a. General derivation of the governing equation

This section presents a general derivation of the available energetics for a multicomponent, compressible fluid. The specific available energy, ae, is

Its rate of change following the three-dimensional, mass-weighted, mean velocity v is

where the material derivative is

and , where m_j and v_j are the mass and velocity of the jth component. Then the continuity equation is

where ρ is the total density. For the hydrostatic equilibrium state, the force balance is

where Φ is the geopotential. The Gibbs relation in the form dh = Tds + αdp + μ_jdχ_j for the equilibrium state implies that the term in square brackets in (4.2) vanishes. Then, using (4.3) and (4.4), (4.2) becomes

The last term vanishes because the material derivatives of the chemical potentials only contain a gravity contribution and the sum of the differences in concentrations vanish:

This result is a consequence of (3.8) that includes a gravity component to the chemical potentials of the hydrometeors. This assumption is addressed further in appendix B. The specific kinetic energy, ke, and enthalpy equations are

where τ is the viscous stress tensor. Then (4.5), (4.6), and (4.7) sum to give

The equation for the concentration rate of change is

where ρ_j is the density of the jth component, whose velocity relative to the mass-weighted average velocity v is and whose rate of production by chemical reactions and/or phase changes is . The entropy equation is

where is the heating rate due to radiation, conduction, and diffusion (i.e., the convergence of the enthalpy flux of the components with enthalpy h_j relative to the mean velocity). Here is the viscous dissipation and e_ij is the rate of strain tensor. Then, the available energetics equation becomes

The internal sources of available energy are given by the last term in (4.11), where

or, using the definition of the chemical potentials,

For example, in the case of a phase change between water vapor and liquid water, we have

where H is the relative humidity. Thus, inclusion of the geopotential term in the equilibrium chemical potential for the hydrometeors (3.8) correctly handles entropy production during phase changes.

It is of some interest to compare the available energy defined here with the engineering concept of exergy (e.g., Bejan 1997). Typically the exergy of the jth component is (plus for the hydrometeors). Then the available energy may be written in terms of the exergy as or and the flux form of (4.11) is

where the rate of working by the pressure field has dropped from the equation.

b. Partitioning and linearization of the available energy

The available energy (4.1) can be partitioned into the sum of the available energies of each component. Using relations of the form yields . The available energy for each component is now shown to be a positive definite quantity.

The available energy for dry air and water vapor (j = 1, 2) may be partitioned (e.g., Bannon 2005) into available potential and available elastic contributions:

where

and

The specific entropy is , where the potential temperature is with and the caret indicates a normalized departure. The equilibrium potential temperature normalizes the potential temperature departure , while the pressure normalizes the pressure departure .

The available energy for liquid or solid water (j = 3 or 4) is

where the geopotential arises from the definition (3.8). The equilibrium temperature normalizes the temperature departure . The expressions for the gases correctly reduce to those for a solid or liquid by taking κ = 0.

The preceding formulas emphasize the enthalpic nature of the available energy. Their definitions in terms of normalized departures are computationally advantageous to ensure positive definite calculations as well as providing easy derivations of their quadratic nature.

a. Standard atmosphere

The case of the standard atmosphere in Fig. 2 is an example of the minimization of the atmospheric Gibbs function δG_A. This hydrostatic atmosphere extends to a height of 25 km with a surface temperature and pressure of 288.15 K and 1013.25 hPa. The lapse rates are 6.5, 0, −1.0, and −2.8 K km⁻¹ for z < 11, 20, 32, and 50 km, respectively. The potential and elastic contributions are defined by (4.17) and (4.18). The temperature profile is plotted in Fig. 3a, where it is compared to its equilibrium temperature T₀ = 251.95 K. The total energy (TE) is slightly greater than that of its equilibrium atmosphere for both the 25- and 50-km-deep cases (Table 1). The internal energy (IE = U) differences between the atmosphere and its equilibrium are small and change sign between the two cases. For both, the potential energy (PE) is less for the equilibrium atmosphere. This reduction in PE is consistent with a lower center of gravity associated with the lower temperature of the equilibrium atmosphere in the lower troposphere (Fig. 3a). Because of the finite vertical extent of each atmosphere, the ratio of the potential to internal energy is less than that of R/c_υa = 40%. Thus, the total energy is only approximately the enthalpy (cf. Lorenz 1955).

