Spatial variabilities and their relationships of the trends of temperature, water vapor, and precipitation in the North American Regional Reanalysis are examined for each season from March 1979 to February 2007. Results show that warming dominates the domain in the troposphere from the surface to 300 hPa. Water vapor increases at lower levels but does not change much at mid-upper levels. Because of the large increase of water vapor holding capacity of the air at all levels due to the warming, relative humidity has a decreasing trend at all levels. The decrease is small at the surface and largest at midlevels. Precipitation, which corresponds well to ascending motion in trends, both increases and decreases in about half of the domain. Statistical analysis from the very large spatial samples indicates that the precipitation trend positively relates to both specific humidity trend and relative humidity trend. However, temperature trend positively relates to specific humidity trend but negatively relates to relative humidity trend. So, in strong warming places, whether precipitation increases or not depends on whether the decrease of relative humidity becomes a limiting factor; small decrease of relative humidity may still allow precipitation to increase, but large decrease of relative humidity may make precipitation decrease. The uncertain relationship between the trends of precipitation and temperature can also be understood from the nonlinear characteristics of the atmospheric processes.
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 The North American Regional Reanalysis (NARR) is a very useful reanalysis data set for studying the variability and changes of the climate over North America. Its precipitation is assimilated from observations [Mesinger et al., 2006], and it is generally believed that the hydrological cycle in the NARR could be more realistic than other reanalyses and model output. Lu and Zeng  and Lu and Takle  analyzed, respectively, the seasonal and interannual variations of the precipitation in the NARR and their associations with the variations of water vapor and temperature. One purpose of this study is to examine the spatial variabilities (i.e., the vertical profiles and horizontal structures) of the trends of temperature, water vapor, and precipitation in the NARR. The overall characteristics of these trends are compared with previous observed results when they are available, although they may not represent the same region and time period. This provides only a preliminary evaluation of the performance of the NARR in simulating these trends. A stricter evaluation will be made additionally.
 Temperature, water vapor, and precipitation can interact, and thus can be linked with each other through the dynamic, thermodynamic, and radiative processes in the atmosphere-earth system. The second purpose of this study is to explore the relationships among the trends of these hydrological quantities. These relationships have not been fully analyzed in previous studies. One reason for this is that, while observed data of these quantities were all available for the same region and time period, the quantities were regionally averaged for calculating the trends, thereby creating difficulty in assessing relationships among the trends. In this study, the very large spatial samples of the three-dimensional fine-grid data of the NARR are used to investigate the relationships. The domain of the NARR, which contains both ocean and land extending from the tropics to the pole, spans the full range of geographical situations and thus can be used to reveal bulk relationships of the trends of these quantities.
 Warming at the global scale is caused by external forcing. However, spatial differences in the warming, as well as the related spatial differences of the trends of water vapor and precipitation, may be caused by the dynamic and thermodynamic processes in the atmosphere. This study aims to expose whether, affected by the atmospheric circulation, these trends posses certain relationships. Atmospheric circulation includes various components, and their roles can be different from place to place. However, the ultimate effect of the atmospheric circulation is to change local water vapor and temperature, and, for the formation of precipitation, to make the air saturated. The precipitation trend is therefore understood in this study from its linkage with the trends of water vapor and temperature. These results from the NARR can be referred when evaluating the performances of global and regional climate models in simulating the trends of these quantities and their relationships [e.g., Knutson et al. 2006], especially for North America.
 The NARR data set and the calculation of the trends of temperature, water vapor, and precipitation are introduced in section 2, and their spatial variabilities are presented in section 3. In section 4 we discuss relationships among the spatial variations of these trends, and we close with a summary in section 5.
2. Data and Analysis
 The North American Regional Reanalysis (NARR) was developed by the National Centers for Environmental Prediction (NCEP). The data, starting from 1979, are available at 3 h intervals with resolution of 32 km in horizontal and 29 layers in vertical. Temperature, specific humidity, precipitation, and vertical velocity data from March 1979 to February 2007 are used in this study to calculate trends over 28 seasons of spring (MAM), summer (JJA), fall (SON), and winter (DJF). More details of the assimilation procedure in the NARR and evaluation of its reliability are given by Mesinger et al.  and Bukovsky and Karoly .
