Introduction

Societies are vulnerable to precipitation change as precipitation variability has a significant impact on numerous socioeconomic outcomes and human well-being1,2,3,4,5. The pattern and magnitude of precipitation are expected to change with global warming as the concentration of atmospheric water vapor is expected to increase6,7,8. Although the ‘wet-get-wetter, dry-get-drier’ (WGWDGD) paradigm6 applies in many locations, the changes vary by region9, due to various interacting drivers of precipitation changes10,11. Owing to the complexity, and heterogeneity of precipitation change with global warming, building a structural model to represent the underlying physical process of precipitation is necessary to predict the future precipitation change, where contributing factors are parceled out and the mechanism of their interactions is identified.

In previous studies, the atmospheric moisture budget simplifies the analysis of precipitation into the quantification of distinct physical processes9,12,13,14. Standard components in the moisture budget include the dynamic and thermodynamic components and horizontal moisture advection. The dynamic process incorporates contributions from the atmospheric circulation where it is embedded12,15,16,17,18, while the thermodynamic process traces changes in atmospheric water vapor content, which are linked to temperature changes6,19. Horizontal moisture advection plays a role as a moisture source for precipitation20,21.

Studies that associate precipitation change with these individual processes focused on the relative importance of one process over the others11,12,13,17,22 while neglecting possible interplay among them. However, increased water vapor reduces the magnitude of the required vertical motion to generate the same strength of precipitation, which, with the same magnitude of vertical motion, then leads to stronger precipitation23,24, suggesting a significant interplay among moisture budget processes in the precipitation process. While the time-averaged moisture budget analysis does not reveal the interactions among the processes and their dynamic evolution, tracing the propagation of a perturbation from a component through the atmospheric moisture system on a daily or sub-daily basis is essential for a complete explanation of the atmospheric moisture dynamics, and the future precipitation changes.

Results

We build on the established framework of exploring the moisture budget balance with the precipitation (P) expressed as a linear function of the four processes: the vertically-integrated vertical motion or “the dynamic process” (DY), the vertically-integrated moisture profile or “the thermodynamic process” (TH), the vertically-integrated horizontal moisture advection (HA), and evaporation (E). We estimate each process by using daily outputs from 10 coupled ocean-atmosphere general circulation models participating in the Fifth Phase of the Coupled Models of Inter-comparison Projects (CMIP5) for the pre-industrial (PI) and Representative Concentration Pathway 4.5 (RCP4.5) simulations for the present (2006–2025) and the future (2081–2100) (See Methods). The dynamic process is associated with vertical velocity with fixed specific humidity, while the thermodynamic process is related to constant vertical velocity with the atmospheric moisture content associated with ambient temperatures. The fixed specific humidity and vertical velocities are defined from the climatological averages associated with their specific climate; the twenty-year averages for the pre-industrial, and 2006–2100 for the current and future climates (See Methods).

Specifically, we construct a dynamically interactive model of atmospheric moisture where changes in a moisture budget component can have an instantaneous impact on other components and propagate through the atmospheric moisture system. To do so, we extend the conventional moisture budget balance such that the estimation of the system of interactions of the five processes (P, DY, TH, HA, E) is performed by applying the structural vector autoregression (SVAR) where all five processes are regressed on the time-lagged values of itself and all other processes and their time-lagged values (See Methods). The physical links among the processes, i.e. how each process influences one another over time, are specified using the Choleski decomposition. Based on the estimation of the SVAR, we identify the propagation of a perturbation or shock in a process (impulse variable) through the moisture budget system to other processes (response variable) over time by computing the impulse-response function (IRF).

Before we present our main analysis on IRF, we examine the time-averaged hydrological pattern and moisture budget components for the ten model average (Fig. 1). Shown in Fig. 1A1, B1 are the precipitation and evaporation, respectively. The difference of P - E in Fig. 1C1 clearly identifies the moisture convergence with P>E in the Intertropical Convergence Zones (ITCZ), and regions of Hadley Cell subsidence over the subtropical oceans with E>P. The horizontal moisture advection (Fig. 1F1) moves moisture largely poleward away from lower latitudes. As the climate warms by 1.2 °C by 2025, and a little more than 3 °C by 210025, the precipitation in the present (Fig. 1A2) and future (Fig. 1A3) increases by 6.7% and 13.3% compared to the PI, respectively.

Fig. 1: The mean of moisture budget components in the pre-industrial, the present (2006–2025) and the future (2081–2100) over the globe.
figure 1

The panels in the leftmost column (A1–F1) show the moisture budget and its components averaged over the pre-industrial, denoted by the subscript 0. The middle column (A2-F2) shows the same components for the present, denoted by the subscript 1 and difference in P, E and P-E from the PI. The rightmost column (A3–F3) shows the same components as in the middle column for the future, denoted by the subscript 2 and difference in P, E and P-E from the PI. The scale bar at the bottom applies to all the panels, except the panels (A2, A3, B2, B3, C2, C3) use the scale bar in the middle of the figure. Units are mm d−1.

