## Abstract

Understanding the development of the atmospheric energy spectrum across scales is necessary to elucidate atmospheric predictability. In this manuscript, the authors investigate energy transfer between the synoptic scale and the mesoscale using direct numerical simulations (DNSs) of two-dimensional (2D) turbulence transfer under forcing applied at different scales. First, DNS results forced by a single kinetic energy source at large scales show that the energy spectra slopes of the direct enstrophy cascade are steeper than the theoretically predicted −3 slope. Second, the presence of two inertial ranges in 2D turbulence at intermediate scales is investigated by introducing a second energy source in the meso-*α*-scale range. The energy spectra for the DNS with two kinetic energy sources exhibit flatter slopes that are closer to −3, consistent with the observed kinetic energy spectra of horizontal winds in the atmosphere at synoptic scales. Further, the results are independent of model resolution and scale separation between the two energy sources, with a robust transition region between the lower synoptic and the upper meso-*α* scales in agreement with classical observations in the upper troposphere. These results suggest the existence of a mesoscale feedback on synoptic-scale predictability that emerges from the concurrence of the direct (downscale) enstrophy transfer in the synoptic scales and the inverse (upscale) kinetic energy transfer from the mesoscale to the synoptic scale in the troposphere.

## 1. Introduction

Atmospheric flows at synoptic scales are often assumed to show a two-dimensional (2D) behavior because of the fact that horizontal scales are considerably larger than the vertical scales (Charney 1971), besides limitations of the analogy between 2D turbulence and quasigeostrophic turbulence in Earth’s rotating atmosphere (Tung and Orlando 2003a,b). Reducing the flow dimensions from three to two (horizontal) requires conservation of the total enstrophy of the flow, a constraint that was first introduced for predicting the behavior of 2D turbulence by Kraichnan (1967). The theoretical kinetic energy (KE) spectrum of a 2D flow regime displays two inertial ranges: an inverse energy flux associated with a −5/3 slope and a direct enstrophy cascade associated with a −3 slope with a logarithmic correction to account for nonlocal transfers (Kraichnan 1971).

The behavior of 2D turbulence in the inertial range has been studied extensively using numerical and experimental models (e.g., Boffetta and Ecke 2012). In particular, scaling analysis of numerical simulations of 2D turbulence in the inertial range using a single energy source (e.g., forcing) fixed at a specific scale showed that turbulent transfer mechanisms depend strongly on the spectral separation between the scales of the energy source, the scale of friction at large scales, and the viscous dissipation scale (Boffetta and Musacchio 2010). A three-dimensional (3D) context is necessary for a realistic approximation of the atmospheric 2D flow independent of the direction of energy transfer. Prior studies of nonrotating stratified turbulence support the hypothesis that mesoscale turbulence transfers result from a forward anisotropic 3D energy cascade (Lindborg 2006). Rotation does not influence the forward cascade significantly if it is weak (Lindborg 2005), but in the case of strong rotation, energy piles up at the larger scales, a behavior linked to the transition to 2D turbulence (Biferale et al. 2012; Marino et al. 2013). Thus, the two inertial-scale ranges are connected and may interact (Fig. 1), and a strict separation between synoptic and mesoscales would be an unphysical constraint. A universally accepted comprehensive explanation of the physics underlying the horizontal spectrum of atmospheric KE and in particular the observed scaling behavior is still lacking.

The horizontal KE spectra of the atmosphere estimated using aircraft measurements and global radiosonde data from different field experiments (Nastrom and Gage 1985; Cho et al. 1999; Frehlich and Sharman 2010) exhibit a −3 slope in the synoptic range of scales (800–2000 km) and a −5/3 slope in the mesoscales (2–600 km). The ability to reproduce this scaling behavior is a key requirement in the evaluation of weather and climate models (e.g., Skamarock 2004; Hamilton et al. 2008; Evans et al. 2013; Skamarock et al. 2014; Nogueira and Barros 2014). The synoptic-scale flow approaches 2D behavior at large scales, and thus, the −3 slope of the observed KE spectra is consistent with the direct enstrophy cascade in the inertial range of 2D turbulence (Cho and Lindborg 2001; Tung and Orlando 2003a; Vallgren et al. 2011).

