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  • View in gallery

    Initial conditions: vertical slice at the position of the maximum potential temperature perturbation. (a) Thin black lines show the unperturbed zonal velocity (interval: 10 m s−1; negative contours are dashed); thin gray lines show the unperturbed potential temperature (interval: 10 K); colors show the perturbation of the zonal velocity (interval: 0.5 m s−1); and the thick lime line indicates the 2 potential vorticity unit (PVU; 1 PVU = 10−6 K kg−1 m2 s−1) dynamical tropopause. (b) Black lines show the relative humidity (interval: 5%), and shading shows the mixing ratio of water vapor (interval: 2 g kg−1) for the moist simulation.

  • View in gallery

    Horizontal wavenumber spectra of (a) horizontal kinetic energy and (b) available potential energy, averaged in the vertical over and in time over . The solid reference lines have slopes of −5/3 and −3. The wavenumber is given on the lower x axis, and the wavelength is given on the upper x axis.

  • View in gallery

    HKE (black), APE (red), VKE (green), and total (blue) nonlinear spectral fluxes, averaged in the vertical over and in time over , vs total horizontal wavenumber for (a) the dry simulation and (b) the moist simulation. For convenience, the value of any flux at the origin (denoted by ×) has been modified to make it equal to that of the corresponding flux at . The inset is an expanded view of the mesoscale subrange .

  • View in gallery

    Spectral conversion terms (black), (red), (green), and (blue; for the moist simulation only), averaged in the vertical over and in time over , vs total horizontal wavenumber for (a) the dry simulation and (b) the moist simulation. Note that . The plotted spectra are multiplied by to preserve the area in log-linear coordinates.

  • View in gallery

    Horizontal wavenumber spectra of (a),(b) the divergences of the vertical fluxes of HKE , APE , VKE , and pressure , averaged in the vertical over and in time over ; and (c),(d) the corresponding vertical fluxes at the top and the bottom of the upper troposphere, averaged in time over for (left) the dry simulation and (right) the moist simulation. The vertical fluxes in (c) and (d) are scaled by with . Other details are as in Fig. 4.

  • View in gallery

    Spectral diabatic terms (a) and (b) for the moist simulation, averaged in the vertical over and in time over . Dashed lines show the contributions from microphysical processes (MP), dotted lines show the contributions from cumulus parameterization (CU), and solid lines show the total contributions. Other details are as in Fig. 4.

  • View in gallery

    (a) Spectral diabatic contributions of the latent heating (red) and dehumidifying (green) from microphysical processes (solid) and cumulus parameterization (dashed) to the APE budget for the moist simulation, averaged in the vertical over and in time over . (b) The ratio of the contribution from dehumidifying to the contribution from latent heating in microphysical processes, computed directly from the green solid and red solid lines in (a). Other details are as in Fig. 4.

  • View in gallery

    Vertical cross section of zonal-mean latent heating rate at t = 5 days. Color shading denotes the total latent heating rate caused by (a) microphysical processes, (b) the deposition/sublimation of ice (idep), (c) the deposition/sublimation of snow (sdep), and (d) the condensation/evaporation of cloud water (cond).

  • View in gallery

    Spectral diabatic contributions associated with the latent heating from the most important microphysical processes shown in Fig. 8. Other details are as in Fig. 4.

  • View in gallery

    Horizontal wavenumber spectra of the 3D divergence terms (thick) and the adiabatic nonconservative terms (thin) for the (a) dry and (b) moist simulations, averaged in the vertical over and in time over . Other details are as in Fig. 4.

  • View in gallery

    Horizontal wavenumber spectra of the nonlinear transfer terms, the net direct forcings, and the dissipation terms in the spectral (a),(c) HKE and (b),(d) APE budgets for the (top) dry and (bottom) moist simulations, averaged in the vertical over and in time over . Only the mesoscale range corresponding to wavelengths less than about 2000 km is present. Other details are as in Fig. 4.

  • View in gallery

    The eddy frequency (black), nonlinear transfer frequency (red), and net direct forcing frequency (green), computed from spectra averaged in the vertical over and in time over . The asterisk subscript can be h or A, which denotes HKE or APE, respectively.

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Applications of a Moist Nonhydrostatic Formulation of the Spectral Energy Budget to Baroclinic Waves. Part II: The Upper-Tropospheric Energy Spectra

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  • 1 Academy of Ocean Science and Engineering, National University of Defense Technology, Changsha, China
  • | 2 College of Meteorology and Oceanography, People’s Liberation Army University of Science and Technology, Nanjing, China
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Abstract

In this second part of a two-part study, a newly developed moist nonhydrostatic formulation of the spectral energy budget of both kinetic energy (KE) and available potential energy (APE) is employed to investigate the dynamics underlying the mesoscale upper-tropospheric energy spectra in idealized moist baroclinic waves. By calculating the conservative nonlinear spectral fluxes, it is shown that the inclusion of moist processes significantly enhances downscale cascades of both horizontal KE and APE. Moist processes act not only as a source of latent heat but also as an “atmospheric dehumidifier.” The latent heating, mainly because of the depositional growth of cloud ice, has a significant positive contribution to mesoscale APE. However, the dehumidifying reduces the diabatic contribution of the latent heating by 15% at all scales. Including moist processes also changes the direction of the mesoscale conversion between APE and horizontal KE and adds a secondary conversion of APE to gravitational energy of moist species. With or without moisture, the vertically propagating inertia–gravity waves (IGWs) produced in the lower troposphere result in a significant positive contribution to the upper-tropospheric horizontal KE spectra at the large-scale end of the mesoscale. However, including moist processes generates additional sources of IGWs located in the upper troposphere; the upward propagation of the convectively generated IGWs removes much of the horizontal KE there. Because of the restriction of the anelastic approximation, the three-dimensional divergence has no significant contribution. In view of conflicting contributions of various direct forcings, finally, an explicit comparison between the net direct forcing and energy cascade is made.

