Two-Streams Revisited: General Equations, Exact Coefficients, and
Optimized Closures
Abstract
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. 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
reflectance and transmittance with relative errors no greater than $4
\times 10^{-5}$ over a large range of optical
parameters. The optimized coefficients show a dependence on the 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.