LA-UR-04-6228
Effective Transport Kernels for
Spatially Correlated Media,
Application to
Cloudy Atmospheres
Anthony B. Davis
Los Alamos National Laboratory
Space & Remote Sensing Sciences Group (ISR-2)
… with help from many others …
Key References
1.
Davis, A., and A. Marshak, Lévy kinetics in slab geometry: Scaling of transmission
probability, in Fractal Frontiers, M. M. Novak and T. G. Dewey (eds.), World Scientific,
Singapore, pp. 63-72 (1997).
2.
Pfeilsticker, K., First geometrical pathlength distribution measurements of skylight using
the oxygen A-band absorption technique - II, Derivation of the Lévy-index for skylight
transmitted by mid-latitude clouds, J. Geophys. Res., 104, 4101-4116 (1999).
3.
Buldyrev, S. V., S. Havlin, A. Ya. Kazakov, M. G. E. da Luz, E. P. Raposo, H. E. Stanley,
and G. M. Viswanathan, Average time spent by Lévy flights and walks on an interval with
absorbing boundaries, Phys. Rev. E, 64, 41108-41118 (2001).
4.
Kostinski, A. B., On the extinction of radiation by a homogeneous but spatially correlated
random medium, J. Opt. Soc. Am. A, 18, 1929-1933 (2001).
5.
Davis, A. B., and A. Marshak, Photon propagation in heterogeneous optical media with
spatial correlations: Enhanced mean-free-paths and wider-than-exponential free-path
distributions, J. Quant. Spectrosc. Rad. Transf., 84, 3-34 (2004).
6.
Davis, A. B., and H. W. Barker, Approximation methods in three-dimensional radiative
transfer, in Three-Dimensional Radiative Transfer for Cloudy Atmospheres, A. Marshak
and A. B. Davis (eds.), Springer-Verlag, Heidelberg (Germany), to appear (2004).
… and others, as we proceed …
Outline
• Motivation & Background
(atmospheric radiation science only)
• Mean-field transport kernels
–
–
–
–
Heuristic scattering-translation factorization
Directional diffusion: Transport MFP revisited
Spatial impact: Non-exponential tails
Implications for effective medium theories (homogenization)
• Anomalous photon diffusion:
The basic boundary-value problem
– Time-dependent (first, then …)
– Steady-state
• Observational corroborations
– Time-domain lightning observations
– Fine spectroscopy in oxygen absorption lines/bands
• Summary & Outlook
Motivation, 1:
Surrealism
• René Magritte, 1929
Motivation, 2:
State-of-the-Art Conceptual Models
• inside operational cloud remote sensing schemes
(chez NASA et Co.), and
• inside any Global Climate Model’s radiation module
This is a cloud.
Motivation, 3:
Reality!
• from Space Shuttle archive
(courtesy Bob Cahalan)
Approximation theory in atmospheric
radiative transfer: Needs assessment
• Variability: Resolved or not?
– in computational grid
– in observations (pixels)
Large-scale radiation budget estimation:
Unresolved variability effects
• Clear-cloudy separation (’70s - ’80s)
– The cloud fraction enters
– A correlation scale enters: Stochastic RT in Markovian binary media
– The Independent-Column Approximation (ICA) limit for very large
aspect ratios
• Cloudy part gets variable
–
–
–
–
Stephens’ closure-based effective medium theory (1988)
Davis et al.’s parameterization with power-law rescaling (1991)
Cahalan’s ICA-based effective medium theory (1994)
Barker’s Gamma-weighted/2-stream ICA (1996)
• More effective medium theories
– Cairns et al.’s renormalization theory (2000)
– Petty’s “cloudets” (2002): large clumps as scattering entities
• Recent numerical solutions for GCM consumption
– And what about cloud overlap (vertical correlation)?
– The McICA Project (2003-)
Some definitions in 3D Radiative Transfer
I(x,)  I(x, y,z,sin  cos ,sin  sin ,cos )
I  KI  I0
I
I  0  I0  KI0  K 2 I0  ... K n I0  ...
1 K

K    k( x , ;x,)
[] dx d
4
Medium

 1

x

x

 ) 

k( x , ;x,)
= exp[ (x ,x)]  s ( x ) p( x , 

x  x  x  x 2

translate
scatter
position-angle coupling
single - scattering albedo : Pr{scattering | x } =  0 ( x ) =  s (x ) / ( x )  1
  | x ) = 2 p( x ,cos s ) dcos s
phase function : dPr(cos  s = 
free - path PDF : dPr(s  x  x | x, x ) = exp[ (x ,x)]  ( x ) ds
1

