1. Introduction

A basic metric for estimating atmospheric predictability is the average difference (or “error”) within an ensemble of forecasts starting from slightly different initial conditions. It is often useful to estimate the spatial scale of the error field and spectral analysis is the natural tool to do so. For meteorological applications, the error field of interest will vary with latitude and longitude (x, y), which requires a two-dimensional (2D) spectral analysis in wavenumbers (k, l). Statistical predictability theory (Lesieur 2008, chapter 11) is based on the theory of homogeneous, isotropic turbulence, in which spectra are circularly symmetric in k–l space. Statistical predictability theory takes advantage of this circular symmetry to reduce two-dimensional spectra in k–l space to one-dimensional (1D) spectra in the polar coordinates $(\kappa ,\theta )=(\sqrt{k^2+l^2},{{\displaystyle \tan }}^{-1}l/k)$ in which the spectra are independent of θ. In recent studies it has become common in meteorological analyses to reduce 2D error spectra to 1D error spectra by computing spectra in the x direction and then averaging the results over y. The objective of this paper is to show that the latter procedure produces 1D error spectra that, except for power-law spectra, fail to capture important aspects of statistical predictability theory.

Statistical predictability theory began with Lorenz (1969) in which the predictability problem was formulated in terms of the growth of small initial differences in a statistically stationary, homogeneous, isotropic turbulent flow; studies followed using more sophisticated turbulence models (Lesieur 2008, chapter 11). These studies find that the growth of the peak scale and amplitude of an error energy spectrum depends on the energy spectrum of the assumed background turbulent flow. For a background turbulent flow with the power-law spectrum κ^−β, the time scale for the error-spectrum evolution, including the increase of the peak scale ${\kappa }_{\text{peak}}^{-1}$ and amplitude, is inversely proportional to κ_peak for β = 5/3 and constant when β = 3 (Lesieur 2008, 412–413). For the “−5/3” case the inverse dependence of error-scale and amplitude growth rate on scale implies limited predictability since confining an initial error to ever smaller scales implies ever faster error growth rates. For the “−3” case, the peak scale of error energy is determined by the peak scale in the background energy spectrum; confining the initial error to ever smaller scales refines the initial condition without introducing faster error growth implying unlimited intrinsic predictability. Note that in both the 3D and 2D cases, the theoretical error spectra for κ < κ_peak have fixed slopes (κ⁴ in 3D and κ³ in 2D) and therefore grow along with the error amplitude at k_peak (Lesieur 2008, Figs. 11.2–11.3). Thus, the error spectra for κ < κ_peak could be described as growing “up-amplitude,” but the emphasis on this feature would be misplaced since it is the growth of the peak-error scale and amplitude that characterize the evolution of the theoretical error energy spectrum. The determination of the peak scale ${\kappa }_{\text{peak}}^{-1}$ through spectral analysis in a manner consistent with statistical predictability theory is the issue addressed in the following.

To economize the writing, the reduction of a spectrum from 2D to 1D by integrating in θ in polar coordinates in wavenumber space is termed a 1Dκ spectrum while the reduction of a spectrum from 2D to 1D by transforming in x (which gives the transform as a function of wavenumber k and y) and then averaging over y is termed a 1Dk spectrum. The error spectra at a particular wavenumber is said to be “saturated” when it reaches the level of the background energy spectrum.

Figure 5 of Mapes et al. (2008) compares the theoretical 1Dκ error spectra of Rotunno and Snyder (2008) to the 1Dk error spectra at the equator¹ from a high-resolution global meteorological model (their Fig. 5c). The 1Dk error spectra are observed “… to fill up the saturation spectrum vertically (up-amplitude) rather than horizontally (upscale). Large scales grow just as rapidly as small scales, and they do so before small scales saturate.” In other words, the analyzed 1Dk error spectra are essentially constant from k = 0 to k_sat, where k_sat is the wavenumber at which the 1Dk error spectra equal the background energy spectrum. As the error spectra grow with time t, they preserve the “flat-line” shape from k = 0 to k_sat(t) which decreases with t, and so growth of the error spectra is said to be “up-amplitude.” In contrast, the theoretical 1Dκ error spectra increase with κ from κ = 0 to a peak at κ_sat(t) in the case of a “−5/3” background energy spectrum. Since the peak wavenumber decreases with t, the growth of the 1Dκ error spectra is said to be upscale. Further discussion of the case with a “−3” background spectrum is given in section 4.

