• Augier, P., , and E. Lindborg, 2013: A new formulation of the spectral energy budget of the atmosphere, with application to two high-resolution general circulation models. J. Atmos. Sci., 70, 22932308, doi:10.1175/JAS-D-12-0281.1.

    • Search Google Scholar
    • Export Citation
  • Bacmeister, J. T., , S. D. Eckermann, , P. A. Newman, , L. Lait, , K. R. Chan, , M. Loewenstein, , M. H. Proffitt, , and B. L. Gary, 1996: Stratosphere horizontal wavenumber spectra of winds, potential temperature, and atmospheric tracers observed by high altitude aircraft. J. Geophys. Res., 101, 94419470, doi:10.1029/95JD03835.

    • Search Google Scholar
    • Export Citation
  • Bannon, P. R., 2005: Eulerian available energetics in moist atmospheres. J. Atmos. Sci., 62, 42384252, doi:10.1175/JAS3516.1.

  • Bierdel, L., , P. Friederichs, , and S. Bentzien, 2012: Spatial kinetic energy spectra in the convection-permitting limited-area NWP model COSMO-DE. Meteor. Z., 21, 245258, doi:10.1127/0941-2948/2012/0319.

    • Search Google Scholar
    • Export Citation
  • Brune, S., , and E. Becker, 2013: Indications of stratified turbulence in a mechanistic GCM. J. Atmos. Sci., 70, 231247, doi:10.1175/JAS-D-12-025.1.

    • Search Google Scholar
    • Export Citation
  • Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 10871095, doi:10.1175/1520-0469(1971)028<1087:GT>2.0.CO;2.

  • Cho, J. Y. N., , and E. Lindborg, 2001: Horizontal velocity structure functions in the upper troposphere and lower stratosphere: 1. Observations. J. Geophys. Res., 106 (D10), 10 22310 232, doi:10.1029/2000JD900814.

    • Search Google Scholar
    • Export Citation
  • Cho, J. Y. N., and Coauthors, 1999: Horizontal wavenumber spectra of winds, temperature, and trace gases during the Pacific Exploratory Missions: 1. Climatology. J. Geophys. Res., 104, 56975716, doi:10.1029/98JD01825.

    • Search Google Scholar
    • Export Citation
  • Davidson, P. A., 2004: Turbulence: An Introduction for Scientists and Engineers. Oxford University Press, 678 pp.

  • Davis, C. A., 2010: Simulations of subtropical cyclones in a baroclinic channel model. J. Atmos. Sci., 67, 28712892, doi:10.1175/2010JAS3411.1.

    • Search Google Scholar
    • Export Citation
  • Denis, B., , J. Côté, , and R. Laprise, 2002: Spectral decomposition of two-dimensional atmospheric fields on limited-area domains using the discrete cosine transform (DCT). Mon. Wea. Rev., 130, 18121829, doi:10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Dewan, E. M., 1979: Stratospheric wave spectra resembling turbulence. Science, 204, 832835, doi:10.1126/science.204.4395.832.

  • Gage, K. S., 1979: Evidence for a k−5/3 law inertial range in mesoscale two-dimensional turbulence. J. Atmos. Sci., 36, 19501954, doi:10.1175/1520-0469(1979)036<1950:EFALIR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gage, K. S., , and G. D. Nastrom, 1986: Theoretical interpretation of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft during GASP. J. Atmos. Sci., 43, 729740, doi:10.1175/1520-0469(1986)043<0729:TIOAWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gkioulekas, E., , and K. K. Tung, 2005a: On the double cascades of energy and enstrophy in two dimensional turbulence. Part 1. Theoretical formulation. Discrete Contin. Dyn. Syst., 5B, 79102.

    • Search Google Scholar
    • Export Citation
  • Gkioulekas, E., , and K. K. Tung, 2005b: On the double cascades of energy and enstrophy in two dimensional turbulence. Part 2. Approach to the KLB limit and interpretation of experimental evidence. Discrete Contin. Dyn. Syst., 5B, 103124.

    • Search Google Scholar
    • Export Citation
  • Hamilton, K., , Y. O. Takahashi, , and W. Ohfuchi, 2008: The mesoscale spectrum of atmospheric motions investigated in a very fine resolution global general circulation model. J. Geophys. Res., 113, D18110, doi:10.1029/2008JD009785.

    • Search Google Scholar
    • Export Citation
  • Klein, P., , B. L. Hua, , G. Lapeyre, , X. Capet, , S. Le Gentil, , and H. Sasaki, 2008: Upper ocean turbulence from high-resolution 3D simulations. J. Phys. Oceanogr., 38, 17481763, doi:10.1175/2007JPO3773.1.

    • Search Google Scholar
    • Export Citation
  • Knievel, J. C., , G. H. Bryan, , and J. P. Hacker, 2007: Explicit numerical diffusion in the WRF model. Mon. Wea. Rev., 135, 38083824, doi:10.1175/2007MWR2100.1.

    • Search Google Scholar
    • Export Citation
  • Koshyk, J. N., , and K. Hamilton, 2001: The horizontal kinetic energy spectrum and spectral budget simulated by a high-resolution troposphere–stratosphere–mesosphere GCM. J. Atmos. Sci., 58, 329348, doi:10.1175/1520-0469(2001)058<0329:THKESA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kraichnan, R. H., 1967: Inertial ranges in two-dimensional turbulence. Phys. Fluids, 10, 14171423, doi:10.1063/1.1762301.

  • Lilly, D. K., 1983: Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci., 40, 749761, doi:10.1175/1520-0469(1983)040<0749:STATMV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 1999: Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech., 388, 259288, doi:10.1017/S0022112099004851.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 2005: The effect of rotation on mesoscale energy cascade in the free atmosphere. Geophys. Res. Lett., 32, L01809, doi:10.1029/2004GL021319.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 2006: The energy cascade in a strongly stratified fluid. J. Fluid Mech., 550, 207242, doi:10.1017/S0022112005008128.

  • Lindborg, E., 2007: Horizontal wavenumber spectra of vertical vorticity and horizontal divergence in the upper troposphere and lower stratosphere. J. Atmos. Sci., 64, 10171025, doi:10.1175/JAS3864.1.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 2009: Two comments on the surface quasigeostrophic model for the atmospheric energy spectrum. J. Atmos. Sci., 66, 10691072, doi:10.1175/2008JAS2972.1.

    • Search Google Scholar
    • Export Citation
  • Morss, R. E., , C. Snyder, , and R. Rotunno, 2009: Spectra, spatial scales, and predictability in a quasigeostrophic model. J. Atmos. Sci., 66, 31153130, doi:10.1175/2009JAS3057.1.

