Extreme precipitation has been projected to increase more than the mean under future changed climate, but its mechanism is not clear. We have separated the ‘dynamic’ and ‘thermodynamic’ components of the mean and extreme precipitation changes projected in 6 climate model experiments. The dynamic change is due to the change in atmospheric motion, while the thermodynamic change is due to the change in atmospheric moisture content. The model results consistently show that there are areas with small change or decreases in the thermodynamic change for mean precipitation mainly over subtropics, while the thermodynamic change for extreme precipitation is an overall increase as a result of increased atmospheric moisture. The dynamic changes play a secondary role in the difference between mean and extreme and are limited to lower latitudes. Over many parts of mid- to high latitudes, mean and extreme precipitation increase in comparable magnitude due to a comparable thermodynamic increase.
 It has been projected that precipitation extremes could increase by more than the (annual or seasonal) mean and may cause more frequent and severe floods in a future warmed climate [e.g., Cubasch et al., 2001]. Trenberth  argued that enhancement of extreme precipitation is principally caused by enhancement of atmospheric moisture content, which feeds increased moisture to all weather systems. On a global mean basis, annual mean precipitation is constrained by the energy balance between atmospheric radiative cooling and latent heating, which is expected to limit mean precipitation increase to be lower than the rate of atmospheric moisture increase [Allen and Ingram, 2002]. On a regional basis, however, Emori et al.  showed that areas where the changes in extremes are larger than those of the mean are limited to a few regions, rather than the whole globe.
 The mechanism that causes larger increases in extremes than in the mean is not clear. For example, simplifying the argument by Trenberth  to an idealized case by assuming that precipitation increases by the same percentage across all severity of events (following the 'fixed fractional uplift' case of Jones et al. ), the percentage change in extremes and the mean should be the same. To further clarify this relationship between the changes in mean and extreme precipitation under changed climate, we attempt to answer the question, to what degree we can attribute atmospheric moisture increase to the changes in precipitation, both for the mean and extremes.
 Daily mean 500 hPa vertical velocity (ω) is taken as a proxy of the strength of 'dynamic disturbance' at each grid point on each day. For example, in mid-latitudes winter, it is expected to correspond to the phase and strength of extratropical cyclones and anticyclones passing over the grid point of interest. For computational purposes, ω is divided into bins with a uniform width of 50 hPa/day. Figure 1a shows an example of the relative frequency of occurrence for each ω bin, regarded as probability density function (PDF) of ω and denoted by Prω, averaged over a particular region (equatorial central Pacific) obtained from the control run of CCSR/NIES/FRCGC AGCM. It has a peak around ω = 0 and tails for both stronger upward and downward motion regimes. This shape of PDF is qualitatively the same for different regions from different models. Note that the positive ω represents upward motion throughout this study, unlike the usual definition of pressure velocity, since positively correlated precipitation with upward vertical motion, as will be shown below, is to be preferred. Daily mean precipitation is then composited for each ω bin to obtain the expected values of daily precipitation as a function of ω at each grid point (Pω). Figure 1b shows an example of Pω, for the same region and model as in Figure 1a. Precipitation is expected to be small over the downward motion regime (ω < 0), while it is expected to be larger with stronger upward velocity over the upward motion regime (ω > 0). A similar relationship is obtained for different regions from different model results, though the slope in the upward motion regime is dependent on the regions and also on the model used. Although Prω and Pω are dependent on seasons, the annual relationships are used in this initial study.
 The annual mean precipitation for a grid point can be represented by
The change in mean precipitation between the changed and control climate, δ, can be expressed by
The first term of the r.h.s is the change in mean precipitation due to the change in the PDF of ω, that is, due to the change in the strength and/or frequency of dynamic disturbances. Hence, we call this term “dynamic change”. The second term is due to the change in the expected precipitation for given ω. This term can be called non-dynamic or “thermodynamic change” and is taken to represent the change in atmospheric moisture content. The third represents the covariation term. This approach is basically that of Bony et al.  for cloud-radiation analysis over the tropics. Note, however, that we use daily data and averages are taken only over time, while Bony et al.  used monthly data and spatial as well as temporal averaging over the tropics.
 A similar expression can be defined for the change in extreme precipitation. Here, we take multi-year mean of yearly 4th largest (approximately 99th percentile) value at each grid point as an index of extreme precipitation. Hereafter, we call this the 99th percentile precipitation and denote it by P99. The corresponding ω value (ω*99) can be obtained by inverting the relationship of Pω, that is,
For inverting Pω, a linear-interpolation is applied to the values between the representative values of the bins to obtain a continuous function of Pω. Note that this value is generally different from the 99th percentile value of ω. The change in 99th percentile precipitation between changed and control climate can be expressed by
The first term of the r.h.s is due to the change in extreme vertical velocity and is called dynamic change. The second term is due to the change in expected precipitation for extreme ω fixed at the control value and is called thermodynamic change. The third represents the covariation term. Though the first and third terms are expressed with a linear approximation, they are actually evaluated with a finite difference approximation in the following analysis so that the equation holds precisely.
