Impact of Convective Parameterizations on Atmospheric Mesoscale Kinetic Energy Spectra in Global High‐Resolution Simulations

The responses of atmospheric kinetic energy (KE) spectra to three convective parameterizations (CPs) in global high‐resolution simulations are revealed. The results show that the KE spectra exhibit high sensitivity to the CPs, mainly at mesoscales in the middle and upper troposphere. The New Tiedtke scheme produces the steepest mesoscale slope, followed by the Kain‐Fritsch scheme and then the Grell‐Freitas scheme. In general, there is a compensating relationship between latent heat released by the CP and microphysics parameterization (MP). The less latent heat released by the CP is compensated by the more latent heat released by the MP. The shallowest mesoscale spectra for the Grell‐Freitas scheme are related to the strongest downscale cascade dominated by the rotational component of the flow, and this is attributed to more latent heat released from MP enhancing the intensity of vorticity in the troposphere and producing more gravity wave activities in the lower stratosphere.

• The kinetic energy (KE) spectra exhibit high sensitivity to the convective parameterizations (CPs) mainly at mesoscales • The more latent heat released from CP, the steeper KE spectra at mesoscales • The shallowest mesoscale KE spectra generated by the Grell-Freitas scheme are attributed to the strongest RKE downscale cascades

Supporting Information:
Supporting Information may be found in the online version of this article.
The important influence of latent heat release on the mesoscale KE spectrum makes the spectrum an insightful tool to investigate systematic differences in model physics schemes.Although the growing of computer power motivates a trend toward models with explicit convection or based on machine learning, the convective parameterization (CP) remains indispensable for global weather forecast and climate models (Rio et al., 2019).
The current operational weather and climate model configurations still do not entirely resolve the meso-and small-scale convections and thus the forecast results largely depend on the CP.CP can drastically regulate the mean climate and tropical transient activity of a general circulation model (Li et al., 2022).Many studies have focused on the sensitivity of the simulations to different CP schemes (e.g., Fowler et al., 2020;Li et al., 2022).For example, the uncertainties in parameterizing moist convection and its impact on grid-scale clouds is considered as a major part of the differences in the hydrometeors simulated between the general circulation models (GCMs) of Phase 3 and 5 of the Coupled Model Intercomparison Project (Fowler et al., 2020).Some studies have compared the explicit simulation of deep moist convection with implicit simulation where deep convection is parametrized (Polichtchouk et al., 2022;Wedi et al., 2020).The parameterization drastically reduces gravity wave momentum flux associated with tropical convection, as deep modes of latent heating and gravity waves are underestimated (Müller et al., 2018).However, few studies have been devoted to examining the differences in detail from the perspective of KE spectra.Furthermore, the direct comparisons between simulations and observations cannot reveal the multi-scale energy cycle processes involved.Thus, there is a need for the modeling community to tackle the following question: what are the dynamical responses of KE spectra to different convective parameterizations (CPs)?Answering this question will provide physical insights into the behavior of a given CP.
This paper focuses on the atmospheric KE spectrum in global high-resolution simulations.Here, the global high-resolution simulations are implemented with the Model for Prediction Across Scales (MPAS)-Atmosphere (MPAS-A; Skamarock et al., 2012), which has been used to study the vertical resolution requirements in atmospheric simulation (Skamarock et al., 2019) and to understand the atmospheric predictability of the different latitude bands (Judt, 2020).The purpose of this study is to identify major differences in the energy transfer and conversion processes at resolved scales between three widely used deep CP schemes in the MPAS Model.In present global simulations, we focus on the upper troposphere and lower stratosphere, as this range is less affected by topography and the main area for the presence of convectively generated gravity waves (CGGWs).

