1. Introduction
Changes in rain are inexorably tied to changes in atmospheric circulation. In response to global warming, climate model projections show an increase in global-mean precipitation, the rate of which is in balance with the change in atmospheric radiative cooling (O’Gorman et al. 2012; Pendergrass and Hartmann 2014a). This rate of increase, 1%–3% per degree of warming across climate models, is smaller than the rate of increase of moisture in the atmosphere, which roughly follows saturation vapor pressure at ~7% K−1 (Held and Soden 2006). The difference between the rates of increase of moisture and precipitation with warming implies a slowing of the atmospheric overturning circulation (Betts 1998). The weakening circulation in climate model projections manifests as a decrease in spatial variance of convective mass flux (Held and Soden 2006) and the Walker circulation (the antisymmetric component of variance of 500-hPa vertical velocity in the tropics; Vecchi and Soden 2007).
Along with changes in circulation, climate models project substantial changes in the distribution of rainfall, as shown in Fig. 1. The rain frequency distribution (Fig. 1a) shows how often it rains at any particular rain rate. It is displayed on a logarithmic rain-rate scale in order to accommodate the full range of rain rates that can be encountered, which encompasses orders of magnitude. The rain amount distribution (Fig. 1b) shows how much rain falls at a particular rain rate. These calculations are based on the mean of models from phase 5 of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012) and are described in more detail in Pendergrass and Hartmann (2014b). Figures 1c and 1d show the multimodel mean changes in the rain frequency and rain amount distributions in response to a doubling of carbon dioxide in a scenario where carbon dioxide concentrations increase by 1% each year. The rain frequency response to warming (Fig. 1c) is an increase in days with heavy rain, a larger decrease in days with moderate rain, a small (statistically insignificant) increase in days with light rain, and a small (statistically significant) increase in the number of dry days (noted at the top left of Fig. 1c). The rain amount response (Fig. 1d) is an increase in rain falling at heavy rain rates and a smaller decrease in rain falling at moderate rain rates, constituting an increase in the total amount of precipitation.
The CMIP5 multimodel mean distributions of daily (a) rain frequency (with dry-day frequency at top left) and (b) rain amount, during the first 10 yr of 1pctCO2. The response of (c) rain frequency and (d) rain amount to increasing carbon dioxide, calculated as the difference between the 10 yr at the time of carbon dioxide doubling and the first 10 yr and normalized by the change in global-mean surface temperature. Change in dry-day frequency (% K−1) is noted in the top-left corner of (c). Error intervals are the 95% confidence limits according to a Student’s t test. As the distributions are plotted on a logarithmic scale, they are weighted by the rain rate r so that the area under the curve accurately represents the contribution of each rain rate to the total integral. Following Pendergrass and Hartmann (2014b,c), though the r-weighting is implicit to the procedure described there.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
Pendergrass and Hartmann (2014c) found that these changes in the distribution of rainfall in response to warming (as well as those arising in response to El Niño and La Niña phases of ENSO) in models can be well described by two empirically derived patterns, denoted the “shift” and “increase” modes, which are illustrated in Fig. 2. Each mode describes a simple adjustment to the climatological distribution of rain. A combination of the shift and increase modes (chosen with an algorithm to optimize the fit to the change in rain amount distribution) captures most of the response in most climate model simulations of global warming and the entire change in some models.
The rain (left) frequency and (right) amount responses to (a),(b) an increase mode of 0.9% (purple), (c),(d) a shift mode of 3.3% (turquoise), (e),(f), a shift mode of 3.3% and increase mode of 0.9% (magenta), and (g),(h) equal magnitude shift and increase of 3.3% (orange). The color scheme corresponds to these modes throughout the paper. The initial distribution is shown in Fig. 3.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
The “increase” mode (Figs. 2a,b) characterizes an increase in the frequency of rain by the same fraction at all rain rates. The bell shape of this mode simply follows the climatological distribution of rain frequency. While the change in rain amount is characterized by a similar bell-shaped pattern, it occurs at higher rain rates (Fig. 2b). The total amount of rain is the product of the rain frequency and rain rate, such that an increase in rain frequency at higher rain rates has a larger impact on the total precipitation than it does at lower rain rates. An increase in rain frequency implies a reduction in the number of dry days. In the global mean, it rains about half of the time, such that a 1% increase at all rain rates is associated with a 0.5% reduction in dry days.
