Earth Syst. Dynam., 4, 455–465, 2013 Open Access www.earth-syst-dynam.net/4/455/2013/ Earth System doi:10.5194/esd-4-455-2013 © Author(s) 2013. CC Attribution 3.0 License. Dynamics

A simple explanation for the sensitivity of the hydrologic cycle to surface temperature and solar radiation and its implications for global climate change

A. Kleidon and M. Renner Max-Planck-Institut für Biogeochemie, Jena, Germany

Correspondence to: A. Kleidon ([email protected])

Received: 30 July 2013 – Published in Earth Syst. Dynam. Discuss.: 14 August 2013 Revised: 15 October 2013 – Accepted: 25 October 2013 – Published: 5 December 2013

Abstract. The global hydrologic cycle is likely to increase 1 Introduction in strength with global warming, although some studies in- dicate that warming due to solar absorption may result in a The hydrologic cycle plays a critical role in the physical different sensitivity than warming due to an elevated green- functioning of the earth system, as the phase changes of liq- house effect. Here we show that these sensitivities of the uid water to vapor require and release substantial amounts of hydrologic cycle can be derived analytically from an ex- heat. Currently, as climate is changing due to the enhanced tremely simple surface energy balance model that is con- greenhouse effect and surface warming, we would expect the strained by the assumption that vertical convective exchange hydrologic cycle to change as well. The most direct effect of within the atmosphere operates at the thermodynamic limit such surface warming is that the saturation vapor pressure of of maximum power. Using current climatic mean conditions, near-surface air would increase, which should enhance sur- this model predicts a sensitivity of the hydrologic cycle of face evaporation rates if moisture does not limit evaporation. 2.2 % K−1 to greenhouse-induced surface warming which is For current surface conditions, the saturation vapor pressure the sensitivity reported from climate models. The sensitivity of air would on average increase at a rate of about 6.5 % K−1. to solar-induced warming includes an additional term, which However, climate model simulations predict a mean sen- increases the total sensitivity to 3.2 % K−1. These sensitiv- sitivity of the hydrologic cycle (or, hydrologic sensitivity) ities are explained by shifts in the turbulent fluxes in the to global warming of about 2.2 % K−1 (Allen and Ingram, case of greenhouse-induced warming, which is proportional 2002; Held and Soden, 2006; Allan et al., 2013), with some to the change in slope of the saturation vapor pressure, and in variation among models. This sensitivity is also reported for terms of an additional increase in turbulent fluxes in the case climate model simulations of the last ice age (Boos, 2012; Li of solar radiation-induced warming. We illustrate an impli- et al., 2013), and is commonly explained in terms of radiative cation of this explanation for geoengineering, which aims changes in the atmosphere (Mitchell et al., 1987; Takahashi, to undo surface temperature differences by solar radiation 2009). management. Our results show that when such an interven- Some studies on the sensitivity of the hydrologic cy- tion compensates surface warming, it cannot simultaneously cle compared the response to elevated concentrations of compensate the changes in hydrologic cycling because of the carbon dioxide (CO2) with the sensitivity to absorbed so- differences in sensitivities for solar vs. greenhouse-induced lar radiation. For instance, Andrews et al. (2009) report surface warming. We conclude that the sensitivity of the hy- a hydrologic sensitivity from the Hadley Centre climate −1 drologic cycle to surface temperature can be understood and model of 1.5 % K for a doubling of CO2, while the sim- predicted with very simple physical considerations but this ulated sensitivity for a temperature increase due to ab- needs to reflect on the different roles that solar and terrestrial sorbed solar radiation was 2.4 % K−1. The study by Bala radiation play in forcing the hydrologic cycle. et al. (2008) compared the effects of doubled CO2 to a geoengineering scheme that reduces solar radiation. They

Published by Copernicus Publications on behalf of the European Geosciences Union. 456 A. Kleidon and M. Renner: Hydrologic cycling and global climatic change also found different hydrologic sensitivities for greenhouse- solar terrestrial induced and solar-radiation-induced changes in surface tem- radiation radiation 4 perature. Govindasamy et al. (2003), Lunt et al. (2008) and Rs σ Ta Tilmes et al. (2013) report similar effects, namely, that the hydrologic cycle reacts differently to surface temperature dif- atmospheric temperature Ta ferences when the warming results from an enhanced green- house effect or enhanced absorption of solar radiation at the surface. radiative Strictly speaking from a viewpoint of saturation vapor heat convective exchange engine exchange w pressure, we would not expect such a difference in hydro- Rl = kr (Ts - Ta) logic sensitivity to surface temperature that would depend sensible latent heat on whether the surface temperature difference was caused by heat flux H flux λ E differences in solar or terrestrial radiation. However, when surface temperature Ts we focus on the surface energy balance rather than the sat- uration vapor pressure, it is quite plausible to expect such a difference in sensitivity. After all, the primary cause for sur- Fig. 1. Schematic illustration of the simple energy balance model face heating is the absorption of solar radiation, while the that is used to describe the strength of the hydrologic cycle through exchange of terrestrial radiation as well as the turbulent heat the rate of surface evaporation, E, with the main variables and fluxes fluxes generally cool the surface. When the surface warms used here. After Kleidon and Renner (2013). because of changes in the atmospheric greenhouse effect, then the rate of surface heating by absorption of solar ra- 2 Model description diation remains the same, so that the total rate of cooling by terrestrial radiation and turbulent fluxes remains the same as We use the approach of Kleidon and Renner (2013), which well. In case the warming is caused by an increase in the describes a thermodynamically consistent global steady state absorption of solar radiation, then the overall rate of cool- of the surface–atmosphere system in which the hydrologic ing by terrestrial radiation and turbulent fluxes needs to in- cycle is represented by evaporation (which balances precip- crease. Hence, we should be able to infer such differences in itation, E = P , in steady state). The layout of the model as the hydrologic sensitivity by considering the surface energy well as the main fluxes is shown in Fig. 1. The model uses balance. the surface and global energy balance to describe the surface In this paper, we show that hydrologic sensitivities can temperature, Ts, as well as the (atmospheric) radiative tem- be predicted by simple surface energy balance considera- perature, Ta. The surface is assumed to be an open water sur- tions in connection with the assumption that convective mass face, all absorption of solar radiation is assumed to take place exchange within the atmosphere operates at the thermody- at the surface, and it is assumed that the atmosphere is opaque namic limit of maximum power (Kleidon and Renner, 2013). for terrestrial radiation so that all radiation emitted to space This approach will be briefly summarized in the next section, originates from the atmosphere. Atmospheric dynamics, and while the detailed thermodynamic derivations of the maxi- particularly the turbulent heat fluxes, are not explicitly con- mum power limit, a fuller description of the assumptions and sidered, but rather inferred from the thermodynamic limit of limitations as well as the comparison to observations can be generating convective motion. The important point to note is found in the appendix and in Kleidon and Renner (2013). The that for convective exchange to take place in a steady state, analytic solution of this model will then be used to derive motion needs to be continuously generated against inevitable analytical expressions of the hydrologic sensitivity to sur- frictional losses. This kinetic energy is generated out of heat- face temperature in Sect. 3 for differences in the atmospheric ing differences akin to a heat engine (as shown in Fig. 1). The greenhouse effect as well as for differences in absorption of conversion of heat to kinetic energy by this heat engine is solar radiation. These sensitivities are compared to the sen- thermodynamically constrained, and such a thermodynamic sitivities obtained from numerical climate model studies. We limit sets the limit to the turbulent exchange at the surface. A provide a brief explanation of these differences from an en- brief derivation of this limit from the laws of thermodynam- ergy balance perspective in Sect. 4, discuss the limitations of ics is provided in the Appendix. We will refer to this limit and our approach, and illustrate one implication of our interpre- the associated state of the surface energy balance as the state tation for geoengineering approaches to global warming. We of maximum power, with power being the physical measure close with a brief summary, in which we also point out defi- of the rate at which work is being performed. We then mea- ciencies in the concept of radiative forcing that is often used sure the strength of the hydrologic cycle by the value of E at in analyses of global warming and possible extensions of our this maximum power state. approach to other aspects of global climatic change. In the model, the surface energy balance is expressed as

