Variational Data Assimilation in the Mathematical Model of the Black Sea Dynamics

V. I. Agoshkov, V. P. Shutyaev, E. I. Parmuzin✉, N. B. Zakharova, T. O. Sheloput, N. R. Lezina

Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russian Federation

✉ e-mail: eparmuzin@gmail.com

Abstract

Purpose. In order to simulate the sea hydrothermodynamics, the problem of variational assimilation of the sea surface temperature data is solved. The data assimilation permits to adjust the numerical model calculations to the measurement data obtained in the environment under study.

Methods and Results. The mathematical model of hydrothermodynamics of the Black and Azov seas developed at the Institute of Numerical Mathematics, RAS, and represented in the sigma coordinate system is considered. The distinctive feature of the model consists in applying the splitting method to physical processes and spatial coordinates that can significantly simplify the variational data assimilation algorithm. The problem of variational assimilation of the sea surface temperature data is formulated. A cost functional has been introduced; it includes the control function – heat flux at the sea upper boundary and satellite observations of the sea surface temperature. The necessary condition for the functional minimum is reformulated through the optimality system including the direct and adjoint problems, and the control condition. Using the variational assimilation of the satellite-derived observations, the algorithm for solving the stated problem was developed. It takes into account the observational errors’ covariance matrix calculated based on the statistical characteristics of the sea surface temperature observational data. The algorithm implies a sequential solution of the optimality system in the iterative process with the specially selected iterative parameter. The results of numerical solution of this problem are represented by the example of the Black and Azov seas.

Conclusions. The results of numerical modeling with the observational data assimilation and without it are compared; efficiency of the observational data assimilation procedures is shown. Influence of the sea surface temperature assimilation upon the other system parameters is investigated. It is shown that when assimilating the sea surface temperature, only temperature in the upper layers is affected, whereas, provided that the depth is sufficient, the profile in the lower layers remains practically unchanged. The impact on the other system parameters is either minimal or not manifested at all.

Keywords

mathematical model, variational assimilation, numerical algorithm, observations, hydrothermodynamics, sea surface temperature

Acknowledgements

The work was supported by the Russian Science Foundation project No 19-71-20035 (problem statement, algorithm development, numerical experiments) and the grant of the President of the Russian Federation No MK-3228.2018.5 (processing of observation data and their preparation for model calculations).

Original russian text

Original Russian Text © V.I. Agoshkov, V.P. Shutyaev, E.I. Parmuzin, N.B. Zakharova, T.O. Sheloput, N.R. Lezina, 2019, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 35, Iss. 6, pp. 585–599 (2019)

For citation

Agoshkov, V.I., Shutyaev, V.P., Parmuzin, E.I, Zakharova, N.B., Sheloput, T.O. and Lezina, N.R., 2019. Variational Data Assimilation in the Mathematical Model of the Black Sea Dynamics. Physical Oceanography, 26(6), pp. 515-527. doi:10.22449/1573-160X-2019-6-515-527

