1. Introduction
The dynamical origin of the kinetic energy (KE) spectrum on the mesoscale range (wavelengths from a few kilometers up to several hundred kilometers) has been a mystery to atmospheric scientists for more than three decades (Gage and Nastrom 1986; Gkioulekas and Tung 2006; Sun et al. 2017; Craig and Selz 2018). Understanding the dynamics underlying the mesoscale KE spectrum is not only of academic interest, but it is central to more applied aspects of numerical weather prediction, including error growth (Lorenz 1969; Durran and Gingrich 2014; Weyn and Durran 2017) and the accuracy of numerical forecasting systems (Skamarock 2004; Frehlich and Sharman 2008; Ricard et al. 2013; Skamarock et al. 2014).
In a seminal paper, Nastrom et al. (1984) considered KE spectra based on flight-track data taken in the course of the Global Atmospheric Sampling Program (GASP), motivated in part by the expectation of a mesoscale energy gap (Fiedler and Panofsky 1970). Instead of finding lower energies on the mesoscale than found on the smaller convective and larger baroclinic scales, the atmospheric KE spectrum showed a power-law shape with a −3 exponent in wavenumber on large scales, shallowing to an approximate −5/3 slope on the mesoscales. Subsequent studies by Nastrom and Gage (1985) and Gage and Nastrom (1986, p. 733) emphasized “the remarkable degree of universality in spectral amplitude and spectral shape over the entire range.” Power-law KE spectra are a characteristic property of self-similar flow, such as fully developed, homogeneous, and isotropic turbulence (Kolmogorov 1941; Obukhov 1949). Such flows can be assessed statistically, rendering the mathematical description of the flow and the associated KE spectrum tractable. The −3 spectral slope on the large synoptic scales has been found to be in good agreement with dimensional considerations in the framework of barotropic quasi-two-dimensional (Kraichnan 1967) or quasigeostrophic (Charney 1971) turbulence theory. The discovery of a universal −5/3 spectral slope for the mesoscale KE spectrum was thus groundbreaking, since it paved the way for statistical theories treating the mesoscales as an inertial subrange governed by a universal dynamical process.
The mesoscales cover a wide spectral range and encompass processes with different underlying dynamics, so attributing the observed spectral slope to a single dynamical mechanism is difficult (Bühler et al. 2014). The established theories involve gravity waves (VanZandt 1982; Dewan 1997), two-dimensional vortical flow (Gage 1979; Lilly 1983), and stratified turbulence (Lindborg 2006), and have been tested in numerical modeling frameworks of varying complexity (Koshyk et al. 1999; Tung and Orlando 2003; Lindborg 2005; Takahashi et al. 2006; Kitamura and Matsuda 2010; Brune and Becker 2013; Waite and Snyder 2013). However, over the last few decades, no consensus about the dynamical mechanism underlying the mesoscale KE spectrum could be reached (Gkioulekas and Tung 2006). The lack of a compelling theory to describe the mesoscale KE spectrum might be related to the questionable applicability of simplified statistical frameworks to the highly complex real atmospheric flow (Zhang et al. 2007; Waite and Snyder 2013; Sun et al. 2017; Weyn and Durran 2017). Indeed, there might be no universal mesoscale dynamical process that governs the observed spectral shape.
In the present paper we want to analyze the variability of the mesoscale KE spectrum. Previous studies evaluating measurement data have found variations related to season and latitude (total energy is smaller in summer than in winter and increases with latitude; Nastrom and Gage 1985; Cho et al. 1999), and to underlying terrain (total energy is up to 10 times larger over mountainous terrain than over the ocean; Nastrom et al. 1987; Jasperson et al. 1990). Dependence of the mesoscale KE spectrum on region, height, and season was also found in several numerical studies (Koshyk et al. 1999; Skamarock 2004; Bierdel et al. 2016). However, the attractive paradigm of a universal atmospheric KE spectrum has received much more attention than the significant variability found in the GASP data (Lindborg 1999; Tung and Orlando 2003; Skamarock 2004), and thus far no systematic examination of the variability of the mesoscale KE spectrum has been conducted.
In this study we quantify the variability in kinetic energy on different scales and examine how the shape of the mesoscale KE spectrum is related to particular dynamical processes. Because simulated spectra can be sensitive to the model configuration (e.g., horizontal and vertical resolution and strength of dissipation; Koshyk and Hamilton 2001; Skamarock 2004; Augier and Lindborg 2013; Brune and Becker 2013), we employ routine operational analysis data to reduce the impact of the forecasting system.
