Corresponding author: K. E. Kunkel, National Climatic Data Center, 151 Patton Avenue, Asheville, NC 28801, USA. (email@example.com)
 Probable maximum precipitation (PMP) is the greatest accumulation of precipitation for a given duration meteorologically possible for an area. Climate change effects on PMP are analyzed, in particular, maximization of moisture and persistent upward motion, using both climate model simulations and conceptual models of relevant meteorological systems. Climate model simulations indicate a substantial future increase in mean and maximum water vapor concentrations. For the RCP8.5 scenario, the changes in maximum values for the continental United States are approximately 20%–30% by 2071–2100. The magnitudes of the maximum water vapor changes follow temperature changes with an approximate Clausius-Clapeyron relationship. Model-simulated changes in maximum vertical and horizontal winds are too small to offset water vapor changes. Thus, our conclusion is that the most scientifically sound projection is that PMP values will increase in the future due to higher levels of atmospheric moisture content and consequent higher levels of moisture transport into storms.
 Climate change can be described in terms of the temporal evolution of the full probability density function (pdf) of variables that characterize the state of the atmosphere and the climate system. An important set of these variables have been designated as essential climate variables [GCOS, 2009]. Changes in the tails of the pdfs of some of these variables receive particular attention for climate change impacts and risk assessment.
 Increases in heavy precipitation events have been documented in many regions of the globe [IPCC, 2012] with substantial variations in the spatial distribution of statistically significant trends [Bonin et al., 2011; Kunkel et al., 2013]. Similarly, most areas of the U.S. are projected to see increases through the 21st century [IPCC, 2012], including areas that did not have statistically significant trends in the 20th century. Given these observed and projected changes, precipitation-sensitive information and applications would benefit from incorporation of best estimates of future changes, based on observed trends, model projections, or a combination of these.
 One informational product used for planning, probable maximum precipitation (PMP), is defined as the greatest accumulation of precipitation for a given duration meteorologically possible for a design watershed or a given storm area at a particular location at a particular time of year [WMO, 2009]. A better term for this concept might be potential maximum precipitation (PMP) to avoid any confusion that such an amount is probable, but instead, is potentially possible. A principal application for PMP values is the design of infrastructure for water retention (dams) or routing, where failure would be catastrophic.
 PMP values translate into a return period very much longer than the longest return periods traditionally used in applied climatology products, such as the 100 year return period amount (the amount that in a stationary climate has a 1% chance of occurring any given year or on average once in every 100 years). For example, the 24 h, 100 year return period amount for Urbana, IL, is about 175 mm, but the 24 h PMP amount ranges from about 225 mm for a 51,800 km2 area to over 900 mm for a 26 km2 area [Schreiner and Riedel, 1978].
 The long lifetimes of dams and similar structures ensure that they will experience the impacts of future climate change. We contend that in any future assessment of dam safety risk or other infrastructure where failure can lead to catastrophic consequences, ignoring climate change–induced new probabilities of extreme events, is likely to lead to a false sense of security. In this paper, we discuss the factors influencing PMP estimates for a range of time and space scales and whether any statements can be made about future changes in these factors.
2 Estimation of PMP
 PMP values are, in principle, most dependent upon atmospheric moisture, transport of moisture into storms, persistent upward motion, and strong winds where orographic uplift is important [WMO, 2009; Trenberth et al., 2003]. The general approach, using data and physical judgment, is to estimate the precipitation that would occur if all the relevant factors in a particular place and situation achieved their optimum values simultaneously and remained in place for the specified duration over the basin area. We review the factors below.
a.Convergence and vertical motion
 In past analyses, estimation of the maximum value of horizontal low-level wind convergence and upward motion was not considered to have a robust theoretical basis. Instead, the lengthy and numerous data records of precipitation have been used to identify historical extreme observed storm precipitation (Pstorm). Pstorm values serve as indirect measures of maximum low-level moisture convergence and persistent upward motion.
