## 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 *n* ~ 100 to several thousand. The existence of the *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

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 *n _{T}* such as to determine the slope of the spectrum in favor of the

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

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

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

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

*ξ*and

_{mn}*δ*are the spectral amplitudes of horizontal vorticity

_{mn}*ξ*and horizontal divergence

*δ*, while

*n*is the total and

*m*is the zonal wavenumber. The spherical harmonics

*Y*are real and normalized. Furthermore,

_{nm}**∇**is the horizontal gradient operator in spherical geometry,

**e**

_{z}is the unit vector in the vertical direction, and

*a*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

_{e}*ξ*and

*δ*: that is,The spectrum of enstrophy times

*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 towhere

**F**=

**v**× (

*f*+

*ξ*)

**e**

_{z}and

*B*=

**v**

^{2}/2 in (8). Likewise,

*B*= 0 in (8). The corresponding tendencies due to adiabatic conversion (AC) and momentum diffusion (MD) are denoted as

*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 *ξ _{nm}* in (1) to compute the rotational horizontal wind

**v**

^{rot}. The spectral tendency due to the quasi-2D horizontal advection

**F**=

**v**

^{rot}× (

*f*+

*ξ*)

**e**

_{z}and

*B*= (

**v**

^{rot})

^{2}/2 in (8). These definitions can be applied also to (5) in order to compute the corresponding spectral enstrophy tendency

*n*in question and ending at the truncation wavenumber. We will thus refer toas 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

## 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

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

### b. Spectral energy tendencies and fluxes

Figure 2 presents the individual spectral tendencies as defined above and normalized by *K _{n}* 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).

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.

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 *μ*, which correspond to the corresponding spectral fluxes in the inertial regimes, give rise to separate spectral energies of approximately *μ*^{2/3}(*n*/*a _{e}*)

^{−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 (

*n*/

_{T}*a*)

_{e}^{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.

*u*denote the horizontal velocity at horizontal scale

_{ℓ}*ℓ*. The condition that the horizontal shear production is on the order of the dissipation yieldsFor convenience we use height

*z*as the vertical coordinate and assume the anelastic approximation. Furthermore, we denote the APE at scale

*ℓ*as

*denotes its horizontal variation at scale*

_{ℓ}*ℓ*, and

*ℓ*

_{z}is the vertical scale corresponding to

*ℓ*. Hence, the spectral flux of APE scales likeAlternatively we may estimate the spectral flux due to the adiabatic generation of KE at scale

*ℓ*. For this purpose we assume that the pressure perturbation at scale

*ℓ*is hydrostatic,

*p*

_{ℓ}~

*g*

*ℓ*

_{z}

*ρ*

_{ℓ}, and that the corresponding density perturbation is anelastic,

*u*

_{ℓ}in (13) or (14) with the help of (12) yields the well-known scale-dependent aspect ratio for ST, that is,It approaches unity at the transition to Kolmogorov turbulence, also known as the Ozmidov scale,

*ℓ*= 2

*πa*/

*n*. Furthermore, we compute

*u*

_{ℓ}from the energy spectrum asThe 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.

Comparing the L100 to the L30 run indicates higher values of Fr_{h} 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

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

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 *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.

The question remains as to what determines the mesoscale spectral *n* = 200 (or horizontal wavelengths around 210 km). Figure 7b shows corresponding estimates of *ℓ*_{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 *ℓ*_{z} ≈ 80 m for the L100 run and *ℓ*_{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.

### 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.

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

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

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

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

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

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.

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

*p*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 bywhere

*l*

^{2}= 1.4 × 10

^{7}m

^{2}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 × 10

^{6}m

^{2}. The strain tensor is defined aswith the transposed tensor being indicated by the exponent T and the unit vector in vertical direction defined as

**e**

_{z}. The idemfactor is denoted by the symbol

**I**, |

**S**|

^{2}is the squared Frobenius norm of

**S**, and

*S*

_{0}= 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 toThis scaling is relevant when resolved gravity waves become dynamically unstable. To account for the usual linear stability criterion, the offset Ri

_{0}in (A4) is increased from zero in the boundary layer to a value of 0.25 from about 600 hPa on.

*K*

_{0}(see Fig. A1b). The hyperdiffusion is given in stress-tensor formulation according towhere

**v**

_{f}is a filtered horizontal wind defined as [cf. to (1)]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.

# APPENDIX B

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

**v**

^{2}/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

**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 toafter using the continuity equation. Noting thatand invoking the usual kinematic boundary conditions,

*ω*= 0 for

*p*= 0 and

*p*

_{00}, where

*p*

_{00}is a reference surface pressure, it is readily shown thatHence, 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).

According to (B8), APE is generated via the differential heating only if *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

Since

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^{1}

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