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# Global warming precipitation accumulation increases above the current-climate cutoff scale

Edited by Kerry A. Emanuel, Massachusetts Institute of Technology, Cambridge, MA, and approved November 28, 2016 (received for review September 14, 2016)

## Significance

Large accumulations of rainfall over a precipitation event can impact human infrastructure. Unlike precipitation intensity distributions, probability distributions for accumulations at first drop slowly with increasing size. At a certain size—the cutoff scale—the behavior regime changes, and the probabilities drop rapidly. In current climate, every region is protected from excessively large accumulations by this cutoff scale, and human activities are adapted to this. An analysis of how accumulations will change under global warming gives a natural physical interpretation for the atmospheric processes producing this cutoff, but, more importantly, yields a prediction that this cutoff scale will extend in a warmer climate, leading to vastly disproportionate increases in the probabilities of the very largest events.

## Abstract

Precipitation accumulations, integrated over rainfall events, can be affected by both intensity and duration of the storm event. Thus, although precipitation intensity is widely projected to increase under global warming, a clear framework for predicting accumulation changes has been lacking, despite the importance of accumulations for societal impacts. Theory for changes in the probability density function (pdf) of precipitation accumulations is presented with an evaluation of these changes in global climate model simulations. We show that a simple set of conditions implies roughly exponential increases in the frequency of the very largest accumulations above a physical cutoff scale, increasing with event size. The pdf exhibits an approximately power-law range where probability density drops slowly with each order of magnitude size increase, up to a cutoff at large accumulations that limits the largest events experienced in current climate. The theory predicts that the cutoff scale, controlled by the interplay of moisture convergence variance and precipitation loss, tends to increase under global warming. Thus, precisely the large accumulations above the cutoff that are currently rare will exhibit increases in the warmer climate as this cutoff is extended. This indeed occurs in the full climate model, with a 3 °C end-of-century global-average warming yielding regional increases of hundreds of percent to >1,000% in the probability density of the largest accumulations that have historical precedents. The probabilities of unprecedented accumulations are also consistent with the extension of the cutoff.

Occurrences of intense precipitation are projected to increase (1⇓⇓⇓⇓⇓⇓–8) associated with higher atmospheric moisture content (9, 10) under global warming. Measures of precipitation intensity, coarse-grained to 1- to 5-d intervals, exhibit end-of-century increases on the order of 20% for wettest annual 5-d rainfall (8) or 10% in average wet-day intensity (11) or 17% °C in the 99.9th to 99.999th percentiles of daily precipitation (12) in business-as-usual anthropogenic forcing scenarios. Associated with this, substantial increases in frequency of high-rain-rate events can occur (13⇓–15), and the return times of events exceeding a given threshold decrease (16).

Time-integrated accumulation, the amount of precipitation that falls during a single event, is of concern for many societal impacts (17). Because more intense precipitation could, in principle, yield shorter event durations (10), the expected change in accumulation probabilities is unclear. Here, we derive a stochastic prototype from a fundamental climate model equation. This leads to an explanation of key properties of the probability density function (pdf) of accumulations noted in station observations (18⇓–20)—why the pdf of accumulation size drops slowly with increasing size over many orders of magnitude before reaching a cutoff scale, after which the pdf drops rapidly for very large accumulations. From the theory, we show that physical balances creating this behavior regime imply extremely high sensitivity for the very largest events under climate change. This motivates an evaluation of the accumulation distribution and its changes under global warming in a global climate model, the Community Earth System Model [CESM1 (21)]. The fact that the integrated precipitation during the event plays a key physical role makes accumulation a natural variable for examining these impacts.

## Moisture Equation and Dynamics of Accumulation

The vertically integrated moisture equation for a climate model may be written

The time-integrated accumulation

This moisture loss plays a role in terminating events, so it is useful to define a running accumulation

We now introduce two cases of a stochastic prototype model based on simplifications of Eq. **1**. Case 1 is sufficiently simplified to permit analytic solutions that guide understanding of the physics, whereas Case 2 is integrated numerically in time and then diagnosed as for a climate model.

For Case 1, we make use of the one-to-one monotonic relationship Eq. **3** between time and the running accumulation, using ^{−2} of water). Precipitation during an event is an order of magnitude larger than climatological moisture convergence

For Case 2 of the stochastic approximation to Eq. **1**, we approximate the precipitation dependence on

In both cases, the presence of a threshold for onset and termination of precipitation creates a first-passage problem (28, 29) in which trajectories in the precipitating regime above the threshold evolve by a competition between fluctuating moisture convergence and water loss by precipitation until the moisture first falls below the threshold that terminates the event. For Case 2, trajectories are computed numerically for a long time series that enters and exits events as for a climate model. For Case 1, one has a simple Fokker–Planck equation (30) corresponding to Eq. **4** for the evolution of the pdf *SI Text*), and we return to the properties of the first-passage problem schematically in *Physical Mechanisms and Implications for Robustness* to discuss how the analytic solution relates to more complex cases.

