the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Understanding the dependence of mean precipitation on convective treatment and horizontal resolution in tropical aquachannel experiments
Peter Knippertz
Yvonne Ruckstuhl
Robert Redl
Tijana Janjic
Corinna Hoose
The Intertropical Convergence Zone (ITCZ) is a key circulation and precipitation feature in the tropics. There has been a large spread in the representation of the ITCZ in global weather and climate models for a long time, the reasons for which remain unclear. This paper presents a novel approach with which we disentangle different physical processes responsible for the changeable behavior of the ITCZ in numerical models. The diagnostic tool is based on a conceptual framework developed by Emanuel (2019) and allows for physically consistent estimates of convective mass flux and precipitation efficiency for simulations with explicit and parameterized convection. We apply our diagnostic tool to a set of tropical aquachannel experiments using the ICOsahedral Nonhydrostatic (ICON) model with horizontal grid spacings of 13 and 5 km and with various representations of deep and shallow convection. The channel length corresponds to the Earth's circumference and has rigid walls at 30^{∘} N/S. Zonally symmetric sea surface temperatures are prescribed.
All experiments simulate an ITCZ at the Equator coinciding with the ascending branch of the Hadley circulation and descending branches at 15^{∘} N/S with subtropical jets and easterly trade wind belts straddling the ITCZ. With explicit deep convection, however, rainfall in the ITCZ increases and the Hadley circulation becomes stronger. Increasing horizontal resolution substantially reduces the rainfall maximum in the ITCZ, while the strength of the Hadley circulation changes only marginally. Our diagnostic framework reveals that boundarylayer quasiequilibrium (BLQE) is a key to physically understanding those differences. At 13 km, enhanced surface enthalpy fluxes with explicit deep convection are balanced by increased convective downdrafts. As precipitation efficiency is hardly affected, convective updrafts and rainfall increase. The surface enthalpy fluxes are mainly controlled by mean surface winds, closely linked to the Hadley circulation. These links also help understand rainfall differences between different resolutions. At 5 km, the wind–surfacefluxes–convection relation holds, but additionally explicit convection dries the midtroposphere, which increases the import of air with lower moist static energy into the boundary layer, thereby enhancing surface fluxes. Overall, the different model configurations create little variations in precipitation efficiency and radiative cooling, the effects of which are compensated for by changes in dry stability. The results highlight the utility of our diagnostic tool to pinpoint processes important for rainfall differences between models, suggesting applicability for climate model intercomparison projects.
 Article
(2869 KB)  Fulltext XML
 BibTeX
 EndNote
Moist convection is of paramount importance in the tropics because it controls the distribution of water vapor, clouds and rainfall (Webster, 2020). Also, its importance lies in multiscale interactions with other processes, ranging from turbulence and microphysical processes via radiation and surface fluxes to largescale circulations such as the Hadley cells that comprise the overturning circulation between 30^{∘} N and 30^{∘} S with an ascending branch at the Equator. One of the examples that illustrate the complexity of processes associated with moist convection is the socalled Intertropical Convergence Zone (ITCZ) (Schneider et al., 2014). Over oceans, the ITCZ is collocated with lowlevel convergence and upperlevel divergence of the Hadley circulation accompanied by lowlevel easterly trade winds on the flanks (Johnson et al., 1999; Schwendike et al., 2014). Thermodynamic contrasts between the ocean and the air and surface winds modulate surface enthalpy fluxes, of which enhancement increases rainfall by transporting moisture and heat from the ocean into the air (Raymond et al., 2006; Paccini et al., 2021). Cumulonimbus clouds and clearsky, moist columns in the tropics trap outgoing longwave radiation, and the moist columns increase the shortwave absorption, while the dry columns and shallow clouds in the subtropics enhance net longwave cooling compared to the tropics (Bony et al., 2015; Lau et al., 2020). An important parameter to characterize atmospheric behavior in the tropics is precipitation efficiency, the fraction of rain produced for a given amount of condensate. It has been shown that precipitation efficiency is linked to the ratio of cirrus to deep convective clouds (Stephens, 2005). The area fraction of these two cloud types modulates outgoing longwave radiation, which in turn controls the Earth's energy budget (Lindzen et al., 2001; Hartmann and Michelsen, 2002; Mauritsen and Stevens, 2015).
Climatologically, the location of the ITCZ is slightly shifted into the Northern Hemisphere (Webster, 2020). However, stateoftheart general circulation models (GCMs) still struggle to accurately represent many characteristics of the ITCZ including the doubleITCZ problem leading to excessive rainfall in the Southern Hemisphere (Fiedler et al., 2020; Tian and Dong, 2020). Even in an idealized aquaplanet configuration, which avoids complexities associated with the land–sea distribution and orography, the spatial and temporal distributions of mean precipitation are sensitive to the type of numerical model (Stevens and Bony, 2013; Rajendran et al., 2013; Landu et al., 2014; Benedict et al., 2017), vertical and horizontal resolution (Li et al., 2011; Retsch et al., 2017, 2019), and representation of convection (Möbis and Stevens, 2012; Nolan et al., 2016; Retsch et al., 2019; RiosBerrios et al., 2022).
Most of the current weather and climate models employ parameterizations for shallow and deep convection. The former plays an important role for the exchange between the boundary layer (BL) and the free troposphere, particularly in relatively dry areas (Schlemmer et al., 2017; Naumann et al., 2019; Sakradzija et al., 2020), and the latter is key for rainfall generation and vertical energy transport through latent heat release and mixing with ambient air (Emanuel, 1994; Bechtold, 2017; Webster, 2020). Explicitly representing convection on the model grid and thus avoiding convection parameterization is thought to be promising to reduce errors by permitting multiscale interactions between convection and largescale circulations (Randall, 2013; Palmer and Stevens, 2019; Tomassini, 2020), but it requires high model resolution. Given specific purposes and computational resources, a horizontal grid spacing of < 10 km can be selected to resolve deep convection (Weisman et al., 1997; Hong and Dudhia, 2012; Prein et al., 2015) with some extreme limit of 100 m (Kwon and Hong, 2017; Jeevanjee, 2017). Current global weather models use horizontal grid spacing of about 10 km with parameterized deep and shallow convection (Becker et al., 2021; Gehne et al., 2022). It is now feasible and affordable to conduct regional to global simulations with explicit deep convection (Satoh et al., 2017; Stevens et al., 2019; Schär et al., 2020; Wedi et al., 2020). These convectionpermitting models show some promising results, particularly in the tropics where baroclinic instability is of little relevance for weather systems. Explicit convection captures spatial and temporal variability of tropical rainfall more realistically compared to parameterized convection (Stevens et al., 2020). Windinduced surface exchange of heat and moisture is also improved, as shown for the tropical Atlantic Ocean by Paccini et al. (2021). Moreover, explicit deep convection performs better in terms of convectively coupled equatorial waves (Judt and RiosBerrios, 2021) and gravity wave momentum fluxes, which are often triggered by convection in the tropics and subtropics (Stephan et al., 2019).
Despite these many improvements, convectionpermitting models do not always guarantee alleviating the longstanding ITCZ problem (Zhou et al., 2022). Furthermore, models with explicit deep convection do not outperform those with parameterizations in every aspect. Parameterized deep convection is in better agreement with observations than explicit deep convection in terms of mean rainfall distribution (Wedi et al., 2020). Furthermore, Becker et al. (2021) demonstrated that their new convection parameterization scheme, which improves the coupling of convection to mesoscale dynamics, outperformed explicit deep convection in terms of both mean and intensity of rainfall over tropical Africa. Jung and Knippertz (2023) showed that the representation of equatorial waves does not deviate much between explicit and parameterized deep convection when using a global forecast model. These results indicate that resolving (deep) convection does not automatically improve the multiscale interactions in the atmosphere and does not necessarily reduce the bias in tropical rainfall. In fact, it is crucial to accurately represent physical processes and links between them.
A general problem in this context is that it is far from trivial to disentangle the reasons for the difference in performance when switching from parameterized to explicit deep convection, which often includes changes in horizontal resolution, since convection couples and interacts with so many physical processes. To tackle this problem, we here propose an innovative diagnostic tool based on a conceptual framework developed by Emanuel (2019). This framework is built around boundarylayer quasiequilibrium (BLQE), the weak temperature gradient approximation, and mass and energy conservation. BLQE describes a balance of moist entropy in the subcloud layer. The balance is achieved between surface enthalpy fluxes, which transport warm, moist air into the subcloud layer, and convective downdrafts and environmental subsidence, which transport cool, dry air from the free troposphere into the subcloud layer (Emanuel et al., 1994; Raymond, 1995). The weak temperature gradient approximation neglects horizontal temperature advection, implying a balance between diabatic heating and vertical advection (Sobel et al., 2001). Emanuel's 2019 framework considers processes on a timescale longer than that associated with the redistribution of energy by internal gravity waves. A key parameter of the conceptual model is precipitation efficiency that summarizes the collective effects of turbulent and microphysical processes. Despite its relative simplicity, the framework is able to explain fundamental characteristics of the tropical atmosphere such as the exponential relationship between rainfall and column relative humidity (Bretherton et al., 2004), convective selfaggregation (Bretherton et al., 2005), and the horizontal structures of the Walker and Hadley circulations. We refer to Emanuel (2019) for further demonstrations of atmospheric phenomena in his framework.
The goal of our study is to disentangle the physical processes contributing to differences in the ITCZ when modifying the model configuration. The modifications include changes in horizontal resolution and the representation of deep and shallow convection. To avoid the complexity associated with continents, orography, zonal asymmetries and influences of the extratropics on the tropical conditions, we conducted a set of tropical aquachannel experiments with timeconstant, Equatorsymmetric sea surface temperatures (SSTs). The simulations are as realistic as possible by including a latitudedependent Coriolis parameter and a diurnal cycle in solar irradiance. Section 2 explains further details of the model and experimental design. Section 3 describes the largescale behavior of the aquachannel experiments and differences due to model configuration. Section 4 presents our diagnostic approach based on Emanuel (2019). Results from applying the new approach to the tropical aquachannel simulations are shown and discussed in Sect. 5. Conclusions are given in Sect. 6.
2.1 Model
We use version 2.6.3 of the ICOsahedral Nonhydrostatic (ICON) model (Zängl et al., 2015) in the numerical weather prediction (NWP) configuration. The model solves the fully compressible nonhydrostatic atmospheric equations of motion on an icosahedral–triangular ArakawaC grid. Radiation is computed using the Rapid Radiative Transfer Model (RRTM) (Mlawer et al., 1997). A singlemoment microphysical scheme is used to predict cloud water, rain water, cloud ice and snow (Seifert, 2008). A turbulent kinetic energy scheme is used for the representation of turbulent mixing and surfacetoatmosphere transfer (Raschendorfer, 2001; Mellor and Yamada, 1982). Our model configuration closely follows the operational setup, including a full nonlinear Coriolis parameter, but some aspects are different for the specific purpose of our study. The surface of the entire model domain is covered by water (aquaplanet or aquachannel simulation) to exclude the complexity associated with topography, and the diurnal cycle has fixed equinoctial insolation over the whole simulation period. Zonally symmetric SSTs are prescribed with a maximum of 27 ^{∘}C at the Equator dropping to approximately 5 ^{∘}C at 60^{∘} N/S. This SST distribution has been used in other studies and is called the “Qobs” profile (Neale and Hoskins, 2000). There is no feedback of the atmosphere on the ocean and the underlying water surface, effectively making the ocean an indefinite energy source.
2.2 Simulation setup
To spin up our aquachannel simulations, we adapt the modeling practice used in Bretherton and Khairoutdinov (2015). First, we conduct a global aquaplanet simulation with a horizontal grid spacing of 40 km and a time step of 300 s. The initialization of the 40 km aquaplanet run follows the Qobs case of Neale and Hoskins (2000). The number of vertical levels is 90 with the model top at 75 km. Deep and shallow convection are parameterized using a bulk mass flux scheme (Bechtold et al., 2008; Tiedtke, 1989). The 40 km global aquaplanet experiment is run for 120 simulation days (dashed black line in Fig. 1), after which the grid spacing is reduced to 26 km with a time step of 225 s and the simulation is continued for another 90 d. After that, the model domain is restricted to a channel geometry between 30^{∘} N and 30^{∘} S, and the horizontal grid spacing is reduced to 13 km with a time step of 112.5 s (dotted black line in Fig. 1). The domain encloses the entire globe and forms a closed ring in the zonal direction. Closed walls are introduced at the latitudinal boundaries where virtual potential temperature, water vapor mixing ratio, air density, and zonal and vertical winds are prescribed by zonally and temporally averaging them at 30^{∘} N and 30^{∘} S from the 26 km aquaplanet simulation. The prescribed variables at the closed walls are timeinvariant and zonally constant but vertically variant. Except for the aforementioned quantities, all other variables are set to zero at the walls. The setup for the aquachannel run is identical to the aquaplanet runs except for the simulation geometry and the horizontal resolution. The coarser aquaplanet simulations thus serve to obtain the boundary conditions and to spin up the aquachannel run with reduced computational cost. The total simulation period of the 13 km aquachannel run is 102 d, consisting of spinup at the beginning of 62 d and the analysis period of 40 d. Finally, the grid spacing is reduced to 5 km at day 314 with a time step of 45 s (dotted pink line in Fig. 1). The boundary conditions of the highresolution run are identical to the 13 km run. The analysis period of the final run is 40 d, starting at day 317 (solid pink line in Fig. 1), after a 3 d spinup. The output time step of the 13 and 5 km aquachannel runs is hourly.
