1. Backward modeling of thermal convection: a new numerical approach applied to plume reconstruction Evgeniy Tantserev Collaborators: Marcus Beuchert and Yuri Podladchikov

2. Overview  Introduction  Thermal convection problem  Forward and Backward Heat Conduction Problem  Pseudo-parabolic approach to solve BHCP  Forward and backward modeling of thermal convection problem for high Rayleigh number  Forward and backward modeling of thermal convection problem for low Rayleigh number: different techniques  Conclusions

3.  Geological past is reconstructed using present day observations. It is an inverse problem known to be ill- posed.  Numerous new modeling directions became feasible due to the growth of computer power. Most of these new modeling attempts are “forward” in time because they deal with irreversible processes.  However, geological structures often formed by instabilities. Instabilities are often easier to simulate inverse (reverse) in time.  Practical numerical recipes and mathematical understanding of time “inversion” of instabilities is in great and urgent need in the geodynamics. ”Approach your problem from the right end and begin with answers. Then one day, perhaps you will find the final question.” From ”The Hermit Clad in Crane Feathers” in the Chinese Maze Murders, by R. Van GulikIntroduction

4. Well-posed problem ExistenceUniquenessStability Ill-posed problem Non- existence Non- uniqueness Non- stability For the well-posedness should be all three conditions For the ill-posedness enough the existence of one conditionIntroduction

5. Introduction Mantle plumes are among the most spectacular features of mass and heat transport from the mantle to the Earth’s surface. Forward modeling requires starting from generic initial temperature distributions in the mantle and follows the evolution of arising mantle plumes Temperature

6. Thermal convection problem λ is a non-dimensional activation parameter The thermal convection of mantle plumes is mainly driven by two processes: advection and thermal diffusion. Initial profile of the temperature: steady- state distribution with initial perturbation with added noise of maximum amplitude 1 %. As boundary conditions on the top we put temperature T=-0.5 and on the bottom T=0.5. And we have periodic boundary conditions on the sides.

7. boundary conditions initial distribution of temperature final distribution of temperature The forward problem is to find the final distribution of temperature (for time t f ) for given heat conduction law, boundary conditions and initial distribution of temperature. Forward Heat conduction – well-posed problem

8. boundary conditions The Final distribution of temperature The inverse problem is to find the initial distribution of temperature (for time t0) for given heat conduction law, boundary conditions and final distribution of temperature Initial distribution of temperature, which we need to obtain! Final distribution of temperature which we know from experimental data Backward Heat conduction – ill-posed problem

9. Initial distribution of temperature, which we need to obtain Final distribution of temperature which we know from experimental data The pseudo-parabolic reversibility method Regularized BHCP – well-posed problem

10. 1.Heat diffusion and viscous flow problem – FEM using Galerkin method 2.Advection problem - method of backward characteristics Forward modeling of plumes

11. Reverse modeling for this case was done, using the same code as for forward but with negative time steps. This problem is relatively stable because for high Ra number we have domination of advection process over the diffusion process Reverse modeling of thermal convection problem for high Rayleigh number

12. Forward problem TRM is highly unstable for this case due to increasing influence of diffusion term 1.Time Reverse Method change sign of the time-step from positive to negative one and use the same code Forward and Backward modeling of thermal convection for low Rayleigh number

13. 2. Using Tichonov Regularization We use TRM but every 3rd time step we change sign of time step from negative to positive one and solving forward heat diffusion problem we regularize the solution of thermo-convective problem Forward and Backward modeling of thermal convection for low Rayleigh number

14. PPB approach let us to evaluate temperature distribution for a longer backward time 3. Pseudo-parabolic approach (PPB) with regularized parameter ε = 10^-2 Forward and Backward modeling of thermal convection for low Rayleigh number

15. 1.Time Reverse method 2. Using Tichonov’s Regularization 3. Pseudo-parabolic approach with regularized parameter e = 10^-2 Forward and Backward modeling for low Rayleigh number

16. The diagram of different methods for low Rayleigh number Forward and Backward modeling for low Rayleigh number

17. Conclusions  For high Ra number (in our case 9*10^6) backward modeling of mantle plumes is relatively stable.  For low Ra number(2*10^5) we need to apply additional techniques to model backward process  Time reverse method for this case of Rayleigh number is highly unstable, method based on Tichonov’s regularization is more stable and pseudo-parabolic method is the most stable in time reverse restoration of the temperature profile  PPB approach is perspective method for restoration of temperature profile of mantle plumes

18. Thank you for your attention!

19. The quasi-reversibility method of Lattes and Lions. Elliptic approximation of Tiba Pseudo-parabolic approximation Reverse methods for the solution of Backward Heat Conduction Problem These regularized Backward Heat Conduction Problems are well-posed. Inverse I: Heat conduction – the worse case

20. Applications of the reversibility’s methods to the solution of Backward Heat Conduction Problem The diagrams show which method works better for analytical solutions for different frequencies and regularized parameters, that is differences between analytical solution regularized problem and original problem in the norm smaller fixed constant for longest time. Inverse I: Heat conduction – the worse case