1. Introduction
a. Why potential evapotranspiration?
Potential evapotranspiration (PET), a basic land climate variable (e.g., Hartmann 1994), is the rate at which a given climate is trying to evaporate water from the soil–vegetation system. In other words, for given atmospheric and radiative conditions, PET is the surface evapotranspiration (ET) rate that would hold if the soil and vegetation were well watered. Synonymous and near-synonymous concepts include reference evapotranspiration, potential evaporation, evaporative demand, and pan evaporation. Critically, PET may be thought of as the water required to maintain a garden or irrigated crop, or the water “price” a plant must pay to maintain open stomata. A higher-PET climate is thus a more arid, evaporative climate. Therefore, in this study we attempt to understand how local PET will scale with global greenhouse warming, using global climate models (GCMs) as well as basic physical principles.
PET is also of interest because it is a key factor explaining other hydrologic and climatic quantities. Several prominent conceptual models of land hydrology, including the Palmer Drought Severity Index (PDSI) (Palmer 1965) and the Budyko and Miller (1974) ecohydrologic theory, take precipitation (water supply) and PET (water demand) as climate-supplied forcings and give soil moisture, actual ET (latent heat) flux, runoff, and/or drought index as land-generated responses. In these sorts of frameworks, understanding precipitation and PET changes is necessary for understanding other land hydroclimatic changes. In particular, recent studies using the PDSI to warn of widespread drought increases with future greenhouse warming (e.g., Dai 2013; Burke et al. 2006) cite systematic global PET increases as the main driver of their alarming results. Understanding the nature, magnitude, and pattern of these projected increases is the motive of the present work.
Additionally, PET is a more natural choice than actual ET for the evaporative component of land “aridity” metrics because changes in actual ET often just reflect supply (precipitation) changes. For example, the well-known study of Seager et al. (2007) uses precipitation minus actual ET (P − E) to quantify modeled aridification due to greenhouse warming in a subtropical terrestrial region where the model precipitation declines a great deal. The model ET (the Seager et al. E) in this area also significantly declines, not surprisingly. However, the analysis, by its nature, interprets the ET decline as if it is some other factor helping to offset or mitigate the precipitation decline. In fact, the model climate is probably becoming more evaporative, not less, owing to warming and (presumably) cloud-cover and relative-humidity reduction, and this should not mitigate but aggravate the local ecological effect of the precipitation reduction, even though the actual evaporative flux necessarily decreases owing to the supply decrease (e.g., Brutsaert and Parlange 1998). To avoid this type of pitfall, the aridity of a climate is usually quantified using the ratio P/PET of annual water supply to annual water demand, or similar (e.g., Budyko and Miller 1974; Middleton and Thomas 1997; Mortimore 2009), which has the additional advantage of being dimensionless. Then P/PET < 0.05 is defined as hyperarid, 0.05 < P/PET < 0.2 as arid, 0.2 < P/PET < 0.5 as semiarid, and so forth. Feng and Fu (2013) show that global climate models project systematic future decreases in P/PET (i.e., aridification) over most of the earth’s land, again owing to the (projected) systematic PET increases that we attempt to understand in this work.
b. Quantifying PET
The first term in the numerator of (5) is known as the radiative term, and the second is called the aerodynamic term. Note that in the latter we have rewritten [e*(Ta) − ea] the vapor pressure deficit appearing in (4), as e*(Ta)(1 − RH), where RH is the near-surface relative humidity. This allows changes in the vapor pressure deficit to be separated into constant-RH changes in e* (from Ta changes), and constant-Ta changes in RH. [Henceforth we are dropping the (Ta) and simply writing e* for e*(Ta), since Ts has been eliminated.]
Many of the input variables in (5) will change with significant greenhouse warming. Most immediately, the surface net radiation Rn will tend to increase (absent any cloud feedbacks) because of the extra longwave emitters in the atmosphere, sending more longwave energy back at the surface. This alone would tend to increase PET (5). However, the warming itself will also directly change PET through e* and Δ, which both increase with Ta by the Clausius–Clapeyron law. Constant-RH increases in e* will increase PET by widening the vapor pressure deficit, especially where and when RH is low. [The discussion in the review paper of Roderick et al. (2009) omitted this mechanism.] Increases in Δ may increase or decrease PET depending on the magnitudes of various terms in (5). It is not clear a priori whether the radiation changes or these direct-warming changes will dominate.
