radiative constraints on the hydrological cycle in an idealized radiative… · 2016-05-09 ·...

Radiative Constraints on the Hydrological Cycle in an Idealized Radiative–ConvectiveEquilibrium Model

KEN TAKAHASHI

Atmospheric and Oceanic Sciences Program, Princeton University, Princeton, New Jersey

(Manuscript received 31 March 2008, in final form 17 June 2008)

ABSTRACT

The radiative constraints on the partitioning of the surface energy budget and, hence, on the strength ofthe hydrological cycle are analyzed in an idealized one-dimensional radiative–convective equilibrium modelformulated in terms of the energy budgets at the top of the atmosphere, the subcloud layer, and the freeatmosphere, which enables it to predict both surface relative humidity and the air–sea temperature differ-ence. Using semigray radiative transfer, a semianalytical solution was obtained that explicitly shows how thesurface latent heat flux (LHF) is related to the radiative properties of the atmosphere. This solution was alsoused in conjunction with a full radiative transfer code and was found to provide reasonably realisticquantitative estimates.

In the model the LHF is fundamentally constrained by the net longwave flux divergence above the levelof condensation by lifting (LCL) and by the atmospheric absorption of shortwave radiation, with only aweak indirect control by near-surface moisture. The latter implies that the Clausius–Clapeyron relation doesnot directly constrain the strength of the hydrological cycle. Under radiative perturbations, the changes inLHF are determined by the changes in the net longwave fluxes at the LCL, associated mainly with thechanges in the longwave transmissivity, and by the changes in shortwave absorption by the atmosphere (e.g.,by increased water vapor).

Using a full radiative transfer model with interactive water vapor feedback with the semianalyticalsolution indicates a rate of change in LHF with greenhouse forcing of around 2 W m�2 K�1 of surfacewarming, which corresponds to the Planck feedback (�3.2 W m�2 K�1) multiplied by a coefficient of orderone that, to first approximation, depends only on the relative magnitudes of the net longwave radiationfluxes at the LCL and the top of the atmosphere (i.e., on the shape of the vertical profile of the net long-wave flux).

1. Introduction

The question of how the hydrological cycle willchange in the context of global climate change is ofconsiderable societal and scientific interest. One funda-mental aspect of this cycle is its global mean (i.e., globalmean surface evaporation and precipitation), which isintrinsically tied to the global atmospheric and surfaceenergy budgets (Allen and Ingram 2002, hereafterAI02). Although the global average of the hydrologicalcycle does not have as much practical importance as itsregional manifestations, it is a fundamental aspect ofclimate and a quantitative and predictive theory for itwould provide guidance for the diagnosis of biases in

climate models, help to sort out possible discrepancieswith observational estimates (e.g., Wentz et al. 2007),and help the interpretation of model simulations of cli-mate regimes significantly different from the presentone (e.g., AI02; Held and Soden 2006, hereafter HS06).

Comprehensive climate models project an increase inthe global-mean atmospheric moisture content associ-ated with global warming that is well explained by thechanges in the saturation specific humidity, which is astrong function of temperature, for constant relativehumidity (HS06). The global mean hydrological cyclealso increases, but at a smaller percentage rate thanmoisture (AI02; HS06), which indicates that the resi-dence time of water vapor also increases with thewarming, which in turn can be interpreted as a slowingdown of the motions that transport moisture vertically(Betts 1998; HS06; Vecchi and Soden 2007). However,a mechanistic theory for the changes in the residencetime (or in moisture-transporting motions) based on

Corresponding author address: Ken Takahashi, NOAA/Geo-physical Fluid Dynamics Laboratory, 201 Forrestal Rd., Prince-ton, NJ 08540–6649.E-mail: [email protected]

JANUARY 2009 T A K A H A S H I 77

DOI: 10.1175/2008JAS2797.1

© 2009 American Meteorological Society

JAS2797

moist dynamics is lacking, so it is not possible at presentto predict the changes in both the global hydrologicalcycle and the residence time given the changes in mois-ture from this perspective.

Another approach considers the global hydrologicalcycle in the context of the global energy budget of thetroposphere (AI02). This brings the complex physics ofradiative transfer into the problem, which makes it dif-ficult to produce a simple conceptual theory; so, forexample, AI02 take an empirical approach by which thebehavior of the radiative fluxes is fit to observed (ormodeled) changes in temperature, and changes in thesensible heat fluxes (SHFs) are neglected. One of themain difficulties with a theoretical approach based onthe radiative transfer physics is the great complexity inthe detailed interactions between radiation at differentwavelengths and the atmospheric gases. However, tothe extent that these complexities are not essential forunderstanding the qualitative relation between the netradiative energy fluxes and the surface energy budget,realism in the radiative transfer can be sacrificed in afavor of a less realistic but theoretically simpler repre-sentation of the radiative physics. For example, thesemigray approximation, in which the atmosphere isassumed to be transparent to shortwave radiation andthe longwave emissivities of greenhouse gases are in-dependent of frequency, makes the problem math-ematically tractable and exact solutions might be foundin some cases (e.g., Goody and Yung 1989; Weaver andRamanathan 1995).

Another simplifying approximation is to consider aradiative–convective equilibrium model in which large-scale horizontal dynamics (e.g., associated with differ-ential heating and rotation) are ignored. This type ofmodel has been successful in improving our under-standing of the processes controlling the vertical ther-mal structure of the atmosphere and its sensitivity toradiative perturbations (see Ramanathan and Coakley1978 for a review), as well as providing insights intotropical climate (e.g., Sarachik 1978; Betts and Ridgway1989, hereafter BR89). Although it is not obvious thatthis is an adequate approximation for a differentiallyheated rotating planet [however, there is some evi-dence that it might be; see O’Gorman and Schneider(2007), hereafter OS07)], the understanding of this sim-pler system is likely a prerequisite to understanding thegeneral case.

In this study, these two approaches are combinedinto an idealized radiative–convective equilibriummodel with semigray radiation transfer (described insection 2), which is exposed to a wide range of long-wave (“greenhouse”) forcings (section 3). After some

simplifying assumptions, this approach succeeds in pro-viding concrete constraints based on the assumed ra-diative physics, which are expressed in a semianalyticalsolution (section 4). This solution is then used in com-bination with a full radiative transfer model to explorethe quantitative predictions in a more realistic setting(section 5). The implications of the results, including acomparison to previous similar studies, are discussed insection 6. The last section summarizes the main resultsand conclusions.

