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The entropy budget of an atmosphere in radiativeonve tive equilibrium is analyzed here. The di erential heating of the atmosphere, resulting from surfa e heat uxes and tropospheri radiative ooling, orresponds to a net entropy sink. In statisti al equilibrium, this entropy sink is balan ed by the entropy produ tion due to various irreversible pro esses su h as fri tional dissipation, di usion of heat, di usion of water vapor and irreversible phase hanges. Determining the relative ontribution of ea h individual irreversible pro ess to the entropy budget an provide important information on the behavior of onve tion. We dis uss the entropy budget of numeri al simulations with a loud ensemble model. In these simulations, we nd that the dominant irreversible entropy sour e in moist onve tion is asso iated with irreversible phase hanges and di usion of water vapor. In addition, a large fra tion of the fri tional dissipation in moist onve tion results from falling pre ipitation, and turbulent dissipation a ounts for only a small fra tion of the entropy produ tion. These hara teristi s will be present whenever the onve tive heat transport is mostly due to the latent heat transport. In su h ases, moist onve tion is better des ribed as an atmospheri dehumidi er than as a heat engine. The amount of work available to a elerate onve tive updrafts and downdrafts is mu h smaller than predi ted by earlier studies whi h assume that moist onve tion behaves mostly as a perfe t heat engine. 2 1 Introdu tion Carnot (1824) re ognized that the generation of onve tive motion by louds is similar to the produ tion of me hani al work by a steam engine. Sin e then, the se ond law of thermodynami s has been applied to various problems within the atmospheri s ien es, su h as the general ir ulation of the atmosphere (Peixoto et al. 1991, Peixoto and Oort 1992), radiative heat transfer (Li et al. 1994, Li and Chylek 1994), hurri ane dynami s (Emanuel 1986, Bister and Emanuel 1998), dust devils (Renn o 1998), and moist onve tion (Renn o and Ingersoll 1996, hen eforth RI, and Emanuel and Bister 1996, hen eforth EB). Clausius' formulation of the se ond law an be written as S = QT + Sirr (1) Sirr 0: (2) Here, S is the entropy hange asso iated with a physi al transformation, Q is the external heating, T is the temperature of the system, and Sirr is the amount of entropy that is produ ed be ause of the irreversible nature of the transformation. Entropy is a state fun tion of the system: S depends only on the initial and nal states. The irreversible nature of physi al transformations is imposed by requiring the irreversible entropy produ tion to be positive. Equation (1) is referred to as the entropy budget whi h onsists of analyzing the various pro esses o urring in a given system and determining their e e t on the entropy of the system. The se ond law allows one to quantify the irreversibility asso iated with physi al transformations in terms of their irreversible entropy produ tion. We fo us here on the four me hanisms responsible for the bulk of the irreversible entropy produ tion in moist onve tion: fri tional dissipation, evaporation of water vapor, di usion of heat, and di usion of water vapor. Among the di erent sour es of irreversibility, fri tional dissipation should re eive spe ial attention. Be ause of the ontinuous loss of me hani al energy due to fri tional dissipation, me hani al work must be ontinuously produ ed to maintain the atmospheri ir ulation. Carnot's heat engine analogy provides a helpful paradigm: the work required to maintain the atmospheri ow is related to the atmospheri heat transport from the warm part of the globe to the older regions. By determining the individual ontribution of fri tional dissipation in the entropy budget, one an obtain important information on 3 the behavior of onve tion. This is the approa h followed in RI and EB who attempt to determine the work performed by onve tive systems from an analysis of the entropy budget, and use this estimate to onstru t a theory for the verti al velo ity, Conve tive Available Potential Energy (CAPE) and intermitten y of moist onve tion. In both RI and EB, moist onve tion is assumed to behave mostly as a perfe t heat engine where me hani al work is dissipated through a turbulent energy as ade a ting on onve tive updrafts and downdrafts. A rst diÆ ulty with this theory has been dis ussed by Pauluis, Balaji and Held (2000, hen eforth PBH) who show that a signi ant amount of fri tional dissipation o urs in the shear zone surrounding falling hydrometeors. In numeri al simulations, this pre ipitation-indu ed dissipation is found to be larger than the dissipation asso iated with the turbulent as ade from the onve tive s ales to smaller s ales. Most of the work performed by onve tion is used in lifting water rather than in a elerating the updrafts and downdrafts. However, PBH also observe that the total work performed by the atmosphere is signi antly smaller than the expe ted for a Carnot y le This leads us to ask whether or not moist onve tion behaves as a perfe t engine. We do not question here the fa t that the produ tion of me hani al work in the atmosphere is asso iated with a heat transport from a warm sour e to a old sink. However, moist onve tion does not a t solely as a heat engine: it also plays an essential role as an atmospheri dehumidi er. The as ent of moist air in deep onve tive towers results in a removal of water vapor through ondensation and pre ipitation. In radiativeonve tive equilibrium, this dehumidi ation is balan ed by a moistening of dry air asso iated with various pro esses su h as surfa e evaporation, entrainment of tropospheri air into the planetary boundary layer, mixing of louds into the environment and reevaporation of rain and snow. This moistening is inherently an irreversible pro ess, asso iated at the mi rophysi al s ale with irreversible phase hanges and di usion of water vapor, and therefore results in a net produ tion of entropy. This important aspe t of moist onve tion is dis ussed in greater detail in the ompanion paper (Pauluis and Held, 2000; hen eforth PH). These two aspe ts of onve tion, heat engine and atmospheri dehumidi er, are in ompetition whi h ea h other in terms of irreversible entropy produ tion. To the extent that onve tion behaves as an atmospheri dehumidi er, its ability to fun tion as a heat engine is redu ed. The entral 4 issue regarding the entropy budget of moist onve tion an thus be rephrased as to whether onve tion a ts more as a heat engine or as an atmospheri dehumidi er. As des ribed by RI and EB, this question is dire tly related to other issues su h as CAPE or the verti al velo ity. We use a loud ensemble model (CEM) to simulate radiativeonve tive equilibrium and analyze the entropy budget. Although these models were initially developed for ase studies of onve tion ( Klemp et al. 1978, Lipps and Hemler 1982), the in rease in omputer power makes it possible to simulate large enough domains over a long enough time to a hieve radiativeonve tive equilibrium. Many re ent studies have used CEMs to investigate statisti al properties of onve tion (Tao et al. 1987, Held et al. 1993, Xu and Randall, 1999, Tao et al. 1999). These models usually treat water through bulk parameterizations whi h separate between water vapor, loud water, and preipitation, and provide enough information on thermodynami pro esses for an analysis of the entropy budget. Se tion 2 dis usses general aspe ts of the entropy budget of an atmosphere in radiativeonve tive equilibrium. In parti ular, we emphasize how the entropy budget relates to me hani al work, and how the latter is redu ed in the presen e of other irreversible pro esses. A fundamental di eren e between dry and moist onve tion is that irreversible phase hanges and di usion of water vapor de rease the amount of work performed by the atmosphere. Se tion 3 analyzes the entropy budget of numeri al simulations of radiativeonve tive equilibrium with a loud resolving model. In parti ular, radiativeonve tive equilibriums in dry and moist atmospheres are ompared. It is shown that, in the moist ase, irreversible phase hanges and di usion of water vapor are the dominant entropy sour es. It is also found that