## 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.

*δ*TE between the system and its isothermal reference state by the difference in entropy

*δS*between the two:The specific availability iswhere

*u*,

*s*, and

*α*are the specific internal energy, entropy, and volume;

*T*is the temperature; and

*p*is the pressure. Here

*μ*is the specific chemical potential of the

_{j}*j*th component of mass

*m*and

_{j}*j*th component (i.e., the mass of the

*j*th 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,

*V*and its isothermal reference state

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

*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)Unlike the total mass difference (1.3) that vanishes, the mass difference of an individual component

*j*th component isThe specific entropy in the equilibrium state is a constant

*φ*is the geopotential. The result (2.3) follows where the difference in mass and potential energy are

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 *T*_{tp} = 273.16 K with vapor pressure *e*_{tp} = 6.11 hPa and dry air pressure *p*_{oo} = 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.

Budget of the moist standard atmosphere for the mass *M*, potential energy *U*, total energy (TE = *U* + *S*, and available energy (AE). Values for the equilibrium atmosphere with temperature

## 3. Oceanic available energy

*S*and

*W*refer to the salt and water, respectively, and

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 *S _{A}* = 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

*T*

_{0}= 3.56° ± 0.01°C and

^{−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

*T*

_{0}and

^{−3}MJ m

^{−2}.

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

## 4. Conclusions

The fundamental energy relation (1.1) provides a unique method for the determination of the equilibrium temperature

## Acknowledgments

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

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