Backward modeling of thermal convection: a new numerical approach applied to plume reconstruction

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
Introduction

• 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 Gulik
4
Introduction
For the well-posedness should be all three
conditions
For the ill-posedness enough the existence of one
condition
5
Introduction
Mantle plumes are among the most spectacular
features of mass and heat transport from the
mantle to the Earths 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
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 .

? is a non-dimensional activation parameter
As boundary conditions on the top we put
temperature T-0.5 and on the bottom T0.5. And
we have periodic boundary conditions on the
sides.
7
Forward Heat conduction well-posed problem
boundary conditions
initial distribution of temperature
The forward problem is to find the final
distribution of temperature (for time tf ) for
given heat conduction law, boundary conditions
and initial distribution of temperature .
8
Backward Heat conduction ill-posed problem
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
9
Regularized BHCP well-posed problem
The pseudo-parabolic reversibilitymethod
Initial distribution of temperature, which we
need to obtain
Final distribution of temperature which we know
from experimental data
10
Forward modeling of plumes
1.Heat diffusion and viscous flow problem FEM
using Galerkin method
2.Advection problem - method of backward
characteristics
11
Reverse modeling of thermal convection problem
for high Rayleigh number
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
12
Forward and Backward modeling of thermal
convection for low Rayleigh number
Forward problem
1.Time Reverse Method change sign of the
time-step from positive to negative one and use
the same code
TRM is highly unstable for this case due to
increasing influence of diffusion term
13
Forward and Backward modeling of thermal
convection for low Rayleigh number
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
14
Forward and Backward modeling of thermal
convection for low Rayleigh number
3. Pseudo-parabolic approach (PPB) with
regularized parameter e 10-2
PPB approach let us to evaluate temperature
distribution for a longer backward time
15
1.Time Reverse method
Forward and Backward modeling for low Rayleigh
number
2. Using Tichonovs Regularization
3. Pseudo-parabolic approach with regularized
parameter e 10-2
16
Forward and Backward modeling for low Rayleigh
number
The diagram of different methods for low Rayleigh
number
17
Conclusions

• For high Ra number (in our case 9106) backward
  modeling of mantle plumes is relatively stable.
• For low Ra number(2105) 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
  Tichonovs 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.
Inverse I Heat conduction the worse case
Reverse methods for the solution of Backward Heat
Conduction Problem
Elliptic approximation of Tiba
Pseudo-parabolic approximation
These regularized Backward Heat Conduction
Problems are well-posed.
20
Inverse I Heat conduction the worse case
Applications of the reversibilitys 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.