To illustrate the modeling approach, Fig. 1 depicts the evolution of the probability density distribution of precipitable water (PW) in the equatorial belt (20^{∘} N/S) over the entire run time from day 0 to 357. In the beginning of the 40 km aquaplanet simulation, PW is distributed narrowly around 40 kg m^{−2}, but by day 50 the distribution has widened with a broad dry maximum around 25 kg m^{−2} and a narrower secondary maximum near 55 kg m^{−2}. After that, the bimodal shape remains stable, even when the grid spacing is reduced from 40 to 26 km on day 120. The moist maximum corresponds to the actual ITCZ region, while the dry maximum represents the large area of subsidence in the cooler outer tropics with relatively few intermediate values of PW in between. Such a rapid evolution into a stable bimodal structure was seen in other aquaplanet simulations with the Coriolis force (e.g., Arnold and Randall, 2015; Khairoutdinov and Emanuel, 2018), as the largescale circulation redistributes moisture from the relatively homogeneous initial conditions.
When the 13 km aquachannel experiment begins on day 210, a considerable change can be observed. The range of PW slowly decreases due to a reduction of the moist columns and an increase in the frequency of dry areas, despite little change in magnitude. This drift slows down but still continues into the investigation period after day 274, suggesting that a full equilibrium has not been reached yet. Towards the end of the coarseresolution aquachannel simulation around day 314, there are some indications of a bimodal distribution again yet much closer to each other than in the global simulation before day 210. The reason for this behavior lies in the prescribed properties at the closed walls. While in the global configuration the Hadley cells span over 30^{∘} N/S, in the aquachannel configuration the model creates its own limited Hadley circulation away from the walls with subsidence around 15^{∘} N/S (described in Sect. 3.2). The narrower overturning circulation reduces the amount of moisture converging into the ITCZ (not shown). It is also conceivable that the suppression of exchange with the higher latitudes reduces moisture uptake through surface fluxes triggered by dry intrusions from the midlatitudes (Bretherton and Khairoutdinov, 2015).
Throughout the entire period of the 5 km aquachannel run (day 314–357), the bimodal distribution persists with the frequency density of PW confined between 20 and 50 kg m^{−2}. The remarkably smooth transition from the 13 to 5 km runs indicates that a change in horizontal resolution creates almost no distortion of the fields, which is also observed when reducing the grid spacing from 40 to 26 km (dashed black line in Fig. 1). This allows us to have relatively few days of spinup for the highresolution run. In summary, the PW evolution over the entire simulation period exhibits smooth transitions not only from the coarse to high resolutions but also from the aquaplanet and aquachannel geometries.
Our original intention to prescribe zero meridional wind and constant zonal and vertical winds from the 26 km aquaplanet experiment at the rigid walls was to simulate a Hadley circulation with descending branches near 30^{∘} N/S as in the global runs, but the model develops its own Hadley circulation rather than connecting its dynamical fields with the boundaries. We suspect that a possible reason is the suppression of eddy transport between the tropics and extratropics at the boundaries, forcing the model to develop its own subtropical jets internally. Ultimately, this also leads to distortions in the fields of cloud, radiation and surface fluxes in the outer tropics. We presume that a wider channel, a twoway nested channel within a global domain or an aquaplanet would simulate jets at a more realistic location and may affect many aspects, particularly associated with tropical–extratropical interactions. However, the channel geometry suppresses these interactions and thus reduces complexity. Furthermore, the advantage of having jets at a more realistic location does not outweigh the merit of our configuration that is still able to reproduce a complex structure of dynamics and thermodynamics of the tropical atmosphere with affordable computational resources. To give as little weight as possible to the artifacts from the channel approach, we restrict our analysis to an equatorial belt between 20^{∘} N and 20^{∘} S (corresponds to the area used in Fig. 1). We are confident that our analysis for this area can give useful insights into how convective treatment and model resolution affect ITCZ processes, at least in a qualitative sense.
We experimentally modify the representation of deep and shallow convection in the aquachannel configuration. At 13 km, the different treatments of deep and shallow convection are described in the following way (see Table 1): (a) an experiment named P13 where the deterministic deep and shallow convection schemes are turned on (already shown in the context of Fig. 1), (b) E13 where both deep and shallow convection schemes are turned off, (c) S13 where only the deep convection scheme is turned off and (d) SS13 where the standard deterministic shallow convection scheme (Bechtold et al., 2008; Tiedtke, 1989) is replaced by a stochastic scheme (Sakradzija et al., 2015, 2020). In the stochastic shallow convection scheme, the shallowcloud ensemble is represented based on the theory of Craig and Cohen (2006). The number of new clouds is set using a Poisson distribution and the lifetime average mass flux using a Weibull distribution. In the stochastic scheme, there are two constraints: the mass flux closure of the deterministic scheme to constrain the ensemble average mass flux and the surface Bowen ratio to control the average mass flux per cloud (Sakradzija and Hohenegger, 2017). Note that the four experiments with the horizontal grid spacing of 13 km share the same aquaplanet runs as spinup and that we modify the convective treatment when the channel geometry is introduced (dotted black line at day 210 in Fig. 1). Other than the different representations of convection, the setups remain identical among the coarseresolution aquachannel experiments.
At day 314, the horizontal grid spacing is further reduced to 5 km, creating two highresolution runs (Table 1): P5 with the deterministic deep and shallow convection schemes and E5 with explicit deep and shallow convection. Both highresolution simulations are initialized with P13 at day 314, but spinup is 2 d longer for E5 than P5, the analysis period of which begins at day 317 (solid pink bar in Fig. 1). The boundary conditions for the highresolution runs are identical to the coarseresolution ones.
In the following sections, we analyze the last 40 d of each aquachannel experiment, e.g., after spinup (day 274–314 for the coarseresolution runs). To compare the experiments with different horizontal resolutions, model grids are coarsened on a 0.2^{∘} latitude–longitude grid, using a conservative remapping.
3.1 Precipitation
The latitudinal distributions of zonally and timeaveraged precipitation are shown in Fig. 2a. At 13 km, all experiments show a distinct ITCZ with high mean precipitation concentrated between 5^{∘} N and 5^{∘} S, where the SST maximum is prescribed. Explicit deep convection (E13, S13, and SS13) yields greater mean precipitation in the ITCZ than parameterized deep convection (P13) by about 35 %. Between 5^{∘} N and 5^{∘} S the time and zonally averaged precipitation is 7.28, 9.76, 9.76 and 9.64 mm d^{−1} for P13, E13, S13 and SS13, respectively. P13 also produces a narrower, more pointy rainfall distribution. The treatment of shallow convection does not appear to have a large influence on the mean ITCZ structure. The 5 km experiments are generally drier and show indications of a broader – even double – ITCZ. As observed for the coarseresolution runs (13 km) but to a lesser extent, E5 (7.5 mm d^{−1}) has higher mean rainfall in the ITCZ than P5 (6.72 mm d^{−1}). The shape of mean rainfall is fairly symmetric in E5, yet rainfall clearly favors the Northern Hemisphere in P5. These runs are initialized with P13, which produces a disturbance similar to the Madden–Julian Oscillation in the last 20 d with higher rainfall in the Northern Hemisphere than in the Southern Hemisphere (not shown). This initial asymmetry appears to have a longlasting effect in both runs (see ITCZ broadening around 7.5^{∘} N) but particularly in P5. We plan to conduct a more detailed analysis on internal variability in the future.
Outside the ITCZ, the overall rainfall amount and the differences between the experiments are relatively small. At around 10^{∘} N/S rainfall decreases to about 1 mm d^{−1}, and beyond this it slightly increases again with latitude. This pattern of rainfall in the outer tropics is also observed in other aquaplanet, aquachannel and aquapatch simulations (e.g., Nolan et al., 2016; RiosBerrios et al., 2022). It is due to rainfall embedded in filaments of high PW being sheared off from the ITCZ into the outer tropics (see the “Video supplement”).
The rainfall intensity distribution further underlines the substantial sensitivities to convective treatment and resolution (Fig. 2b). Comparing the results among the 13 km runs, light and moderate rains (< 30 mm d^{−1}) occur more frequently in P13 than the others, which produce extreme rainfall rates of 200 mm d^{−1} and more, leading to the overall larger precipitation in the ITCZ. Going from 13 to 5 km, the frequency of intense rainfall decreases; E5 shows a discernible reduction compared to E13, while P5 shows a small decrease compared to P13. This leads to a smaller difference in rainfall intensity between P5 and E5 compared to the coarseresolution runs. This resolution dependency differs from Becker et al. (2021), who showed that rainfall intensity over tropical Africa is not dependent on resolution but on convective treatment (see their Fig. 3).
Correspondingly, the “Video supplement” depicts that largescale systems of precipitation with weak intensity are formed with parameterized deep convection (P13 and P5), whereas intense, localized storms are formed with explicit deep convection (E13, E5, S13 and SS13). The much higher intensities also lead to a more wiggly zonal average, as evident in Fig. 2a. To initiate deep convection explicitly, the model needs to develop instability on a grid box scale. The larger the grid box (or the coarser the grid resolution), the more instability can be accumulated over time, which in turn produces more intense rainfall (Weisman et al., 1997) and occasionally intense grid point storms (Giorgi, 1991; Scinocca and McFarlane, 2004). Meanwhile, a convection parameterization scheme triggers convection by perturbing temperature and humidity profiles at low levels, which allows P13 and P5 to produce light and moderate rain more frequently. This difference in rainfall intensity between parameterized versus explicit deep convection was also observed in realistic simulations (Pante and Knippertz, 2019; Judt and RiosBerrios, 2021).
3.2 Dynamical structure
Figure 3 shows a cross section of the time mean meridionalheight mass stream function and zonal wind. The meridionalheight mass stream function is calculated by integrating the meridional wind from the surface level to a certain altitude. Volumetric flux is conserved along a line of a constant meridionalheight mass stream function. P13 features a largely Equatorsymmetric tropospheredeep Hadley circulation with lowlevel convergence and corresponding upperlevel divergence in the ITCZ (Fig. 3a). The remaining small asymmetries, which occur despite the symmetric nature of our simulation setup, are a further indication that the simulations may not have fully reached equilibrium or that there can be spontaneous symmetry breaking through internal variability. The descending branches occur around 15^{∘} N/S, which is narrower than the climatological Hadley circulation in the real atmosphere (Webster, 2020) and in global aquaplanet simulations (not shown). The narrower Hadley circulation in the aquachannel experiments is because the exchange between the tropics and extratropics is suppressed at the closed walls of the tropical channel (discussed in Sect. 2.2). Strong westerly uppertropospheric jets occur at the outer edges of these narrow Hadley cells, reaching an average speed of 30 m s^{−1}. These in principle resemble the subtropical jets of the real atmosphere but shifted closer to the Equator and weaker. The lowlevel easterly trade wind belt starts at about 14^{∘} N/S and reaches about 2 km, above which westerlies dominated. This creates a considerable westerly shear for the ITCZ convection (see the “Video supplement”).
The other experiments (Fig. 3b–f) generally produce similar largescale dynamical structures to P13. However, the strength of the overturning circulation and accompanying jets depends on convective treatment and on horizontal resolution. At 13 km, explicit deep convection increases the maximum value of the mass stream function to 1.84×10^{11}, 1.81×10^{11} and 1.82×10^{11} kg s^{−1} for E13, S13 and SS13, respectively, compared to P13 (1.3×10^{11} kg s^{−1}). This tendency is also present in the highresolution runs, leading to maximum values of the mass stream function of 1.36×10^{11} and 1.67×10^{11} kg s^{−1} for P5 and E5, respectively. The magnitude of the simulated circulation is in agreement with other aquaplanet studies (Medeiros et al., 2016; RiosBerrios et al., 2020), but P13 is at the lower end of the range found in these studies. The stronger largescale circulation with explicit deep convection at 13 km is accompanied by stronger trade winds, with an increase in surface horizontal wind speed to about 4 m s^{−1}. This largely agrees with Paccini et al. (2021), who showed an increase in surface winds from a parameterized lowresolution run to an explicit highresolution run using the ICONNWP in a realistic setup. In contrast, the trade wind speed is not much influenced by the convective treatment at 5 km, although the change in the largescale circulation is considerable. Additionally, lowlevel zonal winds exhibits asymmetry, possibly due to the longterm memory that is also evident in the ITCZ. The asymmetry in surface winds at 5 km is shown clearly in Sect. 5.1.1.
One interesting aspect is that the runs with explicit deep convection (E13, S13, SS13 and E5) exhibit equatorial easterlies in the midtroposphere up to 5–7 km, while P13 and P5 exhibit westerlies there. Possibly, the explicit deep convection produces more upward convective momentum transport. This mechanism may also weaken the westerlies in the upper troposphere, leading to an overall much enhanced horizontal wind shear towards the outer tropics. The vertical shear in contrast is reduced in the ITCZ with potential consequences for intense, shortlived rainfall (see the “Video supplement”) because the sufficient shear is needed to generate longlived organized systems (Wu and Moncrieff, 1996). There are also some subtle differences in the strength and depth of the trade wind layer, supporting the idea that vertical momentum transport may play a role.
The coarseresolution runs with explicit deep convection (E13, S13 and SS13) generate a bimodal structure in the mass stream function, indicating a secondary shallow circulation that diverges polewards at around 7.5 km (Fig. 3b–d). Such a shallow meridional circulation is observed in the eastern Pacific, but the flow diverges at lower altitudes than in our experiments (Zhang et al., 2004).
This section presents our diagnostic tool based on Emanuel's 2019 framework, which we apply in Sect. 5 to output from the aquachannel experiments to further explore the discussed differences mainly in rainfall. Amongst the three equations in the original framework, we only use the formulation of convective updraft mass flux, which can be directly related to precipitation. We refer to Emanuel (2019) for the complete derivation of the conceptual framework.
