  We observe, at low temperature, the appearance of propagating events that
originate from the inter-conversion between adjacent four and five coordinated
water molecules in supercooled liquid water, resulting in the migration of the
coordination number five in a string-like fashion, creating rotational jumps
along the way. The length of the connected events increases with lowering
temperature. Each propagation event is terminated by a 3-coordinated species,
present in a small number at large supercooling, which interacts cooperatively
with 5-coordinated molecules to annihilate both the species. We find that these
growing dynamical correlations manifest in a divergent-like growth of a
non-linear density response function, \times4(t), which is given by a four
point time correlation function (FPTCF). The locus of the maximum of
\times4(t), when plotted against the time t* of maximum, exhibits a sharp peak
precisely at the temperature where the static response functions (specific
heat, isothermal compressibility) also show similar sharp, divergent-like,
peak. While the decay of population fluctuation time correlation function of
both 4- and 5- coordinated species slows down dramatically and a step-like
feature of the relaxation becomes evident, the lifetime itself of 5-corrdinated
species remains short. These results suggest a new molecular mechanism of low
temperature anomalies and of the liquid-liquid transition in terms of
initiation, growth and termination of propagating jump and inter-conversion
events.
