• Antonov, J. I., , R. A. Locarnini, , T. P. Boyer, , A. V. Mishonov, , and H. E. Garcia, 2006. Salinity. Vol. 2, World Ocean Atlas 2005, NOAA Atlas NESDIS 62, 182 pp.

  • Bannon, P. R., 2012: Atmospheric available energy. J. Atmos. Sci., 69, 37453762.

  • Bohren, C. F., , and B. A. Albrecht, 1998: Atmospheric Thermodynamics. Oxford, 402 pp.

  • Dutton, J. A., 1973: The global thermodynamics of atmospheric motion. Tellus, 25, 89110.

  • Feistel, R., 2008: A Gibbs function for seawater thermodynamics for −6° to 80°C and salinity up to 120 g kg−1. Deep-Sea Res. I, 55, 16391671.

    • Search Google Scholar
    • Export Citation
  • Gibbs, J. W., 1873: On the equilibrium of heterogeneous substances. Trans. Conn. Acad. Art Sci., 3, 108248.

  • Huang, R. X., 2005: Available potential energy in the world's oceans. J. Mar. Res., 63, 141158.

  • Karlsson, S., 1990: Energy, entropy and exergy in the atmosphere. Ph.D. dissertation, Chalmers University of Technology, 121 pp.

  • Locarnini, R. A., , A. V. Mishonov, , J. I. Antonov, , T. P. Boyer, , and H. E. Garcia, 2006: Temperature. Vol. 1, World Ocean Atlas 2005, NOAA Atlas NESDIS 61, 182 pp.

  • McDougall, T. J., , and P. M. Barker, 2011: Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox. SCOR/IAPSO Working Group 127, 28 pp.

  • Oort, A. H., , S. C. Ascher, , S. Levitus, , and J. P. Peixoto, 1989: New estimates for the available potential energy in the world ocean. J. Geophys. Res., 94 (C3), 31873200.

    • Search Google Scholar
    • Export Citation
  • Oort, A. H., , L. A. Anderson, , and J. P. Peixoto, 1994: Estimates of the energy cycle of the oceans. J. Geophys. Res., 99 (C4), 76657688.

    • Search Google Scholar
    • Export Citation
  • Reif, F., 1965: Fundamentals of Statistical and Thermal Physics. McGraw-Hill, 651 pp.

  • Tailleux, R., 2013: Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech., 45, 3558.

  • View in gallery

    (a) The atmospheric availability (thick solid curve) for a 25-km-deep moist standard atmosphere as a function of the reference temperature Tr. The tilted solid line is the difference in total potential energy between the atmosphere and the reference atmosphere; the dashed line is that for the reference temperature times the difference in entropy . The thin vertical line corresponds to the equilibrium temperature T0 where the availability is a minimum. (b) The thick solid curve denotes the mass of water vapor Mrυ in the reference atmosphere as a function of reference temperature. The horizontal dashed line is the mass of water vapor Mυ in the atmosphere. Their difference, δMυ = MυMrυ, denotes the amount of water in the atmosphere precipitated into the surface ice reservoir in the reference state. The reference surface vapor pressure , denoted by the thin solid curve, increases following the Clausius–Clapeyron relation for water vapor and ice with Tr until the reference atmosphere is sufficiently warm to contain all the atmosphere's water in a subsaturated state. The asterisks on the abscissas denote T0 = 256.83 ± 0.01 K.

  • View in gallery

    Vertical profiles of the horizontal mean WOA05 (a) temperature, (b) absolute salinity, and (c) specific available energy as a function of depth. The dashed curves correspond to the equilibrium values T0 = 3.56°C and SA0 = 34.90 g kg−1.

  • View in gallery

    The oceanic availability (thick solid curve) for the horizontal mean WOA05 as a function of the reference temperature. The tilted solid line is the difference in total potential energy between the ocean and the reference ocean; the dashed line is that for the reference temperature times the difference in entropy . The thin vertical line with asterisk corresponds to the equilibrium temperature T0 = 3.56°C where the availability is a minimum.