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The global patterns and magnitude of the time-averaged shares of the dynamic (Fig. 1D1) and the thermodynamic (Fig. 1E1) components are identical when averaged, since the twenty-year time period over which the terms (ω, ∂q/∂p) where ω is the vertical velocity, q is specific humidity, and p is the atmospheric pressure, are being averaged is also used for the climatological average of the PI \((\overline{\omega },\overline{\partial q/\partial p})\) (See Methods). Conceptually, it shows that upward vertical velocity and high moisture content combine to produce the regions of precipitation. The precipitation change in the present (Fig. 1A2) and future (Fig. 1A3) continue to be provided primarily by the dynamic (Fig. 1D2, D3) and thermodynamic components (Fig. 1E2, E3), again with almost equal time-averaged share because the twenty-year averages of each of these terms are not all that different from the climatological averages being utilized (2006–2100).

According to our analysis based on the estimated IRF shown in Fig. 2, the daily changes in a precipitation event after perturbations in the dynamic, the thermodynamic, the HA, and evaporation present a sharp contrast to their time-averaged counterparts. A perturbation on the dynamic component (Fig. 2A–C) has the largest impact on the precipitation on the same day and decreases on subsequent days, implying that the role of the dynamic component is to remove atmospheric moisture, and subsequently, its impact becomes rapidly smaller over time. A perturbation on the thermodynamic component (Fig. 2D–F) has a much smaller instantaneous impact on precipitation than that of the dynamic component, and its impact lasts for two days (same day and the next day), implying that the moisture buildup continues to contribute to the precipitation a day later. The dominating impact of the dynamic process and the supporting role of the thermodynamic process on precipitation may explain the discrepancy between the observed changes in precipitation magnitude and the Clausius Clapeyron (CC) increase in moisture availability (thermodynamic component), suggesting that the CC does not control precipitation directly on a daily basis. The largest impact of the horizontal moisture advection on precipitation is delayed and occurs the next day, with minimal impact on the same day (Fig. 2G–I). This implies that the horizontally advected moisture builds up moisture in the atmosphere for the next day’s precipitation, which is consistent with findings of previous research13,26. A perturbation on evaporation has its maximum impact on the first day and its impact drops substantially from the next day onward (Fig. 2J–L).

Fig. 2: The time-evolution of the impact of the four moisture budget components on precipitation over the globe.
figure 2

Each panel shows the impulse response functions (IRF) for up to seven days since the initial perturbation in impulse variable, averaged over models and orthogonalization schemes where the impulse variables are the dynamic (A–C), the thermodynamic (D–F), horizontal advection (HA) (G–I), and evaporation (J–L) and the response variable is precipitation in all panels. The IRF tracks precipitation change over time after there is an increase in the impulse variable by the amount of its standard deviation (See equation 8 in Methods). The vertical line of the error bar passing through the point estimate (circle) represents the 95% confidence interval, which is based on the spread over models and orthogonalization schemes in the SVAR regression (See Methods). The small interval around the estimates suggest that the estimates of IRF have relatively small spread. For each impulse variable, the impact on precipitation over land and ocean are separately presented, as well as the impact over the whole globe. The IRFs are estimated for the pre-industrial (red), the present (light blue), and the future (dark blue). Units on the vertical axis are mm d−1.

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Comparing the time-averaged (Fig. 1) and the instantaneous impact (Fig. 2), the time-averaged moisture budget hides the connection and timing of the component impacts on precipitation. The dominating impact of the dynamic component on precipitation is hidden when the dynamic component is time-averaged, and daily interactions with other components are ignored. In daily interaction, a large thermodynamic component may not result in precipitation when not accompanied by condensation via vertical motion. The impact of the HA on precipitation is highest on a day after the perturbation (Fig. 2G–I), without a significant immediate impact, revealing the delayed moisture-buildup process. Therefore, the time-averaged decomposition of moisture budget components may fail to attribute changes in precipitation to each underlying component.

There are a few caveats for interpreting the results. Note that the IRF measures the impact of one-time perturbation in an impulse variable, with the magnitude of perturbation equal to the standard deviation of the variable, on itself and other components (See Methods). Hence, most of the significant impact largely disappears in a day or two (Fig. 2). If heavy precipitation continues over several days, multiple perturbations over several components can be responsible for it. Note also that although the SVAR framework does not diagnose the origin of the perturbation, the factors outside of the moisture budget equation are incorporated into the evolution of perturbations. For example, extreme precipitation is often influenced by Rossby wave dynamics27 and atmospheric rivers28,29. These external mechanisms influence precipitation through the perturbation.

The overall relative contribution of each component on precipitation can be alternatively quantified by comparing the forecast error variance decomposition (FEVD), also estimated from the SVAR (See Methods). The FEVD decomposes the variance of the forecast error into the contributions from the perturbation of specific components. Here, we compute the FEVD of perturbation from each component to precipitation. We find that the largest contribution to the variability of precipitation comes from the dynamic component, 47.6 percent, which is even greater than the precipitation’s contribution to itself, 44.4 percent. The variance of the forecast error due to the thermodynamic component is 3.5 percent of the total variance. The rest of the variance is from HA (1.2 percent) and evaporation (3.2 percent). Therefore, the FEVD results confirm that the dynamic component has a dominating influence on precipitation, with a much smaller contribution coming from the thermodynamic component.