    • Search Google Scholar
    • Export Citation
  • Nastrom, G. D., , and K. S. Gage, 1985: A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci., 42, 950960, doi:10.1175/1520-0469(1985)042<0950:ACOAWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pauluis, O., , and I. M. Held, 2002: Entropy budget of an atmosphere in radiative–convective equilibrium. Part I: Maximum work and frictional dissipation. J. Atmos. Sci., 59, 140149, doi:10.1175/1520-0469(2002)059<0140:EBOAAI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Peng, J., , L. Zhang, , Y. Luo, , and Y. Zhang, 2014a: Mesoscale energy spectra of the mei-yu front system. Part I: Kinetic energy spectra. J. Atmos. Sci., 71, 3755, doi:10.1175/JAS-D-13-085.1.

    • Search Google Scholar
    • Export Citation
  • Peng, J., , L. Zhang, , Y. Luo, , and C. Xiong, 2014b: Mesoscale energy spectra of the mei-yu front system. Part II: Moist available potential energy spectra. J. Atmos. Sci., 71, 14101424, doi:10.1175/JAS-D-13-0319.1.

    • Search Google Scholar
    • Export Citation
  • Plougonven, R., , and C. Snyder, 2007: Inertia–gravity waves spontaneously generated by jets and fronts. Part I: Different baroclinic life cycles. J. Atmos. Sci., 64, 25022520, doi:10.1175/JAS3953.1.

    • Search Google Scholar
    • Export Citation
  • Ricard, D., , C. Lac, , S. Riette, , R. Legrand, , and A. Mary, 2013: Kinetic energy spectra characteristics of two convection-permitting limited-area models AROME and Meso-NH. Quart. J. Roy. Meteor. Soc., 139, 1327–1341, doi:10.1002/qj.2025.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., 2004: Evaluating mesoscale NWP models using kinetic energy spectra. Mon. Wea. Rev., 132, 30193032, doi:10.1175/MWR2830.1.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., , and J. B. Klemp, 2008: A time-split nonhydrostatic atmospheric model for weather research and forecasting applications. J. Comput. Phys., 227, 34653485, doi:10.1016/j.jcp.2007.01.037.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp. [Available online at http://www.mmm.ucar.edu/wrf/users/docs/arw_v3_bw.pdf.]

  • Smith, S. A., , D. C. Fritts, , and T. E. Vanzandt, 1987: Evidence for a saturated spectrum of atmospheric gravity waves. J. Atmos. Sci., 44, 14041410, doi:10.1175/1520-0469(1987)044<1404:EFASSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Snyder, C., , and R. S. Lindzen, 1991: Quasigeostrophic wave-CISK in an unbounded baroclinic shear. J. Atmos. Sci., 48, 7686, doi:10.1175/1520-0469(1991)048<0076:QGWCIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Takahashi, Y. O., , K. Hamilton, , and W. Ohfuchi, 2006: Explicit global simulations of the mesoscale spectrum of atmospheric motions. Geophys. Res. Lett., 33, L12812, doi:10.1029/2006GL026429.

    • Search Google Scholar
    • Export Citation
  • Terasaki, K., , H. Tanaka, , and M. Satoh, 2009: Characteristics of the kinetic energy spectrum of NICAM model atmosphere. SOLA, 5, 180183, doi:10.2151/sola.2009-046.

    • Search Google Scholar
    • Export Citation
  • Tulloch, R., , and K. S. Smith, 2009: Quasigeostrophic turbulence with explicit surface dynamics: Application to the atmospheric energy spectrum. J. Atmos. Sci., 66, 450467, doi:10.1175/2008JAS2653.1.

    • Search Google Scholar
    • Export Citation
  • Tung, K. K., , and W. W. Orlando, 2003: The k−3 and k−5/3 energy spectrum of atmospheric turbulence: Quasigeostrophic two-level model simulation. J. Atmos. Sci., 60, 824835, doi:10.1175/1520-0469(2003)060<0824:TKAKES>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Waite, M. L., , and C. Snyder, 2009: The mesoscale kinetic energy spectrum of a baroclinic life cycle. J. Atmos. Sci., 66, 883901, doi:10.1175/2008JAS2829.1.

    • Search Google Scholar
    • Export Citation
  • Waite, M. L., , and C. Snyder, 2013: Mesoscale energy spectra of moist baroclinic waves. J. Atmos. Sci., 70, 12421256, doi:10.1175/JAS-D-11-0347.1.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Initial conditions. (a) Vertical slice of the unperturbed jet, showing zonal velocity u (thin black contours with a contour interval 10 m s−1), potential temperature θ (thin gray contours with a contour interval 10 K), and PV (shadings; ). Negative u is shown by dashed lines. (b) The mixing ratio of water vapor (shading; g kg−1) and the modified potential temperature (thin gray lines with contour interval 10 K) for the RH60 case. Thick dashed line in both panels indicates the 2-PVU (1 PVU = 10−6 K kg−1 m2 s−1) dynamical tropopause.

  • View in gallery

    (a) Time series of domain-averaged available potential energy per unit volume for three cases and (b) vertical profiles of domain-averaged heating rate for the RH60 case at t = 3, 4, 5, 6, and 7 days.

  • View in gallery

    Horizontal wavenumber spectra of (a),(c) rotational (thick lines) and divergence (thin lines) kinetic energy and (b),(d) horizontal kinetic energy per unit volume averaged in the vertical over the lower stratosphere () and time over (top) t = 4–7 and (bottom) t = 10–13 days . The solid reference lines have slopes of − and −3, respectively. The wavenumber is given as the lower x axis and the wavelength is given as the upper x axis.

  • View in gallery

    Zonal wavenumber spectra of horizontal kinetic energy per unit volume averaged in the vertical over the lower stratosphere () in y over and in time over t = 4–7 days. The smooth thin solid black curve is the Lindborg (1999) spectrum scaled by the vertical average of over the lower stratosphere. Other details are as in Fig. 3.

  • View in gallery

    Horizontal wavenumber spectra of moist available potential energy per unit volume averaged in the vertical over the lower stratosphere () and in time over (a) t = 4–7 and (b) t = 10–13 days. Other details are as in Fig. 3.

  • View in gallery

    Horizontal wavenumber spectra of vertical kinetic energy per unit volume, averaged in the vertical over the lower stratosphere and in time over (a) t = 4–7 and (b) t = 10–13 days. The reference spectrum (dashed line) has a slope of −3. Other details are as in Fig. 3.

  • View in gallery

    The nonhydrostatic degree , computed from conversion spectra averaged in the vertical over the lower stratosphere and in time over (a) t = 4–7 and (b) t = 10–13 days.

  • View in gallery

    Cumulative total (), horizontal kinetic energy (), available potential energy (), vertical kinetic energy (), and pressure () vertical flux divergence terms, averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for the (a) dry and (b) RH60 cases. Note that . The inset is an expanded view of the mesoscale subrange .

  • View in gallery

    Cumulative spectral conversions , , and , averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for the (a) dry and (b) RH60 cases. Other details are as in Fig. 8.

  • View in gallery

    HKE (black), APE (red), VKE (green), and total (blue) nonlinear spectral fluxes averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for (a) the dry case and (b) the RH60 case. The value of any flux at the origin (denoted by × on the y axis) has been modified to make it equal to that of the corresponding flux at . Other details are as in Fig. 8.

