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A fully implicit model of the three-dimensional thermohaline ocean circulation. (English) Zbl 1051.86004
Summary: In this paper, a fully implicit numerical model of the three-dimensional thermohaline ocean circulation is presented. With this numerical model it is possible to follow branches of steady states in parameter space and monitor their linear stability. Also, transient flows can be computed allowing much larger time steps than those possible with explicit schemes. By using recently developed solvers for linear systems of equations and for generalized eigenvalue problems, results for reasonable spatial resolution can be obtained. Bifurcation diagrams and transient flows are computed for typical flows in a single hemispheric basin situation, with focus on (i) the performance of the methodology and (ii) the new type of information which can be obtained on these flows.

86A05 Hydrology, hydrography, oceanography
65P30 Numerical bifurcation problems
65F10 Iterative numerical methods for linear systems
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[1] Botta, E.F.F.; Dekker, K.; Notay, Y.; van der Ploeg, A.; Vuik, C.; Wubs, F.W.; de Zeeuw, P.M., How fast the Laplace equation was solved in 1995, Appl. numer. math., 24, 439, (1997) · Zbl 0891.65112
[2] Botta, E.F.F.; Wubs, F.W., Matrix renumbering ILU: an effective algebraic multilevel ILU-preconditioner for sparse matrices, SIAM J. matrix anal. appl., 20, 1007, (1999) · Zbl 0937.65057
[3] E. F. F. Botta, F. W. Wubs, and A. van der Ploeg, A fast linear-system solver for large unstructured problems on a shared-memory computer, in Proceedings of the Conference on Algebraic Multilevel Iteration Methods with Applications, edited by O. Axelsson and B. PolmanUniversity of Nijmegen, Nijmegen, The Netherlands, 1996, pp. 105-116.
[4] Broecker, W.S., The great Ocean conveyor, Oceanography, 4, 79, (1991)
[5] Bryan, F.O., High-latitude salinity effects and interhemispheric thermohaline circulations, Nature, 323, 301, (1986)
[6] Bryan, K., Accelerating the convergence to equilibrium of Ocean-climate models, J. phys. oceanogr., 14, 666, (1984)
[7] Cessi, P.; Ierley, G.R., Symmetry-breaking multiple equilibria in quasi-geostrophic, wind-driven flows, J. phys. oceanography, 25, 1196, (1995)
[8] Cessi, P.; Young, W.R., Multiple equilibria in two-dimensional thermohaline circulation, J. fluid mech., 241, 291, (1992) · Zbl 0755.76040
[9] Christodoulou, K.N.; Scriven, L.E., Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem, J. sci. comput., 3, 355, (1988) · Zbl 0677.65032
[10] Dijkstra, H.A., Surface tension driven cellular patterns in three-dimensional boxes—part II: A bifurcation study, Microgravity sci. technol., 24, 415, (1995)
[11] Dijkstra, H.A., Nonlinear physical oceanography, (2000) · Zbl 0964.86003
[12] Dijkstra, H.A.; Molemaker, M.J., Symmetry breaking and overturning oscillations in thermohaline-driven flows, J. fluid mech., 331, 195, (1997) · Zbl 0898.76035
[13] Dijkstra, H.A.; Molemaker, M.J.; van der Ploeg, A.; Botta, E.F.F., An efficient code to compute nonparallel flows and their linear stability, Comput. fluids, 24, 415, (1995) · Zbl 0848.76056
[14] Dukowicz, J.K.; Smith, R.D., Implicit free-surface method for the bryan – cox-semtner Ocean model, J. geophys. res., 99, 7991, (1994)
[15] Gent, P.R.; McWilliams, J.C., Isopycnal mixing in Ocean circulation models, J. phys. oceanogr., 20, 150, (1990)
[16] Goldhirsch, I.; Orszag, S.A.; Maulik, B.K., An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices, J. sci. comput., 2, 33, (1987) · Zbl 0666.65033
[17] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996) · Zbl 0865.65009
[18] Greatbatch, R.J.; Zhang, S., An interdecadal oscillation in an idealized Ocean basin forced by constant heat flux, J. climate, 8, 82, (1995)
[19] I. Gustafsson, A class of first order factorization methods, BIT, 18, 142, 1978. · Zbl 0386.65006
[20] Haney, R.L., Surface thermal boundary conditions for Ocean circulation models, J. phys. oceanogr., 4, 241, (1971)
[21] Huck, T.; de Verdiere, A.Colin; Weaver, A.J., Interdecadal variability of the thermohaline circulation in box-Ocean models forced by fixed surface fluxes, J. phys. oceanogr., 29, 865, (1999)
[22] H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in, Applications of Bifurcation Theory, edited by, P. H. Rabinowitz, Academic Press, New York, 1977. · Zbl 0581.65043
[23] Killworth, P.D., A two-level wind and buoyancy driven thermocline model, J. phys. oceanogr., 15, 1414, (1985)
[24] Large, W.G.; McWilliams, J.C.; Doney, S.C., Oceanic vertical mixing: A review and a model with nonlocal boundary layer parameters, Rev. geophys., 32, 363, (1994)
[25] Maier-Reimer, E.; Mikolajewicz, U.; Hasselman, K., Mean circulation of the Hamburg LSG OGCM and its sensitivity to the thermohaline surface forcing, J. phys. oceanogr., 23, 731, (1993)
[26] Manabe, S.; Stouffer, R.J., Two stable equilibria of a coupled Ocean-atmosphere model, J. climate, 1, 841, (1988)
[27] Manabe, S.; Stouffer, R.J., Century-scale effects of increased CO_2 on the Ocean-atmosphere system, Nature, 364, 215, (1993)
[28] J. Marotzke, Instabilities and multiple steady states of the thermohaline circulation, in Ocean Models in Climate Problems, edited by D. L. T. Anderson and J. WillebrandKluwer Academic, Dordrecht/Norwell, MA, 1989, pp. 501-511.
