We use version 2.6.3 of the ICOsahedral Nonhydrostatic (ICON) model in the numerical weather prediction (NWP) configuration. The model solves the fully compressible nonhydrostatic atmospheric equations of motion on an icosahedral–triangular Arakawa-C grid. Radiation is computed using the Rapid Radiative Transfer Model (RRTM) . A single-moment microphysical scheme is used to predict cloud water, rain water, cloud ice and snow . A turbulent kinetic energy scheme is used for the representation of turbulent mixing and surface-to-atmosphere transfer . Our model configuration closely follows the operational setup, including a full non-linear 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 . There is no feedback of the atmosphere on the ocean and the underlying water surface, effectively making the ocean an indefinite energy source.

To spin up our aquachannel simulations, we adapt the modeling practice used in . 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 . 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 . The 40 km global aquaplanet experiment is run for 120 simulation days (dashed black line in Fig. ), 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. ). 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 time-invariant 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 spin-up 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. ). The boundary conditions of the high-resolution 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. ), after a 3 d spin-up. The output time step of the 13 and 5 km aquachannel runs is hourly.

To illustrate the modeling approach, Fig.  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 kgm-2, but by day 50 the distribution has widened with a broad dry maximum around 25 kgm-2 and a narrower secondary maximum near 55 kgm-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.,, as the large-scale 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 coarse-resolution 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. ). 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 .

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 kgm-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. ). This allows us to have relatively few days of spin-up for the high-resolution 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 two-way 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. ). 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 ): (a) an experiment named P13 where the deterministic deep and shallow convection schemes are turned on (already shown in the context of Fig. ), (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 is replaced by a stochastic scheme . In the stochastic shallow convection scheme, the shallow-cloud ensemble is represented based on the theory of . 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 . Note that the four experiments with the horizontal grid spacing of 13 km share the same aquaplanet runs as spin-up and that we modify the convective treatment when the channel geometry is introduced (dotted black line at day 210 in Fig. ). Other than the different representations of convection, the setups remain identical among the coarse-resolution aquachannel experiments.

At day 314, the horizontal grid spacing is further reduced to 5 km, creating two high-resolution runs (Table ): P5 with the deterministic deep and shallow convection schemes and E5 with explicit deep and shallow convection. Both high-resolution simulations are initialized with P13 at day 314, but spin-up is 2 d longer for E5 than P5, the analysis period of which begins at day 317 (solid pink bar in Fig. ). The boundary conditions for the high-resolution runs are identical to the coarse-resolution ones.

In the following sections, we analyze the last 40 d of each aquachannel experiment, e.g., after spin-up (day 274–314 for the coarse-resolution runs). To compare the experiments with different horizontal resolutions, model grids are coarsened on a 0.2∘ latitude–longitude grid, using a conservative remapping.

The latitudinal distributions of zonally and time-averaged precipitation are shown in Fig. a. 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 mmd-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 coarse-resolution runs (13 km) but to a lesser extent, E5 (7.5 mmd-1) has higher mean rainfall in the ITCZ than P5 (6.72 mmd-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 long-lasting 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 mmd-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.,. 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. b). Comparing the results among the 13 km runs, light and moderate rains (< 30 mmd-1) occur more frequently in P13 than the others, which produce extreme rainfall rates of 200 mmd-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 coarse-resolution runs. This resolution dependency differs from , 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 large-scale 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. a. 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 and occasionally intense grid point storms . 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 .

Figure  shows a cross section of the time mean meridional-height mass stream function and zonal wind. The meridional-height 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 meridional-height mass stream function. P13 features a largely Equator-symmetric troposphere-deep Hadley circulation with low-level convergence and corresponding upper-level divergence in the ITCZ (Fig. a). 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 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. ). Strong westerly upper-tropospheric jets occur at the outer edges of these narrow Hadley cells, reaching an average speed of 30 ms-1. These in principle resemble the subtropical jets of the real atmosphere but shifted closer to the Equator and weaker. The low-level 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. b–f) generally produce similar large-scale 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 1.84×1011, 1.81×1011 and 1.82×1011 kgs-1 for E13, S13 and SS13, respectively, compared to P13 (1.3×1011 kgs-1). This tendency is also present in the high-resolution runs, leading to maximum values of the mass stream function of 1.36×1011 and 1.67×1011 kgs-1 for P5 and E5, respectively. The magnitude of the simulated circulation is in agreement with other aquaplanet studies , but P13 is at the lower end of the range found in these studies. The stronger large-scale 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 ms-1. This largely agrees with , who showed an increase in surface winds from a parameterized low-resolution run to an explicit high-resolution run using the ICON-NWP 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 large-scale circulation is considerable. Additionally, low-level zonal winds exhibits asymmetry, possibly due to the long-term memory that is also evident in the ITCZ. The asymmetry in surface winds at 5 km is shown clearly in Sect. .

