December  2011, 15(3): 769-788. doi: 10.3934/dcdsb.2011.15.769

Robust control problems for primitive equations of the ocean

1. 

Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States

Received  February 2009 Revised  April 2010 Published  February 2011

In this article, we study some robust control problems associated with the primitive equations of the ocean and related to data assimilation in oceanography. We prove the existence and uniqueness of solutions to these control problems.
Citation: T. Tachim Medjo. Robust control problems for primitive equations of the ocean. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 769-788. doi: 10.3934/dcdsb.2011.15.769
References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325. doi: 10.1007/BF00271794.

[2]

V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation, Comput. Math. Math. Phys., 37 (1997), 348-358.

[3]

A. Belmiloudi and F. Broissier, A control method for assimilation of surface data in a linearized Navier-Stokes-type problem related to oceanography, SIAM. J. Control Optim., 35 (1997), 2183-2197. doi: 10.1137/S0363012995286137.

[4]

A. Bennett, "Inverse Methods in Physical Oceanography," Cambridge University Press, 1994.

[5]

T. Bewley, R. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392. doi: 10.1016/S0167-2789(99)00206-7.

[6]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

[7]

F. X. Le Dimet and V. P. Shutyaev, On Newton methods in data assimilation, Russ. J. Numer. Anal. Math. Modelling, 15 (2000), 419-434. doi: 10.1515/rnam.2000.15.5.419.

[8]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," Series Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.

[9]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980.

[10]

C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460. doi: 10.1016/j.na.2004.12.005.

[11]

C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chin. Ann. of Math. B, 23 (2002), 1-16. doi: 10.1142/S025295990200002X.

[12]

C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131.

[13]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159.

[14]

F. X. Le Dimet and V. Shutyaev, On data assimilation for quasilinear parabolic problems, Russian J. Numer. Anal. Math. Modelling, 16 (2001), 247-259.

[15]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002.

[16]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Computational Mechanics Advance, 1 (1993), 3-54.

[17]

J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII), Computational Mechanics Advance, 1 (1993), 55-120.

[18]

J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII), Math. Pures et Appl., 73 (1995), 105-163.

[19]

J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday, Nonlinear Anal. Ser. A: Theory Methods, 40 (2000), 439-482. doi: 10.1016/S0362-546X(00)85026-9.

[20]

T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean, Nonlinear Anal., 68 (2008), 3550-3564 doi: 10.1016/j.na.2007.03.046.

[21]

T. Tachim Medjo, Optimal control of the primitive equations of the ocean with state constraints, Accepted in Nonlinear Anal., Ser. A: Theory Methods, 2010.

[22]

T. Tachim Medjo and R. Temam, A small eddy correction algorithm for the primitive equations of the ocean, in "Mathematical Modeling, Simulation, Visualization and E-Learning: Proceedings of the Bellagio International Conference" (D. Konate, editor), Springer, (2007), 107-150.

[23]

T. Tachim Medjo and R. Temam, A two-grid finite difference method for the primitive equations of the ocean, Nonlinear Anal., 69 (2008), 1034-1056. doi: 10.1016/j.na.2008.02.044.

[24]

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

[25]

J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992.

[26]

O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339. doi: 10.1111/j.2153-3490.1981.tb01755.x.

[27]

O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory, Q. J. R. Meteorol. Soc., 113 (1987), 1311-1328. doi: 10.1256/smsqj.47811.

[28]

E. Tziperman and W. C. Thacker, An optimal-control/adjoint approach to studying the oceanic general circulation, Journal of Physical Oceanography, 19 (1989), 1471-1485. doi: 10.1175/1520-0485(1989)019<1471:AOCEAT>2.0.CO;2.

[29]

W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986.

[30]

C. Wunsch, "The Ocean Circulation Inverse Problem," Cambridge University Press, 1996. doi: 10.1017/CBO9780511629570.

[31]

D. Zupanski, A general weak constraint applicable to operational 4D-var data assimilation system, Mon. Weather Rev., 125 (1993), 2274-2292. doi: 10.1175/1520-0493(1997)125<2274:AGWCAT>2.0.CO;2.

show all references

References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325. doi: 10.1007/BF00271794.

[2]

V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation, Comput. Math. Math. Phys., 37 (1997), 348-358.

[3]

A. Belmiloudi and F. Broissier, A control method for assimilation of surface data in a linearized Navier-Stokes-type problem related to oceanography, SIAM. J. Control Optim., 35 (1997), 2183-2197. doi: 10.1137/S0363012995286137.