  • View in gallery

    Cumulative 3D divergence terms , , and ; cumulative diffusion terms , , and ; and cumulative total adiabatic nonconservative term , averaged in the vertical over the lower stratosphere and in time over t = 4–7 days, vs total horizontal wavenumber for the (a) dry and (b) RH60 cases. Note that . Other details are as in Fig. 8.

  • View in gallery

    Cumulative spectral conversion , averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for the dry (green) and RH60 (red) cases. Other details are as in Fig. 8.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 23 23 4
PDF Downloads 24 24 3

Applications of a Moist Nonhydrostatic Formulation of the Spectral Energy Budget to Baroclinic Waves. Part I: The Lower-Stratospheric Energy Spectra

View More View Less
  • 1 College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing, China
© Get Permissions
Full access

Abstract

The authors investigate the mesoscale dynamics that produce the lower-stratospheric energy spectra in idealized moist baroclinic waves, using the moist nonhydrostatic formulation of spectral energy budget of kinetic energy and available potential energy by J. Peng et al. The inclusion of moist processes energizes the lower-stratospheric mesoscale, helping to close the gap between observed and simulated energy spectra. In dry baroclinic waves, the lower-stratospheric mesoscale is mainly forced by weak downscale cascades of both horizontal kinetic energy (HKE) and available potential energy (APE) and by a weak conversion of APE to HKE. At wavelengths less than 1000 km, the pressure vertical flux divergence also has a significant positive contribution to the HKE; however, this positive contribution is largely counteracted by the negative HKE vertical flux divergence. In moist baroclinic waves, the lower-stratospheric mesoscale HKE is mainly generated by the pressure and HKE vertical flux divergences. This additional HKE is partly converted to APE and partly removed by diffusion. Another negative contribution to the mesoscale HKE is from the forcing of a visible upscale HKE cascade. Besides the conversion of HKE, however, the three-dimensional divergence also has a significant positive contribution to the mesoscale APE. With these two direct APE sources, the lower-stratospheric mesoscale also undergoes a much stronger upscale APE cascade. These results suggest that both downscale and upscale cascades through the mesoscale are permitted in the real atmosphere and the direct forcing of the mesoscale is available to feed the upscale energy cascade.

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

The authors investigate the mesoscale dynamics that produce the lower-stratospheric energy spectra in idealized moist baroclinic waves, using the moist nonhydrostatic formulation of spectral energy budget of kinetic energy and available potential energy by J. Peng et al. The inclusion of moist processes energizes the lower-stratospheric mesoscale, helping to close the gap between observed and simulated energy spectra. In dry baroclinic waves, the lower-stratospheric mesoscale is mainly forced by weak downscale cascades of both horizontal kinetic energy (HKE) and available potential energy (APE) and by a weak conversion of APE to HKE. At wavelengths less than 1000 km, the pressure vertical flux divergence also has a significant positive contribution to the HKE; however, this positive contribution is largely counteracted by the negative HKE vertical flux divergence. In moist baroclinic waves, the lower-stratospheric mesoscale HKE is mainly generated by the pressure and HKE vertical flux divergences. This additional HKE is partly converted to APE and partly removed by diffusion. Another negative contribution to the mesoscale HKE is from the forcing of a visible upscale HKE cascade. Besides the conversion of HKE, however, the three-dimensional divergence also has a significant positive contribution to the mesoscale APE. With these two direct APE sources, the lower-stratospheric mesoscale also undergoes a much stronger upscale APE cascade. These results suggest that both downscale and upscale cascades through the mesoscale are permitted in the real atmosphere and the direct forcing of the mesoscale is available to feed the upscale energy cascade.

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

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 dependence (where is the total horizontal wavenumber), whereas at scales larger than ~500 km, the spectra show a dependence. The observed −3 power law regime is well explained by quasigeostrophic turbulence theory (Charney 1971), but the dynamics producing the mesoscale spectral behavior remain the subject of a great deal of controversy (e.g., Lindborg 2005, 2007; Tulloch and Smith 2009).

Two completely different explanations for the mesoscale − spectra have been put forward. One is the inverse energy cascade due to two-dimensional (2D) turbulence (Kraichnan 1967) or quasi-two-dimensional (stratified) turbulence (Gage 1979; Lilly 1983), which assumes that the energy is transferred upscale through the mesoscale. The other is the direct energy cascade due to nonlinearly interacting internal gravity waves (IGWs) (Dewan 1979; Smith et al. 1987), quasigeostrophic dynamics (Tung and Orlando 2003; Gkioulekas and Tung 2005a,b), surface quasigeostrophic dynamics (Tulloch and Smith 2009), or anisotropic turbulence with strong stratification (Lindborg 2006). However, as stated in Waite and Snyder (2013, hereafter WS2013), “[i]mplicit in all of these proposed theories for the mesoscale spectrum is the assumption that it can be idealized as a turbulent inertial subrange—that is, that kinetic energy is transferred to successively smaller and smaller scales in a conservative manner (e.g., Davidson 2004). In particular, these cascade theories require that the mesoscale not be a significant source or sink of [kinetic energy]. However, many physical phenomena in mesoscale meteorology have the potential to add or remove kinetic energy.” These physical phenomena include the release of latent heat, the vertical propagation of IGWs, boundary layer drag, radiative cooling, etc. These processes may even impact the direction of energy cascade, provided that their strengths are adequate.

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 − spectra. Hamilton et al. (2008) examined the dependence of the global-model simulated mesoscale spectra on moist convection parameterization and concluded that the mesoscale spectra in the troposphere were correlated with the precipitation behavior. Waite and Snyder (2009, hereafter WS2009) investigated the mesoscale spectra in dry baroclinic waves simulated by the Advanced Research version of the Weather Research and Forecasting Model (ARW). Their results suggested that the pressure vertical flux divergence associated with vertically propagating IGWs also plays a significant role in the lower-stratospheric energy spectra. Theoretically, spectral energy budget analysis using high-resolution outputs of numerical simulations can provide enough insight into the dynamics of atmospheric energy spectra. However, previous formulations of the spectral energy budget have some drawbacks, which severely limit the outcome. First, the vertical flux is not separated from the energy cascade, resulting in the nonlinear spectral flux not being defined in a conservative way and hence the characteristics of energy cascade through the mesoscale cannot be exactly determined (e.g., Koshyk and Hamilton 2001; Brune and Becker 2013). Second, most spectral energy budget formulations were formulated in a dry, hydrostatic framework (e.g., WS2009) and are thus unsuitable for investigating the dynamics of energy spectra of moist atmosphere. Recently, Augier and Lindborg (2013, hereafter AL2013) developed a new formulation of the spectral energy budget of the atmosphere, which considers both kinetic energy and available potential energy. The most important improvement of this formulation is that the vertical flux is exactly separated from the energy cascade. The authors provided a simple method to obtain the spectral transfer term from the nonlinear term. However, this formulation is still restricted to a hydrostatic atmosphere and cannot fully consider the effects of moist species. Inspired by the innovative work of AL2013, Peng et al. (2014b, hereafter PZ2014b) developed a formulation of the spectral energy budget for a general, moist, nonhydrostatic atmosphere. The application of this moist, nonhydrostatic formulation to the mei-yu front system, a typical mesoscale convective system, showed that it is a convenient tool to investigate the mesoscale dynamics of atmospheric energy spectra (PZ2014b). Considering the differences between moist baroclinic waves and the mei-yu front system, we extend PZ2014b in this study to investigate spectral energetics in idealized moist baroclinic waves.

