Ocean Dynamics

, Volume 60, Issue 3, pp 535–554 | Cite as

Capturing the residence time boundary layer—application to the Scheldt Estuary

  • Sébastien BlaiseEmail author
  • Benjamin de Brye
  • Anouk de Brauwere
  • Eric Deleersnijder
  • Eric J. M. Delhez
  • Richard Comblen


At high Peclet number, the residence time exhibits a boundary layer adjacent to incoming open boundaries. In a Eulerian model, not resolving this boundary layer can generate spurious oscillations that can propagate into the area of interest. However, resolving this boundary layer would require an unacceptably high spatial resolution. Therefore, alternative methods are needed in which no grid refinement is required to capture the key aspects of the physics of the residence time boundary layer. An extended finite element method representation and a boundary layer parameterisation are presented and tested herein. It is also explained how to preserve local consistency in reversed time simulations so as to avoid the generation of spurious residence time extrema. Finally, the boundary layer parameterisation is applied to the computation of the residence time in the Scheldt Estuary (Belgium/The Netherlands). This timescale is simulated by means of a depth-integrated, finite element, unstructured mesh model, with a high space–time resolution. It is seen that the residence time temporal variations are mainly affected by the semi-diurnal tides. However, the spring–neap variability also impacts the residence time, particularly in the sandbank and shallow areas. Seasonal variability is also observed, which is induced by the fluctuations over the year of the upstream flows. In general, the residence time is an increasing function of the distance to the mouth of the estuary. However, smaller-scale fluctuations are also present: they are caused by local bathymetric features and their impact on the hydrodynamics.


Residence time Boundary layer Parameterisation X-FEM Finite elements Diagnostic Adjoint modelling Local consistency 



Sébastien Blaise is a Research fellow with the Belgian Fund for Research in Industry and Agriculture (FRIA). Richard Comblen and Eric Deleersnijder are, respectively, a Research fellow and a Research associate with the Belgian National Fund for Scientific Research (FNRS). Eric Delhez is an honorary Research Associate with the Belgian National Fund for Scientific Research. Anouk de Brauwere is a postdoctoral researcher at the Research Foundation Flanders (FWO-Vlaanderen). The present study was carried out within the scope of the project “A second-generation model of the ocean system”, which is funded by the Communauté Francaise de Belgique, as Actions de Recherche Concertées, under contract ARC 04/09-316, and the project “Tracing and Integrated Modelling of Natural and Anthropogenic Effects on Hydrosystems” (TIMOTHY), an Interuniversity Attraction Pole (IAP6.13) funded by the Belgian Federal Science Policy Office (BELSPO). This work is a contribution to the development of SLIM, the Second-generation Louvain-la-Neuve Ice-ocean Model ( The authors are indebted to Emmanuel Hanert and Olivier Gourgue for their useful comments.


