## 1. Introduction

Projections of future climate change and its regional impacts generally rely upon “before” and “after” snapshots of climate parameters in order to compare the climate states between one time period and another. Implicit in this method of presentation is an assumption that parameters will vary in a quasi-linear sense from the current state to the final state, albeit with possible internal variations on interannual to decadal time scales (e.g., Giorgi 2005). However, given the complexity of the climate system—including the regional and global influence of positive and negative feedback loops along with the interaction of multiple physical subprocesses—it has been recognized that the evolution from one state to another does not necessarily need to follow a quasi-linear path (e.g., Alley et al. 2003; Rial et al. 2004). Instead, the long-term evolution of the system can be nonlinear. For instance, the evolution of the system can contain 1) increased sensitivity to anthropogenic forcing, such that trends (positive or negative) increase with increased forcing; 2) decreased sensitivity to anthropogenic forcing, such that trends decrease with increased forcing; and 3) “turning point” behavior in which the short-term initial trends are of the opposite sign as the longer-term trends. Previous examples of such nonlinear behavior in historical and projected climate parameters include accelerated melting of arctic sea ice (Winton 2006; Serreze and Francis 2006), increasing sensitivity of extreme event occurrences to increasing temperatures (Good et al. 2006), increasing trends in the rate of sea level rise (Church and White 2006), and discontinuous shifts in ecosystem communities (Burkett et al. 2005).

Identification of parameters that may experience nonlinear responses to global climate variations is further complicated when examining spatial–temporal fields with an eye toward characterizing regional climate change impacts. While such studies can be carried out a posteriori once a given region has been selected for analysis, it is of interest to see whether there are methods for identifying such regions a priori from spatiotemporal data taken from global and/or regional climate projections. If so, it suggests the possibility for identifying “hot spot” regions in which nonlinear behavior may be prevalent and targeting analytical studies to understand this behavior.

Here we develop a methodology for identifying such nonlinear evolutions and apply it to projected changes in soil moisture across the United States arising from increased anthropogenic greenhouse gas emissions/concentrations. It is important to note that these results should be considered plausible projections of climate change in these regions, not necessarily forecasts of such change. At the same time, we highlight these projections because they contain highly nonlinear long-term evolutions that have important consequences for local and regional adaptation and planning activities in response to global climate change. In addition, the analysis demonstrates the utility of the methodology for identifying such nonlinear evolutions, which can be applied more generally to other climate parameters in other regions of the world.

## 2. Data

For this study, we use coupled atmosphere–ocean–land surface model output produced from NCAR’s Community Climate System Model, version 3 (CCSM3; Collins et al. 2006), run under two basic emission scenarios. In the first, four T85 (approximately 1.4°) resolution simulations of the CCSM3 are forced by projected increases in greenhouse gas (GHG) concentrations, sulfate aerosols, and solar activity associated with the A2 emissions scenario from the Intergovernmental Panel on Climate Change (IPCC) Special Report on Emission Scenarios (SRES; Nakićenović et al. 2000). For this scenario, atmospheric CO_{2} concentrations reach approximately 525 ppm by volume by 2050 and 800 ppm by 2100. In the second, a 400-yr control simulation of the CCSM3 is forced with present-day concentrations of GHGs, sulfate aerosols, and solar activity. For comparison with the output from the A2 model simulations, the 400-yr control simulation is partitioned into four separate 100-yr segments. All data from the model runs are taken from the Program for Climate Model Diagnoses and Intercomparison (PCMDI) and are made available through the World Climate Research Programme’s (WCRP’s) Coupled Model Intercomparison Project (CMIP3) multimodel dataset. For this investigation, we will be examining parameters related to the near-surface and subsurface hydrologic cycle. Primary interest will be upon changes in total soil moisture content, with peripheral investigations of contributing moisture fluxes including precipitation, evaporation, and total runoff. Throughout this paper, we will be examining longer-term changes in various fields. Hence, we first compute the yearly averages for each field at each grid point. We then apply a 20-yr box-filter average to each field at each grid point to arrive at 20-yr running mean values. All figures will be based upon these 20-yr running mean gridpoint values unless noted otherwise.