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

Following PZ2014b, the governing equations for a moist, fully compressible, nonhydrostatic atmosphere, expressed in the height coordinate on an f plane and without the additional large-scale forcing effects, can be written as
e1
e2
e3
e4
e5
where is horizontal velocity vector, is vertical velocity, is horizontal gradient operator, is the vertical unit vector, is the gravitational acceleration, is the density of dry air, and denote the mixing ratios (mass per unit mass of dry air) of water vapor, cloud, rain, ice, and any other hydrometeors, respectively; is the total mixing ratio; is the Exner pressure, is pressure, and is the reference surface pressure; is the modified potential temperature, where and are the gas constants for dry air and water vapor, respectively, and is the potential temperature; represents the combined diabatic contributions, and represents the combined contributions of diffusion; denotes the dissipation of , with being any of these variables, u, w, etc.; and and represent the diabatic contributions (e.g., microphysics, radiation) to and , respectively. Primes denote the perturbations from the time-invariant, hydrostatically balanced, dry reference states and . Note that the modified potential temperature defined here is different from the virtue potential temperature, which is generally defined by .

b. Formulation of the spectral energy budget

Energy spectra are computed using 2D discrete cosine transform (DCT; Denis et al. 2002) at each vertical level. Let be the DCT of field , where denotes the horizontal wave vector. For clarity, define for two scalar fields of a and b and for two vector fields of and , where an asterisk denotes the complex conjugate. To make our spectra have the dimension of energy rather than energy per unit mass as in most studies, we prefer to consider spectral energy per unit volume. The HKE spectrum per unit volume is defined by , where denotes the density of the dry reference state. It can then be naturally decomposed into spectra of horizontally rotational and divergent kinetic energy (RKE and DKE, respectively), given by and with and . The VKE spectrum per unit volume is defined by . The moist APE spectrum per unit volume is defined by , with the dimensional prefactor and the Brunt–Väisälä frequency . A new formulation of the spectral energy budget, suitable for the moist, fully compressible, nonhydrostatic atmosphere, was presented in PZ2014b. With slight rearrangements, it can be outlined as follows:
e6
e7
e8
In the above equations, , , and are spectral transfer terms due to nonlinear interactions; is HKE vertical flux, is VKE vertical flux, and is APE vertical flux; is pressure vertical flux; and , , and are the spectral tendencies due to the three-dimensional (3D) divergence (henceforth 3D divergence terms). The term represents the spectral conversion of APE to HKE, the term represents the spectral conversion of APE to VKE, and the term represents the spectral conversion of APE to other forms of energy; and are spectral tendencies due to diabatic processes; the terms , , and are the diffusion terms; and , , and are the adiabatic nonconservative terms. Considering that the term named is the vertical divergence of the corresponding vertical flux, henceforth is named the vertical flux divergence. For example, the term is referred to as the HKE vertical flux divergence. Detailed expressions for the above terms are given in the appendix, and the derivation of the equations is given in PZ2014b.

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

The total horizontal wavenumber is defined by . One-dimensional (1D) spectra as a function of are constructed by averaging over wavenumber bands (as in, e.g., WS2009; PZ2014b), where is the width of band, is the horizontal grid spacing, and with and being the numbers of grid points in the x and y directions, respectively. For example, the 1D spectrum of HKE per unit volume is defined as
e9
The 1D spectra of any term in Eqs. (6)(8) are defined similarly.
The nonlinear spectral flux of the HKE is defined as
e10
and the spectral fluxes of VKE and APE are defined similarly. Following AL2013, when Eqs. (6)(8) are summed as in Eq. (10) over all the corresponding terms with total horizontal wavenumber , we obtain
e11
e12
e13

Note that each of these terms is an accumulation of a corresponding term in Eqs. (6)(8). For example, is the cumulative HKE, is the cumulative HKE vertical flux divergence, is the cumulative 3D divergence term, is the cumulative pressure vertical flux divergence, is the cumulative conversion of APE to HKE, and is the cumulative diabatic term. Note that it is the negative derivative of each cumulative quantity with respect to (i.e., the negative slope of the corresponding curve in Figs. 812) that indicates the local contribution of the corresponding term at a given wavenumber.

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 . Lateral boundary conditions are periodic in the zonal direction 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 and there are 180 layers in the vertical with an approximately uniform spacing of in the troposphere. Advection is handled with the fifth-order horizontal and third-order vertical upwind-based schemes. Explicit sixth-order numerical diffusion (Knievel et al. 2007) is used to suppress subgrid-scale noises. The configuration of other physical parameterizations (i.e., Rayleigh damping, vertical mixing, microphysics, and cumulus scheme) is identical to WS2013. No additional physical parameterizations (e.g., radiation scheme, surface fluxes, boundary layer model) are utilized.

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.

Fig. 1.
Fig. 1.

Initial conditions. (a) Vertical slice of the unperturbed jet, showing zonal velocity u (thin black contours with a contour interval 10 m s−1), potential temperature θ (thin gray contours with a contour interval 10 K), and PV (shadings; ). Negative u is shown by dashed lines. (b) The mixing ratio of water vapor (shading; g kg−1) and the modified potential temperature (thin gray lines with contour interval 10 K) for the RH60 case. Thick dashed line in both panels indicates the 2-PVU (1 PVU = 10−6 K kg−1 m2 s−1) dynamical tropopause.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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 () of the moist cases are equal to that of the dry case. (The reason behind this adjustment was explained in WS2013.) After this adjustment, the actual RH of the initial moist jet is no longer uniform. It is higher than the prescribed value in the troposphere but lower than the prescribed value in the stratosphere: the maximum RH is ~40% for RH30 and ~80% for RH60 near the surface and the minimum RH is ~17% for RH30 and ~35% for RH60 at the height of 20 km. Figure 1b shows water vapor mixing ratio and the modified potential temperature of the basic state for the RH60 case. The moist jets are perturbed with the dry fastest-growing normal mode.

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 . Figure 2a presents the time series of domain-averaged APE; that is, , where the integration is performed from to over the model domain. Obviously, increases with increasing water vapor (Bannon 2005). The rapid decrease of occurs at the stage when the baroclinic wave grows rapidly (see Fig. 2 in WS2013). Also plotted in Fig. 2 is the vertical profiles of the domain-averaged diabatic contribution for the RH60 case at 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 peaks at ~8 km, with a maximum of at 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.