There is, however, some disagreement in the literature (e.g., Xia et al. 2011; Vallgren et al. 2011; Waite and Snyder 2009) over the direction of net energy transfer in the mesoscales where the slope of the observed spectra is −5/3. Waite and Snyder (2009) showed that model simulations of dry mesoscale dynamics in the absence of topography produce −5/3 spectra associated with a net forward energy cascade. However, this result does not rule out the coexistence of an inverse energy flux produced by moist convection as in realistic numerical weather simulations (Nogueira and Barros 2014; Sun et al. 2017). The hypothesis, first proposed by Lilly (1989), that a source of energy in the mesoscales is required to develop and maintain the inverse energy transfer flux was explored by Cencini et al. (2011) using a single direct numerical simulation (DNS) of 2D turbulence with modified viscosity (hyperviscosity) and two energy sources with a large-scale gap (wavenumber ratio of 7/240). Whereas the specific range of scales, the scale gap, and the transition scale range of the spectrum in Cencini et al. (2011) are different from the observed atmospheric spectra, they showed that an inertial range developed between the two energy sources with coexisting and overlapping direct enstrophy and inverse energy fluxes dominating the KE budget in vicinity of the energy sources at the large (−3 spectral slope) and small scales (−5/3 spectral slope), respectively.

The objective of this study is to investigate the behavior of energy transfer in 2D turbulence between the meso-*α* (~200–600 km) and synoptic (~1000–4000 km) scales toward elucidating the dependence of energy transfer mechanisms on the spectral separation of the two energy sources vis-à-vis the observed scaling behavior in the atmosphere (e.g., Nastrom and Gage 1985). First, the impact of the relative scale of forcing (e.g., location of a single energy source; DNS series S in Table 1) on turbulence transfer is demonstrated. Second, the hypothesis that mesoscale inverse energy transfer significantly impacts enstrophy transfer in the synoptic range is tested using simulations with two energy inputs (DNS series D in Table 2) and for different model resolutions (DNS series H in Table 3). Finally, the adaptive behavior of the enstrophy and inverse energy cascades in the inertial range to scale separation between the two energy inputs is examined (DNS series G in Table 4).

## 2. DNS experiments

The numerical experiments are based on DNSs of the incompressible Navier–Stokes equations:

where *u* is the velocity field, *x* is the space variable, *t* is time, *v* is kinematic viscosity, is the forcing term, *l* and *j* are tensor indices assuming values of 1 and 2 for 2D flow, and finally, the superscript *n* differentiates the large- and small-scale forcing terms taking values of 1 and 2, respectively. The equations are solved in nondimensional form, and a detailed description of the solver can be found in Bhushan and Warsi (2007). Specifically, the DNSs are conducted using a pseudospectral solver with a 3/2 dealiasing rule in Fourier space on a double periodic domain using a grid defined by collocation points. The numerical scheme consists of three steps. First, a second-order Adams–Bashforth scheme is implemented to solve the convective terms with *δt* time interval. Next, the pressure correction is applied to the velocity field. The third step incorporates the dissipation terms. Defining the Fourier transform of the velocity field as , where *k* is the wavenumber and the caret indicates the Fourier component, Eqs. (1a) and (1b) reduce to

where *m* = 1, 2 is an index variable, the asterisk represents convolution over all wavenumbers, and is the Kronecker delta (Pope 2000). The forcing term is defined as

where is a constant nominal energy input rate; and are, respectively, the forcing wavenumber and the radial width of the source in wavenumber space; and is the velocity magnitude. Equation (3) defines the energy input (forcing) and the range of wavenumbers over which it acts in Eq. (2). Thus, the effective energy input is . As the square of the wavenumber appears in the dissipation term in Eq. (2), the effective dissipation is a function of both kinematic viscosity and the number of collocation points *N*_{c} used in the numerical simulation. Following a simple heuristic analysis (Kraichnan 1967; Borue 1993; Eyink 1996; Boffetta and Musacchio 2010), the inverse energy flux and enstrophy cascade directions can be qualitatively determined from

where and are, respectively, the energy and enstrophy transfer rates to small scales; likewise, and are the energy and enstrophy transfer rates to the larger scales; and represent, respectively, the characteristic smallest and the largest scale of the system; and finally, is the forcing scale. In a dissipative system, and can be associated with small- and large-scale dissipation characteristic lengths, respectively, ( and ) as in Boffetta and Musacchio (2010). Although it is not possible to determine exactly how and are dependent on model physical parameters (such as viscosity), the dominant direction of cascades can be qualitatively deduced in the limits of the equations. The length-scale ratio is constant if and are fixed; however, and increase as the forcing scale changes from the larger to the smaller scales. That is, first, is very small and energy accumulates at the larger scales, and then is very small and all enstrophy goes to the dissipation range.

A total of 21 DNSs were conducted to test single- versus dual-energy-source behavior and scale separation as a function of effective energy input and grid resolution (viscosity). The specific DNS configurations and the scaling analysis results for each case are presented in Tables 1–4. In the S series (Table 1), the energy input location shifts from the larger scales to smaller scales and the effective energy input is constant. In the D series (Table 2), the large-scale energy input is fixed at with the secondary energy source at , and the effective energy input ratio decreases as the second energy source is reduced from D1 through D6 by reducing the nominal energy input ratio . Four simulations referred to as the H series, in all respects similar to S2 but with increased grid resolution and lower viscosity, were conducted to assess the sensitivity of the DNSs to grid resolution and viscosity as summarized in Table 3. A final set of five simulations, the G series in Table 4, was conducted to examine the sensitivity of the KE spectrum scaling behavior in the inertial range to changes in the scale separation.