The framework assumes boundarylayer quasiequilibrium (BLQE; Raymond, 1995), the weak temperature gradient approximation (Sobel et al., 2001) and energy and mass conservation, and it neglects horizontal advection of moist static energy in the BL. Using these assumptions, the framework formulates convective updraft mass flux M_{u} as follows:
where M_{u} is the convective upward mass flux of water vapor in $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1}}$; ϵ_{p} is the precipitation efficiency; F_{h} is the surface enthalpy flux; $\dot{Q}$ is the radiative cooling; h_{b} and h_{m} are the moist static energy of the BL and the free troposphere, respectively; and S is the dry stability. Moist static energy is defined as $h=\mathit{\varphi}\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}{c}_{\mathrm{p}}T\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}{L}_{\mathrm{v}}{q}_{\mathrm{v}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{L}_{\mathrm{f}}{q}_{\mathrm{i}}$, with ϕ being geopotential, c_{p} the specific heat at constant pressure, T the temperature, L_{v} the latent heat of vaporization, L_{f} the latent heat of freezing, q_{v} the specific humidity and q_{i} the specific ice content. Equation (1) demonstrates that the convective updraft mass flux increases with increasing surface enthalpy flux, with decreasing the vertical difference in moist static energy, with decreasing radiative cooling and with increasing dry static stability. The complete set of the conceptual framework for M_{u} (Emanuel, 2019) is included in Appendix A.
One important parameter of the framework is ϵ_{p}, which represents the fraction of all the condensate that reaches the ground as precipitation. Microphysical processes are not treated explicitly but formulated through one constant value of ϵ_{p}. Also, ϵ_{p} is used to parameterize convective downdraft mass flux M_{d} as a function of M_{u} in the following way: ${M}_{\mathrm{d}}=(\mathrm{1}{\mathit{\u03f5}}_{\mathrm{p}}){M}_{\mathrm{u}}$. For ϵ_{p}=1, all condensate precipitates, such that there is no evaporation and thus no downdraft mass flux. For ϵ_{p}=0, all condensate eventually evaporates again, such that downdraft and updraft mass fluxes balance.
For simplicity, Emanuel (2019) assumed that the average of the tropospheric $\dot{Q}$ can be approximated with the radiative cooling at the top of the BL in order to couple the budget of h and the largescale thermodynamic balance (see details in Appendix A). Raymond et al. (2015) suggested using a lower tropospheric quasiequilibrium instead of the entire tropospheric adjustment because when h_{b} increases from its equilibrium, the lower troposphere responds to the deviation on a convective timescale. We thus average h in the lower troposphere between 0.5 and 5 km to obtain a typical value h_{m}, and the same layer is considered for $\dot{Q}$, which is an averaged quantity, and S, which represents a slope of dry static energy. For computing h_{b}, we average h from the lowest atmospheric level of 10 m to an approximate BL top of 500 m. We tested alternatives for the BL in the range from 0.4 to 1.5 km and for the troposphere from 4 to 9.5 km, and we found the main findings to be rather insensitive to the exact choice of altitudes (not shown).
To relate this conceptual framework to our physical output, we need to find a relation between the modeled precipitation (either explicit or parameterized) and convective mass flux. We assume that precipitation rate Pr is directly proportional to M_{u} and ϵ_{p}:
with 〈q_{v}〉 being the column specific humidity. The notation 〈X〉 indicates the massweighted column mean quantity, $\int \mathit{\rho}X\phantom{\rule{0.125em}{0ex}}\mathrm{d}z/\int \mathit{\rho}\phantom{\rule{0.125em}{0ex}}\mathrm{d}z$. Precipitation can be related to the water vapor concentration at the subcloud layer or the average specific humidity of the subcloud layer rather than 〈q_{v}〉. We tested different choices of the thermodynamic variable in Eq. (2), but it does not influence our main results – only their scale in magnitude.
Using Eqs. (1) and (2), we have two unknowns, M_{u} and ϵ_{p}, because the other quantities can be readily obtained from the model output and we can solve for them. In principle M_{u} could be calculated from w for each simulation, but vertical motions of explicit and parameterized convection contain different processes. Parameterized convection assumes a profile of w through convective adjustment (Tiedtke, 1989), whereas explicit w is computed from the dynamical core. Therefore, comparing these two motions directly from the model output is a comparison of apples and oranges. The same principle applies to ϵ_{p}, which is related to M_{u}. In our diagnostic tool, M_{u} and ϵ_{p} are not obtained directly from vertical motion but indirectly using other consistent quantities. In other words, F_{h}, h_{b}−h_{m}, $\dot{Q}$, S, Pr and 〈q_{v}〉 are fed into the two independent Eqs. (1 and 2) to estimate M_{u} and ϵ_{p}. In this way, the estimates are physically consistent across the experiments with different convective treatments.
Section 3 showed substantial differences in the mean state, mainly due to the horizontal resolution and the deep convective treatment. Section 4 presented a diagnostic tool to compare these differences in a fair manner. Here we apply the diagnostic tool to averaged fields over the analysis period of 40 simulation days, with a particular focus on mean rainfall. Given the zonal symmetry of our tropical channel, we mostly consider zonal means. In the following, we discuss the different aspects of the conceptual model one after another: surface enthalpy fluxes (Sect. 5.1), vertical structure of moist static energy (Sect. 5.2), radiative cooling (Sect. 5.3), dry stability (Sect. 5.4), precipitation efficiency and convective mass flux (Sect. 5.5), and finally meridional advection in the BL (Sect. 5.6).
5.1 Surface enthalpy fluxes
The time and zonal mean of surface enthalpy fluxes is shown in Fig. 4a. P13 has F_{h} maxima in the trades and a local minimum in the ITCZ (black line in Fig. 4a), similar to the situation over realworld tropical oceans but confined to a narrower latitudinal stretch. The coarseresolution experiments with explicit deep convection (E13, S13 and SS13) share similar latitudinal distributions, but F_{h} increases compared to P13, in particular between 10^{∘} N and 10^{∘} S (by 20 %–25 %). Note that shallow convection is represented by the deterministic shallow convection scheme for S13, by the stochastic shallow convection scheme for SS13 and explicitly for E13 (Sect. 2.2). At 13 km, therefore, the main difference in F_{h} is due to the treatment of deep convection rather than shallow convection. The dependence of F_{h} on convective treatment is consistent with that of the Hadley circulation and thus surface winds described in Sect. 3.2. The highresolution experiments exhibit a similar F_{h} distribution and dependence on convective treatment to the coarseresolution ones. However, F_{h} is enhanced less strongly between 10^{∘} N and 10^{∘} S with explicit convection and has the deeper local minima in the ITCZ, leading to an increase from P5 to E5 (by 11 %). The resolution dependence is more complex. From P13 to P5, F_{h} is reduced in the ITCZ but enhanced in the trade wind zone, leading to an overall small increase by 2 %. From E13 to E5, F_{h} is systematically reduced by 11 %. The difference between the runs remains smaller beyond 15^{∘} N/S than in the inner tropics. To investigate what controls these differences in surface enthalpy fluxes, we conduct a more detailed analysis. We decompose surface fluxes into their contributing factors (Sect. 5.1.1) and examine their statistical distribution (Sect. 5.1.2).
5.1.1 Decomposition of surface fluxes
In a standard air–sea bulk formula, surface enthalpy fluxes can be written as
where ρ_{s} is the air density at the lowest model level, L_{v} is the latent heat of vaporization, c_{p} is the specific heat at constant pressure, c_{E} and c_{H} are the surface exchange coefficients for latent and sensible heat, respectively, ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ is the surface horizontal wind speed, and Δq and ΔT are the air–sea moisture and temperature contrasts. For our analysis, we define $\mathrm{\Delta}q={q}_{*}\left(\mathrm{SST}\right){q}_{\mathrm{v}}\left({z}_{\mathrm{1}}\right)$ and $\mathrm{\Delta}T=\mathrm{SST}T\left({z}_{\mathrm{1}}\right)$, where q_{*}(SST) is the saturated specific humidity for a given SST and z_{1} indicates the lowest model level of the atmosphere, which equals to 10 m in our case. Here we begin with partitioning F_{h} into surface sensible and latent heat fluxes to examine the importance of thermodynamic variables, i.e., Δq and ΔT as well as ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ for mean F_{h}.
Figure 5a–c shows zonally and timeaveraged values of the individual terms of Eq. (3). Figure 5a reveals that ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ mirrors the patterns in F_{h} (Fig. 4a) with maxima in the trade winds and with minima at the Equator and in the area of the subsiding branches of the Hadley cells (Fig. 3). Winds then increase again further away from the Equator. P13 shows considerably weaker ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ by about 1 m s^{−1} than the other coarseresolution runs (E13, S13, and SS13) out to about 10^{∘} from the Equator. The treatment of shallow convection (S13 and SS13) appears to have a rather small influence on ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$. In P5 and E5, the pattern of ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ exhibits asymmetry with stronger trade winds in the Northern Hemisphere and a deeper local minimum as evident in F_{h} (Fig. 4a). The difference in ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ between these two runs is arguably small, albeit with stronger Hadley circulation in E5 than in P5 (Fig. 3e and f). This indicates that the surface wind speed is less sensitive to convective treatment at 5 km. Meanwhile, the differences in ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ due to resolution, e.g., between P13 and P5 and between E13 and E5, reflect those in surface enthalpy fluxes (Fig. 4a).
The moisture contrast shows a much smoother latitudinal distribution and considerable contrasts between all simulations, mainly due to convective treatment (Fig. 5b). In P13 and P5, Δq is almost constant around 6.60 g kg^{−1} within 15^{∘} N/S and then sharply falls off towards higher latitudes as q_{*} quickly drops at these latitudes. The explicit treatment of deep convection (E13, S13, SS13 and E5) appears to allow for more vigorous downdrafts injecting dry air from the midtroposphere into the BL. In contrast to other fields discussed so far, the treatment of shallow convection also plays a significant role. S13, which uses the same shallow convection scheme as P13 but no parameterization of deep convection, shows only slightly enhanced Δq, particularly in the trade wind zone, where shallow mixing is important. The change to the stochastic treatment (SS13) from the deterministic treatment (S13) has little effect in the moist ITCZ area but further enhances Δq in the trades, eventually lining up with E13 at around 10^{∘} N/S. In E5 Δq closely follows that in E13, but there is a discernible decrease by around 0.5 g kg^{−1} in the trade wind zone in the Northern Hemisphere.
Figure 5c shows that the latitudinal structure of ΔT is complex in response to different convective treatments and resolutions. All simulations have a local maximum at the Equator, probably related to cool downdrafts from convection, but some have prominent maxima near the subsiding branches of the Hadley cells before all runs show a dropoff towards higher latitudes. E13 shows the overall smallest ΔT, possibly because it produces deeper convective downdrafts, leading to more adiabatic warming during the descent. E5 closely follows this structure with a marginal increase in ΔT. The two coarseresolution simulations with parameterized shallow convection (S13 and SS13) largely agree with E13 near the Equator but show considerably larger ΔT in the outer tropics, in particular S13. The reasons for this are not entirely clear. P13 has relatively high ΔT at the Equator compared to the other coarseresolution runs and intermediate values in the outer tropics. Finally, P5 has the highest ΔT at the Equator with a maximum of 2.3 K, meaning that the air near the surface is much colder in P5 than in the other runs. P5 shows the lowest rainfall intensity (Fig. 2b) and thus probably a higher fraction of subcloud evaporation, which cools and moistens the BL, leading to considerably high ΔT (Fig. 5c) as well as low Δq (Fig. 5b). However, these differences in ΔT have little impact on the surface enthalpy fluxes, since the surface sensible heat flux contributes only about 10 %. (In the latitudinal belt of 20^{∘} N/S the time and domain mean of the latent heat flux accounts for 95.5–113.6 W m^{−2}, while the surface sensible heat flux is 9.2–11.1 W m^{−2}.)
Surface enthalpy fluxes can be modulated by mean winds and thermodynamics, as well as local perturbations of those components. To quantify this, the time and zonal mean of surface enthalpy fluxes (Fig. 4a) are separated into mean contribution and local perturbation contribution by surface horizontal wind speed and thermodynamic variables. Assuming X is a temporally and spatially varying variable, we define $X=\mathit{\left\{}X\mathit{\right\}}+{X}^{\prime}$, where {X} indicates the horizontal mean (latitude and longitude) and X^{′} the anomaly from the horizontal mean, and $X=\left[X\right]+{X}^{*}$, where [X] indicates the time mean and X^{*} the anomaly from the time mean. If a field Y is a product of X_{1} and X_{2}, i.e., Y=X_{1}X_{2}, then $\mathit{\left\{}Y\mathit{\right\}}=\mathit{\left\{}{X}_{\mathrm{1}}\mathit{\right\}}\mathit{\left\{}{X}_{\mathrm{2}}\mathit{\right\}}+\mathit{\left\{}{X}_{\mathrm{1}}^{\prime}{X}_{\mathrm{2}}^{\prime}\mathit{\right\}}$ for the longitudinal mean (similarly for the time mean).
