1. Introduction
It is challenging for coupled global climate models to produce realistic simulations of precipitation regional patterns, temporal variations, and statistics such as frequency and intensity of rainfall (e.g., Covey et al. 2003; Trenberth et al. 2003; Meehl et al. 2005). Many climate models still leave much to be desired in simulating realistic precipitation statistics, although considerable progress is being made. Energy balance places constraints on the global-mean rainfall, but spatiotemporal patterns have more subtle constraints and hence can be difficult to model. Occurrence probability of precipitation is one major characteristic of rainfall that climate models have struggled to capture. Many weather and climate models tend to precipitate too frequently at low intensities, even when the simulated mean values are reasonable (Chen et al. 1996; Osborn and Hulme 1998; Dai et al. 1999; Trenberth et al. 2003; Dai and Trenberth 2004; Sun et al. 2006). This problem may be due in substantial part to issues in the convection parameterization schemes and their interactions with the large-scale dynamics in the models.
Retrieved statistics of rainfall, similar to those reported in, for example, Bretherton et al. (2004), Peters and Neelin (2006, hereafter PN06), and Neelin et al. (2009, hereafter NPH09) can be used to constrain climate models and convective parameterizations. Bretherton et al. (2004) analyzed satellite microwave retrievals on daily and monthly time scales and found an exponential relationship between conditionally averaged precipitation and column-relative humidity. Examining conditional averages of microwave retrievals of precipitation on column water vapor (CWV)—both essentially instantaneous in time—PN06 noted a rapid precipitation increase beyond a threshold value of CWV, much as one might expect from onset of convective conditional instability in a deep convection parameterization. Holloway and Neelin (2009, hereafter HN09) used in situ observations to evaluate this relationship to onset of conditional instability and further showed that inclusion of substantial entrainment in the convective instability calculation was important to correctly obtain the pickup in precipitation. CWV was shown to be a reasonable proxy variable for the effect of environmental lower-tropospheric moisture on conditional instability of an entraining plume, for which deep convective instability typically occurs only for sufficiently moist environment [as also noted in, e.g., Brown and Zhang (1997); Kuang and Bretherton (2006); Del Genio and Wu (2010)]. NPH09 used precipitation and CWV from satellite retrievals and tropospheric temperature from reanalysis to provide a quantification of the role of tropospheric temperature in governing the onset boundary for strong deep convection and to examine related convective transition properties.
In analyzing a complex system, such as deep convection, interacting with the large scale, guidance from simpler prototypes can be useful. PN06 noted that the statistics have sufficient similarities to certain aspects of continuous phase transitions and related critical phenomena that this analogy could be used to suggest a set of interrelated properties to seek in the observations. These include clusters, power-law spatial and temporal correlations, and power-law event size distributions in measures of the smaller-scale convection (such as precipitation or cloud water) occurring near the onset of conditional instability at a critical value, wc, of large-scale CWV. A natural next step was a model whose relationship to atmospheric prognostic equations could be more easily seen and in which the relationship to observed probability density function (PDF) of CWV could be quantitatively examined. Stechmann and Neelin (2011) showed that a prognostic water vapor equation stochastically forced across a parameterized precipitation onset exhibits properties including power-law ranges in temporal correlation and event size distribution and that reasonable matches to the observed estimates of PDFs of CWV arise straightforwardly from a Fokker–Planck equation in which precipitation acts as the drift term. Many of these properties can be interpreted in terms of a first-passage process (Stechmann and Neelin 2014) with stochastic forcing across thresholds for precipitation onset/termination. The forcing across this threshold can occur substantially by large-scale processes, suggesting that it should be possible to capture these PDFs reasonably well in climate models, even without a stochastic convective parameterization [for review see, e.g., Neelin et al. (2008)], as will be addressed in part here.
In pragmatically assessing climate models against such statistics, one of the more important characteristics is to correctly capture the transition from shallow to deep convection. Sahany et al. (2012, hereafter SNHN12) showed that the pickup in precipitation and the location of the onset of deep convection were simulated reasonably well with a relatively high-resolution (0.5°) version of the Community Atmosphere Model (CAM3.5), primarily due to stronger entrainment rates included in the Neale–Richter (Neale et al. 2008) modified version of the Zhang and McFarlane convection scheme used in the model. The stronger entrainment led to the deep convective plumes becoming more sensitive to the ambient humidity of the environment (as represented by the model gridbox average moisture), such that instability for deep convection occurs only at higher free-tropospheric water vapor.
