#### 5.1. Environmental conditions by rain rate

Many studies have found a strong relationship between tropospheric water vapour and precipitation in observations (Bretherton *et al.*, 2004; Peters and Neelin, 2006; Holloway and Neelin, 2009) and CSRMs (Grabowski, 2003; Derbyshire *et al.*, 2004; Bretherton *et al.*, 2005). However, parametrized convection usually lacks this strong relationship; in particular, models with parametrized convection tend to have too much light rain falling in relatively dry tropospheric conditions (Thayer-Calder andRandall, 2009, their figure 4). The sensitivity of convection to free-tropospheric moisture has been shown to depend largely on entrainment within convective parametrizations and simple plume models, with larger entrainment rates yielding a greater, more-realistic sensitivity to moisture by helping to suppress deep convection in regions with drier free-tropospheric air (Derbyshire *et al.*, 2004; Holloway and ???Neelin, 2009).

To investigate the distribution of subsaturation (the relevant quantity to assess the impact of entrainment drying on convective plumes), Figure 3 shows the saturation deficit, defined as *q*_{s} − *q*, where *q* is the specific humidity and *q*_{s} is the saturation specific humidity, for the different model runs and ECMWF analyses at different precipitation rates. (Note that we use TRMM precipitation for the ECMWF panel, not the precipitation in the analyses themselves, since the satellite observations should be closer to truth than the operational analyses; ECMWF column water vapour agrees well with TMI column water vapour, as discussed above in section 2, and Holloway and ???Neelin (2009) showed that column water vapour is highly correlated with lower-free-tropospheric moisture in tropical radiosonde data). Standard-error values are nearly all below 0.1 g kg^{−1}, with a few values reaching 0.6 in the highest precipitation bins because of the limited sample numbers. The 4 km *3Dsmag* run shows significant subsaturation at mid-levels for light rain rates (below 1.0 E−1 mm h^{−1}) but much lower values at moderate and high rain rates, which is overall very similar to the ECMWF analyses. In contrast, the 12 km *param* run shows much drier free-tropospheric air at low and moderate rain rates (below 3.0 E−1 mm h^{−1}). On the other hand, the 4 km *2Dsmag* run shows even drier lower-free-tropospheric air for nearly all precipitation rates. The 12 km *3Dsmag* explicit convection run also shows a fairly dry lower free troposphere, although it has a fairly small gradient of saturation deficit with increasing precipitation rate at these levels, which agrees more with ECMWF.

Since the 4 km *2Dsmag* run still produces a precipitation distribution similar to observations, it seems unlikely that the sensitivity of convection to free-tropospheric moisture is the main process lacking in the 12 km *param* convection that would be needed to reproduce a realistic precipitation distribution. However, it is a likely candidate for other problems, shown in both the 12 km *param* and 4 km *2Dsmag* runs but not the 4 km *3Dsmag* run, in large-scale organization, including MJO propagation (Holloway, Woolnough, and Lister, 2012; pers. comm.). These composites of subsaturation are also important in understanding the implications of the different precipitation distributions for large-scale waves and circulations (discussed in section 6 below). One feature of Figure 3 is that the 12 km *param* model is even closer to saturation than ECMWF and all of the other models for rain rates greater than 2.0 E+0 mm h^{−1} and vertical levels in range 600–400 hPa; this unique behaviour in the parametrized convection run at high rain rates is discussed further below as an indicator of a possible mechanism for its lack of heavy rain.

Moist static energy (*h*) and saturation moist static energy (*h*_{s}) profiles are shown in Figure 4, with corresponding profiles of saturation deficit and relative humidity shown in Figure 5, for the overall mean and five selected rain rates for the 12 km *param*, 4 km *3Dsmag* and 4 km *2Dsmag* models. Standard-error values are all below 0.2 kJ kg^{−1} for Figure 4 and below 0.1 g kg^{−1} and 1% for Figure 5, with values approaching 1 kJ kg^{−1}, 0.5 g kg^{−1} and 7%, respectively, at 1.0 E+1. Note that we exclude the 12 km *3Dsmag* model from further analysis for visual clarity and because that model was primarily run as confirmation of our hypothesis that explicit versus parametrized convection made a much bigger difference in behaviour than horizontal resolution.