Durran et al. (2013) compute 1Dk (including ensemble averaging) error spectra in a case study of predictability using a limited-area forecast model. Similar to the conclusion in Mapes et al. (2008), Durran et al. (2013, Abstract) find, “There is no evidence of small-scale perturbations developing rapidly and transferring their influence upscale. Instead, the large-scale perturbations appear to grow more rapidly during the first 12 h than those at the smallest resolved scales.” The same conclusion is reached in Weyn and Durran (2017, Abstract) in the context of a simulation of an idealized mesoscale convective system: “Both small- and large-scale errors grow primarily up in amplitude at all scales rather than through an upscale cascade between adjacent scales.” Most recently, Lloveras et al. (2021), in the context of numerical simulations of idealized baroclinic waves with moist convection, note w.r.t their Fig. 16 the difference in the shapes of the low-wavenumber parts of the 1Dk (constant with k) and 1Dκ (increasing with κ) error spectra but conclude: “Nevertheless, the error growth using both computational methods is primarily up-amplitude, with relative errors growing at approximately the same rate at all scales, rather than through an upscale cascade.”

In the present paper, we show that 1Dκ and 1Dk error spectra are generally not the same and, moreover, they differ systematically over the low wavenumbers. Statistical predictability theory has 1Dκ error spectra increasing with κ up to κ_peak for both “−5/3” and “−3” background spectra (Lesieur 2008, chapter 11), where the subscript “peak” signifies the wavenumber at which the error spectrum is a maximum. Here we show that an error field with a circularly symmetric 2D error spectrum has a 1Dκ spectrum that increases with κ up to κ_peak at the low wavenumbers, while the 1Dk spectrum computed from the same error field is constant-with-k up to k_peak.

In section 2, the analytical/computational methods are developed for comparing 1Dκ and 1Dk spectra. Simple examples based on arbitrary homogeneous, isotropic random functions are discussed in section 3. The relevance of these examples to analyses of error spectra in a high-resolution global model (Judt 2018, 2020) is discussed in section 4. Suggestions are made in section 5 for how to compute local 1Dκ error spectra, which is of interest when the error fields are spatially inhomogeneous as in most realistic applications. Conclusions are summarized in section 6.

2. Discrete Fourier analysis

The analysis here is similar to that of Durran et al. (2017) but with emphasis on the relation between 1Dκ and 1Dk error spectra.

3. Comparison of 1Dκ and 1Dk

a. Simple examples

Consider the real, random function f_n,m on a square grid shown in Fig. 1a (see the appendix). Its power spectrum ${\left|F_{k^{\prime },l^{\prime }}\right|}^2/N^2$ is shown in Fig. 1b. The power spectra S(κ) and S^1D(k′) are shown in Fig. 1c. One observes that S^1D(k′) is nearly independent of k′ for $k^{\prime }<k_{\text{peak}}^{\prime }$. The reason for this is clear in light of (8). Substitution of (8) into (10) gives

$S^{1\mathrm{D}}(k)={\displaystyle \sum _{l=1}^{N_y}{\left|F_{k,l}\right|}^2{(N_x{N}_y)}^{-1}},$(12)

or in terms of (k′, l′) and for N_x = N_y = N,

$S^{1\mathrm{D}}(k^{\prime })={\alpha }_{k^{\prime }}{\displaystyle \sum _{l^{\prime }-N/2}^{N/2-1}{\left|F_{k^{\prime },l^{\prime }}\right|}^2{(N)}^{-2}},$(13)

for 0 ≤ |k′| ≤ N/2. The flat-line shape of S^1D(k′) for $k^{\prime }<k_{\text{peak}}^{\prime }$ is a result of the summation over l′ of the nearly circularly symmetric ${\left|F_{k^{\prime },l^{\prime }}\right|}^2/N^2$.

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(a) Random function f_n,m = f(x_n, y_m), (b) its 2D power spectrum ${\left|F_{k^{\prime },l^{\prime }}\right|}^2/N^2$, and (c) the reduction of the 2D power spectrum to 1D through S(κ) in (9) and through S^1D(k′) in (11). The real random function f_n,m is generated following the procedure outlined in the appendix for the Gaussian distribution in (15).

