Geometric Structure of Pseudo-plane Quadratic Flows

TL;DR

This paper investigates the geometric structure of three-dimensional quadratic flows in stratified fluids, revealing invariant conic types and disproving previous conjectures about baroclinic solutions, with implications for turbulence modeling.

Contribution

It provides the first complete set of exact solutions for 3D quadratic flows in stratified fluids, highlighting their geometric properties and correcting earlier assumptions.

Findings

Steady quadratic flows have invariant conic types in non-rotating frames.

Three baroclinic solutions with non-aligned vertical structures are identified.

The topology of quadratic flows is richer than that of high-degree polynomial flows.

Abstract

Quadratic flows have the unique property of uniform strain and are commonly used in turbulence modeling and hydrodynamic analysis. While previous application focused on two-dimensional homogeneous fluid, this study examines the geometric structure of three-dimensional quadratic flows in stratified fluid by solving a steady-state pseudo-plane flow model. The complete set of exact solutions reveals that steady quadratic flows have invariant conic type in non-rotating frame and non-rotatory vertical structure in rotating frame. Three baroclinic solutions with vertically non-aligned structure disprove an earlier conjecture. The rich topology of quadratic flows stands in contrast to the depleted geometry of high-degree polynomial flows. A paradox in the steady solutions of shallow-water reduced-gravity models is also explained.