Corresponding author address: Lifeng Zhang, College of Meteorology and Oceanography, PLA University of Science and Technology, Zhong Hua Men Wai, Nanjing 211101, China. E-mail: zhanglif@yeah.net

Abstract

In this second part of a two-part study, a newly developed moist nonhydrostatic formulation of the spectral energy budget of both kinetic energy (KE) and available potential energy (APE) is employed to investigate the dynamics underlying the mesoscale upper-tropospheric energy spectra in idealized moist baroclinic waves. By calculating the conservative nonlinear spectral fluxes, it is shown that the inclusion of moist processes significantly enhances downscale cascades of both horizontal KE and APE. Moist processes act not only as a source of latent heat but also as an “atmospheric dehumidifier.” The latent heating, mainly because of the depositional growth of cloud ice, has a significant positive contribution to mesoscale APE. However, the dehumidifying reduces the diabatic contribution of the latent heating by 15% at all scales. Including moist processes also changes the direction of the mesoscale conversion between APE and horizontal KE and adds a secondary conversion of APE to gravitational energy of moist species. With or without moisture, the vertically propagating inertia–gravity waves (IGWs) produced in the lower troposphere result in a significant positive contribution to the upper-tropospheric horizontal KE spectra at the large-scale end of the mesoscale. However, including moist processes generates additional sources of IGWs located in the upper troposphere; the upward propagation of the convectively generated IGWs removes much of the horizontal KE there. Because of the restriction of the anelastic approximation, the three-dimensional divergence has no significant contribution. In view of conflicting contributions of various direct forcings, finally, an explicit comparison between the net direct forcing and energy cascade is made.

Corresponding author address: Lifeng Zhang, College of Meteorology and Oceanography, PLA University of Science and Technology, Zhong Hua Men Wai, Nanjing 211101, China. E-mail: zhanglif@yeah.net

1. Introduction

The mesoscale dynamics producing the lower-stratospheric energy spectra, with a focus on the importance of moist processes, were investigated with numerical simulations of idealized moist baroclinic waves in a companion paper by Peng et al. (2015, hereafter Part I). In Part I, a newly developed formulation of the spectral energy budget by Peng et al. (2014) was employed to perform a detailed spectral budget analysis of kinetic energy (KE) and available potential energy (APE). Diagnostic results demonstrated that the lower-stratospheric mesoscale in idealized moist baroclinic waves is not only governed by energy cascade but also significantly influenced by the direct forcings from the vertical fluxes caused by the vertical convection itself and the convectively generated inertia–gravity waves (IGWs) and from the three-dimensional (3D) divergence. Including moist processes can even change the direction of the conversion between the KE and APE, as well as the direction of energy cascade through the lower-stratospheric mesoscale. These results, together with Waite and Snyder (2013, hereafter WS2013), present a challenge to pure cascade theories based on the assumption that atmospheric mesoscale can be idealized as a turbulent inertial subrange (e.g., Gage 1979; Lilly 1983; Dewan 1979; Smith et al. 1987; Tung and Orlando 2003; Gkioulekas and Tung 2005a,b; Tulloch and Smith 2009; Lindborg 2006). In this Part II, we continue to investigate the dynamics responsible for mesoscale energy spectra in the upper troposphere, where the latent heating from moist processes occurs.

The importance of moist processes in establishing the upper-tropospheric mesoscale KE spectrum in moist baroclinic waves has been, to some extent, explored by WS2013. However, there are still many aspects that are unclear and require further investigation before a full understanding of the physical dynamics underlying the mesoscale energy spectra in the upper troposphere is possible. WS2013 diagnosed the buoyancy flux spectrum and the cross-spectrum of the latent heating and potential temperature (i.e., the heating spectrum) and found that the buoyancy flux spectrum, as well as the heating spectrum, in the moist simulation with relatively high moisture exhibited a positive peak at scales of around 800 km and a plateau throughout the mesoscale. Therefore, two possible scenarios were proposed by WS2013 (p. 1244) to explain how moist processes act on the upper-tropospheric mesoscale: “1) [KE] is injected mainly at large scales, thereby enhancing the downscale energy cascade; 2) [KE] is directly injected at mesoscale length scales, which directly energizes the mesoscale.” However, only diagnosing the buoyancy flux spectrum and the heating spectrum cannot completely reveal these two possible scenarios. First, based on WS2013, a quantitative diagnosis of the nonlinear spectral fluxes should be made to clarify whether the energy cascade in the upper-tropospheric mesoscale is still downscale in moist baroclinic waves and, if so, whether and to what extent it is enhanced. Second, moist processes release latent heating that will excite IGWs; the vertical propagation of the convectively generated IGWs will transport much of the energy injected by latent heating to the lower stratosphere, which may result in the net forcing at the upper-tropospheric mesoscale actually being small. Therefore, a further comparison between the net forcing and the energy cascade should be made to ascertain the significance of the direct forcing there. Third, only the influence of total latent heating on energy spectra has been investigated; therefore, further investigation should be made to quantify the contribution of the various microphysical processes on the mesoscale energy spectra. Moreover, besides acting as a source of latent heating, moist processes also act as an “atmospheric dehumidifier” (Pauluis and Held 2002), which is ignored in previous studies.

