Interscale transfer in two-dimensional compact vortices

The property of transfer between different scales of motion in evolving two -dimensional compact vortices is studied here, and a general mathematical f ramework is developed to describe the transfer between scales inside compac t structures. This new approach is applied to the case of an axisymmetric a dvection which represents the leading-order (large time) approximation for Lundgren's family of two-dimensional vortices, It is also generalized to pa ssive scalar advection by non-axisymmetric velocity fields, It is shown tha t scale interactions generated by an axisymmetric advection are essentially local and dominated by distant triadic interactions: in the case of an evo lving spiral vortex sheet this result is confirmed even when non-axisymmetr ic corrections are included. A physical interpretation of the results is gi ven, which can be summarized by saying that locality of scale interactions is caused by the uniformity of shear at a given scale and is therefore incr easingly natural at small lengthscales. Local interactions are shown to ari se in axisymmetric advection but to be uncommon in non-axisymmetric advecti on.