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December  2021, 26(12): 6377-6385. doi: 10.3934/dcdsb.2021023

Global attractors of two layer baroclinic quasi-geostrophic model

Yanhong Zhang

School of Mathematics, Lanzou City University, Lanzhou, 730070, China

Received  July 2020 Published  December 2021 Early access  January 2021

Fund Project: The work was supported in part by the National Science Foundation of China under Grant 11761044

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.

Keywords: Baroclinic quasi-geostrophic model, semigroup; absorbing set, condition (C), uniform compactness, global attractor.
Mathematics Subject Classification: 37C70.
Citation: Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6377-6385. doi: 10.3934/dcdsb.2021023
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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

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.  Google Scholar

[2]

M. Hernandez, K. W. Ong and S. Wang, Baroclinic instability and transitions in a quasi-geostrophic two-layer channel model, arXiv preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math., 87 (1965), 861–866. doi: 10.2307/2373250.  Google Scholar

[8]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

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