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Available Energy of Geophysical Systems

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  • 1 Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

An alternative derivation of the available energy for a geophysical fluid system is presented. It is shown that determination of the equilibrium temperature of the system by the minimization of an energy availability function is equivalent to that found by the vanishing of the entropy difference between the fluid and its equilibrium state. Applications to the atmosphere and the ocean are presented.

Corresponding author address: Peter R. Bannon, Department of Meteorology, 521 Walker Building, The Pennsylvania State University, University Park, PA 16802. E-mail: bannon@ems.psu.edu.

Abstract

An alternative derivation of the available energy for a geophysical fluid system is presented. It is shown that determination of the equilibrium temperature of the system by the minimization of an energy availability function is equivalent to that found by the vanishing of the entropy difference between the fluid and its equilibrium state. Applications to the atmosphere and the ocean are presented.

Corresponding author address: Peter R. Bannon, Department of Meteorology, 521 Walker Building, The Pennsylvania State University, University Park, PA 16802. E-mail: bannon@ems.psu.edu.

1. Introduction

The theories of the maximum energy available for conversion to kinetic energy developed by Karlsson (1990) and Bannon (2012) differ in the determination of the reference temperature of the isothermal reference atmosphere. [Tailleux (2013) provides a general review of available energy.] Karlsson (1990) uses a reference temperature that minimizes the entropy difference between the atmosphere and its reference atmosphere. Bannon (2012) argues that the reference temperature is that which minimizes an energy availability function (formerly a generalized Gibbs function). To be internally consistent with the argument in Bannon (2012, section 2), the entropy difference between the atmosphere and its equilibrium should vanish at the equilibrium temperature. Then the transition from the atmosphere to its equilibrium state is, in principle, achievable by an isentropic process.

The purpose of this paper is to demonstrate the equivalency of the two formulations. In particular, we prove that the energy availability of a system is related to the difference in total potential energy δTE between the system and its isothermal reference state by the difference in entropy δS between the two:
e1.1
The specific availability is
e1.2
where u, s, and α are the specific internal energy, entropy, and volume; T is the temperature; and p is the pressure. Here μj is the specific chemical potential of the jth component of mass mj and is the concentration of the jth component (i.e., the mass of the jth component per total mass). The summation convention is assumed for repeated indices. The generalized Gibbs function of Bannon (2012) is here renamed availability for notational convenience and to emphasize its applicability to systems other than the atmosphere. A reference value is denoted by a subscript r. Unsubscripted variables are functions of position x = (x, y, z) and time t but the reference entropy, pressure, and chemical potentials are only functions of height z. The symbol δ defines a finite difference of the system from its isothermal reference state. For example, . The total potential energy is the sum of the internal and potential energies . The total mass of the system and that of its reference are the same:
e1.3
but the volumes of the two may differ. Here the last volume integral is over the volume consisting of the union of the volumes of the system V and its isothermal reference state . For example, the volume of an ocean with a tilted sea surface will differ from its reference state that is dynamically dead with a sea surface corresponding to a geopotential. Similarly, the volume for an atmospheric system confined to lie below an isobaric surface will differ from its reference state also confined below the same isobaric surface. Then the difference in total entropy is , where the respective densities vanish in the appropriate regions of the total volume .
The implications of the energy relation (1.1) are straightforward. If the entropy difference is negative , then it is impossible for the system to attain its reference state. If the difference is positive , then the availability is less than the difference in total potential energy . Thus, the maximum available energy (AE) is that for the temperature where there is no entropy difference:
e1.4
It is noted that the equilibrium temperature and the available energy are thermodynamic state variables of the geophysical system.

Sections 2 and 3 present general proofs of the relation (1.1) for a moist atmosphere with hydrometeors and for an ocean of seawater. Specific examples are presented. The paper concludes in section 4 with some general comments.

