1. Introduction
The maintenance of the general circulation requires that kinetic energy be continuously generated to balance frictional dissipation. To do so, the atmosphere acts as a heat engine that converts internal energy into kinetic energy through a combination of warm, moist air expansion and cold, dry air compression. As with any heat engine, the ability of the atmosphere to produce kinetic energy is closely tied to an energy transport from a warm source, corresponding to the energy flux at the earth’s surface, to a cold sink due to the radiative imbalance of the troposphere. Quantitative estimates of the kinetic energy generated by the atmospheric circulation can be obtained by analyzing the entropy budget of the atmosphere. The differential heating of the atmosphere corresponds to a net sink of entropy, which must be balanced by the entropy production due to various irreversible processes taking place in the atmosphere. By identifying the contribution of frictional dissipation to the total irreversible entropy production, one can determine the kinetic energy generated and dissipated by the atmospheric circulation. Rennó and Ingersoll (1996) and Emanuel and Bister (1996) argue that if frictional dissipation is the sole irreversible source in the atmosphere, the entropy budget analysis yields a kinetic energy production and dissipation that is similar to the work produced by a Carnot cycle. However, Pauluis and Held (2002a, b) show that in a moist atmosphere, diffusion of water vapor and irreversible phase transitions account for a large portion of the total entropy production—roughly two-thirds in the simulations discussed in Pauluis and Held (2002a). The irreversible entropy source associated with these moist processes reduces the amount of entropy that can be generated by frictional dissipation. As a result, the amount of kinetic energy produced in a moist atmosphere is much lower than what would be expected from a Carnot cycle with the same energy transport.
The results of Pauluis and Held (2002a, b) indicate that the amount of kinetic energy produced by the atmospheric circulation is directly influenced by the intensity of the hydrological cycle and the atmospheric transport of water vapor. The present paper is motivated by the need to address how moist processes affect the generation of available potential energy (APE). In the atmospheric sciences, early discussions of the maintenance of the general circulation were based on the APE budget rather than on the entropy budget. Lorenz (1955) defines the APE as the difference in total static energy (internal plus geopotential) between the current state of the atmosphere and that of an idealized reference state, defined as the state that minimizes the static energy of the atmosphere after a sequence of reversible adiabatic transformations. A reversible transformation here corresponds to a process where there is no irreversible entropy production, and an adiabatic transformation is one that does not involve any external energy source or sink. Hence, the entropy of an air parcel is conserved by reversible adiabatic transformations. By construction, the sum of APE and kinetic energy is conserved under reversible adiabatic redistribution of the air parcels, and the APE itself can be interpreted as an energy reservoir that can be converted into kinetic energy by atmospheric motions. The small-amplitude formulation of the APE derived in Lorenz (1955, 1967) yields a quadratic expression that is commonly used in various discussions of the energetics of atmospheric circulation.
Lorenz (1955, 1967) and Dutton and Johnson (1967) show that external energy sources can act as net sources of APE. In a statistically steady atmosphere, this production of APE is converted into kinetic energy by the circulation, and is then dissipated by friction. At first glance, the descriptions of the maintenance of the atmospheric circulation in terms of the entropy and APE budgets are identical. However, a key problem arises from the fact that the original concept of APE and the corresponding discussion of the maintenance of the atmospheric circulation have been focused on dry circulations.1 Moisture, if it is taken into consideration at all, only enters as a prescribed external energy source, accounting for the “latent heat release” associated with phase transitions of water. From a conceptual point of view, latent heat release is not an external process, but corresponds to an internal energy conversion. Thermodynamically, the external energy source corresponds to the evaporation of water at the earth’s surface, and not to the condensation of water vapor in the troposphere. In practice, the main limitation of the dry framework lies in that it requires knowledge of the distribution of latent heat release, which itself depends on the circulation, in order to determine the source of APE. Furthermore, in light of the new understanding of the role of moist processes in the entropy budget that has emerged in recent years (see Goody 2000, 2003; Pauluis and Held 2002a, b; Pauluis et al. 2000), a dry framework seems inadequate to properly address the maintenance of the general circulation in the earth’s atmosphere.
Lorenz (1978) and Randall and Wang (1992) show that the concept of APE can be applied to a moist atmosphere as long as the transformations leading to the reference state include reversible adiabatic phase transitions and conserve the total water content of each individual parcel. A fundamental difficulty here is that the reference state cannot be derived analytically, but must be determined from an iterative minimization procedure—a graphical one in Lorenz (1978), and a computational algorithm in Randall and Wang (1992). This raises a second difficulty: if the reference state can only be obtained iteratively, how does one determine its evolution, which is necessary to compute the rate of change of APE? Lorenz (1978) discusses this issue, but still relies on a graphical technique to estimate the change of APE. More recently, Bannon (2004, 2005) derives the effects of moisture in the context of various formulations for available energy, and obtains a general formula for the sources and sinks of available energy. However, the reference states used in both papers differ from that used in the Lorenz APE; in Bannon (2004) the reference state corresponds to one where parcels are brought back adiabatically to their initial pressure, while Bannon (2005) uses an arbitrary state of constant temperature and composition. By using these alternative forms for the available energy, Bannon (2004, 2005) is able to avoid the problems associated with changes in the reference state pressure, but the results in Bannon (2004, 2005) cannot be directly applied to the Lorenz APE.
The primary purpose of the present paper is to derive an analytic expression for the rate of change of the Lorenz APE in a moist atmosphere. In section 2a, it is shown that changes in the static energy of the reference state can be related to a change of the parcel entropy, of total humidity, and of its pressure in the reference state. The central result here is that any reorganization of the reference state that results from either changes in entropy or total water content of air parcels does not affect the reference state enthalpy or the APE. Boer (1976) derives a similar result in the dry framework, but this derivation makes use of the analytic expression for the reference state in a dry atmosphere and cannot be straightforwardly extended to the moist case. Section 2b discusses the APE changes resulting from the addition and removal of water to the atmosphere, which occurs, for example, when water evaporates at the earth’s surface or when condensed water precipitates. It is shown here that, in a hydrostatic atmosphere, the change in static energy due to a source or sink of water includes two contributions: a first term given by the enthalpy of the added water enthalpy, and a second term due to the compression of the air below the added mass. This second term integrated over the atmospheric column is equal to the geopotential energy of the water. These new findings make it possible to obtain an expression for the sources and sinks of APE even in the absence of an analytic formula for pressure in the reference state. This expression is equally valid for dry and moist atmospheres and does not require the knowledge of the evolution of the reference state.
In section 3, the sources and sinks of APE are evaluated for various atmospheric processes such as external energy sources, frictional dissipation, surface evaporation, diffusion of sensible heat, diffusion of water vapor, precipitation, and reevaporation. External energy sources generate an amount of APE given by their net energy input multiplied by an efficiency factor equal to the Carnot efficiency of a perfect heat engine acting between the parcel’s temperature and its temperature in the reference state. The efficiency factor for evaporation is slightly lower, mainly due to a water loading contribution. For earthlike conditions, the efficiency factor can be as high as one-third. For a surface energy flux of 100 W m−2, this would correspond to a generation of APE of more than 30 W m−2. Such a large APE source is incompatible with either observed or theoretical estimates of the rate at which kinetic energy is dissipated in the atmosphere. It is argued here that such a large source of APE by the surface flux must be balanced by a comparable loss due to the diffusion of water vapor from unstable to stable air masses.
2. Time tendency of the available potential energy












