## 1. Introduction

Observational studies (Nastrom and Gage 1985; Cho et al. 1999) have shown that horizontal wavenumber spectra of horizontal velocity components and potential temperature in both the upper troposphere and lower stratosphere exhibit an obvious spectral transition at the mesoscale [~(20–2000 km)]. At the lower end of the mesoscale, where wavenumbers correspond to wavelengths less than ~500 km, the spectra exhibit a

Two completely different explanations for the mesoscale −

WS2013 noted that “[o]ver the last decade, the mesoscale energy spectrum has been reproduced in a variety of numerical simulations of the atmosphere, using both global (Koshyk and Hamilton 2001; Takahashi et al. 2006; Hamilton et al. 2008) and regional (Skamarock 2004; Skamarock and Klemp 2008) models.” More recently, both general circulation models (GCMs) (e.g., Terasaki et al. 2009) and mesoscale numerical weather prediction (NWP) models (e.g., Bierdel et al. 2012; Ricard et al. 2013; Peng et al. 2014a, hereafter PZ2014a) have successfully reproduced quite realistic mesoscale spectra. Such comprehensive atmosphere models have provided a convenient framework for studying the physical mechanisms producing the mesoscale −

Idealized baroclinic waves are well known to exhibit a variety of realistic mesoscale structures, including fronts, jets, and IGWs (e.g., Snyder and Lindzen 1991; Plougonven and Snyder 2007, hereafter PS2007; WS2009), and therefore are commonly used as a convenient prototype for midlatitude atmosphere. WS2013 compared the upper-tropospheric energy spectra in baroclinic wave simulations with and without moisture. Their simulations explored the importance of moist processes in establishing the upper-tropospheric mesoscale kinetic energy spectrum, especially its divergence component. Moist processes release latent heat and directly energize the upper-tropospheric mesoscale via positive buoyancy flux. Based on WS2013, there are at least two aspects that need to be investigated. First, the dynamics of the lower-stratospheric energy spectra in moist baroclinic waves have not been considered yet. What is the dependence of the lower-stratospheric energy spectra on moist processes? Besides the contributions from the energy cascade and the direct forcing of the convectively generated IGWs, are there any other significant sources or sinks of kinetic energy and available potential energy in the lower stratosphere? Can the inclusion of moist processes change the direction of the energy cascade through the lower-stratospheric mesoscale? In Part I of this study presented here, we examine these questions. Second, a quantitative diagnosis of the spectral energy budget of kinetic energy and available potential energy should be made in the upper troposphere. By doing this, it is possible to clarify whether and to what extent moist processes enhance mesoscale energy cascade, the significance of the net direct forcing (IGWs will transport much of the energy injected by latent heating to the lower stratosphere), and the effects of moist species in the upper troposphere. The detailed spectral energy budget analysis of the upper troposphere with this new moist nonhydrostatic formulation will be presented in Peng et al. (2014, manuscript submitted to *J. Atmos. Sci.*, hereafter Part II).

The remainder of the paper is organized as follows. The new moist, nonhydrostatic formulation of the spectral energy budget developed by PZ2014b is outlined in section 2. This formulation is an extension of that of AL2013 into a general, moist, nonhydrostatic atmosphere. For consistency, terminologies and notations similar to those of AL2013 are adopted here. A brief description of the baroclinic wave simulations is presented in section 3. In section 4, we present the energy spectra, including horizontal kinetic energy (HKE) spectra, vertical kinetic energy (VKE) spectra, and available potential energy (APE) spectra, and quantify the dependence of these spectra on the degree of humidity. In section 5, we employ this moist, nonhydrostatic formulation of the spectral energy budget to investigate the dynamics of the lower-stratospheric energy spectra. Conclusions are given in section 6.

## 2. Methodology

### a. Governing equations

*f*plane and without the additional large-scale forcing effects, can be written aswhere

**u**,

*w*, etc.; and

### b. Formulation of the spectral energy budget

*a*and

*b*and

### c. One-dimensional total horizontal wavenumber spectra and cumulative summation over total horizontal wavenumbers

*x*and

*y*directions, respectively. For example, the 1D spectrum of HKE per unit volume is defined asThe 1D spectra of any term in Eqs. (6)–(8) are defined similarly.

