The scintillation index of plane and spherical light waves as well as of a Gaussian beam, propagating in the clear-water weakly turbulent ocean, is revealed. The results are of utmost importance for underwater optical communications and sensing. An analysis of the threshold between the weak and strong regimes of oceanic turbulence is made, with the accent on the contribution from salinity-induced turbulence. It is found that strong oceanic turbulence can occur at distances as short as several meters, in striking contrast with atmospheric studies for which the typical distances are on the order of a kilometer.