1. Introduction
It is a well-known fact that the kinetic energy spectrum of the upper troposphere shows a −3 slope in the macroscale (between 3000 and 1000 km; Boer and Shepherd 1983) and a 
When data from a numerical model are used to compute the kinetic energy spectrum, the same −3 and
There are different methods to compute the spatial spectra for a limited-area model. Because of the limited area, the (horizontal) data field is not periodic and thus the Fourier transform cannot be applied directly. If the data field is one-dimensional, then the removal of the linear trend and the use of window functions are a common solution to this problem. If the data field is two-dimensional, then there are two main methods for dealing with the issue: Errico (1985) proposed to remove a linear trend piecewise in all directions before applying the Fourier transform. Denis et al. (2002) proposed using the cosine transform instead of the Fourier transform because it also works with aperiodic fields. Other ways to compute a spectrum include using the (fast) wavelet transform (Bacmeister et al. 1996) or eigenmodes of the spherical Laplacian operator (Park and Cheong 2013).
The aim of this paper is to find the effective resolution of a high-resolution regional climate model and to analyze the kinetic energy spectra of 10-yr climate simulations. This represents a multiyear time scale that goes down to convection-resolving grid scale. This allows for a study of variations between seasons in a climatological sense. Section 2 starts off with a description of the analyzed data and the Consortium for Small-Scale Modeling (COSMO) model in Climate Mode [COSMO-CLM (CCLM)], followed by an explanation of the methods used to compute the energy spectra and effective resolutions. In section 3 our findings will be presented and discussed. In the last section a summary of our results is given.
2. Data and methods
a. The regional climate model
Hindcast high-resolution climate simulations of the years 1991–2000 were performed for this study. They were conducted for two areas with different resolutions (see Fig. 1). The first simulation has a resolution of 4.4 km and covers an area over central Europe. The second one has a resolution of 1.3 km and covers the region of mid-Germany and Luxembourg. ERA-40 data were used for boundary conditions to force a simulation over Europe with a resolution of 18 km (Keuler et al. 2012), in which the 4.4-km run was nested. The 1.3-km run was then nested in the 4.4-km run. The 18-km domain covers an area of 4500 km × 5000 km centered over Germany; that is, it covers all of Europe and northern Africa (Hollweg et al. 2008). The horizontal grid size for the 4.4-km run was 254 × 254, for the 1.3-km run it was 220 × 220. To avoid the influence from the boundary, 20 grid points were cut off from each side, resulting in 214 × 214 and 180 × 180 grid points (black squares in Fig. 1). The vertical grid has 40 levels (10 levels below 1 km and 10 levels above 10 km), is terrain following near the ground (level 40), and changes into a pressure-following system at the upper levels (from levels 9 to 1). Output from the simulations was available for every hour. To avoid spinup effects, the first 72 h were dropped, which left 87 600 h for the evaluation.
(a) The domain of the 4.4-km grid (red) and the 1.3 km grid (blue). (b) The topography of the 1.3-km grid. (c) As in (b), but for 4.4 km. (d) The topography of the 4.4-km grid for the 1.3-km subdomain. The black squares in (b)–(d) show the smaller subdomains over which the spectra are computed.
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
The CCLM is a nonhydrostatic model and is used as the community model for German climate research. It is a modified version of the COSMO model (Steppeler et al. 2003) used by the Climate Limited-Area Modeling (CLM)-Community (Rockel et al. 2008; www.clm-community.org).
The model grid was centered around the origin of a rotated coordinate system (rotated north pole at longitude = 180, latitude = 40), thus resulting in an almost equidistant grid. The horizontal grid size was 0.4° and 0.0118°, respectively (in the rotated coordinate system), resulting in 4.4- and 1.3-km spacing, respectively. For more details see Gutjahr and Heinemann (2013).
b. Kinetic energy spectrum
To compute spectra of a two-dimensional field, we used the method described in Errico (1985), which we denote with E2 (Errico, two-dimensional), and the method described in Denis et al. (2002), which we denote with D2 (Denis, two-dimensional). The coefficients are summed up in the two-dimensional wavenumber space over rings to produce a one-dimensional spectrum. A more detailed description of both methods is given in the following.
Although 
Similarly to 
A reason for summing up over annuli (instead of other geometrical forms) is that the spectrum of a two-dimensional homogeneous wave (i.e., sine/cosine pattern along one axis and homogeneous along the other axis) rotates on a circle in spectral space, when the wave direction is changed (in fact, the angle of wave rotation corresponds to the angle of the circle rotation in spectral space).3
To compute the spectra of a one-dimensional field, four different methods are used. The first two methods are D2 and E2, which work also for the one-dimensional fields and will be called D1 and E1, respectively, in this one-dimensional case. The third method, DD1, first detrends the data (subtractions of the linear regression line) and then applies method D1 to the detrended data. The fourth method uses the Fourier transform (F1). First, the data are detrended, then a Gaussian bell taper is used to damp 10% of the data at each end. After this preparation the Fourier transform is applied.
