On transit-time distributions in unsteady circulation models

In a diffusive geophysical flow, there is not a single timescale or unique pathway for passive scalar transport from the reservoir’s surface into the interior because of irreversible diffusive mixing processes. Instead, there is a range of pathways and hence a transit-time distribution (TTD) since last surface contact. We explore the issues that arise when considering TTDs for unsteady flows and discuss approaches to finding the TTD in numerical general circulation models. In particular, three complementary approaches are possible: First, the forward tracer equation can be used to simulate boundary impulse responses (BIRs). This approach is computationally efficient for the case where information on the TTD is needed at many field points or many field times. Second, the adjoint tracer equation can be used to find the TTD. This method is efficient when the TTD is required at a few field points and field times, but requires an adjoint tracer model. Third, BIR integrations can be used as statistical surrogates of TTDs, exploiting the fact that BIRs and TTDs have identical statistics due to a property of the underlying Green’s function. If an estimate of the ensemble-mean TTD is required, to within an error on the order of the typical fluctuation amplitude, a single realization of the BIR serves as well as a single realization of the TTD. BIR and TTD ensembles give estimates of the ensemble-mean moments of the TTD to the same level of accuracy. Computing ensembles of BIRs instead of ensembles of TTDs is efficient for cases with few surface sources and few field times. Illustrations are presented for barotropic double-gyre circulations.