## 1. Introduction

*k*is the wavenumber. Most investigators have tried to explain this observation by cascade hypotheses, falling into two categories: the upscale energy cascade hypothesis (Gage 1979; Lilly 1983) and the downscale energy cascade hypothesis (Dewan 1979). There is strong observational and numerical evidence (Cho and Lindborg 2001; Augier and Lindborg 2013; Deusebio et al. 2014) supporting the downscale energy cascade hypothesis. The question is if a downscale energy cascade is produced by interacting gravity waves (Dewan 1979), strongly stratified turbulence (Lindborg 2006), or quasigeostrophic turbulence (Tung and Orlando 2003). An important tool to discriminate between different hypotheses is to make a Helmholtz decomposition of the energy spectrum:

*χ*is the velocity potential,

*ω*is the intrinsic frequency. The relation (1) follows directly from the linear vertical vorticity equation and holds for each Fourier mode of a wave field, irrespectively of the vertical structure of the field. For waves with

*f*we should have

Lindborg (2007) used a method to differentiate second-order longitudinal and transverse horizontal structure functions calculated using data taken from a great number of commercial flights to estimate the relative magnitude of *r*. The longitudinal velocity component is in the direction of the separation vector **r** between the two points and the transverse velocity component is perpendicular to **r**. The ratio

Results from general circulation models (GCMs) show quite different results. Hamilton et al. (2008) found that rotational energy was about 4 times larger than divergent energy at mesoscales near the tropopause, while Skamarock et al. (2014) obtained rotational and divergent energy of the same order in the upper troposphere (8.5–10.5 km). Divergent energy was more than 5 times larger in the stratosphere (16–18 km). Brune and Becker (2013) found that divergent energy was greater than rotational energy in the troposphere as well as in the stratosphere at the very highest resolved wavenumbers. Based on simulations of baroclinic waves Waite and Snyder (2013) argued that the

Callies et al. (2014) argued that the mesoscale spectra are generated by weakly nonlinear inertia–gravity waves. However, they did not introduce any assumption of an energy cascade, in contrast to Dewan (1979), who first introduced the interacting wave hypothesis, explicitly assuming that wave interactions are associated with a downscale energy cascade giving rise to a *R* was slightly larger than unity at wavenumbers corresponding to wavelengths between 20 and 200 km and used the theory of linear inertia–gravity waves to interpret this observation. From the observation that *R* was just slightly larger than unity at mesoscales, Callies et al. (2014) drew the conclusion that the mesoscale spectrum is produced by linear or weakly nonlinear inertia–gravity waves with frequencies very close to *f*, in accordance with relation (1).

*r*= 20 km, where the structure function show an approximate

Using data from a GCM Bierdel et al. (2016) have recently carried out an extensive study in which (2) and (3) and the mathematically equivalent formulas derived by Bühler et al. (2014) were tested. It was concluded that these formulas give quite accurate results, with typical relative errors of 10% or less. The errors were also found to decrease with decreasing scale or increasing wavenumber. Strong support was thus provided for the use of these formulas in calculating divergent and rotational energy spectra at mesoscales.

Recently, Callies et al. (2016) carried out a renewed spectral analysis of wind and temperature data from the MOZAIC and the 2008 Stratosphere–Troposphere Analyses of Regional Transport (START08) datasets, using altitude to discriminate between stratospheric and tropospheric data. They found that *R* is slightly larger than unity in the lower stratosphere, while different results were obtained from the two datasets in the upper troposphere. From the MOZAIC data it was found that *R* ≈ 0.5 in the upper troposphere, consistent with the result by Lindborg (2015), while *R* was found to be slightly larger than unity when the START08 data was used. Despite the fact that only 15 flight segments were used from the START08 data, Callies et al. (2016) argued that the result from this dataset is likely to be more reliable because of better instrumentation. In analyzing the aircraft data Callies et al. (2014) and Callies et al. (2016) use linear wave theory (e.g., Gill 1982) of inertia–gravity waves according to which the minimum frequency is equal to *f*, which follows from the linear dispersion relation derived under the assumption of a constant *N*. Callies et al. (2016) developed a simple model in which *N* is constant in the upper troposphere as well as in the lower stratosphere but has a singular jump over the tropopause. The dispersion relation is supposed to be valid in each region but with different values of *N*. From the observation *R* ≈ 0.5 it follows that