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(a) Temperature and (b) available energy as a function of height z for a 25-km-deep standard atmosphere. In (a) the thick solid and solid curves are the soundings for the atmosphere and its equilibrium atmosphere, respectively, with T₀ = 251.95 K. In (b) the solid, dashed, and thick solid curves denote the potential, elastic, and total contributions. The dotted vertical line is the zero line.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-059.1

Table 1.

Traditional energetics of two standard atmospheres of different depths. Values for the equilibrium atmosphere are in parentheses. The total energy is the sum of the internal and potential energies .

The available energy (AE) for both atmospheres resides primarily (Table 2) in the available potential energy (APE) with about 10% in the available elastic energy (AEE). For the 25 km deep atmosphere, the available energy of 11.45 MJ m⁻² is much less than its total energy (Table 1) of 2.4733 GJ m⁻². Thus, only 0.46% is available for conversion into kinetic energy; 0.48% for the 50 km deep atmosphere. In contrast the available energy agrees well with the difference in total energy between the atmosphere and its equilibrium state. This important feature indicates that, in achieving its equilibrium state, the atmosphere’s total energy is reduced by an amount AE that can, in principle, be realized as kinetic energy. In this subsection and the next, the equilibrium temperature is determined to within an accuracy of 0.01 K so as to confirm the agreement between the change in total energy and the available energy. Elsewhere an accuracy of 0.1 K is used.

Table 2.

Available energetics of the two standard atmospheres. is the difference of the total energies of the atmosphere and its equilibrium atmosphere.

The vertical distribution of the available energy (Fig. 3b) is bimodal with a maximum at the surface, a secondary maximum at the tropopause, and a midtropospheric minimum near where the equilibrium temperature and the sounding (Fig. 3a) intersect. The secondary peak lies at the height of the change in the lapse rate. The elastic energy is significant in the stratosphere and its contribution dominates that of the potential energy above 20 km. This behavior holds for all of the cases analyzed and reflects that the elastic energy (4.18) is inversely proportional to the square of the pressure field.

b. Idealized baroclinic zones

The idealized baroclinic zones are based on a generalized compressible Eady base state (appendix C). The available energetics is summarized in Table 3 as a function of the total meridional temperature gradient . The results display a clear monotonic trend in all variables. As the amplitude of the baroclinity () increases, the available energies AE, APE, and AEE increase, while the equilibrium temperature (T₀) decreases slightly. For each case, the difference in total energy between the atmosphere and its equilibrium state is the available energy.

Table 3.

Available energetics of the idealized baroclinic zone as a function of horizontal temperature gradient with no surface winds.

The spatial variation of the available energy (Fig. 4a) exhibits a bimodal variation with larger values where the difference between the temperature and the equilibrium temperature is larger. The dominance of the temperature variation, rather than pressure variation, reflects the dominance of the available potential energy contribution (4.17) to the total available energy. The efficiency factor also reflects this distribution (Fig. 4b). The abrupt transition at 11 km reflects the change to a zero lapse rate in the lower stratosphere.

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Cross section of the (a) specific available energy and (b) the efficiency factor N = (T − T₀)/T for the idealized baroclinic zone with = 30 K. In each panel, the thick solid line is the zero efficiency isopleth.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-059.1

The effect of a surface pressure gradient associated with a mean geostrophic wind is small (Table 4) with a slight increase in available energy as the wind transitions from westerly to easterly. Henceforth, the surface wind is set to zero.

Table 4.

Available energetics of the idealized baroclinic zone with and a surface pressure gradient associated with a uniform geostrophic wind with a speed of 10 m s⁻¹.

The effect of sloping topography on the available energy is readily included in this Eulerian formulation. Table 5 provides a comparison for topography sloping linearly upward toward the pole (henceforth poleward) or toward the equator (henceforth equatorward) to reach a maximum height of 3 km (appendix C). The poleward case contains more available energy. This result is consistent with the linear Eady model analysis of Mechoso (1980) that indicates greater growth rates for unstable baroclinic waves for terrain sloping with the isentropes.

Table 5.

Available energetics of the idealized baroclinic zones over topography sloping upward toward the pole or toward the equator. The horizontal temperature gradient is . The total change in topography is 3 km.

