The Spherical Harmonic Discrete Ordinate Method: Application to 3D Radiative Transfer in Boundary Layer Clouds

K. Franklin Evans, Program in Atmospheric and Oceanic Sciences, University of Colorado, Boulder

Overview

• The Spherical Harmonic Discrete Ordinate Method (SHDOM) is a highly efficient and flexible 3D atmospheric radiative transfer model. The Fortran code, documentation, and a submitted journal article are available.
• SHDOM can compute unpolarized, monochromatic and broadband (with a k-distribution), shortwave and longwave radiative transfer. The medium properties (extinction, phase function, etc.) are specified at each grid point, and the surface albedo may vary as well. The delta-M and untruncated single-scattering methods give accurate fluxes and radiances even with highly peaked phase functions.
• The model capabilities and some useful results are illustrated by computing 3D radiative transfer in a marine boundary layer cloud produced by an LES model simulation of a FIRE stratocumulus case. Broadband shortwave and remote sensing calculations are analyzed.
• The results indicate that the solar beam interacting with the cloud geometry has important effects on fluxes and radiances, which can only be captured by a 3D model.

SHDOM Algorithm

• The source function (not radiance field) is represented by a spherical harmonic series in angle and a discrete grid in space. Discrete ordinates are used during the solution method.
• Adaptive spherical harmonics: only the number of terms needed at each grid point are stored, saving memory. Spherical harmonics result in a faster computation of the scattering integral.
• Adaptive grid: additional points are added where the source function changes most rapidly (see Fig. 1). This results in higher accuracy for a given number of grid points, especially for highly variable media.

Solution method (iterates 1 to 4):

1. Transform source function from spherical harmonics to discrete ordinates.
2. Integrate RTE source function back along discrete ordinates to get radiance.
3. Transform radiance field back to spherical harmonics.
4. Compute source function from radiance.

Validation of SHDOM

• Use doubling-adding in 1D and backward Monte Carlo in 2D and 3D.
• Comparison for a 3D gaussian extinction field (Fig. 2). Peak optical depth of 2; Mie phase functions for effective radius 10 micron at wavelengths of 1.65 micron and 10.7 micron.
• SHDOM accuracy improves with increasing angular and spatial resolution (see table).

LES Model Cloud

• Large Eddy Simulation of FIRE stratocumulus case (from Moeng at NCAR); cloud layer 64x64x17 points (spacing 55 m x 55 m x 25 m).
• Effective radius derived from LWC (N=50 droplets/cm^3) and optical properties computed with Mie theory (Fig. 3) and (Fig. 4).

Broadband Modeling

• Sun angle: theta0=60 and theta0=15.
• Surface albedo: 0.06
• Base grid 65 x 65 x 26 = 110000 points, 340000 max total grid points.
• Nmu=8 X Nphi=16 discrete ordinates.
• Fu and Liou correlated k-distribution -- 6 bands, 54 k's.
• Simulate 3D, 2D independent scan (IS), and 1D independent pixel (IP) approximations.
  CPU: 30 hrs (3D), 11 hrs (IS), 9 hrs (IP) (SGI Challenge)

Remote Sensing Modeling

• Wavelengths: 0.83 and 1.65 micron; no gas absorption.
• Nmu=12 X Nphi=24 discrete ordinates.
• Radiance output: 64 X 64 points X 21 angles.
• 3D, independent scan (IS), and independent pixel (IP).
  CPU: 3.5 hrs, 1.0 hrs, 0.5 hr

Results

• For this 3D LES stratocumulus cloud the independent pixel approximation (IPA) works well for domain average broadband heating rate and fluxes (Fig. 5), but there are significant differences in the pattern of heating (Fig. 6).
• The 2D independent scan approximation (ISA) gives much better results for heating than the IPA (Fig. 7). The ISA allows solar direct beam and horizontal transfer effects, which can give downwelling fluxes much greater the incident solar flux (Fig. 8).
• In broken and thinner overcast clouds direct beam effects causes the downwelling flux to have much more variability than the smooth upwelling flux (Fig. 9). This has important implications for sampling errors in aircraft and ground based measurements of ``anomalous'' solar absorption in clouds (Fig. 10 and Fig. 11).
• There are substantial differences in nadir radiance between 3D and IPA simulations. For low sun the peak 3D reflectances are higher than IPA, while for high sun they are lower (Fig. 12 and Fig. 13). Reflectance histograms (Fig. 14) imply that the 3D effects can cause significant errors in remote sensing the distribution of optical depth (important for domain average properties using IPA).
• It is well known that the independent pixel approximation gives large errors in point-by-point radiance. Recently, Marshak et al. (1995) suggested that a smoothing procedure could be used on IPA radiances to simulate 3D results. Unfortunately, this was based on the ``uniform pixel approximation'' and is not correct for realistic vertically inhomogeneous clouds (Fig. 15, Fig. 16, and Fig. 17).

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