## 3. Methods

To illustrate the method for identifying long-term nonlinear behavior within projections of various climate parameters, we use three artificial time series as shown in Figure 1. These time series are constructed such that they contain 1) a long-term linear trend, 2) normally distributed random perturbations (with a 20-yr box-filter average applied), and 3) a long-term nonlinear time evolution. The combination of these three parameters produces three distinct types of behavior. One time series represents (or symbolizes) increased sensitivity to anthropogenic forcing, such that the onset of the anthropogenic response occurs later in the simulation. Another time series represents decreased sensitivity to anthropogenic forcing, such that the onset of the response occurs fairly early in the simulation, after which the trends decrease with time. The final time series represents “turning point” behavior in which the short-term trends are of the opposite sign as the longer-term trends.

To isolate the long-term nonlinear behavior, we first detrend the data by removing the linear trend connecting the first and last 20-yr period (Figure 1b); this procedure identifies the departures of the time series evolution from the long-term trend implied by taking the difference of the two endpoints. For the time series considered here, the long-term nonlinear behavior in each of the time series is very similar, despite the differences in the overall evolution of the three. This similarity is by construction—the exact same long-term nonlinear time evolution was used to construct the three time series; only the long-term trends and the normally distributed random perturbations vary between them. We highlight again that despite containing the same nonlinear evolution the overall evolutions of the three time series differ significantly. By removing the trend, however, we can begin to identify the underlying similarity between the time series.

In addition, presenting the time series in this manner highlights another characteristic of the long-term nonlinear evolution, namely, its *sign*. In each of the time series shown here, the sign is considered to be negative because the initial slope of the (long term) nonlinear evolution is negative (equivalently, the initial long-term nonlinear evolution produces negative deviations from the hypothetical quasi-linear evolution). By comparing the sign of the long-term nonlinear evolution with the sign of the linear trend, we now see how similar nonlinear behavior can produce significantly different overall time series evolutions. For instance, in the case of the late-onset time series, the sign of the trend (positive) is opposite that of the long-term nonlinear evolution, thereby producing a delayed response within the overall time evolution because of an offset in the initial behavior of the two. In comparison, for the early-onset time series the sign of the trend (negative) is the same as the long-term nonlinear evolution, thereby producing a large initial response within the overall time evolution followed by a subsequent dampening of the initial trend.

One way to quantify the nonlinear departures of the full time series from their (hypothetical) quasi-linear evolutions is to examine the power spectra of these departures. While the variability associated with these departures can be found across a range of frequencies, we are interested in the long-term nonlinear variations, which we define to be those that are contained within the lowest-order mode of variability. To illustrate the selection procedure used to identify these long-term nonlinear variations, each time series shown in Figure 1a is detrended and decomposed into its respective power spectrum via a Fourier transform. Then the spectrum is normalized by the total variance for the time series (Figure 1c). In addition, a stochastic model is created in which 10 000 ensembles of 100-yr time series are randomly generated using a first-order Markov model with *r* = 0.50 lag-1 autocorrelation. Each 100-yr time series is then put through a 20-yr box filter and Fourier transformed to arrive at the distribution of power spectra expected for stochastic, red noise behavior. For the three artificial time series shown here, it is apparent that the lowest-order modes of variability differ significantly from those expected by chance. In addition, this figure emphasizes that the lowest-order modes of variability within the power spectra can help identify the presence of, and quantify the magnitude of, the long-term nonlinear evolutions seen in the detrended time series.