Thus, there are compelling reasons to conduct a completely spectral budget analysis of KE and APE in the upper troposphere of moist baroclinic waves. The remainder of the paper is organized as follows. Section 2 reviews the moist, nonhydrostatic formulation of the spectral energy budget and numerical simulations described in Part I that will be employed here. Section 3 discusses the energy cascade, spectral energy conversion, and spectral energy vertical transportation. Section 4 discusses spectral diabatic influences from moist processes. The contributions from both the latent heating and the dehumidifying are considered, and the contributions of the various microphysical processes to the mesoscale energy spectra are also quantified. Section 5 analyzes the effects of the adiabatic nonconservative processes and 3D divergence. Section 6 further analyzes the net direct forcing and makes an explicit comparison with the energy cascade. Conclusions and discussion are given in section 7.

2. Methodology

a. Spectral energy budget equations

The key points of Part I are that the spectral energy budget equations for KE and APE are formulated in a moist, nonhydrostatic framework, and the vertical fluxes are exactly separated from the energy cascade as in Augier and Lindborg (2013). Horizontal wavenumber spectra are computed by using two-dimensional discrete cosine transform (DCT; Denis et al. 2002). Let be the DCT of field at a given height level, where is the horizontal wave vector. Define the total horizontal wavenumber . In the height coordinate system, the formulation of the spectral energy budget, which is suitable for the moist, compressible, nonhydrostatic atmosphere, can be written as
e1
e2
e3
where is the horizontal KE (HKE) spectrum, is the vertical KE (VKE) spectrum, and is the moist APE spectrum; , , and are the nonlinear transfer terms; , , and are the vertical fluxes of HKE, VKE, and APE; is the pressure vertical flux, which, to some extent, corresponds to the vertical flux of IGW energy; , , and are the 3D divergence terms; represents the spectral conversion of APE to HKE, represents the spectral conversion of APE to VKE, and represents the spectral conversion of APE to other forms of energy; and are the spectral diabatic terms; , , and are the diffusion terms; and , , and are the adiabatic nonconservative terms. Square brackets indicate the one-dimensional horizontal spectra. Detailed expressions for the above terms are given in appendixes A and B, and the derivation of the equations is given in Peng et al. (2014). It should be noted that the spectral energy budget formulation presented here is a raw one rather than a cumulative one, as in Part I.

b. Simulations of baroclinic waves

For the purpose of this study, we repeat the numerical simulations of baroclinic waves in Part I with only some minor modifications. Simulations are performed with the Advanced Research dynamical core of the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008) on an plane in a rectangular channel with periodic boundary conditions in and rigid and symmetric boundary conditions in . The domain size is in , , and directions, respectively. The numbers of horizontal grid points are , and the vertical grid employs 180 layers. The horizontal resolution is .

The initial conditions consist of a zonally uniform, dry baroclinic jet, its fastest-growing normal mode with small amplitude, and—for the moist simulation—a suitable moisture profile. A dry jet with a maximum velocity of 58 m s −1 at a height of about 8.5 km (Fig. 1a) is constructed from a potential vorticity (PV) inversion approach following previous studies (e.g., Plougonven and Snyder 2007, hereafter PS2007; WS2013). For the moist simulation, water vapor is initialized after the PV inversion stage. As a modification to Part I and WS2013, a more complex relative humidity (RH) profile, as described in appendix C, is employed to initialize water vapor. This complex RH profile ensures that the initial RH of the lower troposphere is identical to that of the case starting from a uniform RH of 60% (RH60) in Part I, while the stratosphere is dry enough (Fig. 1b). The motivation of this modification is only to make the RH profile more reasonable. Both the dry and moist jets are perturbed with the dry fastest-growing normal mode (Fig. 1a, shading), which is computed with an iterative breeding procedure similar to PS2007 and is scaled so that its maximum perturbation in potential temperature is 2 K (Davis 2010; WS2013).

Fig. 1.
Fig. 1.

Initial conditions: vertical slice at the position of the maximum potential temperature perturbation. (a) Thin black lines show the unperturbed zonal velocity (interval: 10 m s−1; negative contours are dashed); thin gray lines show the unperturbed potential temperature (interval: 10 K); colors show the perturbation of the zonal velocity (interval: 0.5 m s−1); and the thick lime line indicates the 2 potential vorticity unit (PVU; 1 PVU = 10−6 K kg−1 m2 s−1) dynamical tropopause. (b) Black lines show the relative humidity (interval: 5%), and shading shows the mixing ratio of water vapor (interval: 2 g kg−1) for the moist simulation.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

After initialization, the simulations are integrated forward for 16 days. As one of our foci is on determining the contribution of different microphysical processes to the spectral energy budget, we use the WRF single-moment 6-class microphysics scheme (WSM6; Hong and Lim 2006), which explicitly predicts water vapor, cloud water, cloud ice, rain, snow, and graupel. Other physical parameterizations are identical to Part I, including the cumulus scheme, Rayleigh damping, vertical mixing, and numerical diffusion.

