82
denotes the unit vector nonnal to the coastline, the impenneability condition Uen = 0, when
applied to (A2), leads to
g h V1'/en = (-f e x U + F )en ,
Z
(AI6)
which is indeed a purely local boundary condition for 1' /.
The alternative to the rigid lid approximation is to consider the ocean surface as free. As a
result, 1' / is a prognostic variable of the model, i.e., a variable that may be computed from an
evolution equation, namely (AI). This could allow studying phenomena such as tides or the
inverted barometer effect (ponte, 1993). To overcome the severe limitations of the time step due
to the presence of fast-propagating Poincare waves, the split-explicit method (Gadd, 1978;
Madaia, 1981) may be used, as is done in our OGCM, as well as in other models (Blumberg
and Mellor, 1987; Beckers, 1991; Killworth et al., 1991). The split explicit technique consists
in integrating the external and internal modes - according to a mainly explicit time stepping -
with different time steps, each time increment being selected according to the fastestpropagating phenomenon encountered in the mode considered. Even though the external time
step is 10-100 times smaller than that of the internal mode, the treatment of the external mode
generally requires 10% of the total computer cost, because the external mode equations are twodimensional, and thus much cheaper to deal with than the internal mode, which is threedimensional. A recent variant of the free-surface approach is to integrate the external mode
equations by means of a semi-implicit method, allowing a much longer time step to be used
(Casulli and Cheng, 1992; Casulli and Catani, 1994; Dukowicz and Smith, 1994).
It is conceivable that, in the future, the streamfunction approach will cease to be the favourite
option. Whether or not all models will switch to the free surface fonnulation is far from clear. It
is however worth stressing that Dukowicz and Smith (1994), in the scope of their eddyresolving World Ocean model, identified no less than 6 advantages of the implicit free surface
solution method.
References
Adam, P. (1990) Modelisation Mathematique Tridimensionnelle d'un Ecosysteme Marin -
Application Ii la Region du Detroit de Bering, M~moire de Fin d'Etudes, Facult~ des Sciences
Appliqu~es, Universit~ de Liege, 159 pp.
Arakawa, A. and Lamb, V.R. (1977) Computational design of the basic dynamical processes of
the UCLA general circulation model, Methods in Computational PhYSics, Vol. 17, Academic
Press, 337 pp.
Batteen, M.L. and Han, Y.-I. (1981) On the computational noise of finite-difference schemes used
in ocean models, Tellus, 33, 387-396.
Beckers, I.-M. (1991) Application of the GHER 3D general circulation model to the Western
Mediterranean, J. Mar. Syst., I, 315-332.
Beckers, J.-M. (1992) Analytical linear numerical stability conditions for an anisotropic 3D
advection-diffusion equation, SIAM J. Numer. Anal., 29, 701-713.
Beckers, J.-M. and Deieersnijder, E. (1993) Stability of a FBTCS scheme applied to the
propagation of shallow-water inertia-gravity waves on various space grids, J. Comput. Phys.,
108, 95-104.
Bedford, K.W. Dingman, I.S. and Yeo, W.K. (1987) Preparation of estuary and marine model
equations by generalized filtering methods, Three-Dimensional Models of Marine and Estuarine
Dynamics (J.C.J. Nihoui and B.M. Jam art, Editors), Elsevier, pp. 113-125.
Bermett, A.F. (1992) Inverse Methods in Physical Oceanography, Cambridge Univ. Press, 346 pp.
denotes the unit vector nonnal to the coastline, the impenneability condition Uen = 0, when
applied to (A2), leads to
g h V1'/en = (-f e x U + F )en ,
Z
(AI6)
which is indeed a purely local boundary condition for 1' /.
The alternative to the rigid lid approximation is to consider the ocean surface as free. As a
result, 1' / is a prognostic variable of the model, i.e., a variable that may be computed from an
evolution equation, namely (AI). This could allow studying phenomena such as tides or the
inverted barometer effect (ponte, 1993). To overcome the severe limitations of the time step due
to the presence of fast-propagating Poincare waves, the split-explicit method (Gadd, 1978;
Madaia, 1981) may be used, as is done in our OGCM, as well as in other models (Blumberg
and Mellor, 1987; Beckers, 1991; Killworth et al., 1991). The split explicit technique consists
in integrating the external and internal modes - according to a mainly explicit time stepping -
with different time steps, each time increment being selected according to the fastestpropagating phenomenon encountered in the mode considered. Even though the external time
step is 10-100 times smaller than that of the internal mode, the treatment of the external mode
generally requires 10% of the total computer cost, because the external mode equations are twodimensional, and thus much cheaper to deal with than the internal mode, which is threedimensional. A recent variant of the free-surface approach is to integrate the external mode
equations by means of a semi-implicit method, allowing a much longer time step to be used
(Casulli and Cheng, 1992; Casulli and Catani, 1994; Dukowicz and Smith, 1994).
It is conceivable that, in the future, the streamfunction approach will cease to be the favourite
option. Whether or not all models will switch to the free surface fonnulation is far from clear. It
is however worth stressing that Dukowicz and Smith (1994), in the scope of their eddyresolving World Ocean model, identified no less than 6 advantages of the implicit free surface
solution method.
References
Adam, P. (1990) Modelisation Mathematique Tridimensionnelle d'un Ecosysteme Marin -
Application Ii la Region du Detroit de Bering, M~moire de Fin d'Etudes, Facult~ des Sciences
Appliqu~es, Universit~ de Liege, 159 pp.
Arakawa, A. and Lamb, V.R. (1977) Computational design of the basic dynamical processes of
the UCLA general circulation model, Methods in Computational PhYSics, Vol. 17, Academic
Press, 337 pp.
Batteen, M.L. and Han, Y.-I. (1981) On the computational noise of finite-difference schemes used
in ocean models, Tellus, 33, 387-396.
Beckers, I.-M. (1991) Application of the GHER 3D general circulation model to the Western
Mediterranean, J. Mar. Syst., I, 315-332.
Beckers, J.-M. (1992) Analytical linear numerical stability conditions for an anisotropic 3D
advection-diffusion equation, SIAM J. Numer. Anal., 29, 701-713.
Beckers, J.-M. and Deieersnijder, E. (1993) Stability of a FBTCS scheme applied to the
propagation of shallow-water inertia-gravity waves on various space grids, J. Comput. Phys.,
108, 95-104.
Bedford, K.W. Dingman, I.S. and Yeo, W.K. (1987) Preparation of estuary and marine model
equations by generalized filtering methods, Three-Dimensional Models of Marine and Estuarine
Dynamics (J.C.J. Nihoui and B.M. Jam art, Editors), Elsevier, pp. 113-125.
Bermett, A.F. (1992) Inverse Methods in Physical Oceanography, Cambridge Univ. Press, 346 pp.