## 1. Introduction

Three different factors have been considered to account for the global precipitation increase being smaller than the increase in the atmospheric moisture in response to global warming. First, the net change of radiative heating in the troposphere, or the availability of energy in the troposphere, constrains the increase of precipitation (Hartmann and Larson 2002; Allen and Ingram 2002). Second, it has been suggested that the reduction in convective mass flux contributes to the smaller increase rate in precipitation in a warmer climate (Held and Soden 2006), and the reduction in convective mass flux is consistent with the weakening of tropical Walker circulation in the climate models (Vecchi et al. 2006). Third, the reduction of surface wind is considered to be responsible for muted increase in global evaporation (Mitchell et al. 1987).

It is understood that an increase in the atmospheric water vapor only means the increase in the atmospheric water reservoir and does not automatically imply an increase in precipitation. The increase in global precipitation, however, must be balanced by the same amount of change in the surface evaporation in the equilibrium state. This motivates us to consider the surface evaporation change, instead of precipitation itself, when seeking an explanation for a much smaller increase rate than that implied by the Clausius–Clapeyron (C–C) relation (6.5% K^{−1}) in the global precipitation response to global warming. Here we propose a mechanism responsible for the muted increase in the hydrological cycle in climate models from the perspective of the surface energy balance. We also wish to discuss the physical consistency among the proposed mechanism and the existing mechanisms—that is, the reduction in convective mass flux and the atmospheric energy constraint.

## 2. Data and method

The data used in this study are derived from outputs of the control (1 × CO_{2}) and 2 × CO_{2} equilibrium experiments made with 11 Coupled Model Intercomparison Project phase 3 (CMIP3) coupled slab ocean–atmospheric general circulation climate models (Table 1). We calculate the differences between 1 × CO_{2} and 2 × CO_{2} experiments, and the statistics are based on the average over the global ocean only. The mean values and the changes induced by doubling CO_{2} are based on the average over 20–50 yr, depending on the available data of each model (Table 1). Daily mean data of zonal and meridional winds, instead of monthly mean, are used in calculating the change of surface wind speed.

*P*) and that of evaporation (

*E*) over the global ocean and land surface is where 〈 〉

_{g,l,o}denote the averages over the global surface, the global land surface, and the global ocean surface, respectively; Δ represents the perturbation derived from global warming simulations;

*a*≃ 0.3 is the fraction of the land surface; and

_{l}*L*is the latent heat of evaporation. We found that arithmetic mean of

*a*〈Δ

_{l}L*E*〉

_{l}for the 11 CMIP3 models is about 0.56 W m

^{−2}, much smaller than 4.84 W m

^{−2}, the mean of (1 −

*a*)

_{l}*L*〈Δ

*E*〉

_{o}. Also it is known that the evaporation from the global ocean accounts for about 86% of the total evaporation (Wentz et al. 2007). Therefore, (1) can be approximated as The results plotted in Fig. 1a confirm that the change in global precipitation is mainly associated with the change in the surface evaporation over the global ocean, as suggested by (2). Equation (2) also enables us to account for the change of global precipitation by examining the main factors that influence the change of evaporation over the global ocean only.

Furthermore, it is necessary to distinguish the global ocean from the land surface in analyzing the constraints of the global hydrological cycle based on the surface energy balance. Because of the limited availability of soil moisture, the dominant terms that regulate the change in evaporation over land are expected to be different from that for the change in evaporation over the oceans (e.g., Lu and Ji 2006). As a result, the physical insight would be blurred if we combine the surface energy balance over the global ocean and the land surface together.