2. Atmospheric available energy

The availability for the atmosphere can be expressed (Bannon 2012) as the sum of the contributions of its components as
e2.1
The numeric subscripts j = 1, 2, 3, and 4 (or alphabetical subscripts j = d, υ, l, and i) correspond to the dry air, water vapor, liquid water, and ice, respectively. The differences of the internal energy, entropy, and work term are (with no summation convention implied)
e2.2
e2.3
e2.4
Unlike the total mass difference (1.3) that vanishes, the mass difference of an individual component may not vanish because of a difference in the distribution among the various phases. The entropy relation (2.3) is derived as follows. The entropy difference for the jth component is
e2.5
The specific entropy in the equilibrium state is a constant apart from a contribution owing to the exponential decay with height of the pressure field in the isothermal, hydrostatic, equilibrium atmosphere. Then, following Dutton (1973), we have , where φ is the geopotential. The result (2.3) follows where the difference in mass and potential energy are and .
For a moist atmosphere consisting of dry air and water vapor, the availability (2.1) is, using (2.2)(2.5),
e2.6
where is the specific enthalpy. If the mass of dry air and water vapor is conserved, and (2.6) reduces to (1.1) with . If the equilibrium atmosphere were to be supersaturated at the temperature , then a portion of the vapor is assumed (Bannon 2012) to reside in an ice (liquid if is above freezing) reservoir at the surface. In this case, the vapor entropy is that at saturation with respect to ice (denoted with an asterisk) and the loss of vapor is compensated by a gain in ice so that the total water content is conserved. Then the last term in (2.6) becomes
e2.7
Equality of the chemical potentials at equilibrium between phases implies that . Then the relation (1.1) again holds but with the inclusion of the ice contributions to the differences in total energy and entropy. It is noted that, by similar arguments, the relation (1.1) holds for an atmosphere initially containing water and ice hydrometeors.

The fundamental relation (1.1) provides an unambiguous determination of the available energy of a system. An atmospheric example (Fig. 1) is that for a 25-km-deep moist standard atmosphere with a surface temperature and pressure of 288.15 K and 1013.25 hPa. The lapse rates are 6.5, 0, and −1.0 K km−1 for z > 0, 11, and 20 km, respectively. The water vapor distribution is that of a surface relative humidity of 70% with an e-folding height of 3 km. The thermodynamic parameters are standard (Bohren and Albrecht 1998). Figure 1 and Table 1 provide visual and quantitative evidence of the relation (1.1). Evaluation of the variables requires definition of reference values. The potential energy is defined to be zero at the surface of the atmosphere. The enthalpies of dry air and ice are defined to be zero at a temperature 100°C below the triple point. The entropy is defined relative to a zero entropy at the triple point temperature Ttp = 273.16 K with vapor pressure etp = 6.11 hPa and dry air pressure poo = 1000 hPa. Inspection of Table 1 indicates that the equilibrium atmosphere has, relative to the moist atmosphere, (i) a decrease in potential energy despite a slight increase in that of the water, (ii) a decrease in total internal energy despite an increase in that of the dry air, and (iii) a decrease in total energy despite an increase in that of the dry air. The total energy difference is equal to the available energy. The AE contributions of the dry air and water vapor are both positive with that of the dry air dominating the two.

Fig. 1.
Fig. 1.

(a) The atmospheric availability (thick solid curve) for a 25-km-deep moist standard atmosphere as a function of the reference temperature Tr. The tilted solid line is the difference in total potential energy between the atmosphere and the reference atmosphere; the dashed line is that for the reference temperature times the difference in entropy . The thin vertical line corresponds to the equilibrium temperature T0 where the availability is a minimum. (b) The thick solid curve denotes the mass of water vapor Mrυ in the reference atmosphere as a function of reference temperature. The horizontal dashed line is the mass of water vapor Mυ in the atmosphere. Their difference, δMυ = MυMrυ, denotes the amount of water in the atmosphere precipitated into the surface ice reservoir in the reference state. The reference surface vapor pressure , denoted by the thin solid curve, increases following the Clausius–Clapeyron relation for water vapor and ice with Tr until the reference atmosphere is sufficiently warm to contain all the atmosphere's water in a subsaturated state. The asterisks on the abscissas denote T0 = 256.83 ± 0.01 K.

