It revealed that non-integer Hausdorff dimension was important to complex motion through Lorenz attractor.Quasi geostrophic motion models were often explained the different phenomenon of atmospheric motion,which were first-order approximate to original atmospheric motion equations.The existence of global attractor under three-wave effect and upper bound of Hausdorff dimension were discussed by the semi-group operator method,meanwhile the topology of long-term mode was described.The results showed that,dH(Hausdorff dimension) was ≤ 2.994 and ≤ 2.9996 with and without forced heating effect in turn,under initial situation that one and two waves of atmospheric circulation in zonal and meridional direction,respectively.Our study proved that non-integer Hausdorff dimension existed in Quasi geostrophic motion models as the same as original atmospheric motion equations.
LIU Chun
,
ZHOU Wenlin
,
LI Xiumei
,
WANG Yanbo
. Hausdorff-Dimension Estimates of Global Attractors about Quasi-Geostrophic Model on Three Waves[J]. Plateau Meteorology, 2016
, 35(1)
: 251
-259
.
DOI: 10.7522/j.issn.1000-0534.2014.00141
[1]Abidi H, Hmidi T.2008.On the global well-posedness of the critical quasi-geostrophic equation[J].SIAM J Math Anal, 40(1):167-185.
[2]Bennett A F, Kloedent P E.1980.The simplified quasi-geotropic equations:existence and uniqueness of strong solutions[J].Mathematka, 27:287-311.
[3]Foias C, Manley O, Rosa R, et al.1972.Navier-Stokes Equations and Turbulence[M].Cambridge:Cambridge University Press.
[4]Charney J G.1947.The dynamics of long waves in a baroclinic westerly current[J].Meteor, 4:135-163.
[5]Constantin P, Majda A, Tabak E.1994.Singular front formation in a model for quasigeostrophic flow[J].Phys Fliuds, 6:9-11.
[6]Constantin P, Nie Q, Schorghofer N.1998.Nonsingular surface quasi-geostrophic flow[J].Phys Lett A, 241(3):168-172.
[7]Dudis J J, Davis S H.1971.Energy stability of the Ekman boundary layer[J].Fluid Mech, 47(2):381-403.
[8]Li J, Chou J.2003.The global analysis theory of climate system and its applications[J].Chinese Sci Bull, 48(10):1034-1039.
[9]Kaplan J L, Yorke J A.1979.Chaotic behavior of multidimensional difference equations[M]//Functional differential equations and approximation of fixed points.Springer Berlin Heidelberg, 204-227.
[10]Mu M.1986.Global classical solutions of initial-boundary value problems for nonlinear vorticity equation and its applications[J].Acta Math Scientia, 6:201-218.
[11]Mu M.1987.Global classical solutions of initial-boundary value problems for generalized vorticity equations[J].Scientia Sinica, 30:359-371.
[12]Mitchil K E, Dutton J A.1981.Bifurcation from stationary to periodic solutions in a low-order model of forced, dissipative barotrophic flow[J].Atmos Sci, 38:690-716.
[13]Wittwer P.2002.On the structure of stationary solutions of the Navier-Stokes-equations[J].Communications in Mathematical Physics, 226(3):455-474.
[14]Temam R.1983.Navier-Stokes Equations and Nonlinear Functional Analysis[M].Society for Industry and Applied Mathematics.
[15]Wang S.1992a.On the 2-D model of large-scale atmospheric motion:well-posedness and attractors[J].Nonlinear Anal Theo Math Appl, 18(1):17-60.
[16]Wang S.1992b.Attractors for the 3-D baroclinic quasi-geotropic equations of large-scale atmosphere[J].Math Anal Appl, 165(1):266-286.
[17]Wolansky G.1998.Existence, uniqueness, and stability of stationary barotrophic flow with forcing and dissipation[J].Comm Pure Appl Math, 41(1):19-46.
[18]Wolansky G.1989.The Barotrophic corticity equation under forcing and dissipations:bifurcations of nonsysmmetric responses and multiplicity of solutions[J].SIAM J Appl Math, 49(6):1585-1607.
[19]Vickroy J G, Dutton J A.1979.Bifurcation and catastrophe in a simple, forced, dissipative quasi-geostrophic flow[J].Atmos Sci, 36:42-52.
[20]陈忠明, 闵文彬, 崔春光.2004.西南低涡研究的一些新进展[J].高原气象, 23(6):1-5.Chen Zhongming, Min Wenbin, Cui Chunguang.2004.New advances in southwest China vortex research[J].Plateau Meteor, 23(6):1-5.
