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Sensitivity of Functionals of the Solution to the Variational Assimilation Problem to the Input Data on the Heat Flux for a Model of Sea Thermodynamics

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Abstract

For the mathematical model of the sea thermodynamics developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, the problem of variational assimilation of observational data in order to recover heat fluxes on the sea surface is considered. The sensitivity of functionals of the solution to the input data on the heat flux in this problem is studied, and the results of numerical experiments for the model of Black Sea dynamics are presented.

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REFERENCES

  1. G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems (Kluwer, Dordrecht, 1995).

    Book  MATH  Google Scholar 

  2. J. L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles (Dunod, Paris, 1968).

    MATH  Google Scholar 

  3. Y. K. Sasaki, “An objective analysis based on the variational method,” J. Meteor. Soc. Jpn. 36, 77–88 (1958).

    Article  Google Scholar 

  4. V. V. Penenko and N. N. Obraztsov, “Variational initialization method for the fields of meteorological elements,” Meteorol. Gidrol., No. 11, 1–11 (1976).

  5. V. V. Penenko, Methods for Numerical Modeling of Atmospheric Processes (Gidrometeoizdat, Leningrad, 1981) [in Russian].

    Google Scholar 

  6. F.-X. Le Dimet and O. Talagrand, “Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects,” Tellus A 38, 97–110 (1986).

    Article  Google Scholar 

  7. V. I. Agoshkov, Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics (Inst. V-ychisl. Mat. Ross. Akad. Nauk, Moscow, 2003) [in Russian].

    MATH  Google Scholar 

  8. K. Mogensen, M. A. Balmaseda, A. T. Weaver, M. Martin, and A. Vidard, “NEMOVAR: A variational data assimilation system for the NEMO ocean model,” ECMWF Tech. Memo., No. 120 (2009).

  9. V. V. Penenko, Doctoral Dissertation in Mathematics and Physics (Inst. Numer. Math. Math. Geophys., Sib. Branch, Russ. Acad. Sci., Novosibirsk, 2021).

  10. F.-X. Le Dimet, H. E. Ngodock, B. Luong, and J. Verron, “Sensitivity analysis in variational data assimilation,” J. Meteorol. Soc. Jpn. B 75 (1), 245–255 (1997).

    Article  Google Scholar 

  11. F.-X. Le Dimet, I. M. Navon, and D. N. Daescu, “Second-order information in data assimilation,” Mon. Weather Rev. 130 (3), 629–648 (2002).

    Article  Google Scholar 

  12. F.-X. Le Dimet and V. Shutyaev, “On deterministic error analysis in variational data assimilation,” Nonlinear Processes Geophys. 12, 481–490 (2005).

    Article  Google Scholar 

  13. I. Gejadze, F.-X. Le Dimet, and V. P. Shutyaev, “On analysis error covariances in variational data assimilation,” SIAM J. Sci. Comput. 30 (4), 1847–1874 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Gejadze, F.-X. Le Dimet, and V. P. Shutyaev, “On optimal solution error covariances in variational data assimilation problems,” J. Comput. Phys. 229, 2159–2178 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  15. I. Gejadze, V. P. Shutyaev, and F.-X. Le Dimet, “Analysis error covariance versus posterior covariance in variational data assimilation,” Q. J. R. Meteorol. Soc. 139, 1826–1841 (2013).

    Article  Google Scholar 

  16. V. I. Agoshkov, E. I. Parmuzin, and V. P. Shutyaev, “Observational data assimilation in the problem of Black Sea circulation and sensitivity analysis of its solution,” Izv. Atmos. Ocean. Phys. 49 (6), 592–602 (2013).

    Article  Google Scholar 

  17. V. P. Shutyaev and F.-X. Le Dimet, “Sensitivity of functionals of variational data assimilation problems,” Dokl. Math. 99 (3), 295–298 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. V. Alekseev and V. B. Zalesny, “Numerical model of large-scale ocean dynamics,” in Computational Processes and Systems (Nauka, Moscow, 1993), pp. 232–253 [in Russian].

    Google Scholar 

  19. G. I. Marchuk, V. P. Dymnikov, and V. B. Zalesny, Mathematical Models in Geophysical Hydrodynamics and Numerical Methods for Their Implementation (Gidrometeoizdat, Leningrad, 1987) [in Russian].

