Effective Transport Kernels for Spatially Correlated Media,

Application to Cloudy Atmospheres

Anthony B. DavisLos Alamos National Laboratory

Space & Remote Sensing Sciences Group (ISR-2)

… with help from many others …

LA-UR-04-6228

Key References1. 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

• René Magritte, 1929

Motivation, 1: Surrealism

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,Ω) = I(x,y,z,sinθ cosφ,sinθ sinφ,cosθ)

I = KI + I0

I =I0

1− K= I0 + KI0 + K 2I0 + ...+ K nI0 + ...

K =4 π

∫∫Medium

∫∫∫ k( ′ x , ′ Ω ;x,Ω) [⋅] d ′ x d ′ Ω

k( ′ x , ′ Ω ;x,Ω) = exp[−τ ( ′ x ,x)]translate

1 2 4 4 3 4 4 σ s( ′ x ) p( ′ x , ′ Ω • Ω)scatter

1 2 4 4 4 3 4 4 4 δ ′ Ω − ′ x − x′ x − x

⎛

⎝ ⎜

⎞

⎠ ⎟

1

′ x − x2

position -angle coupling1 2 4 4 4 4 3 4 4 4 4

single - scattering albedo : Pr{scattering | ′ x } = ϖ0( ′ x ) = σ s( ′ x ) /σ ( ′ x ) ≤1

phase function : dPr(cosϑ s = ′ Ω • Ω | ′ x ) = 2π p( ′ x ,cosϑ s) dcosϑ s

free - path PDF : dPr(s = ′ x − x | x, ′ x ) = exp[−τ ( ′ x ,x)] σ ( ′ x ) ds

optical distance :

τ ( ′ x ,x) = τ (x, ′ x ) = ′ x − x σ (ξx + (1−ξ ) ′ x )0

1

∫ dξ

τ ( ′ x , ′ Ω ,s = ′ x − x ) = τ ( ′ x ,x = ′ x + ′ Ω s) = σ ( ′ x + ′ Ω δ)0

s

∫ dδ

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

Directional diffusion

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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phase function (uniform case) : dPr(cosϑ s) = 2π p(cosϑ s) dcosϑ s

asymmetry factor : g = E[cosϑ s] = cosϑ s-1

+1

∫ dPr(cosϑ s)

Start with cosθ0 =1 (θ0 = 0)

E[cosθ1] = E[cosϑ s] = g

cosθn = cosθn−1 cosϑ s − sinθn−1 sinϑ s cosϕ s, for n ≥1

E[cosθn ] = E[cosθn−1cosϑ s] = E[cosθn−1]E[cosϑ s] = gE[cosθn−1]

E[cosθn ] = gn (by induction), an exponential decay in n

Directional diffusion: Spatial impact

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phase function (uniform case) : dPr(cosϑ s) = 2π p(cosϑ s) dcosϑ s

asymmetry factor : g = E[cosϑ s] = cosϑ s-1

+1

∫ dPr(cosϑ s)

free - path PDF (uniform case) : dPr(s) = exp[−σs]σds

Mean - Free Path (MFP) : l = E[s] = s0

∞

∫ dPr(s) = 1/σ

Start with z0 = 0, also x0 = y0 = 0, and θ0 = 0

E[z1] = E[z0 + s0cosθ0] = 0 +E[s] ×1 = l

E[z2] = E[z1 + s1cosθ1] = l + E[scosϑ s] = l + E[s]E[cosϑ s] = l + l g

E[z3] = E[z2 + s2cosθ2] = l + l g + l g2, etc.

E[z∞] = l gn

n≥0

∑ =l

1− g = "transport" MFP l t ... without diffusion approximation!

After n* ≈ (1–g)–1 scatterings, directional memory is lost.

Directional diffusion and its spatial impact illustrated in 2D

-12 -6 0 6 12

0

3

6

9

12

15

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xn, n = 0,…,16

n0

1

1616

2

12

∞16

01

2

n =

E(zn) = Σ gi

n

0

cos-1(gn)gn = E(cosθ

n)

∞

(c)

mean-free-path (mfp)l = E(s) = 1

transport mfplt = l/(1-g) = 6

g = 5/6

0

0.01

0.02

0.03

0

20

40

60

80

100

120

140

-180 -120 -60 0 60 120 180scattering angle, θ

Prob(dθ) = [(1−g2)/(1+g2−2gcosθ)] (dθ/360)for asymmetry factor g = E(cosθ) = 5/6

Prob{-20< θ≤20}≈ 0.70 —>

… x 112= 78 events

(b) Scattering Phase Function

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Total path ( n >> 1) : Ln = si0

n

∑ ≈ l × n = l t × [(1− g)n]# isotropic scat' s n/n∗

1 2 4 3 4

Effective (i.e., mean) transport kernels: the actual photon free-path distributions

€

Pr{s ≥ ′ x − x | x, ′ x } = exp[−τ (x, ′ x )], or

Pr{s ≥ X | x,Ω} = exp[−τ (x,x + ΩX)], hence

E[s | x,Ω] = s d ds( )exp[−τ (x,x + Ωs)]

0

∞

∫ ds.