The “shift” mode (Figs. 2c,d) characterizes a movement of the distribution of rain to higher rain rates but with no net increase in the total rain amount. It is defined as a shift of the rain amount distribution (Fig. 2d); the corresponding change in the rain frequency distribution can also be obtained (Fig. 2c). A larger decrease in the frequency of light rain events is needed to offset the smaller increase in the frequency of strong rain events on total precipitation; hence, the shift mode is associated with an increase in the number of dry days. For a 1% increase in the shift mode, the total number of dry days increases by about 0.5%.
Pendergrass and Hartmann (2014b) determined that the shift and increase mode magnitudes that optimally capture the change in the multimodel mean rain amount distribution in Fig. 1d are a shift mode of 3.3% K−1 and an increase mode of 0.9% K−1. Figures 2e and 2f show the change in rain frequency and amount distributions for this combination of shift and increase modes. The response of the shift mode is larger than the increase mode, such that there is a modest increase in the frequency of dry days.
Not all of the change in the distribution of rain in climate models is captured by the shift and increase modes. Pendergrass and Hartmann (2014c) identified two additional aspects of the changing distribution of rain common to many models: the light rain mode and the extreme mode. The light rain mode is the small increase in rain frequency just below 1 mm day−1 visible in Fig. 1c, also evident in Lau et al. (2013). The extreme mode represents additional increases in rain at the heaviest rain rates, beyond what is captured by the shift and increase modes. It is crucial for capturing the response of extreme precipitation to warming.
Changes in moisture, circulation, and the distribution of rain in response to warming are related. Indeed, the changes in the intensity of extreme rain events in climate model projections of global warming can be linearly related to changes in moisture and vertical velocity in most models and regions (Emori and Brown 2005; O’Gorman and Schneider 2009; Chou et al. 2012). This motivates us to consider whether we can understand the changing distribution of rain in terms of the changes in moisture and vertical velocity distributions, constituting a physically based, rather than empirically derived, approach.
One might assume that changes in the distribution of rain are complex. The distribution of rain (particularly the global distribution) is generated by a number of different types of precipitating systems, each of which is driven by somewhat different mechanisms and might respond differently to external forcing. For example, it would not be surprising if midlatitude cyclones and tropical convection responded differently to global warming. On the other hand, we expect many aspects of the response to warming to be fairly straightforward—warming along with moistening at a relative humidity that stays constant on surfaces of constant temperature (Romps 2014).
In this study, we approach the relationships among changes in moisture, vertical velocity, and rain by examining the response to straightforward changes of simple statistical distributions. We develop two heuristic models that predict the distribution of rain from moisture and vertical velocity distributions. We will see that despite the potential for complexity among these relationships, we can recover many aspects of the changes in rainfall and vertical velocity we see in climate models in an idealized setting.
In section 2, we introduce the first model, in which distributions of moisture and vertical velocity are prescribed. We use the model to explore how the distribution of rain responds to warming and moistening and to changes in the strength and asymmetry (or skewness) of the vertical velocity distribution. Then, in section 3, we introduce a second model that predicts the vertical velocity distribution in order to understand its changes in concert with those of the distribution of rain. In section 4, we show that climate model simulations also have increasing skewness of vertical velocity with warming. Finally, we consider the implications of the increasing skewness of vertical velocity on convective area in section 5 and conclude our study in section 6.
2. The first model: Prescribed vertical velocity
We know rain is a result of very complex processes, many of which are parameterized rather than explicitly modeled in climate models. At the most basic level, rain is regulated by two processes: 1) the moisture content, which is tied to the temperature structure, assuming constant relative humidity, and 2) the magnitude of upward vertical velocity. Instead of considering variability in space, consider a distribution that captures the structure of all regions globally. Furthermore, neglect concerns about the vertical structure of the motion or the structure of the atmosphere and consider only the vertical flux of moisture through the cloud base.
