• Achatz, U., 2007: The primary nonlinear dynamics of modal and nonmodal perturbations of monochromatic inertia–gravity waves. J. Atmos. Sci., 64, 7495.

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  • Akmaev, R. A., 2007: On the energetics of mean-flow interactions with thermally dissipating gravity waves. J. Geophys. Res., 112, D11125, doi:10.1029/2006JD007908.

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    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • Becker, E., 2009: Sensitivity of the upper mesosphere to the Lorenz energy cycle of the troposphere. J. Atmos. Sci., 66, 647666.

  • Becker, E., , and C. McLandress, 2009: Consistent scale interaction of gravity waves in the Doppler spread parameterization. J. Atmos. Sci., 66, 14341449.

    • Search Google Scholar
    • Export Citation
  • Brune, S., , and E. Becker, 2013: Indications of stratified turbulence in a mechanistic GCM. J. Atmos. Sci., 70, 231247.

  • Fritts, D. C., , and T. J. Dunkerton, 1985: Fluxes of heat and constituents due to convectively unstable gravity waves. J. Atmos. Sci., 42, 549556.

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    • Export Citation
  • Fritts, D. C., , L. Wang, , J. Werne, , T. Lund, , and K. Wan, 2009: Gravity wave instability dynamics at high Reynolds numbers. Part II: Turbulence evolution, structure, and anisotropy. J. Atmos. Sci., 66, 11491171.

    • Search Google Scholar
    • Export Citation
  • Hamilton, K., , Y. Takahashi, , and W. Ohfuchi, 2008: 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
  • Hines, C. O., , and C. A. Reddy, 1967: On the propagation of atmospheric gravity waves through regions of wind shear. J. Geophys. Res., 72, 10151034.

    • 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.

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

  • McLandress, C., , W. E. War, , V. I. Fomichev, , K. Semeniuk, , S. R. Beagley, , N. A. McFarlane, , and T. G. Shepherd, 2006: Large-scale dynamics of the mesosphere and lower thermosphere: An analysis using the extended Canadian Middle Atmosphere Model. J. Geophys. Res., 111, D17111, doi:10.1029/2005JD006776.

    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 1983: Baroclinic instability of the summer mesosphere: A mechanism for the quasi-two-day wave? J. Atmos. Sci., 40, 262270.

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    • Export Citation
  • Shaw, T. A., , and E. Becker, 2011: Comments on “A spectral parameterization of drag, eddy diffusion, and wave heating for a three-dimensional flow induced by breaking gravity waves.” J. Atmos. Sci., 68, 24652469.

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    • Export Citation
  • Waite, M. L., , and P. Bartello, 2006: The transition from geostrophic to stratified turbulence. J. Fluid Mech., 568, 89108.

  • Waite, M. L., , and C. Snyder, 2009: The mesoscale kinetic energy spectrum of a baroclinic life cycle. J. Atmos. Sci., 66, 883901.

  • Waite, M. L., , and C. Snyder, 2013: Mesoscale energy spectra of moist baroclinic waves. J. Atmos. Sci., 70, 12421256.

  • Waite, M. L., , and C. Snyder, 2014: Comments on “Indications of stratified turbulence in a mechanistic GCM.” J. Atmos. Sci., 71, 854857.

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    (a) Spectral KE, as well as spectral tendencies of KE owing to (b) horizontal advection and (c) energy deposition (adiabatic conversion plus vertical advection). The spectral tendencies are normalized by the KE shown in (a). Results are from the middle-atmosphere model version described in Becker (2009) that is run here with double horizontal resolution and adjusted diffusion parameters to reproduce the general circulation along with realistic gravity wave drag and dissipation. See BB13 for definitions of KE and spectral tendencies.

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Reply to “Comments on ‘Indications of Stratified Turbulence in a Mechanistic GCM’”

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  • 1 Leibniz-Institute of Atmospheric Physics, University of Rostock, Kühlungsborn, Germany
  • 2 Institute of Oceanography, University of Hamburg, Hamburg, Germany
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Corresponding author address: Erich Becker, Leibniz-Institute of Atmospheric Physics, University of Rostock, Schlossstr. 6, 18225 Kühlungsborn, Germany. E-mail: becker@iap-kborn.de

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-12-025.1.

Corresponding author address: Erich Becker, Leibniz-Institute of Atmospheric Physics, University of Rostock, Schlossstr. 6, 18225 Kühlungsborn, Germany. E-mail: becker@iap-kborn.de

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-12-025.1.

