotation, solar beam, and topography etc. Because energy is deposited in the atmosphere by molecules absorbing photons, energy must inject and propagate upward from the smallest scales.

“Atmospheric Turbulence: A molecular Dynamics Perspective” by Adrian F. Tuck, Oxford, 2008

Earth Feeds on Negative Entropy Flux

Energy balance:

“Cold” photons + more entropy

“Hot” photons

TS = 5800 KSun

TE = 280 KEarth

2.7 KUniverse

H3/41/4~ -E 1- R

0 41

4S E E

R SF F E T

F

4 4

3 3S sE

SS E s E

F cFH c F

T T T T

0 01 13 4

3 4 3S E S

S E E

S Sc T TH

T T T

Negative entropy flux:

Importance of Cloud Albedo and Fraction

Radiation Balance:

S

4σT4

SA1

S

(1-A)S

44

A = Global albedo

c0 fAf)A(1A

0A = Clear sky albedo cA= Cloud albedo

f = Cloud fraction

“What an organism feeds upon is negative entropy (flux).”

The same likely holds for Earth system …

Bottom-Up vs. Top-Down

2.7 KUniverse

Negative Entropy Flux

Energy balance:

“Cold” photons + more entropy

“Hot” photons

TS = 5800 KSun

TE = 280 KEarth

2.7 KUniverse

H3/41/4~ -E 1- R

0 41

4S E E

R SF F E T

F

4 4

3 3S sE

SS E s E

F cFH c F

T T T T

0 01 13 4

3 4 3S E S

S E E

S Sc T TH

T T T

Negative entropy flux:

Paltrige (1978)

Entropy flux in Katrina

Traditional theory cannot explain rain formation adequately:

* Spectral broadening?

* Production of embryonic raindrops ?

Valley of Death or Mountain of Life

Rain initiation has been an outstanding puzzle with two fundamental problems of spectral broadening & formation of embryonic raindrop

dr 1

~dt r

4dr~ ar + br

dt

Valley of Death Mountain of Life

The new theory considers rain initiation as a statistical barrier crossing process. Only those “RARE SEED” drops crossing over the barrier grow into raindrops.

Parameterization problem in GCMs is similar: Issues well recognized, efforts made, & progress realized;

now is the time to for us to be a SEED that accelerate and crosses over the barrier !

In turbulent clouds, consider the droplet system in the region of embryonic raindrops as a

dynamical balance between three physical processes: condensation, evaporation and

collection.

Consider the formation of embryonic raindrops as a barrier-crossing phenomenon like

nucleation.

Basic Idea

Dynamic Equilibrium

Consider the nucleation of a water droplet containing g molecules, whereby single molecules are exchanged with surrounding water vapor, and each step can be represented by the equilibrium:

g 1 g+1A + A = A

Ag = a drop of size g; A1 = a monomer

Detailed Balance

Under equilibrium, detailed balance gives:

g g g+1 g+1β n = γ n

g (s-1) = rate of monomer condensation on the g-dropg (s-1) = rate of monomer evaporation of the g-dropng (cm-3) = constrained equilibrium concentration of g-drop

32

2 1

... g

g g-1

ig 1 1

i=11 i+1

nnn βn = n = n

n n n γ

Kinetic Potential

Under equilibrium, ng can also be expressed in Boltzmann form

where “w/kT” is the reduced thermodynamics potential for droplet formation from vapor. Comparison of the two ng expressions yields the kinetic potential

g 1

w gn = n exp -

kT

g-1

i

i=1 i+1

w gβΦ g = -ln

γ kT

Advantages of Kinetic Potential

The kinetic potential is equivalent to the reduced thermodynamic potential in nucleation theory. However, the kinetic potential is a more general concept in that it is based on rate constants, and well defined even in the absence of equilibrium condition.

Next, we will use the kinetic potential to study the rain initiation.

g-1 g-1

i i

i=1i=1 i+1 i+1

β βΦ g = -ln = - ln

γ γ

Droplet Growth Rates

The growth of cloud droplets is modeled as a sum of condensation and collection processes:

con colg g gβ = β + β

congβ = Condensation rate;

colgβ = Collection growth rate

Long Collection Kernel

61k(R,r) = k R (10 m R 50 m)

(R > 50 m)32k(R,r) = k R

2

R rK(R,r) = Eπ R + r V - V

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.

