## 1. Introduction

The energy spectrum of the atmosphere and its underlying physical mechanisms remain active research topics despite decades of observations and scientific research. Long-range passenger aircraft have collected velocity and temperature measurements since the 1970s. These measurements indicate an energy spectrum varying as

The abovementioned mechanisms for the −3 and −5/3 spectra are based on idealized models. Several full-physics models have successfully captured the observed transition of the spectrum slope from −3 to −5/3 (e.g., Skamarock 2004; Hamilton et al. 2008; Skamarock et al. 2014), yet only limited discussion has been given to explain the mechanism(s) behind it. None of these earlier studies looked into the details of the growth process of the mesoscale energy spectrum. Recently, Waite and Snyder (2013) found that moist processes could energize the mesoscale and thus help the transition of the slope. A similar full-physics idealized baroclinic wave simulation in Sun and Zhang (2016) also found this distinct transition of the simulated kinetic energy spectrum at a scale of ~400 km in their moist experiment. Interestingly the dry experiment in their study maintains the steep −3 slope all the way to the mesoscale in the upper troposphere. This result emphasizes the critical role of moist convection in the creation of the shallower −5/3 slope. Compared to the dry experiment, moist convection generates many gravity wave–like signals at the upper levels of the troposphere (Wei and Zhang 2014, 2015; Wei et al. 2016), which might be responsible for the spectrum transition from −3 to −5/3 at those levels. More recently, Durran and Weyn (2016) shows that a −5/3 spectrum evolves and reaches to scales comparable to observations in their simulations of convective cloud systems.

Motivated by these recent findings, this study aims to investigate the mechanisms responsible for the −5/3 slope in Durran and Weyn (2016), especially the contributions of moist convection and internal gravity waves generated by convective systems. This study confirms their finding of an approximate −5/3 spectrum and provides new information for each specific height level. The remainder of the paper is organized as follows: Section 2 introduces the model setup for our simulation. Section 3 gives a brief overview of the simulation and analyzes the evolution of the mesoscale energy spectrum through a spectral energy budget. Further discussion and concluding remarks are given in section 4.

## 2. Methodology

The Weather Research and Forecast (WRF) Model, version 3.6.1, (Skamarock et al. 2008) is used in an idealized mode for this study. The domain size of the simulation is 800 km × 800 km, with a horizontal grid spacing of 2 km. The model top is fixed at 20 km. To get an accurate calculation of the kinetic energy spectra budget, we have 200 layers in the vertical direction. The vertical grid spacing is approximately 100 m, although the vertical layers are not equally spaced in the WRF Model. The Coriolis force is neglected, unless otherwise stated. Periodic lateral boundary conditions are implemented here to facilitate the spectral analysis. To further simplify both the model and the interpretation, we use a free-slip bottom boundary condition in the simulation. No PBL or surface scheme is used in the current simulation. Also, no cumulus or radiation parameterizations are used. Near the upper boundary, an absorbing layer, as described by Klemp et al. (2008), is applied for the uppermost 5 km of the model domain to reduce artificial reflection of gravity waves. This sponge layer has proved successful in the idealized squall-line simulation done by Klemp et al. (2008). The Morrison scheme is used for the microphysics parameterization (Morrison et al. 2009). The time step for the integration of the model is 3 s.

Figure 1 shows our initial sounding profile for the simulations. It is based on Weisman and Klemp (1982), except that we fix the mixing ratio below 1 km at 14 g kg^{−1} and cap the RH at 75% for any level above 1.4 km. A unidirectional horizontally uniform background wind profile is specified in which the zonal winds linearly increase from −10 m s^{−1} at the surface to 10 m s^{−1} at a height of 5 km and remain 10 m s^{−1} at higher levels (Fig. 2a).

Seven localized warm bubbles with a positive temperature anomaly of 3 K are put into the initial condition to initiate convection. These warm bubbles are aligned from north to south at the domain center, with a horizontal radius of 10 km and a vertical radius of 1.5 km. Their centers lie at 1.5 km above the surface, with a horizontal distance of 20 km away from each other. The warm bubbles, each evolving into a convective cell, then interact with each other under the wind shear. The evolution of these cells will be briefly introduced in the next section. A 20-member ensemble is produced through perturbing the water vapor mixing ratio in the lowest 1 km with a Gaussian white noise of 0.5 g kg^{−1} to reduce case dependency in the statistics. All the simulations are integrated for 6 h, with fields output every 30 s. The output fields are interpolated to constant height levels with a vertical interval of 50 m to facilitate the calculation of the spectra.