Schreiner and Riedel  developed the most recent estimates of PMP values for much of the U.S. east of the Rocky Mountains. They based their analysis on a set of 55 extreme storms occurring at scattered locations during the period 1878–1972 whose precipitation totals constitute the Pstorm set. These storms are assumed to approximate a maximum for precipitation and vertical motion lasting for a given duration. This set of the most extreme precipitation events was gleaned from the pooled data of the entire observing network representing hundreds of thousands of station-years of data. As such, the implied return period of events of this magnitude for any individual location is much longer than the 95 years over which these events occurred. All of the storms used in this eastern U.S. analysis were warm season events. The western U.S. is different, as most of the most extreme events occur during fall or winter [e.g., Corrigan et al., 1999].
b.Atmospheric water vapor
 A second component entering into the empirical estimation of PMP is the maximum atmospheric total column water vapor (precipitable water, PW) that is possible for a given location and season. The maximum possible value, PWmax, is estimated as the observed maximum historical precipitable water [Schreiner and Riedel, 1978].
 A U.S. climatology of extreme PW values from 50 years of radiosonde observations (www.crh.noaa.gov/unr/?n=pw), representing a first-order approximation and a distribution of 50-year recurrence interval value, indicates maximum PW values of nearly 75 mm are found in summer for stations near the Gulf Coast. Since the radiosonde network is relatively sparse in space (400 km mean spacing) and time (12 h interval between observations) and the period of record is short in duration (only 50 years), these extreme PW values probably underestimate PWmax, even in a stationary climate. Although transient values within the cores of storms may be higher than the atmospheric conditions sampled by the radiosonde network [Holloway and Neelin, 2010], such transient values are most likely not representative of the inflow regions of storms.
 The observed record of precipitation, extending back to the late 19th century, is considerably longer than the record of radiosonde observations. Thus, the PW estimate for a climatology of maximum observed “storm events,” PWstorm, has previously been based upon the climatology of maximum surface dewpoint temperatures persisting for a minimum duration of 12 h. Since the atmosphere during torrential rains of several hours duration (such as in tropical cyclones) typically approaches a pseudoadiabatic temperature state, this temperature-humidity profile has been assumed as a limiting extreme for PWmax [Schreiner and Riedel, 1978]. We believe this is an appropriate equilibrium assumption for linking PWmax to PMP. A criticism of this assumption was presented by Chen and Bradley , whose analysis of extreme events in the central U.S. indicated a 7% overestimate of PWmax using the above criteria. Their conclusion could be a result of surface humidity, upper-level dryness, or storm dynamics peculiar to that geographical region or undersampling by the short-term data record. But any current overestimates would apply equally to future changes and our interest in this study is in changes relative to current values.
 Given the durations (6–72 h) for which PMP estimates are made, sustained high moisture flow into storms is necessary. One meteorological type is “open” precipitating systems fed by persistent large-scale winds and oceanic moisture sources far upwind. For example, on the west coast of the U.S., atmospheric rivers of deep tropical moisture from the Pacific Ocean [Dettinger, 2011] can extend inland and create intense precipitation over the upwind slopes of the mountain ranges. For the eastern half of the U.S., the major source of moisture is the Atlantic Ocean/Caribbean Sea/Gulf of Mexico.
 Some extreme rains over days are associated with “closed” precipitating systems which depend on more localized sources of water. The best example is a stationary tropical cyclone over a coastline next to a warm ocean; such events have produced historic PMP events, reflecting a strong positive feedback system involving surface wind, evaporative moisture supply, and precipitation and condensational heating to sustain the energy of the winds [Price, 1981]. However, these isolated storms also experience negative feedbacks limiting the supply of moisture and the lifetime of the system, including subsidence of dry air around the storm periphery, and wind-induced evaporative cooling of the ocean beneath, reducing the oceanic source moisture [Schade, 2000].
 For shorter time scales and smaller basins, some mesoscale convective systems may be relevant to PMP. These systems are normally propagating and are partially “open” as they traverse humid air masses. Their lifetime is often limited by the negative feedback of subsidence of cooled, dry air to the surface. Such systems may be sustained in a region of weak (or negative) upper-level inertial stability, which encourages the divergent branch of the convective circulation [Coniglio et al., 2010].
