  We use $^2$H NMR to study the rotational motion of supercooled water in
silica pores of various diameters, specifically, in the MCM-41 materials C10,
C12, and C14. Combination of spin-lattice relaxation, line-shape, and
stimulated-echo analyses allows us to determine correlation times in very broad
time and temperature ranges. For the studied pore diameters, 2.1-2.9 nm, we
find two crossovers in the temperature-dependent correlation times of liquid
water upon cooling. At 220-230 K, a first kink in the temperature dependence is
accompanied by a solidification of a fraction of the confined water, implying
that the observed crossover is due to a change from bulk-like to
interface-dominated water dynamics, rather than to a liquid-liquid phase
transition. Moreover, the results provide evidence that $\alpha$ process-like
dynamics is probed above the crossover temperature, whereas $\beta$
process-like dynamics is observed below. At 180-190 K, we find a second change
of the temperature dependence, which resembles that reported for the $\beta$
process of supercooled liquids during the glass transition, suggesting a value
of $T_g\!\approx\!185$ K for interface-affected liquid water. In the
high-temperature range, $T\!>\!225$ K, the temperature dependence of water
reorientation is weaker in the smaller C10 pores than in the larger C12 and C14
pores, where it is more bulk-like, indicating a significant effect of the
silica confinement on the $\alpha$ process of water in the former 2.1 nm
confinement. By contrast, the temperature dependence of water reorientation is
largely independent of the confinement size and described by an Arrhenius law
with an activation energy of $E_a\!\approx\!0.5\ $eV in the low-temperature
range, $T\!<\!180 $K, revealing that the confinement size plays a minor role
for the $\beta$ process of water.
