Frequency spectrum of the geomagnetic field harmonic coefficients from dynamo simulations

The construction of geomagnetic, archaeomagnetic or palaeomagnetic field models requires some prior knowledge about the actual field, which can be gathered from the statistical properties of the field over a variety of length-scales and timescales. However, available geomagnetic data on centennial to millennial periods are too sparse to infer directly these statistical properties. We thus use high-resolution numerical simulations of the geodynamo to test a method for estimating the temporal power spectra (or equivalently the autocovariance functions) of the individual Gauss coefficients that describe the geomagnetic field outside the Earth's fluid outer core. Based on the spectral analysis of our simulations, we argue that a prior for the observational geomagnetic field over decennial to millennial periods can be constructed from the statistics of the field during the short satellite era. The method rests on the assumption that time-series of spherical harmonic coefficients can be considered as realizations of stationary and differentiable stochastic processes, namely order 2 autoregressive (AR2) processes. In the framework of these processes, the statistics of Gauss coefficients are well constrained by their variance and one or two timescales. We find that the time spectra in the dynamo simulations of all Gauss coefficients but the axial dipole are well approximated by the spectra of AR2 processes characterized by only one timescale. The process parameters can simply be deduced from instantaneous estimates of the spatial power spectra of the magnetic field and of its first time derivative. Some deviations of the Gauss coefficients statistics from this minimal model are also discussed. Characterizing the axial dipole clearly requires a more sophisticated AR2 process, with a second distinct timescale.