Note that each of these terms is an accumulation of a corresponding term in Eqs. (6)–(8). For example,

To close this section, we compare the formulation presented here with that in AL2013. First, as shown in the introduction, the one in AL2013 was derived in pressure coordinates and based on the hydrostatic assumption, while the one developed here is derived in height coordinates and does not make the hydrostatic assumption. As a result, there exists a spectral budget equation of VKE [Eq. (12)] in the present formulation, in addition to the conversion between APE and HKE. Second, the present formulation explicitly embodies the effects of 3D divergence. It will be shown in section 5d that the 3D divergence of flow has an important contribution to the APE budget in the lower stratosphere. Third, the APE defined in AL2013 is dry and only the latent heat release can be taken into account. In fact, moist convection acts not only as a source of latent heat but also as an “atmospheric dehumidifier” (Pauluis and Held 2002). The APE defined here is based on the modified potential temperature and embodies the effect of water-vapor distribution, resulting in both of these two effects of moist convection being taken into account in the APE budget. Moreover, the gravitational potential energy of moist species is also considered in the present formulation. The effects of moist species will be next focus in Part II of this study.

## 3. Numerical simulation of baroclinic waves

The present work is motivated by WS2013. We consider the mesoscale dynamics underlying the lower-stratospheric energy spectra. For consistency, the simulations presented here are configured, initialized, and run almost exactly as in WS2013 except for some minor modifications to suit our purposes. The configuration of the model, initialization and experimental design, and experimental results are briefly outlined as follows.

### a. The model

The numerical model used in this research is the ARW Model, version 3.2 (Skamarock et al. 2008), which solves the equations of motion for a fully compressible, nonhydrostatic atmosphere. The generic baroclinic wave test case available in the ARW code provides the starting point for constructing the idealized baroclinic wave simulations. All our simulations are performed on an *f* plane with *x* and rigid and symmetric in the meridional direction *y*. Therefore, there is no net energy exchange through the lateral boundaries. The model domain has a zonal extension of 4000 km, a meridional width of 10 000 km, and a vertical depth of 30 km. The horizontal grid spacing is

### b. Initialization and experimental design

The initial conditions consist of a baroclinic, zonal jet and its fastest-growing normal mode. The dry baroclinic jet is constructed following the potential vorticity (PV) inversion approach of previous studies on baroclinic waves (e.g., PS2007; WS2009; WS2013). The procedure of PV inversion used here is an improved version of PS2007, in which the variation of PV in the stratosphere is taken into account (shading in Fig. 1a). (This improved version was developed by Riwal Plougenvan.) Thus, a more realistic potential temperature profile can be obtained. The initial dry baroclinic jet yielded by this improved procedure is shown in Fig. 1a. The fastest-growing normal mode with small amplitude is obtained by a breeding procedure similar to PS2007. Following Davis (2010) and WS2013, the maximum potential temperature perturbation of the mode is rescaled to 2 K.

Following WS2013, for the moist cases, water vapor is initialized after the PV inversion stage, starting from uniform relative humidity (RH) of 30% or 60%; the corresponding moist simulations will be referred to as RH30 and RH60 cases, respectively. Subsequently, the potential temperature is adjusted down slightly so that the virtual potential temperature fields (

After the initialization, the simulations are then run for 16 days, with fields output every 3 h. To handle grid staggering, all ARW output fields are interpolated to a common grid using the ARW postprocessing utility—ARWpost (http://www.mmm.ucar.edu/wrf/users/download/). And then the spectra and spectral budget are computed with these processed output fields. Derivatives are calculated with numerical differences based on three-point, Lagrangian interpolation.

### c. Experimental results

As expected, the simulated baroclinic waves here are very similar to those of WS2013 in spite of some minor modifications. Since many aspects of the simulated baroclinic waves were shown in WS2013 [e.g., the time series of mass-weighted average eddy kinetic energy (their Fig. 2), the evolutions of the mass-weighted average VKE and the domain-averaged precipitation rate (their Fig. 3)], we will focus here on the APE and the diabatic contribution *t* = 3, 4, 5, 6, and 7 days. It is shown that in RH60 (Fig. 2b), during the stage with the strong precipitation (*t* = 4–7 days), positive diabatic contribution mainly takes place below the height of 12 km and the domain-averaged diabatic contribution *t* = 5 days. Therefore, we refer to *z* = 12–15 km as the lower stratosphere, where nearly no direct forcing from diabatic contribution occurs and the vertical propagation of convectively generated gravity waves is the key.

Following WS2013, the simulated baroclinic waves are divided into three phases: the early phase (*t* = 4–7 days), the intermediate phase (*t* = 7–10 days), and the late phase (*t* = 10–13 days). The early phase is characterized by strong convection and concomitant diabatic contribution, while the late phase is characterized by much weaker precipitation and convection. Next, we analyze the energy spectra averaged in the vertical over the lower stratosphere and in time over these two phases of interest.