In the turbulence scheme used in ICON (Raschendorfer, 2001; Mellor and Yamada, 1982), the turbulent exchange coefficients are proportional to the turbulent kinetic energy, and so we expect the coefficients to depend on surface wind speed (as well as vertical stability). This creates an overall more than linear dependence of the surface fluxes on wind speed. For simplicity, we combine the coefficients and surface wind speed together, i.e., ${c}_{\mathrm{E}}{\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ for the surface latent heat flux and ${c}_{\mathrm{H}}{\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ for the surface sensible heat flux. Here, we derive turbulent exchange coefficients from the other variables in Eq. (3). For simplicity, we set the air density in Eq. (3) to a constant value of 1.2 kg m^{−2}. Then, the surface latent and sensible heat fluxes vary with ${c}_{\mathrm{E}}{\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and Δq, and ${c}_{\mathrm{H}}{\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and ΔT, respectively. For example, a zonal mean of the surface latent heat flux can be expressed by the longitudinal mean and its fluctuation as $\mathit{\left\{}{F}_{\mathrm{latent}\phantom{\rule{0.125em}{0ex}}\mathrm{heat}}\mathit{\right\}}={\mathit{\rho}}_{s}{L}_{\mathrm{v}}\mathit{\left\{}{c}_{\mathrm{E}}{\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}\mathit{\right\}}\mathit{\left\{}\mathrm{\Delta}q\mathit{\right\}}+{\mathit{\rho}}_{s}{L}_{\mathrm{v}}\mathit{\left\{}\right({c}_{\mathrm{E}}{\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}{)}^{\prime}\mathrm{\Delta}{q}^{\prime}\mathit{\}}$. Thus, the zonal and time mean of F_{h} can be expressed as
The first and second terms on the righthand side in each row are from the surface latent and sensible heat fluxes, respectively. In the first row [{X_{1}}][{X_{2}}] indicates the contributions of time and zonally averaged surface wind speed combined with the turbulent coefficients and thermodynamic effects to the time and zonal mean of F_{h}. In the second row $\left[\mathit{\right\{}{X}_{\mathrm{1}}^{\prime}{X}_{\mathrm{2}}^{\prime}\mathit{\left\}}\right]$ indicates a product of local fluctuations which is averaged over time and longitude, i.e., the covariance which indicates the contributions of local perturbations. In the last row $\left[\mathit{\right\{}{X}_{\mathrm{1}}{\mathit{\}}}^{*}\mathit{\left\{}{X}_{\mathrm{2}}{\mathit{\}}}^{*}\right]$ indicates the time mean of a product of temporal fluctuations of spatial mean. Equation (4) represents fields that are averaged over longitude first and then time. Averaging, which is carried out over time and then longitude, is also tested (not shown) and does not change the results that are shown below.
Figure 5d and e show latitudinal variations of contributions of the timezonal mean values and the covariance terms (the terms in the first and second rows of Eq. 4, respectively). Overall, the contributions of the timezonal mean values (Fig. 5d) match the patterns of the actual F_{h}, while the covariance terms (Fig. 5e) fluctuate around zero. The temporal anomalies of the zonal mean components (the terms in the last row of Eq. 4) are very small (not shown), compared to the mean and covariance terms. Therefore, the decomposition analysis indicates that the timezonal mean of ${c}_{\mathrm{E}}{\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and Δq is the main contributor to shape the mean F_{h} for all experiments.
In summary, the mean surface properties (Fig. 5a and b) are dominant over the covariance and anomaly terms to shape the mean F_{h} (Fig. 4a). At 13 km, the substantial differences between explicit and parameterized deep convection are found in the surface wind speed (Fig. 5a), which shape the main differences in surface enthalpy fluxes (Fig. 4a). The variations in the moisture contrast (Fig. 5b) create additional minor differences in surface enthalpy fluxes, e.g., among E13, S13 and SS13. For these coarseresolution runs, surface enthalpy fluxes are controlled by surface winds, which are in fact closely coupled to the largescale circulation (Fig. 3a–d), which becomes stronger with explicit deep convection. The resolution dependence exhibits similar relations among surface enthalpy fluxes, surface winds and largescale circulation between P13 and P5 and between E13 and E5. The changes in the moisture contrast are subtle due to the horizontal resolution, except for E5 that reduces moisture contrast in the Northern Hemisphere, which offsets the asymmetry in the local maxima of surface enthalpy fluxes there. Surprisingly, the links we discussed so far do not apply to the sensitivity to the treatment of convection at 5 km. The surface wind speed (Fig. 5a) is similar between E5 and P5. Meanwhile, the moisture contrast (Fig. 5b) increases from P5 to E5, mainly in the inner tropics, contributing to the enhanced surface fluxes in E5. This indicates that a modest change in resolution substantially alters the relation between surface fluxes and surface properties due to convective treatment.
5.1.2 Statistical distribution
Previously, the latitudinal distributions of mean F_{h} were examined. We here construct statistical distributions of surface horizontal wind speed, thermodynamic disequilibrium and surface fluxes (Hsu et al., 2022) to provide a complementary view. This does not require considering the turbulent exchange coefficients and allows us to examine how dependent surface enthalpy fluxes are on surface wind speed and thermodynamic disequilibrium. Specifically, we ignore the surface sensible heat flux, which accounts for only about 10 % of surface enthalpy fluxes, and focus on the surface latent heat flux. Surface latent heat flux is grouped by bins of ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and Δq to outline distributions of the variables and the surface flux in one figure. We sample the surface latent heat flux by bins of ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and Δq at every output time step of 1 h and in every 0.2^{∘} latitude–longitude grid box. The bin size for sampling is 1 m s^{−1} for ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and 1 g kg^{−1} for Δq as in Hsu et al. (2022). We focus on the area between 10^{∘} N and 10^{∘} S where large differences in surface enthalpy fluxes are observed (Fig. 4a). The results, however, do not change much when considering the area between 20^{∘} N and 20^{∘} S.
Figure 6 depicts twodimensional histograms of ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and Δq, as well as corresponding values of the surface latent heat flux (contour). For P13, the density distributions of both ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and Δq are positively skewed with an extensively long tail for the former (Fig. 6a). The bin of ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ of 1–2 m s^{−1} and Δq of 6–7 g kg^{−1} contains the maximum frequency density of 15 % (colored dot). Contour lines demonstrate that the corresponding surface latent heat flux is more strongly dependent on ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ than on Δq. The maximum frequency density (colored dot) is located between the contour lines of 50–100 W m^{−2}. A similar pattern is observed for P5 (Fig. 6c). However, the maximum frequency density is slightly reduced to 12.5 %, the Δq distribution is broader with the upper limit extending to 12 g kg^{−1} and the tail of the ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ distribution becomes shorter, compared to P13. This shorter tail may be associated with downward momentum transport in the ITCZ (Fig. 3a and e).
E13 (Fig. 6b) exhibits the largest contrast to P13, showing relatively evenly distributed ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ and Δq. The maximum density (colored dot) accounts for 6.0 %, which is less than half of that for P13 in the bin of ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ of 3–4 m s^{−1} and Δq of 7–8 g kg^{−1}, showing a greater surface wind speed and greater moisture contrast (seen also in Fig. 5a and b). The surface latent heat flux (contour lines) increases strongly with increasing ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$, while to a lesser extent but noticeably it increases with growing Δq. As expected from the high F_{h} shown in Fig. 4a, the maximum density bin is located between the contour lines of 100–150 W m^{−2}. The histogram for E5 (Fig. 6d) shares similarities to that for E13. However, the ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ distribution is restricted to 10 m s^{−1}, and the upper limit for Δq is extended to 12 g kg^{−1} as seen in the differences between P13 and P5. The maximum frequency density increases to 8.8 % in the bin of ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ of 2–3 m s^{−1} and Δq of 7–8 g kg^{−1}.
The distributions of S13 and SS13 (Fig. 6e and f) show some intermediate features between E13 and P13. The maximum frequency density (colored dot) lies between the contour lines of 100–150 W m^{−2}, consistent with small differences in F_{h} among the explicit deep convection runs (Fig. 4a). The distributions are concentrated at the highest frequency density of 10 % for S13 and 8.4 % for SS13, which are both in the Δq bin of 6–7 g kg^{−1} as in P13.
5.2 Vertical difference in moist static energy
Figure 4b shows the vertical difference in h between the BL and the lower troposphere described by h_{b}−h_{m} (Eq. 1). This contrast is key for BLQE, as it determines the reduction of h in the BL through convective downdrafts and largescale subsidence. In the moist ITCZ region h_{b}−h_{m} has minimum with relatively small differences among the six simulations and P13 showing the smallest values. Then h_{b}−h_{m} increases markedly in the trade wind belt with a much larger dependence on convective treatment and resolution, followed by a gradual falloff towards higher latitudes. In the trade wind area, h_{b}−h_{m} is smallest for E13 among the coarseresolution runs, indicating deep mixing and conditions closer to moist neutrality. S13 shows much increased contrasts, suggesting that here deep mixing may be suppressed at the cost of more subtle shallow mixing. SS13 lies in the middle between these two extremes. P13 shows a fundamentally different behavior with a much slower falloff towards higher latitudes. The highresolution runs show similar patterns of h_{b}−h_{m} as in the coarseresolution runs, but with relatively smoother distributions in h_{b}−h_{m} in the inner tropics. In the ITCZ, h_{b}−h_{m} slightly increases from 13 to 5 km grid spacing. In the outer tropics, P5 exhibits overall smaller h_{b}−h_{m} than P13, while h_{b}−h_{m} for E5 closely matches that for E13 but with a more pronounced asymmetry between the hemispheres.
Profiles of h provide a deeper insight into the h_{b}−h_{m} patterns. The solid and dashdotted lines in Fig. 7 show h profiles below 8 km at characteristic latitudes for the ITCZ (0^{∘}), trades (8^{∘} N/S) and subsidence areas (15^{∘} N/S). Overall, h shifts to lower values from the Equator to higher latitudes, following the prescribed SST pattern. The dotted lines in Fig. 7 show corresponding profiles of dry static energy ($s=\mathit{\varphi}\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}{c}_{\mathrm{p}}T$) with hardly any difference between the experiments. Therefore, lower h with increasing latitude is largely equivalent to drier air.
First, we discuss h profiles for the coarseresolution runs in the ITCZ (Fig. 7a). The BL top for explicit deep convection (E13, S13 and SS13) is at 500 m but shallower for P13 (∼ 400 m or one model level lower), while h_{b} differs little, changing by 0.1 %–0.3 % only (see Table 2). Note that the BL height is fixed for our diagnostic tool to the layer of 10–500 m, but calculating h_{b} with alternating BL heights below or slightly above 500 m does not impact the results. Among the coarseresolution runs, E13 shows the lowest value of h_{b}, possibly related to more frequent and more intense convective downdrafts in line with more intense rainfall (Fig. 2b). In the lower free troposphere, more distinct differences are evident, specifically between 1 and 3 km. E13 has again the overall lowest values, such that downdrafts can more effectively reduce h_{b}. Retsch et al. (2019) also found a drier lower troposphere for their explicit deep convection cases than for parameterized ones. S13 and SS13 show enhanced values relative to E13 around 2 km, while P13 has higher h throughout most of the layer up to 5 km. Applying a vertical average over 0.5 to 5 km, we obtain h_{m}, which varies between 327.4 and 328.9 kJ kg^{−1} (Table 2). Given that differences within and above the BL are largely consistent between the coarseresolution runs, h_{b}−h_{m} in the ITCZ increases by 3 %–6 % from P13 to S13, SS13 and E13, as also evident from Fig. 4b.
At 5 km, changes in h profiles due to convective treatment are largely consistent with the results at 13 km (Fig. 7a), showing that E5 produces lower h, specifically above the BL, and a higher BL top in E5 (500 m) than in P5 (270 m or two model levels lower). The drier h in the lower troposphere is the main reason for the increased h_{b}−h_{m} in E5 compared to P5 (Fig. 4b). When we switch our focus to resolution dependence (P13 versus P5 or E13 versus E5), the higher resolution dries the atmospheric column, which is consistent with a drier ITCZ (Fig. 2a). This lower h is more pronounced in the lower troposphere than in the BL, leading to reduced h_{m} for P5 and E5 than for P13 and E13 (Table 2). Thus, h_{b}−h_{m} in the ITCZ is larger for the highresolution runs (Fig. 4b).
In the trade wind belt, differences in h_{b}−h_{m} among the experiments are larger than in the ITCZ (Fig. 4b). This corresponds with more marked differences in the vertical profiles of h (Fig. 7b). At 13 km, h profiles are shifted to lower values for the explicit deep convection runs (E13, S13 and SS13) than P13. The variations of h_{b} among the coarseresolution experiments are systematic, decreasing from P13 to S13 or SS13 to E13 by 1.13–2.49 kJ kg^{−1} (see Table 2). Meanwhile, h_{m} varies little among the explicit deep convection runs (E13, S13 and SS13; colored vertical bars in Fig. 7b) but decreases from P13 to other coarseresolution runs by 1.8–2.23 kJ kg^{−1} (Table. 2). SS13 exhibits an intermediate behavior of h between S13 and E13, such that SS13 is similar to S13 in the BL, to E13 between 0.5–1.5 km and again to S13 above 1.5 km. This intermediate behavior indicates that the stochastic scheme for shallow convection (SS13) mixes the air between the BL and lower troposphere more efficiently than the deterministic version (S13) but not as deeply as the explicit one (E13). Consequently, this results in some unsystematic behavior of h_{b}−h_{m} from one to another, with E13 showing the lowest value, then P13 and SS13, and finally with S13 showing the highest value (Fig. 4b).
For the highresolution runs, the effect of convective treatment on the h profile is consistent with that for the coarseresolution runs, again with the drier h profile for E5 than for P5 in the trade wind belt (Fig. 7b). Compared to the coarseresolution runs, both P5 and E5 moisten the lower troposphere, which may be associated with the broad ITCZ (Fig. 2a), while the BL is moistened only for E5 possibly through convective mixing, which transports higher h into the BL. Despite the coherent effect of convective treatment on the h profile between the coarse and highresolution runs, the resulting h_{b}−h_{m} (Fig. 4b) decreases from parameterized to explicit deep and shallow convection at 13 km but increases at 5 km.
