# The boundary layer of the residence time field

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## Abstract

The residence time of a tracer in a control domain is usually computed by releasing tracer parcels and registering the time when each of these tracer parcels cross the boundary of the control domain. In this Lagrangian procedure, the particles are discarded or omitted as soon as they leave the control domain. In a Eulerian approach, the same approach can be implemented by integrating forward in time the advection–diffusion equation for a tracer. So far, the conditions to be applied at the boundary of the control domain were uncertain. We show here that it is necessary to prescribe that the tracer concentration vanishes at the boundary of the control domain to ensure the compatibility between the Lagrangian and Eulerian approaches. When we use the Constituent oriented Age and Residence time Theory (CART), this amounts to solving the differential equation for the residence time with boundary conditions forcing the residence time to vanish at the open boundaries of the control domain. Such boundary conditions are likely to induce the development of boundary layers (at outflow boundaries for the tracer concentration and at inflow boundaries for the residence time). The thickness of these boundary layers is of the order of the ratio of the diffusivity to the velocity. They can however be partly smoothed by tidal and other oscillating flows.

## Keywords

Residence time Advection–diffusion Diagnostic Adjoint modelling Boundary layer## Notes

### Acknowledgments

E. Deleersnijder is a Research Associate with the Belgian National Fund for Scientific Research (F.N.R.S.). The authors thank Philippe Ruelle for pointing out very useful references. This paper is MARE publication n° 77. Part of this work was supported by the French Community of Belgium (RACE, ARC-05/10-333).

## References

- Arneborg L (2004) Turnover times for the water above sill level in Gullmar Fjord. Cont Shelf Res 24:443–460CrossRefGoogle Scholar
- Bolin B, Rodhe H (1973) A note on the concepts of age distribution and residence time in natural reservoirs. Tellus 25:58–62Google Scholar
- Braunschweig F, Martins F, Chambel P, Neves R (2003) A methodology to estimate renewal time scales in estuaries: the Tagus Estuary case. Ocean Dyn 53(3):137–145CrossRefGoogle Scholar
- Deleersnijder E, Tartinville B, Rancher J (1997) A simple model of the tracer flux from the Muruoa lagoon to the Pacific. Appl Math Lett 10(5):13–17CrossRefGoogle Scholar
- Deleersnijder E, Campin J-M, Delhez EJM (2001) The concept of age in marine modelling: I. Theory and preliminary model results. J Mar Syst 28:229–267CrossRefGoogle Scholar
- Delhez EJM (2006) Transient residence and exposure times. Ocean Sci 2:1–9. (SRef-ID: 1812-0792/os/2006-2-1)Google Scholar
- Delhez EJM, Campin J-M, Hirst AC, Deleersnijder E (1999) Toward a general theory of the age in ocean modelling. Ocean Model 1:17–27CrossRefGoogle Scholar
- Delhez EJM, Heemink AW, Deleersnijder E (2004) Residence time in a semi-enclosed domain from the solution of an adjoint problem. Estuar Coast Shelf Sci 61:691–702CrossRefGoogle Scholar
- Dyer KR (1998) Estuaries: a physical introduction, 2nd edn. Wiley, New York, p 210Google Scholar
- Holzer M, Hall TM (2000) Transit-time and tracer-age distributions in geophysical flows. J Atmos Sci 57(21):3539–3558CrossRefGoogle Scholar
- Hunter JR, Craig PD, Phillips HE (1993) On the use of random walks models with spatially variable diffusivity. J Comput Phys 106:366–376Google Scholar
- Jay DA, Geyer WR, Montgomery DR (2000) An ecological perspective on estuarine classification. In: Hobbie JE (ed) Estuarine science—a synthetic approach to research and practice. Island, Washington DC, pp 149–175Google Scholar
- Monsen NE, Cloern JE, Lucas LV (2002) A comment on the use of flushing time, residence time and age as transport time scales. Limnol Oceanogr 47(5):1545–1553Google Scholar
- Pietrzak J, Deleersnijder E, Schroeter J (eds) (2005) The second international workshop on unstructured mesh numerical modelling of coastal, shelf and ocean flows (Delft, The Netherlands, September 23–25, 2003). Ocean Model 10:1–252 (Special Issue)Google Scholar
- Soetaert K, Herman PMJ (1996) Estimating estuarine residence times in the Westerschelde (The Netherlands) using a box model with fixed dispersion coefficients. Hydrobiologia 311:215–224CrossRefGoogle Scholar
- Spitzer F (1976) Principles of random walk. Graduate texts in mathematics, vol. 34. Springer, Berlin Heidelberg New York, p 408Google Scholar
- Takeoka H (1984) Fundamental concepts of exchange and transport time scales in a coastal sea. Cont Shelf Res 3:311–326CrossRefGoogle Scholar
- Tartinville B, Deleersnijder E, Rancher J (1997) The water residence time in the Mururoa atoll lagoon: a three-dimensional model sensitivity analysis. Coral Reefs 16:139–203CrossRefGoogle Scholar
- Visser AW (1997) Using random walk models to simulate the vertical distribution of particles in a turbulent water column. Mar Ecol Prog Ser 158(1):275–281CrossRefGoogle Scholar
- Zimmerman JTF (1976) Mixing and flushing of tidal embrayments in the Western Dutch Wadden Sea, Part I: distribution of salinity and calculation of mixing time scales. Neth J Sea Res 10:149–191CrossRefGoogle Scholar
- Zimmerman JTF (1988) Estuarine residence times. In: Kjerfve B (ed) Hydrodynamics of estuaries, vol 1. CRC Press, Boca Raton, FL, pp 75–84Google Scholar