In addition, RH might change in either direction, through a common theoretical expectation for RH is that it should remain roughly constant (e.g., Held and Soden 2000), as generally observed thus far (e.g., Held and Soden 2006). This is the main motivation for considering constant-RH e* changes separately from changes in RH.
Finally, raw observations indicate that |u| decreased in most land areas over the past several decades (McVicar et al. 2012), in sufficient magnitude to overcome the concurrent Ta increases in (5) and explain the widespread observations of declining pan evaporation, that is, declining PET (McVicar et al. 2012; Roderick et al. 2009; Wang et al. 2012). However, it is still unclear whether this terrestrial wind “stilling” is a measurement artifact, as it does not appear in reanalyses (e.g., Pryor et al. 2009; McVicar et al. 2008) or marine observations (McVicar et al. 2012), and some of the pan-evaporation declines themselves are also raw and unadjusted for observing-system changes. Even if real, it is highly unclear whether the stilling is due to global warming (McVicar et al. 2012), and it may have reversed course after about 1998 (Wang et al. 2012). Therefore, in this study we take the future model output of |u| at face value, which contains no such systematic declines. However, if any real global stilling trend of the proposed magnitude were to continue unabated into future decades, PET would presumably continue declining and the conclusions of our study (as well as those mentioned above) would not apply.
Other, non-Penman methods of estimating PET are also in use, as mentioned at the beginning of this subsection. The Thornthwaite (1948) method and other temperature-proxy methods empirically relate PET to Ta alone, for a given location and time of year. This simplicity has encouraged their frequent use for variability in the current climate (e.g., Palmer 1965), which has led some studies to use them, or models containing them, to assess future climate change (e.g., Wehner et al. 2011; Price and Rind 1994); see also references in Lofgren et al. (2011). However, within a given climate (especially during warm, high-PET parts of the year), anomalous warmth is associated with anomalous sunshine (higher Rn), and often also with anomalous low RH, significantly enlarging the positive response of (5). By contrast, future climate change should warm Ta without the sunshine and RH changes that might accompany a similar warm anomaly in year-to-year variability. Thus one would expect from (5) that an empirically determined dependence of PET on Ta from year-to-year variation would overestimate the greenhouse climate change response. Indeed, several studies (e.g., McKenney and Rosenberg 1993; Hobbins et al. 2008) have found that the same long-term climatic changes can imply large increases in the Thornthwaite PET but much smaller increases, or even slight decreases in Penman–Monteith PET. Similarly, negative PDSI responses to future global climate model output are 2–3 times stronger using the default Thornthwaite PET than using Penman–Monteith PET (A. Dai 2012, personal communication) Thus, studies that use a simple temperature-proxy method to assess future PET changes may be severely flawed.
Other studies of future climate change (Lofgren et al. 2011; Arora 2002) simply estimate PET as Rn/Lυ, which we will call the energy-only method. While this works reasonably well for spatial differences in the present climate (Budyko and Miller 1974), one would presume that it underestimates future PET increases because it does not include the independent physical effects of Ta through the Clausius–Clapeyron law, discussed above.
Still other studies of PET change in global climate models (e.g., Rind et al. 1990) have directly used an internal land-model field that is also called “potential evapotranspiration.” However, this field is (quite confusingly) not the same concept as what we have been discussing: it is what would instantaneously start evaporating if the surface were to be suddenly wettened, without any chance to cool down the skin temperature Ts and establish energy balance (3) with Rn. In other words, this field is directly computed using the bulk LH Eq. (2), where rs is still the well-watered “open” stomatal resistance but Ts is now the actual skin-temperature output of the model instead of the well-watered skin temperature used above, which is often much cooler. Indeed, Rind et al. (1990) (and references therein) found that this model “PET” achieved summertime climatological values averaged over the United States of ~40 mm day−1 in the climate models of their day. [The observed summertime PET maxima in, e.g., Hartmann (1994) are almost an order of magnitude lower.] So this quantity, while interesting perhaps, is not the object of our study (and also it is not publicly archived by any of the GCMs in the current phase of the Coupled Model Intercomparison Project).