Collection Growth Rate

dm

= k(R,r)m(r)n(r)drdt

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

Application of the Long kernel yields the growth rate of the radius R (10 m R 50 m):

21

dm= k Lm

dt

col 2g 1

dgβ = = k νLg

dt

Relationship between Effective Evaporation Rate and Condensation Rate

Effective evaporation rate is introduced to consider the complex droplet interactions and competition for water vapor such that a typical droplet size distribution is maintained by detailed balance (constrained equilibrium):

1

con cong+1 g g

g g g

n β β 1= exp -

n γ γ α

1

cong g

1γ exp β

α

Relating to L and N

The liquid water content of the droplet system is given by

cong+1 g

νNγ = exp β

L

g

N gL = νgn dg = ν gexp - dg = Nνα

α α

L

αNν

Examination of Kinetic Potential

Substituting into kinetic potential equation of the effective evaporation rate and collection growth rate and a typical value of condensation rate, we calculated the kinetic potential as a function of L and N:

exp

coni i

coni

iNL

2g-1 g-1

1

i=1 i=1i+1

β β k νLΦ g = - ln - ln

γ β

Droplet Concentration Effect on Kinetic Potential

This figure shows the kinetic potential as a function of the droplet radius at different values of droplet concentration N calculated from the above equation for the kinetic potential. The dashed lines are without collection.

L = 0.5 g m-3

Rain initiation is a barrier-crossing processlike nucleation.

Both critical radiusand potential barrier increases with increasingdroplet concentration.

The results suggest increasing aerosols inhibitrain by enhancing thebarrier height and criticalradius.

Analytical Expressions for Critical Radius

At the critical point, the forward and backward rates are in balance:

con colβ + β = γ

conνNγ = exp β

L

col 2g 1β = k νLg

1/62 1/62

c con 2 2

3 ν N Nr = β 0.99

4π κ L L

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!

How do clouds respond to climate forcings?

Low clouds

High clouds

Middle clouds

This figure shows that is an increasing function of

well described by Eq. (1), permitting incorporation in models.

Dependence of on Relative Dispersion

2/32

1/32

1+ 2εβ =

1+ ε

Data from marine and continental clouds

= Standard deviation/mean

(1)

3

e 2

r n(r)drr =

r n(r)dr

Hansen & Travis (1974, Space Sci. Rev) introduced effective radius re to describe light scattering by a cloud of particles

Effective radius and Its Parameterization

Cloud radiative properties are parameterized using liquid water (path) and re , and very sensitive to re (Slingo 1989, 1990).

re is further parameterized as

1/3 1/3

ew

3 Lr = β

4πρ NThe value of has been incorrectly assumed to be a constant!

dr 1~

dt r4dr

~ ar + brdt

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

Cloud optical depth is given by

1

0

0

w e w e

3LWP 3H Lτ = =

2ρ r 2ρ r

3/13/21

2

3NL

H

w

Substitution of re = (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.

g

gR

12

1

RFARFF mstGM 4

8.0

4

1

Major Cloud Radiative Properties

Fluctuations and interactions in turbulent clouds lead usto 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 explainthermodynamic properties

Knew Newton’s mechanicsfor 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

Most Probable Distribution w.r.t. Hamiltonian Variable

* N xn x = exp -

α α

In the last talk, we have shown that the most probable distribution with respect to x is

Under the assumption of conserved liquid water content, x is the droplet mass. In other words, the most probable Distribution can be written as

g

N gn = exp -

α α

Note the difference in the value of in the two equations

Droplet Concentration Effect on Kinetic Potential

This figure shows the kinetic potential as a function of the droplet radius at different values of droplet concentration N calculated from the above equation for the kinetic potential. The dashed lines are without collection.

L = 0.5 g m-3

Rain initiation is a barrier-crossing processlike nucleation.

Both critical radiusand potential barrier increases with increasingdroplet concentration.

The results suggest increasing aerosols inhibitrain by enhancing thebarrier height and criticalradius.

Analytical Expressions for Critical Radius

At the critical point, the forward and backward rates are in balance:

con colβ + β = γ

conνNγ = exp β

L

col 2g 1β = k νLg

1/62 1/62

c con 2 2

3 ν N Nr = β 0.99

4π κ L L

Estimation of Condensation Rate Constant

The star denotes that L and N are sampled in drizzling clouds.(Liu et al. 2003, GRL)

1/3 1/3

6w

3 Lr = 1.12

4πρ N

Mean radius of the 6th moment is given

According to Liu and Daum (2004, JAS), when r6 = rc, rain starts:

1.126 4

*con 2 3

w *

k Lβ =

νρ N

Revisiting Precipitation Staircase:“Kinetic Potential” Landscape

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 xper unit volume, n(x) = distribution with respect to x, (x) = n(x)/N = probability that a droplet of x occurs.