While illustrative by nature, this methodology can also be applied to actual projections of long-term climate variations. To do so requires some additional considerations. To identify particular regions in which long-term nonlinear variations are projected to occur within spatial–temporal fields we need to look at the amount of gridpoint variance found in the lowest-order mode of variability, as a fraction of total variance. For this study we have four ensemble members within our climate model projections. Therefore, we want to calculate the fractional variance within the lowest-order mode of variability in the (detrended) ensemble-mean gridpoint time series. Significance of the results is then tested in two ways. First, four ensemble members of 100-yr time series are randomly generated using a first-order Markov model with *r* = 0.50 lag-1 autocorrelation (for comparison, within the CCSM3 control simulations the lag-1 autocorrelation for gridpoint soil moisture averaged over the United States—the region investigated below—is *r* = 0.39). Each 100-yr time series is then put through a 20-yr box filter, and the four “ensemble” time series are averaged together and Fourier transformed to arrive at the power spectrum expected by a stochastic, red noise process. This procedure is then repeated 10 000 times to generate the 90% confidence level for comparison with the simulated gridpoint data. Based upon this analysis, fractional amounts of lowest-order variance above 31% are considered significantly different from chance at the 90% level. As an additional qualitative check on significance, we consider the signs of the nonlinear variations in each of the separate ensemble-member time evolutions; gridpoint values are considered consistent (and hence significant) only if the sign of the nonlinear variation agrees in at least three of the four members. While subjective, this consistency threshold is similar to the 80% threshold used in discussions of precipitation trends in the IPCC Fourth Assessment Report (AR4; Solomon et al. 2007).

We will also need to identify whether there are significant deviations from linear behavior within given regions, as opposed to specific grid points. To do so, the ensemble-mean time series at each grid point within a given region is detrended and decomposed into its respective power spectrum via a Fourier transform. Then each gridpoint spectrum is normalized by the total variance at the given grid point. The magnitude of each normalized spectrum is then averaged across the grid points in each region. To test for significance, a stochastic model is created in which four ensemble members of 100-yr time series are randomly generated for a given “grid point” using the first-order Markov model with *r* = 0.50 lag-1 autocorrelation. Each 100-yr time series is then put through a 20-yr box filter; the four “ensemble” time series are averaged together and Fourier transformed to arrive at a power spectrum for each “grid point.” To reproduce the degrees of freedom within a given region, however, we do not create independent sets of random time series for each grid point in a region. Instead, following the method of Bretherton et al. (Bretherton et al. 1999), we determine the estimated spatial degrees of freedom (DOFs) for a given region using the eigenvalues from an empirical orthogonal function (EOF) decomposition of the original yearly data for the region, as found in the CCSM3 control simulation. We then create independent sets of random yearly time series for only the number of spatial DOFs within each region; these are then averaged together to arrive at an estimate of the power spectrum expected by stochastic, red noise processes across the region as a whole. This procedure is then repeated 10 000 times to generate the 90% confidence level for comparison with the simulation data. For consistency, when computing the 90% confidence interval on the area-average gridpoint power spectrum values for the model projections, we assume the same spatial DOFs for the region, not the number that would be present if each grid point were assumed to vary independently.

## 4. Results

Figure 2 shows the projected changes in ensemble-mean gridpoint soil moisture values between the periods 2080–99 and 2000–19, as found in the CCSM3 A2 simulations. These trends are qualitatively consistent with those found in multimodel ensemble-mean projections (Solomon et al. 2007) including decreasing trends across the southern portion of the domain and increasing trends over the northern portion; however, the CCSM3 appears to have slightly stronger drying along the interior northwest, where the IPCC projections show a more westward shift of positive soil moisture anomalies.

To identify possible long-term nonlinear variations in the gridpoint time series of soil moisture within these model projections, Figure 3 shows the amount of gridpoint variance found in the lowest-order mode of variability, as a fraction of total variance, from both the A2 and control simulations. Within the A2 simulation (Figure 3a), there are two apparent “hot spots” of nonlinear responses to anthropogenic forcing. These include the Ohio River basin stretching from western Pennsylvania into Arkansas. The other is centered over the interior southwestern United States—predominantly over Arizona and New Mexico—stretching into Utah and Nevada. Also shown is the sign of the nonlinear evolution, with dots indicating regions in which the sign is the same as the overall sign of the soil moisture trend. Across most of the United States, the sign of the nonlinear evolution is positive and is opposite that of the overall trend. This combination produces a time series evolution in which soil moisture anomalies either remain relatively constant or increase slightly during the early stages of anthropogenic forcing, then subsequently experience relatively large decreases during the latter stages of this forcing (similar to but of opposite sign as the “late onset” evolution seen in Figure 1a).