In the subsidence region (Fig. 7c), the BL height is 500 m in all experiments, and h_{b} is fairly similar across the runs (Table. 2). Among the coarseresolution runs, E13 shows the largest value of h_{m}, indicating more moisture columns than the others as profiles for dry static stability show no substantial differences. This is the opposite of the results by Retsch et al. (2019), who showed that the lower troposphere in the subsidence region is drier with explicit deep convection than parameterized deep convection. Given that the strength of the Hadley circulation is largely comparable between the runs with explicit deep convection (discussed in Sect. 3.2), lowertropospheric moisture is presumably dominated by local mixing rather than largescale subsidence effects. Likewise, S13 and SS13 fundamentally differ in that vertical mixing, mainly in the lower atmospheric layer, is more efficient for the stochastic version (SS13) (Sakradzija et al., 2020). Accordingly, h_{m} for SS13 increases from S13 by more than 0.5 kJ kg^{−1}, which is less than 0.5 kJ kg^{−1} in other latitudinal regions (Table. 2). Thus, we see a systematic change of h_{b}−h_{m} in the subsidence region, representing some effects of local mixing. The highresolution runs show visually identical h profiles (Fig. 7c) with small differences in h_{b} and h_{m} (Table 2).
In summary, h_{b} and h_{m} in the ITCZ and the trade wind belt show some systematic changes due to convective treatment, with h profiles in the former being more sensitive to horizontal resolution. However, subtle changes in h_{b} and h_{m} result in complex patterns of h_{b}−h_{m}. The impact of shallow convective treatment is evident in profiles of h, particularly in the trade wind belt. The stochastic version of shallow convection (SS13) exhibits an intermediate behavior of h between the deterministic (S13) and explicit version (E13), reflected in the latitudinal distribution of h_{b}−h_{m} (Fig. 4b).
5.3 Radiative cooling
Figure 4d shows the time and zonal mean of radiative cooling in the lower troposphere (0.5–5 km). In P13 and P5 the latitudinal distribution of $\dot{Q}$ is overall the flattest, with it marking its minimum in the ITCZ and larger values in the outer tropics (Fig. 4d). E13, S13, SS13 and E5 show a similar pattern but less cooling in the ITCZ. The explicit deep convection runs have stronger cooling in the outer tropics, which is particularly true for S13. One exception is E5 which is more consistent with the parameterized deep convection runs beyond 10^{∘} N/S. In the following, we discuss this result in the context of total cloud cover (Fig. 8) and the latitudinalheight distribution of the radiative temperature tendency (Fig. 9).
Figure 8 shows that amongst all runs, E13 has the overall highest cloud cover, peaking at about 80 % at the Equator and falling off gradually to about 50 % at around 15^{∘} N/S, beyond which there is a slight increase again. Consequently, E13 exhibits net radiative cooling in the troposphere with a marked contrast between the ITCZ region and the outer tropics (Fig. 9b). In the ITCZ, radiative cooling is generally reduced, and in the outer tropics there are signatures of shallow and congestus clouds, forming a trimodal structure (Johnson et al., 1999; Khairoutdinov et al., 2009), consistent with the highest cloud cover relative to the other runs (Fig. 8).
P13 show the largest contrast to E13 with reduced cloud cover, especially in the outer tropics (Fig. 8), and with some differences in the pattern of radiative cooling (Fig. 9a). In the ITCZ, radiative cooling is generally reduced, and there is even a slight warming below the tropopause, likely related to longwave absorption by optically thick cirrus (Senf et al., 2020). This is consistent with the fact that for P13 cloud ice in the upper troposphere is spread over a 1 km deeper layer than in the other coarseresolution experiments (not shown). Note that the neartropopause warming is not included in $\dot{Q}$ (Fig. 4d), which is averaged over 0.5–5 km. In the outer tropics, radiative cooling increases and is quite homogeneous across most of the free troposphere, decreasing gently above about 9 km with no indication of the trimodal structure (Fig. 9a). The top of the BL stands out as an area of enhanced cooling associated with longwave emission from the top of shallow clouds into the relatively dry free troposphere above it.
S13 exhibits similar features to E13 in terms of cloud cover and radiative cooling (including $\dot{Q}$) in the ITCZ, but there are marked differences in the outer tropics. Cloud cover in S13 (47.7 %) is intermediate between P13 (39.8 %) and E13 (53.2 %) (Fig. 8). While the freetropospheric cooling by radiation is consistent with E13, S13 (Fig. 9c) reveals that radiative cooling above the BL is substantially enhanced and very concentrated, creating a gap in cooling above that. This leads to the largest $\dot{Q}$ of all runs in the outer tropics (Fig. 4d). SS13 shows the cloud cover closest to P13 among the other coarseresolution runs (Fig. 8). Freetropospheric radiative cooling for SS13 (Fig. 9d) remains similar to S13 and E13, with slightly enhanced cooling at around 3 km, but the BL top cooling in the outer tropics is much reduced. This is due to the fact that the stochastic version (SS13) allows for efficient mixing between the BL and the lower troposphere and for efficient BL convective heating (Sakradzija et al., 2020; Senf et al., 2020). Consequently, this leads to overall similar $\dot{Q}$ to E13 (Fig. 4d), despite the different vertical structures of radiative temperature tendency.
The highresolution runs (P5 and E5) show differences in total cloud cover and radiative cooling, largely consistent with the differences between P13 and E13. The total cloud cover is reduced from E5 to P5, particularly in the outer tropics (Fig. 8). P5 is characterized by net warming below the tropopause (Fig. 9e), which is slightly shifted to the Northern Hemisphere where the maximum mean rainfall is (Fig. 2a). The distribution of radiative cooling for E5 (Fig. 9f) exhibits a trimodal structure, but net cooling is generally weaker than that for E13, leading to reduced $\dot{Q}$ for E5 in the outer tropics compared to other explicit deep convection runs (Fig. 4d).
5.4 Dry stability
In the conceptual framework, the effects of radiative cooling need to be considered relative to the dry stability S (Eq. 1), which is shown in Fig. 4e. For P13 and P5, S is nearly constant between 15^{∘} N and 15^{∘} S with a value of 14.2 and 14.1 $\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}}$, respectively, beyond which it drops slightly. The other experiments exhibit some noteworthy differences. In the ITCZ, explicit deep convection appears to create a somewhat lower stability by 0.2–0.3 $\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}}$ compared to parameterized deep convection at the same resolution (Table 2). This may be related to the fact that the convection scheme triggers convection a little more easily and can therefore more effectively stabilize the atmosphere in a convectively active region. This subtle difference is also evident from Fig. 7a. (The slope of s in the lower troposphere represents S.) In the outer tropics, the treatment of shallow convection has some effect on S, increasing S by about 0.1 $\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}}$ from one experiment to another (Table 2): E13 likely produces the least shallow mixing and is thus least stable, followed by SS13 and S13. Finally, the largest stability is found in P13, which parameterizes both shallow and deep convection. With increasing resolution from 13 to 5 km, S decreases systematically. As in the ITCZ, differences are overall subtle, which is also evident from profiles of s (Fig. 7b and c).
In the conceptual model, the ratio of radiative cooling in the lower troposphere $\dot{Q}$ and dry stability S are considered (second term in Eq. 1). As radiation and stability in all runs largely compensate for each other, the ratio is almost identical and also relatively constant over latitudes (Fig. 4f). In contrast, the effect of surface enthalpy fluxes and vertical contrast of h leads to substantial variations in the first term in Eq. (1) (Fig. 4c). This demonstrates that despite all the details discussed above, the thermodynamic balance between radiation and stability hardly drives differences between the simulations.
5.5 Convective updraft mass flux and precipitation efficiency
Up to this point, we have examined the distributions of each term on the righthand side in Eq. (1). To conclude our diagnostic framework, here we discuss M_{u}, ϵ_{p} and 〈q_{v}〉, which are directly related to rainfall (Eq. 2), and then link this discussion to the entire framework. We start with quantities in the ITCZ. Surprisingly, ϵ_{p} (Fig. 4h) is quite similar across the runs with a maximum value of 0.63–0.657 (Table 3). Note that the timeaveraged quantities are taken into account here, but timely varying Pr and ϵ_{p} can be strongly correlated (Narsey et al., 2019; Muller and Takayabu, 2020). Furthermore, ϵ_{p} can depend significantly on how convection is treated in models (Li et al., 2022), but the different convective treatments do not alter ϵ_{p} in the ITCZ in our case. Meanwhile, 〈q_{v}〉 (Fig. 4i) marginally decreases from parameterized to explicit convection and from the coarse to high resolutions (Table. 3), indicating that P13 has the moistest atmosphere and E5 is the driest in the ITCZ, as also seen in profiles of h (Fig. 7a). Given the almost identical ϵ_{p} and the marginal changes in 〈q_{v}〉, M_{u} must substantially differ among the runs to match the differences in rainfall (Eq. 2). Correspondingly, the latitudinal distribution of M_{u} (Fig. 4g) closely matches that of mean rainfall (Fig. 2a). For example, the coarseresolution runs show that M_{u} in the ITCZ increases from parameterized to explicit deep convection by 30 %–38 % (Table. 3), which is largely consistent with the mean rainfall increase (about 35 %). For the highresolution runs, M_{u} values increase by 16 % from P5 to E5, which is larger than the mean rainfall increase of 11 %. This gap is noticeable at around 3^{∘} N where the mean rainfall increases by 3 % from P5 to E5 (Fig. 2a), while M_{u} increases by 10 % (Fig. 4g). This is in fact compensated for by the decrease of 6 % in 〈q_{v}〉 (Fig. 4i).
In the outer tropics, ϵ_{p} sharply decreases, reaches a minimum at around 9–12^{∘} N/S and beyond this increases again with latitude (Fig. 4h). The differences among the runs are substantial in the outer tropics, with ϵ_{p} varying between 0.221 and 0.310 (highest in P13). Mean values of 〈q_{v}〉 slightly vary, with P13 marking the greatest value of 2.77 g kg^{−1} (Fig. 4i). From the ITCZ to the outer tropics M_{u} sharply decreases to about 0.015 $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1}}$ and beyond the minimum slightly increases again with latitude (Fig. 4g). These patterns with a minimum and marginal increase in ϵ_{p} and M_{u} are also observed in Pr (Fig. 2a), yet the differences in Pr in the outer tropics do not vary as much as those in ϵ_{p} or 〈q_{v}〉 due to the generally low values of the contributing variables there.
We summarize the results by focusing on the ITCZ where mean rainfall varies most due to the treatment of convection and horizontal resolution. Our results indicate that in the ITCZ, the difference in M_{u} from one experiment to another most closely matches that in Pr, with almost identical ϵ_{p} and marginal changes in 〈q_{v}〉. When revisiting all input variables in Eq. (1) for our diagnostic tool (Fig. 4), M_{u} is shaped by ${F}_{\mathrm{h}}/({h}_{\mathrm{b}}{h}_{\mathrm{m}}$), which describes BLQE, while variations in $\dot{Q}$ and S largely compensate for each other (Fig. 4c and f). Given almost constant ϵ_{p} in the ITCZ, an increase in M_{u} increases M_{d}, through ${M}_{\mathrm{d}}=(\mathrm{1}{\mathit{\u03f5}}_{\mathrm{p}}){M}_{\mathrm{u}}$, which carries low h from the lower troposphere into the BL to balance enhanced F_{h}. At 13 km, the treatment of deep convection produces the main differences in M_{u} by 30 %–38 % between the explicit and parameterized versions, which is associated with substantial differences in F_{h} (20 %–28 %). This change in F_{h} is associated with changed ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ (Sect. 5.1), which in turn is closely linked to the Hadley circulation (Sect. 3.2). Note that these links are not unidirectional but multidirectional interactions in the sense that we cannot disentangle whether a stronger circulation leads to more rainfall or vice versa. From 13 to 5 km, the results share similar importance for mean rainfall, which is again controlled by M_{u}. Decreased M_{u} with increasing resolution is associated with the combined effects of increased h_{b}−h_{m} and suppressed F_{h}, which is shaped by ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ through the largescale circulation. This indicates that the thermodynamics at low altitudes become important at higher resolution. The difference in M_{u} between P5 and E5 is again associated with that in F_{h}, which in fact changes due to Δq rather than ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$. Furthermore, a marginal change in 〈q_{v}〉 has a small contribution to mean rainfall, showing again the importance of thermodynamic properties at low altitudes because substantial changes in 〈q_{v}〉 are found below 6 km, inferring from h profiles (Fig. 7).
5.6 Meridional advection
Equation (1) is obtained by neglecting the horizontal advection in the BL. Here we test the sensitivity of the diagnostic tool when the advection term is included. A scale analysis for h budget of the BL (Eq. A3) reveals that the BL radiative cooling term can safely be ignored while the advection term is not fully negligible ($\mathrm{d}{\dot{Q}}_{\mathrm{b}}\sim $ 1 W m^{−2}, $\mathrm{d}\phantom{\rule{0.125em}{0ex}}\mathit{\rho}{\mathit{V}}_{\mathrm{h}}\cdot \mathrm{\nabla}h\sim $ 10 W m^{−2}, F_{h}∼ 100 W m^{−2}). With this, Eq. (1) can be expressed as
where d is the BL height and V_{h} is the horizontal velocity. Here, only meridional advection is taken into account because BL meridional and zonal gradients are in the order of 10 and 0.1 $\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{km}}^{\mathrm{1}}$, respectively, while BL meridional and zonal winds are comparable in magnitude. The advection term is calculated by integrating the meridional advection of h from the lowest atmospheric layer at 10 m to the BL top at 500 m and assuming an air density of 1.2 kg m^{−2}.
Figure 10 shows the latitudinal distribution of BL meridional advection and the impact of including it on precipitation efficiency and convective updraft mass flux. Parameterized deep and shallow convection (Fig. 10a) produces subtle meridional advection in the BL with nearzero values at the Equator that increases away from the Equator, leading to an average of 4.4 and 3.3 W m^{−2} in the ITCZ for P13 and P5, respectively (Table 3). The small advection term in the ITCZ decreases M_{u} by about 0.0005 $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1}}$ and increases ϵ_{p} by about 0.01 compared to the advectionfree diagnostic tool (Table 3). At around 10^{∘} N/S, BL meridional advection reaches a maximum of 14 W m^{−2} and beyond that decreases with increasing latitude, leading to a small increase in ϵ_{p} by 0.02 and a small decrease in M_{u} by 0.001 $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1}}$.