Therefore, in this study we use (5) to quantify and understand the PET response to future greenhouse warming.
2. Methods
The Penman–Monteith potential evapotranspiration (5) is usually many times larger in magnitude during the day than at night because of both Rn and the vapor pressure deficit. Thus, daytime climate changes may affect time-integrated PET much more than nighttime climate changes, so it is desirable to examine diurnally resolved climate and PET. In the recent fifth phase of the Coupled Model Intercomparison Project (CMIP5) (Taylor et al. 2012), subdaily surface output is conveniently accessible for the first time, at 3-hourly resolution. Sixteen of the CMIP5 global climate models archive all of the necessary information (surface energy budget terms and near-surface temperature, moisture, and wind) at this resolution for years 2081–99 in the business-as-usual representative concentration pathway 8.5 (RCP8.5) scenario and 1981–99 in the historical scenario. However, in three of these, the meteorological fields are given at, say, 10 m above the soil surface, instead of 10 m above the canopy top (M. Watanabe 2013, personal communication), making them inapplicable to (1) and (2) [and thus (5)] in forest areas. So, we use the remaining 13 models, which we list in Table 1 along with any model-specific exceptions to our procedures. We use output from run 1 (“r1i1p1” in CMIP5 filenames) only.
CMIP5 models analyzed in this study.
A prominent version of (5) is the recent American Society of Civil Engineers (ASCE) standardized reference evapotranspiration equation (Allen et al. 2005), which was explicitly developed for the purpose of standardizing the computation of reference or potential ET for all users. The development included the systematic intercomparison and testing of numerous operationally used Penman-type methods. Our full method closely based on Allen et al. is given in section a of the appendix. Briefly, we fix CH and rs as universal constants corresponding to “alfalfa” values as specified by Allen et al. (2005), with CH ≈ 5.7 × 10−3, and rs varying between 30 s m−1 (day) and 200 s m−1 (night). (We will see in section 5 that our conclusions are not very sensitive to these vegetation parameters.) We also compute (Rn − G) as LH + SH (3) because the models do not output G, and we let e*, Δ, and ρa depend on Ta as specified in Allen et al.
Using these procedures and values, for each of the 13 CMIP5 models in Table 1 we compute Penman–Monteith PET (5) for every model grid cell that is at least 80% ice-free land and for every 3-h interval in the 19-yr epochs 1981–99 and 2081–99 (except those that fall on 29 February in models that use the full Gregorian calendar). For each interval in the calendar year, we average over the 19 years to obtain a diurnally and annually varying PET climatology of each epoch. Averaging over the calendar then gives annual-mean climatologies of PET. These are shown for 1981–99 in Fig. 1 along with their multimodel means and appear quite reasonable with higher modeled PET in sunnier, lower-RH, and/or warmer locations. As an additional reality check, Fig. 2 plots these against the corresponding model climatologies of actual ET; each dot is one grid cell. In almost all of the models, our computed PET is a fairly clean, efficient upper bound on the model’s actual ET, as expected from the definition. That is, the most well-watered model grid cells are actually evapotranspiring at rates quite close to our independently computed PET. This success is a rather pleasant surprise considering the very different origins of the two quantities, the models’ use of full Monin–Obukhov surface layer dynamics for CH, and the potentially large contrast between ASCE-standard alfalfa and the vegetation specified in the model grid cells.
1981–99 climatological annual-mean Penman–Monteith PET (mm day−1) for each CMIP5 model in Table 1. Last panel is the mean over all applicable models (omitting locations where less than half of the models were analyzed).
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
1981–99 climatological annual-mean actual ET (vertical, 0–6 mm day−1) vs PET (horizontal, 0–13 mm day−1) for each model, where each dot is one grid cell. Red lines are 1:1 (actual ET = PET).
