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    Tropospheric KE spectra for the (a),(c) L100 and (b),(d) L30 runs at representative model layers around (a),(b) 220 hPa (~11 km) and (c),(d) 520 hPa (~5 km). The solid gray curves show the total KE; the black dashed and dashed–dotted curves are the spectra due to the rotational and nonrotational flow, respectively. The thin solid black lines indicate the −3 and slopes.

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    Normalized spectral KE budget in the L100 run at (a) 220 hPa (~11 km) and (b) 518 hPa (~5 km) for the total budget (gray solid curves), vertical advection (gray dashed–dotted curves), momentum diffusion (gray dotted curves), horizontal advection (black dotted curves), and adiabatic conversion (black dashed curves).

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    Spectral KE fluxes in the L100 run at (a),(b) 220 hPa (~11 km) and (c),(d) 518 hPa (~5 km). (a),(c) The quasi-2D enstrophy (dashed–dotted curves) and energy (solid curves) fluxes due to horizontal advection for all wavenumbers. (b),(d) The mesoscales, with a correspondingly adjusted vertical scale, comparing the quasi-2D energy fluxes (solid curves) and the corresponding fluxes due to the complete horizontal advection (dotted curves) and the adiabatic conversion (dashed curves).

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    Horizontal Froude number as a function of horizontal wavenumber and altitude as estimated from (a) the L100 and (b) the L30 runs. The unit is 10−4, and the contour interval is 5 × 10−5. The vertical coordinate is the hybrid coordinate times the global-mean surface pressure (986 hPa) and extends from the lower troposphere to the tropopause region.

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    Spectral fluxes due to (a),(d) the quasi-2D horizontal advection of enstrophy (contour interval = 100 m2 s−2 day−1) and (b),(e) the complete horizontal advection of KE (contour interval = 0.1 m2 s−2 day−1), as well as due to (c),(f) adiabatic conversion (contour interval = 0.1 m2 s−2 day−1). The top (bottom) panels display the results from the L100 (L30) run.

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    Weighted difference between the spectral fluxes due to horizontal advection and adiabatic conversion |(FKHAFKAC)/FKHA| for (a) the L100 and (b) the L30 runs (contour interval = 0.5). A value close to zero (dark gray) indicates that both fluxes are approximately identical and positive.

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    (a) Horizontally and temporally averaged dissipation rate and (b) vertical scale corresponding to a horizontal scale of 210 km as deduced from (16). The black and gray curves give the results from the L100 and L30 runs, respectively. A dissipation rate of 1 W kg−1 corresponds to a sensible heating rate of about 0.1 K day−1. Typical values at 200 hPa are thus 0.001 and 0.003 K day−1 for the L100 and L30 runs.

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    Vertical exchange of mesoscale KE in terms of the combined spectral flux due to adiabatic conversion and vertical advection for (a) the L100 and (b) the L30 runs. Contours are drawn for 0, ±0.1, ±0.2, 0.3, 0.4, … , 0.8, 1, 1.2 m2 s−2 day−1.

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    Tunable parameters of the hyperdiffusion: (a) spectral filter f(n) and (b) the prescribed diffusion coefficient K0 as a function of hybrid coordinate multiplied by a reference surface pressure of 1013 hPa.

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Indications of Stratified Turbulence in a Mechanistic GCM

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  • 1 Leibniz-Institute of Atmospheric Physics, University of Rostock, Kühlungsborn, Germany
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Abstract

The horizontal kinetic energy spectrum and its budget are analyzed on the basis of a general circulation model with simplistic parameterizations of radiative and latent heating. A spectral truncation at total wavenumber 330 is combined with a level spacing of either ~200 m or ~1.5 km from the midtroposphere to the lower stratosphere. The subgrid-scale parameterization consists of a Smagorinsky-type anisotropic diffusion scheme that is scaled by a Richardson criterion for dynamic instability and combined with a stress-tensor-based hyperdiffusion that acts only on the very smallest resolved scales. Simulations with both vertical resolutions show a transition from the synoptic −3 to the mesoscale slope in the upper-tropospheric kinetic energy spectrum. Analysis of the spectral budget indicates that the mesoscale slope can be interpreted as stratified turbulence, as has been proposed in recent works of Lindborg and others, only when a high vertical resolution is applied. In this case, the mesoscale kinetic energy around 300–150 hPa is dominated by the nonrotational flow, and the forward horizontal energy cascade is accompanied by an equally strong forward spectral flux due to adiabatic conversion. This adiabatic conversion mainly results from a vertical potential energy flux that originates in the midtroposphere, where the mesoscale adiabatic conversion is negative. For a conventionally coarse vertical resolution, however, the mesoscale slope in the troposphere is dominated by the rotational flow, and the spectral flux due to adiabatic conversion is not comparable to that associated with horizontal advection.

Current affiliation: Institute of Oceanography, University of Hamburg, Hamburg, Germany.

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

Abstract

The horizontal kinetic energy spectrum and its budget are analyzed on the basis of a general circulation model with simplistic parameterizations of radiative and latent heating. A spectral truncation at total wavenumber 330 is combined with a level spacing of either ~200 m or ~1.5 km from the midtroposphere to the lower stratosphere. The subgrid-scale parameterization consists of a Smagorinsky-type anisotropic diffusion scheme that is scaled by a Richardson criterion for dynamic instability and combined with a stress-tensor-based hyperdiffusion that acts only on the very smallest resolved scales. Simulations with both vertical resolutions show a transition from the synoptic −3 to the mesoscale slope in the upper-tropospheric kinetic energy spectrum. Analysis of the spectral budget indicates that the mesoscale slope can be interpreted as stratified turbulence, as has been proposed in recent works of Lindborg and others, only when a high vertical resolution is applied. In this case, the mesoscale kinetic energy around 300–150 hPa is dominated by the nonrotational flow, and the forward horizontal energy cascade is accompanied by an equally strong forward spectral flux due to adiabatic conversion. This adiabatic conversion mainly results from a vertical potential energy flux that originates in the midtroposphere, where the mesoscale adiabatic conversion is negative. For a conventionally coarse vertical resolution, however, the mesoscale slope in the troposphere is dominated by the rotational flow, and the spectral flux due to adiabatic conversion is not comparable to that associated with horizontal advection.

Current affiliation: Institute of Oceanography, University of Hamburg, Hamburg, Germany.

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

1. Introduction

Since the analysis of aircraftborne measurements by Nastrom et al. (1984) and Nastrom and Gage (1985), it is known that the large-scale spectrum of the horizontal kinetic energy (KE) of the upper troposphere exhibits a transition from a −3 to a law in the mesoscales. Here, the mesoscales correspond to horizontal wavelengths of several hundred down to a few kilometers (Orlanski 1975), equivalent to total horizontal wavenumbers of n ~ 100 to several thousand. The existence of the law in the mesoscales has later been confirmed by another set of aircraft data [Pacific Exploratory Mission (PEM)] analyzed by Cho et al. (1999). There have been several attempts to theoretically assess the energy transformations that might explain the different regimes of the spectrum by involving scaling laws analogous to the ideas of Kolmogorov (1941). Accordingly, the term macroturbulence has been used to characterize those properties of the large-scale atmospheric dynamics that appear to be compatible with concepts of classical turbulence theory and give rise to the KE spectrum (Vallis 2006, chapter 8).