 Precipitation trend (denoted as dP/dt) is calculated for each of the total 349 × 277 grid points contained in the horizontal domain of the NARR. Trends of temperature (dT/dt), specific humidity (dq/dt), relative humidity (dr/dt), and ascending motion [d(−Omega)/dt)] are calculated for each grid point at each level. The spatial variabilities of the trends are reflected through their horizontal structures and vertical profiles averaged over the domain. Spatial relationships are determined through analyzing correlations (R) between the horizontal variations of the trends of two quantities. The fine resolution of the NARR, which provides nearly 100,000 grid points in the domain, enables us to clearly identify correlation between the trends of these quantities. Calculations show that the trends with larger absolute values are generally more significant.
3. Spatial Variabilities of Trends
Figure 1 suggests that the warming is not just in the near-surface air; it prevails in the troposphere up to 300 hPa in all seasons. Warming appears at about 80% of the grid points at levels below 400 hPa in all seasons (Figure 1a). The warming rates averaged over the domain are all about 0.2°C/decade (Figure 1b), and the general feature of the profiles is similar to that in Figure 4 of Lanzante et al. . Angell  indicated from the radiosonde data of North America for 1975–1994 that the trend of the 850–300 hPa annual temperature averaged in 20°N–80°N is also 0.2°C/decade. In the upper troposphere and lower stratosphere at levels above 300 hPa, cooling dominates the domain, and cooling appears in almost the entire domain at 100 hPa. Observed data [e.g., Angell, 1988; Oort and Liu, 1993; Angell, 1999] revealed the cooling, and Angell  showed that the 100–50 mb annual temperature trend is −0.5°C/decade. Studies of the vertical profiles of temperature trends by Thorne et al.  pointed out that the cooling in the upper troposphere and lower stratosphere can be attributed to both the dynamic effects of the atmosphere and the radiative effects of water vapor, ozone, and volcanic gases.
Figure 2a presents the horizontal distribution of the temperature trend at 800 hPa. As suggested from the vertical profiles, temperature has increasing trends over the major portion of the domain. Cooling occurs at high, middle, and low latitudes for winter and spring, but is mainly in low latitudes for summer and fall. Figure 2a also shows a summer warming hole in the central United States, which was found from observations of surface temperature [e.g., Pan et al., 2004; Kunkel et al., 2006]. This hole appears at all levels of the troposphere in the NARR. The trend distributions at different levels also suggest that the warming hole in the central United States might be linked with the warming holes over the Pacific and Atlantic Oceans.
3.2. Water Vapor
Figure 3 shows the vertical profiles of the trends of specific humidity for the four seasons. With the warming in lower levels, increasing trends of water vapor dominate the domain below 900hPa for winter and spring and below 800hPa for summer and fall (Figure 3a). The fractions of the grid points with increasing water vapor trends are all largest at the surface, being about 70∼85% for all seasons. The domain-averaged trends of water vapor are thus maximal at the surface, being about 0.04∼0.14g/kg/decade (Figure 3b) or 1.0∼2.4%/decade (Figure 3c) for all seasons. The increasing trends of water vapor in the surface air are attributed to the enhanced evaporation due to the warming of the surface air. In mid-upper levels, changes of water vapor are weaker, and water vapor trends are represented more by decreases than by increases (Figures 3a–3c), which may result from the dynamic, thermodynamic, and radiative effects of the atmosphere. Figure 3d presents the ratios of the domain-averaged relative trends of water vapor to the averaged temperature trends, which can be regarded as the production rates of water vapor relative to a 1°C of warming. The largest water vapor production rates are all at the surface, being about 6% per °C for all seasons. Ross and Elliott  found from observations of stations over North America during 1973–1993 that the annual trends of surface to 500 hPa precipitable water were generally increasing with a rate of 3–7%/°C. Dai  showed from global observations that the increasing rates of surface specific humidity for the four seasons are all less than, but close to, 7%/°C, the rate projected from the Clausius-Clapeyron relation. The horizontal distributions of the specific humidity trends at 800hPa are presented in Figure 2b. The patterns of the specific humidity trends for the four seasons are in general very similar to those of the temperature trends, indicating a positive relationship between the spatial variations of trends of temperature and specific humidity.
