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  • View in gallery

    Simulation domain and the four analysis regions: upper Mississippi River basin (UMS), southeastern United States (SE), southwestern United States (SW), and Pacific Northwest (PNW).

  • View in gallery

    Normalized frequency of precipitation as a function of daily intensity for 1981–88 in observations and in the RegCM2 and HIRHAM NCEP-driven simulations for each of the analysis regions in Fig. 1: (left) cold season and (right) warm season. Arrows mark the 95th percentile of observed precipitation accumulated from low to high intensity. Straight lines are fits to a log-linear function for each source’s precipitation (identified in the key).

  • View in gallery

    Normalized precipitation as a function of daily intensity in the three RegCM2 simulations for each of the analysis regions in Fig. 1: (left) cold season and (right) warm season. Results are shown for simulations using boundary conditions from the NCEP–NCAR reanalysis (NCEP), HadCM2 contemporary climate (CTRL), and HadCM2 scenario climate (SCEN).

  • View in gallery

    Same as in Fig. 3, but for the HIRHAM simulations.

  • View in gallery

    Change in the percentage contribution of each precipitation intensity category to total precipitation for each of the analysis regions and seasons.

  • View in gallery

    Normalized gamma distributions for precipitation in two different climates, denoted C1 and C2. The point xc is the crossing intensity separating increasing from decreasing contributions to the normalized precipitation distribution.

  • View in gallery

    The percentile separating increases from decreases in precipitation intensity as a function of the gamma distribution shape parameter α for the normalized intensity distribution (solid) and the nonnormalized intensity distribution (dashed).

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A Possible Constraint on Regional Precipitation Intensity Changes under Global Warming

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  • 1 Department of Geological and Atmospheric Sciences, Iowa State University, Ames, Iowa
  • | 2 Department of Agronomy, Iowa State University, Ames, Iowa
  • | 3 Department of Geological and Atmospheric Sciences, Iowa State University, Ames, Iowa
  • | 4 Department of Agronomy, Iowa State University, Ames, Iowa
  • | 5 Danish Meteorological Institute, Copenhagen, Denmark
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Abstract

Changes in daily precipitation versus intensity under a global warming scenario in two regional climate simulations of the United States show a well-recognized feature of more intense precipitation. More important, by resolving the precipitation intensity spectrum, the changes show a relatively simple pattern for nearly all regions and seasons examined whereby nearly all high-intensity daily precipitation contributes a larger fraction of the total precipitation, and nearly all low-intensity precipitation contributes a reduced fraction. The percentile separating relative decrease from relative increase occurs around the 70th percentile of cumulative precipitation, irrespective of the governing precipitation processes or which model produced the simulation. Changes in normalized distributions display these features much more consistently than distribution changes without normalization.

Further analysis suggests that this consistent response in precipitation intensity may be a consequence of the intensity spectrum’s adherence to a gamma distribution. Under the gamma distribution, when the total precipitation or number of precipitation days changes, there is a single transition between precipitation rates that contribute relatively more to the total and rates that contribute relatively less. The behavior is roughly the same as the results of the numerical models and is insensitive to characteristics of the baseline climate, such as average precipitation, frequency of rain days, and the shape parameter of the precipitation’s gamma distribution. Changes in the normalized precipitation distribution give a more consistent constraint on how precipitation intensity may change when climate changes than do changes in the nonnormalized distribution. The analysis does not apply to extreme precipitation for which the theory of statistical extremes more likely provides the appropriate description.

@ Current affiliation: School of Information and Library Science, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina

Corresponding author address: William J. Gutowski Jr., 3010 Agronomy Building, Iowa State University, Ames, IA 50011. Email: gutowski@iastate.edu

Abstract

Changes in daily precipitation versus intensity under a global warming scenario in two regional climate simulations of the United States show a well-recognized feature of more intense precipitation. More important, by resolving the precipitation intensity spectrum, the changes show a relatively simple pattern for nearly all regions and seasons examined whereby nearly all high-intensity daily precipitation contributes a larger fraction of the total precipitation, and nearly all low-intensity precipitation contributes a reduced fraction. The percentile separating relative decrease from relative increase occurs around the 70th percentile of cumulative precipitation, irrespective of the governing precipitation processes or which model produced the simulation. Changes in normalized distributions display these features much more consistently than distribution changes without normalization.