It is general wisdom that, according to Kraichnan’s theory of two-dimensional (2D) turbulence (Kraichnan 1967), the −3 law in the synoptic range and the flattening of the spectrum in the planetary regime are consistent with a forward (downscale) enstrophy cascade and an inverse (upscale) energy cascade, respectively (Lilly 1969). This view was strongly supported by the work of Boer and Shepherd (1983), who analyzed the planetary and synoptic parts of the global KE spectrum of the troposphere on the basis of assimilated observational data with a triangular spectral truncation at total wavenumber 32. They confirmed that the scaling laws of 2D turbulence can well be applied to these scales, which are strongly dominated by the rotational velocity components and geostrophic balance in the extratropics. In particular, they found that the −3 slope of the KE spectrum in the synoptic regime is accompanied by a strong enstrophy flux and a weak energy flux toward the higher wavenumbers. In addition, they identified an upscale energy flux toward planetary wavenumbers. Their findings for the global KE spectrum have later been confirmed in general circulation models (Koshyk et al. 1999a; Takahashi et al. 2006). Since the large-scale extratropical atmospheric flow can approximately be described by quasigeostrophic dynamics, the regime from the planetary to synoptic scales of the atmospheric macroturbulence is often also called geostrophic turbulence (e.g., Salmon 1998, chapter 6).

In contrast to the −3 slope, the origin of the slope in the mesoscale regime has long been controversial (Gage 1979; Dewan 1979; Lilly 1983; Vallis et al. 1997; Koshyk and Hamilton 2001; Tung and Orlando 2003; Smith 2004). Recent reviews can be found in Lindborg (2006) and Gkioulekas and Tung (2006), illustrating that previous explanations differ substantially and are often not compatible. It is, however, clear that any speculation about some kind of a classical forward 3D energy cascade due to isotropic macroturbulence can be ruled out because the ratio of the vertical scale to the horizontal scale is still very small in the mesoscales and the same holds for the corresponding wind components.

Gage (1979) and Lilly (1983) proposed that the mesoscales are subject to an inverse 2D energy cascade, analogously to the planetary regime, with some energy injection at very high wavenumbers. In fact, regional circulation models with very high resolutions revealed the possibility that the small-scale generation of KE due to convection can well lead to an upscale energy flux (Vallis et al. 1997; Lilly et al. 1998). However, the statistical relevance of such events in the global KE spectrum has remained uncertain.

Contrary to this explanation, Tung and Orlando (2003) advanced the finding of Boer and Shepherd (1983) that the enstrophy cascade in the synoptic regime is accompanied by a weak forward energy cascade. They simulated the transition (the “kink”) in the spectrum using a forced-dissipative quasigeostrophic two-layer numerical model with very high horizontal resolution. As a result of the two forward cascades, the energy cascade dominated over the enstrophy cascade from some transition wavenumber nT such as to determine the slope of the spectrum in favor of the law. It is questionable whether the results of Tung and Orlando (2003) can be applied to the real atmosphere (Lindborg 2006). Nevertheless, the conclusion holds that a kink in the KE spectrum can be simulated in a numerical model if only the proper relation of the energy to the enstrophy cascade in the mesoscales is given. Gkioulekas and Tung (2007) revisited this argument. They emphasized that the mesoscales in the real atmosphere cannot be described by quasigeostrophic dynamics and that some other or extended explanation for the law must be sought. Later on, Tulloch and Smith (2009) successfully simulated the Nastrom–Gage spectrum with a slope in the mesoscales on the basis of a surface quasigeostrophic model. As is evident from the debate of Lindborg (2009) and Smith and Tulloch (2009), there is still no consensus whether quasigeostrophic dynamics can be applied to interpret the mesoscale macroturbulence in the troposphere.

Atmospheric general circulation models (GCMs) can nowadays be run at sufficiently high resolutions to resolve a good part of the mesoscales down to horizontal wavelengths shorter than 100 km. Using a comprehensive GCM, Koshyk et al. (1999b) successfully simulated the transition from a −3 slope to a slope in the KE spectrum of the upper troposphere. Later on, Koshyk and Hamilton (2001) made sensitivity tests with different resolutions and gave estimates also for the macroturbulence in the stratosphere and lower mesosphere. They provided a detailed analysis of the KE budget analogously to the approach of Boer and Shepherd (1983), but now also focusing on the terms that give rise to deviations from 2D turbulence. In particular, they found that the contributions from adiabatic conversion and advection due to the nonrotational flow are important for the mesoscale spectral budget at all altitudes. This strongly supported the conclusion that the kink in the upper-tropospheric energy spectrum cannot be explained by 2D or geostrophic turbulence. More recently, Hamilton et al. (2008) used a spectral GCM that was run not only in its comprehensive mode but also as a mechanistic GCM without moisture and moist convection. Since a kink was obtained in either case, the transition from a −3 to a law must be considered as a generic dynamical phenomenon that reflects a net forward energy cascade and involves the nonrotational flow.

The concept of stratified turbulence (ST)—that is, the development of layers of quasi-horizontal motions with strong vertical shear in the presence of strong stratification [Waite and Bartello (2004) and references therein]—was already outlined by Gage (1979) and Lilly (1983), albeit in the context of an upscale 2D energy cascade. Lindborg (2006, 2007) applied the concept of ST to explain the upper-tropospheric law in the mesoscales by a forward horizontal energy cascade. He based his interpretation on aircraftborne wind measurements (Cho et al. 1999), on tank experiments carried out by Billant and Chomaz (2000), and on numerical simulations using a forced-dissipative box model with periodic boundary conditions. He proved that a forward horizontal energy cascade can exist in the presence of strong stratification (low Froude number) for horizontal wavelengths from about 500 km down to at least few kilometers. Lindborg (2006) also confirmed that the combination of inertial and buoyancy forces is pivotal. In fact, Billant and Chomaz (2001) found that in the regime of ST, the ratio of inertial to buoyancy forces should be close to unity. Lindborg (2006) furthermore concluded that in ST, a spectral law of with regard to the horizontal wavenumber is equivalent to a −3 law with regard to the vertical wavenumber.

The scaling criterion for stratified turbulence yields a scale-dependent aspect ratio for the vertical and horizontal scales and velocities (Lindborg 2006). Usually, this requires that the forward horizontal cascade of KE is accompanied by a corresponding cascade of available potential energy with comparable strength (Molemaker and McWilliams 2010). Indeed, the power spectra of potential temperature as measured by Nastrom and Gage (1985) or simulated in Boussinesq-type box models are supportive of such a relation. We shall argue that the same scaling relation for stratified turbulence is obtained by invoking some adiabatic generation of KE in the mesoscale regime, which in turn can be caused by either buoyancy production or nonlocal vertical exchange. Indications of adiabatic generation of KE in the tropospheric mesoscales can already be inferred from the analysis of the spectral KE budget by Koshyk and Hamilton (2001). They noted that such a behavior is reminiscent of the energy deposition due to gravity waves (GWs) in the mesosphere.

