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Linear modes for which the Coriolis acceleration is almost entirely in balance with the Lorentz force are called Magneto-Coriolis (MC) modes. These MC modes are thought to exist in Earth's liquid outer core and could therefore contribute to the variations observed in Earth's magnetic field. The background state on which these waves ride is assumed here to be static and defined by a prescribed magnetic field and zero flow. We introduce a new computational tool to efficiently compute solutions to the related eigenvalue problem, and study the effect of a range of both axisymmetric and non-axisymmetric background magnetic fields on the MC modes. We focus on a hierarchy of conditions that sequentially partition the numerous computed modes into those which are: (i) in principle observable, (ii) those which match a proxy for interannual geomagnetic signal over 1999-2023, and (iii) those which align with core-flows based on recent geomagnetic data. We found that the background field plays a crucial role in determining the structure of the modes. In particular, we found no examples of axisymmetric background fields that support modes consistent with recent geomagnetic changes, but that some non-axisymmetric background fields do support geomagnetically consistent modes. [ABSTRACT FROM AUTHOR]

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x) α (1 + x) β . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.

Perennial water storage in firn aquifers has been observed within the lower percolation zone of the southeast Greenland ice sheet. Spatially distributed seismic and radar observations, made ~50 km upstream of the Helheim Glacier terminus, reveal spatial variations of seismic velocity within a firn aquifer. From 1.65 to 1.8 km elevation, shear‐wave velocity (Vs) is 1,290 ± 180 m/s in the unsaturated firn, decreasing below the water table (~15 m depth) to 1,130 ± 250 m/s. Below 1.65 km elevation, Vs in the saturated firn is 1,270 ± 220 m/s. The compressional‐to‐shear velocity ratio decreases in the downstream saturated zone, from 2.30 ± 0.54 to 2.01 ± 0.46, closer to its value for pure ice (2.00). Consistent with colocated firn cores, these results imply an increasing concentration of ice in the downstream sites, reducing the porosity and storage potential of the firn likely caused by episodic melt and freeze during the evolution of the aquifer. Plain Language Summary: An integrated geophysical analysis of seismic, radar, and borehole measurements has been completed over a firn aquifer in southeast Greenland. We show the stiffness of the aquifer increases at lower elevations, closer to sea level, which leads to a decrease in pore space for the meltwater to be stored. This corresponds to an increase in ice content within the firn at lower elevations, as observed in borehole measurements, and likely caused by the meltwater refreezing within and below the aquifer. Key Points: Characterizing the storage potential of firn aquifers is important for predicting water budgets in a warming worldMultiple geophysical methods are combined to evaluate the elastic properties and porosity of the firn aquiferSeismic velocities detect an increase in ice content in the aquifer downslope, reducing the porosity and storage potential of the firn [ABSTRACT FROM AUTHOR]

MuLTI (Multimodal Layered Transdimensional Inversion) is a Markov chain Monte Carlo implementation of Bayesian inversion for the probability distribution of shear wave velocity (Vs) as a function of depth. Based on Multichannel Analysis of Surface Wave methods, it requires as data (i) a Rayleigh‐wave dispersion curve and (ii) additional layer depth constraints, the latter we show significantly improve resolution compared to conventional unconstrained inversions. Such depth constraints may be drawn from any source (e.g., boreholes, complementary geophysical data) provided they also represent a seismic interface. We apply MuLTI to a Norwegian glacier, Midtdalsbreen, in which ground‐penetrating radar was used to constrain internal layers of snow, ice, and subglacial sediments, with transitions at 2 and 25.5 m, and whose Vs is assumed to be in the range 500–1,700, 1,700–1,950, and 200–2,800 m/s, respectively. Synthetic modeling demonstrates that MuLTI recovers the true model of Vs variation with depth. Significantly, compared to inversions without depth constraints, in this synthetic case MuLTI not only reduces by about a factor of 10 the error between the true and the best fitting model, but also reduces by a factor of 2 the vertically averaged spread of the distribution of Vs based on the 95% credible intervals. We further show that using frequencies above about 100 Hz lead to unconverged solutions due to mode ambiguities associated with fine spatial structures. For our acquired data on Midtdalsbreen, we use 14‐100 Hz data for which MuLTI produces a large‐scale converged inversion. Plain Language Summary: Geophysical inversion is used to infer plausible subsurface features from surface measurements. However, inversions based on data sets acquired with a single geophysical technique often have poor resolution due to many different subsurface models fitting the data within the error tolerance. This study presents a novel method, Multimodal Layered Transdimensional Inversion, MuLTI, for inverting seismic surface wave data with constraints on depths of internal layers to obtain a more accurate and reliable interpretation of the subsurface. Here our depth constraints are drawn from ground‐penetrating radar horizon observations. MuLTI has been tested on an example data set from a glaciated environment to determine the seismic wave velocity of the subglacial sediment, which has important implications for glacier flow dynamics. By constraining the subsurface with ground‐penetrating radar depth horizons, results show the inverted solution being 2 times better resolved and 10 times more accurate within the glaciological subsurface, than without constraints applied. Thus, we demonstrate MuLTI can mitigate poor resolution of an unconstrained inversion, particularly at increased depth. Although we present examples from a glaciated data set, this novel methodology is applicable to any layered subsurface environment. Key Points: Inversions derived from seismic surface waves are underconstrained and nonuniqueAdding depth constraints for the near surface internal structure mitigates the poor resolution and nonuniqueness of surface wave inversionsMuLTI is a Bayesian inversion of seismic surface waves with depth constraints [ABSTRACT FROM AUTHOR]