In all three climate states, the changes in relative magnitudes of the components, and their temporal variation are similar, with all components exhibiting an increase in their impact on precipitation (Fig. 2). In particular, the dynamic component’s influence on precipitation increases for the first day; the thermodynamic, horizontal advection, and evaporation influence increase for up to the first four days. The fact that the increase in the dynamic component does not lead to a more extended period of impact on precipitation in a warmer climate is related to the almost same-day moisture removal by the dynamic component. A longer-lasting moisture supply via the thermodynamic and HA as climate warms may be associated with an extended period of precipitation when it rains. One hypothesis for this result is based on the fact that for extratropical storms, there is often a secondary rainfall event (even after the passage of the cold front) on the following day, associated with the passage of the upper-level trough and resultant instability. This rainfall event is usually light, due to lack of moisture. Our results suggest that with the additional moisture loading of the atmosphere, there may be sufficient moisture available for the instability to trigger an additional period of moderate to heavy rainfall30.

To examine the whole range of precipitation responses to perturbations in the dynamic, thermodynamic, and HA processes, presented in Fig. 3 are the probability densities of the impact of these processes on precipitation based on its frequency over the globe at the time when each process has its maximum impact on precipitation (day 0 for dynamic and thermodynamic processes and day 1 for HA). As climate warms, all three processes show an increase in the spread of probability densities, with the probability of low to medium impact reduced, and the probability of (extreme) large impact increased, implying that extreme or heavy precipitation is more likely in the future. This is in accordance with the increase in precipitation variability in the future8. Among the three components, the dynamic component’s impact on same-day precipitation has the largest increase in spread (Fig. 3A), while the thermodynamic (Fig. 3D) and HA’s (Fig. 3G) impacts on precipitation have a smaller spread in their density in the pre-industrial and a smaller increase in the spread of the density. This implies that the dynamic’s contribution to extreme heavy precipitation increases more than the other components, while its contributions to light precipitation shrink the most. In addition, for light precipitation events (the one associated with the leftmost side of the density), the warmest climate of the future (2081–2100) has the smallest probability mass, implying that light rainfall is less likely in the future. Notice that the thermodynamic component can be negatively associated with precipitation, capturing the cases where moisture build-up failed to develop into precipitation. This may occur when there is no significant weather system, frontal boundary, or topographic lifting to encourage the ascent of moist air to result in precipitation.

Fig. 3: The probability densities of the impacts of moisture budget components on precipitation.
figure 3

The probability densities based on the corresponding IRFs over the globe at a single day of the impact are presented. The impulse variables are the dynamic (A–C), the thermodynamic (D–F), and horizontal advection (HA) (G–I). The response variable is precipitation in all panels. The timings when the IRFs are computed are; day 0 (initial day) for the dynamic and the thermodynamic, and day 1 (a day after the initial perturbation) for HA, i.e., the time when the impulse variables have the strongest impact on precipitation according to Fig 2. The density estimate is smoothed by the kernel density estimation with the Epanechnikov kernel function. The probability densities are estimated for the pre-industrial (red), the present (light blue), and the future (dark blue). The vertical dotted lines of matching color with the density represent the mean of the impact for the three climate states. For each impulse variable, the probability densities are separately estimated over land and ocean. The units on the horizontal axis are mm d−1.

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While the increase in the spread of probability density for the dynamic process is similar over land (Fig. 3B) and ocean (Fig. 3C), the increase in the spread for the thermodynamic is greater over the land (Fig. 3E) than over the ocean (Fig. 3F), and vice versa for the HA (Fig. 3H, I). The primary cause for the thermodynamic difference is that land temperatures by the end of the twentieth century had warmed more than ocean counterparts31,32, and this land-sea warming contrast is expected to continue in the warmer climate33. This implies that a heavy rainfall event increases the moisture at the land surface, and thus the warmer air over land can obtain more moisture, leading to even heavier rainfall. As for horizontal advection, the greater overall availability of moisture over the ocean implies that horizontal advection will be able to provide for additional heavy rainfall there more frequently in the warmer climate.

The changes in the geographical distribution of the impacts of these processes on precipitation as climate warms are presented in Fig. 4. The increase in the spread of the impact of dynamic components on same-day precipitation as climate warms is greatest in the tropics and storm track regions, with a decreasing impact in the subtropics (Fig. 4A, B); the effect intensifies as the climate warms further (Fig. 4B). The changes in the geographical distribution of dynamic process defy the assumption of the stationarity of the upward motion for the WGWDGD paradigm; the northward movement of the dynamics influence and storm track over the North Pacific is consistent with the IPCC conclusion in the last several reports34. These results are related to the discrepancy between the P-E terms in the pre-industrial (Fig. 1C1) and changes in P -E terms in warmer climates (Fig. 1C2, C3).

Fig. 4: The differences in the impacts of the moisture budget components on precipitation based on the IRF, calculated for each respective climate.
figure 4

The impulse variables for the IRFs are the dynamic (A, B), the thermodynamic (C, D), and horizontal advection (HA) (E, F). The response variable is precipitation for all panels. ΔtIRF(x → y, s) indicates the difference of IRFs between two experiment times (t = 1 denotes difference between the present (2006-2025) and the pre-industrial and t = 2 for difference between the future (2081-2100) and the pre-industrial) at s days after one-time perturbation in impulse variable x on the response variable y. The timing of the impacts is the same as in Fig. 2. The units are mm d−1.

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The impact of the thermodynamic component on same-day precipitation for the future climate is somewhat less localized for the present climate (Fig. 4C), though it more clearly emphasizes the tropics and storm track regions with further warming (Fig. 4D) (notice the change of scale). The horizontal advection of moisture (Fig. 4E, F), the weakest influence on precipitation, emphasizes the moisture advection increase to the extratropics for the next day’s precipitation, with the effect becoming increasingly pronounced for the future (Fig. 4F).

