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A fully-implicit model of the global ocean circulation. (English) Zbl 1032.76557
Summary: With the recent developments in the solution methods for large-dimensional nonlinear algebraic systems, fully-implicit ocean circulation models are now becoming feasible. In this paper, the formulation of such a three-dimensional global ocean model is presented. With this implicit model, the sensitivity of steady states to parameters can be investigated efficiently using continuation methods. In addition, the implicit formulation allows for much larger time steps than can be used with explicit models. To demonstrate current capabilities of the implicit global ocean model, we use a relatively low-resolution (4\(^\circ\) horizontally and 12 levels vertically) version. For this configuration, we present: (i) an explicit calculation of the bifurcation diagram associated with hysteresis behavior of the ocean circulation and (ii) the scaling behavior of the Atlantic meridional overturning versus the magnitude of the vertical mixing coefficient of heat and salt.

MSC:
76E20 Stability and instability of geophysical and astrophysical flows
65P30 Numerical bifurcation problems
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
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[1] Alley, R.B.; Marotzke, J.; Nordhaus, W.D.; Overpeck, J.T.; Peteet, D.M.; Pielke, R.A.; Pierrehumbert, R.T.; Rhines, P.B.; Stocker, T.F.; Talley, L.D.; Wallace, J.M., Abrupt climate change, Nature, 299, 2005-2010, (2003)
[2] Kushnir, Y., Interdecadal variations in north atlantic sea surface temperature and associated atmospheric conditions, J. phys. oceanogr., 7, 141-157, (1994)
[3] Moron, V.; Vautard, R.; Ghil, M., Trends, interdecadal and interannual oscillations in global sea-surace temperature, Clim. dyn., 14, 545-569, (1998)
[4] Delworth, T.L.; Mann, M.E., Observed and simulated multidecadal variability in the northern hemisphere, Clim. dyn., 16, 661-676, (2000)
[5] McWilliams, J.C., Modeling the Ocean general circulation, Ann. rev. fluid mech., 28, 215-248, (1996)
[6] R.C. Pacanowski, S.M. Griffies, MOM 3.0, User Manual GFDL/NOAA Report
[7] Rahmstorf, S., Bifurcations of the atlantic thermohaline circulation in response to changes in the hydrological cycle, Nature, 378, 145-149, (1995)
[8] Chassignet, E.P.; Arango, H.; Dietrich, D.; Ezer, T.; Ghil, M.; Haidvogel, D.B.; Ma, C.-C.; Mehra, A.; Paiva, A.M.; Sirkes, Z., DAMEE-NAB: the base experiments, Dyn. atmos. oceans, 32, 155-183, (2000)
[9] Hasselmann, K., An Ocean model for climate variability studies, Prog. oceanogr., 11, 69-92, (1982)
[10] Haidvogel, D.B.; Beckmann, A., Numerical Ocean circulation modelling, (1999), Imperial College Press London, UK · Zbl 0932.76001
[11] Rahmstorf, S., The thermohaline circulation: a system with dangerous thresholds, Clim. change, 46, 247-256, (2000)
[12] Stommel, H., Thermohaline convection with two stable regimes of flow, Tellus, 2, 230-244, (1961)
[13] Ghil, M.; Childress, S., Topics in geophysical fluid dynamics: atmospheric dynamics, dynamo theory, and climate dynamics, (1987), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0643.76001
[14] Dijkstra, H.A., Nonlinear physical oceanography: A dynamical systems approach to the large scale Ocean circulation and el niño, (2000), Kluwer Academic Publishers Dordrecht, The Netherlands
[15] Dijkstra, H.A.; Öksüzŏglu, H.; Wubs, F.W.; Botta, E.F.F., A fully implicit model of the three-dimensional thermohaline Ocean circulation, J. comput. phys., 173, 685-715, (2001) · Zbl 1051.86004
[16] Winton, M.; Sarachik, E.S., Thermohaline oscillations induced by strong steady salinity forcing of Ocean general circulation models, J. phys. oceanogr., 23, 1389-1410, (1993)
