# A local diagnostic energy to study diabatic effects on a class of degenerate Hamiltonian systems with application to mixing in stratified flows

## TL;DR

This paper introduces a local diagnostic energy for degenerate Hamiltonian systems to analyze diabatic effects, with applications to stratified flows, recovering known energy concepts and revealing universal properties in low-Rossby number regimes.

## Contribution

It develops a new local energy-like quantity for degenerate Hamiltonian systems, applicable to geophysical flows, and connects it to existing energy concepts like Available Potential Energy.

## Key findings

- Recovers Holliday and McIntyre's Available Potential Energy in stratified flows.
- Shows the local diagnostic energy has a universal character in low-Rossby number flows.
- Provides a framework for studying diabatic effects on the Available Energy.

## Abstract

In Hamiltonian systems characterized by a degenerate Poisson algebra, we show how to construct a local energy-like quantity that can be used to study diabatic effects on the evolution of the Available Energy of the system, the latter concept formalizing the original idea of Margules'. We calculate the local diagnostic energy for geophysically relevant flows. For the particular case of stratified Boussinesq flows, we show that under moderately general conditions, in inertial frames where the initial distribution of potential vorticity is even around the origin, our framework recovers the Available Potential Energy introduced by Holliday and McIntyre \cite{HollidayM81}, and as such depends only on the mass distribution of the flow. In non-inertial frames, we show that the local diagnostic energy of flows which are, in an appropriate sense, characterized by a low-Rossby number ${\rm Ro}$ ground state, has to lowest order in ${\rm Ro}$, a universal character.