Ergodic dynamics of the coupled quasigeostrophic-flow-energy-balance system. (English) Zbl 1151.86001
J. Math. Sci., New York 151, No. 1, 2677-2688 (2008); translation from Fundam. Prikl. Mat. 12, No. 6, 67-84 (2006).
Summary: The authors consider a mathematical model for the coupled atmosphere-ocean system, namely, the coupled quasigeostrophic-flow-energy-balance model. This model consists of the large-scale quasigeostrophic oceanic flow model and the transport equation for oceanic temperature, coupled with an atmospheric energy-balance model. After reformulating this coupled model as a random dynamical system (cocycle property), it is shown that the coupled quasigeostrophic-energy balance fluid system has a random attractor, and under further conditions on the physical data and the covariance of the noise, the system is ergodic, namely, for any observable of the coupled atmosphere-ocean flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.
MSC:
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
| 37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
| 35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |
| 37H05 | General theory of random and stochastic dynamical systems |
| 37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
References:
| [1] | L. Arnold, Random Dynamical Systems, Springer-Verlag, New York (1998). · Zbl 0906.34001 |
| [2] | F. Chen and M. Ghil, ”Interdecadal variability in a hybrid coupled ocean-atmosphere model,” J. Phys. Oceanography, 26, 1561–1578 (1996). · doi:10.1175/1520-0485(1996)026<1561:IVIAHC>2.0.CO;2 |
| [3] | I. D. Chueshov, J. Duan, and B. Schmalfuß, ”Probabilistic dynamics of two-layer geophysical flows,” Stoch. Dyn., 1, No. 4, 451–475, 2001. · Zbl 1038.60054 · doi:10.1142/S0219493701000229 |
| [4] | I. D. Chueshov and M. Scheutzow, ”Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,” J. Dynam. Differential Equations, 13, 355–380 (2001). · Zbl 0990.37041 · doi:10.1023/A:1016684108862 |
| [5] | H. Crauel and F. Flandoli, ”Hausdorff dimension of invariant sets for random dynamical systems,” J. Dynam. Differential Equations, 10, 449–474 (1998). · Zbl 0927.37031 · doi:10.1023/A:1022605313961 |
| [6] | A. Debussche, ”Hausdorff dimension of a random invariant set,” J. Math. Pures Appl., 9, 967–988 (1998). · Zbl 0919.58044 |
| [7] | T. DelSole and B. F. Farrell, ”A stochastically excited linear system as a model for quasigeostrophic turbulence: Analytic results for one-and two-layer fluids,” J. Atmospheric Sci., 52, 2531–2547 (1995). · doi:10.1175/1520-0469(1995)052<2531:ASELSA>2.0.CO;2 |
| [8] | H. A. Dijkstra, Nonlinear Physical Oceanography, Kluwer Academic, Boston (2000). · Zbl 0964.86003 |
| [9] | J. Duan, H. Gao, and B. Schmalfuß, ”Stochastic dynamics of a coupled atmosphere-ocean model,” Stoch. Dyn., 2, No. 3, 357–380 (2002). · Zbl 1090.86003 · doi:10.1142/S0219493702000467 |
| [10] | Yu. V. Egorov and M. A. Shubin, ”Partial differential equations,” in: Encyclopedia of Mathematical Sciences, Vol. 1, Springer-Verlag, New York (1991). · Zbl 0741.00025 |
| [11] | F. Flandoli and B. Schmalfuß, ”Random attractors for the stochastic 3-D Navier-Stokes equation with multiplicative white noise,” Stochastics Stochastics Rep., 59, 21–45 (1996). · Zbl 0870.60057 |
| [12] | H. J. Gao and J. Duan, ”Averaging principle for quasi-geostrophic motion under rapidly oscillating forcing,” Appl. Math. Mech., in press. · Zbl 1144.76488 |
| [13] | A. Griffa and S. Castellari, ”Nonlinear general circulation of an ocean model driven by wind with a stochastic component,” J. Marine Research, 49, 53–73 (1991). · doi:10.1357/002224091784968567 |
| [14] | G. Holloway, ”Ocean circulation: Flow in probability under statistical dynamical forcing,” in: S. Molchanov and W. Woyczynski, eds., Stochastic Models in Geosystems, Springer-Verlag (1996). · Zbl 0864.76076 |
| [15] | J.-L. Lions and E. Magenes, Problémes aux Limites Non Homogénes et Applications, Vol. 1, Dunod (1968). |
| [16] | G. R. North and R. F. Cahalan, ”Predictability in a solvable stochastic climate model,” J. Atmospheric Sci., 38, 504–513 (1981). · doi:10.1175/1520-0469(1981)038<0504:PIASSC>2.0.CO;2 |
| [17] | G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, Cambridge Univ. Press, Cambridge (1992). · Zbl 0761.60052 |
| [18] | G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univ. Press, Cambridge (1996). · Zbl 0849.60052 |
| [19] | B. Schmalfuß, ”Invariant attracting sets of nonlinear stochastic differential equations,” in: H. Langer and V. Nollau, eds., ISAM Seminar-Gaußig, Vol. 54, Akademie (1989), pp. 217–228. |
| [20] | B. Schmalfuß, ”Long-time behaviour of the stochastic Navier-Stokes equation,” Math. Nachr., 152, 7–20 (1991). · Zbl 0728.60069 · doi:10.1002/mana.19911520102 |
| [21] | B. Schmalfuß, ”A random fixed point theorem and the random graph transformation,” J. Math. Anal. Appl., 225, No. 1, 91–113 (1998). · Zbl 0931.37019 · doi:10.1006/jmaa.1998.6008 |
| [22] | R. Temam, Navier-Stokes Equation and Nonlinear Functional Analysis, CBMS-NSF Regional Conf. Ser. in Appl. Math., Vol. 66, SIAM, Philadelphia (1983). |
| [23] | R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Berlin (1997). · Zbl 0871.35001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