c.Physical synthesis: Linking PMP and atmospheric water vapor
 Traditionally, the calculation of PMP assumes a statistical equilibrium relationship between PMP and PWmax linking the “storm event” data (i.e., Pstorm) with estimates of extreme values:
indicating that PMPest increases proportionately to PWmax as inferred from dewpoint temperature, for a given storm climatology. Ncycles is the number of water replacement cycles for the column during the duration of precipitation, and the underlying assumption is that Ncycles is the same for the PMP calculation as for the storm data.Expression ((1)) can alternatively be expressed as the ratio of time scales
where (τrepl) is the replacement time scale for the water in the column and (τdur) is duration time over which the total precipitation is accumulated. Since air rises over the depth of the rain system H in time (τrepl), its average vertical velocity is W = H/(τrepl). Hence, we can further rewrite any of the above as
where Prate = P/(τdur) is the average precipitation rate (intensity) over the duration (τdur). Thus, these relationships reflect the underlying assumption in PMP estimation that the average vertical motion for the equilibrium assumption is assumed to be the same for the PMP case as for the storm data. Thus, W, as calculated here, is an “efficiency” parameter representing the estimate of maximum persistent upward vertical motion (Wmax) consistent with the column water budget in an extreme precipitation event. Over topography with slope S, one expects W to be proportional to the product of upstream wind and S.
 For the aforementioned example of the point 24 h PMP at Urbana, IL, the maximum PW at this site is roughly 64 mm. This leads to a value of Ncycles of about 15, an approximate replacement time scale of slightly less than 2 h, and an average vertical velocity of about 1.5 m s−1. The values of PMP for basins decrease with increasing basin size because Pstorm values decrease as the size of the area over which precipitation is averaged increases. For the location of Urbana, the 24 h PMP value for an area of 51,800 km2 (the largest area estimated by Schreiner and Riedel  is about 230 mm, about ¼ the point value, and W is similarly reduced to about 0.4 m s−1.
3 Possible Effects of Climate Change on Extreme Precipitation
 The radiative energy imbalance caused by increases in greenhouse gas concentrations is highly likely to continue the increases in ocean heat storage and a rise in sea surface temperatures (SSTs) that have already been observed [Trenberth et al., 2007]. The warming ocean will in turn lead to a rise in evaporation and atmospheric water vapor content, following the Clausius-Clapeyron relationship for saturation water vapor pressure. A probable consequence is the intensification of the hydrologic cycle and PMP over land and ocean. The effect of this intensification on changes in PWmax values over land was investigated by analyzing future (2041–2070 and 2071–2100) and control (1971–2000) simulations from the Coupled-Model Intercomparison Project phase 5 (CMIP5) archive. Seven GCM simulations were examined (listed in supplementary online material). The model data were first regridded to a common grid of 2° latitude by 2.5° longitude, comparable to the largest basin sizes for PMP applications. For each grid point, the maximum value over the entire 30 year period of the 12 h persisting PW (PWmax) was identified. Finally, a multimodel mean map was produced. The analysis was performed for two representative concentration pathways (RCP), the RCP4.5 and the RCP8.5.
 Figure 1 shows the global pattern of maximum PW (top) and its projected percentage changes for 100 years in the future (middle). The analysis reveals projected increases across all grid cells, indicating general global moistening of the atmosphere. The overall global patterns of contemporary PWmax (top) and the absolute magnitudes of the future differences (supplementary online material) are very similar: moisture increases are a maximum in regions where they are currently large. These changes in PW content represent changes in the pattern of latent energy content and are focused in the tropical belt of latitudes, particularly the oceanic ITCZs, western Pacific warm pool, and adjacent Asian monsoon regions.
 The patterns of fractional percentage changes (middle) are quite different from that of absolute changes, indicating somewhat larger changes toward the poles. Over large parts of the Northern Hemisphere, the percentage increases are in the range of 20%–30% by 2071–2100. At high latitudes and over some land areas, particularly Eurasia, the increases are more than 30% by the end of the 21st century. For North America and surrounding ocean areas, there are increases of 20%–30% by 2071–2100 with the greatest increases over the western U.S. (where the actual PWmax values remain relatively low).