[4]

A. Bennett, "Inverse Methods in Physical Oceanography," Cambridge University Press, 1994.

[5]

T. Bewley, R. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392. doi: 10.1016/S0167-2789(99)00206-7.

[6]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

[7]

F. X. Le Dimet and V. P. Shutyaev, On Newton methods in data assimilation, Russ. J. Numer. Anal. Math. Modelling, 15 (2000), 419-434. doi: 10.1515/rnam.2000.15.5.419.

[8]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," Series Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.

[9]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980.

[10]

C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460. doi: 10.1016/j.na.2004.12.005.

[11]

C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chin. Ann. of Math. B, 23 (2002), 1-16. doi: 10.1142/S025295990200002X.

[12]

C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131.

[13]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179. doi: 10.3934/dcds.2007.17.159.

[14]

F. X. Le Dimet and V. Shutyaev, On data assimilation for quasilinear parabolic problems, Russian J. Numer. Anal. Math. Modelling, 16 (2001), 247-259.

[15]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002.

[16]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Computational Mechanics Advance, 1 (1993), 3-54.

[17]

J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII), Computational Mechanics Advance, 1 (1993), 55-120.

[18]

J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII), Math. Pures et Appl., 73 (1995), 105-163.

[19]

J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday, Nonlinear Anal. Ser. A: Theory Methods, 40 (2000), 439-482. doi: 10.1016/S0362-546X(00)85026-9.

[20]

T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean, Nonlinear Anal., 68 (2008), 3550-3564 doi: 10.1016/j.na.2007.03.046.

[21]

T. Tachim Medjo, Optimal control of the primitive equations of the ocean with state constraints, Accepted in Nonlinear Anal., Ser. A: Theory Methods, 2010.

[22]

T. Tachim Medjo and R. Temam, A small eddy correction algorithm for the primitive equations of the ocean, in "Mathematical Modeling, Simulation, Visualization and E-Learning: Proceedings of the Bellagio International Conference" (D. Konate, editor), Springer, (2007), 107-150.

[23]

T. Tachim Medjo and R. Temam, A two-grid finite difference method for the primitive equations of the ocean, Nonlinear Anal., 69 (2008), 1034-1056. doi: 10.1016/j.na.2008.02.044.

[24]

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

[25]

J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992.

[26]

O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339. doi: 10.1111/j.2153-3490.1981.tb01755.x.

[27]

O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory, Q. J. R. Meteorol. Soc., 113 (1987), 1311-1328. doi: 10.1256/smsqj.47811.

[28]

E. Tziperman and W. C. Thacker, An optimal-control/adjoint approach to studying the oceanic general circulation, Journal of Physical Oceanography, 19 (1989), 1471-1485. doi: 10.1175/1520-0485(1989)019<1471:AOCEAT>2.0.CO;2.

[29]

W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986.

[30]

C. Wunsch, "The Ocean Circulation Inverse Problem," Cambridge University Press, 1996. doi: 10.1017/CBO9780511629570.

[31]

D. Zupanski, A general weak constraint applicable to operational 4D-var data assimilation system, Mon. Weather Rev., 125 (1993), 2274-2292. doi: 10.1175/1520-0493(1997)125<2274:AGWCAT>2.0.CO;2.

[1]

Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023

[2]

Ludovic Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic models. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3143-3168. doi: 10.3934/dcds.2022012

[3]

T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171

[4]

May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179

[5]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025

[6]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336

[7]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6377-6385. doi: 10.3934/dcdsb.2021023

[8]

Tsukasa Iwabuchi. On analyticity up to the boundary for critical quasi-geostrophic equation in the half space. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1209-1224. doi: 10.3934/cpaa.2022016

[9]

Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525

[10]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[11]

Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152

[12]

Colin Cotter, Dan Crisan, Darryl Holm, Wei Pan, Igor Shevchenko. Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model. Foundations of Data Science, 2020, 2 (2) : 173-205. doi: 10.3934/fods.2020010

[13]

Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277

[14]

T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119

[15]

Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133

[16]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197

[17]

Eleftherios Gkioulekas, Ka Kit Tung. Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 293-314. doi: 10.3934/dcdsb.2007.7.293

[18]

Tongtong Liang, Yejuan Wang. Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4697-4726. doi: 10.3934/dcdsb.2020309

[19]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093

[20]

Samira Amraoui, Didier Auroux, Jacques Blum, Emmanuel Cosme. Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022058

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]