By extending the previously proposed heuristic parameterization, the author derived an analytical equation computing the overlap areas between the precipitation (or radiation) areas and the cloud areas in a cloud system consisting of cumulus and stratus. The new analytical equation is accurate and much more efficient than the previous heuristic equation, which suffers from the truncation error in association with the digitalization of the overlap areas. Global test simulations with the new analytical formula in an offline mode showed that the maximum cumulus overlap simulates more surface precipitation flux than the random cumulus overlap. On the other hand, the maximum stratus overlap simulates less surface precipitation flux than random stratus overlap, which is due to the increase in the evaporation rate of convective precipitation from the random to maximum stratus overlap. The independent precipitation approximation (IPA) marginally decreases the surface precipitation flux, implying that IPA works well with other parameterizations. In contrast to the net production rate of precipitation and surface precipitation flux that increase when the cumulus and stratus are maximally and randomly overlapped, respectively, the global mean net radiative cooling and longwave cloud radiative forcing (LWCF) increase when the cumulus and stratus are randomly overlapped. On the global average, the vertical cloud overlap exerts larger impacts on the precipitation flux than on the radiation flux. The radiation scheme taking the subgrid variability of water vapor between the cloud and clear portions into account substantially increases the global mean LWCF in tropical deep convection and midlatitude storm track regions.

中文翻译：

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积云和地层的垂直叠置的经济解析方程

通过扩展先前提出的启发式参数化方法，作者得出了一个解析方程，用于计算由积云和地层组成的云系统中的降水（或辐射）区域与云区域之间的重叠区域。与先前的启发式方程相比，新的解析方程式更为精确且效率更高，后者具有与重叠区域数字化相关的截断误差。在离线模式下使用新的分析公式进行的全局测试模拟表明，最大的积云重叠比随机的积云重叠模拟了更多的表面降水通量。另一方面，最大的地层重叠模拟的地表降水通量比随机的地层重叠要少，这是由于对流降水的蒸发速率从无规到最大地层重叠增加。独立的降水量近似值（IPA）略微降低了表面降水通量，这意味着IPA与其他参数设置效果很好。与当积云和地层最大和随机重叠时分别增加的净降水量和地表降水通量相反，当积云和地层随机时，全球平均净辐射冷却和长波云辐射强迫（LWCF）增加重叠。在全球平均水平上，垂直云重叠对降水通量的影响大于对辐射通量的影响。