On the concept of pseudo-energy of T. G. Shepherd.
Abstract
In a recent paper, 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 the special choice of a basic state defined by theta_0(p) = (p) where the potential temperature (p) is the average on an isobaric surface, 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. But 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 a_h = (h - h_r) - T_r (s - s_r) which is the flow energy of a fluid - see Marquet (1991) - can be obtained from the generalized approach of Shepherd if a constant basic state at temperature T_r and pressure p_r is considered. 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 also explained that for a real isothermal basic state made up of an atmosphere at constant temperature T_0 but with a variable pressure, the generalized expression of Shepherd reduces with a good accuracy to the approximate functions introduced by Pearce (1978) or Blackburn (1983) in meteorology, it is moreover exactly the primary result obtained by Thomson (1853) in thermodynamics.