1. Introduction
In this study, we present new insights on large-scale perturbations to the aerial hydrological cycle due to quasi-equilibrium CO2 doubling by employing a novel Lagrangian framework that exploits data from numerical water tracers (WTs) implemented in a global climate model (GCM). This framework offers an alternative perspective to existing theories of hydrological cycle change, which are primarily Eulerian and focus on the local energetic (i.e., radiative and diabatic) and dynamic (moisture flux convergence) processes that drive hydrological cycle perturbations. We begin by reviewing evidence for changes in the hydrological cycle due to CO2-induced warming and discuss how these changes are understood using Eulerian theoretical frameworks. We then consider in depth how our Lagrangian framework, utilizing WTs, offers a complementary lens for viewing perturbations to the hydrological cycle due to CO2 doubling.
The hydrological cycle is, undoubtedly, of great importance to human societies and natural ecosystems. Several lines of observational evidence suggest that the hydrological cycle has shifted relative to its preindustrial state in response to anthropogenic perturbations. Ocean salinities appear to be trending upward in the tropics and downward in the high latitudes, suggesting that the climatological difference between evaporation and precipitation has increased and the hydrological cycle has intensified (Helm et al. 2010; Durack et al. 2012). Surface specific humidity has increased in response to increased surface temperatures (Willett et al. 2007), and terrestrial precipitation over the NH subtropics has declined (Zhang et al. 2007).
GCM studies predict a medley of further hydrologic changes with planetary warming. In the tropics, models predict enhanced precipitation seasonality (Chou et al. 2007) and more heavy precipitation events (Meehl et al. 2005; Hu et al. 2012; Lau et al. 2013). The subtropics are expected to dry (Allan et al. 2010) and widen (Seager et al. 2007; Seidel et al. 2008). The midlatitude storm tracks are expected to shift poleward (Hall et al. 1994; Yin 2005; Bengtsson et al. 2006; Chang et al. 2012) and change in intensity (O’Gorman 2010; Chang et al. 2012), the characteristic length scale of atmospheric eddies is expected to increase (Kidston et al. 2010; Rivière 2011), and moisture transport into the polar regions is expected to increase (Hwang and Frierson 2010). In the high latitudes, precipitable water is projected to increase (Serreze et al. 2012) and the hydrological cycle to intensify (Bengtsson et al. 2011). As land areas become more arid (Dai 2012; Sherwood and Fu 2014), the relative importance of oceanic moisture sources for continental precipitation is expected to increase (Gimeno et al. 2013).
From a global perspective, there are two important principles that govern how the hydrological cycle responds to anthropogenic greenhouse gas emissions. First, tropospheric water vapor increases at a rate dictated by the Clausius–Clapeyron (C-C) equation, a 7% increase in specific humidity per °C of warming (Manabe and Wetherald 1975; Held and Soden 2006). Second, precipitation and evaporation cannot increase at the C-C rate because of energetic constraints. In particular, evaporation is limited by the surface energy budget, while precipitation is limited by the atmospheric longwave radiative cooling rate and the dry static energy (DSE) flux divergence. As a result, both precipitation and evaporation increase with temperature at a slower rate, 1% to 3% per °C of warming, which varies depending on the GCM and emissions scenario (Mitchell et al. 1987; Stephens et al. 1994; Allen and Ingram 2002).
An influential study attributing changes in the hydrological cycle to C-C scaling of atmospheric moisture with temperature is that of Held and Soden (2006). While neither P nor E scales at the C-C rate, Held and Soden (2006) showed that the change in 
Nevertheless, perturbations to the atmospheric moisture divergence, 
Others argue that closer inspection of the spatial pattern of precipitation change reveals inconsistencies with the thermodynamic argument of Held and Soden (2006). Scheff and Frierson (2012b) and Scheff and Frierson (2012a) point out that precipitation does not decrease uniformly within the subtropics in CMIP5 models; instead, the subtropics dry preferentially on their poleward flanks as the subtropical dry zones expand, suggesting that this and the accompanying poleward shift of the storm tracks and jet are a dynamic response to CO2-induced warming. Corresponding changes in precipitation are not a simple result of increased atmospheric moisture, but are rather due to poleward shifts in general circulation features that are common to most GCM predictions and that are not easily explained by a single, unifying theory (see, e.g., Vallis et al. 2014; Lorenz 2014).