a. Closed transformations










































From a physical point of view, Eq. (24) can be interpreted as stating that reorganization of the parcels between different reference states does not produce or consume any kinetic energy. Indeed, consider the evolution of an atmosphere that remains at all times in the reference state corresponding to its current entropy and humidity distribution. In this case, the pressure term in (24) is exactly to the amount of enthalpy that is converted into kinetic energy. However, as by definition the reference state minimizes the total static energy, the air parcels must be reorganized as soon as an infinitesimally small amount of kinetic energy could be produced through such reorganization. It is thus impossible to convert enthalpy into kinetic energy in an atmosphere that remains at all times in its reference state. Similarly, kinetic energy cannot be converted into enthalpy, as it would imply that the reverse transformation would be able to extract kinetic energy.
b. Open transformations












As in the proof given in section 2a, the path between A and C can be subdivided into a sequence of smaller transitions by passing through a set of intermediate reference states Ai. The intermediary states Bi and Di are constructed in a similar way to those in section 2a. The state Bi has the same entropy and total water distribution as Ai+1, but the parcels are located at the same relative position as in Ai. The state Di has the same entropy and total water distribution as Ai, but the parcels are located at the same relative position as in Ai+1. One can then show that because the reference state minimizes the total enthalpy, the change in the total enthalpy that results from the reorganization of the reference state must vanish. In essence, the steps from (12) to (24) can be reproduced for open transformations. The sole difference is that in addition to the contribution of entropy changes Tds, there is also a contribution due to changes in the total water mixing ratio (μυ + gz)drT in Eqs. (12), (15), and (18)–(20).