0 = Rs − Rl − H − λE, (1)

Earth Syst. Dynam., 4, 455–465, 2013 www.earth-syst-dynam.net/4/455/2013/ A. Kleidon and M. Renner: Hydrologic cycling and global climatic change 457 where Rs is the absorbed solar radiation at the surface (which that the net radiation of the surface at a state of maximum is prescribed), Rl the net cooling of the surface by terres- convective power is half of the absorbed solar radiation, Rs. trial radiation, H the sensible heat flux, and λE the latent This partitioning between radiative and turbulent heat heat flux. We use simple, but common formulations for these fluxes at the surface is associated with a characteristic tem- fluxes which are simple enough to obtain analytical results. perature difference, Ts − Ta, which can be used to infer the For the net radiative cooling, we assume a simple linearized associated temperatures. The radiative temperature of the at- form, Rl = kr(Ts − Ta). Here, kr is a linearized radiative “con- mosphere, Ta, follows directly from the global energy bal- ductance” that relates to the strength of the greenhouse ef- ance, eqn. 2, and is unaffected by the partitioning: fect. The sensible and latent heat fluxes are expressed as tur-  1/4 bulent exchange fluxes in the form of H = cp ρw(Ts − Ta) Rs Ta = . (5) and λE = λρw(qsat(Ts) − qsat(Ta)). The heat capacity of σ 3 −3 −1 air is cp ρ = 1.2 × 10 J m K , with a density of about ρ = 1.2 kg m−3; w is a velocity which describes the rate of Surface temperature, Ts, at the maximum power state is vertical mass exchange and is determined below from the derived from the expression of net radiative exchange, thermodynamic maximum power limit; λ = 2.5 × 106 JK−1 Rl,opt = kr(Ts − Ta) = Rs/2, and is given by is the latent heat of vaporization; qsat = 0.622 esat/p is the Rs saturation specific humidity; esat is the saturation vapor pres- Ts = Ta + . (6) sure, and p = 1013.25 hPa is surface air pressure. For the sat- 2kr uration vapor pressure, we use the numerical approximation In Kleidon and Renner (2013), we showed that this model a−b/T of esat(T ) = e0 · e (Bohren and Albrecht, 1998), with reproduces the global evaporation rate as well as poleward e0 = 611 Pa, a = 19.83 and b = 5417 K and temperature T in moisture transport very well. It is important to note, how- K. The global energy balance yields an expression for the ever, that the expression for evaporation given by Eq. (4) temperature Ta: represents the maximum evaporative flux that is achieved by locally generated motion near the surface only. In practice, 0 = R − σ T 4, (2) s a the equilibrium evaporation rate is often corrected by the where σ is the Stefan–Boltzmann constant. Priestley-Taylor coefficient (Priestley and Taylor, 1972) of The strength of the convective heat fluxes are derived from ca. 1.26, which can be understood as the effect of horizontal the assumption that surface exchange is driven mostly by lo- motion that is generated by horizontal differences in absorp- cally generated buoyancy at the surface, and that the power to tion of solar radiation (Kleidon and Renner, 2013). However, as this coefficient simply acts as a multiplier, it does not af- generate motion by dry convection, H · (Ts − Ta)/Ts is max- imized. The Carnot limit has a maximum, because a greater fect the relative sensitivity of evaporation to changes in the surface energy balance. Also note that evaporation driven by value of H is associated with a smaller value of Ts − Ta due to the constraint imposed by the surface energy balance. This local convection by surface heating can already explain more than 70 % of the strength of the present-day hydrologic cy- tradeoff between H and Ts − Ta results in a distinct state of maximum power associated with convective exchange at in- cle (Kleidon and Renner, 2013). We will therefore consider termediate values for these two terms (see also Appendix). only this locally driven rate of evaporation in the following The maximization is achieved by optimizing the vertical ex- derivation of the sensitivities. change velocity w. At maximum power, the optimum value for the vertical exchange velocity, wopt, is given by 3 Results γ R w = s , (3) To derive the hydrologic sensitivity to surface temperature, opt + − s γ 2cp ρ (Ts Ta) we are interested in the expression 1/E dE/dTs. We first note that Ts is not the independent variable of our model, because where γ = 65 Pa K−1 is the psychrometric constant and Ts = Ts(kr,Rs) with the relationship given by Eq. (6), and s = de /dT is the slope of the saturation vapor pressure sat s that solar radiative forcing, Rs, and the greenhouse param- curve. This maximum power state results in an energy par- eter, kr, are our independent variables. We can, however, use titioning at the surface of Eq. (6) to make Ts and kr our independent variables, and Rs our dependent variable. This sounds a bit backward, but is Rs γ Rs s Rs R = H = λE = . (4) mathematically sound and allows us to compute 1/E dE/dT l,opt 2 opt s + γ 2 opt s + γ 2 s analytically.

