  In this study we investigate vortex structures which lead to the maximum
possible growth of palinstrophy in two-dimensional incompressible flows on a
periodic domain. The issue of palinstrophy growth is related to a broader
research program focusing on extreme amplification of vorticity-related
quantities which may signal singularity formation in different flow models.
Such extreme vortex flows are found systematically via numerical solution of
suitable variational optimization problems. We identify several families of
maximizing solutions parameterized by their palinstrophy, palinstrophy and
energy and palinstrophy and enstrophy. Evidence is shown that some of these
families saturate estimates for the instantaneous rate of growth of
palinstrophy obtained using rigorous methods of mathematical analysis, thereby
demonstrating that this analysis is in fact sharp. In the limit of small
palinstrophies the optimal vortex structures are found analytically, whereas
for large palinstrophies they exhibit a self-similar multipolar structure. It
is also shown that the time evolution obtained using the instantaneously
optimal states with fixed energy and palinstrophy as the initial data saturates
the upper bound for the maximum growth of palinstrophy in finite time. Possible
implications of this finding for the questions concerning extreme behavior of
flows are discussed.
