## Abstract

Pauluis et al. argue that frictional dissipation of energy around falling hydrometeors is an important entropy source in the tropical atmosphere. Their calculations suggest that the frictional dissipation around hydrometeors is about ⅓ of the work available from a reversible convective heat engine. Moreover, based on the residual of the energy budget of a numerical model, not shown in their note, the authors argue that irreversible entropy sources due to diffusion of water vapor and phase changes reduce the mechanical work available from the convective heat engine by about ⅔. Pauluis et al. conclude that only a tiny fraction of the energy potentially available from a convective heat engine is used to perform work.

Rennó and Ingersoll show that frictional heating can be easily included in the heat engine framework via increases in the thermodynamic efficiency of a reversible heat engine. It is shown that the effect of any other irreversible process is merely to reduce the thermodynamic efficiency of a reversible convective heat engine. Thus, the framework proposed by Rennó and Ingersoll is valid even when the heat engine is as irreversible as suggested by Pauluis et al. Since irreversible entropy sources reduce the mechanical work available from the convective heat engine, the study of Pauluis et al. implies that the bulk thermodynamic efficiency of the tropical atmosphere is only a tiny fraction of that predicted by the framework proposed by Rennó and Ingersoll. Both theoretical and observational evidence that the calculations performed by Pauluis et al. overestimate the irreversible entropy changes in the real tropical atmosphere is shown. Moreover, evidence that numerical models are highly dissipative when compared with nature is shown. Therefore, the interpretation of Pauluis et al. that the reversible heat engine framework grossly overestimates the rate at which work is performed by tropical convective systems is not agreed with.

## 1. Introduction

Rennó and Ingersoll (1995, 1996), Michaud (1995), and Emanuel and Bister (1996) proposed heat engine theories for atmospheric convection. Their framework assumes that moist convection is a reversible heat engine and predicts the buoyancy, the convective velocity, and the fractional area covered by the convective drafts of an ensemble of convective systems in quasi steady state. In particular, Rennó and Ingersoll (1996) apply their theory to the tropical atmosphere and show that its predictions are consistent with observations. Pauluis et al. (2000) challenge this idea by arguing that frictional dissipation of energy around falling hydrometeors consumes about ⅓ of the work produced by a reversible moist convective heat engine. Moreover, they state that about ⅔ of the energy that could potentially be available if the tropical heat engines were reversible is lost in irreversible entropy sources due to diffusion of water vapor and phase changes. Thus, Pauluis et al. (2000) argue that only a tiny fraction (less than 10%) of the energy available from a reversible heat engine is available to do work in the tropical atmosphere.

The results presented by Pauluis et al. (2000) imply that the heat engine theories for atmospheric convection proposed by Rennó and Ingersoll (1995, 1996), Michaud (1995), and Emanuel and Bister (1996) grossly overestimate the moist convective velocities observed in nature. Moreover, they imply that the theory for the maximum intensity of hurricanes proposed by Emanuel (1986, 1988) overestimates their maximum intensity. However, Emanuel (1988, 1999) shows convincing evidence that the predictions based on reversible thermodynamics are in good agreement with observations. Since the hurricane’s convective heat engine is accompanied by intense convection and large precipitation rates, we believe that they indicate that neither dissipation around falling hydrometeors nor irreversible processes related to water substance is extremely important in nature.

In the idealized model proposed by Rennó and Ingersoll (1996) most of the precipitation falls within convective downdrafts (see Fig. 1 of their article). Indeed, their model assumes that an ensemble of convective systems is in steady state and that the energy available from the heat engine is used to force bulk fluid motion against friction. We believe that this is a good idealization of natural convective systems and that, to a first approximation, the precipitation drag does reversible work on downdrafts. This happens because the pressure drag is at least an order of magnitude larger than the frictional drag around falling hydrometeors. Thus, the work performed by the drag of falling hydrometeors is mostly used to force bulk fluid motion on their wake. Indeed, bulk fluid motion is forced even when the hydrometeors fall through an air mass at rest. In the next section, we present an order of magnitude calculation that supports this idea. We believe that the numerical model used by Pauluis et al. (2000) is inappropriate to verify the above ideas because it does not include frictional drag explicitly. Indeed, their model only includes the effect of water substance on density. The frictional dissipation is indirectly calculated as the work done by the total drag force (including the pressure drag) on falling hydrometeors.

We believe that the results presented by Pauluis et al. (2000) might point to a problem with numerical models. It is well known that cloud-resolving models are highly diffusive when compared with nature. Even large eddy simulation models fail to predict the entropy and velocity fluctuations observed in nature (e.g., Hadfield et al. 1991). Moreover, we do not believe that models with horizontal resolution of about 2 km can properly resolve convective drafts. In nature, the cores of convective drafts have diameters of the order of 100 m. Zipser and LeMone (1980) and LeMone and Zipser (1980) showed that only 10% of the updraft cores observed in the Global Atmospheric Research Programme (GARP) Atlantic Tropical Experiment (GATE) region have diameters in excess of 2 km.

