Core-flow constraints on extreme archeomagnetic intensity changes

Recent studies (Ben-Yosef et al., 2009; Shaar et al., 2011) propose extreme archeomagnetic intensity changes (termed spikes) in the range ~4-5μT/year c.a. 1000 BC in the Near East, around 40 to 50 times larger than values typical of the present-day. In order to investigate whether such extreme changes are consistent with a model of the source region of the magnetic field, namely the fluid flow at the surface of Earth's core, we construct upper bounds for instantaneous magnetic intensity change at an arbitrary site on the Earth's surface. These bounds are constrained by the amount of kinetic energy available to sustain the change, taken here to be a prescribed value for the root-mean-squared surface velocity of 13 km/yr as inferred from the current state of the core. Further, we focus attention on two end-members of optimised core surface flow structure: unrestricted and purely-toroidal. As the derivation of the bounds demands complete knowledge of the geomagnetic field at the core surface, we model the unknown field by means of a Monte Carlo approach, extending to high degree the CHAOS-4 (epoch 2010 AD) and CALS10k1.b (epoch 1000 BC) geomagnetic field models.Using 2000 realisations for each family of stochastic field models, we find that optimised core flows are always large-scale and that they tend to generate a non-dipole, quadrupole-dominated secular variation at the Earth's surface. The dependence of the upper bounds as a function of site location reflects the large-scale structure of the intensity itself: stronger field permits more rapid change. For the site in the Near East, purely-toroidal flows have upper bounds of approximately 0.62 ± 0.02μT/year, whereas unrestricted flows increase this bound to 1.20 ± 0.02μT/year. We favour the former as more geophysically sound, on the account of a large body of previous results from core surface flow inversions and consistency with the existence of a stratified layer at the top of the core. Even if we allow for a generous threefold increase in the prescribed rms velocity (and a concomitant threefold increase in the bound), we conclude that the reported occurrences of extreme intensity changes as suggested in the Near East are not compatible with the commonly accepted structure of core-surface flow. However, it may be that an explanation for spikes lies beyond our current perception of core-dynamics and future work would be further motivated by seeking corroborative evidence of rapid intensity change from sites elsewhere on Earth's surface; we therefore also discuss the form that the secular variation would take in the case of simultanenous archeomagnetic spikes

Enhanced magnetic fields within a stratified layer

Mounting evidence from both seismology and numerical experiments on core composition suggests the existence of a layer of stably stratified fluid at the top of Earth’s outer core. In such a layer, a magnetostrophic force balance and suppressed radial motion lead to stringent constraints on the magnetic field, named Malkus constraints, which are a much more restrictive extension of the well known Taylor constraints. Here, we explore the consequences of such constraints for the structure of the core’s internal magnetic field. We provide a new simple derivation of these Malkus constraints, and show solutions exist which can be matched to any external potential field with arbitrary depth of stratified layer. From considerations of these magnetostatic Malkus constraints alone, it is therefore not possible to uniquely infer the depth of the stratified layer from external geomagnetic observations. We examine two models of the geomagnetic field defined within a spherical core, which obey the Taylor constraints in an inner convective region and the Malkus constraints in an outer stratified layer. When matched to a single-epoch geomagnetic potential field model, both models show that the toroidal magnetic field within the outer layer is about 100 times stronger compared to that in the inner region, taking a maximum value of 8 mT at a depth of 70 km. The dynamic regime of such a layer, modulated by suppressed radial motion but also a locally enhanced magnetic field, may therefore be quite distinct from that of any interior dynamo

The observational signature of modelled torsional waves and comparison to geomagnetic jerks