€

We are interested in the analytical properties of

Pr{step ≥ s} = exp[−τ (x,x + Ωs)] = exp[−sσ (x,Ω;s)]

averaged over (x,Ω) and eventually all realizations of

the 3D "disorder," especially when (say) the statistical

moments of σ (x,Ω;s) are only weakly dependent on s.

€

Now let τ (x,x + Ωs) = σ (x,Ω;s)random variable1 2 4 3 4 × s

parameter{

where σ (x,Ω;s) =1

sσ (x + Ωδ) dδ.

0

s

∫

Need for long-range spatial correlations!

0.0

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position, x (arbitrary units)

standard deviation 0.25(a)

0.0 10.0 20.0 30.0 40.0 50.0

wn, s=1wn, s=11wn, s=101

0.0

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histogram (%)

40 bins from 0.0 to 2.0

(b)

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0 200 400 600 800 1000 1200

position, x (arbitrary units)

standard deviation 0.25(a)

0.0 5.0 10.0 15.0 20.0 25.0

Bm, s=1Bm, s=11Bm, s=101

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histogram (%)

15 bins from 0.3 to 1.7

(b)

Synthetic scale-invariant media that are turbulence-like

Three remarkable properties of effective free-path distributions

var[σ(x)] = 0 exponential FPD ( with constantσ = τ(s)/s)

[varσ(x)] > 0 - non exponential FPD ( with the same⟨σ⟩ = ⟨τ(s)⟩/s )as in the above case

001234q −ln⟨T(s)q⟩ for a given step-size s

τ⟨ (s)⟩−ln⟨T (s)⟩

€

We consider

T(s)q = exp(−qσ s)

and

−d

dsT(s) = σ exp(−σ s)

L [some characterisitic function theory] L

1. E[s] = σ −1 ≥ σ −1

: variability increases MFP

2. −d

dsln T(s) generally not ∝ s (only when σ is degenerate, uniform medium)

3. E[sq ] ≥ Γ(q +1) σ −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?

€

Estimate ζ =1

σ

⎛

⎝ ⎜

⎞

⎠ ⎟× ∇ lnσ = ∇

1

σ

Consider extinction σ(x) or “local” (pseudo-)MFP 1/σ(x).

How much does it typically change, on a relative scale, between two discrete transport events (emission or injection, scattering, absorption or escape)?

€

a. ζ >>1: "fast" variability, only σ matters (exponential kernels are OK)b. ζ = O(1) : "resonant" variability, expect 3D RT effects (non - exponential steps)c. ζ <<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).

€

... and maybe

×1/(1− g)

Ω • [⋅], or take a ∇⊥

[⋅]dx over some scale∫

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

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

Barker, H. W., B. A. Wielicki, and L. Parker, 1996: A parameterization for computing grid-averaged solar fluxes for inhomogeneous marine boundary layer clouds - Part 2, Validation using satellite data, J. Atmos. Sci., 53, 2304-2316.

From:

Gamma distributions capture many cloud optical depth scenarios.

Expectations for Earth’s cloudy atmosphere, 2: Effective transport kernels are power-law

10-5

10-4

10-3

10-2

10-1

100

0.1 1.0 10.0

Ensemble-averaged Free-Path Distributions (FPDs) for Gamma-distributed optical distances (fixed ⟨σ⟩)

1/213/2248∞

⟨σ⟩s

( )a

0.0

0.2

0.4

0.6

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- - ( ) Ensemble averaged Free Path Distributions FPDs - ( for Gamma distributed optical distances fixed⟨l⟩)

3/2248∞

s/⟨l ⟩

(b)

0.0

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Gamma Probability Density Functionswith ⟨σ⟩s = ⟨τ(s)⟩ = 1, a = 1/ [varτ(s)]

1/213/2248∞

τ(s) = σs( steps )is constant

( )a

€

p(σ ; σ ,ν ) =1

Γ(ν )

ν

σ

⎛

⎝ ⎜

⎞

⎠ ⎟

ν

σ ν −1 exp[−νσ

σ ],

where ν =1

σ 2 σ 2

−1, yields l =

1

σ =

ν

ν −1

1

σ

and T(s) = exp[−σ s] =1

1+ σ s ν( )ν =

1

1+ s l (ν −1)[ ]ν .