## 4. Energy spectra in the lower stratosphere

### a. Horizontal kinetic energy spectra

Figure 3 presents the simulated lower-stratospheric spectra of RKE, DKE, and HKE in the early phase (

During the early phase, both RKE spectrum and DKE spectrum (Fig. 3a) exhibit a comparably strong dependence on moist processes. The RKE spectrum for the dry case shows a distinct shallowing at wavelengths less than around 500 km. Similar findings were reported in WS2009. The slope of RKE spectrum of the dry case is approximately −4.8 for wavelengths

At the late phase, the differences between the dry and moist cases become smaller, but the kinetic energy spectra still exhibit similar dependence on moisture (Figs. 3c and 3d). The RH60 case has more energy than the dry case at all scales, while the RH30 case has slightly more energy than the dry case mainly at wavelengths less than 700 km. At the wavelength of 500 km, the HKE of the RH60 case is approximately 4 and 6 times larger than the RH30 and dry cases, respectively; these differences are much smaller than those at the early phase, but still significant. The RKE spectrum by this time has a slope of −3.5 for RH60, with no apparent shallowing in the mesoscale range (

Moist processes can enhance both the lower-stratospheric mesoscale DKE and RKE, resulting in moist spectra being in better agreement with observations than the dry one. In Fig. 4, we make further comparisons between 1D zonal wavenumber (*y* and *z* and then averaging results in *y* over the most energetic part of the domain (i.e.,

### b. Available potential energy spectra

Only the spectra of potential temperature were considered in WS2013. However, in order to quantitatively analyze spectral energy cycles, further investigation of APE spectra is needed. Figure 5 presents the simulated lower-stratospheric APE spectra. During both phases, the shape of the APE spectra closely resembles that of the HKE spectra at nearly all scales, and the ratio of mesoscale HKE to APE is approximately 2. These results are in a remarkably good agreement with observations (e.g., Gage and Nastrom 1986). Moist enhancement is extended throughout the whole mesoscale in RH60 but is still restricted to small scales in RH30. As a result, the APE spectrum for RH60 has a much higher level than those for the dry and RH30 cases and naturally is closer to the Lindborg (1999) reference spectrum.

During the early phase (Fig. 5a), the APE spectrum of RH60 develops a slope of approximately −2.7 at the larger end of the mesoscale range (

### c. Vertical kinetic energy spectra

Additional analysis of VKE spectrum per unit volume was carried out, and the findings are presented here. On the whole, the VKE spectra for the moist cases exhibit much flatter-scale dependence than the corresponding HKE spectra. During both phases (Fig. 6), the VKE spectra for the moist cases show almost no wavenumber dependence (i.e., a flat spectrum) at the mesoscale, which is consistent with the stratospheric VKE spectra derived from aircraft observations by Bacmeister et al. (1996) and similar to other model studies (e.g., Bierdel et al. 2012; Ricard et al. 2013). However, for the dry case the VKE spectrum for wavelengths larger than 800 km is always slantwise and approaches a slope of around −3 at the larger end of the mesoscale. As the convection levels off (Fig. 6b), the level of the moist VKE spectra decays; even so, it is much stronger than that in the dry case. In addition, the moist spectra by this time are much closer to one another than they are to the dry spectrum.

## 5. Spectral energy budget for the lower stratosphere

In this section, the formulation of the spectral energy budget represented by Eqs. (11)–(13) is employed to investigate the dynamics of the lower-stratospheric energy spectra. To highlight the effects of moist processes, we will focus on the early phase of the dry and RH60 cases, when convection is the strongest and the simulated energy spectra are the closest to the reference spectrum.

### a. Spectra of vertical flux divergence and vertical propagation of energy

The theoretical framework presented here via Eqs. (6)–(8) suggests that moist processes can influence both HKE spectrum and APE spectrum via the direct forcing of latent heating [i.e.,

For the dry case (Fig. 8a), the cumulative HKE vertical flux divergence

For the RH60 case (Fig. 8b), the total vertical flux divergence is dominated by the pressure and HKE vertical flux divergences and is much more significant than that in the dry case. The rapid decrease of the cumulative total vertical flux divergence