The paper is structured as follows: In section 2, the data and the methods used for calculating spectra are described. The results of this analysis are presented in section 3. We first discuss the variability of the upper-tropospheric mesoscale KE spectrum, including the dependence on season (section 3a) and the contributions of the rotational and divergent winds. Second, the dynamical processes setting the amount of KE in the small-scale (section 3b) and large-scale (section 3c) parts of the mesoscale range are investigated. Finally, four example cases are discussed to illustrate the weather regimes associated with different spectral shapes (section 3d). The paper concludes with a summary and discussion in section 4.
2. Data and methods
Routine operational analyses over Germany provided by the German Weather Service [Deutscher Wetterdienst (DWD)] are used to evaluate the variability of the KE spectrum on the mesoscales. These analyses are produced with the Consortium for Small-Scale Modeling (COSMO) numerical weather prediction model (Baldauf et al. 2011) configured over Germany (COSMO-DE). The model domain contains Nx = 421 and Ny = 461 grid points in the longitudinal and latitudinal directions, respectively. COSMO-DE uses a rotated regular grid where the origin is shifted to 50°N, 10°E. With the horizontal grid spacing of 
In addition to the KE spectra of the full wind field, spectra along wet lines and dry lines are considered separately. A straight line between two arbitrary grid points in the domain can be computed with Bresenham’s line algorithm (Bresenham 1965). The atmospheric fields at the points along the line then provide a one-dimensional nonperiodic representation of the data. At each time step, a line is calculated from every fifth grid point in the domain to every other fifth grid point, which results in a total of about 
Correlation analyses of time series are used to investigate the relationship between variables. However, care must be taken because meteorological time series show high autocorrelations over several days and are thus nonstationary, which may lead to spurious correlations (Farnum and Stanton 1989). A nonparametric way to solve this problem is first differencing of the data in time, that is, from a time series 
3. Results
a. Variability of the spectra
First we consider the distribution of KE spectra at 300 hPa over the whole period. Spectra between 500 and 200 hPa were also examined but show similar results, so the spectra displayed here are a good representation of the upper-tropospheric dynamics. To assess the spectral variability, the logarithmic energy axis is divided into bins. The population of each bin is counted and normalized such that the sum equals 1. This relative variability is displayed as color shading in Fig. 1a. Solid, dotted, and dashed black lines depict the mean spectrum averaged over the whole period, winter (December–February), and summer (June–August), respectively.
(left) Distribution of spectral KE from 3 years of COSMO-DE analyzed as a function of horizontal wavelength. Spectra are averaged over the whole period (solid line), winter (dotted), and summer (dashed). The colored shading indicates the logarithm of frequency of occurrence of the energy level over the whole period. Black lines above the spectra indicate a −3 and −5/3 slope for reference. (right) Relative variability of the spectra during the whole period (black), winter (blue) and summer (red). For further explanations, see text.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0155.1
The first and perhaps most important finding of the present paper follows directly from Fig. 1a: the mesoscale KE spectrum is highly variable over the analysis period. The spectral energy varies over at least one to two orders of magnitude on all wavelengths. In agreement with previous studies, the mean spectrum averaged over the whole dataset features an approximate −3 spectral slope and transitions to a slope steeper than −5/3 (see comparison with solid black lines above the spectra). However, this structure is not representative of individual realizations of the flow. Indeed, within the period of time under consideration, the slope of the KE spectrum estimated over the entire range of scales varies from −1.67 to −3.14 with a mean slope of −2.38. From Fig. 1a it can also be seen that the mean KE spectrum contains more energy on large scales in winter and on small scales in summer. Rather than search for universality, we will proceed to examine the variability of the spectra.
To assess the relative variability as a function of wavelength, the spread between the 1st and 99th percentiles (i.e., the width of the colored band in Fig. 1a) is considered. The fraction of the 99th percentile over the 1st percentile is a relative (dimensionless) measure of the variability of the spectra at each wavelength. This variability is shown in Fig. 1b, averaged over the whole period (black) and also separated into winter (blue) and summer (red) months. The variability is, in general, u shaped, featuring high values at the largest and smallest scales. While the relative variability on the small scales is clearly highest in summer, the large scales are highly variable through the whole time period. The overall variability on large scales is thus greater than on small scales. At around 100-km horizontal wavelength, all curves (overall mean, summer, and winter) show a clear minimum. This minimum in spectral variability is consistent with the picture of the mesoscales (Craig and Selz 2018) as a less active range in between the large and small scales that are directly forced by baroclinic and convective instabilities, respectively.