Both atmospheric water vapor content and global-mean precipitation are expected to increase under global warming, but the changes in spatiotemporal distribution of rain rates are more important for societal impacts. Allan and Soden (2008) found that heavy rain events in satellite observations increased during warm periods and decreased during cold periods, but at a rate higher than that predicted by models. Chou et al. (2009) noted a general tendency for tropical precipitation anomalies in climate models under global warming to follow the “rich get richer” effect (Chou and Neelin 2004; Held and Soden 2006) of increased precipitation in regions of climatological moisture convergence, although with regional-scale departures. O’Gorman and Schneider (2009) found that, although theoretically the intensity of precipitation extremes is expected to increase (e.g., within high percentiles of daily rainfall) with increase of column water vapor under global warming, there is significant disagreement in regard to tropical rainfall extremes among climate models from phase 3 of the Coupled Model Intercomparison Project (CMIP3). For both precipitation extremes and their fractional changes under global warming, the intermodel scatter in the tropics was found to be larger than that in extratropics. Muller et al. (2011) and Romps (2011) used convection-resolving models under idealized radiative–convective equilibrium and found that intense precipitation increases with warming at close to the rate expected from Clausius–Clapeyron scaling. Sun et al. (2007), Chou et al. (2012), and O’Gorman (2012) each reported increases in simulated intensity of extreme precipitation events (for instance, in upper percentiles of daily precipitation) over many regions under global warming conditions, although the latter found strong disagreement between climate models on the rate of increase over the tropics.
Changes to the temperature profile under global warming will also prove relevant to our results. Santer et al. (2005) found amplification of surface temperature changes in the tropical upper troposphere in both observations and climate models over monthly time scales. However, over decadal time scales, while the model behavior was similar, there was lack of general agreement among the observational datasets. Chou et al. (2013) used model output from the CMIP3 and CMIP5 archives and found an increase in gross moist static stability over the tropics under global warming conditions.
Here we analyze output from uncoupled and coupled versions of the National Center for Atmospheric Research (NCAR) Community Climate System Model, version 4 (CCSM4), for both present-day and global warming conditions for the high emission scenario representative concentration pathway 8.5 (RCP8.5) to investigate tropical precipitation transition statistics simulated by the model. This is the first time such precipitation transition statistics have been analyzed for a coupled model, both in the current climate and under global warming. Analysis for the uncoupled model (CAM3.5) is presented in section 2 (which includes methodological details and caveats on aspects of the retrievals for readers who might want to reproduce these figures as process-oriented diagnostics for their climate model of choice). Analysis for the coupled model (CCSM4), including changes under global warming, is presented in section 3. We analyze several aspects of precipitation transition statistics, including the sharp pickup (PN06 and NPH09), probability of occurrence of CWV for precipitating grid points, and the convective onset boundary on an empirical temperature–CWV thermodynamic surface for the historical and RCP8.5. The simulated precipitation transition statistics for the historical period are compared with the corresponding retrievals from the Tropical Rainfall Measuring Mission (TRMM; Kummerow et al. 2000) Microwave Imager (TMI) processed by Remote Sensing Systems (RSS) with the Hilburn and Wentz (2008) algorithm [an updated version of Wentz and Spencer (1998)], and temperature profiles from the ERA-40 dataset (Uppala et al. 2005). The precipitation transition statistics under global warming are then analyzed to help understand simulated changes in the properties of deep convection compared with those of present day. Behavior tends to be similar for different tropical ocean basins (NPH09, SNHN12), so examples from the eastern and western Pacific are presented. In section 4 and the appendix, an entraining plume model similar to HN09 has been used for buoyancy computations to explain the characteristics of the convective onset boundary on the empirical thermodynamic surface for historical and global warming conditions.
2. Deep convective transition characteristics from high-resolution CAM simulations
a. Precipitation pickup and estimation of onset boundary
Figure 1a shows conditional average precipitation rate as a function of CWV binned at 0.3-mm intervals for a range of bulk tropospheric temperatures 
(a) Conditional average precipitation as a function of CWV for different bins of 
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
We emphasize, as noted in NPH09 and SNHN12, that one should be cautious in interpreting the microwave retrievals at high water vapor and precipitation. The retrieval estimates cloud water in the column, and the retrieved precipitation rate is based on empirical relationships between cloud water and columnar rain rate (Hilburn and Wentz 2008; Wentz and Spencer 1998). Fewer occurrences of very high rain rate and CWV cases suggest that calibration will tend to be less reliable at high values. For instance, Seo et al. (2007) showed that the TMI-derived rain rates are underestimated by more than 50% at precipitation radar (PR)-derived rain rates of greater than 25 mm h−1. Furthermore, CWV is not directly retrieved above rain rates of about 15 mm h−1, and 25 mm h−1 is considered an extreme upper bound on the algorithm’s ability to retrieve rain (Wentz and Spencer 1998). Thus, the apparent curvature in the retrievals at high CWV may well be associated with saturation of cloud water or retrieval error and does not constitute a basis for comparison to the model. Rather, it is the location of the strong pickup that is of leading interest for model validation (and properties near this onset). PN06 checked the effects of using cloud water rather than precipitation retrievals and obtained very similar values for the estimated wc. See HN09 for verification of the onset dependence for the Atmospheric Radiation Measurement (ARM) Program (Stokes and Schwartz 1994; Mather et al. 1998) in situ (optical rain gauge and radiosonde) observations over the tropical western Pacific.