 The results for 2041–2070 and for the RCP4.5 simulations (supplementary online material) indicate increases for 2041–2070 of roughly half of the 2071–2100 results and for RCP4.5 about half of the results of the RCP8.5 simulations, in approximate correspondence to the difference in greenhouse radiative forcing. The fractional changes in mean water vapor concentrations (not shown) are larger, but only by a small amount, than the changes in the maximum values shown in Figure 1. The maximum values of PW typically occur in July or August in most of the contiguous U.S., except along the west coast, where a fall (either September or October) maximum is simulated (results not shown).
 The increases in PWmax are a robust result in the model simulations and have a strong theoretical basis, the Clausius-Clapeyron equation, linking the increases to increasing temperature. The PWmax increases are large and, if incorporated into PMP estimates, would have major implications for design of dams and other long-lived and critical runoff control structures. An important question then is whether any other meteorological factors may change in ways that offset, or add to, the expected changes in PMP attributed to an increase in PW. The key issue is whether the vertical motion “efficiency” variable W changes in the future. From equation ((3)), logarithmic differentiation equates the difference in fractional changes of PW and PMP to that of W. Over resolved sloping topography, the fractional change of W would be proportional to that of the upslope wind component.
 Although previous assessments of PMP assumed that there is no theoretical basis for determining a maximum value of vertical motion, there are conceptual simplifications for the space and time scales of PMP. Spatially, the scales of PWmax and PMP are rather large away from sharp topography. The relatively long durations of PMP applications (several hours to days) are also long compared to the time scale of transient convective elements. It follows that an idealized subsynoptic scale model of intense, persistent rain events can consist of a steady state, two-dimensional flow of saturated (moist adiabatic) atmosphere columns converging toward the precipitation zone.
 Following the discussion in section 2b, these examples illustrate the type of situations that may result in PMP events:
Radial inflow of high PW air into a slow-moving tropical cyclone and ascent in the inner wall rainband (e.g., the U.S. 24 h rainfall record of 1092 mm at Alvin, TX, during Hurricane Claudette in July 1979 [Hebert, 1980]; this value is close to the 24 h PMP value of approximately 1200 mm).
Flow of moist air toward an extratropical cyclone front that is stationary as a result of synoptic-scale flow (e.g. Illinois state record 24 h rainfall of 430 mm at Aurora on 18 July 1996) [Changnon and Kunkel, 1999].
Sustained low-level jet sustaining a mesoscale convective system (e.g., the Nashville, TN flood of 1–2 May 2010 with 48 h rainfall exceeding 400 mm) [Moore et al., 2012].
Upslope advection of moist air masses by synoptic-scale winds encountering mountainous topography (e.g., the 6–7 November 2006 event in Washington and Oregon, where 3 day rainfall exceeded 700 mm) [Neiman et al., 2008]. The persistence of topographically forced PMP events is then due to the synoptic-scale wind system.
 How will climate warming affect these types of meteorological phenomena? Knutson et al.  assessed the state of knowledge regarding future projections of tropical cyclones. They indicated rainfall rates were likely to increase due to the general increase in water vapor concentrations based on theoretical considerations and high-resolution climate models. They estimated increases of +20% near the tropical cyclone center by the late 21st century under the A1B emissions scenario. For stationary fronts, studies of extratropical cyclones in the CMIP5 models find mixed changes [Colle et al., 2013] in the eastern U.S. with roughly equal areas of increases and decreases; thus, there does not appear to be a compelling reason to expect large changes in the maximum vertical motion produced by extratropical cyclones. Regarding atmospheric rivers, Dettinger  finds that extreme atmospheric river episodes in the western U.S. actually increase in a multimodel ensemble from CMIP3.
 To further explore these characteristics in the CMIP5 simulations, fractional changes (%) in three relevant modeled variables were analyzed: the 30 year maximum values of (a) 6 h upward motion (ωmin where ω = dP/dt and P = pressure), (b) 6 h horizontal wind speed, and (c) daily precipitation. The results for ωmin for RCP8.5 for 2071–2100 are shown in Figure 1 (bottom). Note that these values are relevant to the largest scales for which PMP estimates are provided, as the model resolution is not sufficient to resolve small-scale upward motion and intense precipitation. The differences between 2071–2100 and 1971–2000 over the contiguous U.S. are mostly positive, and both the positive and negative magnitudes are mostly less than 10%, considerably smaller than the water vapor increases. Thus, the model simulations do not show changes in maximum upward motion that could negate the increases in water vapor. Globally, the largest changes in upward motion are increases of greater than 20% at tropical latitudes while the largest areas of decreases of more than −10% are mostly in subtropical latitudes. The changes at mid and high latitudes are mixed in sign and mostly less than 10% in magnitude.