## 3. Results

### a. Overview of the simulation

Before we start the spectral analysis, we would like to first take a look at the evolution of our simulation. Figure 3 visualizes the development of the convective cells in one member of our ensemble. Initially (0–2 h), each warm bubble evolves into a convective cell of similar scale, with a strong embedded updraft. After 2 h, the convective cells start to interact with each other under the vertical wind shear. While the convective cells in the middle of the line get weaker, the cells at both ends of the line become stronger and “eat” all the other cells gradually. At the end of the 6-h simulation, two supercell-like systems form at both ends of the line. Slightly different from our expectations, these cells do not organize into a squall line in almost all the ensemble members. This is likely because of the relatively deep vertical wind shear in our simulation. Another possible reason is that the initial line of warm bubbles is perpendicular to the wind shear direction. Sensitivity runs of adding the warm bubbles at different zonal locations and/or adding vertical shear in the meridional direction are more favorable for the formation of a squall line. As a result of this weak organization of the convective cells, the cold pools are also relatively weak in our simulation. The gray shading in Fig. 3 depicts the anomaly virtual potential temperature less than −0.5°C. Compared to Skamarock et al. (1994), both the range and absolute value of the cold-pool temperatures are smaller, which implies a weaker convective system in our simulation. Despite these differences between our simulated convective system and previous studies, their effects on the spectrum are not critical, as we will show later in further sensitivity experiments.

Strong gravity waves can be generated by these convection cells. Figure 4a shows a south–north cross section over the domain center at 2 h for the same ensemble member as shown in Fig. 3. The location of each convective cell can be identified by the 25-dB*Z* reflectivity line. The region with vertical velocity greater than 0.1 m s^{−1} is shaded in cyan, while the potential temperature is plotted using gray lines. Clear gravity wave signals generated by the convective cells can be found at levels above 12 km (lower stratosphere), where the vertical velocity and the potential temperature show a quadrature phase relationship. In the troposphere, as a result of turbulent motions induced by the convection cells, linear gravity wave signals cannot be easily identified. This result is further supported by the profile of the domain-averaged vertical heat transport (*w* and *T* do not transfer heat, we see negligible transport at levels above 12 km; whereas for the troposphere, there is considerable heat transfer due to convection. Figure 4c also shows the domain-averaged vertical energy flux

The convective cells and the gravity waves they generate have a downgradient effect on the mean flow (Fig. 2b). We see a slight increase of the mean zonal wind (~0.1 m s^{−1}) in the 0–5-km layer; above 5 km, the mean zonal wind decreases a small amount (~0.1 m s^{−1}). This downgradient effect leads to the loss of mean kinetic energy, which will be discussed later. The mean meridional wind is also plotted in Fig. 2c. With the symmetric model setting we used, the mean meridional wind should be zero. Our calculation shows a noisy mean ^{−1} (two orders of magnitude smaller than mean zonal wind change) due to numerical error. The accuracy of our calculation demonstrates that the change of the mean zonal wind (~0.1 m s^{−1}), thus the loss of the mean kinetic energy, is due to mixing induced by convection and gravity waves.

### b. Kinetic energy spectra

The spectrum of the kinetic energy is calculated using the 2D discrete cosine transform (DCT) method (Denis et al. 2002; Peng et al. 2014) at each vertical level. More details on this method can be found in the appendix A. For a 2D field with periodic boundary conditions, the DCT gives results that are very close to those obtained using the discrete Fourier transform, and it is more generally applicable to nonperiodic domains. We use a curly bracket here to denote the DCT spectral coefficients of a field *q* as {*q*(**k**)}, where *z* and time *t* is suppressed for clarity.