DOI

10.22449/1573-160X-2019-6-515-527

References

  1. Marchuk, G.I., Paton, B.E., Korotaev, G.K. and Zalesny, V.B., 2013. Data-Computing Technologies: A New Stage in the Development of Operational Oceanography. Izvestiya, Atmospheric and Oceanic Physics, [e-journal] 49(6), pp. 579-591. https://doi.org/10.1134/S000143381306011X
  2. Marchuk, G.I. and Paton, B.E., 2012. The Black Sea as a Simulation Ocean Model. Russian Journal of Numerical Analysis and Mathematical Modelling, [e-journal] 27(1), pp. 1-4. doi:10.1515/rnam-2012-0001
  3. Le Dimet, F.X. and Talagrand, O., 1986. Variational Algorithms for Analysis and Assimilation of Meteorological Observations: Theoretical Aspects. Tellus A: Dynamic Meteorology and Oceanography, [e-journal] 38(2), pp. 97-110. https://doi.org/10.3402/tellusa.v38i2.11706
  4. Ivanov, V.A., 2014. Spatial and Temporal Variability and Monitoring of Hydrophysical Fields of the Black Sea. Izvestiya, Atmospheric and Oceanic Physics, [e-journal] 50(1), pp. 26-34. https://doi.org/10.1134/S000143381306008X
  5. Mogensen, K., Alonso-Balmaseda, M., Weaver, A., Martin, M. and Vidard, A., 2009. NEMOVAR: A Variational Data Assimilation System for the NEMO Ocean Model. Shinfield Park, England: ECMWF, 8 p. doi:10.21957/3yj3mh16iq
  6. Talagrand, O., 2010. Variational Assimilation. In: W. Lahoz, B. Khattatov and R. Menard eds., 2010. Data Assimilation. Berlin, Heidelberg: Springer, pp. 41–67. doi:10.1007/978-3- 540-74703-1
  7. Auroux, D. and Blum, J., 2005. Data Assimilation Methods for an Oceanographic Problem. In: V. Capasso, J. Périaux eds., 2005. Multidisciplinary Methods for Analysis Optimization and Control of Complex Systems. Mathematics in Industry (The European Consortium for Mathematics in Industry), vol. 6. Berlin, Heidelberg: Springer, pp. 180-194. doi:10.1007/3- 540-27167-8_4
  8. Knysh, V.V., Demyshev, S.G., Korotaev, G.K. and Sarkisyan, A.S., 2007. Method and Results of Assimilation of Climatic Data on Temperature, Salinity, and Sea Level into a Numerical Model of the Black. Izvestiya, Atmospheric and Oceanic Physics, [e-journal] 43(3), pp. 363-377. https://doi.org/10.1134/S0001433807030115
  9. Korotaev, G.K., Knysh, V.V., Lishaev, P.N. and Demyshev, S.G., 2018. Application of the Adaptive Statistics Method for Reanalysis of the Black Sea Fields Including Assimilation of the Temperature and Salinity Pseudo-measurements in the Model. Physical Oceanography, [e-journal] 25(1), pp. 36-51. doi:10.22449/1573-160X-2018-1-36-51
  10. Agoshkov, V.I., 2003. Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics. Moscow: INM RAS, 255 p. (in Russian).
  11. Lions, J.L., 1968. Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles. Paris: Dunod Gauthier-Villars, 426 p.
  12. Marchuk, G.I., 1995. Adjoint Equations and Analysis of Complex Systems. Dordrecht: Springer, 468 p. doi:10.1007/978-94-017-0621-6
  13. Marchuk, G.I., 1974. Numerical Methods in Weather Prediction. New York: Academic Press, 288 p.
  14. Marchuk, G.I., Agoshkov, V.I., Shutyaev, V.P., 1996. Adjoint Equations and Perturbation Algorithms in Nonlinear Problems. New York: CRC Press Inc., 288 p.
  15. Marchuk, G.I., Dymnikov, V.P., Zalesny, V.B., Lykossov, V. and Galin, V.Ya, 1984. Mathematical Modeling of the General Circulation of the Atmosphere and Ocean. Leningrad: Gidrometeoizdat, 320 p. (in Russian).
  16. Marchuk, G.I. and Kagan, B.A., 1989. Dynamics of Ocean Tides. Dordrecht: Kluwer Academic Publishers, 327 p.
  17. Zalesny, V.B., Zakharova, N.B., Gusev, A.V., 2011. Four-Dimensional Problem of Variational Initialization of Hydrophysical Fields of the World Ocean. Russian Journal of Numerical Analysis and Mathematical Modelling, [e-journal] 26(2), pp. 209-229. doi:10.1515/RJNAMM.2011.012
  18. Järvinen, H., Andersson, E. and Bouttier, F., 1999. Variational Assimilation of Time Sequences of Surface Observations with Serially Correlated Errors. Tellus A: Dynamic Meteorology and Oceanography, [e-journal] 51(4), pp. 469-488. doi:10.3402/tellusa.v51i4.13963
  19. Le Dimet, F.X., Navon, I.M. and Ştefănescu, R., 2017. Variational Data Assimilation: Optimization and Optimal Control. In: S. Park, L. Xu eds., 2017. Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (Vol. III). Cham: Springer, pp. 1-53. doi:10.1007/978-3-319-43415-5_1
