  The heat capacity of supercooled water, measured down to -37 {\deg}C, shows
an anomalous increase as temperature decreases. The thermal diffusivity, i. e.,
the ratio of the thermal conductivity and the heat capacity per unit volume,
shows a decrease. These anomalies may be associated with a hypothetical
liquid-liquid critical point in supercooled water below the line of homogeneous
nucleation. However, while the thermal conductivity is known to diverge at the
vapor-liquid critical point due to critical density fluctuations, the thermal
conductivity of supercooled water, calculated as the product of thermal
diffusivity and heat capacity, does not show any sign of such an anomaly. We
have used mode-coupling theory to investigate the possible effect of critical
fluctuations on the thermal conductivity of supercooled water, and found that
indeed any critical thermal-conductivity enhancement would be too small to be
measurable at experimentally accessible temperatures. Moreover, the behavior of
thermal conductivity can be explained by the observed anomalies of the
thermodynamic properties. In particular, we show that thermal conductivity
should go through a minimum as temperature is decreased, as Kumar and Stanley
observed in the TIP5P model of water. We discuss physical reasons for the
striking difference between the behavior of thermal conductivity in water near
the vapor-liquid and liquid-liquid critical points.