Torsional Alfven waves involve the interaction of zonal fluid flow and the ambient magnetic field in the core. Consequently, they perturb the background magnetic field and induce a secondary magnetic field. Using a steady background magnetic field from observationally constrained field models and azimuthal velocities from torsional wave forward models, we solve an induction equation for the wave-induced secular variation (SV). We construct time series and maps of wave-induced SV and investigate how previously identified propagation characteristics manifest in the magnetic signals, and whether our modelled travelling torsional waves are capable of producing signals that resemble jerks in terms of amplitude and timescale. Fast torsional waves with amplitudes and timescales consistent with a recent study of the 6 yr ∆LOD signal induce very rapid, small (maximum ∼2 nT/yr at Earth’s surface) SV signals that would likely be difficult to be resolve in observations of Earth’s SV. Slow torsional waves with amplitudes and timescales consistent with other studies produce larger SV signals that reach amplitudes of ∼20 nT/yr at Earth’s surface. We applied a two-part linear regression jerk detection method to the SV induced by slow torsional waves, using the same parameters as used on real SV, which identified several synthetic jerk events. As the local magnetic field morphology dictates which regions are sensitive to zonal core flow, and not all regions are sensitive at the same time, the modelled waves generally produce synthetic jerks that are observed on regional scales and occur in a single SV component. However, high wave amplitudes during reflection from the stress-free CMB induce large-scale SV signals in all components, which results in a global contemporaneous jerk event such as that observed in 1969. In general, the identified events are periodic due to waves passing beneath locations at fixed intervals and the SV signals are smoothly varying. These smooth signals are more consistent with the geomagnetic jerks envisaged by Demetrescu and Dobrica than the sharp ‘V’ shapes that are typically associated with geomagnetic jerks

Recent north magnetic pole acceleration towards Siberia caused by flux lobe elongation

The wandering of Earth’s north magnetic pole, the location where the magnetic field points vertically downwards, has long been a topic of scientific fascination. Since the first in situ measurements in 1831 of its location in the Canadian arctic, the pole has drifted inexorably towards Siberia, accelerating between 1990 and 2005 from its historic speed of 0–15 km yr−1 to its present speed of 50–60 km yr−1. In late October 2017 the north magnetic pole crossed the international date line, passing within 390 km of the geographic pole, and is now moving southwards. Here we show that over the last two decades the position of the north magnetic pole has been largely determined by two large-scale lobes of negative magnetic flux on the core–mantle boundary under Canada and Siberia. Localized modelling shows that elongation of the Canadian lobe, probably caused by an alteration in the pattern of core flow between 1970 and 1999, substantially weakened its signature on Earth’s surface, causing the pole to accelerate towards Siberia. A range of simple models that capture this process indicate that over the next decade the north magnetic pole will continue on its current trajectory, travelling a further 390–660 km towards Siberia

Successive elimination of shear layers by a hierarchy of constraints in inviscid spherical-shell flows

In a rotating spherical shell, an inviscid inertia-free flow driven by an arbitrary body force will have cylindrical components that are either discontinuous across, or singular on, the tangent cylinder, the cylinder tangent to the inner core and parallel to the rotation axis. We investigate this problem analytically, and show that there is an infinite hierarchy of constraints on this body force which, if satisfied, sequentially remove discontinuities or singularities in flow derivatives of progressively higher order. By splitting the solution into its equatorial symmetry classes, we are able to provide analytic expressions for the constraints and demonstrate certain inter-relations between them. We show numerically that viscosity smoothes any singularity in the azimuthal flow component into a shear layer, comprising inner and outer layers, either side of the tangent cylinder, of width O(E2/7) and O(E1/4), respectively, where E is the Ekman number. The shear appears to scale as O(E-1/3) in the equatorially symmetric case, although in a more complex fashion when considering equatorial antisymmetry, and attains a maximum value in either the inner or outer sublayers depending on equatorial symmetry. In the low-viscosity magnetohydrodynamic system of the Earth's core, magnetic tension within the fluid resists discontinuities in the flow and may dynamically adjust the body force in order that a moderate number of the constraints are satisfied. We speculate that it is violations of these constraints that excites torsional oscillations, magnetohydrodynamic waves that are observed to emanate from the tangent cylinder

Characterisation of subglacial water using a constrained transdimensional Bayesian transient electromagnetic inversion