In a first of two main variants, we estimate wc by estimating a power-law fit above a threshold precipitation value of 2.5 mm h−1 and defining wc as the value at 2.5 mm h−1. Applying the threshold helps to keep the fit outside the range of the foot discussed above (although if applied to a new model or dataset this should be verified). Defining wc at a precipitation threshold value greater than zero aims to accurately characterize the rapid onset of strong precipitation. This will tend to be at a slightly higher-CWV value than where one might find the zero of the ramp function in the parameterization, but it avoids attempting to extrapolate a fit that would be prone to error in the presence of the foot. Within this fitting approach, we tested impacts of fitting individual values of β for each 
In the second variant, referred to as the “restricted-range fit,” we specify both an upper and lower bound for conditional average precipitation, 2.5–6 mm h−1, and use a linear fit within this range, defining wc as a value at 2.5 mm h−1 as before. This fit is then iterated to include only a 5 mm CWV range above wc, as a safeguard against possibly noisy points at high water vapor where there are few counts. The range in both precipitation and in water vapor is narrow enough that differences in curvature at high water vapor and high precipitation tend to be excluded. This proves to be highly relevant not only for model to microwave retrieval comparison, but also within the model itself, since curvature of the model conditional average precipitation curve turns out to be substantial in global warming runs discussed below. One could consider using a narrower range, or even interpolating across 2.5 mm h−1, but this would be prone to estimation effects in cases with shorter datasets or at high temperatures where there are fewer counts (a requirement of 10 counts in each point is illustrated here). Figure 1b shows results of this restricted-range fitting method. As expected, the critical values are very similar to those obtained by the first method, although they do tend to move the CAM and TMI values slightly closer, due in large part to fitting over an approximately linear range in both. Effects of increasing the threshold number of data points in a bin from 5 in Fig. 1a to 10 in Fig. 1b are also evaluated, impacting a few of the highest ensemble-average precipitation values for 
We display these values as an onset curve shown on a temperature–water vapor thermodynamic plane (Fig. 1c) similar to that shown in NPH09 and SNHN12. The critical value wc is seen to have a simple dependence on the bulk tropospheric temperature, increasing approximately linearly with 
The estimated wc values in the model are slightly lower than those for observations for the entire range of tropospheric temperatures analyzed, even after the modified procedure is used, and this is consistent with visual inspection of Fig. 1a, where the rapid pickup of precipitation typically occurs at a slightly lower water vapor for each temperature. However, for a given 
Contours of conditional average precipitation in Fig. 1c emphasize that the onset curves lie parallel to the precipitation contours in the region of strong gradient. For temperatures where there are high data counts, the fitting methods give onset curves very similar to a 2.5 mm h−1 precipitation contour. The difference in the estimated wc values from the two fitting methods for the TMI observations averaged over all 
b. Frequency of occurrence (PDFs)
Figure 2 shows the PDF of column water vapor rescaled by the corresponding critical values for each of the 
Log-linear plot of the frequency of occurrence (Np) of CWV rescaled by the corresponding onset threshold values wc
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
Figure 2a is similar to that shown as Fig. 5b in NPH09. As discussed in NPH09, the occurrence frequency features a Gaussian core with a peak near the critical value (around 0.9), and a longer-than-Gaussian tail on each side that over a certain range can resemble an exponential decay. The long tail above wc is associated with much more frequent excursions into the heavy precipitation regime than one would expect from extrapolation of the Gaussian core and thus is of interest for understanding occurrence of extreme events. As emphasized in NPH09, due caution is needed in interpreting microwave retrievals in the high precipitation regime. This is further discussed in section 2c, since these PDFs potentially represent a significant point of comparison for models.
In Fig. 2b we show a similar figure using the 0.5° CAM3.5 precipitation and column water vapor binned by 
c. Caveats and interpretation of model–retrieval comparison
In section 2a, concerns regarding high rain-rate microwave retrievals were treated by showing that the onset boundary for rapid precipitation pickup is robust to exclusion of these. For the PDFs in Fig. 2, this is the first comparison to climate model results, and the high-CWV regime will prove of interest in the global warming behavior (section 3), but the CWV retrievals can have nontrivial errors in the precipitating regime. A brief discussion of caveats and the extent to which the model provides interpretation of retrieval PDF features is appropriate.
Regarding the existence of a Gaussian core with longer-than-Gaussia tails, there are a number of lines of corroborating evidence that these represent physically reasonable behavior regimes. Neelin et al. (2010) discuss how such long tails are typical of a class of tracer advection problems with a maintained gradient (Pierrehumbert 2000; Bourlioux and Majda 2002; Majda and Gershgorin 2013), and can be found in PDFs for chemical tracers in models and independent retrievals. This establishes a simple mechanism that would apply to CWV distributions generally (not just precipitating points.) Lintner et al. (2011) examine the relationship of such tails in water vapor distributions to circulation using a number of datasets.