The key—and gross—simplification of this model is that we will assume that the vertical velocity is independent of the temperature and moisture content, so we can model these as two independent distributions. We know this is not the case—upward velocity is often driven by convection, which occurs where surface temperature is warm—but for now we will see what insight can be gleaned with this assumption.
a. Model description
Last, we must specify the parameters governing the temperature and vertical velocity distributions, which are listed in Table 1 for reference. For temperature (shown in Fig. 3a) we take 
Initial parameter choices for the first model.
The distributions driving the first model, where vertical velocity is prescribed: (a) temperature and moisture and (b) vertical velocity (skewness is noted in the top right corner). The resulting distributions of (c) rain frequency (dry frequency, when rain rate is equal to zero, is noted in the bottom-left corner) and (d) rain amount.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
The rain frequency distribution (shown in Fig. 3c) is calculated numerically following the description in appendix A. It is dry exactly 50% of the time, since the vertical velocity distribution is symmetric about zero. The peak of the rain frequency distribution occurs at just under 10 mm day−1. The rain amount distribution (Fig. 3d) shows how much rain falls in each rain-rate bin. The peak of the rain amount distribution occurs at a rain rate about an order of magnitude larger than for the rain frequency distribution.
These distributions resemble those in observational datasets and climate models to the correct order of magnitude—compare to Figs. 1a and 1b and Pendergrass and Hartmann (2014c)—despite the crude assumptions of our model. The main deficiency of our model compared to climate models is a lack of precipitation at light rain rates and a corresponding overestimation of dry-day frequency. However, climate models underestimate the dry-day frequency by about a factor of 2 compared to GPCP 1DD and TRMM 3B42 merged satellite–gauge gridded daily observational datasets (Pendergrass and Hartmann 2014c). The implications of this discrepancy on the rain amount distribution are nonetheless small because light rain contributes less than heavy rain does to the total precipitation so that distribution of rain amount appears better than rain frequency qualitatively (cf. Figs. 1b and 3d).
The goal in developing this toy model is to explore what happens in response to perturbations: warming and moistening, weakening of the circulation, and introducing skewness to the vertical velocity distribution. We consider these next.
b. Response to warming and moistening
We approximate warming by simply shifting the mean of the temperature distribution 
Experiments with the first model. (left) Prescribed vertical velocity distribution, with the initial distribution in the gray-dashed line and each experiment’s distribution in solid black (skewness noted at top right of each panel). (center) Predicted rain frequency response (change in dry frequency noted at center left). (right) Predicted rain amount response in black, with the fitted shift-plus-increase response in color. Colors correspond to Fig. 2; the magnitude of the fitted shift and increase modes and their errors are listed in Table 2. Each row is one experiment: (a)–(c) warm, (d)–(f) weaken the vertical velocity distribution, (g)–(i) skew the vertical velocity distribution, (j)–(l) warm and skew, and (m)–(o) warm while weakening and skewing the vertical velocity distribution.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
The rainfall distribution response to warming is equivalent to moving the rain frequency distribution to the right by exactly 7% K−1, or having equal shift and increase modes of 7% K−1 (the fitted shift and increase modes are listed in Table 2), as in Figs. 2g and 2h. In contrast to this warming experiment, in climate model simulations of global warming the shift mode response is larger than that of the increase mode, and total precipitation increases more slowly than moisture. This exposes a flaw: circulation also adjusts to changes in climate, which is not captured by this first experiment. In climate model projections, circulation adjusts to satisfy the energetic constraints of the climate system, including the constraint that precipitation (in the global mean) can only increase as much as atmospheric radiative cooling and sensible heat flux allow (e.g., Allen and Ingram 2002).