**u**is the horizontal wind vector, and

The derived mean kinetic energy spectra, averaged over all 20 ensemble members every 2 h and over all the levels between 0 and 15 km, are shown in Fig. 5. Since the resolution of the simulation is 2 km, signals with a wavelength shorter than 15 km (gray shaded area) are not well resolved by the model. The slope of the spectrum in this region falls off quickly because of implicit dissipation in the model. Any results within this range should be treated with caution. We will focus here on the well-resolved range (wavelengths > 15 km). For the first 2 h, the energy spectrum clearly shows a peak at a scale around 20 km, which is the scale of the warm bubble and the convective cells. Note that the spectral decomposition of an isolated feature projects onto all scales and most prominently onto the largest scales; thus, the initial large-scale signal in the spectrum analysis is mainly due to the projection of energy associated with the limited extent of the convective cells. After 2 h, the growth of the spectrum extends to larger scales. For the time period between 4 and 6 h, the energy spectrum approaches a quasi −5/3 slope for scales shorter than 100 km. Although there are slightly different evolutions of the convective cells in the 20 members, the evolution of the kinetic energy spectrum is insensitive to the details of the convective cells. All the 20 members have formed the quasi −5/3 slope as in the ensemble-mean result after 6 h of integration. This is consistent with Durran and Weyn (2016), which shows that the kinetic energy spectrum with a slope close to −5/3 could indeed be built solely from convection. Further examinations with a smaller time interval (not shown) indicate that the kinetic energy spectrum at scales smaller than 100 km becomes quasi steady after 5 h, when it reaches the −5/3 slope. We also did one experiment with slightly different model settings in which the simulation was integrated for 8 h. The supercells in that experiment maintain themselves and bring the spectrum at larger scale (>100 km) closer to the −5/3 reference line, though the change is much slower. The growth process of the quasi −5/3 slope for scales less than 100 km in our ensemble experiments is the focus of this study.

*ζ*and

**are the vertical vorticity and horizontal divergence, respectively. Figure 6 shows the result after the decomposition. Unlike previous studies involving baroclinic waves and moist convection (e.g., Waite and Snyder 2013), the divergent energy spectrum in the present physical situation is not the only component responsible for the shallower −5/3 slope. At the end of the simulation (4–6 h), the rotational kinetic energy also has a quasi −5/3 slope within the wavelength range of 15–100 km. The magnitude of the rotational kinetic energy within the −5/3 slope range (15–100 km) is even slightly larger than that of the divergent kinetic energy. A closer look shows that the ratio of the divergent to the rotational kinetic energy increases with height. In the troposphere, the amplitude of the rotational kinetic energy is stronger because of the mesoscale convective vortices produced by the convective systems (Davis and Weisman 1994). While in the lower stratosphere the divergent kinetic energy dominates over the rotational kinetic energy, as gravity waves are the primary signals there. Analysis of observational datasets in previous studies led to different conclusions with regard to the ratio of divergent to rotational kinetic energy. Callies et al. (2014) conclude that the divergent component of the kinetic energy is slightly stronger for the mesoscale energy spectrum. On the contrary, other studies find that the rotational kinetic energy is more important (Cho et al. 1999; Lindborg 2015). Differences in data analysis and datasets might be responsible for different conclusions (Bierdel et al. 2016). Further study is clearly needed to reach agreement on this.**

*σ*Figure 7 shows the kinetic energy spectrum averaged over 0–4, 6–10, and 12–15 km (lower troposphere, upper troposphere, and lower stratosphere, respectively). The kinetic energy is stronger in the troposphere than it is in the stratosphere as a result of decreasing density with height. What is more interesting is that, although it differs slightly,^{1} an approximate −5/3 slope in the wavelength range 15–100 km does show up at all levels throughout the atmosphere toward the end of our simulation (Fig. 7). The upper troposphere, where aircraft measurements lie, is not the only level that has a spectrum slope of −5/3; the lower troposphere and the lower stratosphere also have such a slope. The present model thus offers an alternative to the surface quasigeostrophic hypothesis in Tulloch and Smith (2006). Since no surface scheme or boundary layer scheme is adopted in our simulation, the creation of the kinetic spectrum is clearly due to the convection systems (diabatic heating, which has a maximum in the upper troposphere and a value close to zero near the surface and above the tropopause). Any boundary process plays at most a secondary role since no PBL or surface scheme is used in our simulations.

### c. Spectral budget analysis

*k*

_{h}after summed over the wavenumber bands in order to preserve the area in this log-linear plot. Even after this multiplication, for an energy spectrum with a −5/3 power-law slope, it can be proven that the tendency term will decrease with decreasing scale, as is shown by the black line for the range of wavelengths smaller than 100 km. To be clear on the sources and sinks for the energy spectrum

**k**, whereas in the figures, we present each term as a function of the horizontal wavenumber

*k*

_{h}.