    Google Scholar 

  20. V. I. Agoshkov, A. V. Gusev, N. A. Diansky, and R. V. Oleinikov, “An algorithm for the solution of the ocean hydrothermodynamics problem with variational assimilation of the sea level function data,” Russ. J. Numer. Anal. Math. Model. 22 (2), 133–161 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  21. V. I. Agoshkov, E. I. Parmuzin, and V. P. Shutyaev, “Numerical algorithm for variational assimilation of sea surface temperature data,” Comput. Math. Math. Phys. 48 (8), 1293–1312 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  22. V. B. Zalesny, N. A. Diansky, V. V. Fomin, S. N. Moshonkin, and S. G. Demyshev, “Numerical model of the circulation of the Black Sea and the Sea of Azov,” Russ. J. Numer. Anal. Math. Model. 27 (1), 95–112 (2012).

    Article  MATH  Google Scholar 

  23. V. Shutyaev, E. Parmuzin, and I. Gejadze, “Stability analysis of functionals in variational data assimilation with respect to uncertainties of input data for a sea thermodynamics model,” Russ. J. Numer. Anal. Math. Model. 36 (6), 347–357 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. N. Tikhonov, “On the solution of ill-posed problems and regularization method,” Dokl. Akad. Nauk SSSR 151 (3), 501–504 (1963).

    MathSciNet  Google Scholar 

  25. D. G. Cacuci, “Sensitivity theory for nonlinear systems: II. Extensions to additional classes of responses,” J. Math. Phys. 22, 2803–2812 (1981).

    Article  MathSciNet  Google Scholar 

  26. V. P. Shutyaev, Control Operators and Iterative Algorithms in Variational Data Assimilation (Nauka, Moscow, 2001) [in Russian].

    Book  MATH  Google Scholar 

  27. N. A. Diansky, A. V. Bagno, and V. B. Zalesny, “Sigma model of global ocean circulation and its sensitivity to variations in wind stress,” Izv. Atmos. Ocean. Phys. 38 (4), 477–494 (2002).

    Google Scholar 

  28. E. A. Lupyan, A. A. Matveev, I. A. Uvarov, T. Yu. Bocharova, O. Yu. Lavrova, and M. I. Mityagina, “See the Sea satellite service—instrument for studying processes and phenomena on the sea surface,” Sovrem. Probl. Distants. Zondir. Zemli iz Kosmosa 9 (2), 251–261 (2012).

  29. N. B. Zakharova, V. I. Agoshkov, and E. I. Parmuzin, “The new method of ARGO buoys system observation data interpolation,” Russ. J. Numer. Anal. Math. Model. 28 (1), 67–84 (2013).

    Article  MATH  Google Scholar 

  30. H. Hersbach et al., “The ERA5 global reanalysis,” Q. J. R. Meteorol. Soc. 146, 1999–2049 (2020).

    Article  Google Scholar 

  31. V. I. Agoshkov, V. P. Shutyaev, E. I. Parmuzin, N. B. Zakharova, T. O. Sheloput, and N. R. Lezina, “Variational data assimilation in the mathematical model of the Black Sea dynamics,” Phys. Oceanogr. 26 (6), 515–527 (2019).

    Article  Google Scholar 

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ACKNOWLEDGMENTS

We are grateful to the referee for useful remarks, which made it possible to improve the presentation of the results and the material.

Funding

This work was supported by the Russian Science Foundation (project no. 20-11-20057, studies in Sections 1 and 2) and the Department of the Moscow Center for Fundamental and Applied Mathematics at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (Agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2022- 286).

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Authors and Affiliations

  1. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, 119333, Moscow, Russia

    E. I. Parmuzin & V. P. Shutyaev

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  1. E. I. Parmuzin
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  2. V. P. Shutyaev
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Correspondence to E. I. Parmuzin or V. P. Shutyaev.

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Translated by E. Chernokozhin

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Parmuzin, E.I., Shutyaev, V.P. Sensitivity of Functionals of the Solution to the Variational Assimilation Problem to the Input Data on the Heat Flux for a Model of Sea Thermodynamics. Comput. Math. and Math. Phys. 63, 623–632 (2023). https://doi.org/10.1134/S0965542523040127

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