# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1701.06647