2. Radiative–convective equilibrium model

The model (see Fig. 1 for a schematic depiction) con-sists conceptually in a hydrostatic atmosphere featuringlarge-scale subsidence everywhere except for a verysmall fraction of the area in which strong ascent asso-ciated with deep moist convection takes place and fromwhich the latent heating is assumed to be efficientlycommunicated elsewhere (e.g., Sarachik 1978). In thevertical direction, there is a well-mixed subcloud layer,in which potential temperature � and specific humidityq are uniform (no special treatment is given to the sur-face layer), overlying an ocean with surface tempera-ture Ts and a free atmosphere with a thermal structureset everywhere to a pseudoadiabat by deep convection(or, alternatively, the lapse rate may be prescribed).There is no stratosphere in this model. Surface pressurep0 is fixed to the value of 1000 hPa. A list of the modelparameters and their reference values is provided inTable 1.

The design of this model was guided by the followingprinciples: On the one hand, we avoided making strongassumptions about factors that directly affect the sur-face energy fluxes and that therefore should be part ofthe solution (e.g., prescribing the surface relativehumidity, the air–sea temperature difference, or the

FIG. 1. Schematic diagram indicating the main variables, verticallayers, and energy fluxes considered in the model.

78 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 66

Bowen ratio). On the other hand, we made enoughsimplifications that the system can be well understood,but only when these are supported by observations, sothat the model remains qualitatively similar to the earthsystem for the intended purposes. Because this model isnot intended to shed light on climate sensitivity (i.e.,changes in �), this model does not explicitly addresschanges in clouds, in contrast to other studies with simi-lar models (e.g., Pierrehumbert 1995; Larson et al.1999). Although there is a possibility that clouds mayhave a radiative effect relevant to the problem at hand,the issue is so complex that it is left open for futureresearch. The cloud radiative effects are therefore ei-ther taken as given or ignored altogether. Physically,the present model is similar to the tropical model ofBR89 but with the advantage of having a tractable long-wave radiative transfer scheme and other simplifica-tions that greatly ease the analysis.

a. Temperature profile

Given the values of � and q, the vertical profile oftemperature can be determined and used to calculatethe longwave fluxes.

In the subcloud layer, extending from the surface tothe level of condensation by lifting (LCL), the lapserate is dry adiabatic. The pressure and temperature atthe LCL (pLCL and TLCL, respectively) is determinedfrom the values of � and q or, alternatively, the physicaldependence of pLCL on � and q is severed and its value

is prescribed for the purpose of determining the ther-mal structure of the atmosphere.

Starting at the LCL, the free atmospheric tempera-ture can be determined either by integrating upwardfollowing a pseudoadiabat or, alternatively, by prescrib-ing the lapse rate �.

b. Energy budget at the top of the atmosphere

The outgoing longwave flux (UTOA) at the top of theatmosphere (TOA; at p � 0) is required to balance thenet shortwave radiative flux (S0) absorbed by the cli-mate system:

UTOA � S0. �1�

To the extent that the atmosphere is optically thick inthe longwave range, the surface air temperature � isclosely constrained by this requirement, to the extentthat the relation between UTOA and � is known.

c. Free atmosphere energy budget

The energy budget for the free atmosphere, extend-ing from the LCL to the TOA, yields the latent heatflux (LHF) plus the absorbed shortwave radiation (�S0,where � is the prescribed fraction of S0 absorbed by thefree atmosphere) by requiring them to balance thelongwave radiative cooling:

LHF �S0 � UTOA � ULCL DLCL, �2�

where U and D stand for the net upward and downwardlongwave fluxes, respectively.

The LHF through the LCL is equal to that at thesurface because all of the latent heat is assumed to bereleased above the LCL (i.e., there is no net condensa-tion in the subcloud layer). The LHF is related to mois-ture through a bulk formula for surface evaporation:

LHF � L�C qs�Ts� � q�, �3�

where C is a prescribed effective transfer coefficientthat physically depends on wind speed, stability, and airdensity. This equation allows q to be determined ifLHF and Ts are known.

The entrainment flux of sensible heat across the LCLcould be parameterized to be proportional to the sur-face sensible heat flux (Betts 1973; BR89). This entrain-ment term is, however, typically an order of magnitudesmaller than the latent heating, and sensitivity testsshow that it only leads to small quantitative differencesin the results, so it is neglected in the present model.

Large-scale vertical transports of sensible heatthrough the LCL are also neglected. On the earth, thenet vertical heat transport by large-scale circulations(e.g., Hadley circulation, midlatitude eddies) is approxi-

TABLE 1. Constants and parameters with their reference values.The upper block contains physical constants; the lower block con-tains prescribed model parameters.

Symbol Parameter Standard value

cp Specific heat at constantpressure

1004 J K�1 kg�1

Rd Dry air constant 287 J K�1 kg�1

R� Water vapor constant 461 J K�1 kg�1

L� Enthalpy of vaporization 2.5 106 J kg�1

g Gravitational acceleration 9.8 m s�2

�0 Atmospheric longwave opticaldepth

6

n Exponent in the formula for thelongwave optical depth

4

C Surface sensible heat andmoisture transfer coefficient

10�2 kg m�2 s�1

S0 Net shortwave incident at theTOA

240 W m�2

� Fraction of S0 absorbed in theatmosphere

0.29

p0 Surface pressure 105 Pa� Prescribed free tropospheric

lapse rate6.5 10�3 K m�1


mately an order of magnitude smaller than the latentheat flux, so this approximation is adequate even in thepresence of large-scale transport processes not consid-ered in the radiative-equilibrium framework.

d. Subcloud layer energy budget

The energy budget for the subcloud layer (neglectingentrainment through the LCL) is

cpC�T � ULCL � DLCL � Usfc Dsfc. �4�

The term on the lhs is the bulk formula approximationfor SHF, where �T � Ts � � and the effective transfercoefficient C is assumed to be equal to that for mois-ture. As shown later, this equation constrains �T. Theabsorption of shortwave radiation in this layer is ne-glected, which is a convenient but not critical approxi-mation.

e. Longwave radiation

The radiative transfer equations for a semigray at-mosphere determine the upward and downward long-wave fluxes (U and D, respectively):

dU

d�� U � �B, �5�

dD

d�� B � D, �6�

where �B � �T 4, with T representing the atmospherictemperature profile. The optical depth (including thediffusivity factor) � is prescribed as a function of pres-sure p alone as

� � �0� p

p0�n

, �7�

where the exponent n is taken to be 4, based on the factthat the scale height for air density is 4 times larger thanthat of water vapor (e.g., Goody and Yung 1989;Weaver and Ramanathan 1995; Frierson et al. 2006).Although pressure broadening might be expected toincrease the exponent to n � 5 (Weaver and Ra-manathan 1995), this does not affect the qualitative be-havior of the model, so n � 4 is used to allow thecomparison with the results from OS07.