The DNS experiments were designed such that the energy input at synoptic scale (primary energy source) corresponds to a constant turbulence intensity for both the S and D series. The turbulence intensity of the second energy source was reduced for the D series with varying from 1/2 to 16. Note that the specific effective energy inputs for the dual-source DNSs per se are not relevant, just their relative magnitude. For the G series, varied from 2 to 32, and the injections that predict similar energy spectra as D4 and D6 are used for discussions.

In dimensional terms, considering that the domain length scale is *L*_{0} = 12 732 km, the smallest grid spacing when *N*_{c} = 2048 is 6 km and for *N*_{c} = 7680 is 1.66 km. The energy injection scales in the *S* series vary from 4000 to 125 km, thus spanning the full range of synoptic and meso-*α* scales. For the D series, the primary energy injection is in the synoptic range at 2000 km, and the second injection is in the upper mesoscale range at 500 km (see Fig. 1). For the G series, the primary energy injection remains in the synoptic range at 2000 km, but the secondary injection is shifted successively toward the lower meso-*α* and meso-*β* scales, respectively, at 250 and 125 km . The simulation time step is for the 2048^{2} grid and 1 for the 7680^{2} grid. Assuming *U*_{0} = 10 m s^{−1}, the temporal resolutions of the simulations are 0.07 and 0.035 h, respectively.

## 3. DNS results

Snapshots of selected S2 and D5 fields at the end of the DNSs are shown in Fig. 2. Turbulence develops faster in D5 in the presence of a second energy source at small scales. Turbulence also develops faster for cases with the single energy source placed at the smaller scales for the single-source S series (not shown).

The temporal evolution of S2 spectra is shown in Fig. 3a. Despite the absence of large-scale friction, the spectral slopes do not change with time, suggesting 2D turbulence is well developed after *T* = 6 for the enstrophy cascade inertial range , which is in agreement with previous results (Vallgren and Lindborg 2011). Energy piles up at large scales as expected, but this does not affect the spectra to the right of the energy injection. Evidence of both energy and enstrophy transfer is present in the KE spectra of S simulations at scales larger and smaller than the scale of the forcing, respectively. The spectral slope in the enstrophy cascade region is, however, much steeper than the observed and theoretical values of −3 for all single-energy-source simulations (Table 1), a behavior that is consistent with the literature for similar simulations (e.g., Lindborg and Alvelius 2000; Vallgren and Lindborg 2011). Results from DNSs with resolutions as high as 16 384^{2} (Boffetta 2007) and later 32 768^{2} (Boffetta and Musacchio 2010) show that the spectral slope right below the source of energy in the enstrophy cascade range is steeper (about −3.35) than the theoretically predicted value (−3) and might be even steeper in the presence of large-scale drag, as suggested by Vallgren and Lindborg (2011). Nevertheless, the enstrophy cascade always forms for higher wavenumbers to the right of the source of energy and reaches quasi-stationary state.

The spectral slopes for the dual-energy-source DNSs (D series) are flatter than S2 (Fig. 3b; Table 2), gradually steepening from slopes less than −5/3 for D1 to values closer to −3 for D5. Overall, the scaling analysis suggests that the energy transfer mechanism in the synoptic scales changes from approaching the behavior of a kinetic energy cascade (e.g., D1) to approaching the behavior of an enstrophy cascade (e.g., D5) as the magnitude of the second energy input decreases relative to the primary source (i.e., as increases). A key implication of these results is that, albeit of smaller magnitude, the second energy source in the upper meso-*α*-scale range is necessary to develop the enstrophy cascade with a −3 spectral slope in the synoptic scales. That is, the second energy source functions as a feedback mechanism between the meso- and synoptic scales.

Figure 4 shows the temporal evolution of the kinetic energy for the S and D simulations. Again, it is apparent that turbulence develops faster when energy input is in the higher wavenumbers, consistent with the results for all D simulations. The final energy in the inertial range is always higher for the D series compared to S2, as expected; however, in the cases of D1, D2, and D3 , more energy is going to the synoptic scales where it lives longer. In the cases of D4, D5, and D6 , which show a scaling behavior closer to the observations in the atmosphere (a −3 slope), the energy is about the same as in S2, which exhibits a much steeper spectral slope. The energy transfer in the synoptic scales is therefore modified significantly by the presence of a feedback from the mesoscales as small as 12.5% (e.g., D5) of the magnitude of the synoptic energy source.

The results only depend on the boundary conditions imposed on the system, that is, the intensity and the scale of the forcing , and do not exhibit sensitivity to increased grid resolution or the kinematic viscosity. This is illustrated by Fig. 5, which shows the energy spectra for simulations H1, H2, H3, and H4 (Table 3) compared against S2 at simulation time *T* = 12, focusing on the enstrophy inertial range between and as per Fig. 3a. The spectra do not change with time after *T* = 8.0, and all spectra show the same slope of −3.8 as for the S2 case in the synoptic range. Thus, increasing the grid resolution and/or viscosity also does not affect the large scales. This is expected since information travels downscale in the enstrophy cascade, and the small scales outside the inertial range should not influence the large scales in stable numerical experiments.

The enstrophy and energy transfers attributed to the nonlinear terms in Eq. (1b) are shown in Fig. 6 for series S and D (see also Boffetta and Musacchio 2010; Pope 2000). The localized jumps that are especially evident in the energy transfer plot (bottom panel) correspond to the position of energy inputs. Detailed examination of S2 and S4 in the light of Eqs. (4a) and (4b) provides insight into the energy transfer direction. In the case of S2, with large-scale separation between the forcing and viscous dissipation scales, the energy input is mostly transferred to larger scales rather than cascading down to the smaller scales. The enstrophy cascade for S4 is significantly larger than for S2. For cases with two energy sources, turbulent transfer in the intermediate wavenumber range depends on the scale separation between the inputs and on the ratio of the first and second nominal energy sources. If the second energy injection is strong (e.g., D2), the inverse energy cascade dominates between the input wavenumbers as long as , and the spectral slope for these cases is closer to −5/3, as expected in 2D turbulence left of the injection. If the second energy injection is weak (e.g., D5), the enstrophy cascade due to the first energy source is dominant, and the spectral slope is closer to −3, consistent with synoptic-scale observations in the upper troposphere (Nastrom and Gage 1985). An important finding is that this transition takes place with a small amount of feedback from the mesoscale (e.g., for D5 with −3 slope) compared to the magnitude of the primary (large scale) energy input, which can be attained by processes such as frontogenesis or atmospheric convective activity in the real atmosphere.

Figure 7a shows the results from the G series simulations (Table 4) along with D4 and D6 as reference to compare simulations with different separation scales between the energy inputs (the scale of the primary energy input is fixed). Note that the scale separation for G3, G4, and G5 is too large with the second energy source applied in the meso-*β*-scale range where vertical motions are significant in the atmosphere, and thus, these specific DNSs are not representative of realistic behavior, and they are used here for sensitivity purposes only. When the scale separation changes, the relative fraction of the energy input that turns to direct enstrophy flux and inverse energy cascade [Eqs. (4a) and (4b)] also changes. This change is very small for the cases of G1 and G2, but a higher amount of energy input is needed for the cases of G3 and G4 to support similar inverse energy flux rate as G1 and G2. This is illustrated in Fig. 7b, which shows the energy transfer for D4, G1, and G3. The results suggest that there is a preferred scale separation range to capture the scaling behavior of upper-tropospheric observations including the transition between −3 and −5/3 spectral slopes [shaded blue in Fig. 7a; see also Fig. 1 in Nastrom and Gage (1985)], which corresponds to the feedback between the meso-*α* and synoptic scales.

## 4. Conclusions

Previously, modeling studies of atmospheric turbulence have focused on either synoptic or mesoscale ranges, neglecting scale interactions between them. Here, it is demonstrated that a second energy injection is necessary to explain the −3 spectral slope between the synoptic scale and the mesoscale (see Fig. 1). The second energy input in the D series simulations is interpreted as a feedback mechanism from the meso-*α* scale to the synoptic scale, and the results suggest that even a small feedback is sufficient to support two-way transfers. Whereas assessing conclusively the implications of this feedback for the direction of net energy transfer under realistic atmospheric forcing is out of the scope of this work, these findings have important implications for explaining the observed energy spectra of atmosphere and to elucidate the impact of this mesoscale feedback on synoptic-scale predictability (Lorenz 1969; Berner et al. 2017), including the treatment of conservation laws across scales in weather and climate models (e.g., Thuburn 2008). The results presented here should be relevant to 2D geophysical turbulence with coexisting forward and inverse transfer such as the case of ocean circulation (e.g., Marino et al. 2015; McWilliams 2017), although the forcing and fluid properties and specific range scales of interest may be different. Future studies of coupled 2D–3D interactions are necessary to elucidate energy transfer mechanisms and scale interactions at the mesoscale.

## Acknowledgments

The simulations were made possible by the use of Mississippi State University High Performance Computing (HPC) facility. The authors thank three anonymous reviewers for insightful comments. The research was supported by the Pratt School of Engineering.

## REFERENCES

*Turbulent Flows.*Cambridge University Press, 771 pp.

^{−3}and k

^{−5/3}energy spectrum of atmospheric turbulence: Quasigeostrophic two-level model simulation

## Footnotes

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