In CAM, the simulation of corresponding core/tail behavior for CWV PDFs for precipitating points suggests that these features are reasonably straightforward to obtain, at least in a model that can realistically simulate the position of the convective onset boundary relative to column saturation (since the long tail exists in the interval between these). It is worth noting that this is an emergent behavior for which the model has never been examined or tuned. The width of the Gaussian core is comparable to that of the retrievals. The PDFs in the model differ somewhat more in terms of the shape for different 
In assessing at what level of detail the retrievals can be trusted, it is useful to consider what aspects of convective-scale or large-scale physics might contribute to the PDF, as well as sources of retrieval error. In the simple model of Stechmann and Neelin (2011), similar distributions could be mimicked by including the following: (i) a CWV-dependent precipitation pickup; (ii) a stochastic representation of large-scale (i.e., larger than grid-scale) moisture convergence; (iii) a stochastic precipitation component assumed to arise from subgrid-scale effects; and (iv) a stochastic transition from shallow to deep convection that represents the effects of degrees of freedom not captured by grid-scale CWV. The Gaussian core depends quantitatively on all four, while the high-CWV tail depends primarily on the first three. They note that the effects of (ii) and (iii) were largely indistinguishable, while Stechmann and Neelin (2014) show that simplifications without (iv) can capture qualitative aspects in some circumstances. In CAM, the convection parameterization is deterministic, and precipitation rate depends on the entraining CAPE with a specified adjustment time scale. One can infer that the shape of the CAM PDF must thus be due to large-scale forcing pushing the column thermodynamic conditions back and forth across the conditional instability onset boundary for deep convection, with occasional events strong enough to push the system farther than typical above this onset despite the 1-h dissipation time scale. Near the onset boundary, the effects of vertical degrees of freedom that affect entraining CAPE but are not controlled for by CWV and 
In observations, each of these physical processes can be expected to play a role, but imperfectly known retrieval errors can also affect the PDF. The procedure for estimating variations of retrieved CWV against radiosondes is limited in part by differences associated with mismatches in spatial and temporal colocation and point observations versus spatial averages. In raining conditions, an estimated 3.7 mm must accordingly be discounted from the 5-mm root-mean-square (RMS) difference between retrievals of spatially averaged water vapor and neighboring sonde points, implying a little over 3 mm from other sources (Wentz and Spencer 1998). Conservatively taking this as translating to a random error standard deviation 0.05 in w/wc would suggest that CAM (with no random error) is simulating an overly wide core. However, Lintner et al. (2011), using independent upward-looking radiometer instrumentation at Nauru, find CWV PDFs for precipitating points with widths comparable to those seen here. Furthermore, most of the remaining 3 mm RMS in the satellite microwave retrieval–radiosonde differences is not random instrument error but, rather, comes from variations of hydrometeors and water vapor at smaller spatial scales than the horizontal and vertical spatial averaging footprint of the retrieval. These thus represent an atmospheric signal of small-scale variations (i.e., of one of the physical effects that contributes to core width in CAM). Overall, until the microwave retrievals can be further calibrated for these purposes, quantitative differences between model and retrieval PDFs should be viewed with caution. The model results for changes in these PDFs discussed in section 3 may motivate such calibration efforts. Lastly, we note an ambiguity in the appropriate spatial scale for comparison. In section 3, we use an approximately 1° version of the model (versus 0.5° in this section). Similar results are obtained at the respective grid scales, presumably because the parameterized convective plumes interact with grid-scale moisture and temperature such that the model behaves similarly at the smallest scales available to it. Systematic assessment across resolution would be desirable in future; here we provide a first assessment of changes in these statistics at the available resolution.
3. Changes in convective transition statistics and onset threshold under global warming
a. Precipitation pickup and critical surface
Precipitation pickup over the tropical eastern Pacific similar to Fig. 1 is shown for 20-yr time intervals from the historical (Fig. 3a) and RCP8.5 (Fig. 3b) runs for the NCAR CCSM4 from the CMIP5 archive. For the historical run, model output is analyzed for the period 1981–2000. For a given 
Precipitation pickup for 1° CCSM4 over the tropical eastern Pacific, similar to Fig. 1a: (a) historical (1981–2000), and (b) EoC (2081–2100) for the RCP8.5 scenario.
