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

IF 4.4 2区 地球科学 Q1 METEOROLOGY & ATMOSPHERIC SCIENCES
Dion J. X. Ho, Robert Pincus
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引用次数: 0

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 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 × 1 0 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.

Abstract Image

再论双流:一般方程、精确系数和优化闭包
双流方程是最简洁的辐射通量传递一般模型,用一个方程分别模拟上行和下行通量;由于半球之间的通量传递,这些通量是耦合的。辐射通量传递方程的标准双流近似法假定,通量传递的比率(耦合系数)随光学深度的变化而不变,并且与上涌辐射和下沉辐射对称。双流闭合是通过对强度场的角分布进行额外假设而得出的,但目前没有一种方法能很好地适用于光学参数空间的所有部分。我们从短波辐射辐射传输方程的多流数值解中确定了双流耦合系数的精确值。由此得出的唯一系数可以精确地重建整个通量剖面,但取决于光学深度。更重要的是,在假定对称的情况下,这些系数通常具有非物理值。我们推导出了双流方程的一般形式,其中的四个耦合系数保证是物理可解释的。重建整个通量剖面需要非定常耦合系数,而数值优化的定常耦合系数(允许解析解)再现了短波反射率和透射率,在很大的光学参数范围内,相对误差不大于 4 × 1 0 - 5 $4\times 1{0}^{-5}$。优化后的系数与太阳天顶角和总光学深度有关,后者越大,系数越小。这就解释了为什么现有的耦合系数往往省略了前者而忽略了后者,往往只对薄大气层或厚大气层有效。
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来源期刊
Journal of Advances in Modeling Earth Systems
Journal of Advances in Modeling Earth Systems METEOROLOGY & ATMOSPHERIC SCIENCES-
CiteScore
11.40
自引率
11.80%
发文量
241
审稿时长
>12 weeks
期刊介绍: The Journal of Advances in Modeling Earth Systems (JAMES) is committed to advancing the science of Earth systems modeling by offering high-quality scientific research through online availability and open access licensing. JAMES invites authors and readers from the international Earth systems modeling community. Open access. Articles are available free of charge for everyone with Internet access to view and download. Formal peer review. Supplemental material, such as code samples, images, and visualizations, is published at no additional charge. No additional charge for color figures. Modest page charges to cover production costs. Articles published in high-quality full text PDF, HTML, and XML. Internal and external reference linking, DOI registration, and forward linking via CrossRef.
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