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August  2020, 25(8): 3183-3198. doi: 10.3934/dcdsb.2020057

Approximation of stationary statistical properties of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation

Bo You 1,, and  Chunxiang Zhao 2,

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Bo You

Received  May 2019 Revised  October 2019 Published  August 2020 Early access  February 2020

Fund Project: This work was supported by the National Science Foundation of China Grant (11401459, 11801427), the Natural Science Foundation of Shaanxi Province (2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088)

The main objective of this paper is to study the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation. We prove the global attractor and stationary statistical properties of the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation converge to those of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation as the time step goes to zero.

Keywords: Global attractor, planetary geostrophic equations, semi-implicit semi-discrete scheme in time, stationary statistical property.
Mathematics Subject Classification: Primary: 35B40, 35B41, 37M25, 37N10, 65M12; Secondary: 35Q35.
Citation: Bo You, Chunxiang Zhao. Approximation of stationary statistical properties of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3183-3198. doi: 10.3934/dcdsb.2020057
References:
[1]

C. S. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233.  doi: 10.1002/cpa.10056.  Google Scholar

[2]

W. Cheng and X. M. Wang, A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model, SIAM J. Numer. Anal., 47 (2008), 250-270.  doi: 10.1137/080713501.  Google Scholar

[3]

W. Cheng and X. M. Wang, A uniformly dissipative scheme for stationary statistical properties of the infinite Prandtl number model, Appl. Math. Lett., 21 (2008), 1281-1285.  doi: 10.1016/j.aml.2007.07.036.  Google Scholar

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C. Foias, M. Jolly, I. Kevrekidis and E. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613.  doi: 10.1088/0951-7715/4/3/001.  Google Scholar

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C. Foias, M. Jolly, I. Kevrekidis and E. Titi, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Phys. Lett. A, 186 (1994), 87-96.  doi: 10.1016/0375-9601(94)90926-1.  Google Scholar

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A. T. Hill and E. Suli, Approximation of the global attractor for the incompressible Navier-Stokes equations, IMA J. Numer. Anal., 20 (2000), 633-667.  doi: 10.1093/imanum/20.4.633.  Google Scholar

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N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597.  doi: 10.1093/imanum/22.4.577.  Google Scholar

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L. P. Kadanoff, Turbulent heat flow: Structures and scaling, Phys. Today, 54 (2001), 34-39.   Google Scholar

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A. Lasota and M. C. Mackey, Chaos, Fractals and Noise, Stochastic Aspects of Dynamics, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

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A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. II., Dover Publications, Inc., Mineola, NY, 2007.  Google Scholar

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J. Pedlosky, The equations for geostrophic motion in the ocean, J. Phys. Oceanogr., 14 (1984), 448-455.   Google Scholar

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J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar

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N. A. Phillips, Geostrophic motion, Reviews of Geophysics, 1 (1963), 123-176.   Google Scholar

[17]

A. Robinson and H. Stommel, The oceanic thermocline and associated thermohaline circulation, Tellus, 11 (1959), 295-308.   Google Scholar

[18]

R. M. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation, Appl. Anal., 70 (1998), 147-173.  doi: 10.1080/00036819808840682.  Google Scholar

[19]

R. M. Samelson, R. Temam and S. Wang, Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential Integral Equations, 13 (2000), 1-14.   Google Scholar

[20]

R. M. Samelson and G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, J. Phys. Oceanogr., 27 (1997), 186-194.   Google Scholar

[21]

J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. Optim., 10 (1989), 1213-1234.  doi: 10.1080/01630568908816354.  Google Scholar

[22]

J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.  doi: 10.1080/00036819008839963.  Google Scholar

[23] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.   Google Scholar
[24]

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

[25]

F. Tone, On the long-time $H^2$-stability of the implicit Euler scheme for the 2D magnetohydrodynamics equations, J. Sci. Comput., 38 (2009), 331-348.  doi: 10.1007/s10915-008-9236-2.  Google Scholar

[26]

F. Tone and X. M. Wang, Approximation of the stationary statistical properties of the dynamical system generated by the two dimensional Rayleigh-Bénard convection problem, Anal. Appl., 9 (2011), 421-446.  doi: 10.1142/S0219530511001935.  Google Scholar

[27]

F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the 2D Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40.  doi: 10.1137/040618527.  Google Scholar

[28]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-009-1423-0.  Google Scholar

[29]

X. M. Wang, Approximating stationary statistical properties, Chin. Ann. Math. Ser. B, 30 (2009), 831-844.  doi: 10.1007/s11401-009-0178-2.  Google Scholar

[30]

X. M. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp., 79 (2010), 259-280.  doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[31]

X. M. Wang, Numerical algorithms for stationary statistical properties of dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 4599-4618.  doi: 10.3934/dcds.2016.36.4599.  Google Scholar

[32]

P. Welander, An advective model of the ocean thermocline, Numerical Algorithms, 11 (1959), 309-318.   Google Scholar

[33]

B. You, Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation, Stochastics, 89 (2017), 766-785.  doi: 10.1080/17442508.2016.1276913.  Google Scholar

[34]

B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128.  doi: 10.1016/j.na.2014.08.018.  Google Scholar