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0070.1

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In a second example, consider the random function f_n,m shown in Fig. 2a with the logarithm of the power spectrum shown in Fig. 2b. In this case the power spectrum follows a power law (“−5/3” in this example). In the appendix, the power spectral density for a power law can be obtained from (A2) as $\tilde{F}{\tilde{F}}^{*}=A^2{\kappa }^{(\alpha -1)}$ with $S(\kappa )=2\pi \kappa \tilde{F}{\tilde{F}}^{*}=2\pi A^2{\kappa }^{\alpha }$. In continuous form the rhs of (8) implies $S^{\text{1D}}(k)=(1/2a){\int }_{-a}^a\tilde{F}{\tilde{\tilde{F}}}^{*}dl=(1/2a){\int }_{-a}^a{A}^2{(k^2+l^2)}^{(\alpha -1)/2}dl$; letting χ = l/k, the integral reduces to $k^{\alpha }(A^2/2a){\int }_{-{\chi }_1}^{{\chi }_1}{(1+{\chi }^2)}^{(\alpha -1)/2}d\chi$ where the latter integral is just a numerical factor. Thus, for a power law, both S(κ) and S^1D(k) have the same power-law dependence. In terms of the discussion in the previous paragraph, in the power-law case $k_{\text{peak}}^{\prime }=0$.

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As in Fig. 1, but for the power spectrum in (A2).

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0070.1

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b. Discussion

1. 1) All calculations of error spectra based on statistical predictability theory treat fields such as that as shown in Fig. 1a with spectral densities that are exactly circularly symmetric (due to ensemble averaging, which is not done in our examples). Hence any comparison of meteorological-model error spectra with statistical predictability theory must be done with 2D spectral densities reduced to 1D spectra, as described in Errico (1985) and done here with the definition of S(κ) in (9).
2. 2) Reducing a nearly circularly symmetric 2D spectrum to 1D using the y-averaged Fourier transforms in x [here defined as S^1D(k′)] leads to spectra like that in Fig. 1c in which the shape of the spectrum is independent of k′ for $k^{\prime }<k_{\text{peak}}^{\prime }$. A power-law spectrum is the exception since $k_{\text{peak}}^{\prime }=0$ (Fig. 2c).
3. 3) Consider the following hypothetical case of upscale error growth from statistical predictability theory: Using (15) with ϕ = 0 generates a circularly symmetric 2D spectrum; letting the error amplitude A(t) increase as $k_{\text{peak}}^{\prime }(t)$ decreases with time produces the S(κ, t) and S^1D(k′, t) shown in Fig. 3. The same data analyzed through S^1D(k′, t) give the impression that the growth in time of the error spectra is “up-amplitude” rather than upscale since S_1D(k′) is constant across scales for $k^{\prime }<k_{\text{peak}}^{\prime }(t)$.

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As in Fig. 1c, but for three Gaussian spectra generated with (15) using ϕ = 0 and k_w = 20 with (A, k_c) = (0.5, 80), (1.0, 40), and (2.0, 80)

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0070.1

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4. Relevance of statistical predictability theory to atmospheric predictability

Using a high-resolution global model, Judt (2018) performed “identical-twin” type experiments in a case study of a specific 3-week period. The study showed that, initially, error growth was tied to moist convection and therefore highly localized, followed by a phase during which the error grew in scale, magnitude, and spatial extent. At approximately 2–3 weeks into the forecasts, the divergence of forecasts that had started from small differences in the initial conditions had led to errors as large as any sample drawn from a climatological distribution (i.e., predictability was lost).

Analysis of the growing error spectra (his Fig. 13) using spherical harmonics showed a good qualitative correspondence to statistical predictability theory. More specifically, at the earliest times, the error growth rate was maximum at the smallest resolved scales, producing spectra that peaked at those scales and fell off toward low wavenumbers (Fig. 4a). After a few hours, the peak of the error spectra began shifting toward larger scales (i.e., up-scale growth; see crosses in Fig. 4a) while the rate of growth decreased. Overall, this behavior is expected for a “−5/3” background spectrum. The characteristics of error growth changed qualitatively after the peak of the error spectra had propagated through the mesoscales. Between day 5 and 10, the error spectra grew at a constant rate while peaking at the scale of the energy-containing eddies (∼4000 km wavelength), as is expected for a “−3” background spectrum (Fig. 4b).