## 2. Frictional dissipation around hydrometeors

Pauluis et al. (2000) argue that, from a macroscopic perspective, estimating the frictional dissipation in the shear zone around falling drops is straightforward. We do not agree with their statement. Indeed, we show that the calculation is straightforward only when all falling drops are in a Stokes regime, that is only when they are smaller than 40 *μ*m (Re ≲ 1). Moreover, we show that the assumption that all falling hydrometeors are in a Stokes regime grossly overestimates the frictional dissipation of energy in the shear zone around precipitation particles.

The equation of motion applied to a falling hydrometeor of unit mass can be written as

where *w* is the hydrometeor falling speed, *g* is the gravity acceleration, *ρ* is the air density, *p* is the pressure, and *ν* ≈ 1.5 × 10^{−5} m^{2} s^{−1} is the kinematic viscosity of air. Scaling the equation above, we get

where primed quantities are of order unit in magnitude, *W* is the order of magnitude of the hydrometeor’s falling speed, *D* is its diameter, Re ≡ *WD*/*ν* is the Reynolds number, Fr ≡ *W*/*gD* is the Froude number, Δ*P* is the pressure drop across the falling hydrometeor. Equation (2) shows that the viscous stress term can be neglected whenever Re ≫ 1. Thus, the viscous stress term is important only when Re ≲ 1, that is, when the falling drops are in a Stokes regime.

Assuming that the hydrometers are falling at their terminal velocity, the vertical acceleration term becomes small when compared to *g* (e.g., Fr ≪ 1); therefore, the inertial acceleration term can be dropped. In this case, Eq. (2) reverted to dimensional form becomes

Equation (3) shows that the gravity acceleration term is balanced by the pressure and viscous acceleration terms. Moreover, it follows from the above discussion that, when Re ≫ 1, falling hydrometeors do mostly reversible work on the environment. This work can potentially be used to force convective circulations by inducing bulk fluid motion on the hydrometeors’ wake.

Next, we estimate the order of magnitude of the frictional drag in the shear zone around typical hydrometeors. Usually, cloud droplets, rain droplets, and graupel and ice crystals range in size as follows: 10–160 *μ* m, (2–12) × 10^{−3} m, and (2–16) × 10^{−3} m, respectively. Their typical fall speeds are 0.04–0.7, 4.5–10, and 0.5–2.5 m s^{−1}, respectively (see Rogers 1979). It follows from the above that the Reynolds number ranges from 0.04 to 2 for cloud droplets, indicating that the friction dissipation term dominates only for small cloud droplets. For rain droplets the Reynolds number is (0.6–10) × 10^{3}, and for graupel and ice crystals 600 ≲ Re ≲ 3.0 × 10^{3}, indicating that the pressure acceleration term is at least 2 orders of magnitude larger than the viscous acceleration term.

The above discussion suggests that the work done by the drag force is not necessarily consumed by friction in the neighborhood of the falling hydrometeors. Consider a hydrometer falling at terminal velocity. Most of the work of the drag force on the hydrometer is used to decrease its potential energy (a small amount of work is used to force oscillations, deform, and generate internal circulations on them). The reaction to the drag force, in turn, does work on the air. Most of this work is either used to force the air down or is consumed by friction in the neighborhood, while a small amount of work is used to force gravity and acoustic waves [part of the hydrometers’ potential energy is also used to do electrical work, when they are charged (see Williams and Lhermitte 1983)]. Thus, it is conceivable that a large portion of the potential energy of an ensemble of hydrometeors is used to force bulk fluid motion on their wake (e.g., to accelerate the convective downdrafts), when Re ≫ 1. Therefore, we do not agree with the statement made by Pauluis et al. (2000) that the estimation of the total dissipation rate due to precipitation is straightforward. Indeed, we believe that, in order for the frictional dissipation by precipitation to be properly calculated, the hydrometeors’ pressure and frictional drag should be explicitly included in the numerical model. Since, the numerical model used by Pauluis et al. (2000) does not explicity calculate the pressure and frictional drag, we question their findings.