The key result for informing climate model analysis is the pdf **S3**) that is too small to see in gauge observations (18, 19) or in the climate model.

Features of this solution are seen in the top pair of curves of Fig. 1. There is a scale-free range over which the power law *s*. A key transition occurs near

The prototype makes clear that the changes are expected to occur disproportionately for the very largest events (i.e., approximately exponential changes in the rare portion of the size distribution above the regionally defined cutoff for the historical period). Dynamical feedbacks will affect this regionally, including the possibility of regional reductions in largest-event probability density if

For Case 2 of the stochastic prototype, results (Fig. 1) exhibit close parallels to the analytic solution Eq. **7** of Case 1, but with the exponent *SI Text* for details). In Case 2, deviations from the form of Eq. **7** as a function of scale may be noted in both the approximate power law and cutoff ranges, but the key point remains of an approximately scale-free range over which the pdf drops slowly, followed by a scale at which the pdf drops quickly. The Case 2 results can be understood as a modification of the Case 1 assumption of uncorrelated noise, as discussed in *Physical Mechanisms and Implications for Robustness* and *SI Text*.

## SI Text

### Background on the First-Passage Process Simple Prototype.

The simple stochastic prototype, Eq. **4**, is used to approximate the moisture equation from the climate model, using the running accumulation **4**, and the noise term W is assumed to be from a Wiener process (Case 1 of the stochastic prototype), the corresponding Fokker–Planck equation has a classic form (29), except that *q* and **6**, repeated here for convenience:**8**), with the second occurrence shifted to maintain the boundary condition. For this case,

This yields an inverse Gaussian distribution for the accumulation pdf^{−3} to 10^{−2} mm (18). In gauge observations (19, 20) and in the climate model analyzed here, the small event portion of the range is not resolved. Thus, the distribution is simply considered over the range greater than the specified minimum accumulation **7**. The normalization constant *s* range dominates the integral when the power law range is long

Note that the time-mean rainfall is not closely related to the changes in the accumulation distribution. The time mean obeys long-term constraints from moisture and energy budgets that can affect the fraction of time spent precipitating. The dry-spell intervals between precipitation events have dynamics with strong parallels to that considered here for accumulation distributions, with the upward drift in moisture toward onset of precipitation driven by mean moisture convergence, including evaporation. For increasing moisture convergence variance, the probability density of the very longest dry spells in the prototype tends to increase consistently with the results shown here for the change in pdf of the largest precipitation accumulations.

### Considerations of Robustness and Relations with Other Systems.

Despite the complexity of the climate model, the simple stochastic prototype was able to provide predictions of its behavior. While the analytic solution for Case 1 and the scaling solution aim at distilling physical insight, the numerical solutions for Case 2 of the stochastic prototype make clear that the slight adjustments to the exponent of the power law range seen in the climate model solutions are easily obtained for assumptions that are realistic in a climate-modeling context. Here, we elaborate briefly on how the form of the regime change at a cutoff scale is robust to these changes. The Case 2 model is solved in the time domain, as for a climate model (note that *D*_{*} thus has different units than *D*). In Case 2, during nonprecipitating intervals, there is an upward drift in *q* due to

Prototypes for anomalous diffusion regimes and associated first-passage problems have been examined in a number of systems (42), so it is worth considering under what circumstances analogies drawn from these cases can be instructive. First-passage problem solutions exist for representation of these cases by fractional Fokker–Planck equations (43⇓–45). A key result is that the properties discussed in the Weiner process case are modified smoothly, with the exponent of the power law range adjusted. Properties are in some cases approximated by a generalized inverse Gaussian distribution as in Eq. **3** where

### Moment Ratio *s*_{M} and Large-Event Cutoff Geographic Dependence in the Climate Model.

In the main text, an event-size scale **3**), the large-event cutoff **4**, adjustments to the proportionality constant occur associated with

Fig. S1 shows the estimate of the large-event cutoff **6** as a spatial distribution for historical and end-of-century simulations, respectively. Fig. 3 shows the ratio of these. The full ensemble of 450 y is used, computing each of **6**. This first computation of such an estimate as a map shows spatial variations of *a*), with values increasing substantially toward the tropics. In many regions, a tendency may be noted for *b*), large-scale spatial patterns remain similar, and the global warming increase in many areas can be seen primarily as a broad-scale increase on the order of

