On exact and approximate energy equations in pressure coordinates

Studies of the atmospheric energy cycle have been common in meteorology since the modern formulation of the concept by Lorenz. Most of these studies have been based on the approximate version of the energy budget equations using pressure as the vertical coordinate given by Lorenz in the same paper. Inexactitude in the formulation of the energy budget equations in general and the isobaric coordinate version in particular arises from several sources: the neglect of topography in the definition of the reference state for available potential energy, the simplification of the lower boundary condition, the definition of mean and eddy components and in particular in the use of the Lorenz approximate equations. An exact version of the energy cycle equations in isobaric coordinates is derived. The equations are exact in the sense that they are derived from the hydrostatic primitive equations without further approximation. It is pointed out that results already obtained indicate that there is a non-trivial difference between the exact and approximate versions of the energy budget equations.