Subglacial water modulates glacier-bed friction and therefore is of fundamental importance when characterising the dynamics of ice masses. The state of subglacial pore water, whether liquid or frozen, is associated with differences in electrical resistivity that span several orders of magnitude; hence, liquid water can be inferred from electrical resistivity depth profiles. Such profiles can be obtained from inversions of transient (time-domain) electromagnetic (TEM) soundings, but these are often non-unique. Here, we adapt an existing Bayesian transdimensional algorithm (Multimodal Layered Transdimensional Inversion – MuLTI) to the inversion of TEM data using independent depth constraints to provide statistical properties and uncertainty analysis of the resistivity profile with depth. The method was applied to ground-based TEM data acquired on the terminus of the Norwegian glacier, Midtdalsbreen, with depth constraints provided by co-located ground-penetrating radar data. Our inversion shows that the glacier bed is directly underlain by material of resistivity 102 Ωm ± 1000 %, with thickness 5–40 m, in turn underlain by a highly conductive basement (100 Ωm ± 15 %). High-resistivity material, 5×104 Ωm ± 25 %, exists at the front of the glacier. All uncertainties are defined by the interquartile range of the posterior resistivity distribution. Combining these resistivity profiles with those from co-located seismic shear-wave velocity inversions to further reduce ambiguity in the hydrogeological interpretation of the subsurface, we propose a new 3-D interpretation in which the Midtdalsbreen subglacial material is partitioned into partially frozen sediment, frozen sediment/permafrost and weathered/fractured bedrock with saline water

Torsional Axisymmetric Core Oscillations Visualiser (TACO-VIS): A Python module for animating torsional wave data for fluid planetary cores

TACO-VIS provides a simple set of Python visualisation tools for 2D flow velocity data from fluid planetary interiors. It is mainly intended for animating torsional wave models for publication and presentation purposes. TACO-VIS is a lightweight module built only upon the common numpy/matplotlib Python packages and is free to be used and modified as the user requires

The inherent instability of axisymmetric magnetostrophic dynamo models

Recent studies have demonstrated the possibility of constructing magnetostrophic dynamo models, which describe the slowly evolving background state of Earth's magnetic field when inertia and viscosity are negligible. Here we explore the properties of steady, stable magnetostrophic states as a leading order approximation to the slow dynamics within Earth's core. For the case of an axisymmetric magnetostrophic system driven by a prescribed α-effect, we confirmed the existence of four known steady states: ±Bd, ±Bq, where Bd is purely dipolar and Bq is purely quadrupolar. Importantly, here we show that in all but the most weakly driven cases, an initial magnetic field that is not purely dipolar or quadrapolar never converges to these states. Despite this instability, we also show that there are a plethora of instantaneous solutions that are quasi-steady, but nevertheless unstable. If the dynamics in Earth's core are reasonably modelled by a strongly driven α-effect, this work suggests that the background state can never be steady. We discuss the difficulties in comparing our magnetostrophic models with geomagnetic timeseries

Taylor state dynamos found by optimal control: axisymmetric examples

Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke - and -effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the - and -effects are invoked, as well as new distributions. One such new -effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported by Wu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection

Archeomagnetic intensity variations during the era of geomagnetic spikes in the Levant

Observational records of rapidly varying magnetic fields strongly constrain our understanding of core flow dynamics and Earth's dynamo. Archeomagnetic analyses of densely sampled artefacts from the Near-East have suggested that the intensity variation during the first millennium BCE was punctuated with two geomagnetic spikes with rates of change of intensity exceeding 1 μT/yr, whose extreme behaviour is challenging to explain from a geodynamo perspective. By applying a new transdimensional Bayesian method designed to capture variations on both long and short timescales, we show that the data considered only at the fragment (thermal-unit) level require a complex intensity variation with no less than six spikes, each with an approximate duration of between 30 and 100 years. However, the nature of the inferred intensity evolution and the number of spikes detected are fragile and highly dependent on the specific treatment of the archeomagnetic data. No spikes are observed when the data are considered only at the level of a group of fragments from the same archeological context, with a minimum of three different artefacts per context. Furthermore, the number of spikes decreases to zero when increasing the error budget for the intensity at the fragment level within reasonable levels of 3–6 μT and the data age uncertainty up to 50 years. Of the six spikes found, the most resilient when increasing the error budget was dated at ~970 BCE. However, we show that even this spike sensitively depends on the age model proposed for data from the Levant archeological site Timna-30 and disappears when considering a single Gaussian age prior distribution for these data and a moderate minimum intensity error. Thus, depending on the choices made, the Near-Eastern data are compatible with a broad range of time-dependence, from six spikes at one extreme to zero spikes on the other. An error of 6 μT at the fragment level produces a spikeless model with strong similarity with the reconstruction from the SHAWQ-Iron Age global model with rates of change of ~0.2–0.3 μT/yr