fri tional dissipation asso iated with pre ipitation is larger than the dissipation resulting from a turbulent as ade to small s ales, onsistent with the nding of PBH. The di eren es between the entropy budget of the physi al atmosphere and of the numeri al model are dis ussed in the Appendix. Se tion 4 presents a non-dimensional analysis of the entropy budget. This aims at determining a onve tive eÆ ien y, whi h is similar to the notion used by Craig (1996). It is shown that this onve tive eÆ ien y depends on two aspe ts of moist onve tion: the relative magnitude of the onve tive transport of latent heat, and the relative magnitude of the fri tional dissipation by pre ipitation. The theories of RI and EB orresponds to the spe i ase of weak latent heat transport by onve tion, but signi antly overestimate 5 the onve tive eÆ ien y when the latent heat transport is large. The last se tion dis usses the impli ations of our analysis of the entropy budget for the behavior of moist onve tion. 2 Entropy budget and fri tional dissipation Consider an atmosphere in radiativeonve tive equilibrium as des ribed s hemati ally in Figure 1. The radiative ooling Qrad of the troposphere is balan ed by surfa e heat ux, whi h is de omposed into the sensible heat ux Qsen and latent heat ux Qlat. This di erential heating destabilizes the air olumn so that onve tion an develop. There is no large-s ale ir ulation in the sense that there is no mass transport through the lateral boundaries of the system. After some time, the system rea hes a statisti al equilibrium where di erential heating is balan ed by onve tive heat transport. At this point, the total internal energy, me hani al energy and entropy of the atmosphere are statisti ally steady. Conservation of energy requires that surfa e uxes balan e radiative ooling Qrad +Qlat +Qsen = 0: (3) The me hani al work W done by onve tion is due to air expansion and an be written as W = Z p iVi (4) where R = R dxdydz denotes the integral over the entire atmospheri domain. Here, p is the total pressure, Vi is the i-th omponent of the velo ity, i = = xi is the partial derivative in the i dire tion, and the onvention of summing over repeated indi es is adopted. Me hani al energy is also removed and onverted into internal energy through fri tional dissipation. The total dissipation per unit area D is given by D = Dk +Dp = Z ij jVi; (5) where ij is the vis ous stress tensor. For pre ipitating onve tion, fri tional dissipation an be de omposed into the dissipation due to pre ipitation Dp o urring in the mi ros opi shear zones surrounding hydrometeors, and the dissipation Dk asso iated with the turbulent energy as ade from the onve tive s ales of motion to the smaller s ales at whi h vis osity an a t. 6 As dis ussed in PBH and EB, the pre ipitation-indu ed dissipation an be estimated from: Dp = Z g VT = Z qtgw; (6) where g is the gravitational a eleration, is the mass of air per unit volume, is the mass of falling hydrometeors per unit volume, qt is mass of total water per unit mass of moist air, VT is the terminal velo ity of the falling hydrometeors, and w is the verti al velo ity of the air. Equation (6) indi ates that Dp is also equal to the geopotential energy imparted to water by the atmospheri ow. For the system onsidered here, one an negle t both the work performed by the atmosphere on its lower boundary (for example by generating surfa egravity waves) and the kineti energy ux at the same boundary (due to a vis ous transfer of kineti energy to the o eans). In this ase, onservation of me hani al energy requires the me hani al work to be dissipated by fri tion W Dp Dk = 0: (7) The entropy per unit mass of moist air is de ned by s = (1 qt)(Cpd lnT Rd ln pd) + qtCl lnT + qvLv T qvRv lnH: (8) In this expression, qv is the spe i humidity for water vapor, Cpd is the spe i heat at onstant pressure of dry air, Cl is the spe i heat of liquid water, T is the temperature of moist air, Rd and Rv are the gas onstants of dry air and water vapor, pd is the partial pressure of dry air, H = e=es is the relative humidity, e is the water vapor pressure and es is the saturation vapor pressure. More details on this de nition of the entropy of moist air an be found in Emanuel (1994) or Iribarne and Godson (1981). Entropy hanges are due to either a heat ex hange with the environment or to an irreversible pro ess, the latter always resulting in a net produ tion of entropy, as required by the se ond law of thermodynami s. In this study, the external heat sour es are limited to radiative ooling and heat ex hange at the Earth's surfa e. The entropy sour e or sink asso iated with external heating or ooling is given by the energy input divided by the temperature at whi h it o urs. Hen e, the surfa e latent and sensible heat uxes are asso iated with an entropy sour e equal to Qlat=Tsurf and Qsen=Tsurf , where Tsurf is the surfa e 7 temperature. Radiative ooling is asso iated with an entropy sink Qrad=Trad where Trad is an e e tive ooling temperature de ned by Qrad Trad = Z frad T ; (9) where frad is the radiative ooling rate per unit volume. The total radiative ooling is equal to Qrad = R frad. In statisti al equilibrium, various entropy sour es and sinks ompensate ea h other. The entropy budget takes the form Qlat +Qsen Tsurf + Qrad Trad + Sirr = 0; (10) where Sirr is the total entropy produ tion by the irreversible pro esses. As the se ond law of thermodynami s requires irreversible pro esses to be an entropy sour e Sirr 0, the di erential heating of the atmosphere must result in a net entropy sinkQlat +Qsen Tsurf + Qrad Trad 0: (11) Be ause of the energy balan e Qlat +Qsen +Qrad = 0, statisti al equilibrium an only be a hieved when the e e tive ooling temperature Trad is lower than the temperature of the heat sour e Tsurf , as dis ussed for example by Lorentz (1967). For a given distribution of heat sour es and sinks, the entropy budget (10) provides a onstraint on the total produ tion of entropy by irreversible pro esses. By estimating the ontribution of individual pro esses one an then obtain spe i information about onve tive a tivity. The ontribution of fri tional dissipation is of parti ular interest as it is related to the work performed by the system and is potentially related to fundamental hara teristi s of onve tive systems su h as CAPE or verti al velo ity. The entropy produ tion due to fri tional heating Sd is given by Sd = Z ij jVi T = Dk +Dp Td = WTd ; (12) where Td is the e e tive temperature at whi h fri tional dissipation o urs. 8 Combining the expressions (3), (7), (10) and (12) yields an estimate of the total me hani al work done by onve tion W = Td(Tsurf Trad) TsurfTrad jQradj Td Snf ; (13) where Snf = Sirr Sd is the irreversible entropy sour e due to me hanisms other than fri tion. The maximum work Wmax whi h an be produ ed by the system for a given distribution of heat sour es and sinks o urs when no other irreversible entropy sour es are present Snf = 0: Wmax = Td(Tin Trad) TinTrad jQradj: (14) The di eren e between the me hani al work in the atmosphere and this theoreti al maximum is due to the produ tion of entropy by the other irreversible pro esses: Wmax W = Td Snf : (15) Hen e, one needs to estimate Snf to determine the me hani al work produ ed by the system. Conversely, it is possible to derive the entropy produ ed by other irreversible pro esses from the total me hani al work and the distribution of heat sour es and sinks. In the present paper, we fo us on four irreversible pro esses: fri tional heating, di usion of heat, di usion of water vapor and irreversible phase hanges. 2 It is rst argued that the entropy produ tion due to mole ular di usion of temperature Sdif is negligible in all ases of interest. The entropy produ tion due to the di usion of sensible heat is given by Sdif = Z Jsen;i iT T 2 ; (16) where Jsen;i is the mole ular ux of sensible heat in the i dire tion. In the surfa e layer, mole ular di usion transports the sensible heat ux Qsen from 2When radiation is treated as part of the system, absorption of short-wave radiation at the Earth's surfa e is the largest irreversible entropy sour e in the limate system (see Li et al. 1994 and Li and Chylek, 1994) for a dis ussion of the irreversible entropy produ tion by radiative transfer). By treating radiative pro esses as external heat sour es and sinks, the entropy produ tion due to radiative transfer and absorption is in luded in the entropy sour es and sinks due to the external heating. 