Fig. 2.
Fig. 2.

(a) Time series of domain-averaged available potential energy per unit volume for three cases and (b) vertical profiles of domain-averaged heating rate for the RH60 case at t = 3, 4, 5, 6, and 7 days.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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 () and the late phase ().

Fig. 3.
Fig. 3.

Horizontal wavenumber spectra of (a),(c) rotational (thick lines) and divergence (thin lines) kinetic energy and (b),(d) horizontal kinetic energy per unit volume averaged in the vertical over the lower stratosphere () and time over (top) t = 4–7 and (bottom) t = 10–13 days . The solid reference lines have slopes of − and −3, respectively. The wavenumber is given as the lower x axis and the wavelength is given as the upper x axis.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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 (measured here and elsewhere by a least squares fit over a given wavelength range) and is −2.0 at smaller wavelengths . With relatively low moisture (RH30), the moist processes slightly energize the RKE only at the scales less than 900 km and the shallowing of RKE spectrum described above persists. With relatively high moisture (RH60), the moist processes markedly energize the RKE throughout the mesoscale (), resulting in the disappearance of the shallowing of RKE spectrum. The DKE spectrum crosses the RKE spectrum at the mesoscale, and the wavelength at which these two components intersect increases with increasing moisture (i.e., 500 km for the dry case, 625 km for RH30, and 800 km for RH60). In general, the DKE spectrum is much shallower than the RKE spectrum. The relatively large contribution of the shallower DKE spectrum to some extent accounts for the transition in the HKE spectrum (Fig. 3b). RH60 has much more HKE than RH30 and the dry case throughout the mesoscale, especially at the wavelengths around 500 km. For example, the HKE at the wavelength 500 km of RH60 is 35 and 121 times larger than the RH30 case and the dry case, respectively. Therefore, we argue that the distinct shallowing of HKE spectra in the dry and RH30 cases may be largely due to the shortage of mesoscale HKE around the transition scale of 500 km.

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 (), while the DKE spectrum develops a much shallower slope of approximately −1.25.

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 () spectra of HKE per unit volume over the early phase and the reference spectrum obtained from the Measurement of Ozone and Water Vapor by Airbus In-Service (MOZAIC) aircraft observations by Lindborg (1999). To facilitate comparison, the reference spectrum has been scaled by the vertical average of over the lower stratosphere. These 1D zonal wavenumber spectra are computed by conducting 1D-DCT along zonal transects at fixed y and z and then averaging results in y over the most energetic part of the domain (i.e., ); these spectra are similar to the spectra above, which are consistent with the findings of Morss et al. (2009). It is shown that the moist HKE spectrum in RH60 is in a much better agreement with Lindborg (1999) reference spectrum than the HKE spectra in the other two cases, although still with a lower amplitude than observations.

Fig. 4.
Fig. 4.

Zonal wavenumber spectra of horizontal kinetic energy per unit volume averaged in the vertical over the lower stratosphere () in y over and in time over t = 4–7 days. The smooth thin solid black curve is the Lindborg (1999) spectrum scaled by the vertical average of over the lower stratosphere. Other details are as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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.

Fig. 5.
Fig. 5.

Horizontal wavenumber spectra of moist available potential energy per unit volume averaged in the vertical over the lower stratosphere () and in time over (a) t = 4–7 and (b) t = 10–13 days. Other details are as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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 () and shallows to a slope of −1.8 at smaller wavelengths (). At the late phase (Fig. 5b), the APE spectrum for RH60 still processes a clear transition to a shallower slope at the mesoscale, although its level by this time is much lower than that during the early phase.

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.

Fig. 6.
Fig. 6.

Horizontal wavenumber spectra of vertical kinetic energy per unit volume, averaged in the vertical over the lower stratosphere and in time over (a) t = 4–7 and (b) t = 10–13 days. The reference spectrum (dashed line) has a slope of −3. Other details are as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

To illuminate the necessity of employing the nonhydrostatic formulation, we show how nonhydrostatic the simulations presented here are. This can be done by estimating the ratio of to , because the hydrostatic assumption implies that or ; that is, only the conversion between the APE and HKE is significant. For brevity, let
e14
which represents the nonhydrostatic degree of the simulated flow at a given wavenumber . The distribution of as a function of wavenumber , computing from the vertically and temporally averaged spectra, for the dry and RH60 simulations are shown in Fig. 7. For the RH60 case at both phases and for the dry case at the late phase, the value of significantly exceeds 0.2 for wavelengths less than 1000 km, which implies that the conversion of APE to VKE is significant in both simulations, especially at the mesocale; therefore, the hydrostatic assumption should not be justified for spectral conversion calculations. Note that the very large value of at wavelengths around 4000 km for the dry case at the early phase (Fig. 7a) should be due to the fact that the corresponding is nearly equal to zero (as shown in Fig. 9a).
Fig. 7.
Fig. 7.

The nonhydrostatic degree , computed from conversion spectra averaged in the vertical over the lower stratosphere and in time over (a) t = 4–7 and (b) t = 10–13 days.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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., and ], which mainly occurs in the upper troposphere (PZ2014a,b) and, as it turns out (not shown), are negligible in the lower stratosphere. They can also enhance convection and in turn influence the energy spectra by the associated vertical flux terms [i.e., , , and ]. Maybe more importantly, they can influence the HKE spectrum via the pressure vertical flux associated with convectively generated IGWs—namely, . A closer look at the latter two physical processes is made next. Figure 8 presents the cumulative total vertical flux divergence , HKE vertical flux divergence , APE vertical flux divergence , VKE vertical flux divergence , and pressure vertical flux divergence , averaged over the lower stratosphere and over the time interval of interest for the RH60 and dry cases. Here, .

Fig. 8.
Fig. 8.

Cumulative total (), horizontal kinetic energy (), available potential energy (), vertical kinetic energy (), and pressure () vertical flux divergence terms, averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for the (a) dry and (b) RH60 cases. Note that . The inset is an expanded view of the mesoscale subrange .

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

For the dry case (Fig. 8a), the cumulative HKE vertical flux divergence (black line) is negative at all wavenumbers and increases as the wavenumber increases (the local HKE vertical flux divergence is negative), which means that the lower-stratospheric HKE is removed by the HKE vertical flux divergence at all scales. The shape of the cumulative APE vertical flux divergence is similar to that of , but with a relatively smaller magnitude. However, the cumulative pressure vertical flux divergence has a somewhat intricate shape. At the planetary and synoptic scales, it reaches its minimum of at the smallest wavenumber of , increases to at , and decreases to at . At the mesoscale (the inset in Fig. 8a), has a convex shape. This indicates the pressure vertical flux divergence has a positive contribution to the HKE at wavelengths less than ~1000 km, which is consistent with the finding of WS2009. However, this positive contribution is largely counteracted by the negative HKE vertical flux divergence, resulting in the total vertical flux term (black dotted line) being negligible at wavelengths between 200 and 1000 km. To verify the robustness of this result, we also compared the pressure vertical flux divergence and the HKE vertical flux divergence during the late phase and found similar results (not shown).