The magnitude of fitted shift and increase modes along with their error (the magnitude of the response that the fitted shift-plus-increase response fails to capture) for each of the experiments shown and discussed here. The precipitation response to a transient CO2 increase in climate models is shown for the CMIP5 multimodel mean as well as for one GCM, MPI-ESM-LR, which is fit the best of all the CMIP5 models (see Pendergrass and Hartmann 2014b for details). The Model 1 experiments are shown in Fig. 4 and discussed in sections 2b–d. Model 2 experiments are shown in Figs. 6–8 and discussed in section 3c.
c. Response to weakening circulation
A weakening of the atmospheric overturning circulation can be effected in our model by reducing the width of the vertical velocity distribution 
Again, there is no change in the dry frequency, and the total amount of rainfall decreases by 4%, the same amount that we weakened the width of the vertical velocity distribution by. Decreasing the width of the vertical velocity distribution results in a shift of the rain frequency distribution to lower rain rates. In fact, narrowing the w distribution by 7% would exactly cancel the effect of warming by 1 K. We can understand this by considering Eq. (2) or (3); warming by 1 K increases q by 7%, whereas widening the vertical velocity distribution increases w by 7%. The effect of either change on r is the same.
We have just seen that neither warming nor changing the strength of the circulation affects the dry frequency or the symmetry between the rates of change of mean and extreme rainfall. Changes analogous to those we see in climate model simulations thus cannot result from either warming at constant relative humidity or weakening circulation alone. But what if the circulation becomes more asymmetric?
d. Response to changing skewness of vertical velocity
The first moment of the vertical velocity distribution, its mean, must be fixed at zero to maintain mass conservation. We have just seen that changing the second moment (standard deviation or variance) does not cause the changes in the distribution of rain that we see in climate models. We now turn to the third moment, skewness, which measures the asymmetry of a distribution. Skewness, a key quantity, is attended to more widely in the parts of atmospheric sciences dealing with turbulence, like boundary layer meteorology. It has also received some limited attention in climate recently. Monahan (2004) discusses skewness of low-level wind speed arising from surface drag. Luxford and Woollings (2012) discuss how skewness arises in geopotential height from kinematic fluctuations of the jet stream. Sardeshmukh et al. (2015) incorporate skewness into a nonlinear model for atmospheric fields including precipitation. In particular, they highlight the skewness in the vertical velocity field.
Skewness can arise in vertical motion from the asymmetric effect of latent heating. To visualize this effect, picture a developing thunderstorm. The cumulus cloud grows because an updraft is heated when water vapor condenses, sustaining or even strengthening the updraft and eventually resulting in rainfall. Over the life of the thunderstorm, some of this rainfall will reevaporate, but there will be a net latent heating of the atmosphere due to the formation of this thunderstorm equal to the amount of rainfall that reaches the ground. There is no corresponding effect of latent heating on subsiding air; it merely warms adiabatically as it sinks.
To incorporate skewness into the vertical velocity distribution, we draw w from a skew-normal distribution generated following Azzalini and Capitanio (1999), instead of from a normal distribution as before. A skew-normal distribution has 3 degrees of freedom, which determine its mean, variance, and asymmetry. When the asymmetry is zero, the skew-normal distribution becomes normal. We adjust the skew-normal distribution so that the mean is always zero to maintain mass conservation, and we maintain a constant variance of the w distribution to eliminate the effects of changing circulation strength. The resulting distribution of w and the response in rain frequency and amount distributions to a 0.2 increase in skewness are shown in Figs. 4g–i.
The responses of the rain frequency and amount distributions to increasing skewness of the vertical velocity have some intriguing features. There is a notable decrease in the frequency of rain for moderate rain rates (Fig. 4h), but the total amount of rain remains essentially constant owing to a slight increase in the frequency of higher rain rates (Fig. 4i). This strongly resembles the shift mode. The magnitude of the strongest updrafts also changes little. Increasing skewness without conserving the mean of w would increase the strength of the strongest updrafts, but the shift of the distribution to maintain mass continuity compensates for this.
To move toward the response of precipitation to global warming in climate models, we simultaneously warm and increase the skewness of the vertical velocity distribution, shown in Figs. 4j–l. The response of the rain frequency and amount distributions to warming and skewing has all the features seen in climate models: a decrease in the total rain frequency and in the frequency of rain falling at moderate rain rates, along with an increase in rain amount focused at the heaviest rain rates. Increasing the skewness of the vertical velocity distribution effects crucial components of the change. It decreases the total frequency of rain events, breaks the symmetry between the changes in mean and extreme rainfall, and allows us to change the magnitude of the shift mode without changing the increase mode.