In the control simulation (Figure 3b), there also appear to be patches of significant nonlinear variability. The most prominent appears to be found over the northwestern United States. Here the sign of the nonlinear variation appears to be negative and is of the same sign as the overall trend. This combination is characteristic of a time series in which there are rapid decreases in soil moisture early in the simulation, followed by decreasing negative trends later in the simulation (similar to the “early onset” evolution seen in Figure 1a). Investigation of individual time series for this region indicates that the overall trend is driven by large soil moisture decreases during the first century or so, with each subsequent century showing a tapering of this decrease (not shown); when separated into “ensemble” members, this decomposition produces an overall trend in the ensemble mean and overall nonlinear trends in each ensemble member that represent a tapering of the soil moisture drawdown. This nonlinear behavior in the control simulation highlights the need to examine not just the model projections for future climate change scenarios but also the control simulations to ensure that any signals identified in the model forcing experiments are not part of the “spinup” of the model system itself.

To better capture the timing of the nonlinear behavior associated with the long-term evolution at each grid point, the phase of the lowest-order mode of variability at each grid point is shown in Figures 3c,d. Here the phase is based upon a cosine function, with cos[2*πt*/(80 yr − *π*)] having a phase of 0 yr This phase relation provides the approximate time of initial increases in the full time series. If the overall trend is negative, however, we want to identify the approximate time of initial decreases; in this case, the phase is shifted forward by (80 yr)(*π*/2*π*) = 40 yr. As an example, in Figure 1b the nonlinear time evolutions of the “late onset” and “reversal” time series are best captured by cos[2*πt*/(80 yr)] = cos[(2*πt*/80 yr − *π*) + *π*], hence both have a phase of approximately (80 yr)(*π*/2*π*) = 40 yr (which matches the initial increases seen around 2050). In comparison, the nonlinear evolution of the “early onset” time series also has a phase of approximately 40 yr; however, because the overall trend is negative, the phase is shifted forward by 40 yr, giving an overall phase of 0 yr; that is, the initial decrease in the overall time series begins at the start of the simulation.

Based upon the gridpoint phase estimates (Figures 3c,d), projected decreases in soil moisture over the Ohio River basin and southwestern United States do not occur until around 2050 under the A2 scenario. In contrast, the phases of the nonlinear changes in the control simulation tend to be spread across time. The one exception is in the northwestern United States where the control simulation indicates relatively early changes in soil moisture, as discussed previously (keeping in mind that a phase of 2090 as presented here is nearly the same as a phase of 2010). Again, this result highlights the need to examine control simulations as well as climate change simulations when determining the nature of the nonlinear evolutions found in a given model system.

To see how the evolutions of soil moisture in the southwestern United States and across the Ohio River basin compare with the hypothetical time series described in Figure 1, we plot the evolution of the area-average ensemble-mean soil moisture variations in these two regions, as produced by the A2 simulations (Figure 4). We also plot the area-average soil moisture evolution for the two regions but with the linear trend in each evolution removed. Finally, we plot the ensemble-mean, area-average power spectrum of soil moisture in each region, using data taken from the A2 and control simulations; in addition, a first-order Markov model is used to estimate the power spectrum expected by stochastic, red noise processes. For these last two plots, we determine the estimated spatial degrees of freedom for each region using the eigenvalues from an EOF decomposition of the original yearly soil moisture data from that region, as found in the control simulation. This analysis indicates that in each region there are approximately 7–8 spatial degrees of freedom. Hence we create eight independent sets of random yearly time series for each region. For consistency, when computing the 90% confidence interval on the mean of the gridpoint power spectrum values for the A2 and control model simulations, we also assume 8 degrees of freedom for each region, not the 50–80 that would be present if each grid point were assumed to vary independently.

As expected, the evolutions of soil moisture variations in the two regions are not quasi-linear functions of the forcing/response. Over the western United States (e.g., the interior southwest), soil moisture maintains a relatively constant value until 2050. After that, soil moisture begins to decline rapidly toward its final value, in agreement with the phase of the gridpoint values seen in Figure 3c. For the eastern United States (e.g., the Ohio River basin), it appears that soil moisture may increase during the initial period of the simulations then decrease toward the final value at the end of the simulation.