The effect of the inclusion of a moist boundary layer (appendix C) on the available energy is summarized in Table 6. The vapor pressure has a surface relative humidity of 70% and decays vertically with a scale height of 3 km. Inclusion of moisture has increased the equilibrium temperature by ~5 K to 265.12 K and increased all available energy components of the dry air. The vapor contributes to the available energy primarily through its elastic component. The distribution of the available energy and the efficiency factor (not shown) is similar to that of Fig. 4 for the dry case. The major effect of the moisture is through the increased equilibrium temperature that moves the zero efficiency isopleth downward. Figure 5 illustrates the effects of water vapor on the minimization of the Gibbs function. For reference temperatures below about 270 K, the reference atmosphere contains an ice reservoir of tens of kilograms per square meter of water. The enthalpy change associated with this phase transformation increases the Gibbs function curve at those temperatures and moves the temperature minimum toward warmer temperatures. The equilibrium atmosphere is saturated with an ice reservoir that is 1.29 cm thick due to precipitation (3.3). The ice contributes the major amount of the difference in total energy between the atmosphere and its equilibrium by its enthalpy of deposition (Table 6). The vertical distributions of the available energies (not shown) are similar to those presented in section 5d, which analyzes observed moist atmospheric soundings.

Table 6.

Available energetics of the moist idealized baroclinic zone with . The equilibrium temperature is T₀ = 265.12 K with a saturated surface vapor pressure of = 3.09 hPa. The ice reservoir is 1.29 cm thick and contributes 30.18 MJ m⁻² to the total energy difference.

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(a) The atmospheric Gibbs function δG_A for the moist baroclinic zone as a function of the reference temperature T_r. The thick solid, solid, and dashed curves denote the total, dry air, and water vapor contributions. (b) The thick solid curve denotes the mass of water vapor M_υr in the reference atmosphere as a function of T_r. The horizontal dashed line is the mass of water vapor M_υ in the atmosphere. Their difference, M_υ − M_υr, denotes the amount of water in the atmosphere precipitated into the water reservoir in the reference state. The reference surface vapor pressure , denoted by the thin solid curve, increases with T_r following the Clausius–Clapeyron relation until the reference atmosphere is sufficiently warm to contain all the atmosphere’s water in a subsaturated state. The asterisks on the abscissas denote the equilibrium temperature T₀ = 265.12 K; the crosses denote the minimum and maximum equilibrium temperature of individual soundings in the ensemble.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-059.1

c. Comparison with the available potential energy of Lorenz

The evaluation of the available potential energy (LAPE) of Lorenz (1955) uses the approximate formula [Lorenz 1955, (10)]

where an overbar denotes a horizontal average and a prime the deviation from that average. The LAPE is identically zero in the preceding horizontally homogeneous situations of section 5a. No adjustment is made here in the calculation of LAPE for isentropes intersecting the surface.

Exact specification of the counterpart to the Lorenz APE in the present approach is thwarted by its nonlinear structure. For definiteness, the available energy of a single atmospheric sounding is defined as available convective energy (ACE). In contrast, the available baroclinic energy (ABE) is defined as the amount of available energy in excess of that needed to bring the system to the lowest equilibrium temperature of the individual soundings in the ensemble of soundings. Then,

where . Conceptually this approach views the determination of the ABE as a two-step process in which the atmosphere collectively adjusts vertically to its state of lowest total energy. A subsequent adjustment of the system then occurs laterally among the columns to the equilibrium temperature . Alternatively, the atmosphere could first adjust to the temperature of the sounding with the warmest ACE temperature . Graphically (Fig. 5) the minimum and maximum temperatures straddle . The difference in the values of the ABE estimated by the two approaches reflects the asymmetry of the parabolic shape of the atmospheric Gibbs function about . The results (Table 7) indicate qualitative agreement between the ABE and LAPE values. The available baroclinic energy increases with the inclusion of moisture and with topography sloping against the slope of the isotherms. In all cases, the LAPE is less than the AE. This inequality indicates that the Lorenz formulation underestimates the amount of available energy.

Table 7.

Available baroclinic energetics of the idealized baroclinic zones with horizontal temperature gradient . The total change in the topography cases is 3 km.