9 the Earth's surfa e at Tsurf into the atmosphere at Tsurf Tbnd. The entropy produ tion asso iated with mole ular di usion near the surfa e an be approximated by Sdif;sf Qsen Tbnd T 2 surf : (17) Mole ular di usion of heat o urs also as an end result of turbulent mixing in the atmosphere. It an be assumed that turbulent mixing is limited to mixing only of detraining updrafts or downdrafts with ambient air. In this ase, the irreversible entropy produ tion asso iated with the sensible heat ux is Sdif;int Qmix Tdt Tdt2 : (18) Here, Tdt is the temperature di eren e between detraining updrafts (or downdrafts) and the environment, Tdt is the averaged temperature where detrainment o urs, and Qmix is the sensible heat transfer asso iated with detrainment. For dry onve tion, the radiative ooling in the environment is balan ed by the sensible heat ux from the updraft: Qmix Qrad. Therefore, as long as Tdt and Tbnd are small in omparison to Tsurf Trad, the entropy produ tion by mole ular di usion of temperature an be negle ted in the entropy budget. Hen e, for dry radiativeonve tive equilibrium, where the only irreversible pro esses are di usion of heat and fri tional heating, fri tional heating a ounts for most of the irreversible entropy produ tion. One an then assume that the total work done by dry onve tion is lose to the theoreti al maximum: W Wmax. For moist onve tion, the heat transferred at the detrainment level an be written as Qmix MupCp Tdt, where Mup is the upward mass transport by onve tion. The radiative ooling in the free troposphere is approximately balan ed by the warming due to subsiden e Qrad +Mup G = 0, where G is the dry stati energy di eren e between the detrainment level and the sub loud layer. Hen e, the di usion of sensible heat at the detrainment level is Qmix QradCp Tdt= G. As the verti al di eren e of dry stati energy is mu h larger than the enthalpy di eren e between the updrafts and the environment G >> Cp Tdt, the sensible heat ux asso iated with detrainment is signi antly smaller than radiative ooling Qmix << Qrad. Hen e, as long as the temperature di eren es a ross the boundary layer or between detraining air and the environment are signi antly smaller than 10 the di eren e between the surfa e temperature and the average temperature at whi h the atmosphere is ooled radiatively, Tbnd << Tsurf Trad and Tdt << Tsurf Trad, the ontribution of the di usion of heat an be negle ted in omparison to the other irreversible entropy sour es. In moist onve tion, there are two additional irreversible pro esses dire tly involving water vapor: irreversible phase hanges and di usion of water vapor. Irreversible phase hanges o ur when liquid water evaporates in unsaturated air or when water vapor ondenses in supersaturated air. (Freezing and melting an also result in an irreversible entropy produ tion, but are not treated expli itly in this paper.) The irreversible entropy produ tion due to the ondensation of M kg of water is given by MRv lnH. The total entropy produ tion due to irreversible phase hanges Sp is given by the integral Sp = Z (C E)Rv lnH Zz=0 Jv;zRv lnH: (19) Here, C and E are the ondensation and evaporation rates per unit volume, and Jv;z is the verti al omponent of the mole ular ux of water vapor. The rst term on the right-hand side of (19) is the produ tion due to irreversible ondensation and reevaporation in the atmosphere. The se ond term is the produ tion due to irreversible evaporation at the surfa e. The irreversible entropy produ tion by mole ular di usion of water vapor is given by Sdv = Z RvJv;i i ln e; (20) where Jv;i is the i-th omponent of the mole ular ux of water vapor. The distin tion between produ tion of entropy by di usion of water vapor and by phase hanges is rather arti ial: the entropy produ tion asso iated with evaporation at relative humidity H = e0=es is the same as would result from di usion of water vapor from a saturated region with e = es to a region where the water vapor is e = e0. In a numeri al model, the partial pressure of water vapor represents the average pressure over a grid box and is not representative of the partial pressure in the vi inity of a water droplet. Hen e, in a model, the distin tion between the ontribution of phase hanges and di usion is arbitrary and depends on spe i assumptions about the mi rophysi s. Indeed, if it is assumed that all air is saturated in the mi ros opi vi inity of all ondensate surfa es, then phase hanges are reversible and diffusion of water vapor is the only irreversibility. However, the total entropy 11 produ tion due to phase hanges and di usion of water vapor is independent of these assumptions, and is treated here as a single entropy sour e, referred to as irreversible entropy produ tion due to moist pro esses. Expressions (20) and (19) require the knowledge of the mole ular ux of water vapor and of the relative humidity, and are for this reason diÆ ult to use for determining Sdv + Sp . However, PH show that the irreversible entropy produ tion by moist pro esses is related to the onve tive transport of latent heat: Sdv + Sp = T 1 vap QlatTsurf Tlat TsurfTlat Wvap : (21) Here, Wvap is the expansion work by water vapor given by Wvap = Z e iVi = Z ( te+ Vi ie): (22) The e e tive temperature of water vapor pressure hanges Tvap is obtained from the relationship Wvap Tvap = Z te+ Vi ie T ; (23) and the e e tive temperature of latent heat release Tlat is given by Qlat Tlat = Z Lv0(C E) T : (24) Here, Lv0 is the latent heat of vaporization at surfa e temperature, so that the latent heat ux at the surfa e is Qlat = R Lv0(C E). We refer the reader to PH for more dis ussion on the relationship (21) and the various quantities introdu ed in this equation. The estimates of the entropy produ tion by moist pro esses in the numeri al simulations of se tion 3 and s alings arguments of se tion 4 are based on (21). For moist onve tion, the irreversible entropy produ tion is the sum of the produ tion due to fri tional heating, di usion of heat, and the moistening e e t. Di usion of heat is small, as seen before. Therefore, we have Sirr D Td + Sdv + Sp : (25) Hen e, the main question is how the irreversible entropy produ tion is split between fri tional heating and moist pro esses, or, from a mi ros opi perspe tive, between di usion of water vapor and di usion of momentum. 12 3 Numeri al simulations A numeri al loud resolving model is used to simulate radiativeonve tive equilibrium and analyze its entropy budget and kineti energy dissipation. The model has been developed by Lipps and Hemler (1982, 1986, 1988) and is used in a version similar to the one used by Held et al. (1993), but with higher horizontal and verti al resolution. Model dynami s is based on an elasti approximation, whi h allows the zonal-mean state of the system to evolve freely in response to onve tion and in ludes a semi-impli it s heme for verti ally propagating sound-waves. It in orporates a bulk mi rophysi s and separates water vapor, loud water, rain and snow. Surfa e sensible and latent heat uxes are obtained from a bulk parameterization assuming an o ean surfa e at a onstant temperature of 298K, with the drag oeÆ ient based on Monin-Obukhov similarity (Garratt 1992). For the experiments des ribed hereafter, the radiative transfer has been repla ed by a Newtonian ooling, relaxing the atmosphere to a uniform temperature of 200K over a time-s ale of 40 days. This provides a simple way to produ e radiative ooling rates of the right magnitude while letting the tropopause level be determined by the dynami s alone. A two dimensional, horizontally periodi domain is represented by a 320x78 grid. Verti al resolution varies from 50 meters near the surfa e to 500 meters at higher levels. Horizontal resolution is 2km. A mean zonal wind pro le is imposed with u(z) = !min(z; z ) with ! = 10 3s 1 and z = 5km. This is ne essary in two-dimensional simulations to avoid a QBO-like os illation due to the absorption of gravity-waves in the stratosphere as dis ussed in Held et al. (1993). A sponge-layer is present in the top 10km of the model to dissipate the gravity waves before they are re e ted at the upper boundary. This model di ers from that used by PBH in that it is two-dimensional, in that the expli it radiative transfer has been repla ed by Newtonian ooling, and in that it uses Monin-Obukhov drag oeÆ ients. We analyze here two experiments: (1) dry onve tion | in the absen e of water | and (2) moist onve tion | whi h in ludes the full mi rophysi al parameterization of the model. The atmosphere rea hes statisti al equilibrium in less than 20 days in the dry ase, and 40-60 days in the moist ase. In the dry experiment, the Newtonian ooling is un hanged. This is not meant as a realisti model of dry radiativeonve tive equilibrium, but as a way of generating a model with a simpler entropy budget. 13 The verti al pro les of temperature and moist stati energy h = CpT + Lvq + gz are shown in Figure 2. In both the dry and moist simulations, the atmosphere is separated into the troposphere, where onve tion is a tive and the temperature distribution orresponds to an adiabati pro le, and the stratosphere, whi h is (approximately) in radiative equilibrium. In the dry ase, there is a strong inversion at the tropopause that is reminis ent of the trade-wind inversion in the subtropi s. The stratospheri temperature minimum in the moist simulation is an artifa t due to the downward heat ux asso iated with the damping of gravity waves by the sponge-layer. Conve tive a tivity di ers greatly between the dry and moist ases. Figure 3 and Figure 4 show snapshots of the onve tive a tivity for the dry and moist ases respe tively. In the dry ase, onve tive ells over the whole domain. The stati energy is well homogenized with di eren es less than 1 kJ kg 1 between the updrafts and downdrafts. Maximum verti al velo ity is about 8 ms 1 in the updraft and about 6 ms 1 for the downdrafts. In ontrast, moist onve tion is more sporadi and tends to organize in a way reminis ent of squall lines. There are on average between 1 and 3 deep onve tive ells rea hing up to the tropopause level. The strongest onve tive events have a verti al velo ity rea hing up to 10 15 ms 1, and are also asso iated with relatively strong downdrafts. In addition to deep onve tive louds, there are also a large number of onve tive louds detraining below 5 km. The horizontal variations of moist stati energy are large, about 15 kJ kg 1, in omparison with the dry ase, indi ating that individual updrafts arry a larger amount of energy in moist onve tion than in dry onve tion. In the dry ase, onve tion redistributes the sensible heat ux from the surfa e through the whole troposphere. Hen e, radiative ooling is balan ed by onve tive transport of sensible heat ex ept in the surfa e layer. For moist onve tion, the radiative ooling is mostly balan ed by diabati heating, as shown in Figure 5. The onve tive transport of sensible heat a ounts for a small residual. The net latent heating an be separated into the latent heat release due to evaporation and the latent ooling due to reevaporation, as shown in Figure 6. The pre ipitation eÆ ien y p de ned as the ratio of the pre ipitation rate at the surfa e to the total ondensation is fairly small p = R C E R C 0:27: (26) 14 3.1 Me hani al energy budget We turn now to the energy and entropy budgets of these simulations. The results of the analysis of the me hani al energy and entropy budget are s hemati ally presented in Figure 7 and Figure 8. All quantities have been divided by the horizontal area of the domain so that they are expressed in W m 2. The energy, me hani al energy and entropy budgets are also shown in Table 1, Table 2 and Table 3. The various approximations used in the model and their onsequen es for the energy and entropy budgets are disussed in greater detail in the Appendix. Within the model's framework, the produ tion of kineti energy at the onve tive s ales is given by the buoyan y ux Wb = Z w gT 0 T + g(Rv Rd 1)qv ql! ; (27) where ql is the mass of ondensed water per unit mass of moist air. Here, an overline indi ates a horizontally averaged quantity while a prime indi ates the departure from the horizontal mean. As dis ussed by PBH, the buoyan y ux does not a ount for the total me hani al work done by onve tion. It does not in lude the work whi h is required to lift water and whi h is dissipated during pre ipitation. In statisti al equilibrium, the fri tional dissipation due to pre ipitation is given by (6). The total me hani al work in the system is W = Dp +Wb = Z w gT 0 T + gRv Rd qv! : (28) The me hani al work an be written as W = WT + Wb where WT is the thermal part of the buoyan y ux WT = Z g wT 0 T ; (29) and Wv is the ontribution related to the verti al transport of water vapor Wv = Z g wRv Rd qv: (30) This de omposition is motivated by the relationship between water vapor transport and entropy produ tion by moist pro esses, as dis ussed in PH. 15 The mean zonal wind u is xed in the simulations. Maintaining this pres ribed zonal wind against mixing by onve tion is equivalent to adding or subtra ting kineti energy to the system. This external work is equal to Wext = Z U zu0w0: (31) The total generation of me hani al energy is thus the sum of three terms: thermal buoyan y ux WT, water vapor buoyan y ux Wv and the external large-s ale kineti energy for ing Wext. The dissipation of me hani al energy asso iated with the turbulent as ade to smaller s ales Dk is omputed in the model by Dk = Z ij jVi (32) where ij is the model's subgrids ale ux of momentum. Hen e, the me hani al energy budget in the numeri al experiments is WT +Wv +Wext = Dp +Dk: (33) The me hani al energy budget of the dry experiment is s hemati ally represented in Figure 7. Conve tion generates WT = 4:8 Wm 2, while eddies extra t Wext = 3:2 Wm 2 from the mean zonal wind. This large value of Wext is an indi ation of the strong mixing asso iated with intense dry onve tion. The fa t thatWext is positive is not self-evident, given that the most unstable normal modes for two-dimensional Benard onve tion in shear ows produ e ounter-gradient momentum uxes. Me hani al work is balan ed by subgrids ale dissipation Dk = 8:0 Wm 2. The results for the moist experiment are shown in Figure 8. Most of the me hani al energy is produ ed by water vapor ux as Wv = 3:0 Wm 2. The thermal buoyan y ux a ounts for only WT = 1:7 Wm 2. Be ause of the intermittent hara ter of moist onve tion, mixing of momentum is mu h weaker than in the dry ase. Only Wext = 0:04 Wm 2 is extra ted from the mean wind. Dissipation is dominated by pre ipitation, whi h a ounts for Dp = 3:7 Wm 2. The subgrids ale dissipation of kineti energy is only Dk = 1:0 Wm 2. 3.2 Energy budget In radiativeonve tive equilibrium, radiativeooling in the troposphere is balan ed by the surfa e heat uxes. However, in the numeri al model, energy 16 is exa tly not onserved. This results primarily from the fa t that fri tional heating is not in luded in the potential temperature equation used in the model. In the Appendix, it is shown that this nononservation an be treated as if spurious heat sour es were present in the model. In both the dry and moist ase, nononservation of energy translates in a spurious heat sink equal to WT. The additional error term in Table 2 is due to the hange in internal energy of the system. In the dry ase, the sensible heat ux at the surfa e is 111:7 Wm 2. It is balan ed by the newtonian ooling of 106:9 Wm 2 and by the spurious ooling of 4:8 Wm 2. The troposphere is warmer in the moist ase than in the dry ase. This explains that the newtonian ooling is signi antly larger, with a net ooling of 157:4 Wm 2. The latent heat ux is 143:3 Wm 2, whi h is mu h larger than the sensible heat ux of 16:5 Wm 2. There is also an additional spurious heat sink of 1:7 Wm 2. 3.3 Entropy budget The nononservation of energy also modi es the entropy budget, by introdu ing additional entropy sour es and sinks asso iated with the spurious heat sour es and sinks. It is shown in the Appendix that these hanges result in only a small error in the total irreversible entropy produ tion in the system. The entropy budget is analyzed by omparing the maximum theoretial work to the work e e tively performed by onve tion and to the other irreversible entropy sour es Wmax =WT +Wv + Te Sdif + Te Sdv + Sp : (34) Nononservation of energy has two onsequen e for equation (34). First, the de nition of the maximum work must be slightly modi ed. We use the expression (57) for the dry ase and expression (61) for the moist ase. Se ond, an e e tive temperature Te is introdu ed. F or the physi al atmosphere, Te is the fri tional temperature Td. In the numeri al model, it is repla ed by the surfa e temperature Tsurf in the dry ase, and by the e e tive temperature of water vapor transport Tv for the moist ase. The latter is de ned by Tv = Z wqv T 1 Z wqv: (35) 17 By operating these hanges, the error indu ed by the spurious numeri al heat sour es are in orporated in Wmax and Te , while the various irreversible entropy sour es an still be ompared through the budget (34). The irreversible entropy produ tion due to di usion of heat Sdif is obtained from (16), after repla ing the mole ular sensible heat ux by the subgrids ale ux. In this ase, the sensible heat transport an be viewed as resulting from subgrids ale eddies. These eddies are able to extra t available potential energy an onvert it into kineti energy, whi h is in turn dissipated through fri tion. The entropy produ tion Sdif in the simulation is therefore partially due to mole ular di usion of heat and partially due to this generation and dissipation of kineti energy by subgrids ale eddies. It is, however, not possible to separate Sdif between these two pro esses without making further assumptions. The entropy produ tion due to moist pro esses Sdv + Sp is obtained through formula (21). The latent heat transport an be obtained dire tly from the model. The work performed by water vapor expansion annot be obtained dire tly, but the ratio Wvap=Tvap in (21) an be approximated by Wv=Tv (see PH). This method allows us to determine the orresponding entropy produ tion without requiring the expli it the knowledge of the di usive ux of water vapor or the relative humidity. We refer the reader to PH for more dis ussion of the relationship (21) and its physi al interpretation. As the spe i humidity of an air is onserved in the absen e of phase hanges and di usion of water vapor, the entropy produ tion due to the mixing of two air masses is equal to the entropy produ tion that would result from the di usion of the water vapor from one air mass to the other. Turbulent motion an only in rease the gradient of spe i humidity, but annot hange the spe i humidity distribution. The nal homogenization between the two air masses must be in ne a omplished through mole ular di usion of water vapor. Subgrids ale eddies annot hange the entropy produ tion due to moist pro esses if they are not asso iated with phase hanges or result in a verti al transport of water vapor. This implies that, although the numeri al model uses a subgrids ale parameterization for the di usion of water vapor, the entropy produ tion for Sdv + Sp omputed in the simulations orresponds indeed to mole ular di usion and irreversible phase hanges. In the dry ase, the maximum work isWmax = 8:0 Wm 2, while the work done by onve tive systems is only WT = 4:8 Wm 2. A dire t estimation of the entropy produ tion due to the subgrids ale di usion of sensible heat 18 results in a loss of Te Sdif = 3:0 Wm 2. As dis ussed earlier , this entropy produ tion is due both to mole ular di usion of heat and to the generation and dissipation of kineti energy by subgrids ale eddies. There is also a small imbalan e in the entropy budget whi h an be attributed to small hanges in the total entropy of the system and to numeri al errors in the averaging method and adve tion s heme. In the moist experiment, the maximum work is Wmax = 13:8 Wm 2. This value is larger than in the dry ase mainly be ause the total heat ux is larger. Less than one third is e e tively performed by onve tion WT+Wv = 4:7 Wm 2. Subgrids ale di usion of heat a ounts only for a small portion of the irreversible entropy produ tion with Te Sdif = 0:6 Wm 2. The irreversible entropy produ tion due to moist pro esses is obtained through (21) and a ounts for most of the entropy produ tion, with Te Sdv + Sp = 8:3 Wm 2. There is an imbalan e 0:2 Wm 2 in the entropy budget (34), whi h an be attributed to numeri al artifa ts. These simulations indi ate that the entropy budget of dry and moist onve tion di er signi antly. On the one hand, dry onve tion an be des ribed as a ting mostly as a heat engine, as most of the irreversibility is asso iated with fri tional dissipation. (As dis ussed earlier, part of the entropy produ tion by subgrids ale di usion of heat should be interpreted as fri tional dissipation by unresolved eddies.) On the other hand, the entropy budget of moist onve tion is more omplex. The largest fra tion of the entropy produ tion is due to irreversible phase hanges and di usion of water vapor. Together, these a ount for about 60% of the total entropy produ tion. Fri tional dissipation still a ounts for a large fra tion of the irreversible entropy produ tion. However, most of it is asso iated with fri tional dissipation o urring in the shear zones surrounding falling hydrometeors, whi h represents about 30% of the total entropy sour e. The amount of kineti energy generated at the onve tive s ales and dissipated through turbulen e only a ounts for a small fra tion of the total entropy produ tion. In the moist ase, most of the work performed by onve tion is due to expansion of water vapor. Even though water vapor a ounts for less than 2% of the mass of the atmosphere, it produ es two-thirds of the total me hani al work by onve tive systems. Consider a par el of moist air as ending in a loud and des ending in the environment. The work produ ed by dry air is the di eren e between the expansion work performed by dry air during as ent and the ompression work exerted on the dry air during des ent. 19 Water vapor, however, expands during its as ent, but does not require any ompression when it falls as ondensed water. This allows a very small quantity of water vapor to perform a large amount work in omparison to dry air. The work performed by water vapor is related to the latent heat transport by onve tion, and the fa t that water vapor a ounts for most of the me hani al work is dire tly related to the importan e of irreversible phase hanges and di usion of water vapor. As dis ussed in PH, the irreversible entropy produ tion by moist proesses and the expansion work by water vapor are related to the latent heat transport. The important ontribution of these two terms in the entropy and me hani al energy budgets ould be dedu ed dire tly from the observation that the onve tive heat transport is mostly due to latent heat, as seen in Figure 5. In su h ase, moist onve tion a ts more as an atmospheri dehumidi er, whi h ontinuously removes water vapor from the atmosphere and where irreversibility is asso iated with the moistening of dry air, than as a heat engine, whi h produ es motion and where irreversibility is the result of fri tional dissipation. 4 Non-dimensional analysis of the entropy budget The entropy budget of moist onve tion is now dis ussed in terms of nondimensional parameters. Of parti ular interest is the estimation of a onve tive eÆ ien y k de ned as the ratio of the turbulent dissipation Dk to the radiative ooling Qrad k = Dk jQradj : (36) The onve tive eÆ ien y measures the amount of me hani al work e e tively used to generate onve tive motion. It is the parameter required in the theory of RI, EB or Craig (1996) to determine the verti al velo ity of onve tive systems. In ontrast to RI and EB, we argue here that k is not equal to the thermodynami eÆ ien y of a prefe t heat engine, but depends ru ially on the latent heat transport by onve tion and on the formation of pre ipitation. The entropy sink asso iated with di erential heating an by measured in terms of the maximum work Wmax. The maximum theoreti al me hani al eÆ ien y max is the ratio between Wmax and the net radiative ooling Qrad. 20 From (14), this non-dimensional oeÆ ient is max = Wmax jQradj = Td 1 Trad 1 Tsurf Tsurf Trad T : (37) The maximum theoreti al eÆ ien y max is approximately equal to the thermodynami eÆ ien y of a Carnot y le. The magnitude of the latent heat transport an be measured by a quantity Wlat introdu ed by PH and de ned as the amount of work that would be performed if the latent heat is transported by a perfe t heat engine: Wlat = QlatTvap( 1 Tlat 1 Tsurf ): (38) Equation (21) indi ates that the latent heat transport is related to the entropy produ tion due to moist pro esses and the expansion work of water vapor. We introdu e here a non-dimensional parameter as the ratio between Wlat and Wmax = Wlat Wmax Tsurf Tlat T Qlat jQradj : (39) This non-dimensional parameter measures the ontribution of latent heat transport relative to the total heat transport. It is shown in PH that the ratio of the expansion work by water vapor to Wlat is approximately given by = Wvap Wlat Hes Hp ; (40) where Hp = z(ln p) 1 is the pressure s ale height, and Hes = z(ln es) 1 is the s ale height for the saturation water vapor pressure. In the tropi s, we have Hp 8 km and Hes 2:5 km, hen e, the value of should be small: 0:2 < < 0:3. Dividing (21) by Wmax and using (39) and (40) yields T Sdv + Sp Wmax (1 ): (41) The total me hani al work by moist onve tion an then be obtained from (25) W jQradj = Dp +Dk jQradj max(1 (1 )): (42) 21 The parameter is the ratio of the total dissipation asso iated with pre ipitation to the me hani al work done by the water vapor expansion = Dp Wvap Dp Wv = R (qv + ql)w R Rv Rd qvw : (43) The transport of ondensed water by air motions is likely to be dominated by the updrafts. Hen e it is expe ted that Rz=z0 qlw > 0, so that the verti al transport of water vapor gives a lower bound on the fri tional dissipation due to pre ipitation. The onstraint implies that 0:6. In the numeri al simulation presented in this paper, it is 1:2. The di eren e between the work due to water vapor expansion and the dissipation asso iated with pre ipitation is the amount of kineti energy that is produ ed by the hydrologi al y le and that is available to produ e atmospheri motion. This an be measured by the parameter de ned as = Wvap Dp Wlat (1 ): (44) When the dissipation due to pre ipitation is larger than the expansion work of water vapor, 1, the hydrologi al y le dissipates more me hani al energy than it produ es and is negative. Both terms in the produ t on the right-hand side are small, and we expe t to be also small, albeit its sign is un ertain. In the simulation we obtain 0:06. The onve tive eÆ ien y k is de ned as the ratio of the fri tional dissipation due to turbulent as ade to the total heat ux. It is given by k = Dk jQradj max( + (1 )): (45) The onve tive eÆ ien y measures the amount of me hani al work e e tively used to generate onve tive motion. It is the parameter whi h is required in the theories of RI, EB or Craig (1996) to determine the verti al velo ity of onve tive systems. For a given distribution of heat sour es and sinks, the onve tive eÆ ien y depends on and . Table 4 and Table 5 show non-dimensional versions of the me hani al energy and entropy budgets, whi h have been obtained by dividing the dimension ounterpart by Wmax. In the previous dis ussion, it was shown that 22 we have 0:2 < < 0:3 and j j 1, and the main unknown is . When is lose to 1 | when the heat transport by onve tion is mostly due to latent heat | moist onve tion will have the following hara teristi s : 1) irreversible phase hanges and di usion of water vapor are the main irreversible entropy sour es, 2) the expansion work of water vapor a ounts for a large fra tion of the total me hani al work in the system, 3) pre ipitation-indu ed dissipation a ounts for a signi ant portion of the total dissipation, and 4) the onve tive eÆ ien y is mu h smaller than the eÆ ien y of a perfe t heat engine. This is the ase in our simulation, with = 0:87. One an onstru t a simple losure by assuming that the expansion work due to water vapor is balan ed by the fri tional dissipation due to pre ipitation: 0: (46) In this ase, the onve tive eÆ ien y is then given by k (1 ) max: (47) One ould thus dedu e the onve tive eÆ ien y by determining the relative importan e of latent heat transport. The theory presented by RI and EB orresponds to the limit ase where = 0. In this ase, the onve tive eÆ ien y is equal to the maximum eÆien y k = max. Conve tion a ts mostly as a heat engine, and the entropy produ tion due to the moist pro esses is small. This requires the e e tive temperature of latent heat release to be lose enough to the surfa e temperature Tsurf Tlat << Tsurf Trad, or equivalently, that most of the onve tive heat transport is due to sensible heat. Although this is theoreti ally possible in the ase of low pre ipitation eÆ ien y and strong saturated downdrafts, it is unlikely for tropi al ondition. Indeed, the verti al transport of sensible heat and latent heat are given respe tively by w0CpT 0 and w0Lvq0, with the prime indi ating a departure from the horizontal mean. For the sensible heat transport to be of the same order of magnitude as the latent heat, the ratio T 0 q0 must be of the order of 2:5K=(g kg 1). For example, if the spe i humidity di eren e between as ending and des ending air is approximately 2 g kg 1, the required temperature di eren e would be 5K. Observation of water vapor ontent and temperature of updrafts and downdrafts ould help re ne this analysis, but it seems very likely that latent heat transport is the dominant heat transport by onve tion. 23 This lends support for a relatively large value of , say 0:5. Unfortunately, this makes it more diÆ ult to obtain a theory for onve tive eÆ ien y. Indeed, for large , the right-hand side of (45) is strongly dependent on both and . In su h a ase, turbulent dissipation a ounts for a very small fra tion of the entropy produ tion and is very sensitive to small hanges in latent heat transport and pre ipitation-indu ed dissipation. This also suggests some aution as to whether or not the onve tive efien y ould be determined from the entropy budget. For instan e, one an onsider an alternative losure by assuming that pre ipitation-indu ed dissipation a ounts for most of the fri tional dissipation: Dp >> Dk. This is equivalent to taking k 0 in (45) and provides a relationship between and = 1  : (48) In this ase, the parameter is thus given latent heat transport = 1 + 1  : (49) For su h a losure, the entropy budget provides a onstraint on the fri tional dissipation due to pre ipitation but annot be used to determine the turbulent dissipation Dk. The buoyan y of the updrafts is limited by the ondensate loading, and the verti al velo ity of the onve tive system should be derived from the mi rophysi al pro esses asso iated with the formation of pre ipitation in onve tive updrafts. The non-dimensional analysis of the entropy budget shows that when the latent heat transport is the main heat transport by onve tion, onve tive eÆ ien y is signi antly smaller than the me hani al eÆ ien y of a perfe t heat engine. In order to determine the the onve tive eÆ ien y of moist onve tion, one has to address the two two questions: 1) what fra tion of the onve tive heat transport is due to latent heat, and 2) how mu h fri tional dissipation is asso iated with pre ipitation. Hen e, onve tive eÆ ien y may be fairly sensitive to mi rophysi al pro esses asso iated with entrainment and detrainment, reevaporation and formation of pre ipitation. 24 5 Con lusion The entropy budget of an atmosphere in radiativeonve tive equilibrium is hara terized by a balan e between an entropy sink due to the di erential heating of the atmosphere and the entropy produ tion due to the various irreversible pro esses asso iated with onve tion. The main issue is to determine the relative ontribution of ea h irreversible pro ess. For moist onve tion, one must essentially distinguish between fri tional dissipation on the one hand, and irreversible phase hanges and di usion of water vapor on the other. This an be re ast as to whether onve tion behaves more as a heat engine or as an atmospheri dehumidi er. This question is important not only for a theoreti al understanding of how the se ond law of thermodynami s applies to a onve tive atmosphere, but also be ause it may give a better insight into the behavior of onve tion. This is the approa h followed by RI and EB, who derive verti al velo ity, intermitten y and CAPE of moist onve tion based on the assumption that the turbulent dissipation of onve tive updrafts is the main irreversible me hanism asso iated with moist onve tion. Our results di er signi antly from EB and RI in that we argue that phase hanges, di usion of water vapor and fri tional dissipation indu ed by pre ipitation, redu e the amount of kineti energy generated at the onve tive s ales. In numeri al simulations of radiativeonve tive equilibrium with a CEM, moist onve tion has four major hara teristi s: (1) irreversible phase hanges and di usion of water vapor are the main irreversible sour e of entropy, (2) most of the fri tional dissipation o urs in the mi ros opi shear zones surrounding falling pre ipitation, (3) water vapor expansion is the main sour e of me hani al work, and (4) turbulent dissipation a ting on onve tive s ales eddies a ounts only for a small fra tion of the total entropy produ tion. These hara teristi s of moist onve tion obtained with our model an be dire tly related to the fa t the onve tive heat transport is mostly due to the latent heat transport. Indeed, PH show that, in an atmosphere in radiativeonve tive equilibrium, the entropy produ tion by phase hanges and di usion of water vapor is related to the latent heat transport by onve tion and the expansion work by water vapor. This strengthens our on den e that the hara teristi s of the entropy budget in the numeri al simulations are relevant for moist onve tion in general and not simply a model artifa t. 25 Observations of the mass transport and water vapor ontent of updrafts and downdrafts ould also provide a de nitive answer to this question. Our general impression is that, ex ept in ase of very intense saturated downdrafts, latent heat transport should be the main form of onve tive heat transport, and that tropi al onve tion mostly behaves as an atmospheri dehumidi er. The entropy budget an be used to determine a onve tive eÆ ien y, dened as the ratio of the generation of kineti energy at the onve tive s ales to the total heat transport by onve tion, in a similar way to the approa h followed by Craig (1996). By performing a non-dimensional analysis of the entropy budget, it is shown that onve tive eÆ ien y depends primarily on how the heat transport is separated between latent and sensible heat, and, to a lower extent, on the magnitude of the pre ipitation-indu ed dissipation. The theories of RI and EB orrespond to the ase where latent heat transport by onve tion is very small. Although this is in theory possible, it seems unlikely for tropi al onve tion. Both RI and EB signi antly overestimate the onve tive eÆ ien y when latent heat transport is taken into a ount. Alternative losure of the entropy budget an be used to determine the onve tive eÆ ien y. However, some un ertainties remain, mostly asso iated with mi rophysi al pro esses, and it is not lear whether a theory as simple as the one proposed by RI and EB ould be obtained for moist onve tion. A fundamental diÆ ulty in establishing su h a theory lies in the un ertainties asso iated with mi rophysi al pro esses whi h an in uen e both the latent heat transport and the pre ipitation-indu ed dissipation. Further resear h in this dire tion ould potentially lead to a quantitative theory for CAPE and verti al velo ity of moist onve tion. Appendix Spurious heat sour es and entropy budget in numeri al models Approximations made in numeri al models result in the fa t that the model's entropy and energy budgets are di erent from the energy and entropy budgets of a physi al atmosphere. For example, fri tional heating is often negle ted in CEMs. These di eren es are usually small in omparison to radiative ooling, but an nevertheless be problemati sin e the model violates the rst law of 26 thermodynami s. One must therefore address the question of whether or not these errors have an impa t on the problem at hand. We dis uss here how errors related to the onservation of energy modify the irreversible entropy produ tion of a system. Consider a numeri al model for whi h some terms in the internal energy equation have been negle ted. This an be treated as if there were a spurious heat sour e Qsp whi h ompensates exa tly for the missing terms. For example, negle ting the fri tional heating is equivalent to Qsp = D. The energy budget in the model is Qsen +Qlat +Qrad +Qsp = 0: (50) For a model based on prognosti equations for momentum and internal energy, entropy is handled impli itly. If an approximation results a spurious heat sour e, it also results in a spurious entropy sour e equal to the spurious heat sour e divided by an e e tive temperature Tsp. Thus, the entropy budget is Qsen +Qlat Tsurf + Qrad Trad + Sirr + Qsp Tsp = 0: (51) Dividing (50) by Tsurf and removing it from the entropy budget (51) yields Sirr;mod = jQradj( 1 Trad 1 Tsurf ) +Qsp( 1 Tsurf 1 Tsp ): (52) The di eren e between the irreversible entropy produ tion of the model and that of the physi al atmosphere for the same radiative ooling is Sirr;mod Sirr = Qsp( 1 Tsurf 1 Tsp ): (53) As long as the spurious heat sour es are small in omparison to the total heat sink jQspj << jQradj, and as long as the e e tive temperature of the spurious heat sour e is lose enough to the surfa e temperature jTsurf Tspj jTsurf Tradj, the relative error in the irreversible entropy produ tion is small. This is the ase when one negle ts the fri tional heating or the expansion work by water vapor in the internal energy equation. We dis uss now the entropy and energy budgets of the model used for this paper. 27 Dry atmosphere For the dry ase, the only di eren e between the model atmosphere and the physi al atmosphere resides in the treatment of fri tional dissipation of kineti energy Dk. This results in a spurious heat sour e Qsp = Dk = WT Qsen +Qrad WT = 0: (54) The e e tive temperature of this spurious heat sour e is equal to the e e tive temperature of fri tional dissipation. Hen e, the entropy budget of the model is Qsen Tsurf + Qrad Trad + Sdif = 0: (55) The energy and entropy budgets (54) and (55) are similar to the energy and entropy budgets of a heat engine for whi h the work WT is exerted on the environment, instead of being dissipated internally in the system. There are di erent ways of ombining the two budgets (54) and (55) to derive a relationship between me hani al work and entropy sour es and sinks. We hoose to subtra t (54) from the entropy budget (55) multiplied by Tsurf to obtain Wmax = WT + Tsurf Sdif : (56) In this budget, the maximum work Wmax that ould be performed by onve tion is de ned by Wmax = Tsurf Trad Trad jQradj: (57) This is equivalent to assuming that the fri tional dissipation o urs at the surfa e temperature in (13) and (14). Moist atmosphere Lipps and Hemler (1982) nd that the energy budget in a moist atmosphere an be written as Qlat +Qsen +Qrad = WT: (58) This results from two approximations. First, fri tional heating Dk + Dp is negle ted. Se ond, a small error in the potential temperature equation leads to not a ounting for the buoyan y ux due to water vapor Wv. Together, 28 they result in a spurious heat sour e Qsp = Dk Dp +Wv = WT. In theentropy budget, the spurious ooling asso iated with omitting Dk+Dp om-pensates for the irreversible entropy produ tion due to fri tional dissipation.The spurious heat sour e asso iated with Wv orresponds to an additionalentropy sour e equal to Wv=Tv. The entropy budget of the numeri al simu-lations an be written asQlat +QsenTsurf + QradTrad + Sdif + Sdv + Sp + WvTv = 0:(59)The energy and entropy budgets an be ombined to obtain a relationshipbetween the work done by onve tion and the di erent entropy sour es andsinks. Subtra ting the entropy budget (59) multiplied by Tv from (58) yieldsWmax = Wv +WT + Tv Sdv + Sp + Tv Sdif ;(60)where the maximum theoreti al work is given byWmax = Tsurf TvTsurf (Qsen +Qlat) + Tv TradTrad jQradj:(61)29 REFERENCESBister, M. and K. A. Emanuel, 1998: Dissipative heating and hurri aneintensity. Meteorol. Atmos. Phys., 65, 233-240.Carnot, S., 1824:Re exions sur la puissan e motri e du feu. 