[35]

B. You and F. Li, Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, Stoch. Anal. Appl., 34 (2016), 278-292.  doi: 10.1080/07362994.2015.1126184.  Google Scholar

[36]

B. You, C. K. Zhong and F. Li, Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1213-1226.  doi: 10.3934/dcdsb.2014.19.1213.  Google Scholar

show all references

References:
[1]

C. S. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233.  doi: 10.1002/cpa.10056.  Google Scholar

[2]

W. Cheng and X. M. Wang, A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model, SIAM J. Numer. Anal., 47 (2008), 250-270.  doi: 10.1137/080713501.  Google Scholar

[3]

W. Cheng and X. M. Wang, A uniformly dissipative scheme for stationary statistical properties of the infinite Prandtl number model, Appl. Math. Lett., 21 (2008), 1281-1285.  doi: 10.1016/j.aml.2007.07.036.  Google Scholar

[4]

C. M. Elliott and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622-1663.  doi: 10.1137/0730084.  Google Scholar

[5]

C. Foias, M. Jolly, I. Kevrekidis and E. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613.  doi: 10.1088/0951-7715/4/3/001.  Google Scholar

[6]

C. Foias, M. Jolly, I. Kevrekidis and E. Titi, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Phys. Lett. A, 186 (1994), 87-96.  doi: 10.1016/0375-9601(94)90926-1.  Google Scholar

[7] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar
[8]

A. T. Hill and E. Suli, Approximation of the global attractor for the incompressible Navier-Stokes equations, IMA J. Numer. Anal., 20 (2000), 633-667.  doi: 10.1093/imanum/20.4.633.  Google Scholar

[9]

N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597.  doi: 10.1093/imanum/22.4.577.  Google Scholar

[10]

L. P. Kadanoff, Turbulent heat flow: Structures and scaling, Phys. Today, 54 (2001), 34-39.   Google Scholar

[11]

A. Lasota and M. C. Mackey, Chaos, Fractals and Noise, Stochastic Aspects of Dynamics, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[12] A. J. Majda and X. M. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511616778.  Google Scholar
[13]

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. II., Dover Publications, Inc., Mineola, NY, 2007.  Google Scholar

[14]

J. Pedlosky, The equations for geostrophic motion in the ocean, J. Phys. Oceanogr., 14 (1984), 448-455.   Google Scholar

[15]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar

[16]

N. A. Phillips, Geostrophic motion, Reviews of Geophysics, 1 (1963), 123-176.   Google Scholar

[17]

A. Robinson and H. Stommel, The oceanic thermocline and associated thermohaline circulation, Tellus, 11 (1959), 295-308.   Google Scholar

[18]

R. M. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation, Appl. Anal., 70 (1998), 147-173.  doi: 10.1080/00036819808840682.  Google Scholar

[19]

R. M. Samelson, R. Temam and S. Wang, Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential Integral Equations, 13 (2000), 1-14.   Google Scholar

[20]

R. M. Samelson and G. K. Vallis, A simple friction and diffusion scheme for planetary geostrophic basin models, J. Phys. Oceanogr., 27 (1997), 186-194.   Google Scholar

[21]

J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. Optim., 10 (1989), 1213-1234.  doi: 10.1080/01630568908816354.  Google Scholar

[22]

J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.  doi: 10.1080/00036819008839963.  Google Scholar

[23] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.   Google Scholar
[24]

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

[25]

F. Tone, On the long-time $H^2$-stability of the implicit Euler scheme for the 2D magnetohydrodynamics equations, J. Sci. Comput., 38 (2009), 331-348.  doi: 10.1007/s10915-008-9236-2.  Google Scholar

[26]

F. Tone and X. M. Wang, Approximation of the stationary statistical properties of the dynamical system generated by the two dimensional Rayleigh-Bénard convection problem, Anal. Appl., 9 (2011), 421-446.  doi: 10.1142/S0219530511001935.  Google Scholar

[27]

F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the 2D Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40.  doi: 10.1137/040618527.  Google Scholar

[28]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-009-1423-0.  Google Scholar

[29]

X. M. Wang, Approximating stationary statistical properties, Chin. Ann. Math. Ser. B, 30 (2009), 831-844.  doi: 10.1007/s11401-009-0178-2.  Google Scholar

[30]

X. M. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp., 79 (2010), 259-280.  doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[31]

X. M. Wang, Numerical algorithms for stationary statistical properties of dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 4599-4618.  doi: 10.3934/dcds.2016.36.4599.  Google Scholar

[32]

P. Welander, An advective model of the ocean thermocline, Numerical Algorithms, 11 (1959), 309-318.   Google Scholar

[33]

B. You, Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation, Stochastics, 89 (2017), 766-785.  doi: 10.1080/17442508.2016.1276913.  Google Scholar

[34]

B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128.  doi: 10.1016/j.na.2014.08.018.  Google Scholar

[35]

B. You and F. Li, Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, Stoch. Anal. Appl., 34 (2016), 278-292.  doi: 10.1080/07362994.2015.1126184.  Google Scholar

[36]

B. You, C. K. Zhong and F. Li, Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1213-1226.  doi: 10.3934/dcdsb.2014.19.1213.  Google Scholar

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