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
3. Model results
a. Full PET change
For each of the 13 models and for the multimodel mean, Fig. 3 maps the raw percentage change in climatological annual-mean PET (5) between the 1981–99 and 2081–99 epochs. At each location PET always or almost always increases; that is, ambient conditions become more evaporative with greenhouse warming. This more careful calculation confirms the similar results of Burke et al. (2006), Dai (2013), and Feng and Fu (2013), who quantified this future PET increase only for the mean (or for a single model), and did not resolve the diurnal cycle. In some models a few largely high-latitude regions do see PET decreases or little change in PET, but these are quite localized, and even in these places most models (and the mean) show increases in PET.
Percent changes in climatological annual-mean PET between 1981–99 and 2081–99, for each model. (Values in a few color-saturated regions greatly exceed those on the scale.) Last panel is the percentage change in the multimodel mean.
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
Furthermore, the magnitude of the projected PET increases is usually in the low double digits of percent, on the order of 10%–45%. In many models certain northern and/or mountainous locations see more than this, but over very broad swaths of land these sorts of values are typical. For the multimodel mean, the first row of Table 2 summarizes this by averaging the percentage change values over various latitude bands (and subsequent rows similarly average subsequent figures). The magnitudes in Fig. 3 agree well with those in Feng and Fu (2013) despite the differing methods. They are also comparable to change magnitudes for annual precipitation P (e.g., Meehl et al. 2007). This further confirms the importance of using P/PET or similar diagnostics when thinking about the land aridity response to global warming, instead of just P (and/or actual evapotranspiration E, which often contains the same information as P as discussed in section 1a).
Results for the multimodel mean, averaged over different latitude bands.
In most models and in the mean, there is also a clear tendency toward greater percentage increases in PET at higher latitudes (as alluded to above and seen in Table 2), and again Feng and Fu (2013) obtain a similar structure. As far as we know, this basic property has not been explicitly noted in the literature before. (We will see in section 4 that the main reason for this is not Arctic amplification of warming.)
Yet despite all of these broad commonalities, the models also disagree a great deal, on both the detailed spatial patterns and the overall magnitude. We will see how these disagreements arise from differences in the climate changes projected by the models.
b. PET changes due to individual factors
Figures 4, 5, 6, and 7 show the percent changes in climatological annual-mean PET (5) that result from perturbing (Rn − G), Ta, RH, and |u| one at a time to 2081–99 levels while keeping the other variables at 1981–99 levels, as explained in detail in section b of the appendix. One can immediately see here and in Table 2 that the always-positive PET change owing to the Ta increase (Fig. 5) dominates the other factors in most locations, and explains most of the overall 10%–45% magnitude in Fig. 3. This is why PET increases are so much more common than decreases. Again, the physical mechanisms here are widening of the vapor pressure deficit by constant-RH increases in e*, and lowering of the saturated Bowen ratio by increases in Δ (plus isobaric lowering of ρa to a small extent). RH also changes, but the resulting PET changes (Fig. 6) are of both signs, inconsistent from model to model, very weakly positive in the multimodel mean, and only sporadically (nowhere, in the mean) negative enough to cancel the Ta-induced increases in Fig. 5. This validates the constant-RH baseline idea and justifies our decision to think of the vapor pressure deficit as e*(1 − RH) rather than the more customary (e* − ea). (An alternative null assumption of constant vapor pressure deficit would imply systematically increasing RH, which we do not see.) However, the RH-driven changes can still be very important locally in some models, explaining the East African PET decrease in BNU-ESM in Fig. 3, for example.
Percent changes in climatological annual-mean PET from setting only the surface radiative energy supply (Rn − G) to 2081–99 levels while leaving all other variables in (5) at 1981–99 levels.
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
Percent changes in climatological annual-mean PET from setting only the ambient air temperature Ta (and thus the saturation vapor pressure e* and its derivative Δ) to 2081–99 levels while leaving all other variables in (5), including RH, at 1981–99 levels.
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
As in Fig. 4, but setting only the relative humidity to 2081–99 levels.
