Presently, much of CO2 fossil-fuel emissions are removed from the atmosphere through natural ocean uptake of CO2. Many schemes have now been proposed by which the accumulation of anthropogenic CO2 in the atmosphere could be slowed with intentional further storage of CO2 in the ocean. Our review of the literature indicates inconsistency in whether ambient ocean carbon uptake is included when accounting for the effectiveness of such schemes. This inconsistency is a consequence of differing choices of atmospheric boundary condition. In the case of one particular form of ocean sequestration, namely direct injection of liquefied CO2 emissions into the ocean interior, this choice is the determination of whether the atmospheric CO2 concentration responsively increases due to leakage of injected carbon from the ocean or retains a specified value. We first show how results of simulations using these two different boundary conditions can be related with the convolution of an atmosphere pulse release. We then use a numerical model to present a more complete analysis of the role of these boundary conditions. Finally, we suggest that a responsive atmospheric CO2 boundary condition is appropriate for predicting future carbon concentrations, but a specified atmospheric CO2 boundary condition is appropriate for evaluating how much CO2 storage should be attributed to an ocean storage project.