## 3. Irreversible heat engines

Rennó and Ingersoll (1995, 1996), Michaud (1995), Emanuel and Bister (1996), Rennó et al. (1998), and Souza et al. (2000) idealized convective systems as reversible heat engines. Here, we show that irreversible processes can be easily included in their framework. An energy equation for a convecting air parcel follows from the dot product of the velocity vector and the equation of motion (Haltiner and Martin 1957). The resulting equation states that following an air parcel in steady state

where **v** is the velocity vector, *g* the gravity acceleration, *α* the specific volume, *p* the pressure, **f** the frictional force per unit mass, and **dl** is an incremental distance along the air parcel’s path. Note that, contrary to Rennó and Ingersoll (1996), we define the frictional force as negative. Rennó and Ingersoll (1996) assume steady state and reversible processes, which imply that their theory provides an upper bound for the intensity of the circulation. Relaxing the assumption that the convective processes are reversible, the first and second laws of thermodynamics applied to moist air can be written as

where *T* is the absolute air temperature, *s* the specific entropy, *dQ* the heat absorbed by the system, *c*_{p} the specific heat of air at constant pressure, *L*_{υ} the specific latent heat of vaporization, and *r* is the water vapor mixing ratio. The inequalities apply when the changes are irreversible. Thus, we can write

where (*T ds*)_{rev} and (*T ds*)_{irr} are associated with reversible and irreversible entropy sources, and (*dw*)_{irr} is the irreversible work (e.g., the work done by friction forces opposing the expansion). Note that, in this case, the heat absorbed by the system is *dQ* = (*T ds*)_{rev} = *T ds* − (*T ds*)_{irr}. It follows from the above that

which shows that, in a real (irreversible) heat engine, a portion of the heat input is used to do irreversible work.

where in the presence of irreversibility the *total* work done by the convective system is *W* = ∮ *dQ* > ∮ *p **dα.* For example, in the presence of friction the total work of expansion is greater than it would be in the absence of this irreversibility. However, even in the presence of irreversibility, it is convenient to define total convective available potential energy (TCAPE) as the work available from the convective heat engine; that is, TCAPE ≡ ∮ *p **dα.*

Recall that Rennó and Ingersoll (1996) assume that

where the subscript “in” denotes an integration along the heat input branch of the convective circulation. The thermodynamic efficiency of an irreversible convective heat engine should be similarly defined; that is,

Equation (7) shows that only a fraction of the heat input [∫_{in} *dQ* = ∫_{in} (*T **ds*)_{rev}] into a real heat engine is used to do reversible work (∮ *p **dα*). Therefore, the thermodynamic efficiency of any real heat engine is smaller than that of a reversible heat engine; that is, *η*_{irr} < *η*_{rev}. Except for this reduction in the thermodynamic efficiency, the framework proposed by Rennó and Ingersoll (1996), Rennó et al. (1998), and Souza et al. (2000) is unchanged when the heat engine is irreversible. Thus, their framework applies to real convection even when they are as irreversible as suggested by Pauluis et al. (2000). In the next section, we use the heat engine framework to check if irreversible processes as large as those proposed by Pauluis et al. (2000) are consistent with observations of natural convection.

## 4. Natural heat engines

It follows from the studies of Rennó et al. (1998) and Souza et al. (2000) that the pressure drop across a convective circulation such as the Hadley–Walker circulation depends solely on the thermodynamics of the convective heat engine. They show that the pressure drop across one of these circulations [see Eq. (13) of Rennó et al. 1998] is given by

where *γ* is the fraction of the total dissipation of mechanical energy consumed by friction at the surface, and the subscripts 0 and ∞ refer to the near-surface air at the upward and downward branches of the convective circulation. The following approximations were made in the derivation of Eq. (13): (i) the heat engine cycle is reversible; (ii) the circulation is in steady state; (iii) the surface is flat; (iv) changes in kinetic and potential energy between ∞ and 0 can be neglected. Items (i) and (ii) imply that Eq. (13) predicts the maximum thermodynamic intensity of a convective circulation. It follows from the discussion in section 3 that Eq. (13) also applies to irreversible convective heat engines. However, in this case, the thermodynamic efficiency is the irreversible efficiency; that is, *η* = *η*_{irr} [see Eq. (12)].

In the previous section, we showed that the criticism of the heat engine framework by Pauluis et al. (2000) merely implies that reversible thermodynamics grossly overestimates the value of *η.* Thus, Eq. (13) with *γ* = 1 predicts the maximum possible intensity of real convective circulations. First, we use Eq. (13) to show that the predictions of reversible thermodynamics are consistent with the observations of systems ranging in size from small- to mesoscale. Then, we use it to show that the intensity of large-scale circulations forced by ensembles of convective systems in quasi radiative–convective equilibrium is also consistent with the predictions of reversible thermodynamics.