Figure 4 displays the vertical profiles of the trends of relative humidity for the four seasons. The domain has equal numbers of grid points with increasing and decreasing trends of relative humidity at the surface, but is dominated by the grid points with decreasing relative humidity at mid-upper levels (Figure 4a). The domain-averaged trends, both the absolute trends (Figure 4b) and the relative trends (Figure 4c), show that relative humidity decreases at all levels, with small decrease at surface and the largest at midlevels. At the surface this suggests that, although warming produces more water vapor, the effect of warming on increasing the water vapor holding capacity of air is slightly stronger. At midlevels, the warming does not bring more water vapor, but increases the water vapor holding capacity of the air. The ratios of the domain-averaged relative trends of relative humidity to the averaged temperature trends (Figure 4d) show that relative humidity has the smallest decreasing rates at surface, which are all about 1% decrease per °C warming in the four seasons. Vincent et al.  found from hourly observations of 75 stations across Canada for 1953–2005 that a decrease of relative humidity accompanies the warming. Dai  also found from global observations that relative humidity possesses small percentage decreasing trends. The horizontal distributions of the relative humidity trends at 800 hPa are presented in Figure 2c. The structure patterns of the relative humidity trends for the four seasons are also similar to that of the temperature trends, but in general with an out-of-phase relationship.
 The distributions of the precipitation trends for the four seasons are displayed in Figure 2d. Increasing and decreasing trends both take about half of the domain. Fractions of the grid points with increasing precipitation trends for spring, summer, fall, and winter are 44%, 54%, 51%, and 43%, respectively. Previous studies from observed data illustrated that precipitation has both increasing and decreasing trends, depending on geographic locations and seasons [e.g., Stuart and Isaac, 1994]. The averaged decreasing trends are stronger than increasing trends for spring and summer, while both trends are equivalent for fall and winter.
 The large spatial variability of precipitation trends may be affected by the temporal-spatial variability of the external (e.g., solar and volcanic) forcing of the atmosphere [Allen and Ingram, 2002], but can ultimately be attributed to the temporal-spatial variability of the atmospheric circulation [e.g., Curtis and Hastenrath, 1999; Polyakov et al., 2003]. The trend of sea level pressure has been used to reflect the effect of the atmospheric circulation [e.g., Mo and Loon, 1985; Polyakov et al., 2003; Wang and Swail, 2006]. Curtis and Hastenrath  studied the trends of sea level pressure as well as zonal wind. The trend of vertical velocity, which is a component of the atmospheric circulation, is analyzed here. Correlations between the spatial variations of the trends of ascending motion and precipitation are positive at all levels, and are very strong at mid-upper levels for all seasons (figures not shown). The differences between the ascending motion trends averaged over the grid points with increasing and decreasing precipitation trends (Figure 5) are small at lower levels and large at mid-upper levels. The ascending motion trend averaged over the grid points with decreasing precipitation trends may be weakly positive at lower levels, but are strongly negative at upper levels. The ascending motion trend averaged over the grid points with increasing precipitation trends is positive and strong at almost all levels for all seasons. The magnitude of vertical motion, which can be evaluated from the vertical velocities that are averaged over the grid points with ascending and descending motions, respectively, is 0.1 Pa/s. The domain-averaged vertical motion trend in Figure 5 has approximate magnitude of 0.001 Pa/s/decade. So, the relative trend of the vertical motion averaged over the entire domain is only about ±1%/decade, which suggests an overall mass conservation with time over the domain through the ascending and descending motion trends.
4. Relationships Among Spatial Variations of the Trends
4.1. Precipitation and Water Vapor
 Forming precipitation requires water vapor and saturated conditions, so both specific humidity and relative humidity are important. Figure 6a presents the profiles of the correlation between spatial variations of the trends of precipitation and specific humidity. Correlations are positive at all levels for all seasons. For winter and spring, the maximal correlation appears around 600 hPa and can reach about 0.4. For summer and fall, the correlation is smaller at 750 and 1000 hPa, but is larger between them and above 750 hPa with values of about 0.3. At mid-upper levels, the domain-averaged specific humidity trend is negative (Figure 3b). The strong positive correlations of the trends of specific humidity and precipitation suggest that at the mid-upper levels, although the averaged specific humidity trend is negative, the precipitation in the places where the specific humidity trend is positive or slightly negative can have increasing trends. The precipitation in the places where the specific humidity trend is strongly negative can have decreasing trends. At lower levels, the domain-averaged specific humidity trend is positive (Figure 3b). The weak positive correlations of the trends of precipitation and specific humidity suggest that the precipitation in the places with large increasing specific humidity trends may have increasing trends.
