# Partial ages: diagnosing transport processes by means of multiple clocks

## Abstract

The concept of age is widely used to quantify the transport rate of tracers - or pollutants - in the environment. The age focuses only on the time taken to reach a given location and disregards other aspects of the path followed by the tracer parcel. To keep track of the subregions visited by the tracer parcel along this path, partial ages are defined as the time spent in the different subregions. Partial ages can be computed in an Eulerian framework in much the same way as the usual age by extending the Constituent oriented Age and Residence Time theory (CART, www.climate.be/CART). In addition to the derivation of theoretical results and properties of partial ages, applications to a 1D model with lateral/transient storage, to the 1D advection-diffusion equation and to the diagnosis of the ventilation of the deep ocean are provided. They demonstrate the versatility of the concept of partial age and the potential new insights that can be gained with it.

## Keywords

Age Advection-diffusion Tracer methods## Notes

### Acknowledgments

Éric Deleersnijder and Éric J.M. Delhez are both honorary research associates with the Belgian Fund for Scientific Research (F.R.S.-FNRS).

This work was supported by the Fondation BNP Paribas through the project FATES (FAst Climate Changes, New Tools To Understand And Simulate The Evolution of The Earth System) in the scope of its *Climate Initiative* programme.

## References

- Beckers JM, Delhez E, Deleersnijder E (2001) Some properties of generalized age-distribution equations in fluid dynamics. SIAM J Appl Math 61:1526–1544CrossRefGoogle Scholar
- Bencala K, Walters R (1983) Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model. Water Resour Res 19(3):718–724CrossRefGoogle Scholar
- Bendtsen J, Hansen JL (2013) A model of life cycle, connectivity and population stability of benthic macro-invertebrates in the North Sea/Baltic Sea transition zone. Ecol Model 267:54– 65CrossRefGoogle Scholar
- Bendtsen J, Gustafsson KE, Soderkvist J, Hansen JLS (2009) Ventilation of bottom water in the North Sea-Baltic Sea transition zone. J Mar Syst 75:138–149CrossRefGoogle Scholar
- Bolin B, Rodhe H (1973) A note on the concepts of age distribution and residence time in natural reservoirs. Tellus 25:58– 62CrossRefGoogle Scholar
- Campin J-M, Goosse H (1999) Parameterization of density-driven downsloping flow for a coarse-resolution ocean model in z-coordinate. Tellus 51:421–430CrossRefGoogle Scholar
- Campin J-M, Fichefet T, Duplessy J-C (1999) Problems with using radiocarbon to infer ocean ventilation rates for past and present climates. Earth Planet Sci Lett 16(1):17–24CrossRefGoogle Scholar
- Chen X (2007) A laterally averaged two-dimensional trajectory model for estimating transport time scales in the Alafia River estuary, Florida. Hydrodynamic Control Aquat Ecosyst Process 75(3):358–370Google Scholar
- Cornaton F (2012) Transient water age distributions in environmental flow systems: the time-marching Laplace transform solution technique. Water Resour. Res 48:W03524CrossRefGoogle Scholar
- de Brye B, de Brauwere A, Gourgue O, Delhez EJM, Deleersnijder E (2012) Water renewal timescales in the Scheldt Estuary. J Mar Syst 94:74–86CrossRefGoogle Scholar
- Deleersnijder E (2015) Can CART’s classical age distribution function be derived from partial age distribution function? Working paper available at http://hdl.handle.net/2078.1/162261, p 17
- Deleersnijder E, Campin J-M (1995) On the computation of the barotropic mode of free-surface world ocean model. Ann Geophys 13:675–688CrossRefGoogle Scholar
- Deleersnijder E, Delhez EJM (2004) Symmetry and asymmetry of water ages in a one-dimensional flow. J Mar Syst 48:61– 66CrossRefGoogle 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
- Deleersnijder E, Delhez EJM, Crucifix M, Beckers JM (2001) On the symmetry of the age field of a passive tracer released into a one-dimensional fluid flow by a point source. Bull Soc R Sci Liége 70:5–21Google Scholar
- Deleersnijder E, Mouchet A, Delhez EJM, Beckers JM (2002) Transient behaviour of water ages in the World Ocean. Math Comput Model 36:121–127CrossRefGoogle Scholar
- Deleersnijder E, Mouchet A, de Brauwere A, Delhez EJM, Hanert E (2014) The concept of partial age, a generalisation of the notion of age: theory, idealised illustrations and realistic applications. JONSMOD 2014 Workshop (Brussels, 12-14 May 2014), available on the web at http://hdl.handle.net/2078.1/153807 or https://publicwiki.deltares.nl/display/JONSMOD/Presentations+2014
- Delhez EJM, Deleersnijder E (2002) The concept of age in marine modelling II. Concentration distribution function in the English Channel and the North Sea. J Mar Syst 31:279– 297CrossRefGoogle 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, Lacroix G, Deleersnijder E (2004) The age as a diagnostic of the dynamics of marine ecosystem models. Ocean Dyn 54:221–231CrossRefGoogle Scholar
