Logo Logo
Help
Contact
Switch language to German
Theory and application of the adjoint method in geodynamics and an extended review of analytical solution methods to the Stokes equation
Theory and application of the adjoint method in geodynamics and an extended review of analytical solution methods to the Stokes equation
The initial condition problem with respect to the temperature distribution in the Earth's mantle is Pandora's box of geodynamics. The heat transport inside the Earth follows the principles of advection and conduction. But since conduction is an irreversible process, this mechanism leads to a huge amount of information getting lost over time. Due to this reason, a recovery of a detailed state of the Earth's mantle some million years ago is an intrinsically unsolvable problem. In this work we present a novel mathematical method, the adjoint method in geodynamics, that is not capable of solving but of circumventing the presented initial condition problem by reformulating this task in terms of an optimisation problem. We are aiming at a past state of the Earth's mantle that approaches the current and thus, observable state over time in an optimal way. To this end, huge computational resources are needed since the 'optimal' solution can only be found in an iterative process. In this work, we developed a new general operator formulation in order to determine the adjoint version of the governing equations of mantle flow and applied this method to the high-resolution numerical mantle circulation code TERRA. For our models, we used a global grid spacing of approx. 30 km and more than 80 million mesh elements. We found a reconstruction of the Earth's mantle at 40 Ma that is, with respect to our modelling parameters, consistent with today's observations, gathered from seismic tomography. With this published fundamental work, we are opening the door to a variety of future applications, e.g. a possible incorporation of geological and geodetic data sets as further constraints for the model trajectory over geological time scales. Where high-resolution numerical models and even the implementation of inversion schemes have become feasible over the past decades due to increasing computational resources, in the community there is still a high demand for analytical solution methods. Restricting the physical parameter space in the governing equations, e.g. by only allowing for a radial varying viscosity, it can be shown that in some cases, the resulting simplified equations can even be solved in a (semi-)analytical way. In other words, in these simplified scenarios, no large scale computational resources or even high-performance clusters are needed but the solution for a global flow system can be determined in minutes even on a standard computer. Besides this apparent advantage, analytical and numerical solutions can even go hand-in-hand since numerical computer codes may be tested and benchmarked by means of these manufactured solutions. Here, we spend a large portion of this work with a detailed derivation of these analytical approaches. We basically start from scratch, having the intention to cover all possible traps and pitfalls on the way from the governing equations to their solutions and to provide a service to future scientists that are stuck somewhere in the middle of this road. Besides the derivation, we also present in detail how such an analytical approach can be used as a benchmark for a high-resolution mantle circulation code. We applied this theory to the prototype for a new high-performance mantle convection framework being developed in the Terra-Neo project and published the results along with a small portion of the derived theory. In an additional chapter of this work, we focus on a detailed analysis of the current state of the Earth's gravitational field that is measured in an unimaginably accurate way by the recent satellite missions CHAMP, GRACE and GOCE. The origin of the link of our work to the gravitational field also lies in the analytical solution methods. It can be shown that due to the effect of flow induced dynamic topography, the Earth's gravity field is highly sensitive to the viscosity profile in the Earth's mantle. We show that even without using any other external knowledge or data set, the gravitational field itself restricts the possible choices for the Earth's mantle viscosity to a well-defined parameter space. Furthermore, in the course of these examinations, we found that mantle processes are not capable of explaining the short wavelength signals in the observed gravity field at all, even with the best-fitting viscosity profile. To this end, we developed a simple crustal model that is only based on topographic data (ETOPO) and the principle of isostasy and showed that even with this very basic approach we can explain the majority of short length-scale features in the observed gravity signal. Finally, in combination with a (simple, static and analytic) mantle flow model based on a density field derived from seismic topography and mineralogy, we found a nearly perfect fit of modelled and observed gravitational data throughout all wavelengths under consideration (spherical harmonic degree and order up to l=100).
geodynamics, geomathematics, inverse problems, adjoint method, benchmark, Stokes equation, analytical solutions, geodesy, gravity field, data assimilation, spherical harmonics, continuum mechanics, propagator matrix
Horbach, André
2020
English
Universitätsbibliothek der Ludwig-Maximilians-Universität München
Horbach, André (2020): Theory and application of the adjoint method in geodynamics and an extended review of analytical solution methods to the Stokes equation. Dissertation, LMU München: Faculty of Geosciences
[img]
Preview
PDF
Horbach_Andre.pdf