Despite the differences of RH in the stratosphere, the main results of the moist simulation here are qualitatively consistent with those of the RH60 case in Part I, which also verifies the robustness of the conclusions in Part I. Following WS2013 and Part I, the life cycle of simulated baroclinic waves can be divided into three distinct stages: the early phase (t = 4–7 days), the intermediate phase (t = 7–10 days), and the late phase (t = 10–13 days). During the early phase, the baroclinic instability saturates, and the maximum convection and diabatic heating occur. Moreover, the maximum latent heating for the moist simulation occurs at t = 5 days; positive latent heating mainly takes place below the height of 12 km and peaks at around 8 km. A more extensive description can be found in Part I. In what follows, we refer to z = 5–10 km as the upper troposphere and focus only on the early phase to highlight the effects of moist processes. All spectra are averaged in the vertical over and in time over . Before conducting a completely spectral budget analysis, let us first take a look at the simulated upper-tropospheric HKE and APE spectra in both cases, shown in Fig. 2. As expected, these spectra are very similar to those shown in WS2013. Both HKE and APE spectra for the dry case (black) develop a slope of approximately −3 for wavelengths larger than around 400 km and fall off rapidly at smaller scales. The inclusion of moist processes significantly enhances the upper-tropospheric mesoscale HKE and APE, resulting in a −3 power law extending through to wavelength 100 km for the moist case (red).

Fig. 2.
Fig. 2.

Horizontal wavenumber spectra of (a) horizontal kinetic energy and (b) available potential energy, averaged in the vertical over and in time over . The solid reference lines have slopes of −5/3 and −3. The wavenumber is given on the lower x axis, and the wavelength is given on the upper x axis.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

3. Energy cascade, conversion, and vertical transportation

a. Energy cascade

The energy cascade can be exactly estimated by calculating the nonlinear spectral flux:
e4
where denotes the largest wavenumber. By definition, . Positive nonlinear spectral flux means downscale energy cascade, and negative spectral flux means upscale energy cascade.

Figure 3a shows the HKE (), APE (), VKE (), and total () nonlinear spectral fluxes as a function of for the dry simulation. Here, . At the large scales more than about 2000 km, the total nonlinear spectral flux (blue) is dominated by positive APE spectral flux (red) with the maximum value of around , which suggests a significant downscale cascade of APE from the planetary scales to the synoptic scales. At the mesoscale, the APE spectral flux is also positive and reaches its local maximum of around , meaning a relatively weak downscale cascade from the large end of the mesoscale toward the smaller scales. As for the HKE spectral flux, it can be divided into three regions: , , and . These three wavelength bands correspond to scales of motion that exhibit downscale, upscale, and downscale cascades of HKE, respectively. The maximum value of the HKE spectral flux at the mesoscale is at . Therefore, at the mesoscale, the downscale cascade of HKE is slightly stronger than the downscale cascade of APE.

Fig. 3.
Fig. 3.

HKE (black), APE (red), VKE (green), and total (blue) nonlinear spectral fluxes, averaged in the vertical over and in time over , vs total horizontal wavenumber for (a) the dry simulation and (b) the moist simulation. For convenience, the value of any flux at the origin (denoted by ×) has been modified to make it equal to that of the corresponding flux at . The inset is an expanded view of the mesoscale subrange .

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

The nonlinear spectral fluxes in the moist simulation, shown in Fig. 3b, are quite different. Over all scales, the HKE spectral flux is comparable to the APE spectral flux. As a result, the total spectral flux can also be divided into three regions, which correspond to motion that exhibits downscale, upscale, and downscale cascades, respectively. The maximum value of the APE spectral flux at the large scale is around , which is less than half of that in the dry simulation. This suggests that including moist processes weakens the downscale cascade of APE through the large scale. On the contrary, the APE spectral flux in the mesoscale has a local maximum value of , which is much larger than that of the dry simulation. Therefore, the inclusion of moist processes significantly enhances the downscale cascade of APE through the mesoscale. Similarly, it also enhances the downscale cascade of HKE through the mesoscale. Specifically, the HKE spectral flux in the mesoscale has a plateau of .

Furthermore, in both simulations, the VKE nonlinear spectral flux (green line) is negligible and therefore not discussed here.

b. Energy conversion

Figure 4 presents horizontal wavenumber spectra of the conversion terms , , , and for the dry and moist simulations. Here, represents the spectral conversion of moist APE to gravitational energy of total moist species (MGE).

Fig. 4.
Fig. 4.

Spectral conversion terms (black), (red), (green), and (blue; for the moist simulation only), averaged in the vertical over and in time over , vs total horizontal wavenumber for (a) the dry simulation and (b) the moist simulation. Note that . The plotted spectra are multiplied by to preserve the area in log-linear coordinates.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

For the dry simulation (Fig. 4a), the energy conversion is dominated by the conversion between APE and HKE: that is, . Positive implies a conversion from APE to HKE at horizontal wavenumber . Therefore, at scales larger than 2000 km, the conversion is from APE to HKE, which is consistent with the large-scale baroclinic instability; while at the mesoscale, the conversion is from HKE to APE. These results are similar to the findings of WS2013 (see their Fig. 11).

For the moist simulation (Fig. 4b), at scales larger than 2000 km, the energy conversion is also dominated by positive . At the mesoscale, however, the energy conversions are quite different from that in the dry simulation. The term is positive for nearly all the mesoscale less than 1000 km, which means that the conversion is mainly from APE to HKE. This result should be due to the direct forcing of APE by the latent heating. The difference between and at the mesoscale is visible, which suggests that the conversion between APE and MGE or VKE is also significant, although not dominant. For example, the ratio of to at is around 0.5, and the ratio of to at is around 0.2.

c. Energy vertical transportation

By definition, the divergence of any vertical flux , averaged in the vertical over the upper troposphere, is given by
e5
where , , and . The terms and are the vertical fluxes at the top and the bottom of the upper troposphere, respectively. Note that indicates a downward vertical flux at wavenumber . Figures 5a and 5b present horizontal wavenumber spectra of the divergences of the vertical fluxes of HKE , APE , VKE , and pressure , averaged in the vertical over and in time over . Figures 5c and 5d present the corresponding vertical fluxes at the top and the bottom of the upper troposphere, averaged in time over . For consistency, the vertical fluxes in Figs. 5c and 5d are scaled by . Figures 5a and 5c correspond to the dry simulation and Figs. 5b and 5d correspond to the moist simulation.
Fig. 5.
Fig. 5.