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Godson, 1981: Atmospheri thermodynami s.Reidel Publishing Company, Doordre ht, 259pp.Klemp, J. B. and R. B. Wilhelmson, 1978: The simulation of three-dimension onve tive storm dynami s, J. Atmos. S i., 35, 1070-1096.Landau, L. D. and E. M. Lifs hitz, 1987: Fluid Me hani s, 2nd edition.Pergamon Press, Oxford.Li, J., P. Chylek and G. B. Lesins, 1994: Entropy in Climate Models.Part I: Verti al stru ture of the atmospheri entropy produ tion. J. Atmos.S i, 51, 1691-1701.Li, J. and P. Chylek, 1994: Entropy in Climate Models. Part II: Hori-zontal stru ture of the atmospheri entropy produ tion. J. Atmos. S i, 51,1702-1708.30 Lipps, F.B. and R.S. Hemler,1982: A s ale analysis of deep moist onve -tion and some related numeri al al ulations. J. Atmos. S i., 39, 2192-2210.Lipps, F.B. and R.S. Hemler, 1986: Numeri al simulation of deep tropi alonve tion asso iated with large-s ale onvergen e. J. Atmos. S i.,43, 1796-2210.Lipps, F. B., and R. S. Hemler, 1988: Numeri al modeling of a line oftowering umulus on day 226 of GATE. J. Atmos. S i, 45, 2428-2444.Lorentz, E. N. , 1967: The nature and theory of the general ir ulation ofthe atmosphere. World Meteorologi al Organization, 161pp.Pauluis, O. and I. M. Held, 2000: Entropy budget of an atmosphere inradiativeonve tive equilibrium. Part II: Latent heat transport and moistpro esses. Submitted to J. Atmos. S i..Pauluis, O. , V. Balaji and I. M. Held, 2000: Fri tional dissipation in apre ipitating atmosphere. Submitted to J. Atmos. S i..Peixoto J. P., A. H. Oort, M. de Almeida, and A. Tome, 1991: Entropybudget of the Atmosphere. J. Geophys. Res. 96, 10981-10988. 115, 425-461.Peixoto, J. P. and A. H. Oort, 1992: Physi s of Climate. Ameri anInstitute of Physi s, New York.Renn o, N. O., 1998: A simple thermodynami al theory for dust devils.J. Atmos. S i., 55, 3244-3252.Renn o, N.O. and A.P. Ingersoll, 1996: Natural onve tion as a heat en-gine: A theory for CAPE. J. Atmos. S i. 53, 572-585Tao, W. -K., J. Simpson and S. -T. Soong, 1987: Statisti al propertiesof a loud ensemble. A numeri al study. J. Atmos. S i, 44, 3175-3187.Tao, W. -K., J. Simpson, C. H. Sui, B. Zhou, K. M. Lau and M. Mon-rie , 1999: Equilibrium states simulated by loud-resolving models. J. At-mos. S i, 56, 3129-3139.Tompkins, A. M. and G. C. Craig, 1998: Radiativeonve tive equilib-rium in a three-dimensional loud-ensemble model. Q. J. R. Meteo. So .,124, 2073-2097.Xu K. -M. D. A. Randall, 1999: A sensitivity study of radiative-onve tive equilibrium in the tropi s with a onve tion resolving model J.Atmos. S i 56, 3385-3399.31 WT + Wv +Wext = Dk + DpDry ase 4.8 + 0.0 + 3.2 = 8.0 + 0.0Moist ase 1.7 + 3.0 + 0.04 = 1.0 + 3.7Table 1: Me hani al energy budget in the numeri al simulations ( f. equation 33)with the buoyan y ux due to the sensible heat transport WT, buoyan y ux dueto water vapor transport Wv, me hani al energy extra ted from the mean shearWext, fri tional dissipation resulting from turbulent as ade Dk, and pre ipitation-indu ed fri tional dissipation Dp. Values are given in Wm 2.Qsen + Qlat + Qrad + Qsp = errDry ase 111.7 + 0 106.9 4.8 = + 0.1Moist ase 16.5 + 143.3 157.4 1.7 = + 0.7Table 2: Energy budget in the numeri al simulations ( f. equations (54) and (58)),with the surfa e sensible heat ux Qsen, surfa e latent heat ux Qlat, radiativeooling Qrad, spurious heat sour e in the model Qsp. The imbalan e err in theenergy budget is mostly due to hanges in the internal energy of the atmosphere.Values are given in Wm 2.Wmax = WT +Wv + Te Sdv + Sp + Te Sdif + errDry ase8.0= 4.8+ 0+ 3.0+ 0.2Moist ase 13.8= 4.7+ 8.3+ 0.6+ 0.2Table 3: Entropy budgets in the numeri al simulations, with maximum workWmax, buoyan y ux due to sensible heat ux WT, buoyan y ux due to wa-ter vapor transport Wv, moistening e e t Tv Sdv + Sp , di usion of heat bysubgrids ale eddies Tv Sdif and imbalan e in the budget err. Values are givenin Wm 2. The e e tive temperature Te is equal to the surfa e temperatureTe = Tsurf = 298:15K in the dry ase, and to the e e tive temperature of thewater vapor transport Te = Tv = 281:3K in the moist ase.32 Entropy budget Wmax = WT +Wv + Td Sdv + Sp + Td SdifNon-dimensional 1 = 1 (1 )+ (1 )+0analysisSimulations1 =0:34+0:60+0:04Table 4: The non-dimensional version of the entropy budget is obtained bydividing the dimensional budget (34) by the maximum theoreti al work Wmax.The non-dimensional parameters , , , k and max are de ned in the text.The value of these non-dimensional quantities in the moist simulations are alsoshown. In the moist ase, we have = 0:82, = 0:27, = 0:06, k = 0:006 andmax = 0:087 .Me h. en. budget WT +Wv = Dk+DpNon-dimensional 1+=kmax + ( + )analysisSimulations0:12 +0:22 = = 0:07 +0:27Table 5: The non-dimensional expression for the me hani al energy budget isobtained by dividing the dimensional budget by Wmax. The non-dimensional pa-rameters , , , k and max are de ned in the text. The value of these non-dimensional quantities in the moist simulations are also shown. In the moist ase,we have = 0:82, = 0:27, = 0:06, k = 0:006 and max = 0:087 .33 ∆Sp c– ∆Sd v–+ Qlat QlatT surf-------------,Qsen QsenT surf-------------,Dk Dprec+Td---------------------------∆SdiffQrad QradT rad-----------, Figure 1: S hemati representation of the energy and entropy budgets of anatmosphere in radiativeonve tive equilibrium. The heat sour es and sinksare radiative ooling Qrad, surfa e sensible heat ux Qsen and surfa e latentheat ux Qlat. The irreversible entropy sour es are fri tional dissipationD=Td, di usion of heat Sdif , di usion of water vapor Sdv and irreversiblephase hanges Sp .34 16018020022024026028030005101520 Temperature (K)height(km)Moist caseDry case 28029030031032033034035036005101520 Moist static energy (kJ kg)height(km)Moist caseDry case Figure 2: Top: Horizontally averaged temperature in the moist ( ontinu-ous line) and dry experiments (dashed line). Bottom: Horizontally aver-aged moist stati energy in the moist ( ontinuous line) and dry experiments(dashed line).35 1002003004005006002468101214height(km)Vertical velocity 1002003004005006002468101214height(km)Vertical velocity 1002003004005006002468101214X(km)height(km)Condensed waterFigure 3: Top and middle: Snapshots of the verti al velo ity eld in the dry(top) and moist experiments (middle) Contours indi ate verti al velo ity of3ms 1, 1ms 1, 1ms 1 and 3ms 1. Dotted lines are for negative values.Bottom: Snapshot of the loud water and pre ipitating water: dotted linesindi ate a pre ipitation water (snow and rain) ontent larger than 0:1g kg1;ontinuous lines indi ate loud water ontent greater than 0:5g kg 1.36 204060801001201402468101214height(km)Vertical velocity 204060801001201402468101214height(km)Vertical velocity13 204060801001201402468101214X(km)height(km)Condensed waterFigure 4: Same as Figure 3 for part of the domain.37 −0.08 −0.06 −0.04 −0.0200.020.040.060.080246810121416 Heating rate (W m)height(km)Diabatic heatingradiative coolingSubgridscale diffusionAdvection Figure 5: Horizontally averaged heating rates.38 −0.1−0.0500.050.10.150246810121416 Heating rate (W m)height(km)Latent heating due to condensationLatent cooling due to reevaporationNet latent heat releaseRadiative cooling Figure 6: Horizontally averaged heating rate asso iated with ondensation,reevaporation, all phase hanges ( ondensation and reevaporation), and ra-diative ooling.39 Wext 3.2W m2–=WT 4.8W m2–=Dk 8.0W m2–=Td∆Sdif 3.0W m2–=Wmax 8.0W m2–=Figure 7: S hemati representation of the me hani al energy and entropybudgets in the dry experiment. Wmax is the maximum theoreti al work de-ned in (57). WT is the buoyan y ux, Wext is the me hani al energy ex-tra ted from the mean shear, Dk is the turbulent dissipation, and Sdif is theirreversible entropy produ tion due to di usion of heat. The entropy budgetfor the numeri al model is given in terms of equation (57). The e e tivetemperature used in the entropy budget is Td = Tsurf = 298:15 K.40 Td∆Sm 8.3W m2–=Td∆Sdif 0.6W m2–=WT 1.7W m2–=Dprec 3.7W m2–=Wvap 3.0W m2–=Dk 1.0W m2–=Wext 0.04W m2–= Wmax 13.8W m2–=Figure 8: S hemati representation of the me hani al energy and entropybudgets in the moist experiment. Wmax is the maximum theoreti al work de-ned by (61). WT is the thermal ontribution to the buoyan y ux, Wv is thebuoyan y ux asso iated with water vapor transport, Wext is the me hani alenergy extra ted from the mean shear, Dk is the turbulent dissipation, Dpis the pre ipitation-indu ed dissipation, Sdif is the irreversible entropy pro-du tion due to di usion of heat, and Sdv + Sp is the irreversible entropyprodu tion due to the moistening e e t. The entropy budget for the numer-i al model is given in terms of equation (61). The e e tive temperature isgiven by Td = Tv = 281:3 K41