  We present a natural framework for studying the persistence problem in
two-dimensional fluid turbulence by using the Okubo-Weiss parameter $\Lambda$
to distinguish between vortical and extensional regions. We then use a direct
numerical simulation (DNS) of the two-dimensional, incompressible
Navier--Stokes equation with Ekman friction to study probability distribution
functions (PDFs) of the persistence times of vortical and extensional regions
by employing both Eulerian and Lagrangian measurements. We find that, in the
Eulerian case, the persistence-time PDFs have exponential tails; by contrast,
this PDF for Lagrangian particles, in vortical regions, has a power-law tail
with an exponent $\theta=2.9\pm0.2$.
