  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.