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 at the mesoscale suggests that there is significant HKE input at these scales. This positive contribution to HKE is not only governed by the pressure vertical flux divergence associated with the vertically propagating IGWs but also influenced by the HKE vertical flux divergence due to the vertical transportation of moist convection. This result is consistent with the finding of AL2013 that the stratosphere is directly forced by an energy flux from the troposphere (their Fig. 3c). Even with moisture, however, the cumulative APE vertical flux divergence is very small. It seems that there is no significant vertical transport of mesoscale APE toward the lower stratosphere.

b. Spectra of energy conversion

Figure 9 presents the cumulative conversions , , and for the dry and RH60 cases. For the dry case (Fig. 9a), the cumulative spectral conversion is negligible at all wavelengths; therefore, there is almost no discernible difference between the cumulative spectral conversions and . The quantity is positive and decreases with increasing wavenumber (the local conversion is positive). This indicates that the conversion is from APE to HKE at all scales. With moisture (Fig. 9b), the conversion is quite different, especially at the mesoscale. For the RH60 case, the vertical motion is much stronger and the conversion of APE to VKE is significant, although small in magnitude, resulting in being more than . This visible difference between and shows once again the necessity of employing the nonhydrostatic formulation. Remarkably, the cumulative conversion or decreases for the wavelengths larger than 2000 km (the local conversion is positive) and increases at the mesoscale (the local conversion is negative). This means that the conversion is from APE to HKE at the planetary and synoptic scales, while it is from HKE to APE at the mesoscale. Thus, owing to the strong baroclinicity of the system considered here, the lower stratosphere is to some extent directly forced by baroclinic instability, which is in contrast to the results calculating from the Atmospheric GCM for the Earth Simulator (AFES) by AL2013 (their Fig. 3c).

Fig. 9.
Fig. 9.

Cumulative spectral conversions , , and , averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for the (a) dry and (b) RH60 cases. Other details are as in Fig. 8.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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., ); thus, they act solely to redistribute energy among different scales on a specific layer.

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 and then drops gradually to zero at the largest wavenumbers. Therefore, the total energy is transferred toward larger wavenumbers. At the large scales with , the total flux is dominated by the APE spectral flux (red curve), which also increases to reach its maximum of at and decreases gradually to zero at the largest wavenumbers. This suggests there is a downscale transfer of APE from wavelengths larger than 4000 km to smaller wavelengths. The HKE spectral flux is negative at wavelengths between 6000 and 3000 km and reaches a minimum of . This corresponds to an upscale transfer of HKE from the synoptic scale to the planetary scale.

Fig. 10.
Fig. 10.

HKE (black), APE (red), VKE (green), and total (blue) nonlinear spectral fluxes averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for (a) the dry case and (b) the RH60 case. The value of any flux at the origin (denoted by × on the y axis) has been modified to make it equal to that of the corresponding flux at . Other details are as in Fig. 8.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

At the mesoscale (inset in Fig. 10a), the HKE spectral flux becomes comparable to the APE spectral flux. The terms and gradually decrease at the mesoscale, meaning that there are weak downscale APE and HKE cascades at these scales in the lower stratosphere.

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 at , decreases to around , and increases again to reach a value around zero at the dissipation scales (). Similar to the dry case, there is also a downscale APE cascade from the planetary scale to the synoptic scale around 3000 km. Remarkably, in contrast to the dry case, the mesoscale is governed by a relatively stronger upscale APE cascade in RH60, which actually starts at the wavelength of ~200 km rather than a weakly downscale APE cascade. As a result, the APE of the synoptic scale () is deposited by both the downscale transfer from the planetary scale, which is related to the large-scale baroclinic instability, and the upscale transfer from the mesoscale, which is to some extent forced by the mesoscale conversion from HKE to APE (Fig. 9b).

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 at , decreases to around , and increases again to reach a value of approximate zero at . This indicates there is also an upscale HKE cascade at the synoptic scale and mesoscale, which actually starts at the wavelength of ~360 km. At the planetary scale, the RH60 case is characterized by a much stronger downscale HKE cascade than the dry case. In addition, at the wavelengths less than 360 km (inset in Fig. 10b), there is also a weak downscale cascade. Remarkably, this finding is consistent with the structure function analysis of aircraft data by Cho and Lindborg (2001), which revealed a downscale kinetic energy cascade at scales below around 100 km in the lower stratosphere.

d. Effects of diffusion and 3D divergence

Owing to the compressibility of the atmosphere, the 3D divergence (i.e., ) does not have to be zero. To quantitatively investigate the effects of the 3D divergence on the energy budget, we computed the cumulative 3D divergence terms , , and , which are plotted in Fig. 11. Also plotted are cumulative diffusion terms , , and , as well as the cumulative total adiabatic nonconservative term . Note that . In the dry case (Fig. 11a), all these terms are negligible, although the 3D divergence has a negative contribution to both HKE and APE. In contrast, these terms, especially and , in the RH60 case become much stronger. In Fig. 11b, the cumulative 3D divergence term has an amplitude that is of the same order as that of the pressure vertical flux divergence (Fig. 8b) and decreases at nearly all scales (the local 3D divergence term is positive). This interesting result suggests that the 3D divergence has a strong positive contribution to APE in the lower stratosphere. Diffusion has a much stronger negative contribution to HKE (black dashed) than to APE (red dashed). In addition, in both cases the cumulative total adiabatic nonconservative term is negligible, which is related to the fact that the dimensional prefactor and the reference density in the lower stratosphere have small variation with height (not shown).

Fig. 11.
Fig. 11.

Cumulative 3D divergence terms , , and ; cumulative diffusion terms , , and ; and cumulative total adiabatic nonconservative term , averaged in the vertical over the lower stratosphere and in time over t = 4–7 days, vs total horizontal wavenumber for the (a) dry and (b) RH60 cases. Note that . Other details are as in Fig. 8.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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 is equal to . Similarly, we also define the other corresponding terms (Table 1) (e.g., , ). Table 1 summarizes the main energy budget terms for the wavenumber range of [i.e., for wavelengths between 4000 and 2000 km (corresponding approximately to the synoptic scale)] and for the wavenumber range of [i.e., for wavelengths between 2000 and 50 km (corresponding to the mesoscale range)].

Table 1.

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 and directly obtained from Figs. 811.

Table 1.