The expression of Eopt is nearly identical to the equilibrium We now use the expression of Eopt in Eq. (4) as the evap- evaporation rate (Slayter and McIlroy, 1961; Priestley and oration rate to derive the hydrologic sensitivity. This expres- Taylor, 1972), a concept that is well established in estimating sion depends on s and Rs, which are both related to our in- evaporation rates at the surface, with the additional constraint dependent variable Ts. The derivative is thus given by www.earth-syst-dynam.net/4/455/2013/ Earth Syst. Dynam., 4, 455–465, 2013 458 A. Kleidon and M. Renner: Hydrologic cycling and global climatic change

To quantify this first term of the sensitivity for present- 1 dE 1 ∂E ds 1 ∂E ∂Rs −2 = + . (7) day conditions, we use Rs = 240 W m and derive a value E dTs E ∂s dTs E ∂Rs ∂Ts −2 −1 for kr = 3.64 W m K indirectly from the observed global −1 mean temperatures, Ts = 288 K and Ta = 255 K and from Since ∂Rs/∂Ts = (∂Ts/∂Rs) , we can also express this as Eq. (6) above. With this radiative forcing and values of  −1 γ = 65 Pa K−1 and s = 111 Pa K−1, we obtain a numerical 1 dE 1 ∂E ds 1 ∂E ∂Ts = + (8) value of this sensitivity of E dTs E ∂s dTs E ∂Rs ∂Rs 1 ∂E ds −1 for which the derivative ∂T /∂R can be directly calculated ≈ 2.2%K (10) s s E ∂s dTs from Eqs. (6) and (5). We refer to Eq. (8) as the hydrologic which matches the mean sensitivity of climate models of sensitivity. 2.2 % K−1 (Allen and Ingram, 2002; Held and Soden, 2006; The hydrologic sensitivity consists of two terms. The first Li et al., 2013). term on the right hand side expresses the dependence of evap- The second term of Eq. (8) is due to a difference in absorp- oration on s, which depends strongly on surface temperature, tion of solar radiation, 1R , and is given by while the second term describes the dependence of evapora- s tion on the solar radiative forcing, which also affects surface 1 ∂E  ∂T −1 4k σ 1/4 · s = r . (11) temperature. 1/4 1/4 E ∂Rs ∂Rs 2σ Rs + kr Rs When a difference in surface temperature, 1T , is caused s This sensitivity depends only on radiative properties and re- by changes in the atmospheric greenhouse effect (i.e., a dif- sults in a sensitivity of ferent value of kr), then the solar radiative heating is a con- stant and 1/E dE/dT = 1/E ∂E/∂s ds/dT . This sensitivity 1 ∂E  ∂T −1 s s · s ≈ −1 represents only a shift in the partitioning between the sensi- 1%K . (12) E ∂Rs ∂Rs ble and latent heat flux, as the overall magnitude of turbulent This sensitivity is about half the value of the first term when fluxes does not change since R does not change. s evaluated using present-day conditions, so that the total hy- If 1T is caused by a difference in R , then 1/E dE/dT s s s drologic sensitivity to surface temperature change caused by consists of two terms, expressing the change of evaporation solar radiation is about 3.2 % K−1 and thus exceeds the above due to a change in s that is caused by the increase in temper- sensitivity to changes in the atmospheric greenhouse effect. ature, but also the overall increase in turbulent fluxes due to These sensitivities are shown graphically in Fig. 2a. The the increase in R . Hence, we would expect different hydro- s relative proportion of this sensitivity to that caused by logic sensitivities to surface temperature, depending on the changes in the atmospheric greenhouse is consistent with the type of radiative change. Changes in the greenhouse effect af- proportions reported by Bala et al. (2008) and Andrews et al. fect the first term of the right hand side of Eq. (8) only, while (2009). In both studies, the authors reported a sensitivity to changes in solar radiation affect both terms of the right hand surface temperature caused by changes in the atmospheric side of Eq. (8) and thus should result in a greater sensitivity. greenhouse of 1.5 % K−1, while the sensitivity to changes The first term in Eq. (8) expresses the change of evapora- in solar radiation was given as 2.4 % K−1. While the mag- tion, E, to surface temperature, T , by altering the value of s: s nitude of the sensitivity is smaller compared to the sensitiv- 1 ∂E ds γ 1 ds ities calculated here and most other climate models (Allen = . (9) E ∂s dTs s + γ s dTs and Ingram, 2002; Held and Soden, 2006; Li et al., 2013), the sensitivity to temperature differences caused by differ- We note that this sensitivity does not involve the rela- ences in solar radiation is about 60 % greater than those due tive change in saturation vapor pressure 1/esat desat/dTs, to differences in the greenhouse effect, which is similar to but rather the relative change in the slope in saturation va- the difference that is estimated here. por pressure 1/s ds/dTs. The proportionality to the slope We will next look at the sensitivities of convective mass 1/s ds/dTs, rather than 1/esat desat/dTs, is due to the fact that exchange that is associated with these differences in hydro- the intensity of the water cycle does not depend on esat(Ts), logic cycling. The sensible and latent heat flux are accom- but rather on the difference of esat(Ts) − esat(Ta), which is plished by convective motion, which exchanges the heated approximated in our model by the slope s. Hence, the sensi- and moistened air near the surface with the cooled and dried tivity of the hydrologic cycle does not follow 1/esat desat/dT , air of the atmosphere. To evaluate the sensitivity of convec- but rather 1/s ds/dT . The sensitivity is further reduced by a tive motion to surface temperature, we evaluate the relative + factor γ /(s γ ), which originates from the energy balance difference in w in response to a difference in Ts, for which (and maximum power) constraint and ensures that E is not we use the expression of wopt as given in Eq. (3): unbound with much higher values for Ts, but converges to an 1 dw 1 ∂w 1 ∂w  ∂T −1 upper limit of Rs/2. = + · s . (13) w dTs w ∂Ts w ∂Rs ∂Rs