One interesting aspect is that the runs with explicit deep convection (E13, S13, SS13 and E5) exhibit equatorial easterlies in the mid-troposphere 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, short-lived rainfall (see the “Video supplement”) because the sufficient shear is needed to generate long-lived organized systems . 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 coarse-resolution 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. b–d). Such a shallow meridional circulation is observed in the eastern Pacific, but the flow diverges at lower altitudes than in our experiments .

The time and zonal mean of surface enthalpy fluxes is shown in Fig. a. P13 has Fh maxima in the trades and a local minimum in the ITCZ (black line in Fig. a), similar to the situation over real-world tropical oceans but confined to a narrower latitudinal stretch. The coarse-resolution experiments with explicit deep convection (E13, S13 and SS13) share similar latitudinal distributions, but Fh 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. ). At 13 km, therefore, the main difference in Fh is due to the treatment of deep convection rather than shallow convection. The dependence of Fh on convective treatment is consistent with that of the Hadley circulation and thus surface winds described in Sect. . The high-resolution experiments exhibit a similar Fh distribution and dependence on convective treatment to the coarse-resolution ones. However, Fh 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, Fh is reduced in the ITCZ but enhanced in the trade wind zone, leading to an overall small increase by 2 %. From E13 to E5, Fh 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. ) and examine their statistical distribution (Sect. ).

Figure b shows the vertical difference in h between the BL and the lower troposphere described by hb-hm (Eq. ). This contrast is key for BLQE, as it determines the reduction of h in the BL through convective downdrafts and large-scale subsidence. In the moist ITCZ region hb-hm has minimum with relatively small differences among the six simulations and P13 showing the smallest values. Then hb-hm 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, hb-hm is smallest for E13 among the coarse-resolution 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 high-resolution runs show similar patterns of hb-hm as in the coarse-resolution runs, but with relatively smoother distributions in hb-hm in the inner tropics. In the ITCZ, hb-hm slightly increases from 13 to 5 km grid spacing. In the outer tropics, P5 exhibits overall smaller hb-hm than P13, while hb-hm 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 hb-hm patterns. The solid and dash-dotted lines in Fig.  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.  show corresponding profiles of dry static energy (s=ϕ+cpT) 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 coarse-resolution runs in the ITCZ (Fig. a). 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 hb differs little, changing by 0.1 %–0.3 % only (see Table ). Note that the BL height is fixed for our diagnostic tool to the layer of 10–500 m, but calculating hb with alternating BL heights below or slightly above 500 m does not impact the results. Among the coarse-resolution runs, E13 shows the lowest value of hb, possibly related to more frequent and more intense convective downdrafts in line with more intense rainfall (Fig. b). 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 hb. 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 hm, which varies between 327.4 and 328.9 kJkg-1 (Table ). Given that differences within and above the BL are largely consistent between the coarse-resolution runs, hb-hm in the ITCZ increases by 3 %–6 % from P13 to S13, SS13 and E13, as also evident from Fig. b.

At 5 km, changes in h profiles due to convective treatment are largely consistent with the results at 13 km (Fig. a), 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 hb-hm in E5 compared to P5 (Fig. b). 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. a). This lower h is more pronounced in the lower troposphere than in the BL, leading to reduced hm for P5 and E5 than for P13 and E13 (Table ). Thus, hb-hm in the ITCZ is larger for the high-resolution runs (Fig. b).

In the trade wind belt, differences in hb-hm among the experiments are larger than in the ITCZ (Fig. b). This corresponds with more marked differences in the vertical profiles of h (Fig. b). 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 hb among the coarse-resolution experiments are systematic, decreasing from P13 to S13 or SS13 to E13 by 1.13–2.49 kJkg-1 (see Table ). Meanwhile, hm varies little among the explicit deep convection runs (E13, S13 and SS13; colored vertical bars in Fig. b) but decreases from P13 to other coarse-resolution runs by 1.8–2.23 kJkg-1 (Table. ). 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 hb-hm from one to another, with E13 showing the lowest value, then P13 and SS13, and finally with S13 showing the highest value (Fig. b).

For the high-resolution runs, the effect of convective treatment on the h profile is consistent with that for the coarse-resolution runs, again with the drier h profile for E5 than for P5 in the trade wind belt (Fig. b). Compared to the coarse-resolution runs, both P5 and E5 moisten the lower troposphere, which may be associated with the broad ITCZ (Fig. a), 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 high-resolution runs, the resulting hb-hm (Fig. b) decreases from parameterized to explicit deep and shallow convection at 13 km but increases at 5 km.