Since *h*_{s} depends only on temperature and pressure, the *h* and *h*_{s} curves together can be used to estimate the undiluted CAPE and convective inhibition for parcels lifted from various levels in the boundary layer. A lifted parcel will approximately conserve *h*, and once it has reached saturation its buoyancy is proportional to its difference from the *h*_{s} curve; undiluted CAPE is proportional to the positive area between this curve and a straight vertical line drawn from a boundary-layer *h* value. However, this does not include the effects of entrainment mixing, which are likely to be very important where there are large saturation deficits.

Comparing the mean values for the three model versions, there is a significantly lower amount of *h* through much of the troposphere for the 4 km *3Dsmag* model and 4 km *2Dsmag* model, consistent with much drier air in the case of the 4 km *2Dsmag* model and cooler, slightly drier air in the case of the 4 km *3Dsmag* model (see Figure 5). This lower mean *h* may simply reflect the larger number of grid points with rain rates below 4.2 E−2 mm h^{−1} for the 4 km model runs, suggesting that these models are more able to produce suppressed convective regions. At upper-tropospheric levels, *h*_{s} shows a slightly less-stable (more-vertical) profile for the 12 km *param* model. There is also a much shorter vertical distance before lifted undiluted near-surface parcels become buoyant for the 4 km *3Dsmag* model (∼1 km compared with ≥2 km for the other models), possibly indicating more-realistic shallow convection. In addition, there is a sharper kink near the freezing level for the 12 km *param* model *h*_{s}.

While mean relative humidity and saturation deficit are very similar for the 12 km *param* model and the 4 km *3Dsmag* model, for the 4 km *3Dsmag* model this deficit is steadily reduced with increasing rainfall, starting at lower levels and gradually being reduced more and more at upper levels (Figure 5). The 12 km *param* model has much larger saturation deficits than the 4 km *3Dsmag* model at light rain rates (1.0 E−1 mm h^{−1} and below) and even slightly larger saturation deficits at moderate rates (3.0 E−1 mm h^{−1}), although this model is extremely close to saturation at higher rain rates compared with the other models. The 4 km *2Dsmag* version is even dryer than the 12 km *param* model in the boundary layer and lower troposphere at nearly all rain rates, consistent with the saturation deficit composite contours in Figure 3. This is likely due to insufficient mixing of the boundary-layer scheme for this version of the model. However, in the middle and upper troposphere at the two highest rain rates, both of the 4 km model versions are significantly drier than the 12 km *param* model.

While the 4 km *2Dsmag* model is unable adequately to simulate large-scale variability such as the MJO, as mentioned above, it is instructive to compare thermodynamic profiles of this model with those of the 12 km *param* and 4 km *3Dsmag* models to make inferences about why the explicit convection runs (the two 4 km runs in this case) are better able to simulate the observed precipitation distribution. For instance, there is a much more consistent temperature structure in the upper troposphere in the 4 km models across the different rain rates, with *h*_{s} increasing somewhat linearly with height. In contrast, the 12 km *param* model has a more vertical *h*_{s} profile in the upper troposphere, which becomes much warmer at high precipitation rates. One possibility is that heavy rainfall in the 12 km *param* model is only possible for very moist lower-tropospheric conditions where the entire region is fairly close to saturation, leading to less drying of convective plumes via entrainment and therefore warmer moist adiabats and increased convective heating which could then quickly stabilize the profile and reduce the likelihood of more heavy rain in that region.

Composites of vertical velocity on precipitation rate are shown in Figure 6 for four model versions (we do not include ECMWF operational analyses because their vertical velocity fields are likely to be more representative of the forecast model physics than the assimilated data). Standard-error values are all below 0.01 Pa s^{−1} for precipitation-rate bins below 2.0 E+0, with a few values reaching 0.7 for the highest precipitation bins. These velocities are the average on the 1° grid of the resolved vertical velocities. Parametrized convection yields no net vertical velocity on the resolved original grid scale, since convective updraughts are exactly balanced by downdraughts and subsidence. Explicit convection has no such imposed balance at any scale (other than mass continuity). Averaging on to the 1° grid should yield small net vertical motion associated with resolved convective motion (including compensating subsidence) in the explicit convection runs (particularly for the 4 km explicit convection runs, which have many more grid boxes in each 1° grid box). This is equivalent to the scale-separation assumption in convective parametrizations. Any differences between the 1° average vertical velocities of the parametrized convection and explicit convection models reflect weaknesses in that assumption or the effect of differently simulated convective processes on the large-scale circulation, not the averaging technique used.

