The variability of the atmospheric power spectrum

大气的动能谱是对每个长度尺度上有多少能量的衡量--它本质上是动能场的空间傅里叶变换。图1显示了Nstrom和Gage(1985)根据飞机数据计算的频谱。最大长度尺度上的运动包含了大部分动能。图1是对数标度，因此线条的梯度对应于动能光谱密度和波长之间的幂函数关系。光谱梯度很重要，因为它与可预测性有关。Lorenz(1969)指出，在某些统计假设下，如果系统的能量具有-3或更浅的光谱梯度，则2D涡度方程--在天气尺度上很好地逼近大气运动--将具有固有的有限可预测性范围。请注意，结合图1，这确实意味着我们将永远无法提前2-3周预测天气。这是一个有争议的问题。Charney(1971)令人信服地解释了图1的范围的起源是由通过斜压不稳定而很好混合的大尺度准2D流动造成的。到目前为止，还没有对-5/3范围提供类似令人信服的解释。对于某些直觉，2D湍流将显示从能量注入尺度到能量注入尺度的高级能量级联，该能量级联具有-5/3的幂定律和-3的低尺度拟能级联。3D涡旋将不会有高尺度的能量转移，相反，它只会有-5/3的低尺度能量级联(见Val-lis 2017，第14章)。图1：数千架飞机测量的结果，这张图显示了中尺度上神秘的-5/3光谱梯度。[纳斯特罗姆1985]是什么导致了意想不到的高能中尺度？1979年，盖奇提出，这是来自风暴等小规模混合的2D高级能量转移的结果。与这种解释相反的是，VanZandt(1982)指出，与重力波相关的能量的下移可能会产生观测结果。使问题变得更加复杂的是，光谱是可变的。地形强迫被证明具有很大的影响，在比海洋大10倍的山脉上具有中尺度能量(Nstrom等人，1987年)。同样，在模型中，降水也被认为能激发中尺度的能量(Selz等人，2019)。甚至有人提出，可能没有通用的动力学机制来支配所观察到的光谱，缺乏令人信服的理论仅仅是因为统计假设不适用于高度复杂的真实大气(Selz等人，2019年)。答案很可能是以上因素的结合，原因取决于纬度(Cho等人，1999)，也可能取决于海拔。在过去的十年里，计算能力持续增长。ECMWF的分析数据现在可以看到-5/3频谱，我们正在接近一个可以更直接地回答我们关于原因的问题的时期。为了给-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。一个令人着急的问题是，观察到的功率谱变化是否与状态相关的可预测性变化相关。如果发现情况属实，预报员可以提前知道需要多少整体成员才能充分探索状态空间。

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.