In the present paper, we test the concept of ST in association with the adiabatic generation of mesoscale KE for the upper troposphere on the basis of a mechanistic spectral GCM with very high horizontal and vertical resolution and an advanced parameterization of subgrid-scale turbulent diffusion. We also combine the high horizontal resolution with a conventional vertical level spacing in order to allow for comparison with previous results from other GCMs, such as the SKYHI model used by Koshyk and Hamilton (2001) and the AGCM for the Earth Simulator (AFES) model used by Takahashi et al. (2006) and Hamilton et al. (2008).

The remainder of this paper is organized as follows. The mechanistic GCM is described in section 2 and in appendix A. Our method to analyze the horizontal KE spectrum and its budget is described in section 3. Results are presented and discussed in section 4, which is followed by our conclusions. Even though a corresponding analysis of the spectral budget of available potential energy is beyond the scope of the present study, the relevant equation that may be used in a GCM with sensible heat as a prognostic variable is derived in appendix B.

2. Model description and definition of experiments

Our simulations are based on the Kühlungsborn Mechanistic General Circulation Model (KMCM). The KMCM consists of a standard spectral dynamical core with a vertical hybrid coordinate. We assume permanent January conditions with prescribed latent heating in the tropics and self-induced condensational heating in the midlatitudes. Radiative heating is parameterized by temperature relaxation, and the surface temperature used in the boundary layer scheme is prescribed. A more detailed description of these methods is given in Körnich et al. (2006).

For the purpose of the present study, we apply two vertical resolutions. The first consists of 100 hybrid levels up to 0.1 hPa (65 km) with a level spacing of about 200 m between the midtroposphere and the lower stratosphere. The second incorporates only 30 hybrid levels up to 0.1 hPa with a level spacing of about 1.5 km in the relevant height range from about 4 to 20 km. Both setups are combined with the same triangular horizontal spectral truncation at a total wavenumber of 330 (T330), which corresponds to a minimum dynamically resolved horizontal wavelength of about 129 km or a horizontal grid spacing of 52 km.

The only subgrid-scale parameterizations of the KMCM consist of horizontal and vertical turbulent diffusions of momentum and sensible heat. The vertical diffusion scheme is standard, but it includes the correct hydrodynamic dissipation (Becker 2003). Its asymptotic vertical mixing length is 30 m for both simulations. A new nonlinear horizontal diffusion scheme based on the ideas of Smagorinsky (1963) was developed and implemented in the KMCM (Becker and Burkhardt 2007). This parameterization was extended by scaling both the horizontal and vertical diffusion coefficients with the Richardson criterion for dynamical instability, such that a self-consistent turbulent damping of resolved GWs in the upper mesosphere became feasible (Becker 2009).

As already noted by Becker and Burkhardt (2007), the Smagorinsky formulation for the horizontal diffusion is insufficient to simulate the kink in the KE spectrum within a spectral GCM. Instead, an unrealistically strong flattening of the spectrum occurs for high wavenumbers (spectral blocking). Although this is not an issue with regard to the conservation laws or the ability to simulate GWs, it interferes with the goal of identifying the conditions under which a transition from a −3 to a law in the KE spectrum is properly simulated. We therefore chose to extend the Smagorinsky-type horizontal diffusion scheme by some hyperdiffusion. The motivation for this derives from Hamilton et al. (2008), who used a GCM with a spectral dynamical core (like the KMCM) and showed that the kink in the spectrum can be simulated when a biharmonic hyperdiffusion is applied. To preserve angular momentum and energy, our new hyperdiffusion is derived from a stress tensor and applied in grid space. It can easily be tuned for any given spectral resolution and affects only the highest total horizontal wavenumbers. In combination with the Smagorinsky scheme, this method allows to suppress the spectral blocking. The complete horizontal diffusion scheme as applied in the present study is described in appendix A.

We performed each one permanent January simulations with the KMCM using the high (L100 run) and coarse (L30 run) vertical resolution. After equilibration of the zonal-mean climatology and the energy spectrum, each model setup was further integrated for 20 days and data were processed from these 20-day time series with a sampling rate of 22.5 min in each case. The zonal-mean model climatology (not shown) is similar in both runs and is comparable to the simulations of Becker and Burkhardt (2007), although some long-term internal variability associated with the zonal index or the annular modes is not filtered out for the given time series. Note that the two simulations only differ with regard to the vertical resolution. In particular, all other adjustable model parameters are identical, including the heights of the lowest model layers.

3. Spectral kinetic energy and budget

In a spectral GCM, the grid-space representation of the horizontal wind can be written as
e1
Here, ξmn and δmn are the spectral amplitudes of horizontal vorticity ξ and horizontal divergence δ, while n is the total and m is the zonal wavenumber. The spherical harmonics Ynm are real and normalized. Furthermore, is the horizontal gradient operator in spherical geometry, ez is the unit vector in the vertical direction, and ae denotes the Earth’s radius. According to (1), the spectral kinetic energy per unit mass at a particular model layer is given in terms of the discrete power spectra of ξ and δ: that is,
e2
The spectrum of enstrophy times is represented analogously as
e3
Our analysis of the spectral kinetic energy and enstrophy budgets follows the method of Boer and Shepherd (1983) and Koshyk and Hamilton (2001). Accordingly, the spectral tendencies are written as
e4
and
e5
where and are computed from the dynamical core of the model. To split these tendencies into contributions from different processes, we consider the prognostic equations for horizontal vorticity and divergence in grid space
e6
with
e7
Here, η represents the vertical hybrid coordinate (Simmons and Burridge 1981), R is the total momentum diffusion, and Φ is the geopotential. The vector F accommodates all contributions to the horizontal wind tendency that cannot be written as a gradient of a scalar. Transformation into spectral space is done as usual according to
e8
where indicates the integration over all solid angles. The spectral tendency in (4) can now be split up by computing and for particular contributions to the horizontal wind tendency. We write
e9
where is the spectral tendency due to horizontal advection (HA); it is obtained by setting F = v × (f + ξ)ez and B = v2/2 in (8). Likewise, is the spectral KE tendency due to vertical advection (VA), computed by setting and B = 0 in (8). The corresponding tendencies due to adiabatic conversion (AC) and momentum diffusion (MD) are denoted as and , respectively. They are calculated by using and B = Φ in the case of AC, and F = R and B = 0 in the case of MD. The four terms on the rhs of (9) should add up to zero in the climatological mean.

The analysis can be extended by retaining only the vorticity coefficients ξnm in (1) to compute the rotational horizontal wind vrot. The spectral tendency due to the quasi-2D horizontal advection is then obtained by setting F = vrot × (f + ξ)ez and B = (vrot)2/2 in (8). These definitions can be applied also to (5) in order to compute the corresponding spectral enstrophy tendency .