3. Experiments

The model was run for values of �0 ranging from 1.5to 500, with uniform intervals in log �0. The equilibriumstate was obtained by allowing a time-dependent ver-sion of the system (2)–(7) (providing the ocean andsubcloud layers with finite heat capacities) to evolve intime until an effectively steady state was achieved (noevidence for multiple equilibria was found). In this set

of experiments, the temperature profile in the free at-mosphere followed pseudoadiabats. The pressure at theLCL (pLCL) was calculated interactively from the val-ues of � and q. These experiments can be interpreted asequilibrium climate change runs under greenhouse gasforcing in which the radiative effect of water vapor isnot interactive. As �0 (i.e., greenhouse gas concentra-tion) increases, the surface air temperature � increasesmonotonically (in these experiments, � ranges from�270 to 310 K). For the purpose of providing a quan-titative description of some results, a reference statecorresponding to � � 288 K (close to the global meantemperature on the earth and corresponding, in thismodel, to �0 � 6) was selected. Around this referencestate, a unit change in �0 approximately results in achange in � of around 1.8 K, but more generally � scalesroughly with the logarithm of �0, although the sensitiv-ity decreases at higher �0 (not shown).

The surface energy fluxes plotted against � exhibitbehavior very similar to the results from a general cir-culation model (GCM) with semigray radiation re-ported by OS07 (Fig. 2). Particularly, LHF increasesrather linearly with � for states colder than the refer-ence temperature of 288 K. For higher �, LHF ap-proaches the surface insolation (1 � �)S0. Consistentwith this, the sum of SHF and surface longwave fluxesapproaches zero. However, the changes with � of bothof these fluxes are significant, as noted by OS07, soSHF cannot be neglected in principle, as suggested byAI02, for estimating the changes in LHF. The subcloud

FIG. 2. Surface net shortwave [(1 � �)S0; black dotted], latentheat (LHF; black), longwave (Usfc � Dsfc; red) and sensible heat(SHF; blue) fluxes (W m�2) against surface air temperature (�; K)as �0 is varied. Results are shown for the runs with a pseudoadi-abatic profile (solid), � � 6.5 K km�1 (dashed), and � � 6.5 Kkm�1 with pLCL � 900 hPa (short dashed), and from a GCM withsemigray radiation (dotted–dashed; adapted from Fig. 3 of OS07).


Fig 2 live 4/C

layer energy budget (4) indicates that the longwavefluxes through the LCL are also going to zero. It will beshown in the next section that this is the key constrainton LHF as �0 is varied.

With semigray radiation, the decrease in longwavefluxes at the LCL could be a direct response to changesin �0, which change the radiative transfer between dif-ferent atmospheric levels. It could also be a result of thechange in the lapse rate in the free atmosphere because,as the climate warms, the upper troposphere becomeswarmer relative to the surface, and this could lead to areduction in the upward flux through the LCL. Alter-natively, because the net longwave flux typically in-creases with height, an increase in pLCL toward p0 as thesurface humidity increases would lead to a reduction inthe fluxes at this level.

The latter two possibilities can be assessed in astraightforward way in this model. First, the experi-ments are repeated with the lapse rate � in the freeatmosphere fixed at the value of 6.5 K km�1. Becausethis eliminates the negative lapse-rate feedback, therange in � is much higher for the same values of �0 thanwith the pseudoadiabat. On the other hand, the relationbetween the energy fluxes and �0 is changed very little(not shown), but because the temperature axis is“stretched,” smaller slopes result in Fig. 2. This is evi-dence that the relation between � and the surface en-ergy fluxes (including the hydrological cycle) resultmostly from individual dependencies on the radiativeproperties of the atmosphere rather than from a directphysical connection between them. In fact, the ex-pected direct effect of warming would be an enhance-ment of the longwave flux through the LCL and, there-fore, a reduction in LHF (see section 4a).

In a third set of experiments, the fixed lapse rate isretained and in addition pLCL is prescribed to 900 hPa.The resulting relation between the surface energyfluxes and � is similar to the second set of experiments(Fig. 2), despite significant variations in pLCL in both ofthe previous sets of runs.1 However, although the over-all qualitative behavior was unchanged, the quantita-tive differences are not negligible. For instance, an in-teractive pLCL results in a slope of LHF with � about30% larger near the reference state.

4. Semianalytical solution

As shown in the previous section, the variations inpLCL and �, which are the only way by which q directly

affects the radiative fluxes in this model, do not affectthe overall qualitative behavior of the surface fluxesrelative to surface air temperature.

Here these variations are explicitly neglected and thevalues of pLCL and � are prescribed, so the temperatureprofile and, therefore, the longwave fluxes are de-coupled from q, which can be diagnosed from (3) onceLHF and Ts are determined. In this way, the hydrologi-cal cycle becomes explicitly independent of the detailsof the moist processes and the system consists now ba-sically in three unknowns (�, Ts, and LHF), for whichthere are three equations from the energy budgets [(1),(2), and (4)], as well as the radiative transfer equations[(5) and (6)]. The critical step for finding a solution tothe system is determining the longwave fluxes as a func-tion of the model parameters and independent vari-ables without having to solve the radiative transferequations numerically every time. Fortunately, thesemigray radiation scheme allows for this to be over-come to a large extent, as shown next.

Under hydrostatic balance and for a constant lapserate �, the temperature profile is given by T � �(p/p0)Rd� /g. Allowing for a dry adiabatic lapse rate in thesubcloud layer, the atmospheric blackbody flux densityprofile is given by

�B��4 × ��p

p0�4Rd�cp

for p pLCL,

�pLCL

p0�4Rd�cp� p

pLCL�4Rd�g

for p� pLCL.

�8�

The vertical distribution (in p coordinates) of �B nor-malized by ��4 is fixed and determined by the values of� and pLCL. Because p and � are uniquely relatedthrough (7) for a given set of parameters, (8) can bedirectly substituted into the semigray longwave radia-tion Eqs. (5) and (6) and, with the approximation �T K

�, the solution for the net upward longwave flux has theform

U � D � e��01��p�p0�n�4��3�T ��p; �0, pLCL, ��4.

�9�

Here, � gives the profile of the net longwave flux, nor-malized by ��4, that would result if the surface emittedat the same temperature � as the surface air (Fig. 3),and the first term on the rhs is the correction that arisesfrom the temperature difference �T. The � profiles canbe determined by prescribing values for � and pLCL,

1 Note that pLCL decreases from �940 hPa near � � 270 K to�900 hPa near the reference state and then approaches p0 astemperature increased further (pLCL � 990 hPa for � � 340 K).


setting Ts� � � ��1/4 and numerically solving the semi-

gray radiation transfer Eqs. (5) and (6) for � � U � D.Equation (9) is the key simplification that allows thismodel to be analytically tractable because it allows theeffects of � and �T on the longwave fluxes to be cleanlyseparated from those associated with changes in theradiative transfer properties of the atmosphere (e.g.,�0). In the current setup, with fixed � and pLCL, once the� profiles are determined as a function of �0, the systemcan be immediately solved, even as the other param-eters are varied.

