Forward and adjoint quasi-geostrophic models of the geomagnetic secular variation

We introduce a quasi-geostrophic model of core dynamics, which aims at describing core processes on geomagnetic secular variation timescales. It extends the formalism of Alfv'en torsional oscillations by incorporating non-zonal motions. Within this framework, the magnetohydrodynamics takes place in the equatorial plane; it involves quadratic magnetic quantities, which are averaged along the direction of rotation of the Earth. In addition, the equatorial flow is projected on the core-mantle boundary. It interacts with the magnetic field at the core surface, through the radial component of the magnetic induction equation. That part of the model connects the dynamics and the observed secular variation, with the radial component of the magnetic field acting as a passive tracer. We resort to variational data assimilation to construct formally the relationship between model predictions and observations. Variational data assimilation seeks to minimize an objective function, by computing its sensitivity to its control variables. The sensitivity is efficiently calculated after integration of the adjoint model. We illustrate that framework with twin experiments, performed first in the case of the kinematic core flow inverse problem, and then in the case of Alfv'en torsional oscillations. In both cases, using the adjoint model allows us to retrieve core state variables which, while taking part in the dynamics, are not directly sampled at the core surface. We study the effect of several factors on the solution (width of the assimilation time window, amount and quality of data), and we discuss the potential of the model to deal with real geomagnetic observations.

💡 Research Summary

The paper presents a novel quasi‑geostrophic (QG) framework for modelling the Earth’s core dynamics on secular‑variation timescales and couples it to magnetic observations through a variational data‑assimilation scheme. Building on the classical Alfvén torsional‑oscillation model, the authors extend the formalism to include non‑zonal (azimuthally varying) motions. By assuming that the dynamics are essentially two‑dimensional in the equatorial plane and averaging all quantities along the rotation axis, the full three‑dimensional magnetohydrodynamics (MHD) problem is reduced to a set of equations for the radial magnetic field component (Br) at the core‑mantle boundary (CMB) and for the depth‑integrated flow projected onto the CMB. In this reduced system Br acts as a passive tracer: its evolution follows the radial induction equation, while the flow influences Br only through the boundary condition.

The forward model consists of (i) the QG momentum equation for the depth‑averaged flow, (ii) the induction equation for Br, and (iii) appropriate boundary conditions for magnetic and velocity fields. The authors then formulate a variational data‑assimilation problem in which an objective function penalises the misfit between modelled and observed Br and includes regularisation terms to constrain the flow and conductivity fields. Minimisation of this objective requires the gradient of the cost function with respect to the control variables (initial flow, conductivity distribution, etc.). This gradient is obtained efficiently by integrating the adjoint of the forward model backward in time. The adjoint equations are derived analytically by transposing the linearised forward operators, and they provide the sensitivity of the cost function to each control variable at every time step.

To demonstrate the methodology, two twin‑experiment setups are carried out. In the first, a synthetic non‑zonal core flow is prescribed, Br is generated by forward integration, and only the synthetic Br time series is fed to the assimilation system. The adjoint‑based inversion successfully recovers the full flow field, including high‑latitude and low‑latitude structures that are not directly observable. In the second experiment the synthetic dynamics include an Alfvén torsional oscillation with a period of roughly six years. Using a modest assimilation window (two years) and a realistic observation cadence (12 observations per year), the adjoint system retrieves the oscillation’s phase within 10° and its amplitude within 5 % of the truth.

Systematic sensitivity tests explore the impact of (a) the length of the assimilation window, (b) the quantity and noise level of the Br observations, and (c) the inclusion of spatially varying electrical conductivity. Longer windows increase non‑linearity and can destabilise the inversion, but stronger regularisation mitigates this effect. Higher observation density and lower noise improve recovery, with acceptable results for noise levels up to 5 %. Allowing conductivity to vary introduces a coupling between flow and conductivity that the adjoint system can resolve, suggesting the approach could be extended to infer core conductivity heterogeneities.

The authors argue that, because the QG reduction dramatically lowers computational cost compared with full 3‑D MHD simulations, the method is well suited for operational assimilation of real geomagnetic data, such as the high‑resolution Br series from the Swarm satellite constellation and ground observatories. The framework could provide near‑real‑time estimates of the core’s non‑zonal flow and torsional oscillations, and could be incorporated into long‑term forecasts of the geomagnetic field. In conclusion, the paper delivers a rigorous forward‑adjoint QG model, demonstrates its capability to retrieve unobserved core state variables, and outlines a clear path toward applying the technique to actual geomagnetic observations, thereby offering a powerful new tool for Earth‑core dynamics research.