Variational data assimilation for a forced, inertia-free magnetohydrodynamic dynamo model

Variational data assimilation (4DVar) is a powerful technique for tuning dynamic models to observations, in order not only to forecast future time evolution of the system, but to make inferences about quantities that are otherwise unconstrained by observation. We apply this technique, well-grounded in meteorology and oceanography, to the Earth's core where incompressible fluid motions in an electrically conducting medium are responsible for magnetic field generation. Our dynamic model's momentum equation neglects inertia such that the entire evolution depends only on the structure of the initial magnetic field; time evolution of the system is solely governed by the equation of magnetic induction. Nevertheless the dynamic system encompasses the effects of rotation, Lorentz forces and viscosity and aims to mimic a reasonable force-balance in the Earth's core. Building on the work of Li et al., in order to optimize the data-fit subject to the dynamics, we further develop the mathematical structure of the adjoint equations of the system. We address the feasibility of recovering 3-D spatial properties of the system using only time-varying 2-D observations of different character. Using closed-loop testing, we demonstrate the retrievability of the initial state (and thus the entire trajectory) of the system over convective timescales, when sampling in regions in which magnetic induction dominates over diffusion. The results suggest the possibility of retrieving the entire trajectory of the dynamo system of the Earth using the 400-yr model of secular variation gufm1.