Spectra and variability obtained by partitioning the wind into its divergent and rotational components are shown in Fig. 2. The left part of the figure shows the well-known result that the divergent and rotational contributions are equal for small scales, while the rotational wind clearly dominates at larger scales. In Fig. 2b the relative variability of the divergent and rotational components shows that both are u shaped, as found for the total wind, with a clear minimum at around 100 km. However, the relative variability associated with the rotational wind is greater than that of the divergent part on all scales.
As in Fig. 1, but with separation of the wind into its divergent and rotational components.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0155.1
The shape of the spectrum and the associated variability might thus be related to the variable dynamical mechanisms driving the small- and large-scale flows, rather than the variability associated with a dynamical mechanism on the mesoscale range. The relationship between the spectra and the small- and large-scale dynamical forcing mechanisms are examined in the following two subsections.
b. Small-scale variability and precipitation rate
The high variability of the KE spectrum on small scales in summer months suggests a relationship between the amount of small-scale KE and cumulus convection. To investigate this hypothesis, we correlate the KE at each wavenumber with the precipitation rate. Since the dataset does not distinguish between convective and stratiform precipitation, we use the total precipitation rate, and since the dynamical response to convective heating is spread over a broad range of scales (Bierdel et al. 2017, 2018), we use the domain-integrated precipitation rate rather than decomposing it with wavenumber. The result is displayed in Fig. 3 for the whole dataset, the winter and summer seasons, and the rotational and divergent wind.
(left) Spearman rank correlation coefficient of the KE and the domain-integrated precipitation rate as a function of horizontal wavelength for the whole dataset (black), winter (blue), and summer (red). (right) As in the left panel, but separated into divergent and rotational wind. All correlations larger than about 0.05 for the full dataset and 0.10 for the winter and summer time series are significant at the 99.9% confidence level.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0155.1
Figure 3a reveals a clear relationship between the spectral energy and the precipitation rate regardless of season. Overall, correlations are highest in summer and lowest in winter. The lowest correlations are generally found on horizontal scales larger than about 300 km, with values ranging from around 0.03 (winter) to 0.15 (summer). For scales smaller than 300 km, correlations increase significantly with deceasing scale. The maxima in correlation take values of up to about 0.2 (winter) and 0.65 (summer) with 
Because the level of small-scale kinetic energy contributes strongly to the spectral slope, we also calculated the correlation of the slope with the domain-integrated precipitation rate. The resulting correlation coefficients are 0.34 for the whole dataset, 0.51 for the summer months, and 0.17 for the winter months. The correlation of the domain-integrated precipitation rate to the total energy is only 0.10, but this is not surprising, since the small scales contribute only a small fraction to the total energy. All correlations are significant at 
Since convective precipitation is associated with strong latent heating in the midtroposphere and thus with divergent outflow at the upper troposphere, it is expected that the KE of the divergent part of the flow is more strongly correlated with precipitation than the rotational part. Indeed, Fig. 3b confirms this hypothesis: correlations of the divergent wind to the KE are higher at all investigated scales, despite the fact that the relative and total variability of the rotational KE is larger (Fig. 2).
To clearly show the impact of moist processes on the kinetic energy spectrum, we computed spectra along dry- and wet lines as described in section 2. Figure 4 shows the mean KE spectrum at 300 hPa separately for wet lines (solid) and dry lines (dashed). The color shading indicates the distribution of the KE spectrum in the summer months and is calculated as explained in section 3a. The number of spectral modes entering Fig. 4 varies with scale and between wet- and drylines. On average there are 1.2 × 107 modes per time step from dry lines and 5.9 × 103 modes per time step from wet lines. As expected, only a few time steps (301 out of 2208) contribute to the largest wet modes.