  1. Alexe M, Sandu A (2009) Forward and adjoint sensitivity analysis with continuous explicit Runge–Kutta schemes. Appl Math Comput 208(2):328–346CrossRefGoogle Scholar
  2. Allen CM (1982) Numerical simulation of contaminant dispersion in estuary flows. In: Royal society of London proceedings series A, vol 381, pp 179–194Google Scholar
  3. Arega F, Armstrong S, Badr A (2008) Modeling of residence time in the East Scott Creek Estuary, South Carolina, USA. Journal of Hydro-environment Research 2(2):99–108CrossRefGoogle Scholar
  4. Arminjon P, Dervieux A (1993) Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J Comput Phys 106(1):176–198CrossRefGoogle Scholar
  5. Black KP, Gay SL (1987) Eddy formation in unsteady flows. J Geophys Res 92(C9):9514–9522CrossRefGoogle Scholar
  6. Blumberg AF, Mellor GL (1987) A description of three-dimensional coastal ocean circulation model. In: Heaps NS (ed) Three dimensional coastal ocean model. American Geophysical Union, Washington, DC, pp 1–16Google Scholar
  7. Bolin B, Rhode H (1973) A note on the concepts of age distribution and residence time in natural reservoirs. Tellus 25:58–62CrossRefGoogle Scholar
  8. Burchard H (2002) Applied turbulence modelling in marine waters. Lecture notes in earth science, vol 100. Springer, HeidelbergGoogle Scholar
  9. Cockburn B, Shu C-W (1998) The Runge–Kutta discontinuous Galerkin method for conservation laws V-multidimensional systems. J Comput Phys 141:191–224CrossRefGoogle Scholar
  10. Combescure A, Gravouil A, Baietto-Dubourg M-C, Elguedj E, Ribeaucourt R, Ferri E (2005) Extended finite element method for numerical simulation of 3d fatigue crack growth. In: Dowson D, Priest M, Dalmaz G, Lubrecht AA (eds) Life cycle tribology—proceedings of the 31st Leeds-Lyon symposium on tribology held at Trinity and All Saints College, Horsforth, Leeds, UK 7th–10th September 2004. Tribology and interface engineering series, vol 48. Elsevier, Amsterdam, pp 323–328Google Scholar
  11. Comblen R, Lambrechts J, Remacle J-F, Legat V (2009) Practical evaluation of five part-discontinuous finite element pairs for the non-conservative shallow water equations. Int J Numer Methods Fluids. doi: 10.1002/fld.2094 Google Scholar
  12. de Brye B, de Brauwere A, Gourgue O, Kärna T, Lambrechts J, Comblen R, Deleersnijder E (2009) A finite-element, multi-scale model of the Scheldt Tributaries, River, Estuary and ROFI. Coast Eng (under revision)Google Scholar
  13. Deleersnijder E (1993) Numerical mass conservation in a free-surface sigma coordinate marine model with mode splitting. J Mar Syst 4:365–370CrossRefGoogle Scholar
  14. 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(3–4):229–267CrossRefGoogle Scholar
  15. Deleersnijder E, Delhez EJ (eds) (2007) Timescale- and tracer-based methods for understanding the results of complex marine models. Estuar Coast Shelf Sci 74:585–776 (special issue)CrossRefGoogle Scholar
  16. Delhez EJM (2006) Transient residence and exposure times. Ocean Sci 2(1):1–9CrossRefGoogle Scholar
  17. Delhez EJM, Deleersnijder E (2006) The boundary layer of the residence time. Ocean Dyn 56:139–150CrossRefGoogle Scholar
  18. Delhez EJ, Lacroix G, Deleersnijder E (2004a) The age as a diagnostic of the dynamics of marine ecosystem models. Ocean Dyn 54(2):221–231CrossRefGoogle Scholar
  19. Delhez EJM, Heemink AW, Deleersnijder E (2004b) Residence time in a semi-enclosed domain from the solution of an adjoint problem. Estuar Coast Shelf Sci 61:691–702CrossRefGoogle Scholar
  20. Egbert GD, Benett AF, Foreman MGG (1994) TOPEX/ POSEIDON tides estimated using a global inverse model. J Geophys Res 99:24,821–24,852CrossRefGoogle Scholar
  21. Elden L (1982) Time discretization in the backward solution of parabolic equations. II. Math Comput 39(159):69–84CrossRefGoogle Scholar
  22. Garabedian PR (1964) Partial differential equations. Wiley, New YorkGoogle Scholar
  23. 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 74:764–776CrossRefGoogle Scholar
  24. Hanert E, Deleersnijder E, Blaise S, Remacle J-F (2007) Capturing the bottom boundary layer in finite element ocean models. Ocean Model 17:153–162CrossRefGoogle Scholar