1. Introduction

Based on simulations of the general circulation of the troposphere and lower stratosphere with a mechanistic GCM run at high resolution, Brune and Becker (2013, hereafter BB13) found a surprising dependence of the global kinetic energy (KE) spectrum on the vertical level spacing and first-order contributions of adiabatic conversion to the spectral KE budget in the mesoscales. For a fine vertical level spacing of about 200 m from the midtroposphere to the stratosphere (L100 run), the relation of spectral fluxes of KE associated with horizontal advection and adiabatic conversion was of order 1 in the regime of the upper-tropospheric − spectral slope, and the nonrotational flow dominated the KE in this case. By invoking the scaling relations for stratified turbulence according to Lindborg (2006), BB13 interpreted this finding to be indicative of stratified turbulence (ST). They argued that ST is therefore the proper picture for the mesoscale branch of the Nastrom–Gage spectrum. However, this interpretation did not apply for a model run with a conventional vertical level spacing of about 1.5 km (L30 run), as is usually employed above the boundary layer in comprehensive GCMs even for very high horizontal resolution (e.g., Hamilton et al. 2008). Though this model setup yielded an even clearer simulation of the − spectral slope, the aforementioned relation of spectral fluxes was not fulfilled and the rotational flow dominated the KE at all scales resolved.

Waite and Snyder (2014, hereafter WS14) revisit these results of BB13. They follow the speculation of BB13 that the simulation of the Nastrom–Gage spectrum with conventionally coarse vertical resolution may be partly spurious, although the reason for this needs to be investigated in the future. WS14 reach conclusions different from that of BB13 concerning the applicability of ST to interpret the spectral KE budget in the case of fine vertical level spacing (L100 run). In particular, they argue that the model results of BB13 cannot be reconciled with the concept of ST in the sense of Lindborg (2006) for mainly two reasons: The nonrotational flow dominated the mesoscale KE spectrum, whereas an equipartition of rotational and nonrotational flow is usually found in ST simulated with high-resolution forced–dissipative Boussinesq models (Lindborg 2006); the estimated vertical Froude number in the L100 run of BB13 was much smaller than one, though order unity is required for ST. Reflecting on their previous results based on baroclinic life cycle simulations with a mesoscale circulation model (Waite and Snyder 2009, hereafter WS09) and previous studies of the interaction between gravity waves and the rotational flow (Waite and Bartello 2006), WS14 propose that the − slope as simulated in the L100 run of BB13 should be associated with charging of the mesoscale KE by gravity waves. This would imply that the Nastrom–Gage spectrum in the upper troposphere results from a net energy deposition of gravity waves generated at lower altitudes, similar to the behavior in the lower stratosphere as found by WS09. In this reply to the comment of WS14, we further discuss these points.

2. Necessary conditions for stratified turbulence

As mentioned above, there are two main arguments that lead WS14 to suggest an alternative interpretation of the L100 run of BB13. First, the dominance of the nonrotational flow in the mesoscale regime of the − slope as found in BB13 appears questionable since previous simulations of ST showed an equipartition of rotational and nonrotational flow. However, corresponding Boussinesq model simulations have not resolved the injection of KE by baroclinic instability and the associated vertical structure of the energy and enstrophy cascades in the troposphere like a GCM. Therefore, distinct vertical exchange terms by the potential energy flux (assuming pressure as the vertical coordinate) and vertical advection as found in BB13 or Koshyk and Hamilton (2001) could not be captured. Furthermore, the scaling relations for ST as described in Lindborg (2006) do not require an equipartition of rotational and nonrotational energy. Rather, the similarity of the forward spectral fluxes of available potential energy and KE by horizontal advection, or that of KE associated with adiabatic conversion, can only be maintained in conjunction with the nonrotational flow. In this respect, ST dominated by the nonrotational flow is conceivable and equipartition of rotational and nonrotational is not a necessary condition. We agree with WS14 that such a nonrotational ST is partly inconsistent with the picture of Lindborg (2006).