1(x)dx

N

X(x)dx x

Define spectral entropy (x))dx(x)ln(-=H

* Maximizing H subject to Eq. (1) and Eq. (2) and using x = aDb, obtain the most probable size distribution nM(D) = N0Db-1exp(-Db)

32

23

1 c+dDDc+n(D)dDDc=E

* Maximizing E subject to Eq. (1) and Eq. (2), obtain the least probably distribution nL(D) = N(D-Db)

The energy change to form a droplet with diameter D is

Systems Theory

Given cloud depth H, liquid water content L, and effective radius (re), optical depth is given by

1

0

0

ew r

LH

2

3

3/13/21

2

3NL

H

w

Substitution of re = (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

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

00R

0 o 0

1- β 1- RR - RE = =

R R + 1- R β

1- g τR =

2 + 1- g τ

00

0

1- g τR =

2 + 1- g τ

Cloud albedo is given by

0 00

o 0

1- β 1- R RR - R =

R + 1- R β

The relative difference in cloud albedo is

Cloud radiative forcing (CRF) is defined as

Clear Sky

Fclear

Clouds

Fcloud

F = net downward radiative flux

Cloud Radiative Forcing-Direvation

0 0ε

o 0

1- β 1- R R(ΔF) = -82.2

R + 1- R β

ΔF ΔR

cloud clearCRF = F - F

A perturbation of cloud albedo DR will lead to a change in CRF:

The neglect of relative dispersion will lead to errors in CRF:

Derivation 1

dr S=

dt Gr

v v w w v

v d v s

L L ρ ρ R TG = -1 +

R T k T D e

1.945 101325

2.11 10273.15v

TD

P

34.18 10 5.69 0.017 273.15dk T 40.167 3.26 10

6 273.152.5 10

T

vL T

2s

17.67 T - 273.15e T = 6.112×10 exp

T - 29.65

Derivation 2

1

1

r rx

r

1(1 )r r x

Introduce a variable x such that:

211

2

dr S

dt G

Substitution into the growth equation leads to

21

2dx Sr x

dt G

Multiplication of Eq. (4) by x and averaging over droplet population leads to .

2 221 2

2

r d x Sx

dt G

(1)

(2)

(4)

(3)

Derivation 3

2 221 2

2

r d x Sx

dt G (5)

(6)

(7)

Division of Eq. (5) by Eq. (1) gives.

2 21

221

2d x dr

rx

,

2 2x 2

0 102

1

ε rε =

r

Substituting , we have.

Derivation 4

Under the steady-state assumption, r1 is given by (Cooper, 1989, JAS),

1m4 w

aGwr

bS N

2v d v

s p v

R T R Lb

e c R PT

2v

p v d

gL ga

c R T R T

22 2w m

0 10

4πρ bSε = ε r N

aGw

Substituting into the expression for

Relationship to CCN Properties

Substituting into the expression for , we have

1k+2

3/2 3-2k/ k+2 2 k+2 -1/ k+2

m

w

2 aGS = 10 w c

k 34πρ kbB ,

2 2

Assuming a power-law CCN spectrum, Twomey (1959)

kk+2

3/2 3k 24k/ k+2 2 k+2 k+2

w

2 aGN = 10 w c

k 34πρ kbB ,

2 2

2 k+1k+2

3/22 k-1 24k/ k+2 k+2 k+22 w

0 10

w

2 aG4πρ bε = ε r 10 w c

k 3G4πρ kbB ,

2 2

-12ε w N

Long Collection Kernel

61k(R,r) = k R (10

(R > 50 32k(R,r) = k R

2

R rK(R,r) = Eπ R + r V - V

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.

Continuous Collection Process

dm

= k(R,r)m(r)n(r)drdt

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

2

w

dR k L= R

dt 4πρ

41

w

dR k L= R

dt 4πρ (R 50

(R > 50

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.

3

e 2

r n(r)drr =

r n(r)dr

Hansen & Travis (1974, Space Sci. Rev) introduced effective radius re to describe light scattering by a cloud of particles

Effective radius and Its Parameterization

Cloud radiative properties are parameterized using liquid water (path) and re , and very sensitive to re (Slingo 1989, 1990).

re is further parameterized as

1/3 1/3

ew

3 Lr = β

4πρ NThe value of has been incorrectly assumed to be a constant!

* Slingo (1989, JAS) developed a scheme that uses liquid water path and re to parameterize radiative properties of clouds.

* Slingo (1990, Nature) found that the top-of-atmosphere forcing of doubling CO2 could be offset by reducing re by ~ 15 – 20% .

* Kiehl (1994, JGR) reported diminishing known biases of the early NCAR CCM2 as a result of assigning difference values of re to maritime and continental clouds.

Introduction of re into climate models has significantly improved cloud parameterizations in climate models and our capability to study indirectaerosol effect.

Parameterization of Cloud Radiative Properties

Two Key Equations

Unlike condensation, the collection growth rate of radius increases with the drop radius R.

0 K KP = K(R,r )n R dR m r n(r)dr = L K(R,r )n R dR

According to the continuous collection process, we have

0 dm

P = n R dR = n R dR K(R,r)m r n(r)drdt

b b

ξ

a a

f x g x dx = f x g x dx

The generalized mean value theorem for integrals is

Two Key Equations

Unlike condensation, the collection growth rate of radius increases with the drop radius R.

0 K KP = K(R,r )n R dR m r n(r)dr = L K(R,r )n R dR

According to the continuous collection process, we have

0 dm

P = n R dR = n R dR K(R,r)m r n(r)drdt

b b

ξ

a a

f x g x dx = f x g x dx

The generalized mean value theorem for integrals is