Centered-in-space, centered-in-time integration has generally been used for the advection of scalars in ocean models. An assessment is made of the implications of centered leapfrog integration in the context of two-dimensional passive tracer advection within a Stommel (1948) gyre. Nonphysical ripples in the tracer field grow to alarming levels in purely advective integrations. Diffusive parameterizations of eddy mixing moderate these ripples, but it is found that Laplacian diffusion greatly reduces the peak amplitude of the tracer field, while biharmonic or weaker Laplacian diffusion allows ripples of large amplitude. Several forward-in-time, upwind-weighted schemes are found to provide better solutions. Smolarkiewicz's (1984) Multi-Dimensional Positive-Definite Advection and Transport Algorithm (MPDATA) scheme is slightly superior for an integration at moderate resolution within which the western boundary current is poorly resolved in typical fashion. Third-order, upwind-based schemes exhibit little sensitivity to the details of multidimensional treatment for this problem of passive tracer advection, with results nearly as good as for MPDATA.