Horizontal wavenumber spectra of (a),(b) the divergences of the vertical fluxes of HKE , APE , VKE , and pressure , averaged in the vertical over and in time over ; and (c),(d) the corresponding vertical fluxes at the top and the bottom of the upper troposphere, averaged in time over for (left) the dry simulation and (right) the moist simulation. The vertical fluxes in (c) and (d) are scaled by with . Other details are as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

For the dry simulation, the divergence of the pressure vertical flux (blue line in Fig. 5a) is positive throughout the mesoscale with a local maximum around wavelength 2000 km, which is mainly because of an upward vertical flux at the bottom layer (blue dashed line in Fig. 5c). This result is similar to that of Waite and Snyder (2009, hereafter WS2009; see their Fig. 12a). It suggests that the pressure vertical flux, associated with upward-propagating IGWs produced in the lower troposphere (Snyder et al. 1993; PS2007; WS2009) has a significant positive contribution to the upper-tropospheric HKE at the mesoscale, especially at the large end of the mesoscale. Besides the positive contribution of the pressure vertical flux, the HKE vertical flux also has a positive contribution to the mesoscale HKE, which is also dominated by the vertical flux at the bottom layer (black dashed line in Fig. 5c). On the contrary, the APE vertical flux has a negative contribution to the mesoscale APE, which is mainly because of a downward vertical flux of APE at the bottom layer (red dashed line in Fig. 5c).

Figure 5b presents the same quantities as Fig. 5a, but for the moist simulation. Comparing Figs. 5a and 5b, we can see that the vertical fluxes through the mesoscale are quite different, which implies that the inclusion of moist processes significantly changes the vertical transportation of energy. Moist processes release latent heating, which will excite IGWs. A strong upward vertical flux of pressure throughout the mesoscale can be seen at the top layer (blue solid line in Fig. 5d), which suggests that the upward propagation of the convectively generated IGWs transport much of HKE into the lower stratosphere. On the other hand, at the bottom layer of the upper troposphere there is still a strong upward vertical flux at the large-scale end of the mesoscale, as in the dry simulation, while the corresponding flux at smaller scales (less than 1000 km) is weak downward (blue dashed line in Fig. 5d). The outward vertical fluxes of pressure at both the top and the bottom of the upper-troposphere account for the significantly negative contribution of the pressure vertical flux divergence at scales less than about 1000 km (blue solid line in Fig. 5b). This broad negative contribution of the pressure vertical flux divergence suggests that the convectively generated IGWs exist throughout the upper-tropospheric mesoscale. At scales between 2000 and 500 km, is still positive. However, at other scales of the mesoscale, is weakly negative, mainly owing to the downward vertical flux at the bottom of the layer for these scales (black dashed line in Fig. 5d). Furthermore, has a positive contribution to the upper-tropospheric APE at scales less than 1000 km (red line in Fig. 5b), corresponding to the downward flux at the top of layer (red solid line in Fig. 5d) and upward flux at the bottom of layer (red dashed line in Fig. 5d) at these scales.

4. Diabatic effects

The quantitative analysis in the above section clearly demonstrates that the inclusion of moist processes does enhance the downscale cascade through the upper-tropospheric mesoscale, which validates scenario 1 proposed by WS2013, and it also changes the direction of the conversion between APE and HKE and the vertical transportation of energy at the upper-tropospheric mesoscale. In what follows, we further quantify the direct forcing of the diabatic influences from moist processes.

a. Spectral diabatic contributions

The spectral diabatic terms and , as source terms in the spectral HKE and APE budget equations, are calculated with the diabatic influences . In the present moist simulation, the diabatic influences are from two aspects: microphysics and cumulus parameterization. Figure 6 presents spectral diabatic terms and for the moist simulation, averaged in the vertical over and in time over . Comparing Figs. 6a and 6b, we can see that is much more significant than , especially at the mesoscale, meaning that the diabatic influences from moist processes mainly act on the APE. Furthermore, for the spectral APE budget, the total diabatic term is dominated by the contribution of microphysics (dashed line), which is consistent with the findings of WS2013 for the heating. However, it is shown that the wavelength around which the spectral diabatic contribution to the spectral APE budget has a local maximum in the mesoscale is 1000 km, which is slightly larger than the scale (800 km) reported by WS2013. This should be because of the modification of the initialization of the dry basic-state jet (e.g., Davies et al. 1991; Wernli et al. 1998).

Fig. 6.
Fig. 6.

Spectral diabatic terms (a) and (b) for the moist simulation, averaged in the vertical over and in time over . Dashed lines show the contributions from microphysical processes (MP), dotted lines show the contributions from cumulus parameterization (CU), and solid lines show the total contributions. Other details are as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

b. Acting as both a latent heating source and an atmospheric dehumidifier

Bannon (2005) noted that the atmospheric available energy increases with increasing water vapor, which implies that the decrease of water vapor should have a negative effect on the APE. Phase changes release the latent heating and reduce the water vapor content of moist air. In other words, moist processes act not only as a source of latent heat but also as an atmospheric dehumidifier (Pauluis and Held 2002). In the present formulation, the spectral diabatic terms are calculated with the combined diabatic influences , which can be expressed as
e6
where and represent the tendencies of potential temperature and water vapor due to moist processes, respectively. The first term on the right-hand side is the influence due to the latent heating, and the second term is the influence due to the dehumidifying. Therefore, both of these two counteracting effects of moist processes are taken into account. Figure 7a presents the spectral diabatic contributions of the latent heating (red) and dehumidifying (green) from microphysical processes (solid) and cumulus parameterization (dashed) to the spectral APE budget. Over all scales, either the latent heating or dehumidifying is dominated by the microphysical processes. The spectral diabatic contribution due to the latent heating (red solid) from the microphysical processes is positive at all scales, with a local maximum around wavelength 1000 km. On the contrary, the diabatic contribution due to the dehumidifying from the microphysical processes (green solid) is always negative, meaning that, to some extent, it reduces the contribution of the latent heating.
Fig. 7.
Fig. 7.