## 2. Condition for weak nonlinearity

*u*is a horizontal velocity component and the angle brackets represent a spatial average. This condition cannot be tested against aircraft data because these data do not contain any information about the vertical structure of the flow. Therefore, we will focus on the Rossby number condition, which is most appropriately formulated by considering the vertical vorticity equation. The vertical vorticity

*f*plane (Charney 1971).

*r*from the unprimed quantities, both at the same horizontal plane; the angle brackets represent a space average; and

*Q*and

*R*are related (Lindborg 2015) as follows:

*k*. If we instead use

## 3. Data analysis

The aircraft data used in the study are reported in the MOZAIC dataset (Marenco et al. 1998), ranging from August 1994 to December 2010, containing data from 25 922 flights. The flight levels are restricted to the altitude range from 9.4 to 12 km. Each segment that we use contains 2048 data points corresponding to a flight distance of approximately 2000 km. The local velocity components parallel and perpendicular to the flight track are calculated at each point. The two orthogonal components are defined in the following way: **x**, and **x**′ are the coordinates of two points along the flight track, and **t** is a unit vector that is perpendicular to **n**, lying in the horizontal plane. Contrary to other spectral studies using aircraft data, we carry out the Fourier transform directly on the given time signal that is sampled with a frequency of 0.25 Hz. Longitudinal and transverse frequency spectra are computed by applying Welch’s method using a Hanning window with 50% overlap. For each segment, the frequency spectra are then transformed to wavenumber spectra, using the average flight speed over the segment. Using (2) and (3),

### a. Global analysis of tropospheric and stratospheric data

To separate between stratospheric and tropospheric data we use a condition based on ozone concentration as in Cho and Lindborg (2001). To classify as stratospheric the ozone concentration has to be larger than 200 ppbv and to classify as tropospheric it has to be smaller than 100 ppbv at all data points in the segments. The condition that we use in this analysis is considerably stronger than the condition used by Cho and Lindborg (2001), since we require that all points of a whole flight segment should meet the threshold condition, while Cho and Lindborg applied the threshold condition only to each pair of points they used in the structure function calculation. The advantage of using a stronger condition is that data taken very close to the tropopause will be excluded, while the drawback is that less data can be used. In this study we also exclude flights near the equator, between 20°S and 20°N. From the total database we identify 5274 tropospheric data segments and 5553 stratospheric segments. In Fig. 1 we see the distribution of ozone concentration for the whole MOZAIC dataset as well as the distribution in the two subsets that we used. In Figs. 2 and 3 we see the pressure altitude distribution and the latitude distribution, respectively, of both the tropospheric and stratospheric that we have used.

Altitude data distribution.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Altitude data distribution.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Altitude data distribution.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Latitude data distribution.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Latitude data distribution.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Latitude data distribution.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

In Fig. 4 we see the divergent and rotational energy spectra in the troposphere and stratosphere. The spectra are premultiplied by

