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The compressible adjoint equations in geodynamics: derivation and numerical assessment


The adjoint method is a powerful means to obtain gradient information in a mantle convection model relative to past flow structure. While the adjoint equations in geodynamics have been derived for the conservation equations of mantle flow in their incompressible form, the applicability of this approximation to Earth is limited, because density increases by almost a factor of two (Dziewonski and Anderson, Phys Earth Planet Inter 25:297–356, 1981) from the surface to the Core Mantle Boundary. Here we introduce the compressible adjoint equations for the conservation equations in the anelastic-liquid approximation. Our derivation applies an operator formulation in Hilbert spaces, to connect to recent work in seismology (Fichtner et al., Phys Earth Planet Inter 157(1–2):86–104, 2006b) and geodynamics (Horbach et al., Int J Geomath 5(2):163–194, 2014), where the approach was used to derive the adjoint equations for the wave equation and incompressible mantle flow. We present numerical tests of the newly derived equations based on twin experiments, focusing on three simulations. A first, termed Compressible, assumes the compressible forward and adjoint equations, and represents the consistent means of including compressibility effects. A second, termed Mixed, applies the compressible forward equation, but ignores compressibility effects in the adjoint equations, where the incompressible equations are used instead. A third simulation, termed Incompressible, neglects compressibility effects entirely in the forward and adjoint equations relative to the reference twin. The compressible and mixed formulations successfully restore earlier mantle flow structure, while the incompressible formulation yields noticeable artifacts. Our results suggest the use of a compressible formulation, when applying the adjoint method to seismically derived mantle heterogeneity structure.

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  • Bello, L., Coltice, N., Rolf, T., Tackley, P.J.: On the predictability limit of convection models of the Earth’s mantle. Geochem. Geophys. Geosyst. 15(6), 2319–2328 (2014). doi:10.1002/2014GC005254

    Article  Google Scholar 

  • Boussinesq, J.: Théorie analytique de la chaleur: mise en harmonie avec la thermodynamique et avec la théorie mécanique de la lumière, vol. 2. Gauthier-Villars, Paris (1903)

    MATH  Google Scholar 

  • Braun, J.: The many surface expressions of mantle dynamics. Nat. Geosci. 3(12), 825–833 (2010). doi:10.1038/ngeo1020

    Article  Google Scholar 

  • Bunge, H.P.: Low plume excess temperature and high core heat flux inferred from non-adiabatic geotherms in internally heated mantle circulation models. Phys. Earth Planet. Inter. 153(1–3), 3–10 (2005). doi:10.1016/j.pepi.2005.03.017

    Article  Google Scholar 

  • Bunge, H.P., Richards, M.A., Baumgardner, J.R.: Effect of depth-dependent viscosity on the planform of mantle convection. Nature 379(6564), 436–438 (1996). doi:10.1038/379436a0

    Article  Google Scholar 

  • Bunge, H.P., Richards, M.A., Baumgardner, J.R.: A sensitivity study of three-dimensional spherical mantle convection at \(10^8\) Rayleigh number: Effects of depth-dependent viscosity, heating mode, and an endothermic phase change. J. Geophys. Res. 102(B6), 11991–12007 (1997). doi:10.1029/96JB03806

  • Bunge, H.P., Richards, M.A., Lithgow-Bertelloni, C., Baumgardner, J.R., Grand, S.P., Romanowicz, B.A.: Time scales and heterogeneous structure in geodynamic earth models. Science 280(5360), 91–95 (1998). doi:10.1126/science.280.5360.91

    Article  Google Scholar 

  • Bunge, H.P., Hagelberg, C.R., Travis, B.J.: Mantle circulation models with variational data assimilation: inferring past mantle flow and structure from plate motion histories and seismic tomography. Geophys. J. Int. 152(2), 280–301 (2003). doi:10.1046/j.1365-246X.2003.01823.x

    Article  Google Scholar 

  • Burstedde, C., Stadler, G., Alisic, L., Wilcox, L.C., Tan, E., Gurnis, M., Ghattas, O.: Large-scale adaptive mantle convection simulation. Geophys. J. Int. 192(3), 889–906 (2013). doi:10.1093/gji/ggs070

    Article  Google Scholar 

  • Colli, L., Bunge, H.P., Schuberth, B.S.A.: On retrodictions of global mantle flow with assimilated surface velocities. Geophys. Res. Lett. (2015). doi:10.1002/2015GL066001

  • Davies, G.F.: Dynamic Earth: Plates, Plumes and Mantle Convection. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  • Davies, D.R., Goes, S., Davies, J.H., Schuberth, B.S.A., Bunge, H.P., Ritsema, J.: Reconciling dynamic and seismic models of Earth’s lower mantle: the dominant role of thermal heterogeneity. Earth Planet. Sci. Lett. 353–354, 253–269 (2012). doi:10.1016/j.epsl.2012.08.016

