An important issue for the interpretation of data from deep-sea cores is the time for tracers to be transported from the sea surface to the deep ocean. Global ocean circulation models can help shed light on the timescales over which a tracer comes to equilibrium in different regions of the ocean. In this note, we discuss how the most slowly decaying eigenmode of a model can be used to obtain a relevant timescale for a tracer that enters through the sea surface to become well mixed in the ocean interior. We show how this timescale depends critically on the choice between a Neumann surface boundary condition in which the flux of tracer is prescribed, a Robin surface boundary condition in which a combination of the flux and tracer concentration is prescribed or a Dirichlet surface boundary condition in which the concentration is prescribed. Explicit calculations with a 3-box model and a three-dimensional ocean circulation model show that the Dirichlet boundary condition when applied to only part of the surface ocean greatly overestimate the time needed to reach equilibrium. As a result regional- "injection" calculations which prescribe the surface concentration instead of the surface flux are not relevant for interpreting the regional disequilibrium between the Atlantic and Pacific found in paleo-tracer records from deep-sea cores. For tracers that enter the ocean through air-sea gas exchange a prescribed concentration boundary condition can be used to infer relevant timescales if the air-sea gas exchange rate is sufficiently fast, but the boundary condition must be applied over the entire ocean surface and not only to a patch of limited area. For tracers with a slow air-sea exchange rate such as 14C a Robin-type boundary condition is more relevant and for tracers such as δ18O that enter the ocean from melt water, a Neumann boundary condition is presumably more relevant. Our three-dimensional model results based on a steady-state modern circulation suggest that the relative disequilibrium between the deep Atlantic and Pacific is on the order of "only" 1200 years or less for a Neumann boundary condition and does not depend on the size and location of the patch where the tracer is injected.
Primeau, F., & Deleersnijder, E. (2009). On the time to tracer equilibrium in the global ocean. Ocean Science, 5(1), 13–28. https://doi.org/10.5194/os-5-13-2009