Alfvén waves are a set of transverse waves that propagate in an electrically conducting fluid in the presence of an ambient magnetic field. Studies of such waves in the Earth's interior are important because they can be used to make inferences about the structure and physical properties of the core that would otherwise remain inaccessible. We produce 1-D forward models of cylindrical torsional Alfvén waves in the Earth's core, also known as torsional oscillations, and study their evolution in a full sphere and an equatorially symmetric spherical shell. Here, we find that travelling torsional waves undergo significant geometric dispersion that increases with successive reflections from the boundaries such that an initial wave pulse becomes unidentifiable within three transits of the core. Low amplitude wakes trail behind sharply defined pulses during propagation, a phenomenon that we interpret using the failure of Huygens’ principle in even dimensions. We investigate the relationship between geometric dispersion and wavelength, concluding that long-wavelength features are more dispersive than short-wavelength features. This result is particularly important because torsional waves that have been inferred in the Earth's core from secular variation are relatively long wavelength, and are therefore likely to undergo significant dispersion within the core. When stress-free boundary conditions on angular velocity are applied, waves are reflected at the equator of the core–mantle boundary with the same sign as the incident wave. Waves that pass through the rotation axis undergo a pseudo-reflection and display a more complicated behaviour due to a phase shift. In an equatorially symmetric shell, we identify a weak reflection at the tangent cylinder due to geometric effects. [ABSTRACT FROM AUTHOR]

In an earlier paper [P. W. Livermore, D. W. Hughes, and S. M. Tobias, “The role of helicity and stretching in forced kinematic dynamos in a spherical shell,” Phys. Fluids 19, 057101 (2007)], we considered the kinematic dynamo action resulting from a forced helical flow in a spherical shell. Although mean field electrodynamics suggests that the resulting magnetic field should have a significant mean (axisymmetric) component, we found no evidence for this; the dynamo action was distinctly small scale. Here we extend our investigation into the nonlinear regime in which the magnetic field reacts back on the velocity via the Lorentz force. Our main result is somewhat surprising, namely, that nonlinear effects lead to a considerable change in the structure of the magnetic field, its final state having a significant mean component. By investigating the dominant flow-field interactions, we isolate the dynamo mechanism and show schematically how the generation process differs between the kinematic and nonlinear regimes. In addition, we are able to calculate some components of the transport coefficient α and thus discuss our results within the context of mean field electrodynamics. [ABSTRACT FROM AUTHOR]

The dynamics of the Earth's fluid core are described by the so-called magnetostrophic balance between Coriolis, pressure, buoyancy and Lorentz forces. In this regime the geomagnetic field is subject to a continuum of theoretical conditions, which together comprise Taylor's constraint, placing restrictions on its internal structure. Examples of such fields, so-called Taylor states, have proven difficult to realize except in highly restricted cases. In previous theoretical developments, we showed that it was possible to reduce this infinite class of conditions to a finite number of coupled quadratic homogeneous equations when adopting a certain regular truncated spectral expansion for the magnetic field. In this paper, we illustrate the power of these results by explicitly constructing two families of exact Taylor states in a full sphere that match the same low-degree observationally derived model of the radial field at the core–mantle boundary. We do this by prescribing a smooth purely poloidal field that fits this observational model and adding to it an expediently chosen unconstrained set of interior toroidal harmonics of azimuthal wavenumbers 0 and 1. Formulated in terms of the toroidal coefficients, the resulting system is purely linear and can be readily solved to find Taylor states. By calculating the extremal members of the two families that minimize the Ohmic dissipation, we argue on energetic ground that the toroidal field in the Earth's core is likely to be dominated by low order azimuthal modes, similar to the observed poloidal field. Finally, we comment on the extension of finding Taylor states within a general truncated spectral expansion with arbitrary poloidal and toroidal coefficients. [ABSTRACT FROM AUTHOR]

Considerations of mean-field theory suggest that small-scale helical flows are an effective means of generating large-scale (mean) magnetic fields, whereas fast dynamo considerations reveal the importance of Lagrangian chaos in the flow for generating small-scale magnetic fields in the limit of high magnetic Reynolds number. We explore these ideas further by considering the kinematic magnetic fields generated by three forced steady flows in a spherical shell that differ both in their helicity and in their stretching properties. The full magnetic induction equation is solved numerically, with no a priori assumptions about the nature of the generated magnetic field. There are two surprising aspects to our results. One is that the most significant mean field is generated by a flow with zero net helicity; the other is that the flow with the “best” stretching properties turns out to be the most inefficient dynamo. Our results, therefore, suggest that it may not be possible to determine the nature of a kinematic-dynamo generated magnetic field simply from the knowledge of certain flow properties. [ABSTRACT FROM AUTHOR]