 Topographically forced vertical motion will be an important, perhaps even dominant, factor in extreme precipitation storms in certain areas such as the West Coast and along the Appalachian Mountains. This uplift will be directly related to the horizontal wind speed integrated from the upwind land/ocean surface to the crest. The CMIP5 models results (supplementary online material) do show areas of decreases in maximum horizontal wind speed over the western U.S. where topographic uplift is important. However, the magnitudes of the decreases are less than 6% almost everywhere and again much smaller than the water vapor increases.
 Model simulations are known to produce more intense precipitation under anthropogenic forcing [e.g., Trenberth, 2011]. Here we examine the most extreme precipitation values. Changes in the 30 year period maximum daily precipitation (Figure 2, top) are consistent with the above results. Increases are generally in the range of 10%–30% over the CONUS and mostly above 20% or similar to the changes in water vapor concentration. The highest daily precipitation accumulation during a 30 year period is extreme, but far less extreme than PMP values. Nevertheless, these results suggest water vapor changes are the dominant control on the magnitude of extreme precipitation, at least at the scale of the resolution of these model values, which is similar to the largest basin scale for which PMP values have been estimated (i.e., 51,800 km2). Globally, large increases in maximum daily precipitation are simulated nearly everywhere. The few areas of spatially coherent decreases (Caribbean, eastern south Pacific, and south Atlantic) are mostly in subtropical areas where there are both decreases in maximum upward motion and smaller (than surrounding areas) increases in maximum precipitable water.
 The changes in the 30 year maximum PW as a function of changes in the temperature of the climatologically warmest month at 850 hPa generally follow the Clausius-Clapeyron relationship (Figure 2, bottom). The individual grid point values at low and midlatitudes cluster around a slope of 6.3% K−1 line, which is the approximate value of the derivative of saturation vapor pressure with respect to temperature at 288 K. The changes at high latitudes are generally somewhat greater than the nominal 6.3% K−1 value. This result also suggests a strong tie to temperature change and the overall robustness of the model PW projections. Note that the simulated temperature changes are generally smaller in the Southern Hemisphere than in the Northern Hemisphere, reflecting the moderating effects of the larger ocean area.
 Climate model simulations indicate a substantial increase in water vapor concentrations during the 21st century will occur. Since the imbalance in the radiative energy budget arising from an increase in greenhouse gases will almost surely be manifested in an increase in ocean heat content, there is high confidence in this model outcome. This increase in ocean heat content in turn will lead to an increase in atmospheric water vapor concentrations. The model simulations indicate that the changes in maximum water vapor concentrations, which are a principal input to PMP estimation techniques, will change by an amount comparable to mean water vapor changes, and ultimately to an accelerated water cycle with heavier extreme rains. The magnitude of the maximum water vapor changes follows approximately a quasi-exponential Clausius-Clapeyron relationship with temperature.
 Conceptual considerations suggest there are no compelling arguments for either increases or decreases of comparable magnitude in other factors used as inputs to PMP, specifically maximum vertical motion and horizontal wind speed. Indeed, model-simulated changes in the maximum values of these variables are too small to offset the water vapor changes. Model simulated-increases in extreme precipitation confirm the dominant role of water vapor in controlling such extremes. We conclude that the most scientifically sound projection is that PMP values will increase in the future and raise the risk of damaging floods. These conclusions apply not only to the U.S. but also globally to almost all other areas.
 This work was partially supported by NOAA through the Cooperative Institute for Climate and Satellites–North Carolina under Cooperative Agreement NA09NES4400006 and by the NOAA Climate Program Office, Climate Observations and Monitoring Program. We acknowledge the World Climate Research Programme's Working Group on Coupled Modeling, which is responsible for CMIP, and we thank the climate modeling groups (listed in Table 1 of the supplementary material) for producing and making available their model output. For CMIP the U.S. Department of Energy's Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. We thank Mike Coniglio for advice on mesoscale convective systems.