Effective Transport Kernels for Spatially Correlated Media, Application to Cloudy Atmospheres

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Effective Transport Kernels for Spatially
Correlated Media,Application to Cloudy
Atmospheres
LA-UR-04-6228

• Anthony B. Davis
• Los Alamos National Laboratory
• Space Remote Sensing Sciences Group (ISR-2)
• with help from many others

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Key References

• 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).
• 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).
• 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).
• 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).
• 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).
• 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
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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

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Motivation, 1 Surrealism

• René Magritte, 1929

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Motivation, 2 State-of-the-Art Conceptual Models

• inside operational cloud remote sensing schemes
  (chez NASA et Co.), and
• inside any Global Climate Models radiation module

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Motivation, 3 Reality!

• from Space Shuttle archive (courtesy Bob Cahalan)

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Approximation theory in atmospheric radiative
transfer Needs assessment

• Variability Resolved or not?
• in computational grid
• in observations (pixels)

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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)
• Cahalans ICA-based effective medium theory
  (1994)
• Barkers Gamma-weighted/2-stream ICA (1996)
• More effective medium theories
• Cairns et al.s renormalization theory (2000)
• Pettys cloudets (2002) large clumps as
  scattering entities
• Recent numerical solutions for GCM consumption
• And what about cloud overlap (vertical
  correlation)?
• The McICA Project (2003-)

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Some definitions in 3D Radiative Transfer
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Directional diffusion
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Directional diffusion Spatial impact
Review
New
After n (1g)1 scatterings, directional
memory is lost.
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Directional diffusion and its spatial impact
illustrated in 2D
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Effective (i.e., mean) transport kernels the
actual photon free-path distributions
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Need for long-range spatial correlations!
Counter- Example
Example
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Synthetic scale-invariant media that are
turbulence-like
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Three remarkable properties of effective
free-path distributions
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.
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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)?
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).
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An illustration with binary media
Exonentials don't fit PDF
The actual PDF is sub-exponential
Mean optical density underestimates MFP
Implications for effective medium theories
will all fail at large-enough scales watch for
correlations over the (actual) MFP.
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Expectations for Earths cloudy atmosphere, 1
Barker et al.s (1996) LandSat Analysis
Gamma distributions capture many cloud optical
depth scenarios.
From
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Expectations for Earths cloudy atmosphere, 2
Effective transport kernels are power-law
Assuming s H (thickness) in previous slide
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Solar photons multiply scattering in the cloudy
atmosphere
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Anomalous diffusion through a finite medium
Time-dependence for transmission
from free space to a finite slab (thickness H)
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Anomalous diffusion through a finite medium
Steady-state transmission
from a half-space to a finite slab (thickness
H)
For a more rigorous approach
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Observations, 1a Differential absorption
spectroscopy at veryhigh resolution
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, doi10.1029/2003JD0
03540.
x-section density pathlength
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Observations, 1b Ground-basedOxygen
Spectroscopy
Cases near the ?2 line are very overcast, and
those near ?1 are for sparse clouds, as expected
from model.
A single cloud layer (?2) with variable
thickness H ? the slope of the linear path vs
optical depth plot.
A complex cloud situation (1lt?lt2) with
multi-layers, some broken power-laws in ??? will
fit the data.
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Observations, 2a FORTÉ data
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Observations, 2b Lévy analysis for FORTÉ
From
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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!

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La Grande Famille

• René Magritte, 1963