Earth Syst. Dynam., 4, 455–465, 2013 www.earth-syst-dynam.net/4/455/2013/ A. Kleidon and M. Renner: Hydrologic cycling and global climatic change 459

a. evaporation The second term in Eq. (13) describes the indirect effect )

7-1 % 6.5 of differences in solar radiation on w through differences in 6 %6 Ts:

5 (% K %  −1 2 ! 1/4 1 ∂w ∂Ts kr 4kr σ 4 % · = kr + · . (15) 4 1/4 7/4 1/4 + 3.2 w ∂Rs ∂Rs 2σ Rs 2σ Rs kr Rs1/4 3 % 2.2 2.2 2.2 − 2 %2 This expression yields a sensitivity of +3.8 % K 1, so that 1 % 1.0 the total sensitivity of convective mass exchange to tempera- 0 %0 ture differences caused by differences in absorption of solar relative sensitivity relative s esat s term 2 GCM radiation is −2.9 % K−1. This sensitivity is noticeably less T R

GCM than the sensitivity to changes in the atmospheric greenhouse

(eqn. 9) (eqn. effect (see also Fig. 2b). (eqn. 11)(eqn. pressure sensitivity

greenhouse In summary, we have shown here that our analytical ex- sensitivity to sensitivity sensitivity to sensitivity

solar radiation solar pressions for the sensitivity of evaporation rate, Eq. (9), can sensitivity to sensitivity sensitivity to sensitivity saturation vapor saturation reproduce the reported mean sensitivity of climate models to greenhouse-induced temperature differences. Due to an addi- b. convective mass exchange

) tional term that relates to changes in absorbed solar radiation

5-1 % +3.8 4 (Eq. 11), the hydrologic sensitivity is greater when the tem- 3 % perature increase is due to an increase in the absorption of (% K 2 1 % solar radiation, which is also consistent with what is reported 0 -1 % from climate model studies. Associated with these changes -2 in the hydrologic cycle are changes in the intensity of verti- -3 % -4 -2.9 cal mass exchange, which depend on the type of change in -5 % the radiative forcing. Hence, our approach appears to repre- -6 -6.7 -6.7 -7 % sent a simple yet consistent way to capture the mean aspects relative sensitivity relative s terms 1 term 2 greenhouseUntitled 1 T R of climate change that are reflected in surface temperature differences. (eqn. 15) (eqn. (eqn. 14) (eqn. greenhouse sensitivity to sensitivity sensitivity to sensitivity

solar radiation solar 4 Discussion sensitivity to sensitivity sensitivity to sensitivity

Fig. 2. Sensitivity of (a) the hydrologic cycle (evaporation E) and Before we interpret our results in more detail, we first dis- (b) convective mass exchange (exchange velocity w) to differences cuss some of the limitations of our approach and evaluate in surface temperature (Ts). Shown are the numerical values for the the extent to which these affect the results. We then interpret relative sensitivities as given in the text for present-day conditions. our results for the hydrologic sensitivity and relate this in- Also included in (a) is the sensitivity of saturation vapor pressure, terpretation to previous explanations. We close with a brief 1/esat desat/dTs, as well as the mean sensitivity to greenhouse dif- discussion of one of the implications of our work for the cli- ferences reported for climate models by Held and Soden (2006) matic impacts of climate geoengineering by solar radiation (“GCM sensitivity”). management.