Figure 8 shows the contribution of all the terms in Eqs. (5) and (10) as a function of horizontal wavenumber

The sum of ^{2} from the large scales.

Compared to the

The

Figure 9e shows that in the upper troposphere, the diabatic heating is the strongest, which is reflected in

In the lower troposphere, the buoyancy production has a smaller effect (Fig. 9b). It is positive at scales larger than 50 km, likely because of the formation of the cold pools that contain organized downdrafts and negative potential temperature anomalies due to evaporative cooling. The negative

Figure 9i shows that, in the lower stratosphere, convection-generated gravity waves inject a significant amount of energy mainly into small scales through the

In summary, although all the levels yield spectrum slopes of approximately −5/3, the underlying physical processes behind them are substantially different. Both the downscale process [e.g.,

### d. Sensitivity experiments

In the above simulations, we use an ensemble of 20 members to reduce the case dependency of our results. Yet all these members use the same model setup and physics schemes. To ensure that our results are robust, various sensitivity runs are also conducted. Figure 10 shows the kinetic energy spectrum for the DOUBLE experiment, where the horizontal size of the domain is doubled to 1600 km × 1600 km. A similar approximate −5/3 spectrum shows up again for this experiment. Moreover, additional experiments containing different model setups (e.g., different boundary conditions, reduced vertical layers, or different shear profiles) all give similar −5/3 spectra (not shown), implying that the −5/3 spectrum generated by the convective systems is not sensitive to the model setup.

The stratified turbulence theory proposed by Lindborg (2006) requires that vertical scales of *U*/*N* (*U* is the horizontal wind, while *N* is the Brunt–Väisälä frequency; *U*/*N* is around 1 km in the troposphere) be well resolved to drive the −5/3 mesoscale energy spectrum. In our simulation with only 40 vertical layers (a vertical grid spacing of ~500 m), the quasi −5/3 spectrum is still very clear. As we mentioned above, with the strong

As for the impact of different model physics, microphysics is the only parameterization scheme used here, and we do not expect significant differences for the kinetic energy spectrum if other microphysics schemes were adopted. Hence, instead of changing any of the physics schemes, a constant Coriolis parameter (*f* = 1.0 × 10^{−4} s^{−1}) is added to the model, which would affect the organization of the convective cells (Skamarock et al. 1994). Consistent with Skamarock et al. (1994), the evolution of the convective cells with the Coriolis effect exhibits significant asymmetries. At later hours of the simulation, the convective cells at the southern part of the domain center become much stronger than that in the control no-Coriolis simulations, while convective cells in the northern part decay. This asymmetry leads to a systematic reorientation, and the convective system moves toward the right of the wind shear. Because of this asymmetry, the convective cells, especially at the southern flank, tend to be more organized and form a quasi-squall-line structure. The intensity of the system is also stronger than that of the no-Coriolis experiment.

Figure 11 shows the kinetic energy spectrum for the experiment with the Coriolis effect. Compared with the no-Coriolis experiment, the energy spectrum is stronger, especially for the later times of the simulation, when the −5/3 spectrum extends to a scale of 400 km at the upper troposphere. The

## 4. Discussion and concluding remarks

Using an ensemble of high-resolution cloud-model simulations, this study explores the kinetic energy spectrum of organized convective systems under vertical wind shear. Our results further confirm a recent finding by Durran and Weyn (2016) showing that convective systems alone could generate a background mesoscale kinetic energy spectrum with a slope proportional to the −5/3 power of the wavenumber. Building upon this result, the present study gives a picture of the growth processes of the −5/3 spectrum in this physical situation. At each specific height level, the physical processes actively contributing to the formation of the kinetic energy spectrum are as follows: 1) conversion from available potential energy to kinetic energy [buoyancy production or the

Sensitivity experiments of varying domain size or boundary conditions all give a similar approximate −5/3 spectrum in our simulations. Thus, our results are very robust in terms of different model settings. The −5/3 spectrum is also not affected by the organization of the convective systems. In the experiment with a constant nonzero Coriolis parameter, the interaction between different convective cells is greatly altered, especially at later times of the simulation. Thus, the forcing terms of the kinetic energy spectrum [e.g., the