There are noticeable differences in the geographical influence of these three components on the same day or next day precipitation (Fig. 4) compared with their effect on longer-term averaged climatological precipitation changes (Fig. 1). This emphasizes that from a structural component point of view, days of heavy rainfall have different relative contributors in a particular region than the time-averaged rainfall. The dynamic process, when time-averaged (Fig. 1D1), also produces regions of divergence and thus drying; in contrast, the same-day rainfall is focused on events of precipitation where the dynamics is most likely producing convergence. This latter response is thus a more direct result concerning the distribution of heavy precipitation.

Not only do these processes in the moisture budget equation modulate the precipitation, but they interact with one another over time. The temporal evolution of these interactions is resolved via the SVAR model, and the results are presented in Fig. 5. Considering only the largest impacts, an increase in the dynamic process of vertical motion results in precipitation that, by lowering the moisture content of the air on the following day (Fig. 5B), reduces horizontal moisture advection (Fig. 5C). The same-day precipitation and evaporation of falling moisture due to an increase in the dynamical component increases the relative humidity of the air, reducing evaporation from the surface on that day (Fig. 5D).

Fig. 5: The interplay among moisture budget components over time based on the IRF, averaged over the globe.
figure 5

Each panel shows the individual IRFs for up to seven days after the initial perturbation, averaged over models and orthogonalization schemes for the corresponding pair of impulse-response variables. Each row (column) corresponds to an impulse (response) variable such that IRF(x → y) indicates the IRF where the impulse variable is x and the response variable is y: Row-wise, for the panels (A–D), (E–H), (I–L) and (M–P), the impulse variable is DY, TH, HA, and E, respectively. Column-wise, for the first, second, third, and fourth columns from the left, the response variable is DY, TH, HA, and E, respectively. For example, panel (E) shows the impact of an increase in the thermodynamic on the dynamic over time. The vertical line of the error bar passing through the point estimate (circle) represents the 95% confidence interval based on the spread over models and orthogonalization schemes. A horizontal line in each panel indicates zero impact. The IRFs are estimated for the pre-industrial (red), the present (light blue), and the future (dark blue). The vertical axis shows the impact on the precipitation and its unit is mm d−1.

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An increase in the thermodynamic component (moisture in the air) increases the dynamic component by providing more available moisture (Fig. 5E)11. Horizontal advection of warm and moist air helps to produce upward motion, often with the passing of an upper air trough, on the day following the maximum precipitation (Fig. 5I). The small negative association between the thermodynamics and HA, maximized a day after the initial perturbation (Fig. 5G, J), suggests that an increase in the thermodynamic component may reduce HA since the thermodynamic process enhances the dynamic process on the first day, which subsequently decreases moisture in the air and thus HA. This indirect connection may be responsible for the small magnitude of the association. Any precipitation that results from horizontal advection on that following day will help increase the relative humidity at low levels, depressing evaporation from the surface (Fig. 5L).

For evaporation from the surface, an increase in this quantity generally means the relative humidity at the surface has been reduced, e.g., by drier air moving in, which would be associated with reduced dynamic instability (Fig. 5M) and reduced horizontal moisture advection (Fig. 5O); given that drier air, say associated with high-pressure systems, often lasts over a region for days, this effect can continue to dampen vertical instability, although in a decreasing manner over time (Fig. 5M).

In a warmer climate, the same general patterns of interplay among the moisture budget processes continue to hold, with some variation in magnitude. The magnitude of the impact of dynamical-induced rainfall on reducing thermodynamics (Fig. 5B) and horizontal moisture advection (Fig. 5C), by lowering the moisture content of the air the following day, increases as climate warms, suggesting more exhaustive removal of the vertically-integrated moisture during these events. This would result in heavier rainfall, and is consistent with a relatively large increase in the spread of the impact of dynamic components on precipitation (Fig. 3A). In a warmer climate, an increase in the moisture content of the air (thermodynamic component) has more of an influence on vertical instability (the dynamic component) (Fig. 5E) for subsequent days as the warming climate may already be closer to moist static instability. This moisture-induced increase in the dynamic component can subsequently lead to more precipitation as climate warms, as shown in Fig. 2A. In sum, these findings imply that in a warmer climate, extreme precipitation is more likely to be sustained by increased moisture over a longer time period while there is greater removal (higher precipitation efficiency) via the increased vertical motion of the atmosphere.

The interactions among the moisture budget components show consistent patterns over land and ocean (Figs. S1, S2), with minor differences. One noticeable difference is that the impact of the thermodynamic on the dynamic is greater over the ocean (Fig. S2E) than over the land (Fig. S1E). This implies that an increase in moisture in an atmosphere that is already conditionally unstable (over the oceans at low latitudes) will more likely be sufficient to trigger rainfall there than over land.

Discussion

The importance of the vertical motion of the atmosphere to heavy precipitation events has been verified in various other studies even for the present day climate13,35,36,37,38. Rainfall anomalies associated with the Asian monsoon seem to be associated primarily with the dynamic process, with the thermodynamic process only important in East Asia16. The dynamic effect on anomalous current events extends to winter in the extratropics13, although here the thermodynamic influence can be important18. Based on 10 models from CMIP3, a study39 found that with the more stable atmosphere associated with the warmer climate (the lapse rate becomes more moist adiabatic) the dynamic contribution tends to weaken the frequency and intensity of lighter precipitation events. This is consistent with decrease in the dynamic’s contribution to the probability of the light precipitation in the warmer climate in Fig. 3A.

