Yangang Liu 1 Robert McGraw 1, Peter Daum 1, and John Hallett 2 Systems Approach to Cloud Droplet Size Distributions: Analogy with Statistical Physical.

Yangang Liu1

Robert McGraw1, Peter Daum1, and John Hallett2

Systems Approach to Cloud Droplet Size Distributions: Analogy with Statistical Physical and Kinetics

Workshop on Clouds and Turbulence, London, 23-25 March 2009

(1= Brookhaven National Laboratory; 2 = Desert Research Institute)

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 1st ed. preface, 1957)

B. J. MasonFormerly Prof. of Cloud PhysicsImperial College of Science and Technology (1948 – 1965)

n(r) (cm-3mm-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

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

Spectral broadening is a long-standing, unsolved problem in cloud physics

“…., it appears unlikely that internal turbulence can cause deviations greater than about 1 mm from the sizes predicted by the theory of condensation in a steady updraft.” (Quote from Mason, 2nd ed, P145, 1971). But, turbulence remains to be a key to this day.

Regular theory

Observation

Conventional theory

Droplet radius

Con

cen

trat

ion

Commonly Used Size Distribution Functions

Q Q 1τ = = =

R kQ k

(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).

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 & Gibbsintroduced 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 distributionLeast 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.

x = Hamiltonian variable, X = total amount of per unit volume, n(x) = droplet number distribution with respect to x, r(x) = n(x)/N = probability that a droplet of x occurs.

Droplet System

(1)

(2)

Consider the droplet system constrained by

ρ(x)dx = 1

X

xρ(x)dx =N

x

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)E=- (x)ln( (x))dx

N xρ(x)dx = X

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 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.

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 α = X/N

This is a general Weibull distribution!

This figure demonstrates that the Weibull distribution from the systems theory well describes observed droplet size distributions.

Observational Verification

2/32

1/32

1+ 2εβ =

1+ ε

• Data from marine and continental clouds

• e = Standard deviation/mean

1/3 1/3

ew

3 Lr = β

4πρ N

Boltzmann

250

300

350

400

450

500

5.2

5.4

5.6

5.8

6

6.2

6.4

0.15 0.2 0.25 0.3 0.35

N

rvN

(cm

-3) rv (µ

m)

July 27, 2005260 m (msl)

Mixing-Dominated Horizontal Regime

h = 260 m

Relative Dispersion e

0

100

200

300

400

500

2

3

4

5

6

7

8

0.2 0.3 0.4 0.5 0.6 0.7

N

rv

N (

cm

-3) rv (µ

m)

July 27, 2005

Mea

n-V

olum

e R

adiu

s (m

m)

Condensation-Dominated Vertical Regime

July 27, 2005

Relative Dispersion e

Dro

plet

Con

cen

trat

ion

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).

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 ?

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.

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 analyticalthreshold function (Liu & Daum, JAS, 2004; Liu et al., GRL, 2004, 2005, 2006).

Kessler scheme

e = Dispersion

Note the importance of dispersion!

Promising Future

Boltzmann

Laughing Buddha

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 E:

“Cold” photons + more entropy

“Hot” photons

TS = 5800 KSun

TE = 280 KEarth

2.7 KUniverse

3/41/4F ~ -E 1- R

How to relate E and R to clouds and further to MEP ?

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 relationshipR = aE g

R = 3.1E1.29

R = E

3/4γ/4F ~ R 1- R

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

MEP determines the optimal albedo and emissivity

*γR=

3+γ

γ

*γE=α

3+γ

Cirrus StratiformR* = 0.30 R* = 0.25E* = 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.

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? ….

2/32

1/32

1+ 2εβ =

1+ ε

1/3 1/3

ew

3 Lr = β

4πρ N

(e = Standard Deviation/Mean Radius)

The theoretical b-e expression can be used to examine the effect of DISPERSION on cloud reflectivity (R) via the known equation R = f ( ( )].b e

Our systems theory well describes the ambient the cloud droplet size

distribution

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

Con

cen

trat

ion

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:

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

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 N

The value of b has been incorrectly assumed to be a constant!

dr 1

~dt r

4dr~ ar + br

dt

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

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

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 = a(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 (t0) is

Using b = a/a0 = b(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 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

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, r(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(-lDb)

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) = Nd(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 = a(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 (t0) is

Using b = b(e) 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 e.

Dispersion Effect on Cloud Optical Depth

Monodisperse cloud albedo is

The difference in cloud albedo is

Using b = b(e) given by the Weibull size distribution, we can obtain the relative difference in cloud albedo as a function of e.

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 xdt 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 e

Relationship to CCN Properties

Substituting into the expression for e, 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 3G 4πρ kbB ,2 2

-12ε w N

Long Collection Kernel

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

mm)

(R > 50 mm)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 mm.

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 mm)

(R > 50 mm)

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 N

The value of b 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