Second, a vertical Froude number of order 1 is required by the scaling relations of ST. This necessary condition can be tested based on Fig. 7 of BB13. WS14 point out that the level spacing of about 200 m as applied in the L100 run of BB13 was too coarse by at least a factor of 5 in order to represent the aspect ratio required by the scaling relations. BB13 noted this limitation too but pretended that the scaling relations may not be taken too seriously, given the aforementioned result for the spectral fluxes of KE, which seemed to support the picture of ST in the L100 run but not in the L30 run. We admit that the interpretation suggested in BB13 is not rigorous to the extent of which deviations from the aspect ratio for ST [Eq. (16) in BB13] are tolerable. Ideally, one would need to repeat the simulation with sufficient vertical resolution and test whether the results converge. This might also help to reduce the dominance of the nonrotational flow in the mesoscale branch of in the upper-tropospheric KE spectrum.

3. Nonlocal vertical exchange by gravity waves

We now come to the alternative interpretation of WS14, suggesting that the shallow slope and dominating amplitude of the mesoscale nonrotational KE in the upper troposphere, as simulated in BB13 for high vertical resolution (L100 run), results from a net transfer of mesoscale KE from the mid- to the upper troposphere (i.e., from energy deposition by resolved gravity waves). Such a nonlocal vertical exchange is well known for the middle atmosphere. In steady state, the gravity wave kinetic energy generated by energy deposition is partly dissipated into heat, and partly converted by negative buoyancy production into gravity wave potential energy, which is subject to thermal dissipation (e.g., Akmaev 2007; Becker and McLandress 2009; Shaw and Becker 2011). In middle-atmosphere climate models that do not resolve gravity waves, the large-scale thermal effect of energy deposition is parameterized along with the momentum deposition. Hines and Reddy (1967) first noted that the energy deposition consists of two terms: assuming pressure as vertical coordinate, the first term is the convergence of the vertical flux of potential energy; the second term is a residual work that formally corresponds to the shear production by gravity waves (i.e., the negative vertical flux of horizontal momentum times the large-scale vertical wind shear). This term is, like the buoyancy production, typically negative in the middle atmosphere for breaking gravity waves. In particular, the energy deposition, not the potential energy flux convergence, vanishes for conservative gravity waves.

This energetics can also be applied to the spectral KE budget: Since the buoyancy production of gravity wave KE vanishes in the conservative case and should be quite small also for gravity wave breaking (Fritts and Dunkerton 1985), the energy deposition corresponds to the spectral tendencies owing to adiabatic conversion plus vertical advection as defined in BB13. The corresponding mesoscale spectral fluxes give the contributions of all scales from some horizontal wavenumber n on. BB13 emphasized in their section 4d that the mesoscale energy deposition (see their Fig. 8a) was indeed significantly positive in the lower stratosphere; that is, here the spectral slope was strongly influenced by some net input of (nonrotational) KE by nonlocal vertical fluxes related to gravity waves excited in the troposphere. In contrast, the energy deposition was negligibly small in the regime of the − spectral slope in the upper troposphere. That is, the vertical exchange of mesoscale KE between the mid- and upper troposphere was basically conservative with approximately opposing contributions by the spectral fluxes owing to adiabatic conversion and vertical advection or, equivalently, opposing contributions from the mesoscale potential energy flux convergence and shear production. Accordingly, the alternative interpretation proposed by WS14 does not apply in this case. Rather, the small mesoscale energy deposition around 200 hPa (cf. Figs. 8a and 5c in BB13) suggests that part of a mesoscale inertial range is crudely simulated. This range may be indicative of ST owing to the similarity of the spectral fluxes of KE by horizontal advection and adiabatic conversion.

Recently, Augier and Lindborg (2013) have performed an analysis of the spectral KE and APE budgets for two different GCMs, one of which simulates the − spectral slope in the mesoscales of the upper troposphere. They have found that in this case the “cumulative inward vertical flux,” which in terms of gravity wave energetics corresponds to the energy deposition plus the rate of change of KE due to gravity wave drag, is negligible against the positive forward spectral KE flux owing to horizontal advection. Although the underlying model employs a coarse vertical resolution, the result of Augier and Lindborg (2013) is basically consistent with the behavior discussed above for the L100 run in BB13.

To further contrast the vertical exchange by the mesoscales within the troposphere as inferred from the L100 run in BB13 to the exchange between the troposphere and middle atmosphere, we present additional results from a vertically extended model version. The main difference with BB13 is a coarser spectral resolution (T240) and full coverage of the middle atmosphere, using 190 levels up to the lower thermosphere with a model spacing of about 600 m between 2 and 100 km. This setup corresponds to the model of Becker (2009) with double horizontal resolution. Figure 1 shows wavenumberaltitude cross sections of the spectral KE (per unit mass), as well as of the spectral tendencies owing to horizontal advection and energy deposition that are normalized by the spectral KE.