Assuming s = H (thickness) in previous slide:

Solar photons multiply scattering in the cloudy atmosphere

Anomalous diffusion through a finite medium: Time-dependence for transmission

€

Start at x0 = 0 and use xn +1 = xn ± s, where s is drawn from the relevant step PDF

± is a Bernouilli coin flip (g = 0 in 1D)

⎧ ⎨ ⎩

xn = ±si, a 1D random walk1

n

∑ (stationary/independent increments)

var[s] < ∞ : xn2 ~ l t

2n, a (standard) diffusion process

var[s] = ∞ : xnα ~ l t

α n, "anomalous" diffusion

with α = minq

{q : E[sq ] = ∞}, the Lévy index

N.B. We require here that α > 1 so that l t = E[s] < ∞.

€

xnα ~ l t

α n a nT

~ H /l t( )α, where H /l t = (1− g)τ

hence pathlength LT

≈ l t nT

~ H × H /l t( )α −1

… from free space to a finite slab (thickness H):

Anomalous diffusion through a finite medium: Steady-state transmission

€

A lesser - known result for (α = 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 α .

€

Transmission (probability) is

Pr{return time ≥ nT

~ H /l t( )α} ~ H /l t( )

−α2

where H /l t = (1− g)τ , hence T(τ ) ~ [(1− g)τ ]−α2 .

… from a half-space to a finite slab (thickness H):

Frisch, U., and H. Frisch, 1995: Universality in escape from half space of symmetrical random walks, in Lévy Flights and Related Topics in Physics, Eds. M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, Springer-Verlag, New York (NY), pp. 262-268.

Buldyrev, S. V., S. Havlin, A. Ya. Kazakov, M. G. E. da Luz, E. P. Raposo, H. E. Stanley, and G. M. Viswanathan, 2001: Av erage time spent by Lévy flights and walks on an interval with absorbing boundaries, Phys. Rev. E, 64, 41108-41118.

For a more rigorous approach:

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2.001.751.501.251.000.750.500.25

log10

H

α

Observations, 1a: Differential absorption spectroscopy at veryhigh resolutionFrom: 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(λ ) I0 = exp[− κ ρ L]

known/not :

? √ √ estimating molecular cross - sections in the laboratory

√ ? √ monitoring amounts of chemical effluent in situ

√ √ ? scattering/reflection diagnostics of media permeated with gas

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

x-se

ctio

nd

ensi

typ

ath

len

gth

€

I(λ ) = I(kλ ) = I0 exp[−kλ L] p(L)0

∞

∫ dL (equivalence "theorem")⇒ L = −d

dkλ

⎛

⎝ ⎜

⎞

⎠ ⎟ln I(kλ )

Observations, 1b: Ground-basedOxygen Spectroscopy

Pfeilsticker, K., 1999: 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.

10 100

1000

101

α2.01.81.61.41.21.0

H'/<ltransp>

100

<LT'>/<ltransp>

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.

Cases near the α=2 line are very overcast, and those near α=1 are for sparse clouds, as expected from model.

A single cloud layer (α=2) with variable thickness H the slope of the linear path vs optical depth plot.

A complex cloud situation (1<α<2) with multi-layers, some broken; power-laws in α1 will fit the data.

Source

FORTÉ

VHF Optical

wf

ws

tphys = ?

tprop = distance / c

tscattphys

due to scattering in clouds

Suszcynsky, D. M., M. W. Kirkland, A. R. Jacobson, R. C. Franz, S. O. Knox, J. L. L. Guillen, and J. L. Green, 2000: FORTÉ Observations of Simultaneous VHF and Optical Emissions from Lightning: Basic Phenomenology, J. Geophys. Res., 105, 2191-2201.

Observations, 2a: FORTÉ data

Observations, 2b: Lévy analysis for FORTÉ

Davis, A. B., D. M. Suszcynski, and A. Marshak, 2000: Shortwave Transport in the Cloudy Atmosphere by Anomalous/Lévy Photon Diffusion: New Diagnostics using FORTÉ 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.arm.gov/docs/documents/technical/conf_0003/davis-ab.pdf.

From:

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

(if spatial correlations sustained over the scale of the MFP)

• 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