Recent Advances in the Theory of Available Potential Energy for Salty Oceans and Moist Atmospheres

Lorenz theory of available potential energy has been the main framework for describing the energy cycle of both the ocean and the atmosphere for over 50 years. However, despite its wide acceptance, many aspects of the theory have remained unclear and somewhat controversial, especially in the case of binary fluids such as a moist atmosphere or salty ocean for which the definition of a suitable reference state remains ambiguous. On the other hand, the original difficulty associated with the global character of Lorenz original APE theory has since been resolved by the realisation that APE theory can actually be formulated in terms of a local principle, as first established over 30 years ago, as well as by the more recent realisation of how to achieve an exact mean/eddy decomposition of the local APE density, at least in the case of a simple Boussinesq fluid. In this work, the APE theory is extended in several ways. First, we show how to reformulate APE theory in terms of a reference state defined in terms of an adiabatic re-arrangement, but which does not not necessarily coincide with a state of minimum potential energy, with arguments given in favour of why this might be preferable over Lorenz’s original construction. This development is motivated by the widely held belief in oceanography that the best choice of density is one that is as ‘neutral’ as feasible, suggesting that the most suitable reference state in the ocean might be one obtained by ordering the fluid parcels according to their ‘neutral’ density value. Second, we introduce a new local definition of APE density for which an exact mean/eddy decomposition is proposed, which also isolates the part of the APE that arises from the thermobaric character of the nonlinear equation of state for seawater. Thermobaricity is a key feature of the oceanic equation of state, responsible for the lack of PV nonconservation, for countering the decrease in buoyancy of Antarctic Bottom Water due to mixing as it cascades to the bottom of the ocean, and for thermobaric instability. The exact mean/eddy decomposition of the local APE density yields an exact description of the energy cycle that is similar in many ways to the standard quasi-geostrophic approximation originally proposed by Lorenz, but which occasionally can strongly differ from the latter. Third, we show that the local APE density can serve as an accurate streamfunction for the steady-state horizontal circulation. Finally, we argue that APE theory should be regarded as a framework for partitioning the potential energy into a ‘work’ and ‘heat’ part, rather than in terms of ‘available’ and ‘unavailable’ potential energy, which helps understanding why Lorenz APE theory should be regarded as the natural extension of the thermodynamic theory of heat engines to the atmosphere and ocean.