Figure 6b shows the profiles of the correlation between spatial variations of the trends of precipitation and relative humidity. Correlations are positive at all levels below 300 hPa for all seasons, and are relatively weaker at surface and stronger (can reach 0.4) at levels of 400–600 hPa. The domain-averaged relative humidity trend is negative at all levels, and the decrease is strongest at midlevels (Figure 4b). The positive correlations suggest that, although the averaged relative humidity trend is negative, precipitation in places where the relative humidity trend is slightly negative can still have increasing trends. Precipitation in places where the relative humidity trend is strongly negative can have decreasing trends.
4.2. Temperature and Water Vapor
Figure 7 shows the profiles of the spatial correlations of the trends of specific humidity and relative humidity with the temperature trend. The specific humidity trend almost always has positive correlation with temperature trend at all levels for all seasons (Figure 7a). As mentioned in section 2, the very large spatial samples could easily make correlations significant. These positive correlations suggest that in places where there is strong warming, specific humidity can have strong increasing trends at lower levels and increasing or weak decreasing trends at mid-upper levels. The strong increasing trends of water vapor at lower levels may be achieved through the strong surface evaporation trends due to the strong warming.
 However, the relative humidity trend almost always has negative correlations with the temperature trend at all levels for all seasons (Figure 7b), suggesting that in the places where there is strong warming, relative humidity tends to have large decreasing trends at all levels. Therefore, in the strong warming places, there are two features in the change of water vapor: specific humidity has increasing or small decreasing trends, but relative humidity has large decreasing trends.
 The Clausius-Clapeyron equation provides a linear relation between lnqs and −1/T, where qs is the saturation vapor pressure at temperature T. Figure 8 displays the profiles comparing the spatial correlation between the trends of lnq and −1/T with the spatial correlation between the trends of q and T. Although lnq is an increasing function of q and −1/T is an increasing function of T, the correlations between the trends of the two cases can be quite different. Figure 8 shows that the relationships between the trends of lnq and −1/T are in general much stronger than the relationships between the trends of q and T at levels below 300 hPa for all seasons, especially at the lower levels; for example, the correlations at the surface in winter can be increased from 0.3 for q and T to 0.9 for lnq and −1/T.
4.3. Precipitation and Temperature
Figure 9 presents the profiles of the spatial correlations between trends of precipitation and temperature. The correlations are in general very weak, compared with those between trends of water vapor and temperature or between trends of precipitation and water vapor. They can be both positive and negative at different levels and for different seasons. The temperature trends in the places with increasing precipitation trends can be both stronger warming and weaker warming or cooling in different levels and seasons. So, there is no relatively fixed relationship between the trends of precipitation and temperature.
 This uncertain relationship is generally attributed to the changes in the atmospheric circulation. As discussed in section 1, the atmospheric circulation involves various components, and they may have large spatial and temporal variations. The uncertain relationship is understood here from the perspective of the change of water vapor, which is affected by the change of atmospheric circulation and can bridge the changes of precipitation and temperature. The above statistical analysis of the spatial variations of trends reveal that in the places where precipitation has increasing trends, both specific humidity and relative humidity have increasing or small decreasing trends. However, in the places where there is strong warming, while specific humidity can have increasing or small decreasing trends, relative humidity has large decreasing trends. So, in the strong warming places, the changes of specific humidity can relate to increases of precipitation, while the changes of relative humidity can relate to decreases of precipitation. Since the air needs to be saturated with water vapor to form precipitation, in order to have a large seasonal total of precipitation, the seasonal mean relative humidity should not be too small. Therefore, for a specific place, whether the precipitation can have an increasing trend depends on whether the decrease of relative humidity there is large and thus becomes a limiting factor. If the decrease of relative humidity is small so that the seasonal mean relative humidity in the future warmed years is still sufficiently large, then the precipitation can have an increasing trend. Otherwise, if the decrease of relative humidity is very large so that the future seasonal mean relative humidity becomes too small to form larger precipitation, then the decrease of relative humidity would limit the increase of precipitation, thus results in a decreasing precipitation trend.
