# Residence and exposure times : when diffusion does not matter

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## Abstract

Under constant hydrodynamic conditions and assuming horizontal homogeneity, negatively buoyant particles released at the surface of the water column have a mean residence time in the surface mixed layer of *h*/*w*, where *h* is the thickness of the latter and *w* ( > 0) is the sinking velocity Deleersnijder (Environ Fluid Mech 6(6):541–547, 2006a). The residence time does not depend on the diffusivity and equals the settling timescale. We show that this behavior is a result of the particular boundary conditions of the problem and that it is related to a similar property of the exposure time in a one-dimensional infinite domain. In 1-D advection–diffusion problem with a constant and uniform velocity, the exposure time—which is a generalization of the residence time measuring the total time spent by a particle in a control domain allowing the particle to leave and reenter the control domain—is also equal to the advection timescale at the upstream boundary of the control domain. To explain this result, the concept of point exposure is introduced; the point exposure is the time integral of the concentration at a given location. It measures the integrated influence of a point release at a given location and is related to the concept of number of visits of the theory of random walks. We show that the point exposure takes a constant value downstream the point of release, even when the diffusivity varies in space. The analysis of this result reveals also that the integrated downstream transport of a passive tracer is only effected by advection. While the diffusion flux differs from zero at all times, its integrated value is strictly zero.

## Keywords

Advection–diffusion Residence time Exposure time CART## Notes

### Acknowledgements

EJMD and ED are honorary research associates with the National Fund for Scientific Research (Belgium). This work was supported by the Interuniversity Attraction Poles Programme TIMOTHY-P6/13 (Belgian Science Policy). This paper is MARE publication n°232.

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