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Spectral error growth in the predictability experiment of Judt (2018). The evolution of the 250 hPa error kinetic energy spectra (a) between 1 and 12 h and (b) between 5 and 10 days as computed via spherical harmonics [same as Figs. 13a,b in Judt (2018)]. (c),(d) The corresponding spectra computed via Fourier transforms in longitude and averaged over latitude (60°S–60°N). The background spectra (black) are multiplied by 2. Crosses in (a) and (b) mark the peaks of the error spectra.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0070.1

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In an attempt to isolate error spectra over finite bands of latitude, Judt (2020) subjected the error fields to zonal Fourier transforms averaged over latitude. The resulting error spectra (his Fig. 5) are quite unlike the error spectra obtained via spherical harmonics and exhibit the “flat-line” shape for low wavenumbers. One of the motivations for the present paper was to understand how such different spectral shapes could be produced by the same physical-space error field. Motivated by the simple examples comparing S(κ)and S^1D(k′), we produced the spectral analysis in Fig. 4, with Figs. 4a and 4b the same as Figs. 13a and 13b of Judt (2018). The new analysis in Figs. 4c and 4d shows the latitudinal average of the zonal Fourier transforms from 60°S to 60°N. In both the “−5/3” and the “−3” cases, the error spectra in Figs. 4c and 4d show flat-line behavior over the low wavenumbers, similar to the simple examples. The close qualitative correspondence of the different reductions of 2D to 1D error spectra between the global model and the simple examples above is, we believe, good prima facie evidence that the 2D global-model error spectra have at least a qualitative similarity to the circularly symmetric 2D error spectra of statistical predictability theory.

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As in Fig. 1, but for the composite field f_n,m given by (14). Gray lines in (c) show 1Dκ spectra for $f_{m,n}^l$ and $f_{m,n}^s$.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0070.1

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5. “Local” spectra for inhomogeneous fields

Real atmospheric flows are not globally homogeneous and isotropic, which leads directly to considering predictability in specific subdomains of the globe, such as latitude bands or regions. One might hope that the tools of statistical predictability theory, which rest on assumptions of homogeneity and isotropy, could still be applied on subdomains where those assumptions are more closely met.

This raises the question of how to evaluate the scale and other spectral characteristics of perturbations on a given subdomain. Indeed, it at least partly motivates the use of 1Dk spectra, for example by separating the tropics, midlatitudes and polar regions as in Judt (2020). If 1Dk spectra are not particularly well suited to predictability studies, as we argue here, then how should we perform spectral analysis of perturbations for limited subdomains of the globe?

The 2D and 1Dκ spectra for the composite field appear in Figs. 5b and 5c, respectively. With different scales in the two halves of the domain, the 1Dκ spectrum lacks the annular structure seen in Fig. 1b. The 1Dκ spectrum (solid line) is a melding of the spectra from $f_{n,m}^l$ and $f_{n,m}^s$ (which are shown as thick gray lines in Fig. 5c): at large wavelengths the spectrum follows that of $f_{n,m}^l$, and at small wavelengths that of $f_{n,m}^s$. As expected, the 1Dk spectrum shown in Fig. 5c is flat at large scales, unlike the 1Dκ spectra for any of f_m,n, $f_{n,m}^l$, or $f_{n,m}^s$.

We seek an analysis technique that will correctly identify local spectra and dominant scales of the composite field. The simplest approach is to multiply the original global field by a spatially localized window function that is confined to the local region of interest, and then compute the 1Dκ spectrum of the resulting windowed, global field. Wong and Skamarock (2016) apply this technique to remove boundary effects when computing spectra for limited-area models, and similar approaches can be found in other fields of geophysics (Wieczorek and Simons 2005).

Figure 6 illustrates this approach applied to the composite f_m,n. The windowed field w_m,nf_m,n is shown in Fig. 6a for the window function:

$w_{m,n}=\{\begin{array}{ll}{{\displaystyle \sin }}^2{y}_n\hfill & \text{if}{y}_n<\pi ,\hfill \\ 0\hfill & \text{if}{y}_n\ge \pi .\hfill \end{array}$

This function, usually written in terms of a cosine with twice the frequency, is known as the Hann window (Press et al. 2007, section 13.4), but we have explored other choices and any reasonable window with a spatial scale that is broad compared to the scale of $f_{m,n}^s$ gives qualitatively similar results (not shown). Taking the 2D FFT of w_m,nf_m,n and then forming the 1Dκ spectrum yields the results in Figs. 6b and 6c. The 1Dκ spectrum of the windowed field accurately reproduces that of the small-scale field $f_{m,n}^s$ that dominates the behavior of the global field in the southern half of the domain, except for an overall shift in amplitude. For fields whose variance is locally spatially homogeneous, that shift can be removed by normalizing the spectrum of the windowed field by $N^2/{\displaystyle {\sum }_{m,n}{w}_{m,n}^2}$, thus accounting for the point-wise reduction of amplitude of the windowed field.