4.1 Limitations As in the case of evaporation, the sensitivity consists of two terms, with the first term representing the direct response of Naturally, we have made a number of assumptions in our w s T to and s. This first term is given by approach. These assumptions relate to the assumption of 1 ∂w s 1 ds 1 (a) the maximum power limit for convective exchange, (b) a = − − . (14) steady state of the energy balances, (c) surface exchange be- w ∂T s + γ s dT T − T s s s a ing caused by local heating, and (d) a simple treatment of Using the values from above, this yields a sensitivity of processes in our model. −6.7 % K−1. The sensitivity is negative, implying that con- The use of the maximum power limit provided a means to vective mass exchange is reduced by a stronger greenhouse constrain the convective exchange in our model. If this limit effect. This sensitivity is consistent with previous interpre- would not have been invoked, the magnitude of the turbulent tations as described by Betts and Ridgway (1989) and Held heat fluxes would be unconstrained, and some form of em- and Soden (2006), and the estimates of about 4–8 % reported pirical treatment of these fluxes would be required, typically by Boer (1993). with an empirically derived value of the drag parameter. The application of a thermodynamic limit to convective exchange www.earth-syst-dynam.net/4/455/2013/ Earth Syst. Dynam., 4, 455–465, 2013 460 A. Kleidon and M. Renner: Hydrologic cycling and global climatic change avoids this empirical parameter. This limit relates closely to rather than 240 W m−2 of solar radiation which is absorbed the hypothesis that atmospheric motion maximizes material at the surface. We used this simplification to keep the model entropy production, noting that in steady state, power equals as simple as possible (otherwise, we would need to account dissipation, and entropy production is described by dissipa- for atmospheric absorption in the expression for Ta). For the tion divided by temperature. This hypothesis was first pro- hydrologic sensitivity, this simplification plays a minor role posed by Paltridge (1975), and has been quite successful, for because the sensitivity is formulated in relative terms, which instance in predicting heat transport in planetary atmospheres is independent of Rs (at least the first term in Eq. 8). In ad- (Lorenz et al., 2001), in deriving an empirical parameter re- dition, we assumed that the atmosphere is mostly opaque for lated to turbulence in a general circulation model (Kleidon terrestrial radiation. This assumption does not hold for all re- et al., 2003), and other applications in climate science (e.g., gions. Particularly in dry and cold regions, the atmosphere Ozawa et al., 2003). Hence, the assumption that atmospheric is more transparent to terrestrial radiation. This would affect motion operates near such a thermodynamic limit, while not our model in which it is assumed that all terrestrial radiation widely recognized, has considerable support. For the deriva- to space originates from the atmosphere (cf. Eq. 2). tion of our sensitivities, this assumption only matters to the Overall, while we made several assumptions and simplifi- extent that it predicts that net radiation does not change in cations in obtaining our results, it would seem that our results the case of greenhouse-induced warming. In other words, are rather robust. These assumptions may need to be revis- the derivation of the hydrologic sensitivity to greenhouse- ited and refined when using this approach at different scales induced warming (Eq. 9) could have been done with the as- or conditions. For instance, when this approach is applied to sumption that net radiation does not change. Likewise, the land, then one would need to account for the additional con- hydrologic sensitivity to solar-induced changes of surface straint of water limitation. When it is applied to the diurnal temperature (Eqs. 9 and 11) could have been derived from cycle, one would clearly need to account for changes in heat the assumption that the ratio between radiative and turbulent storage. These factors can, of course, be included in an exten- cooling remains fixed. Both of these assumptions can then be sion of the approach, but they should nevertheless not affect justified and explained by the maximum power limit. our results at the global scale in the climatic mean. We also assumed that the energy balances of the sur- face and the atmosphere are in a steady state. This assump- 4.2 Interpretation tion ignores the temporal variations on diurnal and seasonal timescales, which result in the dynamics of boundary layer The interpretation of our results is relatively straightforward growth and changes in heat storage. These aspects are most and can be attributed entirely to changes in the surface en- relevant on land, while over the ocean, these aspects are ergy balance. This focus on changes in the surface energy likely to play a minor role due to the large heat capacity of balance is plausible, because after all, convective mass ex- water. Since the sensitivity of the hydrologic cycle is domi- change, the associated transport of sensible and latent heat, nated by the oceans, it would thus seem reasonable to neglect and hence hydrologic cycling is caused by surface heating. these variations. It is important to note that the actual heating of the surface Another assumption that we have made is that the turbu- is solely due to the absorption of solar radiation, Rs, while lent exchange at the surface results only from local surface terrestrial radiation, Rl, cools the surface. In the following, heating. This assumption neglects the fact that the large-scale we explain these changes and illustrate these for an example circulation adds extra turbulence to the surface, thus gener- of a surface warming of 1Ts = 2 K, which is shown in Fig. 3. ating more turbulence at the surface than what would be ex- When the surface warming is entirely caused by an in- pected by local heating alone. This extra contribution would crease of the atmospheric greenhouse effect, Rs is effectively shift the partitioning in the surface energy balance towards unchanged, but the cooling of the surface by terrestrial ra- turbulent heat fluxes. In the framework of the equilibrium diation is less efficient. In our model, this reduced cooling evaporation rate, this shift can be interpreted by the Priestley- efficiency is reflected in a lower value of kr. This lower value Taylor coefficient. We incorporated this effect in Kleidon and of kr , however, does not affect the partitioning of absorbed Renner (2013) by introducing a factor into the formulation of solar radiation into radiative and turbulent cooling, Rl and the sensible and latent heat flux, but we did not use this factor H + λE, at the maximum power state. This is noticeable in here. The reason for omitting this factor is that as long as this Eq. (4), since the partitioning does not depend on the value factor is independent of Ts, the relative sensitivities that we of kr. Hence, Rs and Rl do not change (cf. Fig. 3, blue bars). derived here are not affected as this factor would cancel out. However, because kr is reduced, it requires a greater temper- Hence, this large-scale contribution to turbulent exchange is ature difference, Ts − Ta, to accomplish the same radiative unlikely to result in substantially different sensitivities. cooling flux, Rl. Since Ta is fixed by the global energy bal- In addition, we implemented processes in our approach in ance and is independent of kr, this can only be accomplished a simplified way. We assumed that all absorption of solar ra- by an increase in Ts. This surface warming is then associated diation takes place at the surface, while observations (e.g., with a different partitioning between sensible and latent heat, Stephens et al., 2012) state that it is only about 165 W m−2 because the slope of the saturation vapor pressure curve, s,

Earth Syst. Dynam., 4, 455–465, 2013 www.earth-syst-dynam.net/4/455/2013/ PD +2K (kr) +2K GEO (Rs) Rs (W m-2) 240 240 0.00 245 5.00 235.00 -5.00

Ts (K) 288 290 2.00 290.00 2.00 287.93 -0.07

Ta (K) 255 255 0.00 256.39 1.32 253.73 -1.34 s (Pa K-1) 110.77 124.38 13.61 124.41 13.64 110.34 -0.43 kr (W m-2 3.64 3.44 -0.21 3.64 0.00 3.44 -0.21 K-1) Rl,u (W m-2) 390 401 10.95 401 10.97 390 -0.36

-2 A. Kleidon and M. Renner: Hydrologic cycling and global climatic change 461 Rl,d (W m ) 270 281 10.95 279 8.47 272 2.14

Rl (W m-2) 120 120 0.00 122.5 2.50 117.5 -2.50

H (W m-2) 44 41 -3.19 42 -2.34 44 -0.82

λE (W m-2) 76 79 3.19 80 4.84 74 -1.68 w (mm s-1) 1.12 0.98 -0.14 1.04 -0.08 1.06 -0.06 warming by greenhouse The situation is different when the surface warms due to warming by solar radiation enhanced absorption of solar radiation (Fig. 3, red bars). solar geoengineering In this case, the surface is heated more strongly (Rs is in- creased), so the rate of cooling, Rl + H + λE, is increased 6 5.0 4.8 as well. Apart from the difference in surface temperature and )