## 3. Results

^{−1}in the 11 CMIP3 global warming simulations. Meanwhile, Fig. 1b shows that the sensible heat flux over the global ocean in these models decreases at a rate of about 5.2% K

^{−1}. Because the mean surface sensible heat flux (about 12 W m

^{−2}over the global ocean) is much smaller than the surface latent heat flux (about 100 W m

^{−2}over the global ocean), the 5.2% K

^{−1}reduction is only equivalent to a reduction of 0.6 W m

^{−2}K

^{−1}in the surface sensible flux. Therefore, the change of the surface sensible heat flux by itself is indeed a very small contribution to the change in the surface energy balance (Held and Soden 2006). However, Fig. 1c suggests that the difference between the fractional change of evaporation (or latent heat flux) and that of sensible heat flux—that is, the fractional decrease of the Bowen ratio—is about 7.5% K

^{−1}over the global ocean, close to the C–C relation (6.5% K

^{−1}). Next, we show that the result in Fig. 1c provides a physical explanation of the muted increase in evaporation over the global ocean by relating the fractional change of evaporation (latent heat flux) to the reduction of the surface sensible flux over the global ocean. The perturbation surface energy balance equation is where

*R*is the sum of the downward longwave radiation and the net shortwave radiation at the surface;

_{s}^{f}*T*is the surface temperature and

_{s}*T*is the surface air temperature;

_{a}*F*

_{sh}=

*βc*(

_{p}*T*−

_{s}*T*) is the surface sensible heat flux;

_{a}*F*

_{lh}=

*βL*[

*q*(

*T*) −

_{s}*rq*(

*T*)] is the surface latent heat flux;

_{a}*β*is the turbulent transfer coefficient for sensible and latent heat fluxes;

*q*(

*T*) is the saturated specific humidity at the temperature

*T*(note that

*q*is also a function of pressure, but we omit its dependence on pressure because the variations of pressure over the ocean surface are small both in time and space);

*r*is the relative humidity of the surface air; and Δ represents the perturbation induced by external forcing and climate feedbacks. Hereafter, Δ

*R*is referred to as “the net radiative forcing at the surface.” The three terms labeled I, II, and III on the right-hand side of (3) are the changes in the upward surface longwave radiation, the surface sensible heat flux, and the surface latent heat flux, respectively.

_{s}^{f}*dq*/

*dT*|

_{Ts}≠

*dq*/

*dT*|

_{Ta}(Joshi et al. 2008), we can rewrite the changes in sensible and latent heat fluxes in (3) as where Δ

*F*

_{sh}and Δ

*F*

_{lh}correspond to the terms labeled II and III in (3), respectively.

It is seen that: 1) the change in the turbulent transfer coefficient of the surface fluxes (Δ*β*/*β*) has the same effect on the fractional change of sensible and latent heat fluxes; 2) the change in air–sea temperature difference, Δ(*T _{s}* −

*T*), influences both the sensible and latent heat flux but at different rates; 3) the change in relative humidity (Δ

_{a}*r*) influences latent heat flux only; and 4) the change in saturated specific humidity {[

*dg*/(

*qdT*)]Δ

*T*} would lead to the fractional change of latent heat flux following the C–C relation.

_{s}*F*

_{sh}/

*F*

_{lh}with global warming according to (4):

*dq*/(

*qdT*), implying that the sum of the terms due to changes in air–sea temperature difference Δ(

*T*−

_{s}*T*) and in relative humidity Δ

_{a}*r*in (5) would be small (about 1% K

^{−1}). The smallness of the sum of these two terms could be either due to both being small or due to two large terms canceling each other. Because the CMIP3 archive does not provide the requisite data for

*T*and

_{a}*r*, we could only make estimates of these two terms by using the derived data—that is, the temperature at 2-m height (

*T*

_{2m}) and

*r*at 925 hPa. Table 2 shows that the linear fit and the arithmetic mean of the second term are 2.4% K

^{−1}and 2.7% K

^{−1}, respectively and their counterparts for the third term are 0.3% K

^{−1}and 0.1% K

^{−1}, respectively. It follows that the third term is indeed small, but the second term is somewhat larger than 1% K

^{−1}, as suggested in Fig. 1c (note that the possible inconsistencies in the derived data could overestimate the magnitude of the second term). With that caveat—that is, the second term could be in the order of 2.4–2.7% K

^{−1}—in mind, we tend to conclude that the changes in air–sea temperature difference Δ(