[29] Marotzke, J.; Welander, P.; Willebrand, J., Instability and multiple steady states in a meridional-plane model of thermohaline circulation, Tellus, 40, 162, (1988)
[30] Meijster, A.; Wubs, F.W., Towards an implementation of a multilevel ILU preconditioner on shared-memory computers, Lecture notes in computer science 1823, 109-118, (2000)
[31] Quon, C.; Ghil, M., Multiple equilibria in thermosolutal convection due to salt-flux boundary conditions, J. fluid mech., 245, 449, (1992) · Zbl 0825.76746
[32] Rahmstorf, S., A fast and complete convection scheme for Ocean models, Ocean modelling, 101, 9, (1993)
[33] Saad, Y., Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices, Lin. alg. appl., 34, 269, (1980) · Zbl 0456.65017
[34] Y. Saad, Iterative Methods for Sparse Matrices, PWS-Kent, Boston, 1996. · Zbl 1031.65047
[35] Schmeits, M.J.; Dijkstra, H.A., On the physics of the 9 months variability in the gulf stream region: combining data and dynamical systems analysis, J. phys. oceanogr., 30, 1967, (2000)
[36] Schmitz, W.J., On the interbasin-scale thermohaline circulation, Rev. geophys., 33, 151, (1995)
[37] Sleijpen, G.L.G.; Van der Vorst, H.A., A jacobi – davidson iteration method for linear eigenvalue problems, SIAM J. matrix anal. appl., 17, 410, (1996)
[38] Steward, W.J.; Jennings, A., A simultaneous iteration algorithm for real matrices, ACM trans. math. software, 7, 184, (1981) · Zbl 0455.65028
[39] Stommel, H., Thermohaline convection with two stable regimes of flow, Tellus, 2, 244, (1961)
[40] Te Raa, L.A.; Dijkstra, H.A., Instability of the thermohaline Ocean circulation on interdecadal time scales, J. phys. oceanogr., (2002)
[41] Tett, S.F.B.; Johns, T.C.; Mitchell, J.F.B, Global and regional variability in a coupled AOGCM, Climate dyn., 13, 303, (1997)
[42] Thual, O.; McWilliams, J.C., The catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box models, Geophys. astrophys. fluid dyn., 64, 67, (1992)
[43] Van Dorsselaer, J.J., Computing eigenvalues occurring in continuation methods with the jacobi – davidson QZ method, J. comput. phys., 138, 714, (1997) · Zbl 0899.65033
[44] Vellinga, M., Instability of two-dimensional thermohaline circulation, J. phys. oceanogr., 26, 305, (1996)
[45] Weaver, A.; Marotzke, J.; Cummings, P.F.; Sarachik, E.S., Stability and variability of the thermohaline circulation, J. phys. oceanogr., 23, 39, (1993)
[46] Weaver, A.J.; Hughes, T.M., On the incompatibility of Ocean and atmosphere and the need for flux adjustments, Climate dyn., 12, 141, (1996)
[47] W. Weijer, and, H. A. Dijkstra, Bifurcations of the three-dimensional thermohaline circulation: the double hemispheric case, J. Mar. Res, in press.
[48] P. Welander, Thermohaline effects in the ocean circulation and related simple models, in Large Scale Transport Processes in Oceans and Atmosphere, edited by J. Willebrand and D. L. T. AndersonReidel, Dordrecht, 1986, pp. 163-200.
[49] Winton, M., The role of horizontal boundaries in parameter sensitivity and decadal-scale variability of coarse-resolution Ocean general circulation models, J. phys. oceanogr., 26, 289, (1996)
[50] Winton, M.; Sarachik, E.S., Thermohaline oscillations induced by strong steady salinity forcing of Ocean general circulation models, J. phys. oceanogr., 23, 1389, (1993)
[51] Wood, R.A.; Keen, A.B.; Mitchell, J.F.B.; Gregory, J.M., Changing spatial structure of the thermohaline circulation in response to atmospheric CO_2 forcing in a climate model, Nature, 399, 572, (1999)
[52] Wright, D.G.; Stocker, T.F., A zonally averaged model for the thermohaline circulation, part I: model development and flow dynamics, J. phys. oceanogr., 21, 1713, (1991)
[53] Yin, F.L.; Sarachik, E.S., An efficient convective adjustment scheme for Ocean general circulation models, J. phys. oceanogr., 24, 1425, (1994)
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