  Thermoacoustic instability in turbulent combustion systems emerges from the
complex interplay among the flame, flow, and acoustic subsystems. While the
onset of thermoacoustic instability exhibits global order in system dynamics,
the characteristics of local interactions between subsystems responsible for
this order are not well understood. In this study, we utilize the framework of
synchronization to elucidate the spatiotemporal interactions among heat release
rate fluctuations in the flame, velocity fluctuations in the flow, and acoustic
pressure fluctuations in a turbulent combustor. We examine two forms of
thermoacoustic instability, characterized by phase synchronization and
generalized synchronization of the acoustic pressure and global heat release
rate oscillations. Despite the presence of global synchrony, we uncover a
coexistence of frequency synchrony and desynchrony in the local interaction of
these oscillations within the reaction field. In regions of frequency-locked
oscillations, various phase-locking patterns occur, including phase synchrony
and partial phase synchrony. We discover that the local development of small
pockets of phase synchrony and strong amplitude correlation between these
oscillations is sufficient to trigger global phase synchronization in the
system. Furthermore, as the global dynamics approach generalized
synchronization, these local regions of synchrony expand throughout the
reaction field. Additionally, through coupled analysis of acoustic pressure and
local flow velocity fluctuations, we infer that the spatial region of
flow-acoustic synchrony plays a significant role in governing thermoacoustic
instabilities. Our findings imply that, in turbulent combustors, an intrinsic
local balance between order, partial order, and disorder within the coupled
subsystems sustains the global order during thermoacoustic instability.