The spectral fluxes due to the particular processes are calculated by summing up the corresponding spectral tendencies, starting from the wavenumber n in question and ending at the truncation wavenumber. We will thus refer to
e10
as the forward quasi-2D spectral fluxes at wavenumber n of kinetic energy and enstrophy, respectively. The corresponding sums over all n vanish by definition, indicating that the quasi-2D spectral fluxes do not involve any vertical exchange. This is different for the spectral fluxes associated with the other tendencies defined above. We may nevertheless formally define the forward spectral fluxes due to horizontal advection, vertical advection, adiabatic conversion, and momentum diffusion as
e11

4. Results and discussion

a. Energy spectra

Figure 1 shows the KE spectra for two heights representative of the midtroposphere (518 hPa or ~5 km, Figs. 1c,d) and the upper troposphere (220 hPa or ~11 km, Figs. 1a,b) for the L100 and the L30 run. In the upper troposphere, both cases exhibit the prominent −3 slope in the synoptic regime, as well as a kink in the spectrum followed by an approximate slope in the mesoscales. The rotational flow (black dotted curves) dominates the shallow planetary regime and the synoptic regime, as expected. In the L100 run, the mesoscale regime is clearly dominated by the nonrotational flow (black dashed–dotted curves). For the L30 run, the rotational flow dominates at all resolved scales and the kink in the spectrum occurs at a larger scale (smaller wavenumber). Surprisingly the magnitude of the mesoscale KE spectrum for the L30 run exceeds that of the L100 run by about a factor of 4.

Fig. 1.
Fig. 1.

Tropospheric KE spectra for the (a),(c) L100 and (b),(d) L30 runs at representative model layers around (a),(b) 220 hPa (~11 km) and (c),(d) 520 hPa (~5 km). The solid gray curves show the total KE; the black dashed and dashed–dotted curves are the spectra due to the rotational and nonrotational flow, respectively. The thin solid black lines indicate the −3 and slopes.

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

In the midtroposphere, the rotational flow in the L100 run (Fig. 1c) dominates the spectrum down to the highest resolved wavenumbers with an approximate spectral slope of −3. The L30 run is, again, different and exhibits a kink in the mesoscales that is dominated by the rotational flow (Fig. 1d). This is similar to its behavior in the upper troposphere, though the kink occurs at a smaller scale in the midtroposphere.

The previous successful GCM simulations of the slope in the mesoscales of the upper troposphere used a vertical level spacing comparable to the L30 run and showed either a dominating role of the nonrotational flow (Hamilton et al. 2008) or an approximate equipartition between the rotational and nonrotational flow (Koshyk and Hamilton 2001). Based on measured velocity structure functions, Lindborg (2007) found for both the upper troposphere and lower stratosphere that the vorticity and the divergence spectra give rise to a slope in the mesoscales, with the rotational component being twice as strong as the nonrotational component.

b. Spectral energy tendencies and fluxes

Figure 2 presents the individual spectral tendencies as defined above and normalized by Kn for the mid- and upper troposphere. The deviation of the total tendency (solid gray curves) from zero is generally small against the individual terms, except for the planetary scale, where our time series is too short with respect to the long-term internal variability associated with the zonal index and annular modes.1 Note also that we did not adjust the scale of the plot to the high relative tendencies at the truncation scale. Considering wavenumbers larger than 10, a forward horizontal energy cascade from the synoptic range toward the mesoscales due to horizontal advection (black dotted curves) as well as a strong dissipation of energy at the highest wavenumbers (gray dotted curves) are evident in both layers. This behavior is expected and fully in accordance with the previous analysis of Koshyk and Hamilton (2001). Figure 2 also shows that the contributions from adiabatic conversion (black dashed curves) and vertical advection (gray dashed–dotted curves) are opposite in the mesoscales, as was also found by Koshyk and Hamilton (2001). However, both spectral tendencies change sign from the upper to the midtroposphere in the L100 run: The mesoscale adiabatic conversion is positive in the upper troposphere but negative in the midtroposphere. This is one of the central findings of the present analysis. This effect is not seen when we average from the lower to the upper midtroposphere, as was done in Fig. 6 of Koshyk and Hamilton (2001).

Fig. 2.
Fig. 2.

Normalized spectral KE budget in the L100 run at (a) 220 hPa (~11 km) and (b) 518 hPa (~5 km) for the total budget (gray solid curves), vertical advection (gray dashed–dotted curves), momentum diffusion (gray dotted curves), horizontal advection (black dotted curves), and adiabatic conversion (black dashed curves).

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

Statistically significant curves as shown in Fig. 2 are obtained after averaging about 10 model days (~600 snapshots). Instantaneously derived curves are subject to large fluctuations (not shown), giving rise to temporarily reversed energy cascades by horizontal advection. This may be interpreted as a kind of backscattering of kinetic energy within the resolved scales. Palmer (2001) stressed the idea of a subgrid-scale parameterization that allows for backscattering of kinetic energy from the unresolved to the resolved scales. More recently, Berner et al. (2009) have attributed the inability of a forecast model to simulate the proper energy spectrum of the mesoscales to the lack of a backscattering parameterization. They showed that such a measure can indeed reduce the unrealistic mesoscale steepening of the energy spectrum that is simulated with the integrated forecasting system (model version CY31R1) of the European Centre for Medium-Range Weather Forecasts (ECMWF). Based on the sensitivity experiments presented in Hamilton et al. (2008) for different hyperdiffusion setups and our mechanistic simulations, we argue that the backscattering scheme in the ECMWF model partly offsets a too strong numerical or explicit hyperdiffusion, and that the net mesoscale energy cascade is downscale in either version of the ECMWF model.

The mid- and upper-tropospheric spectral fluxes in the L100 run are shown in Fig. 3. At both heights the quasi-2D enstrophy flux (dashed–dotted curves in Figs. 3a,c) is clearly positive from wavenumber 20 on. This is consistent with a forward enstrophy cascade due to the rotational flow. The quasi-2D horizontal energy flux (solid curves) is negative for wavenumbers smaller than 20, indicating an inverse energy cascade toward the planetary scale. These findings are entirely consistent with the results of Boer and Shepherd (1983) and Straus and Ditlevsen (1999) from analyses of operational datasets. For our present high-resolution mechanistic model, we have to recall that no convection is included. Hence, KE is only injected at the baroclinic scale (total wavenumbers of about 6–15) because of adiabatic conversion. Therefore, the rotational flow reflects a forward enstrophy and an inverse energy cascade with respect to the baroclinic scale in line with the concept of two-dimensional turbulence.

Fig. 3.
Fig. 3.

Spectral KE fluxes in the L100 run at (a),(b) 220 hPa (~11 km) and (c),(d) 518 hPa (~5 km). (a),(c) The quasi-2D enstrophy (dashed–dotted curves) and energy (solid curves) fluxes due to horizontal advection for all wavenumbers. (b),(d) The mesoscales, with a correspondingly adjusted vertical scale, comparing the quasi-2D energy fluxes (solid curves) and the corresponding fluxes due to the complete horizontal advection (dotted curves) and the adiabatic conversion (dashed curves).

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

Figures 3b,d show the spectral energy fluxes for total wavenumbers larger than 30, using a different scale in order to highlight which cascade dominates the mesoscales. A forward energy flux due to horizontal advection (dotted curves) is obvious in both layers. This flux is much stronger than the quasi-2D flux (solid curves). It follows that the forward horizontal cascade that dominates the KE spectrum in the mesoscales of the upper troposphere cannot be attributed to the rotational flow. Figure 3b also indicates that the forward spectral fluxes due to horizontal advection (dotted) and adiabatic conversion (dashed) have similar magnitudes in the upper troposphere. In the midtroposphere, however, both terms have opposite signs (Fig. 3d). This difference will be discussed in the following subsections.