The 4 km *3Dsmag* model and the 12 km *3Dsmag* model are the only model versions to have upward velocity (at least at low levels) at all precipitation rates, suggesting explicit shallow convection at low rain rates. In fact, the 4 km *3Dsmag* model and the 12 km *3Dsmag* model both have percentages of rainfall contributed by the convective parametrization that range from less than 1% at and above rain rates of 1.0 E−1 mm h^{−1} to 5% and 3%, respectively, at 1.0 E−2 mm h^{−1}. In contrast, the 4 km *2DSmag* model, while still having less than 1% of all rainfall contributed by the convective parametrization, has about 4% of its rainfall at 1.0 E−1 mm h^{−1} contributed by convective parametrization and this increases to 36% as the rain rate goes down to 1.0 E−2 mm h^{−1}, showing that much of the rain at very low rain rates is coming from the parametrization. The 12 km *param* model has over 99% of its rainfall contributed by the convective parametrization at all rainfall rates below 3.0 E+0 mm h^{−1}, with this percentage dropping to as low as 86% within the range of highest model rain rates. The differences in the amount of parametrized convection at very low rain rates might also explain why there is relatively less light rain for the 4 km *3Dsmag* and 12 km *3Dsmag* models.

The 12 km *param* model tends to have relatively top-heavy upward velocity profiles for high precipitation values and subsidence at all levels for very low rain rates. For very heavy rainfall, the 12 km *param* model is more top-heavy than any of the explicit convection runs; we know from Figure 4 that it is warmer in the 12 km *param* run in the upper troposphere than for either of the 4 km runs for the highest rain rate, again suggesting that it may be very difficult to maintain heavy rain for long because of very strong stabilization effects. The different precipitation distribution in the 12 km *param* model relative to the other model runs (Figure 2) means that the overall pattern of resolved free-tropospheric vertical velocity in rainy regions will be much weaker than for the explicit convection runs, which will have implications for large-scale waves and circulation (see discussion in section 6 below). Another difference between the 12 km *param* model and the other model runs is that the 12 km *param* model has upward motion through most of the troposphere for rain rates greater than 2.0 E−1 mm h^{−1}, including within the range of its preferred rain rate, whereas the other models have mid-tropospheric subsidence for rain rates below about 4.0 E−1 mm h^{−1}.

#### 5.2. Heating and moistening rates by rain rate

The temperature and moisture budgets for a given large-scale region can be illustrated by writing the total tendency and large-scale advection terms equal to the apparent heat source *Q*_{1}, radiative heating *Q*_{R} and apparent moisture sink *Q*_{2} (Yanai *et al.*, 1973; Ciesielski *et al.*, 1999) as follows:

- (3)

- (4)

where *θ* is potential temperature, *q* is specific humidity, **v** is the vector horizontal velocity, *w* is the vertical velocity, *c*_{p} is the specific heat of dry air at constant pressure, *L* is the latent heat of condensation, *ρ* is the density, Π is the Exner function defined as

*R* is the gas constant for dry air, *p* is the pressure and *p*_{0} = 1000 hPa is the reference pressure. {} denotes the horizontal average at a single level and time over the ‘large scale’ (1° in this case). Although these equations are often used to describe the separation between resolved and parametrized processes, they can equally be used for any area average to distinguish between net advective fluxes into the area and net advective (vertical transport) fluxes, turbulent fluxes and physical processes occurring within the area. Here we apply this budget analysis to the 1° grid boxes. We will therefore use the terms ‘large-scale’ and ‘subgrid’ with this in mind, as described below. Note that we have defined *Q*_{R} as a separate term rather than including it as part of *Q*_{1}, unlike Yanai *et al.* (1973) and some other articles. *Q*_{1} and *Q*_{2} can then be related to net condensation and vertical subgrid transport as

- (5)

- (6)

where *c* is condensation and *e* is evaporation of condensate; only liquid–vapour phase transitions are included in the equations for simplicity, although in the model calculations ice-phase transitions are also accounted for. {′} denotes the anomaly from the horizontal average, {}, defined above.