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
As in Fig. 4, but setting only the wind speed |u| to 2081–99 levels.
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
PET changes owing to the surface energy supply (Rn − G) (Fig. 4) are also usually positive, confirming the physical intuition from section 1b. However, with modal values of less than 10% (e.g., Table 2) they are generally of secondary importance to the Clausius–Clapeyron-driven changes (Fig. 5) just described. This was not clear a priori—in fact, some studies in the literature had used radiation changes alone to infer PET changes, as discussed above in section 1b. As with RH, though, some models have localized regions where radiation-induced change becomes dominant, such as the Amazon Basin in MRI-CGCM3 (and several other models) or the Tibetan Plateau in INM-CM4 (cf. Figs. 3–5).
In contrast, PET responses to |u| changes (Fig. 7) are only rarely important compared to the other changes. In the multimodel mean and in some individual models (the two BCC models, CNRM-CM5, and INM-CM4), they are hardly noticeable, usually no larger than ±5%. Like the RH responses (Fig. 6) they have no strongly preferred sign, although decreases are perhaps slightly more common than increases. This is all in stark contrast to the dominant “stilling” role posited for |u| in the putative recent PET declines, discussed in section 1b.
Finally, subtracting the sum of these attributed pieces (Figs. 4–7) from the full PET change (Fig. 3) gives the residual PET change due to nonlinearities, covariance changes, and changes in neglected inputs such as ps. This residual is shown in Fig. 8 and is quite weak (0%–10%) compared to the Ta-driven or even (Rn − G)-driven changes, though it is usually positive. (The GFDL-CM3 residual at high northern latitudes is a major exception to both of these statements, perhaps because the changes there in Figs. 3–6 are all so large.) In any case, we can clearly claim success in our attribution exercise since the residuals are much smaller than the full changes in Fig. 3 and are close to zero for the multimodel mean.
Residual percentage changes in climatological annual-mean PET between 1981–99 and 2081–99 that remain after subtracting off the pieces attributed to (Rn − G), Ta, RH, and |u| (Figs. 4–7) from the raw change (Fig. 3).
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
Having now examined all of the pieces, we can see that the constant-RH PET response to temperature change (Fig. 5) not only explains the general positivity and low-double-digit magnitude of the full PET change, but is also largely responsible for the high-latitude amplification noted in the previous subsection. The response to (Rn − G) (Fig. 4) is also polar amplified, but the temperature response still seems to contain most of the latitudinal contrast shown in Fig. 3, as can be clearly seen in Table 2. As for the intermodel disagreement in PET change, responsibility seems to lie with almost all of the terms, but disagreement in the Ta-driven term alone is still large, especially in the overall magnitude. [This makes sense given the well-known disagreement between global climate models on the magnitude of warming in response to an emissions scenario, that is, transient climate sensitivity (e.g., Meehl et al. 2007).]
Therefore, we now attempt a detailed quantitative understanding of the structure and magnitude of this model PET response to ambient temperature change as depicted in Fig. 5.
4. Analytic scaling for the PET response to temperature
a. Basic idea
However, the denominator of (5) cannot necessarily increase so fast: although Δ increases at roughly C-C as demonstrated, γ(1 + rsCH|u|) does not depend on Ta at all. This term stops the denominator from fractionally increasing as fast as the numerator and apparently is the key reason why PET always increases with Ta (Fig. 5) despite the ambiguous sign of the Δ-driven response discussed in section 1b. If not for the presence of γ(1 + rsCH|u|), the denominator would increase about as fast as the numerator, and PET might not be very sensitive to Ta at all.
b. Derivation and exposition of the scaling
Here frad is the fraction of the numerator of (5) made up by the radiative term, as in (9). Similarly, faero is the fraction of the numerator made up by the aerodynamic term, and fΔ is the fraction of the denominator of (5) made up by Δ.