Citation: Journal of the Atmospheric Sciences 70, 8; 10.1175/JAS-D-13-023.1

Table 1.

Budget of the moist standard atmosphere for the mass M, potential energy , internal energy U, total energy (TE = U + ), entropy S, and available energy (AE). Values for the equilibrium atmosphere with temperature = 256.83 K are in parentheses.

Table 1.

3. Oceanic available energy

The reference state of the ocean is taken to be isothermal, isohaline, and hydrostatic. The reference salinity is the same regardless of the temperature. Here, the subscripts S and W refer to the salt and water, respectively, and is the absolute salinity. Treating the seawater as a binary fluid, we have where the relative chemical potential is and . Then the specific availability (1.2) may be written as
e3.1
Gibbs (1873) has shown that the total chemical potential (i.e., the sum of the intrinsic chemical potential and the geopotential) is a constant for each component in a system in equilibrium. Then the relative chemical potential in the reference state is a constant.
The differences in internal and potential energy for the ocean are
e3.2
e3.3
Expressions for the differences in enthalpy and entropy are similar in form to (3.2). We also note that the chemical potential contribution to the availability vanishes:
e3.4
because the reference relative chemical potential is a constant and the salinity is conserved.
Then, one finds
e3.5
where is the specific Gibbs function. It is readily shown that the first two terms on the right-hand side combine to yield the difference in internal energy . The general thermodynamic relation for the specific Gibbs function (e.g., Reif 1965) applied to the isothermal, isohaline, hydrostatic reference state implies that . Then the reference specific Gibbs function is , where is a constant surface value. Using this result, the third term in (3.5) yields the difference in potential energy
e3.6
where we have assumed mass conservation. Thus, the proof of (1.1) is demonstrated. It is noted that the proof is readily extended to the case of an ocean containing ice.

An ocean example of (1.1) is that for the horizontal mean temperature and salinity of the World Ocean Atlas 2005 (WOA05) (Locarnini et al. 2006; Antonov et al. 2006) (Fig. 2) and is presented in Fig. 3. The required thermodynamic variables are evaluated with the Gibbs SeaWater (GSW) Oceanographic Toolbox of Thermodynamic Equation of Seawater-2010 (TEOS-10) (McDougall and Barker 2011). The zero point for the entropy and enthalpy of seawater is defined as that at standard atmospheric surface pressure with a temperature of 0°C and an absolute salinity of SA = 35.165 04 (Feistel 2008). The potential energy zero is defined as that at a depth of 5.5 km. The equilibrium temperature and salinity are found to be T0 = 3.56° ± 0.01°C and = 34.90 g kg−1. The specific available energy resides mainly in the upper ocean and thermocline (Fig. 2c) and its profile predominately reflects the temperature difference from the equilibrium temperature. The mean available energy of 353 MJ m−2 is large compared to the available potential energy estimates of Oort et al. (1989, 1994) and Huang (2005) that lie in the range of 0.4–2.3 MJ m−2. This difference is partially due to the restriction of the available potential energy (APE) estimates to baroclinic energy sources that exclude the available energy inherent in the stable stratification associated with the oceanic thermocline. It is noted that knowledge of T0 and is particularly useful for the oceanographer who may assess the available energy from a limited regional field program. Inspection of the ocean budget (Table 2) indicates that the equilibrium ocean has a greater potential energy but less internal energy than the ocean. The difference in total energy is again equal to the available energy. The slight disparity in mass may be rectified by an 8.6-cm increase in the sea surface height of the equilibrium ocean. This volume increase implies, in the theory of Huang (2005), a pressure work contribution against the standard atmospheric surface pressure of −8.9 × 10−3 MJ m−2.

Fig. 2.
Fig. 2.

Vertical profiles of the horizontal mean WOA05 (a) temperature, (b) absolute salinity, and (c) specific available energy as a function of depth. The dashed curves correspond to the equilibrium values T0 = 3.56°C and SA0 = 34.90 g kg−1.

Citation: Journal of the Atmospheric Sciences 70, 8; 10.1175/JAS-D-13-023.1

Fig. 3.
Fig. 3.