The one-dimensional methods D1, E1, DD1, and F1 are used on two-dimensional fields in the following way. The spectra along each column and row (latitude and longitude of the rotated horizontal grid) are computed separately and averaged to get a single spectrum.
An overview of the methods is given in Table 1.
Methods to compute spectra (D=Denis, DD=detrended Denis, E=Errico, F=Fourier transformation, 1 = one dimensional, and 2 = two dimensional).
c. Effective resolution
Skamarock (2004) proposed to identify the effective resolution either by comparing the spectrum of the model with measurements (e.g., Lindborg 1999; Nastrom and Gage 1985) or by comparing the spectra of two simulations for the same region and time, but with different resolutions. In the latter case, it is assumed that the spectrum of the simulation with the finer resolution is following the correct spectrum longer toward small scales and thus can be used to identify the effective resolution of the coarse simulation.
We are following the second approach for the 4.4-km simulation and choose the area for comparison inside the 1.3-km domain. This area is covered with 54 × 54 grid points by the 4.4-km simulation and with 180 × 180 grid points by the 1.3-km simulation. Both spectra are interpolated using piecewise cubic splines to make a comparison possible.
3. Results
a. Method tests
All one-dimensional methods were tested on one-dimensional artificially created data [similar to the testing done by Denis et al. (2002)]. Several cosine waves with different random wavelength and phases were multiplied with a factor and summed up. The factors were chosen so that the sum would have a specific spectral slope. It was found that all methods work correctly for the relevant spectral slopes between
A simple comparison of the methods for CCLM data is shown in Figs. 2 and 3. The two plots show the kinetic energy spectrum of the 1.3- and 4.4-km runs for the model level 15 (at approximately 7-km height) for their respective grids. The kinetic energy spectrum is computed as the mean of the U and V wind component variance spectra. All methods give the same slopes in their respective small scales (<100-km wavelength for the 4.4-km run and <25-km wavelength for the 1.3-km run), but the two-dimensional methods D2 and E2 compute a higher spectrum in those ranges. A difference was of course expected since the integration over the spectra equals the one- or two-dimensional variance. For the respective larger scales (>100- and >25-km wavelengths), all spectra show approximately the same values. An exception to this is the first coefficient, which is strongly influenced by differences between the methods in handling large-scale trends in the dataset. For example the spectra of the methods D1 and DD1 are almost identical, with only the first coefficient being higher for D1 (because the large-scale trends were not removed).
Kinetic energy spectra (mean over 10 years) of the 4.4-km run on level 15 (7-km height). The black reference line has a slope of
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
Kinetic energy spectra (mean over 10 years) of the 1.3-km run on level 15 (~7-km height). The black reference line has a slope of
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
The same comparison of methods for the vertical wind variance spectrum is shown in Fig. 4. The method E1 produces values that are too high for the first coefficients (same problem as with artificial data). Overall, the first coefficient(s) of the spectra should be viewed with caution as the methods compute very different values.
Vertical wind variance spectra (mean over 10 years) of the (top) 4.4- and (bottom) 1.3-km runs on halflevel 15 (~7-km height). The solid black reference line has a slope of
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
b. Spectra of the model
As a first step, the model results were compared to the spectrum found in measurements. A comparison with Lindborg (1999) (and therefore Nastrom and Gage 1985)4 is shown in Fig. 5 for the horizontal wind components and the potential temperature. The flight height of the airplanes for the measurements used by Nastrom and Gage (1985) was between 9 and 14 km. The displayed model levels 13 and 7 have a height of 8.4 and 13.3 km, respectively. The spectra of the 4.4- and 1.3-km runs were computed for their respective grids. The correct variance spectra of the U and V wind components are reproduced by the model (up to an effective resolution), although the 4.4-km spectrum lies a bit too high for long waves. The potential temperature variance spectrum agrees with the findings of Nastrom and Gage (1985) (where the spectrum was located slightly below the wind variance spectrum as well but had the same shape). A different study by Bacmeister et al. (1996) used aircraft measurements in the lower stratosphere (around 20 km) and found a steeper spectral slope of −2.5 (instead of 
Variance spectra (mean over 10 years) of the U and V wind components, and potential temperature for model levels 13 (dashed line, ~8-km height) and 7 (solid line, ~14-km height) for the 4.4-km (red) and 1.3-km (blue) runs. Spectra of the V wind (theta) are shifted one (two) decade to the right. The spectra were computed using the method E1. The black line is the spectrum from Lindborg (1999).