    Article  Google Scholar 

  • Dziewonski, A.M., Anderson, D.L.: Preliminary reference Earth model. Phys. Earth Planet. Inter. 25, 297–356 (1981). doi:10.1016/0031-9201(81)90046-7

    Article  Google Scholar 

  • Fichtner, A., Bunge, H.P., Igel, H.: The adjoint method in seismology: II - Applications: traveltimes and sensitivity functionals. Phys. Earth Planet. Inter. 157(1–2), 105–123 (2006a). doi:10.1016/j.pepi.2006.03.018

  • Fichtner, A., Bunge, H.P., Igel, H.: The adjoint method in seismology: I—theory. Phys. Earth Planet. Inter. 157(1–2), 86–104 (2006b). doi:10.1016/j.pepi.2006.03.016

  • Fletcher, R., Reeves, C.M.: Function minimization by conjugate gradients. Comput. J. 7(2), 149–154 (1964). doi:10.1093/comjnl/7.2.149

    Article  MathSciNet  MATH  Google Scholar 

  • Fournier, A., Hulot, G., Jault, D., Kuang, W., Tangborn, A., Gillet, N., Canet, E., Aubert, J., Lhuillier, F.: An introduction to data assimilation and predictability in geomagnetism. Space Sci. Rev. 155(1–4), 247–291 (2010). doi:10.1007/s11214-010-9669-4

    Article  Google Scholar 

  • French, S.W., Romanowicz, B.A.: Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography. Geophys. J. Int. 199(3), 1303–1327 (2014). doi:10.1093/gji/ggu334

    Article  Google Scholar 

  • Glatzmaier, G.A.: Numerical simulations of mantle convection: time-dependent, three-dimensional, compressible, spherical shell. Geophys. Astrophys. Fluid Dyn. 43(2), 223–264 (1988). doi:10.1080/03091928808213626

    Article  MathSciNet  MATH  Google Scholar 

  • Grand, S.P., van der Hilst, R.D., Widiyantoro, S.: High resolution global tomography: a snapshot of convection in the Earth. GSA Today 7(4), 1–7 (1997)

  • Grüneisen, E.: Theorie des festen Zustandes einatomiger Elemente. Ann. Phys. 344(12), 257–306 (1912). doi:10.1002/andp.19123441202

    Article  MATH  Google Scholar 

  • Hofmeister, A.M.: Mantle values of thermal conductivity and the geotherm from phonon lifetimes. Science 283(5408), 1699–1706 (1999). doi:10.1126/science.283.5408.1699

    Article  Google Scholar 

  • Horbach, A., Bunge, H.P., Oeser, J.: The adjoint method in geodynamics: derivation from a general operator formulation and application to the initial condition problem in a high resolution mantle circulation model. Int. J. Geomath. 5(2), 163–194 (2014). doi:10.1007/s13137-014-0061-5

    Article  MathSciNet  MATH  Google Scholar 

  • Ismail-Zadeh, A., Schubert, G., Tsepelev, I., Korotkii, A.: Inverse problem of thermal convection: numerical approach and application to mantle plume restoration. Phys. Earth Planet. Inter. 145(1–4), 99–114 (2004). doi:10.1016/j.pepi.2004.03.006

    Article  Google Scholar 

  • Jarvis, G.T., McKenzie, D.P.: Convection in a compressible fluid with infinite Prandtl number. J. Fluid Mech. 96(03), 515–583 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Leng, W., Zhong, S.: Controls on plume heat flux and plume excess temperature. J. Geophys. Res. 113(B4), B04408 (2008). doi:10.1029/2007JB005155

  • Liu, L., Spasojevic, S., Gurnis, M.: Reconstructing Farallon plate subduction beneath North America back to the Late Cretaceous. Science 322(5903), 934–938 (2008). doi:10.1126/science.1162921

    Article  Google Scholar 

  • McKenzie, D.P., Roberts, J.M., Weiss, N.O.: Convection in the Earth’s mantle: towards a numerical simulation. J. Fluid Mech. 62(03), 465–538 (1974)

    Article  MATH  Google Scholar 

  • McNamara, A.K., Zhong, S.: Thermochemical structures beneath Africa and the Pacific Ocean. Nature 437(7062), 1136–1139 (2005). doi:10.1038/nature04066

    Article  Google Scholar 

  • Mitrovica, J.X.: Haskell [1935] revisited. J. Geophys. Res. 101(B1), 555–569 (1996). doi:10.1029/95JB03208

    Article  Google Scholar 

  • Murnaghan, F.D.: Finite Deformation of an Elastic Body. Wiley, New York (1951)