3. Models and Experiments
 The models and experiments used in this study are summarized in Table 1. The coupled ocean-atmosphere climate models were obtained from the Program for Climate Model Diagnosis and Intercomparison (PCMDI) data archive established for the Intergovernmental Panel on Climate Change (IPCC) 4th Assessment Report. In addition, time-slice climate change experiments by two atmosphere-only models, CCSR/NIES/FRCGC AGCM [Emori et al., 2005] and HadAM3P [Rowell, 2005] are also used. The daily 500 hPa vertical velocity data for the coupled models are estimated from daily three-dimensional horizontal velocity data using the continuity equation, while those for the atmospheric models are direct output from the models.
Table 1. Models and Experiments
Global Mean Precipitation Change
Climate Change Experiment
1.125° × 1.125°
2.81° × 2.81°
2.50° × 2.00°
2.81° × 2.81°
1.125° × 1.125°
2 × CO2 (20 years)
1.875° × 1.250°
 The temporal correlation at each grid point between daily precipitation and daily 500 hPa vertical velocity (positive upward) is higher than 0.5 over most parts of the globe in all the models examined. This fact supports the validity of the method described in Section 2. The correlation is relatively low over some subtropical regions with strong subsidence and over polar regions, where precipitation from clouds shallower than 500 hPa is expected to be dominant. In the following analysis, the areas where the correlation of at least one model is lower than 0.2 are masked out (shaded gray in Figures 2 and 3).
Figure 2 shows the multi-model ensemble mean of the total, dynamic and thermodynamic changes in annual mean precipitation defined by (1). They are shown in percentage change relative to the annual mean precipitation in the control experiments. Before the ensemble mean is taken, results from each model are scaled by the values of global mean precipitation change listed in Table 1 to exclude the effects of different climate (hydrological) sensitivity of the models and of different scenarios in the case of time-slice runs. Also, all data is spatially interpolated to a T42 (∼2.81°) grid before averaging. The total change (Figure 2a) shows general increase over tropics and mid- to high latitudes and decrease over some subtropical regions, a pattern commonly seen in previous studies [e.g., Cubasch et al., 2001]. The dynamic change (Figure 2b) partly explains the tropical Pacific increase and most of the subtropical decrease, while it is virtually zero over mid- to high latitudes (outside of 40°S to 40°N). The thermodynamic change (Figure 2c) explains almost all the mid- to high latitudes increase and part of the tropical increase. The covariation (figure not shown) is negligible except for an increase of up to 40% over equatorial central Pacific. The dominance of thermodynamic changes in extratropics and dynamic changes in lower latitudes is consistent with previous studies where the dynamic and thermodynamic changes of moisture transport and its convergence are discussed [e.g., Watterson, 1998].
Figure 3 shows multi-model ensemble mean of the total, dynamic and thermodynamic changes in extreme (99th percentile) daily precipitation defined by (2), in percentage relative to control values. The same scaling and interpolation procedures as for the mean precipitation have been applied before the ensemble mean is taken. Because of insufficient sampling of Pω for extreme cases, the separated dynamic and thermodynamic changes contain noise. To reduce this, we have applied a 1-2-1 spatial filter once (Figures 3b and 3c), assuming the noise as random. The comparison between the total changes in mean and extreme precipitation (Figures 2a and 3a) is basically the same as was found by Emori et al.  for the results of CCSR/NIES/FRCGC AGCM. That is, though their overall patterns are similar, there are some areas where the increase in the extreme is larger than that in the mean (or the mean is decreased). The global mean of the total changes in mean and extreme precipitation are 6.0% and 13.0%, respectively (note that data over masked areas are excluded). The pattern of dynamic change in the extreme is quite similar to the dynamic change in the mean (global mean of −4.3%, −4.4%, respectively, Figures 2b and 3b) though some regional difference can be identified. The thermodynamic change in the extreme shows overall increase, which is in remarkable contrast with the corresponding small changes or decreases in the mean particularly over the subtropics (global mean of 17.8%, 9.4%, respectively, Figures 2c and 3c). The difference in the total changes in mean and extreme precipitation can therefore be attributed mainly to the different thermodynamic change. The covariation for extremes is negligible (not shown).
