  Motivated by interest in the geometry of high intensity events of turbulent
flows, we examine spatial correlation functions of sets where turbulent events
are particularly intense. These sets are defined using indicator functions on
excursion and iso-value sets. Their geometric scaling properties are analyzed
by examining possible power-law decay of their radial correlation function. We
apply the analysis to enstrophy, dissipation, and velocity gradient invariants
$Q$ and $R$ and their joint spatial distibutions, using data from a direct
numerical simulation of isotropic turbulence at ${\rm Re}_\lambda \approx 430$.
While no fractal scaling is found in the inertial range using box-counting in
the finite Reynolds number flow considered here, power-law scaling in the
inertial range is found in the radial correlation functions. Thus a geometric
characterization in terms of these sets' correlation dimension is possible.
Strong dependence on the enstrophy and dissipation threshold is found,
consistent with multifractal behavior. Nevertheless the lack of scaling of the
box-counting analysis precludes direct quantitative comparisons with earlier
work based on the multifractal formalism. Surprising trends, such as a lower
correlation dimension for strong dissipation events compared to strong
enstrophy events, are observed and interpreted in terms of spatial coherence of
vortices in the flow. We show that sets defined by joint conditions on strain
and enstrophy, and on $Q$ and $R$, also display power law scaling of
correlation functions, providing further characterization of the complex
spatial structure of these intersection sets.