The long-term nonlinear evolution in each region becomes more apparent when the long-term trends in each ensemble member are first removed, highlighting the large and significant positive initial deviations from linear behavior in both regions. As with the artificial time series behavior, when this initial positive nonlinear behavior is combined with the negative long-term trend, the overall evolution produces a delayed response in both regions, as seen in Figures 4a,d. Finally, results from the power spectrum analysis indicate that the lowest-order modes of soil moisture variability in both regions in the A2 simulations are significantly different from that expected by chance; in addition, they are significantly different from those found in the control simulations.

Overall, in both regions we find that there is a delay in the response of soil moisture to increasing global temperatures, producing a nonlinear evolution of soil moisture in response to anthropogenic forcing. However, the reason for this behavior differs between the two regions. For instance, in the western United States, the soil moisture evolution follows a strongly nonlinear evolution in precipitation, with initial increases in precipitation over the first 20–30 yr, followed by the onset of significant decreases later in the simulation (not shown). Over the eastern United States, the nonlinear evolution in soil moisture is not related directly to changes in precipitation, which tend to increase quasi-linearly as the simulation progresses (not shown). Instead, soil moisture variations are related to small but important changes in the net water cycle, driven by changing balances between precipitation, evaporation, and runoff. Initial increases in precipitation drive the initial increase in soil moisture seen in Figure 4d; however, as evaporation continues to increase and the precipitation trend slackens, net soil moisture fluxes become negative, leading to a drawdown of soil moisture during the latter parts of the simulation (not shown).

We do not argue that these projected variations in soil moisture content and moisture-flux terms are predicted to occur over the coming century; these results are based upon a small number of ensemble runs from one model system and each needs to be tested for reliability/likelihood using other modeling systems (e.g., Tebaldi and Knutti 2007). Instead, we simply highlight that nonlinear type behavior can be a prevalent feature of the climate system and is important to characterize when considering possible short-term and long-term climate responses to anthropogenic forcing (e.g., Blenkinsop and Fowler 2007).

## 5. Conclusions and discussion

Here we have developed one method for identifying systematic long-term, nonlinear evolutions within projections of climate parameters. By examining the power contained within the lowest-order mode of variability of the gridpoint time series, and comparing the power in this mode of variability with that found in both control simulations and stochastic, red noise processes, it is possible to determine whether the nonlinear evolution within model projections under various climate change scenarios is significantly different from that expected by chance. In addition, it is possible to determine whether the evolution is significantly different from that associated with “natural” variability within the model system, even given no forced climate change. In this manner, it is possible to identify regions in which nonlinear behavior of a given climate parameter may be expected, as well as the phase (or timing) of the onset of the nonlinear behavior.

Here we have applied this methodology to projections of soil moisture evolutions for the United States, taken from the CCSM3 A2 and control simulations. Results indicate that across the Ohio River basin and the southwestern United States there are projected delays in the long-term response of soil moisture to anthropogenic forcing. This long-term response, which is associated with overall decreases in soil moisture content, does not begin to occur until 2050, when CO_{2} concentrations have reached approximately 550 ppm and global temperatures have increased by approximately 1°C (in this model system). The hydrologic balances leading to this nonlinear evolution differ across the United States, indicating that nonlinear regional responses to climate change forcing need to be studied systematically and independently (in other words, there is no “smoking gun” responsible for these variations).

At the same time, these results suggest that adaptation and planning strategies developed in response to near-term changes (over the coming decades) may need to differ from those developed in response to long-term changes (over the coming century). In addition, these results suggest that large decreases in soil moisture can be avoided if mitigation strategies are employed that prevent the CO_{2} concentrations and/or global temperatures from reaching this threshold. To arrive at reliable (i.e., likelihood) forecasts of these projected changes will require similar studies of alternative model systems, which are currently underway. We also highlight that, while we applied this method to projections of soil moisture over the United States, it can also be applied to other climate parameters, regions, model systems, and climate change scenarios—doing so will allow one to arrive at estimates of nonlinear variations for climate parameters in other regions of the world.

## Acknowledgments

Dr. Anderson’s research was supported by a Visiting Scientist appointment to the Grantham Institute for Climate Change, administered by Imperial College of Science, Technology, and Medicine. We also acknowledge the modeling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the WCRP’s Working Group on Coupled Modeling (WGCM) for their roles in making available the WCRP CMIP3 multimodel dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy.

