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

1. 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 …

2. 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 …

3. 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

4. Motivation, 1: Surrealism • René Magritte, 1929

5. This is a cloud. 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

6. Motivation, 3: Reality! • from Space Shuttle archive (courtesy Bob Cahalan)

7. Approximation theory in atmospheric radiative transfer: Needs assessment • Variability: Resolved or not? • in computational grid • in observations (pixels)

8. 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-)

9. Some definitions in 3D Radiative Transfer

10. Directional diffusion

11. Directional diffusion: Spatial impact Review New After n* ≈ (1–g)–1 scatterings, directional memory is lost.

12. Directional diffusion and its spatial impact illustrated in 2D

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

14. Need for long-range spatial correlations! Counter- Example Example

15. Synthetic scale-invariant media that are turbulence-like

16. 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.

17. 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).

18. 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.

19. Expectations for Earth’s cloudy atmosphere, 1: Barker et al.’s (1996) LandSat Analysis Gamma distributions capture many cloud optical depth scenarios. From:

20. Expectations for Earth’s cloudy atmosphere, 2: Effective transport kernels are power-law Assuming s = H (thickness) in previous slide:

21. Solar photons multiply scattering in the cloudy atmosphere

22. Anomalous diffusion through a finite medium: Time-dependence for transmission … from free space to a finite slab (thickness H):

23. Anomalous diffusion through a finite medium: Steady-state transmission … from a half-space to a finite slab (thickness H): For a more rigorous approach:

24. 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, doi:10.1029/2003JD003540. x-section density pathlength

25. 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 (1<<2) with multi-layers, some broken; power-laws in  will fit the data.

26. Dtphys = ? VHF Optical Source ws Dtprop = distance / c due to scattering in clouds FORTÉ Dtphys Dtscat wf Observations, 2a: FORTÉ data

27. Observations, 2b: Lévy analysis for FORTÉ From:

28. 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!

29. La Grande Famille • René Magritte, 1963