*T*−

_{s}*T*) and in relative humidity Δ

_{a}*r*would be secondary in contributing to the change of the Bowen ratio. According to (4) and (5), the relative smallness of these two terms also implies that the reduction in sensible heat flux (Fig. 1b) and the departure of the increase in surface latent heat flux from the C–C relation (Fig. 1a) are mainly due to the reduction of the turbulent transfer coefficient term (Δ

*β*/

*β*term). This is because only the Δ

*β*/

*β*term could cause the same amount of the fractional change in both the sensible and latent heat fluxes simultaneously while keeping the fractional change of the Bowen ratio close to the C–C relation, as suggested in (6):

The analysis above suggests that the reduction of the turbulent heat exchange efficiency at the global ocean surface in response to global warming is one of the main factors responsible for the muted increase in global evaporation and precipitation. It is well known that both the increase in the stability of the atmospheric boundary layer (ABL), which decreases the surface drag coefficient, and the decrease in the surface wind could contribute to a reduction of the efficiency in the surface turbulent heat exchange. It is usually believed that the reduction in the surface wind is the main factor responsible for the muted increase in surface evaporation (Mitchell et al. 1987; Wentz et al. 2007). Table 3 shows, however, that the fractional changes in the surface wind speed over both global and tropical oceans, in all of the models that provide daily-mean surface wind components, are very small, with the magnitude less than 1% K^{−1} . This is in fact consistent with the satellite-based observations (Wentz et al. 2007). Although in the deep tropics, the surface wind does reduce significantly in model simulations, consistent with the weakening of the Walker circulation (Vecchi et al. 2006), there are also bands in mid-to-high latitudes with stronger surface wind speed. We also note that the evaporation from the deep tropics (10°S–10°N) accounts for only about 20% of the total evaporation. Therefore, although it might be a contributor, the change in surface wind speed may not be the main cause of the reduction in the efficiency of the surface turbulent heat exchange, which contrasts with the suggestions of previous studies. This leaves the stabilization of the ABL as the main factor for the reduction in the efficiency of the surface turbulent heat exchange and the muted hydrological response to global warming.

The determination of the stability of the ABL, or the intensity of turbulence, involves the coupling between the surface heat (buoyancy) fluxes, the thermal stratification and vertical profile of adiabatic heating, and the production (or the destruction) of turbulence by shear and buoyancy (Garratt 1992). The coupling processes are usually parameterized in GCMs under various turbulence closure assumptions. A thorough analysis of the overall stabilization of the ABL requires a systematic exploration of the changes in the boundary layer processes and is beyond the scope of the current research.

We stress here that the stabilization of the ABL, the reduction of convective mass flux, and the atmospheric energy constraint are dynamically consistent with each other. They represent the different but consistent aspects of the global hydrological cycle response to global warming.

Our explanation, based on the stabilization of the ABL, is dynamically consistent with the finding that the convective mass flux is reduced with global warming (Held and Soden 2006). Although the increases in temperature and water vapor mean an increase of moist entropy in the boundary layer, the static stability increase in the ABL inhibits the growth of convective available potential energy (CAPE). Therefore, the moist convection (or the vertical water vapor transport) would not increase at the same rate as the increase of water vapor in the atmosphere, and the increasing rate of moist convection must be smaller than that would follow the C–C relation.

Our explanation, based on the stabilization of the ABL, would also shed some light on the atmospheric energy constraint on the relationship between the hydrological cycle and global warming (Hartmann and Larson 2002; Allen and Ingram 2002). In the equilibrium state, the global mean of the net radiation at the top of the atmosphere (TOA) has to be zero; therefore, the perturbation of the atmospheric energy budget equation (Allen and Ingram 2002) is identical to the perturbation surface energy balance equation used in this paper. In Allen and Ingram (2002), the increase in the atmospheric radiative cooling rate with the temperature (*k _{T}*) is about 3 W m

^{−2}K

^{−1}, and this largely determines that the precipitation increases at a rate of about 3% K

^{−1}. This constraint, however, is neither a priori nor purely thermodynamic. The atmospheric cooling rate

