  We bring new results in the study the asymptotic behavior of shrinking vortex
pairs obtained by maximization of the kinetic energy in a 2-dimensional lake
over a class of rearrangements. After improving recent results obtained for the
first order asymptotic behavior of such pairs, we focus on second order
asymptotic properties. We show that among all points of maximal depth, the
vortex locates according to an adaptation of the Kirchoff-Routh function, and
we study the asymptotic shape of optimal vortices. We also explore a relaxed
maximization problem with uniform constraints, for which we prove that the
distribution consists of two vortex patches.