The values of � decrease toward the surface (Fig. 3),implying a flux divergence and therefore cooling. Thisis because the lapse rate chosen is smaller than the onecorresponding to pure radiative equilibrium and there-fore requires convective heating for its maintenance(Manabe and Strickler 1964). As �0 is increased, thedivergence (cooling) is centered at progressively lowerpressures, following the level where � � 1 (Fig. 3).In the layer beneath, the increase in optical thicknesswith �0 is faster because the optical thickness per unitmass (gd� /dp) is highest near the surface (7). In thislayer, as �0 increases and the transmission of long-wave fluxes becomes very limited, the values of U andD both approach the local blackbody flux � B, so bothU � D and d(U � D)/d� [from (5) and (6)] becomesmall. Near the surface, however, �T � 0 can maintaina finite U � D.

For a sufficiently optically thick subcloud layer,the transmission of the �T correction to the long-wave flux emitted by the surface through the LCL andTOA in (9) is relatively small, so we have approxi-mately

Usfc � Dsfc � �sfc��4 4��3�T, �10�

ULCL � DLCL � �LCL��4, and �11�

UTOA � �TOA��4, �12�

where

�TOA � ��p � 0; �0, pLCL, �, �13�

�LCL � ��p � pLCL; �0, pLCL, �, and �14�

�sfc � ��p � p0; �0, pLCL, �. �15�

The values of �LCL and �sfc rapidly decrease with log �0(Fig. 4), with the latter always smaller than the former.At the TOA, � also decreases with log �0 (Fig. 4), but ata slower and more uniform rate. The values of � atthese three levels [(13)–(15)] shown in Fig. 4 com-pletely characterize the semigray radiative transferscheme for the idealized radiative–convective equilib-rium model.

From the energy budgets (1), (2), and (4), using thelongwave fluxes (10)–(12), the following solution is ob-tained:

� � � S0

�TOA��1�4

, �16�

�T �S0

�TOA� �LCL � �sfc

4�1�4�S0 ��TOA�3�4 cpC�, and

�17�

LHF � S0�1 � � ��LCL

�TOA�. �18�

FIG. 3. Normalized vertical profiles of net upward LW fluxes�(p, �0, pLCL, �) for different values of �0, with � � 6.5 K km�1,pLCL � 900 hPa (indicated with circles) and assuming Ts � �. Thelevels at which � � 1 are indicated with crosses.

FIG. 4. Normalized net upward LW fluxes (for Ts � �) at thesurface (�sfc), LCL(�LCL), and TOA(�TOA) as a function of �0, for� � 6.5 K km�1 and pLCL � 900 hPa (solid), � � 6.5 K km�1 andpLCL � 950 hPa (dashed), and � � 5 K km�1 and pLCL � 900 hPa(dotted–dashed).


This solution is a very good approximation to the fullmodel with fixed � and pLCL (Fig. 5), with the bestagreement found for values of �0 near the referencestate and higher, as expected.

Equation (18) expresses the requirement that the netenergy flux through the LCL be zero, so the net down-ward shortwave flux (1� �)S0 will be balanced by LHFand (U � D)LCL. Because �TOA and, for realistic (posi-tive) tropospheric lapse rates, �LCL are both 0, then(18) indicates that

LHF � �1 � ��S0. �19�

This upper bound on LHF exists because shortwaveradiation is the only downward energy flux consideredin this model, so it constrains the sum of all of theupward fluxes, including LHF. Exceptions to this mightbe provided by strong entrainment into the subcloudlayer or by a net downward longwave flux through theLCL associated with negative lapse rates.

The sea–air temperature difference �T is constrainedto be positive to the extent that �LCL � �sfc (17), whichresults from the radiative equilibrium lapse rate beinglarger than the dry adiabatic (Weaver and Ramanathan1995). Thus, in this model the surface sensible heat fluxis always upward to balance the longwave radiativecooling in the subcloud layer (BR89). However, even inthe limit of a vanishing C, (17) predicts that �T wouldonly increase by at most a factor of 3 (for the referenceclimate state) because the longwave fluxes play a sig-nificant role in maintaining a small �T.

a. Response to radiative perturbations

For a small radiative perturbation �F at the TOA,the response in surface air temperature is given by(16) as

�

F�

1

4�TOA��3 � �0, �20�

where �0 is the climate sensitivity parameter (for thePlanck feedback alone), which equals 0.3 K W�1 m�2 atthe reference state.

Greenhouse gas perturbations introduced by chang-ing the optical depth �0 (ignoring possible feedbacksfrom S0 and �) lead to changes in LHF through changesin (U�D)LCL (18). This can be expressed using (11) as

LHF � �U � D�LCL�� LCL

�LCL�

4 �

� �. �21�

The two terms in the parentheses on the rhs correspondto changes in two different aspects of the longwaveradiative transfer, each of which depends on �0 in waysthat depend in turn on the details of the radiative trans-fer. In particular, the second term indicates that warm-ing is directly associated with a decrease in LHF be-cause of the associated increase in (U � D)LCL. How-ever, the strong reduction in �LCL with increasing �0(Fig. 4) leads to a net increase in LHF. For instance,near the reference climate in the idealized model, the�LCL and the � term account for 120% and�20% of theincrease in LHF, respectively. This explains the relativeinsensitivity of the relation between �0 and LHF to thelapse rate reported in section 3, because �LCL is insen-sitive to � (Fig. 4), whereas � itself depends strongly on� through �TOA (Fig. 4).

Considering both longwave and shortwave forcingsat the TOA (�FL and �FS, respectively), the responsein LHF in terms of the parameters of the system isgiven by

LHF � �L��0� FL �S��0, �� FS � S0 �,

�22�

where

�L � � ��LCL

��TOA�

�LCL

�TOA� and �23�

�S � �1 � � ��LCL

�TOA� �24�

are coefficients that depend only on �0 and � and aregenerally different from each other (Fig. 6). The factthat �TOA is a monotonic (decreasing) function of �0(Fig. 4) has been used to remove the explicit depen-dence on �0. This will prove useful in section 5 whendealing with a realistic radiative transfer model for

FIG. 5. Semianalytic solution for surface latent (solid), net long-wave (dotted–dashed), and sensible (dashed) heat fluxes vs sur-face air temperature using the semigray radiation for differentvalues of �0 (thick lines; the circles indicate results from the fullmodel) and full radiative transfer with interactive water vaporfeedback for different concentrations of CO2 (thin lines). Bothmodels have � � 6.5 K km�1 and pLCL � 900 hPa.


which �0 is not well defined. Because �TOA, �LCL and �are all positive, in general �S � 1. On the other hand,�L� 1 only for relatively high optical thickness (Fig. 6),at values of �TOA smaller than the reference value ofaround 0.6 (i.e., �0 � 6).

In the special cases in which either only �FL or �FS

is nonzero and �� 0, (20) and (22) can be used touniquely relate �LHF and ��. In the more general casein which �F� �FL �FS, however, (20) and (22) yield

LHF� �0�1��L �S

2 � � ��L� �S

2 �� FL� FS�,

�25�

so only if �L � �S would the relation between �LHFand �� be indifferent to the type of forcing in thismodel. However, because �0 in the real atmosphere isdominated by water vapor, which itself is a strong func-tion of temperature, a pure shortwave forcing wouldalso be associated with changes in the longwave trans-fer (i.e., �0). It will be shown in the next subsection thatthe presence of radiative feedbacks at the TOA mergesthese idealized responses as well.

The coefficient �L(�0), which determines the re-sponse of LHF to a pure longwave forcing, depends onthe relation between �LCL and �TOA, reflecting the op-posing effects on (U � D)LCL of surface warming andthe reduction in longwave transmissivity noted previ-ously. This coefficient (and hence the rate of increase ofLHF) drops to low values as �TOA decreases (equiva-lently, as �0 increases; Fig. 6), as required by (19).

With respect to shortwave absorption, because �

does not enter (16) or (17) explicitly, it is a sort ofwildcard that can be arbitrarily varied (e.g., by changingconcentrations of atmospheric black aerosols) and con-sequently change LHF with no effect on the tempera-tures and other fluxes. As shown later, there arechanges in � in nature associated with changes in watervapor with warming that might have a significant effecton the changes in LHF.

b. Radiative feedbacks

So far, the longwave optical depth �0, the incominginsolation S0, and the atmospheric shortwave absorp-tivity � have been taken as external parameters of thesystem. On the earth, water vapor is the most importantgreenhouse gas and absorber of insolation in the tro-posphere and is strongly tied to the atmospheric tem-perature. Similarly, albedo, particularly the componentassociated with ice and clouds, is also affected by tem-perature.

For radiative purposes, climate models approxi-mately maintain a constant free troposphere relativehumidity (Soden and Held 2006), which implies thatspecific humidity increases with temperature followingapproximately the Clausius–Clapeyron (C–C) relation.The resulting positive radiative feedback is partly offsetby the negative feedback associated with the reductionin the lapse rate �, which was shown previously to havea relatively weak direct effect on LHF, and it is theresulting net positive feedback that is robust amongclimate models (Soden and Held 2006). Thus, the ra-diative feedbacks from water vapor can be crudely in-corporated into the present model by taking �0 and � tobe monotonically growing functions of �. Similarly, adependence of � on � can also be assumed for calcu-lating changes in �TOA. On the other hand, the sign ofthe albedo feedback is less constrained, and we will justassume that an effect of temperature on albedo exists.

With this enhancement of the model, the semiana-lytical solution (16)–(18) remains adequate, but � nowdepends on � through �0. Given S0 and the functionaldependence of �0 on �, (16) can be directly solved for �and, therefore, for �0. This then determines �T andLHF through (17) and (18). The climate sensitivity withfeedbacks is given by (16) as

�

F� � 1

1 � fL � fS��0 � G�0, �26�

where

fL��0��4��TOA

��0

��0

��

��TOA

�

�

�� and �27�

fS � �0

�S0

��28�

FIG. 6. Change in LHF per unit radiative perturbation associ-ated with variations in �0 (�L, solid) and S0 (�S, dashed), plottedagainst �TOA for semigray radiation (thick) and full radiation withinteractive water vapor feedback (thin). The constant value � �0.29 is assumed throughout.


are the longwave (e.g., water vapor, lapse rate) andshortwave (i.e., surface and cloud albedo) feedback pa-rameters, respectively. The feedbacks enhance the re-sponse of � to a radiative forcing by the gain factor G.

The changes in LHF are given by

LHF�G�1� fS��L fS�S� FL

G�1� fL��S fL�L� FS� S0 �. �29�

The radiative feedbacks lead to an amplification of theresponse of LHF to the forcings by the gain factor G.Because the response in � is amplified by the samefactor, this does not alter the ratio of the changes inLHF and � under pure longwave or shortwave forcing.However, the presence of shortwave (longwave) feed-backs also leads to the combination of �L and �S into aneffective coefficient in the case of a pure longwave orshortwave forcing. In the case of positive feedbacks, theeffective coefficients lie between �L and �S.

c. Effect of changes in pLCL

As noted before, besides its role as a greenhouse gas,the low level water vapor affects the equilibrium cli-mate state through its effect on pLCL, which was keptconstant in the idealized model for simplicity. This ef-fect is not large enough to change the qualitative be-havior of the system (cf. the experiments with fixed �but with pLCL fixed and interactive in Fig. 2), but is notnegligible.

From the definition of the LCL, we have

pLCL

pLCL� � RdL�

cpR�TLCL� 1��1� q

q�

�

TLCL

L� �

R��2�,

�30�

where TLCL � � (pLCL/p0)Rd/cp. Whereas moisteningis associated with an increase in pLCL, warming has anopposing effect. This net result of this cancelation isseen more clearly by writing the second term on the rhsof (30) in terms of the surface relative humidity r0 as

� r0

r0 �1 �

�

TLCL� L�

R��2 ��. �31�

For typical values, the coefficient of the �� term isaround�0.2% K�1. On the other hand, the ratio (�r0/r0)/�� can have larger magnitudes and the first term in (31)generally dominates, but its value varies significantlyunder greenhouse forcing. For instance, in the full ra-diative–convective model, the values range from�1.1% to 1.7% K�1 at temperatures below and abovethe reference, respectively.

Increasing pLCL modifies slightly the atmosphericthermal structure and therefore the � profile, but this

effect is small compared to the reduction in �LCL asso-ciated with the downward displacement of the LCLalong the unperturbed � profile, which decreases withincreasing p (Fig. 3). Therefore, (18) predicts that ifpLCL increases, the LW fluxes at the LCL would de-crease and lead to an increase in LHF. Whether thiseffect enhances or reduces the increase in LHF withincreased �0 or S0 will depend on whether r0 increasesor decreases, respectively.

Including the latter effect of the changes in pLCL onLHF results in the addition of the term

��4��

�p�pLCL

pLCL �32�

to the rhs of (29), which can be expressed in terms of qand � using (30). In principle, the resulting equation canbe solved for �LHF similarly to (29), but the complex-ity of the resulting formula is such that it does not pro-vide clear insights. An empirical approach will be usedin the next subsection.

d. Effect of changes in C

If C were increased (e.g., by increasing surface windspeed), the instantaneous response in LHF would be aproportional increase. However, once equilibrium is re-gained, the radiative constraints will prevail, so therewill be an adjustment in Ts and q that will significantlyalter this response (BR89). As indicated in section 4,the adjustment of q to a change in C could be such thatLHF remains unchanged. However, as shown in theprevious subsection, q itself affects the longwave fluxesat the LCL by affecting pLCL, which in turn controlsLHF. Thus, in equilibrium, changes in C can result inchanges in LHF, but the magnitude of this change willbe largely constrained by radiative considerations.

In the general case, C can be expected to respond tochanges in climate, and the simplest assumption is that�C is linearly related to ��. To assess the effect of �Con the �LHF associated with greenhouse forcing,rather than assuming specific relationships between �Cand ��, a number of scenarios with different prescribedincreases in C but the same increase in �0 are comparedto a control case. The two approaches are equivalentfor the equilibrium results but the latter is simpler toimplement. Thus, the full radiative–convective model(with interactive pLCL and pseudoadiabatic lapse rate)was run with a perturbation in �0 and various pertur-bations in C. The results indicate that a fractionalchange in C per degree warming of about 5% K�1 re-sults in an enhancement in the increase in LHF with thewarming of around 1% K�1, relative to the change in


the case in which �C � 0 (about 3.7% K�1; Fig. 7).Thus, the equilibrium effect of changes in C on LHF isabout a factor of 4–5 smaller than would be expectedfrom the instantaneous response. This is even moredramatic in the model of BR89 (which is similar to thepresent one but with more comprehensive physics), inwhich the fractional increase in LHF is 10–20 timessmaller than the fractional increase in C (their Fig. 19).In the present model, around 80% of the effect of �Con �LHF is associated, through (U � D)LCL (21), withthe change in pLCL and the rest is associated with thechange in ��4 (note that �� is not the same for thedifferent �C because of the differences in lapse rateand pLCL).

5. Full radiative transfer

To assess how the behavior of the semianalyticalmodel (16)–(18) from section 4 changes when the semi-gray radiation scheme is replaced with a realistic radia-tive transfer model, the National Center for Atmo-spheric Research (NCAR) Column Radiation Model,based on the physics of the NCAR Community ClimateModel version 3 (CCM3) climate model (Kiehl et al.1996), was used for the radiative calculations in thissection. No greenhouse gases besides CO2 and watervapor were considered. Conceptually, this case is not asstraightforward as with the semigray radiation becausethe effective �0 has both an externally prescribed com-ponent due to CO2 and a component that depends on �itself due to water vapor (cf. section 4b) and, in fact, �0is not well defined in this case. However, the assump-tion about relative humidity described next provides

the additional constraint necessary for overcoming thisdifficulty.

The � profiles were calculated with this radiationmodel as in section 4 [i.e., taking �T � 0 and � � (U �D)/��4; see Eq. (9)] for different values of � and CO2

concentrations, using the same � and pLCL as in thesemianalytic model, but with the humidity distributiondetermined by prescribing the vertical profile of rela-tive humidity r from Manabe and Wetherald (1967):

r � �0.77p�p0 � 0.02

1 � 0.02, for p�p0 � 0.02,

0, otherwise.�33�

The fixed relative humidity assumption is a good ap-proximation for the longwave radiative effects of watervapor in climate models (Soden and Held 2006). Notethat the relative humidity distribution (33) was used forthe radiative calculations only; at the surface, the rela-tive humidity (r0) remained interactive for the calcula-tions associated with LHF.

The CO2 concentration was varied between 2�6 and213 times a representative value of 360 ppm. For eachconcentration, the value of � consistent with UTOA �S0 � 240 W m�2, was determined (Fig. 8). A transitionbetween these climate states can be qualitatively inter-preted as an equilibrium climate change associated withan external perturbation (e.g., anthropogenic) in CO2.

a. Response to CO2 perturbations

Because �0 is not well defined outside of the semigrayradiation context, the results are presented in terms of

FIG. 7. Fractional changes in LHF per degree warming (% K�1)for different changes in C (% K�1).

FIG. 8. Contours of �TOA (solid) and �LCL (dashed) as CO2

concentration and water vapor are varied (the latter by changing� while keeping relative humidity constant). The thick curve de-notes the states for which UTOA � S0 � 240 W m�2.


�TOA, which monotonically decreases as greenhousegases are added to the atmosphere. This approximationassumes that different greenhouse gases change the �profiles in the same way—particularly, that changes in�TOA are associated with unique changes in �LCL and�sfc independently of what greenhouse gases are per-turbed. This is not expected a priori to be an accurateapproximation because different greenhouse gaseshave different vertical distributions and absorptionproperties, but the results indicate that it is adequatebecause the variations in CO2 and water vapor havesimilar effects on �LCL relative to �TOA (the isolines areroughly parallel in Fig. 8). This approximation is par-ticularly good around the states for which S0 � 240 Wm�2 (thick line in Fig. 8) but is inadequate at low CO2

concentrations and high water vapor (low CO2 and high� in Fig. 8), at which �LCL is insensitive to CO2 but notto water vapor, probably reflecting the vertical distri-bution of the latter, which allows it to prevail in thelower troposphere.

The values of �L and �S were estimated for the cli-mates with UTOA � S0 � 240 W m�2 (Fig. 6). Theresults for �S are qualitatively similar to the semigrayradiation case, but �L shows significant differences. Inparticular, the values of �L do not present the largevariations observed with the semigray radiation and re-main between 0.6 and 0.8, except for CO2 concentra-tions greater than 28 times the reference, above whichthe values drop (Fig. 6). Thus, for a large range ofperturbations in CO2 concentrations, �L is approxi-mately uniform and consistently smaller than unity, andLHF behaves rather linearly in this regime (Fig. 5). Thisbehavior is associated with the fact that as CO2 (andwater vapor) is increased, the � profile is rather uni-formly shifted to lower values (Fig. 9), so ��LCL/��TOA

is close to unity, in contrast to the strong deformation inthe profiles observed in the case of the semigray radia-tion (Fig. 3), which results in ��LCL/��TOA larger thanunity at low values of �0.

Near the reference climate (�TOA �0.6), the coeffi-cients �L and �S take values of 0.8 and 0.3, respectively(Fig. 6). The value for fS is taken as approximately 0.2,based on the analysis of cloud and surface albedo short-wave feedbacks from climate models by Colman(2003). Therefore, [(1 � fS)�L fS�S] � 0.7 and, with��1

0 � 3.21 W m�2 K�1 [the mean of the values fromvarious climate models reported by Soden and Held(2006)], (26) and (29) predict that the ratio of the re-sponses in LHF and � to a pure CO2 perturbation withfixed � is �LHF/�� 2.2 W m�2 K�1. This is compa-rable to the results from comprehensive climate models(HS06), but the concomitant increase in atmosphericshortwave absorption results in a smaller rate of in-

crease in LHF (Takahashi 2008, manuscript submittedto J. Climate, hereafter T08), as illustrated next.

b. Changes in shortwave absorption

To estimate the changes in absorption of shortwaveradiation in the clear sky atmosphere, the full-radiationcalculations were made with equinoctial insolation at asolar zenith angle of 38.3° (the average at the equator;Hartmann 1994) at a tropical latitude of 15°, using noradiatively active gases except water vapor and CO2.

The increase in the fraction � of absorbed shortwaveradiation (calculated as the ratio of the surface to theTOA net shortwave fluxes)2 resulting directly fromquadrupling the concentration of CO2 is comparable tothe increase associated with the increase in water vapordue to a warming of 0.2 K (Fig. 10), which is smallcompared to the warming of more than 4 K expectedfor that forcing, based on a variety of comprehensiveclimate models (Randall et al. 2007). The reduction inCO2 concentration by a factor of 4 has a larger directeffect than its increase (Fig. 10), but it is still smallcompared to the water vapor effect.

The water vapor dependence of � is linear on � forfixed relative humidity (Fig. 10) with a slope of ��/��0.23% K�1. Although the atmosphere is optically thinfor shortwave radiation, � increases with � considerablymore slowly than C–C suggests. This is probably be-cause the relevant water vapor absorption bands (in the

2 The contribution of the subcloud layer to � and its variationwith � is an order of magnitude smaller.

FIG. 9. Similar to Fig. 3, but using the full radiative transfermodel for 2�6 and 26 times the reference CO2 concentration (360ppm) and the corresponding values of � consistent with UTOA �240 W m�2.


near-infrared) are already close to saturation, andtherefore the atmosphere is not optically thin at thosefrequencies.

With S0� 240 W m�2, (29) indicates that the increasein clear sky shortwave absorption by water vaporshould lead to a reduction in the rate of increase ofLHF with respect to � by about 0.5 W m�2K�1 (i.e.,from 2.2 to 1.7 W m�2 K�1). The � effect estimated inthis study is relatively small, but in some comprehen-sive climate models it is comparable to the LHF change(T08).

6. Discussion

The simplest version of the radiative–convectiveequilibrium model considered in this study has threeunknowns: the surface air temperature, the sea–air tem-perature difference and the surface latent heat flux.The first of these is solved for from the energy balanceat the top of the atmosphere, as in the original versionsof the radiative–convective models in which this wasthe only variable (see review by Ramanathan andCoakley 1978). In these models, the total convectivevertical energy flux could be diagnosed by requiring itto balance the radiative cooling associated with the pre-scribed convectively neutral lapse rate. The partitioningof this flux between the different components at thesurface could not be determined because it involves atleast one additional variable and therefore requires onemore equation. This extra constraint is provided in thepresent model by considering the energy budget of thesubcloud layer. The additional information is then pro-

vided by the vertical structure of the longwave fluxesand its magnitude at cloud base, which can be calcu-lated based on the surface temperature and the green-house gas distribution, allowing the latent heat flux plusthe atmospheric shortwave absorption to be deter-mined, as well as the sea–air temperature difference.The role of low-level moisture in the solution is indirectand is manifested mainly through its influence on thepressure at the LCL, which is mainly affected by rela-tive humidity, and on the lapse rate in the free tropo-sphere, which affects primarily the surface air tempera-ture.

Another apparently reasonable closure that has beenused in similar models is setting the surface relativehumidity r0 to a fixed value (e.g., Lindzen et al. 1982;Pierrehumbert 2002), which is motivated by the weakobserved variability in r0. The constant relative humid-ity approximation is indeed adequate for explaining thechanges in the water content of the atmosphere (HS06)with temperature because the rate of increase in thesaturation specific humidity with temperature, whichfollows the Clausius–Clapeyron scaling, is large (�7%K�1) compared to the changes in relative humidity.However, surface evaporation is more sensitive tochanges in r0 than moisture content because it dependsapproximately on qs(Ts)(1 � r0). Thus, for r0 � 0.8, amodest fractional change in r0 of 1% per degree warm-ing would be translated into a larger change in evapo-ration (�4%), which would significantly offset the�7% associated with the C–C increase in qs. For thisreason, the models or observational estimates (e.g.,Wentz et al. 2007) that assume constant r0 tend to pro-duce increases in LHF that essentially follow C–C scal-ing, whereas models that predict r0 (e.g., BR89 and thepresent study) have changes in LHF largely unrelatedto C–C.

The fact that changes in LHF relative to temperatureare only weakly related to changes in moisture impliesthat the residence time of moisture will generally alsochange. Comprehensive models for climates close tothe present have an increase in LHF weaker than C–Cassociated with greenhouse forcing, so the residencetime increases. This has been interpreted as a weaken-ing of the motions responsible of transporting the mois-ture vertically (BR89; Betts 1998; HS06; Vecchi andSoden 2007). The present model does not explicitlyconsider vertical motions and no assumption has beenmade relating any such motion to the moisture budget,so it predicts the change in the residence time with noreference to dynamical processes. Furthermore, the re-lation between the strength of the vertical motions andthe moisture transport might not be straightforward.For instance, explicitly writing the vertical moisture

FIG. 10. Fraction (�, in percent) of net shortwave at the TOAabsorbed in the atmosphere as a function of the water vapordistribution (obtained from a fixed relative humidity profile anddifferent values of �) and for three concentrations of CO2.


transport flux as �M�q��, where M is the vertical massflux, � • � indicates a global average, and primes indicatedeviation from it (note that �M� � 0 in equilibrium),allows its fractional changes to be written as

�M�q��

�M�q��

�M�2�

�M�2�

�q�2�

�q�2�

�

�, �34�

where

� ��M�q��

�M�2� �q�2��35�

is the spatial correlation coefficient between M and q.Thus, assuming the changes in the moisture flux and themoisture fluctuations as given, either the strength of thevertical motions (represented by the standard deviationof M) or the correlation ! could adjust to satisfy (34).Preliminary experiments with a GCM in radiative–convective equilibrium with no convective parameter-ization (i.e., moist convection takes place in the re-solved scales) suggest that changes in ! can be as im-portant as those in the strength of M in (34). In fullGCMs, however, the changes in ! are not as importantwhen considering the parameterized convective massfluxes or the resolved zonally asymmetric tropical over-turning (Vecchi and Soden 2007). The former is prob-ably due to the way the convective schemes are con-structed (i.e., ! might be prescribed to some extent),while the latter is perhaps associated with geographicalconstraints on the pattern of both the circulation andmoisture distribution that do not exist in the radiative–convective setting. These matters deserve further study.