### b. Spectra of energy conversion

Figure 9 presents the cumulative conversions

### c. Nonlinear spectral fluxes and energy cascade

Energy cascade is one of the most fundamental issues in atmospheric dynamics and can be evaluated by constructing spectral fluxes. However, in some studies (e.g., Koshyk and Hamilton 2001; Brune and Becker 2013) the spectral fluxes were commonly defined in a nonconservative way and actually included vertical fluxes, resulting in that the energy transfer among scales could not be exactly investigated. On the contrary, the spectral fluxes defined by Eq. (10) are exactly conservative (i.e.,

Figure 10a presents the nonlinear spectral fluxes for the dry case, averaged in the vertical over the lower stratosphere and in time over *t* = 4–7 days. The total spectral flux (blue curve) has a convex shape: it reaches its maximum of

At the mesoscale (inset in Fig. 10a), the HKE spectral flux becomes comparable to the APE spectral flux. The terms

Figure 10b presents the nonlinear spectral fluxes as in Fig. 10a, but for the RH60 case. The total spectral flux is largely dominated by the APE spectral flux at nearly all scales. The large-scale features of the APE spectral flux are quite similar to those of the dry case. However, the features for the other smaller scales are quite different for these two cases. The APE spectral flux increases to reach its maximum of

The leading features of the spectral HKE flux are similar to those of the spectral APE flux, but with relatively smaller magnitude. Particularly, it reaches its maximum of

### d. Effects of diffusion and 3D divergence

Owing to the compressibility of the atmosphere, the 3D divergence (i.e.,

### e. Quantitative comparison

We shall now present a more quantitative comparison among the effects mentioned above at the synoptic scale and mesoscale for the two cases. The amount of HKE added by the pressure vertical flux divergence in the wavenumber range

Energy budget, averaged in the vertical over the lower stratosphere and in time over *t* = 4–7 days, at approximately the synoptic scale and the mesoscale, respectively, for the dry and RH60 cases. All values are in

As already shown, the mesoscale HKE in the dry case is mainly forced by a weak downscale cascade (

For the RH60 case, the energy budget is quite different. At the mesoscale, the lower-stratospheric HKE is mainly deposited by the pressure vertical flux divergence (

## 6. Conclusions

### a. Key findings

Numerical simulations of idealized moist baroclinic waves have been presented for studying the effects of moist processes on the lower-stratospheric energy spectra. These simulations clearly demonstrate that moist processes can enhance the lower-stratospheric mesoscale energy spectra, including the horizontal kinetic energy spectrum, vertical kinetic energy spectrum, and available potential energy spectrum. In contrast to the findings reported by WS2013 that moist processes primarily enhance the divergent part of the upper-tropospheric HKE spectra, we find that not only the divergent part but also the rotational part of the lower-stratospheric HKE spectra exhibits a comparably strong dependence on moist processes. In other words, the inclusion of moist processes energizes both the lower-stratospheric mesoscale DKE and RKE. As a result, the two moist simulations, especially the RH60 case, have much more HKE than the dry case throughout the mesoscale, especially at wavelengths around 500 km. Indeed, the moist HKE spectrum in the RH60 case is in a much better agreement with Lindborg (1999) reference spectrum than that in the dry case. In both moist and dry cases, the shape of the APE spectra closely resembles that of the HKE spectra throughout the mesoscale, and the ratio of mesoscale HKE spectra to APE spectra is approximately 2, which are in a remarkably good agreement with the observations (e.g., Gage and Nastrom 1986). In addition, the moist VKE spectra show almost no wavenumber dependence (i.e., a flat spectrum) at the mesoscale.

On the basis of these nonhydrostatic simulations, a newly developed formulation of spectral energy budget was applied to study the mesoscale dynamics producing the lower-stratospheric energy spectra. In contrast to previous formulations, it is formulated in a moist, nonhydrostatic framework and can explicitly consider the effects of 3D divergence of the flow. The spectral energy budget analysis supports the following conclusions.

In the dry case, the lower-stratospheric mesoscale is forced by weak downscale cascades of both HKE and APE and by a weak conversion of APE to HKE (mainly at the larger end of the mesoscale range). We also find that the pressure vertical flux divergence has a significant positive contribution to the mesoscale subrange between 50 and 1000 km (Fig. 8a), which is consistent with the findings of WS2009. However, this positive contribution is largely counteracted by the negative HKE vertical flux divergence over the same scale range, which was not mentioned in previous studies, including WS2009. As a result, this mesoscale subrange is solely significantly governed by the downscale cascade, and to some extent can be regarded as a turbulent inertial subrange. Further studies should be made to clarify whether such balance between the pressure vertical flux divergence and the HKE vertical flux divergence in the lower-stratospheric mesoscale is robust to other flows.