  25. Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Leetmaa A, Reynolds B, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo KC, Ropelewski C, Wang J, Jenne R, Joseph D (1996) The NCEP/NCAR 40-year reanalysis project. Bull Am Meteorol Soc 77:437–472CrossRefGoogle Scholar
  26. Kuzmin D, Löhner R, Turek S (eds) (2005) Flux-corrected transport. Principles, algorithms and applications. Springer, HeidelbergGoogle Scholar
  27. Liu C-S, Chang C-W, Chang J-R (2008a) The backward group preserving scheme for 1d backward in time advection–dispersion equation. Numer Methods Partial Differ Equ 26(1):61–80CrossRefGoogle Scholar
  28. Liu W-C, Chen W-B, Kuo J-T, Wu C (2008b) Numerical determination of residence time and age in a partially mixed estuary using three-dimensional hydrodynamic model. Cont Shelf Res 28(8):1068–1088CrossRefGoogle Scholar
  29. Luther KH, Haitjema HM (1998) Numerical experiments on the residence time distributions of heterogeneous groundwatersheds. J Hydrol 207(1–2):1–17CrossRefGoogle Scholar
  30. Meyers SD, Luther ME (2008) A numerical simulation of residual circulation in Tampa Bay. Part II: Lagrangian residence time. Estuar Coast 31:815–827CrossRefGoogle Scholar
  31. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefGoogle Scholar
  32. Monsen NE, Cloern JE, Lucas LV, Monismith SG (2002) A comment on the use of flushing time, residence time, and age as transport time scales. Limnol Oceanogr 47:1545–1553CrossRefGoogle Scholar
  33. Nauman EB (1981) Residence time distributions in systems governed by the dispersion equation. Chem Eng Sci 36(6):957–966CrossRefGoogle Scholar
  34. Pawlowicz R, Beardsley B, Lentz S (2002) Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE. Comput Geosci 28:929–937CrossRefGoogle Scholar
  35. Payne LE (1975) Improperly posed problems in partial differential equations. In: Regional conference series in applied mathematics. Society for Industrial and Applied MathematicsGoogle Scholar
  36. Soetaert K, Herman PMJ (1995) Estimating estuarine residence times in the Westerschelde (The Netherlands) using a box model with fixed dispersion coefficients. Hydrobiologia 311:215–224CrossRefGoogle Scholar
  37. Spivakovskaya D, Hemink AW, Deleersnijder E (2007) Lagrangian modelling of multi-dimensional advection–diffusion with space-varying diffusivities: theory and idealized test cases. Ocean Dyn 57:189–203CrossRefGoogle Scholar
  38. Steen RJ, Evers EH, Hattum BV, Cofino WP, Brinkman TUA (2002) Net fluxes of pesticides from the Scheldt Estuary into the North Sea: a model approach. Environ Pollut 116(1):75–84CrossRefGoogle Scholar
  39. Takeoka H (1984) Fundamental concepts of exchange and transport time scales in a coastal sea. Cont Shelf Res 3(3):311–326CrossRefGoogle Scholar
  40. Tartinville B, Deleersnijder E, Rancher J (1997) The water residence time in the Mururoa atoll lagoon: sensitivity analysis of a three-dimensional model. Coral Reefs 16:193–203CrossRefGoogle Scholar
  41. Thuburn J, Haine TWN (2001) Adjoints of nonoscillatory advection schemes. J Comput Phys 171:616–631CrossRefGoogle Scholar
  42. Wang C-F, Hsu M-H, Kuo AY (2004) Residence time of the Danshuei River estuary, Taiwan. Estuar Coast Shelf Sci 60(3):381–393CrossRefGoogle Scholar
  43. White L, Legat V, Deleersnijder E (2008) Tracer conservation for three-dimensional, finite-element, free-surface, ocean modeling on moving prismatic meshes. Mon Weather Rev 136:420–442CrossRefGoogle Scholar
  44. Wyart E, Duflot M, Coulon D, Martiny P, Pardoen T, Remacle J-F, Lani F (2008) Substructuring FE-XFE approaches applied to three-dimensional crack propagation. J Comput Appl Math 215(2):626–638CrossRefGoogle Scholar
  45. 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

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Sébastien Blaise
    • 1
    Email author
  • Benjamin de Brye
    • 1
  • Anouk de Brauwere
    • 1
    • 2
  • Eric Deleersnijder
    • 1
  • Eric J. M. Delhez
    • 3
  • Richard Comblen
    • 1
  1. 1.Center for Systems Engineering and Applied Mechanics (CESAME)Université Catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Analytical and Environmental ChemistryVrije Universiteit BrusselsBrusselsBelgium
  3. 3.MARE, Modélisation et Méthodes MathématiquesUniversité de LiègeLiègeBelgium

Personalised recommendations