Moreover, increases in E and P are limited energetically well below the C-C rate. The latent energy associated with precipitation is a source of heating, and precipitation is limited by the rate at which the atmosphere dissipates this excess energy radiatively (Mitchell et al. 1987; Stephens and Ellis 2008; Pendergrass and Hartmann 2014) and dynamically (Muller and O’Gorman 2011). The surface energy budget also constrains evaporation below the C-C rate. Increased relative humidity, increased static stability at the surface, decreased wind speed, and increased cloud albedo all act to constrain evaporation increases well below the C-C rate (Boer 1993; Richter and Xie 2008; Lorenz et al. 2010).
In this study, we use numerical WTs implemented in a state-of-the-art GCM to present a Lagrangian schema of hydrological cycle change complementary to existing paradigms, namely C-C scaling of moisture and energetic constraints on E or P. So far, only one other study has used numerical WTs to study changes in the aerial hydrological cycle caused by increased CO2: Bosilovich et al. (2005) showed that the cycling rate of water in the atmosphere decreased, and the residence time scale increased. Such changes have also been predicted by other studies, which have shown that increasing atmospheric moisture at a higher rate than evaporation or precipitation requires that atmospheric moisture residence times must increase (see Trenberth 1998; Held and Soden 2006; van der Ent and Savenije 2011). Here, we also examine the implications of this increased atmospheric moisture residence time.
We find that, from a Lagrangian perspective, the changes in precipitation due to changes in evaporation (local and remote) are uniformly positive and have little latitudinal structure, while changes in precipitation due to changes in transport contain much of the spatial structure of the total precipitation change. Furthermore, we find that changes in the local contribution to the precipitation are small, while changes in the remote contribution are much larger. We show that many of these changes can be understood in terms of a decreasing moisture depletion tendency, which decreases precipitation efficiency (thereby decreasing the tendency of atmospheric moisture to precipitate) and increases the atmospheric moisture residence time and the moisture transport length scale, the distance between moisture source and sink regions.
This paper is structured as follows. In section 2, we review the mathematical development presented in Part I and introduce notation for applying it to perturbation studies. In section 3, we describe the setup of our GCM WT experiments, and in section 4, we analyze the results of these experiments using the mathematics developed earlier. In section 6, we analyze several heuristic models of atmospheric moisture transport to support the results of our GCM experiments. We discuss our results, consider further implications of our work, and offer some concluding remarks in section 7.
2. Overview of mathematical framework and perturbation methods
In Part I of this paper (Singh et al. 2016a, hereafter Part I), we developed a linear algebra framework for analyzing the output from experiments employing numerical WTs. We review this framework here, and introduce some additional notation for analyzing perturbations to the mean state, which we will use in our study of the hydrological cycle response to CO2 doubling.
In the ensuing perturbation analysis, we will decompose Eq. (2) into mean state and perturbation quantities to compute how changes in the local evaporation, the divergence of locally evaporated moisture, and the convergence of remotely evaporated moisture all contribute to changes in the precipitation. In terms of notation, we will use the 
3. Model experiments
The control experiment (piC, for preindustrial control), as described in Part I, was run using the fully coupled Community Earth System Model 1.0 (CESM1) with all model components at 1° spatial resolution and with all model parameters set at preindustrial levels. Aerial water was tagged with its region of origin in 10° latitude bands over each ocean basin. Each continent was tagged separately, with Eurasia and North America subdivided into two parts each. There are 49 distinct tagged regions in total, encompassing the entire globe.
