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Downscaling the 2D Bénard convection equations using continuous data assimilation

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.

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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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  6. 6.

    McGregor, J.: Regional climate modelling. Meteorol. Atmos. Phys. 63, 105–117 (1997)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  11. 11.

    Murphy, J.: An evaluation of statistical and dynamical techniques for downscaling local climate. J. Climate 12, 2256–2284 (1999)

    Article  Google Scholar 

  12. 12.

    Bennet, A.: Inverse Methods in Physical Oceanography, p 346. Cambridge University Press, Cambridge, UK (1992)

    Book  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  18. 18.

    Olson, E., Titi, E. S.: Determining modes for continuous data assimilation in 2D turbulence. J. Stat. Phys. 113, 799– 840 (2003)

    Article  Google Scholar 

  19. 19.

    Olson, E., Titi, E. S.: Determining modes and Grashoff number in 2D turbulence. Theor. Comput. Fluid Dyn. 22, 327–339 (2009)

    Article  Google Scholar 

  20. 20.

    Korn, P.: Data assimilation for the Navier-Stokes- α equations. Physica D 238, 1957–1974 (2009)

    Article  Google 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)

    Article  Google Scholar 

  22. 22.

    Azouani, A., Olson, E., Titi, E. S.: Continuous data assimilation using general interpolant observables. J. Nonlinear Sci. 24, 277–304 (2014)

    Article  Google Scholar 

  23. 23.

    Bessaih, H., Olson, E., Titi, E. S.: Continuous assimilation of data with stochastic noise. Nonlinearity 28, 729–753 (2015)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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 

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Correspondence to I. Hoteit.

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Altaf, M.U., Titi, E.S., Gebrael, T. et al. Downscaling the 2D Bénard convection equations using continuous data assimilation. Comput Geosci 21, 393–410 (2017). https://doi.org/10.1007/s10596-017-9619-2

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Keywords

  • Continuous data assimilation
  • Bénard convection equations
  • Dynamical downscaling