Journal of Nonlinear Science

, Volume 27, Issue 3, pp 1065–1087 | Cite as

Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone

  • Aseel FarhatEmail author
  • Evelyn Lunasin
  • Edriss S. Titi


In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional Bénard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress-free boundary conditions on the walls, and the fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data are error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity.


Bénard convection Boussinesq system Continuous data assimilation Signal synchronization Nudging Downscaling 

Mathematics Subject Classification

35Q30 93C20 37C50 76B75 34D06 



The work of A.F. is supported in part by the NSF Grant DMS-1418911. The work of E.L. is supported in part by the ONR Grant N0001416WX01475 and the ONR Grant N0001416WX00796. The work of E.S.T. is supported in part by the ONR Grant N00014-15-1-2333 and the NSF Grants DMS-1109640 and DMS-1109645.


  1. Albanez, D., Nussenzveig-Lopes, H., Titi, E.S.: Continuous data assimilation for the three-dimensional Navier–Stokes-\(\alpha \) model. Asymptot. Anal. 97(1–2), 139–164 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Altaf, M.U., Titi, E.S., Gebrael, T., Knio, O., Zhao, L., McCabe, M.F., Hoteit, I.: Downscaling the 2D Bénard convection equations using continuous data assimilation. Computational Geosciences (COMG) (2015). arXiv:1512.04671
  3. Azouani, A., Titi, E.S.: Feedback control of nonlinear dissipative systems by finite determining parameters: a reaction-diffusion paradigm. Evol. Equ. Control Theory 3(4), 579–594 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Azouani, A., Olson, E., Titi, E.S.: Continuous data assimilation using general interpolant observables. J. Nonlinear Sci. 24(2), 277–304 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bessaih, H., Olson, E., Titi, E.S.: Continuous assimilation of data with stochastic noise. Nonlinearity 28, 729–753 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cao, C., Kevrekidis, I., Titi, E.S.: Numerical criterion for the stabilization of steady states of the Navier–Stokes equations. Indiana Univ. Math. J. 50, 37–96 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Charney, J., Halem, J., Jastrow, M.: Use of incomplete historical data to infer the present state of the atmosphere. J. Atmos. Sci. 26, 1160–1163 (1969)CrossRefGoogle Scholar
  8. Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1988)Google Scholar
  9. Farhat, A., Jolly, M.S., Titi, E.S.: Continuous data assimilation for the 2D Bénard convection through velocity measurements alone. Phys. D 303, 59–66 (2015)MathSciNetCrossRefGoogle Scholar
  10. Farhat, A., Lunasin, E., Titi, E.S.: Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field. J. Math. Fluid Mech. 18(1), 1–23 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Farhat, A., Lunasin, E., Titi, E.S.: A note on abridged continuous data assimilation for the 3D subgrid scale \(\alpha \)-models of turbulence, PreprintGoogle Scholar
  12. Farhat, A., Lunasin, E., Titi, E.S.: Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements. J. Math. Anal. Appl. 438(1), 492–506 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Foias, C., Manley, O., Temam, R.: Attractors for the Bénard problem: existence and physical bounds on their fractal dimension. Nonlinear Anal. Theory Methods Appl. 11, 939–967 (1987)CrossRefzbMATHGoogle Scholar
  14. Foias, C., Mondaini, C., Titi, E.S.: A discrete data assimilation scheme for the solutions of the 2D Navier–Stokes equations and their statistics. SIAM J. Appl. Dyn. Syst. 15(4), 2109–2142 (2000)CrossRefzbMATHGoogle Scholar
  15. Gesho, M., Olson, E., Titi, E.S.: A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations. Commun. Comput. Phys. 19(4), 1094–1110 (2016)MathSciNetGoogle Scholar
  16. Ghil, M., Shkoller, B., Yangarber, V.: A balanced diagnostic system compatible with a barotropic prognostic model. Mon. Weather Rev. 105, 1223–1238 (1977)CrossRefGoogle Scholar
  17. Ghil, M., Halem, M., Atlas, R.: Time-continuous assimilation of remote-sounding data and its effect on weather forecasting. Mon. Weather Rev. 107, 140–171 (1978)CrossRefGoogle Scholar
  18. Jones, D.A., Titi, E.S.: Determining finite volume elements for the 2D Navier–Stokes equations. Phys. D 60, 165–174 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Leunberger, D.: An introduction to observers. IEEE. Trans. Autom. Control 16, 596–602 (1971)CrossRefGoogle Scholar
  20. Lunasin, E., Titi, E.S.: Finite determining parameters feedback control for distributed nonlinear dissipative systems: a computational study, arXiv:1506.03709 [math.AP] (2015)
  21. Markowich, P., Titi, E.S., Trabelsi, S.: Continuous data assimilation for the three-dimensional Brinkman–Forchheimer–Extended Darcy model. Nonlinearity 29(4), 1292–1328 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Nijmeijer, H.: A dynamic control view of synchronization. Phys. D 154, 219–228 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Robinson, J.C.: Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2001)Google Scholar
  24. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, vol. 68. Springer, New York (1997)CrossRefGoogle Scholar
  25. Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, second ed., CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1995)Google Scholar
  26. Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI (2001), Reprint of the 1984 editionGoogle Scholar
  27. Thau, F.E.: Observing the state of non-linear dynamic systems. Int. J. Control 17, 471–479 (1973)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Aseel Farhat
    • 1
    Email author
  • Evelyn Lunasin
    • 2
  • Edriss S. Titi
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of MathematicsUnited States Naval AcademyAnnapolisUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA
  4. 4.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations