Scaling of the geomagnetic secular variation time scale

Summary The ratio of the magnetic power spectrum and the secular variation spectrum measured at the Earth’s surface provides a time scale τsv(l) as a function of spherical harmonic degree l. τsv is often assumed to be representative of time scales related to the dynamo inside the outer core and its scaling with l is debated. To assess the validity of this surmise and to study the time variation of the geomagnetic field $\dot{B}$ inside the outer core, we introduce a magnetic time-scale spectrum τ(l, r) that is valid for all radius r above the inner core and reduces to the usual τsv at and above the core–mantle boundary (CMB). We study τ in a numerical geodynamo model. At the CMB, we find that τ ∼ l−1 is valid at both the large and small scales, in agreement with previous numerical studies on τsv. Just below the CMB, the scaling undergo a sharp transition at small l. Consequently, in the interior of the outer core, τ exhibits different scaling at the large and small scales, specifically, the scaling of τ becomes shallower than l−1 at small l. We find that this transition at the large scales stems from the fact that the horizontal components of the magnetic field evolve faster than the radial component in the interior. In contrast, the magnetic field at the CMB must match onto a potential field, hence the dynamics of the radial and horizontal magnetic fields are tied together. The upshot is τsv becomes unreliable in estimating time scales inside the outer core. Another question concerning τ is whether an argument based on the frozen-flux hypothesis can be used to explain its scaling. To investigate this, we analyse the induction equation in the spectral space. We find that away from both boundaries, the magnetic diffusion term is negligible in the power spectrum of $\dot{B}$. However, $\dot{B}$ is controlled by the radial derivative in the induction term, thus invalidating the frozen-flux argument. Near the CMB, magnetic diffusion starts to affect $\dot{B}$ rendering the frozen-flux hypothesis inapplicable. We also examine the effects of different velocity boundary conditions and find that the above results apply for both no-slip and stress-free conditions at the CMB.