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
For the RCP8.5 run, the 20-yr period corresponding to the end of the century (EoC; 2081–2100) has been analyzed. The mean warming for the region is about 4 K, so the curves for 273–278 K in Fig. 3b are most comparable to those shown for the historical run in Fig. 3a. To facilitate comparison to the onset values for the historical run, curves are included for a larger range of temperatures, displaying bins from 270 K, even though this temperature is now uncommon in the warmer climate. In Fig. 3b it can be seen that the pickup in precipitation shifts to higher values of CWV for warmer temperatures, as expected qualitatively. Quantification of how the onset shifts will be analyzed in more detail by comparing the increase in wc with 
Before doing this, we note that the curvature of precipitation rate at high values of CWV in the warmest two 
We further note that, in addition to a higher conditional average precipitation at high water vapor and temperature in the global warming case, precipitation variance increases strongly. In the historical run, precipitation variance tends to peak or plateau slightly above critical, with peak value tending to increase with temperature (figure not shown). In the global warming case, the corresponding 
We now return to the leading-order question of how the onset of convection changes. Figure 4a shows the precipitation pickup from the historical run for each of the 
Ensemble-average precipitation over the tropical eastern Pacific as a function of column water vapor rescaled by the corresponding wc for each of the 
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
Figure 5a shows the deep convective onset boundary on a temperature–CWV thermodynamic surface, similar to Fig. 3a of NPH09. Important changes in the onset curves can be seen as one moves from current climatic conditions to a warmer climate. The deep convective onset shape under global warming has the same elements as in the historical case: at sufficiently high temperatures, it inclines away from the saturation curve with the separation between the onset of conditional instability and saturation increasing with 
(a) Deep convective onset boundary for the tropical eastern Pacific similar to Fig. 1b for the historical and EoC. Also shown for reference are the corresponding saturation values for the two cases. (b) As in (a), but only the EoC values are retained, and a new curve with wc values is projected from the historical values by a simple conjecture. Specifically, the wc projections are obtained by shifting the historical 
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
One of the simplest conjectures of how this shift in onset boundary under global warming occurs is a version of a constant relative humidity approximation. Defining a saturation fraction as wc divided by the column saturation value for the same temperature, we examine whether adding 4 K (the average increase in 
b. Probability density function of precipitating points
Figure 6 shows the occurrence frequency of precipitating points for the tropical eastern and western Pacific for the historical and RCP8.5 model outputs. The figure is similar to that shown in Fig. 2 for the observations and the uncoupled version of the model run at a higher resolution (0.5° CAM3.5). The curves on the panels are color coordinated such that a given 
As in Fig. 2, but (left) for the historical and (right) for the EoC for tropical (top) eastern and (bottom) western Pacific. As in Fig. 2, the procedure tends to collapse in occurrence statistics of rescaled column water vapor for precipitating points for different values of 
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
For the historical runs, the results are not very different from the uncoupled version, although it is at a coarser resolution (1° vs 0.5°). Comparing the model output for the historical and the RCP8.5 global warming conditions, it can be seen that the shift in the distribution to higher values of 
The increase in frequency of occurrence in the supercritical range (above wc) appears consistent with changes in extreme events in a warmer climate noted in other studies (e.g., O’Gorman 2012), but gives a complementary view on the manner in which they arise. The critical point corresponds to the onset of deep convective conditional instability (Neelin et al. 2008), and in the climate model the convective scheme attempts to keep the system near this, with a time scale of 1 h for removing entraining CAPE. The supercritical range corresponds to dynamical driving strong enough to push the system beyond the typical convective quasi-equilibrium range represented by the Gaussian core. In the warmer climate, especially at the warmer temperatures, this occurs more often. This diagnostic does not yield an answer to exactly why this is occurring but helps to quantify its occurrence. In the model, when this occurs very strongly, it can push the system all the way to saturation (through some layer, which can occur slightly below column saturation), in which case large-scale condensation is activated in addition to the parameterized convective precipitation. This likely accounts for the peak near saturation for 
c. Events constituting the tail of the distribution
To get a sense of the spatial structure of the events that constitute the tail of the distribution, especially for the higher 
Instantaneous snapshots of precipitation, 
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
4. A simple prototype for the shift in the convective onset boundary
The conjecture presented in Fig. 5b using the historical wc values to predict those for the EoC works reasonably well, so it is useful to understand why in more detail. A set of plume buoyancy computations with a prescribed fractional entrainment rate, under idealized changes to the temperature and moisture profile, are carried out similar to a subset of those in SNHN12. Specifically, an air parcel is initialized with the temperature and specific humidity values of the idealized environment (discussed below) at the 1000-hPa level and ascends with a vertically constant fractional entrainment rate of 0.002 hPa−1 (case C2 from SNHN12), otherwise conserving its total water and ice-liquid water potential temperature. For the historical case, multiples of vertically constant temperature perturbations of 0.2 K are added to a base-state temperature profile over Nauru (an ARM observation site over the tropical western Pacific), obtained by averaging profiles conditioned on high values of CWV (>66 mm; see HN09 for details) representative of conditions favoring deep convection. Tropospheric relative humidity is varied in the range of 51%–99% in increments of 2% using a vertically constant profile above 800 hPa, and a blending region up to 950 hPa tapering (6.5% per 50 hPa) to a surface relative humidity value of 85% typical of deep convective cases over Nauru (HN09). Entraining CAPE contours of 100 J kg−1 are used as a measure of the deep convective onset boundary unless the free troposphere reaches saturation, in which case large-scale precipitation is assumed to onset.