 ( x ,x)   (x, x )  x  x   (x  (1  )x ) d
0


optical distance : 
s
 ( x , , s  x  x )   ( x ,x  x  s)
    ( x   ) d

0


Directional diffusion
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
phase function (uniform case)
: dPr(cos  s ) = 2 p(cos s ) dcos s
asymmetry factor : g = E[cos  s ] =
+1
 cos dPr(cos )
s
s
-1
Start with cos  0  1 ( 0  0)
E[cos1 ] = E[cos  s ] = g
cos n  cos n1 cos  s  sin  n1 sin  s cos  s, for n  1
E[cos n ] = E[cos  n1cos s ] = E[cos n1]E[cos  s ] = gE[cos  n1 ]
E[cos n ] = g n (by induction), an exponential decay in n
Directional diffusion: Spatial impact
phase function (uniform case)
: dPr(cos  s ) = 2 p(cos s ) dcos s
asymmetry factor : g = E[cos  s ] =
+1
 cos dPr(cos )
s
s
-1
free - path PDF (uniform case) : dPr(s) = exp[s]ds

Mean - Free Path (MFP) :
= E[s] =
 sdPr(s)
= 1/
0
Start with z0  0, also x 0  y 0  0, and  0  0
E[z1 ] = E[z0 + s0cos0 ] = 0 + E[s] 1 =
E[z2 ] = E[z1 + s1cos1 ] =  E[scos s ] =
E[z3 ] = E[z2 + s2cos 2 ] =
E[z ] =
g
n0
n

1 g
 E[s]E[cos  s ] =  g
 g  g 2 , etc.
= "transport" MFP
t
... without diffusion approximation!
After n* ≈ (1–g)–1 scatterings, directional memory is lost.
-6
n
(coarse-grained)
pdf
120
100
0.02
0
… x 112
= 78 ev ents
Prob{-20< Š20}
­ 0.70 —>
80
60
0.01
40
20
0
-180
x , n = 0,…,16
-12
140
-120
-60
0
60
scattering angle, 
6
120
0
180
histogram (7x16=112 events)
Directional diffusion
and its spatial impact
illustrated in 2D
(c)
Prob(d) = [(1g2)/(1+g22gcos)] (d/360)
for asymmetry factor
g = E(cos) = 5/6
0.03
12
n
0
0.0
0
1
2
…
2
0.5
g = 5/6
6
16
•
16
1.0
n
E(z ) =  g
i
n
0
1.5
•
16
12
n=0
1
2
mean-free-path (mfp)
= E(s) = 1
cos -1(gn)
gn = E(cos )
2.0
transport mfp
= /(1-g) = 6
n
t
15
2.5
Total path ( n >> 1) : Ln   si   n 
n
0
t
 [(1 g)n]
# isotropic scat's n/n 
t
9
(1g)z = z/
zn , n = 0,…,16
3
1
Effective (i.e., mean) transport kernels:
the actual photon free-path distributions
Pr{s  x  x | x, x }  exp[ (x, x )], or
Pr{s  X | x,}  exp[ (x,x  X)], hence
E[s | x,] 

 s  exp[ (x,x  s)] ds.
d
ds
0
Now let  (x,x  s)  (x,;s) 
random variable
s
parameter
1 s
where (x,;s)    (x   ) d .
s 0
We are interested in the analytical properties of

Pr{step  s}  exp[ (x,x  s)]  exp[s(x,;s)]
averaged over ( x,) and eventually all realizations of
the 3D " disorder," especially when (say) the statistical
moments of (x,;s) are only weakly dependent on s.
Need for long-range spatial correlations!
white noise (x)/ 
(a)
2.0
2.0
(b)
standard deviation 0.25
wn, s=1
wn, s=11
wn, s=101
1.5
1.5
1.0
1.0
0.5
0.5
40 bins from 0.0 to 2.0
0.0
0.0
0
200
400
600
800
1000
1200
0.0
10.0
position, x (arbitrary units)
Brownian motion (x)/ 
(a)
20.0
30.0
40.0
50.0
histogram (% )
2.0
2.0
(b)
standard deviation 0.25
Bm, s=1
Bm, s=11
Bm, s=101
1.5
1.5
1.0
1.0
0.5
0.5
15 bins from 0.3 to 1.7
0.0
0.0
0
200
400
600
800
position, x (arbitrary units)
1000
1200
0.0
5.0
10.0
15.0
histogram (% )
20.0
25.0
Synthetic scale-invariant media that are turbulence-like
Three remarkable properties of
effective free-path distributions
We consider
T(s) q  exp(qs)
and