19MB

Abstract

The initial condition problem with respect to the temperature distribution in the Earth's mantle is Pandora's box of geodynamics. The heat transport inside the Earth follows the principles of advection and conduction. But since conduction is an irreversible process, this mechanism leads to a huge amount of information getting lost over time. Due to this reason, a recovery of a detailed state of the Earth's mantle some million years ago is an intrinsically unsolvable problem. In this work we present a novel mathematical method, the adjoint method in geodynamics, that is not capable of solving but of circumventing the presented initial condition problem by reformulating this task in terms of an optimisation problem. We are aiming at a past state of the Earth's mantle that approaches the current and thus, observable state over time in an optimal way. To this end, huge computational resources are needed since the 'optimal' solution can only be found in an iterative process. In this work, we developed a new general operator formulation in order to determine the adjoint version of the governing equations of mantle flow and applied this method to the high-resolution numerical mantle circulation code TERRA. For our models, we used a global grid spacing of approx. 30 km and more than 80 million mesh elements. We found a reconstruction of the Earth's mantle at 40 Ma that is, with respect to our modelling parameters, consistent with today's observations, gathered from seismic tomography. With this published fundamental work, we are opening the door to a variety of future applications, e.g. a possible incorporation of geological and geodetic data sets as further constraints for the model trajectory over geological time scales. Where high-resolution numerical models and even the implementation of inversion schemes have become feasible over the past decades due to increasing computational resources, in the community there is still a high demand for analytical solution methods. Restricting the physical parameter space in the governing equations, e.g. by only allowing for a radial varying viscosity, it can be shown that in some cases, the resulting simplified equations can even be solved in a (semi-)analytical way. In other words, in these simplified scenarios, no large scale computational resources or even high-performance clusters are needed but the solution for a global flow system can be determined in minutes even on a standard computer. Besides this apparent advantage, analytical and numerical solutions can even go hand-in-hand since numerical computer codes may be tested and benchmarked by means of these manufactured solutions. Here, we spend a large portion of this work with a detailed derivation of these analytical approaches. We basically start from scratch, having the intention to cover all possible traps and pitfalls on the way from the governing equations to their solutions and to provide a service to future scientists that are stuck somewhere in the middle of this road. Besides the derivation, we also present in detail how such an analytical approach can be used as a benchmark for a high-resolution mantle circulation code. We applied this theory to the prototype for a new high-performance mantle convection framework being developed in the Terra-Neo project and published the results along with a small portion of the derived theory. In an additional chapter of this work, we focus on a detailed analysis of the current state of the Earth's gravitational field that is measured in an unimaginably accurate way by the recent satellite missions CHAMP, GRACE and GOCE. The origin of the link of our work to the gravitational field also lies in the analytical solution methods. It can be shown that due to the effect of flow induced dynamic topography, the Earth's gravity field is highly sensitive to the viscosity profile in the Earth's mantle. We show that even without using any other external knowledge or data set, the gravitational field itself restricts the possible choices for the Earth's mantle viscosity to a well-defined parameter space. Furthermore, in the course of these examinations, we found that mantle processes are not capable of explaining the short wavelength signals in the observed gravity field at all, even with the best-fitting viscosity profile. To this end, we developed a simple crustal model that is only based on topographic data (ETOPO) and the principle of isostasy and showed that even with this very basic approach we can explain the majority of short length-scale features in the observed gravity signal. Finally, in combination with a (simple, static and analytic) mantle flow model based on a density field derived from seismic topography and mineralogy, we found a nearly perfect fit of modelled and observed gravitational data throughout all wavelengths under consideration (spherical harmonic degree and order up to l=100).