Scaling of Eliassen-Palm flux vectors

Eliassen-Palm flux is one of the main diagnostics tools for wave propagation and wave-mean flow interaction in atmospheric dynamics and in particular stratosphere-troposphere coupling. Even though the theory has been derived in the 1960s, there is still no consensus about how to display the flux vectors in a plot. This is particularly true where both the troposphere and stratosphere are of importance. Some of the tra-ditional methods are to scale the arrows by either pressure, the exponential of height, the square root of pressure, or even by an arbitrary factor. But the arguments for any of those methods are subjective, and they result in both different amplitudes and direction. Here, we propose an objective way of scaling EP flux vectors, in either linear or logarithmic pressure or height coordinates, which allows for a physically sound representation throughout the entire atmosphere.


| INTRODUCTION
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 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 (F ϕ , F p ) 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.
Abbreviations: AMS83, Andrews et al. (1983); BEH85, Baldwin et al. (1985); EHM80, Edmon et al. (1980); EP, Eliassen-Palm. 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 Reading 2 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).

| VECTOR SCALING
We start from the expression of the EP flux components in pressure coordinates as in Equation (2.1) of AMS83 and All notations are standard, with the convention of overbars and primes denoting zonal averages and their departures. Subscripts refer to partial derivatives and a denotes Earth's radius.
Here, f ϕ is in units of m 2 /s 2 , and, assuming pressure in hPa, f p is in mÁhPa/s 2 . Therefore F ϕ and F p are in units of m 3 /s 2 and m 2 hPa/s 2 respectively. As described by EHM80, if one tries to plot the vector fields (f ϕ , f p ) or (F ϕ , F p ) in a latitude-vertical plot, the directions of the arrows will not visually represent the physical effects of the waves (such as divergence and convergence, and the direction of propagation). To make sure the derivative with respect to the coordinates used on the xand y-axes corresponds to the divergence of F, EHM80 define (their Equation [3.13]) which are in units of m 3 rad and m 3 hPa respectively. With this scaling, the mass weighted divergence of EP flux is simply ∂ ϕFϕ + ∂ pFp , and if one plots the vectors F =F ϕ ,F p À Á in a linear and equal aspect ratio plot with the latitude in radians, the arrows will show the physically correct picture. However, EP fluxes are very rarely plotted with these required specifications: Latitude is usually shown in degrees, and the aspect ratio is usually arbitrary, as even if the plot itself is squared, pressure spans over a 1000 hPa while latitude only extends over 2π radians or 180 maximum. But of course, these numbers can be arbitrary when not plotting the entire atmosphere; if only part of the domain is shown, for instance above 250 hPa and the northern extratropics only, the vectors have to be scaled accordingly. But the most important factor is the scale of the pressure axis, which is often logarithmic rather than linear. Many authors work in log-pressure coordinates z~log(p), which we will discuss in detail in Section 3. First, we need to derive the scaling factors α and β to account for units and plot aspect ratio such that the vector shows both physically correct direction and amplitude (to a constant factor) at any point on the plot. This corresponds to a change of variables, with the derivative giving the necessary scaling coefficients, for example, where x 0 , y 0 are the physical coordinates (latitude, pressure, etc.) and X, Y are the coordinates in the plot, that is, the length along the axis. We will use the convention that X, Y denote the axis length in inches, and x, y ∈ [0, 1] are the fraction along the axes (see Figure 1), and consider the two cases of linear and logarithmic plots. Note that we use inches here (with no loss of generality) as a standard of measuring image size and resolution (e.g., dots per inch) which most plotting software adheres to.

| Linear latitude-pressure plots
For a linear axis, the coordinate transformation is straightforward, and takes the simple form For instance, in a linear latitude-pressure plot, where latitude increases from left to right and pressure decreases from bottom to top, this becomes where Δϕ = max(ϕ) − min(ϕ) and Δp = max(p) − min(p). Now α lin has units of inches per radian and β lin has units of inches per hPa, and the minus sign in β lin comes from the inverted pressure axis. Note that if latitude is inverted, as for example in BEH85, Δϕ = min(ϕ) − max(ϕ) < 0. The scaling of Equations (8) and (9) differs from that proposed in EHM80 only in that it explicitly includes the figure aspect ratio. However, this step is still essential for preserving arrow angles. As described by many subsequent authors, it will yield very small arrows in the stratosphere. But that is only a result of using a logarithmic rather than linear vertical axis, not of any physical shortcoming of the theory (nor physical decrease of wave activity flux), and we propose scaling for a logarithmic axis in the next section.

