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On the concept of pseudo-energy of T. G. Shepherd. A close link with exergy functions

Shepherd (1993) derived a general expression for the available potential energy for compressible, hydrostatic flow, where the sum of this available energy and the kinetic energy is called pseudo-energy. He demonstrated that for a special basic state the small-amplitude limit of the generalized available potential energy reduces to the well-known approximate form of Lorenz (1955) expressed in a pressure vertical coordinate. Other forms of available energies exist in atmospheric energetics and in thermodynamics, where the name exergy has been coined by Rant (1956) to denote the maximum work that can be extracted from any system when it is subject to some constraints (adiabatic transformations or constant total energy for instance). The purpose of this note is to show that the specific available enthalpy function denoted by $a_h = (h - h_r) - T_r \: (s - s_r)$ in Marquet (1991) can be obtained from the generalized approach of Shepherd for another basic state. This special form of pseudo-energy also leads to the global hydrostatic concepts of Dutton (1973) and Pichler (1977). The function $a_h$ only depends on the specific enthalpy $h$ and entropy $s$ at any point, the values $h_r$ and $s_r$ refer to the special dead state at temperature $T_r$ and pressure $p_r$. It is explained that for another choice of the basic state, the generalized expression of Shepherd reduces with a good accuracy to the approximate functions introduced by Pearce (1978) or Blackburn (1983) in meteorology and Thomson (1853, Lord Kelvin) in thermodynamics.