(a) Spectral diabatic contributions of the latent heating (red) and dehumidifying (green) from microphysical processes (solid) and cumulus parameterization (dashed) to the APE budget for the moist simulation, averaged in the vertical over and in time over . (b) The ratio of the contribution from dehumidifying to the contribution from latent heating in microphysical processes, computed directly from the green solid and red solid lines in (a). Other details are as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

To assess the importance of the dehumidifying by moist processes, we further compute the ratio of the contribution from dehumidifying to the contribution from latent heating in microphysical processes, which is shown in Fig. 7b. It is found that, over all scales, this ratio is approximately 0.15, which suggests that it will reduce the diabatic contribution of the latent heating by 15%. Therefore, previous studies, which neglected the dehumidifying by moist processes, to some extent overestimated the effects of moist processes.

c. Further analysis of microphysical processes

The microphysical scheme used in the moist simulation is WSM6, which explicitly predicts six categories of moist species and includes various microphysical processes (i.e., the deposition and sublimation of ice, the deposition and sublimation of snow, and the condensation and evaporation of cloud water).

Figure 8 presents the vertical cross sections of zonal-mean latent heating rates caused by the most important microphysical processes in the present case at t = 5 days, when the diabatic influences are strongest. It is shown that the latent heat release in the upper troposphere is mainly caused by the depositional growth of ice (Fig. 8b) and snow (Fig. 8c), especially the former, while in the lower levels it is mainly caused by the condensation of cloud water (Fig. 8d). The latent heating rate caused by the depositional growth of ice peaks at a height of about 7.5 km, with a maximum of .The spectral diabatic contributions associated with the latent heating from different microphysical processes are shown in Fig. 9. The results suggest the importance of cold cloud processes, especially the depositional growth of cloud ice, on the upper-tropospheric energy spectra.

Fig. 8.
Fig. 8.

Vertical cross section of zonal-mean latent heating rate at t = 5 days. Color shading denotes the total latent heating rate caused by (a) microphysical processes, (b) the deposition/sublimation of ice (idep), (c) the deposition/sublimation of snow (sdep), and (d) the condensation/evaporation of cloud water (cond).

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

Fig. 9.
Fig. 9.

Spectral diabatic contributions associated with the latent heating from the most important microphysical processes shown in Fig. 8. Other details are as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

5. Adiabatic nonconservative processes and 3D divergence

In this section, we briefly investigate the contributions of the 3D divergence and the adiabatic nonconservative processes. Figure 10 presents the horizontal wavenumber spectra of the 3D divergence terms and the adiabatic nonconservative terms for the dry and moist simulations, averaged in the vertical over and in time over . Note that the range of the vertical axis is only a quarter of that in Fig. 3. This suggests that the contributions of these terms are, to some extent, less than the terms above. To facilitate the comparison, the adiabatic nonconservative is divided into two terms: with and .

Fig. 10.
Fig. 10.

Horizontal wavenumber spectra of the 3D divergence terms (thick) and the adiabatic nonconservative terms (thin) for the (a) dry and (b) moist simulations, averaged in the vertical over and in time over . Other details are as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

For the dry simulation (Fig. 10a) at all scales larger than about 500 km, the terms and are small, comparable in magnitude, and opposite in sign; that is, they cancel each other out, and so do the terms and . At scales less than 500 km, similar cancellations can also be found; however, the degree of the cancellations between these terms becomes a little smaller. For the moist simulation (Fig. 10b), these terms have much larger values, especially at the mesoscale, meaning that inclusion of moist processes significantly enhances the 3D divergence of mesoscale motions in the upper troposphere. However, there still exist similar cancellations between these terms as in the dry simulation. Because of the cancellations between these terms, the combined contributions of these terms are still negligible. To clarify the physical reasons behind these cancellations, let us consider the relations
e7
e8
with and . Thus, such cancellations suggest that
e9
that is, the flow in the upper troposphere is, to a large extent, restricted by the anelastic approximation, which is more accurate at larger scales.

Furthermore, there is also another adiabatic nonconservative term , which acts on the APE budget. In the moist simulation, it has a weakly negative contribution to mesoscale APE, in comparison with other physical processes, such as energy cascade and conversion.

6. Net direct forcing and comparison with cascade

To ascertain the extent to which the upper-troposphere mesoscale is directly energized, we further make a quantitative analysis of the net direct forcing (NDF). The NDF spectrum for the spectral budget is calculated by summing all the terms except the nonlinear transfer term and the diffusion term in the corresponding equation. For example, the NDF spectrum for the spectral HKE budget is defined as
e10
Mesoscale spectra of the NDFs of HKE and APE, averaged in the vertical over and in time over , for the dry and moist simulations are plotted as green lines in Fig. 11.
Fig. 11.
Fig. 11.