d. Individual moist soundings

A real data case is afforded by the sounding from the Second Verification of the Origins of Rotation in Tornadoes Experiment [VORTEX2; 2155 UTC 5 June 2009, National Severe Storms Laboratory (NSSL1)] presented in Fig. 6. The sounding exhibits a large potential for deep convection for parcels ascending above about 700 hPa along the 70°C saturated adiabat. The convective available potential energy (CAPE) evaluated from 700 to 170 hPa is 2370 J kg⁻¹. Assuming hydrostatic balance, the available energy may be analyzed in pressure coordinates. The minimization of the atmospheric Gibbs function (not shown) is similar to that depicted in Fig. 5. Table 8 and Fig. 7 present the results of an AE analysis of the profile. The dry air AE (Fig. 7a) exhibits a bimodal distribution with the potential dominating the elastic contribution. In contrast, the water vapor AE (Table 8) is dominated by the elastic contribution. This result implies that the vapor pressure perturbations dominate the potential temperature perturbations (section 4b). The water vapor AE (Fig. 7b) is also bimodal but with a thick 200-hPa layer of near-zero AE. This minimum reflects the impact of the low dewpoints between 700 and 500 hPa in the sounding (Fig. 6). The AE may be compared to the convective available potential energy by dividing by the total mass of the sounding M₀ = 6.845 × 10⁴ kg m⁻², yielding a CAPE of 1541 J kg⁻¹.

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Skew T–logp diagram for the VORTEX2 sounding with the thick solid and dashed–dotted curves denoting the temperature and dewpoint temperature, respectively.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-059.1

Table 8.

Available energetics of the VORTEX2 sounding. The equilibrium temperature is T₀ = 262.1 K, and a surface vapor pressure = 2.4 hPa. The ice reservoir is 6.35 × 10⁻⁵ m thick. The total AE divided by the mass of the sounding yields a CAPE of 1541 J kg⁻¹.

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Available energy for the (a) dry air and (b) water vapor as a function of pressure for the VORTEX2 sounding. The total, potential, and elastic contributions are denoted by the thick solid, solid, and dashed curves, respectively, for each component.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-059.1

A second case is phase III of the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE) sounding analyzed by Randall and Wang (1992; see their Table 1 for the “given sounding”). This tropical sounding (Fig. 8) is relatively moist with the temperature profile close to the saturated adiabat of 70°C. This structure suggests a small value of CAPE. In contrast, the AE analysis of Table 9 and Fig. 9 indicates that the sounding is a potentially large source of kinetic energy. The general features of the AE profiles are consistent with those for the extratropical VORTEX2 sounding. These include a bimodal vertical distribution, a dominant potential contribution by the dry air, and a dominant elastic contribution by the water vapor. With a total mass of M₀ = 9.135 × 10⁴ kg m⁻², the bulk CAPE is 2212 J kg⁻¹. In contrast, Randall and Wang estimate their GCAPE to be 11 J kg⁻¹.

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As in fig. 6, but for the GATE III sounding.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-059.1

Table 9.

Available energetics of the GATE Phase III sounding. The equilibrium temperature is T₀ = 271.4 K with = 5.3 hPa. The ice reservoir is 14.47 × 10⁻⁵ m thick. The total AE divided by the mass of the sounding yields a CAPE of 2212 J kg⁻¹.

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As in fig. 7, but for the GATE III sounding.

Citation: Journal of the Atmospheric Sciences 69, 12; 10.1175/JAS-D-12-059.1

a. Dissipation of kinetic energy

The dissipation term (6.3) includes three distinct processes: viscous dissipation of kinetic energy within the atmosphere and the surface flux of kinetic energy out of the atmosphere by the surface wind stresses and by the precipitation of hydrometeors.

1) Internal dissipation

The loss due to friction can be decomposed into one due to the clear air and one due to hydrometeor drag:

Peixoto and Oort (1991) estimate the large-scale dissipation to be about 2 W m⁻² globally. For the tropics Pauluis et al. (2000) estimate the turbulent cascade by convection to be 1 W m⁻² and the hydrometeor dissipation to be 2–4 W m⁻². Using satellite data, Pauluis and Dias (2012) estimate the dissipation to be 1.8 W m⁻². In the extratropics there is a reduction in precipitation and a lowering of the mean height of hydrometeor formation. Globally we take the hydrometeor dissipation to be 1–3 W m⁻².

2) Kinetic energy loss due to precipitation

Falling hydrometeors will carry their kinetic energy out of the atmosphere at the rate

where P is the precipitation rate, is the surface area of the earth, and the subscript h denotes a hydrometeor. The overbar denotes a representative mean. For a fall speed of 1 m s⁻¹ and a precipitation rate of 1 m yr⁻¹ (), (6.7) yields .

3) Kinetic energy loss due to wind stress

With surface wind speeds exceeding that of the underlying surface, there is a downward transport of kinetic energy out of the atmosphere given by . For a surface wind of 10 m s⁻¹ and bulk aerodynamic drag formulations with a drag coefficient of 10⁻³, the downward transport is .

Thus, the total dissipation rate is about 5–7 W m⁻². Table 10 summarizes the discussion.

Table 10.