3. Sources and sinks of APE
Equation (32) can be used to compute the rate of change of the APE for various atmospheric phenomena. While the APE itself is a property of the distribution of entropy and water vapor of the whole atmosphere, the production of APE can be computed for changes in individual parcels, independent of the transformations taking place in the rest of the atmosphere. This section discusses the APE changes associated with seven atmospheric processes: external energy source, frictional dissipation, surface evaporation, diffusion of heat and of water vapor, precipitation, and reevaporation of condensed water. The APE sources and sinks are also estimated for an idealized radiative–convective equilibrium. The estimated sources and sinks of APE for different processes are shown in Table 1. The discussion here is meant to be illustrative, and the complete computation of the APE sources and sinks are currently under way.
a. External energy sources


For an energy source at the surface, the efficiency factor can be quite large, up to 1/3 for an air parcel at T ∼ 300 K and with a reference temperature Tref ∼ 200 K. The efficiency factor is, however, highly sensitive to the parcel’s properties. In particular, heating in a parcel whose reference level remains near the surface would generate little or no APE. The efficiency factor is negative when the reference temperature is higher than the current temperature. In such a case, heating would reduce the APE, while cooling would increase it. In particular, tropospheric cooling acts as a source of APE when the cooling occurs in parcels whose reference level is lower than their current locations.
b. Frictional dissipation


c. Surface evaporation






For a quantitative example, consider evaporation in an air parcel at the surface, with T = 300 K, H = 0.8, Z = 0, and with a reference state in the upper troposphere corresponding to Tref = 200 K, Href = 1, Zref = 15 000 m. The contribution of the latent heating is (T − Tref/T)Lυ ≈ 810 kJ kg−1, the impact of the irreversible entropy production is RυTref lnH ≈ −20 kJ kg−1, the contribution of the specific heat of liquid water amounts to Cl[T − Tref − Tref ln(T/Tref) ≈ 80 kJ kg−1, and the water loading is g(Z − Zref) ≈ −147 kJ kg−1. The latent heat term dominates (35), though the combined contribution of the other terms cannot be entirely neglected. The overall effect is a change of APE on the order of 720 kJ kg−1 of water vapor added to the atmosphere. The efficiency factor of the latent heat flux, defined as the ratio of the APE production to the surface flux of latent heat, is slightly smaller than that for a sensible heat flux, primarily due to the water loading term. As for an external energy source, the net effect of the evaporation is highly sensitive to a parcel reference state. In particular, evaporation into a “stable” parcel with a reference state at the earth surface would not change the APE.
d. Diffusion of heat


In a dry atmosphere, the reference temperature of an air parcel is determined by its potential temperature. If diffusion takes place between two parcels at the same pressure, the term (Tref,A − Tref,B/T) is proportional to the buoyancy difference between the air parcels. This implies that, outside very limited circumstances, diffusion of heat will be limited to exchange between air parcels with similar reference temperatures.
In a moist atmosphere, the reference temperature is a function of both the entropy and humidity of a parcel. Two parcels can have the same specific volume at a given pressure and still have very different reference temperatures. For example, in the boundary layer, an unstable air parcel, with a reference level in the upper troposphere, can have the same specific volume as a stable parcel with a reference level near the earth’s surface. These two parcels would have the same buoyancy, and can remain close to each other for a long period of time. This makes it possible to have significant diffusion of heat between parcels of different reference temperatures. In these circumstances, Eq. (36) indicates that diffusion of heat could have a fairly large impact on APE.
e. Diffusion of water vapor


f. Precipitation


Precipitation affects the APE both through its effect on the heat capacity [the first term on the right-hand side of (40)] and geopotential (the second term). For precipitation falling out of a convective updraft, one can assume that the reference temperature of the parcel is lower than the current temperature T > Tref while its reference height is higher than the current location Z < Zref. In this case, the reduction in heat capacity reduces the APE, while the reduction in water loading increases the APE. The positive contribution of the water loading term can be seen as the result of the increase in buoyancy due to a reduction of water loading. Its effect is equal to the difference in geopotential energy between the removed water’s current and reference states. A quantitative comparison between the two terms in (40) indicates that the water loading is the dominant effect.
Equation (40) indicates that the removal of the precipitating water results in a net increase of the APE. This increase in APE is not balanced by any equivalent reduction of the kinetic energy, and corresponds to an increase in the sum of the APE and the kinetic energy. In contrast, the sum of APE and kinetic energy in a dry atmosphere can only increase in the presence of external energy sources. Hence, the fact that precipitation can increase the APE is an important way in which dry and moist atmospheres differ.
The effect of precipitation on APE can be better understood by looking at the impact of adding water to a parcel at the earth’s surface, lifting the parcel to a higher level, and removing the water. If the ascent is reversible, it does not change the parcel reference level. The net effect of water loading is a reduction of APE: dAPE = g(zin − zout)drTdmd. This is the amount of work required to lift the water to the level at which it is removed. If the precipitation falls through the air at rest, this is also the amount of kinetic energy dissipated in the microscopic shear zones surrounding the falling precipitation, as discussed in Pauluis et al. (2000).
g. Reevaporation






