Restoring past mantle convection structure through fluid dynamic inverse theory: regularisation through surface velocity boundary conditions

  • Lyudmyla VynnytskaEmail author
  • Hans-Peter Bunge
Original Paper


Mantle convection is governed by coupled, non-linear conservation equations for mass, momentum and energy. In the forward problem the equations relate an initial to a final state of the convective system, while one can pose an inverse problem to restore earlier structure from a final state. Although the ability to restore earlier mantle structure is essential in geodynamics, allowing one to test uncertain parameters of mantle convection models against constraints derived from the geologic record, the convergence properties of the inverse problem are not well understood. Here we show that knowledge of the surface velocity field over the restoration period is crucial for the convergence rate of the restoration problem. With simple mantle convection models, and assuming for reference a given initial and final state, we explore the restoration problem in cases where knowledge of the surface velocity field over the restoration period is available relative to those where it is not. We find convergence of the inverse problem for time periods of \(\sim \)1/3 of a transit time and corresponding to about 50 million years in the Earth’s mantle, if the surface velocity field is known, while it diverges otherwise. For the Earth’s mantle the history of the surface velocity field is known for time periods of \(\sim \)1/2 transit times from reconstructions of past plate motion. Our results suggest that this constraint is of key importance in any attempt to restore past mantle structure.


Mantle circulation Inverse problem Adjoint method Regularisation 

JEL Classification




The work of the first author is supported by a Statoil research grant.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Simula Research LaboratoryLysakerNorway
  2. 2.Ludwig Maximilians UniversityMunichGermany

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