Ocean Dynamics

, Volume 56, Issue 2, pp 86–103 | Cite as

Comparison of free-surface and rigid-lid finite element models of barotropic instabilities

  • Laurent WhiteEmail author
  • Jean-Marie Beckers
  • Eric Deleersnijder
  • Vincent Legat
Original paper


The main goal of this work is to appraise the finite element method in the way it represents barotropic instabilities. To that end, three different formulations are employed. The free-surface formulation solves the primitive shallow-water equations and is of predominant use for ocean modeling. The vorticity–stream function and velocity–pressure formulations resort to the rigid-lid approximation and are presented because theoretical results are based on the same approximation. The growth rates for all three formulations are compared for hyperbolic tangent and piecewise linear shear flows. Structured and unstructured meshes are utilized. The investigation is also extended to time scales that allow for instability meanders to unfold, permitting the formation of eddies. We find that all three finite element formulations accurately represent barotropic instablities. In particular, convergence of growth rates toward theoretical ones is observed in all cases. It is also shown that the use of unstructured meshes allows for decreasing the computational cost while achieving greater accuracy. Overall, we find that the finite element method for free-surface models is effective at representing barotropic instabilities when it is combined with an appropriate advection scheme and, most importantly, adapted meshes.


Finite element method Unstructured meshes Barotropic instabilities Free-surface flow 



Laurent White is a research fellow and Eric Deleersnijder is a research associate with the Belgian National Fund for Scientific Research (FNRS). 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 Française de Belgique, as Actions de Recherche Concertées, under contract ARC 04/09-316. This work is a contribution to the construction of SLIM, the Second-Generation Louvain-la-Neuve Ice-ocean Model ( The authors are indebted to Benoit Cushman-Roisin for the comments he provided during the first stages of the preparation of this paper, and they would also like to thank the two anonymous reviewers for useful suggestions that helped improve this manuscript.


  1. Beckers J-M, Deleersnijder E (1993) Stability of a FBTCS scheme applied to the propagation of shallow-water intertia–gravity waves on various space grids. J Comput Phys 108(1):95–104CrossRefGoogle Scholar
  2. Cockburn B, Karniadakis GE, Shu CW (eds) (2000) Discontinuous Galerkin methods. Theory, computation and applications. In: Lectures notes in computational science and engineering, vol 11. Springer, Berlin Heidelberg New YorkGoogle Scholar
  3. Cushman-Roisin B (1994) Geophysical fluid dynamics. Prentice-Hall, Upper Saddle River, NJGoogle Scholar
  4. Danilov S, Kivman G, Schröter J (2004) A finite-element ocean model: principles and evaluation. Ocean Model 6:125–150CrossRefGoogle Scholar
  5. Dickinson RE, Clare FJ (1973) Numerical study of the unstable modes of a hyperbolic tangent barotropic shear flow. J Atmos Sci 30:1035–1049CrossRefGoogle Scholar
  6. Gresho PM, Chan ST, Lee RL, Upson CD (1984) A modified finite element method for solving the time-dependent, incompressible Navier–Stoked equations. Part 1: theory. Int J Numer Methods Fluids 4:557–598CrossRefGoogle Scholar
  7. Gresho PM, Sani RL (1987) On pressure boundary conditions for the incompressible Navier–Stokes equations. Int J Numer Methods Fluids 7:1111–1145CrossRefGoogle Scholar
  8. Gresho PM, Sani RL (1998) Incompressible flow and the finite element method. Wiley, New YorkGoogle Scholar
  9. Griffies SM, Böning C, Bryan FO, Chassignet EP, Gerdes R, Hasumi H, Hirst A, Treguier A-M, Webb D (2000) Developments in ocean climate modelling. Ocean Model 2:123–192CrossRefGoogle Scholar
  10. Hanert E, Legat V, Deleersnijder E (2003) A comparison of three finite elements to solve the linear shallow water equations. Ocean Model 5:39–58Google Scholar
  11. Hanert E, Le Roux DY, Legat V, Deleersnijder E (2004) Advection schemes for unstructured grid ocean modelling. Ocean Model 7:39–58CrossRefGoogle Scholar
  12. Hanert E, Le Roux DY, Legat V, Deleersnijder E (2005) An efficient Eulerian finite element method for the shallow water equations. Ocean Model 10:115–136CrossRefGoogle Scholar
  13. Howard LN (1964) The number of unstable modes in hydrodynamic stability problems. J Mec 3:433–443Google Scholar
  14. Hart JE (1974) On the mixed stability problem for quasi-geostrophic ocean currents. J Phys Oceanogr 4:349–356CrossRefGoogle Scholar
  15. Hua B, Thomasset F (1984) A noise-free finite element scheme for the two-layer shallow water equations. Tellus 36A:157–165Google Scholar
  16. Killworth PD (1980) Barotropic and baroclinic instability in rotating stratified fluids. Dyn Atmos Ocean 4:143–184CrossRefGoogle Scholar
  17. Killworth PD, Stainforth D, Webb DJ, Paterson SM (1991) The development of a free-surface Bryan–Cox–Semtner ocean model. J Phys Oceanogr 21:1333–1348CrossRefGoogle Scholar
  18. Kuo HL (1949) Dynamic instability of two-dimensional non-divergent flow in a barotropic atmosphere. J Meteorol 6:105–122Google Scholar
  19. Kuo HL (1973) Dynamics of quasigeostrophic flows and instability theory. Adv Appl Mech 13:247–330Google Scholar
  20. Kuo HL (1978) A two-layer model study of the combined barotropic and baroclinic instability in the tropics. J Atmos Sci 35:1840–1860CrossRefGoogle Scholar
  21. Le Roux DY, Staniforth AN, Lin CA (1998) Finite elements for shallow-water equation ocean models. Mon Weather Rev 126(7):1931–1951CrossRefGoogle Scholar
  22. Michalke A (1964) On the inviscid instability of the hyperbolic tangent velocity profile. J Fluid Mech 19:543–556CrossRefGoogle Scholar
  23. Nechaev D, Schröter J, Yaremchuk M (2003) A diagnostic stabilized finite-element ocean circulation model. Ocean Model 5:37–63CrossRefGoogle Scholar
  24. Pain CC, Piggott MD, Goddard AJH, Fang F, Gorman GJ, Marshall DP, Eaton MD, Power PW, de Oliveira CRE (2004) Three-dimensional unstructured mesh ocean modelling. Ocean Model 10:5–33CrossRefGoogle Scholar
  25. Pedlosky J (1964) The stability of currents in the atmosphere and the ocean: part I. J Atmos Sci 2:201–219CrossRefGoogle Scholar
  26. Pedlosky J (1979) Geophysical fluid dynamics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  27. Pietrzak J, Deleersijder E, Schröter J (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 10:1–3 (Editorial in a special issue)CrossRefGoogle Scholar
  28. Schwanenberg D, Kongeter J (2000) A discontinuous Galerkin method for the shallow water equations with source terms. IEEE Comput Sci Eng 11:419–424Google Scholar
  29. White L, Legat V, Deleersnijder E, Le Roux D (2006) A one-dimensional benchmark for the propagation of Poincaré waves. Ocean model, in pressGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Laurent White
    • 1
    • 2
    Email author
  • Jean-Marie Beckers
    • 3
  • Eric Deleersnijder
    • 1
    • 2
  • Vincent Legat
    • 1
  1. 1.Centre for Systems Engineering and Applied Mechanics (CESAME)Université catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.G. Lemaître Institute of Astronomy and Geophysics (ASTR)Université catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Geohydrodynamics and Environmental ResearchUniversité de Liège,LiègeBelgium

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