There are some discrepancies with other studies, potentially due to the temporally-varying method employed here, or due to differing model outputs. Based on the Community Earth System Model Large Ensemble, at mid-to-high latitudes, the dynamic tendency for extreme events per degree of warming is small, and precipitation scales by the CC relationship via the thermodynamic process22. According to a study using the HadCM319, the CC-predicted change in extreme events was in greatest agreement at mid-latitudes, where dynamic components were small. In contrast, our results show the dominating role of the dynamic process on precipitation at a wide range of latitudes (Fig. 4A, B). With respect to our results showing that the dynamic component wields a large impact throughout the extra-tropics, especially with the warmer future climate, there is a tendency for convection to become more active in the extratropics as climate warms and specific humidity increases; the extratropical results in models may well be influenced by the particular convection schemes employed. The expected poleward shift of storm tracks40 will likely produce a dynamic influence on extreme precipitation events at higher latitudes in those models in which it occurs. Notice that neither the Community Earth System Model nor HadCM3 were among the models used in this study (See Methods).

Our statistical model allows us to investigate how these contributions act in conjunction with the temporal variation of the event itself. The contribution of the dynamic process to precipitation is most significant on the day of the precipitation, and that is amplified as climate warms (Fig. 2A). The thermodynamic effect due to the warmer climate maintains an increased influence on precipitation for a longer period of time (about four days) although, in absolute terms, it is still smaller than the dynamic contribution (Fig. 2D). The findings exhibit the sharp contrast between the time-averaged and the instantaneous contributions of the dynamic and the thermodynamic processes to precipitation, which partly explains the disparity in previous studies regarding the roles of the dynamic and the thermodynamic processes in precipitation dynamics.

Although we use daily average rainfall as the target event, the daily data may hide what happens at a sub-daily scale38. In fact, the SVAR model is capable of modeling interactions among moisture budget components for rainfall that varies at a sub-daily scale, such as convective rainfall. However, the difficulty lies in the fact that models use convective parameterizations to produce such events, often without explicit vertical velocities, and thus precise calculations of the dynamic are not possible on that scale. Nevertheless, we expect the dominating role of the dynamic process to continue38, supported by the thermodynamic and the HA on the sub-daily scale; sub-daily variability may come from changes in cloud cover and cloud dynamics which are closely related to the strength of updrafts17, and these are usually embedded in longer-lasting and larger scale circulation systems, as used here41.

We show that the interactions among the moisture budget components enhance or counter the impact of other components on precipitation (Fig. 5). Mentioned in this regard is the result that dynamically-induced rainfall lowers the moisture content of the following day more effectively in a warmer climate, suggesting that it may be associated with a more complete removal of water vapor from the atmospheric column in the models. This would then contribute to heavier rainfall events with the same dynamic forcing, which is something that should be investigated directly with observed data. The influence of the micro process and local factors such as soil moisture, cloud cover, and topographic factors on the interplay among moisture budget components is also worth further investigation.

Methods

Data source

In order to compute the moisture budget components, we extract the following seven variables from each model: precipitation flux, surface upward latent heat flux (to represent evaporation), specific humidity, eastward and northward wind, surface pressure, and omega (ω). The models are part of the coupled ocean-atmosphere general circulation models participating in the Fifth Phase of the Coupled Models of Inter-comparison Projects (CMIP5). The data is archived in the ESGF data portals at http://esgf-node.llnl.gov. We find 10 models in which all seven variables are available in the archive. The models we use include CanESM2, CMCC-CM, CNRM-CM5, FGOALS-g2, GFDL-CM3, GFDL-ESM2M, IPSL-CM5A-LR, MIROC5, MPI-ESM-MR and MRI-CGCM3. The spatial resolution of the outputs is adjusted to the baseline resolution of 2° × 2. 5° if the model is not originally gridded at the resolution. The models (CanESM2, FGOALS-g2, and IPSL-CM5A-LR) with a lower resolution have missing information in some of the grids on the baseline resolution; Fig. S3 shows the distribution of missing grids in CanESM2. The two other models of missing grids, FGOALS-g2 and IPSL-CM5A-LR, have a similar pattern of missing grids, but with more missing information. Note that the coarser resolution only implies that each grid in the model covers a larger spatial area, and thus all areas of the globe are covered by all models we use.

We use the daily outputs of the Representative Concentration Pathway (RCP) 4.5 scenario between 2006 and 2100, which includes long-term, global emissions of greenhouse gases, aerosols, and land-use-land-cover in a global economic framework (hence not just global warming constituents). We choose daily-average simulations over shorter time scales, such as hourly simulations, to avoid multiple counting of the same events which last several hours or more. Moreover, models use convective parameterizations to produce such events, often without explicit vertical velocities and thus the precise calculations of the dynamic are not possible on that scale. On the other end, a longer time scale, such as a monthly series, will hide instantaneous interactions among the moisture budget components. The RCP4.5 is a scenario that stabilizes radiative forcing at 4.5  W m−2 in the year 2100 without ever exceeding that value, providing a platform to explore the climate system response to stabilizing the anthropogenic components of radiative forcing. The RCP4.5 outputs are divided into two non-overlapping periods: the present (2006–2025) and the future (2081–2100). We also use the 20 years of the pre-industrial simulations from the same models to examine the impact of global warming.