In contrast to parameterized deep and shallow convection (P13 and P5), E13 exhibits the largest contribution of BL meridional advection (Fig. 10a). In the ITCZ, the averaged advection is 27.3, consequently reducing M_{u} by 0.003 $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1}}$ but increasing ϵ_{p} by 0.05 (Table 3). At around 7^{∘} N/S the advection term shows local maxima of 32.8 W m^{−2} and then sharply decreases with increasing latitude. The overall large meridional advection term for E13 is consistent with intensified ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ (Fig. 5a) and a greater h change in the BL with latitude (Fig. 7). While the advection term is similar between P13 and P5, it is overall lower for E5 than that for E13, particularly in the ITCZ where the advection term decreases almost by half (Table 3). Furthermore, the peaks are located further away in E5 than in E13. Despite these differences, the resulting changes in ϵ_{p} and M_{u} are at best modest for E5 and E13 (Fig. 10b and c).
Similarly, SS13 shows large BL meridional advection in the ITCZ and has local maxima at around 7^{∘} N/S but overall weaker advection by around 6 W m^{−2} than E13 (Fig. 10a). The advection consideration leads to a decrease in M_{u} by 0.0013 $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1}}$ and an increase in ϵ_{p} by 0.039 in the ITCZ (Table 3), which changes less strongly in the outer tropics (Fig. 10b and c). For S13 meridional advection closely follows that for SS13 (Fig. 10a), leading to averaged advection of 20.9 W m^{−2} in the ITCZ (Table 3). However, the maximum value is located in the ITCZ rather than at around 7^{∘} N/S. This difference between S13 and SS13 is because the minor increase from SS13 to S13 in ${\stackrel{\mathrm{\u203e}}{U}}_{\mathrm{h}}$ in the ITCZ (Fig. 5a) and the slightly reduced meridional h gradient for S13 (Fig. 7a and b). Despite this difference in shape, M_{u} and ϵ_{p} in S13 and SS13 change by almost the same degree when considering meridional advection.
In summary, the meridional advection slightly increases ϵ_{p} and slightly decreases M_{u} for all cases. For example, ϵ_{p} increases in the ITCZ by 8 % from P13 to the other coarseresolution runs (E13, S13 and SS13), whereas it is almost identical without considering the advection term (Sect. 5.5). However, the main differences among the runs are evident in M_{u}, compared to ϵ_{p} and 〈q_{v}〉, meaning that the close association between M_{u} and rainfall remains strong.
Over decades, general circulation models have shown disagreement on tropical rainfall distributions, demonstrating a high level of uncertainty. Idealized modeling frameworks, such as aquaplanet simulations, showed a great sensitivity of tropical rainfall to various factors. This study presented a novel diagnostic tool to identify links between the processes important for rainfall in a fully coupled and physically consistent way. The innovation of our diagnostic tool is the application of the conceptual framework by Emanuel (2019) to output from a numerical model. Amongst other things, the framework assumes mass and energy conservation as well as the boundarylayer quasiequilibrium (BLQE) (Raymond, 1995). BLQE describes the balance of moist static energy in the BL between surface enthalpy fluxes and vertical advection through convective downdrafts and largescale subsidence.
We applied our diagnostic tool to tropical aquachannel experiments using the ICONNWP model. The experiments vary with treatments of shallow and deep convection and with different horizontal grid spacings (13 and 5 km). The channel geometry is designed with a zonal extension as large as the Earth's circumference and a meridional extension between 30^{∘} N and 30^{∘} S where timeinvariant, zonally constant variables are prescribed. The SSTs are prescribed with a zonally symmetric distribution and maximum at the Equator (Neale and Hoskins, 2000).
All experiments show an ITCZ at the Equator and a Hadley circulation with an ascending branch at the Equator and descending branches at 15^{∘} N and 15^{∘} S – somewhat narrower than the Hadley circulation in reality – and accompanying easterly trade winds at the flanks of the ITCZ. The confinement of the Hadley circulation between 15^{∘} N and 15^{∘} S is because the model develops its own internal circulation, at least partly related to suppressed eddy fluxes at the rigid walls. Despite the similar structures among the experiments, there are differences due to changes in convective treatment, mainly deep convection, and in horizontal resolution. From parameterized to explicit deep convection, the maximum precipitation in the ITCZ increases and the Hadley circulation becomes stronger. These changes are more pronounced at 13 km than at 5 km, which shows reduced rainfall in the ITCZ. Figure 11 illustrates how variables are relevant to rainfall change in response to different model configurations. We summarize the changes focusing on the ITCZ region as follows.

Dependence on convective treatment at 13 km. From parameterized (Fig. 11a) to explicit deep convection (Fig. 11b), the rainfall amount increases substantially and the largescale circulation and surface horizontal winds get stronger. Strong surface winds enhance surface enthalpy fluxes by 20 %–28 %. The vertical difference in moist static energy between the BL and the lower troposphere is relatively robust to changing convective treatment. Somewhat surprisingly, precipitation efficiency is little sensitive to the representation of convection with values of 0.633–0.657. In contrast, convective updraft mass flux increases by 30 %–38 % with explicit deep convection. With the constant value of precipitation efficiency, convective updraft mass flux increases proportionally to increasing convective downdraft mass flux, which is balanced by enhanced surface enthalpy fluxes to maintain BLQE. Thus, the rainfall change in response to convective treatment at 13 km is due to the tight links among dynamical fields, surface fluxes and convective mass flux.

Dependence on resolution with the same convective treatment. From the coarse (Fig. 11b) to high resolutions (Fig. 11c), the ITCZ becomes drier, which is slightly further into the Northern Hemisphere. The higher resolution alters the strength of the Hadley circulation and even more so surface winds, which are overall weakened except for the trade wind belt in P5 where winds get intensified. Weakened surface winds suppress surface enthalpy fluxes in the ITCZ. Additionally, the vertical contrast in moist static energy in the ITCZ becomes larger with increasing resolution. The combination of suppressed surface enthalpy fluxes and increased vertical contrast in moist static energy are associated with reduced convective mass flux, while precipitation efficiency changes little due to increasing resolution. This underlines the importance of BLQE to understand rainfall difference due to changing resolution. Thus, both dynamics and thermodynamics become important to understand the sensitivity of mean rainfall to resolution.

Dependence on convective treatment at 5 km. The sensitivity of rainfall to convective treatment at 5 km exhibits some differences from that at 13 km. As for the other cases, explicit convection produces more rainfall than parameterized convection. The difference in rainfall is associated primarily with differences in convective updraft mass flux related to BLQE, while precipitation efficiency remains largely unaffected by convective treatment. At 5 km, however, the differences in surface enthalpy fluxes are not due to differences in surface wind speed, which is relatively similar in the two runs, although the Hadley circulation is stronger in E5 than in P5. Instead, a larger moisture contrast between the ocean surface and the air in E5 enhances surface enthalpy fluxes and convective mass flux. Furthermore, the effects of stronger convective mass flux on rainfall are partially offset by relatively low column averaged humidity, of which the differences in moisture in the lower troposphere contribute the most. Thus, BLQE is still key to understanding the dependence on convective treatment at 5 km, yet the balance is achieved by thermodynamics within and above the BL, while dynamic fields are less involved.
The model configuration changes radiative cooling and dry stability in all latitudes, but these changes compensate for each other, having a very small net effect on convective mass flux. Note that radiative cooling was found to be crucial for radiative convective equilibrium without a largescale circulation (Emanuel, 2019), but this is not the case for our experiments with full physics and dynamics. Moreover, explicit deep convection can produce more delicate distributions of convection, such as deep, shallow and congestus clouds, than parameterized convection, but again mean rainfall is insensitive to a change in radiative cooling associated with these structures. With explicit deep convection, the meridional advection of moist static energy in the BL is not negligible, leading to a slight increase in precipitation efficiency of 0.03–0.05. However, convective updraft mass flux still exhibits the strongest association with rainfall. A caveat of this diagnostic tool is that the effects of entrainment and detrainment are not considered, which might be important for convective updraft mass flux (Zipser, 2003; Möbis and Stevens, 2012). Somewhat indirectly, these effects are included in h_{b}−h_{m} through lowertropospheric h and in ϵ_{p} through indirect effects of subcloud evaporation. Furthermore, the role of thermodynamics in the lower troposphere may become more important when using a slab ocean model (Tompkins and Semie, 2021) or different turbulence and/or microphysics schemes (Lang et al., 2023).
The merit of our diagnostic tool lies in a fair comparison of simulations with different representations of convection to examine the processes potentially linked to rainfall. Since those processes are strongly coupled to each other, it is not trivial to disentangle what processes are ultimately responsible for rainfall. Furthermore, explicit and parameterized convection treats vertical motion differently, so it is inconsistent to compare convective updraft mass flux obtained directly from the modeled vertical wind field. Thus, we emphasize that the diagnostic tool presented here provides a physically consistent, fair comparison between explicit and parameterized convection and helps obtain a quantitative and qualitative view on important links in the system. Although the conclusion of this study may not hold using other simulation geometry such as aquaplanet, the application of the ITCZ diagnostic tool will help gain a deeper understanding of processes responsible for mean rainfall distribution. Lastly, this tool also has potential to specify sources of uncertainty in NWP models and to identify the reasons behind the large spread in ITCZ behavior among different climate models.
In the framework of Emanuel (2019), the largescale vertical velocity at the top of the boundary layer (BL) w is written as
where M_{u} and M_{d} are convective upward and downward mass fluxes of water vapor in $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1}}$, respectively; w_{e} is the environmental vertical velocity away from convection; and ρ is the air density at the top of the BL. Note that Emanuel (2019) uses dimensionless mass flux and vertical velocity fields, but we prefer utilizing them in physical units in order to apply the conceptual model to the simulated fields. Microphysical processes are not treated explicitly but formulated through one constant parameter, the socalled precipitation efficiency ϵ_{p}, which represents the fraction of all condensate that reaches the ground as precipitation. Also, ϵ_{p} can then be used to parameterize M_{d} as a function of M_{u} in the following way: ${M}_{\mathrm{d}}=(\mathrm{1}{\mathit{\u03f5}}_{\mathrm{p}}){M}_{\mathrm{u}}$.
With conservation of moist static energy, h budget for the BL becomes
where V is the threedimensional wind velocity, F_{h} is the surface enthalpy flux, $\dot{Q}$ is the radiative cooling and the subscript b indicates the integral over the depth of the BL. In a wellmixed BL the vertical advection of h occurs at the top of the BL, and boundarylayer quasiequilibrium (BLQE) assumes that the injection of lowh air by convective downdrafts (M_{d}) and largescale subsidence (w_{e}) is balanced by the uptake of high h through surface fluxes (Raymond, 1995). Therefore, the vertical advection can be represented by a simple difference between characteristic values of h for the BL (h_{b}) and the free troposphere (h_{m}), here denoted by h_{b}−h_{m}. In quasiequilibrium, the local time derivative vanishes and Eq. A2 becomes
where d is the BL height, V_{h} is the horizontal wind velocity and ${\dot{Q}}_{\mathrm{b}}$ is the radiative cooling at the top of the BL, which is assumed to be characteristic for the entire BL, i.e., constant. In addition, advection is assumed to be approximately constant throughout the BL. Assuming that d is small, net radiative cooling at the top of the BL and the horizontal advection of h will be small and can be neglected. Then, Eq. (A3) becomes
The weak temperature gradient approximation implies that horizontal advection in the thermodynamic equation can be neglected, and time changes also vanish in quasiequilibrium or steady state, such that thermodynamic balance is between vertical advection and diabatic heating (Sobel et al., 2001). In an ascending region, condensational heating is balanced by adiabatic cooling by an ascending parcel. In a descending region, of which the area fraction is far larger than an ascending region, adiabatic warming by subsidence is balanced by radiative cooling. The thermodynamic balance in the descending region can be formulated as $\mathit{\rho}{w}_{\mathrm{mid}}S=\dot{Q}$, where w_{mid} is the descending motion in the free troposphere and $S\equiv {c}_{\mathrm{p}}\frac{\mathrm{d}T}{\mathrm{d}z}+g$ is closely related to dry static stability, with g the gravitational acceleration. Assuming mass conservation and approximately constant vertical velocity, w_{e} and w_{mid} are approximated to be identical. Thus, using the thermodynamic balance, Eq. (A1) can be further written as
which illustrates the limitation of convection by longwave cooling in the environment.
Using Eqs. (A1), (A4) and (A5), we can then derive a diagnostic expression for M_{u} as
The above formulation of M_{u} is employed by the ITCZ diagnostic tool presented in Sect. 4.
The diagnostic tools can be found at: https://doi.org/10.5281/zenodo.10220388 (Jung, 2023).
Model output is published on Open Data LMU – Physics (https://doi.org/10.57970/P3AHBYBA70) and is available for download (Ruckstuhl et al., 2023).
This paper includes a video supplement, which shows a series of snapshots of precipitable water (shading) and rainfall rate (contour) in the tropical aquachannel simulations. The beginning of the analysis period is referred to as day 0, corresponding to the solid vertical lines in Fig. 1. It is available at https://doi.org/10.5281/zenodo.10360912 (Jung et al., 2023).
The project for the aquachannel runs was initially proposed by PK, CH and TJ. YR and RR set up and performed the simulations with advice from PK, CH and TJ. HJ constructed the diagnostic tool with advice from PK. HJ conducted the analyses and visualized them. HJ and PK interpreted the results. HJ wrote the first version of the manuscript. All authors reviewed and edited it.