The oceanic availability (thick solid curve) for the horizontal mean WOA05 as a function of the reference temperature. The tilted solid line is the difference in total potential energy between the ocean and the reference ocean; the dashed line is that for the reference temperature times the difference in entropy . The thin vertical line with asterisk corresponds to the equilibrium temperature T0 = 3.56°C where the availability is a minimum.

Citation: Journal of the Atmospheric Sciences 70, 8; 10.1175/JAS-D-13-023.1

Table 2.

Budget of the WOA05 mean ocean for the mass M, potential energy , internal energy U, total energy (TE = U + ), and entropy S. The ocean's equilibrium temperature is 3.56°C.

Table 2.

4. Conclusions

The fundamental energy relation (1.1) provides a unique method for the determination of the equilibrium temperature that, as noted by Karlsson (1990), is a state variable of a thermodynamic system. An advantage of the availability (1.2) is that it is independent of any arbitrary constants employed in the definitions of internal energy, potential energy, and entropy. The examples provided here for a constant-volume atmosphere and for an ocean bounded by a constant-pressure atmosphere imply interpretation as Helmholtz or Gibbs functions, respectively. The derivations presented here further suggest that the function is valid for a solid such as Earth's crust. The methodology is readily applicable to other geophysical systems. An equilibrium temperature for the Earth system consisting of the atmosphere, ocean, and pedosphere will be close to that of the ocean because of its greater mass. Applications to three-dimensional global datasets of the atmosphere and ocean are in progress and will be presented in other forums.

Acknowledgments

Raymond G. Najjar kindly provided the horizontal mean WOA05 data.

REFERENCES

  • Antonov, J. I., , R. A. Locarnini, , T. P. Boyer, , A. V. Mishonov, , and H. E. Garcia, 2006. Salinity. Vol. 2, World Ocean Atlas 2005, NOAA Atlas NESDIS 62, 182 pp.

  • Bannon, P. R., 2012: Atmospheric available energy. J. Atmos. Sci., 69, 37453762.

  • Bohren, C. F., , and B. A. Albrecht, 1998: Atmospheric Thermodynamics. Oxford, 402 pp.

  • Dutton, J. A., 1973: The global thermodynamics of atmospheric motion. Tellus, 25, 89110.

  • Feistel, R., 2008: A Gibbs function for seawater thermodynamics for −6° to 80°C and salinity up to 120 g kg−1. Deep-Sea Res. I, 55, 16391671.

    • Search Google Scholar
    • Export Citation
  • Gibbs, J. W., 1873: On the equilibrium of heterogeneous substances. Trans. Conn. Acad. Art Sci., 3, 108248.

  • Huang, R. X., 2005: Available potential energy in the world's oceans. J. Mar. Res., 63, 141158.

  • Karlsson, S., 1990: Energy, entropy and exergy in the atmosphere. Ph.D. dissertation, Chalmers University of Technology, 121 pp.

  • Locarnini, R. A., , A. V. Mishonov, , J. I. Antonov, , T. P. Boyer, , and H. E. Garcia, 2006: Temperature. Vol. 1, World Ocean Atlas 2005, NOAA Atlas NESDIS 61, 182 pp.

  • McDougall, T. J., , and P. M. Barker, 2011: Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox. SCOR/IAPSO Working Group 127, 28 pp.

  • Oort, A. H., , S. C. Ascher, , S. Levitus, , and J. P. Peixoto, 1989: New estimates for the available potential energy in the world ocean. J. Geophys. Res., 94 (C3), 31873200.

    • Search Google Scholar
    • Export Citation
  • Oort, A. H., , L. A. Anderson, , and J. P. Peixoto, 1994: Estimates of the energy cycle of the oceans. J. Geophys. Res., 99 (C4), 76657688.

    • Search Google Scholar
    • Export Citation
  • Reif, F., 1965: Fundamentals of Statistical and Thermal Physics. McGraw-Hill, 651 pp.

  • Tailleux, R., 2013: Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech., 45, 3558.

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