Eroding dipoles and vorticity growth for Euler flows in R3: The hairpin geometry as a model for finite-time blowup

A theory of an eroding ``hairpin'' dipole vortex structure in three dimensions is developed, extending our previous study of an axisymmetric eroding dipole without swirl. The hairpin is proposed as a model of maximal ``self-stretching'' of vorticity. We derive a system of partial differential equations of ``generalized'' form, involving contour averaging of a locally two-dimensional Euler flow. The hairpin is proposed as a structure favouring a rapid stretching of vortex lines, and it is conjectured that an initial condition based upon the hairpin could lead to a a blowup of vorticity in finite time in R<sup>3</sup>. 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 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. Our results also suggest that conservation of kinetic energy as realized in the eroding hairpin excludes a finite time blowup for the corresponding Navier-Stokes model.