Compensated divergent and rotational energy spectra in the (left) upper troposphere and (right) lower stratosphere. The number of segments used are 5274 and 5553 in the stratosphere and troposphere, respectively. To classify as tropospheric, all points in a segment must have an ozone concentration smaller than 100 ppbv, and to classify as stratospheric, all points must have a concentration larger than 200 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Compensated divergent and rotational energy spectra in the (left) upper troposphere and (right) lower stratosphere. The number of segments used are 5274 and 5553 in the stratosphere and troposphere, respectively. To classify as tropospheric, all points in a segment must have an ozone concentration smaller than 100 ppbv, and to classify as stratospheric, all points must have a concentration larger than 200 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Compensated divergent and rotational energy spectra in the (left) upper troposphere and (right) lower stratosphere. The number of segments used are 5274 and 5553 in the stratosphere and troposphere, respectively. To classify as tropospheric, all points in a segment must have an ozone concentration smaller than 100 ppbv, and to classify as stratospheric, all points must have a concentration larger than 200 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Compensated temperature spectra in the troposphere and stratosphere. The same number of segments and the same condition of ozone concentration have been used as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Compensated temperature spectra in the troposphere and stratosphere. The same number of segments and the same condition of ozone concentration have been used as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Compensated temperature spectra in the troposphere and stratosphere. The same number of segments and the same condition of ozone concentration have been used as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

The clean

Near-tropopause (left) rotational and divergent compensated spectra and (right) compensated temperature spectra. The spectra are mean values of 6721 segments with mean ozone concentration larger than 120 ppbv and smaller than 180 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Near-tropopause (left) rotational and divergent compensated spectra and (right) compensated temperature spectra. The spectra are mean values of 6721 segments with mean ozone concentration larger than 120 ppbv and smaller than 180 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Near-tropopause (left) rotational and divergent compensated spectra and (right) compensated temperature spectra. The spectra are mean values of 6721 segments with mean ozone concentration larger than 120 ppbv and smaller than 180 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

*a*is obtained in the troposphere as in the stratosphere. With

*λ*. Such a scale-dependent Rossby number may be defined as

*λ*, where we have used the structure function calculated from aircraft data by Cho and Lindborg (2001) in the lower stratosphere to estimate (15). There is only a minor difference between the two Rossby numbers for

Scale-dependent Rossby numbers as function of *λ*. The blue line is the vorticity Rossby number and the red line is the structure function Rossby number based on the stratospheric structure function measured by Cho and Lindborg (2001).

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Scale-dependent Rossby numbers as function of *λ*. The blue line is the vorticity Rossby number and the red line is the structure function Rossby number based on the stratospheric structure function measured by Cho and Lindborg (2001).

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Scale-dependent Rossby numbers as function of *λ*. The blue line is the vorticity Rossby number and the red line is the structure function Rossby number based on the stratospheric structure function measured by Cho and Lindborg (2001).

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

*g*is the acceleration due to gravity,

*N*is the Brunt–Väisälä frequency. Assuming that

In Fig. 8 we see the average ratio *R* between divergent and rotational energy. The lines are the averages over all data segments and the gray shaded area indicates the maximum deviation from the total mean among mean values calculated from five randomly generated exclusive subsets each containing 1000 segments. In the troposphere, *R* is smaller than unity for all wavenumbers, with a peak, *R* ≈ 0.6, at *R* is smaller than unity, we can conclude that divergent energy is on the same order of magnitude as rotational energy at mesoscales. In the stratosphere, on the other hand, *R* is somewhat larger than unity in most of the mesoscale range. It has a peak, *R* ≈ 1.3, at

Ratio between divergent and rotational energy in the stratosphere and troposphere. The lines are mean values over all segments and the gray shaded areas indicate the maximum deviation from the total mean among mean values from five exclusive subsets, each containing 1000 segments.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Ratio between divergent and rotational energy in the stratosphere and troposphere. The lines are mean values over all segments and the gray shaded areas indicate the maximum deviation from the total mean among mean values from five exclusive subsets, each containing 1000 segments.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Ratio between divergent and rotational energy in the stratosphere and troposphere. The lines are mean values over all segments and the gray shaded areas indicate the maximum deviation from the total mean among mean values from five exclusive subsets, each containing 1000 segments.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