Figures 7 and 8 show the total, subgrid-scale and large-scale heating and moistening averaged for all points below 4.2 E−2 mm h^{−1} and for three higher rain rates for the 12 km *param* and 4 km *3Dsmag* runs. The total tendencies are represented by the first term in (3) and (4), with the large-scale advection represented by the second and third terms. The subgrid-scale terms (and radiation term for temperature) are shown on the right-hand side of (3) and (4), with the subgrid terms further defined by (5) and (6). Standard-error values are all below 0.5 K day^{−1} for Figure 7 and 0.3 g kg^{−1} day^{−1} for Figure 8.

The subgrid terms in Figures 7 and 8, given by (5) and (6), are calculated by first adding the increments to temperature and moisture from the convective parametrization, boundary-layer/large-scale cloud scheme (including vertical subgrid turbulence mixing and surface temperature and moisture fluxes), large-scale rain scheme and horizontal subgrid turbulence mixing (which is very small and is not used for the 12 km *param* model). Note that ‘large-scale’ in ‘large-scale rain’ and ‘large-scale cloud’ refers to the original model grid scale and these schemes are considered ‘subgrid’ processes here. Additionally, the last term in (5) and (6), the vertical transport of heat and moisture from subgrid-scale motions, while partly included in the convective, mixing and surface-flux terms already accounted for in the model increments listed above, will also be partly represented by explicit vertical advection at each model's own grid scale. This is particularly true for the explicit convection runs. To account for this, we estimate these explicit subgrid fluxes from instantaneous anomalies every hour within each 1° large-scale grid box and then terms are averaged over three-hourly periods and added to the other subgrid terms listed above.

The large-scale advection terms are calculated by using the temperature and moisture increments from the advection scheme and then subtracting the explicit subgrid vertical transport term, calculated as described above. The radiation and total tendency terms come directly from model increments, with any small residuals in the budget added to the subgrid term so that the budget is balanced. All terms are averaged onto the 1° grid scale and three-hourly temporal scale.

For heating, the subgrid term for the 4 km *3Dsmag* model is mainly made up of a combination of the large-scale rain scheme and the boundary-layer/large-scale cloud scheme, while for moisture the transport term is also important. The 12 km *param* subgrid heating and moistening is dominated by the convection scheme at all rain rates except in the boundary layer, where the boundary-layer/large-scale cloud scheme is also important.

In most cases, the subgrid terms nearly balance the large-scale advection, although radiation is sometimes important in the heating budget, particularly at lighter rain rates (also note that the subgrid and large-scale advection terms are plotted on a larger scale at high precipitation rates). At very low rain rates (below 4.2 E−2 mm h^{−1} or 1 mm day^{−1}, where deep convection is likely suppressed), the two models are mostly in agreement, although there is slightly more warming and drying above the boundary layer for the 12 km *param* model in the total field increments. At light to moderate rain rates (1.0 E−1 and 3.0 E−1 mm h^{−1}) the total increments show that the 4 km *3Dsmag* run has significant drying and warming in the boundary layer (below 2 km) that is not seen in the 12 km *param* model, as well as more warming and slightly more moistening in the lower free troposphere. This is associated with subgrid increments with significantly more moistening in the free troposphere and drying in the boundary layer as well as mid-tropospheric cooling; these are likely related to more shallow convection and/or evaporation of stratiform rainfall (although the relationship between subgrid terms and total increments can be complicated to explain). The subgrid moistening profiles at rain rates of 1.0 E−1 mm h^{−1} and 3.0 E−1 mm h^{−1} for the 4 km *3Dsmag* run are similar to the suppressed period CSRM integrations in (Woolnough *et al.*, 2010, their figure 8), although the magnitude in the 12 km *param* model is more similar to those profiles at 1.0 E−1 mm h^{−1}; the precipitation in the CSRMs compared in the suppressed period of that study ranges from 4.0 E−2 to 1.0 E−1 mm h^{−1}. Interestingly, the lowest rain-rate bin in our Figure 8, with rain rates lower than those in the suppressed period CSRMs in Woolnough *et al.* (2010), shows strong moistening at levels below 2 km for both model versions, somewhat lower than shown by the CSRMs (at somewhat higher rain rates) in that study.

