Scaling of the geomagnetic secular variation time scales

The ratio of the Lowes spectrum and the secular variation spectrum measured at the Earth's surface provides a time scale $\tau_{\rm sv}(l)$ as a function of spherical harmonic degree $l$. $\tau_{\rm sv}$ 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 $\boldsymbol{\dot B}$ inside the outer core, we introduce a magnetic time-scale spectrum $\tau(l,r)$ that is valid for all radius $r$ above the inner core and reduces to the usual $\tau_{\rm sv}$ at and above the core-mantle boundary (CMB). We study $\tau$ in a numerical geodynamo model. Focusing on the large scales, we find that $\tau \sim l^{-1}$ at the CMB. Just below the CMB, $\tau$ undergo a sharp transition such that the scaling becomes shallower than $l^{-1}$. This transition stems from the magnetic boundary condition at the CMB that ties all three components of $\boldsymbol{\dot B}$ together. In the interior of the outer core, the time variation of the horizontal magnetic field, which dominates $\boldsymbol{\dot B}$, has no such constraint. The upshot is $\tau_{\rm sv}$ becomes unreliable in estimating time scales inside the outer core. Another question concerning $\tau$ is whether a scaling 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 $\boldsymbol{\dot B}$. However, $\boldsymbol{\dot B}$ is controlled by the radial derivative in the induction term, thus invalidating the frozen-flux argument. Near the CMB, magnetic diffusion starts to affect $\boldsymbol{\dot B}$ rendering the frozen-flux hypothesis inapplicable.