  A theory of an eroding "hairpin" vortex dipole structure in three dimensions
is developed, extending our previous study of an axisymmetric eroding dipole
without swirl. The hairpin is here similarly proposed as a model to produce
large "self-stretching" of vorticity, with the possibility of finite-time
blow-up. We derive a system of partial differential equations of "generalized"
form, involving contour averaging of a locally two-dimensional Euler flow. We
do not attempt here to solve the system exactly, but point out that
non-existence of physically acceptable solutions would most probably be a
result of the axial flow. Because of the axial flow the vorticity distribution
within the dipole eddies is no longer of the simple Sadovskii type (vorticity
constant over a cross-section) obtained in the axisymmetric problem. Thus the
solution of the system depends upon the existence of a larger class of
propagating two-dimensional dipoles.
  The hairpin model is obtained by formal asymptotic analysis. As in the
axisymmetric problem a local transformation to "shrinking" coordinates is
introduced, but now in a self-similar form appropriate to the study of a
possible finite-time singularity. We discuss some properties of the model,
including a study of the helicity and a first step in iterating toward a
solution from the Sadovskii structure. We also present examples of
two-dimensional propagating dipoles not previously studied, which have a
vorticity profile consistent with our model. Although no rigorous results can
be given, and analysis of the system is only partial, the formal calculations
are consistent with the possibility of a finite time blowup of vorticity at a
point of vanishing circulation of the dipole eddies, but depending upon the
existence of the necessary two-dimensional propagating dipole.
