Eliassen‐Palm通量矢量缩放比例 (含Python/NCL/GrADS代码+测试数据)

Jucker M. Scaling of Eliassen‐Palm flux vectors. Atmospheric Science Letters.:e1020.

1.本文Python代码+数据链接：

Python script and ERA5 data used to produce Figure 2 of Jucker, M. "Scaling of Eliassen-Palm Flux Vectors", Atmos. Sci. Lett., https://doi.org/10.1002/asl.1020

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Eliassen-Palm通量是大气动力学特别是平流层-对流层耦合中波的传播和波-流相互作用的主要诊断工具之一。尽管该理论在20世纪60年代就已经得出，但对于如何在图中显示通量矢量仍未达成共识。在对流层和平流层都很重要的情况下尤其如此。一些传统的方法是按压力、高度指数、压力的平方根或甚至按任意系数来缩放箭头。但任何一种方法的论证都是主观的，它们导致的结果是振幅和方向都不同。在这里，我们提出了一种客观的EP通量矢量的缩放方法，无论是线性的还是对数的压力或高度坐标，都可以在整个大气层中得到物理上合理的表示。

2.NCL代码链接：

https://www.ncl.ucar.edu/Applications/EPflux.shtml

3.GrADS代码链接：

http://www.atmos.rcast.u-tokyo.ac.jp/nishii/programs/index.html

T-N波作用通量

4.其他参考：

https://www.jianshu.com/p/97042d9019c0

Jucker M. Scaling of Eliassen‐Palm flux vectors. Atmospheric Science Letters.:e1020.

Eliassen‐Palm flux (EP flux, Eliassen and Palm, 1960) is omnipresent as a diagnostic tool for wave‐mean flow interaction, and in particular stratosphere‐troposphere coupling. It shows the direction of small amplitude atmospheric waves as vectors and at the same time acceleration (or deceleration) of zonal mean zonal wind via its divergence (Andrews and McIntyre, 1976, 1978). For finite amplitude theory see for example, Nakamura and Zhu (2010). While the divergence is a scalar and therefore easily represented in a figure, the flux itself is a vector field (here denoted F) and great care must be taken in representing the magnitude and direction of the arrows. The main difficulties arise from the exponential decrease of mass with height, and the non‐trivial aspect ratios resulting from plotting the components in degrees latitude—pressure/height space, as this is not a Cartesian coordinate system.

Unfortunately, there is no consensus in literature about how exactly this should be done. For instance, Andrews et al. (1983, henceforth ‘AMS83’) state that ‘in practice it is found that multiplication of F by p−1 keeps magnitudes roughly comparable throughout the middle atmosphere. However, there appears to be no decisive theoretical justification for such a scaling’. They then scale the arrows at each grid point with an undisclosed number, and only consider the arrow direction in their analysis.

Probably the most important effort to use a physically consistent scaling was undertaken by Edmon et al. (1980, henceforth ‘EHM80’), who derived the appropriate expressions for the meridional and vertical components (Fϕ, Fp) for display in pressure coordinates. Their motivation was to find an expression which would assure that the apparent derivatives with respect to latitude and pressure in the figure correctly represent the EP flux divergence and therefore the acceleration of the zonal flow. While EHM80 also briefly discuss the effect of using logarithmic pressure axes, they do not explicitly show the resulting components. Instead, they argue that given the unavoidable non‐conservative dissipation of wave activity along the vertical path spanning many scale heights, efforts to rescale EP flux vectors in log‐p plots would be futile in any case. This is an unfortunate conclusion, and given the number of publications showing EP flux vectors over multiple scale heights, we believe it is worth using a geometrically consistent scaling for more clarity. Dunkerton et al. (1981) expand on EMH80's discussion, and provide more detail on how to plot EP fluxes in z‐space, but apply volume rather than mass weighting in the vertical, again leading to vanishingly small arrows at high altitudes.

Palmer (1981) give somewhat more details about how to include figure aspect ratio, with explicit values for an aspect ratio scaling constant c. However, such a constant value is again only applicable for linear pressure axis (as we will show below). They work in log‐p or z‐coordinates, and manually set the density to a constant value of one, without any physical nor geometric reason. Later, Baldwin et al. (1985, henceforth ‘BEH85’) suggested multiplication with exp(z/H) (they work in z‐coordinates as well), which is the same as AMS83's division by pressure, without compelling geometric or physical arguments. These authors also remove a multiplicative factor of cosine of latitude. Other authors use the inverse square root of pressure (e.g., Taguchi and Hartmann, 2006) and even such influential organisations as NOAA's Physical Sciences Laboratory (former Earth System Research Laboratory)1 and the University of Reading2 recommend using the inverse square root of pressure plus an arbitrary constant multiplication above an arbitrary pressure level.

In this letter, we will derive a geometrically and physically consistent scaling for EP flux vectors, taking into account spherical geometry, the figure aspect ratio and the units of the vector components. It is a simple derivation, but the reasoning is more geometric than physical, which is probably why previous authors came to conclusions such as the one by AMS83 cited above. It is surprising how such arbitrary scaling has been accepted by the research community, when a correct way of displaying scientific data is so important. For instance, we will show that using the square root of pressure is ill‐informed and should only be used with great caution.

This letter is organised as follows: Section 2 derives the scaling for EP flux arrow plots which conserve the direction and amplitude (to a constant factor) in any linear or logarithmic plot with arbitrary aspect ratio. Section 3 describes how to represent EP flux vectors in log‐pressure or z‐coordinates consistently. Section 4 then concludes by showing the differences between our scaling and the most important scalings used in literature as described above. Python code to compute EP fluxes and display them on an arbitrary figure is part of the Python package aostools (Jucker, 2020b).

• https://psl.noaa.gov/data/epflux/img/EP_Flux_Calculation_and_Display.pdf.
• http://www.met.reading.ac.uk/~pn904784/snap/ep\_flux\_calculations.html.
• https://psl.noaa.gov/data/epflux/.
• daily instantaneous data at 00 UTC on 2.5° and 37‐level grid. Data available from Jucker (2020a).