A brinicle is essentially an underwater icicle or ice stalactite.  Their growth and formation has been observed for the first time as part of the new BBC series Frozen Planet.  This link reveals how this was captured on film.  Here is the amazing timelapse footage of the formation of a brinicle:

For information on brinicle formation and structure read this BBC article and wikipedia entry.  A mathematical treatment of their formation can be found in this academic paper:

Martin, Seelye (1974). “Ice stalactites: comparison of a laminar flow theory with experiment”Journal of Fluid Mechanics 63 (1): 51-79. doi:10.1017/S0022112074001017