To fully capture the changes we see in climate model simulations, we weaken the distribution of vertical velocity (decrease 
To recap, we have shown that warming (increasing 
3. The second model: Predicted vertical velocity
We know that precipitation is energetically constrained by total column heating and cooling. Thus, in this model we start with energetics. We prescribe a distribution of nonlatent heating 
a. Model description
Our goal is to predict the distribution of w, which will in turn give us the rainfall from Eq. (2), as in our first model. We begin with the temperature and moisture distributions (again connected by the assumption of saturation; Fig. 5a), except that the tail of the temperature distribution is truncated at a maximum temperature 
(top) The prescribed distributions driving the second model, where vertical velocity is predicted: (a) temperature and moisture and (b) nonlatent heating (mean is noted in the top-right corner). (bottom) The resulting predicted distributions of (c) vertical velocity, (d) rain frequency (dry frequency noted in the bottom-left corner), and (e) rain amount.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
The parameters we use are listed in Table 3. The mean of the nonlatent atmospheric column heating is equal but opposite to the CMIP5 multimodel mean precipitation (88 W m−2), and its standard deviation is dominated by variability in the dry static energy flux convergence on short time scales (following Muller and O’Gorman 2011); we choose a value similar to those we found in climate model integrations.
Initial parameter choices for the second model.
Truncating the temperature distribution is necessary to ensure that the denominator in Eq. (7) never drops to or below zero, which would result in infinite w. The term 
In addition to our choice of 
The distributions of vertical velocity and rain produced by our model with the parameters listed in Table 3 are shown in Figs. 5c–e. As with the first model, the distributions of rain frequency and amount are qualitatively similar to observations and climate model simulations in terms of both the peak magnitudes and overall structure.
Most importantly, the model predicts a skewed distribution of w. To ensure that the skewness was not an artifact of the nonzero mean of the nonlatent heating distribution, we specified 
b. Perturbations about the control climate
Here we explore the responses to the three parameters other than warming: mean nonlatent heating 
In the first experiment, we increase the magnitude of mean nonlatent heating 
Experiments varying parameters other than the mean temperature with the second model, following Fig. 4, but here the (left) vertical velocity distribution is predicted. (a)–(c) Increasing the magnitude of mean nonlatent heating and increasing the width of the nonlatent heating distributions while holding all other parameters constant. (d)–(f) Increasing the magnitude of mean nonlatent heating and decreasing stability. (g)–(i) Narrowing the nonlatent heating distribution (decreasing 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
Also included in Fig. 6c is the combined shift-plus-increase mode fitted to the rain amount response. The fitted shift-plus-increase response is colored orange (following the color scheme shown in Fig. 2), which corresponds to equal magnitudes of shift and increase modes. The magnitudes and error of the fit are listed in Table 2 (and are normalized by 3-K warming to compare with warming experiments, discussed next); the error is the magnitude of the response that the fitted shift-plus-increase response fails to capture. The fitted shift mode is slightly bigger than the fitted increase mode (11% vs 9% K−1).
The response of the vertical velocity and rainfall distributions is essentially the same response we would get from strengthening w in the first model (the opposite of the weakening w experiment in Figs. 4d–f); only here it is achieved in a way that is consistent with energy as well as mass balance. In this experiment, the magnitudes of vertical velocity and rain change, but the shape of their distributions, including of the fraction of events that are rain-producing updrafts, does not.