  20. Moore, A.M., Arango, H.G., Broquet, G., Powell, B.S., Weaver, A.T. and Zavala-Garay, J., 2011. The Regional Ocean Modeling System (ROMS) 4-dimensional Variational Data Assimilation Systems: Part I – System Overview and Formulation. Progress in Oceanography, [e-journal] 91(1), pp. 34-49. doi:10.1016/j.pocean.2011.05.004
  21. Rabier, F., 2005. Overview of Global Data Assimilation Developments in Numerical Weather-Prediction Centres. Quarterly Journal of the Royal Meteorological Society, [e- journal] 131(613), pp. 3215-3233. doi:10.1256/qj.05.129
  22. Trémolet, Y., 2007. Incremental 4D-Var Convergence Study. Tellus A: Dynamic Meteorology and Oceanography, [e-journal] 59(5), pp. 706-718. https://doi.org/10.1111/j.1600- 0870.2007.00271.x
  23. Trémolet, Y., 2008. Computation of Observation Sensitivity and Observation Impact in Incremental Variational Data Assimilation. Tellus A: Dynamic Meteorology and Oceanography, [e-journal] 60(5), pp. 964-978. doi:10.1111/j.1600-0870.2008.00349.x
  24. Akella, S. and Navon, I.M., 2009. Different Approaches to Model Error Formulation in 4D- Var: a Study with High-Resolution Advection Schemes. Tellus A: Dynamic Meteorology and Oceanography, [e-journal] 61(1), pp. 112-128. doi:10.1111/j.1600-0870.2007.00362.x
  25. Zalesny, V.B., Diansky, N.A., Fomin, V.V., Moshonkin, S.N. and Demyshev, S.G., 2012. Numerical Model of the Circulation of the Black Sea and the Sea of Azov. Russian Journal of Numerical Analysis and Mathematical Modelling, [e-journal] 27(1), pp. 95-112. doi:10.1515/rnam-2012-0006
  26. Agoshkov, V.I., Assovskii, M.V., Zalesny, V.B., Zakharova, N.B., Parmuzin, E.I. and Shutyaev, V.P., 2015. Variational Assimilation of Observation Data in the Mathematical Model of the Black Sea Taking into Account the Tide-Generating Forces. Russian Journal of Numerical Analysis and Mathematical Modelling, [e-journal] 30(3), pp. 129-142. https://doi.org/10.1515/rnam-2015-0013
  27. Marchuk, G.I., Dymnikov, V.P. and Zalesny, V.B., 1987. Mathematical Models in Geophysical Hydrodynamics and Numerical Methods for Their Implementation. Leningrad: Gidrometeoizdat, 295 p. (in Russian).
  28. Agoshkov, V.I., Gusev, A.V., Diansky, N.A. and Oleinikov, R.V., 2007. An Algorithm for the Solution of the Ocean Hydrothermodynamics Problem with Variational Assimilation of the Sea Level Function Data. Russian Journal of Numerical Analysis and Mathematical Modelling, [e-journal] 22(2), pp. 133-161. doi:10.1515/RJNAMM.2007.007
  29. Marchuk, G.I., Rusakov, A.S., Zalesny, V.B. and Diansky, N.A., 2005. Splitting Numerical Technique with Application to the High Resolution Simulation of the Indian Ocean Circulation. Pure and Applied Geophysics, [e-journal] 162(8-9), pp. 1407-1429. doi:10.1007/s00024-005-2677-8
  30. Marchuk, G.I., 1989. Methods of Computational Mathematics. Moscow: Nauka, 608 p. (in Russian).
  31. Samarskii, A.A., 1977. Theory of Difference Schemes. Moscow: Nauka, 657 p. (in Russian).
  32. Parmuzin, E.I. and Agoshkov, V.I., 2012. Numerical Solution of the Variational Assimilation Problem for Sea Surface Temperature in the Model of the Black Sea Dynamics. Russian Journal of Numerical Analysis and Mathematical Modelling, [e-journal] 27(1), pp. 69-94. doi:10.1515/rnam-2012-0005
  33. Buongiorno Nardelli, B., Tronconi, C., Pisano, A. and Santoleri, R., 2013. High and Ultra- High Resolution Processing of Satellite Sea Surface Temperature Data over Southern European Seas in the Framework of MyOcean Project. Remote Sensing of Environment, [e- journal] 129, pp. 1-16. doi:10.1016/j.rse.2012.10.012
  34. Zakharova, N.B., 2017. [Problems of Observational Data Processing in the Tasks of Mathematical Modeling of Marine Environments]. In: Applied Aspects of Geology, Geophysics and Geoecology Using Modern Information Technologies: Materials of 4th International Scientific and Practical Conference. Maikop: Izd-vo IP Kucherenko V.O., pp. 211-219.
  35. Zakharova, N.B., 2016. Verification of the Sea Surface Temperature Observation Data. Current Problems in Remote Sensing of the Earth from Space, 13(3), pp. 106-113. doi:10.21046/2070-7401-2016-13-3-106-113 (in Russian).
  36. Agoshkov, V.I., Parmuzin, E.I., Zakharova, N.B. and Shutyaev, V.P., 2018. Variational Assimilation with Covariance Matrices of Observation Data Errors for the Model of the Baltic Sea Dynamics. Russian Journal of Numerical Analysis and Mathematical Modelling, [e- journal] 33(3), pp. 149-160. doi:10.1515/rnam-2018-0013

Download the article (PDF)