Model Configurations and Observations
The global model used in this study is MPAS-A version 7.3 from National Center for Atmospheric Research (Duda et al., 2023;Skamarock et&nbsp;al., 2012).We employ a horizontal grid spacing of 15 km that is a bit larger than that used by present-day global operational numerical weather prediction models.This configuration resolves some of the mesoscale regime in the atmosphere, but convection and gravity waves are still parameterized.The MPAS-A model top is set at 30 km (∼12 hPa) composed of 55 vertical levels with a vertical spacing of 47 m for the lowest levels stretching to 927 m near the top.As a result, the vertical spacing is about 449 ∼ 873 m from 5 to 15 km, roughly corresponding to the upper troposphere and lower stratosphere at midlatitudes.A gravity wave absorbing layer (Klemp et al., 2008) is used in the top 8 km of the simulated atmosphere.The dynamics and physics time steps are both set to 90 s, and the horizontal diffusion length scale equal to the grid size.A fourth-order horizontal filter is employed in these simulations (Skamarock et al., 2014).
The CP schemes of MPAS-A include the New Tiedtke (NT) scheme (Zhang & Wang, 2017), Grell-Freitas (GF) scheme (Fowler et al., 2016) and Kain-Fritsch (KF) scheme (Kain, 2004).Other physics schemes include the WSM6 cloud microphysics scheme (Hong & Lim, 2006), the Noah land surface scheme (Tewari et al., 2004), the Yonsei University planetary boundary layer scheme (Hong et al., 2006), the Monin-Obukhov surface layer (Janji, 2001), and the Rapid Radiative Transfer Model for GCMs scheme for long-and short-wave radiation (Iacono et al., 2008).The namelist file with the settings required to change the CP scheme is provided in Text S1 in Supporting Information S1.All simulations are initialized at 0000 Universal Time Coordinated (UTC) 10 July 2021 from the European Centre for Medium-Range Weather Forecasts high-resolution reanalysis (ERA5; Hersbach et al., 2023aHersbach et al., , 2023b)).Eleven day simulations are produced in all the experiments with output every 6 hr.The fields in these simulations are thought to spin up within 1 day.
The data from satellite FengYun-4A (FY-4A) are used to evaluate the authenticity of MPAS simulations.FY-4 A is a geostationary meteorological satellite, owned and operated by the China Meteorological Administration.
Outgoing longwave radiation (OLR) is the L2 product with a horizontal resolution of 4 km (Xian et al., 2021).

Kinetic Energy Spectra
Prior to KE spectra calculations with model outputs, the velocities and geopotential are interpolated to pressure coordinates vertically and uniform longitude-latitude meshes (0.25° × 0.25° grid) horizontally.The average spectra herein are derived over the 10 day period from 0000 UTC 11 July to 1800 UTC 20 July 2021.The global horizontal KE (HKE) spectra at a given pressure level p are calculated from the global zonal and meridional wind fields based on the spherical harmonic function expansion as follows: where a is the Earth's radius, (m, n) are the zonal and total wavenumbers, and (ζnm, δnm) are the spherical harmonic coefficients of vertical vorticity and horizontal divergence, respectively (e.g., Augier & Lindborg, 2013;Koshyk & Hamilton, 2001;Li, Peng, Zhang, et al., 2023;Malardel & Wedi, 2016).Accordingly, the wavenumber spectra of the rotational KE (RKE) and divergent KE (DKE) are defined as follows (e.g., Augier & Lindborg, 2013;Li, Peng, Zhang, et al., 2023): (2) The pressure vertical velocity (PVV) (ω; Pa/s) is converted from vertical velocity (m/s) based on hydrostatic balance equation.The wavenumber spectrum of PVV is calculated as follows (e.g., Lambert, 1984;Polichtchouk et al., 2022): The vertically integrated spectra between two pressure levels p b and p t are computed as follows: which unit is J/m 2 (e.g., Augier & Lindborg, 2013;Malardel & Wedi, 2016).

Spectral Budget Formulation
The spectral RKE and DKE budget formulation is written as (Li, Peng, Zhang, et al., 2023): where The detailed expressions of spectral transfer terms and spectral conversion term from DKE to RKE are as follows: where u denotes horizontal velocity; u R and u D denote rotational and divergent components of horizontal velocity, respectively.e z is the vertical (upward) unit vector, f is the Coriolis parameter.The subitems representing contribution from relative vorticity are marked with an underline.The vertically integrated spectral fluxes and accumulated spectral conversion can be written as follows: The calculation is divided into three steps: Step 1, the spectral transfers summed over all zonal wavenumbers m at a given total wavenumber l as in Equation 1; Step 2, vertically integrated between two pressure levels p b and p t , and divided by g as in Equation 4; Step 3, summed over all total wavenumbers greater than or equal to l.In this way, the HKE spectral flux is exactly equal to the sum of RKE and DKE ones, and these spectral fluxes are exactly conserved, that is, Π K,R,D [0] = 0.It should be highlighted that, as the most important improvement on previous formulations, constructing conserved spectral fluxes of RKE and DKE is mathematically cumbersome, which was also the initial motivation for Peng et al. (2023) switching to the spectral budget of squared vorticity and divergence.For more details, please refer to Li, Peng, Zhang, et al. (2023).
Furthermore, let ΔΠ D denote the net amount of DKE going into the wavenumber range [n 1 , n 2 ] by spectral transfer, that is,

Simulation Results
Figures 1a-1d show that all three simulations capture many observed cloud features, such as the powerful extratropical cyclone in the south hemisphere and convective cloud systems in the western Pacific and Indian Oceans.
As expected, the impact of CP is especially apparent in the tropics.For example, the cloud distribution appears too extensive over South China Sea and western Pacific Ocean.The NT scheme simulation is closest to the satellite product.The convection simulated by the KF scheme is generally weak, with the largest deviation from the observation.The GF scheme simulation shows some higher clouds than the observation in the tropics with lower OLR (Figure 1d).Based on the above brief qualitative analysis, we think the 15 km MPAS simulated the atmosphere realistically enough for our research purpose.
The diabatic heating from CPs is mainly located in the troposphere (Figures 1e-1g).In the upper troposphere, the diabatic heating of the three schemes is close.However, it varies greatly in the middle and lower tropo-sphere among these three simulations.Among these three simulations, the NT scheme generates the strongest diabatic heating from CP, with two maximus at 550 and 900 hPa (Figure 1e).On the contrary, the diabatic heating from the microphysics parameterization (MP) in the GF simulation is largest in the upper and lower troposphere (Figure 1f).Although partitions of diabatic heating from CP and MP differ significantly among the three simulations, the sum of the two is close.Note that the sum in the GF simulation is still largest between 200 and 400 hPa (Figure 1g).This suggests that there exists a compensating effect between CP and MP and thus different CP schemes not only produce different diabatic heating by themselves, but also can affect the diabatic heating from the MP.This is consistent with the fact that CP and MP in the model can compete for air moisture via grid-scale fields (e.g., Lin et al., 2016;Liu et al., 2018;Mishra & Srinivasan, 2010;Yang et al., 2013).The intrinsic physical processes of the differences among the three CP schemes can be described as follows: If a CP scheme removes too little instability while still allowing for grid-scale upward motion, the MP scheme will respond to the remaining instability, resulting in an overconcentration of latent heating at low levels over the whole grid box.This can lead to excessive low-level cyclogenesis and surface-pressure values that are much too low.Furthermore, the MP scheme's excessive low-level heating might cause a dynamical reaction that feeds back on itself (Kuo et al., 1996;Wang & Seaman, 1997).Low-level winds respond to reduced pressure by increasing moisture convergence and vertical motion, resulting in higher latent heat release (Figure 1g) and enhanced vortex motion.To further verify this hypothesis, we calculate the root mean square (RMS) of the vertical vorticity defined as follows: It is shown that the RMS values in the troposphere computed by setting n 1 = 1 and n 2 = 720 presents a relationship of GF > KF > NT, for example, 3.26 × 10 −6 s −1 for NT, 3.49 × 10 −6 s −1 for KF, and 3.79 × 10 −6 s −1 for GF at 850 hPa.

Kinetic Energy Spectra
Figure 2 shows the HKE, RKE and DKE spectra in these three simulations as a function of total wavenumber n and wavelength (∼2πa/n) at 300, 200, 100, and 50 hPa.Up to total wavenumber 40 there is a remarkable agreement in the HKE spectra among the simulations (Figure 2a).The GF obviously produces more energy than KF and NT at wavenumbers larger than 40 at these levels and this feature is also shown in the distribution of the DKE, consistent with the performance in the latent heating in the upper troposphere (Figure 1g), while the KF converge with NT at higher altitudes.The spectral slopes of the three simulations are close to −3 at 300 hPa, and, at wavenumbers more than 100, steeper than that obtained from a 15 km MPAS simulation with CP at 8.5-10.5 km (near 300 hPa) in Skamarock et al. (2014).All the simulations present the spectral transition characteristic around 600 km at 200 hPa, although all the mesoscale spectral slopes are steeper than −5/3.We also found that the 200 hPa HKE spectra are steeper than that simulated with 3 km global MPAS in Skamarock et al. (2014).This should be mainly attributed to the much lower horizontal resolution employed here, consistent with the dependence of spectral slope on horizontal resolution shown in their Figure 9.In addition, the somewhat higher vertical resolution here may also prefer to produce a steeper spectral slope, because decreasing vertical resolution may lead to insufficient vertical mixing, resulting in a false increasing mesoscale KE (e.g., Brune & Becker, 2013;Skamarock et al., 2019).
As the height increases, the transition scales move toward larger scales and the spectral slope of the shallow curve is close to or even larger than −5/3.This behavior is consistent with the spectra computed from observations (Lindborg, 1999;Nastrom & Gage, 1985) and the full-physics atmospheric simulation with the same model MPAS by Skamarock et al. (2019), where a less-than full transition is evident in the tropospheric spectrum at z = 10 and 16 km.It is also consistent with results from studies using other models (e.g., Hamilton et al., 2008), where the transitions to a shallower mesoscale spectrum occur at horizontal wavelength of scales less than 1,000 km in the troposphere and at progressively longer horizontal wavelengths with increasing height.The contribution RKE and DKE for the HKE difference is related to both the altitude and the scale.At large scales, the HKE spectra basically coincide and are dominated by RKE at all levels.At mesoscales, the HKE difference is dominated by the RKE at 200 hPa (Figure 2b) while by the DKE at 50 and 100 hPa (Figure 2c).
Figure 2d depicts the PVV spectra for these simulations.All the spectra show very similar peak values around total wavenumber 10.However, only GF simulation presents a second peak at near 8 times the mesh spacing.The GF scheme simulation has most energy at synoptic scales and mesoscales, and the KF and NT scheme simulations nearly converge at 100 and 50 hPa.This is consistent with the differences of KE spectra mentioned above, but more obvious.
Figure 3 shows in detail the relative differences of the GF and KF schemes to the NT scheme in different KE modes at isobaric levels, since spectra in the NT scheme generally have the smallest energy.Obviously, the relative differences between the GF scheme and the NT scheme are greater than those between the KF scheme and the NT scheme, which is consistent with the results in Figure 2. Remarkably, these relative differences mainly appear at mesoscales and increase as the wavenumber increases, with the most significant being in the middle and upper troposphere.In this layer, the relative difference in the HKE spectra is dominated by the RKE for both the GF scheme and KF scheme (Figures 3b and 3f), while in the lower stratosphere, there exists another obvious positive deviation dominated by the DKE appear only for the GF scheme (Figure 3g).
Next, we will pay more attention to the relative differences in the PVV between the NT scheme and GF scheme (Figure 3h).They run throughout the troposphere and the lower stratosphere with two peaks near 600 and 200 hPa (i.e., corresponding to the lower troposphere and upper troposphere, respectively).The stronger grid-scale vertical motion in the simulation with GF should be attributed to its larger diabatic heating from the MP (Figure 3f).In the lower stratosphere, the HKE difference is dominated by the DKE, and the altitude and scale of the difference in DKE is consistent with those of the difference in PVV, suggesting that it may be mainly caused by the upward inertial gravity waves, which are excited by diabatic heating and manifested as the strong vertical movement.In the upper troposphere, although the relative difference in vertical motion is more significant, the difference in DKE is not as significant as in RKE (Figures 3f and 3g).The relative difference in the HKE and RKE spectra reaches a maximum at about 200 hPa (Figures 3e and 3f).To further analyze this issue, we conducted spectral budget analysis on the RKE and DKE in the range of 150-400 hPa.

Spectral Energy Budget
To follow the relative differences in KE spectra (Figures 3a-3c), Figure 4 shows the vertically integrated HKE, RKE and DKE spectra and spectral fluxes vertically integrated from 400 to 150 hPa in the three simulations.The results show consistency at wavenumbers less than 40 and inconsistency at larger wavenumbers among the three simulations.At mesoscales, the mesoscale energy of GF simulation is greater than that of KF simulation, which is greater than that of NT simulation.The RKE spectra dominate the HKE spectra, while the DKE spectra differ more obviously at synoptic scales and mesoscales.
Comparing the corresponding spectral fluxes can give more insights into the influences of the convection scheme on the atmospheric energy cycle, since it illustrates the energy transfer processes at different scales.How to understand the spectral flux curves has be explained in detail in the previous studies (e.g., Augier & Lindborg, 2013;Li, Peng, Zhang, et al., 2023;Malardel & Wedi, 2016;Wedi et al., 2020).If the slope of a Π[n] curve is negative at wave number n, energy is deposited at this wavenumber; otherwise, energy is removed from this wavenumber.The energy cascade is downscale for Π[n] > 0 and upscale for Π[n] < 0. A plateau of the Π[n] curve corresponds to an energy transfer through the corresponding scales without deposition or loss of energy at n.
Although the spectral flux curves are very similar as a whole (Figures 4d-4f), they exhibit obvious differences at wavenumbers larger than 20 (insets in Figures 4d-4f) among these three simulations.They sensitively depend on the CP, which is consistent with previous finding (Malardel & Wedi, 2016).Compared to NT and KF, the GF simulation shows the strongest HKE downscale flux (black solid lines) nearly twice the size of that of NT, Figure 3.The vertical distribution of the relative differences of kinetic energy spectra are presented.The upper panels present the relative differences between KF scheme and NT scheme ((KF−NT)/NT).The lower panels present the relative differences between the GF scheme and NT scheme ((GF−NT)/NT).The black lines are at an interval of 0.4.

10.1029/2023GL105513
9 of 12 dominated by the RKE spectral flux, indicating more energy being transferred into mesoscales and smaller scales (Figure 4f).This is consistent with the relative differences in HKE spectra in Figure 3.Moreover, both the differences in spectra and downscale cascade are dominated by RKE.Thus, we deduce that these relative differences in the KE spectra are attributed to the different strength of downscale energy cascade.This reason can be more clearly shown in Figure 4g.The quantitative calculation for the wavenumbers 100-720 shows that the increment in the HKE spectral flux of GF simulation is mainly due to the vorticity-related term.

Summary
The purpose of this study is to identify characteristic responses and systematic errors associated with mesoscale applications of different CP schemes in global atmospheric circulation model, rather than to discard one CP scheme or to favor another.To explore the responses of the atmospheric KE spectra to different CP schemes, global high-resolution simulations have been conducted.We find that there exists a compensating effect in terms of diabatic heating between CP and MP, and the diabatic heating from CP is inconsistent in the three simulations.
The diabatic heating from NT scheme is largest in the middle and lower troposphere, but the diabatic heating from MP scheme is smallest.The convection structure identified by OLR and zonal distribution of precipitation (see the Figure S1 in Supporting Information S1) simulated with NT is closest to the observation.Thus, it seems that the GF scheme is "underactive" compared to the NT scheme.This suggests that the MP will produce more compensation of diabatic heating in the GF simulation, accompanied by more large-scale precipitation and grid-scale vertical motion.
The responses of KE spectra to the different CP scheme are mainly located at mesoscales in the middle and upper troposphere.The NT scheme yields the steepest mesoscale slope of HKE spectra, followed by the KF scheme and in turn the GF scheme.The spectral slope of mesoscale HKE spectra in the GF simulation is closest to the −5/3 at 200 hPa.This means that some models may simulate a −5/3 spectrum at mesoscales due to unrealistic physical processes.For example, Lloveras et al. ( 2022) simulated moist baroclinic waves, which produced a shallow mesoscale slope, but unrealistic clouds and vertical motions.Thus, it is difficult to measure the correctness of the model simulation solely by looking at the slope of the spectrum without fully understanding of the description of the convection activities.In the lower stratosphere, the GF simulation produces stronger HKE dominated by DKE, associated with upward propagation of CGGWs.The corresponding difference in the PVV further confirms this inference.In the upper troposphere, the differences in HKE spectra are dominated by the RKE spectra.
Although the spectral RKE and DKE budget presents similarity in the main features among the three simulations, the different CP schemes induce significant variation in the downscale energy cascade at mesoscales.The GF scheme produces the strongest downscale HKE cascade dominated by RKE at mesoscales, consistent with the differences in HKE spectra.This suggests that the differences in the upper-tropospheric spectra are mainly attributed to the downscale energy cascade dominated by the rotational component.Further decomposition demonstrates that the downscale energy cascade is mainly dominated by the vorticity-related transfer term.
A quantitively comparison indicates the stronger RKE cascade is driven by the enhanced conversion from DKE to RKE.Thus, it can be concluded that the less heat release from CP means the more heat release from MP; it will induce stronger vertical motion of grid scale and enhance DKE, thereby more DKE converted to RKE.This study shows that the amount of latent heat in CP have a significant impact on the dynamical processes of the atmosphere, which are clearly reflected on the atmospheric KE spectra.

Figure 1 .
Figure 1.The L2 production of outgoing longwave radiation (OLR) by Fengyun-4A satellite (a) and simulated OLR with different convective parameterization (CP) schemes at 0000 Universal Time Coordinated 12 July 2021 (b-d).The global mean vertical heating profiles for CP scheme (e), microphysics parameterization scheme (f) and the sum of the two (g) from three simulations averaged from day 11 July to 20 July.

Figure 2 .
Figure 2. (a) The compensated HKE (HKE × n 5/3 ) spectra at 300, 200, 100 and 50 hPa.For clarity, the 200 hPa, 100 hPa, and 50 hPa spectra are shifted one decade, two decades and three decades up, respectively.The same as HKE, but for RKE (b) and DKE (c).(d) The pressure vertical velocity spectra at 300, 200, 100, and 50 hPa, which are not shifted one decade.The k −3 and k −5/3 slopes are given as black lines.The n = 20 and 8Δx = 120 km are plotted as gray dashed lines.

Figure 4
Figure 4 also shows the accumulated conversion from DKE to RKE (cyan dashed lines), which acts as a direct forcing of RKE spectral flux.A quantitatively comparison of local conversion between three simulations at the wavenumber range between left zero and maximum value of the corresponding downscale RKE flux is further made.They are 0.1249 W/m 2 in the NT simulation, 0.1630 W/m 2 in the KF simulation and 0.2063 W/m 2 in the GF simulation.This indicates the enhanced conversion drives stronger RKE spectral flux.

Figure 4 .
Figure 4.The compensated (a) HKE spectra, (b) RKE spectra, and (c) DKE spectra vertically integrated between 400 and 150 hPa temporally averaged from 11 July to 20 July of the three simulations.HKE spectral flux (black solid), DKE spectral flux (red solid) and RKE spectral flux (blue solid) and accumulated conversion from DKE to RKE (cyan dashed) are shown in panels (d) NT simulation, (e) KF simulation, and (f) GF simulation at the same height and time range.The insets are for the wavenumbers more than 20.(g) Quantitative budgets of the spectral transfers and their subitems for the wavenumbers from 100 to 720 of the three simulations.
Similarly, let ΔΠ R,D vor denote the net contribution from relative vorticity going into the wavenumber range [n 1 , n 2 ], and ΔΠ K vor = ΔΠ D vor + ΔΠ R vor.Similarly,  Δ→ denotes the net amount of DKE converted to RKE in the wavenumber range [n 1 , n 2 ].