Further analysis suggests that this consistent response in precipitation intensity may be a consequence of the intensity spectrum’s adherence to a gamma distribution. Under the gamma distribution, when the total precipitation or number of precipitation days changes, there is a single transition between precipitation rates that contribute relatively more to the total and rates that contribute relatively less. The behavior is roughly the same as the results of the numerical models and is insensitive to characteristics of the baseline climate, such as average precipitation, frequency of rain days, and the shape parameter of the precipitation’s gamma distribution. Changes in the normalized precipitation distribution give a more consistent constraint on how precipitation intensity may change when climate changes than do changes in the nonnormalized distribution. The analysis does not apply to extreme precipitation for which the theory of statistical extremes more likely provides the appropriate description.

@ Current affiliation: School of Information and Library Science, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina

Corresponding author address: William J. Gutowski Jr., 3010 Agronomy Building, Iowa State University, Ames, IA 50011. Email: gutowski@iastate.edu

1. Introduction

An important characteristic of daily precipitation climatology is its frequency versus intensity. Changes in the intensity distribution of daily precipitation can affect flood frequency, crop development, water resources, and other water-sensitive human and natural systems. Projected anthropogenic warming of the earth’s climate may alter the intensity of precipitation (e.g., Cubasch et al. 2001; Trenberth et al. 2003), so evaluation of how the intensity distribution might change in the future has relevance to climate change impacts. In this paper, we diagnose the intensity distribution of precipitation produced by two regional climate models (RCMs) simulating present and future scenario climates. The analysis reveals a constraint on their simulated changes in intensity distribution that appears to operate independently of season, location, or other factors controlling precipitation processes. We further analyze the simulation features underlying this constraint and evaluate the conditions under which it would apply more generally.

The analysis uses daily precipitation from two RCMs that simulated the contiguous United States for three 10-yr sets of boundary conditions: 1979–88 reanalysis, general circulation model (GCM) contemporary climate, and GCM future scenario climate with enhanced greenhouse gas concentration. We have studied a variety of hydroclimate issues using this suite of simulations, such as uncertainties in projecting climate change (Pan et al. 2001b), downscaling for simulating surface hydrology (Wilby et al. 2000; Hay et al. 2002; Jha et al. 2004), soil moisture changes (Pan et al. 2001a), extreme precipitation events (Kunkel et al. 2002), a seasonal precipitation deficit in GCM and RCM simulations (Gutowski et al. 2004), and a central U.S. “warming hole” (Pan et al. 2004). Although updated versions exist for the RCMs and GCM used here, we have substantial understanding of the behavior of this well-diagnosed suite, which aids our diagnosis.

Our analysis focuses on daily precipitation. Although shorter time scales also are important for climate change impacts, previous work (Gutowski et al. 2003; Anderson et al. 2007) indicates that the 50-km grid spacing of our RCMs is too coarse to replicate the observed intensity distribution of subdaily precipitation in the middle of the United States. We thus restrict ourselves to the shortest time interval for which previous work indicates confidence in the credibility of our simulations versus observations.

In section 2, we describe the observational data used to evaluate the model output, the models and characteristics of their simulation suite, and our diagnostic methods. Section 3 presents the models’ precipitation intensity distributions versus observations for several U.S. regions, using output from the reanalysis-driven simulations. Section 3 also gives an evaluation of the changes in intensity distribution under the prescribed climate change. In section 4, we present a simple theoretical basis for understanding the simulated changes with analysis of conditions under which the results may apply more broadly. Section 5 gives a discussion of the results.

2. Observations, models, and methods

a. Observations

Observations collected by the U.S. National Climatic Data Center (NCDC) from a cooperative climate observing network provide the precipitation data used to evaluate model performance. We use data extracted from the National Climatic Data Center archives by Eischeid et al. (2000) and further evaluated by Clark and Hay (2004), who summarize NCDC quality control procedures. We used observations as given and made no adjustment for gauge undercatch, which can produce a negative bias in observations of 3%–10% (e.g., Groisman and Legates 1994).

