Scaling of the geomagnetic secular variation time scales

Recent satellite missions provide accurate measurement of the time derivative of the Gaussian coefficients from which the secular variation spectrum can be calculated. The ratio of the magnetic energy spectrum to the secular variation spectrum gives a typical scale τ for the temporal variation of the geomagnetic field as a function of the spherical harmonics degrees l. There is much interest in the scaling of τ with l: τ ~ l ^β. Numerical simulations and the frozen flux hypothesis suggest the simple relation τ ~ l ^-1 while observational studies give a diverse range of value for β. A question here is whether the frozen flux hypothesis is applicable. It is plausible that magnetic diffusion can be neglected inside the outer core. However, the situation in a boundary layer under the core-mantle boundary (CMB) is less clear. A related question is whether τ observed at the Earth's surface is relevant to what is happening in the interior of the outer core as the form of the magnetic field above the CMB is constrained by the boundary conditions at the CMB. Here we use a numerical dynamo model to investigate these questions. We extend the definition of τ to the inside of the outer core. We find that in our simulations the exponent β undergoes a sharp transition just beneath the CMB, magnetic diffusion plays a role in the scaling of τ above the CMB and the frozen flux hypothesis is not applicable here.