-2 4 3.2 the associated differences in the partitioning between the sen- 2.5 2 sible and latent heat flux, the overall magnitude of the turbu- (W m 0.0 0.0 lent fluxes is altered as well. Hence, the sensitivity is greater 0 than in the case of greenhouse warming, which is noticeable -0.8 -2 -1.7 in our example by the increase in sensible and latent heat -2.5 -2.3 (compare red vs. blue bars in Fig. 3). The additional con-

difference -3.2 -4 tribution by the overall increase in turbulent fluxes depends -5.0 -6 on Rs and on the temperature difference, which depends on l

s

E R k H s and r. Consequently, the second term in the sensitivities R R λ depends explicitly on the radiative properties of the system

flux, (Rs, kr, cf. Eqs. 11 and 15). This enhancement of the turbu- lent fluxes favors greater convective mass exchange, so that radiation, radiation, radiation, radiation, net terrestrial net terrestrial sensible heat heat sensible the sensitivity of convective mass exchange is reduced com- absorbed solarabsorbed latent heat flux,heat latent pared to differences caused by a stronger greenhouse effect. Fig. 3. Estimates of the changes in the surface energy balance Our interpretation is quite different from the common ex- components due to a warming of 1Ts = 2 K caused by an in- planation for the hydrologic sensitivity (e.g., Mitchell et al., crease in the atmospheric greenhouse effect (blue, “warming by 1987; Allen and Ingram, 2002; Takahashi, 2009; Allan et al., greenhouse”), by an increase in absorbed solar radiation (red, 2013). The common explanation starts by considering the at- “warming by solar absorption”), and when a greenhouse warm- mospheric energy balance. Surface warming results in a per- ing of 2 K is compensated for by a reduction of solar radiation turbation of this energy balance. It accounts for the extra re- by some geoengineering management (yellow, “solar geoengineer- −2 lease of latent heat, λ1P , the change in radiative cooling ing”). The numbers are obtained using values of Rs = 240 W m −2 −1 of the atmosphere to space, 1Rtoa, the change in radiative and kr = 3.64 W m K for the present-day climate, a reduction −2 −1 fluxes from the surface, 1Rl, and a change in the sensible of kr to kr = 3.44 W m K to get a surface warming of 2 K by −2 heat flux, 1H: changes in the greenhouse effect, an increase of 1Rs = 5 W m to get a surface warming of 2 K by changes in absorption of so- −2 −1 λ1P = 1Rtoa − 1Rl − 1H. (16) lar radiation, and a combined change of kr = 3.44 W m K and − −2 1Rs = 5 W m to implement the solar radiation management The common explanation for the lower sensitivity of precip- effects. itation to surface warming compared to the sensitivity of the saturation water pressure argues that the additional release of has a greater value at a warmer temperature, resulting in a latent heat, λ1P , is constrained by the ability to radiate away greater proportion, s/(s + γ ), of the turbulent cooling be- the additional heat by the term 1Rtoa − 1Rl. The term 1H ing represented by the latent heat flux, thus resulting in a in these considerations is commonly neglected because H is stronger hydrologic cycle (Fig. 2a). In the example shown in quite a bit smaller than the latent heat flux. Fig. 3 the consequence of the warming is reflected merely in This energy balance is, of course, indirectly also obeyed the shift from the sensible heat flux to the latent heat flux, but in our model even though we do not explicitly consider the magnitude of both does not change. it. First, we consider a steady state, so that 1Rs = 1Rtoa, λ1P = λ1E, and, 1/P · dP/dT = 1/E · dE/dT . Since the difference Ts − Ta is enhanced, the turbulent s s heat fluxes are accomplished by less convective mass ex- We first consider the case of greenhouse-induced sur- face warming. In this case, changes in the greenhouse effect change, which results in the negative sensitivity 1/w ∂w/∂Ts (Fig. 2b). Since both sensitivities deal with the intensity of do not change the radiative temperature of the atmosphere convective transport and its partitioning into sensible and la- (which is entirely determined by Rs, Eq. 5), hence, 1Rtoa = 0. tent heat, the sensitivities are expressed only in terms of re- The term Rl does not change either, because the surface heat- ing by solar radiation did not change (1R = 0) and the maxi- lated properties (s, γ , Ts − Ta, cf. Eqs. 9 and 14), but do not s mum power constraint results in an equal partitioning among depend explicitly on radiative properties of the system (Rs, R and H + λE, no matter how strong the greenhouse effect kr). This interpretation is consistent with the general under- l standing of the greenhouse effect, but it emphasizes that the is. Hence, the overall changes in the atmospheric energy bal- atmospheric greenhouse effect acts to reduce the efficiency ance reduce to by which the surface cools through the emission of terres- λ1P = −1H. (17) trial radiation.

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− This implies that the weak, 2.2 % K 1 increase in the strength a. evaporation of the hydrologic cycle can simply be explained by the re- ∆T = 2K duction of the sensible heat flux at the surface. This inter- ∆T = 2K

pretation is identical to what we found for the changes in the (%) 10 surface energy balance: a greenhouse-induced surface warm- ing merely affects the partitioning between sensible and la- 5 B tent heat, but does not affect the magnitude of the turbulent ∆E/E = 4.4% A heat fluxes (see also example in Fig. 3, blue bars). This ex- 0 planation is different to the common explanation, which ne- ∆E/E = -6.4% C glects changes in the sensible heat flux. In our explanation, -1 −5 -1 the hydrologic sensitivity due to greenhouse-induced surface 2.2 % K 3.2% K warming is entirely due to the reduction of H. −10 Difference in Difference Evaporation The changes in the atmospheric energy balance are dif- −4 −3 −2 −1 0 1 2 3 4 ferent if the surface temperature change was caused by Difference in Temperature (K) changes in solar radiation. If absorbed solar radiation in- creases by 1Rs, then the global energy balance requires that b. convective mass exchange 1R = 1R , so that the radiative temperature T must in- toa s a ∆T = 2K crease. The partitioning of energy at the surface changes as ∆T = 2K well. At a state of maximum power, the additional heating 20 (%) of 1Rs results in an equal increase in radiative and turbulent -6.7 % K fluxes of 1Rl = 1Rs/2, and of 1(H + λE) = 1Rs/2. In ad- 10 -1 dition, the increase in surface temperature alters the partition- -2.9% K -1 A ing between H and λE. Hence, in this case, all four terms are 0 going to change in Eq. (16), which is quite a different change ∆w/w = -13.4% than the greenhouse-induced warming. −10 C B ∆w/w = +5.8% Overall, our explanation is quite different to the common