For an air parcel with a reference level lower than the current level, reevaporation increases the APE. Conversely, if the reference level is higher than the current level, for example in the case of reevaporation in the subcloud layer, then reevaporation of precipitation decreases the APE. It should be stressed here that, similar to precipitation in section 3f, reevaporation in an unsaturated downdraft can result in a net increase of not only the APE but also of the sum of the APE and kinetic energy, without any external source of energy. This is impossible in a dry atmosphere in which the sum of the APE and kinetic energy can only be increased by external energy source.
It is useful here to compare the effects of reevaporation and diffusion of water vapor on the APE. Consider a rising updraft in an unsaturated environment. Some condensed water can fall from the updraft into the environment and reevaporate. A similar effect can be achieved if water vapor diffuses directly from the updraft to the environment. From a thermodynamic point of view, reevaporation and diffusion are irreversible and are associated with exactly the same entropy production. Yet, from the point of view of APE, diffusion of water vapor would be associated with a large destruction of APE, as discussed in section 3e, while the reevaporation can potentially increase APE. The difference arises from the fact that diffusion reduces the upward energy transport by the updraft, while reevaporation can increase the net upward energy transport by generating a cold downdraft. This example indicates that the production or destruction of APE is not directly related to the dissipative nature of the process.
h. Radiative–convective equilibrium
To provide a quantitative example, the sources and sinks of APE for idealized radiative–convective equilibrium are considered here. Consider a quasi-steady atmosphere, heated at the surface with a surface latent heat flux of 90 W m−2, and a sensible heat flux of 10 W m−2. This is balanced by a net tropospheric cooling of 100 W m−2. The surface temperature is 300 K, and the tropopause temperature is 200 K. These numbers are illustrative of tropical conditions. The reference state can be constructed by lifting the “unstable” parcels from the boundary layer to the tropopause, while the rest of the atmosphere—the stable parcels—is compressed by the ascent of the unstable parcels. The portion of the atmosphere that rises to the upper troposphere cannot be determined a priori, but it is reasonable to assume here that it corresponds to the mass of the subcloud layer, approximately 50 mb.
For the surface flux, one can assume that the surface energy fluxes are transferred across a thin surface layer to unstable air parcels. In this case, the effective efficiency for the sensible heat flux is 1/3, and the source of APE is (d/dt)APE|sen = 3.3 W m−2. Using the same value as the example in section 3b for the latent heat flux, the source of APE associated with the latent heat flux is (d/dt)APE|E = 26.8 W m−2. As noted in section 3c, the efficiency factor is slightly lower for the latent heat flux than for the sensible heat flux. The effect of the radiative cooling depends on its vertical structure and on the portion of the atmosphere that is unstable. If the mass of unstable parcels corresponds to an atmospheric layer of 50 mb, the reference temperature of a stable parcel is higher by approximately 5 K than its current temperature. For a uniformly distributed radiative cooling, the energy loss in the stable parcels is 95 W m−2, which translates into a source of APE of approximately 2 W m−2. The rest of the radiative cooling, 5 W m−2 acts on unstable parcels, with T ∼ 300 K and Tref ∼ 200 K, and removes about 1.7 W m−2 of APE. The net contribution of the radiative cooling is small, less than (d/dt)APE|rad < 0.5 W m−2. There is a near cancellation between the cooling of the stable and unstable parcels, which results from the choice of a uniform cooling rate and the fact that the mean temperature of the reference state is typically close to the mean atmospheric temperature.
The contribution of internal processes requires further assumptions, and should thus be viewed as tentative until further study. While precipitation is a dissipative mechanism, it is shown in section 3e that it acts as a source of APE when precipitation forms in parcels that are lower than their reference level. In our example, assuming that the precipitation falls on average from 5000 m yields a source of APE given by (d/dt)APE|prec ∼ Pg(zref − z) ∼ 3.6 W m−2, with P the precipitation rate. Reevaporation can act either as a source or as a sink of APE depending on where it occurs. Taking lnH ∼ lnHref ∼ 1 in (42) yields a scale for the rate of change of the APE due to reevaporation (d/dt)APE|re-ev ∼ PRυT ∼ 3 W m−2. The sign of this contribution is unclear, and this value most likely overestimates the net effect of reevaporation.