In the subsidence region (Fig. c), the BL height is 500 m in all experiments, and hb is fairly similar across the runs (Table. ). Among the coarse-resolution runs, E13 shows the largest value of hm, 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 , 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. ), lower-tropospheric moisture is presumably dominated by local mixing rather than large-scale 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) . Accordingly, hm for SS13 increases from S13 by more than 0.5 kJkg-1, which is less than 0.5 kJkg-1 in other latitudinal regions (Table. ). Thus, we see a systematic change of hb-hm in the subsidence region, representing some effects of local mixing. The high-resolution runs show visually identical h profiles (Fig. c) with small differences in hb and hm (Table ).

In summary, hb and hm 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 hb and hm result in complex patterns of hb-hm. 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 hb-hm (Fig. b).

Figure d 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 Q˙ is overall the flattest, with it marking its minimum in the ITCZ and larger values in the outer tropics (Fig. d). 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. ) and the latitudinal-height distribution of the radiative temperature tendency (Fig. ).

Figure  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. b). 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 , consistent with the highest cloud cover relative to the other runs (Fig. ).

P13 show the largest contrast to E13 with reduced cloud cover, especially in the outer tropics (Fig. ), and with some differences in the pattern of radiative cooling (Fig. a). 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 . 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 coarse-resolution experiments (not shown). Note that the near-tropopause warming is not included in Q˙ (Fig. d), 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. a). 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 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. ). While the free-tropospheric cooling by radiation is consistent with E13, S13 (Fig. c) 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 Q˙ of all runs in the outer tropics (Fig. d). SS13 shows the cloud cover closest to P13 among the other coarse-resolution runs (Fig. ). Free-tropospheric radiative cooling for SS13 (Fig. d) 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 . Consequently, this leads to overall similar Q˙ to E13 (Fig. d), despite the different vertical structures of radiative temperature tendency.

The high-resolution 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. ). P5 is characterized by net warming below the tropopause (Fig. e), which is slightly shifted to the Northern Hemisphere where the maximum mean rainfall is (Fig. a). The distribution of radiative cooling for E5 (Fig. f) exhibits a trimodal structure, but net cooling is generally weaker than that for E13, leading to reduced Q˙ for E5 in the outer tropics compared to other explicit deep convection runs (Fig. d).

In the conceptual framework, the effects of radiative cooling need to be considered relative to the dry stability S (Eq. ), which is shown in Fig. e. For P13 and P5, S is nearly constant between 15∘ N and 15∘ S with a value of 14.2 and 14.1 Jkg-1m-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 Jkg-1m-1 compared to parameterized deep convection at the same resolution (Table ). 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. a. (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 Jkg-1m-1 from one experiment to another (Table ): 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. b and c).

In the conceptual model, the ratio of radiative cooling in the lower troposphere Q˙ and dry stability S are considered (second term in Eq. ). As radiation and stability in all runs largely compensate for each other, the ratio is almost identical and also relatively constant over latitudes (Fig. f). In contrast, the effect of surface enthalpy fluxes and vertical contrast of h leads to substantial variations in the first term in Eq. () (Fig. c). This demonstrates that despite all the details discussed above, the thermodynamic balance between radiation and stability hardly drives differences between the simulations.

Up to this point, we have examined the distributions of each term on the right-hand side in Eq. (). To conclude our diagnostic framework, here we discuss Mu, ϵp and 〈qv〉, which are directly related to rainfall (Eq. ), and then link this discussion to the entire framework. We start with quantities in the ITCZ. Surprisingly, ϵp (Fig. h) is quite similar across the runs with a maximum value of 0.63–0.657 (Table ). Note that the time-averaged quantities are taken into account here, but timely varying Pr and ϵp can be strongly correlated . Furthermore, ϵp can depend significantly on how convection is treated in models , but the different convective treatments do not alter ϵp in the ITCZ in our case. Meanwhile, 〈qv〉 (Fig. i) marginally decreases from parameterized to explicit convection and from the coarse to high resolutions (Table. ), indicating that P13 has the moistest atmosphere and E5 is the driest in the ITCZ, as also seen in profiles of h (Fig. a). Given the almost identical ϵp and the marginal changes in 〈qv〉, Mu must substantially differ among the runs to match the differences in rainfall (Eq. ). Correspondingly, the latitudinal distribution of Mu (Fig. g) closely matches that of mean rainfall (Fig. a). For example, the coarse-resolution runs show that Mu in the ITCZ increases from parameterized to explicit deep convection by 30 %–38 % (Table. ), which is largely consistent with the mean rainfall increase (about 35 %). For the high-resolution runs, Mu 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. a), while Mu increases by 10 % (Fig. g). This is in fact compensated for by the decrease of 6 % in 〈qv〉 (Fig. i).

