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Application of the Variational Method to Lava Flow Modelling

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Data-Driven Numerical Modelling in Geodynamics: Methods and Applications

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Abstract

In this chapter, we present an application of the variational data assimilation method to the problem for determination of thermal and dynamic characteristics of lava flow from thermal measurements at lava’s upper surface. Assuming that the temperature and the heat flow are known at the lava’s upper surface, the missing condition at the lower surface of the lava is determined at first, and then the flow characteristics (temperature and flow velocity) are resolved in the entire model domain.

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References

  • Costa A, Macedonio G (2005a) Computational modeling of lava flows: a review. In: Manga M, Ventura G (eds) Kinematics and dynamics of lava flows. Geological Society of America Special Papers 396, Boulder, pp 209–218

    Chapter  Google Scholar 

  • Costa A, Macedonio G (2005b) Numerical simulation of lava flows based on depth-averaged equations. Geophys Res Lett 32:L05304. doi:10.1029/2004GL021817

    Google Scholar 

  • Cutter S, Ismail-Zadeh A, Alcántara-Ayala I, Altan O, Baker DN, Briceño S, Gupta H, Holloway A, Johnston D, McBean GA, Ogawa Y, Paton D, Porio E, Silbereisen RK, Takeuchi K, Valsecchi GB, Vogel C, Wu G (2015) Pool knowledge to stem losses from disasters. Nature 522:277–279

    Article  Google Scholar 

  • Fletscher R (2000) Practical methods of optimization, 2nd edn. Wiley, Chichester

    Book  Google Scholar 

  • Flynn LP, Harris AJL, Wright R (2001) Improved identification of volcanic features using Landsat 7 ETM+. Remote Sens Environ 78:180–193

    Article  Google Scholar 

  • Griffiths RW (2000) The dynamics of lava flows. Annu Rev Fluid Mech 32:477–518

    Article  Google Scholar 

  • Harris AJL, Flynn LP, Matias O, Rose WI, Cornejo J (2004) The evolution of an active silicic lava flow field: an ETM+ perspective. J Volcanol Geotherm Res 135:147–168

    Article  Google Scholar 

  • Harris AJL, Dehn J, Calvari S (2007) Lava effusion rate definition and measurement: a review. Bull Volcanol 70:1–22

    Article  Google Scholar 

  • Hidaka M, Goto A, Umino S, Fujita E (2005) VTFS project: development of the lava flow simulation code LavaSIM with a model for three-dimensional convection, spreading, and solidification. Geochem Geophys Geosyst 6:Q07008. doi:10.1029/2004GC000869

    Article  Google Scholar 

  • Ishihara K, Iguchi M, Kamo K (1989) Numerical simulation of lava flows on some volcanoes in Japan. In: Fink J (ed) Lava flows and domes, vol 2, IAVCEI Proc. Volcanol. Springer, New York

    Google Scholar 

  • Ismail-Zadeh A, Tackley P (2010) Computational methods for geodynamics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Ismail-Zadeh A, Korotkii A, Schubert G, Tsepelev I (2007) Quasi-reversibility method for data assimilation in models of mantle dynamics. Geophys J Int 170:1381–1398

    Article  Google Scholar 

  • Kabanikhin SI (2011) Inverse and ill-posed problems. Theory and applications. De Gruyter, Berlin

    Book  Google Scholar 

  • Korotkii AI, Kovtunov DA (2006) Reconstruction of boundary regimes in an inverse problem of thermal convection of a high viscous fluid. Proc Inst Math Mech Ural Branch Russ Acad Sci 12(2):88–97 (in Russian)

    Google Scholar 

  • Korotkii AI, Starodubtseva YV (2014) Direct and inverse problems for models of stationary reactive-convective-diffusive flow. Proc Inst Math Mech Ural Branch Russ Acad Sci 20(3):98–113 (in Russian)

    Google Scholar 

  • Korotkii A, Kovtunov D, Ismail-Zadeh A, Tsepelev I, Melnik O (2016) Quantitative reconstruction of thermal and dynamic characteristics of lava flow from surface thermal measurements. Geophys J Int. doi:10.1093/gji/ggw117

    Google Scholar 

  • Ladyzhenskaya OA (1969) The mathematical theory of viscous incompressible flow. Gordon and Breach, New York

    Google Scholar 

  • Lions JL (1971) Optimal control of systems governed by partial differential equations. Springer, Berlin/Heidelberg

    Book  Google Scholar 

  • Miyamoto H, Sasaki S (1998) Numerical simulations of flood basalt lava flows: roles of parameters on lava flow morphologies. J Geophys Res 103:27489–27502

    Article  Google Scholar 

  • Neri A (1998) A local heat transfer analysis of lava cooling in the atmosphere: application to thermal diffusion-dominated lava flows. J Volcanol Geotherm Res 81:215–243

    Article  Google Scholar 

  • Nocedal J, Wright SJ (1999) Numerical optimization. Springer, New York

    Book  Google Scholar 

  • Patankar SV, Spalding DB (1972) A calculation procedure for heat and mass transfer in three-dimensional parabolic flows. Int J Heat Mass Transf 15:1787–1806

    Article  Google Scholar 

  • Polak E (1997) Optimization: algorithms and consistent approximations. Springer, Berlin/Heidelberg

    Book  Google Scholar 

  • Short NM, Stuart LM (1983) The heat capacity mapping mission (HCMM) anthology. Scientific and Technical Information Branch, National Aeronautics & Space Administration, Washington, DC

    Google Scholar 

  • Sweby PK (1984) High resolution schemes using flux limiters for hyperbolic conservation laws. J Numer Anal 21:995–1011

    Article  Google Scholar 

  • Temam R (1977) Navier-Stokes equations: theory and numerical analysis. North-Holland, Amsterdam

    Google Scholar 

  • Tikhonov AN, Arsenin VY (1977) Solution of ill-posed problems. Winston, Washington, DC

    Google Scholar 

  • Tsepelev I, Ismail-Zadeh A, Melnik O, Korotkii A (2016) Numerical modelling of fluid flow with rafts: an application to lava flows. J Geodyn. doi:10.1016/j.jog.2016.02.010

    Google Scholar 

  • Van der Vorst HA (1992) BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13(2):631–644

    Article  Google Scholar 

  • Wang Y, Hutter K (2001) Comparison of numerical methods with respect to convectively dominated problems. Int J Numer Methods Fluids 37:721–745

    Article  Google Scholar 

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Ismail-Zadeh, A., Korotkii, A., Tsepelev, I. (2016). Application of the Variational Method to Lava Flow Modelling. In: Data-Driven Numerical Modelling in Geodynamics: Methods and Applications. SpringerBriefs in Earth Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-27801-8_4

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