4. Discussion
In this paper, an analytic formula for the sources and sinks of APE has been derived. A key element of this derivation lies in proving that the contribution due to the reorganization of air parcels in the reference state vanishes. This is a direct consequence of the choice of a reference state that minimizes the total static energy of the atmosphere, and is therefore valid only in the context of the Lorenz APE framework. Despite the widespread use of APE to discuss the maintenance of the atmospheric circulation, this is the first time that an analytic expression for the production and dissipation of APE in a moist atmosphere has been explicitly derived.
The expression for the APE production is such that the contribution of individual processes can be identified, independently of the transformations taking place in other parts of the atmosphere. This has been used here to discuss the contribution of external energy sources, frictional dissipation, diffusion of sensible heat, surface evaporation, precipitation, diffusion of water vapor, and reevaporation of water.
The production of APE by an external energy source is equal to the net heat source multiplied by an efficiency factor. The efficiency factor is equal to the efficiency of a Carnot cycle with an energy source at the current parcel temperature and an energy sink at the parcel temperature in its reference state. For an energy source at the surface, the same energy flux will generate more APE if it occurs in a parcel with low reference temperature than in an air parcel with high reference temperature. This indicates that the generation of APE is maximized for heating in “unstable” air parcels (i.e., parcels whose reference level is situated in the upper troposphere), while heating “stable” parcels (i.e., whose reference level is situated near the surface) has little impact on the APE.
A comparison between “dry” and “moist” processes indicates that the differences are related to the effects of the water loading, the heat capacity of liquid water, and to the chemical potential difference between water vapor and liquid water. These contributions are in general small when an energy source or transport is involved. For example, the efficiency factor associated with a latent heat flux is found to be only slightly smaller than that of a sensible heat flux. The primary difference between dry and moist atmospheres lies not in the expression for the APE sources and sinks, but rather in the relative importance of the different processes taking place.
Indeed, it is argued here that diffusion can potentially play a dominant role in a moist atmosphere. Diffusion of either sensible or latent heat between two parcels at the same level corresponds to a sink of APE that is proportional to the difference between the parcels’ reference temperature. For a dry atmosphere, the direct relationship between potential temperature, density, and reference temperature should prevent diffusion between parcels with large differences in their reference temperature. In contrast, in a moist atmosphere, parcels with the same specific volume and pressure can have a very different entropy and thus reference temperature. For this reason, one expects mixing and diffusion to play a major role in the APE budget of a moist atmosphere.


Acknowledgments
This work was supported by NSF Grant ATM-0545047.
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Schematic diagram for the transformations discussed in section 2a. The figure illustrates a two-step transformation for a single parcel between the initial reference state A and the final reference state C. The horizontal axis is entropy, and the vertical axis is pressure. The thick curve corresponds to parcel properties in the different reference states between A and C (see section 2a for details).
Citation: Journal of the Atmospheric Sciences 64, 7; 10.1175/JAS3937.1

Schematic diagram for the transformations discussed in section 2a. The figure illustrates a two-step transformation for a single parcel between the initial reference state A and the final reference state C. The horizontal axis is entropy, and the vertical axis is pressure. The thick curve corresponds to parcel properties in the different reference states between A and C (see section 2a for details).
Citation: Journal of the Atmospheric Sciences 64, 7; 10.1175/JAS3937.1
Schematic diagram for the transformations discussed in section 2a. The figure illustrates a two-step transformation for a single parcel between the initial reference state A and the final reference state C. The horizontal axis is entropy, and the vertical axis is pressure. The thick curve corresponds to parcel properties in the different reference states between A and C (see section 2a for details).
Citation: Journal of the Atmospheric Sciences 64, 7; 10.1175/JAS3937.1
Sources and sinks of APE in an idealized radiative–convective equilibrium (see the text for details).


It should be stressed here that, while several textbooks discuss the general circulation in terms of APE, none provides an expression for the sources and sinks of APE in a moist atmosphere. They rely instead on a formulation that is only valid in the context of a dry atmosphere.