In the outer tropics, ϵp sharply decreases, reaches a minimum at around 9–12∘ N/S and beyond this increases again with latitude (Fig. h). The differences among the runs are substantial in the outer tropics, with ϵp varying between 0.221 and 0.310 (highest in P13). Mean values of 〈qv〉 slightly vary, with P13 marking the greatest value of 2.77 gkg-1 (Fig. i). From the ITCZ to the outer tropics Mu sharply decreases to about 0.015 kgm-2s-1 and beyond the minimum slightly increases again with latitude (Fig. g). These patterns with a minimum and marginal increase in ϵp and Mu are also observed in Pr (Fig. a), yet the differences in Pr in the outer tropics do not vary as much as those in ϵp or 〈qv〉 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 Mu from one experiment to another most closely matches that in Pr, with almost identical ϵp and marginal changes in 〈qv〉. When revisiting all input variables in Eq. () for our diagnostic tool (Fig. ), Mu is shaped by Fh/(hb-hm), which describes BLQE, while variations in Q˙ and S largely compensate for each other (Fig. c and f). Given almost constant ϵp in the ITCZ, an increase in Mu increases Md, through Md=(1-ϵp)Mu, which carries low h from the lower troposphere into the BL to balance enhanced Fh. At 13 km, the treatment of deep convection produces the main differences in Mu by 30 %–38 % between the explicit and parameterized versions, which is associated with substantial differences in Fh (20 %–28 %). This change in Fh is associated with changed U‾h (Sect. ), which in turn is closely linked to the Hadley circulation (Sect. ). 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 Mu. Decreased Mu with increasing resolution is associated with the combined effects of increased hb-hm and suppressed Fh, which is shaped by U‾h through the large-scale circulation. This indicates that the thermodynamics at low altitudes become important at higher resolution. The difference in Mu between P5 and E5 is again associated with that in Fh, which in fact changes due to Δq rather than U‾h. Furthermore, a marginal change in 〈qv〉 has a small contribution to mean rainfall, showing again the importance of thermodynamic properties at low altitudes because substantial changes in 〈qv〉 are found below 6 km, inferring from h profiles (Fig. ).

Equation () 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. ) reveals that the BL radiative cooling term can safely be ignored while the advection term is not fully negligible (dQ˙b∼ 1 Wm-2, dρVh⋅∇h∼ 10 Wm-2, Fh∼ 100 Wm-2). With this, Eq. () can be expressed as 5Mu=11-ϵp(Fh-dρVh⋅∇hhb-hm-Q˙S), where d is the BL height and Vh 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 Jkg-1km-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 kgm-2.

Figure  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. a) produces subtle meridional advection in the BL with near-zero values at the Equator that increases away from the Equator, leading to an average of 4.4 and 3.3 Wm-2 in the ITCZ for P13 and P5, respectively (Table ). The small advection term in the ITCZ decreases Mu by about 0.0005 kgm-2s-1 and increases ϵp by about 0.01 compared to the advection-free diagnostic tool (Table ). At around 10∘ N/S, BL meridional advection reaches a maximum of 14 Wm-2 and beyond that decreases with increasing latitude, leading to a small increase in ϵp by 0.02 and a small decrease in Mu by 0.001 kgm-2s-1.

In contrast to parameterized deep and shallow convection (P13 and P5), E13 exhibits the largest contribution of BL meridional advection (Fig. a). In the ITCZ, the averaged advection is 27.3, consequently reducing Mu by 0.003 kgm-2s-1 but increasing ϵp by 0.05 (Table ). At around 7∘ N/S the advection term shows local maxima of 32.8 Wm-2 and then sharply decreases with increasing latitude. The overall large meridional advection term for E13 is consistent with intensified U‾h (Fig. a) and a greater h change in the BL with latitude (Fig. ). 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 ). Furthermore, the peaks are located further away in E5 than in E13. Despite these differences, the resulting changes in ϵp and Mu are at best modest for E5 and E13 (Fig. b 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 Wm-2 than E13 (Fig. a). The advection consideration leads to a decrease in Mu by 0.0013 kgm-2s-1 and an increase in ϵp by 0.039 in the ITCZ (Table ), which changes less strongly in the outer tropics (Fig. b and c). For S13 meridional advection closely follows that for SS13 (Fig. a), leading to averaged advection of 20.9 Wm-2 in the ITCZ (Table ). 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 U‾h in the ITCZ (Fig. a) and the slightly reduced meridional h gradient for S13 (Fig. a and b). Despite this difference in shape, Mu and ϵp in S13 and SS13 change by almost the same degree when considering meridional advection.

In summary, the meridional advection slightly increases ϵp and slightly decreases Mu for all cases. For example, ϵp increases in the ITCZ by 8 % from P13 to the other coarse-resolution runs (E13, S13 and SS13), whereas it is almost identical without considering the advection term (Sect. ). However, the main differences among the runs are evident in Mu, compared to ϵp and 〈qv〉, meaning that the close association between Mu and rainfall remains strong.