The first term within the brackets in (13) tells the story laid out in the previous subsection: the numerator of PET (5) scales like C-C [dΔ/(ΔdT) · 1] or about 5–6% °C−1, but the denominator Δ + γ(1 + rsCH|u|) scales closer and closer to C-C the more important Δ is in it [−dΔ/(ΔdT) · fΔ], weakening the net response. Since Δ is an increasing function of Ta, this cancellation should occur more (fΔ should be larger and the denominator should be more C-C-like) in warmer base climates, so the percentage sensitivity of PET to Ta should be less in warmer base climates. We will see in section 4d that this explains the polar-amplified response pattern in Fig. 5. (Similarly, the sensitivity should be greater in windier climates, in which fΔ is reduced.)
c. From instantaneous to annual-mean scaling
Our Eq. (14) may be a theory for PET sensitivity at a particular instant. However, the results from section 3 and Fig. 5 that we wish to understand are about annually averaged PET. So, to test (14) it is not immediately clear what inputs should be used. For example, we might use the annual-mean warming 
d. Testing the annual-mean scaling
To test this scaling theory (17), we compute 
Since dΔ/(ΔdT) is not that dependent on temperature and 
So, in Fig. 9 we map 
For each model grid cell, the PET-weighted annual average 
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
For each model grid cell, the PET-weighted annual average 1981–99 climatological temperature 
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
Figure 11 then maps the entire bracketed term from (17), that is, our scaling estimate of the percentage sensitivity of 
Our scaling estimate 
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
On the other hand, the PET-weighted projected warming 
PET-weighted annual average of climate warming 
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
We are also now in a position to evaluate the source of the high-latitude amplification of the 
Finally, we can confirm this picture by evaluating (17) and comparing to the model 
This estimate is shown in Fig. 13 and is strikingly close to the model response in Fig. 5. In fact, the summary values in Table 2 differ from the actual values on the line above by only about +1% (of the basic state, about 10% of the changes). Thus, we can claim success in understanding the magnitude, structure, and intermodel spread in Fig. 5. The low double-digit percent magnitude of 
Our scaling estimate [the exponential of (17) minus 1] for the percent changes in climatological annual-mean PET from setting only the ambient air temperature Ta to 2081–99 levels while leaving all other variables in (5), including RH, at 1981–99 levels (cf. Fig. 5). The last panel is the estimated percentage change in the multimodel mean given these estimates for each model.
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
5. Sensitivity of results to imposed vegetation
One might wonder whether the above holds for parameter choices in (5) other than the ones presented in section 2 and the appendix. In particular, the transfer coefficient CH and bulk stomatal resistance rs could potentially modulate the Ta-independent term γ(1 + rsCH|u|) in the denominator of (5), and therefore alter 
We first examine the effect of setting rs ≡ 0, that is, neglecting the relatively small but appreciable stomatal resistance of well-watered transpiring leaves, as in many formulations of Penman–Monteith PET including those used by Burke et al. (2006) and Dai (2013), as well as in the case of pan evaporation. This gives an expression more in the spirit of Penman (1948) than Monteith (1981): the denominator of (5) simply becomes Δ + γ. This choice should systematically increase fΔ and thus reduce the percentage change in 
Figure 14 shows the percentage changes in PET from changing Ta in this case. Comparison with the analogous Fig. 5 shows that setting rs ≡ 0 indeed weakens the response, making single-digit- percentage values somewhat more common and values > 30% less common, but the patterns are very close. The at-a-point differences between the two figures are much less than the spatial and model-to-model variations within each figure, and the summary statistics in Table 2 differ by only about 2%–3% (of the basic state).
Percent changes in climatological annual-mean PET from setting only the ambient air temperature Ta to 2081–99 levels while leaving all other variables in (5) at 1981–99 levels, for the version of (5) in which rs = 0 (cf. Fig. 5).