Considering all four panels of Fig. 3 together, we see that in the midtroposphere the relation of energy to the enstrophy cascade in the mesoscales is biased toward the enstrophy cascade. According to the scaling argument proposed by Tung and Orlando (2003), this explains why the mesoscales exhibit a −3 law in the midtroposphere and a law in the upper troposphere: The KE and enstrophy dissipation rates ε and μ, which correspond to the corresponding spectral fluxes in the inertial regimes, give rise to separate spectral energies of approximately and 2μ2/3(n/ae)−3, respectively; if these energies are superposed, then a kink in the total energy spectrum is expected where the two separate contributions are equal, that is, for (nT/ae)4/3 ~ 3(μ/ε)2/3. In the midtroposphere of the L100 run, the relation μ/ε is evidently larger than in the upper troposphere, such that a kink is not observed within the resolved scales.

c. Criteria for stratified turbulence

As mentioned in the introduction, the usual condition for the existence of ST is that the horizontal KE cascade is accompanied by a cascade of available potential energy (APE) of similar magnitude (Molemaker and McWilliams 2010). Here, we follow the idea of Lindborg (2006) and assume that buoyancy forces can be important as well in order to yield the proper scaling for ST. This means that the spectral fluxes due to adiabatic conversion and horizontal advection of KE may have similar magnitudes. In the following we recapitulate these scaling arguments along with the scale-dependent aspect ratio for ST.

Let u denote the horizontal velocity at horizontal scale . The condition that the horizontal shear production is on the order of the dissipation yields
e12
For convenience we use height z as the vertical coordinate and assume the anelastic approximation. Furthermore, we denote the APE at scale as , where is the horizontal-mean potential temperature, Θ denotes its horizontal variation at scale , and is the buoyancy frequency corresponding to . Since vertical displacements associated with mesoscale perturbations are basically isentropic in the inertial range, the potential temperature variation scales like , where z is the vertical scale corresponding to . Hence, the spectral flux of APE scales like
e13
Alternatively we may estimate the spectral flux due to the adiabatic generation of KE at scale . For this purpose we assume that the pressure perturbation at scale is hydrostatic, p ~ gzρ, and that the corresponding density perturbation is anelastic, , and isentropic (see above). This yields for the spectral flux due to adiabatic conversion the same result as for the spectral flux of APE, that is,
e14
The quotient of horizontal shear production and adiabatic conversion (or horizontal advection of APE) equals the squared vertical Froude number. Hence, this number should be of order 1 in the case of ST, that is,
e15
Eliminating u in (13) or (14) with the help of (12) yields the well-known scale-dependent aspect ratio for ST, that is,
e16
It approaches unity at the transition to Kolmogorov turbulence, also known as the Ozmidov scale, . The Ozmidov scale amounts to at most 10 m in the troposphere. When applied to the mesoscale regime, the relation (16) allows us to derive a −3 slope for the kinetic energy spectrum with regard to the vertical wavenumber from a slope with regard to the horizontal wavenumber. Since (16) is very small in this regime, the horizontal Froude number must be consistent with [see also Riley and Lindborg (2008)]
e17
This number can well be estimated from our model. We assume and use = 2πa/n. Furthermore, we compute u from the energy spectrum as
e18
The horizontal Froude numbers based on this estimate are shown in Fig. 4 for both runs. In the troposphere we generally find values not larger than 10−4. This is well below the critical value of 0.02 estimated by Lindborg (2006) in order to separate ST from flows with a large Froude number dominated by classical 3D turbulence.
Fig. 4.
Fig. 4.

Horizontal Froude number as a function of horizontal wavenumber and altitude as estimated from (a) the L100 and (b) the L30 runs. The unit is 10−4, and the contour interval is 5 × 10−5. The vertical coordinate is the hybrid coordinate times the global-mean surface pressure (986 hPa) and extends from the lower troposphere to the tropopause region.

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

Comparing the L100 to the L30 run indicates higher values of Frh in the mesoscales for coarser vertical resolution, as is also evident from Fig. 1 along with (17) and (18). It is tempting to assume that the lower Froude numbers in the L100 run indicate that the mesoscales are more compatible with ST than in the L30 run, provided that both runs exhibit a spectral slope for the regime considered. But this does not necessarily prove that ST is present in the L100 run.

A reliable test of ST combined with a forward spectral flux due to adiabatic conversion is given by a continuous comparison of the relevant spectral fluxes. The fluxes due to horizontal advection of enstrophy or energy and due to adiabatic conversion are shown in Fig. 5 for both runs. First, we note that from 500 to 300 hPa and from n = 70 to 200, the L100 run is characterized by a much larger relation of enstrophy to energy flux than the L30 run. Since the spectral flux due to adiabatic conversion is negative in this regime, ST in combination with a corresponding forward flux is ruled out for both runs. Comparing Figs. 5e,f at about 220 hPa, we see that even here the spectral flux due to adiabatic conversion is either negative (for n ~ 70–150) or only slightly positive, but it is clearly weaker than the flux due to horizontal advection. Hence, for coarse vertical resolution, the slope may either be explained by a cascade of APE or simply by the aforementioned scaling argument of Tung and Orlando (2003).

Fig. 5.
Fig. 5.

Spectral fluxes due to (a),(d) the quasi-2D horizontal advection of enstrophy (contour interval = 100 m2 s−2 day−1) and (b),(e) the complete horizontal advection of KE (contour interval = 0.1 m2 s−2 day−1), as well as due to (c),(f) adiabatic conversion (contour interval = 0.1 m2 s−2 day−1). The top (bottom) panels display the results from the L100 (L30) run.

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

The situation is different in the L100 run, where the adiabatic conversion is less negative in the midtroposphere and more positive higher up compared to the L30 run (cf. Figs. 5c,f). As a result, the spectral fluxes due to horizontal advection and adiabatic conversion are comparable for the range of the slope in Fig. 1a. This criterion is further quantified in Fig. 6 by showing the absolute difference between the spectral fluxes divided by the former. Approximately equally strong spectral fluxes due to horizontal advection and adiabatic conversion show up in the L100 run (Fig. 6a) in a narrow regime of the mesoscales between 300 and 200 hPa. The L100 run with its high vertical resolution indeed resolves this region with several layers. Surprisingly, the L100 run also meets the criterion in a narrow regime around 750 hPa for n > 50, and for n > 250 even farther below. Indeed, at these altitudes the energy spectrum of the L100 run (not shown) displays a shallow slope in the mesoscales for n > 250, comparable to that in the upper troposphere. The L30 run (Fig. 6b) shows an approximate equality of both spectral fluxes only from 900 to 750 hPa, where the vertical level spacing approaches that of the L100 run.

Fig. 6.
Fig. 6.

Weighted difference between the spectral fluxes due to horizontal advection and adiabatic conversion |(FKHAFKAC)/FKHA| for (a) the L100 and (b) the L30 runs (contour interval = 0.5). A value close to zero (dark gray) indicates that both fluxes are approximately identical and positive.

