Local Lagrangian and Eulerian available energetics in moist atmospheres

Margules (1905) articulated the observation that, of the large reservoirs of internal and potential energies in the atmosphere, only a small portion is transformed into the kinetic energy of atmospheric motions. Thus, only part of the energy reservoir is available for transformation. Building on this idea, Lorenz (1955) introduced the concept of available potential energy as the maximum kinetic energy attainable by an adiabatic redistribution of the mass of a hydrostatic atmosphere. He formulated a global theory of available potential energy that has become a cornerstone for describing the general circulation of the atmosphere.

In this study, two formulations of local available energetics are presented that are valid for a compressible, multi-component fluid that allows for frictional, diabatic, and chemical (e.g., phase changes) processes. The Eulerian formulation is shown to be the nonlinear extension of linear available energetics. The available energy is defined relative to an arbitrary isothermal atmosphere in hydrostatic balance with uniform chemical potentials. It is shown that the available potential energy can be divided into available potential, available elastic, and available chemical energies. The first two are shown to be positive definite. The general formulation is applied to the specific case of an idealized, moist, atmospheric sounding with liquid water and ice. The available energy is dominated by available potential energy in the troposphere but available elastic energy dominates in the upper stratosphere. The available chemical energy is significant in the lower troposphere where it dominates the available elastic energy. The total available energy increases with increasing water content.

The Lagrangian formulation chooses the reference state as the initial state of the parcel that need not be a state of rest. It is shown that it reduces to the convective available potential energy (CAPE) for a resting atmosphere and to symmetric available potential energy (SAPE) for symmetric instability when the parcel-theory assumption (that the pressure of the parcel is always that of the environment) is invoked.