  Recent experiments on liquid water show collective dipole orientation
fluctuations dramatically slower then expected (with relaxation time $>$ 50 ns)
[D. P. Shelton, Phys. Rev. B {\bf 72}, 020201(R) (2005)]. Molecular dynamics
simulations of SPC/E water show large vortex-like structure of dipole field at
ambient conditions surviving over 300 ps [J. Higo at al. PNAS, {\bf 98} 5961
(2001)]. Both results disagree with previous results on water dipoles in
similar conditions, for which autocorrelation times are a few ps. Motivated by
these recent results, we study the water dipole reorientation using molecular
dynamics simulations in bulk SPC/E water for temperatures ranging from ambient
300 K down to the deep supercooled region of the phase diagram at 210 K. First,
we calculate the dipole autocorrelation function and find that our simulations
are well-described by a stretched exponential decay, from which we calculate
the {\it orientational autocorrelation time} $\tau_{a}$. Second, we define a
second characteristic time, namely the time required for the randomization of
molecular dipole orientation, the {\it self-dipole randomization time}
$\tau_{r}$, which is an upper limit on $\tau_{a}$; we find that
$\tau_{r}\approx 5 \tau_{a}$. Third, to check if there are correlated domains
of dipoles in water which have large relaxation times compared to the
individual dipoles, we calculate the randomization time $\tau_{\rm box}$ of the
site-dipole field, the net dipole moment formed by a set of molecules belonging
to a box of edge $L_{\rm box}$. We find that the {\it site-dipole randomization
time} $\tau_{\rm box}\approx 2.5 \tau_{a}$ for $L_{\rm box}\approx 3$\AA, i.e.
it is shorter than the same quantity calculated for the self-dipole. Finally,
we find that the orientational correlation length is short even at low $T$.
