Two‐Streams Revisited: General Equations, Exact Coefficients, and Optimized Closures

• Published in: Journal of advances in modeling earth systems Vol. 16; no. 10
• Main Authors: Ho, Dion J. X., Pincus, Robert
• Format: Journal Article
• Language: English
• Published: Washington John Wiley & Sons, Inc 01-10-2024; American Geophysical Union (AGU)
• Subjects: Angular distribution; Approximation; atmospheric fluxes; Boundary conditions; Coefficients; Depth; Downwelling; Fluctuations; Ocean circulation; Optical analysis; optimization; Ordinary differential equations; plane‐parallel atmosphere; Radiation; Radiative transfer; Reflectance; Rivers; Short wave radiation; two‐stream closures; two‐stream equations; Upwelling
• ISSN: 1942-2466; 1942-2466
• DOI: 10.1029/2024MS004504

Two‐Stream Equations are the most parsimonious general models for radiative flux transfer with one equation to model each of upward and downward fluxes; these are coupled due to the transfer of fluxes between hemispheres. Standard two‐stream approximation of the Radiative Transfer Equation assumes that the ratios of flux transferred (coupling coefficients) are both invariant with optical depth and symmetric with respect to upwelling and downwelling radiation. Two‐stream closures are derived by making additional assumptions about the angular distribution of the intensity field, but none currently works well for all parts of the optical parameter space. We determine the exact values of the two‐stream coupling coefficients from multi‐stream numerical solutions to the Radiative Transfer Equation for shortwave radiation. The resulting unique coefficients accurately reconstruct entire flux profiles but depend on optical depth. More importantly, they generally take on unphysical values when symmetry is assumed. We derive a general form of the Two‐Stream Equations for which the four coupling coefficients are guaranteed to be physically explicable. While non‐constant coupling coefficients are required to reconstruct entire flux profiles, numerically optimized constant coupling coefficients (which admit analytic solutions) reproduced shortwave reflectance and transmittance with relative errors no greater than 4×10−5 $4\times 1{0}^{-5}$ over a large range of optical parameters. The optimized coefficients show a dependence on solar zenith angle and total optical depth that diminishes as the latter increases. This explains why existing coupling coefficients, which often omit the former and mostly neglect the latter, tend to work well for only thin or only thick atmospheres.
Plain Language Summary
Two‐Stream Equations are the most parsimonious general models for the propagation of fluxes through a medium and are almost universally used to compute reflectance and transmittance—the ratio of radiation reflected or transmitted by a medium—in climate models. Their accuracy depends on how their coefficients are specified, that is, the two‐stream closure, and no existing closure works well in all situations. We diagnose what these coefficients ought to be from high resolution radiative transfer models for sunlight and in the process formulated a general form of the Two‐Stream Equations that better lends itself to physical interpretation. Although coefficients that vary with the vertical coordinate are required to reconstruct entire flux profiles, numerically optimized constant coefficients reproduced reflectance and transmittance of sunlight to near‐perfect accuracy. The optimized coefficients provide important insights into closure parameterizations and explain why existing two‐stream closures tend to work well for only thin or only thick atmospheres.
Key Points
A general form of the Two‐Stream Equations better lends itself to physical interpretation than the standard form
Numerically optimized coupling coefficients reproduce shortwave reflectance and transmittance accurate to within 10−4 $1{0}^{-4}$
These optimized coefficients explain the accuracy regimes of existing two‐stream closures