The resulting precipitation onset boundary is seen in Fig. 8 as a function of CWV and bulk tropospheric temperature 
Precipitation onset boundary, similar to that shown in Figs. 1c and 5, but for a simplified prototype using idealized perturbations to Nauru profiles. Contours of 100 J kg−1 entraining CAPE computed by a one-dimensional entraining plume model, or free-tropospheric saturation (seen at lower 
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
With this as our prototype for the observed and modeled precipitation onset in the historical case, we can now create a simple prototype for the changes under global warming. For the EoC computations, the CCSM4 temperature profile anomaly (EoC − Hist.) over Nauru (shown as an inset in Fig. 8) is added to the observed base temperature profile used for the historical case. This warmer profile is then used as a base profile to which the same set of temperature and free-tropospheric relative humidity perturbations are added, as in the historical case. In other words, we assume that the natural variations in the warmer climate sample a similar set of perturbation temperature and free-tropospheric relative humidity profiles as in the historical climate, but all are shifted by the average temperature profile anomaly that the model simulates for the global warming case relative to the historical period. The onset boundary for EoC in Fig. 8 exhibits features much like the historical case. The lower-
The entire structure of the onset boundary at EoC shifts to higher values of CWV and 
Overall, this provides an economical explanation of the features of the onset boundary seen in CCSM (Fig. 5a) for both historical and EoC cases. In CCSM and observations, the vertical temperature structures sampled in natural variability tend to be slightly more unstable at warmer temperature for a given relative humidity (see also SNHN12). The prototype allows the different effects of the global warming temperature profile to be seen in a very simple context and also explains why a constant relative humidity assumption works as a reasonable approximation for obtaining the EoC onset boundary from the historical boundary (Fig. 5b). These calculations also provide the caveat that the shift under global warming for the conditional instability portion of the precipitation onset boundary depends on trusting the model simulation of the change in the vertical temperature profile under global warming.
5. Discussion
The recent availability of large datasets for snapshots of precipitation-related variables from remote sensing retrievals, or high time resolution data, such as from the Atmospheric Radiation Measurement Program sites, makes it possible to seek statistics, which might be termed fast-process diagnostics, that can help us to understand and constrain the fast-physics processes that play a major role in the global hydrological cycle. Examples of this include sets of related statistics for the onset of tropical precipitation [such as Bretherton et al. (2004); PN06; NPH09; HN09], which can serve as additional evaluation metrics for climate models. Because the statistics often rely on satellite retrievals that can have their own inherent uncertainties, the comparison to models can also provide a consistency check on observations. Using outputs from both uncoupled and coupled versions of the NCAR CCSM, some of the deep convective onset characteristics are analyzed, including pickup of precipitation and probability distributions for precipitating points as a function of column water vapor. After comparing to statistics from satellite retrievals in current climate, we ask how the model-simulated statistics change under global warming.
a. Simulations for current climate
Analysis of model output from an uncoupled version of the NCAR CCSM, run in an AMIP mode at a fairly high spatial resolution of 0.5°, yields a good match to previously reported deep convective transition statistics using observations (PN06; NPH09). The precipitation pickup is similar to the findings of SNHN12, including the simulation of the dependence of the convective onset boundary in a thermodynamic plane of column water vapor and a bulk measure of tropospheric temperature 
Assessing the probability distributions for precipitating grid points, the model distributions exhibit a Gaussian core just below the critical CWV value, with a longer-than-Gaussian tail that extends above the critical point for the onset of deep convection in agreement with the satellite retrieval datasets. In this regard, the model can be viewed as corroborating the retrieval-based statistics, which, despite validating well against in situ observations at particular locations (HN09), should be viewed with caution in the high water vapor and precipitation range. As expected, the model exhibits a neat cutoff at column saturation such that the longer-than-Gaussian portion is restricted to a range between the Gaussian core and saturation. Degrading the model temperature observations to a grid comparable to temperature reanalysis affects the distribution tails modestly, suggesting this is a useful step when comparing models to observations, but the basic structure of the core–tail behavior is robust to this. NPH09 had been cautious in interpreting the long tails until subsequent studies showed the widespread existence of comparable tails in reanalyses, model simulations, and retrievals of other tracers (Neelin et al. 2010; Lintner et al. 2011). Simple stochastic models can yield such tails under plausible assumptions (Stechmann and Neelin 2011). Here, the CAM simulation of the longer-than-Gaussian tail range provides evidence that this can be straightforward to simulate in a full atmospheric model for this high CWV, high precipitation regime. Of course, this regime can only be captured if the onset of conditional instability occurs sufficiently below column saturation, so the reasonable simulation of the onset boundary as seen in the CAM is a prerequisite.