Citation: Journal of Climate 27, 4; 10.1175/JCLI-D-13-00233.1
We also examine a “smooth” version of (5), in which the 0.5-m vegetation height h and thus the roughness lengths zom and zoh in (A1) are reduced by a factor of 10, setting h to a grasslike 5 cm and halving CH from ≈5.7 × 10−3 to ≈2.8 × 10−3. [The Penman–Monteith formulations used in Burke et al. (2006), Dai (2013), and Feng and Fu (2013) also assume a smoother surface.] This, too, shifts the range of 
Also, in the no-resistance case, adding this smooth vegetation would not appreciably lower the results any further because, in that case, CH does not even appear in the denominator of (5) and thus can no longer affect 
Finally, we examine a “rough”, forestlike PET in which h, zom, and zoh are increased by a factor of 10, setting h to 5 m and tripling CH to ≈1.7 × 10−2, a very large value. In this case, the range of 
There is also the question of whether rs, like (Rn − G), Ta, RH, and |u| (and ps), should have been treated as changing between the two epochs rather than staying fixed. After all, the carbon dioxide increase that causes greenhouse warming may also cause individual plant stomata to close (e.g., Sellers et al. 1996). However, there is still very large uncertainty about the bulk vegetation changes that will occur in concert with this, much larger than the uncertainty in the climate response to carbon dioxide (Huntingford et al. 2013). Almost nothing is known about this bulk response. Furthermore, the percentage sensitivity of Penman–Monteith PET (5) to a percentage change in rs turns out to depend very strongly on the vegetation parameters rs and CH, in contrast to the much weaker dependence just presented in the case of sensitivity to Ta. Therefore, in this study we decided to only scale the PET response to climate change, and not the response to carbon-dioxide-induced plant physiological change.
6. Summary and discussion
Potential evapotranspiration (PET), the rate at which surface water evaporates if available in a given climate, has been projected to increase with future greenhouse warming in most or all locations, driving strong global trends toward drought (e.g., Dai 2013; Burke et al. 2006) and/or aridity (Feng and Fu 2013). In this study, we systematically analyzed the projected response of the Penman–Monteith equation (5), the fundamental physical quantification of PET used by those studies. We found that, at least in the 13 modern global climate models listed in Table 1, the main reason for the projected PET increase is the warming itself (Fig. 5), not the greenhouse-driven increase in surface net radiation (Fig. 4). The warming causes the PET increase by widening the vapor pressure deficit e*(1 − RH) corresponding to a given relative humidity RH, and/or by increasing the local slope Δ := de*/dT of the Clausius–Clapeyron curve, which governs the partitioning between sensible and latent heat fluxes. Changes in RH are not of any strongly preferred sign and are not large enough to alter this.
The magnitude of the projected annual-mean PET increase between 1981–99 and a business-as-usual 2081–99 scenario is usually a low double-digit percentage (Figs. 5 and 3, Table 2), comparable to projections for local precipitation. This is because the numerator of Eq. (5) increases like Clausius–Clapeyron [5–6% °C−1] with constant-RH warming, but in the denominator only the first term Δ increases similarly, while the second term stays fixed. Thus, the net response of (5) to warming is sub-Clausius–Clapeyron, usually about 1.5–4% °C−1 (Fig. 11). The higher values are found in cooler climates where Δ is smaller and thus less important in the denominator of (5) [i.e., 
A key, further advantage of our scaling approach (17) is that a climate model is not even needed for a user to locally compute the sensitivity of PET to future warming. All variables within the square brackets in (17) can be computed during routine calculations of observed present-day Penman–Monteith PET. For example, the values of fΔ = Δ/[Δ + γ(1 + rsCH|u|)] and PET can be noted at each calculation time step and averaged over several years of data collection to obtain seasonally and/or diurnally resolved climatologies, which can then be used to find 
We also note that the PET percentage responses to changes in (Rn − G), RH, and |u|, depicted in Figs. 4, 6, and 7, can also be analytically scaled in the manner demonstrated for Ta in section 4, with similar levels of success. However, the modeled changes in these variables (for input to these scalings) are not as well understood as the modeled warming dTa, so these scalings do not provide as much understanding.
Finally, we are still interested in under what conditions or assumptions this large systematic PET increase with climate warming actually implies a systematic drying out of the land, as suggested by much of the work cited in section 1. To this end, we also have work in progress testing the sensitivity of modeled soil moisture to large changes in global temperature across a very wide range of continental geographies, forcing mechanisms, and land and atmospheric modeling choices.
