Directional solidification of salt water results in a preferred crystallographic orientation, with ice platelets growing with their short dimension (the c axis of the hexagonal close packed (hcp) ice crystal) transverse to the growth direction. In the absence of a mean saltwater flow parallel to the solidification interface, the ice platelets, and hence the c axes, are arbitrarily oriented in the plane transverse to growth. Here we report on laboratory experiments on salt water solidifying horizontally from a vertical chill plate. This geometry yields a convectively driven, organized fluid flow parallel to the solidification interface, which we found causes the platelets to trend parallel to the flow direction. This contrasts with previous field and laboratory observations of ice solidifying vertically from salt water in the presence of an externally forced flow, such as an ocean current, where the c axes lie parallel to the flow direction. The results support the hypothesis that transverse textures result from a hydrodynamic interaction with hcp dendritic platelets but highlight the difference between flow driven internally versus externally to the dendritic mushy layer. They also suggest that convective fluid flow may be the cause for the fabric observed in resolidified cracks in sea ice sheets and perhaps that fluid motion in the Earth's outer core may play a role in the seismic anisotropy of the hcp iron inner core.