Fig. 1.
Fig. 1.

(a) Spectral KE, as well as spectral tendencies of KE owing to (b) horizontal advection and (c) energy deposition (adiabatic conversion plus vertical advection). The spectral tendencies are normalized by the KE shown in (a). Results are from the middle-atmosphere model version described in Becker (2009) that is run here with double horizontal resolution and adjusted diffusion parameters to reproduce the general circulation along with realistic gravity wave drag and dissipation. See BB13 for definitions of KE and spectral tendencies.

Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-13-0281.1

First, we notice the distinct maximum of mesoscale KE in the mid- and upper troposphere, followed by minimum mesoscale KE in the lower stratosphere. Likewise, the horizontal advection strongly decreases with height across the tropopause. The reason is that no KE is injected by baroclinic instability in the stratosphere. In the upper mesosphere, a regime of negative horizontal advection at larger scales along with positive tendencies at smaller scales is found. This is similar to the troposphere and reflects the fact that the upper mesosphere is subject to large-scale baroclinic instability (Plumb 1983; McLandress et al. 2006), giving rise to in situ generation of traveling planetary waves that feed a forward horizontal cascade of KE. It is nevertheless clear that the flattening and charging of the mesoscale branch of the KE spectrum in the mesosphere is caused by energy deposition of resolved gravity waves (Fig. 1c) and that horizontal advection plays only a minor role. The energy deposition is balanced by momentum diffusion (not shown). For the real atmosphere, we expect that the energy deposition will be balanced by negative horizontal advection such as to induce a forward horizontal KE cascade toward smaller and smaller scales, representing nonlinear gravity wave breaking (Achatz 2007; Fritts et al. 2009). The statistical behavior of such a cascade is possibly governed by the scaling relations of ST. Nevertheless, the scales that are directly subject to energy deposition do hardly fulfill these scaling relations.

The middle-atmosphere model version is tuned to give a realistic gravity wave drag and turbulent dissipative heating in the mesosphere. However, since the simulated gravity wave activity is constrained by the resolved scales, the corresponding KE is overestimated. This holds from the troposphere on. In particular, both the adiabatic conversion (not shown) and the energy deposition are larger than the horizontal advection for the mesoscales in the upper troposphere. Hence, the alternative interpretation of WS14 applies already in this altitude regime; that is, the upper troposphere behaves similar to the mesosphere insofar as the mesoscale spectral tendency of energy deposition dominates over that of horizontal advection. This behavior must, however, be contrasted to the L100 model in BB13 where the energy deposition is negligible for wavenumbers larger than 100 between about 400 and 150 hPa (Fig. 8a in BB13).

4. Conclusions

We have discussed the comment of WS14 concerning the previous analysis of BB13 of the spectral KE budget as simulated with a high-resolution mechanistic GCM. WS14 state that two necessary conditions to apply the concept of stratified turbulence (ST) in the sense of Lindborg (2006) were not fulfilled in the L100 run of BB13. We have argued that an equipartition of rotational and nonrotational flow as found in previous simulations of ST may not be strictly required by the scaling relations of ST. The second objection by WS14 is that even a 200-m vertical spacing as used in BB13 (L100 run) is much too coarse to afford ST. This is correct to the extent that the proportionality factors in all scaling relations are of order 1. GCM simulations with higher vertical resolutions are required to clarify whether both shortcomings can possibly be mitigated.

WS14 suggest that the flattening of the KE spectrum in the upper troposphere along with a dominating nonrotational flow is the result of energy deposition by gravity waves generated at lower altitudes, similar to the effect seen in the lower stratosphere as a result of a baroclinic life cycle as simulated by WS09. We have discussed that the conclusion of WS09 is indeed captured by the L100 run of BB13, but only for the lower stratosphere. For the upper troposphere, however, the energy deposition by gravity waves vanished, although the spectral flux of KE owing to adiabatic conversion was comparable to that owing to horizontal advection.