The same convective onset statistics for the 20-yr period 1980–99 from the CMIP5 historical run for CCSM4 at 1° resolution are highly comparable to those of the uncoupled atmospheric model outputs at 0.5° resolution. Thus, the fully coupled model statistics agree well with the uncoupled versions of the model, and these are not highly sensitive to resolution. The occurrence probability of precipitating points shows similarly good agreement with observations, including the longer-than-Gaussian tail related to extremes in convection. The core and the tail each exhibit more variation from one tropospheric temperature bin to another in both uncoupled and coupled simulations than in the observational estimates. While this is a next-order effect, it appears to be an important one for questions of global warming, as discussed below.
b. Changes under global warming
Because the coupled version of the model shows broad agreement with the observations for the present-day conditions in terms of some of the deep convection statistics analyzed in this study, it is tempting to investigate how these convection-related statistics evolve under a warmer climate. Comparing convective onset statistics from years 2081–2100 of CCSM4 output for the CMIP5 RCP8.5 global warming scenario to the historical case suggests a number of points: (i) The abrupt pickup in rainfall and the corresponding critical value of CWV shift to higher values of moisture as the system shifts to warmer temperatures, as expected. The convective onset curve in the temperature–water vapor plane continues to have a dependence more complex than saturation, since this boundary seems to be set by the onset of conditional instability for deep convection (as shown for the historical period in NPH09 and SNHN12). (ii) The way this onset curve temperature dependence changes under global warming differs from a simple extension of the temperature dependence in the historical period because the vertical structure of the temperature change differs. Taking an idealized column-conditional instability calculation for the historical temperature–moisture dependence and adding the tropical mean temperature change profile associated with global warming provides a simple prototype for the end-of-century onset boundary. The global warming case can be roughly approximated by using column-relative humidity at each bulk temperature to shift the curve from the historical period. However, the column calculation shows that this is only an approximation and underlines the caveat that this would be sensitive to the vertical structure of the temperature change. (iii) The critical value of CWV continues to strongly govern the frequency of occurrence of precipitating points under global warming. To a first approximation, the PDF of precipitating points as a function of CWV for a given temperature remains the same when column water vapor is normalized by the critical value (and each temperature is shifted by the mean increase). However, this normalization allows modest changes in the shape of the PDF to be seen, and these are particularly noticeable at water vapor values above critical. (iv) Specifically, the longer-than-Gaussian tail above critical CWV, which occurs in both observations and model in the historical period, tends to have increased probability in the end-of-century period under global warming in the model. This occurs in the high water vapor regime between the Gaussian core just below critical and a cutoff at saturation. For most temperatures, this above-critical probability increase is seen as a modification of the slope of the longer-than-Gaussian tail (although for the eastern Pacific at the highest temperature, the modification is sufficient to yield a secondary peak in the PDF in the supercritical range). (v) As a result, conditional average precipitation pickup curves and precipitation variance as a function of CWV reach substantially higher values than in historical simulations. Synoptic examination of events in which supercritical CWV values occur suggests that these events are not qualitatively different from those simulated in the historical period. It simply more often occurs that points in the center of the simulated storm-like clusters of convecting grid points reach very high water vapor and associated high precipitation rates under global warming.
One can now ask to what extent the comparison of modeled convective onset statistics to observed estimates in the historical period may permit inference as to the trustworthiness of the model-simulated statistics under global warming. First, we have a reasonable understanding of the way conditional instability sets the critical value for the convective onset boundary in the historical period and evidence that the model does a good job at simulating this for the type of temperature variations encountered within historical climate. This boosts confidence in the model prediction of the way this shifts under global warming (although the conditional instability calculation suggests that this shift could be sensitive to the details of the simulated vertical temperature structure). The shift in the convective onset boundary governs leading-order effects in the shifts of frequency of occurrence.
The clearest differences in CWV distributions for precipitating points at EoC occur in the longer-than-Gaussian tail above critical, and this is associated with excursions above the onset of conditional instability (into the range between onset and saturation) happening more readily in the global warming simulation. Since the distributions are examined with respect to CWV normalized by a critical value that has increased roughly proportional to column saturation, any changes in distribution that would be directly due to saturation changes are already taken into account. This would include not only the shift in the mean of the distribution but any stretching that was simply proportional to the change in saturation. The change in the shape of the distribution tails must thus be associated with more complex effects, such as a dynamical response of the model storm systems. The longer-than-Gaussian tail shows more variation when evaluated as a function of tropospheric temperature in the historical period in the model simulations than in the observational estimates. However, there are sufficient caveats on the combination of microwave retrievals and reanalysis temperatures in the observational estimates in this high water vapor, high precipitation range that this does not necessarily imply lower trust in the model-simulated tails. Rather, the importance of the changes in these tails for the extreme event statistics in the global warming simulation should be taken as motivation for acquiring additional observational data in this range and comparing the behavior of additional models.
Acknowledgments
This work was supported in part by National Science Foundation Grant AGS-1102838, National Oceanic and Atmospheric Administration Grants NA11OAR4310099 and NA14OAR4310274, and Department of Energy Grant DE-SC0006739. We thank J. E. Meyerson for graphical assistance and S. Krueger for discussions and clarifications.
APPENDIX
Interpretations of the CCSM4 Convective Onset Boundary from Idealized Plume Buoyancy Computations
Simplified prototype for the onset boundary for the present day (solid black curve marked historical) and global warming (solid purple curve marked EoC). These curves correspond to Fig. 8, except that the moisture axis is given in terms of free-tropospheric relative humidity. The red arrow indicates shift associated with global warming temperature change profile. Temperature changes along the historical and EoC curves have a different vertical profile (as in Fig. 8). Dashed lines for the historical case show calculations that examine the effects of temperature via dry static energy s and qsat in the conditional instability calculation that defines the onset boundary.
