On tracer boundary conditions for geophysical reservoirs: How to find the boundary concentration from a mixed condition

For several reasons, it is important to know the boundary concentration, a Dirichlet boundary condition, of a dynamically passive tracer entering an advective/diffusive flow. For example, the powerful theory of transit time distributions, which gives valuable insight into transport processes, needs this information in order to be applied to the tracer in question. Often, tracers of practical interest have a different kind of boundary condition, however, where a linear combination of the boundary tracer flux and the boundary tracer concentration is known: a Robin, or mixed, boundary condition. Computing the boundary concentration in these cases is therefore an important problem that is addressed here. A type of tracer Green's function called a boundary condition kernel provides the solution. The article explains several generic approaches to finding this function for analytic and discretized cases. Explicit solutions for the boundary condition kernel are given for advective/diffusive flow in a pipe and diffusive transport in a layer with time‐ and space‐varying diffusivity. These examples are applied to chlorofluorocarbon uptake by the ocean, anthropogenic carbon uptake by the ocean, and the statistical properties of sea surface temperature anomalies, but the fundamental ideas apply to tracer uptake by any geophysical reservoir.