The observational analyses in this paper use data for the 1980s. However, to mesh with other analyses we are performing, we required all stations used here to report for the complete period 1950–99, with no more than 7.5% missing or questionable data (i.e., fewer than three unacceptable observations per month, on average). We assume that continuity of record over a 50-yr period implies reliability and thus an acceptable quality level in the data. Using this criterion, we selected stations for four analysis regions in the United States (Fig. 1): upper Mississippi River basin (476 stations), southeastern United States (168 stations), southwestern United States (43 stations), and Pacific Northwest (104 stations). The small number of stations in the Southwest suggests that sampling of observed precipitation there may be deficient, especially since it is a region of relatively infrequent and scattered rainfall. We assume that analyzing a multiyear record sufficiently minimizes potential sampling limitations.

b. Models and simulations

Model output used here comes from contemporary and future scenario periods simulated by the second-generation regional climate model (RegCM2; Giorgi et al. 1993a, b) and the High Resolution Limited Area Model with Hamburg Physics (HIRHAM; Christensen et al. 1996; Christensen et al. 1998). RegCM2 and HIRHAM simulations used the same continental U.S. domain (Fig. 1) as experiment 1 of the Project to Intercompare Regional Climate Simulations (Takle et al. 1999).

HIRHAM computed precipitation using the Tiedtke (1989) mass-flux convection parameterization and the Sundqvist (1978) explicit moisture scheme. The Tiedtke (1989) scheme has three types of convection, shallow, midlevel, and penetrative, each using a different closure assumption. The Sundqvist (1978) scheme predicts cloud liquid water and diagnoses ice phase water. HIRHAM’s land surface used five prognostic layers for temperature, with one layer each for soil moisture and (when present) snow. Planetary boundary layer computations used a local K-type scheme. Radiative transfer computations used the scheme from the European Centre for Medium-Range Weather Forecasts model, cycle 36, with additional ozone, chlorofluorocarbon, and aerosol effects included for climate simulation. Model grid spacing was 0.5° using a rotated Mercator projection with the equator passing through the middle of the simulation domain. The model had 19 layers in the vertical.

RegCM2 computed precipitation using the Grell (1993) convection parameterization and a simplified version (Giorgi and Shields 1999) of the Hsie et al. (1984) explicit moisture scheme. The Grell (1993) scheme is a version of Arakawa and Schubert (1974) convection that uses a single updraft and downdraft to represent cumulus cloud processes. The rate of large-scale convective destabilization determines the cloud’s mass flux and, hence, convective precipitation rate. The simplified version of Hsie et al. (1984) cloud microphysics computes stable precipitation using a prognostic cloud water equation with no explicit ice processes. Cloud water converts to rainwater by an autoconversion process and precipitates immediately. The model also used the Biosphere–Atmosphere Transfer Scheme (BATS) version 1e (Dickinson et al. 1993) land surface model and the Holtslag et al. (1990) nonlocal boundary layer turbulence parameterization. For radiative transfer the Community Climate Model Version 2 (CCM2) radiation package (Briegleb 1992) was used. Model grid spacing was 52 km on a Lambert conformal projection centered at (37.5°N, 100°W) with 14 layers in the vertical.

For reanalysis-driven simulations, the models used initial and lateral boundary conditions from the reanalysis (Kalnay et al. 1996) produced by the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (NCAR), supplemented by observations of surface temperatures in the Gulf of California and the North American Great Lakes. The simulations ran from October 1978 to December 1988 with the first three months considered a spinup period, which we ignore. GCM-driven simulations used output from the Second Hadley Centre Coupled Ocean–Atmosphere GCM (HadCM2; Johns et al. 1997). HadCM2 was one of two models used for the U.S. National Assessment of Climate Change (U.S. Global Climate Change Research Program 2004). The HadCM2 contemporary climate simulation had effective greenhouse gases corresponding roughly to the 1990s. The HadCM2 scenario climate simulation assumed a 1% yr−1 increase of effective greenhouse gas concentrations after 1990. The 10-yr window used from the scenario climate was the decade 2040–49 (Pan et al. 2001b). In this work, ‘‘climate change’’ is the scenario minus contemporary difference between RCM simulations driven for these two HadCM2 periods. The climate change includes the effects of greenhouse warming and possible regional, decadal-scale variability.