- DeVries T, Primeau F (2010) An improved method for estimating water-mass ventilation age from radiocarbon data. Earth Planet Sci Lett 295:367–378CrossRefGoogle Scholar
- Du J, Shen J (2015) Decoupling the influence of biological and physical processes on the dissolved oxygen in the Chesapeake Bay. J Geophys Res Oceans 120(1):78–93CrossRefGoogle Scholar
- England MH (1995) The age of water and ventilation timescales in a global ocean model. J Phys Oceanogr 25:2756–2777CrossRefGoogle Scholar
- England MH, Maier-Reimer E (2001) Using chemical tracers to assess ocean models. Rev Geophys 39 (1):29–70CrossRefGoogle Scholar
- Ginn TR (1999) On the distribution of multicomponent mixtures over generalized exposure time in subsurface flow and reactive transport: foundations, and formulations for groundwater age, chemical heterogeneity, and biodegradation. Water Resour Res 35(5):1395–1407CrossRefGoogle Scholar
- Gnanadesikan A, Russell JL, Zeng F (2007) How does ocean ventilation change under global warming? Ocean Sci 3(1):43–53CrossRefGoogle Scholar
- Goode DJ (1996) Direct simulation of groundwater age. Water Resour Res 32:289–296CrossRefGoogle Scholar
- Gourgue O, Deleersnijder E, White L (2007) Toward a generic method for studying water renewal, with application to the epilimnion of Lake Tanganyika. Estuar Coast Shelf Sci 74:628– 640CrossRefGoogle Scholar
- Haine TWN, Hall TM (2002) a generalized transport theory: water-mass composition and age. J Phys Oceanogr 32:1932–1946CrossRefGoogle Scholar
- Hall TM, Haine TWN (2004) Tracer age symmetry in advective-diffusive flows. J Mar Syst 48:51–59CrossRefGoogle Scholar
- Hall TM, Plumb RA (1994) Age as a diagnostic of stratospheric transport. J Geophys Res-Atmos 99:1059–1070CrossRefGoogle Scholar
- Holzer M, Hall TM (2000) Transit-time and tracer-age distributions in geophysical flows. J Atmos Sci 57:3539–3558CrossRefGoogle Scholar
- Khatiwala S, Primeau F, Hall T (2009) Reconstruction of the history of anthropogenic CO
_{2}concentrations in the ocean. Nature 462(7271):346–349CrossRefGoogle Scholar - Kumar A, Dalal D (2010) Analysis of solute transport in rivers with transient storage and lateral inflow: an analytical study. Acta Geophysica 58(6):1094–1114CrossRefGoogle Scholar
- Lewandowski R (1997) Analyse mathématique et océnographie, masson Edition. No. 39 in Recherches en mathématiques appliquées. ParisGoogle Scholar
- Liu Z, Wang H, Guo X, Wang Q, Gao H (2012) The age of Yellow River water in the Bohai Sea. J Geophys Res Oceans 117(C11):C11006CrossRefGoogle Scholar
- Meier HEM (2007) Modeling the pathways and ages of inflowing salt- and freshwater in the Baltic Sea. Estuar Coast Shelf Sci 74:610–627CrossRefGoogle Scholar
- Mercier C, Delhez EJM (2007) Diagnosis of the sediment transport in the Belgian Coastal Zone. Estuar Coast Shelf Sci 74:670–683CrossRefGoogle Scholar
- Munk W (1966) Abyssal recipes. Deep Sea Res Oceanogr Abstr 13:707–730CrossRefGoogle Scholar
- Plus M, Dumas F, Stanisiére J-Y, Maurer D (2009) Hydrodynamic characterization of the Arcachon Bay, using model-derived descriptors. 100 Years of Research within the Bay of Biscay XI International Symposium on Oceanography of the Bay of Biscay 29(8):1008–1013Google Scholar
- Primeau F (2005) Characterizing transport between the surface mixed layer and the ocean interior with a forward and adjoint global ocean transport model. J Phys Oceanogr 35:545–564CrossRefGoogle Scholar
- Ren Y, Lin B, Sun J, Pan S (2014) Predicting water age distribution in the Pearl River Estuary using a three-dimensional model. J Mar Syst 139(0):276–287CrossRefGoogle Scholar
- Rukavicka J (2011) On generalized Dyck paths. Electron J Comb 18(1):P40Google Scholar
- Runkel R, Chapra S (1993) An efficient numerical solution of the transient storage equations for solute transport in small streams. Water Resour Res 29(1):211–215CrossRefGoogle Scholar
- Shaffer G, Sarmiento J (1995) Biogeochemical cycling in the global ocean: 1. a new, analytical model with continuous vertical resolution and high-latitude dynamics. J Geophys Res 100:2659–2672CrossRefGoogle Scholar
- Takeoka H (1984) Fundamental concepts of exchange and transport time scales in a coastal sea. Cont Shelf Res 3:311–326CrossRefGoogle Scholar
- Thiele G, Sarmiento J (1990) Tracer dating and ocean ventilation. J Geophys Res 95(C6):9377–9391CrossRefGoogle Scholar
- Villa M, Lopez-Gutierrez J, Suh K-S, Min B-I, Periañez R (2015) The behaviour of 129i released from nuclear fuel reprocessing factories in the North Atlantic Ocean and transport to the Arctic assessed from numerical modelling. Mar Pollut Bull 90:15– 24CrossRefGoogle Scholar
- Wagner B, Harvey J (1997) Experimental design for estimating parameters of rate-limited mass transfer: analysis of stream tracer studies. Water Resour Res 32(8):2441–2451Google Scholar
- Zimmerman JTF (1976) Mixing and flushing of tidal embayments 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