In the second experiment, we again increase the magnitude of mean nonlatent heating but now hold the width of the nonlatent heating distribution constant and instead decrease stability S. We determine the decrease in S required to balance the increase in 
In the third experiment, we narrow the distribution of nonlatent heating by decreasing 
The response of the vertical velocity distribution is a negligible change in width but an increase in skewness, which we can understand as follows. The narrowing 
The response of the rain frequency and amount distributions to changing 
To summarize the effect of perturbing parameters other than temperature in this model: increasing 
c. Response to warming
Next, we explore the response of the vertical velocity and rainfall distributions to warming. We increase 
Experiments warming while varying one other parameter with the second model, following Fig. 6: (a)–(c) increasing stability, (d)–(f) increasing the magnitude of mean nonlatent heating, and (g)–(i) narrowing the nonlatent heating distribution [decreasing 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
In the first experiment, we balance warming by increasing S. Stability also changes in climate model simulations of global warming; specifically, dry static stability increases with warming in the tropics (e.g., Knutson and Manabe 1995) and subtropics and midlatitudes (e.g., Frierson 2006; Lu et al. 2007). We determine effects of changing T on energy and mass balance and the increase in S needed to balance it by linearizing Eq. (B4) for energy and mass balance about perturbations in S and T, shown in appendix C. This linearization shows that one degree of warming is balanced by a 7% increase in stability, where the factor of 7% arises from the moistening associated with the warming. The distributions of vertical velocity and moisture that result from warming by 3 K and increasing stability by 21% are shown in Figs. 7a–c. The increased stability decreases the magnitude of vertical velocity for a given atmospheric column heating so that the vertical velocity is weakened [its standard deviation decreases, as in Held and Soden (2006) and Vecchi and Soden (2007)], and the distribution of rainfall is exactly unchanged. The skewness of vertical velocity is also unchanged. In this model, the dry frequency is just the fraction of the time that the atmospheric column heating is negative; since atmospheric column heating does not change in this experiment, neither does the dry frequency. The trade-off between warming and stability here is similar to the trade-off between warming and the width of the vertical velocity distribution in our first model.
In the second experiment, we warm while increasing the magnitude of mean nonlatent heating 
In the third experiment, warming is balanced by narrowing of the nonlatent heating distribution (decreasing 
In two final experiments, we emulate the changes seen in climate models: we warm and also increase the magnitude of nonlatent atmospheric column heating 
Experiments warming, increasing the magnitude of the nonlatent heating distribution by the value from climate models, 1.1 W m−2 K−1, while varying one other parameter with the second model, following Fig. 6: (a)–(c) increasing stability and (d)–(f) narrowing the nonlatent heating distribution (decreasing 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
First, we warm, increase mean 
Finally, we warm, increase mean 
To summarize, in our second model, the atmosphere can respond in three ways to warming: 1) increasing the stability S, which weakens the circulation w but has no effect on rain, 2) increasing the total precipitation 
In this simple model, if we warm and increase mean latent heating 
4. Comparison with the response to warming in climate models
The two heuristic models above show that increasing skewness of the vertical velocity distribution coincides with key characteristics of the changing distribution of rainfall that we see in climate models. Does skewness of the vertical velocity distribution increase with warming in climate models?
To address this question, we calculate statistics of daily average 500-hPa pressure vertical velocity and their change in three warming experiments in the CMIP5 archive (Table 4). We calculate the area-weighted global-average moments from years 2006–15 and 2090–99 in the RCP8.5 scenario and years 1–10 and 61–70 in the transient 1% yr−1 carbon dioxide increase scenario (
Standard deviation, skewness, and kurtosis of 500-hPa pressure vertical velocity from CMIP5 models and their response to warming (normalized by global-mean surface temperature change).
All climate model simulations have increasing skewness of vertical velocity, consistent with our expectations from the heuristic models along with the changing distribution of rain in climate models. The magnitude of increase in skewness varies widely across models, from less than 1% to 27% K−1. Note that the models with the biggest increases in skewness (GFDL-ESM and IPSL-CM5A models) also have a large extreme mode (Pendergrass and Hartmann 2014b). While we have touched on the extreme mode in our second heuristic model, much about it remains to be investigated.
The variance of vertical velocity decreases in all but one of the climate model simulations. Decreasing variance of vertical velocity at 500 hPa is consistent with Held and Soden (2006) and Vecchi and Soden (2007), though their metrics were slightly different from ours and the magnitude of changes shown here is smaller. Additionally, the change in vertical velocity strength at 500 hPa is expected to underestimate the weakening of the total vertical overturning circulation because the strongest motion is above 500 hPa and shifts upward with warming (Singh and O’Gorman 2012).
We include the changes in kurtosis in Table 4, the fourth moment of the distribution. Larger kurtosis corresponds to a fatter tail and a narrower peak of the distribution; a normal distribution has a kurtosis of 3 (e.g., DeCarlo 1997). In all climate models, kurtosis of vertical velocity is initially greater than Gaussian, and it increases with warming. Our second model predicts an increase in kurtosis along with the increases in skewness. Interestingly, the GFDL models have by far the largest increases in kurtosis with warming (they also have large extreme modes).