RCM simulations were continuous for each of the 10-yr driving periods. However, due to storage problems that produced gaps in output archives for the suite of simulations, we have restricted analyses to periods for which output from both models is available: 1981–88 for reanalysis-driven runs and final 9 yr for each GCM-driven run. Pan et al. (2001b) give further details of the models and simulations and discuss general features of the precipitation output and its change under enhanced greenhouse warming.

c. Diagnostic methods

We assigned simulated precipitation rates to the grid box center in our diagnoses. We treated daily precipitation at all model grid points and observation sites as individual samples. There are, however, divergent views on this approach. Like us, Skelly and Henderson-Sellers (1996) use simulated precipitation to represent a point sample. On the other hand, Osborn and Hulme (1998) argue that simulated daily precipitation events should be viewed as area averages for each grid box, since parameterizations of surface fluxes and precipitation processes typically are developed to represent statistics of an area rather than a point. One might argue that parameterizations often assume homogeneous, isotropic behavior on an infinite plane so that one point is statistically the same as any other, and thus, samples represent points. However, for the resolution of our RCMs, the issue may be secondary, for Mearns et al. (1995) note that a 60-km RCM grid box is sufficiently homogeneous for one observation in a grid box to be a representative sample of daily precipitation. In other words, the grid box daily precipitation may be viewed as comparable to a single-point observation. Note that this behavior simply means that we assume sufficient sampling of daily precipitation in both the observations and the simulations; it does not guarantee that the samples have the same statistical properties.

We defined a precipitation event as a nonzero precipitation record for one day at one location that was not classified as missing or questionable (cf. Clark and Hay 2004). For each of our analysis regions (Fig. 1), we constructed histograms of precipitation intensity, combining precipitation events from all grid points or all observation sites. All histogram bin widths easily satisfied minimum width criteria suggested by Wilks (1995) for avoiding excessively fine and potentially noisy gradations in precipitation intensity. We used two types of histograms: precipitation frequency versus intensity and precipitation amount versus intensity. The latter is equivalent to multiplying the frequency of events in each intensity bin by the bin’s average intensity. Both perspectives serve our further purpose of arriving at a theoretical description that helps us understand the changes in precipitation intensity distribution under climate change.

The time series analyzed may have different numbers of samples due to missing data or different numbers of sampling points, so to aid comparisons between observations and each of the models we normalized both histogram types. The frequency versus intensity histograms were normalized by dividing the event count for each bin by the total number of precipitation events in all samples contributing to the histogram. We normalized precipitation amount versus intensity by dividing the amount in each bin by the total precipitation accumulated at all sampling sites in an analysis region.

We stratified our records into warm season (April–September) and cold season (October–March) to distinguish influences of seasonal precipitation mechanisms on the models’ performance versus observations and on their climate change behavior. For results presented here, the number of precipitation events for different seasons and analysis regions ranged from 9809 in the Southwest cold season observations to over 700 000 in upper Mississippi River basin warm season observations.

3. Simulated precipitation intensity distributions

a. Comparison with observations

Table 1 shows the average precipitation rate and frequency of precipitation events for each season and analysis region. Biases in precipitation rates (not shown) are in the range (−37%, +43%). These are not small, but they are within the ±50% precipitation bias reported by Giorgi et al. (2001) for regional climate models in general. The models generally produce too many precipitation days, primarily because the models produce too many days with light precipitation, as indicated in Table 1 by the columns comparing the percentage of days with precipitation exceeding 2.5 mm and the percentage of days with any amount of precipitation. For precipitation rates greater than 2.5 mm day−1 (the lowest bin), the models typically produce precipitation frequencies in the range 10%–25% among the different analysis regions and seasons, which is roughly in agreement with the observations (Table 1). The excessive light precipitation is consistent with our earlier analysis focusing on RegCM2’s performance in the northern plains (Gutowski et al. 2003) and with other climate simulations (cf. Mearns et al. 1995; Chen et al. 1996; Giorgi and Marinucci 1996). The models are also consistent with these earlier studies in producing too few high-intensity precipitation days versus observations (Fig. 2). However, they agree with observed frequency versus intensity for moderate amounts (1–5 cm day−1).

Figure 2 also shows the 95th percentile for observed precipitation. For most seasons and regions the models agree fairly well with observations out to about the 95th percentile, though the models’ frequencies at this intensity tend to be less. The shortfall appears to be due in part to resolution, which limits the strength of upward motions that produce precipitation, either by direct uplift or by helping to trigger convection (e.g., Jones et al. 1995; Gutowski et al. 2003). The resolutions used also limit the models’ ability to depict small-scale convective events (e.g., Gutowski et al. 2003; Biasutti et al. 2006).