    MATH  Google Scholar 

  • Oeser, J., Bunge, H.P., Mohr, M.: Cluster design in the earth sciences: TETHYS. In: High Performance Computing and Communications, Lecture Notes in Computer Science, pp. 31–40. Springer, Berlin (2006). doi:10.1007/11847366_4

  • Ritsema, J., Deuss, A., van Heijst, H.J., Woodhouse, J.H.: S40RTS: a degree-40 shear-velocity model for the mantle from new Rayleigh wave dispersion, teleseismic traveltime and normal-mode splitting function measurements. Geophys. J. Int. 184(3), 1223–1236 (2011)

    Article  Google Scholar 

  • Schuberth, B.S.A., Bunge, H.P., Ritsema, J.: Tomographic filtering of high-resolution mantle circulation models: Can seismic heterogeneity be explained by temperature alone? Geochem. Geophys. Geosyst. 10(5) (2009a). doi:10.1029/2009GC002401

  • Schuberth, B.S.A., Bunge, H.P., Steinle-Neumann, G., Moder, C., Oeser, J.: Thermal versus elastic heterogeneity in high-resolution mantle circulation models with pyrolite composition: high plume excess temperatures in the lowermost mantle. Geochem Geophys Geosyst 10(1) (2009b). doi:10.1029/2008GC002235

  • Serrin, J.: Mathematical principles of classical fluid mechanics. In: Fluid Dynamics I/Strömungsmechanik I, Encyclopedia of Physics/Handbuch der Physik, pp. 125–263. Springer, Berlin (1959). doi:10.1007/978-3-642-45914-6_2

  • Seton, M., Müller, R.D., Zahirovic, S., Gaina, C., Torsvik, T., Shephard, G., Talsma, A., Gurnis, M., Turner, M., Maus, S., Chandler, M.: Global continental and ocean basin reconstructions since 200 Ma. Earth Sci. Rev. 113(3–4), 212–270 (2012)

    Article  Google Scholar 

  • Shephard, G.E., Bunge, H.P., Schuberth, B.S.A., Müller, R.D., Talsma, A.S., Moder, C., Landgrebe, T.C.W.: Testing absolute plate reference frames and the implications for the generation of geodynamic mantle heterogeneity structure. Earth Planet. Sci. Lett. 317–318, 204–217 (2012). doi:10.1016/j.epsl.2011.11.027

    Article  Google Scholar 

  • Simmons, N.A., Myers, S.C., Johannesson, G., Matzel, E.: LLNL-G3Dv3: global P wave tomography model for improved regional and teleseismic travel time prediction. J. Geophys. Res. 117(B10), B10302 (2012)

    Article  Google Scholar 

  • Stacey, F.D., Davis, P.M.: High pressure equations of state with applications to the lower mantle and core. Phys. Earth Planet. Inter. 142(3–4), 137–184 (2004)

    Article  Google Scholar 

  • Tackley, P.J.: Dynamics and evolution of the deep mantle resulting from thermal, chemical, phase and melting effects. Earth Sci. Rev. 110(1–4), 1–25 (2012). doi:10.1016/j.earscirev.2011.10.001

    Article  Google Scholar 

  • Turcotte, D.L., Oxburgh, E.R.: Mantle convection and the new global tectonics. Annu. Rev. Fluid Mech. 4(1), 33–66 (1972)

    Article  Google Scholar 

  • Vynnytska, L., Bunge, H.P.: Restoring past mantle convection structure through fluid dynamic inverse theory: regularisation through surface velocity boundary conditions. Int. J. Geomath. 6(1), 83–100 (2014). doi:10.1007/s13137-014-0060-6

    Article  MathSciNet  MATH  Google Scholar 

  • Weismüller, J., Gmeiner, B., Ghelichkhan, S., Huber, M., John, L., Wohlmuth, B., Rüde, U., Bunge, H.P.: Fast asthenosphere motion in high-resolution global mantle flow models. Geophys. Res. Lett. 42(18), 2015GL063727 (2015). doi:10.1002/2015GL063727

  • Zhong, S., Liu, X.: The long-wavelength mantle structure and dynamics and their implications for large-scale tectonics and volcanism in the Phanerozoic. Gondwana Res. (2015). doi:10.1016/

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The authors are grateful to Barbara A. Romanowicz and an anonymous reviewer for constructive suggestions. We would like to thank Willi Freeden for the competent handling of the manuscript.

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Correspondence to Siavash Ghelichkhan.

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Ghelichkhan, S., Bunge, HP. The compressible adjoint equations in geodynamics: derivation and numerical assessment. Int J Geomath 7, 1–30 (2016).

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  • Adjoint method
  • Inverse problem
  • Geodynamics
  • Anelastic-liquid approximation
  • Computational methods

Mathematics Subject Classification

  • 86–08
  • 86A22