As in Fig. 1a, but color shading indicates the variability of the KE spectra in summer months only. Black and blue solid lines indicate spectra averaged over summer months, separated into wet and dry segments, respectively (see section 2). The dashed lines show linear fits to the wet and dry spectra.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0155.1
From Fig. 4 the difference of KE in wet and dry regions is clearly visible at scales smaller than 100 km and it essentially vanishes at the largest horizontal scales. This is consistent with the variation in the correlation coefficient across scales seen in Fig. 3. Power-law fits for the dry and wet spectra are also plotted in Fig. 4, showing slopes of −2.55 for the dry case and a much shallower value of −2.01 for the wet case.
c. Large-scale variability and potential vorticity
The variation of the KE at larger scales suggests a relationship with the synoptic flow over Germany, that is, with the geostrophically balanced dynamics. This would imply that the KE would be related to pressure gradients, which are in turn associated with potential vorticity (PV). Since the COSMO-DE analyses dataset does not contain PV, it is instead derived from the fifth major global reanalysis produced by the European Centre for Medium-Range Weather Forecasts (ERA5; Hersbach and Dee 2016). PV is indicative of the large-scale circulation; thus, it is unlikely to differ significantly in the limited-area COSME-DE model. PV fields are also available at 300 hPa every 3 h for the entire period. As for precipitation, a PV time series is constructed by integrating the PV field over the COSMO-DE domain. This time series is correlated with the kinetic energy, and the results are shown in Fig. 5 for the whole dataset and for separations into summer and winter seasons and divergent and rotational winds, as before.
As in Fig. 3, but with the KE correlated with the domain-integrated potential vorticity, taken from ERA5. All correlations larger than about 0.05 for the full dataset and 0.10 for the winter and summer time series are significant at the 99.9% confidence level.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0155.1
Contrary to the correlation with precipitation rate, correlations between the KE spectrum and the PV generally increase with increasing scale. The correlations decrease almost to zero on scales smaller than about 100 km over the whole time period, as well as in summer and in winter (Fig. 5a). The maximum correlations are found on the largest analyzed scales (around 1000 km) and equal 0.29 (whole time series), 0.35 (summer), and 0.26 (winter) with 
To further confirm this result, correlations of the rotational and divergent KE to the PV can be considered (Fig. 5b). These show a similar increase of the correlation of the rotational part with increasing scale, while the divergent part shows a correlation close to zero on all analyzed scales. This is again consistent with the hypothesis that the largest analyzed scales are influenced by downscale cascades from geostrophic turbulence. However, the correlation coefficient of the rotational KE at the largest scales is less than that for the total KE, which points to some source of error, since at those scales the rotational and total energies (and thus the correlations) should be almost equal (Fig. 2). This discrepancy is not present at smaller scales. One possible explanation for this problem is aliasing, where unresolved modes that are larger than the domain size project onto the largest resolved modes. This may influence the two correlations in different ways.
d. Four example cases
The correlation analysis in the preceding subsections showed that small-scale KE varies with the precipitation rate, while large-scale KE is related to the 300-hPa domain-integrated PV. The spectral shape (i.e., amplitude and slope) of the mesoscale part of the spectrum will likely be influenced by both of these dynamical processes, whose relative strengths vary with season, location, and the daily weather. To illustrate this variability, four example cases that show the range of observed spectral shapes will now be presented.
In each of the four panels of Fig. 6, the left part shows a snapshot of the KE spectrum at 300 hPa for the time noted above the figure (solid black line). The color shading indicates the variability of the spectrum over the whole time period as in Fig. 1. A least squares fit computed over the whole range of scales is depicted as a dashed black line, and the estimated slope m as well as the total KE E is given in the upper right of each figure. Note that the total KE is calculated only from the displayed modes, so modes 0 and 1 are omitted. The right figure in each of the four panels shows the domain of the COSMO-DE model with isolines corresponding to the 300-hPa geopotential and blue shading showing the precipitation rate.
Four extreme examples of the mesoscale KE spectrum. (a) Small total KE, (b) high total KE, (c) steep slope, and (d) shallow slope. The left parts show the KE spectrum for the time (solid), the fitted slope (dashed), and the variability over the entire dataset (color shading as in Fig. 1a). The right parts show a map of the COSMO-DE domain, plotted with 300-hPa geopotential contours, with a line spacing of 200 m2 s−2 (black solid lines), and precipitation rate (darker blue shading).
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0155.1
Figure 6a shows a KE spectrum with exceptionally low KE over the entire scale range. As expected from the correlation analysis, the low energy on small scales is associated with a low precipitation rate (weak convective activity) and the low large-scale KE is consistent with the relatively uniform geopotential gradient corresponding to a zonal flow that projects only weakly onto the wavelengths included in the spectrum. The combination of low small- and large-scale KE results in low mesoscale KE. In Fig. 6b a case with remarkably high total KE over the entire scale range is depicted. As expected the high small-scale KE coincides with intense precipitation (organized and convective) over large parts of the domain. Meanwhile, the strongly varying gradients of the geopotential (indicating high PV anomaly) are related to high spectral amplitudes on the large scales.