For later reference, Fig. 2 also shows results of fitting precipitation frequency versus intensity to a log-linear distribution of the form
i1525-7541-8-6-1382-e1
where F is frequency, x is intensity, and β and C are constants. Table 1 shows β and the explained variance for each fit. Although slope factors β tend to be lower for simulations than corresponding observations, all fits explain at least 90% of the variance for each case except for the simulations in the Southwest warm season.

b. Climate change

Figures 3 and 4 show normalized precipitation as a function of intensity for each of the analysis regions and seasons. Distributions versus intensity are similar to frequency versus intensity (Fig. 2) except that the maximum often occurs for more intense precipitation than one of the lowest categories, as one might expect for a distribution function that is the product of precipitation frequency times intensity. Symbols at the smallest amounts in the figures are single-occurrence events, which is why the lower ends of many of the distributions slope upward with increasing intensity.

In Figs. 3 and 4, precipitation intensities from NCEP-driven and GCM-driven simulations for contemporary climate tend to lie fairly close to each other compared to corresponding distributions for the scenario climate. Regions where this relationship is less robust are regions where the intensity distribution changes little with climate change (e.g., the southwestern United States). These regions also tend to have less total rainfall than the other regions. The overall implication of the results is that the GCM driving for contemporary climate does not introduce substantial distortion of the precipitation intensity distribution compared to the observation-based (reanalysis) driving.

For most regions and seasons, the scenario simulation has more precipitation in most intensity categories. Increases are especially noticeable at higher intensities. In addition, the scenario simulations sometimes produce daily precipitation in high-intensity categories that have no occurrences in the simulations with reanalysis or GCM contemporary boundary conditions. For many of the regions and seasons in Figs. 3 and 4, the more intense precipitation has relatively larger increases, so that an increase in precipitation intensity accompanies an increase in total precipitation. Consequently, Table 2 shows that for most of the regions and seasons simulated here, the relative increase in precipitation is larger than the relative increase in the number of rain days, yielding an increase in the average intensity of daily precipitation.

A related issue is how the increase in intensity manifests itself in the distribution of precipitation change versus intensity. We consider changes in the precipitation intensity distribution by looking at relative changes, that is, changes in the normalized distribution. Note that total precipitation in a bin may increase under the warming scenario, but its relative contribution to total precipitation may decrease. Bins with positive change in the normalized distribution thus not only have greater precipitation in the scenario climate, but they contribute relatively larger amounts to the total.

Figure 5 shows the changes in normalized precipitation distribution for each region, season, and model. The general pattern of change is fairly simple and roughly the same for all regions except the Southwest: under the warming scenario, the portion of precipitation from intensities less than about 2 cm day−1 decreases while the portion from higher intensities increases. Except for the southwestern United States there is typically just one major transition separating decreasing and increasing intensity categories. The HIRHAM changes in the southwestern United States differ from all others in that total precipitation decreases. HIRHAM’s cold season distribution change in Fig. 5 is consistent with the theoretical model presented in the next section, but its warm season change is not, despite its similarity to the RegCM2 curve. We discuss this behavior in the next section.

In addition to the common pattern of intensity change in Fig. 5, Table 2 shows that the transition percentile between decreasing and increasing relative contribution to the total occurs typically in the range 68%–84%. This is noteworthy because there are substantial differences in the precipitation climatologies of these regions. The Pacific Northwest has a winter maximum governed by synoptic dynamics interacting with topography. The upper Mississippi River basin has a spring–summer maximum with substantial contribution by mesoscale convective systems. The southwestern United States receives much of its precipitation from monsoon processes that occur primarily in the warm season. The southeastern United States has a relatively small amplitude annual cycle with contributions by both synoptic and mesoscale dynamics. Despite all these differences, as well as differences in overall precipitation amounts, changes, and the model producing the change, for regions showing increased precipitation, the transition between decreasing and increasing relative contributions occurs in a fairly narrow range of percentiles.

4. A simple theoretical model

Results in section 3 suggest that a simple exponential decay approximates fairly well precipitation frequency as a function of intensity. Precipitation amount as a function of intensity then varies as
i1525-7541-8-6-1382-e2
where x is intensity of daily precipitation, and po and β are parameters describing details of the distribution. The relationship (2) is one form of a more general relationship often used to describe precipitation versus intensity (e.g., Wilks 1995):
i1525-7541-8-6-1382-e3
with the restriction α ≥ 1. The total precipitation during the period described by (3) is
i1525-7541-8-6-1382-e4
and the total number of rain days is
i1525-7541-8-6-1382-e5
Here, Γ(α) is the gamma function and the parameter α is a shape parameter.
Normalizing (3) by the total precipitation yields the gamma distribution
i1525-7541-8-6-1382-e6
Our fits to the model output in Fig. 2 suggest α = 2 [i.e., (2)] is an acceptable choice. Fits of observed precipitation to gamma distributions show α in the range 1–5 for most of the contiguous United States (Wilks and Eggleston 1992). Using (6), the precipitation percentile accumulating from zero is
i1525-7541-8-6-1382-e7
where for → ∞, → 100%.