To make a more thorough analysis of the statistical distribution of the estimated values of *R* we should first decide the minimum number of segments that is reasonable to use when calculating *R*. Since the determination of ^{1} For the tropospheric data we use *n* = 527 samples and for the stratospheric data we use *n* = 555 samples and calculate *R* at *R*, *R* by definition is positive definite and the distribution is far from symmetric with respect to the mean we plot the distribution of *R*. For comparison we have in each case also added a normal distribution with the directly calculated standard deviation, *σ*, of *R* from the mean corresponding to one standard deviation can be estimated as *s* = 0.037 and *s* = 0.030, for the troposphere and stratosphere, with corresponding relative deviations of *R* of 4% and 3%, respectively. With *n* = 100, that is, 1000 flight segments, the corresponding values are 9% and 7%. If we assume that

Probability density functions of

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Probability density functions of

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Probability density functions of

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

### b. Global analysis separating between land and ocean

In analyzing the Global Atmospheric Sampling Program (GASP) data Nastrom and Gage (1985) made a separation between data taken over land and ocean and found that the magnitude of mesoscale spectra taken over land exceeded the magnitude of spectra taken over ocean by about 25%. Spectra from an area over the western United States showed 2–4-times-larger magnitudes than global averaged. These observations indicate that upward-propagating gravity waves, induced by orography or convection, may contribute to energizing mesoscale motions close to the tropopause. On the other hand, Callies et al. (2016) reported that they found no difference in the ratio *R* between areas with low and high orographic forcing.

Upward-propagating gravity waves are expected to have frequencies considerably larger than *f*, since the group velocity of inertia gravity waves with frequencies close to *f* is almost horizontal (Gill 1982). Upward-propagating waves are therefore expected to give rise to a larger magnitude of divergent and temperature spectra but not of the rotational spectrum. To investigate this we identify a great number of segments where all points in each segment are over either land or ocean. Given the large difference between tropospheric and stratospheric data that we saw in the previous section it would be preferable to separate between the troposphere and stratosphere also in this case. However, in order to obtain a sufficiently large number of segments we have to relax the condition on ozone concentration that we use to distinguish between tropospheric and stratospheric data. In this case we use a condition on mean concentration. To classify as tropospheric the mean concentration of ozone over a segment has be lower than 120 ppbv and to classify as stratospheric it the mean concentration has to be larger than 180 ppbv. The number of (land, ocean) segments are (2010, 5269) for the upper troposphere and (7293, 2656) for the lower stratosphere. Also in this case we exclude all data between 20°S and 20°N. In Fig. 10 we see the ratios between spectra calculated over land and ocean for the tropospheric data and stratospheric data. Overall, there is no dramatic difference between land and ocean. In the upper troposphere, the land–ocean ratio is around 1.4 for the divergent and the temperature spectra and close to unity for the rotational spectrum at

Land–ocean spectral ratio for the (left) upper troposphere and (right) lower stratosphere. The number of (land, ocean) segments are (2010, 5269) for the upper troposphere and (7293, 2656) for the lower stratosphere. In this case we use a weaker condition to distinguish between tropospheric and stratospheric segments. To classify as tropospheric, the mean ozone concentration over the segment has to be smaller than 120 ppbv, and to classify as stratospheric, it has to be larger than 180 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Land–ocean spectral ratio for the (left) upper troposphere and (right) lower stratosphere. The number of (land, ocean) segments are (2010, 5269) for the upper troposphere and (7293, 2656) for the lower stratosphere. In this case we use a weaker condition to distinguish between tropospheric and stratospheric segments. To classify as tropospheric, the mean ozone concentration over the segment has to be smaller than 120 ppbv, and to classify as stratospheric, it has to be larger than 180 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Land–ocean spectral ratio for the (left) upper troposphere and (right) lower stratosphere. The number of (land, ocean) segments are (2010, 5269) for the upper troposphere and (7293, 2656) for the lower stratosphere. In this case we use a weaker condition to distinguish between tropospheric and stratospheric segments. To classify as tropospheric, the mean ozone concentration over the segment has to be smaller than 120 ppbv, and to classify as stratospheric, it has to be larger than 180 ppbv.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