Golden and Bluestein (1994) observed waterspouts under precipitating cumulus clouds with tops at about 3.0 km, which corresponds to *η*_{rev} ≈ 0.1. Based on the thermodynamic sounding at Key West displayed in their article, we estimate that the convective updrafts are about 1 K warmer than the downdrafts. Assuming that both are saturated, that *η* = *η*_{rev}, that *γ* = 1, and that the wind is in cyclostrophic balance, the predicted pressure drop is Δ*p* ≈ 6.5 hPa, and the tangential wind speed is *υ* ≈ 25 m s^{−1}. Since the minimum wind speed necessary to create a spray ring is 20 m s^{−1}, this prediction is in good agreement with observations. If the irreversible processes associated with water substance were as important as suggested by Pauluis et al. (2000), the thermodynamic efficiency would be at least an order of magnitude smaller; that is, *η* = *η*_{irr} ≲ 0.01. In this case, the predicted pressure drop is Δ*p* ≲ 0.7 hPa, and the tangential wind speed is *υ* ≲ 8 m s^{−1}. These values are only a small fraction of the observed values. Therefore, the suggestion by Pauluis et al. (2000) that irreversible processes are of paramount importance is not consistent with observations of waterspouts. The same argument holds for the prediction of the maximum intensity of hurricanes. However, the reader might argue that convective vortices belong to a special class of convective systems that are thermodynamically efficient. Since, Eq. (13) applies to any moist convective circulation, next we apply it to a large-scale tropical circulation, that is, the Hadley circulation.

The observational data presented on pages 135–160 of Peixoto and Oort (1992) show that in June–July–August Δ*p* ≈ 10 hPa, Δ*T* ≈ 5 K, and that Δ*r* ≈ 5 × 10^{−3} kg kg^{−1} between 5°N and 30°S (the extension of the Hadley circulation). Taking *η* = *η*_{rev} ≈ 0.1, the approximate value of the thermodynamic efficiency of the circulation, Eq. (13) predicts Δ*p* ≈ 10 hPa when we take *γ* ≈ 0.5, indicating that half of the dissipation of mechanical energy occurs near the surface. Assuming that *η* = *η*_{irr} ≲ 0.01, as suggested by Pauluis et al. (2000), we get Δ*p* ≲ 2 hPa. This result is inconsistent with observations. It is interesting that a global circulation model simulation also shows that the Hadley cell extends from 5°N to about 30°S and Δ*p* ≈ 10 hPa. However, in contrast with observations, the model gives Δ*T* ≈ 15 K, and Δ*r* ≈ 9 × 10^{−3} kg kg^{−1}. These values are more than double the observed values. Applying Eq. (13) to the model data with *γη* ≈ 0.05, we get Δ*p* ≈ 24 hPa. In order to obtain the model pressure drop, we have to assume *γη* ≈ 0.025. We found a similar result when trying to compute the pressure drop across the Walker circulation. Thus, it seems that the numerical model is much more dissipative than nature.

## 5. Conclusions

We show that the computation of the frictional dissipation by precipitation is not as simple as suggested by Pauluis et al. (2000). Therefore, we do not agree with their scaling analysis of the frictional dissipation in a precipitating atmosphere. We also show observational evidence that the numerical calculations performed by Pauluis et al. (2000) overestimate the irreversible entropy changes in the real tropical atmosphere. Moreover, we show evidence that numerical models are highly dissipative when compared with nature. Therefore, we do not agree with the interpretation of Pauluis et al. (2000) that the reversible heat engine framework grossely overestimates the rate at which work is performed by tropical convective systems.

We speculate that Pauluis et al. (2000) overestimate the frictional dissipation by precipitation because their numerical model is highly dissipative and does not explicitly simulate the effects of falling hydrometeors. Indeed, the energy equation [see Eq. (4)] shows that when the effects of water substance on density are implicitly included in the numerical model, and the model atmosphere is in hydrostatic balance, the weight of water substance is balanced by the vertical component of the pressure gradient force. Thus, water substance does not drag the air downward, and it is physically inconsistent to assume that hydrometeors cause frictional dissipation. Since, in this case, the hydrometeors’ water loading does not force downdrafts, an implicit calculation of the frictional dissipation by them would overestimate the frictional dissipation in nature.

We show that irreversible processes can be easily included in the heat engine framework. Moreover, we show that except for a reduction in the thermodynamic efficiency, the framework proposed by Rennó and Ingersoll (1996) is unchanged by the inclusion of irreversible processes. Therefore, the heat engine framework is general and applies to any real convective system. Indeed, we argue that the heat engine framework is a useful tool for the testing of numerical models.

## Acknowledgments

The author thanks Ms. Margaret S. Rae for reading the manuscript and making many useful suggestions. He also thanks Prof. David Raymond and his students at New Mexico Tech for their comments and suggestions. Finally, the author thanks The University of Arizona’s Department of Atmospheric Sciences and the NSF for supporting this research under Grant ATM-9612674.

## REFERENCES

## Footnotes

*Corresponding author address:* Dr. Nilton O. Rennó, Dept. of Atmospheric Sciences, The University of Arizona, Tucson, AZ 85721-0081.

Email: renno@atmo.arizona.edu