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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.
[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.

[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.

[10]

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

[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.

[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.
[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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. 

[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. 

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.
[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.

[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.

[10]

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

[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.

[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.
[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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. 

[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. 

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

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