At large precipitation rates (2.0 E+0 mm h^{−1}), both models show strong subgrid drying and warming throughout the troposphere, but the total increment for moisture shows significant low-level drying below 2 km and free-tropospheric moistening above 4 km in the 4 km *3Dsmag* run, which is almost absent in the 12 km *param* run. The subgrid heating in the 12 km *param* model is stronger at upper-tropospheric levels (6–12 km) at the highest rain rate than in the 4 km *3Dsmag* model, agreeing with the discussion in section 5.1 suggesting that more upper-tropospheric stabilization for heavy rainfall in the 12 km *param* model might be a cause of it being very difficult for that model to produce very heavy rain.

Note that there are obvious large differences in the vertically integrated drying and heating (which would be found by vertically integrating subgrid terms suitably weighted by density and multiplying by appropriate constants) for the two models, even at the same precipitation rate. Since the subgrid terms represent the effects of surface evaporation as well as conversion between liquid or solid condensate and water vapour, these are partly due to differences in surface evaporation rates and partly due to the advection of condensate and changes in the total condensate for a given rain rate: for instance, at 3.0 E−1 mm h^{−1} the 4 km *3Dsmag* model has about 1.0 E−1 mm h^{−1} (one third) of the rain rate explained not by subgrid moisture tendencies but by the advection of liquid and ice condensate into the 1° box (7.0 E−2 mm h^{−1}) and also a reduction in the condensate reservoir of the box (3.0 E−2 mm h^{−1}), whereas these two terms are negligible in the 12 km *param* model. The mid-tropospheric cooling and moistening in the subgrid terms show the effects of evaporation of these falling hydrometeors.

The subgrid heating profiles at the highest rain rates are more top-heavy than some of the Tropical Ocean Global Atmosphere Program's Coupled Ocean Atmosphere Response Experiment (TOGA-COARE) estimates and Goddard Cumulus Ensemble (GCE) CSRM estimates in Shige *et al.* (2007), although their shapes are similar to TRMM estimates of latent heating in that study over various tropical convective regions. The top-heaviness of the heating profiles agrees with the overall conclusion of Lin *et al.* (2004) that MJO active events have more upper-tropospheric heating, relative to total heating, than mean tropical rainfall. However, it is difficult to compare these profiles directly with most other studies, since these are averages not over regions but rather over rainfall regimes, which occur over many locations and times.

For the *subgrid* heating and moistening terms, the models agree fairly well at the highest precipitation rate but are very different at the lower rain rate of 1.0 E−1 mm h^{−1}. The 4 km *3Dsmag* model exhibits cooling and moistening at mid-levels and warming and drying at low levels, while the 12 km *param* model shows much less, with perhaps slight warming and moistening at low levels for subgrid terms. Looking only at what convection does directly (the subgrid terms), the 4 km *3Dsmag* model would seem to take longer in the transition to deeper convection, while the 12 km *param* model might transition to deep convection more quickly (especially given its tendency to rain at profiles with dryer mid-levels anyway, likely due to small entrainment rates) and therefore might consume CAPE more quickly and not maintain extended heavy precipitation because the CAPE never builds up, similar to arguments from DeMott *et al.* (2007). However, it is not as simple as arguing that rainfall occurs only in moister tropospheric conditions, since the 4 km *2Dsmag* model still gets the rainfall distribution right. There may be other reasons why CAPE is able to build up for longer locally before heavy rain begins. However, there may also be reasons why rainfall in the 4 km models is able to release potential energy from larger regions via circulation feedback, which does not operate to the same degree in the 12 km *param* model.