Although the concept that deep convection is able to generate the −5/3 spectrum resembles Lilly’s hypothesis, the building-up process of the spectrum is not the 2D inverse cascade as proposed by Lilly (1983). For each specific level, the divergence of vertical energy flux is critical, which means there are strong connections between different levels, and it is therefore a three-dimensional process. Moreover, buoyancy production and vertical flux of energy act at all the scales, so the dynamics cannot be described as an inertial-subrange cascade, as also pointed out by Waite and Snyder (2009). In addition, the filling out of the energy spectrum by nonlinear interactions varies greatly between different vertical levels. It goes through a downscale propagation in the lower troposphere to an upscale-like propagation in the lower stratosphere. Both downscale and upscale processes happen at the same time but at different levels. We rarely find any true cascade signal in the simulations [consistent with Durran and Weyn (2016)]. Small-scale convection can even directly interact with the mean flow.

While convection is the ultimate source for the kinetic energy spectrum in our simulations, at high altitudes, where the aircraft observations lie, it is the convection-generated gravity waves that are the primary contributors to

A better understanding of the creation of the atmospheric energy spectrum is beneficial for the study of the atmospheric predictability. It has been proposed that the error growth behavior is closely related to the energy spectrum of the basic flow within which the errors grow (Lorenz 1969; Rotunno and Snyder 2008). For a flow with energy spectra of power-law behavior

We also want to emphasize that convection is not the only explanation for the observed −5/3 spectrum. We cannot rule out all the other hypotheses that have been proposed to explain the spectrum, although we have shown that some of them are not necessary in a moist environment. It is still an open question of how important convection is in the observed −5/3 spectrum of the real atmosphere. Moreover, although the current study clarifies the sources of

## Acknowledgments

The authors thank Chris Snyder and Tiffany Shaw for thoughtful comments on the manuscript. Discussions with Dale Durran, Kerry Emanuel, Raf Ferrari, Joern Callies, and many other researchers on the subject were beneficial. Part of the research was conducted during the first author’s summer visit to NCAR/MMM sponsored by the NCAR/Advanced Study Program Graduate Visitor Program. This research is partially supported by the National Science Foundation under AGS Grants 1114849 and 1305798. Computing was performed at the Texas Advanced Computing Center.

## APPENDIX A

### Discrete Cosine Transform

All the spectrum and budget analysis in this article is calculated using a discrete cosine transform (DCT) method defined as in Denis et al. (2002). A brief introduction of this method is given as follows.

*f*(

*i*,

*j*) of

*N*

_{i}by

*N*

_{j}grid points, the direct and inverse DCT are, respectively, defined as

**k**(

*m*,

*n*) with a single-scale parameter

*m*,

*n*) on a given circle with the origin (

*m*= 0,

*n*= 0) has the same wavenumber. The one-dimensional wavenumber spectrum

Also note here that the discrete cosine transform has no imaginary part; thus, the complex conjugate is not involved here, which is different from the discrete Fourier transform.

## APPENDIX B

### Decomposition of Advection and Pressure Term

*T*(

**k**) term in Eq. (B4) over all the wavenumbers is zero. The proof is as follows. According to Eq. (A5),

*s*represents the horizontal domain,

*l*represents the lateral boundaries of

*s*, and

**n**denotes the unit vector pointing along the outward normal to

*l*. A double periodic lateral boundary condition gives a zero result to the integration along the boundary.

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^{1}

The calculated linear-fit slope varies from −1.6 to −2.1 for wavelengths between 16 and 100 km at different height levels during 4–6 h of our simulations (linear-fit slope is −1.75 for 0–15 km, −1.90 for 0–4 km, −1.61 for 6–10 km, and −2.05 for 12–15 km). The steeper slope of −2 mainly lies in the stratosphere, especially where the gravity wave signal is relatively weak. For the constant-Coriolis experiment shown in Fig. 12, the slope range is much smaller (from −1.6 to −1.8) as a result of more organized convection (linear-fit slope is −1.69 for 0–15 km, −1.72 for 0–4 km, −1.66 for 6–10 km, and −1.63 for 12–15 km).

^{2}

In classical turbulence theory, an energy cascade often refers to the transfer of energy from larger scales of motion to smaller scales, also called a direct energy cascade. If *T*(**k**) is negative at relative larger scales and positive at small scales, then this is consistent with the cascade picture. Otherwise, it is not.