explanation of the hydrologic sensitivity. Yet, our explana- in Difference Exchange −20 tion is simple, physically based, consistent with the atmo- −4 −3 −2 −1 0 1 2 3 4 spheric energy balance, and predicts the right value of the Difference in Temperature (K) sensitivities. Fig. 4. Illustration of the contrary effects of greenhouse vs. solar- induced changes on (a) evaporation, E, and (b) convective mass 4.3 Implications exchange, w. The sensitivity of evaporation to a surface warm- ing of 1Ts = 2 K by an elevated greenhouse effect results in an in- An important implication of our interpretation of the hydro- crease of 2.2 % K−1, resulting in a change from point A to B (blue logic sensitivity is that the forcing of the surface cannot be line). A geoengineering response aimed at compensating this in- simply lumped into a single, radiative forcing concept. The crease in surface temperature by reducing solar radiation would notion of a “radiative forcing” combines the changes in solar change evaporation at a rate of 3.2 % K−1, resulting in a change and terrestrial radiation into one variable. However, as these from point B to C (red line). Likewise, the surface warming would − sensitivities show, solar radiation plays a very different role decrease the vertical exchange velocity, w, by −6.7 % K 1 (point A than terrestrial radiation. The strength of hydrologic cycling to B), while the geoengineering response would increase it at a rate −1 as well as convective mass exchange react quite differently of 2.9 % K (point B to C). Hence, while the geoengineering re- if the surface is warmed due to stronger heating by solar ra- sponse may undo differences in surface temperature, it cannot com- pensate changes in the hydrologic cycle and vertical mass exchange diation or due to a weaker cooling by a stronger greenhouse at the same time. effect. An immediate consequence of this notion is that cli- mate geoengineering cannot simply be used to undo global warming (see also Bala et al., 2008 and Tilmes et al., 2013). This can be illustrated using the sensitivities given above. by 4.4 % with a warming of 2 K. This increase is shown by We consider the case of surface warming of 2 K caused the arrow in Fig. 4a from the original climatic state “A” to by an enhanced greenhouse effect, as before, except that the state in which the surface is heated by 2 K (point “B”). we look at the relative sensitivities rather than the absolute The convective mass exchange would be reduced following changes in the surface energy balance. Since this increase the sensitivity 1/w ∂w/∂Ts (Eq. 14), which has a value of of surface temperature is caused by the greenhouse effect, −6.7 % K−1. With a 2 K warming, the convective mass ex- the hydrologic cycle would be strengthened by the sensitivity change would be reduced by 13.4 % (Fig. 4b). 1/E ∂E/∂s ds/dTs (Eq. 9 and blue bars in Fig. 3). This sen- If this surface warming is reduced by a reduction of ab- sitivity has a value of 2.2 % K−1, so that E would increase sorbed solar radiation (cf. yellow bars in Fig. 3), as proposed

Earth Syst. Dynam., 4, 455–465, 2013 www.earth-syst-dynam.net/4/455/2013/ A. Kleidon and M. Renner: Hydrologic cycling and global climatic change 463 by some geoengineering schemes, then the value of Rs would analytical expressions yield sensitivities that are consistent change and we need to consider both terms in the sensitiv- with those found in rather complex climate models. We con- ities of evaporation (Eq. 8) and convective mass exchange clude that the hydrologic sensitivity to surface warming can (Eq. 13). The hydrologic cycle would be reduced not at a rate be explained in simple, physical considerations of the surface of 2.2 % K−1 as in the case above, but rather at the combined energy balance. value of 3.2 % K−1, which includes the additional sensitivity An important implication of our results is that geoengi- given by Eq. (11). Hence, with a cooling of 2 K that would neering approaches to reduce global warming are unlikely be necessary to undo the surface warming, the strength of the to succeed in restoring the original climatic conditions. Be- hydrologic cycle would be reduced in total by 6.4 %. This is cause of the difference in hydrologic sensitivities to solar vs. shown in Fig. 4a by the arrow from point B to C. Overall, greenhouse induced surface warming, the changes in hydro- the warming would be undone, but the strength of the hydro- logic cycling and convective mass exchange do not compen- logic cycle at this state of geoengineering (point C) would sate even if surface temperature changes are compensated by be weaker by 2 % compared to the original state (point A). solar radiation management. This example emphasizes the Likewise, the sensitivities of convective mass exchange do different roles that solar and terrestrial radiation play in the not compensate either. The cooling of 2 K by the reduction surface energy balance and challenges the frequently used ra- of absorbed solar radiation follows the weaker sensitivity of diative forcing concept, which lumps these two components −2.9 % K−1, so that the convective mass exchange would in- together. It would seem insightful to extend our study in the crease by only 5.8 % (see arrow from point B to C in Fig. 4b). future to other aspects of global climatic change, in which the Hence, overall, the greenhouse warming by 2 K and the geo- different roles in solar and terrestrial radiation are explicitly engineering cooling by 2 K would weaken convective mass considered in a thermodynamically consistent way. exchange by 6.8 % (compare point A and C in Fig. 4b, which is also seen in the yellow bars in Fig. 3). Hence, such intervention by geoengineering may undo Appendix A surface warming, but it cannot undo differences in hydro- logic cycling and convective mass exchange at the same time. Thermodynamic limits What this tells us is that it is important to consider the differ- The first and second law of thermodynamics set a fundamen- ent roles of solar and terrestrial radiation separately in future tal direction as well as limits to energy conversions within studies on the strength of the hydrologic cycle and global cli- any physical system. We apply it here to derive the limit to matic change (see also Jones et al., 2013). how much kinetic energy can be derived from the differen- tial radiative heating between the surface and the atmosphere. The following derivation summarizes Kleidon and Renner 5 Summary and conclusions (2013). To derive the limit, we consider a heat engine as marked In this study we showed that the sensitivity of the hydro- in Fig. 1 that is driven by the sensible heat flux, H . In the logic cycle to surface temperature can be quantified using steady-state setup used here, the first law of thermodynamics a simplified surface energy balance and the assumption that requires that the turbulent heat fluxes in and out of the engine, convective exchange near the surface takes place at the limit H and H , are balanced by the generation of kinetic energy, of maximum power. This model yields analytical expres- out G: sions for the hydrologic sensitivity and shows that it does not scale with the saturation vapor pressure, but rather with 0 = H − Hout − G. (A1) its slope. The hydrologic sensitivity scales with the slope of the saturation vapor pressure curve because hydrologic cy- The second law of thermodynamics requires that the entropy cling relates to the differences in saturation vapor pressure of the system does not decrease during the process of gen- between the temperatures at which evaporation and conden- erating kinetic energy. This requirement is expressed by the sation takes place. This difference is approximated by the entropy fluxes associated with the heat fluxes H and Hout slope. The actual sensitivity is then further reduced by a fac- that enter and leave the heat engine at the temperatures of the tor γ /(s + γ ), which originates from the surface energy bal- surface and the atmosphere: ance constraint. Our analytical expressions also show that H H surface warming caused by increases in absorbed solar ra- − out ≥ 0. (A2) diation result in a greater sensitivity of the hydrologic cy- Ts Ta cle than warming caused by an increased greenhouse effect. This greater sensitivity for warming due to solar radiation is In the best case, the entropy balance equals zero, which then simply explained by the requirement for a greater total cool- allows us to express the flux Hout as a function of H, Ts, and ing rate by radiative and turbulent fluxes. Even though our Ta: approach is highly simplistic and omits many aspects, the www.earth-syst-dynam.net/4/455/2013/ Earth Syst. Dynam., 4, 455–465, 2013 464 A. Kleidon and M. Renner: Hydrologic cycling and global climatic change