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Projected changes in total soil moisture content under the CCSM3 A2 emissions scenario. Changes calculated as the difference between the ensemble-mean soil moisture content averaged from 2080 to 2099 and from 2000 to 2019. Values presented as a fraction of the interannual standard deviation of the 20-yr running mean gridpoint values for the full period (2000–99).

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

Projected changes in total soil moisture content under the CCSM3 A2 emissions scenario. Changes calculated as the difference between the ensemble-mean soil moisture content averaged from 2080 to 2099 and from 2000 to 2019. Values presented as a fraction of the interannual standard deviation of the 20-yr running mean gridpoint values for the full period (2000–99).

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

Projected changes in total soil moisture content under the CCSM3 A2 emissions scenario. Changes calculated as the difference between the ensemble-mean soil moisture content averaged from 2080 to 2099 and from 2000 to 2019. Values presented as a fraction of the interannual standard deviation of the 20-yr running mean gridpoint values for the full period (2000–99).

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

The fractional amount of (detrended) variance in soil moisture contained within the lowest-order mode of variability, as found in the (a) A2 emissions scenario simulations and (b) control simulations. Positive (negative) values represent grid points in which the lowest-order mode of variability produces positive (negative) initial deviations from quasi-linear behavior. Lowest gridpoint value shown is ±0.31, which represents the 90% confidence limit based upon 10 000 realizations of a stochastic model for a red noise process generated by a first-order Markov model with *r* = 0.50 lag-1 autocorrelation. In addition, only gridpoint values in which three of the four ensemble members have the same sign for the lowest-order mode of variability are plotted here. Dots indicate grid points in which the sign of lowest-order mode of variability is the same as the long-term trend in soil moisture. Boxes indicate regions of further investigation. Also shown is the phase of the lowest-order mode of soil moisture variability, as found in the (c) A2 emissions scenario simulations and (d) control simulations. Values plotted with respect to year of initial increase in soil moisture. For regions that experience negative long-term trends in soil moisture, values are shifted forward by 40 yr to represent year of initial decrease in soil moisture. See text for details.

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

The fractional amount of (detrended) variance in soil moisture contained within the lowest-order mode of variability, as found in the (a) A2 emissions scenario simulations and (b) control simulations. Positive (negative) values represent grid points in which the lowest-order mode of variability produces positive (negative) initial deviations from quasi-linear behavior. Lowest gridpoint value shown is ±0.31, which represents the 90% confidence limit based upon 10 000 realizations of a stochastic model for a red noise process generated by a first-order Markov model with *r* = 0.50 lag-1 autocorrelation. In addition, only gridpoint values in which three of the four ensemble members have the same sign for the lowest-order mode of variability are plotted here. Dots indicate grid points in which the sign of lowest-order mode of variability is the same as the long-term trend in soil moisture. Boxes indicate regions of further investigation. Also shown is the phase of the lowest-order mode of soil moisture variability, as found in the (c) A2 emissions scenario simulations and (d) control simulations. Values plotted with respect to year of initial increase in soil moisture. For regions that experience negative long-term trends in soil moisture, values are shifted forward by 40 yr to represent year of initial decrease in soil moisture. See text for details.

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

The fractional amount of (detrended) variance in soil moisture contained within the lowest-order mode of variability, as found in the (a) A2 emissions scenario simulations and (b) control simulations. Positive (negative) values represent grid points in which the lowest-order mode of variability produces positive (negative) initial deviations from quasi-linear behavior. Lowest gridpoint value shown is ±0.31, which represents the 90% confidence limit based upon 10 000 realizations of a stochastic model for a red noise process generated by a first-order Markov model with *r* = 0.50 lag-1 autocorrelation. In addition, only gridpoint values in which three of the four ensemble members have the same sign for the lowest-order mode of variability are plotted here. Dots indicate grid points in which the sign of lowest-order mode of variability is the same as the long-term trend in soil moisture. Boxes indicate regions of further investigation. Also shown is the phase of the lowest-order mode of soil moisture variability, as found in the (c) A2 emissions scenario simulations and (d) control simulations. Values plotted with respect to year of initial increase in soil moisture. For regions that experience negative long-term trends in soil moisture, values are shifted forward by 40 yr to represent year of initial decrease in soil moisture. See text for details.