Citation: Journal of Climate 27, 24; 10.1175/JCLI-D-13-00747.1
For a nonentraining plume, rising from specified temperature and relative humidity at 1000 mb, the buoyancy at any given level in the free troposphere would only be affected by the temperature profile of the environment relative to the moist adiabat specified by the 1000 mb values. The value of q or r above the starting level would be irrelevant to the buoyancy. In contrast, an entraining plume is also affected by the moisture of the environment, as different values of environmental h are mixed into the parcel during ascent. As a result, higher moisture tends to yield greater conditional instability for a given temperature profile and entrainment rate. When the temperature profile is changed slightly, the relative humidity needed to achieve the onset of conditional instability for an entraining plume changes accordingly, and the idealized calculations aim to develop intuition for this.
For certain changes in temperature profile, the onset of conditional instability occurs at lower relative humidity for higher temperatures, as occurs when sampling temperatures in present-day observations (Fig. 1c), or for similar sampling of the model in both present-day and end-of-century conditions (Fig. 5a). In the idealized calculation, this is illustrated by adding vertically constant temperature perturbations to the base-state profile, which is more destabilizing in terms of the change of the temperature profile than either perturbations of a moist adiabat or the temperature change associated with global warming in CCSM. Increasing temperature would yield increased conditional instability unless r decreases, so the curve of constant entraining CAPE that marks the stability boundary must slope toward lower r with higher T (black solid curve in Fig. A1).
If the temperature change has a different vertical profile, the net effect of the respective tendencies due to the temperature profile and to r can be different. In particular, if the temperature increases sufficiently with height, as occurs for the global warming temperature change profile, then the effect is closer to neutral in terms of requiring little change in r to achieve the onset of conditional instability. In Fig. A1, the shift associated with adding the global warming temperature change profile to the base profile is indicated by the red arrow. The slope of this line is still slightly negative (i.e., to obtain the same value of entraining CAPE, r must decrease slightly). However, this decrease is small enough that one could plausibly consider using a constant relative humidity approximation, with suitable caveats. In particular, modest differences in the vertical profile of the global warming temperature change could affect this significantly.
Within the warmer climate, the convective onset boundary simulated in CCSM has similar slope to that in the historical period (Fig. 5), but the entire onset structure is shifted. In the idealized calculation, the global warming case uses a new base-state profile [i.e., the historical base state plus global warming change profile from CCSM (black dot on the EoC curve in Fig. A1)]. Variations about this within the warmer climate are represented by again using the vertically constant profile added to the new base-state profile, as described in section 4. This results in a curve (purple curve in Fig. A1) with similar slope in the r–
To further develop intuition regarding the impacts of different effects of the environmental profile on the stability boundary for entraining plumes, consider two additional idealized calculations for the historical case (dashed blue lines in Fig. A1). It is of interest to distinguish between 1) the direct effects of the environmental temperature profile on stability, largely via the comparison between the environmental temperature and that of the entraining parcel lifted from 1000 mbar, and 2) the effects that occur via qsat in Eq. (A1) and Eq. (A3), which depend entirely on the impact of the entrained air on the parcel. To visualize the effects associated with the direct effect of temperature on s, this calculation is repeated with no temperature perturbation applied in qsat(T), referred to as the s-only case. Initial parcel properties at 1000 mbar are still as described in section 4. Dry static energy increases then tend to decrease conditional instability unless r increases, so the s-only case stability boundary must slope toward higher r with higher T. The complementary qsat-only case repeats the idealized calculation with no temperature perturbation applied in any part of the calculation except the environmental qsat. The temperature increases in this case increase environmental moisture and thus the buoyancy of entraining parcels unless r decreases, so the stability boundary must slope toward lower r with higher T. This negative slope is larger in magnitude than the positive slope of the s-only case, indicating that this effect is stronger. As a result, in the full calculation where both temperature effects on s and qsat are included, the stability boundary has negative slope in the free-tropospheric relative humidity–temperature plane. In a CWV–temperature plane, this corresponds to the onset boundary angling away from saturation as a function of temperature, as seen in Fig. 8.
In summary, this idealized conditional instability calculation provides a succinct prototype for the convective onset boundary, as seen in both the current climate and simulated global warming climate and for the shift between these. At a given environmental relative humidity, environmental temperature changes yield competing tendencies between the effects of s and qsat on the stability of an entraining parcel. Slight differences in the profile of temperature change can yield substantial differences in the slope of the convective onset boundary in a temperature–moisture or temperature–relative humidity plane. For the vertical profile of global warming temperature change in CCSM4, the onset boundary for current climate is shifted in a manner that is sufficiently similar to what would be obtained from a constant relative humidity assumption that the latter can be used as reasonable approximation to attain the future onset boundary from the onset boundary in the historical period.
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