The "Poor Man's Navier-­Stokes Equations:" What are they; How do we obtain them; What might we do with them ?

The poor man's Navier­Stokes (PMNS) equations (McDonough and Huang, Int. J.
Numer. Meth. Fluids, 2004) comprise a discrete dynamical system of relatively
low order, but quite high co-dimension, that can be derived directly from the
N.­S. equations via a single-mode Galerkin approximation. Hence, they are even
simpler than typical low-order shell models, and no specific physical
symmetries are imposed by their derivation. But unlike discretizations of the
Lorenz equations (to which they might seem related), the PMNS equations retain
the "structural symmetries" of the full viscous, incompressible N.­S. equations
(i.e., all equations of the PMNS system are of the same form), and the time
series they generate are in good correspondence with those observed in
laboratory experiments and computed with direct numerical simulation, in
contrast to those of the Lorenz equations and the related Henon map.
In this presentation we will outline the PMNS equations derivation and from
this conjecture their possible relationship to a pseudo differential operator
of the N.­S. equations. We will briefly indicate the connection of their
bifurcation parameters to those of physical fluid flows and show how the PMNS
equations might be employed as part of subgrid-scale models for large-eddy
simulation of turbulence. With regard to this application, we demonstrate that
the PMNS equations can exhibit a k-5/3 inertial subrange wavenumber spectrum,
a la Kolmogorov, as well as generally interesting and complicated bifurcation
diagrams in which typical well-known bifurcation sequences are seen to be
embedded, and intriguing--almost art-like--phase portraits.