At least one of the (co)authors is a member of the editorial board of Weather and Climate Dynamics. The peerreview process was guided by an independent editor, and the authors also have no other competing interests to declare.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
Hyunju Jung thanks Kerry Emanuel for the email exchange that initiated the use of his framework and Ron McTaggartCowan for suggesting the surface flux diagnostic framework. We also thank Oriol Tinto Prims for resolving technical issues regarding the simulations and three anonymous reviewers for their thoughtful comments that improved our paper. The research leading to these results has been performed within the subproject “B6” of the Transregional Collaborative Research Center SFB/TRR 165 “Waves to Weather” (Craig et al., 2021), http://www.wavestoweather.de (last access: 29 November 2023), funded by the German Research Foundation (DFG).
This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. SFB/TRR 165).
This paper was edited by Martin Singh and reviewed by three anonymous referees.
Arnold, N. P. and Randall, D. A.: Globalscale convective aggregation: Implications for the MaddenJulian Oscillation, J. Adv. Model. Earth Sy., 7, 1499–1518, https://doi.org/10.1002/2015MS000498, 2015. a
Bechtold, P.: Atmospheric moist convection, Meteorological Training Course Lecture Series, ECMWF, https://www.ecmwf.int/en/elibrary/79885atmosphericmoistconvection (last access: 29 November 2023), 2017. a
Bechtold, P., Köhler, M., Jung, T., DoblasReyes, F., Leutbecher, M., Rodwell, M. J., Vitart, F., and Balsamo, G.: Advances in simulating atmospheric variability with the ECMWF model: From synoptic to decadal timescales, Q. J. Roy. Meteor. Soc., 134, 1337–1351, https://doi.org/10.1002/qj.289, 2008. a, b
Becker, T., Bechtold, P., and Sandu, I.: Characteristics of convective precipitation over tropical Africa in stormresolving global simulations, Q. J. Roy. Meteor. Soc., 741, 4388–4407, https://doi.org/10.1002/qj.4185, 2021. a, b, c
Benedict, J. J., Medeiros, B., Clement, A. C., and Pendergrass, A. G.: Sensitivities of the hydrologic cycle to model physics, grid resolution, and ocean type in the aquaplanet Community Atmosphere Model, J. Adv. Model. Earth Sy., 9, 1307–1324, https://doi.org/10.1002/2016MS000891, 2017. a
Bony, S., Stevens, B., Frierson, D. M., Jakob, C., Kageyama, M., Pincus, R., Shepherd, T. G., Sherwood, S. C., Siebesma, A. P., Sobel, A. H., Watanabe, M., and Webb, M. J.: Clouds, circulation and climate sensitivity, Nat. Geosci., 8, 261–268, https://doi.org/10.1038/ngeo2398, 2015. a
Bretherton, C. S. and Khairoutdinov, M. F.: Convective selfaggregation feedbacks in nearglobal cloudresolving simulations of an aquaplanet, J. Adv. Model. Earth Sy., 7, 1765–1787, https://doi.org/10.1002/2015MS000499, 2015. a, b
Bretherton, C. S., Peters, M. E., and Back, L. E.: Relationships between water vapor path and precipitation over the tropical oceans, J. Climate, 17, 1517–1528, https://doi.org/10.1175/15200442(2004)017<1517:RBWVPA>2.0.CO;2, 2004. a
Bretherton, C. S., Blossey, P. N., and Khairoutdinov, M.: An energybalance analysis of deep convective selfaggregation above uniform SST, J. Atmos. Sci., 62, 4273–4292, https://doi.org/10.1175/JAS3614.1, 2005. a
Craig, G. C. and Cohen, B. G.: Fluctuations in an equilibrium convective ensemble. Part I: Theoretical formulation, J. Atmos. Sci., 63, 1996–2004, https://doi.org/10.1175/JAS3709.1, 2006. a
Craig, G. C., Fink, A. H., Hoose, C., Janjić, T., Knippertz, P., Laurian, A., Lerch, S., Mayer, B., Miltenberger, A., Redl, R., Riemer, M., Tempest, K. I., and Wirth, V.: Waves to Weather: Exploring the limits of predictability of weather, B. Am. Meteorol. Soc., 102, E2151–E2164, https://doi.org/10.1175/BAMSD200035.1, 2021. a
Emanuel, K.: Inferences from simple models of slow, convectively coupled processes, J. Atmos. Sci., 76, 195–208, https://doi.org/10.1175/JASD180090.1, 2019. a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q
Emanuel, K. A.: Atmospheric convection, Oxford University Press on Demand, ISBN 9780195066302, 1994. a
Emanuel, K. A., David Neelin, J., and Bretherton, C. S.: On largescale circulations in convecting atmospheres, Q. J. Roy. Meteor. Soc., 120, 1111–1143, https://doi.org/10.1002/qj.49712051902, 1994. a
Fiedler, S., Crueger, T., D’Agostino, R., Peters, K., Becker, T., Leutwyler, D., Paccini, L., Burdanowitz, J., Buehler, S. A., U. Cortes, A., Dauhut, T., Dommenget, D., Fraedrich, K., Jungandreas, L., Maher, N., Naumann, A. K., Rugenstein, M., Sakradzija, M., Schmidt, H., Sielmann, F., Stephan, C., Timmreck, C., Zhu, X., and Stevens, B.: Simulated tropical precipitation assessed across three major phases of the coupled model intercomparison project (CMIP), Mon. Weather Rev., 148, 3653–3680, https://doi.org/10.1175/MWRD190404.1, 2020. a
Gehne, M., Wolding, B., Dias, J., and Kiladis, G. N.: Diagnostics of tropical variability for numerical weather forecasts, Weather Forecast., 37, 1661–1680, https://doi.org/10.1175/WAFD210204.1, 2022. a
Giorgi, F.: Sensitivity of simulated summertime precipitation over the western United States to different physics parameterizations, Mon. Weather Rev., 119, 2870–2888, https://doi.org/10.1175/15200493(1991)119<2870:SOSSPO>2.0.CO;2, 1991. a
Hartmann, D. L. and Michelsen, M. L.: No evidence for iris, B. Am. Meteorol. Soc., 83, 249–254, https://doi.org/10.1175/15200477(2002)083<0249:NEFI>2.3.CO;2, 2002. a
Hong, S.Y. and Dudhia, J.: Nextgeneration numerical weather prediction: Bridging parameterization, explicit clouds, and large eddies, B. Am. Meteorol. Soc., 93, ES6–ES9, https://doi.org/10.1175/2011BAMS3224.1, 2012. a
Hsu, C.W., DeMott, C. A., Branson, M., Reeves Eyre, J., and Zeng, X.: Ocean Surface Flux Algorithm Effects on Tropical IndoPacific Intraseasonal Precipitation, Geophys. Res. Lett., 49, e2021GL096968, https://doi.org/10.1029/2021GL096968, 2022. a, b
Jeevanjee, N.: Vertical velocity in the gray zone, J. Adv. Model. Earth Sy., 9, 2304–2316, https://doi.org/10.1002/2017MS001059, 2017. a
Johnson, R. H., Rickenbach, T. M., Rutledge, S. A., Ciesielski, P. E., and Schubert, W. H.: Trimodal characteristics of tropical convection, J. Climate, 12, 2397–2418, https://doi.org/10.1175/15200442(1999)012<2397:TCOTC>2.0.CO;2, 1999. a, b
Judt, F. and RiosBerrios, R.: Resolved Convection Improves the Representation of Equatorial Waves and Tropical Rainfall Variability in a Global Nonhydrostatic Model, Geophys. Res. Lett., 48, e2021GL093265, https://doi.org/10.1029/2021GL093265, 2021. a, b
Jung, H.: hyunjujung/ITCZdiagnostic: ITCZ diagnostic tool for WCD paper, Zenodo [code], https://doi.org/10.5281/zenodo.10220388, 2023.
Jung, H. and Knippertz, P.: Link between the timespace behavior of rainfall and 3D dynamical structures of equatorial waves in global convectionpermitting simulations, Geophys. Res. Lett., 50, e2022GL100973, https://doi.org/10.1029/2022GL100973, 2023. a
Jung, H., Knippertz, P., Ruckstuhl, Y., Redl, R., Janjic, T., and Hoose, C.: Video Supplement for “Understanding the dependence of mean precipitation on convective treatment and horizontal resolution in tropical aquachannel experiments”, Zenodo [video], https://doi.org/10.5281/zenodo.10360912, 2023.
Khairoutdinov, M. F. and Emanuel, K.: Intraseasonal variability in a cloudpermitting nearglobal equatorial aquaplanet model, J. Atmos. Sci., 75, 4337–4355, https://doi.org/10.1175/JASD180152.1, 2018. a
Khairoutdinov, M. F., Krueger, S. K., Moeng, C.H., Bogenschutz, P. A., and Randall, D. A.: Largeeddy simulation of maritime deep tropical convection, J. Adv. Model. Earth Sy., 1, 15, https://doi.org/10.3894/JAMES.2009.1.15, 2009. a
Kwon, Y. C. and Hong, S.Y.: A massflux cumulus parameterization scheme across grayzone resolutions, Mon. Weather Rev., 145, 583–598, https://doi.org/10.1175/MWRD160034.1, 2017. a
Landu, K., Leung, L. R., Hagos, S., Vinoj, V., Rauscher, S. A., Ringler, T., and Taylor, M.: The dependence of ITCZ structure on model resolution and dynamical core in aquaplanet simulations, J. Climate, 27, 2375–2385, https://doi.org/10.1175/JCLID1300269.1, 2014. a
Lang, T., Naumann, A. K., Buehler, S. A., Stevens, B., Schmidt, H., and Aemisegger, F.: Sources of Uncertainty in MidTropospheric Tropical Humidity in Global StormResolving Simulations, J. Adv. Model. Earth Sy., 15, e2022MS003443, https://doi.org/10.1029/2022MS003443, 2023. a
Lau, W. K., Kim, K.M., Chern, J.D., Tao, W.K., and Leung, L. R.: Structural changes and variability of the ITCZ induced by radiation–cloud–convection–circulation interactions: inferences from the Goddard Multiscale Modeling Framework (GMMF) experiments, Clim. Dynam., 54, 211–229, https://doi.org/10.1007/s0038201905000y, 2020. a
Li, F., Collins, W. D., Wehner, M. F., Williamson, D. L., Olson, J. G., and Algieri, C.: Impact of horizontal resolution on simulation of precipitation extremes in an aquaplanet version of Community Atmospheric Model (CAM3), Tellus A, 63, 884–892, https://doi.org/10.1111/j.16000870.2011.00544.x, 2011. a
Li, R. L., Studholme, J. H., Fedorov, A. V., and Storelvmo, T.: Precipitation efficiency constraint on climate change, Nat. Clim. Change, 12, 642–648, https://doi.org/10.1038/s4155802201400x, 2022. a
Lindzen, R. S., Chou, M.D., and Hou, A. Y.: Does the earth have an adaptive infrared iris?, B. Am. Meteorol. Soc., 82, 417–432, https://doi.org/10.1175/15200477(2001)082<0417:DTEHAA>2.3.CO;2, 2001. a
Mauritsen, T. and Stevens, B.: Missing iris effect as a possible cause of muted hydrological change and high climate sensitivity in models, Nat. Geosci., 8, 346–351, https://doi.org/10.1038/ngeo2414, 2015. a
Medeiros, B., Williamson, D. L., and Olson, J. G.: Reference aquaplanet climate in the Community Atmosphere Model, Version 5, J. Adv. Model. Earth Sy., 8, 406–424, https://doi.org/10.1002/2015MS000593, 2016. a
Mellor, G. L. and Yamada, T.: Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys., 20, 851–875, https://doi.org/10.1029/RG020i004p00851, 1982. a, b
Mlawer, E. J., Taubman, S. J., Brown, P. D., Iacono, M. J., and Clough, S. A.: Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlatedk model for the longwave, J. Geophys. Res.Atmos., 102, 16663–16682, https://doi.org/10.1029/97JD00237, 1997. a
Möbis, B. and Stevens, B.: Factors controlling the position of the Intertropical Convergence Zone on an aquaplanet, J. Adv. Model. Earth Sy., 4, M00A04, https://doi.org/10.1029/2012MS000199, 2012. a, b
Muller, C. and Takayabu, Y.: Response of precipitation extremes to warming: what have we learned from theory and idealized cloudresolving simulations, and what remains to be learned?, Environ. Res. Lett., 15, 035001, https://doi.org/10.1088/17489326/ab7130, 2020. a
Narsey, S., Jakob, C., Singh, M. S., Bergemann, M., Louf, V., Protat, A., and Williams, C.: Convective precipitation efficiency observed in the tropics, Geophys. Res. Lett., 46, 13574–13583, https://doi.org/10.1029/2019GL085031, 2019. a
Naumann, A. K., Stevens, B., and Hohenegger, C.: A moist conceptual model for the boundary layer structure and radiatively driven shallow circulations in the trades, J. Atmos. Sci., 76, 1289–1306, https://doi.org/10.1175/JASD180226.1, 2019. a
Neale, R. B. and Hoskins, B. J.: A standard test for AGCMs including their physical parametrizations: I: The proposal, Atmos. Sci. Lett., 1, 101–107, https://doi.org/10.1006/asle.2000.0022, 2000. a, b, c
Nolan, D. S., Tulich, S. N., and Blanco, J. E.: ITCZ structure as determined by parameterized versus explicit convection in aquachannel and aquapatch simulations, J. Adv. Model. Earth Sy., 8, 425–452, https://doi.org/10.1002/2015MS000560, 2016. a, b
