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Global attractors of two layer baroclinic quasi-geostrophic model
School of Mathematics, Lanzou City University, Lanzhou, 730070, China |
We study the dynamics of a two-layer baroclinic quasi-geostrophic model. We prove that the semigroup $ \{S(t)\}_{t\geq 0} $ associated with the solutions of the model has a global attractor in both $ {{\dot H}_{p}}^1(\Omega) $ and $ {{\dot H}_{p}}^2(\Omega) $. Also we show that for any viscosity $ \mu>0 $, there is an open and dense set of forcing $ \mathcal G\subset{{\dot H}_{p}}^0(\Omega) $ such that for each $ G = (g_1, g_2)\in \mathcal G $, the set $ S(G, \mu) \subset {{\dot H}_{p}}^4(\Omega) $ of the steady state problem is non–empty and finite.
References:
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M. Cai,
A analytic study of the baroclinic adjustment in a quasigeostrophic two-layer channel model, Journal of the Atmospheric Sciences, 49 (1992), 1594-1605.
doi: 10.1175/1520-0469(1992)049<1594:AASOTB>2.0.CO;2. |
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M. Hernandez, K. W. Ong and S. Wang, Baroclinic instability and transitions in a quasi-geostrophic two-layer channel model, arXiv preprint. |
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Q. Ma, S. Wang and C. Zhong,
Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.
doi: 10.1512/iumj.2002.51.2255. |
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T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Volume 119 of Mathematical Surveys and Monographs, Providence, RI: American Mathematical Society, 2005.
doi: 10.1090/surv/119. |
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M. Mak,
Equilibration in nonlinear Baroclinic instability, J. Atomspheric Sciences, 42 (1985), 2764-2782.
doi: 10.1175/1520-0469(1985)042<2764:EINBI>2.0.CO;2. |
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J. Pedlosky,
Finite-amplitude baroclinic waves, J. Atomspheric Sciences, 27 (1970), 15-30.
doi: 10.1175/1520-0469(1970)027<0015:FABW>2.0.CO;2. |
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S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math., 87 (1965), 861–866.
doi: 10.2307/2373250. |
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R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
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show all references
References:
| [1] |
M. Cai,
A analytic study of the baroclinic adjustment in a quasigeostrophic two-layer channel model, Journal of the Atmospheric Sciences, 49 (1992), 1594-1605.
doi: 10.1175/1520-0469(1992)049<1594:AASOTB>2.0.CO;2. |
| [2] |
M. Hernandez, K. W. Ong and S. Wang, Baroclinic instability and transitions in a quasi-geostrophic two-layer channel model, arXiv preprint. |
| [3] |
Q. Ma, S. Wang and C. Zhong,
Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.
doi: 10.1512/iumj.2002.51.2255. |
| [4] |
T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Volume 119 of Mathematical Surveys and Monographs, Providence, RI: American Mathematical Society, 2005.
doi: 10.1090/surv/119. |
| [5] |
M. Mak,
Equilibration in nonlinear Baroclinic instability, J. Atomspheric Sciences, 42 (1985), 2764-2782.
doi: 10.1175/1520-0469(1985)042<2764:EINBI>2.0.CO;2. |
| [6] |
J. Pedlosky,
Finite-amplitude baroclinic waves, J. Atomspheric Sciences, 27 (1970), 15-30.
doi: 10.1175/1520-0469(1970)027<0015:FABW>2.0.CO;2. |
| [7] |
S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math., 87 (1965), 861–866.
doi: 10.2307/2373250. |
| [8] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
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