We assume that the gamma distribution describes precipitation versus intensity for contemporary and future climates. We further assume that the shape parameter α is constant under the climate change studied here. However, most of the analysis presented here does not require α = 2. Finally, guided by Table 1, we assume initially that the number of rain days does not change, or N = constant.

Figure 6 shows schematically how precipitation versus intensity governed by the gamma function might appear for two different climates. The difference between the two curves in Fig. 6 has roughly the same variation with intensity as the differences in Fig. 5. The crossing point xc in Fig. 6 separates intensities with increased contribution to the normalized distribution from those with decreased contribution. Using (6), this point occurs for climate states C1 and C2 when
i1525-7541-8-6-1382-e8
so that
i1525-7541-8-6-1382-e9
Note that this has a single solution.
Suppose now that precipitation amount changes between the two climates, so that PC1 = P and PC2 = (1 + δP) P, where δP is the fractional change in total precipitation from a reference climate, here C1. Then, using (4) and (5) to substitute for βC1 and βC2 in (9) and using the assumption that N = constant and the identity Γ(α) = (α − 1) Γ(α − 1),
i1525-7541-8-6-1382-e10
Substituting (10) into (7), the percentile at the crossing point xc is
i1525-7541-8-6-1382-e11
that is, (xc) is proportional to the incomplete gamma function with y = x/β and upper limit xc/β. In (10) and (11), β has the value of the reference climate C1.
Exact, analytic evaluations of (11) do not appear to exist for arbitrary α, but they do exist for α = integer. Following Fig. 2 and using α = 2,
i1525-7541-8-6-1382-e12
One can proceed similarly for other positive-integer values of α, although the case α = 1 poses problems for computing the number of rain days using (5) and thus does not appear to be physically realizable in the present context.
For |δP| ≪ 1,
i1525-7541-8-6-1382-e13
and then
i1525-7541-8-6-1382-e14
For very small changes in precipitation, the crossing percentile is thus a simple number with no reference to the climatic state (e.g., P or N), except for possible dependence on α. However, numerical computation of (11) to lowest order in δP for other α (Fig. 7) shows that (xc) is insensitive to α over a range of values consistent with observational analysis (e.g., Wilks and Eggleston 1992).

The behavior of (9)(14) offers an explanation for the relatively simple functional form for the changes in normalized intensity distribution appearing in Fig. 5 for nearly all regions and seasons in both models. The similarity between the behavior of (9)(14) and the simulated changes includes the distribution change in HIRHAM’s southwestern United States in the cold season, because δP < 0 for this case. The one change in Fig. 5 that departs substantially from this behavior is HIRHAM’s warm season change in the southwestern United States, which also has δP < 0 but shows a distribution change corresponding to increasing precipitation. This may be a consequence of relatively infrequent precipitation in HIRHAM’s southwestern United States in the warm season: this case has the smallest percentage of days with more than 2.5 mm of precipitation. Furthermore, its low seasonal average precipitation (<1 mm day−1) suggests a temporally varying shape parameter α during the season (Groisman et al. 1999), undermining the assumption that α is unchanging, or the model may simply have behavior that is contrary to the assumptions here, such as adherence to a gamma distribution: this case has the smallest R2 (Table 1) of the fits to the distribution in (1).

The behavior of (9)(14) also suggests why the crossing point between relative increase and relative decrease is roughly the same for all cases despite substantial differences in precipitation climatologies. Adherence of the precipitation intensity distribution to a gamma function poses a substantial constraint on how climate change alters the intensity distribution that is largely independent of baseline climate. For the situation analyzed here, if the overall precipitation increases but the number of rain days remains the same, then there must be more days with relatively high intensity precipitation and fewer days with low-intensity precipitation. The crossing point in the precipitation intensity spectrum is a consequence of an increase in high-intensity but low-frequency events and a decrease in low-intensity, high-frequency events. The shape of gamma distribution governs how this balance occurs. Note that this analysis shows an outcome of adherence to a gamma distribution. It does not give a physical basis for why the gamma distribution describes well precipitation amount versus intensity.