There are a few differences in patterns of precipitation changes with previous works. For example, in their Fig. 12.22, IPCC AR534 shows an increase in precipitation over most of the Asian region in the period of 2180–2200, relative to the current period (1986–2005), especially in the central and northeast (NE) Asia, with some drying to the southeast (SE). Similar results are found in ref. 42. On the other hand, the precipitation shown in Fig. 1A2, A3 indicates decreases in precipitation over SE Asia, with increases further north, relative to the pre-industrial (Fig. 1A1); the precipitation reduction is greater in the models used in our paper than in the full set in CMIP5, and the increases to the north are smaller. The reduction in precipitation over SE Asia in the future despite the increased moisture availability due to warming is associated with the poleward extension of the subtropical zones, or northward retreat of the polar-front zones42. The difference with other studies can be due to (i) the use of RCP4.5 for its simulation of the 21st century, while the others34,42 refer to RCP8.5, and (ii) the use of the model’s pre-industrial simulation as a basis for comparison, instead of the current day model values.

Decomposition of moisture budget

The atmospheric moisture budget equation defines the equilibrium state of atmospheric moisture content. According to the equation, the moisture budget can be decomposed into distinct physical components as follows.

$$P=-[q\nabla V]-[V\nabla q]-[\partial q/\partial t]+E$$
(1)

where P is precipitation and E is evaporation, q is specific humidity, V = (u, v) is the horizontal wind vector, and ∇ = i(∂/∂x) + j(∂/∂y), i.e., changes concerning the horizontal directions. Brackets denote mass-weighted vertical integrals. The first and the second terms in equation (1) are the (vertically-integrated) vertical moisture advection (VA) and horizontal moisture advection (HA), respectively. The third term is the change in atmospheric moisture storage. The quantity of all components is expressed in the unit of millimeters per day mm d−1. We find that the storage term is an order of magnitude smaller than other moisture budget components (by the order of magnitude 10−3), which is consistent with the findings from previous studies43,44. Therefore, we ignore the moisture storage term in the analysis for simplicity.

By the mass continuity, the VA is expressed as  − [ω∂q/∂p], where ω is the vertical velocity, and p is the atmospheric pressure, i.e., the product of vertical stability of the atmosphere and vertical distribution of specific humidity. As strong vertical transport of moisture is a source of extreme precipitation events10,45, the VA is an essential determinant of precipitation distribution. Therefore, we decompose the VA to identify the individual role of vertical stability (DY or “dynamic component”) and specific humidity (TH or “thermodynamic component”).

Previous studies12,13,16,46,47 often use the linear approximation of VA term in order to examine the contribution of the dynamic and the thermodynamic processes separately. For example, a study by 16 uses the linear approximation of the moisture budget equation to examine the contributions of the dynamic and the thermodynamic components to extreme rainfall over East Asia. A work by13 applies the linearly approximated version of vertical advection of moisture to January 2014 precipitation in the southern UK to examine the physical drivers of this extreme event.

Following these studies, we linearly approximate the VA by the sum of the two terms: (i) the dynamic component which corresponds to change in vertical velocity ω, but fixing ∂q/∂p at the climatology mean and (ii) the thermodynamic component, which corresponds to change in specific humidity, with ω fixed at the climatology mean. The TH is associated with saturation vapor pressure whose change with respect to temperature tends to obey the Clausius-Clapeyron equation6,12. The remaining terms in the approximation are the product of changes in time-mean specific humidity and vertical velocity, and the covariance term of the transient eddy. Both terms are the second-order terms of linear approximation and are small relative to the first-order terms (the dynamic and the thermodynamic) as in12,16 and thus are ignored in the analysis. As a result, the linearly approximated VA term is written as follows.

$$-[\omega \partial q/\partial p]\approx -[\omega \overline{\partial q/\partial p}]-[\overline{\omega }\partial q/\partial p]=DY+TH$$
(2)

where the over-bar indicates the climatological mean for each climate; the twenty-year period for the pre-industrial, and the 2006-2100 period for both the current and the future.

Substituting equation (2) to equation (1), the precipitation is a linear function of the dynamic component, thermodynamic component, horizontal advection of moisture, and evaporation:

$$P \,\, \approx\, -[\omega \overline{\partial q/\partial p}]-[\overline{\omega }\partial q/\partial p]-[V\nabla q]+E\ =\, DY+TH+HA+E$$
(3)

Equation (3) is the building block of the dynamic system of moisture evolution and resulting precipitation12,13.

Figure 1 presents the means of each moisture budget component in the pre-industrial and their difference with the present and the future. Since some models have missing grid information (See Data Source and Fig. S3), the means based on all model outputs may generate a non-generic pattern on the map of the means where missing are present (shown in Figs. S4 and S5). Therefore, the map in Fig. 1 is adjusted to a coarser resolution to smooth out the means.

Structural Vector Autoregression (SVAR)

The dynamic system of atmospheric moisture should incorporate the key features of the moisture budget components: the physical components influence each other both instantaneously and inter-temporally. For example, horizontal moisture convergence causes a mass of air to rise and decreases the stability of the air mass. The dependency among physical components may be extended over time as the atmospheric moisture and wind convergence of the previous days can set the current states of the physical process.