The KE spectrum depicted in Fig. 6c features an exceptionally steep slope of −3.14. The steep slope on the mesoscales arises from enhanced large-scale KE tied to a strong PV anomaly and a reduced small-scale KE associated with an almost complete absence of precipitation. This relationship is further confirmed in Fig. 6d, where an atmospheric state that features a remarkably shallow KE spectrum with a slope of −1.74 is shown: the shallow slope appears to be tied to low large-scale energy owing to weak gradients of geopotential coinciding with increased small-scale KE associated with enhanced convective activity and precipitation. While the steep and shallow spectral slopes are found in cases in which either the small or the large scales are exceptionally energetic and the other end of the spectral range is characterized by low energies, intermediate slopes are found in cases in which large- and small-scale KE are both either high or low.
4. Summary and discussion
The nature of dynamical processes controlling the atmospheric kinetic energy spectrum on the mesoscale range has long been debated. In this study we have investigated the variability of the mesoscale KE spectrum in slope and amplitude together with the dynamical processes contributing to this variability, using three years of analysis data from a kilometer-scale numerical weather prediction system. The main findings of this study are as follows:
The mesoscale KE spectrum is highly variable.
Enhanced small-scale [smaller than O(100) km] KE is associated with high precipitation rates.
Enhanced large-scale [O(1000) km] KE is associated with high potential vorticity.
The KE spectral slope and amplitude on the mesoscale range is influenced by the combination of these small- and large-scale dynamical processes.
The mesoscale KE spectrum is found to vary systematically with environmental factors and therefore the present study does not support the hypothesis of a universal mesoscale KE spectrum. Maxima in variability found at the small and large ends of the mesoscale range are correlated with the precipitation rate (and latent heat release) and potential vorticity (i.e., strength of pressure gradients), respectively. The coexistence of these two dynamical processes, whether by interaction or superposition, influences the mesoscale part of the KE spectrum. The upscale impact of small-scale dynamical features or the downscale impact of the large-scale flow configuration have a stronger or weaker impact on the intermediate mesoscales depending on the prevailing weather regime. In addition to the significant variability of the mesoscale KE spectra, a minimum in variability is found in the center of the mesoscale range at around 100-km horizontal wavelength. This “gap” in variability—as opposed to the energy gap proposed by Fiedler and Panofsky (1970)—potentially allows for a definition of the mesoscales as the scale range that is minimally affected by variations of the small- and large-scale flow associated with changing weather regimes (Craig and Selz 2018).
If the atmospheric mesoscales are characterized as a relatively passive gap between two highly variable regimes, then it is not surprising that it has proved difficult to identify a single dominant mechanism leading to a universal −5/3 energy spectrum. It may be that alternative theoretical frameworks are required, for example, one that explicitly models the dynamics of scale interactions rather than treating them in a purely statistical manner (Achatz et al. 2017).
This view of the mesoscale gap may also shed light on the processes of error growth that impose intrinsic limits on predictability. Previous studies evaluating specific wintertime situations showed that small-amplitude large-scale perturbations dominate the evolution of the forecast error (Durran et al. 2013; Durran and Gingrich 2014). However, in summertime areas of deep moist convection are found to be primary sources of error growth (Selz and Craig 2015). These findings are consistent with the hypothesis put forward in the present study that mesoscale flow is—depending on the prevailing weather regime—more influenced by large- or small-scale processes (and the associated errors). This understanding might help with the design of future-generation ensemble prediction systems, where (spectral) regions of enhanced sensitivity can be identified and for which the up- or downward impact of associated errors on mesoscale flow can be accounted.
While the current study is limited to three years of COSMO-DE analyses and a domain size of about 1000 km, further investigation of the variability of the mesoscale KE spectrum is desirable. This includes investigations of measurement data, as well as the consideration of longer time periods, larger domains, and analyses produced with other forecasting systems in other geographical regions.
Acknowledgments
The research leading to these results has partly been done within the subproject “A1—Upscale Impact of Diabatic Processes from Convective to Near-Hemispheric Scale” of the Transregional Collaborative Research Center SFB/TRR 165 “Waves to Weather” funded by the German Research Foundation (DFG). The authors thank the German Weather Service (DWD) and the European Centre for Medium-Range Weather Forecasts (ECMWF) for making COSMO-DE analyses and ERA5 data available. The authors are grateful to three anonymous reviewers for their helpful comments.
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