In the RH60 case, the lower-stratospheric HKE at the mesoscale is mainly deposited by the pressure and HKE vertical flux divergence. This added mesoscale HKE is partly converted to APE at the same wavenumber and partly removed by diffusion. Another negative contribution to the mesoscale HKE comes from the nonlinear spectral flux, which shows that the lower-stratospheric mesoscale (with wavelengths larger than 360 km) undergoes a visible upscale HKE cascade. However, besides the conversion of the mesoscale HKE, another important positive contribution to the mesoscale APE in the lower stratosphere is from the 3D divergence term. With these two remarkable mesoscale APE sources, the lower-stratospheric mesoscale also undergoes a much stronger upscale APE cascade.

Differences of the spectral energy budget between the dry and RH60 simulations clearly reveal that the inclusion of moist processes can change the direction of the conversion between the HKE and APE and the direction of energy cascade through the lower-stratospheric mesoscale; besides the direct forcings of the HKE by the moist convection itself and the convectively generated IGWs, it also adds a significant positive forcing—that is, the 3D divergence term—to mesoscale APE.

### b. Implications for the dynamics underlying mesoscale energy spectra

Our results suggest the following dynamics for how the inclusion of moist processes enhances the lower-stratospheric mesoscale energies. Moist processes release the latent heating that enhances the vertical convection and excites inertia–gravity waves. The upper-tropospheric HKE is vertically transported to the lower stratosphere through the moist convection and the vertical propagation of convectively generated IGWs, especially through the latter. This added mesoscale HKE in the lower stratosphere is partly used to energize the mesoscale HKE spectra, partly upscale transferred to the synoptic scale, and partly converted to the mesoscale APE. With the positive contributions from the conversion of HKE and the forcing of the 3D divergence, the lower-stratospheric mesoscale APE spectrum is energized and a remarkable upscale APE cascade occurs. In addition, it should be clarified that the present results show a clear upscale cascade of HKE and APE in the large-scale end of the mesoscale range, not the entire mesoscale. For HKE below ~360 km and for APE below ~200 km, there is still a downscale cascade, which is to some extent consistent with the structure function analysis of aircraft data by Cho and Lindborg (2001).

The upscale cascade of both HKE and APE presented here, associated with intermittent moist convection, suggests that both downscale and upscale cascades through the mesoscale are permitted in the real atmosphere and the direction of energy cascade depends on possible energy sources. In contrast to the upscale cascade theory of Lilly (1983), which posits that the energy is injected at small scales, our results show that the small-scale energy source for the upscale cascade is not necessary and that the direct forcing of the mesoscale is also available to feed the upscale energy cascade through the lower-stratospheric mesoscale.

### c. Possible mechanism for the upscale energy cascade

One of the key findings of our study is the clear upscale energy cascade over the large-scale end of the mesoscale range in the RH60 simulation. The question is, then, what mechanism would drive this upscale energy cascade? Our tentative answer to this question is that it can be explained by geostrophic adjustment process. There are at least three direct evidences supporting this conjecture.

First, as shown in Fig. 3a, the DKE spectrum is of the same order of magnitude as the RKE spectrum in the mesoscale range. This fact rules out the motions at these scales properly being described within the QG approximation (Lindborg 2007). In other words, the unbalanced motions at these scales are significant.

Positive *t* = 4–7 days, for the dry and RH60 simulations. It is shown that in the RH60 simulation there is a significant positive conversion of DKE to RKE over the wavenumber range with upscale energy cascade, while the corresponding conversion in the dry simulation is negligible.

Despite these direct evidences, the correctness of this conjecture needs to be further confirmed.

We thank the editor, Dr. Ming Cai, and one anonymous reviewer for their detailed and constructive comments, which were very helpful for improving our manuscript and for a more physics-based explanation on our findings. We thank Dr. Michael L. Waite at University of Waterloo for his assistance in configuring the model and implementing the experimental design. We are grateful to Dr. Riwal Plougonven of LMD Paris for providing the initialization codes for idealized baroclinic waves. These codes are available from Dr. Riwal Plougonven via e-mail: riwal.plougonven@polytechnique.org. This research is supported by the National Natural Science Foundation of China (Grant 41375063) and by the National Natural Science Foundation for Young Scientists of China (Grant 41205074).

# APPENDIX

## Detailed Expressions of the Terms in the Spectral Energy Budget Equations

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