  Rotating spiral waves with a central core composed of phase-randomized
oscillators can arise in reaction-diffusion systems if some of the chemical
components involved are diffusion-free. This peculiar phenomenon is
demonstrated for a paradigmatic three-component reaction-diffusion model. The
origin of this anomalous spiral dynamics is the effective non-locality in
coupling, whose effect is stronger for weaker coupling. There exists a critical
coupling strength which is estimated from a simple argument. Detailed
mathematical and numerical analyses are carried out in the extreme case of weak
coupling for which the phase reduction method is applicable. Under the
assumption that the mean field pattern keeps to rotate steadily as a result of
a statistical cancellation of the incoherence, we derive a functional
self-consistency equation to be satisfied by this space-time dependent
quantity. Its solution and the resulting effective frequencies of the
individual oscillators are found to agree excellently with the numerical
simulation.
