Exact expressions for available potential energy and available potential vorticity

TL;DR

This paper derives exact analytical expressions for available potential energy and vorticity, improving understanding and computational efficiency in stratified fluid dynamics by reformulating equations in a novel coordinate system.

Contribution

It introduces new exact formulas for APE and APV based on first principles, aligning with prior perturbation results and facilitating numerical implementation.

Findings

Derived exact APE and APV expressions from first principles.

Reformulated equations in terms of coordinate label z−η for clarity.

Expressions are simple, intuitive, and suitable for numerical computation.

Abstract

Exact analytical expressions for available potential energy density (APE) and available potential vorticity (APV) are derived from first principles. These APE and APV expressions align with previously known quantities found using perturbation expansions in Holliday & McIntyre (1981) and Wagner & Young (2015), respectively. The key is to recast the equations of motion and their conservation laws in terms of the coordinate label $z-\eta$, where z is a fluid parcels height at the current time, and $\eta$ is the isopycnal deviation from the parcels current height after adiabatic rearrangement of all parcels into the no-motion state. In addition to their intuitive appeal and simplicity, the new APE and APV expressions are easily implemented in numerical computations of Boussinesq dynamics with non-constant stratification.