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Computational Geosciences

, Volume 21, Issue 3, pp 393–410 | Cite as

Downscaling the 2D Bénard convection equations using continuous data assimilation

  • M. U. Altaf
  • E. S. Titi
  • T. Gebrael
  • O. M. Knio
  • L. Zhao
  • M. F. McCabe
  • I. HoteitEmail author
Original Paper

Abstract

We consider a recently introduced continuous data assimilation (CDA) approach for downscaling a coarse resolution configuration of the 2D Bénard convection equations into a finer grid. In this CDA, a nudging term, estimated as the misfit between some interpolants of the assimilated coarse-grid measurements and the fine-grid model solution, is added to the model equations to constrain the model. The main contribution of this study is a performance analysis of CDA for downscaling measurements of temperature and velocity. These measurements are assimilated either separately or simultaneously, and the results are compared against those resulting from the standard point-to-point nudging approach (NA). Our numerical results suggest that the CDA solution outperforms that of NA, always converging to the true solution when the velocity is assimilated as has been theoretically proven. Assimilation of temperature measurements only may not always recover the true state as demonstrated in the case study. Various runs are conducted to evaluate the sensitivity of CDA to noise in the measurements, the size, and the time frequency of the measured grid, suggesting a more robust behavior of CDA compared to that of NA.