Paccini, L., Hohenegger, C., and Stevens, B.: Explicit versus parameterized convection in response to the Atlantic Meridional Mode, J. Climate, 34, 3343–3354, https://doi.org/10.1175/JCLID200224.1, 2021. a, b, c
Palmer, T. and Stevens, B.: The scientific challenge of understanding and estimating climate change, P. Natl. Acad. Sci., 116, 24390–24395, https://doi.org/10.1073/pnas.1906691116, 2019. a
Pante, G. and Knippertz, P.: Resolving Sahelian thunderstorms improves midlatitude weather forecasts, Nat. Commun., 10, 3487, https://doi.org/10.1038/s41467019110814, 2019. a
Prein, A. F., Langhans, W., Fosser, G., Ferrone, A., Ban, N., Goergen, K., Keller, M., Tölle, M., Gutjahr, O., Feser, F., Brisson, E., Kollet, S., Schmidli, J., van Lipzig, N. P. M., and Leung, R.: A review on regional convectionpermitting climate modeling: Demonstrations, prospects, and challenges, Rev. Geophys., 53, 323–361, https://doi.org/10.1002/2014RG000475, 2015. a
Rajendran, K., Kitoh, A., and Srinivasan, J.: Effect of SST variation on ITCZ in APE simulations, J. Met. Soc. Jpn., 91, 195–215, https://doi.org/10.2151/jmsj.2013A06, 2013. a
Randall, D. A.: Beyond deadlock, Geophys. Res. Lett., 40, 5970–5976, https://doi.org/10.1002/2013GL057998, 2013. a
Raschendorfer, M.: The new turbulence parameterization of LM, COSMO Newsletter, 1, 89–97, http://www.cosmomodel.org/content/model/documentation/newsLetters/newsLetter01/newsLetter_01.pdf (last access: 29 November 2023), 2001. a, b
Raymond, D., Fuchs, Ž., Gjorgjievska, S., and Sessions, S.: Balanced dynamics and convection in the tropical troposphere, J. Adv. Model. Earth Sy., 7, 1093–1116, https://doi.org/10.1002/2015MS000467, 2015. a
Raymond, D. J.: Regulation of moist convection over the west Pacific warm pool, J. Atmos. Sci., 52, 3945–3959, https://doi.org/10.1175/15200469(1995)052<3945:ROMCOT>2.0.CO;2, 1995. a, b, c, d
Raymond, D. J., Bretherton, C. S., and Molinari, J.: Dynamics of the intertropical convergence zone of the east Pacific, J. Atmos. Sci., 63, 582–597, https://doi.org/10.1175/JAS3642.1, 2006. a
Retsch, M.H., Hohenegger, C., and Stevens, B.: Vertical resolution refinement in an aquaplanet and its effect on the ITCZ, J. Adv. Model. Earth Sy., 9, 2425–2436, https://doi.org/10.1002/2017MS001010, 2017. a
Retsch, M. H., Mauritsen, T., and Hohenegger, C.: Climate change feedbacks in aquaplanet experiments with explicit and parametrized convection for horizontal resolutions of 2,525 up to 5 km, J. Adv. Model. Earth Sy., 11, 2070–2088, https://doi.org/10.1029/2019MS001677, 2019. a, b, c, d
RiosBerrios, R., Medeiros, B., and Bryan, G.: Mean Climate and Tropical Rainfall Variability in Aquaplanet Simulations Using the Model for Prediction Across ScalesAtmosphere, J. Adv. Model. Earth Sy., 12, e2020MS002102, https://doi.org/10.1029/2020MS002102, 2020. a
RiosBerrios, R., Bryan, G. H., Medeiros, B., Judt, F., and Wang, W.: Differences in Tropical Rainfall in Aquaplanet Simulations With Resolved or Parameterized Deep Convection, J. Adv. Model. Earth Sy., 14, e2021MS002902, https://doi.org/10.1029/2021MS002902, 2022. a, b
Ruckstuhl, Y., Redl, R., Knippertz, P., Jung, H., and Janjic, T.: Output data of idealised tropical channel simulations of an aqua planet at 5km and 13km resolution with ICON for different convection conducts, LudwigMaximiliansUniversität München [data set], https://doi.org/10.57970/P3AHBYBA70, 2023. a
Sakradzija, M. and Hohenegger, C.: What determines the distribution of shallow convective mass flux through a cloud base?, J. Atmos. Sci., 74, 2615–2632, https://doi.org/10.1175/JASD160326.1, 2017. a
Sakradzija, M., Seifert, A., and Heus, T.: Fluctuations in a quasistationary shallow cumulus cloud ensemble, Nonlin. Processes Geophys., 22, 65–85, https://doi.org/10.5194/npg22652015, 2015. a
Sakradzija, M., Senf, F., Scheck, L., Ahlgrimm, M., and Klocke, D.: Local impact of stochastic shallow convection on clouds and precipitation in the tropical Atlantic, Mon. Weather Rev., 148, 5041–5062, https://doi.org/10.1175/MWRD200107.1, 2020. a, b, c, d
Satoh, M., Tomita, H., Yashiro, H., Kajikawa, Y., Miyamoto, Y., Yamaura, T., Miyakawa, T., Nakano, M., Kodama, C., Noda, A. T., Nasuno, T., Yamada, Y., and Fukutomi, Y.: Outcomes and challenges of global highresolution nonhydrostatic atmospheric simulations using the K computer, Prog. Earth Planet. Sci., 4, 1–24, https://doi.org/10.1186/s4064501701278, 2017. a
Schär, C., Fuhrer, O., Arteaga, A., Ban, N., Charpilloz, C., Di Girolamo, S., Hentgen, L., Hoefler, T., Lapillonne, X., Leutwyler, D., Osterried, K., Panosetti, D., Rüdisühli, S., Schlemmer, L., Schulthess, T. C., Sprenger, M., Ubbiali, S., and Wernli, H.: Kilometerscale climate models: Prospects and challenges, B. Am. Meteorol. Soc., 101, E567–E587, https://doi.org/10.1175/BAMSD180167.1, 2020. a
Schlemmer, L., Bechtold, P., Sandu, I., and Ahlgrimm, M.: Uncertainties related to the representation of momentum transport in shallow convection, J. Adv. Model. Earth Sy., 9, 1269–1291, https://doi.org/10.1002/2017MS000915, 2017. a
Schneider, T., Bischoff, T., and Haug, G. H.: Migrations and dynamics of the intertropical convergence zone, Nature, 513, 45–53, https://doi.org/10.1038/nature13636, 2014. a
Schwendike, J., Govekar, P., Reeder, M. J., Wardle, R., Berry, G. J., and Jakob, C.: Local partitioning of the overturning circulation in the tropics and the connection to the Hadley and Walker circulations, J. Geophys. Res.Atmos., 119, 1322–1339, https://doi.org/10.1002/2013JD020742, 2014. a
Scinocca, J. F. and McFarlane, N. A.: The variability of modeled tropical precipitation, J. Atmos. Sci., 61, 1993–2015, https://doi.org/10.1175/15200469(2004)061<1993:TVOMTP>2.0.CO;2, 2004. a
Seifert, A.: A revised cloud microphysical parameterization for COSMOLME, COSMO Newsletter, 7, 25–28, 2008. a
Senf, F., Voigt, A., Clerbaux, N., Hünerbein, A., and Deneke, H.: Increasing Resolution and Resolving Convection Improve the Simulation of CloudRadiative Effects Over the North Atlantic, J. Geophys. Res.Atmos., 125, e2020JD032667, https://doi.org/10.1029/2020JD032667, 2020. a, b
Sobel, A. H., Nilsson, J., and Polvani, L. M.: The weak temperature gradient approximation and balanced tropical moisture waves, J. Atmos. Sci., 58, 3650–3665, https://doi.org/10.1175/15200469(2001)058<3650:TWTGAA>2.0.CO;2, 2001. a, b, c
Stephan, C. C., Strube, C., Klocke, D., Ern, M., Hoffmann, L., Preusse, P., and Schmidt, H.: Gravity waves in global highresolution simulations with explicit and parameterized convection, J. Geophys. Res.Atmos., 124, 4446–4459, https://doi.org/10.1029/2018JD030073, 2019. a
Stephens, G. L.: Cloud feedbacks in the climate system: A critical review, J. Climate, 18, 237–273, https://doi.org/10.1175/JCLI3243.1, 2005. a
Stevens, B. and Bony, S.: What are climate models missing?, Science, 340, 1053–1054, https://doi.org/10.1126/science.1237554, 2013. a
Stevens, B., Satoh, M., Auger, L., Biercamp, J., Bretherton, C. S., Chen, X., Düben, P., Judt, F., Khairoutdinov, M., Klocke, D., Kodama, C., Kornblueh, L., Lin, S.J., Neumann, P., Putman, W. M., Röber, N., Shibuya, R., Vanniere, B., Vidale, P. L., Wedi, N., and Linjiong, Z.: DYAMOND: The DYnamics of the atmospheric general circulation modeled on nonhydrostatic domains, Prog. Earth Planet. Sci., 6, 61, https://doi.org/10.1186/s406450190304z, 2019. a
Stevens, B., Acquistapace, C., Hansen, A., Heinze, R., Klinger, C., Klocke, D., Rybka, H., Schubotz, W., Windmiller, J., Adamidis, P., Arka, I., Barlakas, V., Biercamp, J., Brueck, M., Brune, S., Buehler, S. A., Burkhardt, U., Cioni, G., CostaSuros, M., Crewell, S., Crueger, T., Deneke, H., Friederichs, P., Henken, C. C., Hohenegger, C., Jacob, M., Jakub, F., Kalthoff, N., Köhler, M., van Laar, T. W., Li, P., Löhnert, U., Macke, A., Madenach, N., Mayer, B., Nam, C., Naumann, A. K., Peters, K., Poll, S., Quaas, J., Röber, N., Rochetin, N., Scheck, L., Schemann, V., Schnitt, S., Seifert, A., Senf, F., Shapkalijevski, M., Simmer, C., Singh, S., Sourdeval, O., Spickermann, D., Strandgren, J., Tessiot, O., Vercauteren, N., Vial, J., Voigt, A., and Zängl, G.: The added value of largeeddy and stormresolving models for simulating clouds and precipitation, J. Met. Soc. Jpn., 98, 395–435, https://doi.org/10.2151/jmsj.2020021, 2020. a
Tian, B. and Dong, X.: The doubleITCZ bias in CMIP3, CMIP5, and CMIP6 models based on annual mean precipitation, Geophys. Res. Lett., 47, e2020GL087232, https://doi.org/10.1029/2020GL087232, 2020. a
Tiedtke, M.: A comprehensive mass flux scheme for cumulus parameterization in largescale models, Mon. Weather Rev., 117, 1779–1800, https://doi.org/10.1175/15200493(1989)117<1779:ACMFSF>2.0.CO;2, 1989. a, b, c
Tomassini, L.: The interaction between moist convection and the atmospheric circulation in the tropics, B. Am. Meteorol. Soc., 101, E1378–E1396, https://doi.org/10.1175/BAMSD190180.1, 2020. a
Tompkins, A. M. and Semie, A. G.: Impact of a mixed ocean layer and the diurnal cycle on convective aggregation, J. Adv. Model. Earth Sy., 13, e2020MS002186, https://doi.org/10.1029/2020MS002186, 2021. a
Webster, P. J.: Dynamics of the Tropical Atmosphere and Oceans, WileyBlackwell, ISBN 9780470662564, 2020. a, b, c, d
Wedi, N. P., Polichtchouk, I., Dueben, P., Anantharaj, V. G., Bauer, P., Boussetta, S., Browne, P., Deconinck, W., Gaudin, W., Hadade, I., Hatfield, S., Iffrig, O., Lopez, P., Maciel, P., Mueller, A., Saarinen, S., Sandu, I., Quintino, T., and Vitart, F.: A baseline for global weather and climate simulations at 1 km resolution, J. Adv. Model. Earth Sy., 12, e2020MS002192, https://doi.org/10.1029/2020MS002192, 2020. a, b
Weisman, M. L., Skamarock, W. C., and Klemp, J. B.: The resolution dependence of explicitly modeled convective systems, Mon. Weather Rev., 125, 527–548, https://doi.org/10.1175/15200493(1997)125<0527:TRDOEM>2.0.CO;2, 1997. a, b
Wu, X. and Moncrieff, M. W.: Collective effects of organized convection and their approximation in general circulation models, J. Atmos. Sci., 53, 1477–1495, https://doi.org/10.1175/15200469(1996)053<1477:CEOOCA>2.0.CO;2, 1996. a
Zängl, G., Reinert, D., Rípodas, P., and Baldauf, M.: The ICON (ICOsahedral Nonhydrostatic) modelling framework of DWD and MPIM: Description of the nonhydrostatic dynamical core, Q. J. Roy. Meteor. Soc., 141, 563–579, https://doi.org/10.1002/qj.2378, 2015. a
Zhang, C., McGauley, M., and Bond, N. A.: Shallow meridional circulation in the tropical eastern Pacific, J. Climate, 17, 133–139, https://doi.org/10.1175/15200442(2004)017<0133:SMCITT>2.0.CO;2, 2004. a
Zhou, W., Leung, L. R., and Lu, J.: Linking largescale doubleITCZ bias to localscale drizzling bias in climate models, J. Climate, 35, 4365–4379, https://doi.org/10.1175/JCLID220336.1, 2022. a
Zipser, E. J.: Some views on “hot towers” after 50 years of tropical field programs and two years of TRMM data, in: Cloud Systems, Hurricanes, and the Tropical Rainfall Measuring Mission (TRMM), edited by: Tao, W.K. and Adler, R., Springer, 49–58, https://doi.org/10.1007/9781878220639_5, 2003. a
 Abstract
 Introduction
 Aquachannel experiments
 Overview of aquachannel experiments
 ITCZ diagnostic tool
 Application
 Conclusions
 Appendix A: Emanuel's 2019 framework
 Code availability
 Data availability
 Video supplement
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Aquachannel experiments
 Overview of aquachannel experiments
 ITCZ diagnostic tool
 Application
 Conclusions
 Appendix A: Emanuel's 2019 framework
 Code availability
 Data availability
 Video supplement
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References