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

The question remains as to what determines the mesoscale spectral slope throughout the mid- and upper troposphere in the L30 run. Ultimately answering this question requires an evaluation of the spectral budget of the APE, which is beyond the scope of the present study. Nevertheless, we can take advantage of the aspect ratio in (16) in order to estimate the vertical resolution necessary to resolve ST for horizontal wavenumbers around n = 200 (or horizontal wavelengths around 210 km). Figure 7b shows corresponding estimates of z for the L100 (black curve) and the L30 (gray curve) run, using the time-averaged horizontal-mean profiles of the buoyancy frequency and dissipation from the respective runs. While the two profiles (not shown) are almost identical, the dissipation profiles (Fig. 7a) are not. In particular, the dissipation in the mid- and upper troposphere is about a factor of 5 stronger in the L30 run. Nevertheless, the strengths of the Lorenz energy cycle, which is measured by the globally integrated dissipation and dominated by the contribution from the boundary layer, differ only by about 5%, with the stronger cycle even in the L100 run. From Fig. 7b we estimate for the upper troposphere z ≈ 80 m for the L100 run and z ≈ 200 m for the L30 run, corresponding to a necessary level spacing of 40 and 100 m, respectively. With a vertical level spacing of about 1500 m, the L30 run has a too coarse vertical resolution by about a factor of 15. The L100 run with a vertical level spacing of 200 m in the upper troposphere represents a major improvement, but for a proper representation of ST the vertical resolution should be still finer.

Fig. 7.
Fig. 7.

(a) Horizontally and temporally averaged dissipation rate and (b) vertical scale corresponding to a horizontal scale of 210 km as deduced from (16). The black and gray curves give the results from the L100 and L30 runs, respectively. A dissipation rate of 1 W kg−1 corresponds to a sensible heating rate of about 0.1 K day−1. Typical values at 200 hPa are thus 0.001 and 0.003 K day−1 for the L100 and L30 runs.

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

d. Vertical exchange of mesoscale kinetic energy

Figures 5a,d show that the enstrophy cascade is generally strongest in the midtroposphere. Since this cascade reflects the injection of enstrophy and KE by baroclinic Rossby waves, we would expect that the forward energy cascade accompanying the enstrophy cascade should be strongest in the midtroposphere, too. This is the case in the L30 run. Nevertheless, the height dependence is less pronounced for the energy cascade (Fig. 5e) than for the enstrophy cascade (Fig. 5d). For the L100 run, the energy flux due to horizontal advection changes more smoothly with height and does not reflect the enstrophy flux at all for n > 100 (see Figs. 5a,b). A possible explanation for this behavior is that the energy that is generated at the baroclinic scale and partly subject to a forward horizontal energy cascade is vertically redistributed by the tropospheric mesoscales. Such a nonlocal vertical exchange is well known from internal gravity waves (Lindzen 1973; Becker 2004; Shaw and Becker 2011).

To apply the picture of gravity wave energetics, we must further analyze the spectral tendency (or flux) due to adiabatic conversion. For this purpose we consider the horizontal momentum equation in pressure coordinates, which roughly applies to the vertical hybrid coordinates used in GCMs in the midtroposphere and becomes accurate higher up. Taking the horizontal mean, the adiabatic conversion can be split into a buoyancy production plus the convergence of a vertical potential energy flux. This is further discussed in appendix B [(B1)(B3)], where also the budget for the horizontal-mean APE based on the approximation of Lorenz (1955) is given in (B8). Applying the spectral decomposition to the horizontal wind in (B1) and considering only the mesoscales (say, n > 100), one is tempted to conclude that the buoyancy production term vanishes for the present mechanistic model, simply because no mechanism for generating APE at the mesoscales—such as convection, for instance—is included. However, if there is a forward spectral flux of APE, some buoyancy production of KE can well exist at the mesoscales. The vertical distributions of the spectral fluxes due to adiabatic conversion in Fig. 5c,f are indicative of a nonlocal vertical exchange of KE within the mesoscales rather than a buoyancy production. This is consistent with the analysis of Koshyk and Hamilton (2001, their Fig. 7b), who also found a positive adiabatic conversion in the mesoscales of the upper troposphere that they could attribute to a nonlocal vertical exchange rather than buoyancy production. From the opposite signs of the mesoscale adiabatic conversion in the mid- and upper troposphere in Figs. 5c,f, we can furthermore conclude that there is an upward mesoscale potential energy flux (or pressure flux when using height z as a vertical coordinate) between the two height regions. This situation is reminiscent of the vertical transport of energy from the lower to the middle atmosphere by internal gravity waves.

Assuming that buoyancy production is negligible at the mesoscales, the complete mesoscale KE tendency due to vertical exchange is given by the adiabatic conversion plus vertical advection. In the single-column picture of gravity wave energetics, this quantity represents the energy deposition (e.g., Shaw and Becker 2011). It is depicted in Fig. 8 for the two runs. We see that in the region of ST in the L100 run (for n > 150 and around 250 hPa in Fig. 8a), the energy deposition is very small against the contribution from adiabatic conversion. In other words, the KE deposited by the mesoscale vertical potential energy flux convergence is largely neutralized by vertical advection. Resorting again to the single-column energetics of gravity waves, this means that the vertical energy exchange is basically conservative, which is consistent with the assumption that it takes place within the inertial range of ST. Nevertheless, in the lower stratosphere and in the lower troposphere, there is a notable net energy deposition. The former is likely due to the breakdown of resolved gravity waves. The latter indicates that the mesoscales transfer kinetic energy also downward, such that the majority of the net dissipation in the atmosphere occurs in the lower troposphere [see also the estimates given in Becker and Burkhardt (2007)]. Figure 8 furthermore reveals that in the midtroposphere, the energy deposition is much stronger for the coarse vertical resolution (Fig. 8b), while the L100 run (Fig. 8a) even exhibits a reversed sign from about 400 to 180 hPa for n < 150. The weaker mesoscale energy flux toward the stratosphere and the stronger adiabatic conversion plus vertical advection in the midtropospheric mesoscales may contribute to the increased importance of the energy cascade relative to the enstrophy cascade in the L30 run.

Fig. 8.
Fig. 8.

Vertical exchange of mesoscale KE in terms of the combined spectral flux due to adiabatic conversion and vertical advection for (a) the L100 and (b) the L30 runs. Contours are drawn for 0, ±0.1, ±0.2, 0.3, 0.4, … , 0.8, 1, 1.2 m2 s−2 day−1.

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

There is another qualitative difference between the L100 and L30 run that is not evident from Fig. 8. Though both runs exhibit a strong cancelation of the mesoscale spectral fluxes due to adiabatic conversion and vertical advection, the latter mainly affects the nonrotational part of the spectral energy budget in the L100 run, while the rotational mesoscale flow is affected by the vertical advection in the L30 run. This difference likely induces the strong increase of spectral enstrophy flux for high wavenumbers in the L30 run (see Fig. 5d).

