The variability of the atmospheric power spectrum

大气的动能谱是衡量每个长度尺度上有多少能量的尺度，它本质上是动能场的空间傅里叶变换。图1显示了Nastrom和Gage（1985）根据飞机数据计算出的光谱。最大长度尺度上的运动包含了大部分动能。图1是在对数尺度上，因此线条的梯度对应于动能谱密度与波长之间的幂律关系。由于光谱梯度与可预见性的联系，它是重要的。Lorenz（1969）表明，在某些统计假设下，二维涡度方程——在天气尺度上很好地近似大气运动——如果系统的能量具有-3或更浅的光谱梯度，则具有内在有限范围的可预测性。请注意，根据图1，这确实意味着我们永远无法提前2-3周以上预测天气。一个有争议的问题。Charney（1971）令人信服地解释了图1中-3范围的起源，这是由斜压不稳定混合的大规模准二维流动造成的。到目前为止，还没有提供类似的令人信服的解释-5/3范围。直觉上，二维湍流将显示能量注入尺度上的高级能量级联，它具有-5/3幂律和-3的低尺度熵级联。3D湍流不会有高级能量的转移，相反，它只会有一个-5/3的低级能量级联（见vallis 2017，第14章）。图1：数千次飞机测量的结果，这张图显示了中尺度上神秘的-5/3光谱梯度。[Nastrom 1985]是什么导致了意想不到的高能中尺度？1979年，Gage提出这是小规模混合（如风暴）的二维高级能量转移的结果。与这种解释相反，VanZandt（1982）指出，与重力波相关的能量的低尺度运动可以产生这种观测结果。使问题进一步复杂化的是，频谱是可变的。地形强迫已被证明对山脉上空的中尺度能量有很大影响，其影响比海洋上空的中尺度能量大10倍（Nastrom et al . 1987）。同样，降水也被认为可以激活模式中的中尺度，例如（Selz et al, 2019）。甚至有人认为，可能没有普遍的动力机制来控制观测到的光谱，缺乏令人信服的理论仅仅是由于统计假设不适用于高度复杂的真实大气（Selz et al 2019）。答案很可能是上述因素的结合，其中原因占主导地位取决于纬度（Cho et al . 1999），也可能取决于海拔。在过去的十年中，计算能力持续增长。ECMWF的分析数据现在可以看到-5/3的频谱，我们正在接近一个可以以更直接的方式回答我们关于原因的问题的时期。因此，为了为-5/3的起源提供证据，建议在分析中对功率谱进行彻底的分类。一个诱人的问题是，观察到的功率谱变化是否与依赖状态的可预测性的变化相关。如果发现是这样的话，预报员就可以提前告诉我们需要多少个集合成员来充分探索状态空间。

The kinetic energy spectrum of the atmosphere is a measure of how much energy is at each length scale - It is essentially the spatial Fourier transform of the kinetic energy field. Figure 1 shows the spectrum calculated by Nastrom and Gage (1985) from aircraft data. Motions on the largest length scales contain most of the kinetic energy. Figure 1 is on a log scale, so the gradients of the lines correspond to a power law relationship between spectral density of kinetic energy and wavelength. The spectral gradient is significant due to its link with predictability. Lorenz (1969) showed that, under certain statistical assumptions, the 2D vorticity equations -a good approximation to atmospheric motion on synoptic scales- will have an intrinsically finite range of predictability if the energy of the system has a -3 or shallower spectral gradient. Note that taken with figure 1, that re-ally does imply we will never be able to predict the weather more than 2-3 weeks in advance.A controversial question.The origin of the -3 range of figure 1 was convincingly explained by Charney (1971) as resulting from a large scale quasi-2D flow that is well-mixed via baroclinic in-stability. A similarly convincing explanation for the -5/3 range has not been provided so far. For some intuition, 2D turbulence will display an upscale energy cascade from the energy injection scale that has a -5/3 power law and a downscale enstrophy cascade of -3. 3D turbu-lence will have no transfer of energy upscale, instead it will just have a -5/3 downscale energy cascade (see Val-lis 2017, ch 14).Figure 1: The result of thousands of aircraft measurements, this fig-ure shows the mysterious -5/3 spectral gradient at the mesoscales. [Nastrom 1985]What could be causing the unexpectedly energetic mesoscale? In 1979, Gage suggested that it was the re-sult of 2D upscale energy transfer from small scale mix-ing such as storms. In opposition to this explanation, VanZandt (1982) pointed out a downscale motion of en-ergy associated with gravity waves could produce the observation. Further complicating the issue, the spec-trum is variable. Orographic forcing was shown to have a large effect with mesoscale energy over mountains up to 10 times larger than over ocean (Nastrom et al 1987). Similarly, precipitation has been seen to energize the mesoscales in models, (Selz et al 2019) for example. It has even be suggested that there may be no universal dynamical mechanism that governs the observed spec-trum, the lack of compelling theory simply being due to the statistical assumptions not applying to the highly complex real atmosphere (Selz et al 2019). The answer is likely to be a combination of the above with which cause dominates depending on latitude (Cho et al 1999), and possibly altitude.In the last decade computing power has continued to grow. ECMWF analysis data can now see the -5/3 spec-trum and we are approaching a period where our ques-tions about the cause can be answered in a more direct manner. To provide evidence for the origin of the -5/3, a thorough categorization of the power spectrum in analysis istherefore proposed. A tantalizing question is whether observed variation in the power spectrum correlates with changes in the state-dependent predictability. If this was found to be the case, forecasters could tell in advance how many ensemble members would be needed to sufficiently explore the state-space.