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
The height dependence of the kinetic energy spectra is shown in Fig. 6. For the kinetic energy spectra, a decrease with height is present in the troposphere for wavelengths smaller than 200 km. The 1.3-km vertical wind variance spectra show also a decrease with height in the troposphere for wavelengths smaller than 10 km. For larger wavelengths the spectra do not change much in the troposphere. A relatively sharp transition is found at around 10-km height for the smallest scales (best visible in the 1.3-km kinetic energy spectra; Fig. 6). This feature is not present in the spectra of every time step, but it occurs only for certain periods (25%–50% of the time). Further analysis showed no dependence on the season or on the daily cycle (or a combination of both).
Kinetic energy spectra as a function of height (mean over 10 years) computed with method E2 for the 4.4- and 1.3-km domains.
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
For the vertical wind variance spectra, Bacmeister et al. (1996) found a spectral slope of zero (i.e., no slope) down to the wavelength of 10 km, where it changed to a −2.5 slope. The vertical wind variance spectrum from the CCLM simulations also shows this zero slope up to a wavelength of 5–25 km depending on height and the resolution (see Fig. 4 and 7). The natural steepening of the slope at a wavelength of 10 km is difficult to distinguish from the expected false steepening of the slope due to the effective resolution (which could be different for the vertical wind). But a comparison of the 4.4- and 1.3-km spectra indicates that the natural change of the slope for wavelengths smaller than 10 km (in the stratosphere) seems to be at least partially present in the model. This is consistent with the findings of Langhans et al. (2012), where the COSMO model produced a similar vertical wind variance spectrum.
Vertical wind variance spectra as a function of height (mean over 10 years) computed with method E2 for the 4.4- and 1.3-km domains.
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
Both the kinetic energy (Fig. 6) and the vertical wind variance spectra (Fig. 7) show the largest changes with height near the ground and in the stratosphere, where the spectrum is smallest. For wavelengths smaller than 25 km, these vertical changes are most pronounced. The variance spectra of the vertical wind show a maximum at a height of 1 km, which is the typical height of the convective boundary layer. This indicates that there might be a relation to summertime convective activity.
Comparisons between spectra over flat terrain with those over mountainous terrain showed the influence of the topography in an increase of the spectral power over mountains.
Data concerning a seasonality of the kinetic energy spectrum are sparse. Nastrom and Gage (1985) reported a spectrum with lower intensity in summer and higher intensity in winter. Park and Cheong (2013) found the opposite for NCEP data. For our 10-yr simulation with 1.3-km resolution, we find different variations dependent on the height (see Fig. 8). Near the tropopause (levels 5–12), higher values occur during summer and smaller values during winter with the maximum around May–July and the minimum around February–March. The summertime increase near the tropopause is most pronounced for wavelengths smaller than 10 km and is likely caused by deep convection. For other heights, the seasonal variation of the spectrum depends on the wavelength range. For the stratosphere (levels 1–4) and below 1 km (levels 31–40), a decrease in summer is present for wavelengths larger than 10 km. For the troposphere (levels 13–30), a slight increase occurs during summer for wavelengths smaller than 30 km.
Kinetic energy spectra as a function of month (from 1 = January to 12 = December) computed with method D2 for the 1.3-km domain. The mean was taken for each month over all 10 years and over the levels 1–4 (16–26 km), 5–12 (9–15 km), 13–30 (1–8 km), and 31–40 (10 m–1 km).
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
The vertical wind variance spectrum depends also on the season (Fig. 9). Again, a large seasonal variation is found for the stratosphere for large scales and for the troposphere for small scales. Since the vertical wind is computed at different levels than the horizontal wind (half levels), the heights have been chosen differently compared to the kinetic energy spectra. For small scales (<5–10 km), the convective activity can be seen in summer for the whole troposphere (200 m–15 km). Similar results are found for the 4.4-km run (not shown).
Vertical wind variance spectra as a function of month (from 1 = January to 12 = December) computed with method D2 for the 1.3-km domain. The mean was taken for each month over all 10 years and over the halflevels 2–5 (16–21 km), 6–12 (9–15 km), 13–36 (200 m–9 km), and 37–41 (0–140 m).