Horizontal wavenumber spectra of the nonlinear transfer terms, the net direct forcings, and the dissipation terms in the spectral (a),(c) HKE and (b),(d) APE budgets for the (top) dry and (bottom) moist simulations, averaged in the vertical over and in time over . Only the mesoscale range corresponding to wavelengths less than about 2000 km is present. Other details are as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

For the dry simulation, as already shown, the direct forcing of the mesoscale HKE spectra in the upper troposphere is dominated by negative (i.e., the conversion from HKE to APE), positive , and positive . As a result, the corresponding NDF (green line in Fig. 11a) is positive at wavelengths less than 800 km and crosses zero several times at larger scales. To mesoscale APE, apart from the significant, positive forcing from the conversion of HKE, there is also a significant, negative forcing from the term , resulting in its NDF (green line in Fig. 11b) being negative at scales less than about 1000 km and positive at the rest of the mesoscale.

For the moist simulation, the direct forcing of mesoscale HKE spectra is also mainly from , , and . However, the contributions of these terms are quite different in magnitude and sign, especially at the small-scale end of the mesoscale. As a result, NDF (green line in Fig. 11c) is much more significant than that in the dry simulation. The very large positive NDF at the large-scale end of mesoscale is mainly due to the positive contributions from and , especially the former, while at smaller scales, the positive contribution from is largely counteracted by the negative contributions from and , resulting in NDF being relatively small. Moist processes release the latent heating, which has a significant positive contribution to mesoscale APE. However, the release of the latent heating will, in turn, change the modified potential temperature perturbation and enhance vertical convection, which results in much more conversion from mesoscale APE to other forms of energy, including KE and MGE. Thus, as shown in Fig. 11d (green line), NDF of APE is negative at nearly all of the mesoscale, except its largest scales.

Also plotted in Fig. 11 are mesoscale spectra of the nonlinear transfer terms (red lines) and the diffusion terms (green lines). On the whole, in both the spectral HKE and APE budgets and for both simulations, the nonlinear transfer term (i.e., the gradient of the nonlinear spectral flux) and NDF are comparable in magnitude and opposite in sign at most of the mesoscale. This robustly suggests that NDF at the mesoscale is as important as the energy cascade, and therefore the turbulent inertial theories based on a pure energy cascade are apparently inappropriate for the atmosphere. Comparing the dry (Figs. 11a,b) and moist (Figs. 11c,d) simulations, we see that moist processes enhance both the downscale cascade and NDF through the upper-tropospheric mesoscale. Although the latent heating from moist processes directly injects much energy into the upper-tropospheric mesoscale, NDF there is not always positive. In the moist simulation, the mesoscale upper-tropospheric HKE spectrum (Fig. 11c) at scales larger than about 500 km is mainly energized by the enhanced NDF and weakened by the enhanced downscale cascade, while at the smaller scales it is energized by the enhanced downscale cascade and mainly weakened by the dissipation. On the other hand, the upper-tropospheric APE spectrum (Fig. 11d) throughout the mesoscale is energized by the enhanced downscale cascade and weakened by the enhanced NDF.

Following WS2009, the significance of the deviations of and from zero can be further evaluated by comparing their characteristic inverse time scales (i.e., and ) with the eddy frequency (, where denotes the density of the dry reference state). Using the vertically and temporally averaged spectra shown above, the terms , , , , , and for the dry and moist simulations are computed and plotted in Fig. 12. Both and in the moist simulation are larger than in the dry simulation, especially at the small-scale end of the mesoscale; this is consistent with the findings of WS2013 that the inclusion of moist processes energizes the upper-tropospheric mesoscale. For the dry simulation, and have values on the order of 1 day−1 for wavelengths smaller than about 400 km, which are on the same order as or greater than . This suggests that the nonlinear transfer and NDF terms at these scales in the spectral HKE budget are dynamically significant, which is consistent with the findings of WS2009. In addition, similar results can also be found in the spectral APE budget. For the moist simulation, for wavelengths smaller than about 400 km, and and at the whole mesoscale also have values on the order of 1 day−1. These values are also on the same order as or greater than the corresponding eddy frequencies, which implies that the corresponding terms are also dynamically significant. By contrast, although the NDF is significantly enhanced by moist processes, has values on the order of 0.5 day−1, which is smaller than .

Fig. 12.
Fig. 12.

The eddy frequency (black), nonlinear transfer frequency (red), and net direct forcing frequency (green), computed from spectra averaged in the vertical over and in time over . The asterisk subscript can be h or A, which denotes HKE or APE, respectively.

Citation: Journal of the Atmospheric Sciences 72, 10; 10.1175/JAS-D-14-0359.1

7. Conclusions and discussion

In this Part II, a newly developed moist nonhydrostatic formulation of the spectral energy budget of both KE and APE is employed to investigate the dynamics underlying the mesoscale upper-tropospheric energy spectra in idealized moist baroclinic waves. From the results of the completely spectral energy budget analysis presented in this paper, we summarize and conclude as follows:

  1. The energy cascade has been exactly estimated by calculating the conservative nonlinear spectral flux. It is shown that moist processes enhance the downscale cascades of both HKE and APE through the upper-tropospheric mesoscale, validating scenario 1 proposed by WS2013. Including moist processes also changes the direction of the mesoscale conversion between APE and HKE and adds a secondary mesoscale conversion of APE to MGE: in the dry simulation, the mesoscale conversion is from HKE to APE, while in the moist simulation, it is mainly from APE to HKE.
  2. In both dry and moist simulations, an upward vertical flux of pressure, associated with vertically propagating IGWs produced in the lower troposphere, results in a significant positive contribution to the upper-tropospheric HKE at the large-scale end of the mesoscale. Including moist processes generates additional sources of IGWs, which are located in the upper troposphere; the upward propagation of the convectively generated IGWs will transport much of the HKE into the lower stratosphere.
  3. Moist processes act not only as a source of latent heat but also as an “atmospheric dehumidifier”. The latent heating has a significant positive contribution to mesoscale APE, with a peak at the large-scale end of the mesoscale, consistent with the findings by WS2013. However, dehumidifying reduces the diabatic contribution of the latent heating by 15% at all mesoscale length scales. This suggests that previous studies, including WS2013, that neglected dehumidifying by moist processes, to some extent, overestimated the effects of moist processes.
  4. The contribution of various microphysical processes on the mesoscale energy spectra has been further quantified. The corresponding results suggest the importance of cold cloud processes, especially the depositional growth of cloud ice.
  5. The upper troposphere is, to a large extent, restricted by the anelastic approximation. As a result, the 3D divergence has no significant contribution to its mesoscale energy spectra, in contrast to the lower stratosphere, where the 3D divergence has a significant positive contribution to mesoscale APE spectra (see Part I).
  6. Finally, an explicit comparison between the NDF and the nonlinear term is made. On the whole, in both the spectral HKE and APE budgets and for both simulations, these two terms are comparable in magnitude and opposite in sign at most of the mesoscale. This robustly suggests that the NDF at the mesoscale is as important as the energy cascade. Moist processes enhance the NDF at the upper-tropospheric mesoscale, but the enhanced NDF does not always energize all mesoscale length scales, which is different from scenario 2 proposed by WS2013.

The results present here and in Part I clearly demonstrate the different effects of moist processes on spectral energy budgets in the lower stratosphere and upper troposphere. With moist processes, in Part I we have shown that there is a clear upscale (inverse) energy cascade over the large-scale end of the mesoscale range in the lower stratosphere, whereas no such inverse cascade is obtained in the upper troposphere. The question is, then, what are possible explanations for this difference? Recently, Augier and Lindborg (2013) have shown that the interactions between the rotational modes account only for the upscale HKE flux, while the interactions involving the divergent part of the velocity field produce the downscale energy cascade (e.g., Deusebio et al. 2013). Thus, this difference should be because the inclusion of moist processes primarily enhances the divergent part of the velocity field in the upper troposphere (Fig. 6 in WS2013), while it significantly enhances both the divergent and rotational parts of the velocity field in the lower stratosphere (Fig. 3a in Part I). That is to say, the enhancement of the rotational flow accounts for the upscale energy cascade in the lower stratosphere. However, this raises another question of why the enhancement of the rotational flow in the upper troposphere is not as significant as that in the lower stratosphere when moist processes are considered. Dynamically, moist processes influence the rotational flow via their effects on the direct conversion of unbalanced (divergent) flow to balanced (rotational) flow, which is related to the geostrophic adjustment process. Thus, it seems more likely that this conversion is less significant in the upper troposphere than in the lower stratosphere for the present simulation of idealized moist baroclinic waves. To verify the robustness of this conjecture, the spectral budget of divergence and rotational kinetic energy in the mesoscale is currently in progress.

Acknowledgments

We thank the editor, Dr. Ming Cai, and two anonymous reviewers for helpful comments on the manuscript. We also thank Riwal Plougonven of LMD Paris for providing the initialization codes for idealized baroclinic waves and Michael L. Waite (University of Waterloo) for his assistance in setting up the idealized simulations. This research is supported by the National Natural Science Foundation of China (Grant 41375063) and partly supported by the National Natural Science Foundation for Young Scientists of China (Grant 41205074).

APPENDIX A

Definitions of HKE, VKE, and APE Spectra

In the height coordinate system, define the perturbation variables as deviations from a time-invariant, hydrostatically balanced, dry reference state as follows:
ea1
where is the Exner pressure, is the pressure, is a reference surface pressure, is the gas constant for dry air, is the specific heat of dry air at constant pressure, is the density of dry air; and is a modified moist potential temperature, with and representing the potential temperature and water vapor mixing ratio (mass per unit mass of dry air), respectively, and is the gas constant for water vapor.
Let for two scalar fields a and b and for two vector fields and , where the asterisk denotes the complex conjugate, and denotes the real part. The HKE, VKE, and APE spectra per unit volume are expressed as
ea2
ea3
and
ea4
where is the horizontal velocity vector, is the vertical velocity, is the dimensional prefactor with the Brunt–Väisälä frequency . The resultant two-dimensional spectra are averaged over annuli of constant on the kxky plane to obtain the one-dimensional spectra, where the annuli have width [ is the horizontal grid spacing, and , with and the number of points for the two dimensions of the horizontal domain (e.g., WS2013)]. For example, the one-dimensional spectrum of HKE is defined as
ea5
The one-dimensional spectra of any other terms in appendix B are defined in the corresponding way.

APPENDIX B

Detailed Expressions of the Terms in Spectral Energy Budget Equations

Let denote the dissipation of (with a placeholder for the fields u, w, etc.); is the total mixing ratio, where denote the mixing ratio of cloud, rain, ice, and any other hydrometeors, respectively. The tendency of the modified potential temperature is given by
eb1
where represents the combined diabatic influences, and represents the combined dissipations.
Detailed mathematical expression of the terms of the spectral energy budget equations in section 2a are given as
eb2
eb3
eb4
eb5
eb6
eb7
eb8
eb9
where
eq1
eq2
eq3
eq4
eq5
eq6
eq7
eq8
eq9
eq10
and
eq11

APPENDIX C

Initial Relative Humidity Profile for the Moist Simulation

The more complex initial relative humidity (RH) for the moist simulation is given by Tan et al. (2004):
ec1

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