Atmospheric water has a complex behaviour partly due to the influence of precipitation. Consequently, it is challenging to explain properties of water such as the scale-dependence of its variance, for which a range of spectral exponents has been identified in observational data. Here, a precipitating quasi-geostrophic (PQG) model is explored as a possible prototype for contributing to understanding of water spectra, in an idealised setting. Geostrophic turbulence is examined in numerical simulations, where precipitation is included to explore its effect on the water spectrum, but where phase changes are neglected to allow corresponding theoretical analysis. The water spectral exponent is seen to range from approximately −1.4 to approximately −5 depending on the rainfall speed parameter, , which indicates a significant influence of precipitation on the water spectrum. The limiting values of this range are explained through asymptotic analyses for large and small values of . To obtain this theoretical understanding of the model, a key observation is that water can be written as a linear combination of two other tracers (equivalent potential temperature and a moist variable M), which themselves have theoretically tractable spectra. These two other tracers are linked to distinct modes of the PQG equations–the vortical mode and a moist mode – and the analysis here highlights the usefulness of wave or mode decompositions for understanding water in a saturated domain.

中文翻译：

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大气水在准地转湍流降水中的光谱

大气水具有复杂的行为，部分原因是降水的影响。因此，解释水的特性（例如其方差的尺度依赖性）具有挑战性，在观测数据中已经确定了一系列光谱指数。在这里，在理想化的环境中，将沉淀准地转 (PQG) 模型作为有助于理解水光谱的可能原型进行了探索。在数值模拟中检查地转湍流，其中包括降水以探索其对水谱的影响，但忽略相变以进行相应的理论分析。水谱指数的范围从大约 -1.4 到大约 -5 取决于降雨速度参数，这表明降水对水谱有显着影响。该范围的极限值通过对 的大值和小值的渐近分析来解释。为了获得对模型的这种理论理解，一个关键的观察结果是水可以写成两个其他示踪剂（等效潜在温度和潮湿变量 M）的线性组合，它们本身具有理论上易于处理的光谱。这另外两个示踪剂与 PQG 方程的不同模式（涡流模式和潮湿模式）相关联，这里的分析强调了波或模式分解对于理解饱和域中的水的有用性。为了获得对模型的这种理论理解，一个关键的观察结果是水可以写成两个其他示踪剂（等效潜在温度和潮湿变量 M）的线性组合，它们本身具有理论上易于处理的光谱。这另外两个示踪剂与 PQG 方程的不同模式（涡流模式和潮湿模式）相关联，这里的分析强调了波或模式分解对于理解饱和域中的水的有用性。为了获得对模型的这种理论理解，一个关键的观察结果是水可以写成两个其他示踪剂（等效潜在温度和潮湿变量 M）的线性组合，它们本身具有理论上易于处理的光谱。这另外两个示踪剂与 PQG 方程的不同模式（涡流模式和潮湿模式）相关联，这里的分析强调了波或模式分解对于理解饱和域中的水的有用性。