*k*is also dependent on the stability of the ABL in the global climate model. This may be proved by the work of Lindzen et al. (1982) in which they developed a one-dimensional radiative–convective climate model with a fixed turbulent mixing coefficient—that is, the fixed stability of the ABL and surface wind and the fractional change of precipitation (from their Tables 1 and 2) is about 5%–7% K

_{T}^{−1}, which is close to the C–C relation. By applying the same atmospheric energy budget equation as in Allen and Ingram (2002) to the Lindzen et al. (1982) model, we find when the stability of the ABL is fixed,

*k*is about 6–7 W m

_{T}^{−2}K

^{−1}, much larger than the estimate in Allen and Ingram (2002). Therefore, the atmospheric energy constraint on global hydrological cycle is also a dynamic constraint and largely determined by the changes in the stability of the ABL and the turbulent transfer between the ABL and the surface.

## 4. Concluding remarks

We found that both the global precipitation and evaporation in the CMIP3 global warming simulations increase at 1%–3% K^{−1}, much smaller than the rate suggested from the C–C relation (6–6.5% K^{−1}). However, the reduction of surface sensible heat flux over the global ocean (5.2% K^{−1}) matches the difference between the fractional increase of evaporation and the C–C relation, implying that the fractional decrease of the Bowen ratio over the global ocean closely follows the C–C relation. The analysis suggests that the stabilization of the ABL in response to global warming is the main factor responsible for the simultaneous reduction of the surface sensible flux and the muted increase in the surface latent heat. Because the stabilization of the ABL causes the same amount of fractional change in both the sensible and latent heat fluxes, the fractional decrease of the Bowen ratio closely follows the C–C relation. This result contrasts with the thinking that the changes in the air–sea temperature difference, in surface air relative humidity, or in surface wind, are the main factors leading to the smaller increase in evaporation in response to global warming.

We also point out that the stabilization of the ABL, the reduction of convective mass flux, and the atmospheric energy constraint are dynamically consistent with each other. They represent the different but consistent aspects of the global hydrological cycle response to global warming.

The stabilization of the ABL may explain the trend discrepancy between the evaporation estimated from the satellite data and the GCM simulations. It was estimated from satellite data that the evaporation increased at about 6% K^{−1}, a rate close to the C–C relation during the past two decades (Wentz et al. 2007). The constant value of the stability parameter in the bulk formula used in Wentz et al. (2007), implying no change in the ABL stability, would give rise to an evaporation change that has to be close to the C–C relation based on our analysis. We expect that the discrepancy between the estimate from the satellite data and the model simulations could be, at least partly, reconciled if we take the stabilization of the ABL into consideration.

Uncertainties still exist in the projection of both the global temperature and the hydrological cycle change in the future by the models. Our analysis suggests the way that the atmospheric boundary layer changes with global warming is important to the hydrological cycle response to future warming. It should be noted that the boundary layer parameterization in coupled climate models is far from perfect (Delage 1997; Mahrt 2008), and that the long-term observations of the ABL over the global ocean are still limited. Therefore, further observation and modeling studies on atmospheric boundary layers, especially the surface heat and moisture transport, may be essential for reducing uncertainties in the global hydrological cycle change.

We thank two anonymous referees and Dr. William J. Gutowski Jr. for their useful comments. We also acknowledge the modeling groups for making their simulations available for analysis, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) for collecting and archiving the CMIP3 model output, and the WCRP’s Working Group on Coupled Modelling (WGCM) for organizing the model data analysis activity. The WCRP CMIP3 multimodel dataset is supported by the Office of Science of the U.S. Department of Energy. This work is supported by the NOAA CPO Grant GC06-038.

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The CMIP3 models, affiliated research centers, and the length of data.

Statistics of the contributions to the fractional decrease of the Bowen ratio from the changes of air–sea temperature difference and of near-surface relative humidity—the second and third terms in RHS of (5)—in the 11 CMIP3 models.

The fractional change (% K^{−1}) of surface wind speed per 1 K surface warming over the global and tropical ocean surface in six CMIP3 models.