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

Changes in area-average soil moisture for (a) the western United States and (d) the eastern United States. Only grid points with significant long-term nonlinear evolutions are included—see Figure 3 for area-average regions and significant grid points. Shown here are 20-yr running means based upon the A2 emissions scenarios, with solid lines indicating the ensemble-mean evolution and the dashed lines indicating the 90% confidence interval for the mean, based upon a two-tailed *t* test with 4 ensemble members. Solid black line indicates the linear trend connecting the first 20-yr period and last 20-yr period. (b), (e) Same as (a) and (d) except for detrended 20-yr running means calculated such that the linear trend connecting the first 20-yr period and last 20-yr period from each ensemble member has been removed. Solid lines indicate the ensemble-mean evolution and the dashed lines indicate the 90% confidence interval for the mean, based upon a two-tailed *t* test with 4 ensemble members. (c), (f) The area-average power spectrum for the detrended time series at each grid point within the given region, taken from the A2 emissions scenario (blue) and control scenario (red). Dumbbell shows the 90% confidence interval for the area-average power spectrum, based upon two-tailed *t* test with 8 DOFs; confidence interval only shown for the lowest-order mode of variability. The black line gives the 90% confidence level estimated from 10 000 realizations of a first-order Markov model with *r* = 0.50 lag-1 autocorrelation and with 8 DOFs. See text for details.

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

Changes in area-average soil moisture for (a) the western United States and (d) the eastern United States. Only grid points with significant long-term nonlinear evolutions are included—see Figure 3 for area-average regions and significant grid points. Shown here are 20-yr running means based upon the A2 emissions scenarios, with solid lines indicating the ensemble-mean evolution and the dashed lines indicating the 90% confidence interval for the mean, based upon a two-tailed *t* test with 4 ensemble members. Solid black line indicates the linear trend connecting the first 20-yr period and last 20-yr period. (b), (e) Same as (a) and (d) except for detrended 20-yr running means calculated such that the linear trend connecting the first 20-yr period and last 20-yr period from each ensemble member has been removed. Solid lines indicate the ensemble-mean evolution and the dashed lines indicate the 90% confidence interval for the mean, based upon a two-tailed *t* test with 4 ensemble members. (c), (f) The area-average power spectrum for the detrended time series at each grid point within the given region, taken from the A2 emissions scenario (blue) and control scenario (red). Dumbbell shows the 90% confidence interval for the area-average power spectrum, based upon two-tailed *t* test with 8 DOFs; confidence interval only shown for the lowest-order mode of variability. The black line gives the 90% confidence level estimated from 10 000 realizations of a first-order Markov model with *r* = 0.50 lag-1 autocorrelation and with 8 DOFs. See text for details.

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1

Changes in area-average soil moisture for (a) the western United States and (d) the eastern United States. Only grid points with significant long-term nonlinear evolutions are included—see Figure 3 for area-average regions and significant grid points. Shown here are 20-yr running means based upon the A2 emissions scenarios, with solid lines indicating the ensemble-mean evolution and the dashed lines indicating the 90% confidence interval for the mean, based upon a two-tailed *t* test with 4 ensemble members. Solid black line indicates the linear trend connecting the first 20-yr period and last 20-yr period. (b), (e) Same as (a) and (d) except for detrended 20-yr running means calculated such that the linear trend connecting the first 20-yr period and last 20-yr period from each ensemble member has been removed. Solid lines indicate the ensemble-mean evolution and the dashed lines indicate the 90% confidence interval for the mean, based upon a two-tailed *t* test with 4 ensemble members. (c), (f) The area-average power spectrum for the detrended time series at each grid point within the given region, taken from the A2 emissions scenario (blue) and control scenario (red). Dumbbell shows the 90% confidence interval for the area-average power spectrum, based upon two-tailed *t* test with 8 DOFs; confidence interval only shown for the lowest-order mode of variability. The black line gives the 90% confidence level estimated from 10 000 realizations of a first-order Markov model with *r* = 0.50 lag-1 autocorrelation and with 8 DOFs. See text for details.

Citation: Earth Interactions 13, 1; 10.1175/2008EI269.1