Several two-dimensional models have been proposed in
recent years that contain aspects of the underlying
dynamics of the three-dimensional incompressible
Euler equations while being more tractable. This
presentation will introduce a new model in this
class that is more directly inspired by fully
three-dimensional solutions as well as a new
conditional restriction upon Euler [Gibbon et al.
(2006)] that shows that only certain symmetrical
alignments are allowed if there is to be a
singularity. The model will be similar to [Gibbon et
al. (1999)], but an equation for curvature of vortex
lines in the plane instead of stretching out of the
plane will be the equation in addition to vorticity.
The importance of studying models such as this has
been highlighted by a new pseudospectral calculation
of collapsing Euler vortices [Hou and Li (2006)]
that has called into question the long-term
conclusions of singular behavior described earlier
in [Kerr (1993), Kerr (2005)]. Because similar
differences between well-resolved numerical
solutions using similar initial conditions have
appeared before, it seems that ultimately numerical
solutions cannot resolve this issue and can only act
as a guide for analytic work. Guidelines for
comparing results from high resolution calculations
using different numerical methods will be discussed.
[Gibbon et al.
(1999)] J.D. Gibbon, A. Fokas, and C.R. Doering,
“Dynamically stretched vortices as solutions of the
Navier-Stokes equations”, Physical D
[Gibbon et al.
(2006)] Gibbon, J.D., D.D. Holm, R.M. Kerr, I
Roulstone, “Quaternions and particle dynamics in the
Euler fluid equations”, Nonlinearity
[Hou and Li (2006)]
T.Y. Hou and R. Li, “Dynamic depletion of vortex
stretching and non-blowup of the 3-D incompressible
Euler equations”, Accepted J. Nonlin. Sci. (2006).
[Kerr (1993)] R.M.
Kerr, “Evidence for a singularity of the
threedimensional, incompressible Euler equations”,
[Kerr (2005)] R.M.
Kerr, “Velocity and scaling of collapsing Euler
vortices”, Phys. Fluids