A two-dimensional numerical simulation is used to investigate the circulation of brine in the thawed porous bottom sediments which overlie ice-bearing permafrost on the continental shelf of the Artic Ocean. The brine motion is assumed to be governed by Darcy's Law and is effective in transporting salt to the ice-bearing sediments and therefore effective in determining their thaw rate. The simulations are initiated by assuming that the initial brine concentration in the thawed sediments matches the average annual salinity of the overlying sea water. A diffusive layer along the phase boundary develops at the base of the thawed sediments and thickens to the point where it becomes convectively unstable. The system is followed in time to observed the development of a stable convective flow pattern. Simulations were run at Rayleigh numbers of 1750 and 17,500. In the 1750 Rayleigh number run the boundary layer and convective plumes had characteristic thicknesses of about 2 meters in a 20-m thick thaw layer. The high Rayleigh number run had thicknesses of only a few tenths of a meter. The high Rayleigh number run is more consistent with the observations, but the results are at sufficient variance with the oservations to suggest there is some not yet identified process affecting the transport of brine to the phase boundary.