### c. Analysis of low-latitude data

We identify 2800 segments for which all data points lie between 20°S and 20°N. In this case there is no need to distinguish between tropospheric and stratospheric data using a condition on ozone concentration, since all low-latitude data are tropospheric. In Fig. 11 we see rotational and divergent spectra and temperature spectra averaged over all low-latitude segments. The spectra are similar to the tropospheric spectra seen in Figs. 4 and 5. The rotational spectrum displays a relatively clean *R* ≈ 0.5, or even somewhat smaller, similar to the averaged result for troposphere at higher latitudes. The magnitude of the rotational spectrum is comparable, but somewhat larger, than we found previously. Since the low-latitude spectra are very similar to the higher-latitude spectra the Rossby number is considerably larger. If we use the low-latitude mean value of *f*, we find that Ro is about 4 times larger for the low-latitude data (i.e., Ro ≈ 10 at

Low-latitude (left) divergent and rotational compensated spectra and (right) compensated temperature spectra. The spectra are mean values over 2800 segments between 20°S and 20°N. All segments can be classified as tropospheric.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Low-latitude (left) divergent and rotational compensated spectra and (right) compensated temperature spectra. The spectra are mean values over 2800 segments between 20°S and 20°N. All segments can be classified as tropospheric.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

Low-latitude (left) divergent and rotational compensated spectra and (right) compensated temperature spectra. The spectra are mean values over 2800 segments between 20°S and 20°N. All segments can be classified as tropospheric.

Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0063.1

## 4. Conclusions

We have found that the ratio *R* between divergent and rotational energy is smaller than unity, *R* ≈ 0.5, at mesoscales in the upper troposphere, consistent with the result of Callies et al. (2016) when they used the MOZAIC dataset. As we discussed in the introduction, from this observation we conclude that linear theory of inertia–gravity waves is not likely to hold. A convergence test shows that at least 1000 flight segments are needed in order to obtain reasonably converged results, while 100 is not enough. Thus, 15 segments, as used by Callies et al. (2016) in analyzing the START08 tropospheric data, is far from enough. In the stratosphere, we obtain a ratio that is somewhat larger than unity, around 1.3, at

Based on the condition (13), it can be argued that weakly nonlinear wave theory cannot provide a general explanation of the mesoscale energy spectra in the upper troposphere and lower stratosphere. Globally, the Rossby number associated with mesoscale motions is on the order of unity and at low latitudes it is considerably larger than unity. The similarity between the low-latitude spectra and the globally averaged spectra in the upper troposphere suggests that system rotation is not a crucial factor in the generation of the spectra. As pointed out by one of the reviewers, the Rossby number corresponding to scales larger than 100 km is smaller than unity and *R* > 1 for the lower-stratospheric data at these scales. It is therefore possible that linear waves may play a role in the generation of the

The strong condition we used on ozone concentration to distinguish between tropospheric and stratospheric data has revealed that there are significant differences between mesoscale spectra in the upper troposphere and lower stratosphere. In the lower stratosphere the

*N*to convert temperature spectra to average potential energy (APE) spectra, we found that APE/KE ~ 0.5, both in the troposphere and the stratosphere. This result is also consistent with the results of Callies et al. (2016). However, as shown by Bühler et al. (2014), the energy of inertia–gravity waves, conforming to the assumptions of linear theory, is partitioned as

*R*< 1. In the stratosphere, we found that

Globally, we found that there is no dramatic difference between land and ocean. In the upper troposphere, however, divergent and temperature spectra have somewhat larger magnitude at

Given the observations that the Rossby number is on the order of unity, divergent and rotational energy are on the same order of magnitude, that APE is about half the magnitude of total kinetic energy, and that the wavenumber spectra display a clean *N*. As shown in this study, rotational energy is on the same order of magnitude as divergent energy at *N* would be restricted to the divergent component this explanation cannot be valid. We find it likely that the transition can only be explained by taking the whole stratospheric energy budget into consideration, including ozone production and depletion as well as radiation. It should be pointed out that the wave hypothesis put forward by Callies et al. (2014) and Callies et al. (2016) cannot explain this, since it does not make any predictions whatsoever regarding the shape of the spectra. Callies et al. (2016) only point out that waves can produce a power-law spectrum, while it is not explained why the spectrum is observed to be of Kolmogorov type: *b* is a parameter having the dimension