Keywords

Continuous data assimilation Bénard convection equations Dynamical downscaling 

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References

  1. 1.
    Timbal, A., Dufour, A., McAvaney, B.: An estimate of future climate change for Western France using a statistical downscaling technique. Climate Dyn. 20, 807–823 (2003)Google Scholar
  2. 2.
    Hewitson, B. C., Crane, R. G.: Consensus between GCM climate change projections with empirical downscaling: precipitation downscaling over South Africa. Int. J. Climatol. 26, 1315–1337 (2006)CrossRefGoogle Scholar
  3. 3.
    Gutzler, D. S., Robbins, T. O.: Climate variability and projected change in the western United States: regional downscaling and drought statistics. Clim. Dyn. 37, 835–849 (2011)CrossRefGoogle Scholar
  4. 4.
    Jha, S. K., Mariethoz, G., Evans, J., McCabe, M. F., Sharma, A.: A space and time scale-dependent nonlinear geostatistical approach for downscaling daily precipitation and temperature. Water Resour. Res. 51(8), 6244–6261 (2015)CrossRefGoogle Scholar
  5. 5.
    Guiterrez, J. M., San-Martin, D., Brands, S., Manzanas, R., Herrera, S.: Reassessing statistical downscaling techniques for their robust application under climate change conditions. J. Climate 26, 171–188 (2013)CrossRefGoogle Scholar
  6. 6.
    McGregor, J.: Regional climate modelling. Meteorol. Atmos. Phys. 63, 105–117 (1997)CrossRefGoogle Scholar
  7. 7.
    Liu, P., Tsimpidi, A. P., Hu, Y., Stone, B., Russell, A. G., Nenes, A.: Differences between downscaling with spectral and grid nudging using WRF. Atmos. Chem. Phys. 12, 3601–3610 (2012)CrossRefGoogle Scholar
  8. 8.
    Feser, F., Barcikowska, M.: The influence of spectral nudging on typhoon formation in regional climate models. Environ. Res. Lett. 1, 014024 (2012)CrossRefGoogle Scholar
  9. 9.
    Lo, C. J., Yang, Z. L., Pielke, R. A.: Assessment of three dynamical climate downscaling methods using the weather research and forecasting (WRF) model. J. Geophys. Res. 113, D09112 (2012)Google Scholar
  10. 10.
    Wilby, R., Wigley, L. T. M. L.: Downscaling general circulation model output: a review of methods and limitations. Prog. Phys. Geog. 21, 530—548 (1997)CrossRefGoogle Scholar
  11. 11.
    Murphy, J.: An evaluation of statistical and dynamical techniques for downscaling local climate. J. Climate 12, 2256–2284 (1999)CrossRefGoogle Scholar
  12. 12.
    Bennet, A.: Inverse Methods in Physical Oceanography, p 346. Cambridge University Press, Cambridge, UK (1992)CrossRefGoogle Scholar
  13. 13.
    Altaf, M. U., Butler, T., Mayo, T., Luo, X., Dawson, C., Heemink, A. W., Hoteit, I.: A comparison of ensemble Kalman filters for storm surge assimilation. Mon. Wea Rev. 142, 2889– 2914 (2014)CrossRefGoogle Scholar
  14. 14.
    Altaf, M. U., Ambrozic, M., McCabe, M. F., Hoteit, I.: A study of reduced-order 4DVAR with a finite element shallow water model. Int. J. Numer. Methods Fluids 80, 631–647 (2016)CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Daley, R.: Atmospheric Data Analysis. Cambridge Atmospheric and Space Science Series, Cambridge University Press, Cambridge (1991)Google Scholar
  17. 17.
    Henshaw, W. D., Kreiss, H. O., Ystrom, J.: Numerical experiments on the interaction between the large and small scale motion of the Navier Stokes equations. SIAM J. Multiscale Modeling Simulation 1, 119–149 (2003)CrossRefGoogle Scholar
  18. 18.
    Olson, E., Titi, E. S.: Determining modes for continuous data assimilation in 2D turbulence. J. Stat. Phys. 113, 799– 840 (2003)CrossRefGoogle Scholar
  19. 19.
    Olson, E., Titi, E. S.: Determining modes and Grashoff number in 2D turbulence. Theor. Comput. Fluid Dyn. 22, 327–339 (2009)CrossRefGoogle Scholar
  20. 20.
    Korn, P.: Data assimilation for the Navier-Stokes- α equations. Physica D 238, 1957–1974 (2009)CrossRefGoogle Scholar
  21. 21.
    Hayden, K., Olson, E., Titi, E.S.: Discrete data assimilation in the Lorenz and 2D Navier–Stokes equations. Physica D 240, 1416–1425 (2011)CrossRefGoogle Scholar
  22. 22.
    Azouani, A., Olson, E., Titi, E. S.: Continuous data assimilation using general interpolant observables. J. Nonlinear Sci. 24, 277–304 (2014)CrossRefGoogle Scholar
  23. 23.
    Bessaih, H., Olson, E., Titi, E. S.: Continuous assimilation of data with stochastic noise. Nonlinearity 28, 729–753 (2015)CrossRefGoogle Scholar
  24. 24.
    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, 1–23 (2016)CrossRefGoogle Scholar
  25. 25.
    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, 492–506 (2016)CrossRefGoogle Scholar
  26. 26.
    Gesho, M., Olson, E., Titi, E. S.: A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations. Communications in Computational Physics 19, 1094–1110 (2016)Google Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    Ghil, M., Shkoller, B., Yangarber, V.: A balanced diagnostic system compatible with a barotropic prognostic model. Mon. Wea. Rev. 105, 1223–1238 (1977)CrossRefGoogle Scholar
  29. 29.
    Ghil, M., Halem, M., Atlas, R.: Time-continuous assimilation of remote-sounding data and its effect on weather forecasting. Mon. Wea. Rev. 106, 140–171 (1978)Google Scholar
  30. 30.
    Hoke, J., Anthes, R.: The initialization of numerical models by a dynamic relaxation technique. Mon. Wea. Rev. 104, 1551–1556 (1976)CrossRefGoogle Scholar
  31. 31.
    Aswatha, C. J., Gowda, G., Sridhara, S. N., Seetharamu, K. N.: Buoyancy driven heat transfer in cavities subjected to thermal boundary conditions at bottom wall. Journal of Applied Fluid Mechanics 5, 43–53 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • M. U. Altaf
    • 1
  • E. S. Titi
    • 2
  • T. Gebrael
    • 3
  • O. M. Knio
    • 1
  • L. Zhao
    • 4
  • M. F. McCabe
    • 1
  • I. Hoteit
    • 1
    Email author
  1. 1.King Abdullah University of Science and TechnologyThuwalSaudi Arabia
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.American University of BeirutBeirutLebanon
  4. 4.Georgia Institute of TechnologyAtlantaUSA

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