d
T(s)  exp(s)
ds
[some characterisitic function theory
1. E[s]  1  
1
]
: variability increases MFP
d
2.  ln T(s) generally not  s (only when is degenerate, uniform medium)
ds
3. E[s ]  (q  1) 
q
q
: exponential PDF underestimates high  order moments
For 2.-3., using a very different approach, see:
Kostinski, A. B., 2001: On the extinction of radiation by a homogeneous but
spatially correlated random medium, J. Opt. Soc. Am. A, 18, 1929-1933.
Variability scales of 3D-transport interest?
Consider extinction (x) or “local” (pseudo-)MFP 1/(x).
How much does it typically change, on a relative scale, between
two discrete transport events (emission or injection, scattering,

absorption or escape)?
1/(1 g)
1 
1
Estimate     ln   
 



... and maybe  [], or take a  

 []dx over some scale

a.   1: "fast" variability,
only  matters (exponential kernels are OK)

b.   O(1) : "resonant" variability, expect 3D RT effects (non - exponential steps)

c.   1: "slow" variability,  apply 1D RT locally (then average as needed)
N.B. Extreme cases are well-known in stochastic RT theory for binary
Markovian media, respectively, the limits of:
a. “atomistic” mixing (i.e. optical homogeneity using mean values);
c. linear mixing by volume fraction (a.k.a. the ICA/IPA in atmospheric work).
An illustration with binary media:
Implications for effective medium theories:
* will all fail at large-enough scales;
* watch for correlations over the (actual) MFP.
Expectations for Earth’s cloudy atmosphere, 1:
Barker et al.’s (1996) LandSat Analysis
Gamma distributions capture many cloud optical depth scenarios.
From:
Barker, H . W., B. A. Wieli cki, and L. Parker, 1996: A parameterization for computing gridaveraged solar fluxes for inhomogeneous marine boundary layer clouds - Part 2, Validation using
satelli te data, J. Atmos. Sci., 53, 2304-2316.
Expectations for Earth’s cloudy atmosphere, 2:
Effective transport kernels are power-law
(a)
Gamma Probability Density Functions
with s = (s) = 1, a = 1/var[(s)]



1


p(;  , ) 
],
   1 exp[
( )   

dPa/d(s) from Eq. (62)
2.0
1/2
1
3/2
2
4
8
•
1.5
1.0
where  
0.5
0.0
0.0
0.5
1.0
1.5
2.0
(s) = s
(step s is constant)
2.5

Assuming s = H (thickness) in previous slide:
3.0
1
2
 1
2

, yields
and T(s)  exp[s] 
1

1
1  s  




1
 1 
1
 .
1 s ( 1)
Solar photons multiply scattering in the
cloudy atmosphere
Anomalous diffusion through a finite medium:
Time-dependence for transmission
s is drawn from the relevant step PDF
Start at x 0  0 and use x n 1  x n  s, where 
 is a Bernouilli coin flip ( g = 0 in 1D)
x n   si , a 1D random walk (stationary/independent increments)
n
1
var[ s]   : x n
2
var[ s]   : x n

~
~
2
t
n, a (standard) diffusion process

t
n, "anomalous" diffusion
with  = min {q : E[sq ] = }, the Lévy index
q
N.B. We require here that  > 1 so that
t
= E[s]  .
… from free space to a finite slab (thickness H):
xn ~

tn
hence pathlength
n
T
L
~ H /
T

t

t
 , where H /
n
T
~ H  H /
 (1 g)
t
 1
t

Anomalous diffusion through a finite medium:
Steady-state transmission
A lesser - known result for (  = 2) diffusion in a semi - infinite medium :
Pr{" return time"  n} ~ 1/ n
Frisch and Frisch (1995) generalize to any PDF for
s, hence any  .
Frisch, U., and H. Frisch, 1995: Universality in escape from half space of symm etrical random
walks, in LŽ
v y Flights and Related Topics in Physics, Eds. M. F. Shlesinger, G. M. Zaslavsky,
and U. Frisch, Springer-Verlag, New York (NY), pp. 262-268. 0.0
… from a half-space
to a finite slab (thickness H):
T(H)
-0.5
Pr{return time  n
where H /
T

t
log
~ H /
 } ~ H / t 
-2.0
 (1 g) , hence T( ) ~ [(1 g) ] 2 .
For a more rigorous approach:
-1.5
 2

t
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
-1.0
10
Transmission (probability) is