GCMs are still not capable of resolving the whole mesoscale range in the horizontal and the vertical resolution is still not sufficiently fine to fully simulate stratified turbulence. Different GCMs (e.g., Hamilton et al. 2008; Skamarock et al. 2014) give quite different results for *R*. However, a common feature of the results from different GCMs and observational results is that *R* is generally larger in the stratosphere as compared to the troposphere. This result may be interpreted in light of the forward-cascade hypothesis. In the troposphere it must be assumed that baroclinic instability is the main forcing mechanism, generating motions at the 1000-km scale carrying vertical vorticity. In the stratosphere, on the other hand, the forcing mechanism are waves propagating from the troposphere (Augier and Lindborg 2013), also at the 1000-km scale. From the 1000-km scales, energy is transferred to smaller scales by strong interactions involving both rotational and divergent modes. During this process, a state develops in which rotational and divergent energy are on the same order of magnitude. However, at the 100-km scale there is still a footprint of the forcing, giving an *R* which is larger than unity in the stratosphere and smaller than unity in the troposphere. It is only a matter of time before high-resolution GCMs will tell us if this interpretation is correct.

## Acknowledgments

We thank the principal investigators of the MOZAIC program for making their database available to us through contact information that can be found at http://www.iagos.org. The MOZAIC program is funded by the European Communities with strong support from Airbus Industries and its partners, Air France, Lufthansa, Austrian Airlines, and Sabena. We thank three reviewers for extensive and useful criticism.

## APPENDIX A

### Derivation of Condition for Weak Nonlinearity for a Random Field

In this appendix we derive a global condition for weak nonlinearity for a random axisymmetric (or horizontally isotropic), homogeneous, incompressible field with Gaussian statistical distribution. The condition that we derive is stronger than (6). By axisymmetry we mean invariance under rotations with respect to vertical axes and reflections in vertical planes (parity invariance). That the derivation is carried out for such idealized conditions should not be mistaken for an assumption regarding nature. The condition of Gaussianity can only be warranted in the limit of very weak nonlinearity, since deviations from linearity will lead to deviations from Gaussianity. Neither can it be taken for granted that reflectional invariance with respect to vertical planes is realistic in presence of strong system rotation. We may consider the condition that we derive as a global condition on an initial random field in a numerical simulation that we would like to be marginally influenced by nonlinear effects in its future evolution. The analysis is similar to the analysis of Lindborg and Riley (2007), who derived a condition on the average Richardson number for weak nonlinearity of internal gravity waves in the ocean.

*u*and

*υ*are the velocity components in the

*x*and

*y*directions, respectively. We note that

*a*,

*b*,

*c*, and

*d*represent velocity components or their derivatives. A proof that fourth-order statistical moments of velocities and velocity derivatives can be factorized in this way if they have jointly Gaussian statistical distributions is given by Frisch (1995). The assumption of statistical homogeneity will be used to eliminate spatial derivatives of statistical moments, for example,

## APPENDIX B

### One-Dimensional and Two-Dimensional Enstrophy Spectra

*r*from the point where the unprimed quantity is measured and the two points are in the same horizontal plane. That the correlations are only functions of

*r*is a consequence of axisymmetry, sometimes referred to as horizontal isotropy. The two correlations are related by

*R*,

*r*to

*x*to make it clear that (B2) is the one-dimensional transform. The one-dimensional vorticity spectrum

*Q*. The relation between

*r*to

*x*in (B1), multiplying both sides with

*x*, transforming both sides and noting that

*R*. It is straightforward to show that the relation between the one-dimensional and two-dimensional rotational spectra can be written as

*k*. If

*a*and

*b*is

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The average length of the 15 tropospheric segments from the START08 data used by Callies et al. (2016) was 195 km (J. Callies 2017, personal communication).