[21]丑纪范.2002.大气科学中的非线性与复杂性[M].北京:气象出版社.Chou Jifan.2002.Nonlinear and Complexity of Atmospheric Sciences[M].Beijing:China Meteorological Press.
[22]高普云.2005.非线性动力学-分叉、混沌与孤立子[M].长沙:国防科技大学出版社.Gao Puyun.2005.Nonlinear Dynamics, Bifurcation, Chaos and Soliton[M].Changsha:National University of Defense Technology Press.
[23]郝柏林.1983.分岔、混沌、奇怪吸引子、湍流及其它-关于确定论系统中的内在随机性[J].物理学进展, 3(3):335-416.Hao Bolin.1983.Bifurcation, chaos, strange attractor, turbulence and other-on the inherent randomness in deterministic system[J].Progress in Physics, 3(3):335-416.
[24]胡隐樵.1999.非平衡态大气热力学的研究[J].高原气象, 18(3):306-319.Hu Yinqiao.1999.Research of atmospheric thermodynamics of non-equilibrium state[J].Plateau Meteor, 18(3):306-319.
[25]黄海洋, 郭柏灵.2006.大尺度大气方程组解和吸引子的存在性[J].中国科学:地球科学, 36(4):392-400.Huang Haiyang, Guo Bolin.2006.Existence of solutions and attractors for large scale atmospheric equations[J].Science in China:Earth Sciences, 36(4):392-400.
[26]李炳熙.1984.高维动力系统的周期轨道:理论和应用[M].上海:上海科学技术出版社.Li Bingxi.1984.Periodic orbits of high dimensional dynamical systems:Theory and Applications[M].Shanghai:Shanghai science and Technology Press.
[27]李建平, 丑纪范.1998a.大气运动方程的定性理论[J].大气科学, 22(4):348-360.Li Jianping, Chou Jifan.1998.The qualitative theory on the dynamical equations of atmospheric motion and its application[J].Chinese J Atoms Sci, 22(4):348-360
[28]李建平, 丑纪范.1998b.大气动力学方程组的定性理论及其应用[J].大气科学, 22(4):443-453.Li Jianping, Chou Jifan.1998.The qualitative theory of the dynamical equations of atmospheric motion and its applications[J].Chinese J Atmos Sci, 22(4):443-453.
[29]李建平, 丑纪范.2003.非线性大气动力学的进展[J].大气科学, 27(4):653-673.Li Jianping, Chou Jifan.2003.Advances in nonlinear atmospheric dynamics[J].Chinese J Atmos Sci, 27(4):653-673.
[30]刘春, 刘思波, 张春辉, 等.2014.一个Navier-Stokes算子方程广义能量的存在性[J].高原气象, 33(2):467-473.Liu Chun, Liu Sibo, Zhang Chunhui, et al.2014.Existence of generalized energy about a Navier-Stokes operator equation[J].Plateau Meteor, 33(2):467-473.DOI:10.7522/j.issn.1000-0534.2012.00190.
[31]刘式适, 刘适达.1991.大气动力学[M].北京:北京大学出版社.Liu Shikuo, Liu Shida.1991.Atmospheric Dynamics[M].Beijing:Dekker Press.
[32]刘式适, 柏晶瑜, 陈华.2000.青藏高原大地形作用下的Rossby波[J].高原气象, 19(3):331-338.Liu Shikuo, Bai Jingyu, Chen Hua.2000.Rossby wave affected by large-scale topography of Qinghai-Xizang Plateau[J].Plateau Meteor, 19(3):331-338.
[33]刘晓冉, 李国平.2007.热力强迫的非线性奇异惯性重力内波与高原低涡的联系[J].高原气象, 26(2):225-232.Liu Xiaoran, Li Guoping.2007.Nonliear singular inertial gravitational wave forced by diabatic heating and its association with the Tibetan Plateau vortex[J].Plateau Meteor, 26(2):225-232.
[34]谢志辉, 丑纪范.1999.大气动力学方程组全局分析的研究进展[J].地球科学进展, 14(2):133-139.Xie Zhihui, Chou Jifan.1999.Progress in the global analysis to the atmospheric dynamical equations[J].Advance in Earth Sciences, 14(2):133-139.
[35]朱抱真.1991.大气和海洋的非线性动力学概论[M].北京:海洋出版社.Zhu Baozhen.1991.Introduction to the Nonlinear Dynamics of the Atmosphere and Oceans[M].Beijing:China Ocean Press.