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As in Fig. 5, but for the windowed field w_n,mf_n,m. The gray line in (c) shows only the spectrum for $f_{m,n}^s$.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0070.1

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For a “real-world” example of windowing, we applied the Hann window with bounds from 10°S to 10°N to the high-resolution simulation of Judt (2018) (Fig. 7). As expected from the examples in section 3, this method yields a slope for the background spectrum that is very close to that seen in 1Dk spectra computed over 10°S–10°N [Fig. 5a in Judt (2020)]. This windowing method also captures the upscale evolution of the error in the tropics, with a decrease of the peak wavenumber for the 1Dκ spectra similar to that seen in the global 1Dκ error spectra in Fig. 4. In contrast, the 1Dk spectra for the errors [also displayed in Fig. 5a of Judt (2020)] show little indication of upscale evolution, again consistent with the examples in section 3.

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As in Figs. 4a and 4b, but for the windowed field where the error field is multiplied by a Hann window function with bounds from 10°S to 10°N. Because of the artifacts at the largest scales (see text), only spectral peaks at wavenumbers > 10 are considered for marking with crosses.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0070.1

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The 1Dκ error spectra in Fig. 7, which are based on horizontal velocities on the sphere, exhibit an artifact at the largest scales that does not appear in the idealized, planar example of Fig. 6. Specifically, the error spectra bend upward for scales comparable to or larger than the scale of the window (roughly wavenumber 10 in this case). The cause of this artifact is not obvious to us, but it does not limit the ability of the windowed 1Dκ spectra to capture the upscale error evolution.

6. Conclusions

Measures of error growth in meteorological forecast models are primary tools for assessing atmospheric predictability. The spatial patterns of an error field are often analyzed through Fourier decomposition to determine the prominent error-growth length scales. It has become common² to take the latitudinal average of the Fourier transform in the zonal direction to reduce 2D error spectra to 1D (1Dk spectra). These 1Dk error spectra are roughly independent of scale from large to small scales (basically a “flat line”) and maintain this form as the error spectra grow with time. Such growth of the error spectra is said to be “up-amplitude.” In contrast, statistical predictability theory predicts error spectra that are circularly symmetric in 2D wavenumber space with peak amplitude at a finite scale. These theoretical 2D spectra are reduced to 1D through integration around annuli in wavenumber space as a function of wavenumber magnitude κ (1Dκ spectra). These error spectra increase from nearly zero at large scales toward a peak at smaller scales and maintain this form as the error spectra grow with time. Since the smaller-scale peak wavenumber decreases with time (in the “−5/3” case), such growth of the error spectra is said to be upscale. The purpose of this note is to show that the two methods (1Dk and 1Dκ) of reducing 2D spectra to 1D are not the same, except in the special case of a power-law spectrum.

We show that 1Dk spectrum from a hypothetical error field typical of statistical predictability theory exhibits the “flat-line” low-wavenumber signature seen in many such analyses in the literature. A hypothetical case of an error spectra growing in amplitude with a decreasing peak scale, illustrates the “up-amplitude” versus upscale descriptions. The relevance of statistical predictability theory was shown in Fig. 13 of Judt (2018) and partly reproduced here as Figs. 4a and 4b. Using the same dataset, Figs. 4c and 4d show that 1Dk-type error spectra exhibit the low-wavenumber flat-line shapes while their 1Dκ-type counterparts in Figs. 4a and 4b do not.

Finally, we suggest how 1Dκ spectra can be computed on spatial subdomains, which is a useful diagnostic tool for error fields that have spatially inhomogeneous statistics.

Although the analyst is at liberty to choose either 1Dk or 1Dκ error spectra, only the latter forms a basis for comparison of meteorological-model error spectra and the predictions of statistical predictability theory. If one chooses the former, then a dynamical interpretation must be based on an error equation resulting from a consistent application of a Fourier decomposition and averaging of the governing equations. We are unaware of any such equation in the literature.

Data availability statement.

The output from the global high-resolution simulations in Figs. 4 and 7 can be made available upon request.