5. Summary and conclusions

We have applied a high-resolution mechanistic GCM with a standard spectral dynamical core and realistic large-scale dynamics in order to analyze the kinetic energy (KE) in the mesoscales of the troposphere. Our main question has been whether the concept of stratified turbulence (ST) can be applied when a slope is simulated. According to estimates of Lindborg (2006), ST requires a relatively high vertical resolution that was presumably not met in previous high-resolution GCM simulations [see also the discussion by Lindzen and Fox-Rabinovitz (1989)]. To illuminate the role of vertical resolution for the ability of a GCM to simulate an approximate slope in the mesoscales, we have combined a T330 spectral resolution with two different vertical resolutions: In our L100 run, the level spacing is about 200 m between 4- and 20-km height, whereas our L30 run uses a more conventional level spacing of about 1.5 km. Both runs have the highest model layer at about 60 km, the same level spacing in the boundary layer, and identical model parameters otherwise.

The present version of the mechanistic GCM does not include any parameterization of gravity waves or convection. The subgrid-scale parameterizations basically consist of a standard vertical diffusion scheme with the discretization given in Becker (2003) and a harmonic horizontal diffusion scheme based on Smagorinsky’s generalized mixing-length ansatz. Since the harmonic Smagorinsky scheme is not sufficient to ensure a reasonably high wavenumber part of the energy spectrum (Becker and Burkhardt 2007), it has been completed by a weak hyperdiffusion that effectively damps only wavenumbers larger than about 300 (see appendix A). Such a model setup allows for simulating a kink in the upper-tropospheric energy spectrum that is well separated from the scales directly affected by the hyperdiffusion. Measuring the strength of the Lorenz energy cycle in the two simulations showed that the hyperdiffusion accounts for only about 1% of the global-mean dissipation, while about 10% is due to the Smagorinsky scheme and the majority (about 1.5 W m−2) is due to the vertical momentum diffusion. It would be worthwhile to dispense with a nonphysical hyperdiffusion and advance the Smagorinsky scheme, for instance, by introducing the constraint of scale invariance according to the theory of Germano et al. (1991). Such a scheme is currently being developed.

Our simulations confirm the behavior of the rotational flow in the planetary and synoptic regime as found by Boer and Shepherd (1983): In accordance with the concept of two-dimensional turbulence, the rotational flow is characterized by a forward enstrophy cascade and an inverse energy cascade away from the energy injection at the baroclinic scale (around total horizontal wavenumber 10). Furthermore, both the L100 and L30 simulations show a transition from an enstrophy to an energy inertial range in the mesoscales of the upper troposphere. This slope is dominated by the nonrotational flow only for the high vertical resolution. In contrast, the mesoscale spectral energy in the midtroposphere is dominated by the rotational flow for both vertical resolutions. Here, the enstrophy inertial subrange extends into the mesoscales for high vertical resolution, whereas a kink followed by a slope is found in the conventional case.

The scaling law for ST defines a scale-dependent aspect ratio of vertical and horizontal scales. An estimate of the vertical scale corresponding to a horizontal wavenumber of about 200 (210-km wavelength) yields larger values in the upper troposphere for the coarse vertical resolution, which is caused by a stronger dissipation in that run. However, the conventional vertical-level spacing is too coarse by a factor of about 15 in order to meet the ST aspect ratio, that is, 100 m would be required for horizontal wavenumber 200. The L100 run represents already a major improvement, but its 200-m vertical level spacing is still not sufficient.

The ST aspect ratio can be obtained by assuming either a cascade of available potential energy (APE) or a forward spectral flux of KE due to adiabatic conversion. Analysis of the spectral KE fluxes reveals that only for the fine vertical level spacing is the forward flux due to horizontal advection accompanied by a forward flux due to adiabatic conversion of comparable strength.

Based on these findings, we argue that a considerably high vertical resolution is required to simulate the kink in the upper-tropospheric energy spectrum as a result of ST. Such a high vertical resolution was not applied in previous successful GCM simulations of the mesoscale energy spectrum (Koshyk and Hamilton 2001; Takahashi et al. 2006; Hamilton et al. 2008). In this respect, our L30 run corresponds to these previous model setups. A rather high vertical resolution was applied by Watanabe et al. (2008) for gravity wave simulations from the troposphere to the mesosphere. In their model, a shallow mesoscale slope developed in the upper-tropospheric KE spectrum; however, the horizontal resolution was not sufficient to resolve a good part of the regime.

In forced-dissipative Boussinesq models with periodic boundary conditions as applied by Lindborg (2006), the generation of APE is ultimately due to the forcing of the rotational modes at scales that are not very much separated from the regime of ST. Furthermore, a forward cascade of APE is typically found, allowing for reconciling the spectral slope with ST. In a GCM or in the real atmosphere, the mesoscales are widely separated from the energy injection scale, which basically corresponds to baroclinic Rossby waves. The present mechanistic GCM indicates that ST in the upper troposphere is possibly accompanied by a forward spectral flux of KE due to adiabatic conversion. This is different from the dynamics in idealized box models. A simple consideration of the horizontal-mean kinetic energy equation shows that adiabatic conversion in the mesoscales can be due to either buoyancy production or a vertical potential energy flux convergence (assuming pressure as a vertical coordinate). The latter appears to apply to the upper troposphere in the present L100 run. The source for the vertical energy flux convergence lies in the mesoscales of the midtroposphere, where the adiabatic conversion is negative. Thus, there is a nonlocal vertical energy transfer within the mesoscales from the mid- to the upper troposphere, which is to some extent analogous to the vertical energy transfer from the lower to the middle atmosphere by internal gravity waves. The question remains whether this mechanism also applies in comprehensive high-resolution GCMs or the real atmosphere.

A second region of ST presumably exists in the lower-tropospheric mesoscales. Here, the level spacing of the L30 run approaches the high vertical resolution of the L100 run. Accordingly, both runs exhibit a shallow spectrum and similar strengths of the mesoscale spectral fluxes due to horizontal advection and adiabatic conversion.

The question remains why a conventional vertical resolution combined with a very high horizontal resolution leads to a slope in the mesoscales even more clearly than a setup with high vertical resolution, despite the fact that the aspect ratio and the spectral flux due to adiabatic conversion are much less consistent with ST than for high vertical resolution. In the future, the study of extreme cases, such as upper-tropospheric vertical level spacing of 3000 or 30 m, may shed more light on this behavior, which is not yet understood within the framework of our analysis. Ultimately, a combined spectral analysis of both the KE and APE budgets is required in order to clarify to what extent the mesoscale energy spectra simulated in a GCM are consistent with ST. However, such an investigation is beyond the scope of the present study. In appendix B we have presented the APE budget to be evaluated when using pressure as a vertical coordinate, which approximately applies to GCMs with a vertical hybrid coordinate from the midtroposphere on.