To further illuminate the last point, we have added results from a middle-atmosphere model version with a somewhat reduced spatial resolution and the tuning so as to simulate realistic large-scale effects of resolved gravity waves in the mesosphere. As a result of the different setting, the upper troposphere now shows a presumably artificial dominance of the mesoscale energy deposition over the horizontal advection in the spectral KE budget. Hence, for this new simulation, the mesoscale KE in the upper troposphere and in the mesosphere is strongly influenced by a balance between energy deposition and momentum diffusion such that the alternative interpretation of WS14 applies to both altitudes’ regimes.

Summarizing, the L100 run of BB13 did not clearly fulfill all aspects of ST for the mesoscales in the upper troposphere. Nevertheless, the simulation showed an approximate equivalence of the forward spectral fluxes owing to horizontal advection and adiabatic conversion along with no net charging of the upper-tropospheric KE by gravity waves. Such a behavior is consistent with the scaling relations of ST, but different from the spectral KE budget in the middle atmosphere. As shown by Waite and Snyder (2013), one may expect other complications when moisture processes come explicitly into play.

Acknowledgments

We would like to thank M. Waite and C. Snyder for their worthwhile comment on our previous article. Some valuable hints by E. Lindborg and an anonymous reviewer are gratefully acknowledged.

REFERENCES

  • Achatz, U., 2007: The primary nonlinear dynamics of modal and nonmodal perturbations of monochromatic inertia–gravity waves. J. Atmos. Sci., 64, 7495.

    • Search Google Scholar
    • Export Citation
  • Akmaev, R. A., 2007: On the energetics of mean-flow interactions with thermally dissipating gravity waves. J. Geophys. Res., 112, D11125, doi:10.1029/2006JD007908.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • Becker, E., 2009: Sensitivity of the upper mesosphere to the Lorenz energy cycle of the troposphere. J. Atmos. Sci., 66, 647666.

  • Becker, E., , and C. McLandress, 2009: Consistent scale interaction of gravity waves in the Doppler spread parameterization. J. Atmos. Sci., 66, 14341449.

    • Search Google Scholar
    • Export Citation
  • Brune, S., , and E. Becker, 2013: Indications of stratified turbulence in a mechanistic GCM. J. Atmos. Sci., 70, 231247.

  • Fritts, D. C., , and T. J. Dunkerton, 1985: Fluxes of heat and constituents due to convectively unstable gravity waves. J. Atmos. Sci., 42, 549556.

    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., , L. Wang, , J. Werne, , T. Lund, , and K. Wan, 2009: Gravity wave instability dynamics at high Reynolds numbers. Part II: Turbulence evolution, structure, and anisotropy. J. Atmos. Sci., 66, 11491171.

    • Search Google Scholar
    • Export Citation
  • Hamilton, K., , Y. Takahashi, , and W. Ohfuchi, 2008: 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
  • Hines, C. O., , and C. A. Reddy, 1967: On the propagation of atmospheric gravity waves through regions of wind shear. J. Geophys. Res., 72, 10151034.

    • 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.

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

  • McLandress, C., , W. E. War, , V. I. Fomichev, , K. Semeniuk, , S. R. Beagley, , N. A. McFarlane, , and T. G. Shepherd, 2006: Large-scale dynamics of the mesosphere and lower thermosphere: An analysis using the extended Canadian Middle Atmosphere Model. J. Geophys. Res., 111, D17111, doi:10.1029/2005JD006776.

    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 1983: Baroclinic instability of the summer mesosphere: A mechanism for the quasi-two-day wave? J. Atmos. Sci., 40, 262270.

    • Search Google Scholar
    • Export Citation
  • Shaw, T. A., , and E. Becker, 2011: Comments on “A spectral parameterization of drag, eddy diffusion, and wave heating for a three-dimensional flow induced by breaking gravity waves.” J. Atmos. Sci., 68, 24652469.

    • Search Google Scholar
    • Export Citation
  • Waite, M. L., , and P. Bartello, 2006: The transition from geostrophic to stratified turbulence. J. Fluid Mech., 568, 89108.

  • Waite, M. L., , and C. Snyder, 2009: The mesoscale kinetic energy spectrum of a baroclinic life cycle. J. Atmos. Sci., 66, 883901.

  • Waite, M. L., , and C. Snyder, 2013: Mesoscale energy spectra of moist baroclinic waves. J. Atmos. Sci., 70, 12421256.

  • Waite, M. L., , and C. Snyder, 2014: Comments on “Indications of stratified turbulence in a mechanistic GCM.” J. Atmos. Sci., 71, 854857.

    • Search Google Scholar
    • Export Citation
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