As already shown, the mesoscale HKE in the dry case is mainly forced by a weak downscale cascade (; units: , similarly hereafter) and by a conversion from APE (). On the whole, both the pressure vertical flux and the HKE vertical flux have negative contributions to mesoscale HKE ( and ). However, at wavelengths between 1000 and 50 km, the pressure vertical flux divergence has a positive contribution (Fig. 8a), which is in the same order of magnitude as the contribution from the weak downscale cascade (Fig. 10a); this positive contribution, however, is largely counteracted by the negative HKE vertical flux divergence. The only source of mesoscale APE is the positive forcing by a 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 () and the 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 flux (), which suggests that the stratospheric mesoscale is forced by a visible upscale HKE cascade. However, besides the conversion from 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 is forced by a much stronger upscale APE cascade (). At the synoptic scale, the positive contributions to the lower-stratospheric HKE are from the pressure vertical flux divergence (), the HKE vertical flux divergence (), the upscale transfer from the mesoscale (0.13), and the weak downscale transfer from the larger scales (0.06). The positive contributions to the lower-stratospheric APE are from the forcing of the 3D divergence (), the upscale transfer from the mesoscale (0.40), and the strong downscale transfer from the larger scales (0.85). In addition, the synoptic scale also undergoes a strong conversion of APE to other forms of energy (), primarily generating the HKE ().

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.

Second, the root-mean-square (RMS) Rossby number corresponding to the wavenumber range with the upscale energy cascade is ~0.1. Such small but nonnegligible Rossby number suggests, on the one hand, that Earth’s rotation is important at these scales and, on the other hand, that the magnitude of vertical vorticity is also significant, although typically much smaller than the Coriolis parameter . The RMS Rossby number is defined by (Klein et al. 2008)
e15
where is the RMS of the vertical vorticity. Following Lindborg (2009), corresponding to a given horizontal wavenumber range can be calculated from the RKE spectrum as
e16
As already shown, the upscale energy cascade mainly occurs over the range of wavelengths between 200 and 2000 km, which corresponds to the range of horizontal wavenumbers from to . Using the vertically and temporally averaged shown in Fig. 3a, the RMS Rossby number at this wavenumber range is computed and found to be ~0.1 for the RH60 simulation.
Third, the energy conversion at these scales is from divergent (unbalanced) to rotational (balanced) components. In the spectral energy budget, the conversion between DKE and RKE appears as the cross-spectrum of horizontal divergence and vertical vorticity, which, for a given wave vector , can be expressed as
e17

Positive represents the conversion of DKE to RKE at wave vector . The cumulative conversion is constructed analogously to . In Fig. 12, we have plotted cumulative conversions , averaged in the vertical over the lower stratosphere and in time over 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.

Fig. 12.
Fig. 12.

Cumulative spectral conversion , averaged in the vertical over the lower stratosphere and in time over t = 4–7 days vs total horizontal wavenumber for the dry (green) and RH60 (red) cases. Other details are as in Fig. 8.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0306.1

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

Acknowledgments

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

Detailed mathematical expression of the terms in the spectral energy budget equations in section 2b are given as follows:
ea1
ea2
ea3
ea4
ea5
ea6
ea7
ea8
ea9
ea10
and
ea11

REFERENCES

  • Augier, P., , and E. Lindborg, 2013: A new formulation of the spectral energy budget of the atmosphere, with application to two high-resolution general circulation models. J. Atmos. Sci., 70, 22932308, doi:10.1175/JAS-D-12-0281.1.

    • Search Google Scholar
    • Export Citation
  • Bacmeister, J. T., , S. D. Eckermann, , P. A. Newman, , L. Lait, , K. R. Chan, , M. Loewenstein, , M. H. Proffitt, , and B. L. Gary, 1996: Stratosphere horizontal wavenumber spectra of winds, potential temperature, and atmospheric tracers observed by high altitude aircraft. J. Geophys. Res., 101, 94419470, doi:10.1029/95JD03835.

    • Search Google Scholar
    • Export Citation
  • Bannon, P. R., 2005: Eulerian available energetics in moist atmospheres. J. Atmos. Sci., 62, 42384252, doi:10.1175/JAS3516.1.

  • Bierdel, L., , P. Friederichs, , and S. Bentzien, 2012: Spatial kinetic energy spectra in the convection-permitting limited-area NWP model COSMO-DE. Meteor. Z., 21, 245258, doi:10.1127/0941-2948/2012/0319.

    • Search Google Scholar
    • Export Citation
  • Brune, S., , and E. Becker, 2013: Indications of stratified turbulence in a mechanistic GCM. J. Atmos. Sci., 70, 231247, doi:10.1175/JAS-D-12-025.1.

    • Search Google Scholar
    • Export Citation
  • Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 10871095, doi:10.1175/1520-0469(1971)028<1087:GT>2.0.CO;2.

  • Cho, J. Y. N., , and E. Lindborg, 2001: Horizontal velocity structure functions in the upper troposphere and lower stratosphere: 1. Observations. J. Geophys. Res., 106 (D10), 10 22310 232, doi:10.1029/2000JD900814.

    • Search Google Scholar
    • Export Citation
  • Cho, J. Y. N., and Coauthors, 1999: Horizontal wavenumber spectra of winds, temperature, and trace gases during the Pacific Exploratory Missions: 1. Climatology. J. Geophys. Res., 104, 56975716, doi:10.1029/98JD01825.

    • Search Google Scholar
    • Export Citation
  • Davidson, P. A., 2004: Turbulence: An Introduction for Scientists and Engineers. Oxford University Press, 678 pp.

  • Davis, C. A., 2010: Simulations of subtropical cyclones in a baroclinic channel model. J. Atmos. Sci., 67, 28712892, doi:10.1175/2010JAS3411.1.

    • Search Google Scholar
    • Export Citation
  • Denis, B., , J. Côté, , and R. Laprise, 2002: Spectral decomposition of two-dimensional atmospheric fields on limited-area domains using the discrete cosine transform (DCT). Mon. Wea. Rev., 130, 18121829, doi:10.1175/1520-0493(2002)130<1812:SDOTDA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Dewan, E. M., 1979: Stratospheric wave spectra resembling turbulence. Science, 204, 832835, doi:10.1126/science.204.4395.832.

  • Gage, K. S., 1979: Evidence for a k−5/3 law inertial range in mesoscale two-dimensional turbulence. J. Atmos. Sci., 36, 19501954, doi:10.1175/1520-0469(1979)036<1950:EFALIR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gage, K. S., , and G. D. Nastrom, 1986: Theoretical interpretation of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft during GASP. J. Atmos. Sci., 43, 729740, doi:10.1175/1520-0469(1986)043<0729:TIOAWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Gkioulekas, E., , and K. K. Tung, 2005a: On the double cascades of energy and enstrophy in two dimensional turbulence. Part 1. Theoretical formulation. Discrete Contin. Dyn. Syst., 5B, 79102.

    • Search Google Scholar
    • Export Citation
  • Gkioulekas, E., , and K. K. Tung, 2005b: On the double cascades of energy and enstrophy in two dimensional turbulence. Part 2. Approach to the KLB limit and interpretation of experimental evidence. Discrete Contin. Dyn. Syst., 5B, 103124.

    • Search Google Scholar
    • Export Citation
  • Hamilton, K., , Y. O. Takahashi, , and W. Ohfuchi, 2008: The mesoscale spectrum of atmospheric motions investigated in a very fine resolution global general circulation model. J. Geophys. Res., 113, D18110, doi:10.1029/2008JD009785.