For Part II, a second experiment was branched from the preindustrial control simulation in which atmospheric CO2 was doubled from its preindustrial concentration of approximately 290 ppm to 580 ppm. All other atmospheric variables, including concentrations of trace gases, aerosol forcing, and total solar irradiance, were held at preindustrial levels. The simulation was allowed to run without WTs for 270 years to approach a quasi-equilibrium state. For a final 30 years, WTs with the same spatial configuration as for piC were introduced. We refer to the final 30 years of this run, years 270 to 300, as the equilibrium CO2-doubling experiment (Eqm2×CO2). The net TOA energetic imbalance in Eqm2×CO2 is less than 0.1 W m−2. Climatologies are created from 30 years of model output with WTs. All perturbation quantities are given as the difference between the CO2-doubling experiment and the control, Eqm2×CO2–piC.
4. Aerial hydrological cycle perturbations resulting from evaporation changes versus transport changes
a. Decomposing the change in P
Perturbation analysis of the contribution of 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
The change in precipitation, 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
The key features of this decomposition can be summarized as follows:
The change in precipitation due to changes in evaporation is always positive, and is relatively spatially homogeneous in the annual mean. This can only be true if the change in evaporation is (mostly) positive, which it is (see section 5).
The change in precipitation due to changes in the transport (both changes in the convergence of remotely evaporated moisture and the divergence of locally evaporated moisture) is spatially inhomogeneous compared to the change due to increasing evaporation. Broadly, this term is positive over the equatorial Pacific, negative over the subtropics and portions of the midlatitudes, and moderately positive over the high latitudes.
Over the tropics and subtropics, the change in the precipitation due to the transport term is greater than that due to the evaporation term.
In the annual mean, increased precipitation in the middle and high latitudes is due to increases in both the transport and evaporation terms. Intensified precipitation in the winter hemisphere, however, is dominated by the evaporation term, which is more seasonally variable in the middle and high latitudes.
We continue considering why moisture transport changes with CO2 doubling in section 5. First, however, we will show how the change in the difference between precipitation and evaporation, 
b. Decomposing the change in 
This decomposition (shown in Fig. 3) reveals that both components of 
Zonal mean change in the precipitation minus evaporation, 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
It is important to note that the perturbation matrix operator framework presented here is Lagrangian, and is not equivalent to the traditional Eulerian framework where the moisture convergence term 
5. Changes in locally sourced precipitation versus changes in remotely converged precipitation
Perturbation analysis of changes in the aerial hydrological cycle with CO2 doubling, with the total change in precipitation decomposed as in Eq. (7). Shown are (a)–(c) the total change in precipitation (for comparison purposes), together with (d)–(f) the (local) change in evaporation, (g)–(i) the change in the divergence of locally evaporated moisture, and (j)–(l) the change in the convergence of remotely evaporated moisture. All quantities are shown (left) in the annual average and averaged over (middle) DJF and (right) JJA; all are in units of mm day−1.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
Zonal mean over ocean regions of the annually averaged change in precipitation with CO2 doubling and its perturbation decomposition as in Fig. 4.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
The change in evaporation is positive nearly everywhere (Figs. 4d–f), while the change in the local divergence 
While 
Over the middle and high latitudes, the remote contribution increases both annually and seasonally, and increases slightly in the respective winter hemisphere, compared to the annual mean. This increased convergence of remotely evaporated moisture into the middle and high latitudes is consistent with increased poleward transport of latent heat in a warmer climate (Held and Soden 2006; Hwang and Frierson 2010).
a. The change in the local contribution to the precipitation
The change in the local contribution to the precipitation, equal to the sum of the first two terms on the RHS of Eq. (7), is substantially smaller in magnitude than the change in the remote contribution (see Fig. 6; also cf. Figs. 7a–c and 4j–l). On average, the change in the local contribution makes up less than 20% of the total precipitation change in each tagged region.
The change in precipitation due to the change in the local contribution to the precipitation 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
(a)–(c) The change in the local precipitation contribution 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
Generally, the local contribution decreases in the tropics and subtropics over both land and ocean, and increases in the middle and high latitudes (Figs. 7a–c). There are, however, many exceptions. Over the oceans, the local contribution decreases over the Atlantic from 60°N to 40°S in the annual mean; on the other hand, the local contribution increases over the equatorial Pacific (0°–10°S) and Indian (0°–10°N) basins. For land regions, the local contribution decreases over South America, Africa, southern Eurasia, and southern North America, while it increases over the remaining land areas.