The results with the full radiative transfer model(section 5) are in quantitative agreement with thosefrom comprehensive climate models, but because therewas no a priori expectation that the present modelwould behave similarly to a global climate model, it ispossible that this agreement is fortuitous. However,OS07 indicated that a radiative–convective version oftheir semigray radiation GCM provided a good repre-sentation of the latter under a broad range of forcings,suggesting that the earth might approximately behaveas a radiative–convective equilibrium model regardingthe global hydrological cycle. The question then arisesof what are the fundamental differences between theglobal energy budgets of a radiative–convective worldand a rotating planet with differential heating. If theglobal mean vertically varying energy equation wereobtained for the latter, a few terms involving spatialcovariances would appear that are not in the presentradiative–convective model. One of these terms is the

vertical transport of dry static energy by the large-scalecirculation, but, as mentioned in the introduction, thisterm is small (less than 10 W m�2 globally) compared tothe latent heat flux. Another one is associated with thelogarithmic dependence of UTOA on specific humidity,which implies that spatial variations in the latter canhave significant effects on the spatial mean of theformer (Pierrehumbert et al. 2006). The relevance ofthe present model to the global mean hydrological cycleis therefore contingent upon the relative smallness ofthese covariance terms.

The effects of clouds on the hydrological cycle werenot considered in the present model. Although the as-sociated feedbacks at the TOA are considered in (26)and (29), changes in clouds can potentially also affectthe �L coefficient (23) by modifying the shape of thevertical profile of the longwave fluxes. This remains tobe addressed, although the fact that comprehensive cli-mate models under greenhouse forcing show robustchanges in LHF plus atmospheric absorption of short-wave (T08), despite significant differences in the treat-ment of clouds, suggests that this effect is probablysmall, at least in current climate models.

7. Summary and conclusions

A one-dimensional radiative–convective equilibriummodel has been formulated in terms of the energy bud-gets at the top of the atmosphere, the free atmosphere,and the subcloud layer. The surface relative humidityand the air–sea temperature difference are part of thesolution. Using semigray radiative transfer and pre-scribing the pressure at the LCL and the free atmo-spheric lapse rate (which does not qualitatively changethe behavior of the system), semianalytical solutions forsurface air temperature, the sea–air temperature differ-ence, and the surface latent heat flux (LHF) are ob-tained in terms of the radiative properties of the systemalone (i.e., longwave optical depth, shortwave absorp-tivity, net incoming shortwave). This solution then al-lows the surface humidity to be diagnosed for a givenwind speed and other parameters controlling the sur-face moisture flux.

The semianalytical solution shows that forcingthrough increased greenhouse gases leads to an in-crease in LHF through a reduction in the net longwaveflux through the LCL due to the increased opacity ofthe low-level atmosphere, despite an opposing directeffect from the increased surface air temperature,which enhances the longwave flux at the LCL. On theother hand, the same change in the longwave fluxthrough the LCL is associated with a decrease in thelongwave cooling in the subcloud layer, so a reduction


in the sea–air temperature difference is required to re-duce the surface longwave and sensible heat fluxes andmaintain the energy balance in this layer.

In the simplest version of the model, the rate of in-crease in LHF relative to � under greenhouse forcing isgiven by �LHF/�� 1

0 �L, where ��10 � 3.2 W m�2/K

is the Planck feedback and �L is a coefficient that de-pends only on the relative magnitudes of the net long-wave fluxes at the LCL and the TOA. When a fullradiative transfer model with an interactive water vaporlongwave radiative effect (assuming fixed relative hu-midity for radiative purposes only), �L took a value ofaround 0.8, which leads to �LHF/�� 2.6 W m�2 K�1

under greenhouse forcing. Radiative feedbacks modifythe effective coefficient and, using an estimate of short-wave feedbacks, the effective coefficient and thechange in LHF are reduced to 0.7 and 2.2 W m�2 K�1,respectively. The increase in clear-sky shortwave ab-sorption associated with the increase in water vaporreduces the latter further to 1.7 W m�2 K�1, close towhat comprehensive climate models predict for theglobal hydrological cycle (HS06).

The prescription of the pressure at the LCL (pLCL)and the lapse rate in the free atmosphere makes LHFinsensitive to the specific humidity at the surface, sothe Clausius–Clapeyron relation has no bearing onLHF in this case. More generally, pLCL is sensitive tothe surface relative humidity r0. However, in the caseof greenhouse forcing, increases in temperature andmoisture have opposing effects on r0 and the resultingeffect on LHF through pLCL is subtle. On the otherhand, a change in wind speed leads to a more robustincrease in r0, so pLCL is increased, which results in adecrease in the longwave flux at the LCL and, there-fore, in an increase in LHF. However, at most 25% ofthe relative change in wind speed translates into achange in LHF.

Acknowledgments. The author thanks Drs. Isaac M.Held and Gabriel A. Vecchi for useful discussions andDrs. Held and Geoffrey K. Vallis for comments on themanuscript. Discussions with Drs. L. E. Back, D. S.Battisti, P. Caldwell, D. M. W. Frierson, P. A.O’Gorman, O. M. Pauluis, G. Roe, and M. Zhao, aswell as the comments from two anonymous reviewers,are also appreciated. This research was supported bythe National Oceanic and Atmospheric Administration(NOAA) Climate and Global Change Postdoctoral Fel-lowship Program, administered by the University Cor-poration for Atmospheric Research, under AwardNA06OAR4310119 from the NOAA Climate Pro-grams Office, U.S. Department of Commerce, with ad-ditional funding from Award NA17RJ2612 from

NOAA, U.S. Department of Commerce. The authorthanks the NOAA Geophysical Fluid Dynamics Labo-ratory for hosting him during the development of thisstudy. The statements, finding, conclusions, and recom-mendations are those of the author and do not neces-sarily reflect the views of the NOAA Climate ProgramOffice or the U.S. Department of Commerce.

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radiative constraints on the hydrological cycle in an idealized radiative… · 2016-05-09 ·...

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