| Logarithmic latitude-pressure or latitude-height plots
Most studies concerned with stratosphere-troposphere coupling plot EP flux vectors in both the troposphere and the stratosphere, and therefore utilise a logarithmic pressure axis, or equivalently log-pressure (z) coordinates. Using the same scaling as for linear axes will not work, and has led to arbitrary methods for re-scaling to make the vectors visible in the stratosphere. However, these arbitrary scalings are ill-informed, and it is relatively simple to derive a consistent scaling for logarithmic axes. For the derivation, assume y ∈ [0, 1] the position along the y-axis as in (7), and p 0 = p(y = 0) and p 1 = p(y = 1). For instance, p 0 = 1,000 hPa and p 1 = 1 hPa. Then, at any given pressure p along the (log 10 ) logarithmic y-axis, y p ð Þ = log 10 p 0 ð Þ−log 10 p ð Þ log 10 p 0 ð Þ−log 10 p 1 ð Þ = log 10 p 0 =p ð Þ log 10 p 0 =p Now we can directly apply Equation (6): F I G U R E 1 Definition of variables used to compute the scaling for the vector components. X and Y are the total width and height in inches, x and y are the normalised figure coordinates running from 0 on the left/bottom to 1 on the right/top. x 0 and y 0 are the physical coordinates, as for instance latitude in degrees or pressure in hPa Finally, Again, the units of β log are inches per hPa. It is now obvious why AMS83 and others find that dividing the arrows by pressure is a good way to display EP flux arrows: It's the required factor when changing variables from p to log(p). While previous authors failed to find a physical reason, there is a geometric reason why this is true, and there is an additional scaling factor of Y/ln(p 0 /p 1 ) which has to be included. It also becomes clear that scaling by the square root of pressure as in for example, Taguchi and Hartmann (2006) has neither a physical nor a geometric basis.

| F P VERSUS F Z
It is often convenient to work with the log-pressure coordinate z = − Hln(p/p 0 ), as z has units of meters and is close to geometric height in the atmosphere. Using z instead of p has the consequence that the EP flux components (1)-(3) change form to (Andrews et al., 1987, Equation 3.5.3) and f ϕ z and f ϕ are the same as the change of variable from p to z cancels out in the second term, and we can drop the superscript. However, the units of f z are m 2 /s 2 compared to hPaÁm/s 2 for f p . Most importantly, both terms in Equation (15) are now multiplied by density ρ 0 = ρ s exp (−z/H). Closely following EHM80 and Dunkerton et al. (1981), we want to plot the vectors F in such a way that they visually represent wave propagation, but also that their divergence corresponds to zonal mean acceleration. For this, we need to find Δ z such that where dm is the mass element of a zonally symmetric portion of the atmosphere and takes the form dm = ρ 0 2πa 2 cosϕdϕdz: Equations (16) and (17) are the counterparts of Equations (3.11) and (3.10) in EHM80 in z-coordinates. The density ρ 0 appears explicitly on the right hand side of Equation (16) to make sure the units of Δ z are m 3 , just like Δ in EHM80. Similarly, ρ 0 must be included in the definition of dm. Further following EHM80 yields the vector componentŝ for which the mass weighted divergence of EP flux is now ∂ ϕF z ϕ + ∂ zFz . We note that considering mass instead of volume introduces the factor of 1/ρ 0 which is missing in Dunkerton et al. (1981), and assures that the vector components do not become exponentially smaller with height. Similarly, AMS83 convert (F ϕ , F p ) to (F ϕ z = F ϕ p/ p 0 , F z = −F p /p0) with z = − ln(p/p 0 ). These factors come from replacing θ p with θ z (with H = 1). However, such scaling still produces rapidly decaying vector sizes with height, and AMS83 still have to 'multiply them by scalar normalizing factors, which differ from one grid point to another'. The missing physical explanation from those arguments is that dϕdp includes mass weighting via the pressure element dp, whereas dϕdz does not, and the density has to be explicitly taken out of Δ z in Equation (16). This in turn causes a division by ρ 0 to appear in Equation (18), which ascertains that the vectors do not become vanishingly small in the stratosphere. There is also no need to artificially set the density to unity throughout the domain as done by some authors.
Naturally, vertical propagation of waves throughout the atmospheric column does not happen purely conservatively, such that dissipation will decrease wave activity and we cannot expect EP flux arrows to remain of constant size over multiple scale heights. However, the exponential effect of mass dominates the effect of nonconservative dissipation and in practice EP flux arrows scaled with our method will be of comparable amplitude (Figure 2).

| EXAMPLES
This letter shows that the most widely used scaling via dividing by pressure happens to be the correct scaling for a logarithmic axis (with additional constant factors, see Equation (12)). We will now compare the arrows resulting from our derivation to some of the most widely referenced methods. We note that it is very rare for authors to give details on whether or not they included figure aspect ratio for their plots, and we assume here that they do (the arrow direction depends on the ratio between X and Y in, e.g., Equation (7)). Many graphical packages also include automatic scaling algorithms, but these 'intelligent' functions cannot guess the physical meaning of the axis dimensions, and often fail in representing both the angles and amplitudes of the arrows. Figure 2 shows a selection of plots which we reconstructed from the PSL website 3 (panels a,c,e), from  Palmer (1981). (d) Uses scaling as proposed by Andrews et al. (1983), without their manual point-by-point scaling. 9f) Replicates fig. 6E from Baldwin et al. (1985). Panels (b), (d) and (f) have the same aspect ratio as in the respective papers, and (a), (c) and (e) have the same aspect ratio as the PSL online plotting tool. Note that only the methods of Andrews et al. (1983) and Palmer (1981) yield arrows parallel to the correct scaling Palmer (1981) (panel b), AMS83 (panel d), and BEH85 (panel f). All panels replicate the figure aspect ratios as well as the meridional and vertical range. All plots except panel b) use the time period from BEH85, that is, January 5 to March 3, 1979, while panel b) uses the same date as in the original work, that is, February 21, 1979, which is within the date range of all other plots. The plots are slightly different as we use ERA5 data 4 rather than the original datasets as they were not available to the author. In addition, AMS83 analyse model data which were not available for this study, which is why the same data were used as for the other plots.
In all panels, the red arrows show the scaling derived with Equations (4), (8) and (12) (as well as (18) for z-coordinates), whereas the hollow black arrows show the scaling as described in the respective paper. Probably the most important point to make from Figure 2 is that only the methods of Palmer (1981) and AMS83 (both scaled by 1/p, see Table 1) preserve the correct direction of the arrows, but neither has visible arrows throughout the vertical range. The methods of BEH85 and ESRL alter both the arrow direction and scale, and therefore physical analyses using those methods might lead to different interpretations. In contrast, all plots confirm that the newly derived method shows useful and accurate information throughout the entire plot domain, and is desirable over the other tested alternatives.

| CONCLUSIONS
In this letter, we show a derivation of the scaling of EP flux vectors for consistent plotting, and provide Python code for use (Jucker, 2019). In addition to presenting the derivation for plots with arbitrary aspect ratio, we also discuss the difference between working with F z and F p , and how to scale these vectors on logarithmic pressure axes. Explicitly, the proposed scaling is summarised in Table 2. Given how simple the scaling is, we hope that this letter is useful as a reference for other authors struggling with the way to plot EP arrows for the analysis of atmospheric dynamics and in particular stratosphere-troposphere coupling.  (Jucker, 2020a) and are part of the Python package aostools (Jucker, 2020b).

CONFLICT OF INTEREST
The author declares no conflict of interest.  F(ϕ, p) F ( ϕ, log(p)) F(ϕ, z)

ORCID
Note: X/Y is the figure aspect ratio with X the length of the x-axis and Y the length of the y-axis (typically in inches).F ϕ ,F p À Á are given in Equation (4), andF z ϕ ,F z in Equation (18). ϕ 0,1 are the left, right latitude limits in degrees, p 0,1 are the bottom, top pressure limits, and z 0,1 the bottom, top z limits. Note that the shown option scales F x with figure aspect ratio X/Y; another possibility is to scale F x by X and F y by Y as in equations (8), (9), (12).