## Precipitation Event-Size Change Under Global Warming

Motivated by the behavior predicted by the theory, we carried out an ensemble of 15 CESM1 simulations with the required high-resolution time output to assess these distributions under historical estimates of radiative forcing by greenhouse gases and aerosols and under Representative Concentration Pathway 8.5 (33) (*Materials and Methods*). Precipitation accumulation distributions for various regions in Fig. 1 consistently exhibit long power law ranges with exponent **7** with

As expected, the power-law range is for each region followed by a large-event cutoff, above which the pdf of large accumulations drops steeply. Were the power law range to extend indefinitely, the mean and variance of the accumulation would diverge, an indicator of how important the physics of the cutoff is in limiting large events. Under global warming, the power-law portion of the distribution remains essentially unchanged, while, in many regions, the large-event cutoff changes, extending the power law range to slightly larger values on the

Referencing future probabilities to current climate probabilities of a given accumulation, as in Fig. 2, is informative, but the end-of-century extension of the distribution yields accumulations that are unprecedented in the historical period. A direct measure of the change in distribution extent is thus useful for interpretation and for estimating statistical significance by geographic region. The change in distribution in Fig. 1 is very close to a simple rescaling, even for curves where the cutoff is more complex than exponential, as occurs for the lower four curves. This is indicated in Fig. 1 by overlaying curves interpolated for each historical distribution onto the corresponding end-of-century distribution, with

In addition to providing a simple hypothesis for future changes in corresponding observed distributions, the result that the distribution changes are governed by the shift in the cutoff can greatly aid in significance testing and displaying changes at a regional level. Fig. 3 uses an estimate *Materials and Methods* and Eq. **9**) to indicate the simulated spatial distribution of these changes under future climate. The ratio of *Materials and Methods*), and thus of

The regions for which distributions are averaged in Fig. 1 are indicated in Fig. 3, chosen to sample regions of relatively modest change, such as the Midlatitude North American region, as well as regions of large change, such as a box in the Indian region and a box covering E. China and the neighboring ocean area including Taiwan. Factors rescaling the box-average distributions in Fig. 1 bracket the Clausius–Clapeyron scaling, from slightly below (1.14 for Australia and Midlatitude North America) to substantially above (1.46 for E. China). It is worth emphasizing that, even for regions where the rescaling in **7** yields a fractional increase in the pdf of approximately

## Physical Mechanisms and Implications for Robustness

This distinct behavior for the largest events is due to the time-dependent dynamics that yields the cutoff scale. Standard prototypes for changes in extreme events (17, 35) consider stationary distributions of a climate variable that in a warmer climate becomes shifted due to a change in mean or whose width changes due to a change in variability. For non-Gaussian distributions (Fig. 4*A*), such as precipitation rate (14, 15, 35, 36) or water vapor (27), the change in mean and variance are typically linked. In addition to affecting occurrences of extreme values (red), these mechanisms create changes throughout the probability distribution (decreases in blue; increases in light red). In the behavior for accumulations found here (Fig. 4*B*), changes are primarily in the probability of the most extreme accumulation range. How does this occur?

A scaling argument from the theory here points to the essential features of the climate physics that create this sensitivity and provides a sense of the robustness. Probability distribution solutions of Eq. **6** during the event have the form*SI Text*), so Eq. **8** is sufficient to show the key physical factors.

Fig. 4*C* illustrates the collision of two temporal-dependences in Eq. **8** that yield the different behavior ranges. In early time, the spread of the distribution width like

## Discussion

In the physical mechanism for changes in precipitation accumulation presented here, one of the essential ingredients, an increase in the variability of moisture convergence, is in common with the prevailing discussion of changes in precipitation intensity under warming. The other ingredients—an integrating variable and a threshold for event termination—dictate not a general broadening of the accumulation distribution, but a shift in the cutoff that limits very large events. These ingredients imply that, for integrated accumulations, a corollary of the rich-get-richer effect applies, especially to the largest events. This might be termed the biggest-get-more-frequent or the biggest-get-bigger, because successively larger increases in frequency occur for successively larger accumulation categories above the historical cutoff scale as the cutoff value increases. Geographic patterns of the changes should be viewed with caution from a single climate model, and caveats apply to climate model simulation of extreme precipitation events (6, 14, 27)—these results motivate evaluation over a wider ensemble of climate models, despite the need for high time-resolution output, and over observations more extensive than the set so far examined. The simplicity of the mechanism and of the resulting distributions for accumulation in Fig. 1 suggest that precipitation accumulation provides a natural coordinate for evaluating statistics of extreme precipitation change. The success seen in Fig. 1 in approximating changes in the pdf, even for unprecedented events, by a suitable rescaling of the historical distributions suggests that this can be exploited to evaluate scenarios for potential impacts.