 The results of individual models are similar to the ensemble means described above. The largest model difference is over lower latitudes (30°S–30°N), where models respond differently to the various projected (or prescribed, in time-slice runs) sea surface temperature (SST) changes. The inter-model standard deviation of the total change either in the mean or in extreme is typically 10–30% (relative to control values) over the lower latitudes. Over the mid- to high latitudes, the inter-model standard deviation is typically smaller than 10%, suggesting that the present results are more robust for higher latitudes.
5. Concluding Discussion
 The thermodynamic change in extreme precipitation has shown an overall increase (Figure 3c), which is due to the expected precipitation for given vertical motion, Pω, increasing in the strong upward motion regime, regardless of geographical location. As suggested in Section 2, we consider that this is caused by increased atmospheric moisture content. This relation is supported by the fact that models with larger increases in global mean precipitable water (column integrated water vapor) tend to give larger global mean thermodynamic increases in extreme precipitation (the inter-model correlation coefficient is 0.85, which is significant at the 5% level). It is also interesting to note that increased Pω at a given ω means increased latent heating for a given upward motion. We found that, at least over the lower latitudes, this is balanced by increased adiabatic cooling due to enhanced dry static stability.
 The thermodynamic change in mean precipitation (Figure 2c) has areas of modest change or decreasing values, mainly over the subtropics, unlike what was found for the extreme precipitation. This is because Pω decreases or changes little in the downward and/or weak upward motion regime over these areas, in spite of the increased atmospheric moisture content. Although Pω is small over the weak/downward motion regime and its change is also small, this is not negligible when integrated over whole ω range (i.e., annual mean), because of the large statistical weight for this regime. It is this mechanism that keeps the percentage increase in global annual mean precipitation lower than that in global mean extreme precipitation. The decrease can be partly understood by considering the moisture budget of atmospheric column. When atmospheric moisture is increased, moisture divergence will be increased for a given lower tropospheric wind divergence, which would act to reduce Pω in the downward motion regime, unless surface evaporation increases in compensation.
 Over many parts of the mid- to high latitudes and tropics, the thermodynamic increase in the mean is as high as that in the extreme, resulting in the total increase in the both being comparable. These areas roughly correspond to areas with a mean upward motion in the control climate. Important exceptions are some tropical land areas (e.g., Amazon) and northern North Atlantic including the UK/Europe area. Further analysis is needed to clarify these regional details.
 The dynamic terms play a secondary role in making difference between mean and extreme precipitation changes. The dynamic changes suggest that the frequency of strong upward motion is decreased over many parts of subtropics and is increased over the equatorial Pacific. Though this seems to be related to the changes in SST and atmospheric stability, more work is needed to clarify this.
 The seasonal breakdown of the present analysis is desired for future work. The spread of composited data in constructing Pω (error bars in Figure 1b) should be smaller in a seasonal relationship than in the annual, especially for higher latitudes, where Pω can be considerably seasonally dependent. However, we have confirmed that seasonality does not seriously affect the annual results of the present study. For the extreme precipitation, it is because extremely strong upward motions (∼ω*99) occur mostly accompanying extreme precipitation events (∼P99) in wet seasons and seldom occur in dry seasons. That is, for an extremely strong upward motion, the annual Pω approximately represents that of the wet seasons alone.
 The present analysis relies on the modeled Pω and PDF of ω. At present, there seems to be no good way to validate these functions, since daily vertical velocities of re-analysis data seem strongly dependent on the models used in the re-analysis and does not always correspond well to the observed daily precipitation. The quantitative validation of this analysis remains for future work. Nevertheless, the result of this study is qualitatively robust among all the models examined.
 This work was done whilst the first author (SE) was at the Hadley Centre as a visiting scientist. We thank people at the Hadley Centre as well as the K-1 Japan project members for support and discussion. Thanks are extended to Ian Watterson, Isaac Held, Julia Slingo, Chris Ferro, Richard Jones, Myles Allen, William Ingram, Pardeep Pall and an anonymous reviewer for helpful comments and discussion. We also acknowledge the international modeling groups for providing their data for analysis, the PCMDI for collecting and archiving the model data, the JSC/CLIVAR Working Group on Coupled Modelling (WGCM) and their Coupled Model Intercomparison Project (CMIP) and Climate Simulation Panel for organizing the model data analysis activity, and the IPCC WG1 TSU for technical support. The IPCC Data Archive at Lawrence Livermore National Laboratory is supported by the Office of Science, U.S. Department of Energy. This work was partially supported by the Research Revolution 2002 (RR2002) of the Ministry of Education, Sports, Culture, Science and Technology of Japan, by the Global Environment Research Fund (GERF) of the Ministry of the Environment of Japan, and by the U.K. Department of the Environment, Food and Rural Affairs (Contract PECD/7/12/37). The model calculations of MIROC3.2 and CCSR/NIES/FRCGC AGCM were made on the Earth Simulator. The GFD-DENNOU Library was used for the drawings.