Analyzing individual snapshots from our simulation yields strong spectral fluxes due to horizontal advection that can be either forward or inverse (section 4b). Nevertheless, a weak forward flux shows up when averaging over a few model days or longer; that is, the net energy cascade in the mesoscales is forward. This finding is in line with all results on macroturbulence in large-scale circulation models or forced-dissipative Boussinesq models of which we are aware. We also stress that a net forward cascade is required by the energetics of the Lorenz energy cycle (Lorenz 1967): In the real atmosphere, the kinetic energy injected by baroclinic waves is ultimately dissipated by molecular friction; that is, the energy is permanently cascaded from the synoptic scale to the microscale. The accompanying (molecular) frictional heating yields the necessary entropy production that balances the loss of entropy due to the large-scale differential heating. These elementary constraints on the energetics of the general circulation as conceived by Lorenz (1967) would not apply if there was an inverse net energy cascade in the mesoscales. Note also that the kinetic energy cascaded upward toward the planetary scale by the quasigeostrophic flow is ultimately transferred to smaller scales and dissipated by molecular friction in the boundary layer.

The dynamics of the mesoscales is intimately linked to the intensity by which the large-scale weather systems convert APE into KE and feed the forward cascades of KE and APE. In this respect, high-resolution climate models with appropriate resolutions, both horizontally and vertically, and physically consistent subgrid-scale schemes will become increasingly important in the future. In the present study, we have emphasized the role of the ratio of vertical to horizontal resolution in order simulate the mesoscale KE spectrum in the upper troposphere consistently with the concept of ST.

Acknowledgments

For valuable discussion we thank Heiner Körnich. We are indebted to Erik Lindborg for inspiration and valuable comments on the manuscript. The helpful comments by an anonymous reviewer are gratefully acknowledged. This study was partly founded by the Deutsche Forschungsgemeinschaft under Grant BE 3208/2-1 and by the Leibniz Graduate School ILWAO.

APPENDIX A

Extension of the Smagorinsky Scheme by a Stress-Tensor-Based Hyperdiffusion

The Smagorinsky-type horizontal diffusion applied in the present model version is the same as in Becker (2009). When extending this scheme with an additional hyperdiffusion, the complete horizontal wind tendency on a particular model layer due to horizontal diffusion can be written as
ea1
Here, Δp is the pressure thickness of the particular model level in the hybrid coordinate system. For convenience, the level index is neglected in (A1). Furthermore, the symbol denotes the horizontal gradient operator in spherical coordinates at constant hybrid coordinate. The Smagorinsky-type horizontal diffusion coefficient is given by
ea2
where l2 = 1.4 × 107 m2 is a prescribed squared horizontal mixing length. It is constant with height and smoothly reduced in the lowest five model layers to a value of 4 × 106 m2. The strain tensor is defined as
ea3
with the transposed tensor being indicated by the exponent T and the unit vector in vertical direction defined as ez. The idemfactor is denoted by the symbol I, |S|2 is the squared Frobenius norm of S, and S0 = 10−5 s−1 is a minimum horizontal shear to ensure continuity of derivatives. The factor α = 8.6 is a tunable parameter in order to enhance the horizontal diffusion coefficient for the small local Richardson number according to
ea4
This scaling is relevant when resolved gravity waves become dynamically unstable. To account for the usual linear stability criterion, the offset Ri0 in (A4) is increased from zero in the boundary layer to a value of 0.25 from about 600 hPa on.
The additional horizontal hyperdiffusion in (A1) is proportional to a prescribed diffusion coefficient K0 (see Fig. A1b). The hyperdiffusion is given in stress-tensor formulation according to
ea5
where vf is a filtered horizontal wind defined as [cf. to (1)]
ea6
The filter function f(n) is displayed in Fig. A1a. It smoothly starts to deviate from zero at wavenumber 270 and becomes unity at the truncation wavenumber.
Fig. A1.
Fig. A1.

Tunable parameters of the hyperdiffusion: (a) spectral filter f(n) and (b) the prescribed diffusion coefficient K0 as a function of hybrid coordinate multiplied by a reference surface pressure of 1013 hPa.

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-025.1

The stress-tensor formulation of the horizontal momentum diffusion automatically fulfills, unlike to numerical filters or other hyperdiffusion schemes, the angular momentum conservation law for arbitrary control volumes. In the same way, it also fulfills the hydrodynamic energy conservation law due to including the horizontal shear production (or frictional heating) term
ea7
in the thermodynamic equation of motion. Even though the shear production of the hyperdiffusion [second term on right-hand side of (A7)] is not positive definite, our experiments showed that negative shear production rates due to this term are always negligible and that the complete horizontal shear production is dominated by the frictional heating term from the Smagorinsky scheme [first term on right-hand side of (A7)], which is positive definite.

APPENDIX B

Kinetic and Available Potential Energy Budgets of the Primitive Equations in Pressure Coordinates

Computation of the kinetic energy and sensible heat budgets per unit mass are straightforward for various geophysical flows. For the primitive equations in pressure coordinates, we get for the horizontal-mean budgets the following:
eb1
eb2
Here, KE = v2/2 denotes the horizontal kinetic energy and ω is the pressure velocity. The global horizontal mean is indicated by an overbar, and deviations from the mean are written as and for the geopotential and temperature, respectively. The symbol R denotes the (turbulent) momentum diffusion, and the diabatic heating Q includes all external (radiative, latent, sensible) contributions, as well as the frictional heating corresponding to R. The symbols have their usual meanings otherwise. The contribution of the horizontal gradient of the geopotential to the KE budget—that is, the adiabatic conversion—has been decomposed according to
eb3
after using the continuity equation. Noting that
eb4
and invoking the usual kinematic boundary conditions, ω = 0 for p = 0 and p00, where p00 is a reference surface pressure, it is readily shown that
eb5
Hence, any net generation of KE is due to extracting APE from the sensible heat reservoir through adiabatic expansion. Nevertheless, the divergence of the vertical geopotential flux on the right-hand side of (B1) may account for a significant vertical exchange of kinetic energy, which is pivotal for the energetics of gravity waves. An equivalent statement holds in the z vertical coordinate system concerning the vertical pressure flux (Shaw and Becker 2011).
As noted by Molemaker and McWilliams (2010), the definition of a local APE is not straightforward. However, the approach of Lorenz (1955) to define the APE for the entire atmosphere has led to an approximate formula that has proven to work quite well in various applications. We define the local APE corresponding to Lorenz (1955) as
eb6
To derive the corresponding budget in the global horizontal mean, we define the squared buoyancy frequency of the horizontal mean state as
eb7
multiply the thermodynamic equation underlying (B2) with , and apply the horizontal average. After a few manipulations, the budget of can be written as
eb8
Comparing (B8) to (B1) confirms that is exchanged with via the buoyancy production term.

According to (B8), APE is generated via the differential heating only if and Q are positively correlated. This result is fully consistent with the Lorenz energy cycle, since the overall generation of APE by differential heating must be associated with a reduction of the atmospheric entropy. This reduction in turn is balanced by the entropy production due to frictional heating, which generates unavailable potential energy.

A closed budget of is obtained only if . The reason is that (B6) is only an approximate representation of the true APE and is subject to the simplifying assumption (Lorenz 1955). Hence, within this limit APE is not exchanged vertically.

Since can be written as a power spectrum of the horizontally varying temperature—that is, the prognostic thermodynamic variable of a spectral GCM—the spectral budget can be evaluated analogously to that of KE by taking advantage of the spectral transform method. In the future, such an analysis may give further insight into the mesoscale energy transformations.

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1

Long-term simulations at coarser resolution showed a perfect balance in the spectral budget.

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