## Materials and Methods

Precipitation accumulations in CESM are computed as the integrated precipitation from the first exceedance of a small threshold (0.4 mm/h) to the first drop below the threshold. These have not previously been computed in climate models—the high time-resolution data are not normally saved. Augmented output (for 30S to 50N) from an ensemble of runs with the fully coupled National Center for Atmospheric Research CESM1 (Version 1.0.5) (21) under historical estimates of radiative forcing by greenhouse gases and aerosols and under Representative Concentration Pathway (RCP) 8.5 (33) with precipitation saved at all time steps (30-min intervals) is used to compute the accumulation distributions in Fig. 1. The ensemble of 15 simulations of 30 y each are initiated from different atmospheric initial conditions for the historical and RCP8.5 simulations (37) to yield different instances of atmospheric and climate internal variability. CESM output data used in this study are available from the authors following guidelines in the CESM data plan (www.cesm.ucar.edu/management/docs/data.mgt.plan.2011.pdf).

For the accumulation distributions and ratios in Figs. 1 and 2, the latitude–longitude ranges for the six averaging boxes are: Midlatitude North America (37N to 50N, 240E to 288E), Africa (20S to 18N, 10E to 44E), Australia (28S to 16S, 120E to 152E), South American region (20S to 0, 290E to 320E), India (15N to 25N, 70E to 90E), and E. China region (20N to 30N, 110E to 130E), which includes ocean regions surrounding Taiwan, the East China Sea, and the southwest islands of Japan. Accumulation distributions are computed for each spatial point and then averaged within these domains. As seen in Fig. 3, the latter two regions provide examples within the long band across Southeast Asia exhibiting substantial increases in

The bin size in Figs. 1 and 2 has a value of 0.2 in

For analysis of spatial distributions of the cutoff (Fig. 3), we use an estimator of the cutoff **7** (*SI Text*), this is related to the large-event cutoff

The Clausius–Clapeyron scaling of

A bootstrap procedure (40) is used to establish confidence intervals in Figs. 1 and 2 and to mask small changes in Fig. 3. The fifteen 30-y runs for each of historical and end-of-century are broken into 5-y segments, for a total of 90 segments, which are considered independent for these precipitation statistics. A set of 1,000 bootstrap replications is constructed by random picks with replacement from the set of segments to create 1,000 artificial ensembles of 90 segments. Statistics for each replication are computed exactly as on the original ensemble of 90 segments, for historical and end-of-century respectively. For Fig. 3,

In Fig. 4*A*, a gamma distribution pdf, commonly used for daily rainfall intensity distributions (36), is used to illustrate the typical occurrence of changes throughout the distribution (14) as mean and variance increase (linear axes; shape parameter = 2; scale parameter = 3 and 4, schematizing historical and EoC cases, respectively). In Fig. 4*C*, the pdf as a function of moisture and **8** is schematized as shading with black contours for an idealized case for which the effect of drift toward the event-termination threshold due to precipitation loss can be seen within the diagram. The no-drift case (dropping the drift term) is shown as a single contour for reference. The white arrow shows the difference to the corresponding contour in the full case, indicating the growing importance of the drift term.

For the Case 2 stochastic prototype, solutions in Fig. 1 are for ^{−1} and ^{−1} with an increased value for ^{2} h^{−1} yield shorter/longer cutoff cases shown, chosen to illustrate cutoffs similar to E. China historical and end of century CESM1 results, respectively.

## Acknowledgments

We thank J. E. Meyerson for graphical assistance; O. Peters for discussions; and several colleagues, two reviewers, and the editor for comments on the manuscript. Computations were carried out at the National Center for Atmospheric Research with high-performance computing support from Yellowstone(ark:/85065/d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by NSF. This work was supported in part by Department of Energy Grant DESC0006739; National Science Foundation Grants AGS-1102838 and AGS-1540518; National Oceanic and Atmospheric Administration Grant NA14OAR4310274; and Office of Naval Research Grant N00014-12-1-0744 (to S.N.S.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: neelin{at}atmos.ucla.edu. ↵

^{2}Present address: Centre for Atmospheric Sciences, Indian Institute of Technology Delhi, New Delhi 110016, India.↵

^{3}Present address: Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY 14853-1901.

Author contributions: J.D.N. designed research; J.D.N., S.S., S.N.S., and D.N.B. performed research; S.S. carried out climate model analysis; S.N.S. collaborated on theory; D.N.B. ran the Community Earth System Model simulations; J.D.N., S.S., and D.N.B. analyzed data; and J.D.N. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1615333114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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