Zonal mean over ocean regions of the annually averaged change in the local contribution to the precipitation and its perturbation decomposition as in Fig. 7.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
The change in the export fraction, 
The change in the export fraction 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
b. The change in the remote contribution to the precipitation
In Fig. 10, we show the relative change in the local contribution to total precipitation, 
(a)–(c) The change in the fraction of the total precipitation originating remotely 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
Decomposition of the remote moisture convergence term, 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
Zonal mean over ocean regions of the annually averaged change in the remote moisture convergence and its perturbation decomposition as in Fig. 11.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
Changes in the remote moisture convergence due to changes in the partitioning of moisture between remote regions, 
Another decomposition of the 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
Zonal mean over ocean regions of the annually averaged change in the ocean-to-ocean remote moisture convergence as in Fig. 13.
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
6. Understanding moisture transport changes in Eqm2×CO2 using heuristic models
Several lines of evidence point to an increased distance between moisture source and sink regions in Eqm2×CO2 relative to piC. First, the fraction of locally evaporated moisture that is exported increases globally (Fig. 9), and the fraction of the precipitation due to remotely sourced moisture increases nearly globally (Fig. 10). Second, the convergence of remotely evaporated moisture shifts toward longer distances between source (evaporation) and sink (precipitation) regions: interbasin moisture convergence increases everywhere (Figs. 13b and 14) and the part of the intrabasin moisture convergence that originates as evaporation from an adjacent latitude band decreases almost everywhere (Figs. 13c and 14), while the part that originates as evaporation from more distant latitude bands increases (Figs. 13d and 14).
The global extent of these changes suggests that a simple, robust mechanism is responsible. In the ensuing analysis, we show that this increase in the length scale of moisture transport corresponds to an increase in the moisture residence time, which arises from a decline in the tendency of atmospheric moisture to precipitate.
a. Adjustment time scales in a one-box model
If Q increased at the C-C rate with a fixed moisture depletion tendency coefficient γ, then the precipitation, 
b. The length scale in a spatially continuous model
7. Discussion and conclusions
In this study, we have used a Lagrangian matrix operator framework to consider perturbations in the aerial hydrological cycle due to quasi-equilibrium CO2 doubling. This work presents a different perspective, complementary to Eulerian frameworks, for evaluating hydrological cycle changes by utilizing the information made available by numerical WTs implemented in a GCM. Our analysis synthesizes the two governing principles of CO2-induced hydrological cycle change: that atmospheric moisture increases at the C-C rate with temperature (approximately 7% °C−1), and that energetic constraints on E and P cap their increase well below the C-C rate. Both of these principles are linked by the moisture depletion tendency γ (Trenberth 1998; Bosilovich et al. 2005), which must decline if both are to be satisfied simultaneously. A decline in the moisture depletion tendency γ corresponds to an increase in moisture residence time and, as we have shown, an increase in the length scale of moisture transport.
If E, P, and specific humidity q all increased at the same rate with increasing temperature, and the circulation remained relatively fixed, γ would remain constant and much of the change in precipitation would arise from the evaporation change itself, that is, 
A corollary of this finding is that dynamic shifts in general circulation features, particularly the expansion of the subsiding regions of the Hadley circulation and the poleward shift in the storm track that accompanies it, are expected with this increase in the advective length scale of moisture transport. Widening of the subtropics corresponds to a larger source region from which water can evaporate along the way to the deep tropics, which is consistent with an increased transport length scale. Additionally, increasing the transport length scale will also push the midlatitude precipitation maximum poleward, since water from subtropical source regions will travel further. In other words, increasing the advective length scale of moisture transport produces hydrological cycle perturbations with CO2-induced warming that are consistent with the dynamical perturbations observed in a wide range of GCMs.
