Thermodynamic and vortic fine structures of real Schur flows

TL;DR

This paper investigates the thermodynamic and vorticity structures of real Schur flows, deriving governing equations and proving Lie invariances in multiple dimensions, advancing understanding of flow symmetries and invariants.

Contribution

It introduces a detailed analysis of the thermodynamic and vorticity structures of RSFs, including derivation of governing equations and proof of Lie invariances in higher dimensions.

Findings

Derived complete set of equations for viscous and driven 2C2Dcw1C3D flows.

Proven Lie invariances of vorticity 2-forms in any dimension $d \,\geq\, 3$.

Identified invariances of entropic and vorticity quantities, including Ertel potential vorticity forms.

Abstract

A two-component-two-dimensional coupled with one-component-three-dimensional (2C2Dcw1C3D) flow may also be called a real Schur flow (RSF), as its velocity gradient is uniformly of real Schur form, the latter being the intrinsic local property of any general flows. The thermodynamic and `vortic' fine structures of RSF are exposed and, in particular, the complete set of equations governing a (viscous and/or driven) 2C2Dcw1C3D flow are derived. The Lie invariances of the decomposed vorticity 2-forms of RSFs in $d$-dimensional Euclidean space $\mathbb{E}^d$ for any interger $d\ge 3$ are also proven, and many Lie-invariant fine results, such as those of the combinations of the entropic and vortic quantities, including the invariances of the decomposed Ertel potential vorticity (and their multiplications by any interger powers of entropy) 3-forms, then follow.