  Turbulence organization, long conceptualized in terms of spatial
coherent-structures, has resisted clear description. A major limitation has
been the lack of tools to identify instantaneous spatial organization, while
unravelling the superposition of structures. To address this, we present a
generalized correlation framework, using: (i) correlation measures to identify
instantaneous vector-field patterns, and (ii) a Helmholtz-decomposition based
structure-disentanglement paradigm. After examples using canonical flows, we
apply these methods to homogeneous isotropic turbulence fields. We show that
high kinetic energy ($E_k$) regions manifest as interspersed, localized,
velocity-jets, contrary to the prevalent view of high $E_k$ regions as large
swirling structures (eddies). High enstrophy ($\omega^2$) regions form small
vorticity-jets, invariably surrounded by swirling-velocity. The jet-like and
swirling-velocity structures are spatially exclusive. Decomposing the
Biot-Savart contributions from different levels and regions of the
vorticity-field reveals the organization of velocity-field structures. High
$E_k$ jets are neither self-induced (due to their low vorticity contents), nor
induced by strong vorticity, being almost entirely induced, non-locally, by the
permeating intermediate range (rms level) vorticity. High $\omega^2$ swirls,
instead, are a superposition of self-induced swirling-velocity along with a
background-induced flow. Moreover, intermediate vorticity dominantly induces
the velocity-field everywhere. This suggests that turbulence organization could
emerge from non-local and non-linear field interactions, dominated by
permeating intermediate vorticity, leading to an alternative description of
turbulence, contrary to the notion of a strict structural hierarchy. The tools
presented can be readily applied to generic vector and scalar fields associated
with diverse phenomena.