As our results show, increasing evaporation without changing moisture transport results in a nearly uniform pattern of increased precipitation, with a notable maximum in the middle and high latitudes in winter. In fact, much of the increased precipitation in the extratropics in Eqm2×CO2 is due to increased evaporation in the subtropics and midlatitudes that is transported poleward in a manner similar to that in the mean state. In the tropics, however, changes in evaporation cannot explain the pattern of strengthening equatorial precipitation and decreasing subtropical precipitation. In this regard, changes in moisture transport are essential for creating the distinct spatial pattern of precipitation change associated with CO2-induced warming.
Given that the advective length scale of moisture transport is 
The change in precipitation due to 
The local and remote contributions to the precipitation perturbation are each impacted differently by this advective length scale increase. Surprisingly, we find that the local contribution to the precipitation changes very little in the Eqm2×CO2 experiment relative to piC, despite the fact that the export fraction increases almost everywhere. This can be explained, however, by a compensating increase in the local evaporation: even though more locally evaporated moisture is transported away from where it evaporates as the advective length scale of moisture transport increases, local evaporation increases sufficiently to counteract this. As a result, the change in the local contribution makes up less than 20% of the total precipitation perturbation. This leaves the change in the convergence of remotely evaporated moisture (the remote contribution) as the dominant component of the total precipitation perturbation.
These changes in the local and remote contributions to the precipitation are not uniform among the basins. Overall, the Atlantic basin dries the most 
Over the continents, the relative contribution of the local and remote portions of the precipitation perturbation varies by region. Mostly, the local contribution to the precipitation declines, with the exception of the high latitudes, where the local contribution increases modestly. The remote contribution, on the other hand, increases over all land regions primarily because ocean-to-land moisture convergence increases. In general, the sum of these two competing tendencies determines whether precipitation increases or decreases over a given landmass. Over the polar regions, where both the local and remote contributions increase, precipitation always increases. Over most other landmasses, the increase in the remote contribution wins out over the declining local contribution, and precipitation increases year-round. Exceptions are over South America, and, seasonally, over southern North America and Australia, where the declining local contribution is not sufficiently compensated for by the increasing remote contribution, and precipitation declines.
Our findings have important implications for isotope studies. Our GCM WT experiments confirm that the advective length scale of moisture transport changes with temperature. In warmer temperatures, as in the Eqm2×CO2 experiment, the advective length scale increases relative to piC; similarly, in cooler temperatures, the advective length scale is expected to decrease. From the perspective of moisture from the subtropics that is transported poleward, both the temperature at which it evaporated and the time spent in the aerial hydrological cycle impact the isotopic signature; both of these components, however, depend on temperature in both a local and global sense. This suggests, then, that back-of-the-envelope calculations of state variables from isotope data are ill advised, as these secondary effects of temperature and its impact on transport may not be easily distilled into a simple model. Thus, we contend that isotope-enabled GCM studies are essential for interpreting isotope proxy records, particularly those in which global temperatures are very different from that in the present climate.
We conclude by pointing out the major caveat associated with this and any study using numerical WTs in GCMs, namely that the findings we present here are limited by the ability of our GCM to model the aerial hydrological cycle with fidelity. Thus, the conclusions presented here should be verified with additional GCM studies, including those that are isotope-enabled.
Acknowledgments
HKAS thanks colleagues Dennis Hartmann and Chris Bretherton for feedback; and acknowledges support from the U.S. DOE CSGF. CMB acknowledges support from the National Science Foundation (NSF) through Grant PLR-1342497. All authors acknowledge high-performance computing support from Yellowstone (ark:/85065/d72d3xhc) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the NSF.
APPENDIX
Time Scales and Length Scales in an n-Box Model
Schematic of the n-box model used to study moisture transport length scales. The length of each box is L and the moisture content is 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
In Fig. A2, we present some distributions of 
The value of 
Citation: Journal of Climate 29, 18; 10.1175/JCLI-D-16-0293.1
Suppose that the evaporation is allowed to vary between boxes such that 
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