To incorporate the interplay among the moisture budget components, we employ the vector autoregression (VAR) model to the moisture budget components. The standard univariate autoregression, where a variable is regressed on past values of itself, is a special case of the VAR. The VAR extends this univariate autoregression to the system of regressions where all variables are predicted by their own lagged values, all other variables, and their lagged values. This model is pioneered by Sims48 and has been applied in estimating the impact of economic shocks49,50, modeling brain network51,52, and predicting disease53,54.

For the dynamic system of five physical components of the moisture budget (HA, VA, TH, P, E) in a given grid, the reduced-form VAR estimation model is specified as follows:

$${y}_{t}=v+{B}_{1}{y}_{t-1}+....+{B}_{k}{y}_{t-k}+{\mu }_{t},{\mu }_{t} \sim N(0,{\Sigma }_{\mu })$$
(4)

where

$${y}_{t} = {(H{A}_{t},D{Y}_{t},T{H}_{t},{P}_{t},{E}_{t})}^{{\prime} }\\ {B}_{i} = \,{5} \, {{\rm{x}}} \, {5} \, {{\rm{coefficient}}} \, {{\rm{matrices}}}\,\\ v = {({v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5})}^{{\prime} }\\ {\mu }_{t} = {({\mu }_{t}^{HA},{\mu }_{t}^{DY},{\mu }_{t}^{TH},{\mu }_{t}^{P},{\mu }_{t}^{E})}^{{\prime} }\\ E({\mu }_{t}) = 0,E({\mu }_{t}{\mu }_{t}^{{\prime} })={\Sigma }_{\mu },E({\mu }_{t}{\mu }_{s}^{{\prime} })=0,\forall s \, \ne \, t$$

The maximum likelihood estimator (MLE) of the coefficients in B can be found by equation-by-equation OLS where we choose the optimal lag length k by applying the Alkaline information criteria. Although the MLE estimates are consistent and asymptotically efficient, they are biased since the reduced-form error terms are serially correlated. Therefore, in the system of lagged variables in equation (4), individual coefficients in B do not measure the marginal effect of a shock or perturbation in a component onto itself or other components. For example, a coefficient on the dynamic does not measure its sole marginal effect on precipitation when a perturbation to the dynamic component is associated with a corresponding perturbation to the thermodynamic. Hence, we need to find an identifying restriction for independent perturbations in the error terms.

In an ideal framework, we have a structural representation of VAR, called SVAR, where the error terms are serially uncorrelated as follows:

$$A{y}_{t}=v+{B}_{1}{y}_{t-1}+....+{B}_{k}{y}_{t-k}+{\epsilon }_{t},{\epsilon }_{t} \sim N(0,I)$$
(5)

The matrix A defines the structure of contemporaneous perturbations among the five components. Pre-multiplying equation (5) by A−1, we obtain the reduced-form VAR55

$${A}^{-1}A{y}_{t}={A}^{-1}{B}_{1}{y}_{t-1}+....+{A}^{-1}{B}_{k}{y}_{t-k}+{A}^{-1}{\epsilon }_{t}$$

which is simplified as:

$${y}_{t}={F}_{1}{y}_{t-1}+...+{F}_{k}{y}_{t-k}+{\mu }_{t},{\mu }_{t} \sim N(0,{\Sigma }_{\mu })$$
(6)

where Fi = A−1Bi and \({\Sigma }_{\mu }={A}^{-1}{A}^{-1{\prime} }\).

Since the variance-covariance matrix, Σμ, is symmetric, the system in equation (6) is not identified unless we impose further restriction on matrix A (there are 25 unknowns in the elements of Σμ but only 15 equations due to symmetry).

Here, we apply the Cholesky decomposition for identification purpose where we assume that matrix A is a lower triangular56. This restriction limits the instantaneous correlations among the five physical components by imposing the orthogonalization or “ordering” of the error. Simply put, a component lying higher or earlier in an order has instantaneous effects on those lower or later in the order, and the lower component only affects those above them with a lag (a day in our sample). Since we are interested in the impact of other components on precipitation, we select orderings that put precipitation at the end of the order. Regarding the rest of the orderings, we fully exploit all possible combinations of orderings (4! of them).

An example of ordering and its error structure shows how an ordering defines the error term structure and interactions among the components. In this example, we place HA at the top of the ordering: HA affects all of the components in the system instantaneously, while the other variables do not have an instantaneous effect on HA, but with a delay of one day. The next on the list is DY, followed by TH and E. The one at the bottom of the order is P. In this case, the perturbation in DY can affect TH, E, and P instantaneously, but the latter can only affect DY a day later. This ordering strategy yields the following error terms structure:

$${\mu }_{t}={A}^{-1}{\epsilon }_{t}=\left(\begin{array}{ccccc}{\delta }_{11}&0&0&0&0\\ {\delta }_{21}&{\delta }_{22}&0&0&0\\ {\delta }_{31}&{\delta }_{32}&{\delta }_{33}&0&0\\ {\delta }_{41}&{\delta }_{42}&{\delta }_{43}&{\delta }_{44}&0\\ {\delta }_{51}&{\delta }_{52}&{\delta }_{53}&{\delta }_{54}&{\delta }_{55}\end{array}\right)\left(\begin{array}{c}{\epsilon }_{t}^{HA}\\ {\epsilon }_{t}^{DY}\\ {\epsilon }_{t}^{TH}\\ {\epsilon }_{t}^{E}\\ {\epsilon }_{t}^{P}\end{array}\right)$$
(7)