We are now in a position to reconcile the differing magnitudes of the shift and increase modes with warming that we see in climate model simulations. For the multimodel mean, moistening occurs at about 6%–7% K−1, and global mean precipitation increases at 1.5% K−1. The multimodel mean rain amount response has an increase mode of 1% K−1 and a shift mode of 3.3% K−1. MPI-ESM-LR, whose response is best captured by the shift and increase modes, has an increase mode of 1.3% K−1 and a shift mode of 5.7% K−1.
We relate the shift and increase modes to changes in moisture and circulation as follows (and shown in Fig. 4 as well as listed in Table 2): moistening at 7% K−1 results in equal magnitudes of shift and increase modes. This is countered by a narrowing of the vertical velocity distribution that is not quite as large, bringing the net magnitudes of both the shift and increase modes down. Finally, an increase in skewness of the vertical velocity distribution results in a shift mode with no corresponding increase mode. The combination of these three changes results in a shift mode that is larger than the increase mode seen in the climate model response to warming.
While the heuristic models developed here capture some important aspects of the response of rainfall and vertical velocity to warming seen in climate models, the cost of its simplicity is the number of assumptions that must be made. Assumptions for our idealized relationship between moisture, vertical velocity, and rain rate include the following: that all moisture is removed whenever there is upward motion, that the vertical structure of the atmosphere is fixed, and that relative humidity does not change. Our models do not accommodate any unresolved processes, parameterized in climate models, which can alter the relationship between rainfall and vertical velocity. This idealized framework also does not address the differing direct and temperature-mediated responses of precipitation and circulation to greenhouse gas forcing. Finally, aggregating over all locations and seasons convolves many different processes, and the relationships we explore here may not hold for all of them. Nonetheless, while we anticipate that our heuristic models do not capture the behavior of every relevant process that contributes to the responses of rainfall and vertical velocity to global warming, we think these models are useful for understanding a substantial portion of the response in many regions of most climate models.
5. Convective area
The spatial manifestation of the distribution of rain and vertical velocity is convective area, by which we mean the area with upward motion and the cloudiness and rainfall that accompany it. The fraction of time that vertical motion is upward and the fraction of time that it is raining in the heuristic models presented here is analogous to the fraction of the area in a domain where rain is occurring. The literature is currently unsettled about how the change in convective area and frequency of upward motion are expected to change with warming. Johnson and Xie (2010) argue that the convectively active fractional area of the tropics changes little relative to the area above an absolute SST threshold, which increases by 45% over the twenty-first century in the experiments they analyze, though this study focused on monthly mean precipitation, rather than daily data. In contrast, Vecchi and Soden (2007) report a decrease in the number of grid points with upward motion in GFDL CM2.1 simulations of global warming in the tropics. Other recent studies focusing on monthly to seasonal mean precipitation find a decrease in the area of the ITCZ with warming (Neelin et al. 2003; Huang et al. 2013; Wodzicki and Rapp 2016). Byrne and Schneider (2016) examine the width of the ITCZ over a wide range of climates in a gray-radiation climate model and find different responses in different climate states. In CMIP5 model simulations, the frequency of dry days has a small but significant increase (see Fig. 1a or Pendergrass and Hartmann 2014b).
The heuristic models shown here reproduce the increase in dry frequency seen in the CMIP5 models and thus also the decrease in convective area. Figure 9 shows a schematic of the tropical overturning circulation to aid in interpreting its response to changes in the distribution of vertical velocity. The initial distribution has a region of ascent that is narrower than the region of descent, analogous to the circulation in the tropical atmosphere (Fig. 9a). Because the region of ascent is narrower and mass is conserved, the ascending motions are stronger than corresponding descending ones. Decreasing the standard deviation of the vertical velocity distribution decreases the magnitude of both upward and downward motion (weakening the circulation), with no change in area of either region (Fig. 9b). Increasing the skewness of vertical velocity increases the magnitude of upward motion while decreasing its area and decreases the speed of descent while increasing its area (Fig. 9c). When the decrease in standard deviation and increasing skewness occur together, both contribute to weakening the descending motion, but they have competing effects on the magnitude of ascent, resulting in little change in updraft strength (Fig. 9d).