    • Search Google Scholar
    • Export Citation
  • Klein, P., , B. L. Hua, , G. Lapeyre, , X. Capet, , S. Le Gentil, , and H. Sasaki, 2008: Upper ocean turbulence from high-resolution 3D simulations. J. Phys. Oceanogr., 38, 17481763, doi:10.1175/2007JPO3773.1.

    • Search Google Scholar
    • Export Citation
  • Knievel, J. C., , G. H. Bryan, , and J. P. Hacker, 2007: Explicit numerical diffusion in the WRF model. Mon. Wea. Rev., 135, 38083824, doi:10.1175/2007MWR2100.1.

    • Search Google Scholar
    • Export Citation
  • Koshyk, J. N., , and K. Hamilton, 2001: The horizontal kinetic energy spectrum and spectral budget simulated by a high-resolution troposphere–stratosphere–mesosphere GCM. J. Atmos. Sci., 58, 329348, doi:10.1175/1520-0469(2001)058<0329:THKESA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kraichnan, R. H., 1967: Inertial ranges in two-dimensional turbulence. Phys. Fluids, 10, 14171423, doi:10.1063/1.1762301.

  • Lilly, D. K., 1983: Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci., 40, 749761, doi:10.1175/1520-0469(1983)040<0749:STATMV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 1999: Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech., 388, 259288, doi:10.1017/S0022112099004851.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 2005: The effect of rotation on mesoscale energy cascade in the free atmosphere. Geophys. Res. Lett., 32, L01809, doi:10.1029/2004GL021319.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 2006: The energy cascade in a strongly stratified fluid. J. Fluid Mech., 550, 207242, doi:10.1017/S0022112005008128.

  • Lindborg, E., 2007: Horizontal wavenumber spectra of vertical vorticity and horizontal divergence in the upper troposphere and lower stratosphere. J. Atmos. Sci., 64, 10171025, doi:10.1175/JAS3864.1.

    • Search Google Scholar
    • Export Citation
  • Lindborg, E., 2009: Two comments on the surface quasigeostrophic model for the atmospheric energy spectrum. J. Atmos. Sci., 66, 10691072, doi:10.1175/2008JAS2972.1.

    • Search Google Scholar
    • Export Citation
  • Morss, R. E., , C. Snyder, , and R. Rotunno, 2009: Spectra, spatial scales, and predictability in a quasigeostrophic model. J. Atmos. Sci., 66, 31153130, doi:10.1175/2009JAS3057.1.

    • Search Google Scholar
    • Export Citation
  • Nastrom, G. D., , and K. S. Gage, 1985: A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci., 42, 950960, doi:10.1175/1520-0469(1985)042<0950:ACOAWS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pauluis, O., , and I. M. Held, 2002: Entropy budget of an atmosphere in radiative–convective equilibrium. Part I: Maximum work and frictional dissipation. J. Atmos. Sci., 59, 140149, doi:10.1175/1520-0469(2002)059<0140:EBOAAI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Peng, J., , L. Zhang, , Y. Luo, , and Y. Zhang, 2014a: Mesoscale energy spectra of the mei-yu front system. Part I: Kinetic energy spectra. J. Atmos. Sci., 71, 3755, doi:10.1175/JAS-D-13-085.1.

    • Search Google Scholar
    • Export Citation
  • Peng, J., , L. Zhang, , Y. Luo, , and C. Xiong, 2014b: Mesoscale energy spectra of the mei-yu front system. Part II: Moist available potential energy spectra. J. Atmos. Sci., 71, 14101424, doi:10.1175/JAS-D-13-0319.1.

    • Search Google Scholar
    • Export Citation
  • Plougonven, R., , and C. Snyder, 2007: Inertia–gravity waves spontaneously generated by jets and fronts. Part I: Different baroclinic life cycles. J. Atmos. Sci., 64, 25022520, doi:10.1175/JAS3953.1.

    • Search Google Scholar
    • Export Citation
  • Ricard, D., , C. Lac, , S. Riette, , R. Legrand, , and A. Mary, 2013: Kinetic energy spectra characteristics of two convection-permitting limited-area models AROME and Meso-NH. Quart. J. Roy. Meteor. Soc., 139, 1327–1341, doi:10.1002/qj.2025.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., 2004: Evaluating mesoscale NWP models using kinetic energy spectra. Mon. Wea. Rev., 132, 30193032, doi:10.1175/MWR2830.1.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., , and J. B. Klemp, 2008: A time-split nonhydrostatic atmospheric model for weather research and forecasting applications. J. Comput. Phys., 227, 34653485, doi:10.1016/j.jcp.2007.01.037.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp. [Available online at http://www.mmm.ucar.edu/wrf/users/docs/arw_v3_bw.pdf.]

  • Smith, S. A., , D. C. Fritts, , and T. E. Vanzandt, 1987: Evidence for a saturated spectrum of atmospheric gravity waves. J. Atmos. Sci., 44, 14041410, doi:10.1175/1520-0469(1987)044<1404:EFASSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Snyder, C., , and R. S. Lindzen, 1991: Quasigeostrophic wave-CISK in an unbounded baroclinic shear. J. Atmos. Sci., 48, 7686, doi:10.1175/1520-0469(1991)048<0076:QGWCIA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Takahashi, Y. O., , K. Hamilton, , and W. Ohfuchi, 2006: Explicit global simulations of the mesoscale spectrum of atmospheric motions. Geophys. Res. Lett., 33, L12812, doi:10.1029/2006GL026429.

    • Search Google Scholar
    • Export Citation
  • Terasaki, K., , H. Tanaka, , and M. Satoh, 2009: Characteristics of the kinetic energy spectrum of NICAM model atmosphere. SOLA, 5, 180183, doi:10.2151/sola.2009-046.

    • Search Google Scholar
    • Export Citation
  • Tulloch, R., , and K. S. Smith, 2009: Quasigeostrophic turbulence with explicit surface dynamics: Application to the atmospheric energy spectrum. J. Atmos. Sci., 66, 450467, doi:10.1175/2008JAS2653.1.

    • Search Google Scholar
    • Export Citation
  • Tung, K. K., , and W. W. Orlando, 2003: The k−3 and k−5/3 energy spectrum of atmospheric turbulence: Quasigeostrophic two-level model simulation. J. Atmos. Sci., 60, 824835, doi:10.1175/1520-0469(2003)060<0824:TKAKES>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Waite, M. L., , and C. Snyder, 2009: The mesoscale kinetic energy spectrum of a baroclinic life cycle. J. Atmos. Sci., 66, 883901, doi:10.1175/2008JAS2829.1.

    • Search Google Scholar
    • Export Citation
  • Waite, M. L., , and C. Snyder, 2013: Mesoscale energy spectra of moist baroclinic waves. J. Atmos. Sci., 70, 12421256, doi:10.1175/JAS-D-11-0347.1.

    • Search Google Scholar
    • Export Citation
Save