Using the orthogonalization in equation (7), we compute the impacts of an independent perturbation in the error term of a component on the temporal paths of all of the components of the system by estimating impulse-response functions (IRFs); The IRF traces the impact of a perturbation in a component on the current and future values of all the components over time55. Based on the IRF in the example above, we can identify the contribution of horizontal advection of moisture to the precipitation over time. Here, the impulse is the horizontal advection of moisture and the response is precipitation. Based on the same ordering, we can set the dynamic component as an impulse and the thermodynamic as a response, and estimate the impact of perturbation in the dynamic component on the thermodynamic component over time. In other word, for a given ordering, multiples IRFs can be estimated.

Regarding the magnitude of perturbation, as a perturbation of size one does not always make sense, we normalize the size of the perturbation to be one standard deviation of the impulse variable rather than one unit. Then, according to the IRF, the marginal effect of a perturbation in component j at time t on component i at time t + s, for s = 0, 1, 2, . . . is equal to

$${\tau }_{i,j}^{t,s}=\frac{\partial {y}_{t+s}^{i}}{\partial {\epsilon }_{t}^{j}}{\sigma }_{j}$$
(8)

where σj denotes the standard deviation of j.

We estimate the IRF in each grid for all possible orthogonalizations (orderings). Given a pair of impulse-response variables, we may obtain different marginal effects with different orthogonalizations. Then, for a given grid, the impact of an impulse variable on a response variable is based on the average value of the IRFs from the orderings in which the impulse variable is of a higher order than the response variable, which allows the instantaneous influence by the impulse variable on the response variable.

Note that our results based on the IRF are not significantly influenced by the presence of models with coarser resolution. Since the IRF is estimated for each grid, the individual estimation of the IRF is independent of the number of grids. In addition, aggregation of the IRF over regions (land vs. ocean in Figs. 2 and 3) and the globe (Figs. 2 and 5) and the distribution of means of IRF (Fig. 4) are not significantly influenced by the presence of the models of coarse grids, because (i) the missing information on some grids in certain models is almost uniformly distributed across latitudes and longitudes (Fig. S3), and thus does not bias the aggregated values over one region over the others, and (ii) although there may be a notable difference in the magnitude of moisture budget components in a given grid across different models, the estimated IRF based on the moisture budget components, i.e., the standardized marginal impact of one component on another in (8), at individual grid does not seem to be largely different across different model outputs, which is evident from the small error bars associated with the IRF estimates in Figs. 2 and 5 that are based on the spread over models.

Forecast error variance decomposition (FEVD)

Forecast error variance decomposition (FEVD) is a part of the SVAR analysis that decomposes the variance of the forecast error into the contributions from specific perturbations. Starting from equation (6), the variance of variable i is given by

$$var({y}_{it})=\sum\limits_{k = 1}^{n}\sum\limits_{j = 0}^{\infty }{F}_{ik}^{{j}^{2}}var({\mu }_{kt})$$
(9)

where \({\sum}_{j = 0}^{\infty }{F}_{ik}^{{j}^{2}}\) is the variance of variable i generated by a perturbation or increase in variable k. Therefore, the fraction of variance of variable i explained by variable k is equal to:

$$\sum\limits_{j = 0}^{\infty }{F}_{ik}^{{j}^{2}}var({\mu }_{kt})/var({y}_{it})$$
(10)

Spatial correlation

Another consideration for the SVAR estimation with the spatial data is the possibility of spatial correlation among the moisture budget components of nearby grids. We test this possibility by including precipitation and moisture budget components of the neighboring grids in the SVAR regression. Specifically, the eight nearest grids surrounding the grid are considered neighbors. We measure the contribution from neighbors by the IRF and the forecast error variance decomposition (FEVD). The experiments were conducted for randomly selected 100 grids over geographically separated regions and different models.

We find that the impact of moisture budget components of neighboring grids on precipitation of the grid in question is an order of magnitude smaller than those from the grid. The reason for this finding can be that the current grid resolution of 2o × 2.5o, is coarse enough to minimize the spatial correlation. Therefore, the omitted variable bias from not including neighboring grids is not significant. On the other hand, the downside of including the neighbors in the SVAR regression is the significant increase in computation time of estimation as the number of variables significantly increases; including information from one neighboring grid will introduce five variables and their multiple time-lagged terms. Hence, we do not use the information from the neighboring grids in the SVAR regression.

Residual from the moisture budget equation

The moisture budget equation in equation (1) may not hold with equality based on model outputs, due to various sources of errors that do not have a clear physical interpretation57. To examine the consistency of the model output regarding the moisture budget equation, we define the size of this “residual” value, denoted by R, for each grid as follows:

$$R=P-E+[q\nabla V]+[V\nabla q]$$
(11)

and compute R for all models and climate states. Then we compute the area-weighted mean of the residual value over the globe for each climate state. We find that R is equal to −0.24, −0.17, and −0.28 mm d−1 for the pre-industrial, the present, and the future, respectively. Given that the global average precipitation varies from 2.7–3.2 mm d−1 in the different climate regimes, the residuals represent imbalances of 5–10%, which is comparable to a relatively small residual magnitude in other studies58.