[17] Large, W.G.; McWilliams, J.C.; Doney, S.C., Ocean vertical mixing: a review and a model with a nonlocal boundary layer parameterization, Rev. geophys., 32, 363-403, (1994)
[18] North, G.R.; Cahalan, R.F.; Coakley, J.A., Energy balance climate models, Rev. geophys. space phys., 19, 19-121, (1981)
[19] Te Raa, L.A.; Dijkstra, H.A., Instability of the thermohaline Ocean circulation on interdecadal time scales, J. phys. oceanogr., 32, 138-160, (2002)
[20] Weijer, W.; Dijkstra, H.A., Bifurcations of the three-dimensional thermohaline circulation: the double hemispheric case, J. mar. res., 59, 599-631, (2001)
[21] Keller, H.B., Numerical solution of bifurcation and nonlinear eigenvalue problems, () · Zbl 0581.65043
[22] Steward, W.J.; Jennings, A., A simultaneous iteration algorithm for real matrices, ACM trans. math. software, 7, 184-198, (1981) · Zbl 0455.65028
[23] Brezinski, C., Projection methods for systems of equations, (1997), North-Holland Publishing Co Amsterdam · Zbl 0867.65009
[24] Seydel, R., Practical bifurcation and stability analysis: from equilibrium to chaos, (1994), Springer-Verlag New York, USA · Zbl 0806.34028
[25] Botta, E.F.F.; Wubs, F.W., MRILU: an effective algebraic multi-level ILU-preconditioner for sparse matrices, SIAM J. matrix anal. appl., 20, 1007-1026, (1999) · Zbl 0937.65057
[26] Adcroft, A.; Hill, C.N.; Marshall, J.C., A new treatment of the Coriolis terms in C-grid models at both high and low resolutions, Month. weather rev., 127, 1928-1936, (1999)
[27] Vellinga, M., Multiple equilibria of the thermohaline circulation as a side effect of convective adjustment, J. phys. oceanogr., 28, 305-319, (1998)
[28] Edwards, N.R.; Shepherd, J.G., Multiple thermohaline states due to variable diffusivity in a hierarchy of models, Ocean modeling, 3, 67-94, (2001)
[29] K.E. Trenberth, J.G. Olson, W.G. Large, A global ocean wind stress climatology based on ECMWF analyses, Tech. rep., National Center for Atmospheric Research, Boulder, CO, U.S.A., 1989
[30] S. Levitus, R. Burgett, T. Boyer, World Ocean Atlas 1994, Volume 3: Salinity., NOAA Atlas NESDIS 3, US Department of Commerce, Washington DC, 1994, 0-99
[31] England, M.H., Representing the global-scale water masses in Ocean general circulations models, J. phys. oceanogr., 23, 1523-1552, (1993)
[32] J.M. Oberhuber, The Budget of Heat, Buoyancy and Turbulent Kinetic Energy at the Surface of the Global Ocean, Max Planck Institute fur Meteorologie Hamburg report nr. 15, Hamburg, Germany, 1988
[33] Tziperman, E.; Toggweiler, J.R.; Feliks, Y.; Bryan, K., Instability of the thermohaline circulation with respect to mixed boundary conditions: Is it really a problem for realistic models?, J. phys. oceanogr., 24, 217-232, (1994)
[34] Weaver, A.J.; Hughes, T.M., On the incompatibility of Ocean and atmosphere and the need for ux adjustments, Clim. dyn., 12, 141-170, (1996)
[35] Bryan, F., Parameter sensivity of primitive equation Ocean general circulation models, J. phys. oceanogr., 17, 970-985, (1987)
[36] Welander, P., The thermocline problem, Phil. trans. roy. soc. London A, 270, 415-421, (1971)
[37] L.A. Te Raa, Internal variability of the thermohaline ocean circulation, Ph.D. Thesis, Utrecht University, Utrecht, The Netherlands, 2003
[38] Gnanadesikan, A., A simple predictive model of the structure of the oceanic pycnocline, Science, 283, 2077-2081, (1999)
[39] Marotzke, J.; Welander, P.; Willebrand, J., Instability and multiple steady states in a meridional-plane model of thermohaline circulation, Tellus, 40, 162-172, (1988)
[40] Marotzke, J.; Scott, J.R., Convective mixing and the thermohaline circulation, J. phys. oceanogr., 29, 2962-2970, (1999)
[41] Gent, P.R.; Willebrand, J.; McDougall, T.J.; McWilliams, J.C., Parameterizing eddy-induced tracer transports in Ocean circulation models, J. phys. oceanogr., 25, 463-474, (1995)
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