Continuing to higher order in δP,
i1525-7541-8-6-1382-e15
and
i1525-7541-8-6-1382-e16a
i1525-7541-8-6-1382-e16b
The crossing intensity xc and the crossing percentile (xc) both increase with positive precipitation change. This behavior may provide a partial explanation for crossing percentiles in Table 2 that exceed the 59% given by (14), though the dependence of (xc) on δP in (16b) is relatively weak.
Relaxing the assumption of no change in rain days (or rain-day frequency), assume that NC1 = N and NC2 = (1 + δN) N, where δN is the fractional change in rain days. Then instead of (10)
i1525-7541-8-6-1382-e17
so that
i1525-7541-8-6-1382-e18
and
i1525-7541-8-6-1382-e19
Increases in rain days counteract the effect of increased precipitation. Sensitivities of xc and (xc) to δN have the same magnitude as their sensitivities to δP. To at least order δP and δN, the crossing intensity and percentile both again make no reference to the initial climate except for the shape parameter α.
The analysis has focused on the normalized precipitation distribution (6). Without normalizing the distribution, using (3) and again the assumption of constant α when climate changes, the crossing point now is
i1525-7541-8-6-1382-e20
For the previous case with PC1 = P, PC2 = (1 + δP) P, and δN = 0, we now have
i1525-7541-8-6-1382-e21
instead of (10), the result for the normalized distribution. Then, the precipitation percentile for the crossing point is
i1525-7541-8-6-1382-e22
Again, the crossing intensity and percentile refer to the initial climate only through the shape parameter α. The difference between the crossing percentile for the normalized distribution (11) and the result in (22) for the nonnormalized distribution for the same conditions is simply the upper limit of the integrals, which differs between (10) and (21) by the factor (α − 1)/α, to the lowest order in δP. This difference affects the crossing percentile and, perhaps more important, its sensitivity to changing α (Fig. 7). Here, (xc) is less sensitive than (c) to its initial climate, indicating that changes in the normalized precipitation distribution, as opposed to the nonnormalized distribution, give a more consistent and therefore more useful constraint on how precipitation intensity may change when climate changes.

5. Discussion

A diagnosis of daily precipitation change versus intensity categories from two regional climate simulations of the United States shows that increased precipitation under a global warming scenario results in more intense daily precipitation, a well-recognized feature of such climate change (e.g., Cubasch et al. 2001 and references therein). In addition, for the simulations examined here, the intensity spectrum for precipitation change shows a relatively simple pattern (Fig. 5), whereby nearly all high-intensity daily precipitation contributes a larger fraction of the total precipitation, and nearly all low-intensity precipitation contributes a reduced fraction. Also, the crossing percentile between relative decrease and relative increase occurs around the 70th percentile of cumulative precipitation, irrespective of the governing precipitation processes or which model produced the simulation. These two features, the common crossing percentile and pattern of change, would not be discernable in prior studies that resolved the precipitation intensity spectrum with only a few bins or that examined only changes in gamma-function parameters but not changes in the gamma function itself. Equally important, changes in normalized distributions display these features much more readily than distribution changes without normalization (Fig. 7).

Further analysis suggests that this consistent response in precipitation intensity may be a consequence of the intensity spectrum’s adherence to a gamma distribution. Under the gamma distribution, when the total precipitation or number of precipitation days changes, there is a single transition between precipitation rates that contribute relatively more to the total and rates that contribute relatively less. The crossing percentile in the limit of small changes is 59%. The behavior is roughly the same as the results of the numerical models and is insensitive to characteristics of the baseline climate, such as average precipitation, frequency of rain days, and the shape parameter of the precipitation’s gamma distribution.

The analysis here assumes that the gamma distribution’s shape parameter α does not change when climate changes. Small changes in α may occur under global warming scenarios (e.g., Wilby and Wigley 2002; Watterson 2005), which might account for some of the differences in the crossing percentile between (14) and those in Table 2. Also, using an unchanging shape parameter assumes implicitly that the underlying precipitation processes, whatever they are, do not change substantially as the climate changes. This suggests that there may be regions that present a stiffer challenge to the constraint posed here, such as regions where precipitation processes change because of shifts in storm tracks or the location of the intertropical convergence zone. One also should note that although the gamma distribution is well established as a descriptive statistic for the intensity distribution of precipitation (e.g., Wilks 1995), there does not appear to be an equally well established physical basis for why a gamma distribution fits observed behavior so well.