Dissipation of kinetic energy. In a steady state the total dissipation rate equals the rate of conversion of available energy to kinetic energy.

b. Generation of available energy

The generation rate (6.5) includes surface fluxes of exergy into the atmosphere and interior diabatic processes. The interior terms are weighted by the efficiency factor that modulates the impact of a particular process in enhancing the available energy relative to the equilibrium isothermal atmosphere (Fig. 1) at the temperature T₀. A contour plot (not shown) of the efficiency factor would have a zero contour along the T₀ = 255 K isotherm that runs from the middle tropical troposphere downward toward the poles (with additional zero isopleths in the upper stratosphere and lower mesosphere). For the same magnitude temperature difference , the efficiency would be greater above the zero isotherm than below. For example, at the surface of the tropics , while at the polar tropopause the efficiency is . Endothermic processes below the zero contour where the efficiency is positive would increase the available energy; exothermic processes above would also increase it.

1) Radiation

The radiative generation of available energy is

where the heating rate is expressed in terms of a warming rate . The sum of the clear-sky shortwave and longwave warming is relatively constant with height. Then

where the scale height is H_s = 8 km and is a nondimensional integrated efficiency factor weighted by the exponential decay of density with height:

where is the surface temperature and the lapse rate is Γ = 6.5 K km⁻¹. This mean efficiency varies from = 2.1% for the tropics ( = 300 K and = 16 km) to 0.5% for midlatitudes ( = 288 K and = 12 km) to −1.2% for the polar regions ( = 273 K and = 8 km). Then, for a mean = 1%, the radiative processes contribute .

The effect of clouds on the radiative generation could be significant. Low-level cloud bases, where the efficiency is positive generally, absorb longwave radiation emitted from the surface, leading to a positive AE generation. Emission of longwave radiation to space from high-level cloud tops, where the efficiency is negative, will also generate positive AE.

2) Dissipation

There is a feedback of the viscous dissipation (in which kinetic energy is converted into thermal energy) to act as an available energy generation process. Assuming that all of the conversion occurs close to the surface in the boundary layer, then the feedback is positive with a maximum positive efficiency. One finds

3) Precipitation exergy flux

The generation of exergy by precipitation is

where again . The exergy for liquid or solid water (j = 3 or 4) is

Because exergy is a positive definite quantity, precipitation always carries exergy out of the atmosphere and thus acts as a sink of AE. Taking , the liquid water exergy is . The geopotential contribution in (6.13) arises from the definition (3.8). The mean continental elevation is about 1 km but land covers only 30% of the earth’s surface, so globally. Then, the precipitation flux is . We note that the exergy of ice is about half that of liquid precipitation and that the geopotential contribution can be significant locally.

4) Evaporation exergy flux

The evaporation exergy generation (e.g., Pauluis 2007) is

where the net evaporation is taken to equal the net precipitation and H is the relative humidity. This input is composed of three contributions. The first cancels the precipitation sink (to first order) but without the geopotential term:

The second is due to the enthalpy of evaporation:

The third is

for H = 50% and H₀ = 100%. Then the total evaporative generation is .

5) Dry air exergy flux

The turbulent transport of exergy from the surface may be estimated in a manner analogous to that for the enthalpy. The bulk aerodynamic formula for the surface enthalpy flux is , where the right-hand bracket on the temperature denotes a difference across the interface between the air and the underlying surface. For a mean enthalpy flux of 24 W m⁻² (Trenberth et al. 2009), the mean air – surface temperature difference is using C_H = 10⁻³. By analogy, the exergy flux is defined by , where the exergy for dry air is

Near the surface because the dry air pressures of the atmosphere and its equilibrium atmosphere are, to a very good approximation, equal to each other. Then, using the mean air – surface temperature difference estimated above, one finds, using C_ex = 10⁻³, and .

c. Irreversible entropy production

The last term in (6.4) represents irreversible entropy production processes (e.g., the subcloud evaporation of raindrops). It is positive definite and hence is a sink of available energy. The term is called lost work in exergy theory (e.g., Bejan 1997). Pauluis and Held (2002) estimate an irreversible entropy production of 8 W m⁻² that includes a surface contribution due to irreversible evaporation. The latter process is contained here in the third contribution of the surface evaporation (6.17). Thus, we take .

Table 11 summarizes the discussion on the available energy generation budget.

Table 11.

Generation of available energy. In a steady state, the total rate of generation of available energy less the lost work equals the rate of conversion of available energy to kinetic energy.