formulate a hypothesis, namely, that the surface-energy par- Ta H = H · . (A3) titioning would operate near this maximum power limit. This out T s hypothesis is very closely related to the proposed princi- When combined with Eq. (A2), this yields the well-known ple of Maximum Entropy Production (MEP, Ozawa et al., Carnot limit of the power generated by a heat engine: 2003; Kleidon et al., 2010), noting that in steady state, P = D, and entropy production is described by D/T . A more com- T − T G = H · s a . (A4) plete discussion on this relationship is given in Kleidon and Ts Renner (2013). In this paper, we assume that natural pro- cesses operate at this thermodynamic limit to derive the ana- In steady state, this generated power is dissipated by friction, lytical expressions. so that G = D. While the Carnot limit provides a constraint on how much power can be generated, it does not directly provide a limit on the value of H. Such a limit is obtained when we consider Acknowledgements. We thank two anonymous reviewers, that the temperature difference, T − T , is not independent of R. P. Allan, Amilcare Porporato and Michael Roderick for con- s a structive comments on this manuscript. A. Kleidon acknowledges H, but rather constrained by the energy balance (Eq. 1). This financial support from the Helmholtz Alliance “Planetary Evolu- temperature difference can be expressed using the expression tion and Life”. This research contributes to the “Catchments As for the net radiative exchange flux, Rl, from above as Organized Systems (CAOS)” research group funded by the German Science Foundation (DFG). Rs − H − λE Ts − Ta = . (A5) kr Edited by: M. Sivapalan Hence, the surface energy balance demands that the tempera- ture difference, Ts − Ta, decreases with an increasing value of H. When combined, this yields an expression for the Carnot References limit of Allan, R. P., Liu, C., Zahn, M., Lavers, D. A., Koukouvagias, E., and R − H − λE G = H · s . (A6) Bodas-Salcedo, A.: Physically consistent responses of the global kr atmospheric hydrologic cycle in models and observations, Surv. Geophys., doi:10.1007/s10712-012-9213-z, in press, 2013. Due to the contrasting effects of H on G in the two terms of Allen, M. R. and Ingram, W. J.: Constraints on future changes in the right hand side, the Carnot limit has a maximum value at climate and the hydrologic cycle, Nature, 419, 224–232, 2002. intermediate values of H. Andrews, T., Forster, P. M., and Gregory, J. M.: A surface energy We obtain this maximum in the Carnot limit when we first perspective on climate change, J. Climate, 22, 2557–2570, 2009. use the formulation of the sensible and latent heat flux from Bala, G., Duffy, P. B., and Taylor, K. E.: Impact of geoengineering above in terms of the vertical velocity, w to express the tem- schemes on the global hydrologic cycle, P. Natl. Acad. Sci. USA, perature difference: 105, 7664–7669, 2008. Betts, A. K. and Ridgway, W.: Climatic equilibrium of the atmo- Rs spheric convective boundary layer over a tropical ocean, J. At- Ts − Ta = , (A7) kr + cp ρw(1 + s/γ ) mos. Sci., 46, 2621–2641, 1989. Boer, G. J.: Climate change and the regulation of the surface mois- and then combine this expression with the formulation of the ture and energy budgets, Clim. Dynam., 8, 225–239, 1993. sensible heat flux, which yields: Bohren, C. F. and Albrecht, B. A.: Atmospheric Thermodynamics, Oxford Univ. Press, New York, 1998. cp ρw G = · R2. (A8) Boos, W. R.: Thermodynamic scaling of the hydrologic cycle of the
2 s Ts kr + cp ρw(1 + s/γ ) Last Glacial Maximum, J. Climate, 25, 992–1006, 2012. Govindasamy, B., Caldeira, K., and Duffy, P. B.: Geoengineering This equation has a maximum value with respect to w, which Earth’s radiation balance to mitigate climate change from a qua- can be derived analytically when we neglect that Ts in the drupling of CO2, Global Planet. Change, 37, 157–168, 2003. denominator depends on H as well. The maximization is Held, I. M. and Soden, B. J.: Robust responses of the hydrological achieved by ∂G/∂w = 0. This yields the expression of the cycle to global warming, J. Climate, 19, 5686–5699, 2006. optimum exchange velocity (Eq. 3) and results in an optimal Jones, A. D., Collins, W. D., and Torn, M. S.: On the additivity of radiative forcing between land use change and greenhouse gases, energy partitioning as given by Eq. (4). Geophys. Res. Lett., 40, 4036–4041, doi:10.1002/grl.50754, This maximum power limit describes the upper thermody- 2013. namic limit by which convective motion can be generated to Kleidon, A. and Renner, M.: Thermodynamic limits of hydrologic sustain the sensible heat flux out of local radiative heating by cycling within the Earth system: concepts, estimates and implica- absorption of solar radiation at the surface. It does not nec- tions, Hydrol. Earth Syst. Sci., 17, 2873-2892, doi:10.5194/hess- essarily imply that this limit is achieved. This would rather 17-2873-2013, 2013.

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