 The nonlinear characteristics of the atmosphere, as reflected in Figure 8, can also be used to understand the uncertain relation between the trends of precipitation and temperature. Mathematically, if precipitation has nonlinear relations with temperature and water vapor, the change of precipitation may relate to both the changes of temperature and water vapor and the means of temperature, water vapor, and precipitation. Taking spring as an example, Figure 10 shows the profiles of the spatial correlations of precipitation trend with temperature trend as well as the climatic means (i.e., the means of the 28 springs) of specific humidity, temperature, and precipitation. The correlations of precipitation trend with the three climatic means are all negative, and their magnitudes are all greater than that of the correlation with temperature trend. This means that the dependence of the precipitation trend on the temperature trend is strongly affected by the spatial differences of the mean climate state. Compared with the grid points having decreasing precipitation, the grid points having increasing precipitation tend to appear in the places with less water vapor and lower temperature at all levels, which means the high-latitude places in the NARR domain. This is consistent with the conclusion from the previous observed data that, with the warming, precipitation tends to increase in high latitudes [Houghton, 1997]. Calculations indicate that this nonlinear feature (that the spatial difference of precipitation trend depends on the difference of mean climate) is also true at local scales.
 The warming and the associated trends of water vapor and precipitation in the North American Regional Reanalysis are evaluated in this study. Results show that warming has not occurred merely in the surface air; it has occurred in the troposphere from the surface to 300 hPa in all seasons. However, water vapor increased only in lower levels through the enhanced evaporation due to the warming. In mid-upper levels, water vapor has decreasing trends. On the other hand, the water vapor holding capacity of the air increased at all levels due to the warming. The total effect of the warming is that relative humidity decreased at all levels below 300hPa. The decrease is small at surface and maximal at 600–800 hPa levels for all seasons. Precipitation has both increasing and decreasing trends, with each taking about half of the domain in all seasons. Trend analysis of vertical velocity, which is part of the atmospheric circulation, shows that corresponding to the increase (decrease) of precipitation, ascending motion increased (decreased) at mid-upper levels and did not change much in lower levels.
 The relationships among the spatial variations of the trends of these quantities are studied through analyzing the very large spatial samples of the NARR. Results show that in the places with increasing precipitation trends, specific humidity has increasing or small decreasing trends, and these can be found in the strong warming places. In the places with the increasing precipitation trends, relative humidity also has small decreasing trends, but what are found in the strong warming places are large decreasing trends of relative humidity. Therefore, in the strong warming places, whether precipitation increases or not depends on whether the decrease of relative humidity becomes a limiting factor. If relative humidity decreases slightly, precipitation can still increase. If relative humidity decreases too much, precipitation may decrease. The weak and uncertain relation between the trends of precipitation and temperature can also be understood from the nonlinear effect of the atmosphere. The relation between trends about water vapor and temperature can be much improved by linking them with the Clausius-Clapeyron relation. Allen and Ingram  pointed out that the warming is dominated by the anthropogenic forcing that increases steadily, while changes of precipitation may be dominated by the natural forcing that vary on shorter time scales.
 Some fundamental characteristics of the trends of these quantities in the NARR are revealed, including the cooling in the upper troposphere and lower stratosphere, the summer warming hole in the central United States, the warming rate of about 0.2°C/decade at all mid-lower levels for all seasons, the increase of surface water vapor with temperature at rate about 6%/°C, and the small decrease of surface relative humidity. These characteristics are, in general, consistent with the results of previous studies from observations, although they may be for different regions and periods. The major purpose of this study is to analyze the spatial structures of the trends of these quantities in the NARR, and explore the relationships among these trends. The detailed quantitative evaluation of the trends in the NARR needs to be made with observations for the same region and time period.
 The North American Regional Reanalysis is distributed by the NCEP/EMC. The two anonymous reviewers and the editor are thanked for their constructive comments and suggestions. The first author (EL) thanks John M. Wallace and Raymond W. Arritt for their helpful discussions. This work was supported by USDA National Research Initiative grant 20063561516724.