The analysis assumes adequate sampling to characterize the precipitation intensity distribution. Inadequate sampling may be why HIRHAM’s warm season distribution change for the southwestern United States does not fit the theoretical model. Another possibility not explored here is that the HIRHAM model’s changes for this period simply do not adhere to the assumptions of the theoretical model, such as constant shape parameter α. Such behavior may also be occurring in scattered regions where observations have shown increases in relatively intense precipitation while overall precipitation has been relatively constant (Easterling et al. 2000; Alpert et al. 2002).

The analysis does not rest on any assumption about changes in the length of wet or dry periods, nor does it make any predictions for how the number of consecutive days with or without precipitation might change. Equally important, the analysis does not apply to extreme precipitation, such as precipitation above the 99.9th percentile (e.g., Groisman et al. 2005), for which the theory of statistical extremes more likely provides the appropriate description (e.g., Leadbetter et al. 1983; Meehl et al. 2000; Wilson and Toumi 2005).

Acknowledgments

We thank Martyn Clark for supplying the Cooperative Observing Network precipitation dataset, and the reviewers for their comments that helped improve this paper. This work was supported by National Oceanic and Atmospheric Administration Grant NA16GP15822, Department of Energy Grant DEFG0201ER63250, National Science Foundation Grants ATM-0450148 and ATM-0633567, the Electric Power Research Institute, and the Iowa State University Freshman Honors Program. The National Center for Atmospheric Research provided computing support for the RegCM2 simulations.

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Fig. 1.
Fig. 1.

Simulation domain and the four analysis regions: upper Mississippi River basin (UMS), southeastern United States (SE), southwestern United States (SW), and Pacific Northwest (PNW).

Citation: Journal of Hydrometeorology 8, 6; 10.1175/2007JHM817.1

Fig. 2.
Fig. 2.

Normalized frequency of precipitation as a function of daily intensity for 1981–88 in observations and in the RegCM2 and HIRHAM NCEP-driven simulations for each of the analysis regions in Fig. 1: (left) cold season and (right) warm season. Arrows mark the 95th percentile of observed precipitation accumulated from low to high intensity. Straight lines are fits to a log-linear function for each source’s precipitation (identified in the key).

Citation: Journal of Hydrometeorology 8, 6; 10.1175/2007JHM817.1

Fig. 3.
Fig. 3.

Normalized precipitation as a function of daily intensity in the three RegCM2 simulations for each of the analysis regions in Fig. 1: (left) cold season and (right) warm season. Results are shown for simulations using boundary conditions from the NCEP–NCAR reanalysis (NCEP), HadCM2 contemporary climate (CTRL), and HadCM2 scenario climate (SCEN).

Citation: Journal of Hydrometeorology 8, 6; 10.1175/2007JHM817.1

Fig. 4.
Fig. 4.

Same as in Fig. 3, but for the HIRHAM simulations.

Citation: Journal of Hydrometeorology 8, 6; 10.1175/2007JHM817.1

Fig. 5.
Fig. 5.

Change in the percentage contribution of each precipitation intensity category to total precipitation for each of the analysis regions and seasons.

Citation: Journal of Hydrometeorology 8, 6; 10.1175/2007JHM817.1

Fig. 6.
Fig. 6.

Normalized gamma distributions for precipitation in two different climates, denoted C1 and C2. The point xc is the crossing intensity separating increasing from decreasing contributions to the normalized precipitation distribution.

Citation: Journal of Hydrometeorology 8, 6; 10.1175/2007JHM817.1

Fig. 7.
Fig. 7.

The percentile separating increases from decreases in precipitation intensity as a function of the gamma distribution shape parameter α for the normalized intensity distribution (solid) and the nonnormalized intensity distribution (dashed).

Citation: Journal of Hydrometeorology 8, 6; 10.1175/2007JHM817.1

Table 1.

Properties of observed (OBS) and simulated precipitation for 1981–88 for each of the analysis regions in Fig. 1: average precipitation rate, the percentage of days reporting precipitation (parentheses: percentage of days with more than 2.5-mm precipitation), the slope factor β in (1), and the explained variance R2 of the curve fits using (1).

Table 1.
Table 2.

Properties of simulated changes for each of the analysis regions in Fig. 1: change in average precipitation rate, change in frequency of precipitation days, and percentile of transition between relative decrease and relative increase in precipitation.

Table 2.
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