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
When comparing the kinetic energy spectra for different years, a slight variability of the height of the spectrum (i.e., overall variance) was visible, but no trend was present and no change in the shape of the spectrum was found.
c. Effective resolution
In Fig. 10 (top) kinetic energy spectra for 1.3, 4.4, and 18 km are shown. The spectra are computed as before, on their respective grids, without boundary influence zones (18 km—all of Europe; 4.4 km—mid-Europe; and 1.3 km—mid-Germany). The 18-km spectrum follows the expected −3 slope until around 100 km, but then it falls off. In contrast the 4.4-km spectrum captures the transition to the expected
(top) Kinetic energy spectra (mean over 10 years) on level 15 (~7-km height) for 1.3, 4.4, and 18 km over their respective domains computed with method E1. The solid black reference line has a slope of
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
To determine the effective resolution, we follow the method described in section 2c, which computes the spectra over the same domain (mid-Germany). In a range of 20% tolerance for a lower bound and a 50% tolerance for the upper bound around the 1.3-km spectrum, the 4.4-km spectrum is regarded as being similar to the 1.3-km spectrum. The lower bound is used to determine the point at which the 4.4-km spectrum falls off. The first values of the 4.4-km spectrum (large wavelengths) are ignored in this procedure as they often lie beyond the boundary. Figure 10 (bottom) illustrates this method for the kinetic energy spectra at about 1-km height computed with E2. In this example the 1.3-km spectrum follows the 
If a 10% tolerance for the lower bound is used, then the uppermost six levels have to be excluded from the analysis (same reasons as above). Values of the effective resolution lie between 7 and 13
Bierdel et al. (2012) reported an effective resolution of 4–5
In Fig. 11 the difference of the 1.3- and 4.4-km spectra is plotted for both two-dimensional methods. The difference between both methods for wavelengths between 240 and 100 km stems most likely from large-scale trends. While the method E2 removes those trends, the method D2 does not and yields an increase in the first cosine coefficients and thus an increase in the first part of the spectrum. Overall, the difference plot shows the benefit of the high-resolution simulations for wavelengths smaller than 50 km. Particularly at heights below 1 km, the 1.3-km run benefits from the more detailed topography and a better resolution of the related boundary layer processes.
Difference of kinetic energy spectra as a function of height (1.3 km minus 4.4 km over the same domain) for methods (top) E2 and (bottom) D2.
Citation: Journal of Climate 29, 3; 10.1175/JCLI-D-15-0540.1
4. Conclusions and summary
In this paper we used the one- and two-dimensional cosine and the Fourier transforms to compute kinetic energy spectra. The spectra were computed for a dataset of 10 years of regional climate simulations, which was run with COSMO-CLM at 4.4- and 1.3-km resolution. The spectra show many features, which are in agreement with observations:
steeper slope in the lower stratosphere, and
vertical wind variance spectrum with a zero slope.
The effective resolution was found to be 7–10 (one-dimensional methods) and 5–7 (two-dimensional methods) times the horizontal spacing (for the 4.4 km simulation) depending on the height of the wind field used. Using the same methodology as Bierdel et al. (2012), an effective resolution of 5.6
It should be noted that at least for the COSMO model, the effective resolution seems to be mainly restricted by the choice of the numerical schemes. Tests with the same model but using an improved numerical scheme (Ogaja and Will 2015, manuscript submitted to Meteor. Z.) were able to increase the effective resolution by a factor of 2
The applied methods to compute the spectra (Errico 1985; Denis et al. 2002) do not show much difference. But a big difference exists between one-dimensional and two-dimensional methods. The two-dimensional methods generally yield higher values for spectral intensity. Also, the effective resolution is better if two-dimensional methods are used to compute the spectra instead of one-dimensional methods.
Acknowledgments
The authors thank the CLM-Community, Klaus Keuler, and Kai Radtke (BTU Cottbus) for providing the CCLM 18-km data. The 1.3- and 4.4-km simulations were funded by the Research Initiative Rhineland-Palatinate and carried out within the “Global Change” project of the University of Trier by Lukas Schefczyk and Oliver Gutjahr (University of Trier), whom we thank as well for some help concerning its usage.
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Because the dimensions of our fields were equal, we only describe the case for 
When computing the cosine transform (for large data fields), this formula should not be used. Similarly to the fast Fourier transform, there are fast algorithms that reduce the computing time for the (discrete) cosine transform from 
This holds true only for a grid with equal spacing in both directions. If the spacing for the north–south direction is different from the spacing in the west–east direction, then a corresponding elliptic shape should be used.
The Lindborg spectrum (although derived by the use of structure functions and MOZAIC data) fits the U and V wind variance spectrum from Nastrom and Gage (1985) [which was derived with the Fourier transform and the Global Assimilation and Prognosis (GASP) dataset] quite well.
