A Variational Approach for Modeling the Inductive Response of Earth's Mantle to Time‐Varying External Geomagnetic Source

Abstract A mathematical and numerical framework is developed for modeling Earth's magnetic field and optimizing the trajectory of the whole system using geomagnetic observations. The conventional approach is adopted to model Earth's magnetic field in the source‐free region via the Gauss coefficient, for which the fast dynamical component for the mantle‐induced field in the mantle region is further constrained by the underlying physical laws of magnetic induction. A spectral approach is adopted to represent the total geomagnetic field. It thereafter yields a rapid convergence of the solution and a smooth and stable continuation of the geomagnetic field from the observational points to all space. Central to the mathematical development of this work is an adjoint formulation for optimizing the trajectory of the coupled dynamical system for an observation time window in variable magnetic boundary conditions, partially determined by other components. We demonstrate the effectiveness of our framework by optimizing a representative geomagnetic system comprising the primary and secondary components of the internal geomagnetic field and a simplified external field using a set of synthetic observations and show that all components of the system can be accurately determined in a variety of space weather conditions. We intend to construct the framework to model the geomagnetic field in a broad region from Earth's interior to the magnetosphere and describe the nonlinear nature and the dynamical balance of Earth's magnetic system in different magnetic weather conditions by utilizing a realistic space current model.