A schematic showing the effects of changing width and skewness of the vertical velocity distribution. An (a) initial skewed distribution of w, is perturbed by (b) decreasing its standard deviation, (c) increasing its skewness, and (d) both decreasing standard deviation and increasing skewness together.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0097.1
6. Conclusions
We have introduced two idealized models relating the distributions of rain and vertical velocity. In both models, temperature (and thus moisture, assuming constant relative humidity) is prescribed, and the distribution of rainfall is predicted. In the first model, the distribution of vertical velocity is also prescribed and can be varied; mass conservation is respected. In the second model, the distribution of nonlatent atmospheric column heating is prescribed, the distribution of vertical velocity is predicted, and both mass and energy are conserved. Some key assumptions made by both models are that relative humidity is fixed within and between climate states and that stability is constant within each climate state.
Both of these models show that increasing skewness, or asymmetry, of the vertical velocity distribution is necessary to recover important characteristics of the changing distribution of rain with warming predicted by climate models: dry-day frequency increases, and extreme precipitation increases at a rate faster than the increase in mean precipitation. In the context of shift and increase modes of change of the distribution of rain, an increase in skewness is necessary to achieve the larger shift mode than increase mode seen in climate model projections. The second model, where the distribution of vertical velocity is predicted, shows how the asymmetric influence of latent heating creates skewness in the vertical velocity distribution. Experiments with this model show that this skewness increases in response to warming, along with the adjustments needed to maintain mass and energy balance. In addition to an increase in skewness, the standard deviation of the vertical velocity distribution also decreases, consistent with the weakening circulation found in climate model simulations of global warming.
The models developed here capture salient aspects of the changing distributions of rain and vertical velocity with simple thermodynamic relationships, implying that we do not need to resort to complex dynamical explanations for these aspects of the changing distribution of rain. The idealized relationships between the distributions of vertical velocity and precipitation explored here hopefully form a basis for understanding the richer and more complex interactions in climate models and in the real world.
Acknowledgments
We thank Clara Deser, Ben Sanderson, Brian Rose, Flavio Lehner, and two anonymous reviewers for their feedback. NCAR’s Advanced Studies Program postdoctoral research fellowship provided funding for AGP. EPG acknowledges support from the National Science Foundation through Grant AGS-1264195.
APPENDIX A
Numerical Solutions
a. Normal and skew-normal distributions
We calculate the value of the normal distribution at points that are evenly spaced in percentile space—5000 points for Model 1 and 10 000 for Model 2. For the temperature distribution, any values of 
Calculating the skew-normal distribution is similar to a joint distribution because the algorithm of Azzalini and Capitanio (1999) calls for operating on two normal distributions. We start with normal distributions 
b. Frequency and amount distributions
We use logarithmically spaced bins for the rain frequency and amount distributions and choose 250 of them to obtain stable fits of the shift-plus-increase modes. Details of the calculation and further examples of rain amount and rain frequency distributions can be found in Pendergrass and Hartmann (2014c). We use 50 linearly spaced bins for 
c. Model 2 parameters
To calculate the parameters in the second model, there are two steps: the initial setup to find a balanced state and variation of parameters about this state.
To set up the model initially, the challenge is meeting energy and mass balance. We accomplish this numerically by specifying all parameters other than 
To find a new balanced state due to small variations in T and S around the initial balanced state, we use the linearizations in appendix C. This is done in three different ways. Whenever possible, we use the linearization alone to find new values of T and S, or of the new LHS of the energy/mass balance equation. When necessary, we resolve for a new 
APPENDIX B
Conservation of Mass and Energy
It is also possible to arrive at Eq. (B3) by starting from the mass conservation constraint [Eq. (B1)], substituting Eq. (5), exploiting the independence of q and 
APPENDIX C
Linearization of Energy and Mass Balance about T and S
a. Linearization in T
b. Linearization in S
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With the introduction of 