-2.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
log H
10
Buldyrev, S. V., S. Havlin, A. Ya. Kazakov, M. G. E. da Luz, E. P. Raposo, H. E. Stanley, and
G. M. Viswanathan, 2001: Average time spent by LŽvy fli ghts and walks on an interval with
absorbing boundaries, Phys. Rev. E, 64, 41108-41118.
2.5
Observations, 1a:
Differential absorption
spectroscopy at very
high resolution
x-section
density
pathlength
From: Min Q.-L., L. C. Harrison, P. Kiedron, J. Berndt,
and E. Joseph, 2004: A high-resolution oxygen A-band
and water vapor band spectrometer, J. Geophys. Res.,
109, D02202, doi:10.1029/2003JD003540.
I( ) I0  exp[   L]
?   estimating molecular cross - sections in the laboratory


known/not :  ?  monitoring amounts of chemical effluent in situ


  ? scattering/reflection diagnostics of media permeated with gas

d 
I( )  I(k )  I0  exp[k L] p(L) dL (equivalence " theorem" )  L   ln I(k )
dk 
0
Observations, 1b:
Ground-based
Oxygen Spectroscopy
Cases near the =2 line are very
overcast, and those near =1 are for
sparse clouds, as expected from model.

<LT'>/< transp>

1000





100
10
1
10
Pfeilsticker, K., 1999: First Geometrical Pathlength Distribution Measurements of Skylight
Using the Oxygen A-band Absorption Technique - II, Derivation of the LŽvy-Index for Skyli ght
Transmitted by Mid-Latitude Clouds, J. Geophys. Res., 104, 4101-4116.
A single cloud layer (=2) with
variable thickness H  the slope of
the linear path vs optical depth plot.
A complex cloud situation (1<<2)
with multi-layers, some broken;
power-laws in  will fit the data.
Min, Q.-L., L. C. Harrison, and E. E. Clothiaux, 2001: Joint statistics of photon path length and
cloud optical depth: Case studies, J. Geophys. Res., 106, 7375-7385.
100
H'/< transp>
Observations, 2a: FORTÉ data
Dtphys = ?
Source
VHF
Optical
ws
Dtprop = distance / c
FORTÉ
Dtphys
due to
scattering
in clouds
Dtscat
wf
Suszc ynsky, D. M., M. W. Kirkland, A. R. Jacobson, R. C. Franz, S. O. Knox, J. L. L. Guill en,
and J. L. Green, 2000: FORTƒ Observations of Simultaneous VHF and Optical Emissions from
Lightning: Basic Phenomenology, J. Geophys. Res., 105, 2191-2201.
Observations, 2b:
Lévy analysis for FORTÉ
From:
Davis, A. B., D. M. Suszc ynski, and A. Marshak, 2000: Shortwave Transport in the Cloudy
Atmosphere by Anomalous/LŽvy Photon Diffusion: New Diagnostics using FORTƒ Lightning
Data, in Proceedings of 10th Atmospheric Radiation Measurement (ARM) Science Team
Meeting, 03/13-17, 2000, San Antonio (Tx), U.S. Dept. of Energy, on-line at
http://www.ar m.gov/docs/documents/technical/conf_0003/davis-ab.pdf.
Summary & Outlook
• Diverse modeling approaches to unresolved variability
–
–
–
–
Analytical (effective medium parameters in 2-stream theory)
Semi-analytical (gamma-weighted/2-stream ICA)
New transport theories (stochastic RT, anomalous photon diffusion)
Numerical solutions for GCM consumption (McICA project)
• Effective transport kernels
– Actual MFPs longer than expected from mean extinction
– Never exponential except for uniform media
– Always sub-exponential
spatial correlations sustained over the scale of the MFP)
(if
• Power-law tails in the effective transport kernel
– Anomalous photon diffusion (APD) theory
– Supporting observational evidence
• Reconcile climate-scale computations and observations
– US DOE Atmospheric Radiation Measurement (ARM) program, etc.
– Need realistic yet tractable models, such as APD, to interpret data
– Get the cloud physics/dynamics right!
La Grande Famille
• René Magritte, 1963