Acknowledgments
The lead author would like to thank A. Dai for providing results on drought projections using different PET methods, and A. Swann for a key suggestion on section 4b. The authors also acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups (listed in Table 1) for producing and making available their model output. For CMIP the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. This work was supported by NSF Grants AGS-0846641 and AGS-0936059.
APPENDIX
Detailed Methods
a. Parameter and procedural choices for the Penman–Monteith equation
Allen et al. (2005) provide parameters for two different reference vegetation types: short clipped grass, and alfalfa (with the expectation that “crop coefficients” will be determined for conversion of the resulting potential evapotranspiration output to values suitable for other vegetation). We use the alfalfa values, reasoning that natural vegetation is closer to alfalfa in roughness and leafiness than it is to short clipped grass. Similarly, procedures are standardized separately for hourly and for daily calculation time steps; we use the hourly procedures on the 3-hourly model intervals. For meteorological variables, the model output is given as synoptic “snapshots” every 3 h, so for each interval we average the initial and final values of Ta, specific humidity qa, and |u| to estimate 3-h means, analogous to the hour means used by Allen et al. [Note that the raw output includes the wind components u and υ but not the speed |u|, so |u| has to be computed as 
With these choices of time step and vegetation type, the ASCE standardized procedures for variables in (5), and our few departures from them, are given as follows. A constant ps is hydrostatically estimated from the surface elevation, but for simplicity we directly use the 3-hourly ps output from the model, averaged like Ta and qa above. The e* is computed from the (3)-h-mean Ta using the empirical form e*(T) = 610.8 exp[17.27T/(T + 237.3)], where e* is in pascals and T is in degrees Celsius, and Δ using its derivative. Also, ρa is computed from the dry-air ideal gas law, using the (3)-h-mean Ta multiplied by 1.01 to account for virtual effects. A number of standardized methods are given to compute ea from measurements; we directly use the 3-h-mean model qa above, multiplying by ps/ε to convert the units (nearly identical to their method 1). RH can then be computed as ea/e*. (In a few models this RH can occasionally slightly exceed 1, presumably due to interpolation; in these cases we set RH = 1 to avoid unphysical negative values of the aerodynamic term.) The Lυ is idealized as a constant 2.45 × 106 J kg−1. A field estimation method for Rn and a simple parameterization of G are given, but we simply compute (Rn − G) from the model-output actual turbulent heat fluxes SH and LH using (3), which is still valid. These fluxes are already provided as 3-h means over our intervals, so there is no need for averaging. Then rs is set at 30 s m−1 (“open”) during the day and 200 s m−1 (“closed”) at night, where “day” and “night” are defined as Rn > 0 and Rn < 0. We use (Rn − G) > 0 and (Rn − G) < 0 instead; this is justified since Allen et al. (2005) parameterize G as a small positive fraction of Rn.
b. Determining the PET responses to individual variables
We would like to isolate the PET changes owing to changes in the individual inputs (Rn − G), Ta, RH, and |u|. However, we cannot simply give (5) the 2081–99 time series for one of these and the 1981–99 time series for all other variables because the differing synoptic histories of the two epochs would destroy any interinput correlations other than the diurnal and annual cycles, adding an artificial change to the result. So, for each of these four inputs, we compute diurnally and annually varying climatologies for each model (as for PET), further smooth them with a 7-day running mean that respects the diurnal cycle, difference the two epochs (divide them, in the case of |u|), and perturb each year of the 1981–99 input time series by this diurnally and annually varying difference (factor), creating an input time series with the climatological properties of 2081–99 but the synoptic history of 1981–99. These can then be used one at a time in (5) to isolate the responses to (Rn − G), Ta, RH, and |u|. [When we perturb (Rn − G), we still use the original 1981–99 (Rn − G) series to define day and night for setting rs. Global warming may accomplish many feats, but it certainly will not transmute night into day! Consistent with this, when computing the 2081–99 PET in section 2, we subtract our diurnally and annually varying climatological difference from each year of the 2081–99 (Rn − G) series before it is used to define night and day.]
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