We consider several tracer advection schemes in the context of a simple sector configuration of a Bryan-Cox-Semtner type ocean model. The schemes are centered-in-time centered-in-space, donor cell upwind differencing, flux corrected transport (FCT), QUICKEST and MPDATA. While we discuss the transport of temperature and salinity, and apply alternative schemes to this transport within our model, we concentrate primarily on a problem which involves passive tracer advection in which vertical gradients are high and concentrations asymptote to zero near the sea surface, as is the case in biogeochemical ocean modeling. We find that flux correction (available optionally with the MPDATA scheme, as well as the FCT scheme) is required in order to produce acceptable results on this challenging problem. A method of supercycling of passive tracers relative to dynamic variables, whereby the time step for passive tracers is larger that that used for the other prognostic variables of the model, is introduced in order to apply these more expensive flux limited schemes efficiently. We also present a general method for the second-order accurate application of one-dimensional forward-in-time advection schemes in three dimensions.